Handbook of Quantitative Finance and Risk Management (v. 1-3)

Handbook of Quantitative Finance and Risk Management

Editors Cheng-Few Lee, Rutgers University, USA Alice C. Lee, State Street Corp., USA John Lee, Center for PBBEF Research, USA

Advisory Board Ivan Brick, Rutgers University, USA Stephen Brown, New York University, USA Charles Q. Cao, Penn State University, USA Chun-Yen Chang, National Chiao Tung University, Taiwan Wayne Ferson, Boston College, USA Lawrence R. Glosten, Columbia University, USA Martin J. Gruber, New York University, USA Hyley Huang, Wintek Corporation, Taiwan Richard E. Kihlstrom, University of Pennsylvania, USA E. H. Kim, University of Michigan, USA Robert McDonald, Northwestern University, USA Ehud I. Ronn, University of Texas at Austin, USA



Disclaimer: Any views or opinions presented in this publication are solely those of the authors and do not necessarily represent those of State Street Corporation. State Street Corporation is not associated in any way with this publication and accepts no liability for the contents of this publication.

Cheng-Few Lee  Alice C. Lee  John Lee Editors

Handbook of Quantitative Finance and Risk Management

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Editors Cheng-Few Lee Rutgers University Department of Finance and Economics 94 Rockafeller Road New Brunswick, NJ 08854-8054, Janice H. Levin Bldg. USA [email protected]

John Lee Center for PBBEF Research North Brunswick, NJ USA [email protected]

Alice C. Lee State Street Corp. Boston, MA USA [email protected]

ISBN 978-0-387-77116-8 e-ISBN 978-0-387-77117-5 DOI 10.1007/978-0-387-77117-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010921816 c Springer Science+Business Media, LLC 2010  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Quantitative finance and risk management is a combination of economics, accounting, statistics, econometrics, mathematics, stochastic process, and computer science and technology. This handbook is the most comprehensive handbook in quantitative finance and risk management, which integrates theory, methodology, and application. Due to the importance of quantitative finance and risk management in the finance industry, it has become one of the most popular subjects in business schools and departments of mathematics, operation research, and statistics. In addition, the finance industry has many job opportunities for people with good training in quantitative finance and risk management. Thus, a handbook should have a broad audience and be of interest to academics, educators, students, and practitioners. Based on our years of experience in industry, teaching, research, textbook writing, and journal editing on the subject of quantitative finance and risk management, this handbook will review, discuss, and integrate theoretical, methodological and practical issues of quantitative finance and risk management. This handbook is organized into five parts as follows: Part I. Overview of Quantitative Finance and Risk Management Research Part II. Portfolio Theory and Investment Analysis Part III. Options and Option Pricing Theory Part IV. Risk Management Part V. Theory, Methodology, and Applications Part I of this handbook covers three chapters: they are “Chapter 1. Theoretical Framework of Finance,” “Chapter 2. Investment, Dividend, Financing, and Production Policies,” and “Chapter 3. Research Methods of Quantitative Finance and Risk Management.” Part II of this handbook covers 18 chapters of portfolio theory and investment analysis. Part III of this handbook includes 21 chapters of options and option pricing theory. Part IV of this handbook includes 23 chapters of theory and practice in risk management. Finally, Part V of this handbook covers 44 chapters of theory, methodology, and applications in quantitative finance and risk management. In the preparation of this handbook, first, we would like to thank the members of advisory board and contributors of this handbook. In addition, we note and appreciate the extensive help from our Editor, Ms. Judith Pforr, our research assistants Hong-Yi Chen, Wei-Kang Shih and Shin-Ying Mai, and our secretary Ms. Miranda Mei-Lan Luo. Finally, we would like to thank the Wintek Corporation and the Polaris Financial Group for the financial support that allowed us to write this book. There are undoubtedly some errors in the finished product, both typographical and conceptual. We invite readers to send suggestions, comments, criticisms, and corrections to the author Professor Cheng-Few Lee at the Department of Finance and Economics, Rutgers University at Janice H. Levin Building Room 141, Rockefeller Road, Piscataway, NJ 08854-8054. New Brunswick, NJ Boston, MA North Brunswick, NJ

Cheng-Few Lee Alice C. Lee John Lee v

About the Editors

Cheng-Few Lee is Distinguished Professor of Finance at Rutgers Business School, Rutgers University and was chairperson of the Department of Finance from 1988 to 1995. He has also served on the faculty of the University of Illinois (IBE Professor of Finance) and the University of Georgia. He has maintained academic and consulting ties in Taiwan, Hong Kong, China, and the United States for the past three decades. He has been a consultant to many prominent groups, including the American Insurance Group, the World Bank, the United Nations, The Marmon Group Inc., Wintek Corporation, and Polaris Financial Group. Professor Lee founded the Review of Quantitative Finance and Accounting (RQFA) in 1990 and the Review of Pacific Basin Financial Markets and Policies (RPBFMP) in 1998, and serves as managing editor for both journals. He was also a co-editor of the Financial Review (1985– 1991) and the Quarterly Review of Economics and Business (1987–1989). In the past 36 years, Dr. Lee has written numerous textbooks ranging in subject matters from financial management to corporate finance, security analysis and portfolio management to financial analysis, planning and forecasting, and business statistics. In addition, he edited a popular book entitled Encyclopedia of Finance (with Alice C. Lee). Dr. Lee has also published more than 170 articles in more than 20 different journals in finance, accounting, economics, statistics, and management. Professor Lee was ranked the most published finance professor worldwide during the period 1953–2008. Professor Lee was the intellectual force behind the creation of the new Masters of Quantitative Finance program at Rutgers University. This program began in 2001 and has been ranked as one of the top ten quantitative finance programs in the United States. These top ten programs are located at Carnegie Mellon University, Columbia University, Cornell University, New York University, Princeton University, Rutgers University, Stanford University, University of California at Berkley, University of Chicago, and University of Michigan. Alice C. Lee is currently a Director in the Model Validation Group, Enterprise Risk Management, at State Street Corporation. Most recently, she was an Assistant Professor of Finance at San Francisco State University. She has more than 20 years of experience and has a diverse background, which includes academia, engineering, sales, and management consulting. Her primary areas of teaching and research are corporate finance and financial institutions. She is coauthor of Statistics for Business and Financial Economics, 2e (with Cheng F. Lee and John C. Lee) and Financial Analysis, Planning and Forecasting, 2e (with Cheng F. Lee and John C. Lee). In addition, she has co-edited other annual publications including Advances in Investment Analysis and Portfolio Management (with Cheng F. Lee). John C. Lee is a Microsoft Certified Professional in Microsoft Visual Basic and Microsoft Excel VBA. He has a bachelor and masters degree in accounting from the University of Illinois at Urbana-Champaign. John has more than 20 years’ experience in both the business and technical fields as an accountant, auditor, systems analyst, and as a business software developer. He has authored a book on how to use MINITAB and Microsoft Excel to do statistical analysis; this book is vii

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a companion text to Statistics of Business and Financial Economics, of which he is one of the co-authors. John has been a senior technology officer at the Chase Manhattan Bank and assistant vice president at Merrill Lynch. He is currently Director of the Center for PBBEF Research.

About the Editors

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I

v

Overview of Quantitative Finance and Risk Management Research

1

Theoretical Framework of Finance : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Discounted Cash-Flow Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 M and M Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Markowitz Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Option Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Futures Valuation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 6 10 10 12 14 15 22 22

2

Investment, Dividend, Financing, and Production Policies: Theory and Implications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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2.1 2.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investment and Dividend Interactions: The Internal Versus External Financing Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Interactions Between Dividend and Financing Policies . . . . . . . . . . . . . . . . . . 2.4 Interactions Between Financing and Investment Decisions . . . . . . . . . . . . . . . 2.5 Implications of Financing and Investment Interactions for Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Implications of Different Policies on the Beta Coefficient . . . . . . . . . . . . . . . . 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2A Stochastic Dominance and its Applications to Capital-Structure Analysis with Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2A.2 Concepts and Theorems of Stochastic Dominance . . . . . . . . . . . . . . . 2A.3 Stochastic-Dominance Approach to Investigating the Capital-Structure Problem with Default Risk . . . . . . . . . . . . . . . . . . . 2A.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 25 28 30 34 36 36 38 38 38 39 40

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Contents

Research Methods in Quantitative Finance and Risk Management : : : : : : : : : : 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Econometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Other Disciplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II

41 41 41 43 46 48 49 50

Portfolio Theory and Investment Analysis

4

Foundation of Portfolio Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5: 3 Cheng-Few Lee, Alice C. Lee, and John Lee 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Risk Classification and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Portfolio Analysis and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4 The Efficient Portfolio and Risk Diversification . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Determination of Commercial Lending Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.6 The Market Rate of Return and Market Risk Premium . . . . . . . . . . . . . . . . . . . 66 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5

Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Joseph E. Finnerty, and Hong-Yi Chen 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Measurement of Return and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Utility Theory, Utility Functions, and Indifference Curves . . . . . . . . . . . . . . . 5.4 Efficient Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Capital Asset Pricing Model and Beta Forecasting : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Graphical Approach to the Derivation of the Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mathematical Approach to the Derivation of the Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Market Model and Risk Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Growth Rates, Accounting Betas, and Variance in EBIT . . . . . . . . . . . . . . . . . 6.6 Some Applications and Implications of the Capital Asset Pricing Model . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 6A Empirical Evidence for the Risk-Return Relationship . . . . . . . . . . . . . Appendix 6B Anomalies in the Semi-strong Efficient-Market Hypothesis . . . . . . . . Index Models for Portfolio Selection : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Single-Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Multiple Indexes and the Multiple-Index Model . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 71 77 91 91 93 93 93 96 97 100 104 105 105 106 109 111 111 111 118 121 122

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Appendix 7A A Linear-Programming Approach to Portfolio-Analysis Models . . . . 122 Appendix 7B Expected Return, Variance, and Covariance for a Multi-index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Performance-Measure Approaches for Selecting Optimum Portfolios : : : : : : : : Cheng-Few Lee, Hong-Yi Chen, and Jessica Shin-Ying Mai 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Sharpe Performance-Measure Approach with Short Sales Allowed . . . . . . . . 8.3 Treynor-Measure Approach with Short Sales Allowed . . . . . . . . . . . . . . . . . . . 8.4 Treynor-Measure Approach with Short Sales Not Allowed . . . . . . . . . . . . . . . 8.5 Impact of Short Sales on Optimal-Weight Determination . . . . . . . . . . . . . . . . 8.6 Economic Rationale of the Treynor Performance-Measure Method . . . . . . . . 8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 8A Derivation of Equation (8.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 8B Derivation of Equation (8.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 8C Derivation of Equation (8.15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Creation and Control of Speculative Bubbles in a Laboratory Setting : : : : James S. Ang, Dean Diavatopoulos, and Thomas V. Schwarz 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Bubbles in the Asset Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Portfolio Optimization Models and Mean–Variance Spanning Tests : : : : : : : : : : Wei-Peng Chen, Huimin Chung, Keng-Yu Ho, and Tsui-Ling Hsu 10.1 Introduction of Markowitz Portfolio-Selection Model . . . . . . . . . . . . . . . . . . . 10.2 Measurement of Return and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Efficient Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Mean–Variance Spanning Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Alternative Computer Program to Calculate Efficient Frontier . . . . . . . . . . . . 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Combining Fundamental Measures for Stock Selection : : : : : : : : : : : : : : : : : : : : Kenton K. Yee 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Bayesian Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Triangulation in Forensic Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Bayesian Triangulation in Asset Pricing Settings . . . . . . . . . . . . . . . . . . . . . . . 11.5 The Data Snooping Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Using Guidance from Theory to Mitigate Data Snooping . . . . . . . . . . . . . . . . 11.7 Avoiding Data-Snooping Pitfalls in Financial Statement Analysis . . . . . . . . . 11.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 11A Proof of Theorem 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11A.1 Generalization of Theorem 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 125 128 130 132 132 133 133 133 134 135

137 139 140 145 161 163

165 166 166 172 175 182 184

185 187 189 190 194 195 197 199 200 201 201

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12 On Estimation Risk and Power Utility Portfolio Selection : : : : : : : : : : : : : : : : : : Robert R. Grauer and Frederick C. Shen 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Multiperiod Investment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Alternative Ways of Estimating the Joint Return Distribution . . . . . . . . . . . . . 12.6 Alternate Ways of Evaluating Investment Performance . . . . . . . . . . . . . . . . . . 12.7 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 International Portfolio Management: Theory and Method : : : : : : : : : : : : : : : : : : Wan-Jiun Paul Chiou and Cheng-Few Lee 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Overview of International Portfolio Management . . . . . . . . . . . . . . . . . . . . . . . 13.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Forming the Optimal Global Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 The Benefits of International Diversification Around the World . . . . . . . . . . . 13.6 The Optimal Portfolio Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

14 The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model: A Case of World Equity Fund Market : : : : : : : : : : : : : : : : : : Chin W. Yang, Ken Hung, and Jing Cui 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Le Châtelier Principle in the Markowitz Investment Model . . . . . . . . . . . 14.4 An Application of the Le Châtelier Principle in the World Equity Market . . . 14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 205 206 206 208 210 216 217 218

221 222 226 226 227 229 232 233 235 235 236 236 237 245 245

15 Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints : : : : Darinka Dentcheva and Andrzej Ruszczy´nski 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Stochastic Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 The Dominance-Constrained Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Optimality and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247

16 Portfolio Analysis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Jack Clark Francis 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Inputs for Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The Security Analyst’s Job . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Four Assumptions Underlying Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . 16.5 Different Approaches to Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 A Portfolio’s Expected Return Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 The Quadratic Risk Formula for a Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 The Covariance Between Returns from Two Assets . . . . . . . . . . . . . . . . . . . . .

259

247 248 249 252 254 256 257 257

259 259 259 260 260 261 261 262

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16.9 Portfolio Analysis of a Two-Asset Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.10 Mathematical Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.11 Calculus Minimization of Risk: A Three-Security Portfolio . . . . . . . . . . . . . . 16.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262 265 265 266 266

17 Portfolio Theory, CAPM and Performance Measures : : : : : : : : : : : : : : : : : : : : : : Luis Ferruz, Fernando Gómez-Bezares, and María Vargas 17.1 Portfolio Theory and CAPM: Foundations and Current Application . . . . . . . 17.2 Performance Measures Related to Portfolio Theory and the CAPM: Classic Indices, Derivative Indices, and New Approaches . . . . . . . . . . . . . . . . . . . . . . . 17.3 Empirical Analysis: Performance Rankings and Performance Persistence . . . 17.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

18 Intertemporal Equilibrium Models, Portfolio Theory and the Capital Asset Pricing Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Stephen J. Brown 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Intertemporal Equilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Relationship to Observed Security Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Intertemporal Equilibrium and the Capital Asset Pricing Model . . . . . . . . . . . 18.5 Hansen Jagannathan Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Are Stochastic Discount Factors Positive? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 274 277 280 280 283 283 283 284 285 285 286 286 287

19 Persistence, Predictability, and Portfolio Planning : : : : : : : : : : : : : : : : : : : : : : : : Michael J. Brennan and Yihong Xia 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Detecting and Exploiting Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Stock Price Variation and Variation in the Expected Returns . . . . . . . . . . . . . . 19.4 Economic Significance of Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Forecasts of Equity Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19A The Optimal Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19B The Unconditional Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19C The Myopic Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19D The Optimal Buy-and-Hold Strategy . . . . . . . . . . . . . . . . . . . . . . . . . .

289

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies : : : : : Lan-chih Ho, John Cadle, and Michael Theobald 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Theory of Alternative Portfolio Insurance Strategies . . . . . . . . . . . . . . . . . . . . 20.3 Empirical Comparison of Alternative Portfolio Insurance Strategies . . . . . . . 20.4 Recent Market Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Implications for Financial Market Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319

289 290 296 298 303 314 314 315 316 317 317

319 319 324 329 331 332 332

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Contents

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Reto Francioni, Sonali Hazarika, Martin Reck, and Robert A. Schwartz 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Microstructure’s Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Perfectly Liquid Environment of CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 What Microstructure Analysis Has to Offer: Personal Reflections . . . . . . . . . 21.5 From Theory to Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Deutsche Börse: The Emergence of a Modern, Electronic Market . . . . . . . . . 21.7 Conclusion: The Roadmap and the Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 21A Risk Aversion and Risk Premium Measures . . . . . . . . . . . . . . . . . . . . 21A.1 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.2 Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 21B Designing Xetra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.1 Continuous Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.2 Call Auction Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.3 Electronic Trading for Less Liquid Stocks . . . . . . . . . . . . . . . . . . . . . . 21B.4 Xetra’s Implementation and the Migration of Liquidity to Xetra Since 1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part III

333 333 334 335 339 344 345 347 347 349 349 349 350 350 351 351 352

Options and Option Pricing Theory

22 Options Strategies and Their Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 355 :: Cheng Few Lee, John Lee, and Wei-Kang Shih 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 22.2 The Option Market and Related Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 22.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 22.4 Risk-Return Characteristics of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 22.5 Examples of Alternative Option Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 22.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 23 Option Pricing Theory and Firm Valuation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee, Joseph E. Finnerty, and Wei-Kang Shih 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Basic Concepts of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Factors Affecting Option Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Determining the Value of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.5 Option Pricing Theory and Capital Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.6 Warrants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Applications of the Binomial Distribution to Evaluate Call Options : : : : : : : : : : Alice C. Lee, John Lee, and Jessica Shin-Ying Mai 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 What Is an Option? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 The Simple Binomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4 The Generalized Binomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 24.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

377 377 377 380 384 387 390 391 392 393 393 393 393 395 397 397

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25 Multinomial Option Pricing Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee and Jack C. Lee 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Multinomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 A Lattice Framework for Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 25A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Two Alternative Binomial Option Pricing Model Approaches to Derive Black-Scholes Option Pricing Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee and Carl Shu-Ming Lin 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 The Two-State Option Pricing Model of Rendleman and Bartter . . . . . . . . . . 26.3 The Binomial Option Pricing Model of Cox, Ross, and Rubinstein . . . . . . . . 26.4 Comparison of the Two Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 26A The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Normal, Lognormal Distribution and Option Pricing Model : : : : : : : : : : : : : : : : Cheng Few Lee, Jack C. Lee, and Alice C. Lee 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3 The Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.4 The Lognormal Distribution and Its Relationship to the Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.5 Multivariate Normal and Lognormal Distributions . . . . . . . . . . . . . . . . . . . . . . 27.6 The Normal Distribution as an Application to the Binomial and Poisson Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.7 Applications of the Lognormal Distribution in Option Pricing . . . . . . . . . . . . 27.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bivariate Option Pricing Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee, Alice C. Lee, and John Lee 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 The Bivariate Normal Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3 American Call Option and the Bivariate Normal CDF . . . . . . . . . . . . . . . . . . . 28.4 Valuating American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5 Non-Dividend-Paying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.6 Dividend-Paying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Displaced Log Normal and Lognormal American Option Pricing: A Comparison : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Ren-Raw Chen and Cheng-Few Lee 29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 The American Option Pricing Model Under the Lognormal Process . . . . . . . 29.3 The Geske-Roll-Whaley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 29A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 399 399 402 406 406 406 409 409 409 415 417 418 418 419 421 421 421 422 422 423 425 426 428 428 429 429 429 430 431 433 433 438 438 439 439 439 440 442 442 443

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30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model : George Chalamandaris and A.G. Malliaris 30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 The ITÔ Process and Financial Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 ITÔ’S Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4 Stochastic Differential-Equation Approach to Stock-price Behavior . . . . . . . 30.5 The Pricing of an Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.6 A Reexamination of Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.7 Extending the Risk-Neutral Argument: The Martingale Approach . . . . . . . . . 30.8 Remarks on Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 30A An Alternative Method To Derive the Black–Scholes Option-Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.1 Assumptions and the Present Value of the Expected Terminal Option Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.2 Present Value of the Partial Expectation of the Terminal Stock Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.3 Present Value of the Exercise Price under Uncertainty . . . . . . . . . . . . 31 Constant Elasticity of Variance Option Pricing Model: Integration and Detailed Derivation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Y.L. Hsu, T.I. Lin, and C.F. Lee 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The CEV Diffusion and Its Transition Probability Density Function . . . . . . . 31.3 Review of Noncentral Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 The Noncentral Chi-square Approach to Option Pricing Model . . . . . . . . . . . 31.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 31A Proof of Feller’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

447 447 447 451 452 454 455 458 463 465 465 466 466 467 469 471 471 471 473 474 478 478 478

32 Stochastic Volatility Option Pricing Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee and Jack C. Lee 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Nonclosed-Form Type of Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Review of Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Closed-Form Type of Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 32A The Market Price of the Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

481

33 Derivations and Applications of Greek Letters: Review and Integration : : : : : : Hong-Yi Chen, Cheng-Few Lee, and Weikang Shih 33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Delta () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 Theta .‚/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4 Gamma ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.5 Vega ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.6 Rho ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.7 Derivation of Sensitivity for Stock Options Respective with Exercise Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.8 Relationship Between Delta, Theta, and Gamma . . . . . . . . . . . . . . . . . . . . . . . 33.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

491

481 481 485 485 489 489 489

491 491 494 496 498 500 501 502 503 503

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34 A Further Analysis of the Convergence Rates and Patterns of the Binomial Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : San-Lin Chung and Pai-Ta Shih 34.1 Brief Review of the Binomial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2 The Importance of Node Positioning for Monotonic Convergence . . . . . . . . . 34.3 The Flexibility of GCRR Model for Node Positioning . . . . . . . . . . . . . . . . . . . 34.4 Numerical Results of Various GCRR Models . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 34A Extrapolation Formulas for Various GCRR Models . . . . . . . . . . . . . . 35 Estimating Implied Probabilities from Option Prices and the Underlying : : : : : Bruce Mizrach 35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2 Black Scholes Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.3 Empirical Departures from Black Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.4 Beyond Black Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.5 Histogram Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.6 Tree Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.7 Local Volatility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.8 PDF Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.9 Inferences from the Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.10 Jump Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Are Tails Fat Enough to Explain Smile : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Ren-Raw Chen, Oded Palmon, and John Wald 36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4 Data and Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 36A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.1 The Derivation of the Lognormal Model Under No Rebalancing . . . 36A.2 Continuous Rebalancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.3 Smoothing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.4 Results of Sub-Sample Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Option Pricing and Hedging Performance Under Stochastic Volatility and Stochastic Interest Rates : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Gurdip Bakshi, Charles Cao, and Zhiwu Chen 37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2 The Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.3 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.4 Empirical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 37A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

505 505 506 507 507 510 513 513 515 515 516 517 518 518 520 522 522 524 526 528 528 531 531 532 533 537 541 541 542 542 543 543 544

547 547 549 556 557 571 571 572

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38 Application of the Characteristic Function in Financial Research : : : : : : : : : : : H.W. Chuang, Y.L. Hsu, and C.F. Lee 38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2 The Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.3 CEV Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4 Options with Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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39 Asian Options : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Itzhak Venezia 39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.2 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

583

40 Numerical Valuation of Asian Options with Higher Moments in the Underlying Distribution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Kehluh Wang and Ming-Feng Hsu 40.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2 Definitions and the Basic Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3 Edgeworth Binomial Model for Asian Option Valuation . . . . . . . . . . . . . . . . . 40.4 Upper Bound and Lower Bound for European Asian Options . . . . . . . . . . . . . 40.5 Upper Bound and Lower Bound for American Asian Options . . . . . . . . . . . . . 40.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 The Valuation of Uncertain Income Streams and the Pricing of Options : : : : : : Mark Rubinstein 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Uncertain Income Streams: General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Uncertain Income Streams: Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 41A The Bivariate Normal Density Function . . . . . . . . . . . . . . . . . . . . . . . . 42 Binomial OPM, Black-Scholes OPM and Their Relationship: Decision Tree and Microsoft Excel Approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : John Lee 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Call and Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 One Period Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Two-Period Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Using Microsoft Excel to Create the Binomial Option Trees . . . . . . . . . . . . . . 42.6 Black-Scholes Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.7 Relationship Between the Binomial OPM and the Black-Scholes OPM . . . . 42.8 Decision Tree Black-Scholes Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 42A Excel VBA Code: Binomial Option Pricing Model . . . . . . . . . . . . . .

575 575 576 577 581 581

583 584 586 586

587 587 588 589 591 593 594 602 602 605 605 606 608 611 613 613 614

617 617 617 618 621 622 624 625 626 626 627 627

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Part IV

Risk Management

43 Combinatorial Methods for Constructing Credit Risk Ratings : : : : : : : : : : : : : : : 639 :: Alexander Kogan and Miguel A. Lejeune 43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 43.2 Logical Analysis of Data: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 43.3 Absolute Creditworthiness: Credit Risk Ratings of Financial Institutions . . . 643 43.4 Relative Creditworthiness: Country Risk Ratings . . . . . . . . . . . . . . . . . . . . . . . 648 43.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 Appendix 43A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 44 The Structural Approach to Modeling Credit Risk : : : : : : : : : : : : : : : : : : : : : : : : Jing-zhi Huang 44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Structural Credit Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.3 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 An Empirical Investigation of the Rationales for Integrated Risk-Management Behavior : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Michael S. Pagano 45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.2 Theories of Risk-Management, Previous Research, and Testable Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.3 Data, Sample Selection, and Empirical Methodology . . . . . . . . . . . . . . . . . . . . 45.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

665 665 665 668 671 671 675 675 677 685 689 694 694

46 Copula, Correlated Defaults, and Credit VaR : : : : : : : : : : : : : : : : : : : : : : : : : : : : Jow-Ran Chang and An-Chi Chen 46.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

697

47 Unspanned Stochastic Volatilities and Interest Rate Derivatives Pricing : : : : : : Feng Zhao 47.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.2 Term Structure Models with Spanned Stochastic Volatility . . . . . . . . . . . . . . . 47.3 LIBOR Market Models with Stochastic Volatility and Jumps: Theory and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.4 Nonparametric Estimation of the Forward Density . . . . . . . . . . . . . . . . . . . . . . 47.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 47A The Derivation for QTSMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 47B The Implementation of the Kalman Filter . . . . . . . . . . . . . . . . . . . . . . Appendix 47C Derivation of the Characteristic Function . . . . . . . . . . . . . . . . . . . . . . .

713

697 698 703 710 711

713 716 723 734 746 746 748 750 751

xx

48 Catastrophic Losses and Alternative Risk Transfer Instruments : : : : : : : : : : : : : Jin-Ping Lee and Min-Teh Yu 48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.2 Catastrophe Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.3 Catastrophe Equity Puts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.4 Catastrophe Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.5 Reinsurance with CAT-Linked Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Chuang-Chang Chang, Pei-Fang Hsieh, and Hung-Neng Lai 49.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 49A The Correlations Between the Standard Wiener Process Generated from a Bank’s Net Interest Income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 49B The Risk-Adjusted Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 49C The Discrete Version of the Risk-Adjusted Process . . . . . . . . . . . . . . 50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets C.H. Ted Hong 50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.2 Model Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.3 Default Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.4 Prepayment Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.5 Delinquency Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50A Default and Prepayment Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50B General Model Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50C Default Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50D Prepayment Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 The Effect of Default Risk on Equity Liquidity: Evidence Based on the Panel Threshold Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Huimin Chung, Wei-Peng Chen, and Yu-Dan Chen 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 51A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Put Option Approach to Determine Bank Risk Premium : : : : : : : : : : : : : : : : : : : Dar Yeh Hwang, Fu-Shuen Shie, and Wei-Hsiung Wu 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Evaluating Insurer’s Liability by Option Pricing Model: Merton (1977) . . . . 52.3 Extensions of Merton (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Applications for Merton (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

753 753 753 757 760 763 764 766 767 767 768 771 775 777 777 778 778 778 779 779 780 782 792 797 800 802 802 803 803 805 807 807 808 812 815 815 816 819 819 820 820 823

Contents

xxi

52.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 52A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 52B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Keiretsu Style Main Bank Relationships, R&D Investment, Leverage, and Firm Value: Quantile Regression Approach : : : : : : : : : : : : : : : : : : : : : : : : : : Hai-Chin Yu, Chih-Sean Chen, and Der-Tzon Hsieh 53.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Data and Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Empirical Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

825 826 826 827 829 829 831 831 836 840 841

54 On the Feasibility of Laddering : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Joshua Ronen and Bharat Sarath 54.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

843

55 Stock Returns, Extreme Values, and Conditional Skewed Distribution : : : : : : : Thomas C. Chiang and Jiandong Li 55.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2 The AGARCH Model Based on the EGB2 Distribution . . . . . . . . . . . . . . . . . . 55.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.4 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.5 Distributional Fit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.6 The Implication of the EGB2 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

853

56 Capital Structure in Asia and CEO Entrenchment : : : : : : : : : : : : : : : : : : : : : : : : Kin Wai Lee and Gillian Hian Heng Yeo 56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.2 Prior Research and Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.3 Data and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 56A Variables Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

863

57 A Generalized Model for Optimum Futures Hedge Ratio : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Jang-Yi Lee, Kehluh Wang, and Yuan-Chung Sheu 57.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.2 GIG and GH Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.3 Futures Hedge Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.4 Estimation and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 57A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

873

843 845 849 851 851

853 854 855 856 859 859 861 862

863 864 865 867 871 871 872

873 876 877 879 880 880 881

xxii

58 The Sensitivity of Corporate Bond Volatility to Macroeconomic Announcements : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Nikolay Kosturov and Duane Stock 58.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.2 Theory and Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.3 Data and Return Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.4 Descriptive Statistics of Daily Excess Returns . . . . . . . . . . . . . . . . . . . . . . . . . 58.5 OLS Regressions of Volatility and Excess Returns . . . . . . . . . . . . . . . . . . . . . . 58.6 Conditional Variance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.7 Alternative GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 58A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Raw Material Convenience Yields and Business Cycle : : : : : : : : : : : : : : : : : : : : : Chang-Wen Duan and William T. Lin 59.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.2 Characteristics of Study Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Sheng-Syan Chen, Cheng-Few Lee, and Han-Hsing Lee 60.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60.2 The Traditional Theory of Optimal Capital Structure . . . . . . . . . . . . . . . . . . . . 60.3 Optimal Capital Structure in the Contingent Claims Framework . . . . . . . . . . 60.4 Recent Development of Capital Structure Models . . . . . . . . . . . . . . . . . . . . . . 60.5 Application and Empirical Evidence of Capital Structure Models . . . . . . . . . 60.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Actuarial Mathematics and Its Applications in Quantitative Finance : : : : : : : : : Cho-Jieh Chen 61.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Actuarial Discount and Accumulation Functions . . . . . . . . . . . . . . . . . . . . . . . 61.3 Actuarial Mathematics of Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Actuarial Mathematics of Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Actuarial Premiums and Actuarial Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Applications in Quantitative Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 The Prediction of Default with Outliers: Robust Logistic Regression : : : : : : : : : Chung-Hua Shen, Yi-Kai Chen, and Bor-Yi Huang 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Literature Review of Outliers in Conventional and in Logit Regression . . . . . 62.3 Five Validation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Source of Data and Empirical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

883 883 884 886 886 897 899 903 910 912 913 915 915 917 919 921 922 930 931 933 933 934 936 941 948 950 950 953 953 953 955 958 959 961 963 963 965 965 966 967 969 969 973 976

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63 Term Structure of Default-Free and Defaultable Securities: Theory and Empirical Evidence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 979 Hai Lin and Chunchi Wu 63.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979 63.2 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980 63.3 Bond Pricing in Dynamic Term Structure Model Framework . . . . . . . . . . . . . 980 63.4 Dynamic Term Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 63.5 Models of Defaultable Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 63.6 Interest Rate and Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 63.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 64 Liquidity Risk and Arbitrage Pricing Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1007 Umut Çetin, Robert A. Jarrow, and Philip Protter 64.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 64.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009 64.3 The Extended First Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 64.4 The Extended Second Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 64.5 Example (Extended Black–Scholes Economy) . . . . . . . . . . . . . . . . . . . . . . . . . 1015 64.6 Discontinuous Supply Curve Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 64.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 Appendix 64A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018 65 An Integrated Model of Debt Issuance, Refunding, and Maturity : : : : : : : : : : : : 1025 Manak C. Gupta and Alice C. Lee 65.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025 65.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026 65.3 Operationalizing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 65.4 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032 65.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 Part V

Theory, Methodology, and Applications

66 Business Models: Applications to Capital Budgeting, Equity Value, and Return Attribution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1041 ::: Thomas S. Y. Ho and Sang Bin Lee 66.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041 66.2 The Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 66.3 Simulation Results of the Capital Budgeting Decisions . . . . . . . . . . . . . . . . . . 1045 66.4 Relative Valuation of Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 66.5 Equity Return Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 66.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 Appendix 66A Derivation of the Risk Neutral Probability . . . . . . . . . . . . . . . . . . . . . . 1052 Appendix 66B The Model for the Fixed Operating Cost at Time T . . . . . . . . . . . . . . 1052 Appendix 66C The Valuation Model Using the Recombining Lattice . . . . . . . . . . . . 1053 Appendix 66D Input Data of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054

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67 Dividends Versus Reinvestments in Continuous Time: A More General Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1055 Ren-Raw Chen, Ben Logan, Oded Palmon, and Larry Shepp 67.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 67.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 67.3 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 67.4 Expected Bankruptcy Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058 67.5 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 67.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 68 Segmenting Financial Services Market: An Empirical Study of Statistical and Non-parametric Methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1061 Kenneth Lawrence, Dinesh Pai, Ronald Klimberg, Stephen Kudbya, and Sheila Lawrence 68.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 68.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 68.3 Evaluating the Classification Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 68.4 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 68.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 68.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 69 Spurious Regression and Data Mining in Conditional Asset Pricing Models : : : 1067 Wayne Ferson, Sergei Sarkissian, and Timothy Simin 69.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 69.2 Spurious Regression and Data Mining in Predictive Regressions . . . . . . . . . . 1068 69.3 Spurious Regression, Data Mining, and Conditional Asset Pricing . . . . . . . . . 1069 69.4 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 69.5 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 69.6 Results for Predictive Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 69.7 Results for Conditional Asset Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 69.8 Solutions to the Problems of Spurious Regression and Data Mining . . . . . . . 1086 69.9 Robustness of the Asset Pricing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 69.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 70 Issues Related to the Errors-in-Variables Problems in Asset Pricing Tests : : : : : 1091 Dongcheol Kim 70.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 70.2 The Errors-in-Variables Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092 70.3 A Correction for the Errors-in-Variables Bias . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 70.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 70.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108 71 McMC Estimation of Multiscale Stochastic Volatility Models : : : : : : : : : : : : : : : 1109 German Molina, Chuan-Hsiang Han, and Jean-Pierre Fouque 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109 71.2 Multiscale Modeling and McMC Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 71.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 71.4 Empirical Application: FX Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 71.5 Implication on Derivatives Pricing and Hedging . . . . . . . . . . . . . . . . . . . . . . . . 1118

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71.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119 Appendix 71A Proof of Independent Factor Equivalence . . . . . . . . . . . . . . . . . . . . . . 1119 Appendix 71B Full Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120 72 Regime Shifts and the Term Structure of Interest Rates : : : : : : : : : : : : : : : : : : : : 1121 Chien-Chung Nieh, Shu Wu, and Yong Zeng 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121 72.2 Regime-Switching and Short-Term Interest Rate . . . . . . . . . . . . . . . . . . . . . . . 1122 72.3 Regime-Switching Term Structure Models in Discreet Time . . . . . . . . . . . . . . 1126 72.4 Regime-Switching Term Structure Models in Continuous Time . . . . . . . . . . . 1128 72.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 73 ARM Processes and Their Modeling and Forecasting Methodology : : : : : : : : : : 1135 Benjamin Melamed 73.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 73.2 Overview of ARM Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 73.3 The ARM Modeling Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139 73.4 The ARM Forecasting Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140 73.5 Example: ARM Modeling of an S&P 500 Time Series . . . . . . . . . . . . . . . . . . 1145 73.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1149 74 Alternative Econometric Methods for Information-based Equity-selling Mechanisms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1151 Lee Cheng-Few and Yi Lin Wu 74.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1151 74.2 The Information Contents of Equity-Selling Mechanisms . . . . . . . . . . . . . . . . 1152 74.3 Alternative Econometric Methods for Information-Based Equity-Selling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153 74.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162 75 Implementation Problems and Solutions in Stochastic Volatility Models of the Heston Type : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1165 Jia-Hau Guo and Mao-Wei Hung 75.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165 75.2 The Transform-Based Solution for Heston’s Stochastic Volatility Model . . . 1165 75.3 Solutions to the Discontinuity Problem of Heston’s Formula . . . . . . . . . . . . . 1168 75.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171 76 Revisiting Volume vs. GARCH Effects Using Univariate and Bivariate GARCH Models: Evidence from U.S. Stock Markets : : : : : : : : : : : : : : : : : : : : : : 1173 Zhuo Qiao and Wing-Keung Wong 76.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173 76.2 The Mixture of Distribution Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 76.3 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 76.4 Empirical Findings in NYSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 76.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179 Appendix 76A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1180

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77 Application of Fuzzy Set Theory to Finance Research: Method and Application : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1183 Shin-Yun Wang and Cheng Few Lee 77.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183 77.2 Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184 77.3 Applications of Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1190 77.4 A Example of Fuzzy Binomial OPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194 77.5 An Example of Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196 77.6 Fuzzy Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 77.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199 78 Hedonic Regression Analysis in Real Estate Markets: A Primer : : : : : : : : : : : : : 1201 Ben J. Sopranzetti 78.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1201 78.2 The Theoretical Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1201 78.3 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 78.4 The Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 78.5 Empirical Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 78.6 The Semi-Log Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204 78.7 The Box-Cox Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 78.8 Problems with Hedonic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 78.9 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206 78.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 79 Numerical Solutions of Financial Partial Differential Equations : : : : : : : : : : : : : 1209 Gang Nathan Dong 79.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209 79.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209 79.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210 79.4 Finite Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210 79.5 Finite Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217 79.6 Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218 79.7 Empirical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219 79.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221 80 A Primer on the Implicit Financing Assumptions of Traditional Capital Budgeting Approaches : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1223 Ivan E. Brick and Daniel G. Weaver 80.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223 80.2 Textbook Approaches to NPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224 80.3 Theoretical Valuation of Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226 80.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228 80.5 Personal Tax and Miller Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229 80.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 81 Determinants of Flows into U.S.-Based International Mutual Funds : : : : : : : : : 1235 Dilip K. Patro 81.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235 81.2 Motivation and Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236

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81.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237 81.4 Methodology and Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238 81.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253 Appendix 81A Econometric Analysis of Panel Data . . . . . . . . . . . . . . . . . . . . . . . . . . 1253 82 Predicting Bond Yields Using Defensive Forecasting : : : : : : : : : : : : : : : : : : : : : : : 1257 Glenn Shafer and Samuel Ring 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257 82.2 Game-Theoretic Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260 82.3 Defensive Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265 82.4 Predicting Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269 82.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271 83 Range Volatility Models and Their Applications in Finance : : : : : : : : : : : : : : : : : 1273 Ray Yeutien Chou, Hengchih Chou, and Nathan Liu 83.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273 83.2 The Price Range Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274 83.3 The Range-Based Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276 83.4 The Realized Range Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278 83.5 The Financial Applications and Limitations of the Range Volatility . . . . . . . . 1279 83.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280 84 Examining the Impact of the U.S. IT Stock Market on Other IT Stock Markets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1283 Zhuo Qiao, Venus Khim-Sen Liew, and Wing-Keung Wong 84.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283 84.2 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284 84.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 84.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289 Appendix 84A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290 85 Application of Alternative ODE in Finance and Economics Research : : : : : : : : 1293 Cheng-Few Lee and Junmin Shi 85.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293 85.2 Ordinary Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294 85.3 Applications of ODE in Deterministic System . . . . . . . . . . . . . . . . . . . . . . . . . 1295 85.4 Applications of ODE in Stochastic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297 85.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300 86 Application of Simultaneous Equation in Finance Research : : : : : : : : : : : : : : : : 1301 Carl R. Chen and Cheng Few Lee 86.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301 86.2 Two-Stage and Three-Stage Least Squares Method . . . . . . . . . . . . . . . . . . . . . 1302 86.3 Application of Simultaneous Equation in Finance Research . . . . . . . . . . . . . . 1305 86.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306

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87 The Fuzzy Set and Data Mining Applications in Accounting and Finance : : : : : 1307 Wikil Kwak, Yong Shi, and Cheng-Few Lee 87.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307 87.2 A Fuzzy Approach to International Transfer Pricing . . . . . . . . . . . . . . . . . . . . 1307 87.3 A Fuzzy Set Approach to Human Resource Allocation of a CPA Firm . . . . . 1312 87.4 A Fuzzy Set Approach to Accounting Information System Selection . . . . . . . 1316 87.5 Fuzzy Set Formulation to Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 87.6 A Data Mining Approach to Firm Bankruptcy Predictions . . . . . . . . . . . . . . . 1324 87.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 88 Forecasting S&P 100 Volatility: The Incremental Information Content of Implied Volatilities and High-Frequency Index Returns : : : : : : : : : : : : : : : : : 1333 Bevan J. Blair, Ser-Huang Poon, and Stephen J. Taylor 88.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333 88.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334 88.3 Methodology for Forecasting Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 88.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338 88.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344 89 Detecting Structural Instability in Financial Time Series : : : : : : : : : : : : : : : : : : : 1345 Derann Hsu 89.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345 89.2 Genesis of the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345 89.3 Problems of Multiple Change Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347 89.4 Here Came the GARCH and Its Brethrens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348 89.5 Examples of Structural Shift Analysis in Financial Time Series . . . . . . . . . . . 1349 89.6 Implications of Structural Instability to Financial Theories and Practice . . . . 1352 89.7 Direction of Future Research and Developments . . . . . . . . . . . . . . . . . . . . . . . 1353 89.8 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354 90 The Instrument Variable Approach to Correct for Endogeneity in Finance : : : 1357 Chia-Jane Wang 90.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357 90.2 Endogeneity: The Statistical Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358 90.3 Instrumental Variables Approach to Endogeneity . . . . . . . . . . . . . . . . . . . . . . . 1358 90.4 Validity of Instrumental Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1361 90.5 Identification and Inferences with Weak Instruments . . . . . . . . . . . . . . . . . . . . 1364 90.6 Empirical Applications in Corporate Finance . . . . . . . . . . . . . . . . . . . . . . . . . . 1366 90.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368 91 Bayesian Inference of Financial Models Using MCMC Algorithms : : : : : : : : : : 1371 Xianghua Liu, Liuling Li, and Hiroki Tsurumi 91.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371 91.2 Bayesian Inference and MCMC Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371 91.3 CKLS Model with ARMA-GARCH Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374 91.4 Copula Model for FTSE100 and S&P500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376 91.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1380

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92 On Capital Structure and Entry Deterrence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1381 Fathali Firoozi and Donald Lien 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1381 92.2 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382 92.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384 92.4 Capital Structure and Entry Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 92.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 93 VAR Models: Estimation, Inferences, and Applications : : : : : : : : : : : : : : : : : : : : 1391 Yangru Wu and Xing Zhou 93.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 93.2 A Brief Discussion of VAR Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 93.3 Applications of VARs in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393 93.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 94 Signaling Models and Product Market Games in Finance: Do We Know What We Know? : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1399 Kose John and Anant K. Sundaram 94.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399 94.2 Supermodularity: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1400 94.3 Supermodularity in Signaling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1400 94.4 Supermodularity in Product Market Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403 94.5 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406 94.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 95 Estimation of Short- and Long-Term VaR for Long-Memory Stochastic Volatility Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1409 Hwai-Chung Ho and Fang-I Liu 95.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409 95.2 Long Memory in Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410 95.3 VaR Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411 95.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414 96 Time Series Modeling and Forecasting of the Volatilities of Asset Returns : : : : 1417 Tze Leung Lai and Haipeng Xing 96.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 96.2 Conditional Heteroskedasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 96.3 Regime-Switching, Change-Point and Spline-GARCH Models of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421 96.4 Multivariate Volatility Models and Applications to Mean–Variance Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424 96.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425 97 Listing Effects and the Private Company Discount in Bank Acquisitions : : : : : 1427 Atul Gupta and Lalatendu Misra 97.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427 97.2 Why Acquiring Firms May Pay Less for Unlisted Targets . . . . . . . . . . . . . . . . 1428 97.3 Sample Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1430 97.4 Event Study Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431 97.5 Findings Based on Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433

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97.6 Cross-Sectional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439 97.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1442 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443 98 An ODE Approach for the Expected Discounted Penalty at Ruin in Jump Diffusion Model (Reprint) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1445 Yu-Ting Chen, Cheng-Few Lee, and Yuan-Chung Sheu 98.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445 98.2 Integro-Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446 98.3 Explicit Formula for ˆ – ODE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448 98.4 The Constant Vector Q: Second Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453 98.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458 Appendix 98A Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458 Appendix 98B Toolbox for Phase-Type Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 1462 Appendix 98C First Order Derivative of ˆ at Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1462 99 Alternative Models for Estimating the Cost of Equity Capital for Property/Casualty Insurers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1465 Alice C. Lee and J. David Cummins 99.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465 99.2 Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466 99.3 Model-Specification and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467 99.4 Data Description and Cost of Equity Capital Estimates . . . . . . . . . . . . . . . . . . 1470 99.5 Evaluations of Simulations and Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476 99.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1480 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1481 100 Implementing a Multifactor Term Structure Model : : : : : : : : : : : : : : : : : : : : : : : 1483 Ren-Raw Chen and Louis O. Scott 100.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483 100.2 A Multifactor Term Structure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483 100.3 Pricing Options in the Multifactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485 100.4 Calibrating a Multifactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 100.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488 101 Taking Positive Interest Rates Seriously : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1489 Enlin Pan and Liuren Wu 101.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1489 101.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1490 101.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1491 101.4 The Hump-Shaped Forward Rate Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494 101.5 Fitting the US Treasury Yields and US Dollar Swap Rates . . . . . . . . . . . . . . . 1495 101.6 Extensions: Jumps in Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498 101.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1500 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1500 Appendix 101A Factor Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1501 Appendix 101B Extended Kalman Filter and Quasilikelihood . . . . . . . . . . . . . . . . . . 1502

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102 Positive Interest Rates and Yields: Additional Serious Considerations : : : : : : : : 1503 Jonathan Ingersoll 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503 102.2 A Non-Zero Bound for Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503 102.3 The Cox–Ingersoll–Ross and Pan–Wu Term Structure Models . . . . . . . . . . . . 1504 102.4 Bubble-Free Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506 102.5 Multivariate Affine Term-Structure Models with Zero Bounds on Yields . . . 1511 102.6 Non-Affine Term Structures with Yields Bounded at Zero . . . . . . . . . . . . . . . . 1514 102.7 Non-Zero Bounds for Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516 102.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 Appendix 102A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 102A.1 Derivation of the Probability and State price for rT D 0 for the PW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 102A.2 Bond Price When rt D 0 Is Accessible for Only the Risk-Neutral Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519 102A.3 Properties of the Affine Exponentially Smoothed Average Model . . 1520 102A.4 Properties of the Three-Halves Power Interest Rate Process . . . . . . . 1521 103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1523 Cheng-Few Lee, Dilip K. Patro, and Bo Liu 103.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523 103.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524 103.3 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526 103.4 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527 103.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534 103.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553 104 A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1555 Michael Mania, Marina Santacroce, and Revaz Tevzadze 104.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555 104.2 Some Basic Definitions, Conditions, and Auxiliary Facts . . . . . . . . . . . . . . . . 1556 104.3 Backward Semimartingale Equation for the Value Process . . . . . . . . . . . . . . . 1558 104.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1564 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565 105 The Density Process of the Minimal Entropy Martingale Measure in a Stochastic Volatility Model with Jumps (Reprint) : : : : : : : : : : : : : : : : : : : : : 1567 Fred Espen Benth and Thilo Meyer-Brandis 105.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567 105.2 The Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1568 105.3 The Minimal Entropy Martingale Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569 105.4 The Density Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1571 105.5 The Entropy Price of Derivatives and Integro-Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573 105.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1574 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575

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106 Arbitrage Detection from Stock Data: An Empirical Study : : : : : : : : : : : : : : : : : 1577 Cheng-Der Fuh and Szu-Yu Pai 106.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577 106.2 Arbitrage Detection: Volatility Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579 106.3 Arbitrage Detection: Mean Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583 106.4 Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586 106.5 Conclusions and Further Researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1590 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1591 107 Detecting Corporate Failure : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1593 Yanzhi Wang, Lin Lin, Hsien-Chang Kuo, and Jenifer Piesse 107.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593 107.2 The Possible Causes of Bankruptcy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594 107.3 The Methods of Bankruptcy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594 107.4 Prediction Model for Corporate Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596 107.5 The Selection of Optimal Cutoff Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603 107.6 Recent Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604 107.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604 108 Genetic Programming for Option Pricing : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1607 N.K. Chidambaran 108.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607 108.2 Genetic Program Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608 108.3 Black–Scholes Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611 108.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613 108.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614 109 A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions in Option Pricing, Review, and Integration : : : : : : : : : : : : : : 1615 Ren-Raw Chen and Cheng-Few Lee 109.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615 109.2 The CEV Diffusion and Its Transition Density . . . . . . . . . . . . . . . . . . . . . . . . . 1616 109.3 The CEV Option Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619 109.4 Computing the Non-Central Chi-Square Probabilities . . . . . . . . . . . . . . . . . . . 1622 109.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623 Appendix 109A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1627 Author Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1685 Subject Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1709

Contents

List of Contributors

James S. Ang, Florida State University, Tallahassee, FL, USA Gurdip Bakshi, University of Maryland, College Park, MD, USA Hamid Beladi, University of Texas at San Antonio, San Antonio, TX, USA Fred Espen Benth, University of Oslo and Agder University College, Kristiansand, Norway Bevan J. Blair, Ingenious Asset Management, London, UK Michael J. Brennan, University of California at Los Angeles, Los Angeles, CA, USA Ivan Brick, Rutgers University, Newark, NJ, USA Stephen J. Brown, New York University, New York, NY, USA John Cadle, University of Birmingham, Birmingham, UK Charles Cao, Department of Finance, Smeal College of Business, Pennsylvania State University, University Park, PA, USA Umut Çetin, Technische Universität Wien, Vienna, Austria George Chalamandaris, Athens University of Economics and Business, Athens, Greece Chuang-Chang Chang, National Central University, Taipei, Taiwan, ROC Jow-Ran Chang, National Tsing Hua University, Hsinchu, Taiwan, ROC An-Chi Chen, KGI Securities Co. Ltd., Taipei, Taiwan, ROC Carl R. Chen, University of Dayton, Dayton, OH, USA Chih-Sean Chen, Chung Yuan University, Taoyuan County, Taiwan, ROC Cho-Jieh Chen, University of Alberta, Edmonton, AB, Canada Hong-Yi Chen, Rutgers University, Newark, NJ, USA Ren-Raw Chen, Fordham University, New York, NY, USA Sheng-Syan Chen, National Taiwan University, Taipei, Taiwan, ROC Wei-Peng Chen, Shih Hsin University, Taipei, Taiwan, ROC Yi-Kai Chen, National University of Kaohsiung, Kaohsiung, Taiwan, ROC Yu-Dan Chen, National Chiao Tung University, Hsinchu, Taiwan, ROC Yu-Ting Chen, National Chao Tung University, Hsinchu, Taiwan, ROC Zhiwu Chen, Yale University, New Haven, CT, USA

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Thomas C. Chiang, Drexel University, Philadelphia, PA, USA N. K. Chidambaran, Fordham University, New York, NY, USA Wan-Jiun Paul Chiou, Shippensburg University, Shippensburg, PA, USA Heng-chih Chou, Ming Chuan University, Taipei, Taiwan, ROC Ray Y. Chou, Academia Sinica, Taipei, Taiwan, ROC H.W. Chiang, National Taiwan University, Taipei, Taiwan, ROC Huimin Cheng, National Chiao Tung University, Hsinchu, Taiwan, ROC San-Lin Cheng, National Taiwan University, Taipei, Taiwan, ROC Jing Cui, Clarion University of Pennsylvania, Clarion, PA, USA J. D. Cumming, Temple University, Philadelphia, PA, USA Darinka Dentcheva, Stevens Institute of Technology, Hoboken, NJ, USA Dean Diavatopoulos, Villanova University, Philadelphia, PA, USA Gang Nathan Dong, Rutgers University, Newark, NJ, USA Chang-Wen Duan, Tamkang University, Taipei, Taiwan, ROC Luis Ferruz, University of Zaragoza, Zaragoza, Spain Wayne Fresón, University of Southern California, Los Angeles, CA, USA Joseph E. Finnerty, University of Illinois at Urbana-Champaign, Champaign, IL, USA Fathali Firoozi, University of Texas at San Antonio, San Antonio, TX, USA Jean-Pierre Fouque, University of California, Santa Barbara, CA, USA Reto Francioni, Deutsche Börse, Frankfurt, Germany Jack Clark Francis, Baruch College, New York, NY, USA Cheng-Der Fuh, National Central University and Academia Sinica, Taipei, Taiwan, ROC Fernando Gómez-Bezares, University of Deusto, Bilbao, Spain Robert R. Grauer, Simon Fraser University, Burnaby, BC, Canada Jia-Hau Guo, Soochow University, Taipei, Taiwan, ROC Atul Gupta, Bentley University, Waltham, MA, USA Manak C. Gupta, Temple University, Philadelphia, PA, USA Chuan-Hsiang Han, National Tsing Hua University, Hsinchu, Taiwan, ROC Sonali Hazarika, Baruch College, New York, NY, USA Hwai-Chung Ho, Academia Sinica and National Taiwan University, Taipei, Taiwan, ROC Keng-Yu Ho, National Taiwan University, Taipei, Taiwan, ROC Lan-chih Ho, Central Bank of the Republic of China, Taipei, Taiwan, ROC Thomas S. Y. Ho, Thomas Ho Company, Ltd, New York, NY, USA C.H. Ted Hong, Beyondbond Inc., New York, NY, USA Tsui-Ling Hseu, National Chiao Tung University, Hsinchu, Taiwan, ROC Der-Tzon Hsieh, National Taiwan University, Taipei, Taiwan, ROC

List of Contributors

List of Contributors

xxxv

Pei-Fang Hsieh, Department of Finance, National Central University, Chung Li City, Taiwan, ROC Derann Hsu, University of Wisconsin–Milwaukee, Milwaukee, WI, USA Ming-Feng Hsu, Tatung University, Taipei, Taiwan, ROC Ying Lin Hsu, National Chung Hsing University, Taichung, Taiwan, ROC Bor-Yi Huang, Department of Business Management, Shih Chien University, Taipei, Taiwan, ROC Dar-Yeh Huang, National Taiwan University, Taipei, Taiwan, ROC Jingzhi Huang, Pennsylvania State University, University Park, PA, USA Ken Hung, Texas A&M International University, Laredo, TX, USA Mao-Wei Hung, National Taiwan University, Taipei, Taiwan, ROC Jonathan E. Ingersoll, Jr., Yale School of Management, New Haven, CT, USA Robert A. Jarrow, Cornell University, Ithaca, NY, USA Kose John, New York University, New York, NY, USA Dongcheol Kim, Korea University Business School, Seoul, Korea Ronald Klimberg, St. Joseph’s University, Philadelphia, PA, USA Alexander Kogan, Rutgers University, Newark, NJ, USA Nikolay Kosturov, University of Oklahoma, Norman, OK, USA Stephen Kudbya, New Jersey Institute of Technology, Newark, NJ, USA Hsien-chang Kuo, National Chi-Nan University and Takming University of Science and Technology, Nantou Hsien, Taiwan, ROC Wikil Kwak, University of Nebraska at Omaha, Omaha, NE, USA Hung-Neng Lai, Department of Finance, National Central University, Chung Li City, Taiwan, ROC Tze Leung Lai, Stanford University, Stanford, CA, USA Kenneth Lawrence, New Jersey Institute of Technology, Newark, NJ, USA Sheila Lawrence, Rutgers University, Newark, NJ, USA Alice C. Lee, State Street Corp., Boston, MA, USA Cheng-Few Lee, Rutgers University, New Brunswick, NJ, USA and National Chiao Tung University, Hsinchu, Taiwan, ROC Han-Hsing Lee, National Chiao Tung University, Hsinchu, Taiwan, ROC Jack C. Lee, National Chiao Tung University, Hsinchu, Taiwan, ROC Jang-Yi Lee, Tunghai University, Taichung, Taiwan, ROC Jin-Ping Lee, Feng Chia University, Taichung, Taiwan, ROC John Lee, Center for PBBEF Research, Hackensack, NJ, USA Kin Wai Lee, Nanyang Technological University, Singapore, Singapore Sang Bin Lee, Hanyang University, Seoul, Korea Miguel A. Lejeune, George Washington University, Washington, DC, USA

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Jiandong Li, Central University of Finance and Economics, P.R. China Liuling Li, Rutgers University, New Brunswick, NJ, USA Donald Lien, University of Texas at San Antonio, San Antonio, TX, USA Venus Khim-Sen Liew, Universiti Malaysia Sabah, Sabah, Malaysia Carle Shu Ming Lin, Rutgers University, New Brunswick, NJ, USA Hai Lin, Xiamen University, Xiamen, Fujian, China Lin Lin, Department of Banking and Finance, National Chi-Nan University, 1 University Rd., Puli, Nantou Hsien, Taiwan 545, ROC T. I. Lin, National Chung Hsing University, Taichung, Taiwan, ROC William T. Lin, Tamkang University, Taipei, Taiwan, ROC Bo Liu, Citigroup Global Market Inc., New York, NY, USA Fang-I Liu, National Taiwan University, Taipei, Taiwan, ROC Nathan Liu, National Chiao Tung University, Hsinchu, Taiwan, ROC Xianghua Liu, Rutgers University, Piscataway, NJ, USA Ben Logan, Bell Labs, USA Jessica Mai, Rutgers University, Newark, NJ, USA A.G. Malliaris, Loyola University Chicago, Chicago, IL, USA Michael Mania, A. Razmadze Mathematical Institute, Georgia and Georgian-American University, Tbilisi, Georgia Benjamin Melamed, Rutgers Business School, Newark and New Brunswick, NJ, USA Thilo Meyer-Brandis, University of Oslo, Oslo, Norway Lalatendu Misra, University of Texas at San Antonio, San Antonio, TX, USA Bruce Mizrach, Rutgers University, New Brunswick, NJ, USA German Molina, Statistical and Applied Mathematical Sciences Institute, NC, USA Chien-Chung Nieh, Tamkang University, Taipei, Taiwan, ROC Michael S. Pagano, Villanova University, Philadelphia, PA, USA Dinesh Pai, Rutgers University, Newark, NJ, USA Szu-Yu Pai, National Taiwan University, Taipei, Taiwan, ROC Oded Palmon, Rutgers University, New Brunswick, NJ, USA Enlin Pan, Chicago Partners, Chicago, IL USA Dilip K. Patro, Office of the Comptroller of the Currency, Washington, DC, USA Jenifer Piesse, University of London, London, UK Ser-Huang Poon, University of Manchester, Manchester, UK Philip Protter, Cornell University, Ithaca, NY, USA Zhuo Qiao, University of Macau, Macau, China Martin Reck, Deutsche Börse, Frankfurt, Germany Samuel Ring, Rutgers University, Newark, NJ, USA

List of Contributors

List of Contributors

xxxvii

Joshua Ronen, New York University, New York, NY, USA Mark Rubinstein, University of California, Berkley, CA, USA Andrzej Ruszczynski, Rutgers University, Newark, NJ, USA Marina Santacroce, Politecnico di Torino, Department of Mathematics, C.so Duca degli Abruzzi 24, 10129 Torino, Italy Bharat Sarath, Baruch College, New York, NY, USA Sergei Sarkissian, McGill University, Montreal, QC, Canada Robert A. Schwartz, Baruch College, New York, NY, USA Thomas V. Schwarz, Grand Valley State University, Allendale, MI, USA Louis O. Scott, Morgan Stanley, New York, NY, USA Glenn Shafer, Rutgers University, Newark, NJ USA Chung-Hua Shen, National Taiwan University, Taipei, Taiwan, ROC Frederick C. Shen, Coventree Inc, Toronto, ON, Canada Larry Shepp, Rutgers University, Piscataway, NJ, USA Yuan-Chung Sheu, National Chao Tung University, Hsinchu, Taiwan, ROC Junmin Shi, Rutgers University, Newark, NJ, USA Yong Shi, University of Nebraska at Omaha, Omaha, NE, USA and Chinese Academy of Sciences, Beijing, China Fu-Shuen Shie, National Taiwan University, Taipei, Taiwan, ROC Pai-Ta Shih, Department of Finance, National Taiwan University, Taipei 106, Taiwan, ROC Wei-Kang Shih, Rutgers University, Newark, NJ, USA Timothy Simin, Pennsylvania State University, University Park, PA, USA Ben J. Sopranzetti, Rutgers University, Newark, NJ, USA Duane Stock, University of Oklahoma, Norman, OK, USA Anant Sunderam, Tuck School, Hanover, NH, USA Stephen J. Taylor, Lancaster University, Lancaster, UK Revaz Tevzadze, Institute of Cybernetics, Georgia and Georgian-American University, Tbilisi, Georgia Michael Theobald, Accounting and Finance Subject Group, University of Birmingham, Birmingham, UK Hiroki Tsurumi, Rutgers University, New Brunswick, NJ, USA María Vargas, University of Zaragoza, Zaragoza, Aragon, Spain Itzhak Venezia, Hebrew University, Jerusalem, Israel John Wald, Pennsylvania State University, University Park, PA, USA Chia-Jane Wang, Manhattan College, New York, NY, USA Kehluh Wang, National Chiao Tung University, Hsinchu, Taiwan, ROC

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Shin-Yun Wang, National Dong Hwa University, Hualien, Taiwán, ROC Yanzhi Wang, Yuan Ze University, Taoyuan, Taiwán, ROC Daniel Weaver, Rutgers University, Piscataway, NJ, USA Wing-Keung Wong, Hong Kong Baptist University, Hong Kong, Kowloon Tong, Hong Kong Donald H. Wort, California State University East Bay, Hayward, CA, USA ChunChi Wu, University of Missouri, Columbia, MO, USA Liuren Wu, Baruch College, New York, NY, USA Shu Wu, The University of Kansas, Lawrence, KS, USA Wei-Hsiung Wu, National Taiwan University, Taipei, Taiwan, ROC Yangru Wu, Rutgers Business School, Newark and New Brunswick, NJ, USA Yi Lin Wu, National Tsing Hua University, Hsinchu, Taiwan, ROC Yihong Xia, Wharton School, Pennsylvania, PA, USA Haipeng Xing, SUNY at Stony Brook, Stony Brook, NY, USA Chin W. Yang, Clarion University of Pennsylvania, Clarion, PA, USA Kenton K. Yee, Columbia Business School, New York, NY, USA Gillian Hian Heng Yeo, Nanyang Technological University, Singapore, Singapore Hai-Chin Yu, Chung Yuan University, Taoyuan, Taiwan, ROC Min-Teh Yu, Providence University, Taichung, Taiwan, ROC Yong Zeng, The University of Missouri at Kansas City, Kansas City, MO, USA Feng Zhao, Rutgers University, Newark, NJ, USA Xing Zhou, Rutgers Business School, Newark and New Brunswick, NJ, USA

List of Contributors

Chapter 1

Theoretical Framework of Finance

Abstract The main purpose of this chapter is to explore important finance theories. First, we discuss discounted cash-flow valuation theory (classical financial theory). Second, we discuss the Modigliani and Miller (M and M) valuation theory. Third, we examine Markowitz portfolio theory. We then move on to the capital asset pricing model (CAPM), followed by the arbitrage pricing theory. Finally, we will look at the option pricing theory and futures valuation and hedging. Keywords Discounted cash-flow valuation r M and M valuation theory r Markowitz portfolio theory r Capital asset pricing model r Arbitrage pricing theory r Option pricing model r Futures valuation and hedging

1.1 Introduction Value determination of financial instruments is important in security analysis and portfolio management. Valuation theories are the basic tools for determining the intrinsic value of alternative financial instruments. This chapter provides a general review of the financial theory that most students of finance would have already received in basic corporate finance and investment classes. Synthesis and integration of the valuation theories are necessary for the student of investments in order to have a proper perspective of security analysis and portfolio management. The basic policy areas involved in the management of a company are (1) investment policy, (2) financial policy, (3) dividend policy, and (4) production policy. Since the determination of the market value of a firm is affected by the way management sets and implements these policies, they are of critical importance to the security analyst. The security analyst must evaluate management decisions in each of these areas and convert information about company policy into price estimates of the firm’s securities. This chapter examines these policies within a financial theory framework, dealing with valuation models.

There are six alternative but interrelated valuation models of financial theory that might be useful for the analysis of securities and the management of portfolios: 1. Discounted cash-flow valuation theory (classical financial theory) 2. M and M valuation theory 3. Capital asset pricing model (CAPM) 4. Arbitrage Pricing Theory (APT) 5. Option-pricing theory (OPT) 6. Futures Valuation and Hedging The discounted cash-flow valuation and M and M theories are discussed in the typical required corporate-finance survey course for both bachelor’s and master’s programs in business. The main purpose of this chapter is to review these theories and discuss their interrelationships. The discounted cash-flow model is first reviewed by some of the basic valuation concepts in Sect. 1.2. In the second section, the four alternative evaluation methods developed by M and M in their 1961 article are discussed. Their three propositions and their revision with taxes are explored, including possible applications of their theories in security analysis. Miller’s inclusion of personal taxes is discussed in Sect. 1.3. Section 1.4 discusses the Markowitz portfolio theory. Section 1.5 includes a brief overview of CAPM concepts. Section 1.6 introduces the Arbitrage Pricing Theory (APT). Sections 1.6 and 1.7 discuss the option-pricing theory and the futures valuation and hedging. Conclusion is presented in Sect. 1.8.

1.2 Discounted Cash-Flow Valuation Theory Discounted cash-flow valuation theory is the basic tool for determining the theoretical price of a corporate security. The price of a corporate security is equal to the present value of future benefits of ownership. For example, for common stock, these benefits include dividends received while the stock is owned plus capital gains earned during the ownership period. If we assume a one-period investment and a world of certain cash flows, the price paid for a share of

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_1, 

3

4

1

stock, P0 , will equal the sum of the present value of a certain dividend per share, d1 (assumed to be paid as a single flow at year end), and the selling price per share P1 : P0 D

d1 C P1 1Ck

(1.1)

in which k is the rate of discount assuming certainty. P1 can be similarly expressed in terms of d2 and P2 : P1 D

d2 C P2 1Ck

(1.2)

If P1 in Equation (1.1) is substituted into Equation (1.2), a two-period expression is derived: P0 D

d2 d1 P2 C C .1 C k/ .1 C k/2 .1 C k/2

(1.3)

It can be seen, then, that an infinite time-horizon model can be expressed as the P0 D

1 X t D1

dt .1 C k/t

MV 0 D

1 X t D1

Dt .1 C k/t

where: PV D present value of the bond; n D the number of periods to maturity; CFt D the cash flow (interest and principal) received in period t; kb D the required rate of return of the bondholders (equal to risk-free rate i plus a risk premium).

1.2.1.1 Perpetuity The first (and most extreme) case of bond valuation involves a perpetuity, a bond with no maturity date and perpetual interest payments. Such bonds do exist. In 1814, the English government floated a large bond issue to consolidate the various small issues it had used to pay for the Napoleonic Wars. Such bonds are called consols, and the owners are entitled to a fixed amount of interest income annually in perpetuity. In this case, Equation (1.6) collapses into the following: PV D

(1.4)

Since the total market value of the firms’ equity is equal to the market price per share multiplied by the number of shares outstanding, Equation (1.4) may be re-expressed in terms of total market value MV 0 : (1.5)

Theoretical Framework of Finance

CF kb

(1.7)

Thus, the valuation depends directly on the periodic interest payment and the required rate of return for the bond. It can be seen that required rates of return, necessitated by a higher rate of inflation or an increase in the perceived risk of the bond, lower the present value, decreasing the bond’s market value. For example, if the stated annual interest payment on the perpetuity bond is $50 and the required rate of return in the market is 10%, the price of the security is stated: PV D $50=0:10 D $500

in which Dt D total dollars of dividends paid during year t. Using this basic valuation approach as a means of expressing the appropriate objective of the firm’s management, the valuation of a firm’s securities can be analyzed in a world of certainty.

If its issuing price had been $1,000, it can be seen that the required rate of return would have been only 5% (kb D CF=PV D $50=$1; 000 D 0:05, or 5%).

1.2.1 Bond Valuation

1.2.1.2 Term Bonds

Bond valuation is a relatively easy process, as the income stream the bondholder will receive is known with a high degree of certainty. Barring a firm’s default, the income stream consists of the periodic coupon payments and the repayments of the principal at maturity. These cash flows must be discounted to the present using the required rate of return for the bond. The basic principles of bond valuation are represented in the equation: n X CFt (1.6) PV D .1 C kb /t t D1

Most bonds are term bonds, which mature at some definite point in time. Thus, Equation (1.6) should be respecified to take this fact into account: PV D

n X t D1

It Pn C .1 C kb /t .1 C kb /n

(1.8)

where: It D the annual coupon interest payment; Pn D the principal amount (face value) of the bond; and n D the number of periods to maturity.

1 Theoretical Framework of Finance

Again, it should be noted that the market price, PV, of a bond is affected by changes in the rate of inflation. If inflation increases, the discount rate must also increase to compensate the investor for the resultant decrease in the value of the debt repayment. The present value of each period’s interest payment thus decreases, and the price of the bond falls. The bondholder is always exposed to interest-rate risk, the variance of bond prices resulting from fluctuations in the level of interest rates. Interest-rate risk, or price volatility of a bond caused by changes in interest-rate levels, is directly related to the term to maturity. There are two types of risk premiums associated with interest-rate risk as it applies to corporate bonds. The bond maturity premium refers to the net return from investing in long-term government bonds rather than the short-term bills. Since corporate bonds generally possess default risk, another of the components of corporate bond rates of return is default premium. The bond default premium is the net increase in return from investing in long-term corporate bonds rather than in long-term government bonds. Additional features of a bond can affect its valuation. Convertible bonds, those with a provision for conversion into shares of common stock, are generally more valuable than a firm’s straight bonds for several reasons. First, the investor receives the potential of positive gains from conversion, should the market price of a firm’s common stock rise above the conversion price. If the stock price is greater than the conversion price, the convertible bond generally sells at or above its conversion value. Second, the bondholder also receives the protection of fixed income payment, regardless of the current price of the stock – assuring the investor that the price of the bond will be at least equal to that of a straight bond, should stock prices fail to increase sufficiently. Third, for any given firm the coupon rate of return from its bonds is generally greater than the dividend rate of return (dividend yield) from its common stock – thus causing a measure of superiority for a convertible bond over its conversion into common stock until stock dividends rise above the bond’s coupon rate. Even then, the convertible bond may be preferred by investors because of the higher degree of certainty of interest payments versus dividends that would decline if earnings fall. A sinking fund provision may also increase the value of a bond, at least at its time of issue. A sinking-fund agreement specifies a schedule by which the sinking-fund will retire the bond issue gradually over its life. By providing cash to the sinking-fund for use in redeeming the bonds, this provision ensures the investor some potential demand for the bond, thus increasing slightly the liquidity of the investment. Finally, the possibility that the bond may be called will generally lower the value relative to a noncallable bond. A call provision stipulates that the bond may be retired by the issuer at a certain price, usually above par or face value. Therefore, in periods of large downward interest movements,

5

a company may be able to retire a high coupon bond and issue new bonds with a lower interest payment requirement. A call feature increases the risk to investors in that their expected high interest payments may be called away from them, if overall interest rate levels decline.

1.2.2 Common-Stock Valuation Common-stock valuation is complicated by an uncertainty of cash flows to the investor, necessarily greater than that for bond valuation.1 Not only might the dividends voted to shareholders each period change in response to management’s assessment concerning the current level of earnings stability, future earnings prospects, or other factors, but the price of the stock may also either rise or fall – resulting in either capital gains or losses, if the shares are sold. Thus, the valuation process requires the forecasting of both capital gains and the stream of expected dividends. Both must also be discounted at the required rate of return of the common stockholders. P0 D

d2 d1 Pn C CC 1Ck .1 C k/2 .1 C k/n

(1.9)

where: P0 D the present value, or price, of the common stock per share; d D the dividend payment per share; k D the required rate of return of the common stockholders; and Pn D the price of the stock in period n when sold. However, Pn can also be expressed as the sum of all discounted dividends to be received from period n forward into the future. Thus, the value at the present time can be expressed as an infinite series of discounted dividend payments: 1 X dt (1.4) P0 D .1 C k/t t D1 in which dt is the dividend payment in period t. Several possibilities exist regarding the growth of dividend payments over time. First, dividends may be assumed to be a constant amount, and the formula for the stock’s valuation is simple Equation (1.7), where CF is the constant dividend and k is the required rate of return of the common stockholder. Second, dividends may be expected to grow at some constant rate, g. In such a case, a dividend at time t is simply the compound value of the present dividend (i.e., 1

This is true because foregoing interest puts the firm into default, while missing dividend payments does not.

6

1

Pt D .1 C g/t d0 ). Under this assumption, if g < k, the valuation equation can be simplified to the following: P0 D

d1 .k  g/

(1.10)

This equation represents the Gordon growth model. Note that a critical condition for this model is that the constant growth of dividends must be less than the constant required rate of return. The zero growth situation is a special case of this model, in which: d1 (1.11) P0 D k Finally, dividends can exhibit a period of supernormal growth (i.e., g is greater than k) before declining to the normal growth situation assumed in the Gordon model (g is less than k). Supernormal growth often occurs during the “takeoff” phase in a firm’s life cycle. That is, a firm may experience a life cycle analogous to that of a product: first, a low-profit introductory phase, then a takeoff phase of high growth and high profits, leveling off at a plateau during its mature stage, perhaps followed by a period of declining earnings. Computer and electronics manufacturers experienced a period of supernormal growth during the 1960s, as did semiconductor firms during the 1970s. Bioengineering firms appear to be the super growth firms of the 1980s. The valuation of a supernormal growth stock requires some estimate of the length of the supernormal growth period. The current price of the stock will then consist of two components: (1) the present value of the stock during the supernormal growth period, and (2) the present value of the stock price at the end of the supernormal growth period: P0 D

n X d0 .1 C gs /t t D1

.1 C k/t

C

dnC1 kgn

.1 C k/n

exclusion of portfolio concepts and the interrelationship with the overall market indexes. Most of the classical models are also static in nature, overlooking the concept of dynamic growth. Nevertheless, a fundamental approach to security valuation – the stream of dividends approach – has evolved from this theory.

1.3 M and M Valuation Theory Modigliani and Miller (M and M 1961) have proposed four alternative valuation methods to determine the theoretical value of common stocks. This section discusses these valuation approaches in some detail. M and M’s four more or less distinct approaches to the valuation of common stock are as follows: 1. The discounted cash-flow approach; 2. the current earnings plus future investment opportunities approach; 3. the stream of dividends approach; and 4. the stream of earnings approach Working from a valuation expression referred to by M and M as the “fundamental principle of valuation”: 1 .d1 C P1 / (1.13) 1Ck M and M further developed a valuation formula to serve as a point of reference and comparison among the four valuation approaches: P0 D

V0 D

1 X t D0

(1.12)

where: gs D supernormal growth rate; n D the number of periods before the growth drops from supernormal to normal; k D the required rate of return of the stockholders; and gn D the normal growth rate of dividends (assumed to be constant thereafter).

1 .Xt  It / .1 C k/t C1

(1.14)

where: V0 D the current market value of the firm; Xt D net operating earnings in period t; and It D new investment during period t. In this context, the discounted cash-flow approach can be expressed: V0 D

1 X t D0

As we can see from our development of the discounted cashflow financial theory, the primary determinant of value for securities is the cash flow received by the investors. Anything that affects the cash flow, such as the dividend policy, investment policy, financing policy, and production policy of the firm, needs to be evaluated in order to determine a market price. Some shortcomings of this approach include the overemphasis on the evaluation of the individual firm to the

Theoretical Framework of Finance

1 .Rt  Ot / .1 C k/t C1

(1.15)

in which Rt is the stream of cash receipts by the firm and Ot is the stream of cash outlays by the firm. This fundamental principle is based on the assumption of “perfect markets,” “rational behavior,” and “perfect certainty” as defined by M and M. Since Xt differs from Rt and It differs from Ot only by the cost of goods sold and depreciation expense, if .Rt  Ot / equals .Xt  It /, then (1.15) is equivalent to (1.14) and the discounted cash-flow approach is an extension of

1 Theoretical Framework of Finance

7

Equation (1.13), the fundamental valuation principle. Hence, the security analyst must be well versed in generally accepted accounting principles in order to evaluate the worth of accounting earnings of .Xt  It /. The investment-opportunities approach seems in some ways the most natural approach from the standpoint of an investor. This approach takes into account the ability of the firm’s management to issue securities at “normal” market rates of return and invest in the opportunities, providing a rate higher than the normal rate of return. From this framework, M and M developed the following expression, which they show can also be derived from Equation (1.14):

in the pre-M and M period. Assuming an infinite time horizon, this approach defines the current market price of a share of common stock as equal to the discounted present value of all future dividends: P0 D

t D0

X0 X It .kt  k/ C k .1 C k/t C1 t D0 1

(1.16)

in which X0 is the perpetual net operation earning and kt is the “higher than normal” rate of return on new investment It . From the expression it can be seen that if a firm cannot generate a rate of return of its new investments higher than the normal rate, k, the price/earnings ratio applied to the firm’s earnings will be equal to 1/k, thus implying simple expansion rather than growth over time. An important variable for security analysis is a firm’s P/E ratio (or earnings multiple), defined as:

Conceptually the P/E ratio is determined by three factors: (1) the investor’s required rate of return .K/; (2) the retention ratio of the firm’s earning, b, where b is equal to 1 minus the dividend payout ratio; and (3) the firm’s expected return on investment .r/. Using the constant-growth model Equation (1.10):

V0 D

1 X

1 .Dt / .1 C k/t C1

(1.19)

With no outside financing, it can be seen that Dt D Xt  It and: 1 X 1 V0 D .Xt  It / .1 C k/t C1 t D0 which is Equation (1.14). With outside financing through the issuance of shares of new common stock, it can be shown that: V0 D

1 X t D0

1 .Dt C Vt C1  mt C1  Pt C1 / .1 C k/t C1

(1.20)

Vt C1  .mt C1 /.Pt C1 / D It  .Xt  Dt / Thus, Equation (1.20) can also be written in the form of Equation (1.14): V0 D

1 X t D0

d1 kg

E1 .1  b/ P0 D k  .br/ P0 1b D E1 k  .br/

(1.18)

in which mt C1 is the number of new shares issued at price Pt C1 . For the infinite horizon, the value of the firm is equal to the investments it makes and the new capital it raises, or:

Market price P=E ratio D Earnings per share

P0 D

1 .dt / .1 C k/t C1

Restating in terms of total market value:

t D0

V0 D

1 X

(1.17)

in which b is the retention rate and E1 is the next period’s expected profit. The P0 =E ratio is theoretically equal to the payout ratio of a firm divided by the difference between the investor’s required return and the firm’s growth rates. In the above equation a direct relationship has been identified between price/earnings ratio and discount cash-flow valuation model. The stream-of-dividends approach has been by far the most popular in the literature of valuation; it was developed

1 .Xt  It / .1 C k/t C1

(1.14)

Given the M and M ideal assumptions, the above result implies the irrelevance of dividends because the market value of the dividends provided to new stockholders must always be precisely the same as the increase in current dividends. This is in direct disagreement with the findings of the discounted cash-flow model, where dividends are a major determinant of value. In this case, dividends have no impact on value, and the firm’s investment policy is the most important determinant of value. Security analysis should concern itself with the future investment opportunities of the firm and forget about dividends. M and M also developed the stream-earnings approach, which takes account of the fact that additional capital must be acquired at some cost in order to maintain the stream of future earnings at its current level. The capital to be raised

8

1

is It and its cost is K percent per period thereafter; thus, the current value of the firm under this approach can be stated:

V0 D

1 X t D0

1 .Xt  It / .1 C k/t C1

which, again, is Equation (1.14). Under none of these four theoretical approaches does the term Dt remain in the final valuation expression and because Xt ; It , and k are assumed to be independent of Dt , M and M conclude that the current value of a firm is independent of its current and future dividend decisions. The amount gained by stockholders is offset exactly by the decline in the market value of their stock. In the short run, this effect if observed when a stock goes ex-dividend – that is, if the market price of the stock falls by the amount of the dividend on the last day the old shareholders are entitled to receive a dividend payment. The stock’s value depends only on the expected future earnings stream of the firm. Security analysts spend much time and effort forecasting a firm’s expected earnings. While the above analysis ignores the case in which external financing is obtained through the issuance of debt, in such a situation M and M’s position then rests upon their indifference proposition with respect to leverage (M and M 1958), discussed elsewhere in this chapter. Since that analysis shows that under a set of assumptions consistent with their “fundamental principal of valuation” the real cost of debt in a world of no taxation is equal to the real cost of equity financing, M and M conclude that the means of external financing used to offset the payment of dividends does not affect their hypothesis that dividends are irrelevant. Prior to Miller and Modigliani’s (1961) article, the classical view held that dividend policy was a major determinant of the value of the corporation and that firms should seek their “optimal payout ratios” to maximize their value. M and M’s conclusions about the irrelevance of dividend policy given investment policy, collided head-on with the existing classical view. The view that the value of the firm is independent of dividend policy also extends into a world with corporate taxes but without personal taxes.

1.3.1 Review and Extension of M and M Proposition I

Theoretical Framework of Finance

cast doubt upon the existence of such an optimal structure. The specific assumptions that they made, consistent with the dividend irrelevance analysis previously outlined, include the following: 1. Capital markets are perfect (frictionless). 2. Both individuals and firms can borrow and lend at the risk-free rate. 3. Firms use risk-free debt and risky equity. 4. There are only corporate taxes (that is, there are no wealth taxes or personal income taxes). 5. All cash flow streams are perpetuities (that is, no growth). Developing the additional concepts of risk class and homemade leverage, M and M derived their well-known Proposition I, both with and without corporate taxes.2 If all firms are in the same risk class, then their expected risky future net operating cash flow .XP / varies only by a scale factor. Under this circumstance, the correlation between two firms’ net operating income (NOI) within a risk class should be equal to 1.0. This implies that the rates of return will be equal for all firms in the same risk class, that is: Rit D

XP it  XP i t 1 Xi t 1

and because XPit D CXP jt where C is the scale factor: Rjt D

CXP jt  CXPjt D Rit CXP jt 1

(1.22)

in which Rit and Rjt are rates of return for the i th and j th firms, respectively. Therefore, if two streams of cash flow differ by only a scale factor, they will have the same distributions of returns and the same risk, and they will require the same expected return. The concept of homemade leverage is used to refer to the leverage created by individual investors who sell their own debt, while corporate leverage is used to refer to the debt floated by the corporation. Using the assumption that the cost of homemade leverage is equal to the cost of corporate leverage, M and M (1958) derived their Proposition I both with and without taxes. However, the Proposition I with taxes was not correct, and they subsequently corrected this result in their 1963 paper. Mathematically, M and M’s Proposition I can be defined: Vj D .Sj C Bj / D Xj =k

The existence of optimal capital structure has become one of the important issues for academicians and practitioners in finance. While classical finance theorists argue that there is an optimal capital structure for a firm, the new classical financial theory developed y M and M (1958, 1963) has

(1.21)

2

(1.23)

In 1985 Franco Modigliani won the Nobel Prize for his work on the life cycle of savings and his contribution to what has become known as the M and M theory, discussed in this section.

1 Theoretical Framework of Finance

9

and Proposition I with taxes can be defined as VjL D

.1  j /Xj Ij D VjU C Bj C  k r

(1.24)

In Equation (1.23), Bj ; Sj , and Vj are, respectively, the market value of debt, common shares, and the firm. Xj is the expected profit before deduction of interest, k the required rate of return or the cost of capital in risk class k. In Equation (1.24), k is the required rate of return used to capitalize the expected returns net of tax for the unlevered firm with long-run average earnings before tax and interest of .Xj / in risk class k. j is the corporate tax rate for the j th firm, Ij is the total interest expense for the j th firm, and r is the market interest rate used to capitalize the certain cash inflows generated by risk-free debt. Bj is total risk-free debt floated by the j th firm, and V L and V U are the market values of the leveraged and unleveraged firms, respectively. By comparing these two equations, we find that the advantages of a firm with leverage will increase that firm’s value by j Bj – that is, the corporate tax rate times the total debt floated by that firm. One of the important implications of this proposition is that there is no optimal capital structure for the firm unless there are bankruptcy costs associated with its debt flotation. If there are bankruptcy costs, then a firm will issue debt until its tax benefit is equal to the bankruptcy cost, thus providing, in such a case, an optimal capital structure for the firm. In addition to the bankruptcy costs, information signaling (see Leland and Pyle 1977, and Ross 1977a) and differential expectations between shareholders and bondholders can be used to justify the possible existence of an optimal structure of a firm. The existence of optimal capital structure is an important issue for security analysts to investigate because it affects the value of the firm and the value of the firm’s securities. Is a firm with a high level of debt more valuable than a similar firm with very little debt? M and M say it doesn’t matter or that the highly leveraged firm is more valuable. The important assumptions used to prove the M and M Proposition I with taxes are that (1) there are no transaction costs, (2) homemade leverage is equal to corporate leverage, (3) corporate debt is riskless, and (4) there is no bankruptcy cost. Overall, M and M’s Proposition I implies that there is no optimal capital structure. If there is a tax structure that systematically provides a lower after-tax real cost of debt relative to the after-tax real cost of equity, the corporation will maximize the proportion of debt in its capital structure and will issue as much debt as possible to maximize the tax shield associated with the deductibility of interest. Stiglitz (1969) extends M and M’s proposition using a general equilibrium state preference framework. He is able to show that M and M’s results do not depend on risk classes, competitive capital markets, or agreement by

investors. The only two fundamental assumptions are that there is no bankruptcy and individuals can borrow at the same rate as firms. Stiglitz (1974) develops the argument that there may be a determinate debt-equity ratio for the economy as a whole, but not for the individual firm.

1.3.2 Miller’s Proposition on Debt and Taxes Miller (1977) argues that although there is no optimal capital structure for an individual firm, in the aggregate case there may be an optimal structure. In balancing bankruptcy cost against tax shelter, an optimal capital structure is derived, just as the classical view has always maintained. The Tax Reform Act of 1986 taxes dividends and longterm capital gains at the same top rate of 28%. This is a major change from the old 50% rate on dividends and 20% rate on long-term capital gains. The new tax bill has also shifted the major tax burden to corporations and away from individuals. Even though the maximum corporate tax rate will decrease to 34% from the current top rate of 46%, corporations will be paying more taxes because of the loss of the Investment Tax Credits and the Accelerated Cost Recovery System depreciation allowances. These changes in the tax code will shift the emphasis of corporate management from retaining earnings in order to generate price appreciation and capital gains to the payout of corporate funds in the form of dividends. In his presidential address at the Annual Meeting of the American Finance Association, Merton Miller (1977) incorporates personal taxes into the Modigliani and Miller (1958, 1963) argument for the relationship between the firm’s leverage and cost of capital. M and M’s Proposition I shows that the value of the leveraged firm equals the value of the unleveraged firm plus the tax shield associated with interest payments, as shown by Equation (1.24): V L D V U C tc B

(1.25)

where: V L D the value of the leveraged firm; V U D the value of the unleveraged firm; tc D the corporate tax rate; and B D the value of the firm’s debt. Miller generalizes the M and M relationship shown in Equation (1.25) to include personal taxes on dividends and capital gains as well as taxes on interest income, to yield:   .1  tc /.1  tps / B VL D VU C 1 .1  tpB /

(1.26)

10

1

Theoretical Framework of Finance

in which tps is the personal tax rate on income from stock and tpB is the personal tax rate on income from bonds.

In the second approach, the maximization problem in Equation (1.30) can be rewritten as

1.4 Markowitz Portfolio Theory

82 9 31=2 ˆ > n n X < X =   5 4 W i R i C 1 Cov Ri ; Rj  p Max L D ˆ > : j D1 i D1 ; i D1 n X

Professors Markowitz, Miller, and Sharpe earned their Nobel Prize in applied economics in 1989. Section 1.3 briefly discussed the M and M theory. In the next section, we will discuss CAPM. In this section we will discuss Prof. Markowitz portfolio theory. In the paper entitled “Markowitz, Miller, and Sharpe: The First Nobel Laureates in Finance,” Lee (1991) has discussed this historical event in details. Markowitz suggests two constrained maximization approaches to obtain the optimal portfolio weight. The first approach is to minimize the risk or variance of the portfolio, subject to the portfolio’s attaining some target expected rate of return, and also subject to the portfolio weight summing to one. The problem can be stated mathematically: Min p2 D

n n X X

Wi Wj ij

(1.27)

Wi E .Ri / D E 

(1.28)

i D1 j D1

Subject to

n X

1:

C2

n X

! Wi  1

(1.31)

i D1

where Ri is the average rate of rates of return of the portfolio given targeted standard deviation of the portfolio P . Essentially Equation (1.31) maximizes the expected rates of return of the portfolio given the targeted standard deviation of the portfolio. In this book, a large portion is dedicated to the portfolio theory and its application. Chapter 10 discusses portfolio optimization models and mean-variance spanning tests. Chapter 12 discusses the estimation risk and power utility portfolio selection. Chapter 13 discusses theory and methods in the international portfolio management. Chapter 17 provides discussion on portfolio theory, CAPM, and the performance measures. Chapter 18 discusses the intertemporal equilibrium models, portfolio theory, and the capital asset pricing model.

1.5 Capital Asset Pricing Model

i D1 

where E is the target expected return and 2:

n X

Wi D 1:0

(1.29)

i D1

The first constraint simply says that the expected return on the portfolio should equal the target return determined by the portfolio manager. The second constraint says that the weights of the securities invested in the portfolio must sum to one. The Lagrangian objective function can be written: Min L D

n n X X

W i W j C 1

i D1 j D1

C2

n X

n X 

Wi E .Ri /  E 



i D1

! Wi  1

(1.30)

i D1

Taking the partial derivatives of this equation with respect to each of the variables, W1 ; W2 ; W3 ; 1 ; 2 and setting the resulting five equations equal to zero yields the minimization of risk subject to the Lagrangian constraints. This system of five equations and five unknowns can be solved by the use of matrix algebra. Equations (1.30) minimizes the portfolio variance given the portfolio’s targeted expected rate of return.

At about the same time as M and M were developing their work, developments in portfolio theory were leading to a model describing the formation of capital asset prices in world of uncertainty: the capital asset pricing model (CAPM). The CAPM is a generalized version of M and M theory in which M and M theory is provided with a link to the market: E.Rj / D Rf C ˇj ŒE.Rm /  Rf

(1.32)

where: Rj D the rate of return for security j ; ˇj D a volatility measure relating the rate of return on security j with that of the market over time; Rm D the rate of return for the overall market (typically measured by the rate of return reflected by a market index, such as the S&P 500); and Rf D the risk-free rate available in the market (usually the rate of return on U.S. Treasury bills is used as a proxy). In the CAPM framework, the valuation of a company’s securities is dependent not only on its cash flows but also on those of other securities available for investment. It is assumed that much of the total risk, as measured by standard

1 Theoretical Framework of Finance

11

deviation of return, can be diversified away by combining the stock of a firm being analyzed with those of other companies. Unless the cash flows from these securities are perfectly positively correlated, smoothing or diversification will take place. Thus, the security return can be divided into two components: a systematic component that is perfectly correlated with the overall market return and an unsystematic component that is independent of the market return: Security return D Systematic return C Unsystematic return (1.33) Since the security return is perfectly correlated with the market return, it can be expressed as a constant, beta, multiplied by the market return .Rm /. The beta is a volatility index measuring the sensitivity of the security return to changes in the market return. The unsystematic return is residual of the relationship of Rj with Rm . As has been previously noted, the standard deviation of the probability distribution of a security’s rate of return is considered to be an appropriate measure of the total risk of that security. This total risk can be broken down into systematic and unsystematic components, just as noted above for security return: Total security risk D Systematic risk C Unsystematic risk (1.34) Diversification is achieved only when securities that are not perfectly correlated with one other are combined. The unsystematic risk components tend to cancel each other as they are all residuals from the relationship of security returns with the overall market return. In the process, the portfolio risk measure declines without any corresponding lowering of portfolio return (see Fig. 1.1). It is assumed in this illustration that the selection of additional securities as the portfolio size is increased is performed in some random manner, although any selection process other than intentionally choosing perfectly correlated securities will suffice. Unsystematic risk is shown to be gradually eliminated until the remaining portfolio risk is completely market related. While for an actual portfolio the systematic risk will not remain constant as securities are added, the intent is to show that the unsystematic-risk portion can be diversified away, leaving the market related systematic portion as the only relevant measure of risk. Empirical studies have shown that a portfolio of about 20 securities not highly correlated with one another will provide a high degree of diversification. Although capital-market theory assumes that all investors will hold the market portfolio, it is neither necessary nor realistic to assume that all investors will be satisfied with the market level of risk. There are basically two ways that investors can adjust their risk level within the CAPM theoretical framework. First, funds for investment can

Fig. 1.1 Diversification process

be divided between the market portfolio and risk-free securities. The capital-market line (CML) is derived assuming such a tradeoff function. This is illustrated in Fig. 1.2, in which point M is the market portfolio and points on the CML below and above M imply lending and borrowing at the risk-free rate. The second way of adjusting the portfolio risk level is by investing in a fully diversified portfolio of securities (that is, the correlation coefficient of the portfolio with the market, rpm is equal to 1.0) that has a weighted average beta equal to the systematic-risk level desired: ˇp D

n X

Wj Bj

(1.35)

j D1

in which Wj is the proportion of total funds invested in security j . In the CAPM, systematic risk as measured by beta is the only risk that need be undertaken; therefore, it follows that no risk premium should be expected for the bearing of unsystematic risk. With that in mind, the relationship between expected return and risk can be better defined through the illustration of the security-market line (SML) in Fig. 1.3 in which Rm and ˇm are the expected return and risk level of the market portfolio. In equilibrium, all securities and combinations of securities are expected to lie along this line. In contrast, only fully diversified portfolios would be expected to fall along the CML, because only with full diversification is total risk equal to systematic risk alone. In addition to the static CAPM developed by Sharpe (1964) and others Merton has discussed intertemporal CAPM. In Chap. 6 we will discuss the static CAPM and beta forecasting and in Chap. 18 we will focus on intertemporal CAPM.

12

1

Theoretical Framework of Finance

bij D a coefficient called a factor loading that quantifies the sensitivity of asset i ’s returns to the movements in the common factor ıQj (and is analogous to the beta in the CAPM); and 2Q i D an error term, or unsystematic risk component, idiosyncratic to the i th asset, with mean zero and variance equal to Q 22 . Moreover, it is assumed that the 2Q i reflects the random influence of information that is unrelated to other assets. Thus, the following condition is assumed to hold:

Fig. 1.2 Capital market line

 ˚ E 2Q i j2Q j D 0

(1.37)

as well as 2i and 2j independence for all i ¤ j . Also, for any two securities i and j :  ˚ E 2Q i ; 2Q j D 0

Fig. 1.3 Security market line

1.6 Arbitrage Pricing Theory 1.6.1 Ross’s Arbitrage Model Specification This section focuses on two related forms of the arbitrage pricing model (APM). The first is the model as originally proposed by Ross (1976). The initial and probably the most prominent assumption made by APM concerns the return-generating process for assets. Specifically, individuals are assumed to believe (homogeneously) that the random returns on the set of assets being considered are governed by a k-factor generating model of the form: rQi D Ei C bi1 ıQ1 C : : : C bi k ıQk C 2Q i .i D 1; : : : ; n/

(1.36)

where: rQi D random return on the i th asset; Ei D expected return on the i th asset; ıQj D j th factor common to the returns of all assets under consideration with a mean of zero, common factors that in essence capture the systematic component of risk in the model;

(1.38)

for all i and j , where i ¤ j . If this last condition did not hold – that is, if there was too strong a dependence between 2Q i and 2Q j – it would be equivalent to simply saying that the k-hypothesized common factors existed. Finally, it is assumed that for the set of n assets under consideration that n is much greater than the number of factors k. Before developing Ross’s riskless arbitrage argument, it is essential to examine Equation (1.36) more closely and draw some implications from its structure. First, consider the effect of omitting the unsystematic risk terms eQi . Equation (1.36) would then imply that each asset i has returns rQ that are an exact linear combination of the returns on a riskless asset (with constant return) and the returns on k other factors or assets (or column vectors) ıQi ; : : : ; ıQk . Moreover, the riskless return and each of the k factors can be expressed as a linear combination of k C 1 other return – for example, r, through rkC1 – in this type of setting. Taking this logic one step further, since any other asset return is a linear combination of the factors, it must also be a linear combination of the returns of the first k C 1 assets. Hence, portfolios composed from the first k C 1 assets must be perfect substitutes for all other assets in the market. Consequently, there must be restrictions on the individual returns generated by the model, as perfect substitutes must be priced equivalently. This sequence of mathematical logic is the core of APT. That is, only a few systematic components of risk exist in the economy, and consequently many portfolios will be close substitutes, thereby demanding the same value. To initiate Ross’s arbitrage argument about APT, it is best to start with the assumption of Equation (1.36). Next, presume an investor who is contemplating an alteration of the currently held portfolio, the difference between any new portfolio and the old portfolio will be quantified by changes

1 Theoretical Framework of Finance

13

in the investment proportions xi .i D 1; : : : ; n/. The xi represents the dollar amount purchased or sold of asset i as a fraction of total invested wealth. The investor’s portfolio investment is constrained to hold to the following condition: n X

xi D 0

(1.39)

i D1

In words, Equation (1.39) says that additional purchases of assets must be financed by sales of others. Portfolios that require no net investment such as x  .xj ; : : : ; xn / are called arbitrage portfolios. Now, consider an arbitrage portfolio chosen in the following manner. First, the portfolio must be chosen to be well diversified by keeping each element, x, of order 1=n in size. Second, the x of the portfolio must be selected in such a way as to eliminate all systematic risk (for each h): xbh

X

xi bih D 0 .h D 1; : : : ; k/

(1.40)

i D1

The returns on any such arbitrage portfolios can be described: Q x rQ D .xE/ C .xb/ıQ1 C : : : C .xbk /ıQk C .x 2/  xE C .xb1 /ıQ1 C : : : C .xbk /ıQk D xE where x rQ D

n P

xi rQi and xE D

i D1

n P

xi Ei . Note that the term

i D1

Q is (approximately) eliminated by the effect of holding a .x 2/ well-diversified portfolio of n assets where n is large. Using the law of large numbers, if  2 denotes the average variance of the eQi terms, and assuming for simplicity that each x, approximately equals 1=n and that the 2i are mutually independent: 1X Q D Var Var.x 2/ 2i n i

! D

2 Var.2Q i / D n2 n2

(1.41)

Thus if n is large the variance of x 2Q will be negligible. Reconsidering the steps up to this point, note that a portfolio has been created that has no systematic or unsystematic risk and using no wealth. Under conditions of equilibrium, it can be stated unequivocally that all portfolios of these n assets that satisfy the conditions of using no wealth and having no risk must also earn no return on average. In other words, there are no free lunches in an efficient market, at least not for any extended period of time. Therefore the expected return on the arbitrage portfolio can be expressed:

x rQ D xE D

n X

xi Ei D 0

(1.42)

i D1

Another way to state the preceding statements and results is through linear algebra. In general, any vector x with elements on the order of 1=n that is orthogonal to the constant vector and to each of the coefficient vectors bh .h D 1; : : : ; k/ must also be orthogonal to the vector of expected returns. A further algebraic consequence of this statement is that the expected return vector E must be a linear combination of the constant vector and the b vectors. Using algebraic terminology, there exist k C 1 weights .0 ; 1 ; : : : ; k / such that: E i D 0 C 1 b i l C : : : C k b i k ;

for all i

(1.43)

In addition, if there exists a riskless asset with return E0 , which can be said to be the common return on all zero-beta assets – that is, bih D 0 (for all h) – then: E 0 D 0 Utilizing this definition and rearranging: E i  E 0 D 1 b i l C : : : C k b i k

(1.44)

The pricing relationship depicted in Equation (1.44) is the central conclusion of the APT. Before exploring the consequences of this pricing model through a simple numerical example, it is best to first give some interpretation to the h , the factor risk premium. If portfolios are formed with a systematic risk of 1 relative to factor h and no risk on other factors, then each h can be interpreted as: h D E h  E 0

(1.45)

In words, each h can be thought of as the excess return or market risk premium on portfolios with only systematic factor h risk. Hence, Equation (1.44) can be rewritten: Ei  E0 D .E 1  E0 /bi1 C : : : C .E k  E0 /bi k

(1.46)

The implications that arise from the arguments concerning APT that have been constructed thus far can be summarized in the following statement: APT yields a statement of relative pricing on subsets of the universe of assets. Moreover, note that the arbitrage pricing model of Equations (1.44) or (1.46) can be tested by examining only subsets of the set of all return. Consequently, the market portfolio plays no special role in APT, since any well-diversified portfolio could serve the same purpose. Hence, it can be empirically tested on any set of data, and the results should be generalizable to the entire market.

14

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Theoretical Framework of Finance

Even though the APT is very general and based on few assumptions, it provides little guidance concerning the identification of the priced factors. Hence empirical research must achieve two goals. 1. Identify the number of factors. 2. Identify the various economics underlying each factor. Chapter 64 will discuss the relationship between the liquidity risk and the arbitrage pricing theory. Fig. 1.4 Theoretical and actual values of a call option

1.7 Option Valuation Option contracts give their holders the right to buy and sell a specific asset at some specified price on or before a specified date. Since these contracts can be valued in relation to common stock, the basic concepts involved have a number of applications to financial theory and to the valuation of other financial instruments. While there are a variety of option contracts – for example, call options, put options, combinations of calls and puts, convertible securities, and warrants – this chapter’s discussion is limited to call options. A call option gives the holder the right to buy a share of stock at a specified price, known as the exercise price, and the basic American option can be exercised at any time through the expiration date. The value of the option at expiration is the difference between the market price of the underlying stock and its exercise price (with a minimum value of zero, of course). While several factors affect the value of an option, the most important factor is the price volatility of the stock – the greater the volatility, the greater the value of the option, other things remaining the same. We will also note that the longer the time left before expiration and the higher the level of interest rates in the market, the greater the option value, all other things held the same. The theoretical value of a call option at expiration is the difference between the market price of the underlying common stock, ps and the exercise price of the option, E, or zero, whichever is greater: C D Max.Ps  E; 0/

(1.47)

When the price of the stock is greater than the exercise price, the option has a positive theoretical value that will increase dollar for dollar with the price of the stock. When the market price of the stock is equal to or less than the exercise price, the option has a theoretical value of zero, as shown in Fig. 1.4. Nevertheless, as long as some time remains before expiration, the actual market price of the option (referred to as the option premium at the time of issue) is likely to be greater than its theoretical value. This increment above the

theoretical value is called the time value or speculative value of the option, and its size will depend primarily on the perceived likelihood of a profitable move on the price of the stock before expiration of the option. The full range of possible values for the market price of the option is from the theoretical value on the low side to the market price of the stock itself on the high side. For the option price to be equal to the stock price, an infinite time to expiration would be implied. For the option price to be equal to the theoretical value only, imminent expiration would be implied. For virtually all options for which the value would be determined, however, the option price would fall somewhere between these two extremes. Because an option costs less than its underlying stock, the percentage change in option price is greater than the percentage change in stock price, given some increase in the market price of the stock. Thus, a leveraged rate of return can be earned by investment in the option rather than the stock. As stock price continues to increase, the difference between the percentage change in option price and the percentage change in stock price will tend to converge. Thus far it has been shown that the value of an option will be a function of the underlying stock price, the exercise price of the option, and the time to maturity. Yet there is still another factor that is probably the single most important variable affecting the speculative value of the option. That is the price volatility of the underlying stock. The greater the probability of significant change in the price of the stock, the more likely it is that the option can be exercised at a profit before expiration. There is another factor affecting the speculative premium for options. This is the level of interest rates in the market – specifically for option analysis, the call money rate charged by brokers for the use of margin in common-stock accounts. As this concept is discussed later, it is sufficient here to point out that the leverage achieved through option investment is similar to that achieved through direct margin purchase of the underlying common stock, but without the explicit interest cost involved in the latter. Thus, the higher the call money rate, the greater the savings from the use of options and the greater the speculative value of the option.

1 Theoretical Framework of Finance

15

To summarize, there are five variables necessary to determine the value of an American call option (ignoring dividends on the common stock): 1. and 2. Stock price–Exercise price: The relationship between these two prices determines whether the option has a positive theoretical value. 3. Time to maturity: The longer the time to maturity, the greater the speculative value of the option because the chances for a profitable movement in the price of the stock are increased. 4. Volatility of stock price: There is a positive relationship between the volatility of the underlying stock price and the speculative value of the option because with greater volatility, there is greater potential for gain on the upside and greater benefit from the downside protection involved with the option. 5. Interest rate: The higher the call money rate for direct margin purchase of common stock, the greater the relative value of being able to achieve equal amounts of leverage through the alternative of option purchase. The factors that affect the value of an option can be written in a functional form: C D f .S, X;  2 ; T,rf /

(1.48)

where: C D value of the option; S D stock price; X D exercise price;  2 D variance of the stock; T D time to expiration; and rf D risk-free rate. The value of the option increases as a function of the value of the stock for a given exercise price and maturity date. The lower the exercise price, the greater the value of the option. The longer the time to maturity, the higher the value of the option. The holder of an option will prefer more variance in the price of the stock to less. The greater the variance (price volatility) the greater the probability that the stock price will exceed the exercise price and thus benefit the holder. Considering two related financial securities – common stock and the option on the common stock – it is possible to illustrate how a risk-free hedged position can be developed. In this way, unprofitable price movements in one of the securities will be offset by profitable price movements in the other. The hedge ratio determines the portion of stock held long in relation to the options in the short position (or vice versa). With a complete hedge, the value of the hedged position can be the same regardless of the stockprice outcome. In efficient financial markets, the rate of return earned on perfectly hedged positions will be the risk-free rate. Consequently, it is then possible to determine

the appropriate option price at the beginning of the period. If the actual market price is above or below this value, arbitrage would then drive the option price toward its correct level. This process and the development of the Black-Scholes (1973) continuous type of option-pricing model will be discussed in Chaps. 23, 24, and 27. Cox et al. (1979) discrete type of binomial option-pricing model will be analyzed in Chaps. 25, 26, and 28.

1.8 Futures Valuation and Hedging A basic assumption of finance theory is that investors are risk averse. If we equate risk with uncertainty, can we question the validity of this assumption? What evidence is there? As living, functional proof of the appropriateness of the risk aversion assumption, there exist entire markets whose sole underlying purpose is to allow investors to display their uncertainties about the future. These particular markets, which primary focus on the future, are called just that, futures markets. These markets allow for the transfer of risk from hedgers (risk-averse individuals) to speculators (risk-seeking individuals). A key element necessary for the existence of futures markets is the balance between the number of hedgers and speculators who are willing to transfer and accept risk. A future contract is a standardized legal agreement between a buyer and a seller, who promise now to exchange a specified amount of money for goods or services at a future time. Of course, there is nothing really unusual about a contract made in advance of delivery. For instance, whenever something is ordered rather than purchased on the spot, a futures (or forward) contract is involved. Although the price is determined at the time of the order, the actual exchange of cash for the merchandise takes place later. For some items the lag is a few days, while for others (such as a car) it may be months. Moreover, a futures contract imparts a legal obligation to both parties of the contract to fulfill the specifications. To guarantee fulfillment of this obligation, a “goodfaith” deposit, also called margin, may be required from the buyer (and the seller, if he or she does not already own the product). To ensure consistency in the contracts and to help develop liquidity, futures exchanges have been established. These exchanges provide a central location and a standardized set of rules to enhance the credibility of these markets and thus generate an orderly, liquid arena for the price determination of individual commodities at distinct points in the future. A substantial increase in the number of types of futures contracts offered by the exchanges has been occurring over the last decade. At the same time, the growth in futures trading volume has been phenomenal. Two explanations can be offered for this increase in futures activity. These increases

16

can be intuitively correlated with the growing levels of uncertainty in many facts of the economic environment – for example, inflation and interest rates. A second view is based on the argument that even though the world has not become any more uncertain, the increased integration of financial and real markets has increased the risk exposure of any given individual. The tremendous growth in the home-mortgage and consumer-debt financial markets has allowed the purchase of more expensive real assets. This increase in the rise of individual financial leverage has increased individual exposure to interest-rate fluctuations, thereby increasing the requirements for risk-sharing across markets or between individuals with varied portfolios. Futures markets have the potential to help people manage or transfer the uncertainties that plague the world today. This section examines the basic types of futures contracts offered and the functioning of futures markets. In addition the uses of financial and index futures are illustrated, and the theoretical pricing concepts related to these financial instruments are discussed. The important terms associated with futures contracts and futures markets are defined and an analysis of futures market follows. A theory of valuation is introduced, and the section closes with a discussion of various hedging strategies and concepts.

1.8.1 Futures Markets: Overview In the most general sense, the term commodity futures is taken to embrace all existing futures contracts. Nevertheless, for purposes of clarity and classification its meaning here is restricted to a limited segment of the total futures markets. Accordingly, futures contracts can be classified into three main types. 1. Commodity futures 2. Financial futures 3. Index futures Within this classification commodity futures include all agriculturally related futures contracts with underlying assets, such as corn, wheat, rye, barley, rice, oats, sugar, coffee, soybeans, frozen orange juice, pork bellies, live cattle, hogs, and lumber. Also within the commodity-futures framework are futures contracts written on precious metals, such as gold, silver, copper, platinum, and palladium, and contracts written on petroleum products, including gasoline, crude oil, and heating oil. Many of the futures contracts on metals and petroleum products have been introduced as recently as the early 1980s. Producers, refineries, and distributors, to name only a few potential users, employ futures contracts to assure a particu-

1

Theoretical Framework of Finance

lar price or supply – or both – for the underlying commodity at a future date. Futures-market participants are divided into two broad classes: hedgers and speculators. Hedging refers to a futuresmarket transaction made as a temporary substitute for a cashmarket transaction to be made at a later date. The purpose of hedging is to take advantage of current prices by using futures transactions. For example, banks and corporations can be hedgers when they use futures to fix future borrowing and lending rates. Futures market speculation involves taking a short or long futures position solely to profit from price changes. If you think that interest rates will rise because of an increase in inflation, you can sell T-bill futures and make a profit if interest rates do rise and the value of T-bills falls. Financial futures are a trading medium initiated with the introduction of contracts on foreign currencies at the International Monetary Market (IMM) in 1972. In addition to futures on foreign currencies, financial futures include contracts based on Treasury bonds (T-bonds), Treasury bills (T-bills), Treasury notes (T-notes), bank certificates of deposit, Eurodollars, and GNMA mortgage securities. These latter types of financial futures contracts are also referred to as interest-rate futures as their underlying asset is an interest-bearing security. While foreign-currency futures arose with the abolition of the Bretton Woods fixed exchange-rate system during the early 1970s, interest-rate futures surged in popularity and number following the change in U.S. monetary policy in October 1979. The effect of the Federal Open Market Committee’s decision to deemphasize the traditional practice of “pegging” interest rates was to greatly increase the volatility of market interest rates. Thus, interest-rate changes have become a highly prominent risk to corporations, investors, and financial institutions. Index futures represent the newest and boldest innovation in the futures market to date. An index-futures contract is in for which the underlying asset is actually a portfolio of assets – for example, the Major Market Index (MMI) includes 20 stocks traded on the NYSE and the S&P index includes 500 stocks. Contracts on more diverse types of indexes include a high-quality bond index, an interest-rate index composed of interest-bearing market securities, and the consumer price index. The S&P 500 index, requiring delivery of the 500 stocks constituting the S&P 500 stock index, would certainly have dampened enthusiasm for this and similar index contracts. Because of this, an index-futures contract is settled on the basis of its cash value when the contract matures. The cash value of the contract is equal to the closing index value on its last trading day multiplied by a dollar amount of $500. Many portfolio managers are taking advantage of index futures to alter their portfolios risk-return distributions.

1 Theoretical Framework of Finance

17

1.8.2 The Valuation of Futures Contracts The discussions of each of the three classifications of futures contracts have pointed out pricing idiosyncrasies and have examined specific pricing models for particular types of contracts. Nevertheless, the underlying tenets of any particular pricing model have their roots in a more general theoretical framework of valuation. Consequently, the focus is now on the traditional concepts of futures contracts valuation.

Ft;T D St .1 C Rf;T t /

1.8.2.1 The Arbitrage Argument An instant before the futures contract matures, its price must be equal to the spot (cash) price of the underlying commodity, or: (1.49) Ft;T D St where: Ft;T D the price of the futures contract at time t, which matures at time T , where T > t and T  t is a very small interval of time; and St D the spot price of the underlying commodity at time t. If Equation (1.49) did not hold arbitrage condition would prevail. More specifically, when t D T at the maturity of the contract, all trading on the contract ceases and the futures price equals the spot price. If an instant before maturity Ft;T < St , one could realize a sure profit (an arbitrage profit) by simultaneously buying the futures contract (which is undervalued) and selling the spot commodity (which is overvalued). The arbitrage profit would equal: St  Ft;T

carry out the arbitrage process, the trader would incur certain costs. For instance, if the spot commodity were purchased because it is undervalued relative to the futures, the trader or arbitrageur would incur an opportunity or interest cost. Any funds he or she tied up in the purchase of the commodity could alternatively be earning some risk-free interest rate Rf through investment in an interest-bearing risk-free security. Therefore, the futures price should account for the interest cost of holding the spot commodity over time, and consequently Equation (1.49) can be modified to:

(1.50)

However, if Ft;T > St is the market condition an instant before maturity, smart traders would recognize this arbitrage condition and sell futures contracts and buy the spot commodity until t D T and Ft;T D St . In fact, the effect of selling the futures and buying the spot would bid their prices down and up, respectively. Thus, the arbitrage process would alleviate any such pricing disequilibrium between the futures contract and its underlying spot commodity.

1.8.2.2 Interest Costs The previous simplified argument demonstrated that the futures and spot prices must be equal an instant before the contract’s maturity. This development assumes no costs in holding the spot commodity or carrying it (storing it) across time. If such a market condition held, Equation (1.49) could be extended to apply to any point of time where t < T . However, by having to buy or sell the spot commodity to

(1.51)

where Rf;T t is the risk-free opportunity cost or interest income that is lost by tying up funds in the spot commodity over the interval T  t.

1.8.2.3 Carrying Costs Since theories on the pricing of futures contracts were developed long before the introduction of financial or index futures, the costs of storing and insuring the spot commodity were considered relevant factors in the price of a futures contract. That is, someone who purchased the spot commodity to hold from time t to a later period T incurs the costs of actually housing the commodity and insuring it in case of fire or theft. In the case of livestock such as cattle or hogs, the majority of this cost would be in feeding. The holder of a futures contract avoids these costs borne by the spot holder, making the value of the contract relative to the spot commodity increase by the amount of these carrying costs. Therefore, Equation (1.51) can be extended: Ft;T D St .1 C Rf;T t / C CT t

(1.52)

where CT t is the carrying costs associated with the spot commodity for the interval T  t.

1.8.2.4 Supply and Demand Effects As for other financial instruments or commodities, the price of a futures contract is affected by expectations of future supply and demand conditions. The effects of supply and demand for the current spot commodity (as well as for the future spot commodity) have not yet been considered in this analysis. If the probability exists that future supplies of the spot commodity might significantly differ from current supplies, then this will affect the futures price. The discussion up to this point has assumed that the aggregate supply of the commodity was fixed over time and that demand remained

18

1

constant; however, for agricultural, financial, and index futures this is a very unrealistic assumption. For instance, if it is expected that the future available supply of wheat for time T will decline because of poor weather, and demand is unchanged, one would then expect the future spot price of wheat to be higher than the current spot price. Furthermore, a futures contract on wheat that matures at time T can also be considered to represent the expected spot price at time T and consequently should reflect the expected change in supply conditions. In a more extreme fashion, if there is no current supply of wheat, then the futures price would reflect only future supply conditions and the expected future spot price at time T . This can be expressed as: ST / Ft;T D Et .e

(1.53)

S T / is the spot price at a future point T exwhere Et .e pected at time t, where t < T . The tilde above ST indicates that the future spot price is a random variable because future factors such as supply cannot presently be known with certainty. Equation (1.53) is called the unbiased-expectations hypothesis because it postulates that the current price of a futures contract maturing at time T represents the market’s expectation of the future spot price at time T . Which of these expressions for the price of a futures contract at time t will hold in the market – the arbitrage pricing relationship in Equation (1.52) or the unbiased-expectations hypothesis in Equation (1.53)? As the markets are assumed to be efficient the answer is that, the market price of the futures contract will take on the minimum value of either of these two pricing relationship, or: S T /; St .1 C Rf;T t / C CT t Ft;T D MinŒEt .e

(1.54)

For any storable commodity on a given day t, the futures price Ft;T will be higher than the spot price St on day tI Ft;T > St . The amount by which the futures price exceeds the spot price .Ft;T St / is called the premium. In most cases this premium is equal to the sum of financial costs St Rf;T t and carrying costs CT t . The condition of Ft;T > St is associated with a commodity market called a normal carryingchange market. In general, the difference between the futures price Ft;T and spot price St is called the basis. Basis D Ft;T  St

(1.55)

1.8.2.5 The Effect of Hedging Demand John Maynard Keynes (1930), who studied the futures markets as a hobby, proposed that for some commodities there was a strong tendency for hedgers to be concentrated on the

Theoretical Framework of Finance

short side of the futures market. That is, to protect themselves against the risk of a price decline in the spot commodity, the spot holder or producer (such as a farmer) would hedge the risk by selling futures contracts on his or her particular commodity. This demand for hedging, producing an abundant supply of futures contracts, would force the market price below that of the expected spot price at maturity (time T ). Moreover, the hedgers would be transferring their price S T / and Ft;T risk to speculators. This difference between Et .e when Ft;T < Et .e S T /, can be thought of as a risk premium paid to the speculators for holding the long futures position and bearing the price risk of the hedger. This risk premium can be formulated as: S T /  Ft;T Et .RP / D Et .e

(1.56)

where Et .RP / is the expected risk premium paid to the speculator for bearing the hedger’s price risk. Keynes described this pricing phenomenon as normal backwardation. When the opposite conditions exist – hedgers are concentrated on the long side of the market and bid up the futures spot pricing Ft;T over the expected future spot S T / – the pricing relationship is called contango (that is, Et .e S T //. To reflect the effect of normal Et .RP / D Ft;T  Et .e backwardation or contango on the current futures price, the S T / term in Equation (1.56) must be adjusted for the efEt .e fects of hedging demand: S T / C E.RP /; St .1 C Rf;T t / C CT t Ft;T D Min ŒEt .e (1.57) Equation (1.57) expresses a broad pricing framework for the value of a futures contract. Over the life of the futures contract the futures price must move toward the cash price, because at the maturity of the futures contract the futures price will be equal to the current cash price. If hedgers are in a net short position, then futures prices must lie below the expected future spot price, and futures prices would be expected to rise over the life of the contract. However, if hedgers are net long, then the futures price must lie above the expected futures spot price and the price of the futures would be expected to fall. Either a falling futures price (normal backwardation) or a rising futures price (contango) determines the boundaries within which the actual futures price will be located. However, numerous other factors can alter and distort the relationship shown by Equation (1.57). For instance, the analysis implicitly assumes that interest rates remain constant from time t to the contract’s maturity date at time T . However, since market interest rates fluctuate, an increasing or decreasing term structure of interest rates would bias the price of the futures contract higher or lower. In fact, the more accurate one’s forecast of future interest rates, the more accurate the current valuation of the futures contract.

1 Theoretical Framework of Finance

Empirical research casts rather strong doubt on the size of the expected risk-premium component of futures prices, particularly for financial and index futures. In fact, the expectation of speculators, along with actual futures contract supply and demand conditions in the pit, can combine to reverse the effect expected by Keynes. This results in part from the makeup of futures’ users, a clear majority of whom are not hedgers as suggested by Keynes. Additionally, a futures contract for which an illiquid level of trading volume exists would put the bid and offer prices for the contract further apart. A seller of such a futures contract would require more than the theoretical fair price as compensation for the risk undertaken. The risk is of prices starting to rise in an illiquid market in which the position cannot be immediately closed out. It costs money to maintain the position; therefore, a premium is required to cover this cost.

19

Table 1.1 The components of basis risk Type of risk Components Expiration-date risk

Futures contracts are not usually available for every month. If a hedger needed a futures contract for July and the only contracts that were available were for March, June, September, and December, the hedger would have to select either the June or September contract. Either of these contracts would have a different price series than a July contract (if one existed). Hence, the hedger cannot form a perfect hedge and is faced with the chance that the basis may change

Location risk

The hedger requires delivery of the futures contract in location Y, but the only futures contracts available are for delivery in location X. Hence, the hedger cannot form a perfect hedge because of the transportation costs from X to Y; this may cause the basis to change

Quality risk

The exact standard or grade of the commodity required by the hedger is not covered by the futures contract. Therefore, the price movement of commodity grade A may be different from the price movement of commodity grade B, which will cause the basis to change and prevent the hedger from forming a perfect hedge

Quantity risk

The exact amount of the commodity needed by the hedger is not available by a single futures contract or any integer multiple thereof. Hence, the amount of the commodity is not hedged exactly; this prevents the hedger from forming a perfect hedge, and the underhedged or overhedged amount is subject to risk

1.8.3 Hedging Concepts and Strategies The underlying motivation for the development of futures markets is to aid the holders of the spot commodity in hedging their price risk; consequently, the discussion now focuses on such an application of futures market. Four methodologies based on various risk-return criteria are examined; moreover, to fully clarify the hedger’s situation some of the common problems and risks that arise in the hedging process are analyzed.

1.8.3.1 Hedging Risks and Costs As mentioned previously, hedging refers to a process designed to alleviate the uncertainty of future price changes for the spot commodity. Typically this is accomplished by taking an opposite position in a futures contract on the same commodity that is held. If an investor owns the spot commodity, as is usually the case (a long position), the appropriate action in the futures market would be to sell a contract (a short position). However, disregarding for the moment the correct number of futures contracts to enter into, a problem arises if the prices of the spot commodity and the futures contract on this commodity do not move in a perfectly correlated manner. This nonsynchronicity of spot and futures prices is related to the basis and is called the basis risk. The basis has been defined as the difference between the futures and spot prices. Basis risk is the chance that this difference will not remain constant over time. Four types of risk contribute to basis risk; these are defined in Table 1.1. These four types of risks prevent the hedger from forming a perfect hedge (which would have zero risk). Even though the hedger

is reducing the amount of risk, it has not been reduced to zero. It is often said that hedging replaces price risk with basis risk. The potential causes of basis risk are not necessarily limited to those identified in Table 1.1. Hence, basis risk is the prominent source of uncertainty in the hedging process. Other potential causes are (1) supply-demand conditions and (2) cross-hedging consequences. Even if the futures contract is written on the exact commodity that the hedger holds, differing supply and demand conditions in the spot market and futures market could cause the basis to vary over time. Occasionally, speculators in the futures market will bid the futures price above or below its equilibrium position, due perhaps to the excitement induced by an unexpected news release. Of course, the market forces of arbitrage will eventually bring the spot and futures prices back in line. The limiting case is at the expiration of the futures contract, when its price must converge to the spot price.

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Consequently, the disequilibrating influence on the basis stemming from supply-demand forces can be alleviated by entering a futures contract that matures on the exact day that the hedger intends to sell the spot commodity. But although most futures contracts are quite flexible, it is unlikely that any contract would correlate so precisely with the hedger’s needs. In some cases (such as for agricultural commodities, where futures contracts are offered that mature each month) the basis risk due to nonsimultaneous maturities is not so great. However, for other commodities, particularly financial instruments, futures contracts maturing 3 months apart are more typically offered. Thus, at the time the hedger needs to sell the spot commodity in the market, any protection in price risk over the hedging period could conceivably be wiped out by a temporary adverse change in the basis. Cross-hedging refers to hedging with a futures contract written on a nonidentical commodity (relative to the spot commodity). Although not often necessary with agricultural futures, cross-hedging is frequently the best that can be done with financial and index futures. Changes in the basis risk induced by the cross-hedge are caused by less-than-perfect correlation of price movements between the spot and futures prices – even at maturity. That is, because the spot commodity and futures contract commodity are different, their respective prices will tend to be affected (even though minutely at times) by differing market forces. While the futures price must equal the price of its underlying spot commodity at the contract’s maturity date, this condition does not necessarily hold when the hedger’s commodity is not the “true” underlying asset. Therefore, even when the liquidation of the spot commodity coincides with the maturity of the futures contract, there is no guarantee of obtaining the original price of the commodity that held at the initiation of the hedge.

1

Theoretical Framework of Finance

where:

St D change in the spot price at time t;

Ft D change in the futures price at time t; a D constant; H D hedged ratio; and et D residual term at time t. Furthermore, the hedge ratio measure can be better understood by defining it in terms of its components: Xf Xs

D

 S; F DH 2  F

(1.59)

where: Xf and Xs D the dollar amount invested in futures and spot;  S; F D the covariance of spot and futures price changes; and 2 D the variance of futures price changes.  F Thus H , the minimum-variance hedge ratio computed in variability, is also a measure of the relative dollar amount to be invested in futures per dollar of spot holdings. In a sense it is a localized beta coefficient similar in concept to the beta of a stock à la capital asset pricing theory. As a measure of hedging effectiveness, Johnson utilizes the squared simple-correlation coefficient between spot and futures price changes, 2 . More formally, Johnson’s hedgingeffectiveness measure can be ascertained by first establishing the following expression: HE D 1 

VH Vu

(1.60)

where: 1.8.3.2 The Johnson Minimum-Variance Hedge Strategy Developed within the framework of modern portfolio theory, the Johnson hedge model (1960) retains the traditional objective of risk minimization but defines risk as the variance of return on a two-asset hedge portfolio. As in the two-parameter world of Markowitz (1959), the hedger is assumed to be infinitely risk averse (that is, the investor desires zero variance). Moreover, with the risk-minimization objective defined as the variance of return of the combined spot and futures position, the Johnson hedge ratio is expressed in terms of expectations of variances and covariances for price changes in the spot and futures markets. The Johnson hedge model can be expressed in regression form as: (1.58)

St D a C H Ft C et

2 Vu D variance of the unhedged spot position D Xs2  S 2  S D variance of spot price changes; and VH D the variance of return for the hedged portfolio D 2 .1  2 /. Xs2  S

By substituting the minimum-variance hedge position in the futures, Xf :  2 XS2  S .1  2 / D 2 HE D 1  2 XS2  S 

(1.61)

In simpler terms then, the Johnson measure of hedging effectiveness is the R2 of a regression of spot-price changes on futures-price changes. To utilize this hedging method, it is necessary to regress historical data of spot-price changes on futures-price changes. The resulting beta coefficient from the regression would be the localized Johnson hedge ratio, and the regression R2 would represent the expected degree of

1 Theoretical Framework of Finance

variance minimization using this hedge ratio over the hedging horizon. “Localized” and “expected” must be emphasized because, first of all, although the Johnson hedge ratio can be re-estimated, it nonetheless is a static measure based on historical data. What held for the past may not hold precisely for the future. Moreover, large price moves may distort this hedge ratio considerably. Hence, R2 is what can be expected based on the past in terms of variance reduction for the total hedge position. It should not be expected to hold exactly.

1.8.3.3 The Howard-D’Antonio Optimal Risk-Return Hedge Strategy The classic one-to-one hedge is a naïve strategy based upon a broadly defined objective of risk minimization. The strategy is naïve in the sense that a hedging coefficient of one is used regardless of past or expected correlations of spotand futures-price changes. Working’s strategy brings out the speculative aspects of hedging by analyzing changes in the basis and, accordingly, exercising discrete judgment about when to hedge and when not to hedge. The underlying objective of Working’s decision rule for hedgers is one of profit maximization. Finally, Johnson (1960), in applying the mean-variance criteria of modern portfolio theory, emphasizes the risk-minimization objective but defines risk in terms of the variance of the hedged position. Although Johnson’s method improves on the naïve strategy of a one-to-one hedge, it however, essentially disregards the return component associated with a particular level of risk. Rutledge (1972) uses both mean and variance information to derive hedge ratio. In a recent paper by Howard and D’Antonio (1984), a hedge ratio and measure of hedging effectiveness are derived in which the hedger’s risk and return are both explicitly taken into account. Moreover, some of the variable relationships derived from their analysis help explain some of the idiosyncrasies of hedging that occur in practice. Using a mean-variance framework, the HowardD’Antonio strategy begins by assuming that the “agent” is out to maximize the expected return for a given level of portfolio risk. With a choice of putting money into three assets – a spot position, a futures contract, and a risk-free asset – the agent’s optimal portfolio will depend on the relative risk-return characteristics of each asset. For a hedger, the optimal portfolio may contain a short futures position, a long futures position, or no futures position at all. In general, the precise futures position to be entered into will be determined by (1) the risk-free rate, (2) the expected returns and the standard deviations for the spot and futures positions, and (3) the correlation between the return on the spot position and the return on the futures.

21

Howard and D’Antonio arrive at the following expressions for the hedge ratio and the measure of hedging effectiveness: Hedge ratio H D

.œ  ¡/ ” .1  œ¡/

(1.62)

and s Hedging effectiveness HE D

1  2œ¡ C œ2 1  ¡2

(1.63)

where:   D f =s D relative variability of futures and spot returns; ’ D r f =.r s  i / D relative excess return on futures to that of spot; ” D Pf =Ps D current price ratio of futures to spot; œ D ˛= D .r f =f /=Œ.r s  i /=s D risk-to-excessreturn relative of futures versus the spot position; Ps ; Pf D the current price per unit for the spot and futures respectively; ¡ D simple correlation coefficient between the spot and futures returns; s D standard deviation of spot returns; f D standard deviation of futures returns; r s D mean return on the spot over some recent past interval; r f D mean return on the futures over some recent past interval; and i D risk-free rate. By analyzing the properties of œ these authors discern some important insights for the coordinated use of futures in a hedge portfolio. Numerically, œ expresses the relative attractiveness of investing in futures versus the spot position. When œ < 1; œ D 1, and œ > 1, the futures contract offers less, the same, and more excess return per unit of risk than the spot position, respectively. Since this analysis is being undertaken from a hedger’s point of view, it is assumed œ < 1. An assumption that œ > 1 would inappropriately imply that theoretically it is possible to hedge the futures position with the spot asset. From a practitioner’s perspective it is also important to note that even when œ ¤ ¡, a hedged position using the futures may not provide a real net improvement in the riskreturn performance of a portfolio. Unless HE is significantly greater than 1, other factors such as transaction costs, taxes, the potential for margin calls, and liquidity may negate the overall benefit of hedging with futures. This point, along with the previous results about hedging, helps explain why certain futures contracts highly correlated with their underlying

22

assets are not used extensively as hedging vehicles as might be expected. This section has focused on the basis concepts of futures markets. Important terms were defined and basic models to evaluate futures contracts were discussed. Finally, hedging concepts and strategies were analyzed and alternative hedging ratios were investigated in detail. These concepts and valuation models can be used in security analysis and portfolio management related to futures and forward contracts. Please see Chap. 57 for a generalized model for optimum futures hedge ratio.

1.9 Conclusion This chapter has reviewed and summarized alternative valuation theories – discounted cash flow, M and M, CAPM, APT, OPT, and futures valuation – and the Markowitz portfolio theory that are basic to introductory courses in financial management or investments. These theories can directly and indirectly become guidelines for further study of security analysis and portfolio management. Derivations and applications of these valuation models to security analysis and portfolio management are studied in detail in later parts of this handbook.

References Barnea, A., R. A. Haugen, and L. W. Senbet. 1981 “Market imperfections, agency problems, and capital structure: a review.” Financial Management 10, 7–22. Beranek, W. 1981. “Research directions in finance.” Quarterly Journal of Economics and Business 21, 6–24. Black, F. and M. Scholes. “The pricing of options and corporate liabilities.” Journal of Political Economy (May–June, 1973): 673–654. Brealey, R. and S. Mers. 1988. Principles of corporate finance, McGraw-Hill, New York. Brigham, E. F. 1988. Financial management: theory and practice, 4th ed., Dryden Press, Hinsdale. Copeland, T. E. and J. F. Weston. 1988. Financial theory and corporate policy, 3rd ed., Addison-Wesley, New York. Cox, J., S. A. Ross, and M. Rubinstein. 1979. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 7, 229–263. Cox, J. C. and M. Rubinstein. 1985. Option markets, Prentice-Hall, Englewood Cliffs, NJ. DeAngelo, H. and R. W. Masulis. 1980. “Optimal capital structure under corporate and personal taxation.” Journal of Financial Economics 8, 3–29. Fama, E. F. and M. H. Miller. 1972. Theory of finance, Holt, Rinehart and Winston, New york. Galai, D. and R. W. Masulis. 1976. “The option pricing model and the risk factor of stock.” Journal of Financial Economics 3, 53–81. Haley, C. W. and L. D. Schall. 1979. Theory of financial decision, 2nd ed., McGraw-Hill, New York. Howard, C. T. and L. J. D’Antonio. 1984. “A risk-return measure of hedging effectiveness.” Journal of Financial and Quantitative Analysis 19, 101–112.

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Theoretical Framework of Finance

Hsia, C. C. 1981. “Coherence of the modern theories of finance.” The Financial Review, 16, 27–42. Jensen, M.C. and W.H. Meckling. 1978. “Can the corporation survive?” Financial Analysts Journal, 34, 31–37. Johnson, L. L. 1960. “The theory of hedging and speculation in commodity futures.” Review of Economic Studies 27, 139–151. Keynes, J. M. 1930. A treatise on money, Vol 2. Macmillan & Co., London. Lee, C. F. 1983. Financial analysis and planning: theory and application. a book of readings. Addison-Wesley, MA. Lee, C. F. 1985. Financial analysis and planning: theory and applications. Addison-Wesley, MA. Lee, C. F. 1991. “Markowitz, Miller, and Sharpe: the first Nobel laureates in finance,” Review of Quantitative Finance and Accounting 1 209–228. Lee, C. F. and J. E. Finnerty. 1990. Corporate finance: theory, method, and applications. Harcourt Brace Jovanovich, Orlando, FL. Lee, C. F. and J. C. Junkus. 1983. “Financial analysis and planning: an overview.” Journal of Economics and Business 34, 257–283. Leland, H. and D. Pyle. 1977. “Informational asymmetries, financial structure, and financial intermediation.” Journal of Finance 32, 371–387. Mao, H. C. F. 1969. Quantitative analysis of financial decisions, The Macmillan Company, New York. Markowitz, H. 1959. Portfolio selection, Wiley, New York. Miller, M. H. 1977. “Debt and taxes.” Journal of Finance 32, 101–175. Miller, M. H. 1988. “The Modigliani–Miller proposition after 30 years.” Journal of Economic Perspectives 2, 99–120. Miller, M. H. and F. Modigliani. 1961. “Dividend policy growth and the valuation of share.” Journal of Business 34, 411–433. Modigliani, F. and M. Miller. 1958. “The cost of capital, corporation finance and the theory of investment.” American Economic Review 48, 261–297. Modigliani, F. and M. Miller. 1963. “Corporate income taxes and the cost of capital: a correction.” American Economic Review 53, 433–443. Pogue, G. A. and K. Lull. 1974. “Corporate finance: an overview.” Sloan Management Review, 15, 19–38. Reilly, F. K. 1985. Investment analysis and portfolio management, 2nd ed., Dryden Press, Hinsdale. Ross, S. 1976. “The arbitrage theory of capital asset pricing.” Journal of Economic Theory, 13, 341–360. Ross, S. A. 1977a. “The determination of financial structure: the incentive signalling approach.” Bell Journal of Economics 8, 23–40. Ross, S. 1977b. “Return, risk and arbitrage.” in Risk and return in finance, I Friend and J. L. Bicksler. (Eds.). vol. 1, Ballinger, Cambridge, MA. Ross, S. 1978. “Mutual fund separation in financial theory – the separating distributions.” Journal of Economic Theory 15, 254–286. Rutledge, D. J. S. 1972. “Hedgers’ demand for futures contracts: a theoretical framework with applications to the United States soybean complex.” Food Research Institute Studies 11, 237–256. Stiglitz, J. E. 1969. “A re-examination of the Modigliani-Miller theorem.” The American Economic Review 54, 784–793. Stiglitz, J. E. 1974. “On the irrelevance of corporate financial policy.” The American Economic Review 54, 851–866. Taggart, R. A Jr. 1980. “Taxes and corporate capital structure in an incomplete market.” Journal of Finance 35, 645–659. Van Horne, J. C. 1985. Financial management and policy, 6th ed., Prentice-Hall, Englewood Cliffs, NJ. Weston, J. F. 1981. “Developments in finance theory.” Financial Management 10 (Tenth Anniversary Issue), 5–22. Weston, J. F. and T. E. Copeland. 1986. Managerial finance, 8th ed., Dryden Press, Hinsdale. Working, H. 1953. “Hedging reconsidered.” Journal of Farm Economics 35, 544–561.

Chapter 2

Investment, Dividend, Financing, and Production Policies: Theory and Implications

Abstract The purpose of this chapter is to discuss the interaction between investment, financing, and dividends policy of the firm. A brief introduction of the policy framework of finance is provided in Sect. 2.1. Section 2.2 discusses the interaction between investment and dividends policy. Section 2.3 discusses the interaction between dividends and financing policy. Section 2.4 discusses the interaction between investment and financing policy. Section 2.5 discusses the implications of financing and investment interactions for capital budgeting. Section 2.6 discusses the implications of different policies on the beta coefficients. The conclusion is presented in Sect. 2.7.

investment and financing policy is discussed; the role of the financing mix in analyzing investment opportunities will receive detailed treatment. Again, the recognition of risky debt will be allowed, so as to lend further practicality to the analysis. The recognition of financing and investment effects are covered in Sect. 2.5, where capital-budgeting techniques are reviewed and analyzed with regard to their treatment of the financing mix. In this section several numerical comparisons are offered, to emphasize that the differences in the techniques are nontrivial. Section 2.6 will discuss implication of different policies on systematic risk determination. Summary and concluding remarks are offered in Sect. 2.7.

Keywords Investment policy r Financing policy r Dividend policy r Capital structure r Financial analysis r Financial planning, Default risk of debt r Capital budgeting r Systematic risk

2.2 Investment and Dividend Interactions: The Internal Versus External Financing Decision

2.1 Introduction This chapter discusses the three-way interaction between investment, financing, and dividend decisions. As shown in financial policy literature, there exists a set of ideal conditions under which there will be no interaction effects between the areas of concern. However, the ideal conditions imposed by academicians to analyze the effects dividend and financing policy have on the investment decision and the value of the firm are not realistic, when one considers financial management in the real world. Thus, an overview of the interactions of financing, investment, and dividend decisions is important for those concerned with financial analysis and planning. This chapter sequentially addresses the three interaction effects. In Sect. 2.2 the relation between investment and dividend policy is explored through a discussion of internal vs. external financing, with the emphasis on the use of retained earnings as a substitute for new equity, or vice versa. The interaction between corporate financing and dividend policies will be covered in Sect. 2.3, where default risk on debt is recognized. Then in Sect. 2.4, the interaction between

Internal financing consists primarily of retained earnings and depreciation expense, while external financing is comprised of new equity and new debt, both long and short term. Decisions on the appropriate mix of these two sources for a firm are likely to affect both the payout ratio and the capital structure of the firm, and this in turn will generally affect its market value. In this section an overview of internal and external financing is provided, with the discussion culminating in a summary of the impacts that earnings retention or earnings payout (with or without supplemental financing from external sources) can have on a firm’s value and on planning and forecasting.

Internal Financing Changes in equity accounts between balance-sheet dates are generally reported in the statement of retained earnings. Retained earnings are most often the major internal source of funds made available for investment by a firm. The cost of these retained earnings is generally less than the cost associated with raising capital through new common-stock issues.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_2, 

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24

Table 2.1 Payout ratio – composite for 500 firms

2

Investment, Dividend, Financing, and Production Policies: Theory and Implications

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973

0.58 0:567 0:549 0:524 0:517 0:548 0:533 0:547 0:612 0:539 0:491 0:414

It follows that retained earnings, rather than new equity, should be used to finance further investment if equity is to be used and the dividend policy (dividends paid from retained earnings) is not seen to matter. The availability of retained earnings is then determined by the firm’s profitability and the payout ratio, the latter being indicative of dividend policy. Thus we find that the decision to raise funds externally may be dependent on dividend policy, which in turn may affect investment decisions. The payout ratios indicated in Table 2.1 show that, on average, firms in the S&P 500 retained more than 50% of their earnings after 1971 rather than pay them out in dividends. This is done because of the investment opportunities available for a firm, and indicates that retained earnings are a major source of funds for a firm.

External Financing External financing usually takes one of two forms, debt financing or equity financing. We leave the debt vs. equity question to subsequent sections, and here directly confront only the decision whether to utilize retained earnings or new common stock to finance the firm. We have previously shown that the market value of the firm is unaffected by such factors if dividend policy and capital structure are irrelevant. It should also be clear that dividend policy and capital structure can affect market values. Therefore, the consideration of an optimal internal-external financing combination is important in the field of financial management. This optimal combination is a function of the payout ratio, the debt-to-equity ratio, and the interaction between these two decision variables. From this it can be shown (and this will be the topic of discussion in the following paragraphs) that different combinations of internal and external

1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985

0.405 0:462 0:409 0:429 0:411 0:38 0:416 0:435 0:422 0:399 0:626 0:525

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997

0.343 0:559 0:887 0:538 0:439 0:418 0:137 0:852 0:34 0:346 0:327 0:774

1998 1999 2000 2001 2002 2003 2004 2005 2006

0.234 0.435 0.299 0.383 0.04 0.259 0.311 0.282 0.301

financing will have different effects on the growth rate of earnings per share, of dividends per share, and, presumably, of the share price itself. Higgins (1977) considered the amount of growth a firm could maintain if it was subject to an external equity constraint and sought to maintain certain debt and payout ratios. He was able to show the sustainable growth rate in sales, S , to be: S D

p.1  d /.1 C L/ .rr/.ROE/ D t  p.1  d /.1 C L/ 1  .rr/.ROE/

(2.1)

where p, profit margin on sales; d , dividend payout ratio; L, debt-to-equity ratio; t, total asset-to-sales ratio; rr, retention rate; ROE, return on equity. Higgins (1977) used 1974 U.S. manufacturing firms’ composite financial statement data as an example to calculate S  . Using the figures p D 5:5%; d D 33%; L D 88% and t D 73%, he obtained: S D

.0:055/.1  0:33/.1 C 0:88/ D 10:5% 0:73  .0:055/.1  0:33/.1 C 0:88/

Using this method we can calculate sustainable growth rate in sales for financial analysts and planning models. From this we can see that a firm with many valuable investment opportunities may be forced to forego some of these opportunities due to capital constraints, and the value of the firm will not be maximized as it could have been. Dividend and capital-structure decisions relate directly to investment decisions. While it may not be reasonable to assume that the firm could not issue new equity, the question is at what cost it can be raised under such a constraint. In an even more practical vein, Higgins also incorporated inflation into his model, acknowledging the fact that most depreciation methods that serve to make depreciation a source of funds are founded on historical costs, and not the

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

replacement values the firm must pay to sustain operations at their current level. Introducing the new variables defined below, it is possible to define sustainable growth in real and nominal terms, the former being the item of interest here. Let c, nominal current assets to nominal sales; f, nominal fixed assets to real sales; j, inflation rate. So real sustainable growth Sr is Sr D

.1 C j /p.1  d /.1 C L/  jc : .1 C j /c C f  .1 C j /p.1  d /.1 C L/ (2.2)

Using figures from the manufacturing sector and an inflation rate of 10%, which at the time of writing was approximately the actual inflation rate then prevailing, it was found that real sustainable growth was only a third of that of the nominal figure; this serves to further emphasize the importance of the interaction between dividend policy and investment decisions, since the former acts as a constraint on the latter. Higgins (1977) paper has recently been generalized by Johnson (1981) and Higgins (1981). The usual caveat associated with the dividend-irrelevance proposition is that investment policy is unaffected; that is, new equity is issued to replace those retained earnings that are paid out. Here we emphasize new equity with the intent of avoiding financing-policy questions. Knowing that the flotation costs involved with new issues could be avoided by employing retained earnings, the effect such a strategy has on firm value is largely an empirical question. Any such tests directed toward this issue must also recognize that there may be a preference for dividends that may dominate or mitigate the flotation-cost effects. With these factors in mind, Van Horne and McDonald (1971) ran cross-sectional regressions on samples of utility stocks and electronics manufacturers to see whether dividend payouts and the rates of new issues of common stocks had significant effects on priceearnings ratios. The utility industry results indicate that, at the lower end of the new-equity issue spectrum, the dividendpreference effect overshadowed the flotation-cost effect, a result that appeared to shift in direction when higher levels of new equity financing were considered. Further analysis into the dividend-preference question has been performed by Litzenberger and Ramaswamy (1979). Using a generalized capital-asset-pricing model, they examined this trade-off between dividend preference and tax effects, all in an effort to justify the contention of an optimal dividend policy. The results presented by Van Horne and McDonald were admittedly tentative (especially since the electronics industry sample yielded few corroborating results), and most of the t-statistics associated with the utility sample new-issue coefficients were quite low. The strongest statement that could be made following this analysis is that there appears to be little detrimental effect on firm valuation from following a strategy of main-

25

taining a high payout rate and financing further investment with new-equity issues.

2.3 Interactions Between Dividend and Financing Policies If we were able to hold the capital structure of firms constant, then we would be able to determine the advantage or disadvantage of internal financing relative to external financing. Van Horne and McDonald, as briefly mentioned before, were able to empirically test for the effects created by either policy, providing the substance for a major part of the following discussion. If we were to go into more depth and specifically consider firms that issue risky debt while maintaining shareholder’s limited-liability status, and still keeping to the chosen internal or external equity plan, then the interactions between financing and investment decisions can affect the relative positions of stock and bondholders. Black (1976) argued that a possible strategy a firm could follow to transfer economic resources from bondholders to stockholders is to pay as generous dividends as possible. In this way the internal-external financing plan is predetermined, as paying large dividends jeopardizes the bondholders’ position as assets are siphoned away from the firm, all to the gain of the shareholders. In this section the interactions between dividend and financing policy will be analyzed in terms of (1) cost of equity capital and (2) the default-risk viewpoint of debt.

Cost of Equity Capital and Dividend Policy1 Van Horne and McDonald (1971) chose to develop a cross-sectional model to test for the significance of dividend payouts in the valuation process in the electric utility industry. The use of the electric utility industry proved to be operationally less difficult than other industries because it was desired to hold interfirm differences constant, and independent variable selection to this end was relatively straightforward. Since the authors were interested in capitalization rates, year-end P/E ratios were selected to be the dependent variables.1 This value was thought to be a function of three variables: growth in assets, dividend payout, and financial risk. The latter variable is considered sufficient for this sample due to the homogeneity of the other aspects of the included firm’s risks. The model can be written out and more carefully defined by the following; P =E D a0 C a1 g C a2 p C a3 R C u;

(2.3)

26

2

Investment, Dividend, Financing, and Production Policies: Theory and Implications

where g, compound growth of assets over eight previous years; p, dividend payout ratio on an annual basis; Lev, interest charges/[operating revenues – operating expenses]; u, error term. While the growth and risk factors do not correspond exactly to what most capital-market theory tells us is relevant, research with respect to this particular industry lends credence to these variables as defined here (see Malkiel 1970). Upon regressing the data for the 86 companies included in the sample, all three independent variables are found to be statistically significant. Of the 86 firms, all of which paid dividends, 37 firms also had new equity issues. Testing to see whether the 37 firms that issued new equity came from the same population as those that did not raise new equity, the researchers found no essential difference. Since we know there are nontrivial flotation costs associated with the issuance of new equity, the finding of no difference between the sub samples implies one of two things: either the costs associated with new issues are relatively too small compared to the total costs of the firm to be detected (despite their large absolute size, or size relative to the new issue by itself), or a net preference for dividends by holders of electric utility stocks exists that acts to offset the aforementioned expenses. The main thrust of this paper was to assess the impact of new equity flotation costs on firm value. With the number of firms issuing new equity, Van Horne and McDonald were able to calculate new-issue ratios – that is, new issues/total shares outstanding at year-end – and separate these firms into four groups, leaving those firms issuing new equity in a separate classification, as indicated in the upper portion of Table 2.2. Adding dummy variables to Equation (2.3), the effect of new issue rates could be analyzed: P =E D a0 C a1 .g/ C a2 p C a3 .Lev/ C a4 .F1 / Ca5 .F2 / C a6 .F3 / C a7 .F4 /:

(2.4)

where g, compound growth rate; Lev, financial risk measured by times interest earned, and F1 ; F2 ; F3 ; F4 , dummy variables representing levels of new equity financing. We would expect negative coefficients on the dummy variables if the flotation costs were to be relevant, but in fact all coefficients were positive, though only one was significant (possibly due to small sample size and small relative differ-

Table 2.2 New-issue ratios of electric utility firms

ences). Empirical results are indicated in the lower portion of Table 2.2. However, by replacing the dummy variable of Group E for any one of the four dummy variables discussed above, Van Horne and McDonald found that the estimate of its coefficient is negative. One question pertaining to the figures presented here stems from the interpretation of the significance of the dummy variables. As it turns out, the class B dummy-variable coefficient is greater than the class A coefficient, supposedly because of a preference for dividend payout, while actually that payout was lower in class B. The question is whether the dummy that is intended to substantiate the dividend-preference claim through the new-issue ratio actually tells us that investors instead attach a higher value to higher earnings retention. The finding of higher P/E ratios resulting from earnings retention would be consistent with the Litzenberger-Ramaswamy framework cited earlier, and the new-issue effect is somewhat confounded with the dividend effect. From the data we can see that the coefficients did decrease as the new-issue ratios increased, though they were not significant, and, not surprisingly, new-issue ratios were rather highly negatively correlated with dividend payout ratios. From this we can say, although with some hesitation, that external equity appears to be a more costly alternative compared to internal financing when pushed to rather extreme limits. Over more moderate ranges, or those more closely aligned with the industry averages, this claim cannot be made with the same degree of certainty, since we cannot be certain that the payout ratio does not have a positive influence on share price, and therefore an inverse relationship to the cost of equity capital.

Default Risk and Dividend Policy With the development of the Option Pricing Model, Black and Scholes (1973) have made available a new method of valuing corporate liabilities or claims on the firm. Chen (1978) has reviewed some recent developments in the theory of risky debt, and examines in a more systematic fashion the determinants of the cost of debt capital. Both Reseck (1970) and Hellwig (1981) have shown that the Modigliani

F dummy variable grouping A

B

C

D

New-issue ratio interval 0 0.001–0.05 0.05–0.1 0.1–0.15 Number of firms in interval 49 16 11 6 Mean dividend payout ratio 0:681 0:679 0:678 0:703 Dummy variable coefficient 1:86 3:23 1:26 0:89 Dummy variable t -statistic 1:33 2:25 0:84 0:51 From Van Horne and McDonald (1971), Reprinted by permission

E 0.15 and up 4 0.728 N.A. (N.A.)

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

and Miller (1958 and 1963) arbitrage process that renders capital structure irrelevant is generally invalidated if the debt under consideration is risky and the shareholders enjoy limited liability. From this and the 1963 M&M article, we can show that there do exist optimal structures for firms under the more realistic conditions mentioned above. In the option-pricing framework the two claimants of the firm, the debt holders and the equity holders, are easily seen to have conflicting interests; thus one group’s claim can only be put forth at the expense of the other party. The assumption that the total firm value is unchanged is utilized to highlight the wealth-transfer effects, but, as we see in the following section, that need not be the case. It is now necessary to find a way to value corporate debt when default risk is introduced. If the total value of the firm is known or given (this value should be reasonably well known or approximated prior to debt valuation as the former is an upper bound on the latter), then once the debt value is established the total equity value falls out as the residual. It is also required that we know how to perform the transfer of wealth (since the stated goal of corporate finance is to maximize the shareholders’ wealth), and learn of any possible consequences that could result from such attempts. Merton (1974) sought to find the value of risky corporate debt, assuming that the term structure of riskless interest rates was flat and known with certainty. This was an indirect goal, since the true emphasis lay in finding a way to value the equity of the firm as a continuous option. The further assumption of consol-type debt was also employed in an attempt to avoid the transactions-cost arguments involved with rolling over debt at maturity. Invoking Modigliani and Miller’s Proposition I (M&M 1958) and allowing for risky debt, we find the value of the stock to be the difference between the value of the firm as a whole and the value of the total debt financing employed to support that whole. Explicitly, S DV 

C .1  L/; r

(2.5)

where S , total value of the firm’s stock; V , total firm value; C , constant coupon payment on the perpetual bonds; r, riskless rate of interest; L, a complicated risk factor associated with possible default on the required coupon payment. The last term of Equation (2.5) is more often represented by B, the value of the total debt claims outstanding against the firm, stated in a certainty-equivalent form. It should be apparent from this equation that by introducing a greater probability of default on the debt while maintaining firm value, the equity holders gain at the expense of the bondholders. Rendleman (1978) took Merton’s model and adapted it to allow for the tax deductibility of interest charges. As in the Merton article, debt is assumed to be of perpetual type, consols – a justifiable assumption as most firms roll over

27

their debt obligations at maturity. The tax benefit introduced is assumed to be always available to the firm. In instances where the interest expense is greater than earnings before taxes, the carryback provision of the tax code is used to obtain a refund for the amount of the previously unused tax shield and, in the event the three-year carryback provision is not sufficient to make use of the tax shield, Rendleman suggests that a firm could sell that advantage to another firm by means of a merger or an acquisition, when the other firm involved could use the tax benefit. Thus the coupon (net) to be paid each time period is given by C.1  T /. At first glance it is apparent that the equity holders gain from this revision, as they are alleged to in the 1963 taxcorrected model of M&M. But something else is at work as well: Since the firm is subject to a lower debt-service requirement, it can build a larger asset base, which serves to act as insurance in the case of default probabilities owing to the small net coupon payment. In short, the risk premium associated with debt is reduced and the value of the stock, given in Equation (2.5) can be rewritten as shown below: S DV 

C .1  T / .1  L /; r

(2.6)

where all the variables are as defined before and L is less than the L given before because of the lessened default risk. It has been argued above that the bondholders benefit when interest is tax-deductible, excluding the possibility here that an overzealous attempt to lever the firm is quickly undertaken when the tax-deductibility feature is introduced, and Equations (2.5) and (2.6) seem to indicate that the value of the shareholders’ claim also increases, although we have not yet provided any theoretical justification for such a statement. Rendleman’s analysis gave us the L-value, which is the clue to this seeming problem. Management can act in a way to jeopardize the bondholders’ claims by issuing more debt and thus making all debt more risky, or by undertaking projects that are riskier than the average project undertaken by the firm. This is the subject of the next section. Another possibility is to pay large dividends so as to deplete the firm of its resources, thus paying off the shareholders but hurting the bondholders. Black (1976) goes so far as to suggest that the firm could liquidate itself, pay out the total as a dividend, and leave the bondholders holding the proverbial bag. Bond covenants, of course, prevent this sort of action (hence, agency theory), and if a firm could not sell its growth opportunities not yet exploited in the market, this may not maximize the shareholder’s wealth, either, but within bounds it does seem a reasonable possibility. In their review article, Barnea, Haugen, and Senbet (BHS, 1981) discuss the issues related to market imperfections, agency problems, and the implications for the consideration of optimal capital structures.2 The authors make use of M&M’s valuation theory and option-pricing theory to

28

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

reconcile the differences between academicians and practitioners about the relevance of financing and dividend policies. They arrive at the conclusion that, without frictionless capital markets, agency problems can give rise to potential costs. These costs can be minimized through complex contractual arrangements between the conflicting parties. Potential agency costs may help to explain the evolution of certain complexities in capital structure, such as conversion privileges of corporate debt and call provisions. If these agency costs are real, then financial contracts that vary in their ability to reduce these costs may very well sell at different equilibrium prices or yields, even if the financial marketplaces are efficient. An optimal capital structure can be obtained when, for each class of contract, the costs associated with each agency problem are exactly balanced by the yield differentials and tax exposures. Overall, BHS show that optimal capital structures can exist and that this is still consistent with the mainstream of classical finance theory.

2.4 Interactions Between Financing and Investment Decisions Myers (1974) has analyzed in detail the interactions of corporate financing and investment decisions and the implications therein for capital budgeting. He argues that the existence of these interaction effects may be attributable to the recognition of transaction costs, corporate taxes, or other market imperfections. Ignored in his analysis is the probability of default on debt obligations, or, as could otherwise be interpreted, changes in this default risk. As alluded to in the previous section, if we consider possible effects of default risk on the firm’s investment and financing decisions, then further analysis must be performed to determine how these interactions can affect the wealth positions of shareholders and bondholders. In this section we will analyze this issue by considering both the risk-free debt case and the risky debt case. Chapter 92 and Chap. 60 discuss the theoretical and empirical issues of capital structure.

2.4.1 Risk-Free Debt Case Following Myers (1974), the basic optimization framework is presented in accordance with well-accepted mathematical programming techniques as discussed in the literature of capital-rationing. It is presented as a general formulation; it should be considered one approach to analyzing interactions and not the final word per se. Specific results derived by Chambers et al. (1982), will be used later to demonstrate the importance of considering alternative financing mixes when evaluating investment opportunities.

We identify a firm Q, which faces several investment opportunities of varying characteristics. The objective is to identify those projects that are in the stockholders’ interest (i.e., they maximize the change in the firm’s market value ex-dividend at the end of the successive time periods t D 0; 1; : : : ; T) and undertake them in order of their relative values. We specify relative values so that project divisibility remains possible, as was assumed by Myers. Required is a financing plan that specifies the desired mix of earnings retained, debt outstanding, and proceeds from the issuance of new equity shares. Let dV be a general function of four factors in a direct sense: (i) the proportion of each project j accepted, xj ; (ii) the stock of debt outstanding in period t; yt ; (iii) the total cash dividends paid in period t, Dt ; and (iv) the net proceeds from equity issued in period t; Et . For future use let Z be denoted as the debt capacity of the firm in period t, this being defined as a limit on yt imposed internally by management or by the capital markets, and Ct is the expected net after-tax cash flow to the firm in time period t. The problem can now be written as: Maximize dV .xj ; yt ; Dt ; Et / D W; subject to the constraints; .a/ Uj D xj  1  0 .j D 1; 2; : : : ; J /I .b/ UtF D yt  Zt  0 .t D 1; 2; : : : ; T /I

(2.7)

.c/ UtC D Ct Œyt yt 1 .1C.1  /r/ CDt Et  0; where £ and r are the tax rates and borrowing rates, respectively. Both are assumed constant for simplicity, but in actuality both could be defined as functions of other variables. Constraints (a) and (b) specify the percentage of the project undertaken cannot exceed 100%, and the debt outstanding cannot exceed the debt capacity limit. Constraint (c) is the accounting identity, indicating that outflows of funds equal inflows, and could be interpreted as the restriction that the firm maintain no excess funds. Constraint (b) can be used to investigate the interaction between the firm’s financing and investment decisions by examining the necessary conditions for optimization. If we assign the symbols Lj ; Lft , and Lct to be the shadow prices on constraints (a)–(c), respectively, and let Aj D dW=dxj , Ft D dW=dyt ; Zjt D dZ t =dxj , and Cjt D dCt =dxj , we can rewrite Equation (2.7) in Lagrangian form as the following: Max W 0 D W  Lj .xj  1/  Lf t .yt  Zt / Lct fCt  Œyt  yt 1 .1 C .1  t/r/ g C.Dt  Et /: (2.70 )

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

The necessary first-order conditions for the optimum are shown as Equations (2.8) through (2.11), with accompanying explanations. For each project: Aj C

T X

ΠLf t Zjt C Lct Cjt  Lj  0

(2.8)

t D0

This can be interpreted as follows: The percentage of a project undertaken should be increased until its incremental cost exceeds the sum of the incremental value of the project, the latter consisting of the added debt capacity and the value of the cash flows generated by that project. The incremental increase in value of the firm obtained by increasing xj to this maximum point is termed the Adjusted Present Value, APV (“adjusted” because of the consideration of interaction effects). We can examine each of these effects in turn. For the debt constraint in each period: Ft  Lf t C Lct  Lc;t 1 Œ1 C .1  /r  0:

(2.9)

For the constraint implied on dividends: dW  Lct  0I dDt

(2.10)

29

tax purposes. This procedure will be further investigated in the following section when we compare it with other wellaccepted capital-budgeting techniques. When dividend policy is not irrelevant, the standard argument arises as to whether the preference for dividends, given their general tax status, outweighs the transaction costs incurred by the firm when dividends are paid and new financing must be raised. In this framework, dividends are viewed as another cash flow and, as such, the issue centers on whether the cash flows from the project plus any increases in the debt capacity of the firm are adequate to cover the cost of financing, through whatever means. Although we are unable to discern the effect of dividend policy on the value of the firm, it is not outside the solution technique to incorporate such efforts by including the related expenses as inputs to the numerical-solution procedure. It is generally assumed that these interaction effects exist, so if we are to consider disregarding them, as some would insist, we must know what conditions are necessary for lack of dependence of financing and investment decisions. As Myers points out, only in a world of perfect markets with no taxes is this the case. Otherwise, the tax deductibility of the interest feature of debt suggests that the APV method gives a more accurate assessment of project viability than does the standard NPV method.

and for the new equity constraint: dW C Lct  0: dEt

(2.11)

While Equations (2.8)–(2.11) are all of interest, the focus is on Equation (2.8), which tells us that the Net Present Value (NVP) rule commonly put forth should be replaced by the APV rule, which accounts for interaction effects. Specifically considering the financing constraint, Equation (2.9), we would like to be able to find the value of this constraint, which is most easily done by assuming for the moment that dividend policy is irrelevant. Combined with Equations (2.8) and (2.9) and the definition of APV we obtain3; APV j D Aj C

T X

Zjt Ft

(2.12)

t D0

which, in the spirit of the M&M with-tax firm-valuation model, tells us that the value of a project is given by the increase in value that would occur in an unlevered firm, plus the value of the debt the project is capable of supporting. This follows from the firms ability to deduct interest expenses for

Risky Debt Case Rendleman (1978) not only examined the risk premiums associated with risky debt, but also considered the impact that debt financing could have on equity values, with taxes and without. The argument is to some extent based on the validity (or lack thereof) of the perfect-market assumption often invoked, which, interestingly enough, turns out to be a double-edged sword. Without taxes, the original M&M article claims, the investment decision of a firm should be made independent of the financing decision. But the financing base of the firm supports all the firm’s investment projects, not some specific project. From this we infer that the future investments of a firm and the risk premiums embodied in the financing costs must be considered when the firm takes on new projects. If, for example, the firm chooses to take on projects of higherthan-average risk, then this may have an adverse effect on the value of the outstanding debt (to the gain of the shareholders), and the converse holds true as well. It follows that the management of a firm should pursue more risky projects to transfer some of the firm’s risk from the shareholders to

30

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

the bondholders, who do not receive commensurate return for that risk. If the bondholders anticipate this action on the part of management, then it is all the more imperative that management takes the action because the bondholders are requiring and receiving a risk premium, for which they are not incurring the “standard” or market-consensus level of risk. Myers (1977) presents an argument in which a firm should issue no risky debt. The rationale for this strategy is that a firm possesses certain real options, or investment opportunities, that unfold over time. With risky debt some of these investment projects available only to the firm of interest may not be undertaken if it is in the interest of the shareholders to default on a near-term scheduled debt payment. In this way risky debt induces a suboptimal investment policy and firm value is not maximized as it would be if the firm issued no risky debt. One problem here is that we do not have strict equivalence between equity-value maximization and firm-value maximizations, so investment policy cannot be thought of entirely in terms of firm-value maximization. This is a quirk associated with the option-pricing framework when applied to firm valuation, and it clouds the determination of what is suboptimal in the finance area.4 Although we concede that we aren’t entirely sure as to how we should treat the interaction effects of financing and investment policy, we can state with some degree of certainty that, even in the absence of the tax deductibility of interest, these two finance-related decisions are interdependent and, as a result, the financial manager should remain wary of those who subscribe to the idea that financing decisions do not matter. Allowing the tax deductibility of interest, it is ironic to note that the conclusions are not nearly as clear-cut as before. If the firm undertakes further, more risky projects, the value of debt may actually increase if the firm does not issue further debt. The shareholders gain from the new project only on its own merits. If no additional debt financing is raised, it is impossible to obtain a larger tax shield and the debt may actually become more secure as a result of the larger asset base. If, however, the firm does issue more debt and in that way acts to jeopardize the currently outstanding debt, the number of considerations multiply, and analysis becomes exceedingly difficult because the value of the project by itself, plus the value of the added tax shield, needs to be considered in light of the possible shifting of wealth due to transfers of risk among claimants of the firm. Thus it seems that, when we allow the real world to influence the model, as it well should, the only thing we can say for certain about financing and investment decisions is that each case must be considered separately.

2.5 Implications of Financing and Investment Interactions for Capital Budgeting This section is intended to briefly review capital-budgeting techniques, explain how each in turn neglects accounting for financing influences in investment-opportunity analysis, and discuss the method by which they do incorporate this aspect of financial management into the decision process. We will draw heavily on the work of Chambers et al. (1982), in making particular distinctions between methods most often covered in corporate-finance textbooks and presumably used in practice, presenting numerical comparisons derived from varying sets of circumstances. Chambers, Harris, and Pringle (CHP) examined four standard, discounted cash-flow models and considered the implications of using each as opposed to the other three. By way of simulation, they were able to deal with differences in financing projects as well as with possible differences in the risk underlying the operating cash flows generated by each project. The problem inherent in this and any project evaluation (not withstanding other difficulties) is in concentrating on the specification of the amount of debt used to finance the investment project. Project debt is therefore defined as the additional debt capacity afforded the firm as a result of accepting the project, and can alternatively be described as the difference between the firm’s optimal debt level with the project and without it. Conceptually this is a fairly concrete construct, but it still leaves some vague areas when we are actually performing the computations. It is essential to arrive at some value estimate of the estimated cash flows of a project. The CHP analysis considered the following four methods: (i) the Equity Residual method; (ii) the “after-tax Weighted Average Cost-ofCapital” method; (iii) the “Arditti-Levy” weighted cost-ofcapital method; and (iv) the Myers “Adjusted Present Value” method. For simplicity, Table 2.3 contains the definitions of the symbols used in the following discussion, after which we briefly discuss the formulation of each method, and then elaborate on the way in which each method incorporates interacting effects of financing and investment decisions.

Equity-Residual Method The equity-residual method is formulated in Equation (2.13) in a manner that emphasizes the goal of financial management, the pursuance of the interests of shareholders:

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

Table 2.3 Definitions of variables

31

Rt D Pretax operating cash revenues of the project during period t; Ct D Pretax operating cash expenses of the project during period t; dept D Additional depreciation expense attributable to the project in period t; £c D Applicable corporate tax rate; I D Initial net cash investment outlay; Dt D Project debt outstanding during period t; NP D Net proceeds of issuing project debt at time zero; rt D Interest rate of debt in period t; ke D Cost of the equity financing of the project; kw D After-tax weighted-average cost of capital (i.e., debt cost is after-tax); kAL D Weighted average cost of capital – debt cost considered before taxes; ¡ D Required rate-of-return applicable to unlevered cash-flow series, given the risk class of the project r, ke , and ¡ are all assumed to be constant over time

NPV.ER/ D

N X Œ. Rt  Ct  dept  rD t /.1  c / C Dt  . Dt  Dt C1 /

.1 C ke /t

t D1

The formula presented above can be interpreted as stating that the benefit of the project to the shareholders is the present value of the cash flows not going to pay operating expenses or to service or repay debt obligations. With these flows identified as those going to shareholders, it is appropriate, and rather easy, to discount these flows at the cost of equity. The only difficulty involved is identifying this cost of equity, a problem embodied in all capital-budgeting methods.

 ŒI  NP :

(2.13)

2.5.1 Arditti and Levy Method The Arditti-Levy method is most similar to the after-tax weighted-average cost-of-capital method. This formulation can be written as: NPV.AL/ D

N X Œ. Rt  Ct  dept  rD t /.1  c / C dept C rD t I .1 C kAL /t tD1

(2.15)

After-Tax, Weighted-Average, Cost-of-Capital Method The after-tax, weighted-average, cost-of-capital method, depicted in Equation (2.14), has two noticeable differences from the formulation of the equity-residual method: NPV D

N X . Rt  Ct  dept /.1  c / C dept t D1

.1 C kw /t

I:

(2.14)

First, no flows associated with the debt financing appear in the numerator, or as relevant cash flows. Second, the cost of capital is adjusted downward, with kw being a weighted average of debt and equity costs, the debt expense accounted for on an after-tax basis. In that way debt financing is reckoned with in an indirect manner. With the assumption that r and ke are constant over time, kw can be affected only by the debtto-equity ratio, a problem most often avoided by assuming a fixed debt-to-equity ratio.

It was necessary to restate the after-tax, weighted-average, cost-of-capital formula in this manner because the tax payment to the government has an influence on the net cash flows, and for that reason the cash-flow figures would be misleading. To rectify this problem the discount rate must now be adjusted as the interest tax shield is reflected in the cash flows and double counting would be involved. While the after-tax, weighted-average cost of capital recognized the cost of debt in the discount rate, it was akin to the equity-residual method in considering only returns to equity. The Arditti and Levy formulas imply that a weighted average discount rate including the lower cost of debt could only (or best) be used if all flows to all sources of financing were included; hence the term rDt is found at the end of the first term. This can be rationalized if one considers the case of a firm where there is one owner. The total cash flow to the owner is the relevant figure, and the discount rate applicable is simply a weighted average of the two individual required rates of return.

32

2

Table 2.4 Application of four capital budgeting techniques

Investment, Dividend, Financing, and Production Policies: Theory and Implications

Inputs: (1) ke D 0:112 (2) r D 0:041 (3) c D 0:46 (4)  D 0:0802 (5) w D 0:6 Method NPV results Discount rates 1. Equity-residual 2. After-tax WACC 3. Arditti-Levy WACC 4. Myers APV

$230:55 270:32 261:67 228:05

ke D 0:112 kw D 0:058 kAL D 0:069 r D 0:041 and  D 0:0802

From Chambers et al. (1982). Reprinted by permission

2.5.2 Myers Adjusted-Present-Value Method This method, derived in Sect. 2.4, is closely related to the Arditti-Levy method except for the exclusion of the interestexpense flows to the bondholders. In treating the financing mix, Myers implicitly assumes that the tax shield, created by the interest payments and afforded to the equity holders, has the same risk for the equity holders as the coupon or interest payments have for the bondholders. Instead of aggregating all factors and attempting to arrive at a suitable weightedaverage discount rate, Myers found it less difficult to leave operating- and financing-related flows separated, and to discount each at an appropriate rate. This formulation can be written as: APV D

N X . Rt  Ct  dept /.1  c / C dept t D1

I C

.1 C  /t N X c rD t : .1 C r /t t D1

(2.16)

This formulation appears to be closely related to the Modigliani and Miller with-tax firm-valuation model, and well it should, given Myers’ motivation for this work. We choose not to discuss it here, but the reader should be aware that Myers’ emphasis was not solely on the tax advantage of debt, as the last term in the above equation tends to imply. The four methods are obviously comparable, but usually give different figures for the net present value. The equityresidual, after-tax, weighted-average cost-of-capital and the Arditti-Levy weighted-average, cost-of-capital formulations are comparable if the value of the debt outstanding remains a constant proportion of the remaining cash flows; this is often taken to mean a constant debt ratio. This will not guarantee that the Myers APV method will yield the same results, although the Myers method is equivalent to the other three only if the project life can be described as one period, or as infinite with constant perpetual flows in all periods. By way of numerical examples we now address the task of determining the factors that create the differences in the net-presentvalue figures generated by each method. The first example involves the simplest case, where a $1,000 investment generates operating flows of $300 per

year for all 5 years of the project’s life. Debt financing comprises $600 of the project’s total financing and the principal is repaid over the five time periods in equal amounts. The discount and tax rates employed are included below in Table 2.4 with the results for each of the four methods. Because the Arditti-Levy method recognizes the acquisition of the debt capital and uses a lower discount rate, as does the after-tax WACC, these two methods give the highest netpresent-value figures. The equity-residual method recognizes only the $400 outflow at time zero, but the higher discount suppresses the net-present-value figure. The Myers APV method, though discounting financing-related flows at the cost of debt, also attains a low net-present-value figure because the majority of the flows are discounted at a higher unlevered cost-of-equity rate. Basically we can speak of three differences in these methods that create the large discrepancies in the bottom-line figures. The risk factor, reflected in discount rates that may vary over time, is a major element, as was evidenced in the example shown above. The pattern of debt and debt payments will also be an important factor, particularly so when the debt repayment is not of the annuity form assumed earlier. Finally, the recognition and valuation of the debt tax shields will play an important role in net-present-value determination, especially when the constant-debt-repayment assumption is dropped and the interest expenses and associated tax shields grow larger. In the CHP study the valuation models described earlier were employed in a simulation procedure that would allow the assessment of the investment proposal. The inputs to the capital-budgeting procedures were varied across simulations, and the effects of the changes in each input were scrutinized from a sensitivity-analysis viewpoint. They considered, but did not dwell upon, the effects of changing discount rates over time or by some scaling factor on the bottom-line valuation figures; further discussion was deferred primarily because of the multitude of possible combinations that would be of interest. Compounding the problem, one would also be interested in different debt-equity combinations and the effects these would have with changing discount rates, as the weighting scheme of the appropriate discount rates plays an integral part in the analysis. In avoiding this aspect of the analysis, the projects evaluated in the forthcoming discussion will be viewed as being of equivalent risk, or as coming

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

Table 2.5 Inputs for simulation

Project

Net cash inflows per year

33

Project life

1 2 3 4

$300 per year 5 years $253.77 per year 5 years $124.95 per year 20 years $200 per year, years 1–4 5 years $792.58 in year 5 For each project the initial outlay is $1,000 at time t D 0, with all subsequent outlays being captured in the yearly flows Debt schedule Market value of debt outstanding remains a constant proportion of the project’s market value Equal principal repayments in each year Level debt, total principal repaid at termination of project r D 0:041 Inputs: ke D 0:112 .M &M / D 0:0986 kw D 0:085 c D 0:46 M D 0:085 W D 0:3 From Chambers et al. (1982). Reprinted by permission

from the same risk class. Lest the reader feel shortchanged, it is suggested that one examine each method and verify for him or herself what changes would be produced in the netpresent-value figures with varying debt schedules. More in tune with the basic theme of this chapter, we go on to consider the effects that changing financing mixes, debt-payment patterns, and project lives have on the figures attained for each of the four methods considered, the first (financing mix) being the major issue involved. In confronting mix effects, we are required to select a model for valuing debt in the Myers APV method because ¡ (the unlevered cost of equity capital) is unobservable. In this case we must choose between numerous alternatives, the most notable being those of Modigliani and Miller (1963), where debt provides an interest-tax shield, and that of Miller (1977), where the inclusion of personal taxes on investor interest income has the effect of perfectly offsetting the tax shield the firm receives, rendering the debt tax advantage moot. In the simulation results of CHP presented in the following paragraphs, the Myers cost of unlevered equity capital is computed using each method, with subscripts denoting the particular method used. Four projects of varied cash-flow patterns and lives are to be presented and valued, each of which will be simulated with three different debt schedules. Brief descriptions of the projects and debt schedules can be found in Table 2.5, along with the fixed inputs to be used in the actual computations. The projects’ cash flows, as listed in the table, were actually manipulated in a predetermined way, so that the more interesting cases would be presented. Project 1 is simply a base figure, while project 2 has a cash-flow pattern that makes the net present value 0, when using the after-tax, weightedaverage cost of capital. Project 3 has a longer life, and thus will serve as a method of determining the effects of increasing project life. Finally, Project 4 has four level payments

with a larger final payment in year 5, intended to simulate an ongoing project with terminal value in year 5. The results of the simulations are presented in Table 2.6. For purposes of comparison, the net-present-value figures of the after-tax, weighted-average cost of capital are reported first, for two reasons. The after-tax weighted-average cost of capital is probably the best-known technique of capital budgeting, and its net present value figures are insensitive to the debt schedule. The initial weights of the debt and equity used to support the project are all that are used in calculating the weighted discount rate, rendering the repayment pattern of debt irrelevant. As indicated earlier, the after-tax weighted-average cost of capital, Arditti-Levy weighted-average cost of capital, and the equity-residual methods are all equivalent if the debt ratio is held constant. It is also of interest here that the Myer’s APV figures, using the Miller method for determining the cost of capital, are constant over debt schedules; they are, for all intents and purposes, the same as the after-tax, weightedaverage cost-of-capital figures, a finding that is reasonable if one believes that the cost of debt capital is the same as that of unlevered equity. Of all the methods cited above, the equityresidual method is the most sensitive to the debt schedule employed, with both principal and interest payments included in the cash flows, a feature compounded by the higher discount rates included in the computations. The two methods that include the interest tax shield, the Arditti-Levy method and the Myers APV (M&M) method are also sensitive to the amount of debt outstanding, the interest tax shield being of greater value the longer the debt is outstanding. In all cases the latter method gives the lowest net-present-value figures due to the treatment of the interest tax shield. In further simulations CHP showed (and it should be no surprise) that as higher levels of debt are employed and higher tax rates are encountered, the magnitude of the

34

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

Table 2.6 Simulation results

Net-present-value under alternative debit schedule Project

Capital budgeting

Constant debt ratio

Equal principal

Level debt

1

After-tax WACC Arditti-levy WACC Equity-residual Myers APV (M&M) Myers APV (M) After-tax WACC Arditti-Levy WACC Equity-Residual Myers APV (M&M) Myers APV (M) After-tax WACC Arditti-Levy WACC Equity Residual Myers APV (M&M) Myers APV (M) After-tax WACC Arditti-Levy WACC Equity Residual Myers APV (M&M) Myers APV (M)

182 182 182 160 182 0 0 0 18 0 182 182 182 138 182 182 182 182 155 182

182 179 167 157 182 0 1 3 19 0 182 169 128 119 182 182 174 147 146 182

182 187 202 166 182 0 7 32 10 0 182 186 194 150 182 182 182 183 156 182

2

3

4

From Chambers et al. (1982). Reprinted by permission

differences is amplified, and the method employed in the capital-budgeting decision takes on greater and greater importance. Even so it was argued that changes associated with changes in the inputs of the longest-lived project, where the changes in the net-present-value figures were the most pronounced, were not that great when compared with the outcomes associated with changing the estimates of the cash flows by as little as 5%. The importance of this finding is that projects that are of short duration and are financed with relatively little debt are not as sensitive to the capitalbudgeting technique employed. But, in the case of longerlived projects, the method selected can have serious implications for the acceptance or rejection of a project, particularly when higher levels of debt financing are employed. Analysts undoubtedly possess their own views as to which method is stronger conceptually and, in the event of capitalbudgeting procedures, should be aware of the debt policy to be pursued. Even in light of these views, it may be prudent to use the Myers APV method with the M&M unlevered-equity cost-determination method as the first screening device for a project or set of projects. Since this method yields the most conservative figures, any project that appears profitable following this analysis should be undertaken, and any project failing this screening can be analyzed using the methods chosen by the financial manager, if further analysis is thought to be warranted.

In this book, Chap. 56 discusses the capital structure and CEO entrenchment in Asia. Moreover, Chap. 60 discusses the theory and application of alternative methods to determine optimal capital structure. Finally, Chap. 92 provides discussion on the capital structure and entry deterrence.

2.6 Implications of Different Policies on the Beta Coefficient Investment, financing, dividend, and production policy are four important policies in financial management decision. In previous sections, we have discussed investment, financing and dividend policies. In this section, we will discuss the impacts of financing, production and the dividend policy on beta coefficient determination.

Impact of Financing Policy on Beta Coefficient Determination Suppose that the security is a share in the common stock of a corporation. Let us assume that that this corporation increases the proportion of debt in its capital structure, all other relevant factors remaining unchanged. How would you

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

expect this change to affect the firm’s beta? We have seen that an increase in the level of debt leads to an increase in the riskiness of stockholders’ future earnings. To compensate for this additional risk, stockholders will demand a higher expected rate of return. Therefore, from Equation (2.17), beta must rise for this company’s stock. We see that, all other things equal, the higher the proportion of debt in a firm’s capital structure, the higher the beta of shares of its common stock.

B.1  c / (2.17) ˇL D ˇU 1 C S where ˇL is the leveraged bet; ˇU is the unlevered operating beta; B is the amount of debt; S is the amount of equity; and c is the corporate tax rate. When the market model is used to estimate a firm’s beta, the resulting estimate of the beta is the market assessment of both operating and financial risk. This is called leveraged beta. Hamada (1972) and Rubinstein (1973) suggest that Equation (2.17) can be modified to calculate the unleveraged beta. This beta is an estimate of the firm’s operating or business, risk.

Impact of Production Policy on Beta Coefficient Determination In this section, we will first discuss production policy. Then we will discuss implications of different policies on the beta coefficient determination. Production policy refers to how a company uses different input mix to produce its products. The company’s production process can be classified into either capital intensive or labor intensive, which depend on whether capital labor ratio (K=L Ratio) is larger or smaller than one. We can use either Cobb-Douglas production function or Variable Elasticity of Substitution (VES) production function to show how capital and labor affect the change of the beta coefficient. The Cobb-Douglas production function in two factors, capital .K/ and labor .L/ can be defined as follows. (2.18) Q D K a Lb where Q is firm’s output, a and b are positive parameters. Lee et al. (1990) have derived the theoretical relationship between beta coefficient and the capital labor ratio in terms of the Cobb-Douglas production functions as follows in Equation (2.18). ˇD

.1 C r/Cov.e; Q RQ m / Q Var.Rm / f Œ1  .1  E/b g

(2.19)

35

where r, the risk-free rate; RQ m , return on the market portfolio; e, Q random price disturbances with zero mean; E D .@P =@Q/.Q=P /, an elasticity constant; b, contribution of labor to total output; D 1 cov.Qv; RQ m /, and , the market price of systematic risk. In addition, the VES production function in two factors capital .K/ and labor .L/ can be defined as follows in Equation (2.19). Q D K ˛.1s/ ŒL C .  1/K ˛s

(2.20)

where Q is firm’s output and ˛; s, and  are parameters with the following constraints: ˛ > 0; 0<s<1 0  s  1; L=K > .1  /=.1  s/3 : Lee et al. (1995) have derived the theoretical relationship between beta coefficient and the capital labor ratio in terms of the VES production functions as follows. ˚ .1 C r/ C ov.e; Q RQ m /  Œˆ.1  E/˛s 1 w.pQ/1 .1  /K C ov.Qv; RQ m / ˇD Var.RQ m / fˆ  Œˆ.1  E/˛s  w .pQ/1 .1  /K (2.21) where p, expected price of output; , reciprocal of the price elasticity of demand, 0    1I w, expected wage rate; vQ , random shock in the wage rate with zero mean; ˆ D 1  cov.e; Q RQ m /, and r; RQ m ; e; Q E; ;  are as defined in Equation (2.19).

Impact of Dividend Policy on Beta Coefficient Determination Impact of payout ratio on beta coefficient is one of the important issues in finance research. By using dividend signaling theory, Lee et al. (1993) have derived the relationship between the beta coefficient and pay out ratio as follows: ˇi D ˇpi Œ1 C F .di /

(2.22)

ˇi , the firm’s systematic risk when the market is informationally imperfect and the information asymmetry can be resolved by dividends; ˇpi , the firm’s systematic risk when market is informationally perfect;  , a signaling cost incurred

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

if firm’s net after-tax operating cash flow X falls below the promised dividend DI di , firm’s dividend payout ratio; F .di /, cumulative normal density function in term of payout ratio. Equation (2.21) implies that beta coefficient is a function of payout ratio. In sum, this section has shown that how financing, production, and dividend policy can affect the beta coefficient. This information discussed in this section is important for calculating the cost of capital equity.

2.7 Conclusion In this chapter we have attacked many of the irrelevant propositions of the neoclassical school of finance theory, and in so doing have created a good news-bad news sort of situation. The good news is that by claiming that financial policies are important, we have justified the existence of academicians and a great many practicing financial managers. The bad news is that we have made their lot a great deal more difficult as numerous tradeoffs were investigated, the more general of these comprising the title of the chapter. In the determination of dividend policy, we examined the relevance of the internal-external equity decision in the presence of nontrivial transaction costs. While the empirical evidence was found to be inconclusive because of the many variables that could not be controlled, there should be no doubt mind that flotation costs (incurred when issuing new equity to replace retained earnings paid out) by themselves have a negative impact on firm value. But if the retained earnings paid out are replaced whole or in part by debt, the equity holders may stand to benefit because the risk is transferred to the existing bondholders – risk they do not receive commensurate return for taking. Thus, if the firm pursues a more generous dividend-payout policy while not changing the investment policy, the change in the value of the firm depends on the way in which the future investment is financed. The effect that debt financing has on the value of the firm was analyzed in terms of the interest tax shield it provides and the extent to which the firm can utilize that tax shield. In Myers’ analysis we also saw a that a limit on borrowing could be incorporated so that factors such as risk and the probability of insolvency would be recognized when making each capital-budgeting decision. When compared to other methods widely used in capital budgeting, Myers’ APV formulation was found to yield more conservative benefit estimates. While we do not wish to discard the equityresidual, after-tax weighted cost-of-capital method or the Arditti-Levy weighted cost-of-capital method, we set forth Myers’ method as the most appropriate starting point when a firm is first considering a project, reasoning that if the project

was acceptable following Myers’ method, it would be acceptable using the other methods – to an even greater degree. If the project was not acceptable following the APV criteria, it could be reanalyzed with one of the other methods. The biases of each method we hopefully made clear with the introduction of debt financing. In Sect. 2.6 we have discussed how different policies can affect the determination of beta coefficient. In essence, this chapter points out the vagaries and difficulties of financial management in practice. Virtually no decision concerning the finance function can be made independent of the other variables under management’s control. Profitable areas of future research in this area are abundant; some have already begun to appear in the literature under the heading “simultaneous-equation planning models.” Any practitioner would be well advised to stay abreast of developments in this area.

Notes 1. Alderson (1984) has applied this kind of concept to the problem of corporate pension funding. He finds grounds for the integration of pension funding with capital structure decisions. In doing so, he argues that unfunded vested liabilities are an imperfect substitute for debt. 2. BHS (1981) discusses the relationship between debt capacity and the existence of optimal capital structure. 3. If the dividend policy is irrelevant, then Lct D Lc;t1 D 0; if dW=dyt D Ft is positive, then the constraints U will always be binding, Equation (2.7b) will be strict equalities, and Lft for all t. Substituting Lct D 0 and Lft D Ft into Equation (2.8), we obtain Equation (2.12). 4. This statement can be explained by the agency theory, which was developed by Jensen and Meckling (1976) and was extended by Barnea et al. (1981). According to the agency theory and the findings of Stiglitz (1972), the manager might not act out of the equityholders’ best interest. For his own benefit, the manager might shift the wealth of a firm from the equityholders to bondholders.

References Alderson, M. J. 1984. Unfunded pension liabilities and capital structure decisions: a theoretical and empirical investigation, PhD dissertation, The University of Illinois, Urbana-Champaign Arditti, F. D. 1980. “A survey of valuation and the cost of capital.” Research in Finance 2, 1–56 Arditti, F. D. and Y. C. Peles. 1977. “Leverage, bankruptcy, and the cost of capital.” in Natural resources, uncertainty and general equilibrium systems: essays in memory of Rafael Lusky, A. Blinder and P. Friedman (Eds.). Academic Press, New York

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications Arditti, F. D. and H. Levy. 1977. “The weighted average cost of capital as a cutoff rate: a critical analysis of the classical textbook weighted average.” Financial Management 6(3), 24–34 Ang, J. S. and J. H. Chua. 1982. “Mutual funds: different strokes for different folks.” The Journal of Portfolio Management 8, 43–47 Barnea, A., R. A. Haugen and L. W. Senbet. 1981. “Market imperfections, agency problems, and capital structure: a review.” Financial Management 10, 7–22 Baron, D. P. 1975. “Firm valuation, corporate taxes, and default risk.” Journal of Finance 30, 1251–1264 Baumol, W. and B. G. Malkiel. 1967. “The firm’s optimum debtequity combination and the cost of capital.” Quarterly Journal of Economics 81(4), 546–578 Black, F. 1976. “The dividend puzzle.” Journal of Portfolio Management, 5–8 Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81(3) 637–654 Chambers, D. R., R. S. Harris, and J. J. Pringle. 1982. “Treatment of financing mix in analyzing investment opportunities.” Financial Management 11, 24–41 Chen, A. H. 1978. “Recent development in cost of debt capital.” Journal of Finance 33, 863–877 Conine, T. E. Jr. 1980. “Debt capacity and the capital budgeting decision: a comment.” Financial Management, 20–22 DeAngelo, H. and L. DeAngelo. 2006. “The irrelevance of the MM dividend irrelevance theorem.” Journal of Financial Economics 79, 293–315 Fama, E. F. and M. H. Miller. 1972. The theory of finance, Holt, Rinehart and Winston, New York Gahlon, J. M. and R. D. Stover. 1979. “Debt capacity and the capital budgeting decision: a caveat.” Financial Management 8, 55–59 Gupta, M. C. and A. C. Lee. 2006. “An integrated model of debt issuance, refunding, and maturity.” Review of Quantitative Finance and Accounting 26(2), 177–199 Gupta, M. C., A. C. Lee, and C. F. Lee. 2007. Effectiveness of dividend policy under the capital asset pricing model: a dynamic analysis. Working Paper, Rutgers University Hamada, R. S. 1972. “The effect of the firm’s capital structure on the systematic risk of common stocks.” Journal of Finance 27(2), 435–452 Hanoch, G. and H. Levy. 1969. “The efficiency analysis of choices involving risk.” Review of Economic Studies 36, 335–346 Hellwig, M. F. 1981. “Bankruptcy, limited liability, and the ModiglianiMiller theorem.” The American Economic Review 71, 155–170 Higgins, R. C. 1977. “How much growth can a firm afford?” Financial Management 6(3), 7–16 Higgins, R. C. 1981. “Sustainable growth under inflation.” Financial Management, 36–40 Hong, H. and A. Rappaport. 1978. “Debt capacity and the capitalbudgeting decision: a caveat.” Financial Management, 7–11 Jean, W. 1975. “Comparison of moment and stochastic-dominance ranking methods.” Journal of Financial and Quantitative Analysis 10, 151–162 Jensen, M. and W. Meckling. 1976. “Theory of the firm: managerial behavior, agency costs, and ownership structure.” Journal of Financial Economics 3, 305–360 Johnson, D. J. 1981. “The behavior of financial structure and sustainable growth in an inflationary environment.” Financial Management 10, 30–35 Kim, E. H. 1978. “A mean-variance theory of optimal capital structure and corporate debt capacity.” Journal of Finance 33, 45–64 Kim, E. H. 1982. “Miller’s equilibrium, shareholder leverage clienteles, and optimal capital structure.” Journal of Finance 37, 301–319 Kim, E. H., J. J. McConnell, and P. R. Greenwood. 1977. “Capitalstructure rearrangements and me-first roles in an efficient capital market.” Journal of Finance 32, 811–821

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Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York Lee, C. F., W. Chunchi, and D. Hang. 1993. “Dividend policy under conditions of capital markets and signaling equilibrium.” Review of Quantitative Finance and Accounting 3, 47–59 Lee, C. F., S. Rahman, and K. Thomas Liaw. 1990. “The impacts of market power and capital-labor ratio on systematic risk: a Cobb-Douglas approach.” Journal of Economics and Business 42(3), 237–241 Lee, C. F., K. C. Chen, and K. Thomas Liaw. 1995. “Systematic risk, wage rates, and factor substitution.” Journal of Economics and Business 47(3), 267–279 Levy, H. and M. Sarnat. 1972. Investment and portfolio analysis, Wiley, New York Litzenberger, R. H. and K. Ramaswamy. 1979. “The effect of personal taxes and dividends on capital-asset prices: theory and empirical evidence.” Journal of Financial Economics 7, 163–195 Litzenberger, R. H., and H. B. Sosin. 1979. “A comparison of capitalstructure decision of regulated and nonregulated firms.” Financial Management 8, 17–21 Malkiel, B. G. 1970. “The valuation of public-utility equity.” Bell Journal of Economics and Management Science 1, 143–160 Martin, J. D. and D. F. Scott. 1976. “Debt capacity and the capital budgeting decision.” Financial Management 5(2), 7–14 Martin, J. D. and D. F. Scott Jr. 1980. “Debt capacity and the capital budgeting decision: a revisitation.” Financial Management 9(1), 23–26 Merton, R. C. 1974. “On pricing of corporate debt: the risk structure of interest rates.” Journal of Finance 29, 449–470 Miller, M. H. 1977. “Debt and taxes.” Journal of Finance 32, 261–275 Modigliani, F. and M. H. Miller. 1958. “The cost of capital, corporate finance, and theory of investment.” American Economic Review 48(3), 261–297 Modigliani, F. and M. H. Miller. 1963. “Corporate income taxes and the cost of capital: a correction.” American Economic Review 53(3), 433–443 Myers, S. C. 1974. “Interaction of corporate financing and investment decision.” Journal of Finance 29, 1–25 Myers, S. C. 1977. “Determinants of corporate borrowing.” Journal of Financial Economics 5, 147–175 Porter, R. B. and J. E. Gaumwitz. 1972. “Stochastic dominance vs. mean-variance portfolio analysis: an empirical evaluation.” American Economic Review 57, 438–446 Rendleman, R. J. Jr. 1978. “The effects of default risk on firm’s investment and financing decisions.” Financial Management 7(1), 45–53 Reseck, R. W. 1970. “Multidimensional risk and the Modigliani-Miller hypothesis.” Journal of Finance 25, 47–51 Rhee, S. G. and F. L. McCarthy. 1982. “Corporate debt capacity and capital-budgeting analysis.” Financial Management 11, 42–50 Rubinstein, M. E. 1973. “A mean-variance synthesis of corporate financial theory.” Journal of Finance 28, 167–187 Stiglitz, J. 1972. “Some aspects of pure theory of corporate finance: Bankruptcies and take-overs.” Bell Journal of Economics and Management Science 3, 458–482 Tuttle, L., and R. H. Litzenberger. 1968. “Leverage, diversification, and capital-market effects on a risk-adjusted capital-budgeting framework.” Journal of Finance 23, 427–443 Van Horne, J. C. and J. D. McDonald. 1971. “Dividend policy and new equity financing.” Journal of Finance 26, 507–519

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Appendix 2A Stochastic Dominance and its Applications to Capital-Structure Analysis with Default Risk

EF U.X / > EG U.X /

where EF U.X / and EG U.X / are expected utilities. Mathematically, they can be defined as: Zx

2A.1 Introduction EF U.X / D Mean-variance approaches were extensively used in the literature to derive alternative finance theories and perform related empirical studies. Besides mean-variance approaches, there is a more general approach, stochasticdominance analysis, which can be used to choose a portfolio, to evaluate mutual-fund performance, and to analyze the optimal capital-structure problem. Levy and Sarnat (1972), Porter and Gaumwitz (1972), Jean (1975), and Ang and Chua (1982) have discussed the stochastic-dominance approach to portfolio choice and mutual-fund performance evaluation in detail. Baron (1975), Arditti and Peles (1977), and Arditti (1980) have used the theory of stochastic dominance to investigate the optimal capital-structure question. In this appendix we will discuss only how stochastic-dominance theory can be used to analyze the issue of optimal capital structure with default risk.

2A.2 Concepts and Theorems of Stochastic Dominance

(2A.1)

U.X /f .X /dx

(2A.2a)

U.X /g.X /dx

(2A.2b)

x

Zx EF U.X / D x

where U.X / D the utility function, X D the investment dollar-return variable, f .X / and g.X / are probability distributions of X . It should be noted that the above-mentioned first-order stochastic dominance does not depend upon the shape of the utility function with positive marginal utility. Hence, the investor can either be risk-seeking, risk neutral, or risk-averse. If the risk-aversion criterion is imposed, the utility function is either strictly concave or nondecreasing. If utility functions are nondecreasing and strictly concave, then the second-order stochastic-dominance theorem can be defined. Mathematically, the second-order dominance can be defined as: Zx ŒG.T /  F .t/ dt > 0; where G.t/ ¤ F .t/ for some t: x

The expected utility rule can be used to introduce the economics of choice under uncertainty. However, this decision rule has been based upon the principle of utility maximization, where either the investor’s utility function is assumed to be a second-degree polynomial with a positive first derivative and a negative second derivative, or the probability function is assumed to be normal. A stochastic-dominance theory is an alternative approach of preference orderings that does not rely upon these restrictive assumptions. The stochastic-dominance technique assumes only that individuals prefer more wealth to less. An asset is said to be stochastically dominant over another if an individual receives greater wealth from it in every (ordered) state of nature. This definition is known as first-order stochastic dominance. Mathematically, it can be described by the relationship between two cumulativeprobability distributions. If X symbolizes the investment dollar return variable and F .X / and G.X / are two cumulativeprobability distributions, then F .X / will be preferred to G.X / by every person who is a utility maximizer and whose utility is an increasing function of wealth if F .X /  G.X / for all possible X , and F .X / < G.X / for some X . For the family of all monotonically nondecreasing utility functions, Levy and Sarnat (1972) show that first-order stochastic dominance implies that:

(2A.3) Equation (2A.3) specifies a necessary and sufficient condition for an asset F to be preferred over a second asset G by all risk-averters. Conceptually, Equation (2A.3) means that in order for asset F to dominate asset G for all risk-aversion investors, the accumulated area under the cumulative-probability distribution G must be greater than the accumulated area for F for any given level of wealth. This implies that, unlike first-order stochastic dominance, the cumulative density functions can cross. Stochastic dominance is an extremely important and powerful tool. It is properly founded on the basis of expected utility maximization, and even more important, it applies to any probability distribution. This is because it takes into account every point in the probability distribution. Furthermore, we can be sure that, if an asset demonstrates second-order stochastic dominance, it will be preferred by all risk-aversion investors, regardless of the specific shape of their utility functions. Assume that the density functions of earnings per share (EPS) for both firm A and firm B are f .X / and g.X /, respectively. Both f .X / and g.X / have normal distributions and the shape of these two distributions are described in Fig. 2A.1. Obviously the EPS of firm A will dominate the EPS of firm B if an investor is risk-averse because they both

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

39

Fig. 2A.1 Probability distributions of asset F and asset G

offer the same expected level of wealth .f D g / and because the variance of g.X / is larger than that of f .X /. Based on both the first-order and the second-order stochastic-dominance theorems mentioned earlier, a new theorem needed for investigating capital structure with default risk can be defined as: Theorem 2A.1. Let F , G be two distributions with mean values 1 and 2 respectively, such that, for some X0 < 1, F  G for X  X0 (and F < G for some X1 < X0 ) and F G for some X X0 , then F dominates G (for concave utility functions) if and only if 1 2 . The proof of this theorem can be formed in either Hanoch and Levy (1969) or Levy and Sarnat (1972). Conceptually, this theorem states that if two cumulative distributions, F and G, intersect only once, then if F is below G to the left of the intersection point and has a higher mean than does G, the investment with cumulative distribution F dominates that with cumulative return distribution G on a second-degree stochastic-dominance basis.

earnings X, before taxes and financial charges, such that, in any state of nature that occurs, both firms have the same earnings. In addition, it is assumed that this random variable X is associated with a cumulative-distribution function F .X /. Firm A is assumed to be financed solely by equity, while firm B is financed by both debt and equity. The market value V1 of firm A equals E1 , the value of its equity, while the market value V2 of firm B equals the market value of its equity .E2 / plus the value of its debt .D2 /. Debt is assumed to sell at its par value and to carry a gross coupon rate of r .r > 0/. The firm can generally use the coupon payments as a tax shield and therefore the after-tax earnings for firms A and B can be defined as X.1T / and .X rD2 /.1T /, respectively (where T is the corporate-profit tax rate.) If an investor purchases a fraction of firm A, this investment results in dollar returns of:  0; if X  0; Y1 D (2A.4) ˛X.1  T / if X > 0; with cumulative probability function: 

2A.3 Stochastic-Dominance Approach to Investigating the Capital-Structure Problem with Default Risk The existence of default risk is one of the justification of why an optimal capital structure might exist for a firm. To analyze this problem. Baron (1975), Arditti and Peles (1977), and Arditti (1980) have used the stochastic-dominance theorem described in the previous sections to indicate the effects of debt-financing bonds on relative values of levered and unlevered firms. Baron analyzed the bonds in terms of default risk, tax rates, and debt levels. Consider two firms, or the same firm before and after debt financing, with identical probability distribution of gross

G1 .Y / D

F .0/ F .Y =.1  T /

if Y D 0; if Y > 0:

(2A.5)

If the investor purchased a fraction ’ of the equity and ’ of the debt of firm B, this dollar return would be: 8 if X  0; ˆ <0 Y2 D ˛.1  k/X.1  T / if 0 < X.1  T / < D2 CrD2 ; ˆ : ˛.X.1  T / C T rD2 if x.1  T / D2 C rD2 ; (2A.6) where k represents the percentage of liquidation cost. Since these bonds comprise a senior lien on the earnings of the firm, failure to pay the promised amount of D2 CrD2 to bondholders can force liquidation of the firm’s assets.

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

The cumulative distribution associated with Equation (2A.7) is: (

if Y D0; if 0 < y  ˛.1  k/ŒD2 CrD2 ; F ŒY =˛.1T /  T rD2 if Y  ˛ŒD2 C rD2 :

F .0/

G2 .Y / D F ŒY =˛.1k/.1T /

(2A.7) Comparing Equation (2A.5) with Equation (2A.7), it can be found that: G1 .Y / < G2 .Y / if 0 < Y < ˛.D2 C rD2 /; (2A.8) G2 .Y / G2 .Y / if Y ˛.D2 C rD2 /;

a firm should issue. However, these results have indicated the possibility of interior (noncorner) optimal capital structure. By using a linear utility function, Arditti and Peles (1977) show that there are probability distributions for which an interior capital structure for a firm can be found. In addition, they also make some other market observations, which are consistent with the implications of their model. These observations are (i) firms with low income variability, such as utilities, carry high debt-equity ratios; and (ii) firms that have highly marketable assets and therefore low liquidation costs, such as shipping lines, seem to rely heavily on debt financing relative to equity.

(2A.9)

where ˛.D2 CrD2 / is the critical income level of bankruptcy. Equations (2A.8) and (2A.9) imply that the levered firm cannot dominate the unlevered firm on a first-degree or second-degree stochastic-dominance basis. From the theorem of stochastic dominance that we presented in the previous section, it can be concluded that the unlevered firm cannot dominate the levered firm because the expected return of the levered firm is generally higher than that of the unlevered firm (otherwise why levered?), and because the G2 .Y / and G1 .Y / curves intersect only once. Consequently, a general statement using stochastic dominance cannot be made with respect to the amount of debt that

2A.4 Summary In this appendix we have tried to show the basic concepts underlying stochastic dominance and its application to capitalstructure analysis with default risk. By combining utility maximization theory with cumulative-density functions, we are able to set up a decision rule without explicitly relying on individual statistical moments. This stochastic-dominance theory can then be applied to problems such as capitalstructure analysis with risky debt, as was shown earlier.

Chapter 3

Research Methods in Quantitative Finance and Risk Management

Abstract The main purpose of this chapter is to discuss important quantitative methods used to do the research in quantitative finance and risk management. We first discuss statistics theory and methods. Second, we discuss econometric methods. Third, we discuss mathematics. Finally, we discuss other methods such as operation research, stochastic process, computer science and technology, entropy, and fuzzy set theory. Keywords Statistics r Econometrics r Mathematics r Operation research r Stochastic process r Computer science and technology r Entropy r Fuzzy set theory

3.1 Introduction Quantitative finance and risk management research requires finance theories, finance policies, and methodology. Finance theories and finance policies have been discussed in Chaps. 1 and 2. This chapter briefly discusses the alternative methodologies used in quantitative finance and risk management research. In Sect. 3.2 we will review statistics theory and methodology used in quantitative finance and risk management research. Econometric and mathematics used in quantitative finance and risk management research will be presented in Sects. 3.3 and 3.4, respectively. In addition, in Sect. 3.5 we will briefly review the methods used in quantitative finance and risk management such as operation research, stochastic process, computer science and technology, entropy, and fuzzy set theory. Finally, our conclusion will be presented Sect. 3.6.

3.2 Statistics Statistics is important in developing finance theories and in implementing empirical models. It is the prerequisite for studying financial economics, especially for researchers who encounter much uncertainty. The concepts of distribution,

such as binomial distribution, multinomial distribution, normal distribution, log-normal distribution, and non-central Chi-square distribution, help researchers to develop alternative option pricing models. In this book, Chap. 24 discusses the applications of the binomial distribution to evaluate call options. Chapter 25 discusses the multinomial options pricing model. It extends the binomial option pricing model to multinomial option pricing model. Then we derive the multinomial option pricing model and apply it on the limiting case of Black and Scholes model. Finally, we introduce a lattice framework for option pricing model and its application to option valuation. In Chaps. 27 and 28, normal distribution, lognormal distribution, and bivariate normal distribution option pricing models are discussed. Chapter 27 introduces normal distribution, lognormal distribution, and their relationship. Multivariate normal and lognormal distributions are also examined. Finally, both normal and lognormal distributions are applied to derive Black–Sholes formula under the assumption that the rate of stock price follows a lognormal distribution. Chapter 28 presents the American option pricing model on stock with dividend payments and without dividend payments. A Microsoft excel program for evaluating the American option pricing model is also presented. Chapter 29 compares the American option prices with one known dividend under two alternative specifications of the underlying stock price: displace log normal and logmal processes. Many option pricing models follow the standard assumption of the Black-Scholes model (1973) in which the stock price follows a log normal process. However, in order to reach a closed form solution for the American option price with one known dividend, Roll (1977), Geske (1979), and Whaley (1981) assume a displaced lognormal process for the cum-dividend stock price, which results in a lognormal process for the ex-dividend stock price. We compare two alternative pricing results in this paper. Chapter 31 describes the non-central Chi-square type of OPM. It reviews the renowned Constant Elasticity of Variance (CEV) option pricing model and gives the detailed derivations. There are two purposes of this Chap. 10. First, it shows the details of the formulas needed in deriving the option pricing and bridges the gaps in deriving the necessary formulas for the

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model. Second, it uses a result by Feller to obtain the transition probability density function of the stock price at time T given its price at time t with t < T . In addition, some computational considerations are given that will facilitate the computation of the CEV option pricing formula. Chapter 37.1 analyzes the convergence rate and pattern of binomial distribution. It extends the generalized CoxRoss-Rubinstein (hereafter GCRR) model of Chung and Shih (2007). We provide a further analysis of the convergence rates and patterns based on various GCRR models. The numerical results indicate that the GCRR-XPC model and the GCRR-JR (p D 1=2) model (defined in Table 34.1) outperform the other GCRR models for pricing European calls and American puts. Our results confirm that the node positioning and the selection of the tree structures (mainly the asymptotic behavior of the risk-neutral probability) are important factors for determining the convergence rates and patterns of the binomial models. Chapter 35 uses mixture distribution to estimate implied probabilities from options prices. It examines a variety of methods for extracting implied probability distributions from option prices and the underlying. The chapter first explores non-parametric procedures for reconstructing densities directly from options market data. It then considers local volatility functions, both through implied volatility trees and volatility interpolation. It then turns to alternative specifications of the stochastic process for the underlying. A mixture of log normal model is estimated and applied to exchange rate data, and it is illustrated how to conduct forecast comparisons. Finally, the chapter turns to the estimation of jump risk by extracting bipower variation. Chapter 57 uses generalized hyperbolic distribution to develop a generalized model for optimum futures hedge ratio. It proposes the generalized hyperbolic distribution as the joint log-return distribution of the spot and futures. Using the parameters in this distribution, it derives several most widely used optimal hedge ratios: minimum variance, maximum Sharpe measure, and minimum generalized semivariance. Under mild assumptions on the parameters, it finds that these hedge ratios are identical. Regarding the equivalence of these optimal hedge ratios, the analysis in this chapter suggests that the martingale property plays a much important role than the joint distribution assumption. Multivariate statistics such as factor analysis, and discriminant analysis are also important in performing empirical study in quantitative finance and risk management. This naturally leads to the use of regression analysis, which includes the single equation and multiple equation models. Maximum likelihood methods are also important in statistical estimation. Moreover when the estimation requires fewer distributional assumptions about the variables, nonparametric estimation is preferred. Chapter 65 uses nonparametric methods in examining segmenting financial services market. It analyzes segmentation of financial markets

based on the general segmentation bases. In particular, it identifies potentially attractive market segments for financial services using a customer dataset. It develops a multigroup discriminant model to classify the customers into three ordinal classes: prime customers, highly valued customers, and price shoppers based on their income, loan activity, and demographics (age). The multi-group classification of customer segments uses both classical statistical techniques and a mathematical programming formulation. This study uses the characteristics of a real dataset to simulate multiple datasets of customer characteristics. The results of the proposed experiments show that the mathematical programming model in many cases consistently outperforms standard statistical approaches in attaining lower APparent Error Rates (APER) for 100 replications in both high and low correlation cases. Researchers use Bayesian inference in which the observations are used to update the probability that their hypothesis is true. Chapter 11 discusses Bayesian triangulation in asset pricing. It surveys some relevant academic research on the subject, including work about the combining of forecasts (Bates and Granger 1969), the BlackLitterman model (1991, 1992), the combining of Bayesian priors and regression estimates (Pastor 2000), model uncertainty and Bayesian model averaging (Hoeting et al. 1999; Cremers 2002), the theory of competitive storage (Deaton and Laroque 1992), and the combination of valuation estimates (Yee 2007). Despite its wide-ranging applicability, the Bayesian approach is not a license for data snooping. The second half of this chapter describes common pitfalls in fundamental analysis and comments on the role of theoretical guidance in mitigating these pitfalls. Chapter 91 has proposed Bayesian inference of financial models using Monte Carlo Markov Chian (MCMC) algorithm. It introduces Gibbs sampler and Markov Chain Monte Carlo (MCMC) algorithms. The marginal posterior probability densities obtained by the MCMC algorithms are compared to the exact marginal posterior densities. This chapter presents two financial applications of MCMC algorithms: the CKLS model of the Japanese uncollaterized call rate and the Gaussian copula model of S&P 500 and FTSE 100. The concept of stochastic dominance is important in portfolio optimization problems. Chapter 15 considers the problem of constructing a portfolio of finitely many assets whose return rates are described by a discrete joint distribution. This chapter presents a new approach to portfolio selection based on stochastic dominance. The portfolio return rate in the new model is required to stochastically dominate a random benchmark. It formulates optimality conditions and duality relations for these models and constructs equivalent optimization models with utility functions. Two different formulations of the stochastic dominance constraint: primal and inverse, lead to two dual problems that involve

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von Neuman–Morgenstern utility functions for the primal formulation and rank dependent (or dual) utility functions for the inverse formulation. This study also discusses the relations of our approach to value at risk and conditional value at risk. Numerical illustration is provided. Characteristics function and spectrum analysis are important in option pricing research. Chapter 37 has applied characteristics function to derive generalized option pricing model. Chapter 38 discusses the application of the characteristic function in financial research. It introduces the application of the characteristic function in financial research. It considers the technique of the characteristic function useful for many option pricing models. Two option pricing models are derived in details based on the characteristic functions. Copula distribution is also important in doing research in the default risk. Chapter 46 provides a new approach to estimate future credit risk on target portfolio based on the framework of CreditMetricsTM by J.P. Morgan. However, it adopts the perspective of factor copula and then brings the principal component analysis concept into factor structure to construct a more appropriate dependence structure among credits. In order to examine the proposed method, this chapter uses real market data instead of virtual ones. It also develops a tool for risk analysis that is convenient to use, especially for banking loan businesses. The results show that people assume dependence structures are normally distributed will indeed lead to risks underestimate. On the other hand, the proposed method in this chapter captures better features of risks and shows the fat-tail effects conspicuously even though assuming the factors are normally distributed.

discussion of stochastic model fitting and explains the fitting approach that motivates the introduction of ARM processes. It then continues with a review of ARM processes and their fundamental properties, including their construction, transition structure, and autocorrelation structure. Next, the chapter outlines the ARM modeling methodology by describing the key steps in fitting an ARM model to empirical data. It also describes in some detail the ARM forecasting methodology and the computation of the conditional expectations that serve as point estimates (forecasts of future values) and their underlying conditional distributions from which confidence intervals are constructed for the point estimates. Finally, the chapter concludes with an illustration of the efficacy of the ARM modeling and forecasting methodologies through an example utilizing a sample from the S&P 500 Index. Chapter 76 has proposed univariate and bivariate GARCH analyses for the volume versus GARCH effects. It tests for the generalized autoregressive conditional heteroscedasticity (GARCH) effect on U.S. stock markets for different periods and re-examines the findings in Lamoureux and Lastrapes (1990) by using alternative proxies for information flow, which are included in the conditional variance equation of a GARCH model. It also examines the spillover effects of volume and turnover on the conditional volatility using a bivariate GARCH approach. Our findings show that volume and turnover have effects on conditional volatility and that the introduction of volume/turnover as exogenous variable(s) in the conditional variance equation reduces the persistence of GARCH effects as measured by the sum of the GARCH parameters. The results confirm the existence of the volume effect on volatility, consistent with the findings by Lamoureux and Lastrapes (1990) and others, suggesting that the earlier findings were not a statistical fluke. The findings in this chapter also suggest that, unlike other anomalies, the volume effect on volatility is not likely to be eliminated after discovery. In addition, this study rejects the pure random walk hypothesis for stock returns. The bivariate analysis also indicates that there are no volume or turnover spillover effects on conditional volatility among the companies, suggesting that the volatility of a company’s stock return may not necessarily be influenced by the volume and turnover of other companies. Chapter 82 has developed defense forecasting methods to forecast stock price. This study introduces a new method of forecasting, defensive forecasting, and illustrates its potential by showing how it can be used to predict the yields of corporate bonds in real time. Defensive forecasting identifies a betting strategy that succeeds if forecasts are inaccurate and then makes forecasts that will defeat this strategy. The theory of defensive forecasting is based on the game-theoretic framework for probability introduced by Shafer and Vovk in 2001. This study frames statistical tests as betting strategies. It rejects a hypothesis when a strategy for betting against

3.3 Econometrics Econometric models are the necessary research tools in the research of quantitative finance and risk management. The major econometric methods used in the modern finance research include but not limited to the following subjects: linear regression models, time series modeling, multiple equations models, generalized methods of moments, and panel data models. Many of the aforementioned subjects also contain well developed sub-subjects. For example, the development of the concepts of stationarity, autoregressive and moving average model, and cointegration, within the framework of time series are widely applied in the finance research. Multiple equation models include the application of vector autoregression (VAR), simultaneous equation models, and seemingly uncorrelated regression. Chapter 73 discusses the ARM processes and their properties. It also presents AMR modeling and forecasting methodology that try to capture a strong statistical signature of the empirical data. This chapter starts with a high-level

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it multiplies the capital it risks by a large factor. The study proves each classical theorem, such as the law of large numbers, by constructing a betting strategy that multiplies the capital it risks by a large factor if the theorem’s prediction fails. Defensive forecasting plays against strategies of this type. Chapter 86 discusses the application of simultaneous equation in finance research. This chapter introduces the concept of simultaneous equation system and the application of such system to finance research. The study first discusses the order and rank condition of identification problem. It then introduces the two-stage least squares (2SLS) and three-stage least squares (3SLS) estimation method. The pro and cons of these estimations methods are summarized. Finally, the results of a study on executive compensation structure and risk-taking are used to illustrate the difference between single equation and simultaneous equation method. Chapter 89 has proposed a method to detect structural instability of financial time series. It reviews the literature that deals with structural shifts in statistical models for financial time series. The issue of structural instability is at the heart of capital asset pricing theories on which many modern investment portfolio concepts and strategies rely. The need to better understand the nature of structural evolution and to devise effective control of the risk in real world investments has become ever more urgent in recent years. Since the early 2000s, concepts of financial engineering and the resulting product innovations have advanced in an astonishing pace, but unfortunately often on a faulty premise and without proper regulatory keep-ups. The developments have threatened to destabilize worldwide financial markets. This chapter explains the principle and concepts behind these phenomena. Along the way it highlights developments related to these issues over the past few decades in the statistical, econometric, and financial literature. It also discusses recent events that illuminate the dynamics of structural evolution in the mortgage lending and credit industry. Chapter 93 discusses theoretical VAR model and its applications. Chapter 96 uses GARCH and SplineGARCH to model and forecast volatilities and asset returns. Dynamic modeling of asset returns and their volatilities is a central topic in quantitative finance. Herein this chapter reviews basic statistical models and methods for the analysis and forecasting of volatilities. It also surveys several recent developments in regime-switching, change-point, and multivariate volatility models. Chapter 99 uses combined forecasting methods to estimate cost of equity capital for property/casualty insurers. This chapter estimates the cost of equity capital for Property/Casualty insurers by applying three alternative asset pricing models: the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), and a unified CAPM/APT model (Wei 1988). The in-sample forecast ability of the models is evaluated by applying the mean squared error method, the Theil U2 (1966) statistic, and the

Granger (1974) conditional efficiency evaluation. Based on forecast evaluation procedures, the APT and Wei’s unified CAPM/APT models perform better than the CAPM in estimating the cost of equity capital for the PC insurers and a combined forecast may outperform the individual forecasts. Chapter 103 uses Box–Cox functional forms transformation in closed country fund research. This chapter proposes a generalized functional form CAPM model for international closed-end country funds performance evaluation. It examines the effect of heterogeneous investment horizons on the portfolio choices in the global market. Empirical evidences suggest that there are some empirical anomalies that are inconsistent with the traditional CAPM. These inconsistencies arise because the specification of the CAPM ignores the discrepancy between observed and true investment horizons. A comparison between the functional forms for share returns and NAV returns of closed-end country funds suggests that foreign investors may have more heterogeneous investment horizons versus their U.S counterparts. Market segmentation and government regulation does have some effect on the market efficiency. No matter which generalized functional model is used, the empirical evidence indicates that, on average, the risk-adjusted performance of international closed-end fund is negative even before the expenses. The fixed effect panel data model and the random effect panel data model are also important in much empirical finance research. Chapter 51 uses a panel threshold model to examine the effect of default risk on equity liquidity. This chapter sets out to investigate the relationship between credit risk and equity liquidity. It posits that as the firm’s default risk increases, informed trading increases in the firm’s stock and uninformed traders exit the market. Market-makers widen spreads in response to the increased probability of trades against informed investors. Using the default likelihood measure calculated by Merton’s method, this chapter investigates whether financially ailing firms do indeed have higher bid-ask spreads. The panel threshold regression model is employed to examine the possible non-linear relationship between credit risk and equity liquidity. Since high default probability and worse economic prospects lead to greater expropriation by managers, and thus greater asymmetric information costs, liquidity providers will incur relatively higher costs and will therefore offer higher bid-ask spreads. This issue is further analyzed by investigating whether there is any evidence of increases in the vulnerability in equity liquidity of firms with high credit risk. The results in this chapter show an increase in the default likelihood might precipitate investors’ loss of confidence of equities trading and thus cause a decrease in liquidity, as particularly evident during the Enron crisis period. Chapter 69 discusses the spurious regression and data mining in conditional asset pricing models. Stock returns are not highly autocorrelated, but there is a spurious regression bias in predictive regressions for

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stock returns similar to the classic studies of Yule (1926) and Granger (1974). Data mining for predictor variables reinforces spurious regression bias because more highly persistent series are more likely to be selected as predictors. In asset pricing regression models with conditional alphas and betas, the biases occur only in the estimates of conditional alphas. However, when time-varying alphas are suppressed and only time-varying betas are considered, the betas become biased. The analysis shows that the significance of many standard predictors of stock returns and time-varying alphas in the literature is often overstated. There are many other econometric methods discussed in this book as well. For example, Chap. 43 uses combinatorial methods for constructing credit risk ratings. This study uses a novel combinatorial method, the Logical Analysis of Data (LAD) to reverse-engineer and construct credit risk ratings that represent the creditworthiness of financial institutions and countries. The proposed approaches are shown to generate transparent, consistent, self-contained, and predictive credit risk rating models, closely approximating the risk ratings provided by some of the major rating agencies. The scope of applicability of the proposed method extends beyond the rating problems discussed in this study, and can be used in many other contexts where ratings are relevant. The proposed methodology is applicable in the general case of inferring an objective rating system from archival data, given that the rated objects are characterized by vectors of attributes taking numerical or ordinal values. Chapter 50 uses dynamic econometric loss model in the default study of U.S. subprime markets. It applies not only the static loan and borrower characteristic variables such as loan terms, Combined-Loan-To-Value ratio (CLTV), Fair Isaac credit score (FICO), as well as dynamic macroeconomic variables such as HPA to projects defaults and prepayments, but also the spectrum of delinquencies as an error correction term to add an additional 15% accuracy to the model projections. In addition to the delinquency attribute findings, this chapter also shows that cumulative HPA and the change of HPA contribute various dimensions that greatly influence defaults. Another interesting finding is a significant long-term correlation between HPI and disposable income level (DPI). Since DPI is more stable and easier for future projections, it suggests that HPI will eventually adjust to coincide with DPI growth rate trend and that HPI could potentially experience as much as a 14% decrease by the end of 2009. Chapter 62 uses robust logistic regression to study the default prediction with outliers. This chapter suggests a Robust Logit method, that extends the conventional logit model by taking outlier into account, to implement forecast of defaulted firms. This study employs five validation tests to assess the in-sample and out-of-sample forecast performances, respectively. With respect to in-sample forecasts, the Robust Loigt method proposed in this chapter is substantially

superior to the logit method as it uses all validation tools. With respect to the out-of-sample forecasts, the superiority of Robust Logit is less pronounced. Issues related with errorsin-variables problems in asset pricing tests is discussed in Chap. 70. This chapter develops a new method of correcting the EIV problem in the estimation of the price of beta risk within the two-pass estimation framework. The study performs the correction for the EIV problem by incorporating the extracted information about the relationship between the measurement error variance and the idiosyncratic error variance into the maximum likelihood estimation under either homoscedasticity or heteroscedasticity of the disturbance term of the market model. The evidence shows that after correcting for the EIV problem, market beta has significant explanatory power for average stock returns, and that the firm size factor becomes much less important. This chapter also reexamines the explanatory power of beta and the firm size for average stock returns using the proposed correction. If no correction for the EIV problem is implemented, then, consistent with recent empirical studies, there is no significant relation between beta and expected returns, especially when firm size is included as an additional explanatory variable. With the EIV correction, both the price and significance of betas increase. This chapter documents that the EIV correction diminishes the impact of the firm size variable. Nevertheless, firm size remains a significant force in explaining the crosssection of average returns. While my results suggest a more important role for market beta, the evidence is still consistent with a misspecification in the capital asset pricing model. Chapter 72 discusses term structure models that incorporate Markov regimes shifts. Both discrete-time and continuoustime regimes switching term structure models are discussed. In this chapter we briefly review models of the term structure of interest rates under regime shifts. These models retain the same tractability of the single-regime affine models on one hand, and allow more general and flexible specifications of the market prices of risk, the dynamics of state variables as well as the short-term interest rates on the other hand. These additional flexibilities can be important in capturing some silent features of the term structure of interest rates in empirical applications. The models can also be applied to analyze the implications of regime-switching for bond portfolio allocations and the pricing of interest rate derivatives. Chapter 78 discusses the hedonic regression analysis in real estate markets. This chapter provides an overview of the nature and variety of hedonic pricing models that are employed in the market for real estate. It explores the history of hedonic modeling and summarizes the field’s utilitytheory-based, microeconomic foundations. It also provides empirical examples of each of the three principal methodologies and a discussion of and potential solutions for common problems associated with hedonic modeling. Chapter 92 discusses the instrumental variable approach to correct for

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endogeneity in corporate finance. The theoretical literature on the link between an incumbent firm’s capital structure (financial leverage, debt/equity ratio) and entry into its product market is based on two classes of arguments: the “limited liability” arguments and the “deep pocket” arguments. However, these two classes of arguments provide contradictory predictions. This study provides a distinct strategic model of the link between capital structure and entry that is capable of rationally producing both of the existing contradictory predictions. The focus is on the role of beliefs and the cost of adjusting capital structure, two of the factors that are absent in the existing models. The beliefs may be exogenously given or endogenized by a rational expectations criterion.

obviously predates economic theory. However, a formal model showing how to make the most of the power of diversification was not devised until 1952, a feat for which Harry Markowitz eventually won the Nobel Prize in economics. Markowitz portfolio shows that as assets are added to an investment portfolio the total risk of that portfolio — as measured by the variance (or standard deviation) of total return — declines continuously, but the expected return of the portfolio is a weighted average of the expected returns of the individual assets. In other words, by investing in portfolios rather than in individual assets, investors could lower the total risk of investing without sacrificing return. In the second part, it introduces the mean-variance spanning test that follows directly from the portfolio optimization problem. Chapter 12 provides discussion on estimation risk and power utility portfolio selection. It examines additional ways of estimating joint return distributions that allow us to explore the stationary/nonstationary tradeoffs implicit in “expanding” versus "moving" window estimation methods, the benefits of alternative methods of allowing for the “memory loss” inherent in the moving-window EPAA, and the possibility that weighting more recent observations more heavily may improve investment performance. Chapter 13 discusses the theory and method in international portfolio management. This chapter investigates the impact of various investment constraints on the benefits and asset allocation of the international optimal portfolio for domestic investors in various countries. The empirical results indicate that local investors in less developed countries, particularly in East Asia and Latin America, benefit more from global diversification. Though the global financial market is becoming more integrated, adding constraints reduces but does not completely eliminate the diversification benefits of international investment. The addition of portfolio bounds yields the following characteristics of asset allocation: a reduction in the temporal deviation of diversification benefits, a decrease in time-variation of components in optimal portfolio, and an expansion in the range of comprising assets. Our findings are useful for asset management professionals to determine target markets to promote the sales of national/international funds and to manage risk in global portfolios. In the research of derivatives or options pricing, the use of the Ito calculus and the derivation of the ordinary and partial differential equations are important in arriving at the closed form solution for option pricing model. Applications of Ito calculus in deriving option pricing model model can be found in Chapter 30. It basically develops certain relatively recent mathematical discoveries known generally as stochastic calculus, or more specifically as Itô’s Calculus, and illustrates their application in the pricing of options. The mathematical methods of stochastic calculus are illustrated in alternative derivations of the celebrated Black-Scholes-Merton model. The topic is motivated by a desire to provide an

3.4 Mathematics The mathematical methodology used in finance has its root dated back to research in economics. Financial researchers are interested in equilibrium analysis, optimization problems, and dynamic analysis. The use of linear and matrix algebra, real analysis, multivariate calculus, constrained and unconstrained optimization, nonlinear programming, and optimal control theory are indispensable. Research in portfolio theory and its applications frequently uses constrained maximization. Chapter 5 discusses risk aversion, capital asset allocation, and Markowiatz portfolio selection model. It introduces utility function and indifference curve. Based on utility theory, it derives the Markowitz’s model and the efficient frontier through the creation of efficient portfolios of varying risk and return. It also includes methods of solving for the efficient frontier both graphically and mathematically, with and without explicitly incorporating short selling. Chapter 7 demonstrates how index model can be used for portfolio selection. It discusses both single-index model and multiple-index portfolio selection model. It uses constrained maximization instead of minimization procedure to calculate the portfolio weights. We find that both single-index and multi-index models can be used to simplify the Markowitz model for portfolio section. Chapter 8 discusses Sharpe measure, Treynor measure, and optimal portfolio selection. Following Elton et al. (1976) and Elton et al. (2007), it introduces the performancemeasure approaches to determine optimal portfolios. We find that the performance-measure approaches for optimal portfolio selection are complementary to the Markowitz full variance-covariance method and the Sharpe index-model method. The economic rationale of the Treynor method is also discussed in detail. Chapter 10 discusses portfolio optimization models and mean-variance spanning tests. It introduces the theory and the application of computer program of modern portfolio theory. The notion of diversification is the age-old adage “don’t put your eggs in one basket,”

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intuitive understanding of certain probabilistic methods that have found significant use in financial economics. Chapter 32 uses Ito calculus in stochastic volatility option pricing model. It assumes that the volatility of option price model is stochastic instead of deterministic. The chapter applies such assumption to the nonclosed-form solution developed by Scott (1987) and the closed-form solution of Heston (1993). In both cases, it considers a model in which the variance of stock price returns varies according to an independent diffusion process. For the closed form option pricing model, the results are expressed in terms of the characteristic function. Chapter 33 uses traditional calculus method to derive Greek letters and discuss their applications. It introduces the definitions of Greek letters. It also provides the derivations of Greek letters for call and put options on both dividendspaying stock and non-dividends stock. It then discusses some applications of Greek letters. Finally, it shows the relationship between Greek letters, one of the examples can be seen from the Black-Scholes partial differential equation. Chapter 37 applies Ito calculus to option pricing and hedging performance under stochastic volatility and stochastic interest rate. Recent studies have extended the Black-Scholes model to incorporate either stochastic interest rates or stochastic volatility. But, there is not yet any comprehensive empirical study demonstrating whether and by how much each generalized feature will improve option pricing and hedging performance. This chapter fills this gap by first developing an implementable option model in closed-form that admits both stochastic volatility and stochastic interest rates and is parsimonious in the number of parameters. The model includes many known ones as special cases. Based on the model, both delta-neutral and single-instrument minimumvariance hedging strategies are derived analytically. Using S&P 500 option prices, we then compare the pricing and hedging performance of this model with that of three existing ones that respectively allow for (1) constant volatility and constant interest rates (the Black-Scholes), (2) constant volatility but stochastic interest rates, and (3) stochastic volatility but constant interest rates. Overall, incorporating stochastic volatility and stochastic interest rates produces the best performance in pricing and hedging, with the remaining pricing and hedging errors no longer systematically related to contract features. The second performer in the horse-race is the stochastic volatility model, followed by the stochastic interest rates model and then by the Black-Scholes. Chapter 44 uses Ito calculus in the structural approach to modeling credit risk. This chapter presents a survey of recent developments in the structural approach to modeling of credit risk. It first reviews some models for measuring credit risk based on the structural approach. This study then discusses the empirical evidence in the literature on the performance of structural models of credit risk. Chapter 85 has demonstrated the applications of alternative ODE in finance and economics research. This chapter introduces ordinary

differential equation (ODE) and its application in finance and economics research. There are various approaches to solve an ordinary differential equation. This study mainly introduces the classical approach for a linear ODE and Laplace transform approach. It illustrates the application of ODE in both deterministic and stochastic systems. As application in deterministic systems, the study applies ODE to solve a simple Gross Domestic Product (GDP) model and an investment model of a firm that is well studied in Gould (1968). As an application in stochastic systems, this chapter employs ODE in a jump-diffusion model and derives the expected discounted penalty given that the asset value follows a phase-type jump diffusion process. In particular, this study examines capital structure management model and derives the optimal default-triggering level. For related results, please see Chen et al. (2007). Chapter 98 discusses an ODE approach for the expected discounted penalty at ruin in a jump-diffusion model. Under the assumption that the asset value follows a jump diffusion, this chapter shows the expected discounted penalty satisfies an ODE and obtains a general form for the expected discounted penalty. In particular, if only downward phase-type jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Levy measure), closed form solutions for the expected discounted penalty can be obtained. In this chapter, Leland’s structural model with jumps is analyzed. For earlier and related results, see Gerber and Landry (1998), Hilberink and Rogers (2002), Mordecki (2002), Kou and Wang (2004), Asmussen et al. (2004) and others. Chapters 101 and 102 use the stochastic differential equation approach to derive a theoretical model of positive interest rate. Chapter 101 presents a dynamic term structure model in which interest rates of all maturities are bounded from below at zero. Positivity and continuity combined with no arbitrage result in only one functional form for the term structure with three sources of risk. This chapter casts the model into a state-space form and extracts the three sources of systematic risk from both the U.S. Treasury yields and the U.S. dollar swap rates. It analyzes the different dynamic behaviors of the two markets during credit crises and liquidity squeezes. Chapter 102 explores what types of diffusion and drift terms forbid negative yields, but nevertheless allow any yield to be arbitrarily close to zero. This chapter shows that several models have these characteristics; however, they may also have other odd properties. In particular the square root model of Cox-Ingersoll-Ross has such a solution, but only in a singular case. In other cases, bubbles will occur in bond prices leading to unusually behaved solutions. Other models, such as the CIR three-halves power model, are free of such oddities.

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Fuzzy set theory is also important in quantitative research risk management research. Chapter 77 discusses fuzzy set theory and its applications to finance research. Chapter 87 discusses the fuzzy set and data mining applications in accounting and finance research. To reduce the complexity in computing tradeoffs among multiple objectives, this chapter adopts a fuzzy set approach to solve various accounting or finance problems such as international transfer pricing, human resource allocation, accounting information system selection, and capital budgeting problems. A solution procedure is proposed to systematically identify a satisfying selection of possible solutions that can reach the best compromise value for the multiple objectives and multiple constraint levels. The fuzzy solution can help top management make a realistic decision regarding its various resource allocation problems as well as the overall firm’s strategic management when environmental factors are uncertain. The purpose of last part of the chapter is to test the effectiveness of a multiple criteria linear programming (MCLP) approach to data mining for bankruptcy prediction using financial data. Data mining applications have recently been receiving more attention in general business areas, but there is a need to use more of these applications in accounting and finance areas when dealing with large amounts of both financial and nonfinancial data. The results of the MCLP data mining approach in our bankruptcy prediction studies are promising and may extend to other countries. Chapter 61 discusses the actuarial mathematics and its applications in quantitative finance. First, it introduces the traditional actuarial interest functions and uses them to price different types of insurance contracts and annuities. Based on the equivalence principle, risk premiums for different payment schemes are calculated. After premium payments and the promised benefits are determined, actuarial reserves are calculated and set aside to cushion the expected losses. In a similar method, it uses actuarial mathematics are used to price the risky bond, the credit default swap, and the default digital swap, while interest structure is not flat and the first passage time is replaced by the time of default instead of the future life time. Moreover, product market games and signaling model in finance is discussed in Chap. 94. A crucial set of assumption in a large class of models of financial market signaling and product market/financial market interactions are related to super- or submodularity of the maximand. The assumption of one or the other can produce result reversals or indeterminacy. To the extent that data or underlying economics cannot unambiguously choose between super- and submodularity, the theoretical robustness and empirical testability of such models becomes questionable. Based on three examples drawn from recent finance literature, this study discusses the role that the assumptions play and raises issues of empirical identifiability in testing model predictions.

3.5 Other Disciplines In addition to statistics, econometrics, and mathematics, there are many other disciplines used in the finance research. Operation research model is one example. Linear programming models and quadratic programming are used in cash flow matching and portfolio immunization. Chapter 14 discusses the Le Châtelier principle in the Markowitz quadratic programming investment model using a case of world equity fund market. Due to limited numbers of reliable international equity funds, the Markowitz investment model is ideal in constructing an international portfolio. Overinvestment in one or several fast-growing markets can be disastrous as political instability and exchange rate fluctuations reign supreme. This chapter applies the Le Châtelier principle to the international equity fund market with a set of upper limits. Tracing out a set of efficient frontiers, this study inspects the shifting phenomenon in the mean-variance space. The optimum investment policy can be easily implemented and risk minimized. Chapter 16 discusses quadratic programming in the portfolio analysis. Markowitz portfolio analysis delineates a set of highly desirable portfolios; these portfolios have the maximum return over a range of different risk levels. Conversely, Markowitz portfolio analysis can find the same set of desirable investments by delineating portfolios that have the minimum risk over a range of different rates of return. The set of desirable portfolios is called the efficient frontier. Portfolio analysis analyzes rate of return statistics, risk statistics (standard deviations), and correlations from a list of candidate investments (stocks, bonds, and so forth) to determine which investments in what proportions (weights) enter into the efficient portfolios. Markowitz’s portfolio theory can be analyzed further to determine interesting asset pricing implications. The role of risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVar) in risk management are the interest of both academia and practitioners. Chapter 95 demonstrates the estimation of short- and long-term VaR for long-memory stochastic volatility models. The phenomenon of long-memory stochastic volatility (LMSV) has been extensively documented for speculative returns. This chapter investigates the effect of LMSV for estimating the value at risk (VaR) or the quantile of returns. The proposed model allows the return’s volatility component to be short- or longmemory. This chapter derives various types of limit theorems that can be used to construct confidence intervals of VaR for both short and long-term returns. For the latter case, the results are particularly of interest to financial institutions with exposure to long-term liabilities, such as pension funds and life insurance companies, which need a quantitative methodology to control market risk over longer horizons.

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Computer science is another important discipline in quantitative finance and risk management research. The program design, data structures, and algorithms help finance researchers implement their models with large data sets. The concepts of neural networks have also been applied in security analysis and bankruptcy prediction literature. In empirical implementation of the derivatives pricing model, numerical analysis is required to compute the fair value of the derivatives. Chapter 39 uses numerical methods in evaluating Asian options. It introduces Monte Carlo simulations, diverse mathematical and numerical techniques designed to estimate the distribution of underlying asset, then uses arbitrage arguments, binomial tree models, and the application of insurance formulas in the evaluation of options. Chapter 40 demonstrates that numerical valuation of Asian options with higher moments in the underlying distribution. It develops a modified Edgeworth binomial model with higher moment consideration for pricing European or American Asian options. If the number of the time steps increases, our numerical algorithm is as precise as that of Chalasani et al. (1999), with lognormal underlying distribution for benchmark comparison. If the underlying distribution displays negative skewness and leptokurtosis, as often observed for stock index returns, our estimates are better and very similar to the benchmarks in Hull and White (1993). The results show that the modified Edgeworth binomial model proposed in this chapter can value European and American Asian options with greater accuracy and speed given higher moments in their underlying distribution. Chapter 79 shows numerical solutions of financial partial differential equations. This chapter provides a general survey of important PDE numerical solutions and studies in detail of certain numerical methods specific to finance with programming samples. These significant numerical solutions for financial PDEs include finite difference, finite volume, and finite element. Finite difference is simple to discretize and easy to implement; however, explicit method is not guaranteed stable. Finite volume has an advantage over finite difference that it does not require a structured mesh. If a structured mesh is used, as in most cases of pricing financial derivatives, the finite volume and finite difference method yield the same discretization equations. Finite difference method can be considered a special case of finite element method. In general, the finite element method has better quality in approximation to exact solution when compared to the finite difference method. Since most PDEs in financial derivatives pricing have simple boundary condition, implicit method of finite difference is preferred to the finite element method in applications of financial engineering due to low cost of implementation.

Moreover, in constructing stochastic or probabilistic financial models as opposed to the traditional static and deterministic models, the Monte Carlo method is required to compute the results. Chapters 71 and 91 show how Monte Carlo Markov Chain (MCMC) method is utilized in finance research. Chapter 71 proposes to use Monte Carlo Markov Chain methods to estimate the parameters of Stochastic Volatility Models with several factors varying at different time scales. The originality of this approach, in contrast with classical factor models, is the identification of two factors driving univariate series at well-separated time scales. This is tested with simulated data as well as foreign exchange data. Other methodologies used in quantitative finance and risk management research also include entropy methods. Chapters 104 and 105 show how entropy methods can be used to finance research. Chapter 104 expresses the density of the minimal entropy Martingale measure in terms of the value process of the related optimization problem and show that this value process is determined as the unique solution of a semiMartingale backward equation. This chapter considers some extreme cases when this equation admits an explicit solution. Chapter 105 derives the density process of the minimal entropy Martingale measure in the stochastic volatility model proposed by Barndorff-Nielsen and Shephard. The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman-Kac representation of this function is provided. The dynamics of the processes determining the price and volatility are explicitly given under the minimal entropy Martingale measure, and this study derives a Black & Scholes equation with integral term for the price dynamics of derivatives. It turns out that the minimal entropy price of a derivative is given by the solution of a coupled system of two integro-partial differential equations.

3.6 Conclusion This chapter discusses the methodology in quantitative finance and risk management research. The list of the subjects that contribute to the modern finance research include statistics, econometrics, mathematics, and other disciplines such as operation research models and computer science. This list is definitely not exhaustive as finance research requires a broad range of methodological tools from various disciplines. Detailed applications of the aforementioned subjects are discussed in the latter part of this handbook.

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References

Heston, S. 1993. “A closed-form solution for options with stochastic volatility with application to bond and currency options.” The Review of Financial Studies 6(2), 327–343. Hilberink, B. and L. C. G. Rogers. 2002. “Optimal capital structure and endogenous default.” Finance and Stochastics 6, 237–263. Hoeting, J., D. Madigan, A. Raftery, and C. Volinsky. 1999. “Bayesian model averaging: a tutorial.” Statistical Science 14(4), 382–417. Hull, J. and A. White. 1993. “Efficient procedures for valuing European and American path-dependent options.” The Journal of Derivatives 1, 21–31. Kou, S.G., H. Wang. 2004. “Option pricing under a double exponential jump diffusion model.” Management Science 50, 1178–1192. Lamoureux, C. G. and W. D. Lastrapes. 1990. “Heteroskedasticity in stock returns data: volume versus GARCH effects.” Journal of Finance 45(1), 221–29. Mordecki, E. 2002. “Optimal stopping and perpetual options for Levy processes.” Finance and Stochastics 6, 473–493. Pastor, L. 2000. “Portfolio selection and asset pricing models.” Journal of Finance 55(1), 179–223. Roll, R. 1977. “An analytical valuation formula for unprotected American call options on stocks with known dividends.” Journal of Financial Economics 5, 251–258. Scott, L. 1987. “Option pricing when the variance changes randomly: theory, estimation, and an application.” Journal of Finance and Quantitative Analysis 22(4), 419–438. Shafer, G. and V. Vovk. 2001. Probability and finance: it’s only a game, Wiley, New York. Theil, H. 1966. Applied economic forecasting, North-Holland, Amsterdam. Wei, K. C. J. 1988. “An asset-pricing theory unifying the CAPM and APT.” Journal of Finance 43(4), 881–892. Whaley, R. 1981. “On the valuation of American call options on stocks with known dividends.” Journal of Financial Economics 9, 207–211. Yee, K. 2007. “Using accounting information for consumption planning and equity valuation.” Review of Accounting Studies 12(2–3), 227–256. Yule G. U. 1926. “Why do we sometimes get nonsense correlations between time series? A study in sampling and the nature of time series.” Journal of the Royal Statistical Society 89, 1–64

Asmussen, S., F. Avram, M. R. Pistorius. 2004. “Russian and American put options under exponential phase-type Levy models.” Stochastic Processes and their Applications 109, 79–111. Bates, J. and C. Granger. 1969. “The combination of forecasts.” Operational Research Quarterly 20, 451–468. Black, F. and R. Litterman. 1991. “Asset allocation: combining investor views with market equilibrium.” Journal of Fixed Income 1(2), 7–18. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–659. Black, F. and R. Litterman. 1992. “Global portfolio optimization.” Financial Analysts Journal (September/October) 28–43. Chalasani, P., S. Jha, F. Egriboyun, and A. Varikooty. 1999. “A refined binomial lattice for pricing American Asian options.” Review of Derivatives Research 3(1), 85–105. Chen, Y. T., C. F. Lee, and Y. C. Sheu. 2007. “An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model.” Finance and Stochastics 11, 323–355. Chung, S. L. and P. T. Shih. 2007. “Generalized Cox–Ross–Rubinstein binomial models.” Management Science 53, 508–520. Cremers, K. 2002. “Stock return predictability: a Bayesian model selection perspective.” Review of Financial Studies 15(4), 1223–1249. Deaton, A. and G. Laroque. 1992. “On the behavior of commodity prices.” Review of Economic Studies 59, 1–23. Elton, E.J., M.J. Gruber, S.J. Brown, and W.N. Goetzmann. 2007. Modern portfolio theory and investment analysis, Wiley, New Jersey. Elton, E.J., M.J. Gruber, and M. W. Padberg. 1976. “Simple criteria for optimal portfolio selection.” Journal of Finance 31, 1341–1357. Gerber, H. U., B. Landry. 1998. On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurances: Mathematics and Economics 22, 263–276. Geske, R. 1979. “A note on an analytical valuation formula for unprotected American call options on stocks with known dividends.” Journal of Financial Economics 7, 375–380. Gould, J. P. 1968. “Adjustment costs in the theory of investment of the firm.” The Review of Economic Studies 1, 47–55. Granger, C. W. J. and P. Newbold. 1974. “Sourious Regressions in Econometrics.” Journal of Econometrics 2, 111–120.

Research Methods in Quantitative Finance and Risk Management

Chapter 4

Foundation of Portfolio Theory Cheng-Few Lee, Alice C. Lee, and John Lee

Abstract In this chapter, we first define the basic concepts of risk and risk measurement. Using the relationship of risk and return, we introduce the efficient-portfolio concept and its implementation. Then the concept of dominance principle and performance measures are also discussed and illustrated. Finally, the interest rate and market rate of return are used as measurements to show how the commercial lending rate and the market risk premium can be calculated. Keywords Risk r Systematic risk r Unsystematic risk r Return r Efficient frontier r Efficient portfolio r Performance measure r Risk premium r Sharpe measure r Jensen measure r Treynor measure

4.1 Introduction This chapter focuses on the various types of risk. An understanding of the sources of risk and their determination is very useful for doing meaningful security analysis and portfolio management; thus, discussing risk prior to an exploration of portfolio-selection models is essential. In this chapter methods of measuring risk are defined by using basic statistical methods, and the concepts and applications of the dominance principle and portfolio theory are discussed in some detail. A discussion of the different types of risk and their classification is followed by a review of the concepts and applications of portfolio analysis. The dominance as well as the necessity of using performance measures as a means to comC.-F. Lee () Rutgers University, New Brunswick, NJ, USA e-mail: [email protected] A.C. Lee State Street Corp., Boston, MA, USA e-mail: [email protected] J. Lee Center for PBBEF Research, Hackensack, NJ, USA e-mail: [email protected]

pare performance among different portfolios are treated. The determination of the commercial lending rate in accordance with risk and return concepts is then analyzed, and the final section of this chapter concerns calculations of the market rate of return and the market risk premium.

4.2 Risk Classification and Measurement Risk is defined as the probability of success or failure. To be able to measure risk, it is necessary to define the range of outcomes and the probability that these outcomes will occur. A probability distribution may be determined either subjectively or objectively; a subjective determination is made by the individual about the likelihood of various future outcomes, while an objective determination involves measuring past data that are associated with the frequency of certain types of outcomes. In either case, a subjective or objective determination requires a quantitative measure of risk. The measure most commonly used is the standard deviation or the variance of the possible returns. In considering two securities, A and B, it is possible to use a subjective approach, an objective approach, or a combination of the two approaches to indicate the expected range of outcomes and the probability that each outcome will occur. Figure 4.1 illustrates the distribution of return possibilities for these two securities. While the standard deviation is stated in rates of return, the variance is stated in terms of the rate of return squared. As it is more natural to discuss rates of return rather than rates of return squared, risk is usually measured with the standard deviation of returns. Nevertheless, for statistical purposes it is usually more convenient to use the variance rather than the standard deviation. Either risk measure is appropriate because the standard deviation is merely a simple mathematical transformation of the variance. It is the job of the security analyst to provide estimates of a financial asset’s risk. Supplying estimates of a security variance or standard deviation is one method of estimating the total risk of a security.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_4, 

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Fig. 4.1 Probability distributions between securities A and B

Why do securities have differing levels of total risk? What factors cause the returns of securities to vary? Table 4.1 defines various types of risk and in so doing offers answers to these questions, and in analyzing the risks defined in Table 4.1 the following paragraphs provide further illustration. At any point a security or a portfolio could be subject to a number of these risks. In general, the various risks are not mutually exclusive, nor are they additive. By mutually exclusive is meant that the fact that a portfolio contains a certain

kind of risk does not mean that it also contains other types of risk. By additive is meant that if a portfolio contains a number of different types of risk, the total risk of the portfolio is simply the sum of the individual risks. Additionally, some the risks defined in Table 4.1 are easily quantified or measured – for example using beta to measure systematic risk – while other types of risk may not have well-defined quantitative risk measures such as management risk or political risk. Finally, it should be noted that the types of risk listed on Table 4.1 are mutually exclusive.

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Table 4.1 Types of risk Risk type

Description

Call risk

The variability of return caused by the repurchase of the security before its stated maturity

Convertible risk

The variability of return caused when one type of security is converted into another type of security

Default risk

The probability of a return of zero when the issuer of the security is unable to make interest and principal payments – or for equities, the probability that the market price of the stock will go to zero when the firm goes bankrupt

Interest-rate risk

The variability of return caused by the movement of interest rates

Management risk

The variability of return caused by bad management decisions; this is usually a part of the unsystematic risk of a stock, although it can affect the amount of systematic risk

Marketability risk

The variability of return caused by the commissions and price concessions associated with selling an illiquid asset

Political risk

The variability of return caused by changes in laws, taxes, or other government actions

Purchasing-power risk

The variability of return caused by inflation, which erodes the real value of the return

Systematic risk

The variability of a single security’s return caused by the general rise or fall of the entire market

Unsystematic risk

The variability of return caused by factors unique to the individual security

4.2.1 Call Risk

4.2.2 Convertible Risk

An investor who purchases a security, whether it is a debt instrument or equity, usually has a pretty good idea of his or her investment horizon – that is, how long he or she intends to hold the security. If the issuer calls the bond or repurchases the stock at a point in time prior to the end of the investor’s investment horizon, the return earned by the investor may be less than expected. The investor is facing call risk (see Table 4.1 for a detailed definition). Generally all corporate bonds and preferred stocks are callable at some point during the life of the security. Investors do not like this call feature, because bonds are most often called after interest rates have fallen, so that the issues can replace them with a new issue of bonds at a lower interest rate. When a bond is called, instead of receiving the expected interest flows for the remainder of his or her investment horizon, the investor must reinvest the proceeds received from the issuer at a lower rate of interest. It is this reinvestment at a new lower rate of interest that causes the yield or return of the bond to be different than what the investor expected. For this reason, in periods of high interest rates, when interest rates are expected to drop to lower levels in the future, bonds that are call protected for a certain number of years usually sell at a higher price than callable bonds. Thus, a deferred call feature can cause an interest spread or differential to develop vis-à-vis a similar issue with no call feature or an issue that is immediately callable. The size of the interest differential varies with the investor’s expectations about the future movement of interest rates. Whether a bond is callable immediately after it is issued or after a deferred period, the actual call of the bond involves the payment of a premium over the par value of the bond by the issuer.

If a bond or a preferred stock is convertible into a stated number of shares of common stock of the corporation issuing the original security, the rate of return of the investment may vary because the value of the underlying common stock has increased or decreased. This kind of investment is facing convertible risk, as described in Table 4.1. A convertible security normally has a lower coupon rate, or stated dividend (in the case of preferred stocks), because investors are willing to accept a lower contractual return from the company in order to be able to share in any rise in the price of the firm’s common stock. The size of the yield spread between straight and convertible securities is dependent on the future prospects of the individual firm and the general level of interest rates. The difference between the value paid for the conversion right and the actual returns experienced over the life of the security increase the variability of returns associated with the investment.

4.2.3 Default Risk The quality of the issue or the chances of bankruptcy for the issuer have a significant impact on the rate of return of a security. A lower-quality issue will sell at a lower price and thus offer a higher yield than a similar security issued by a higherquality issuer. This is true in all segments of the securities markets. Hence, most investors hold diversified portfolios in order to reduce their exposure to any one security going into default or the issuer going bankrupt. The risk relating to default or bankruptcy is called default risk, as described in Table 4.1.

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4.2.4 Interest-Rate Risk As the general level of interest rates changes, the value of an individual security also changes. As interest rates rise, the value of existing securities falls, and vice versa. Additionally, longer-term securities are more affected by a given change in the general level of interest rates than are shorter-term securities. The fluctuation of security returns caused by the movement of interest rates is called interest-rate risk, as described in Table 4.1. For example, Bond A has a coupon of 7% and a maturity of 5 years. If interest rates go from 7 to 8%, the price of A will fall $6.75 for each $100 of par value, whereas the price of B will fall only $4.00 for each $100 of par value. Another factor that affects the amount of price change for a given interest rate changes is the size of the coupon. The lower the coupon rate of an issue, the more widely its price will change for a given change in interest rates. For example, Bond C is a 20-year bond with a percent coupon and a market value of $68 when the general level of interest rates are 7%. Bond D is a 20-year bond with 7% coupon, so its market value is par $100. If interest rates go from 7 to 8%, the value of Bond C will fall to $60.38, a decrease in value of 11.3% based on its market price [(60.38–68)/68], whereas Bond D’s value will fall by 9.8% [(90.2–100)/100]. In general, a shorter-maturity, higher-coupon security is subjected to less interest-rate risk than a long-term, low coupon security.

4.2.5 Management Risk Management risk, as indicated in Table 4.1, is caused by errors of a firm’s managers when they make business and financing decisions. Business risk refers to the degree of fluctuation of net income associated with different types of business operations. This kind of risk is related to different types of business and operating strategies. Financial risk refers to the variability of returns associated with leverage decisions. How much of the firm should be financed with equity and how much should be financed with debt? It should be noted that both business risk and financial risk are not necessarily constant over time. Both risks can be affected by the fluctuations of the business cycle or by changes in government policy. Jensen and Meckling have presented a theory, called agency theory, that deals with the problem of management errors and their impact on security owners. An agentprincipal relationship exists when decision-making authority is delegated. The modern corporation is a good example of this phenomenon. The owners or shareholders of the

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firm delegate the decision-making authority to the firm’s managers, who are in reality employees or agents of the owners. Other things being equal, the managers make decisions that satisfy the needs of the managers rather than the desires of the owners unless the firm’s owners can protect themselves from management decisions. Some have argued that it is the function of financial arrangements using bond covenants, options, and so on, to narrow the divergence of goals between a firm’s owners and its managers. In any case, it is the possibility of management error or divergent goals that is one of the causes of variability of return for securities.

4.2.6 Marketability (Liquidity) Risk The marketability risk (or liquidity risk) of a security (see the description in Table 4.1) affects the rate of return received by its owner. Marketability is made up of two components: (1) the volume of securities that can bought or sold in a short period of time without adversely affecting the price, and (2) the amount of time necessary to complete the sale of a given number of securities. Other things being equal, the less marketable a security, the lower its price or the higher its yield. A highly liquid security – for example, IBM shares traded on the New York Stock Exchange (NYSE) – can be purchased or sold in large quantities in a very short time. Millions of shares of IBM are traded every day on the NYSE. The stock of a small firm traded over the counter (OTC) may trade only a few hundred shares a week and is said to be illiquid. One good measure of the marketability of a security is the spread between the bid and ask prices. The bid price is the current price at which the security can be sold and the ask price is the current price at which the security can be bought. The bid price is always lower than the ask price. The bid-ask spread is the cost of selling the asset quickly. That is to say, it is the amount of margin required by the market maker to stand ready to buy or sell reasonable amounts of the security quickly. The more illiquid the security the wider the bid-ask spread.

4.2.7 Political Risk International political risk (as described in Table 4.1) stems from the political climate and conditions of a foreign country. For instance, if a government is unstable, as is the case in some South American countries, there may be wild fluctuations in interest rates due to a lack of confidence by the population and the business community. There may even be some chance of sabotage of plant and equipment or other acts

4 Foundation of Portfolio Theory

of terrorism. Contracts may not be upheld or enforceable. These are all factors that may cause the rate of return on certain securities to vary. Domestic political risk takes the form of laws, taxes, and government regulations. As the government changes the tax law, the returns of securities can be greatly affected. For example, the Tax Reform Act of 1986 probably had a very important role in the bull market of January to August 1987, when the Dow-Jones Industrial Average went from less than 1,900 to above 2,700.

4.2.8 Purchasing-Power Risk Purchasing-power risk, as described in Table 4.1, is related to the possible shrinkage in the real value of a security even though its nominal value is increasing. For example, if the nominal value of a security goes from $100 to $200, the owner of this security is pleased because the investment has doubled in value. But suppose that, concurrent with the value increase of 100%, the rate of inflation is 200% – that is, a basket of goods costing $100 when the security was purchased now costs $300. The investor has a “money illusion” of being better off in nominal terms. The investment did increase from $100 to $200; nevertheless, in real terms, whereas the $100 at time zero could purchase a complete basket of goods, after the inflation only two-thirds of a basket can now be purchased. Hence, the investor has suffered a loss of value.

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4.3 Portfolio Analysis and Application Essential to adequate diversification is a knowledge of the primary concepts of portfolio analysis and their application. This section focuses on basic analysis and applications; more advanced portfolio theory and methods will be discussed in the next chapters. A portfolio can be defined as any combination of assets or investments. Portfolio analysis is used to determine the return and risk for these combinations of assets. Portfolio concepts and methods are employed here to formally develop the idea of the dominance principle and some other measures of portfolio performance.

4.3.1 Expected Return on a Portfolio The rate of return on a portfolio is simply the weighted average of the returns of individual securities in the portfolio. For example, 40% of the portfolio is invested in a security with a 10% expected return (security A); 30% is invested in security B with a 5% expected return; and 30% is invested in security C with a 12% expected return. The expected rate of return on this portfolio can be expected: RN p D Wa RN a C Wb RN c C Wc RN c D .0:4/.0:1/ C .0:3/.0:5/ C .0:3/.0:12/ D 0:091

4.2.9 Systematic and Unsystematic Risk In Table 4.1, total risk was defined as the sum of systematic and unsystematic risk. Total risk is also equal to the sum of all the risk components just discussed. However, the importance and the contribution to total risk depends on the type of security under consideration. The total risk of bonds contains a much larger fraction of interest-rate risk than the total risk of a stock. Each of the types of risk discussed may contain a systematic component that cannot be diversified away and an unsystematic component that can be reduced or eliminated, if the securities are held in a portfolio. It is assumed throughout this text that rational investors are average to risk, whatever its source; hence a knowledge of various risk-management techniques is essential. Toward that end this chapter now focuses on the management of risk through diversification.

in which Wa ; Wb , and Wc are the percentages of the portfolio invested in securities A, B, and C, respectively. The summation of these weights is equal to one. Rp represents the expected rate of return for the portfolio and is the weighted average of the securities’ expected rate of return. In general, the expected return on an n-asset portfolio is defined by Equation (4.1): RN p D

n X

RN i Wi

(4.1)

i D1

where: n P i D1

Wi D 1 Wi D the proportion of the individual’s investment allocated to security i I and RN i D the expected rate of return for security i:

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4.3.2 Variance and Standard Deviation of a Portfolio

Var.W1 R1t C W2 R2t / D

t D1

The riskiness of a portfolio is measured by the standard deviation (or variance) of the portfolio. To calculate the standard deviation of a portfolio, we should first identify the covariance among the securities within the portfolio. The covariance between two securities used to formulate a portfolio can be defined as:

Cov.W1 R1 ; W2 R2 / D

N X .W1 R1t  W1 RN 1 /.W2 R2t  W2 RN 2 / N 1 tD1

D W1 W2

N X .R1t  RN 1 /.R2t  RN 2 / N 1 tD1

D W1 W2 Cov.R1 ; R2 /

N P

(4.2)

D Œ.0:4/.0:1/ C .0:6/.0:05/ .0:4/.0:15/ C .0:6/.0:1/ 2 =2 CŒ.0:4/.0:15/ C .0:6/.0:1/ .0:4/.0:15/ C .0:6/.0:1/ 2 =2 CŒ.0:4/.0:2/ C .0:6/.0:15/ .0:4/.0:15/ C .0:6/.0:1/ 2 =2 .0:12  0:12/2 .0:07  0:12/2 C D 2 2 .0:17  0:12/2 C 2 Varportfolio D 0:0025

The riskiness of a portfolio can be measured by the standard deviation of returns, as indicated in Equation (4.3):

where: R1t D the rate of return for the first security in period tI R2t D the rate of return for the second security in period tI N N R1 and R2 D average rates of return for the first security and the secondsecurity; respectivelyI and Cov.R1 ; R2 / D the covariance between R1 and R2 : The covariance as indicated in Equation (4.2) can be used to measure the covariability between two securities (or assets) when they are used to formulate a portfolio. With this measure the variance for a portfolio with two securities can be derived: Var.W1 R1t C W2 R2t / 2  N P .W1 R1t C W2 R2t /  .W1 RN 1 C W2 RN 2 / D N 1 tD1

Œ.W1 R1t CW2 R2t /.W1 RN 1 CW2 RN 2 / 2 N 1

p D

v u N uP N 2 u t t D1 .Rpt  Rp / N 1

where p is the standard deviation of the portfolio’s return and RN p is the expected return of the n possible returns. Figure 4.2 illustrates possible distributions for two portfolios, assuming the same expected return but different levels of risk. Since portfolio B’s variability is greater than that of portfolio A, investors regard portfolio B as riskier than portfolio A.

4.3.3 The Two-Asset Case To explain the fundamental aspect of the risk-diversification process in a portfolio, consider the two-asset case. Following Equation (4.3):

Example 4.1 provides further illustration. Example 4.1. If you have two securities, you can use Equation (4.2) to calculate the portfolio variance as follows: Security 1

Security 2

W1 D 40% t D1 RN 1t D 10% D2 15% D3 20% RN 1 D 15%

W2 D 60% t D1 RN 2t D 5% D2 10% D3 15% RN 1 D 10%

(4.3)

Fig. 4.2 Probability distributions of returns for two portfolios

4 Foundation of Portfolio Theory

s p D D

N P t D1

s

N P tD1

59

If 12 D 1, Equation (4.7) reduces from:

.Rpt RN p /2 N 1 ŒW21 .R1t RN 1 /2 CW22 .R2t RN 2 /2 C2W1 W2 .R1t RN 1 /.R2t RN 2 /

@p D ŒW12 12 C .1  W1 /2 22 C 2W1 .1  W1 /12 1 2 1=2 @W1

Œ2W1 12  2.1  W1 /22 C 2.1  2W1 /12 1 2 R W1 Œ12 22  2 12 1 2  Œ12  12 1 2 D ŒW12 12 C .1  W1 /2 22 C 2W1 .1  W1 /12 1 2

N 1

q D W21 Var.R1t / C W22 Var.R2t / C 2W1 W2 Cov.R1 ; R2 / (4.4)

where W1 C W2 D 1. By the definitions of correlation coefficients between R1 and R2 .12 /, the Cov R1 ; R2 can be rewritten: (4.5) Cov.R1 ; R2 / D 12 1 2 where 1 and 2 are the standard deviations of the first and second security, respectively. From Equations (4.4) and (4.5), the standard deviation of a two-security portfolio can be defined: p p D qVar.W1 R1t C W2 R2t / D W12 12 C .1  W1 /2 22 C 2W1 .1  W1 /12 1 2 (4.6) Example 4.2 provides further illustration.

to:

2 .2  1 / .2  1 /.2  1 / 2 D 2  1

W1 D

Returning to the Examples for securities 1 and 2, since the standard deviations are equal and the securities are perfectly positively correlated, Equation (4.7a) indicates that there is no value for W1 that will minimize the portfolio variance. The weight from W1 that gives the minimum-variance portfolio is 0.05/(0.05–0.05) or 0.05/0, which is undefined. If 12 D 1, Equation (4.7) reduces to: W1 D

Example 4.2. For securities 1 and 2 used in the previous example, applying Equation (4.6) should give the same results for portfolio variance. Security 1

Security 2

12 D 0:0025 W1 D 0:4 12 D C1

22 D 0:0025 W2 D 0:6

(4.7a)

2 .2 C 1 / .1 C 2 /.1 C 2 / 2 .2 C 1 /

(4.7b)

However, if the correlation coefficient between 1 and 2 is 1, then the minimum-variance portfolio must be divided equally between security 1 and security 2 – that is: 0:05 0:05 C 0:05 D 0:5

W1 D

p p D p.0:4/2 .0:0025/C.0:6/2 .0:0025/C2.0:4/.0:6/.1/.0:05/.0:05/ D 0:0025

p D 0:05 or Var portfolio D 0:0025, the same answer as for Example 4.1. If 12 D 1:0 Equation (4.6) can be simplified to the linear expression: p D W1 1 C W2 2

As an expanded form of Equation (4.6), a portfolio can be written: 31=2 2 n n n1 X X X p D 4 W12 12 C 2 Wi Wj ij i j 5 i D1

2

where W2 D .1  W1 /. Since Equation (4.6) is a quadratic equation, some value of W1 minimizes p . To obtain this value, differentiate Equation (4.6) with respect to W1 and set this derivative equal to zero. Solving for W1 : 2 .2  12 1 / W1 D 2 1 C 22  212 1 2

D4

n n X X

i D1 j Di C1

31=2 Wi Wj Cov.Ri t ; Rjt /j5

(4.8)

j D1 i D1

where: (4.7)

A condition needed for Equation (4.7) is that W1  1; that is, no more than 100% of the portfolio can be in any single security, and negative positions (short positions) can be maintained on any security.

Wi and Wj D the investor’s investment allocated to security i and securityj; respectivelyI ij D the correlation coefficient between security i and security j I and n D the number of securities included in the portfolio:

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Since Equation (4.8) has n securities, there are n variance terms (that is, W12 12 ) and .n2  n/ covariance terms (that is, Wi Wj 12 i j ). If n D 200, Equation (4.8) will have 200 variance terms and 39,800 covariance terms, and any practical application will require a great amount of information as well as many computations. This issue will be discussed in more detail in Chap. 5; in the meanwhile, Example 4.3 provides further illustration. Example 4.3. Consider two stocks, A and B: RN A D 10; RN B D 15; A D 4, and B D 6. (1) If a riskless portfolio could be formed from A and B, what would be the expected return of Rp ? (2) What would the expected return be if AB D 0? Solution 1=2  1. p D WA2 A2 C.1  WA /2 B2 C2WA .1  WA /AB A B If we let AB D 1 p D ŒWA A  .1  WA /B p D 0 D 4WA  6.1  WA /RB WA D 3=5 so RN p D WA RN A C .1  WA /RB



2 3 .10/ C .15/ D 5 5 D 12% 2. If we let AB D 0 i1=2 h p D WA2 A2 C .1  WA /2 B2 i @p 1h D 2WA A2 C 2.1  WA /B2 .1/ @WA 2h i1=2 D0

WA2 A2 C .1  WA /2 B2 WA A2  .1  WA /B2 D 0   WA D B2 =A2 C B2 D B2 = 42 C 62 D 9=13 4 9 .10/ C .15/ RN P D 13 13 D 11:54%

4.4.1 The Efficient Portfolio By definition, a portfolio is efficient, if there are no other portfolios having the same expected return at a lower variance of returns. Moreover, a portfolio is efficient if no other portfolio has a higher expected return as the same risk of returns. This suggests that given two investments, A and B, investment A will be preferred to B if: E.A/ > E.B/ and Var.A/ D Var.B/ or E.A/ D E.B/ and Var.A/ < Var.B/ where: E.A/ and E.B/ D the expected returns of A and B: and Var (A) and Var (B) D their respective variances or risk. The mean returns and variance of every investment opportunity can be calculated and plotted as a single point on a mean-standard deviation diagram, as shown in Fig. 4.3. All points below curve EF represent portfolio combinations that are possible. Point D represents a portfolio of investments with a return RD and risk D . All points above EF are combinations of risk and returns that do not exist. Point B would therefore represent risk and return that cannot be obtained with any combination of investments. The EF curve is also called the efficient frontier because all points below the curve are dominated by a point found on the curve. For instance, suppose a firm is willing to assume a maximum level of risk D . It can obtain a return of RD with portfolio D or move to point C on the frontier and receive a higher return RC with that portfolio. Therefore, C dominates D because it would be preferred to D. For the same level of risk, it has a higher return. A similar argument could be made in terms of risk. If the firm wants to achieve a return of RA , it will select portfolio

4.4 The Efficient Portfolio and Risk Diversification Utilizing the definitions of standard deviation of expected return of a portfolio discussed previously, this section discussed the concepts of the efficient portfolio and risk diversification. Fig. 4.3 The efficient frontier in portfolio analysis

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A over D, because A represents the same return at a smaller level of risk or standard deviation: A < B . Therefore, point D is not efficient but points A and C are. A decision maker could, therefore, select any point on the frontier and be secure in knowing that a better portfolio is not available. Example 4.4 further illustrates this concept. Example 4.4. To show how the portfolio concepts and methods discussed in this section can be used to do practical analysis, monthly rates of return for January 1980 to December 1984 for Pennzoil and Coca Cola are used as examples. The basic statistical estimates for these two firms are average monthly rates of return and the variance-covariance matrix. The average monthly rates of return for Pennzoil (PZ) and Coca Cola (CK) are 0.0093 and 0.0306, respectively. The variances and covariances are listed in the following table. Variance-covariance matrix PZ CK Pennzoil Coke

0.0107391 0.000290

0.000290 0.0495952

From Equation (4.7), we have: 0:0495952 C 0:000290 0:0107391 C 0:0495952 C 0:00058 0:0498852 D 0:069143 D 0:8189

W1 D

W2 D 1:0  0:8189 D 0:1811 Using the weight estimates and Equations (4.2) and (4.3): E.RN P / D .0:8189/.0:0093/ C .0:1811/.0:0306/ D 0:01316 P2 D .0:8189/2.0:0107391/ C .0:1811/2.0:0495952/

4.4.2 Corporate Application of Diversification The effect of diversification is not necessarily limited to securities but may have wider applications at the corporate level. Frequently, managers will justify undertaking many product lines because of the effects of diversification. Instead of “putting all of the eggs in one basket,” the investment risks are spread out among many lines of services or products in hope of reducing the overall risks involved and maximizing returns. To what degree this diversification takes place in other types of corporate decisions depends on the decision maker’s preference for risk and return. The overall goal is to reduce business risk fluctuations of net income. However, it should be noted that investors can do their own homemade diversification, which generally reduces the need of corporate diversification. This type of corporate diversification can be taken to the multinational level. For example, General Motors has overseas divisions throughout the world. Although these divisions all produce the same product, autos and auto parts, GM’s status as a multinational corporation allows it to take advantage of the diversifying effects of different exchange rates and political and economic climates.

4.4.3 The Dominance Principle The dominance principle has been developed as a means of conceptually understanding the risk/return tradeoff. As with the efficient-frontier analysis, we must assume an investor prefers returns and dislikes risks. For example, as depicted in Fig. 4.4, if an individual is prepared to experience risk associated with A , he or she can obtain a higher expected return with portfolio A.RN A / than with portfolio B.RN B /. Thus A dominates B and would be preferred. Similarly, if an individual were satisfied with a return of RN B , he or she would select portfolio B over C because the risks associated with B

C2.0:8189/.0:1811/.0:000290/ D 0:0088 P D 0:0940 when 12 is less than 1.00 it indicates that the combination of the two securities will result in a total risk less than their added respective risks. This is the diversification effect. If 12 were equal to 1.00, this would mean that the combination of the two securities has no diversification effect at all. The correlation coefficient of 12 D 0:0125659 indicates that a portfolio combining Pennzoil and Coca Cola would show a diversification effect and a reduction in risk.

Fig. 4.4 The dominance principle in portfolio analysis

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are less .B < C /. Therefore using the dominance principle reinforces the choice of an efficient portfolio and is also a method in determining it. Figure 4.4 makes clear that points A and B are directly comparable because they have a common standard deviation, A . Points B and C are directly comparable because of a common return, RN B . Now consider portfolio D. How does its risk versus return compare with the other portfolios shown in Fig. 4.4? It is difficult to say because the risk and return are not directly comparable using the dominance principle. This is the basic limitation of the dominance principle – that portfolios without a common risk or return factor are not directly comparable.

4.4.4 Three Performance Measures For such a situation it becomes necessary to use some other performance measure. There are basically three important portfolio performance measures taught in investment courses: (1) the Sharpe, (2) the Treynor, and (3) the Jensen measures. The Sharpe measure (SP) (Sharpe 1966) is of immediate concern. Given two of the portfolios depicted in Fig. 4.4, portfolios Band D, their relative risk-return performance can be compared using the equations: SPD D

RN D  Rf RN B  Rf and SPB D D B

where: SPD ; SPB RN D ; RN B Rf D ; B

D Sharpe performance measuresI D the average return of each portfolioI D risk-free rateI and D the respective standard deviation on risk of each portfolio:

Because the numerator is the average return reduced by the risk-free rate, it represents the average risk premium of each portfolio. Dividing the risk premium by the total risk per portfolio results in a measure of the return (premium) per unit of risk for each portfolio. The Sharpe performance measure equation will therefore allow a direct comparison of any portfolio, given its risk and returns. Consider Fig. 4.5; portfolio A is being compared to portfolio B. If a riskless rate exists, then all investors would prefer A to B because combinations of A and the riskless asset give higher returns for the same level of risk than combinations of the riskless asset and B. The preferred portfolio lies on the ray passing through Rf that is furthest in the counterclockwise direction (the ray that has the greatest slope). Example 4.5 further illustrates this concept.

Fig. 4.5 Combinations of portfolio and the risk-free investment

Example 4.5. An insurance firm is trying to decide between two investment funds. From past performance it was able to calculate the average returns and standard deviations for these funds. The current T-bill rate is 9.5% and the firm will use this as a measure of the risk-free rate.

Average return R (percent) Standard deviation  (percent)

Smyth fund

Jones fund

18 20

16 15

Risk-free rate D Rf (percent) D 9:5 Using the Sharpe performance measure, the risk-return measurements for these two firms are: 0:18  0:095 D 0:425 0:20 0:16  0:095 D 0:433 D 0:15

SPSmyth D SPJones

It is clear that the Jones fund has a slightly better performance and would be the better alternative of the two. The Sharpe measure looks at the risk-return decision from the point of view of an investor choosing a portfolio to represent the majority of his or her investment. An investor choosing a portfolio to represent a large part of his or her wealth would likely be concerned with the full risk of the portfolio, and the standard deviation is a measure of that risk. On the other hand, if the risk level of the portfolio is already determined by the investor and what is important is to evaluate the performance of the portfolio over and above the total market performance, perhaps the proper risk measure would be the relationship between the return on the portfolio and the return on the market, or beta. All combinations of a riskless asset and a risky portfolio lie on a straight line con-

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necting them. The slope of the line connecting the risky asset A and the risk-free rate is .RN A  Rf /=A . Here, as in the Sharpe measure, an investor would prefer the portfolio on the most counterclockwise ray emanating from the riskless asset. This measure, called the Treynor measure (TP), developed by Treynor in 1965, examines differential return when beta is the risk measure. Example 4.6 provides further illustration.

SPA D 0:84

The Treynor performance measure uses the beta coefficient (systematic risk) instead of total risk for the j th portfolio .j / as a risk measure. Applications of the TP are similar to the SP as discussed previously.1 Jensen (1968, 1969) has proposed a measure referred to as the Jensen differential performance index (Jensen’s measure). The differential return can be viewed as the difference in return earned by the portfolio compared to the return that the capital asset pricing line implies should be earned. Consider the line connecting the riskless rate and the market portfolio. A manager could obtain any point along this line by investing in the market portfolio and mixing this with the riskless asset to obtain the desired risk level. If the constructed portfolio is actively managed, then one measure of performance is the difference in return earned by actively managing the portfolio, versus what would have been earned if the portfolio had been passively constructed of the market portfolio and the riskless asset to achieve the same risk level. The slope of the line connecting the riskless asset and the market portfolio is .RN M  Rf /=ˇM , and the intercept must be the riskless rate. The beta on the market portfolio is one; therefore, the CAPM equation results in:

SPB D 0:73

RN P D Rf C .RN M  Rf /ˇP :

Example 4.6. Rank the portfolios shown in the table based on the Sharpe measure. Assume Rf D 8%. If Rf D 5%, how does the order change? Portfolio

Return (percent)

Risk (percent)

A B C D E

50 19 12 9 8:5

50 15 9 5 1

Solution Sharpe measure SPM D

RN M  Rf 

SPC D 0:44 SPD D 0:20 SPE D 0:50 Ranked by the Sharpe measure, A > B > E > C > D. The Sharpe measure indicates that portfolio A is the most desirable: it has the highest return per unit of risk. For Rf D 5 percent SPA D 0:90 SPB D 0:933 SPC D 0:77 SPD D 0:80 SPE D 0:35

Jensen’s measure (JM) is the differential return of the managed portfolio’s actual return less the return on the portfolio of identical beta that lies on the line connecting the riskless asset and the market portfolio. Algebraically, Jensen’s measure is expressed: JM D RN P  ŒRf C .RN M  Rf /ˇp Figure 4.6 depicts portfolio rankings for the Jensen measure, and Example 4.7 provides further illustration. Example 4.7. Rank the portfolio in the table according to Jensen’s measure:

The order changes to E > B > A > D > C. E is now the best portfolio as it has the highest return per unit of risk. The Treynor measure can be expressed by the following: TP D

RN j  Rf ˇj

where: RN j D average return of j th portfolioI Rf D ris  free rateI and ˇj D beta coefficient for j th portfolio:

Fig. 4.6 Jensen’s measure for portfolio rankings

1

Discussion of the Treynor measure adapted from Treynor (1965). Adapted by permission.

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1. 2. 3. 4.

C.-F. Lee et al.

Assuming RM D 10 percent and Rf D 8 percent Assuming RM D 12 percent and Rf D 8 percent Assuming RM D Rf D 8 percent Assuming RM D 12 percent and Rf D 4 percent

Portfolio

Ri (percent)

(percent)

ˇi

A B C D E

50 19 12 9 8:5

50 15 9 5 1

2.5 2.0 1.5 1.0 0.25

Solution

2.

3.

4.

D

ŒRN P  Rf ŒRN M  Rf  P m

D SPP  SPm .commom constant/l If the Jensen measure (JM) is divided by ˇP , it is equivalent TM to the Treynor measure plus some constant common to all portfolios: ŒRN P  Rf ŒRN M  Rf ˇP JM D  ˇP ˇP ˇP D TMP  ŒRN M  Rf

Ri D Rf Cˇi .RM Rf /

1.

ŒRN P  Rf ŒRN M  Rf .pm / JM D  p P P m m

JM D .Ri Rf /ˇi .RM Rf /

JMA D 37 percent JMB D 7 percent JMC D 1 percent JMD D 2 percent JME D 0 percent Ranked by the Jensen measure A > B > C > E > D. JMA D 32 percent JMB D 3 percent JMC D  2 percent JMD D 3 percent JME D 0:5 percent The rank change to A > B > E > C > D. JMA D 42 percent JMB D 11 percent JMC D 4 percent JMD D 1 percent JME D 0:5 percent The rank is A > B > C > D > E. JMA D 26 percent JMB D 1 percent JMC D 4 percent JMD D 3 percent JME D 2:5 percent The rank now is A > E > B > D > C.

Interrelationship among Three Performance Measure. It should be noted that all three performance measures are interrelated. For instance, if pm D pm =p m D 1, then the Jensen measure divided by p becomes equivalent to the Sharpe measure. Since ˇp D pm =m2 and pm D pm =p m the Jensen measure (JM) must be multiplied by 1=p to derive the equivalent Sharpe measure:

D TMP  commom constant Example 4.8 provides further illustration.

Example 4.8. Continuing with the example used for the Sharpe performance measure in Example 4.5, assume that in addition to the information already provided, the market return is 10%, the beta of the Smyth Fund is 0.8, and the Jones Fund beta is 1.1. Then, according to the capital asset pricing line, the implied return earned should be: RN Smyth D 0:095 C .0:10  0:095/.0:8/ D 0:099 RN Jones D 0:095 C .0:10  0:095/.1:1/ D 0:1005 Using the Jensen measure, the risk-return measurements for these two firms are: JMSmyth D 0:18  0:099/ D 0:081 JMJones D 0:16  0:1005 D 0:0595 From these calculations, it is clear that the Smyth Fund has a better performance and would be the better alternative of the two. Note that this is the opposite of the results determined from the Sharpe performance measure in Example 4.5. Computing the Treynor measure would reinforce the Jensen results. More analysis of these performance measures will be undertaken in detail in next chapters.

4.5 Determination of Commercial Lending Rate This section concerns a process for estimating the lending rate a financial institution would extend to a firm or the borrowing rate a firm would think is reasonable based on economic, industry, and firm-specific factors.

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As shown previously, part of the rate of return is based on the risk-free rate. The risk-free rate Rf must first be forecasted for three types of economic conditions – boom, normal, and poor. The second component of the lending rate is the risk premium .RP /. This can be calculated individually for each firm by examining the change in earnings before interest and taxes (EBIT) under the three types of economic conditions. The EBIT is used by the lender as an indicator of the ability of the potential borrower to repay borrowed funds. Table 4.2 has been constructed based on the methods discussed previously. In total there are nine possible lending rates under the three different economic conditions. The construction of these lending rates is shown in Table 4.3. This table shows that during a boom the risk-free rate is set at 12%, but the risk premium can taken on different values. These is a 40% chance that it will be 3.0%, a 30% chance it will be 5.0%, and a 30% chance it will be 8.0%. The products of the RP probabilities and the Rf probability are the joint probabilities of occurrence for the lending rates computed from these parameters. Therefore, there is a 10% chance that a firm will be faced with a 15% lending rate during a boom, a 7.5% chance of a 17% rate, and a 7.5% chance of an 18%

Table 4.2 Possible lending rates Economic conditions Rf (percent)

Probability

Boom

12.0

0.25

Normal

10.0

0.50

Poor

8.0

0.25

rate. This process applies for the other conditions, normal and poor, as well. Based upon the mean and variance Equations (4.1) and (4.2) it is possible to calculate the expected lending rate and its variance. Using the information provided in Table 4.3, the weighted average can be calculated: R D.0:100/.15%/ C .0:075/.17%/ C .0:075/.20%/ C .0:200/.13%/ C .0:150/.15%/ C .0:150/.18%/ C .0:100/.11%/ C .0:075/.13%/ C .0:075/.16%/ D15:1% With a standard deviation of: h  D .0:100/.15  15:1/2 C .0:075/.17  15:1/2 C .0:075/.20  15:1/2 C .0:200/.13  15:1/2 C .0:150/.5  15:1/2

EBIT ($ millions)

Probability

Rp (percent)

2.5 1.5 0.5 2.5 1.5 0.5 2.5 1.5 0.5

0.40 0.30 0.30 0.40 0.30 0.30 0.40 0.30 0.30

3 5 8 3 5 8 3 5 8

Table 4.3 Construction of actual lending rates Economic conditions

(A) Rf (percent)

(B) Probability

(C) Rp (percent)

(D) Probability

.B  D/ Joint probability of occurrence

.A C C/ Lending rate (percent)

Boom

12

0.25

Normal

10

0.50

Poor

8

0.25

3.0 5.0 8.0 3.0 5.0 8.0 3.0 5.0 8.0

0.40 0.30 0.30 0.40 0.30 0.30 0.40 0.30 0.30

0.100 0.075 0.075 0.200 0.150 0.150 0.100 0.075 0.075

15 17 20 13 15 18 11 13 16

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Fig. 4.7 Probability of Xi in the intervals ˙1; ˙2; ˙3

C .0:150/.18  15:1/2 C .0:100/.11  15:1/2 C .0:075/.13  15:1/2 i1=2 C .0:075/.16  15:1/2 D.0:001 C 0:271 C 1:801 C 0:882 C 0:0015 C 1:2615 C 1:681

away. The remaining risk is systematic risk that which influences all risky assets. The market rate of return can be calculated using one of several types of market indicator series, such as the DowJones Industrial Average, the Standard and Poor (S&P) 500, or the New York Stock Exchange Index, using the following equation: It  It 1 D Rmt It 1

C 0:331 C 0:061/ D2:51% If this distribution is indeed approximately normal, the mean and standard deviation can be employed to make some statistical inferences. Figure 4.7 makes it clear that 68.3% of the observations of a standard normal distribution are within one standard deviation of the mean, 95.4% are within three. Since the average lending rate is assumed to be normally distributed with a mean of 15.1% and a standard deviation of 2.51%, it is clear that almost all (99.7%) of the lending rates will lie in the range of 7.57–22.63%, because 7.57% is three standard deviations below 15.1 and 22.63% is three standard deviations above the mean. It is also clear that 68.3% of the rates will lie in the range of 12.59–17.61%.

4.6 The Market Rate of Return and Market Risk Premium The market rate of return is the return that can be expected from the market portfolio. This portfolio is of all risky assets – that is, stocks, bonds, real estate, coins, and so on. Because all risky assets are included, the market portfolio is a completely diversified portfolio. All unsystematic risks related to each individual asset would, therefore, be diversified

(4.9)

where: Rmt D market rate of return at time tI It D market index at tI and It 1 D market index at t  1: This equation calculates the percent change in the market index during period t and the previous period t  1. This change is the rate of return an investor would expected to receive in t had he or she invested in t  1. A risk-free investment is one in which the investor is sure about the timing and amount of income streams arising from that investment. However, for most types of investments, investors are uncertain about the timing and amount of income of their investments. The types of risks involved in investments can be quite broad, from the relatively riskless T-bills to highly risky speculative stocks. The reasonable investor dislikes risks and uncertainty and would, therefore, require an additional return on his investment to compensate for this uncertainty. This return, called the risk premium, is added to the nominal risk-free rate. The risk premium is derived from several major sources of uncertainty or risk, as was discussed at the beginning of this chapter. Table 4.4 illustrates this concept. In this table the market rate of return using the S&P 500 was calculated using

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Table 4.4 Market returns and T-bill by quarters Year

Month

S&P 500

2005 2006

Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1; 248:29 1; 280:08 1; 280:66 1; 294:87 1; 310:61 1; 270:09 1; 270:20 1; 276:66 1; 303:82 1; 335:85 1; 377:94 1; 400:63 1; 418:30 1; 438:24 1; 406:82 1; 420:86 1; 482:37 1; 530:62 1; 503:35 1; 455:27 1; 473:99 1; 526:75 1; 549:38 1; 481:14 1; 468:36 1; 378:55 1; 330:63 1; 322:70 1; 385:59 1; 400:38 1; 280:00 1; 267:38 1; 282:83 1; 164:74 968:75 896:24 887:88

2007

2008

Equation (4.9) to devise average monthly returns. Monthly T-bill rates are listed in the column B. The T-bill rate was deducted from the market return rate .Rm  Rf / to devise the risk premium. In the last 2 months of 2007 and for most months of 2007 the market was in decline, with low returns resulting in each month. This allowed the T-bill investors to obtain a higher than the market return and resulted in negative risk premiums. The second half of 2006 demonstrated an increasing market level and higher market returns. In October 1982 the return was 3.15%, the highest in the past 10 months. This al-

(A) Market return (percent) 2:55 0:05 1:11 1:22 3:09 0:01 0:51 2:13 2:46 3:15 1:65 1:26 1:41 2:18 1:00 4:33 3:25 1:78 3:20 1:29 3:58 1:48 4:40 0:86 6:12 3:48 0:60 4:75 1:07 8:60 0:99 1:22 9:21 16:83 7:48 0:93

(B) T-bill rate (percent)

(A–B) Risk premium (percent)

0.34 0.36 0.37 0.38 0.39 0.39 0.41 0.42 0.41 0.40 0.41 0.41 0.41 0.42 0.42 0.41 0.40 0.39 0.40 0.40 0.36 0.32 0.31 0.25 0.27 0.17 0.14 0.12 0.12 0.15 0.15 0.14 0.14 0.07 0.04 0.01

2:21 0:32 0:74 0:84 3:48 0:38 0:10 1:71 2:05 2:75 1:23 0:85 0:99 2:60 0:58 3:92 2:86 2:17 3:60 0:89 3:22 1:16 4:71 1:11 6:38 3:65 0:74 4:64 0:95 8:75 1:14 1:08 9:35 16:90 7:53 0:94

lowed market rates to leap beyond the riskless T-bill rate, and the result was a positive risk premium. During the period of fluctuation in the level of stock market prices, the first half of 2007 revealed a fluctuated risk premium. Theoretically it is not possible for a risk premium required by investors to be negative. Taking on risk involves some positive cost. Nevertheless, using short-run estimators as in Table 4.4 may result in negative figures because they reflect the fluctuations of the market. The basic problem with using actual market data to assess risk premiums is the difference between expected returns (which are always positive)

68

and realized returns (which may be positive or negative). It becomes evident that investors’ expectations will not always be realized.

4.7 Conclusion This chapter has defined the basic concepts of risk and risk measurement. The efficient-portfolio concept and its implementation was demonstrated using the relationships of risk and return. The dominance principle and performance measures were also discussed and illustrated. Finally, the interest rate and market rate of return were used as measurements to show how the commercial lending rate and the market risk premium can be calculated. Overall, this chapter has introduced uncertainty analysis assuming previous exposure to certainty concepts. Further application of the concepts discussed in this chapter as related to security analysis and portfolio management are explored in later chapters.

References Ben-Horin, M. and H. Levy. 1980. “Total risk, diversifiable risk and non-diversifiable risk: a pedagogic note.” Journal of Financial and Quantitative Analysis 15, 289–295. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Bowman, R. G. 1979. “The theoretical relationship between systematic risk and financial (accounting) variables.” Journal of Finance 34, 617–630. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Evans, J. L. and S. H. Archer. 1968. “Diversification and the reduction of dispersion: an empirical analysis.” Journal of Finance 23, 761–767.

C.-F. Lee et al. Francis, J. C. and S. H. Archer. 1979. Portfolio analysis, Prentice-Hall, Englewood Cliffs, NJ. Ibbotson, R. G. and R. A. Sinquefield. 1976. “Stocks, bonds, bills, and inflation: simulations of the future (1976–2000).” Journal of Business 49, 313–338. Jensen, M. C. 1968. “The performance of mutual funds in the period 1945–1964.” Journal of Finance 23, 389–416. Jensen, M. C. 1969. “Risk, the pricing of capital assets, and the evaluation of investment portfolios.” Journal of Business 42, 167–185. Lee, C. F. and S. N. Chen. 1981. “The sampling relationship between Sharpe’s performance measure and its risk proxy: sample size, investment horizon and market conditions.” Management Science 27(6), 607–618. Lee, C. F. and S. N. Chen. 1984. “On the measurement errors and ranking of composite performance measures.” Quarterly Review of Economics and Business 24, 6–17. Lee, C. F. and S. N. Chen. 1986. “The effects of the sample size, the investment horizon and market conditions on the validity of composite performance measures: a generalization,” Management Science 32(11), 1410–1421. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, New Jersey. Markowitz, H. M. 1959. Portfolio selection: efficient diversification of investments, Wiley, New York, NY. Modigliani, F. and G. A. Pogue. 1974. “An introduction to risk and return.” Financial Analysis Journal 30, 69–86. Robicheck, A. A. and R. A. Cohn. 1974. “The economic determinants of systematic risk.” Journal of Finance 29, 439–447. Schall, L. D. 1972. “Asset valuation, firm investment, and firm diversification.” Journal of Business 45, 11–28. Sharpe, W. F. 1966. “Mutual fund performance.” Journal of Business 39, 119–138. Thompson, D. J. 1976. “Sources of systematic risk in common stocks.” Journal of Business 49, 173–188. Tobin, J. 1958. “Liquidity preference as behavior toward risk.” Review of Economic Studies 25, 65–86. Treynor, J. 1965. “How to rate management of investment funds.” Harvard Business Review 43, 63–75. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Chapter 5

Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model Cheng-Few Lee, Joseph E. Finnerty, and Hong-Yi Chen

Abstract In this chapter, we first introduce utility function and indifference curve. Based on utility theory, we derive the Markowitz’s model and the efficient frontier through the creation of efficient portfolios of varying risk and return. We also include methods of solving for the efficient frontier both graphically and mathematically, with and without explicitly incorporating short selling. Keywords Markowitz model r Utility theory r Utility functions r Indifference curve r Risk averse r Short selling r Dyl model r Iso-return line r Iso-variance ellipse r Critical line r Lagrange multipliers

5.1 Introduction In this chapter, we address basic portfolio analysis concepts and techniques discussed in the Markowitz portfolioselection model and other related issues in portfolio analysis. Before Harry Markowitz (1952, 1959) developed his portfolio-selection technique into what is now modern portfolio theory (MPT), security-selection models focused primarily on the returns generated by investment opportunities. The Markowitz theory retained the emphasis on return, but it elevated risk to a coequal level of importance, and the concept of portfolio risk was born. Whereas risk had been considered an important factor and variance an accepted way of measuring risk, Markowitz was the first to clearly and rigorously show how the variance of a portfolio can be reduced through the impact of diversification. He demonstrated that by combining securities that are not perfectly positively correlated into a portfolio, the portfolio variance can be reduced. The Markowitz model is based on several assumptions concerning the behavior of investors:

1. A probability distribution of possible returns over some holding period can be estimated by investors. 2. Investors have single-period utility functions in which they maximize utility within the framework of diminishing marginal utility of wealth. 3. Variability about the possible values of return is used by investors to measure risk. 4. Investors use only expected return and risk to make investment decisions. 5. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance. 6. Return is desirable; risk is to be avoided. It follows, then, that a security or portfolio is considered efficient if there is no other investment opportunity with a higher level of return at a given level of risk and no other opportunity with a lower level of risk at a given level of return.

5.2 Measurement of Return and Risk This section focuses on the return and risk measurements utilized in applying the Markowitz model to efficient portfolio selection.

5.2.1 Return Using the probability distribution of expected returns for a portfolio, investors are assumed to measure the level of return by computing the expected value of the distribution. E.RP / D

H.-Y. Chen () and C.-F. Lee Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

n X

Wi E .Ri /

(5.1)

i D1

where:

J.E. Finnerty University of Illinois, Urbana-Champaign, IL, USA e-mail: [email protected] C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_5, 

n X

Wi D 1:0I

i D1

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n D the number of securities; Wi D the proportion of the funds invested in security i ; Ri , RP D the return on i th security and portfolio p; and E . / D the expectation of the variable in the parentheses.

Since rAA and rBB D 1:0 by definition, terms can be simplified and rearranged, yielding:

Thus, the return computation is nothing more than finding the weighted average return of the securities included in the portfolio. The risk measurement to be discussed in the next section is not quite so simple, however. For only in the case of perfect positive correlation among all its components is the standard deviation of the portfolio equal to the weighted average standard deviation of its component securities.

For a three-security portfolio, the variance of portfolio can be defined:

5.2.2 Risk Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation. The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio combination of securities is equal to the weighted average covariance of the returns on its individual securities: n n X     X Wi Wj Cov Ri ; Rj Var Rp D

(5.2)

i D1 j D1

Covariance can also be expressed in terms of the correlation coefficient as follows:   Cov Ri ; Rj D rij i j D ij

(5.3)

Where rij D correlation coefficient between the rates of return on security i; Ri , and the rates of return on security j; Rj , and i , and j represent standard deviations of Ri and Rj respectively. Therefore: n n X   X Wi Wj rij i j Var Rp D

(5.4)

i D1 j D1

For a portfolio with two securities, A and B, the following expression can be developed: B B X   X Var Rp D Wi Wj rij i j i DA j DA

D WA WA rAA A A C WA WB rAB A B CWB WA rBA B A C WB WB rBB B B

  Var Rp D WA2 A2 C WB2 B2 C 2WA WB rAB A B

(5.5)

C P C   P Var Rp D Wi Wj rij i j i DA j DA

D WA WA rAA A A C WA WB rAB A B CWA WC rAC A C C WB WA rAB B A CWB WB rBB B B C WB WC rBC B C CWC WA rCA C A C WC WB rCB C B CWC WC rC C C C Again simplifying and rearranging yields:   Var Rp D WA2 A2 C WB2 B2 C WC2 C2 C 2.WA WB rAB A B CWA WC rAC A C C WB WC rBC B C / (5.6) Thus, as the number of securities increases from three to two, there is one more variance term and two more covariance terms. The general formula for determining the number of terms that must be computed (NTC) to determine the variance of a portfolio with N securities is NTC D N variancesC

N2  N covariances 2

For the two-security example two variances were involved  ı A2 and B2 , and 22  2 2 D .4  2/=2 D 1 covariances, AB , or three computations. ı For the three-security case, three  variances and 32  3 2 D .9  3/=2 D 3 covariances were needed, AB ; AC , and BC , or six computations. ıWith  four securities, the total number would be 4 C 42  4 2 D 4 C .16  4/=2 D 4 C 6 D 10 computations. As shown from these examples, the number of covariance terms increases by N  1, and the number of variance terms increases by one, as N increases by 1 unit. If rAB D rAC D rBC D 1, then the securities are perfectly positively correlated with each other, and Equation (5.6) reduces to:   Var Rp D .WA A C WB B C WC C /2

(5:60 )

This implies that the standard deviation of the portfolio is equal to the weighted average standard deviation of its component securities. In other words: Standard deviation of Rp D WA A C WB B C WC C

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

Table 5.1 Portfolio size and variance computations Number of securities Number of Var and Cov terms 2 3 4 5 10 15 20 25 50 75 100 250 500

3 6 10 15 55 120 210 325 1; 275 2; 850 5; 050 31; 375 125; 250

Table 5.1 clearly illustrates the tremendous estimation and computational load that exists using the Markowitz diversification approach. Later chapters will illustrate how this problem can be alleviated.

5.3 Utility Theory, Utility Functions, and Indifference Curves In this section, utility theory and functions, which are needed for portfolio analysis and capital asset models, will be discussed in detail. Utility theory is the foundation for the theory of choice under uncertainty. Following Henderson and Quandt (1980), cardinal and ordinal theories are the two major alternatives used by economists to determine how people and societies choose to allocate scarce resources and to distribute wealth among one another over time.1

5.3.1 Utility Functions Economists define the relationships between psychological satisfaction and wealth as “utility.” An upward-sloping relationship, as shown in Fig. 5.1, identifies the phenomena of increasing wealth and increasing satisfaction as being directly related. These relationships can be classified into linear, concave, and convex utility functions. 1

A cardinal utility implies that a consumer is capable of assigning to every commodity or combination of commodities a number representing the amount or degree of utility associated with it. An ordinary utility implies that a consumer needs not be able to assign numbers that represent (in arbitrary unit) the degree or amount of utility associated with commodity or combination of commodity. The consumer can only rank and order the amount or degree of utility associated with commodity.

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In Fig. 5.1a, for each unit change in wealth, there is a linear utility function and equal increase in satisfaction or utility. A doubling of wealth will double satisfaction, and so on. If an investor’s utility function is a linear utility function, we call this kind of investor a risk-neutral investor. This is probably not very realistic: a dollar increase in wealth from $2 to $1 is probably more important than an increase from $2 million to $1 million, because the marginal utility diminishes with increased wealth. In Fig. 5.1b the concave utility function shows the relationship of an increase in wealth and a less-than-proportional increase in utility. In other words, the marginal utility of wealth decreases as wealth increases. As mentioned above, the $1 increase from $2 to $1 of wealth is more important to the individual than the increase from $1,000,001 to $1 million. Each successive increase in wealth adds less satisfaction as the level of wealth rises. We call an investor with a concave utility function a risk-averse investor. Finally, Fig. 5.1c is a convex utility function, which denotes a more than proportional increase in satisfaction for each increase in wealth. Behaviorally, the richer you are the more satisfaction you receive in getting an additional dollar of wealth. Investors with convex utility functions are called risk-seeking investors. The utility theory primarily used in finance is that developed by Von Neumann and Morgenstern (VNM 1947). VNM define investor utility as a function of rates of return or wealth. Mao (1969) points out that the VNM utility theory is really somewhere between the cardinal and ordinal utility theories. The function associated with the VNM’s utility theory in terms of wealth can be defined: U D f .w ; w / where w indicates expected future wealth and w represents the predicted standard deviation of the possible divergence of actual future wealth from w . Investors are expected to prefer a higher expected future wealth to a lower value. Moreover, they are generally risk averse as well. That is, they prefer a lower value of w to a higher value, given the level of w .2 These assumptions imply that the indifference curves relating w and w will be upward sloping, as indicated in Fig. 5.2. In Fig. 5.2, each indifference curve is an expected utility isoquant showing all the various combinations of risk and return that provide an equal amount of expected utility for the investor. In explaining how investment decisions or portfolio choices are made, utility theory is used here not to imply that individuals actually make decisions using a utility curve, but rather as an expository vehicle that helps explain how 2

Technically, these conditions can be represented mathematically by @U =@w > 0 and @U =@w < 0.

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Fig. 5.1 Utility functions

utilities of the various possible returns. The weights are the probabilities of occurrence associated with each of the possible returns. It is calculated by the following formula: E .U / D

n X

U .wi /Pi

(5.7)

i

where: E .U / D expected utility; U .wi / D the utility of the i th outcome wi ; and Pi D the Probability of the i th outcome. Example 5.1 provides further illustration. Example 5.1. Given investments A and B as shown in the table, determine the utilities of A and B for the given utility functions. Fig. 5.2 Indifference curves of utility functions

A Outcome wi 10 5 1

Probability 2/5 2/5 1/5

B Outcome wi 9 3

Probability investors presumably act. In general, humans behave as if 2/3 more is better than less (the utility curve is upward slop1/3 ing) and marginal utility is decreasing (the utility curve is concave). In an uncertain environment it becomes necessary to ascertain how different individuals will react to risky situ- 1. U.w/ D wi ations. The risk is defined as the probability of success or 2. U.w/ D w2i failure. Alternatively, risk could be described as variability 3. U.w/ D w2i  wi of outcomes, payoffs, or returns. This implies that there is a distribution of outcomes associated with each investment Solution decision. What is needed is a linkage between utility or expected utility and risk. Expected utility has been defined as 1. For U.w/ D wi the numerical value assigned to the probability distribution associated with a particular portfolio’s return. This numeriUtility A D 25 .10/ C 25 .5/ C 15 .1/ D 6 51 cal value is calculated by taking a weighted average of the Utility B D 23 .9/ C 13 .3/ D 7

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

73

Fig. 5.3 Risk-neutral investors and fair games

2. For U.w/ D w2i Utility A D 25 .100/ C 25 .25/ C 15 .1/ D 50 51 Utility B D 23 .81/ C 13 .9/ D 57 3. U.w/ D w2i  wi (use results from 1 and 2) Utility A D50 51  6 15 D 44 Utility B D57  7 D 50 In all three cases, B has the higher degree of utility because it has a higher expected value as well as a smaller dispersion than A. Linear utility function and risk. It is useful now to consider how the shape of an individual’s utility function affects his or her reaction to risk. Assume that an individual who has $5,000 and whose behavior is a linear utility function (Fig. 5.1a) is offered a chance to gain $10,000 with a probability of half or to lose $10,000 with a probability of half. What should he or she pay for such an opportunity? The answer is nothing, for as shown in Fig. 5.3, this individual would be no better or worse off accepting or rejecting this opportunity. If he rejected the offer, his wealth would be $5,000 with utility U1 ; if he paid nothing for the opportunity, his wealth would remain as $5,000 with Utility U1 . Any payment for this chance would reduce his wealth and therefore be undesirable. This is so because the expected value of the fair game is zero: 1 2

.10; 000/ C 12 .10; 000/ D 0

Figure 5.3 illustrates this linear utility function concept. In the following section we will analyze the implication of concave utility function. Concave utility function and risk. Now consider an individual whose behavior is a concave utility function. If this individual participates and wins, her utility is shown by point UW in Fig. 5.4. But if she loses, her position is shown by UL . The expected value of this fair game, having a 50% chance of winning and a 50% chance of losing, is shown by point A.3 The utility of the fair game is UF . A comparison of UF with her initial position, Ui , shows that the investor should not accept this fair game. As shown in Fig. 5.4 the utility of winning .UW  Ui / is less than the utility of losing .Ui  UL /. Therefore, the utility of doing nothing is greater than the expected utility of accepting the fair game. In fact, the individual should be willing to pay up to the difference between the utility of winning and the utility of losing .Ui  UL /  .UW  Ui / to avoid being involved in this situation. Alexander and Francis (1986) theoretically analyze this issue in more detail. Hence investors with concave utility functions are said to be risk averse. That is, they would reject a fair game because the utility derived from winning is less than the utility lost should they lose the game. In other words, the expected utility of participating in a fair game is negative. Example 5.2 will provide illustration of linear, concave, and convex utility function cases for different investment decisions. 3

Following Fama (1970) and Alexander and Francis (1986:177), a fair game means that the expected returns, given information set  , equal the expected returns without the information set. Note that this does not mean the expected returns are zero or positive – they could be negative.

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For a risk-averse investor QUi > E (UF) Þ E(UF)-Ui < 0 \Reject a fair game

Uw -Ui =

Ui -UL =

E(UF)

E(Ul) Concave Utility Function Linear Utility Function

Fig. 5.4 Risk-averse investors and fair games

E.UF / D 12 .Uw C UL / is the expected utility of the fair game under a concave utility function. E.U` / is the expected utility of the fair game under a linear utility function.

This implies the investor is risk averse and would reject a fair gamble. 3.

u.w/ D e 2w u .w/ D 2e 2w 00 u .w/ D 4e 2w < 0 0

Example 5.2. Given the following utility functions for four investors, what can you conclude about their reaction towards a fair game? 1. 2. 3. 4.

u.w/ D w C 4 u.w/ D w  12 w2 (quadratic utility function) u.w/ D e 2w (negative exponential utility function), and u.w/ D w2  4w

Evaluate the second derivative of the utility functions according to the following rules. 00

u .w/ < 0 implies risk averse 00 u .w/ D 0 implies risk neutral 00 u .w/ > 0 implies risk seeker Solution 1.

u.w/ D w C 4 u0 .w/ D 1 00 u .w/ D 0 This implies risk neutrality; it is indifference for the investor to accept or reject a fair gamble.

2.

u.w/ D w  12 w2 u0 .w/ D w2 00 u .w/ D 2w < 0 .We assume wealth is nonnegative:/

This implies risk adversity; the investor would reject a fair gamble. 4.

u.w/ D w2  4w u .w/ D 2w  4 00 u .w/ D 2 > 0 0

This implies risk preference; the investor would seek a fair gamble. The convex utility function is not realistic in real-world decisions; therefore, it is not further explored at this point. The following section discusses the implications of alternative utility functions in terms of indifference curves.

5.3.2 Risk Aversion and Utility Values But when risk increases along with return, the most attractive portfolio is not obvious. How can investors quantify the rate at which they are willing to trade off return against risk? By assuming that each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of those portfolios. Higher utility values are assigned to portfolios with more attractive risk-return

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

profiles. Portfolios receive higher utility scores for higher expected returns and lower scores for higher volatility. Assigning a portfolio with expected return and variance of returns, the following utility score: U D E.r/ 

1

A s2 2

Where U D utility E.r/ D expected return on the asset or portfolio A D coefficient of risk aversion s 2 D variance of returns Example: Consider three investors with different degrees of risk aversion: A1 D 2:0; A2 D 3:0, and A3 D 4:0, all of whom are evaluating the three portfolios in Table 5.2. Because the risk-free rate is assumed to be 5%, utility score implies that all three investors would assign a utility score of 0.05 to the risk-free alternative. Table 5.2 presents the utility scores that would be assigned by each investor to each portfolio. The portfolio with the highest utility score for each investor appears in bold. Notice that the high-risk portfolio, H, would be chosen only by the investor with the lowest degree of risk aversion, A1 D 2:0, while the low-risk portfolio, L. would be passed over even by the most risk-averse of our three investors. All three portfolios beat the risk-free alternative for the investors with levels of risk aversion given in Table 5.2.

5.3.3 Capital Allocation Across Risky and Risk-Free Portfolios From the previous discussion we know that the riskier investments offer higher average returns, while less risky investments offers lower average returns. Therefore, a rational investor makes optimal portfolio choice between risky and risk-free securities instead of making all-or-nothing choices from these investment classes. They can and do construct their portfolios using securities from all asset classes. For example, some of the portfolios may be in risk-free Treasury bills, some in high-risk stocks.

75

The most straightforward way to control the risk of the portfolio is through the fraction of the portfolio invested in Treasury bills and other safe money market securities versus risky assets. The capital allocation decision is an example of an asset allocation choice – a choice among broad investment classes, rather than among the specific securities within each asset class. Most investment professionals consider asset allocation the most important part of portfolio construction. Example 5.3. Assume that the total market value of a private fund is $500,000, of which $100,000 is invested in a risk-free asset for practical purposes. The remaining $400,000 is invested in risky securities, where $240,000 in equities (E) and $160,000 in long-term bonds (B). Under such assumption, the risky portfolio consists 60% of E and 40% of B, and the weight of the risky portfolio in the mutual fund is 80%. Suppose that the fund manager wishes to decrease risk by reducing the allocation to the risky portfolio from 80 to 70% and not change the proportion of each asset in the risky portfolio. The risky portfolio would then total only 0:7 $500; 000 D $350; 000, requiring the sale of $50,000 of the original $400,000 of risky holdings, with the proceeds used to purchase more shares in risk-free asset. Total holdings in the risk-free asset will increase $500; 000 .10:7/ D $150; 000, the original holdings ($100,000) plus the new contribution ($50,000). To leave the proportions of each asset in the risky portfolio unchanged. Because the weights of E and B in the risky portfolio are 60 and 40%, respectively, the fund manager should sell 0:6 $50; 000 D $30; 000 of E and 0:4 $50; 000 D $20; 000 of B. After the sale, the proportions of each asset in the risky portfolio are in fact unchanged: E W WE D

240; 000  30; 000 D 0:6 400; 000  50; 000

B W WB D

160; 000  20; 000 D 0:4 400; 000  50; 000

5.3.4 Indifference Curves Indifference (utility-function) curves are abstract theoretical concepts. They cannot as a practical matter be used to actually measure how individuals make investment deci-

Table 5.2 Utility scores of alternative portfolios for investors with varying degrees of risk aversion Investor Utility score portfolio L Utility score portfolio L Utility score portfolio L risk aversion (A) E.r/ D :07I  D :05 E.r/ D :09I  D :1 E.r/ D :13I  D :2 2.0

:07 

3.0

:07 

4.0

:07 

1 2 1 2 1 2

 2  :052 D :0675  3  :052 D :0663 2

 4  :05 D :0650

1  2  :12 2 :09  12  3  12 :09  12  4  :12

:09 

D :0800

:13 

D :0750

:13 

D :0700

:13 

1 2 1 2 1 2

 2  :22 D :0900  3  :22 D :0700  4  :22 D :0500

76 Fig. 5.5 Indifference curves for various types of individuals

C.-F. Lee et al.

a

b

Risk-Averse investor d

c

Risk-Seeking Investor sions – or any other decisions for that matter. They are, however, useful tools for building models that illustrate the relationship between risk and return. An investor’s utility function can be utilized conceptually to derive an indifference curve, which shows individual preference for risk and return.4 An indifference curve can be plotted in the risk-return space such that the investor’s utility is equal all along its length. The investor is indifferent to various combinations of risk and return, hence the name indifference curve. Various types of investor’s indifference curves are shown in Fig. 5.5. In the same level of satisfaction, the risk-averse investor requires more return for an extra unit of risk than the return he asks for previous one unit increase of risk. Therefore, the indifference curve is convex. Figure 5.5a presents two indifference curves that are risk-averse and U1 > U2 (higher return and lower risk). For a risk-neutral investor, the indifference curve is a straight line. In the same level of satisfaction, the investor requires the same return for an extra unit of risk as the return he asks for previous one unit increase of risk. Figure 5.5b presents two indifference curves that are

4

Risk-Neutral Investor

By definition, an indifference curve shows all combinations of products (investments) A and B that will yield same level of satisfaction or utility to consume. This kind of analysis is based upon ordinal rather than cardinal utility theory.

Level of Risk-Aversion

risk-neutral, and U1 > U2 . For a risk-seeking investor, the indifference curve is a concave function. In the same level of satisfaction, the investor requires less return for an extra unit of risk than the return he asks for previous one unit increase of risk. Figure 5.5c presents two indifference curves that are risk-seeking and U1 > U2 . The more risk-averse individuals will claim more premiums when they face uncertainty (risk). In Fig. 5.5d, when facing 0 , investor 1 and investor 2 have the same expected return, E .R0 /. However, when the risk adds to 1 , the expected return E .R1 / investor 1 asks is higher than the expected return E .R2 / investor 2 asks. Thus, investor 1 is more risk-averse than investor 2. Therefore, in return-risk plane, the larger slope of indifference curve that the investor has, the higher risk-averse level the investor is. Later in this chapter, different levels of risk and return are evaluated for securities and portfolios when a decision must be made concerning which security or portfolio is better than another. It is at that point in the analysis that indifference curves of hypothetical individuals are employed to help determine which securities or portfolios are desirable and which ones are not. Basically, values for return and risk will be plotted for a number of portfolios as well as indifference curves for different types of investors. The investor’s optimal portfolio will then be the one identified with the highest level of utility for the various indifference curves.

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

Investors generally hold more than one type of investment asset in their portfolio. Besides securities an investor may hold real estate, gold, art, and so on. Thus, given the measures of risk and return for individual securities developed, the measures of risk and return may be used for portfolios of risky assets. Risk-averse investors hold portfolios rather than individual securities as a means of eliminating unsystematic risk; hence the examination of risk and return will continue in terms of portfolios rather than individual securities. Indifference curves can be used to indicate investors’ willingness to trade risk for return; now investors’ ability to trade risk for return needs to be represented in terms of indifference curves and efficient portfolios, as discussed in the next section.

5.4 Efficient Portfolios Efficient portfolios may contain any number of asset combinations. Two examples are shown, a two-asset combination and a three-asset portfolio; both are studied from graphical and mathematical solution perspectives. The degree to which a two-security portfolio reduces variance of returns depends on the degree of correlation between the returns of the securities. This can be best illustrated by expressing the variability in terms of standard deviation (the square root of the variance): p D

q

WA2 A2 C WB2 B2 C 2WA WB rAB A B

(5.8)

First, assume that rAB D 1:0, which would mean that securities A and B are perfectly positively correlated. Then: p D

q

WA2 A2 C WB2 B2 C 2WA WB A B

or:

q p D

.WA A C WB B /2

so: p D WA A C WB B

(5.9)

With the securities perfectly positively correlated, the standard deviation of the portfolio combination is equal to the weighted average of the standard deviation of the component securities. Since the correlation coefficient cannot be greater than 1.0, the weighted average represents the highest possible values of the portfolio standard deviation as discussed previously in this chapter. In this case, there is no diversification taking place. With any correlation coefficient less than 1.0, there will be a diversification effect, and this effect will be larger the lower the value of the correlation coefficient. The ultimate diversification impact occurs if rAB D 1:0, perfect negative correlation.

p D

77

q

WA2 A2 C WB2 B2  2WA WB A B

This can be reduced to: q p D .WA A  WB B /2 D jWA A  WB B j

(5.10)

Example 5.3 provides further illustration. Example 5.4. Two-security portfolio diversification. Assume that RA D 10 percent; RB D 8 percent; A D 4 percent, and B D 3 percent. Since the values for both RA and A are greater than those for RB and B , respectively, there is no dominant choice between the two securities. Higher return is associated with higher risk. Additionally, assume that rAB D 1:0 and that the securities are equally weighted in the portfolio. If follows then that: q .0:5/2 .4/2 C .0:5/2 .3/2 C .2/ .0:5/ .0:5/ .1:0/ .4/ .3/ p D .0:25/ .16/ C .0:25/ .9/ C .2/ .0:25/ .12/ p D 4 C 2:25 C 6 p D 12:25

p D

D 3:5

Notice that this is the same result that would have been achieved using Equation (5.9), p D .0:5/ .4/ C .0:5/ .3/ D 2 C 1:5 D 3:5 percent. If rAB D 0:5 there still is positive correlation, but as it is not perfect positive correlation, there is a diversification effect. q .0:5/2 .4/2 C .0:5/2 .3/2 C .2/ .0:5/ .0:5/ .0:5/ .4/ .3/ p D 4 C 2:25 C 3 p D 9:25 D 3:04

p D

This reduction of p from 3.50 to 3.04 is the effect of diversification. Notice that the first two terms under the square root radical were unaffected and that the third term was only half the size it was with no diversification. The diversification impact from a lower correlation coefficient occurs in the covariance term only. If security A and B were independent of one another, that is, rAB D 0, it is clear that the covariance term would equal zero. p p D p 4 C 2:25 C .2/ .0:5/ .0:5/ .0/ .4/ .3/ D p4 C 2:25 C 0 D 6:25 D 2:5 and the diversification impact reduces p to 2.50. If security A and B were perfectly and negatively correlated, that is, AB D 1, substituting the numbers from the example, first into Equation (5.8), gives

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p p D p 4 C 2:25 C .2/ .0:5/ .0:5/ .1:0/ .4/ .3/ D p6:25  6 D 0:25 D 0:5 and into Equation (5.10): p D .0:5/ .4/  .0:5/ .3/ D 2  1:5 D 0:5 with the appropriate weighting factors, the p may be reduced to zero if there is negative correlation. For example, if WA D 3=7 and WB D 4=7 in this problem: p D .3=7/ .4/  .4=7/ .3/ D0

Fig. 5.6 The relationship between correlation and portfolio risk

5.4.1 Portfolio Combinations Assume that the actual correlation coefficient is 0.50. By varying the weights of the two securities and plotting the combinations in the risk-return space, it is possible to derive a set of portfolios (see Table 5.3) that form an elliptical curve (see Fig. 5.6). The amount of curvature in the ellipse varies with the degree of correlation between the two securities. If there were perfect positive correlation, all of the combinations would lie on a straight line between points A and B. If the combination were plotted with the assumption of perfect negative correlation, the curvature would be more pronounced and one of the points .WA D 0:43; WB D 0:57/ would actually be on the vertical axis (a risk of zero). This process could be repeated for all combination of investments; the result is graphed in Fig. 5.7. In Fig. 5.7 the area within curve XVYZ is the feasible opportunity set representing all possible portfolio combinations. The curve YV represents all possible efficient portfolios and is the efficient frontier. The line segment VX is on the feasible opportunity set, but not on the efficient frontier; all points on VX represent inefficient portfolios. The portfolios and securities that lie on the frontier VX in Fig. 5.7 would not be likely candidates for investors to hold. This is so because they do not meet the criteria of maximizing expected return for a given level of risk or minimizing risk Table 5.3 Possible risk-return combination .RA D 0:10; RB D 0:08; A D 0:04; B D 0:03; AB D 0:5/ Portfolio WA WB RP (percent) P (percent) 1 2 3 4 5

1:00 0:75 0:50 0:25 0:00

0:00 0:25 0:50 0:75 1:00

10:0 9:5 9:0 8:5 8:0

4:00 3:44 3:04 2:88 3:00

Fig. 5.7 The minimum-variance set

for a given level of return. This is easily seen by comparing the portfolio represented by points X and X 0 . As investors always prefer more expected return than less for a given level of risk, X 0 is always better than X . Using similar reasoning, investors would always prefer V to X because it has both a higher return and a lower level of risk. In fact, the portfolio at point V is identified as the minimum-variance portfolio, because no other portfolio exists that has a lower variance. For each of the combinations of individual securities and inefficient portfolios in Fig. 5.7 there is a corresponding portfolio along the efficient frontier that either has a higher return given the same risk or a lower risk given the same return. However, points on the efficient frontier do not dominate one another. While point Y has considerably higher return than point V , it also has considerably higher risk. The opti-

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

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number of securities making up the portfolios will result in a function similar to that illustrated in Fig. 5.9. As shown, the additional reduction in portfolio variance rapidly levels off as the number of securities is increased beyond five. An earlier study by Evans and Archer (1968) shows that the percentage of diversification that can be achieved with randomly selected, equally weighted portfolios level off rapidly beyond a portfolio size of about 15.

5.4.2 Short Selling

Fig. 5.8 Indifference curves and the minimum-variance set

mal portfolio along the efficient frontier is not unique with this model and depends upon the risk/return tradeoff utility function of each investor. Portfolio selection, then, is determined by plotting investors’ utility functions together with the efficient-frontier set of available investment opportunities. No two investors will select the same portfolio except by chance or if their utility curves are identical. In Fig. 5.8, two sets of indifference curves labeled U and U 0 are shown together with the efficient frontier. The U curves have a higher slope, indicating a greater level of risk aversion. The investor is indifferent to any combination of Rp and p along a given curve, for example, U1 ; U2 , or U3 . The U 0 curve would be appropriate for a less risk-averse investor – that is, one who would be willing to accept relatively higher risk to obtain higher levels of return. The optimal portfolio would be the one that provides the highest utility – a point in the northwest direction (higher return and lower risk). This point will be at the tangent of a utility curve and the efficient frontier. The tangency point investor in Fig. 5.8 is point X 0 ; for the risk-averse it is point Y . Each investor is logically selecting the optimal portfolio given his or her risk-return preference, and neither is more correct than the other. Individual investors sometimes find it necessary to restrict their portfolios to include a relatively small number of securities. Mutual-fund portfolios, on the other hand, often contain securities from more than 500 different companies. To give some idea of the number of securities that is necessary to achieve a high level of diversification, Levy and Sarnat (1971) considered a naïve strategy of equally weighted portfolios – that is, of dividing the total investment into equal proportions among component securities. Plotting the variance of the portfolios developed in this manner against the

Short selling (or “going short”) is a very regulated type of market transaction. It involves selling shares of a stock that are borrowed in expectation of a decline in the security’s price. When and if the price declines, the investor buys an equivalent number of shares of the same stock at the new lower price and returns to the lender the stock that was borrowed. The Federal Reserve Board requires short selling customers to deposit 50% of the net proceeds of such short sales with the brokerage firm carrying out the transaction. Another key requirement of a short sale, set by the Securities and Exchange Act of 1934, is that the short sale must occur at a price higher than the preceding sale – or at the same price as the preceding sale, if that took place at a higher price than a preceding price. This is the so-called uptick or zero-tick rule. It prevents the price of a security from successively falling because of continued short selling. Relaxing the assumption of no short selling in this development of the efficient frontier involves a modification of the analysis of the previous section. The efficient frontier analyzed in the previous sections was bounded on both ends by Y and the minimum variance portfolio V , respectively, as shown in Fig. 5.7. Point Y is called the maximum-return portfolio, as there is no other portfolio with a higher return. This point is normally an efficient security or portfolio with the greatest level of risk and return. It could also be a portfolio of securities, all having the same highest levels of risk and return. Point Z is normally a single security with the lowest level of return, although it could be a portfolio of securities, all having the same low level of return. The Black (1972) model is identical to the Markowitz model except that it allows for short selling.5 That is, the nonnegativity constraint on the amount that can be invested in each security is relaxed, WA ? 0. A negative value for the weight invested in a security is allowed, tantamount to allowing a short sale of the security. The new efficient frontier that can be derived with short selling is shown in Fig. 5.10a.

5

Most texts do not identify the Markowitz model with restrictions on short sale. Markowitz (1952), in fact, excluded short sales.

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Fig. 5.9 Native diversification reduces risk to the systematic level in a randomly selected portfolio

The major difference between the frontier in Fig. 5.10a (short selling) and Fig. 5.10b (no short selling) is the disappearance of the end points Y and Z. An investor could sell the lowest-return security .Y /. If the number of short sales is unrestricted, then by a continuous short selling of X and reinvesting in Y the investor could generate an infinite expected return. Hence the upper bound of the highest-return portfolio would no longer be Y but infinity (shown by the arrow on the top of the efficient frontier). Likewise the investor could short sell the highest-return security U and reinvest the proceeds into the lowest-yield security X , thereby generating a return less than the return on the lowest-return security. Given no restriction on the amount of short selling, an infinitely negative return can be achieved, thereby removing the lower bound of X on the efficient frontier. But rational investors will not short sell a high-return stock and buy a low-return stock and buy a low-return stock. The portfolios on VZ 0 always dominate those of VX, as shown in Fig. 5.10a. Whether an investor engages in any of this short-selling activity depends on the investor’s own unique set of indifference curves. Hence, short selling generally will increase the range of alternative investments from the minimum-variance portfolio to plus or minus infinity. However, in the Black model with short selling, no provision was made for the SEC margin requirement. Dyl (1975) imposed the margin requirement on short selling and added it to the Markowitz development of the efficient frontier. The Dyl model. Dyl introduced short selling with margin requirements by creating a new set of risky securities, the ones sold short, which are negatively correlated with the existing set of risky securities. These new securities greatly enhance the diversification effect when they are placed in portfolios. The Dyl model affects the efficient frontier in two ways: (1) If the investor were to combine in equal weight any long position in a security or portfolio with a short position in a security or a portfolio, the resulting portfolio would yield zero return and zero variance and (2) any combination of unequal weighted long or short positions would yield

Fig. 5.10 (a) The efficient frontier with short selling. (b) Efficient frontiers with and without short selling and margin requirements

portfolios with higher returns and lower risk levels. Overall, these two effects will yield an efficient frontier that dominates the Markowitz efficient frontier. Figure 5.10b compares the Dyl and Markowitz efficient frontiers.

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

Even though the inclusion of short selling changes the location and boundaries of the efficient frontier, the concavity of the curve is still intact. This is important in that it preserves the efficient frontier as the locus of optimal portfolios for risk-average investors. As long as the efficient frontier remains concave, by using indifference curves it will be possible to locate the optimal portfolio for each investor. Techniques for calculating the efficient frontier with short selling. Since there are thousands of securities from which the investor can choose for portfolio formation, it can be very costly and time consuming to calculate the efficient frontier set. One way of determining the optimal investment proportions in a portfolio is to hold the return constant and solve for the weighting factors (W1 : : : ; Wn for n securities) that minimize the variance, given the constraint that all weights sum to one and that the constant return equals the expected return developed from the portfolio. The optimal weights can then be obtained by minimizing the Lagrange function C for portfolio variance. C D



n P Wi Wj rij i j C 1 1  Wi i D1 j  D1 i D1  n P C2 E   Wi E .Ri / n P n P

(5.11)

i D1

in which 1 and 2 are the Lagrange multipliers, E  and E .Ri / are targeted rate of return and expected rate of return for security i , respectively, ij is the correlation coefficient between Ri and Rj , and other variables are as previously defined. Function C has n C 2 unknowns: W1 ; : : : ; Wn ; 1 , and 2 . By differentiating C with respect to Wi ; 1 and 2 and equating the first derivatives to zero, nC2 equations can be derived to solve for n C 2 unknowns. As shown in the empirical example later in this chapter, matrix algebra is best suited to this solution process. By using this approach the minimum variance can be computed for any given level of expected portfolio return (subject to the other constraint that the weights sum to one). In practice it is best to use a computer because of the explosive increase in the number of calculations as the number of securities considered grows. The efficient set that is generated by the aforementioned approach [Equation (5.11)] is sometimes called the minimum-variance set because of the minimizing nature of the Lagrangian solution. Thus far no specific distribution for measuring returns has been assumed. In most cases specific distribution of returns will be needed when applying the portfolio model. The normal and log normal are two of the most commonly used distribution. The normal distribution. As you undoubtedly are aware, the normal (probability) distribution is a bell-shaped curve centered on the mean of a given population or sample distri-

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bution. The area under the curve is an accumulation of probabilities that sum to one. With half of the area lying to the left and half to the right of the mean, probability statements may be made about the underlying sample that makes up the distribution. Within the scope of the Markowitz model, the normal distribution may be applied because of the use of the meanvariance assumptions. The formation of probability statements from the underlying sample is a result of the standard deviation (square root of the variance) quantifying the spread of the distribution. If the sample is normally distributed, approximately 68% of the observations will lie within one standard deviation on either side of the mean, approximately 95% will lie within two standard deviations, and approximately 99% will lie within three standard deviations. This ability to ascertain intervals of confidence for the observations allows the utilization of the normal distribution to predict ranges within which the portfolio returns will lie. Utilizing the standard normal distribution (mean of zero, standard deviation of one), any set of mean-variance data can be standardized to develop probability statements about returns on a given portfolio. Standardization is an algebraic operation in which the mean is subtracted from a given return and this difference is divided by the standard deviation. The resultant metric is then standard normal and can be compared with tabulated values to calculate probabilistic quantities of occurrence. Within this framework, any hypothesized level of return can be assigned a probability of occurrence given a probabilistic value of occurrence that is based upon the nature of the sample. For expository purposes, suppose the mean return on a particular investment is 10% for a given period, and that historically these returns have a variance of 0.16%. It is of interest to know what the probabilities are of obtaining a 15% or greater return, or a return less than or equal to 8%. These probabilities may be evaluated by using the normal distribution, as stated in notation below: ˇ

ˇ P X 0:15 ˇRp D 0:1; p2 D 0:0016 ˇ

ˇ P X  0:18 ˇRp D 0:1; p2 D 0:0016 by standardizing the probability statements to z values:

0:15  0:1 D 1:25 P z 0:04

0:08  0:1 P z D 0:5 0:04

(5.12)

These standardized values can then be compared with the tabulated z values found in tables of the standardized normal distribution. For z 1:25 the probability is 10.56%

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(50% – 39.44%), and for z D 0:05 the probability is 0.3085. Therefore this investment has an 11.51% chance of obtaining a 15% or more return, and a 30.85% chance of obtaining 8% or less. It is extremely difficult to make probabilistic statements when using the normal distribution. Given that the return data utilized are ex post, the predictive ability of the normal distribution based on historical data is limited. The past does not always predict the future; hence, this problem is somewhat alleviated by an assumption that the security under consideration is in a static state. Nevertheless, as shown by research and common sense, this assumption is rather bold, and the use of the normal distribution should be limited to comparisons of various past portfolio returns. Another difficulty with the normal distribution is that if the distribution of the sample is skewed in any way, the standard deviation will not properly reflect the equivalent areas under the curve on either side of the mean. This problem will be discussed and solved in the next section through the application of the log normal distribution. A variable is log normally distributed if a logarithm of this variable is normally distributed. The log normal distribution. The log normal distribution discussed in the next chapter will explore further of this section. The approach outlined in the last section for delineating the efficient frontier assumes that the security-return data are normally distributed. In reality, most financial researchers would agree that security-return data tend to be positively skewed. This skewness can be a serious problem in accurately developing the efficient frontier with the Markowitz model because of the assumption that only the first two moments of the return distribution, mean and variance, are important. One reason for returns being positively skewed is the inability of the investor to lose more than 100% of his or her investment, effectively creating a lower bound to portfolio returns. This is called the limited-liability constraint. But since capital gains and dividends could conceivably be infinite, the upper tail of the distribution of returns has no upper bounds. The range of probable returns is spread towards the positive side and therefore contains a potential bias when it is utilized in developing statistical estimates. If the return distributions are significantly skewed, the efficient frontier can be more accurately determined with the use of logarithmically transformed holding-period returns. That is, each holding-period return for security i with holding period T .1 C RiT / is transformed by computing its natural logarithm, ln.1 C RiT /. The logarithmic transformation converts a data set of discretely compounded returns into continuously compounded returns. The distribution of discrete time returns will be more positively skewed the larger the differencing interval used to measure the returns; that is,

C.-F. Lee et al.

if skewness exists in the return distribution, annual data will be more positively skewed than monthly data, which will be more skewed than weekly data. The continuous compounding implied in the logarithmically transformed data will virtually eliminate any positive skewness existing in the raw return data when rates of return are log normally distributed. In practice, it is possible to directly derive the efficient frontier using means and variances of logarithmically transformed data. Under this circumstance, assume that utility functions use continuous returns and variances, and then use the ln.1 C Rp / transformation on discrete data. To deal with the untransformed log normally distributed data, Elton et al. (1976) use the mean and variance of log normal distributed as defined in Equations (5.13) and (5.14) to derive the efficient frontier. In other words, they first delineate the efficient frontier by using the means and variances of the untransformed data and then use Equations (5.13) and (5.14) to determine the subset of the ŒE.1 C Rp /; .1 C Rp / efficient frontier in terms of log-transformed data, ln.1 C Rp /. E.1 C Rp / D eE .rp /C1=2

2

(5.13)

where rp D ln.1 C Rp / and  is the standard deviation of rp .

2

2  2 .1 C Rp / D e2E .rp /C e  1

(5.14)

  where rp D ln.1 C Rp / and  2 D Var rp . Use of the logarithmic transformation can also be shown to eliminate less desirable portfolios in the lowest-return segment of the efficient frontier computer using the raw data.6 Before moving on to an example, however, it will be useful to briefly summarize the portfolio-selection process. Again, the Markowitz model of portfolio selection is a mathematical approach for deriving optimal portfolios; that is, portfolios that satisfy the following conditions. 1. The least risk for a given level of expected return (minimum-variance portfolios). 2. The greatest expected return for a given level of risk (efficient portfolios). How does a portfolio manager apply these techniques in the real world? The process would normally begin with a universe of securities available to the fund manager. These securities would be determined by the goals and objectives of the mutual fund. For example, a portfolio manager who runs a mutual fund specializing in health-care stocks would be required to select securities from the universe of health-care stocks. This would greatly reduce the analysis of the fund manager by limiting the number of securities available. 6

See Baumol (1963).

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

The next step in the process would be to determine the proportions of each security to be included in the portfolio. To do this, the fund manager would begin by setting a target rate of return for the portfolio. After determining the target rate of return, the fund manager can determine the different proportions of each security that will allow the portfolio to reach this target rate of return. The final step in the process would be for the fund manager to find the portfolio with the lowest variance given the target rate of return. The next section uses a graphical approach to derive the optimal portfolios for three securities.

5.4.3 Three-Security Empirical Solution To facilitate a realistic example, actual data have been taken from a set of monthly returns generated by the Dow-Jones 30 Industrials. This example focuses on the returns and risk of the first three industrial companies – AXP (American Express), XOM (Exxon Mobil), and JNJ (Johnson & Johnson) – for the period January 2001–September 2007. The data used are tabulated in Table 5.4. Both graphical and mathematical analyses are employed to obtain an empirical solution. Graphical analysis. To begin to develop the efficient frontier graphically, it is necessary to move from the three dimensions necessitated by the three-security portfolio to a twodimensional problem by transforming the third security into an implicit solution from the other two.7 To do this it must be noted that since the summation of the weights of the three securities is equal to unity, then implicitly: W3 D 1  W1  W2

(5.15)

Additionally, the above relation may be substituted into Equation (5.1):   E Rp D W1 E .R1 / C W2 E .R2 / C W3 E .R3 / D W1 E .R1 / C W2 E .R2 / C E .R3 /

Table 5.4 Data for three securities Company E .ri / i2 JNJ AXP XOM 7

0:0053 0:0055 0:0126

0:0455 0:0614 0:0525

D ŒE .R1 /  E .R3 / W1 C ŒE .R2 / E .R3 / W2 C E .R3 / Finally, inserting the values for the first and second securities yields:   E Rp D .0:0053  0:0126/ W1 C .0:0055  0:0126/ W2 C 0:0126

(5.17)

D 0:0073W1  0:0071W2C0:0126 As can be seen, Equation (5.17) is a linear function in two variables and as such is readily graphable. Since given a certain level of portfolio return the function will solve jointly for the weights of securities 1 and 2, it solves indirectly for the weight of security 3. The variance formula shown in Equation (5.2) is converted in a similar manner by substituting in Equation (5.15) as follows: p2 D

3 3 X X

    Wi Wj Cov Ri ; Rj D Var Rp

i D1 j D1

D W12 11 C W22 22 C W32 33 C 2W1 W2 12 C2W1 W3 13 C 2W2 W3 23 D W12 11 C W22 22 C .1  W1  W2 /2 33 12 C2W1 W2 C 2W1 .1  W1  W2 / 13 C2W2 .1  W1  W2 / 23 D .11 C 33  213 / W12 C .233 C 212 213  223 / W1 W2 C .22 C 33  223 / W22 C .233 C 213 / W1 C .233 C 213 / W2 C 33

(5.18)

Inserting the covariances and variances of the three securities from Table 5.4:

D W1 E .R1 / CW2 E .R2 / C .1  W1  W2 / E .R3 / W1 E .R3 /  W2 E .R3 /

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(5.16)   Cov Ri ; Rj 12 D 0:0009 23 D 0:0010 13 D 0:0004

The process of finding the efficient frontier graphically described in this section was originally developed by Markowitz (1952). Francis and Archer (1979) have discussed this subject in detail.

p2 D Œ0:0455 C 0:0525  2 .0:0004/ W12 C Œ2 .0:0525/ C2 .0:0009/  2 .0:0004/  2 .0:0010/ W1 W2 CŒ0:0614 C 0:0525  2 .0:0010/ W22 CŒ2 .0:0525/ C 2 .0:0004/ W1 CŒ2 .0:0525/ C 2 .0:0010/ W2 C 0:0525 D 0:0972W12 C 0:104W1 W2 C 0:1119W22 0:1042W1  0:103W2 C 0:0525

(5.19)

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Minimum-risk portfolio. Part of the graphical solution is the determination of the minimum-risk portfolio. Standard partial derivatives are taken of Equation (5.18) with respect to the directly solved weight factors as follows: @p2 @W1

D 2 .11 C 33  213 / W1 C .233 C 212 213  223 / W2 C .233 C 213 / D 0

@p2 @W2

D .233 C 212  213  223 / W1

(5.20)

C2 .22 C 33  223 / W2 C .223  233 / D 0 When these two partial derivatives are set equal to zero and the unknown weight factors are solved for, the minimum risk portfolio is derived. Using the numeric values from Table 5.4:

Table 5.5 Iso-return lines Target return W2 1:0 0:5 0:0 0:5 1:0

0:0082 W1 1:5753 1:0890 0:6027 0:1164 0:3699

0:01008 W1 1:3178 0:8315 0:3452 0:1411 0:6274

0:01134 W1 1:1452 0:6589 0:1726 0:3137 0:8000

other to solve for W2 . For example, if W2 D 0 and   values E Rp D 0:0082, then W1 can be solved as follows (see Table 5.5 for final results). 0:0082 D 0:0073W1  0:0071 .0/ C 0:0126 0:0073W1 D 0:0126  0:0082 D 0:01154 W1 D 0:6027

In a similar fashion, other points are calculated and are listed in Table 5.5. A line may be drawn between the points to de@p2 D 2Œ0:0455 C 0:0525  2 .0:0004/ W1 C Œ2 .0:0525/ velop the various iso-expected return function lines. @W1 There are a multitude of possible return functions, but C2 .0:0009/  2 .0:0004/  2 .0:0010/ only these three lines .IR1 ; IR2 and IR3 ) in terms of the data listed in Table 5.5 are shown in Fig. 5.11. W2 C Œ2 .0:0525/ C 2 .0:0004/ In Fig. 5.11 vertical axis and horizontal axis represent W1 D 0:1944W1 C 0:104W2  0:1042 D 0 and W2 , respectively. Each point on the iso-expected return line of Fig. 5.11 rep@p2 D Œ2 .0:0525/ C 2 .0:0009/  2 .0:0004/  2 .0:0010/ resents a different combination of weights placed in the three @W2 securities. The issue now is to determine the portfolio that W1 C 2Œ.0:0614/ C 0:0525  2 .0:0010/ lies on the iso-expected return line with the lowest variance. W2 C Œ2 .0:0010/  2 .0:0525/ (5.21) Table 5.6 shows the portfolio variances as we move along the iso-expected return line. For example, moving along the D 0:104W1 C 0:2238W2  0:103 D 0 iso-expected return line associated with an expected return of 0.82%, note that the minimum variance portfolio is asBy solving these two equations simultaneously the weights sociated with weights of 0.2577 in security 2 and 0.3521 in of the minimum-risk portfolio are derived. This variance rep- security 1. resents the lowest possible portfolio-variance level achievable, given variance and covariance data for these stocks. Iso-variance ellipses. As shown in Table 5.6 the minimumThis can be represented by the point V of Fig. 5.8. This variance portfolio can be found by moving along the isosolution is an algebraic exercise that yields W1 D 0:3857 expected return line until the minimum-variance portfolio is and W2 D 0:2810 and, therefore, through Equation (5.15), reached. To better visualize this, examine the variance of a portfolio as the proportions in security 1 and security 2 W3 D 0:3333. are changed. A plot of these points will allow the family of The iso-expected return line. The variance and return iso-variance ellipses to be traced. The iso-variance ellipses equations have been derived, and it is now time to com- are the ellipses that have the same variance on every point plete the graphing procedure. To begin the graphing, the iso- of this ellipse curve. It should be noted that an ellipse is expected return function lines must be delineated given var- an egg-shaped circle of points with a common center and ious levels of expected return. The iso-expected return line orientation. The minimum-risk portfolio variance is the cenis a line that has the same expected return on every point ter of all possible ellipses as it has the least risk. It will be of the line. Utilizing Equation (5.17), three arbitrary returns desirable, then, to find this minimum risk value, as no other are specified: 0.0082, 0.01008, 0.01134 (these correspond weighting scheme of these three securities will develop a to 70, 80, and 90% annual returns of Exxon Mobil). These lesser risk. The solutions of Equation (5.21) have determined three monthly returns are then set equal to Equation (5.17), that the weights associated with the minimum-risk portfolio and graphing is now possible by setting W1 equal to zero or are W1 D 0:3857; W2 D 0:2810, and W3 D 0:3333. Substi-

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Fig. 5.11 Iso-return lines

Table 5.6 Portfolio variance along the iso-return line

0.0082 w2 1 0:75 0:5 0:25 0 0:0795 0:151 0:25 0:2577 0:5 0:75 1

w1 1:5753 1:3322 1:0890 0:8459 0:6027 0:5254 0:4559 0:3596 0:3521 0:1164 0:1267 0:3699

0.01008 Var 0.1806 0.1225 0.0771 0.0447 0.0250 0.0214 0.0194 0.0182 0.0182 0.0242 0.0431 0.0748

w1 1:3178 1:0747 0:8315 0:5884 0:3452 0:2679 0:1983 0:1021 0:0946 0:1411 0:3842 0:6274

0.01134 Var 0.1618 0.1091 0.0693 0.0423 0.0281 0.0263 0.0258 0.0268 0.0269 0.0383 0.0626 0.0998

w1 1:1452 0:9021 0:6589 0:4158 0:1726 0:0953 0:0257 0:0705 0:0780 0:3137 0:5568 0:8000

Var 0.1564 0.1074 0.0713 0.0479 0.0374 0.0368 0.0373 0.0397 0.0400 0.0549 0.0829 0.1238

Note: Underlined variances indicate minimum variance portfolios

tuting this information listed in Table 5.4 into Equation (5.2), the variance of the minimum-risk portfolio is: n n X     X Wi Wj Cov Ri Rj Var Rp D i D1 j D1

be completed by taking Equation (5.18) andholding one of  , constant. portfolio variance Var R the weights, say W p  2 Bring the Var Rp to the right-hand side of the equation and notice that the equation is of a quadratic form and can be solved using the quadratic formula:

D W12 11 C W22 22 C W32 33 C 2W1 W2 12

W1 D

C2W2 W3 23 C 2W1 W3 13 D .0:3857/2 .0:0455/ C .0:2810/2 .0:0614/ C .0:3333/2 .0:0525/ C2 .0:3857/ .0:2810/ .0:0009/ C2.0:2810/.0:3333/.0:0010/ C 2.0:3857/ .0:3333/.0:0004/ D 0:017934 Note that the variance of the minimum-risk portfolio can be used as a base for graphing the iso-variance ellipses. It can

b ˙

p b 2  4ac 2a

(5.22)

where: a D all coefficients of W12 ; b D all coefficients of W1 ; and c D all coefficients that are not multiplied by W1 , or W12 : or a D 11 C 33 C 13 ; b D .233 C 212  213  223 / W2  233 C 213 ; and c D .22 C33  223 / W22 C .233 C 223 / W2 C 33  Var Rp .

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Substituting the numbers from the data of Table 5.4 into Equation (5.18) yields: Var.Rp / D 0:0972W21 C 0:104W1 W2 C 0:1119W22 0:1042W1  0:103W2 C 0:0525 0 D 0:0972W21 C 0:104W1 W2 C 0:1119W22 0:1042W1  0:103W2 C 0:0525Var.Rp / where: a D 0:0972I b D 0:104W2  0:1042I and c D 0:1119W22  0:103W2 C 0:0525  Var.Rp /: When these expressions are plugged into the quadratic formula: p  .0:104W2  0:1042/ ˙ b 2  4ac W1 D (5.23) 2 .0:0972/ where b 2 D .0:104W2  0:1042/2 4ac D 4f.0:0972/Œ0:1119W22  0:103W2 C0:0525  Var.Rp / g This is a solution for the two points of W1 (W11 and W12 ) for a given portfolio variance and weight of the second security .W2 /. When selecting the level of Var .Rp / it is best to choose a value that is slightly larger than the minimumrisk value. This assures the calculation of a possible portfolio, since no portfolio of these three securities may have less risk. Additionally, an initial selection of a value for W2 close to the minimum-risk portfolio W2 will be desirable. In sum, Equation (5.23) can be used to construct the isovariance ellipse for given Var .Rp / in terms of arbitrary W2 . By jointly considering iso-expected return lines and isovariance ellipses, efficient portfolios can be identified that are defined as the portfolios with the minimum variance given the expected rate of return. This task can be done by computer, as mentioned in the last section. To obtain the weights of an efficient portfolio, the following set of instructions must be entered into the computer. 1. Find the portfolio weights that minimize portfolio variance, subject to the target expected rate-of-return constraint. For this case, target rates of return are 0.82, 1.008, or 1.1134%. The sum of the portfolio weights for all stocks in the portfolio must be equal to one. Mathematically, this instruction is defined in Equation (5.15).

2. The expected return and variance for a three-security portfolio is defined in Equations (5.16) and (5.18), respectively. 3. The estimated expected return, variance, and covariance as defined in Table 5.4 should be entered into the computer for estimation. Using E.Rp / D 0:82 percent as an example, W1 D 0 and W2 D 0:6197. From the relationship W3 D 1  W1  W2 D 1  0  0:6197 D 0:3803. In other words, the point H represents three portfolio weights for the three stocks. The computer substitutes weights, variances, and covariances (listed in Table 5.4) into Equation (5.18) to obtain portfolio variance. The portfolio variance consistent with the weights of point H is computed. The computer now moves by some predetermined distance either northeast or southwest along the 0.82% iso-expected return line. From this kind of search, the minimum variance associated with this 0.82% iso-expected return line is identified to be 0.0182, as indicated in Table 5.6. By a similar procedure, the minimum variances associated with 1.008 and 1.134% iso-expected return lines are identified to be 0.0258 and 0.0368, respectively. When minimum variances associated with predefined iso-expected return are identified, the optimal weights associated with expected returns equal to 1.008 and 1.134% in terms of data indicated in Table 5.4 are also calculated. These results are indicated in Table 5.6. Using the minimum variances associated with isoexpected returns (0.82, 1.008, and 1.134%), three isovariance ellipses can be drawn as IV 1 ; IV 2 , and IV 3 . When Var.Rp / D 0:0182 and W2 D 0:3, the quadratic formula from Equation (5.23) above yields two roots. p .0:0730/C

W11 D D 0:4247

.0:0730/2 .0:0052/ 2.0:0972/

p .0:0730/2 W12 D .0:0730/ 2.0:0972/ .0:0052/ D 0:3263 To solve for W11 and W12 we need b 2 > 4ac. Solving for the variance: .0:104W2  0:1042/2  4f.0:0972/ > Œ0:1119W22 0:103W2 C 0:0525  Var.Rp / g If W2 and Var .Rp / are not selected so that b 2 > 4ac, Equation (5.23) will require taking the square root of a negative number, and mathematically the solution will be an imaginary number. Solving W2 in terms of an imaginary number is not meaningful; therefore we only consider the case when b 2 > 4ac.

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

The variance chosen was the minimum variance of the portfolio associated with E.Rp / D 0:0082. Actually, any variance might have realistically been chosen as long as it exceeded the minimum-risk portfolio variance. Table 5.7 lists various W11 and W12 values for given levels of W2 and Var .Rp /. In addition, Table 5.7 presents the value of b 2 and 4ac to check whether or not the root is a real number. It should be noted that all possible variances are higher than the minimum-risk portfolio variance. Data from Table 5.7 are used to draw three iso-variance ellipses, as indicated in Figs. 5.12 and 5.13. The critical line and efficient frontier. After the iso-expected return functions and iso-variance ellipses have been plotted, it is an easy task to delineate the efficient frontier. By definition, the efficient portfolio is the portfolio with the highest return for any given risk. In Fig. 5.13, the efficient portfolios are those where a given iso-expected return line is just tangent to the variance ellipse. MRPABC is denoted as Table 5.7 Various W1 given W2 and Var.Rp / W2 Var.Rp / b b2 0:28 0:29 0:30 0:31 0:32 0:28 0:29 0:30 0:31 0:32 0:28 0:29 0:30 0:31 0:32

Fig. 5.12 Iso-variance ellipses

0:0182 0:0182 0:0182 0:0182 0:0182 0:0258 0:0258 0:0258 0:0258 0:0258 0:0368 0:0368 0:0368 0:0368 0:0368

0:0751 0:0740 0:0730 0:0720 0:0709 0:0751 0:0740 0:0730 0:0720 0:0709 0:0751 0:0740 0:0730 0:0720 0:0709

0:0056 0:0055 0:0053 0:0052 0:0050 0:0056 0:0055 0:0053 0:0052 0:0050 0:0056 0:0055 0:0053 0:0052 0:0050

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the critical line; all portfolios that lie between points MRP and C are said to be efficient, and the weights of these portfolios may be read directly from the graph. From the graph, portfolio weights for the portfolios that minimize variances, given a 0.82, 1.008, or 1.134% expected rate of return are listed in Table 5.6. It is possible, given these various weights, to calculate the E.Rp / and the variances of these portfolios, as indicated in Table 5.8. The efficient frontier is then developed by plotting each risk-return combination, as shown in Fig. 5.14. In this section a step-wise graphical approach has been employed to obtain optimal weights for a three-security portfolio. In the next section a mathematical optimization approach is used to calculate the optimal weight for a threesecurity portfolio. Mathematical analysis. The same efficient frontier can be obtained mathematically through the use of the Lagrangian multipliers. The Lagrangian method allows the minimization

C 0:0142 0:0138 0:0135 0:0131 0:0128 0:0066 0:0062 0:0059 0:0055 0:0052 0:0044 0:0048 0:0051 0:0055 0:0058

4 ac 0:0055 0:0054 0:0052 0:0051 0:0050 0:0026 0:0024 0:0023 0:0021 0:0020 0:0017 0:0019 0:0020 0:0021 0:0023

W11 0:4385 0:4325 0:4247 0:4149 0:4025 0:6707 0:6652 0:6594 0:6534 0:6470 0:8268 0:8213 0:8157 0:8099 0:8039

W12 0:3339 0:3293 0:3263 0:3254 0:3272 0:1017 0:0965 0:0916 0:0870 0:0827 0:0543 0:0596 0:0647 0:0696 0:0742

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Fig. 5.13 Iso-variance ellipses and iso-return lines

Table 5.8 Weights, Rp , and p for efficient points Portfolio W1 W2 W3 A B C

0:3521 0:1983 0:3902

0:2577 0:1510 0:6507

0:3902 0:0795 0:8252

E.Rp /

Var.Rp /

0:0082 0:01008 0:01134

0:0182 0:0258 0:0368

 .Rp / 0:1349 0:1606 0:1918

Portfolios A, B, and C represent expected returns of 0.82, 1.008, and 1.134%, respectively Fig. 5.14 The efficient frontier for the data of Table 5.8

or maximization of an objective function when the objective function is subject to some constraints. One of the goals of portfolio analysis is minimizing the risk or variance of the portfolio, subject to the portfolio attaining some target expected rate of return, and also subject to the portfolio weights’ summing to one. The problem can be stated mathematically: Min p2 D

n n X X i D1 j D1

Wi Wj ij

Subject to

(a)

n P

Wi E.Ri / D E 

i D1

where E  is the target expected return and

(b)

n P i D1

Wi D 1:0

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

The first constraint simply says that the expected return on the portfolio should equal the target return determined by the portfolio manager. The second constraint says that the weights of the securities invested in the portfolio must sum to one. The Lagrangian objective function can be written: C D

n n X X





Wi Wj Cov Ri Rj C 1 1 

i D1 j D1

"

C2 E  

n X

# Wi E .Ri /

n X

! Wi

i D1

(5.11)

i D1

Taking the partial derivatives of this equation with respect to each of the variables, W1 ; W2 ; W3 ; 1 ; 2 and setting the resulting five equations equal to zero yields the minimization of risk subject to the Lagrangian constraints. This system of five equations and five unknowns can be solved by the use of matrix algebra. Briefly, the Jacobian matrix of these equations is 2

211 6 221 6 6 231 6 4 1 E .R1 /

212 213 222 223 232 233 1 1 E .R2 / E .R3 /

3 2 3 2 3 W1 0 1 E .R1 / 6W2 7 6 0 7 1 E .R2 / 7 7 6 7 6 7 6 7 6 7 1 E .R3 / 7 7 6W3 7 D 6 0 7 4 1 5 4 1 5 5 0 0 E 0 0 2

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W A1 K 3 2 3 W1 0:9442 6 W2 7 6 0:6546 7 6 7 6 7 6 W3 7 D 6 0:5988 7 6 7 6 7 4 1 5 4 0:1937 5 2 20:1953 2

(5.26)

Using the matrix operation on the other two arbitrary returns yields the results shown in Table 5.8. As shown in the composition of Portfolios A, B, and C, the weights of security 3 are negative, implying that this security should be sold short in order to generate an efficient portfolio. It should be noted that the results of Table 5.8 are similar to those of Table 5.6. Therefore, both graphical and mathematical methods can be used to calculate the optimal weights. With the knowledge of the efficient-portfolio weights, given that E.Rp / is equal to 0.00106, 0.00212, and 0.00318, the variances of the efficient portfolios may be derived from plugging the numbers into Equation (5.2). Taking the square root of the variances to derive the standard deviation, the various risk-return combinations can be plotted and the efficient frontier graphed, as shown in Fig. 5.13 in the previous section. As can be seen from comparing this figure to the graphical derivation of the efficient frontier derived graphically, the results are almost the same.

(5.24) Therefore, it is possible to premultiply both sides of the matrix Equation (5.24), AW D K, by the inverse of A (denoted A1 ) and solve for the W column. This is possible because all values in the A and K matrices are known or arbitrarily set. The first problem is the inversion of the matrix of coefficients. First, derive the determinant of matrix A. Next, after developing the signed, inverted matrix of cofactors for A, divide the cofactors by the determinant, resulting in the inversion of the original matrix. Finally, premultiply the column vector for the investment weights. Plugging the data listed in Table 5.4 and E  D 0:00106 into the matrix above yields 2

0:0910 6 0:0036 6 6 6 0:0008 6 4 1 0:0053

0:0018 0:1228 0:0020 1 0:0055

0:0008 0:0020 0:1050 1 0:0126

3 2 3 2 3 1 0:0053 W1 0 6 7 6 7 1 0:0055 7 7 6 W2 7 6 0 7 7 6 7 6 7 1 0:0126 7 6W3 7 D 6 0 7 7 6 7 6 7 5 4 1 5 4 1 5 0 0 2 0 0 0:00106 (5.25)

When matrix A is properly inverted and postmultiplied by K, the solution vector A1 K is derived:

5.4.4 Portfolio Determination with Specific Adjustment for Short Selling By using a definition of short sales developed by Lintner (1965) the computation procedure for the efficient frontier can be modified. Lintner defines short selling as putting up an amount of money equal to the value of the security sold short. Thus the short sale is a use rather than a source of funds to the short seller. The total funds the investor invests short, plus the funds invested long, must add up to the original investment. The proportion of funds invested short is jXi j, since Xi < 0. The constraint in the minimization problem concerning the weights of the individual securities needs to be modified to incorporate this fact. Additionally, the final portfolio weight (output of the matrix inversion) must be rescaled so the sum of the absolute value of the weights equals one. By defining short sales in this manner, the efficient frontier does not extend to infinity (as shown in Fig. 5.10a) but resembles the efficient frontier in Fig. 5.10b. Monthly rates of return for Johnson & Johnson, American Express, and Exxon Mobile are used to perform the analysis in this section. The sample period is from January

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2001 to September 2007. The Markowitz model determines optimal asset allocation by minimizing portfolio variance using a constrained optimization procedure: 



Min Var Rp D

3 3 X X

Wi Wj ij

(5.2)

Security JNJ AXP XOM

Monthly E.Ri /

i2

12

13

23

0.0053 0.0055 0.0126

0.0455 0.0614 0.0525

0.0009

0.0004

0.0010

i D1 j D1

Subject to: (a) (b)

3 P i D1 3 P

Substituting these values into the matrix: 2 3 2 0:0910 6 0:0018 6 6 6 0:0008 6 4 0:0053 1

Wi E .Ri / D E (5.1) 

jWi j D 1:0 (5.1)

i D1

Where the E  is the investors’ desired rate of return, and where the absolute value of the weights jWi j allows for a given W to be negative (sold short) but maintains the requirement that all funds are invested or their sum equals one. The Lagrangian function is Min L D

n n X X

W i W j C 1

i D1 j D1

C2

n X

!

n X 

Wi E .Ri /  E



0:0018 0:0614 0:0020 0:0055 1

E  D 0:013

By solving using the identity-matrix technique the weights for the three securities can be obtained: Johnson & Johnson D 0:0406 American Express D 0:0146 Exxon Mobile D 1:0552



i D1

By using the following relationship to rescale these weights so that the second constraint for the sum of the absolute values of the weights to equal one is satisfied:

Wi  1

i D1

Again, derivatives with respect to Wi ’s and ’s are found. Setting these equations equal to zero leaves the following system of equations in matrix form: C  

X K 2 3 3 2 W1 0 211 212 21n E .R1 / 1 6 6 221 7 7 6 222 22n E .R2 / 1 7 7 6 W2 7 6 0 7 6 6 6 7 6 : 7 :: :: :: :: 7 6 :: 7 6 :: 7 6 : 7 : : : :7 6 : 7D 6 : 7 6 : 7 6 6 7 7 6 6 2n1 2n2    2nn E .Rn / 1 7 6 Wn 7 6 0 7 7 6 6 7 6 7 4 E .R1 / E .R2 /    E .Rn / 0 0 5 4 1 5 4 E 5 1 1  1 0 0 2 1 2

3 3 2 0:0008 0:0053 1 0 W1 7 7 6 6 0:0020 0:0055 1 7 7 6 W2 7 6 0 7 7 7 6 7 6 0:0525 0:0126 1 7 6 W3 7 D 6 0 7 7 6 7 7 6 0:0126 0 0 5 4 1 5 4 E 5 2 1 1 0 0

3

To solve this matrix:

WjAj D

jAj jAj C jBj C jC j

(5.27)

The resealed absolute weights are: 0:0406 D 0:0366 0:0406 C 0:0146 C 1:0552 0:0146 WAXP D D 0:0131 0:0406 C 0:0146 C 1:0552 1:0552 WXOM D D 0:9503 0:0406 C 0:0146 C 1:0552 WJNJ D

The return on this portfolio is: CX C1 CX IX X

DK D C1 K D C1 K D C1 K

Rp D .0:0366/ .0:0053/ C .0:0131/ .0:0055/ C .0:9503/ .0:0126/ D 1:22%

where C1 is the inverse of matrix C and I is the identity matrix. The solution to the above formula will give the weights in terms of E  . By using Johnson & Johnson, American Express, and Exxon Mobile a three-security portfolio will be formed and optimal weights will be solved for.

The variance: p2 D .0:0366/2 .0:0455/ C .0:0131/2 .0:0614/ C .0:9503/2 .0:0525/ C 2 .0:0366/ .0:0131/ .0:0009/ C 2 .0:0366/ .0:9503/ .0:0004/ C 2 .0:0131/ .0:9503/ .0:0010/ D 0:0475

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

5.4.5 Portfolio Determination Without Short Selling The minimization problem under study can be modified to include the restriction of no short selling by adding a third constraint: Wi 0; i D 1; : : : ; N The addition of this nonnegativity constraint precludes negative values for the weights (that is, no short selling). The problem now is a quadratic programming problem similar to the ones solved so far, except that the optimal portfolio may fall in an unfeasible region. In this circumstance the next best optimal portfolio is elected that meets all of the constraints. An example of this situation is shown in Fig. 5.15.8 In Fig. 5.15a the optimal portfolio is located in the region of positive values for weights Wi , and thereby satisfies the

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constraint Wi 0. In Fig. 5.15b the optimal portfolio shown by Wi falls in a region where W1 are negative, and so the constraint is not satisfied. The next best optimal portfolio is shown by B, and this is the solution of the problem that satisfies the nonnegativity constraint. Additional discussion of the problems is presented in later chapters. Most recently, Lewis (1988) has developed a simple algorithm for the portfolioselection problem. His method is based on an interactive use of the Markowitz critical-line method for solving quadratic programs. In ı Fig. 5.15a, b the vertical axis is defined as .RN p  Rf / p , this is a Sharpe performance mean, as defined in Chap. 4, and can be regarded as the objective function for a portfolio selection.

5.5 Conclusion This chapter has focused on the foundations of Markowitz’s model and on derivation of efficient frontier through the creation of efficient portfolios of varying risk and return. It has been shown that an investor can increase expected utility through portfolio diversification as long as there is no perfect positive correlation among the component securities. The extent of the benefit increases as the correlation is lower, and also increases with the number of securities included. The Markowitz model can be applied to develop the efficient frontier that delineates the optimal portfolios that match the greatest return with a given amount of risk. Also, this frontier shows the dominant portfolios as having the lowest risk given a stated return. This chapter has included methods of solving for the efficient frontier both graphically and through a combination of calculus and matrix algebra, with and without explicitly incorporating short selling. The next chapter will illustrate how the crushing computational load involved in implementing the Markowitz model can be alleviated through the use of the index portfolio, and how the tenets of the Markowitz efficient frontier are still met.

References Alexander, G. J. and J. C. Francis. 1986. Portfolio analysis, PrenticeHall, Englewood Cliffs, NJ. Baumol, W. J. 1963. “An expected gain-confidence limit criterion for portfolio selection.” Management Science 10, 171–182. Fig. 5.15 Optimal portfolio in feasible and unfeasible regions ı In Fig. 5.15a and b the vertical axis is defined as .Rp  Rf / p , which is the ı Sharpe performance measure as defined in Chap. 4. .Rp  Rf / p is the objective function for portfolio optimization; it will be explored in Chap. 8.

8

Bertsekas, D. 1974. “Necessary and sufficient conditions for existence of an optimal portfolio.” Journal of Economic Theory 8, 235–247. Black, F. 1972. “Capital market equilibrium with restricted borrowing.” Journal of Business 45, 444–455. Blume, M. 1970. “Portfolio theory: a step toward its practical application.” Journal of Business 43, 152–173.

92 Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Brealey, R. A. and S. D. Hodges. 1975. “Playing with portfolios.” Journal of Finance 30, 125–134. Breen, W. and R. Jackson. 1971. “An efficient algorithm for solving large-scale portfolio problems.” Journal of Financial and Quantitative Analysis 6, 627–637. Brennan, M. J. 1975. “The optimal number of securities in a risky asset portfolio where there are fixed costs of transaction: theory and some empirical results.” Journal of Financial and Quantitative Analysis 10, 483–496. Cohen, K. and J. Pogue. 1967. “An empirical evaluation of alternative portfolio-selection models.” Journal of Business 46, 166–193. Dyl, E. A. 1975. “Negative betas: the attractions of selling short.” Journal of Portfolio Management 1, 74–76. Elton, E. J. and M. Gruber. 1974. “Portfolio theory when investment relatives are log normally distributed.” Journal of Finance 29, 1265–1273. Elton, E. J. and M. Gruber. 1978. “Simple criteria for optimal portfolio selection: tracing out the efficient frontier.” Journal of Finance 13, 296–302. Elton, E. J., M. Gruber, and M. E. Padberg. 1976. “Simple criteria for optimal portfolio selection.” Journal of Finance 11, 1341–1357. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Evans, J. and S. Archer. 1968. “Diversification and the reduction of dispersion: an empirical analysis.” Journal of Finance 3, 761–767. Fama, E. F. 1970. “Efficient capital markets: a review of theory and empirical work.” Journal of Finance 25, 383–417. Feller, W. 1968. An introduction to probability theory and its application, Vol. 1, Wiley, New York. Francis, J. C. and S. H. Archer. 1979. Portfolio analysis, Prentice-Hall, Englewood Cliffs, NJ. Gressis, N., G. Philiippatos, and J. Hayya. 1976. “Multiperiod portfolio analysis and the inefficiencies of the market portfolio.” Journal of Finance 31, 1115–1126. Henderson, J. and R. Quandt. 1980. Microeconomic theory: a mathematical approach, 3d Edition, McGraw-Hill, New York. Lee, C. F. and A. C. Lee 2006. Encyclopedia of finance, Springer, New York.

C.-F. Lee et al. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Levy, H. and M. Sarnat. 1971. “A note on portfolio selection and investors’ wealth.” Journal of Financial and Quantitative Analysis 6, 639–642. Lewis, A. L. 1988. “A simple algorithm for the portfolio selection problem.” Journal of Finance 43, 71–82. Lintner, J. 1965. “The valuation of risk assets and the selection of risky investments in stock portfolio and capital budgets.” Review of Economics and Statistics 47, 13–27. Mao, J. C. F. 1969. Quantitative analysis of financial decisions, Macmillan, New York. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd ed., Wiley, New Jersey. Markowitz, H. M. 1952. “Portfolio selection.” Journal of Finance 1, 77–91. Markowitz, H. M. 1959. Portfolio selection. Cowles Foundation Monograph 16, Wiley, New York. Markowitz, H. M.1976. “Markowitz revisited.” Financial Analysts Journal 32, 47–52. Markowitz, H. M. 1987. Mean-variance analysis in portfolio choice and capital markets. Blackwell, Oxford. Martin, A. D., Jr. 1955. “Mathematical programming of portfolio selections.” Management Science 1, 152–166. Merton, R. 1972. “An analytical derivation of efficient portfolio frontier.” Journal of Financial and Quantitative Analysis 7, 1851–1872. Mossin, J. 1968. “Optimal multiperiod portfolio policies.” Journal of Business 41, 215–229. Ross, S. A. 1970. “On the general validity of the mean-variance approach in large markets,” in Financial economics: essays in honor of Paul Cootner, W. F. Sharpe and C. M. Cootner (Eds.). Prentice Hall, Englewood Cliffs, NJ, pp. 52–84. Sharpe, W. F. 1970. Portfolio theory and capital markets, McGraw-Hill, New York. Von Neumann, J. and O. Morgenstern. 1947. Theory of games and economic behavior, 2nd Edition, Princeton University Press, Princeton, NJ. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Chapter 6

Capital Asset Pricing Model and Beta Forecasting Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort

Abstract In this chapter, using the concepts of portfolio analysis and the dominance principle, we derive the capital asset pricing model (CAPM). Then we show how total risk can be decomposed into systematic risk and unsystematic risk. Finally, we discuss the determination of beta and introduce different methods for forecast beta coefficient. Keywords Capital asset pricing model r Beta r Capital market line r Market risk premium r Security market line r Market model r Capital labor ratio r Efficient market hypothesis r Market beta r Accounting beta r Systematic risk r Unsystematic risk

6.1 Introduction One of the important financial theories is the capital asset pricing model, commonly referred to as CAPM, which constitutes the major topic of this chapter; in addition, alternative methods for forecasting beta (systematic risk) are discussed. Using the concepts of basic portfolio analysis and the dominance principle discussed in the last two chapters, two alternative methods are employed to derive the CAPM. The discussion concerns how the market model can be used to decompose (that is, divide) risk into two components, systematic and unsystematic risk. In addition, applications of the beta coefficient and procedures for forecasting a beta coefficient. A graphical approach is first utilized to derive the CAPM, after which a mathematical approach to the derivation is developed that illustrates how the market model can be used to decompose total risk into two components. This is followed J.E. Finnerty () University of Illinois at Urbana-Champaign, Champaign, IL, USA e-mail: [email protected] C.-F. Lee Rutgers University, New Brunswick, NJ, USA D.H. Wort University of California, Hayward, CA, USA

by a discussion of the importance of beta in security analysis and further exploration of the determination and forecasting of beta. The discussion closes with the applications and implications of the CAPM. The appendix offers empirical evidence of the risk-return relationship.

6.2 A Graphical Approach to the Derivation of the Capital Asset Pricing Model Following the risk-return tradeoff principle and the portfolio diversification process, Sharpe (1964), Lintner (1965), and Mossin (1966) have developed an asset pricing model that can determine both the market price of a portfolio and the price of an individual security. They focus on the pricing determination of those parts of security risk that can be eliminated through diversification as well as those that cannot. Systematic risk is that part of total risk that results from the common variability of stock prices and the subsequent tendency of stock prices to move together with the general market. The other portion of total risk is unsystematic risk, the result of variables peculiar to the firm or industry; for example, a labor strike or resource shortage. Systematic risk, also referred to as market risk, reflects the swings of the general market. Some stocks and portfolios can be very sensitive to movements in the market, while others show more independence and stability. The universally accepted notion for the measure of a stock’s or a portfolio’s relative sensitivity to the market based upon its past record is the Greek letter beta .“/. The estimation, application, and forecasting of beta are discussed in the following sections.

6.2.1 The Lending, Borrowing, and Market Portfolios The next chapters will assume that the efficient frontier was constructed with risky assets only. If there are both risk-free

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Fig. 6.1 The capital market line

and risky assets, investors would then have the choice of investing in one or the other, or in some combination of the two. The concept of the risk-free asset is generally proxied by a government security, such as the Treasury bill (T-bill). T-bills are backed by the Federal government and are default free, hence considered riskless. An investor’s portfolio can be composed of different combinations of riskless and risky assets. Figure 6.1 is a graph of different sets of portfolio opportunities; it includes the risk-free asset with a return of Rf . Since the riskless asset has zero risk, it is represented by a point on the vertical axis. With the additional alternative to invest in risk-free assets that yield a return of Rf , the investor is able to create new combinations of portfolios that combine risk-free assets with the risky assets. The investor is thus able to achieve any combination of risk and return that lies along the line connecting Rf and a tangent point Mp the market portfolio. All portfolios along the line Rf Mp C are preferred to the risky portfolio opportunities on the curve AM p B. Therefore, the points on the line Rf Mp C represent the best attainable combinations of risk and return. At point Rf the investor has all available funds invested in the riskless asset and expects to receive the return of Rf . The portfolios along the line Rf Mp are lending portfolios and contain combinations of investments in the risk-free asset and investments in a portfolio of risky assets Mp . These are called lending portfolios because investors are, in effect, lending money to the government at the risk-free rate, when they invest in the risk-free asset. At point Mp the investor wants only risky assets and has put his wealth into the risky-asset portfolio, which is called the market portfolio. The investor at this point is neither lending nor borrowing. An alternative view is that at this point the investor may be lending and borrowing equal amounts that just offset each other. At Mp investors receive a rate of return Rm and undertake risk m .

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If it is assumed that the investor can borrow money at the risk-free rate Rf and invest this money in the risky portfolio Mp , he will be able to derive portfolios with higher rates of return but with higher risks along the line extending beyond Mp C . The portfolios along line Mp C are borrowing portfolios. They are so called because they contain a negative amount of the risk-free asset. The negative amount invested in the risk-free asset can be viewed as borrowing funds at the risk-free rate and investing in risky assets. The borrowing for investment is called margin and is controlled by the government. Therefore, the new efficient frontier becomes Rf Mp C and is referred to as the capital market line (CML). The capital market line describes the relationship between expected return and total risk.

6.2.2 The Capital Market Line An illustration and explanation of the capital market line have already been provided; the equation for the CML is    p  E Rp D Rf C E .Rm /  Rf m

(6.1)

where: Rf D the risk-free rate; Rm D return on market portfolio Mp ; Rp D return on the portfolio consisting of the risk-free asset and portfolio Mp ; and p m D the standard deviations of the portfolio and the market, respectively. The implementation of Equation (6.1) can be explained graphically. An investor has three choices in terms of investments. He may invest in the riskless asset Rf in the market portfolio Mp or in any other efficient portfolio along the capital market line, such as portfolio P of Fig. 6.1. If the investor puts his money into the riskless asset he can receive a certain return of Rf , if his investments are put into the market portfolio he can expect an average return of Rm and risk of m , if he invests in portfolio P he can expect an average return  of Rp with  risk of p . The difference between Rm and Rf Rm  Rf is called the market risk premium. The investor of portfolio P only needs to take on risk of p  so his risk premium is Rp  Rf . This is less than the market risk premium because the investor is taking on a smaller amount of risk, p < m . By geometric theory, triangles Rf PF and Rf Mp D are similar. Consequently, they are directly proportional. Thus:  p    E Rp  Rf D E .Rm /  Rf m

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At equilibrium all investors will want to hold a combination   of the risk-free asset and the tangency portfolio Mp . Since the market is cleared at equilibrium – that is, prices are such that the demand for all marketable assets equals their supply – the tangency portfolio Mp must represent the market portfolio. An individual security’s proportional makeup .Xi / in the market portfolio will be the ratio of its market value (its equilibrium price times the total number of shares outstanding) to the total market value of all securities in the market, or: Xi D

Market value of individual asset Market value of all assets

(6.2)

Thus, at equilibrium, prices are such that supply is equated to demand, and all securities in the market will be represented in the market portfolio according to their market value. Given the capital market line (CML), assuming that investors are interested only in mean and variance of return, the CML will dominate all other attainable portfolios and securities because of the dominance principle, as discussed in Chap. 4. This implies that they will attempt to put some portion of their wealth into the market portfolio of risky assets, depending on their individual risk preference. Since it has been established that the market portfolio is the only relevant portfolio of risky assets, the relevant risk measure of any individual security is its contribution to the risk of the market portfolio. The beta coefficient relates the covariance between security and market to the market’s total variance, and it is the relevant market-risk measure. This will be explored in the next two sections. Notationally, beta is ˇi D

im Cov .Ri ; Rm / D m2 Var .Rm /

Example 6.1 provides further illustration about risk/return tradeoffs.

at too large a premium. C is an asset with expected negative return – for example, a lottery ticket. The price of the lottery ticket is too high to be justified by the expected value of winning the jackpot. Even though the jackpot may be large, the probability of winning it is very small; hence, its expected value is small. Therefore, the cost of the ticket is greater than the expected value of winning, which yields a negative return.

6.2.3 The Security Market Line: The Capital Asset Pricing Model An asset’s systematic risk with the market, ˇ, is the only relevant risk measure of capital asset pricing for the individual asset and the portfolio. Consequently, a derivation of the relationship between systematic risk and return can be made where the expected linear relationship between these two variables is referred to as the security market line (SML), illustrated by Fig. 6.2. To derive the graphical picture of the SML it is necessary to list a few assumptions concerning the investors and the securities market. 1. Investors are risk averse. 2. The CAPM is a one-period model because it is assumed that investors maximize the utility of their end-of-period wealth. 3. All investors have the same efficient frontier – that is, they have homogeneous expectations concerning asset returns and risk. 4. Portfolios can be characterized by their means and variances.

Example 6.1. Describe the kinds of assets that could characterize the risk/return tradeoffs depicted by A, B, and C in the figure.

Solution A is a correctly priced asset, perhaps a stock or a portfolio. B is an overpriced asset, perhaps a bond selling

Fig. 6.2 The capital asset pricing model showing the security market line

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5. There exists a risk-free asset with a return Rf , the rate at which all investors borrow or lend. The borrowing rate is equal to the lending rate. 6. All assets are marketable and perfectly divisible, and their supplies are fixed. 7. There are no transaction costs. 8. Investors have all information available to them at no cost. 9. There are no taxes or regulations associated with trading. All individual assets or portfolios will fall along the SML. The position of an asset will depend on its systematic risk or beta. The required risk premium for the market is Rm  Rf , and ˇm is equal to one. Given the SML, the return on a risky asset is equal to:   E .Ri / D Rf C ˇi E .Rm /  Rf

(6.3)

E.Ri / D the expected rate of return for asset i ; Rf D the expected risk-free rate; ˇi D the measure of normalized systematic risk (beta) of asset i , and E.Rm / D the expected return on the market portfolio. (Note: This security market line is also generally called the capital asset pricing model (CAPM). The mathematical derivation of this model will be shown in the next section.) The relationship between the CML and the SML can be seen by rearranging the definition of the beta coefficient: ˇi D

im Cov .Ri ; Rm / i;m i m i;m i D 2 D D 2 Var .Rm / m m m

(6.4)

where: i D standard deviation of a security’s rate of return; m D standard deviation of the market rate of return; i;m D the correlation coefficient of Ri and Rm ; and im im D : i m If im D 1, then Equation (6.3) reduces to:  i  E .Rm /  Rf E .Ri / D Rf C m

(6:30 )

If i;m D 1, this implies that this portfolio is an efficient portfolio. If i is an individual security, it implies that the returns and risks associated with the asset are perfectly correlated with the market as a whole. There are several implications: 1. Equation (6.3) is a generalized case of Equation (6.30 ). 2. The SML instead of the CML should be used to price an individual security or an inefficient portfolio. 3. The CML prices the risk premium in terms of total risk, and the SML prices the risk premium in terms of systematic risk.

Example 6.2 provides further illustration. Example 6.2. Suppose the expected return on the market portfolio is 10% and that Rf D 6 percent. Further, if that you were confronted with an investment opportunity to buy a security with return expected to be 12% and with ˇ D 1:2. Should you undertake this investment? Solution Ri D a C bˇi for the market portfolio with ˇ D 1 we have 0:10 D a C .b 1/ D a C b and for the risk-free rate wit ˇ D 0, we have 0:06 D a C .b 0/ D a Solving these equations for a and b yields: a D 0:06 b D 0:04 For the security with ˇ D 1:2 the expected return given the SML is Ri D 0:06 C 0:04.1:2/ D 0:108 Since 12% > 10:8%, the security is undervalued; that is, the security should be purchased since its return is above the equilibrium return of 10.8% for that level of the risk.

6.3 Mathematical Approach to the Derivation of the Capital Asset Pricing Model Using the assumptions about efficient and perfect markets stated earlier in the chapter, we can show how Sharpe derived the capital asset pricing model (CAPM). Sharpe (1964) used a general risky asset that did not lie along the CML and dubbed it i . Risk and return for the possible combinations of security i with the market portfolio M are shown in Fig. 6.3. The average return and standard deviation for any I-M combination can be approached in the Markowitz fashion for a two-asset case:   .i/ E Rp Dwi E .Ri / C .1  wi /E.Rm /

1=2   .ii/  Rp D w2i i2 C .1  wi /2 m2 C 2 .1  wi / wi im (6.5)

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along the ı security market line. It should be noted that the term im m2 represents the beta coefficient for the regression of Ri vs. Rm so that Equation (6.9) can be rewritten as   E .Ri / D Rf C E .Rm /  Rf ˇi

(6.10)

which is the formula for the CAPM.

6.4 The Market Model and Risk Decomposition

Fig. 6.3 The opportunity set provide by combinations of risky asset i and the market portfolio M

in which wi represents excess demand for i or demand greater than its equilibrium weight in portfolio M. The changes in mean and standard deviation as the proportion wi changes are represented by:   @E Rp D E .Ri /  E .Rm / @wi i1=2 h   1=2 w2i i2 C .1  wi /2 m2 C 2wi .1  wi / im @ Rp D h i @wi  2wi i2  2m2 C 2wi m2 C 2im  4wi im (6.6)

when wi D 0, the i th security is held in proportion to its total market value, and there is no excess demand for security i . This is the key insight to Sharpe’s argument, for when wi D 0, it is possible to equate the slope of the curve iMi0 with the capital market line, thus obtaining an expression for the return on any risky security i . At equilibrium when wi D 0, the slope along the iMi0 curve will be equal to   @E Rp   D @ Rp

@E .Rp / @wi @ .Rp / @wi

D

E .Ri /  E .Rm / 2 im m m

(6.7)

To use the capital asset pricing model the market model must be employed to estimate the beta (systematic risk). In addition, the market model can be used to do risk decomposition. Both the market model and risk decomposition are discussed in this section.

6.4.1 The Market Model As noted previously, the total risk of a portfolio or security can be conceived as the sum of its systematic and unsystematic risks. The equation that expresses these concepts states that the return on any asset at time t can be expressed as a linear function of the market return at time t plus a random error component. Thus, the market model is expressed as: Ri;t D ˛i C ˇi Rm;t C ei;t

(6.11)

where Ri;t D the return of the i th security in time t; ˛i D the intercept of the regression; ˇi D the slope; Rm;t D the market return at time t; and ei;t D random error term.

The slope of the capital market line at point M is E .Rm /  Rf m

6.4.2 Risk Decomposition (6.8)

Setting Equation (6.7) equal to Equation (6.8) and rearranging the terms to solve for E .Ri / gives the equation for the security market line or CAPM:  im  E .Ri / D Rf C E .Rm /  Rf m2

(6.9)

This represents the return on any risky asset i . At equilibrium, every risky asset will be priced so that it lies

The regression model of Equation (6.11), which describes the risky asset’s characteristics relative to the market portfolio, is also called the characteristic line. Since beta is the slope coefficient for this regression (market model) it demonstrates how responsive returns for the individual earning assets are to the market portfolio. By using the market model, the total variance for security i i2 can be represented by and decomposed into: (6.12) i2 D ˇi2 m2 C ei2

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in which ˇi2 m2 is the systematic-risk component of total risk and ei2 is the unsystematic component. The CAPM as developed here is expressed in terms of expected values. Since expected values are not directly measured, we must transform the CAPM into an expression that uses observable variables. Assuming that on average expected returns E .Ri;t / for a security equal realized returns, returns can be expressed as: Ri;t D E .Ri;t / C ˇi ŒRm;t  E .Rm;t / C ei;t

(6.13)

in which ei;t is a random error term. Assuming that the expected value of the error term is 0, that it is uncorrelated with the term ŒRm;t  E .Rm;t / and that Cov.ei;t ; ei;t 1 / D 0, the expression for E .Ri;t / can be substituted from the CAPM Equation (6.10) into Equation (6.13):        Ri;t D Rf;t C E Rm;t  Rf;t ˇi C Rm;t  E Rm;t ˇi C ei;t

Simplifying this equation:   Ri;t  Rf;t D ˛i C ˇi Rm;t  Rf;t C ei;t

(6.14)

where ˛i is the intercept. All variables of Equation (6.14) can be estimated from observed data. Equation (6.14) is called the risk-premium version of the market model. It is similar to the market model indicated in Equation (6.11), except that instead of using the total returns Ri and Rm , it uses the riskpremium portion of the returns, or Ri  Rf and Rm  Rf . Example 6.3 provides further illustration. Example 6.3. To show how Equations (6.11) and (6.12) can be used, we will use monthly return data from Exxon Mobil and American Express. The time period covers January 2000 to November 2007. Their average return, beta coefficient, total variance, and residual variance are as indicated in the table below. We can see that American Express with ˇ D 1:26176 is more sensitive to the fluctuations of the market than Exxon Mobil with ˇ D 0:64647.

Exxon Mobil American Express

Return

Beta

Residual variance

Total variance

0.0126

0.64647

0.00214

0.00275

0.0055

1.26176

0.00138

0.00377

Both the magnitude of total variance and beta of American Express were larger than those of Exxon Mobil, although the average rates of return for Exxon Mobil were higher than American Express. Hence, Exxon Mobil was a more desirable security, if that was the only choice the investor had.

6.4.3 Why Beta Is Important for Security Analysis Implications and applications of beta coefficients in security analysis will be discussed in this section. Beta is used as a measurement of risk: it gauges the sensitivity of a stock or portfolio relative to the market, as indicated in Equation (6.11). A beta of 1.0 is characteristic of a broad market index such as the NYSE index or the S&P 500. A beta of 2.0 indicates that the stock will swing twice as far in either direction than a fall or rise in the market average. If the market gains 15%, a security of beta D 2 is expected to gain 30%. Conversely, should the market fall 15%, the stock is expected to fall 30%. A beta of 0.50 indicates that the stock is more stable than the market and will move only half as much as the market. For example, if the market loses 20%, a stock with beta D 0:50 will lose only 10% of the market, and if the market should gain 20%, the stock will gain only 10%. Highbeta stocks have been classified as aggressive while low-beta stocks are referred to as defensive. From the table in Example 6.3 it is clear that Exxon Mobil is an offensive stock while American Express is a defensive stock. The CAPM has illustrated the concept that risks are associated with portfolios where the relevant (systematic) risk of an individual security is dependent upon the security’s effect on portfolio risk. Therefore, the CAPM equation E.Ri / D Rf C ˇi ŒE.Rm /  Rf represents a description of how rates of return are established in the marketplace assuming investors behave according to the assumptions of the model. Thus a stock’s beta measures its contributions to the risk of the portfolio and is therefore a measure of the stock’s riskiness relative to the market. In the previous section it was found that the rates of return of stock i are presumed to bear a linear relationship with the market rate of return as defined in Equation (6.11). Equation (6.11) is a fixed-coefficient market model. Following Fabozzi and Francis (1978), Sunder (1980), and Lee and Chen (1980), a random-coefficient market model can be defined as Equation (6.15a) or Equation (6.15b): Rit D ˛i C ˇit Rmt C eit   Rit  Rft D ˛i0 C ˇit0 Rmt  Rft C eit

(6.15a) (6.15b)

in which ˇit D ˇi C it :it represents the random fluctuation associated with the beta coefficient. Using the randomcoefficient market model, the total risk can be decomposed into three components, as defined in Equation (6.16): i2 D ˇj2 m2 C "i2 C 2 m2

(6.16)

in which 2 m2 represents an interaction risk between the market and the random fluctuation of the ˇ. Equation (6.16)

6 Capital Asset Pricing Model and Beta Forecasting

is a generalized case of Equation (6.12). It contains systematic risk, unsystematic risk, and a risk term that reflects any interaction between the systematic and unsystematic risk. Both fixed-coefficient and random-coefficient market models can be used to do security analysis and portfolio selection, discussed later in this text. The relationship between total risk, market risk, and firmspecific risk can be shown as: Total risk D Market risk C Firm-specific risk Since firm-specific risk can be eliminated by diversification: Relevant risk D Market risk Beta is important for the investment manager because it can be used to (1) select individual stocks for investment, (2) construct portfolios of financial assets with desired levels of risk and return, and (3) evaluate the performance of portfolio managers. Details of the various uses of beta and the CAPM are provided later in this text. Example 6.4 provides further illustration. Example 6.4. Given the SML Ri D 0:06 C 0:08ˇi . What should the expected return of a security be if it has a ˇ twice as great as a similar security returning 18%? Solution Ri D 0:06 C 0:08ˇ D 0:18 Solving for ˇ yields ˇ D 1:5 Therefore, the security’s ˇ is 2 1:5 D 3:0 and the required return is: Ri D 0:06 C 0:08 .3:0/ D 0:30 A 30% return is required.

6.4.4 Determination of Systematic Risk As mentioned above, systematic risk is nondiversifiable because it runs across industries and companies and affects all securities. Previous chapters discussed many factors that determine risk. At this point we are interested only in those management decisions that can be related to the degree of systematic risk a firm exhibits; namely, the operating decisions and the financing decisions of management.

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There are two dimensions of risk that affect a firm’s systematic risk. The first is financial risk, the additional risk placed on the firm and its stock holders due to the firm’s decision to be leveraged – that is, to take on additional debt. The second, business risk, is the riskiness involved with a firm’s operations, if it takes on no debt. A particular firm’s capital structure affects the riskiness inherent in the company’s common stock and thus affects its required rate of return and the price of the stock. A company’s capital-structure policy requires choosing between risk and return. Taking on increasing levels of debt increases the riskiness of the firm’s earning stream, but it usually also results in a higher expected rate of return. High levels of risk tend to lower a stock’s price, but a high level of expected rates of return tends to raise it. Therefore, striking a balance with the optimal capital structure maximizes the price of the stock. Business risk is the risk inherent in a firm’s operations. It can also be defined as the uncertainty inherent in projection of future operating income or earnings before interest and taxes (EBIT). Fluctuations in EBIT can result from a number of factors. On the national level these can be economic factors such as inflationary or recessionary times. At the industry level some factors may be the level of competition between similar industries, natural or man-made catastrophes, labor strikes, price controls, and so on. There are a host of possibilities that affect EBIT by raising or lowering its level. Uncertainty regarding future income flows is a function of the company’s business risk. It may fluctuate among industries, among firms, and cross time. The extent of business risk is dependent on the firm and the Industry. Cyclical industries such as steel production or automobile manufacture have especially high business risks because they are dependent on the strength of the economy. The retail food industry is considered to be quite stable because food is a necessary good that will be purchased regardless of the state of the economy. Business risk is dependent on several factors, the more important of which are listed here: Demand variability : Stability in the levels of demand for the firm’s product results in a reduction of business risk. 1. Sales price variability: Highly volatile prices result in high-risk, volatile markets; therefore, stability of prices results in reduction of business risk. 2. Suppliers’ price variability: Firms whose input prices are highly variable are exposed to higher levels of risk. 3. Output price flexibility relative to input prices: As a result of inflation, a firm that is able to raise its output prices with increasing input costs minimizes its business risk. When a firm uses debt or financial leverage, business risk and financial risk are concentrated on the stockholders. For

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example, if a firm is capitalized only with common equity, then the investors all share the business risk in proportion to their ownership of stock. If, however, a firm is 50% levered (50% of the corporation is financed by debt, the other half by common equity), the investors who put up the equity will then have to bear all business risk and some financial risk. The effect of leverage on return on assets (ROA) and return on equity (ROE) and its effect on the stockholders can be generalized as follows: 1. The use of leverage or debt generally increases ROE. 2. The standard deviation of ROA.ROA / is a measure of business risk while the standard deviation of ROE.ROE / is a measure of the risk borne by stockholders. ROA D ROE if the firm is not levered; otherwise with the use of debt ROE > ROA an indication that business risk is being borne by stockholders. 3. The difference between ROE and ROA is the actual risk stockholders face and a measure of the increased risk resulting from financial leverage. Thus 2 2  ROA : Risk of financial leverage D ROE

It becomes necessary, therefore, to define a company’s sustainable growth rate: P .1  D/ .1 C L/

S D g D S T  P .1  D/ .1 C L/

(6.17)

where: P D the profit margin on all sales; D D the target dividend payout ratio; L D the target debt to equity ratio; T D the ratio of total assets to sales; S D annual sales; and

S D the increase in sales during the year. How is Equation (6.17) derived? Assuming a company is not raising new equity, the cash to finance growth must come from retained profits and new borrowings: Retained profits D Profits  Dividends D Profit margin Total sales  Dividends

6.5 Growth Rates, Accounting Betas, and Variance in EBIT Besides leverage, other financial variables associated with the firm can affect the beta coefficient. These are the growth rate, accounting beta, and variance in EBIT.

D P .S C S/ .1  D/ Further, because the company wants to maintain a target debt-to-equity ratio equal to L, each dollar added to the owners’ equity enables it to increase its indebtedness by $L. Since the owners’ equity will rise by an amount equal to retained profits: New borrowings D Retained profit

6.5.1 Growth Rates

Target debt-to-equity ratio D P .S C S /.1  D/L

The growth rate can be measured in terms of the growth in total assets or the growth in sales. It is determined by the percentage change between two periods. salest  salest 1

100% or salest 1 total assetst  total assetst 1

100% total assetst 1 Another method to measure growth rates is presented by Higgins (1984). Growth and its management present special problems in financial planning. From a financial perspective growth is not always a blessing. Rapid growth can put considerable strain on a company’s resources, and unless management is aware of this effect and takes active steps to control it, rapid growth can lead to bankruptcy.

The use of cash represented by the increase in assets must equal the two sources of cash (retained profits and new borrowings): Uses of cash D Sources of cash Increases in assets D Retained profits C New borrowings

ST D P .S C S /.1  D/ C P .S C S /.1  D/L

S T D P .1  D/.1 C L/S C P .1  D/.1 C L/ S

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S ŒT  P .1  D/.1 C L/ D P .1  D/.1 C L/S P .1  D/.1 C L/

S D S T  P .1  D/.1 C L/ In Equation (6.17) the S =S or g  is the firm’s sustainable growth rate assuming no infusion of new equity. Therefore, a company’s growth rate in sales must equal the indicated combination of four ratios, P, D, L, and T . In addition, if the company’s growth rate differs from g  , one or more of the ratios must change. For example, if a company grows at a rate in excess of g  , then it must either use its assets more efficiently or it must alter its financial policies. Efficiency is represented by the profit margin and asset-to-sales ratio. It therefore would need to increase its profit margin .P / or decrease its asset-to-sales ratio .T / to increase efficiency. Financial policies are represented by payout or leverage ratios. In this case, a decrease in its payout ratio .D/ or an increase in its leverage .L/ would be necessary to alter its financial policies to accommodate a different growth rate. It should be noted that increasing efficiency is not always possible and altering financial policies not always wise (For other available methods to alter growth the reader is referred to Chap. 8 of Lee (1985).)

6.5.2 Accounting Beta The accounting beta can be calculated from earnings-pershare data. Using EPS as an example, beta can be computed as follows: EPSi;t D ˛i C ˇi EPSm;t C ei;t where: EPSi;t D earnings per share of firm i at time t; EPSm;t D earnings per share of market average at time t; and ei;t D error term. The estimate of ˇi is the EPS type of accounting beta.

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The total variance of EBIT can be used to measure the overall fluctuation of accounting earnings for a firm.

6.5.4 Capital-Labor Ratio The capital-labor ratio has an impact on the magnitude of the beta coefficient. In order to examine this impact it is necessary to examine the capital-labor ratio. A production function is a function that can be seen as a function of labor and capital: Q D f .K; L/

(6.18)

where K D capital and L D labor: K=L (the capital-labor ratio) is generally used to measure a firm’s degree of capital intensity. Corporations often choose between increasing their capital intensity through installation of computers, use of robotics in place of labor, or increase in labor inputs. Small industries that specialize in hand-crafted or -tooled goods will have to increase their capital ratio in to increase production. However, it has been the trend in recent years for many growth-oriented firms, whether manufacturers or other members of the business sector, to increase their efficiencies through increased investments in capital. Auto manufacturers are finding that robots are able to assemble cars of high and consistent quality at only a fraction of the cost of workers. Capital intensity results in increased total risks and generally results in an increase in beta. Large investments are often needed to fully automate a plant or to computerize a bank completely. Taking on debt or issuing securities is normally how these capitalization increases are financed. If the capital-labor ratio is greater than one – that is, if K is greater than L – a firm is capital intensive. If the capital-labor ratio is less than one – that is, if K is less than L – then there is a reduction in capital intensity and a shift toward humanresource investment.

6.5.5 Fixed Costs and Variable Costs 6.5.3 Variance in EBIT The variance in EBIT .X / can be defined as: n  P

Xt  X

2

t D1

n1 in which Xt D earnings before interest and taxes in period t, and X D average EBIT.

Business risk is dependent upon the extent a firm builds fixed costs into its operations. If a large percentage of a firm’s costs are fixed, costs cannot decline proportionally when demand falls off. If fixed costs are high, then a slight drop in sales can lead to large declines in EBIT. Therefore, the higher a firm’s fixed costs are, the greater its business risk and, generally, the higher its beta. A firm with a large amount of fixed costs is said to have a large degree of operating leverage. An example of these costs may be the highly skilled workers of an engineering

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firm. The firm cannot hire and fire experienced and highly skilled workers easily; therefore, the workers must be retained and paid during a period of slack demand. Similarly, a firm that is highly leveraged will be characterized by the scenario that small changes in sales will result in large changes in operating income. Variable costs have the opposite effect, because they are adjustable to the firm’s needs. Should a drop in sales occur, variable costs can be lowered to meet the lowered output. The extent to which firms can control their operating leverage is dependent upon their technological needs. Companies that require large investments in fixed asset – such as steel mills, auto manufacturers, and airlines – will have large fixed costs and operating leverages. Therefore, how much a company is willing to undertake operating leverages must come into play during capital-budgeting decisions. If the company is risk averse, it may opt for alternatives with smaller investments and fixed costs. A company having a larger percentage of fixed costs generally implies that it uses more capital-intensive types of technology in production. For example, an auto manufacturer such as General Motors has a higher percentage of fixed costs than a food manufacturer such as General Foods. Therefore, GM’s capital-labor ratio can be expected to be higher than that of General Foods. An implication of this phenomenon is that General Foods has a lower beta than General Motors.

C.-F. Lee et al.

the estimation of beta. They conclude that using both price and accounting information shows promise for better beta estimates. Rosenberg and Marathe (1975) use 54 factors in six categories to estimate betas. Their factors include historical price information, price/earning ratios, financial ratios, and statistical measures associated with the market model. They support the notion that betas are determined by fundamental factors related to firms in addition to market pricing data. Given that the most useful beta is one that can be forecasted correctly, and that the beta is related to the financial and business risk of the firm as well as the degree to which a firm’s businesses co-vary with the total economy, it becomes necessary to determine the best way to forecast beta. The next section addresses this issue.

6.5.7 Market-Based versus Accounting-Based Beta Forecasting Market-based beta forecasts are based on market information alone. Historical betas of firms are used as a proxy for their future betas. This implies that the unadjusted sample beta, ˇOt , is equal to the population value of future beta: ˇt C1 D ˇOt

6.5.6 Beta Forecasting Analysts and investment managers who use the CAPM and beta are interested in whether the beta coefficient and the standard-deviation statistics for different securities and portfolios are stable through time or whether they change in a predictable fashion. If the beta coefficient and standard deviation are stable, then using a beta derived from current and historical price data is fine, because the beta today is the same as the beta in the future. However, if the beta coefficient and standard deviation are unstable or vary through time, the analyst or manager must forecast a beta’s future value before employing it. The available evidence on the stability of beta indicates that the beta on an individual security is generally not stable, while portfolios have stable betas. Hence we are faced with the problem of forecasting future betas in order to use the CAPM for individual securities. Beta forecasting refers to using the historical beta estimates or other historical financial information to forecast future betas. Beaver et al. (1970) argue that if it is possible to find the underlying determinants of beta, such knowledge can be used to help forecast future betas. Beaver et al. use accounting information from the financial statements of firms to help in

(6.19a)

Alternatively, there may be a systematic relationship between the estimated betas for the first period and those of the second period, as shown by Blume (1971): ˇOi;t C1 D a0 C a1 ˇOi;t

(6.19b)

in which ˇOi;t C1 and ˇOi;t estimated beta for the i th firm in period t C 1 and t respectively. Example 6.5 provides further illustration. Example 6.5. If aO 0 D 0:35; aO 1 D 0:80, and, ˇOi;t D 0:12, then the future beta can be either ˇt C1 D 1:2 or ˇOt C1 D 0:35 C .0:80/.1:2/ D 1:31 It is worthwhile to note that Value Line uses Equation (6.19b) to estimate the future beta. Accounting-based beta forecasts rely on the relationships of accounting information such as the growth rate of the firm, EBIT, leverage, and the accounting beta as a basis for forecasting beta. To use accounting information in beta forecasts, the historical beta estimates are first cross-sectionally related to accounting information such as growth rate, variance of EBIT, leverage, accounting beta, and so on:

6 Capital Asset Pricing Model and Beta Forecasting

ˇi D a0 C a1 X1i C a2 X2i C aj Xj i C : : : C am Xmi

103

(6.20)

where Xji is the j th accounting variables for i th firm, and aj is the regression coefficient. Some researchers have found that the historically based beta is the best forecast, while others have found that the accounting-based beta is best. Lee et al. (1986) use composite concepts to show that both accounting and market information are useful for beta forecasting. They find that neither beta forecasts based on market information nor those based on accounting information are conditionally more efficient with respect to each other. It can be inferred, then, that each set of forecasts contains useful information for the prediction of systematic risk. The statistical procedure used by Lee et al. (1986) is the ordinary least-squares method. Ordinary least squares is a statistical procedure for finding the best fitting straight line for a set of points; it seems in many respects a formalization of the procedure employed when fitting a line by eye. For instance, when visually fitting a line to a set of data, the ruler is moved until it appears that the deviations of the points from the prospective line have been minimized. If the predicted value of yi (dependent variable) obtained from the fitted line is denoted as yOi the prediction equation becomes: yOi D ˇO0 C ˇO1 xi where ˇO0 and ˇO1 represent estimates of the true ˇ0 and ˇ1 I xi is an independent variable. Graphically, the vertical lines drawn from the prediction line to each point represent the deviations of the points from the predicted value of y. Thus the deviation of the i th point is yi  yOi , where: SSE D

n X

.yi  yOi /

In order to find the best fit it is necessary to minimize the deviations of the points. A criterion of best fit that is often employed is known as the principle of least squares. It may be stated as follows: Choose the best fitting line as the one that minimizes the sum of squares of the errors (SSE) of the observed values of Y from those predicted. Further, the mean sum of squares of the error (MSSE) of the observed values of Y could be minimized from the predicted mean. Expressed mathematically:

MSSE D

n P

.yi  yOi /2

i D1

n

.yi  yOi /2

i D1

n

Mincer and Zarnowitz (1969) suggest a decomposition of the mean squared error term into three components representing

2    D .yi  yOi /2 C Sy  Sy C 1  2 Sy2

where y, and Sy represent the mean, standard deviation, and correlation coefficient of y, respectively. Using the ordinary least-squares method, Lee et al. (1986) show that it is possible to achieve better forecasts by combining two types of betas. Using an ordinary least-squares estimate of ˇOLS and a Bayesian adjustment procedure of Vasicek (1973) as defined in Equation (6.21):

ˇV D

ˇN C V ˇO V

ˇOOLS

ˇOOLS (6.21)

1 1

C V ˇO V ˇOOLS

where: ˇOLS D the least-squares estimate of a first-period individual beta; V ˇOOLS D variance estimate of ˇOLS ; ˇN D the

cross-sectional mean value of estimated ˇOLS ; O V ˇ D the cross-sectional variance of estimated ˇOLS ; and ˇV D the Bayesian adjusted beta. Equation (6.21) indicates that the Vasicek type of Bayesian adjustment beta is a weighted average of ˇN and ˇOLS . The weights are

2

i D1

n P

bias, inefficiency, and random error. Mathematically, this is represented:

w1 D

1 V ˇO 1 1 C

V ˇO V ˇOOLS



and w2 D

1

V ˇOOLS

1 1 C

V ˇO V ˇOOLS

The authors first use first-period regressions projected forward to obtain forecasts of the second-period betas. The summary of the stepwise regression for the first-period data is given in Table 6.1. Table 6.2 summarizes the overall mean squared errors together with the mean squared error decomposition. Lee et al. (1986) note that for forecasts developed without the Bayesian adjustment, the accounting-based forecast is better, in terms of overall mean squared error. This advantage results entirely from inefficiency in the Mincer-Zarnowitz sense of the market-based forecast. By contrast, the Bayesian-adjusted market-based forecasts suffer far less from this problem and, as a result, the mean squared prediction error for Bayesian-adjusted market-based

104

C.-F. Lee et al.

Table 6.1 Summary of stepwise regression results for first-period data

Dependent variable adjusted R2 Independent variables Intercept Financial leverage Dividend payout Sales Operating income Assets

“O OLS 0.245

“O V 0.236

Coefficients (t -values) 0.911 1.037 (11.92) (12.71) 0.704 0.599 (5.31) (5.81) 0:175 0:108 .3:50/ .2:85/ 0.030 0.018 (3.02) (2.28) 0.011 0.026 (2.18) (2.28) – 0:026 – .2:41/

Source: Lee et al. (1986) Table 6.2 Mean squared error decompositions for forecasts of second-period beta Market-based forecasts Accounting-based forecasts Without Bayesian adjustment Bias Inefficiency Random error Total mean squared error

0.0003 0.0309 0.0581 0.0893

With Bayesian adjustment 0.0000 0.0090 0.0577 0.0667

Without Bayesian adjustmentl 0.0003 0.0015 0.0659 0.0677

With Bayesian adjustment 0.0000 0.0001 0.0638 0.0640

Source: Lee et al. (1986)

forecasts are only marginally larger than the accountingbased forecasts 0:0667 compared to 0.0640. Lee et al. (1986) tested a composite predictor for beta consisting of both a market beta and an accounting beta. Table 6.3 shows the results of using this composite method to forecast beta for both the Bayesian-adjusted and nonadjusted model. In Table 6.3, ˇOLS (2) represents the OLS estimated A and ˇOVA represent accountingbeta in the second period. ˇOLS based beta without and with Bayesian adjustment, respectively. By comparing both MSE (market) of 0.893 and 0.0667 and MSE(accounting) of 0.0677 and 0.0640 with MSE(composite) of 0.0545 and 0.0530, respectively, it can be concluded that on the basis of mean squared error, the composite predictor outperforms either the market- or accounting-information predictor. Of the models used for the composite predictor, it appears that the Bayesian adjustment case is superior. From research it appears, then, that accounting-based and market-based forecasts can be combined to produce a superior composite forecast of beta.

6.6 Some Applications and Implications of the Capital Asset Pricing Model Since its development, the uses of the capital asset pricing model (CAPM) have extended into all areas of corporate finance and investments. The CAPM can be applied to two

aspects of the capital-budgeting problem: (1) determining the cost of capital, and (2) assessing the riskiness of a project under consideration. The CAPM can also be useful in a realestate problem, deciding to lease or buy. The use of the CAPM can be extended into valuation of the entire firm. Because of its impact upon firm valuation, the CAPM has been of great use in the merger-analysis area of financial analysis. The CAPM has also been used to test various financial theories. By including a dividend term and considering its effects, the CAPM can be used to test the effects of the firm’s dividend policy. An area that has received a great deal of attention is the use of the CAPM in testing the efficient-market hypothesis. An application of the CAPM to the capital-budgeting process concerns the valuation of risky projects. If accurate estimates can be made about the systematic risk of a project, then the CAPM can be used to determine the return necessary to compensate the firm for the project’s risk. If the sum of the estimated cash flows discounted by the CAPM-calculated required rate of return is positive, then the firm should undertake the project. Rubinstein (1973) demonstrates how the CAPM can be used to value securities and to calculate their risk-adjusted equilibrium price. First, the CAPM must be converted to using price variables instead of expected return. It may be rewritten: P1  P0 E.Ri / D P0

6 Capital Asset Pricing Model and Beta Forecasting

105

Table 6.3 Results for the estimation of market- and accounting-based composite with and without Bayesian adjustment O Without Bayesian adjustment “O OLS .2/ D a C b1 “O A OLS C b2 “OLS C w “O A OLS 0.399 0.102 MSE D 0:0545 O With Bayesian adjustment “O OLS .2/ D a C b1 “O A V C b2 “V C w Intercept 0:251 0:104

Coefficients Standard errors

Coefficients Standard errors

Intercept

“O A V

0:096 0:134

0.606 0.133 MSE D 0:0530

“O OLS 0:370 0:052

“O V 0:475 0:067

Source: Lee et al. (1986)

where: Ri D the expected returns for the i th firms; P1 D the price of stock in time l; and P0 D the price of stock in the previous period. Thus, the CAPM is redefined:  im  E .P1 /  P0 D Rf:t C E .Rm /  Rf P0 m2

(6.22)

or, rearranging Equation (6.22): P0 D

E .P1 /   1 C Rf C E .Rm /  Rf im2 m

Thus the rate of return used to discount the expected endof-period price contains a risk premium dependent upon the security’s systematic risk. CAPM has also been applied in the analysis of mergers. It has been shown that the risks of a portfolio can be substantially reduced with the inclusion of securities that are not perfectly correlated in terms of returns. This principle also applies with respect to the mergers between firms. The merging of two firms with different product lines, called a conglomerate merger, creates diversification, considered of great benefit. Suppose that one firm sells a product that is recession resistant, then a decrease in earnings of one division of the conglomerate will be offset by the steady earnings of another division. The overall result will be a relatively stable income stream despite shifting trends in the economy.

6.7 Conclusion This chapter has discussed the basic concepts of risk and diversification and how they pertain to the CAPM. The procedures for deriving the CAPM itself were presented, and the CAPM was shown to be an extension of the capital market

line theory. The possible uses of the CAPM in financial management were also indicated. The concept of beta and its importance to the financial manager was introduced. Beta represents the systematic risk of a firm and is a comparison measure between a particular firm’s security or portfolio risk and the market average. Systematic risk was further discussed through an investigation of the beta coefficient and the impact of other important financial variables on the magnitude of the beta coefficient. The statistical method of least squares and its application were introduced, and beta forecasts based on the leastsquares method were compared with those based on market information, accounting information, and a compositepredictor beta forecast composed of both accounting and market information. The composite predictor appears to yield a better forecast than either the market-information or accounting-information forecasts separately.

References Banz, R. W. 1981. “The relationship between return and market value of common stocks.” Journal of Financial Economics 9, 3–18. Basu, S. 1977. “Investment performance of common stocks in relation to their price-earnings ratios: a test of the efficient markets hypothesis.” Journal of Finance 32, 663–682. Beaver, W., P. Kettler, and M. Scholes. 1970. “The association between market determined and accounting determined risk measures.” Accounting Review 45, 654–682. Black, F., M. C. Jensen, and M. Scholes. 1972. “The capital asset pricing model: some empirical tests,” in Studies in the theory of capital markets, M. C. Jensen (Ed.). Praeger, New York, pp. 20–46. Blume, M. E. 1971. “On the assessment of risk.” Journal of Finance 26, 1–10. Blume, M. E. and I. Friend. 1973. “A new look at the capital asset pricing model.” Journal of Finance 28, 19–34. Blume, M. E. and F. Husick. 1973. “Price, beta, and exchange listing.” Journal of Finance 28, 19–34. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Brennan, M. J. 1971. “Capital market equilibrium with divergent borrowing and lending rate.” Journal of Financial and Quantitative Analysis 7, 1197–1205.

106 Douglas, G. W. 1969. “Risk in the equity markets: an empirical appraisal of market efficiency.” Yale Economic Essays 9, 3–45. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Fabozzi, F. J. and J. C. Francis. 1978. “Beta as a random coefficient.” Journal of Financial and Quantitative Analysis 13, 101–116. Fama, E. F. 1968. “Risk, return and equilibrium: some clarifying comments.” Journal of Finance 23, 29–40. Fama, E. F, and J. MacBeth. 1973. “Risk, return and equilibrium: empirical tests.” Journal of Political Economy 31, 607–636. Francis, J. C. 1986. Investments: analysis and management, 4th Edition, McGraw-Hill, New York. Higgins, R. C. 1974. “Growth, dividend policy and cost of capital in the electric utility industry.” Journal of Finance 29, 1189–1201. Higgins, R. C. 1984. Analysis for financial management, Richard D. Irwin, Homewood, IL. Jensen, M. C. 1969. “Risk, the pricing of capital assets, and the evaluation of investment portfolio.” Journal of Business 42, 607–636. Jensen, M. C. 1972. “Capital markets: theory and evidence.” The Bell Journal of Economic and Management Science 3, 357–398. Keim, D. B. 1983. “Size-related anomalies and stock return seasonality: further empirical evidence.” Journal of Financial Economics 11, 13–32. Lee, C. F. 1985. Financial analysis and planning: theory and application, Addison-Wesley, Reading, MA. Lee, C. F. and S. N. Chen. 1980. “A random coefficient model for reexamining risk decomposition method and risk-return relationship test.” Quarterly Review of Economics and Business 20, 58–69. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York. Lee, C. F, P. Newbold, J. E. Finnerty, and C. C. Chu. 1986. “On accounting-based, market-based and composite-based beta predictions: methods and implications.” The Financial Review 21, 51–68. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Lintner, J. 1965. “The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets.” Review of Economics and Statistics 47, 13–37. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, Hoboken, NJ. Merton, R. C. 1980. “On estimating the expected return on the market, an exploratory investigation.” Journal of Financial Economics 8, 323–361. Miller, M. and M. Scholes. 1972. “Rates of return in relation to risk: a reexamination of some recent findings,” in Studies in theory of capital markets, M. C. Jensen (Ed.). Praeger, New York, pp. 47–78. Mincer, J. and V. Zarnowitz. 1969. “The evaluation of economic forecasts,” in Economic forecasts and expectations, J. Mincer (Ed.). National Bureau of Economic Research, New York. Mossin, J. 1966. “Equilibrium in a capital asset market.” Econometria 34, 768–873. Reinganum, M. R. 1981. “Misspecification of capital asset pricing: empirical anomalies based on earnings yields and market values.” Journal of Financial Economics 8, 19–46. Roll, R. 1977. “A critique of the asset pricing theory’s tests-part i: on past and potential testability of the theory.” Journal of Financial Economics 4, 129–176. Roll, R. 1978. “Ambiguity when performance is measured by the securities market line.” Journal of Finance 33, 1051–1069. Rosenberg, B. and V. Marathe. 1975. Tests of the capital asset pricing hypothesis, Working Paper No. 32 of the Research Program in France, Graduate School of Business and Public Administration, University of California, Berkeley.

C.-F. Lee et al. Rosenberg, B. and W. McKibben. 1973. “The prediction of systematic and specific risk in common stocks.” Journal of Finance and Quantitative Analysis 8, 317–333. Rubinstein, M. E. 1973. “A mean-variance synthesis of corporate financial theory.” Journal of Finance 28, 167–168. Sharpe, W. 1964. “Capital asset prices: a theory of market equilibrium under conditions of risk.” Journal of Finance 19, 425–442. Sharpe, W. 1966. “Mutual fund performance.” Journal of Business 39, 119–138. Sunder, S. 1980. “Stationarity of market risk: random coefficients tests for individual stocks.” Journal of Finance 35, 883–896. Vasicek, O. A. 1973. “A note on using cross-sectional information in Bayesian estimation of security betas.” Journal of Finance 28, 1233–1239. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Appendix 6A Empirical Evidence for the Risk-Return Relationship The validity of the CAPM can be borne out partly through observations of actual portfolios held in the marketplace. As discussed in previous chapters, there are two primary relationships between risk and return. First, the rates of return of efficient portfolios are linear functions of their riskiness as measured by their standard deviation. This is illustrated by the capital market line (CML). Second, the rate of return of an individual asset is determined by its contribution of risk to the portfolio, and this is measured by beta, where beta has a linear relationship with the security’s expected rate of return. This is illustrated by the security market line (SML). The performance of mutual funds can be employed to test the explanatory powers of the linear relationship between risk and return of the CML. Mutual funds are professionally managed and therefore the most visible type of portfolio, easily used for comparison testing. One study was performed by Sharpe (1966) to test the performance of a fund and the relationship between its rate of return and risk over time. Sharpe computed average annual returns and the standard deviations of these returns for 34 mutual funds from 1954 to 1963. His model implies that portfolios with higher risks will receive higher returns. This Sharpe found to be true for all 34 funds. He calculated the correlation between average returns and their standard deviations to be 0.836, indicating that more than 80% of the difference in returns was due to differences in risk. Sharpe also found that there was a linear relationship between returns and risks, except in the region of very high risks. Sharpe’s study provides basic support to the contention that the CML explains the relationship between risk and return, both in portfolio theory and in the marketplace. Another study was performed by Jensen (1969). He studied the correlation of beta coefficients (market sensitivity) and the expected return of mutual funds. On the basis

6 Capital Asset Pricing Model and Beta Forecasting

107

of analysis of 115 mutual funds over a 9-year period he was able to conclude that high returns were associated with high volatility or high systematic risks. He also found evidence that beta coefficients are a valid and accurate measure of risk. Both the Sharpe and Jensen studies on the risk and return of mutual funds show that an empirical risk-return relationship does exist among mutual funds (However, Sharpe used the capital market line to perform his empirical tests while Jensen used the security market line, derived by Sharpe 1964.) The second implication of the risk-return relationship is that the risk premium on individual assets depends on the contribution each makes to the riskiness of the entire portfolio. The CAPM is a simple linear model expressed in terms of expected returns and expected risk. In its ex-ante form:   E .Ri / D Rf C E .Rm /  Rf ˇi

(6A.1)

Although many of the aforementioned extensions of the model support this simple linear form, others suggest that it may not be linear, that factors other than beta are needed to explain E .Ri /, or that the Rf is not the appropriate riskless rate. The first step necessary to empirically test the theoretical CAPM is to transform it from expectations (ex-ante) form into a form that uses observed data. On average, the expected rate of return on an asset is equal to the realized rate of return. This can be written: Rit D E .Rit / C ˇi ımt C eit where: ımt D Rmt  E.Rmt /I E.ımt / D 0I eit D a random error termI Cov .eit ; ımt / D 0I Cov .eit ; ei t 1 / D 0I and ˇit D Cov.Rit Rmt /Var.Rmt /: When CAPM is empirically tested it is usually written in the following form: 0 D 0 C 1 ˇp C ept Rpt

where:

1 D Rmt  Rf t 0 Rpt D Rpt  Rf t

(6A.2)

These relationships can be stated as follows. 1. The intercept term 0 should not be significantly different from zero. 2. Beta should be the only factor that explains the rate of return on a risky asset. If other terms, such as residual variance, dividend yields, price/earnings ratios, firm size, or beta squared are included in an attempt to explain return, they should have no explanatory power. 3. The relationship should be linear in beta. 4. The coefficient of beta, “II” should be equal to Rmt  Rft . 5. When the equation is estimated over very long periods of time, the rate of return on the market portfolio should be greater than the risk-free rate. The work of Jensen (1972) provides a comprehensive and unifying review of the theoretical developments and the empirical work done in the field until that year. In his paper he points out that the main result of the original papers in this area is the demonstration that one can derive the individual’s demand function for assets, aggregate these demands to obtain equilibrium prices (or expected returns) solely as a function of potentially measurable market parameters. Thus, the model becomes testable. Let us now summarize the empirical work of Douglas (1969), Black et al. (1972), and Fama and MacBeth (1973). The first published test of the CAPM was by Douglas (1969), who regressed the returns of a large cross-sectional sample of common stocks on their own variances and on their beta coefficients ˇi obtained by market models. His results are in variance with the Sharpe-Lintner-Mossin model, for he found that the return was positively related to the variance of the security but not the covariance with the index of returns. Douglas also summarizes some of the work of Lintner (1965), who estimates beta from a typical market model. Douglas then adds a term for the standard deviation of error (proxy for unsystematic risk). He finds that the coefficient for the unsystematic risk is both positive and significant, the intercept term is higher than the appropriate risk-free rate, and the coefficient for the market risk premium is too low. Black, Jensen, and Scholes observe that cross-sectional tests may not provide direct validation of the CAPM, and they proceed to construct a time-series test, which they consider more powerful. Their results lead them to assert that the usual form of the CAPM does not provide an accurate description of the structure of security returns. Their results indicate that ˇs are non-zero and are directly related to the risk level. Low-beta securities earn significantly more on average than predicted by the model, and high-risk securities earn significantly less on average than predicted by the model. They go on to argue for a two-factor model: Rit D .1  ˇi /Rzt C ˇi .Rmt / C eit

(6A.3)

108

C.-F. Lee et al.

If E.Rz / D 0 then the Sharpe-Lintner-Mossin CAPM would be consistent with this model. However, the cross-sectional term for the intercept is a constant and not equal to zero. They then proceed to look for a rationale of this finding in Black’s zero-beta model. Fama and MacBeth (1973) test (1) a linear relationship between return on the portfolio and the portfolio’s beta, and (2) whether unsystematic risk has an effect between portfolio return and a risk measure in addition to beta. Their basic estimation equation is: 2 RQ it D Q0t C Q1t ˇ i C Q2t ˇ i C Q3t  i2 ./ C eQit

(6A.4)

in which Q0t is the intercept term, ˇ i is the average of the ˇi for all individual securities in portfolio i , and  i ./ is the average of the residual standard deviations from all securities in portfolio j . Although they find that there are variables in addition to the portfolio beta that systematically affect period-byperiod returns (which are apparently related to the average squared beta of the portfolio and the risk factor other than beta), they dismiss the latter as “almost surely proxies,” since “there is no rationale for their presence in our stochastic riskreturn model.” Their results seem to suggest that the SharpeLintner-Mossin model does not hold. The intercept factor, Q0t is generally greater than RF , and Q1t is substantially less than Rm  Rf . This seems to indicate that the zero-beta model is more consistent with the data. Blume and Husick (1973) find empirical evidence to indicate that historical rates of return may sometimes foreshadow changes in future betas, and that stocks with higher transaction costs should yield somewhat higher gross expected returns. Their data indicate that beta is not stationary over time, and that it does change over time as a function of price. Transaction-cost effects appear less important than the informational effects of price in explaining future returns or future betas. Therefore, the return-generating process may be more complex than what has been assumed. Blume and Friend (1973) examine both theoretically and empirically the reasons why the CAPM does not adequately explain differential returns on financial assets. Empirically the risk-return tradeoffs implied by stocks on the New York Stock Exchange for three different periods after World War II cast doubt on the validity of the CAPM either in its S-L-M form or zero-beta form. However, they do confirm the linearity of the relationship for NYSE stocks. Rosenberg and McKibben (1973) observe that predictions of the riskiness of returns on common stocks can be based on fundamental accounting data for the firm and also on the previous history of stock prices. This paper tries to combine both sources of information to provide efficient predictions. A stochastic model of the parameters is built and the mean squared error is used as a criterion for the evaluation of the

forecasting performance of estimators. They conclude that the results “strongly confirm the usefulness of the specific risk predictions based on the accounting descriptors.” Merton (1980) is concerned with the estimation of the expected return on the market. He notes that the current practice for estimating the expected market return adds the historical average realized excess market returns to the current observed interest rate. However, while this model explicitly reflects the dependence of the market return on the interest rate, it fails to account for the effects of changes in the level of market risk. Three models of equilibrium expected market returns are elaborated, and estimation procedures that incorporate the prior restriction that equilibrium expected excess returns on the market must be positive are derived and applied to return data for the period 1926–1978. The following are the principal conclusions of the study. 1. The nonnegativity restriction above should be explicitly included as part of the specifications. 2. Estimators that use realized returns should be adjusted for heteroskedasticity. Roll (1977) directs an attack on the empirical tests of the CAPM. While recognizing that the theory is testable in principle, he asserts that “no correct and unambiguous test to the theory (has) appeared in the literature and there is practically no possibility that such a test can be accomplished in the future.” This conclusion is derived from the mathematical equivalence between the individual return beta linearity and the market portfolio’s mean variance efficiency. Therefore, any valid test presupposes complete knowledge of the market portfolio’s composition. The major results reported by Roll from his theoretical inquiry include the following: 1. The only testable hypothesis is that the market portfolio is mean-variance efficient. 2. All other so-called implications of the CAPM are not independently testable. 3. In any sample there will always be an infinite number of ex-post mean-variance efficient portfolios; betas calculated will satisfy the linearity relation exactly, whether or not the true market portfolio is mean-variance efficient. 4. The theory is not testable unless the exact composition of the true market portfolio is known and used in the tests. 5. Using a proxy for the market portfolio does not solve the problem, for the proxy itself might be mean-variance efficient even when the true market portfolio is not, and conversely. 6. Empirical tests that reject the S-L-M model have results fully compatible with the S-L-M model and a specification error in the measured market portfolio. 7. If the selected index is mean-variance efficient, then the betas of all assets are related to their mean returns by the same linear function (all assets and portfolios fall exactly on the SML).

6 Capital Asset Pricing Model and Beta Forecasting

8. For every ranking of performances obtained with a mean-variance inefficient index, there exists another nonefficient index that reverses the ranking. Roll’s critique is a broad indictment of most of the accepted empirical evidence concerning the CAPM theory.

Appendix 6B Anomalies in the Semi-strong Efficient-Market Hypothesis Three anomalies in the semi-strong efficient-market hypothesis are noteworthy. Four authors – Basu, Banz, Reinganum, and Keirn – deal with these three anomalies: (1) P/E ratios; (2) size effects; and (3) the January effect. Basu (1977) empirically notes that a firm with a low P/E ratio, when adjusted for risk, has an excess return over firms that have a high P/E ratio. If this is true, then there are implications for the market’s efficiency, the validity of the CAPM, or both. However, Basu found that the excess returns, when adjusted for transaction costs, taxes, and so forth, were so much smaller as to be insignificant. Therefore, CAPM and market efficiency were supported. In conjecturing why a difference in P/E ratios could affect the returns of the firm, we believe that the relationship may involve the firm’s ability to raise debt. A low P/E ratio may indicate more difficulty in raising capital than a high P/E ratio. This difficulty in raising capital could result in different lending and borrowing rates for different firms. Therefore, a Brennan version of the standard CAPM pricing model may be more applicable. Reinganum (1981) also empirically tests the P/E effect and finds the same results as Basu. In addition, Reinganum was concerned with the efficiency of the market. In order to see if the market was informationally efficient, Reinganum also looked at returns of firms with neither high nor low P/E ratios. He found these firms to be correctly priced. From these results he conjectures that the market was informationally efficient, but CAPM did not allow for the P/E effect on returns. Therefore an APT model with the P/E effect as one of its factors would be preferable to the CAPM model. Banz (1981) empirically tests the effect of firm size, finding that small-company stock returns were higher than

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large-company stock returns. Banz argues that the P/E ratio serves as a proxy for the size of a firm and not vice versa. His conjecture about why this anomaly exists centers around informational distribution. A small firm’s information distribution is somewhat limited, which causes investors to be wary of buying the stock and depresses the price. Banz suggests the APT valuation model may be more robust than the CAPM in that the APT would be able to capture the size effect by using a P/E ratio as a proxy for the size effect as a factor in the model. Since the distribution of information affects a firm’s ability to raise capital (less information on a firm may cause a firm to pay a premium for capital) the premium would indicate that the lending and borrowing rates of different firms are not the same. A valuation model capturing different lending and borrowing rates was provided by Brennan (1971). The utilization of this model could be implemented where rates differed for large and small firms. If small firms do not borrow in the capital market, then Black’s noborrowing CAPM would be more suitable than the standard CAPM. Keim (1983) empirically tests one of the most baffling anomalies, the January effect. He found that for stocks with excess returns, over 50% of these excess returns were realized in January. In addition, 50% of the January excess return occurs in the first week of January. This phenomenon of excess returns occurring in the month following the taxyear-end has been found empirically in Great Britain also. Although Keim offers no rationale for this phenomenon, others have tried to find tax reasons for this anomaly. The selling of assets in January rather than December to postpone capital gains taxes to the next taxable year has been one suggested rationale. Unlike the P/E and size effects, this anomaly does not have a clear proxy that could be utilized as a factor in an APT model. CAPM does not capture the effect well, and without any theoretical or economical rationale for this effect, any valuation model would be hard pressed to account for it. Many of the authors seem to conclude that the standard CAPM is not working well, and that alternative valuation models should be considered to capture these anomalies. This might imply that security analysis and portfolio management techniques can be used to beat the market.

Chapter 7

Index Models for Portfolio Selection Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort

Abstract In this chapter, we discuss both the single-index model and multiple-index portfolio selection model. We use constrained maximization instead of minimization procedure to calculate the portfolio weights. We find that both single-index and multi-index models can be used to simplify the Markowitz model for portfolio section. Keywords Single-index model r Multi-index model r Market model r Multiple indexes r Linear programming approach r Lagrange multipliers

7.1 Introduction Previously, we have presented and discussed the Markowitz model for delineating the efficient frontier. Numerous examples were shown that indicated the potentially crushing number of computations resulting from the calculations for even a three-security portfolio. This chapter offers some simplifying assumptions that reduce the overall number of calculations through the use of the Sharpe single-index and multiple-index models. The essential difference between the single- and multipleindex models is the assumption that the single-index model explains the return of a security or a portfolio with only the market. The multiple-index model describes portfolio returns through the use of more than one index. The investor may quantify the return on a portfolio by seeking an index that is representative of the market together with indexes that are representative of the industries of which the component securities are members or exhibit some other

D.H. Wort () University of California, Hayward, CA, USA e-mail: [email protected] C.-F. Lee Rutgers University, New Brunswick, NJ, USA J.E. Finnerty University of Illinois at Urbana-Champaign, Champaign, IL, USA

common factor. More is said about the multiple-index model later in the chapter; for now the single-index model is the focus of discussion.

7.2 The Single-Index Model The major simplifying assumption that yields the index model from Markowitz’s portfolio theory is that covariances between individual securities contained in the portfolio are zero. This assumption greatly reduces the number of calculations needed to find the set of efficient portfolios. The use of the index model necessitates additional statistical estimates for the parameters of the index; nevertheless, these additions are minor in comparison to the reduction in the calculation load as a result of ignoring the covariance terms between securities. Suggested by Markowitz (1959), the single-index model was fully developed by Sharpe (1970), who assumed that the covariances could be overlooked. The return of an individual security was tied to two factors – a random effect and the performance of some underlying market index. Notationally: Rit D ai C bi RI t C eit

(7.1)

where: ai and bi D regression parameters for the i th firm; RIt D the tth return of some underlying market index; Rit D the tth return on security i ; and eit D the tth random effect for the i th security. Equation (7.1) is the market model as discussed in the last chapter. This regression makes several assumptions about the random effect term. 1. The expected value of the tth random effect for security i is zero. More explicitly, E .eit / D 0. 2. The variance of the error terms is constant. This amounts to the assumption that the errors are homoscedastic. 3. There is no relationship between the errors and the return on the market: Cov .eif ; RIt / D 0.

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4. The random effects are not serially correlated: E.eit ; eitCn / D 0. 5. The i th security’s random effect is unrelated to any other random effects of any other security: E.eit ; ejt / D 0. The fifth assumption guarantees that the regression coefficients ai and bi are the best unbiased linear estimators of the true parameters. An investigation of some of the results of the previous assumptions is in order. The expected value of the return on security i is equal to the sum of the intercept, the adjusted return on the index, and some random effect. This can be expressed:

returns of two different securities are not interrelated but only connected to the market index, the covariance between the two securities can be derived from the twice-adjusted variance of the market index (once by the b coefficient of the first security with the market and again by the b coefficient of the second security with the market). The investigation starts with a statement about the covariance of the two securities:    ij D E .Rit  ri / Rjt  rj D EŒ..ai C bi RIt C eit /.ai C bi rI //

..aj C bj RIt C ejt /.aj C bj rI //    D E .bi .RIt  rI / C eit / bj .RIt  rI / C ejt   D bi bj E .RIt  rI /2 C bi E ejt .RIt  rI /

E.Ri / D E.ai C bi RIt C eit / D E.ai / C E.bi RIt / C E.eit /

(7.2)

Because ai and bi are constants, and E .eit / is equal to zero: ri D E.Ri / D ai C bi rI

(7.3)

where rI is equal to the mean of the returns on the market index. If the mean of the returns on security i equals ri , then the variance of security i is equal to the expected value of squared deviations from rit . This translates to: i2 D E.Rit  ri /2 D EŒ.ai C bi RIt C eit /  .ai C bi rI / 2 D EŒbi .RIt  rI / C eit 2 D bi2 E .RIt  rI /2 C 2bi EŒeit .RIt  rI / C E.eit /2 (7.4) By the third assumption the covariance of the random effect and the deviation of the index return from its mean are zero; also, the expected value of the squared errors is equal to the variance of the random effect. Thus: i2 D bi2 E.RIt  rI /2 C E.eit /2 D bi2 I2 C ei2

(7.5)

This is equivalent to saying that the variance of the returns on a security is made up of some adjusted quantity of the variance of the market (usually referred to as systematic risk) plus the variance of the random effects exclusive to that particular security (unsystematic risk). The last result to be investigated from these assumptions involves a minor step into abstraction in which the possibility of interaction between two securities and the market index is considered. It is suggested that because variations in the

C bj E.eit .RIt  rI // C E.eit ejt /

(7.6)

Since according to the third and fourth assumptions the last three terms of the last summation are equal to zero: ij D bi bj I2

(7.7)

The number of calculations necessary to utilize the ı Markowitz model is N C .N 2  N / 2, as shown in the last chapter. For a portfolio with a hundred securities, this translates to 5,050 calculations. With the Sharpe singleindex model, only 100 estimates are needed for the various security-regression coefficients and only one variance calculation, the variance of the returns on the market. In addition, 100 estimates of the unsystematic risk ei2 are also needed for the single-index model. Hence, the single-index model has dramatically reduced the input information needed.1

7.2.1 Deriving the Single-Index Model So far only the Sharpe single-index model has been utilized to study the returns of a single security i as determined by its relation to the returns on a market index. Expected return of a portfolio. Now consider the return on a portfolio of n securities. The return of a portfolio of n securities is the weighted summation of the individual returns of the component securities. Notationally: n   X E Rpt D xi E .Ri / i D1

1

Discussion of the single-index model adapted in part from Sharpe (1970). Adapted by permission.

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Where Rpt is the rate of return for a portfolio in period t and Xi is the weight associated with the i th security. Rit D ai C bi RIt C eit Rpt D D

n X i D1 n X

xi .ai C bi RIt C eit / xi .ai C eit / C

i D1

D

n X

n X

xi .bi RIt /

i D1

xi ai C

i D1

n X

.xi bi / .RIt / C

i D1

n X

xi eit

(7.8a)

i D1

It was shown previously that when the number of component securities in a portfolio approaches 15, the unique risk of the component securities is reduced through diversification. In Equation (7.10) the last term, the weighted sum of the random effect variances, approach zero as n increases. So, again, as the number of securities increases, the unsystematic risk is reduced and the remaining risk of the portfolio is the adjusted variance of the market index. Example 7.1 further illustrates this concept. Example 7.1. Given the following information, what should the ˇ of the portfolio .bp / be?

Thus, E.Rpt / D

n X

xi ai C

i D1

n X

.xi bi /E.RIt /

(7.8b)

Variance of a portfolio. To derive the variance of the portfolio p2 , consider first that the mean return of the portfolio is equal to the expected value of the return on the portfolio. Then, following the definition of p2 in previous chapters and Equations (7.5) and (7.7), we can derive P2 as follows. The variance of the portfolio is equal to P2 D EŒ.Rpt  E.Rpt //

(7.9a)

Substituting Equations (7.8a) and (7.8b) into Equation (7.9A), we obtain h i2  Xn Xn 2 P D E .RIT  E.RIT // .xi bi / C xi ei i D1

D D D

Xn

i D1

hXn

i D1

i D1

2 Xn .xi bi / I2 C

i D1

Xn

i D1

j D1

xi bi

.xi xj bi bj /I2 C

i hXn j D1

i

2 xi ei Xn

xj bj I2 C

i D1

Xn i D1

i D1

i D1

xj bj D bp . Hence, the last equation reduces to: n X p2 D bp2 I2 C xi2 ei2 i D1

Solution (7.10):

Substituting related information into Equation

bp D D

I2 P2 0:082 0:041

D 2:0 Therefore, bp D 1:414. Equation (7.8b) implies that the portfolio can be viewed as an investment in n basic securities and a weighted adjusted return in the market, or n X

.xi bi / E.RIt t/

i D1

The return on the market can be decomposed as a combination of the expected return plus some random effect. When this random effect is positive, the atmosphere is bullish and when it is negative, the atmosphere is bearish. Notationally:

D anC1 C enC1;t

xi2 ei2

Because the weighted sum of the bi coefficients is equal n P xi bi D bp , similarly to the coefficient of the portfolio

j D1

xi2 ei2 D 0

RIt D E.RIt / C enC1;t

xi2 ei2

(7.9b)

n P

n X

i D1

This equation indicates that the return of a portfolio may be decomposed into the summation of the weighted returns peculiar to the individual securities and the summation of the weighted adjusted return on the market index. Thus, the portfolio may be viewed as a combination of n basic securities and a weighted adjusted return from an investment in the market index.

Xn

p2 D 0:082 I2 D 0:041

(7.10)

(7.11)

This bit of algebraic maneuvering enables the weighted adjusted investment in the market to be viewed as an investment in an artificial security, the .n C 1/th of an n-security portfolio. The weight for this .n C 1/th security is the sum of the n weights multiplied by their respective related coefficients to the market index. Thus: xnC1 D

n X i D1

xi bi

(7.12)

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The reason for this divergence in notation resulting in the definition of the .n C 1/th security’s return and weight is that Equation (7.8) can be simplified to yield a working model for portfolio analysis. Substituting the last results for XnC1 and RIt into Equation (7.8) yields: Rpt D

n X

D

t D1

relationship n X

xi .ai C eit / C xnC1 .anC1 C enC1;t /

RIt D n˛i C

t D1

i D1 nC1 X

Example 7.2. Given RIt ; Rit and ˛i as indicated in the table n P shown below and the fact that ˇi RIt D 56, using the

n X

ˇi RIt C

t D1

find the values for ˛i ; ˇi and ei . xi .ai C eit /

(7.13)

Solution n X

Because the expected value of the random-effect terms is zero, the summation that results after the application of the expectations operator can be expressed: E.Rpt / D

n X

n P

RIt D 28; so ˇi D

t D1

D

nC1 X

xi ai C xnC1 E.RIt /

D

(7.14)

This yields a formula for the return of a portfolio that is easily applied to portfolio analysis. Before proceeding, however, it is necessary to simplify the variance formula so that it may be used as well. Remembering that the variance of the portfolio is the expected value of the squared deviations from the expected market return, the last results concerning Rpt and E.Rpt / may be applied:

Substitute the values

iD1

n X

n P

ˇi RIt D 56; n D 4

t D1

RIt D n˛i C

n X

ˇi RIt C

t D1

n X

eit

t D1

and solve for ˛i . RIt

Rit

˛i

ˇi RIt

ei

4 6 10 8 28

12 14 20 18 64

2 2 2 2 8

8 12 16 16 56

12  2  8 D 2 14  2  12 D 0 20  2  20 D 2 18  2  16 D 0 0

iD1

iD1

The ˇi RIt column in the table is filled by simply multiplying ˇ.D 2/ by the RIt column, The ei ’s are the amounts such that Rit D ˛ C ˇi Rmt C eit is an equality so Rit D ˛i C ˇi RIt C eit is satisfied, From the last column of the table we know that n P eit D 0. Therefore

iD1

12 xi eit A

iD1 nC1 X

RIt D 64;

t D1

iD1

92 82 3 nC1 = < nC1 X X DE 4 xi .ai C eit /5  xi E .ai / ; :

D Var @

56 28

into the regression line

92 82 3 nC1 = < nC1 X X DE 4 xi .ai C eit /5  xi ŒE .ai / C E .eit / ; :

0

n P t D1

82 3 2 392 nC1 = < nC1 X X   5 4 4 Var Rpt D E xi .ai C eit /  E xi .ai C eit /5 ; :

nC1 X

RIt

D2 xi ai

i D1

iD1

ˇi RIt

t D1 n P t D1

i D1

DE@

eit

t D1

i D1

0

n X

1 xi eit A

i D1

(7.15)

iD1

This result follows from the assumption that the covariances are equal to zero. Additionally, each variance term is only a weighted sum of the errors around the market return. The direct usefulness of the last two conclusions will become apparent when solving for the security weights. Example 7.2 provides further illustration.

˛i D

64  56 C 0 D2 4

In this section, the expected return E.Rpt / and the variance of a portfolio Var.Rpt / in terms of the single-index model has been derived. In the following section, the optimal portfolio selection procedures discussed in previous chapters are used to explore the single-index optimum-portfolio selection model.

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7.2.2 Portfolio Analysis and the Single-Index Model Before beginning the portfolio analysis using the single-index model, it is necessary to explain a derivation of security weights through the use of the Lagrangian calculus maximization discussed in previous chapters. The maximization procedure maximizes a linear combination of the following two equations:2 E.Rp / D

nC1 X

E.Ri /

or

i D1

Max nC1 X   Var Rp D Var xi eit

! (7.16a)

i D1

subject to nC1 P

xi D 1 and

i D1

xnC1 D

n P

(7.16b) xi bi

i D1

The first constraint is equivalent to requiring that the sum of the weights of the component securities is equal to one. The second constraint requires that the weight of the market index within the portfolio returns is equal to the summation of the weighted adjustment factors of the component securities. This requirement is as described in Equation (7.16b). Combining the above two objective functions with the two constraints yields a Lagrangian function: ! n X     xi  1 P D ˆE Rp  Var Rp C 1

C2

n X

!

i D1

xi bi  xnC1

i D1

Dˆ

nC1 X

xi ai 

i D1

C2

nC1 X

Var .eit / C 1

i D1 n X

xi bi  xnC1

!

n X

! xi  1

i D1

(7.16c)

i D1

The only difference between this maximization function and the minimization shown in previous chapters, beyond intent,

2 The maximization of the negative of the variance is equivalent to the minimization of the variance itself. This can be viewed as originating from the negative spectrum of the number line and maximizing towards zero.

is that instead of fixing some arbitrary value of return needed .E  /, an attempt can be made to quantify the level of risk aversion that the investors of the portfolio require, thereby placing the portfolio on the efficient frontier not by desired return but by level of utility, as discussed in previous chapters. This indication of risk aversion is denoted by the Greek letter ˆ (Phi). When low values are exhibited, risk aversion is pronounced; when high values are in evidence, substantial risk taking is allowed. This notion of the level of risk aversion is best pictured in Fig. 7.1. In Fig. 7.1, A represents an investor’s objective function to minimize the risk only – therefore, the aggressiveness to return is zero. C represents an investor’s objective function to maximize returns only – therefore, the aggressiveness to return is infinite. At B an investor’s attitude toward return and risk is between A and C. Figure 7.2 provides further illustration of the approach being utilized. What is depicted is the variation of the objective function P as the parameter denoting risk is varied. Again, notice that the risk-return relation for a low ˆ is lower than that for the more risk-taking, high value of ˆ. Points A and C are as discussed for Fig. 7.1. Lines BD and B 0 D 0 represent the objective function when the risk-aversion parameter is equal to one. Note that BD instead of B 0 D 0 represents the maximization of the objective function. In sum, different values of ˆ generate different optimal objective functions. Consider now the three-security portfolio. In this framework, the preceding objective function expands to: P D ˆx1 a1 C ˆx2 a2 C ˆx3 a3 Cˆx4 a4  x12 Var .e1t / x22 Var .e2t /  x32 Var .e3t /  x42 Var .e4t / C1 x1 C 1 x2 C1 x3  1 C 2 x1 b1 C 2 x2 b2 C 2 x3 b3  2 x4 (7.17) Take note of all terms in the preceding equation that contain a4 and x4 . By utilizing the relations of the individual securities to the market, it has been possible to delete most of the calculations necessitated by the full Markowitz variancecovariance model. The return that will be generated from an optimal portfolio derived through this maximization rests on the estimation of the future expected return and variance of the index. This will lead to the availability of ranging analysis for these estimations, which will produce robust estimates for the security weights highlighted at the end of this section. To proceed with the Lagrangian maximization it is necessary to notice that the above equation has six unknowns – the four weights and the two Lagrangian coefficients. All other values are expected to be known or estimated. To maximize, the partial derivative of the objective function is taken with respect to each of the six variables:

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Fig. 7.1 Level of risk aversion and investors’ investment attitude

Fig. 7.2 Level of risk aversion and objective function

32 D Var .e3t / I and

@P D ˆa1  2x1 12 C 1 C 2 b1 D 0 @x1

42 D Var .e4t / :

@P D ˆa2  2x2 22 C 1 C 2 b2 D 0 @x2

Equation (7.18) is a set of six equations in six unknowns when set equal to zero. These six equations can be rewritten in matrix format:

@P D ˆa3  2x3 32 C 1 C 2 b3 D 0 @x3 @P D ˆa4  2x4 42  2 D 0 @x4 @P D x1 C x2 C x3  1 D 0 1 @P D x1 b1 C x2 b2 C x3 b3  x4 D 0 2 Where: 12 D Var .e1t / I 22 D Var .e2t / I

(7.18)

A 2 212 0 0 0 6 0 222 0 0 6 6 0 0 232 0 6 6 0 0 0 242 6 4 1 1 1 0 b2 b3 1 b1

1 1 1 0 0 0

3 b1 b2 7 7 b3 7 7 17 7 05 0

x D k 2 3 2 3 x1 ˆa1 6x2 7 6ˆa2 7 6 7 6 7 6x3 7 6 7 6 7 D 6ˆa3 7 6x4 7 6ˆa4 7 6 7 6 7 41 5 4 1 5 2 0

(7.19a) Equation (7.19a) has related the individual securities to the index and has discarded the use of numerous covariance terms. The majority of the elements of the matrix indicated in the first matrix of Equation (7.19a) are zero.

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As in previous chapters, to solve for the x column vector of variables the P matrix is inverted and each side of the equation is pre multiplied by A1 . This yields the x column vector of variables on the left-hand side and A1 multiplied by k, the solution column vector, on the right-hand side. Ax D k A1 Ax D A1 k Ix D A1 k

(7.19b)

where I is the identity matrix. The procedure of solving Equations (7.19a) and (7.19b) is discussed in later chapters. A linear-programming approach to solve this kind of model is discussed in the first appendix of this chapter. In order to estimate a single-index type of optimal portfolio the estimates of a market model are needed; a discussion of the market model and beta estimates is therefore needed as well.

7.2.3 The Market Model and Beta Equation (7.1) defines the market model: Rit D ai C bi RIt C eit

(7.1)

From this market model, ai ; bi ; Var .eit / and Var .RIt / can be estimated: all are required for the single-index type of optimal portfolio. So far the regression coefficient has been referred to as bi when, in fact, it is a risk relationship of a security with the market. Quantified as the covariance of security i with the market index divided by the variance of the return of the market, beta is a relational coefficient of the returns of security i as they vary with the returns of the market. Notationally: bi D

i:I D I2

n X

Œ.Rit  rit / .RIt  rIt /

t D1

.RIt  rIt /2

(7.20)

Of course, due to the continuous nature of security returns, the result of the above calculation is an estimate and therefore subject to error. This error can be quantified by the use of standard error estimates, which provide the ability to make interval estimations for future predictions. The following discussion is related to the beta estimate and its forecasting as discussed in previous chapters. Because of the linkage between security returns and the firm’s underlying fundamental nature, beta estimates will vary over time. It is the job of the security analyst to decide how to modify not only the beta estimate but also the period

of time from which the sample of returns is to be drawn. Some popular security-evaluation techniques use a tiered growth model to correspond to the product life-cycle theory. Within the scope of this theory, beta estimates for the relation of that security to the market will vary with respect to the time period chosen for the sample of returns. It is obvious that the returns generated by a firm in its infancy have very little correlation with returns during the growth or maturity phase. It is up to the analyst to judge which phase a company may be in and to adjust the beta estimates accordingly. Assume for expository purposes that all stocks move in perfect alignment with the market, and therefore the security returns all produce beta estimates of 1. It could then be said that any estimation of betas above 1 would indicate positive sampling error, and all estimates below 1 negative sampling error. Future periods would show returns that caused movement from either side of the estimate deviation back towards 1. Blume (1975) shows that the adjustment of a beta in one period could accurately predict the adjustment in subsequent periods. Utilizing regression analysis, Blume demonstrates that a historical beta could be adjusted successfully to predict future levels. Large brokerage houses have taken this theory a little further by applying weighted averages of historic and average betas in adjusting beta estimates. This has the advantage of being simple to calculate, but it has the ability nevertheless to adjust beta back towards an average level (This kind of beta-adjustment process was discussed in the last chapter; the Value Line beta estimate is essentially based on this process, as explored in previous chapters.) Historic beta estimates give the analyst a feeling for how the firm is generating returns to an investor but give very little guidance about the firm’s underlying fundamental nature. It is held that a firm’s balance-sheet ratios over time can indicate how risky the firm is relative to the market. Using seven financial quantities of a firm, Beaver et al. (1970) show that betas can be estimated by multiple regression relating the financial levels to the riskiness of the firm. Using 1. 2. 3. 4. 5. 6. 7.

Dividend payout Asset growth Leverage Liquidity Asset size Earnings volatility, and Accounting beta (relation of earnings of the firm with the earnings of the economy as a whole)

BKS show not only that each of the variables carries a logical relation to the market but also that the multiple regression is significant. Other studies, most notably Rosenberg and Guy (1976a, b), use many more fundamental factors in the regression for estimating betas. Historic betas are of consequence because of their relation with market returns; nevertheless, the sampling period

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tends to crowd out the information contained in returns due to recent developments for the firm. Fundamental betas tend to recognize significant changes in the makeup of a firm – for example, a change in debt structure, or liquidity. The disadvantage related to fundamental betas is that they treat intercompany responsiveness as constant. It is patently obvious that a small firm taking on a large debt will be at much more risk than a Ford or GM doing the same. Because of the advantages and disadvantages apparent in both types of beta estimates, it has been shown that a combination of the estimates is more suitable to risk classification. Rosenberg and McKibben (1973) found that intercompany responsiveness is substantial, so an analysis was undertaken to introduce these variations into the regression. Utilizing a set of dummy variables to capture the advantages of both types of betas, Rosenberg and Marathe (1974, Berkeley Working Paper Series) have described a multiple regression for the beta estimate of a firm that include information such as variability, level of success, relative size, and growth potential. By using a very large and complex model they found that the forecasting ability of the regression analysis is substantial, but they were inconclusive as to whether the benefit of the analysis outweighs the computational cost. Forecasting future beta levels holds its roots in the difficulty of forecasting the fundamental nature of the firm itself. This section has briefly looked over some of the available adjustment processes for beta estimation. While reasonable in its scope, this coverage of the adjustment process is by no means complete. Additional research material is cited at the end of this chapter for use in further study. It should be noted during future readings that the more sophisticated an adjustment process is, the more likely it is that the computational costs will outweigh the improvement of the beta forecast.

7.3 Multiple Indexes and the Multiple-Index Model The previous section reviewed the possibilities of adjusting the beta estimate in the single-index model to capture some of the information concerning a firm not contained in the historic returns. The multiple-index model (MIM) pursues the same problem as beta adjustment, but approaches the problem from a different angle. The multiple-index model tackles the problem of relation of a security not only to the market by including a market index, but also to other indexes that quantify other movements. For example, U.S. Steel has returns on its securities that are related to the market, but due to the declining nature of the American steel industry there is also some relation to the steel industry itself. If an index

C.-F. Lee et al.

could be developed that represented the movement of the U.S. steel industry, it could be utilized in the return analysis, and a more accurate estimate of possible security returns for U.S. Steel could be derived. The assumption underlying the single-index model is that the returns of a security vary only with the market index. The expansion provided by the multiple-index model includes factors that affect a security’s return beyond the effects of the market as a whole. Realistically any index might be used, but well-known, published indexes are usually incorporated. These may include general business indicators, industry-specific indicators, or even self-constructed indexes concerning the structure of the firm itself. The model to be addressed contains L indexes; nevertheless, a few examples of possible indexes of interest are offered. The multi-index model is related to the arbitrage pricing model developed by Ross (1976, 1977), which will be explored in detail in the next chapter. The covariance of a security’s return with other market influences can be added directly to the index model by quantifying the effects through the use of additional indexes. If the single-index model were expanded to take into account interest rates, factory orders, and several industry-related indexes, the model would change to:     I1 C bi2 I2 C    C biM IM C ci Ri D ai C bi1

(7.21)

In this depiction of the multiple-index model, Ij is the actual level of index j , while bij is the actual responsiveness of security i to index j . If the component of the security return is not related to any of the indexes then this index model can be divided into two parts: (1) ai , the expected value of the unique return, and (2) ci which represents the distribution of the random effect. ci has a mean effect of zero and a variance of c2i . This model can be utilized with multiple regression techniques, but if the indexes are unrelated to each other the calculations would be made much simpler. This assumption reduces the number of calculations, as compared with the Markowitz full variance-covariance model, but is obviously more complex than the single-index model. To assure that the indexes are unrelated, the index variables can be orthogonalized (made uncorrelated) by completing inter-regressions on the indexes themselves. Assume there is a hypothetical model that deals with two indexes:   I1 C bi2 I2 C ci Ri D ai C bi1

(7.22)

Suppose the indexes are the market index and an index of wholesale prices. If these two indexes are correlated, the correlation may be removed from either index.

7 Index Models for Portfolio Selection

119

To remove the relation between I1 and I2 , the coefficients of the following equation can be derived by regression analysis: I2 D e0 C e1 I1 C di where:

 2 DE bi1 .I1t C I 1 /2 C 2bi1 bi 2 .I1t C I 1 /.I2t C I 2 / C bi22 .I2t C I 2 /2 C 2bi1 .I1t C I 1 /eit  C2bi 2 .I2t C I 2 /eit C eit2   2 D bi1 E.I1t C I 1 /2 C bi1 bi 2 E .I1t C I 1 /.I2t C I 2 /

e0 and e1 D the regression coefficients; and di D the random error term. By the assumptions of regression analysis, di is uncorrelated with I1 . Therefore:

C bi22 E.I2t C I 2 / C bi1 EŒ.I1t C I 1 /ei   C bi 2 EŒ.I2t C I 2 /ei C E ei2 But by assumption:

  dOi D I2  eO0 C eO1 I1

EŒ.I1t C I 1 /.I2t C I 2 / D 0 EŒ.I1t C I 1 /ei D 0

which is an index of the performance of the sector index without the effect of I1 (the market removed). Defining: I2 D dOi D I2  eO0  eO1 I1

EŒ.I2t C I 2 /ei D 0 and E.I1t C I 1 /2 D 12

an index is obtained that is uncorrelated with the market. By solving for I2 and substituting into Equation (7.22):

E.I2t C I 2 /2 D 22   E ei2 D ei2 I

  I1 C bi2 I2  bi2 eO0  bi2 eO1 I1 C ci Ri D ai C bi1

therefore,

Rearranging:       bi2 eO1 I1 C bi2 I2 C ci Ri D ai  bi2 eO0 C bi1 If the first set of terms in the brackets are defined as ai and the second set of terms are defined as bi1 ; bi2 D bi 2 ; I1 D I1 and ci D ei , the equation can be expressed: Ri D ai C bi1 I1 C bi 2 I2 C ei

(7.23)

in which I1 and I2 are totally uncorrelated: the goal has been achieved. As will be seen later, these simplifying calculations will make the job of determining variance and covariance much simpler. The expected return can be expressed:

D E.ai / C E.bi1 I1t / C E.bi 2 I2t / C E.eit / (7.24)

Since ai ; bi1 and bi 2 are constants, E .ei / D 0 by assumption, where I 1 D E.I1t / and I 2 D E.I2t /. Variance can be expressed:  2 DE .Rit  Ri /2

(7.25)

Covariance between security i and security j can be expressed ij DEŒ.Rit  Ri /.Rjt  Rj / Œwhere Ri DE.Rit / and Rj D E.Rjt /  ˚ DE .ai Cbi1 I1 Cbi 2 I2t C e1t /  .ai C bi1 I 1 C bi 2 I 2 / 

.aj C bj1 I1t C bj 2 I2t C ejt /  .aj C bj1 I 1 C bj 2 I 2 / DEfŒbi1 .I1t  I 1 / C bi 2 .I2t  I 2 / C ei Œbj1 .I1t  I 1 / C .bj 2 .I2t  I 2 / C ejt g D EŒbi1 bj1 .I1t  I 1 /2 C bi 2 bj 2 .I2t  I 2 /2

E.Rit / D E.ai C bi1 I1t C bi 2 I2t C eit /

D ai C bi1 I 1 C bi 2 I 2

2 2 i2 D bi1 1 C bi22 22 C ei2

where Ri D E .Rit /

DEŒ.ai C bi1 I1 C bi 2 I2 C eit /  .ai C bi1 I 1 C bi 2 I 2 / 2 DEŒ.a1 C ai / C bi1 .I1t C I 1 / C bi 2 .I2t C I 2 / C eit 2

D bi1 bj1 12 C bi 2 bj 2 22

(7.26)

since all remaining expected values of the cross-product terms equal zero. The extended results of expected return variance and covariance for a multi-index and more than two indexes can be found in the second appendix of this chapter. One simplifying way of applying the multiple-index model is to start with the basic market model and add indexes to reflect industry-related effects. If the firm has 100% of its operations in one industry, Equation (7.23) can be used to

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represent a two-index model with market index and industry index. In general this approach reduces the number of data inputs to 4N C 2I C 2. These data inputs are (1) the expected return and variance for each stock and market index; (2) the covariance between the individual security and the market index and the industry index; and (3) the mean and variance of each industry index. Although this is a larger number of data inputs than for the simple market model, the accuracy of the estimation of security return increases. So the tradeoff is one of more information (higher cost to use) versus greater accuracy of the forecasted security return. Care must be taken in applying the multiple-index model. It is often the case that the additional information resulting from the application of a higher-complexity model is outweighed by the computational cost increase. In an attempt at making the model as parsimonious as possible, it is necessary to judge the increased information gained by utilizing the more complex model. This can be accomplished by examining the mean square error for the forecast of the actual historic values. Although not within the scope of this text, the ability to judge the accuracy of a forecast is essential and can be acquired from any good statistical forecasting text. Example 7.3 provides further illustration of the single-index model.

Going back to the calculus maximization derivation of optimal portfolio weights, recall that a linear combination of the following two factors is being maximized: nC1   X E .Ri / or E Rp D i D1 nC1 X   Var Rp D Var xi eit

! (7.27a)

i D1

subject to nC1 P

xi D 1 and

i D1 n P

(7.16B)

xi bi D xnC1

i D1









P D ˆE Rp  Var Rp C 1 n

P xi bi  xnC1 C2 i D1

n P

xi  1

i D1

nC1 P 2 xi ai  xi Var .Ri / C 1 i D1 i D1 n

P xi bi  xnC1 C2

Dˆ

nC1 P





n P

xi  1

i D1

i D1

(7.16C) Example 7.3. During the discussion of the single-index model, a method was presented for determining optimal portfolio weights given different levels of risk aversion. In this section a three-security portfolio is examined in which the returns of the securities are related to a market index.

Ri Alpha Beta Var (residual)

JNJ

AXP

XOM

S&P 500

Risk-free

Artificial security

0.0053 0.0046 0.29 0.0019

0.0055 0.0022 1.21 0.0016

0.0126 0.0109 0.65 0.0022

0.0025 – – –

0.0023 – – –

– – – 0.0015

The only difference between this maximization function and the minimization shown in previous chapters, beyond intent, is that instead of fixing some arbitrary value of return needed .E  /, an attempt is made to quantify the level of risk aversion that the holders of the portfolio require, thereby placing the portfolio on the efficient frontier, not by desired return, but by level of utility. Consider again a three-security portfolio. In this framework, the preceding objective function expands to: P D ˆx1 a1 C ˆx2 a2 C ˆx3 a3 C ˆx4 a4  x12 Var .R1 /  x22 Var .R2 /  x32 Var .R3 /  x42 Var .R4 / C 1 x1 C 1 x2 C 1 x3  1 C 2 x1 b1 C 2 x2 b2 C 2 x3 b3  2 x4

Varaiance-covariance matrix

JNJ AXP XOM S&P 500

JNJ

AXP

XOM

S&P 500

0.0020 0.0009 0.0004 0.0004

0.0009 0.0038 0.0010 0.0018

0.0004 0.0010 0.0027 0.0010

0.0004 0.0018 0.0010 0.0015

The securities and the index have the following observed parameter estimates taken from actual monthly returns during the period January 2001 to September 2008 (see table). The table includes the calculation of the beta estimate for each security. Check these figures, remembering that beta is equal to the covariance of security i with the market divided by the variance of the market returns.

(7.17)

Take note of all terms in the preceding equation that contain a4 and x4 . By utilizing the individual securities’ relations to the market, it has been possible to delete most of the calculations necessitated by the Markowitz full variancecovariance model. The return that will be generated from an optimal portfolio derived through this maximization rests on the estimation of the future expected return and variance of the index. By partially differentiating the above objective function with respect to each of the weights and the ’s, it is possible to develop the following six equations in six unknowns:

7 Index Models for Portfolio Selection

121 n n X   X xi xj ij Var Rp D

@P D ˆa1  2x1 12 C 1 C 2 b1 D 0 @x1

i D1 j D1

@P D ˆa2  2x2 22 C 1 C 2 b2 D 0 @x2

Taking the covariance expressions:

@P D ˆa3  2x3 32 C 1 C 2 b3 D 0 @x3

  Var Rp D Œ..0:0652/2 0:0020/C..0:4341/2 0:0038/ C ..1:4994/2 0:0027/ C 2 ..0:0652/

@P D ˆa4  2x4 42 C 1 C 2 b4 D 0 @x4

.0:4341/ 0:0009/

@P D x1 C x2 C x3  1 D 0 @1

C 2 ..0:0652/ .1:4994/ 0:0004/ (7.18)

As shown in Equation (7.18), these last six equations can be transformed into Jacobian Matrix notation. A matrix can be developed by utilizing the set of data in the first table of this problem and a ˆ of 1.0 (denoting moderate risk aversion): A x D k 2 3 2 3 3 0:013 0 0 0 1 1:15 0:0084 x1 0:015 0 0 1 0:907 6x2 7 6 0 60:01277 6 6 7 6 7 7 0 0:006 0 1 0:557 6x3 7 6 0 60:01877 6 7 6 7 D 6 7 0 0 0:004 0 0 7 6x4 7 6 0 60:01257 4 1:0 5 4 5 4 1 1:0 1:0 0 0 0 1:0 5 2 1:15 0:90 0:55 1:0 0 0 0 2

When the P matrix is invented and premultiplies each side of the equation: x D A1 k 3 2 3 2 0:06523 x1 6x2 7 60:434147 7 6 7 6 6x3 7 6 1:49937 7 7 6 7D6 6x4 7 6 0:42381 7 7 6 7 6 41 5 40:005215 0:00129 2 This solution vector shows that investment should short 6:52% in Johnson & Johnson, short 43:41% in American Express, and long 149.94% in Exxon Mobil. Additionally, the weight of x4 is the sum of the weighted adjustments as indicated in Equation (7.16b). The return on this portfolio is the weighted sum of the individual returns: E.Rp / D

n X

xi E.Ri /

D 0:0055 A portfolio has been developed that is efficient within the realm of this model. In the table below are the portfolios derived by varying the utility factor ˆ from 0 (totally risk averse) to 2 (more aggressive risk posture). This last table offers the ability to develop an efficient frontier under the Sharpe single-index model. The figure shows that the portfolios start to trace out an efficient boundary. For a fuller graph, the analyst would continue the calculations with ever higher values of ˆ stretching the efficient frontier.   Var Rp

Portfolio

ˆ

1

0.0

0.5233

0.1531

0.3236

0.0077

0.0013

0.0361

2

0.5

0.2290

0.1405

0.9115

0.0119

0.2223

0.4715

X1

X2

Rp

X3

p

3

1.0

0.0652

0.4341

1.4994

0.0162

0.0055

0.0742

4

1.5

0.3595

0.7278

2.0873

0.0204

0.0109

0.1044

5

2.0

0.6538

1.0214

2.6751

0.0246

0.0185

0.1360

0.03 0.025 Expected Retrun E(R)

@P D x1 b1 C x2 b2 C x3 b3  x4 D 0 @2

C 2 ..0:4341/ .1:4994/ 0:0010/

0.02 0.015 0.01 0.005

i D1

D Œ.0:0652 0:0053/ C .0:4341 0:0055/

0 0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.1600

Risk

C .1:4994 0:0126/ D 0:0162 The variance of the portfolio is the sum of the weighted variance and covariance terms:

7.4 Conclusion This chapter has discussed the essentials of single- and multiple-index models. The theoretical underpinnings of the

122

theories have been explored and numerical examples have been provided. An efficient boundary has been derived under the guidelines of the model and the quantitative analysis of the related parameters has been studied. It has been shown that both single-index and multi-index models can be used to simplify the Markowitz model for portfolio section; remember, however, that the multi-index model is much more complicated than the single-index model.

References Beaver, W., P. Kettler, and M. Scholes. 1970. “The association between market determined and accounting determined risk measures.” The Accounting Review 45, 654–682. Blume, M. 1975. “Betas and their regression tendencies.” Journal of Finance 20, 785–795. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Brenner, M. 1974. “On the stability of the distribution of the market component in stock price changes.” Journal of Financial and Quantitative Analysis 9, 945–961. Elton, E. J., M. J. Gruber, and T. Urich. 1978. “Are betas best?” Journal of Finance 23, 1375–1384. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Fouse, W., W. Jahnke, and B. Rosenberg. 1974. “Is beta phlogiston?” Financial Analysts Journal 30, 70–80. Frankfurter, G. and H. Phillips. 1977. “Alpha-beta theory: a word of caution.” Journal of Financial Management 3, 35–40. Frankfurter, G. and J. Seagle. 1976. “Performance of the sharpe portfolio selection model: a comparison.” Journal of Financial and Quantitative Analysis 11, 195–204. Gibbons, M. R. 1982. “Multivariate tests of financial models, a new approach.” Journal of Financial Economics 10, 3–27. Haugen, R. and D. Wichern. 1975. “The intricate relationship between financial leverage and the stability of stock prices.” Journal of Finance 20, 1283–1292. Jacob, N. 1974. “A limited-diversification portfolio selection model for the small investor.” Journal of Finance 19, 847–856. King, B. 1966. “Market and industry factors in stock price behavior.” Journal of Business 39, 139–140. Latane, H., D. Tuttle, and A. Young. 1971. “How to choose a market index.” Financial Analysts Journal 27, 75–85. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Levy, R. 1974. “Beta coefficients as predictors of return.” Financial Analysts Journal 30, 61–69. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, Hoboken, NJ. Markowitz, H. 1959. Portfolio selection. Cowles Foundation Monograph 16, Wiley, New York. Morgan, I. G. 1977. “Grouping procedures for portfolio formation.” Journal of Finance 21, 1759–1765. Roll, R. 1969. “Bias in fitting the Sharpe model to time series data.” Journal of Financial and Quantitative Analysis 4, 271–289. Rosenberg, B. and J. Guy. 1976a. “Prediction of beta from investment fundamentals.” Financial Analysts Journal 32, 60–72.

C.-F. Lee et al. Rosenberg, B. and J. Guy. 1976b. “Prediction of beta from investment fundamentals, part II.” Financial Analysts Journal 32, 62–70. Rosenberg, B. and W. McKibben. 1973. “The prediction of systematic and specific risk in common stocks.” Journal of Financial and Quantitative Analysis 8, 317–333. Rosenberg, B. and V. Marathe. 1974. The prediction of investment risk: systematic and residual risk, Berkeley Working Paper Series. Ross, S. A. 1976. “Arbitrage theory of capital-asset pricing.” Journal of Economic Theory 8, 341–360. Ross, S. A. 1977. “Return, risk and arbitrage,” in Risk and return in finance, Vol. 1, I. Friend and J. L. Bicksler (Eds.). Ballinger, Cambridge, pp. 187–208. Sharpe, W. F. 1967. “A linear programming algorithm for mutual fund portfolio selection.” Management Science 13, 499–510. Sharpe, W. F. 1970. Portfolio theory and capital markets, McGraw-Hill, New York. Smith, K. V. 1969. “Stock price and economic indexes for generating efficient portfolios.” Journal of Business 42, 326–335. Stone, B. 1973. “A linear programming formulation of the general portfolio selection problem.” Journal of Financial and Quantitative Analysis 8, 621–636. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th ed., Duxbury Press, California.

Appendix 7A A Linear-Programming Approach to Portfolio-Analysis Models Sharpe (1967) developed a simplified portfolio-analysis model designed to be formulated as a linear-programming problem. Jacob (1974) developed a linear-programming model for small investors that delineates efficient portfolios composed of only a few securities. Both of these approaches have as their objectives: (1) a reduction in the amount of data required and (2) a reduction in the amount of computer capability required to solve the portfolio-selection problem. Sharpe approaches the problem of capturing the essence of mean-variance portfolio selection in a linear-programming formulation by the following: 1. Making a diagonal transformation of the variables that will convert the problem into a diagonal form, and 2. Using a piecewise linear approximation for each of the terms for variance. The LP that results from the use of market responsiveness as the risk measure and the imposition of an upper limit on investment in each security is Max P D 

" n X i D1

# xi E.Ri /  .1  /

"

n X

# xi ˇi

i D1

(7A.1)

7 Index Models for Portfolio Selection

subject to:

n P

123

a reasonable linear approximation to the first term in the objective function is provided by:

xi D 1

i D1

!2 " n # n X 1 1 X xi ˇi  2 .Rm / D  2 .Rm / xi ˇi K i D1 K i D1

0  xi  U

where: xi D the fraction of the portfolio invested in security i ; E .Ri / D the expected returns of security i ; ˇi D the beta coefficient of security i ; U D the maximum fraction of the portfolio that may be held in any one security; and  D a parameter reflecting the degree of risk aversion. The  is used in generating the efficient frontier. The value of  corresponds to the rate of substitution of return for risk measured by the individual security ˇ, while the U maximum percentage to be invested in any security greatly reduces the number of securities that will be needed for diversification. Building on this framework, Jacob derives a LP model that allows the small investor explicit control over the number of securities held. Jacob’s LP model incorporates the effects of unsystematic risk as well as systematic risk, the beta in the Sharpe LP. Jacob’s model is to minimize Systematic risk   1  2 Rp D K

Unsystematic risk n X

!2 xi ˇi

 2 .Rm / C

i D1

1 K

2 X n

xi2  2 .ei /

i D1

After division by K and rearrangement of terms, the objective function can be restated:   1 2 2 ZD xi ˇi  .Rm / C  .ei / K i D1 n X

Letting Zi D ˇi  2 .Rm / C .1=K/ 2 .ei /, the problem can be cast as: Minimize Z D 

n X

Xi Zi  .1  /

i D1

n X

Xi E.Ri /

i D1

subject to: n X

Xi D 1:0

0  Xi  1=K

i D 1; 2; : : : ; N

i D1

in which  is varied from zero to one. This problem can be solved by the linear-programming approach suggested by Sharpe (1967). An example of solving this problem can be found in Jacob (1974).

(7A.2) subject to: 1 K 1 K

n P i D1 n P

  xi E .Ri /  E  Rp xi D K

xi D 0 or 1

i D1

where: xi D the investment in security i – all, or nothing; E .Ri / D the expected return on security i equal to ˛i C ˇi Rm K D the desired upper bound on the number of securities the investor is willing to consider, usually in the range of 15–20; ˇi D the measure of systematic risk;  2 .e  i/ D  the measure of unsystematic risk; and  E Rp D the lowest acceptable rate of return the investor is willing to earn on his or her portfolio. An additional simplification is to turn the objective function of Equation (7A.2) into a linear relationship. Since the decision variables .xi / are binary valued, the unsystematicrisk term of the objective

is already linear. Given n function P xi ˇi =K is very close to unity, that the portfolio beta i D1

Appendix 7B Expected Return, Variance, and Covariance for a Multi-index Model Using the orthogonalization technique discussed in the text, the multi-index model (MIM) is transformed into Ri D ai C bi1 I1 C bi 2 I2 C    C biL IL C ci

(7B.1)

in which all Ij are uncorrelated with each other. To interpret the transformed indexes, notice that I2 is now the difference between the actual level of the index and the level it would be, given the level of the other indexes. Also, bi 2 is now the sensitivity of security i ’s return to a change in I2 , given that all other indexes are held constant. It is also convenient, beyond making the indexes uncorrelated, to assume that the covariance of the residuals with the indexes is equal to zero. With this final assumption, the MIM can be recapped as follows. Generalized equation: Ri D ai C bi1 I1 C bi 2 I2 C    C biL IL C ci

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C.-F. Lee et al.

for all securities i D 1 to N , and:

Remembering that:

E.ci / D 0 Var .ci / D ci2 Var .Ij / D Ij2 Cov .Ij ; Ik / D EŒ.Ij  I j /.Ik  I k / D 0; I j D E.Ij /; I ci D E.Ici / 5. Cov.ci ; Ij / D EŒci .Ij  I j / D 0 6. Cov.ci ; cj / D E.ci ; cj / D 0 for all i, j, and k.

1. 2. 3. 4.

The last statement is equivalent to the residuals being unrelated, which is to say that the only reason for common comovement of returns is due to the movement of the indexes. Although this is like ignoring some of a model’s shortcomings, it facilitates a usable model to obtain a quantified idea of future return patterns. The expected value of the model with the multiple indexes is

E



Ii  I i

     Ij  I j D 0 and E Ii  I i ci D 0

the only nonzero term involving index one is expressed: 2 2 2 E.I1  I 1 /2 D bi1 I1 bi1

Because all terms involving ci are zero and E.ci /2 D ci2 : 2 2 2 2 I1 C bi22 I22 C    C biL IL C ci2 i2 D bi1

(7B.3)

The covariance of the returns between security i and j utilizing the multiple-index model can be expressed:    ij D E .Ri  ri / Rj  rj Again, substituting for Ri ; ri ; Rj , and rj :

E.Ri / D E.ai C bi1 I1 C bi 2 I2 C    C biL IL C ci / D E.ai / C E.bi1 I1 / C E.bi 2 I2 / C   

ij DEŒŒ.ai C bi1 I1 C bi 2 I2 C    C biL IL C ci /  .ai C bi1 I 1 C bi 2 I 2 C    C biL I L / 

C E.biL IL / C E.ci /

Œ.aj C bj1 I1 C bj 2 I2 C    C bjL IL C cj / because a and the b’s are constants, and the expected value of the residuals is equal to zero ri D ai C bi1 I 1 C bi 2 I 2 C    C biL I L

(7B.2)

where I j is the expected value of index j . The variance of the returns using multiple indexes is:

 .aj C bj1 I 1 C bj 2 I 2 C    C bjL I L / Again, noting that the ai cancel and combining the terms involving the same b’s: ij D EfŒbi1 .I1  I 1 / C bi 2 .I2  I 2 / C    C biL .IL  I L / C ci

i2 D E.Ri  ri /2 where ri is the expected value of the returns of security i . Substituting Ri and ri from above: i2

DEŒ.ai C bi1 I1 C bi 2 I2 C    C biL IL C ci /    ai C bi1 I 1 C bi 2 I 2 C    C biL I L 2

Rearranging, and noticing that the ai cancel, yields:       i2 D EŒbi1 I1  I 1 C bi2 I2  I 2 C    C biL IL  I L C ci 2

Next the terms in the brackets are squared. To proceed with this, concentrate on the first index, and the rest of the terms involving the other indexes follow directly. The first index times itself and all other terms yields  2    2 I1  I 1 C bi1 bi 2 I1  I 1 I2  I 2 C    EŒbi1      Cbi1 biL I1  I 1 IL  I L C bi1 I1  I 1 .ci /

Œbj1 .I1  I 1 / C bj 2 .I2  I 2 / C    C bjL .IL  I L / C cj g Next multiply out terms, again concentrating on the terms involving bi1 : EŒbi1 bj1 .I1  I 1 /2 C bi1 bi 2 .I1  I 1 /.I2  I 2 / C    C bi1 bjL .I1  I 1 /.IL  I L / C bi1 .I1  I 1 /cj Because the covariance between two indexes is zero, and the covariance between any residual and an index is zero: 2 bi1 bj1 E.I1  I 1 /2 D bi1 bj1 I1

To conclude, remember that the covariance of the residuals is equal to zero and therefore: 2 2 ij D bi1 bj1 I1 C bi 2 bj 2 I22 C    C biL bjL IL

(7B.4)

Chapter 8

Performance-Measure Approaches for Selecting Optimum Portfolios Cheng-Few Lee, Hong-Yi Chen, and Jessica Shin-Ying Mai

Abstract In this chapter, following Elton et al. (Journal of Finance 31:1341–57, 1976; Modern portfolio theory and investment analysis, 7th edn. Wiley, New York, 2006), we introduce the performance-measure approaches to determine optimal portfolios. We find that the performance-measure approaches for optimal portfolio selection are complementary to the Markowitz full variance-covariance method and the Sharpe index-model method. The economic rationale of the Treynor method is also discussed in detail. Keywords Sharpe performance measure approach r Lintner’s method of short sales r Treynor measure r Short sales r Standard method of short sales

8.1 Introduction Previously, we have discussed Markowitz’s (1952, 1959) full variance-covariance approach to determine optimal weights of a portfolio. Moreover, we also utilized Sharpe’s indexmodel approach to simplify Markowitz’s optimal portfolioselection process. This chapter assumes the existence of a risk-free borrowing and lending rate and advances one step further to simplify the calculation of the optimal weights of a portfolio and the efficient frontier. First discussed are Lintner’s (1965) and Elton et al. (1976) Sharpe performancemeasure approaches for determining the efficient frontier with short sales allowed. This is followed by a discussion of the Treynor performance-measure approach for determining the efficient frontier with short sales allowed. The Treynor measure approach is then analyzed for determining the efficient frontier with short sales not allowed. And finally, Dow Jones 30 data from January 2003 through December 2007 are employed to demonstrate how the Treynor method can be applied in the real world. Overall, this chapter

C.-F. Lee, H.-Y. Chen, and J.S.-Y. Mai () Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

relates the performance-measure concepts and methods to the portfolio-selection models, and make more accessible the insights of optimal portfolio selection.

8.2 Sharpe Performance-Measure Approach with Short Sales Allowed In deriving the capital asset pricing model (CAPM), Lintner (1965) suggests a performance-measure approach for determining the efficient frontier discussed in previous chapters. Lintner arrived at this approach through a sequence of logical steps. Following previous chapters, the objective function for portfolio selection can be expressed: Max L D

n X

W i R i C 1

i D1

82 9 31=2 ˆ > n n X < X =   4 Cov Ri ; Rj 5  p ˆ > : j D1 i D1 ; C2

n X

! Wi  1

(8.1)

i D1

where: Ri D average rates of return for security i ; Wi (or  the optimal weight for i th (or j th) security;  Wj / D Cov Ri ; Rj D the covariance between Ri and Rj ; p D the standard deviation of a portfolio; and 1 ; 2 D Lagrangian multipliers. Markowitz’s portfolio selection model minimizes the variance given the targeted expected rate of return. Equation (8.1) maximizes the expected rates of return given targeted standard deviation. If a constant risk-free borrowing and lending rate Rf is subtracted from Equation (8.1):

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_8, 

125

126

C.-F. Lee et al.

Max L0 D

n X 

 Wi Ri  Rf C 1

i D1

82 9 31=2 ˆ > n n X < X =   5 4 Wi Wj Cov Ri ; Rj  p ˆ > : i D1 j D1 ; C2

n X

! Wi  1

(8.2)

i D1

Equations (8.1) and (8.2), both formulated as a constrained maximization problem, can be used to obtain optimum portfolio weights Wi .i D 1; 2;    ; n/. Since Rf is a constant, the optimum weights obtained from Equation (8.1) will be equal to those obtained for Equation (8.2). Previous chapters used the methodology of Lagrangian multipliers; it can be shown that Equation (8.2) can be replaced by a nonconstrained maximization method as follows. Incorporating n P Wi  1 into the objective function by the constant i D1

substituting n X

Rf D .1/ Rf D

! Wi Rf D

i D1

n X

Wi Rf

i D1

into Equation (8.2): Max L0 D

n X

  Wi Ri  Rf C 1

i D1

20 11=2 3 n n   6@X X 7 Wi Wj Cov Ri ; Rj A p 5 (8.2a) 4 i D1 j D1

A two-Lagrangian multiplier problem has been reduced to a one-Lagrangian problem as indicated in Equation (8.2a). By using a special property of the relationship between !1=2 n  n P n    P P Ri  Rf and Wi Wj Cov Ri ; Rj the i D1

i D1 j D1

constrained optimization of Equation (8.2a) can be reduced to an unconstrained optimization problem, as indicated in Equation (8.3).1 n P



Wi Ri  Rf



i D1

Max L D n P i D1

Wi2 i2 C

n P n P

Fig. 8.1 Linear efficient frontier

  where ij D Cov Ri ; Rj . Alternatively, the objective function of Equation (8.3) can be developed as follows. This ratio L is equal to excess average rates of return for the i th portfolio divided by the standard deviation of the i th portfolio. This is a Sharpe performance measure. Following Sharpe (1964) and Lintner if there is   (1965), a risk-free lending and borrowing rate Rf and short sales are allowed, then the efficient frontier (efficient set) will be linear,   as discussed in previous chapters. In terms of return Rp standard-deviation p space, this linear efficient frontier is indicated as line Rf E in Fig. 8.1. AEC represents a feasible investment opportunity in terms of existing securities to be included in the portfolio when there is no risk-free lending and borrowing rate. If there is a risk-free lending and borrowing rate, then the efficient frontier becomes Rf E. An infinite number of linear lines represent the combination of a riskless asset and risky portfolio, such as Rf A; Rf B and Rf E. It is obvious that line Rf E has the highest slope, as represented by Rp  Rf (8.4) ‚D p in which Rp D

Wi Ri ; Rf , and p are defined as in

i D1

Equation (8.2). Thus the efficient set is obtained by maximizn P Wi D 1, Equation (8.4) ing ‚. By imposing the constraint i D1

is expressed:

! n X Rp  Rf C Wi  1 ‚ D p i D1 0

!1=2

.i ¤ j /

Wi Wj ij

i D1 j D1

Since the ratio of Equation (8.3) is homogeneous of degree zero with respect to Wi . In other words, the ratio L is unchanged by any proportionate change in the weight of Wi .

(8.5)

By using the procedure of deriving Equations (8.2a), (8.5) becomes n P

(8.3) 1

n P

‚0 D

  Wi Ri  Rf

i D1

n P

i D1

Wi2 i2

C

n P n P i D1 i D1

.i ¤ j / ij

(8.5a)

8 Performance-Measure Approaches for Selecting Optimum Portfolios

This equation is equivalent to Equation (8.3). This approach is used by Elton and Gruber (1987) to derive their objective function for optimal portfolio selection.2 Following the maximization procedure discussed earlier in previous chapters, it is clear that there are n unknowns to be solved in either Equation (8.3) or Equation (8.5a). Therefore, calculus must be employed to compute n firstorder conditions to formulate a system of n simultaneous equations: 1

dL D0 dW 1

2

dL D0 dW 2

127

Example 8.1. Let R1 D 15% R2 D 12% R3 D 20% 3 D 9% 1 D 8% 2 D 7% r13 D 0:4 r23 D 0:2 r12 D 0:5 Rf D 8% Substituting this information into Equation (8.6a): 15  8 D 64H1 C .0:5/ .8/ .7/ H2 C .0:4/ .8/ .9/ H3 12  8 D .0:5/ .8/ .7/ H1 C 49H2 C .0:2/ .7/ .9/ H3 20  8 D .0:4/ .8/ .9/ H1 C .0:2/ .7/ .9/ H2 C 81H3 Simplifying:

:: :

:: :

7 D 64H1 C 28H2 C 28:8H3

n

dL D0 dW n

4 D 28H1 C 49H2 C 12:6H3 12 D 28:8H1 C 12:6H2 C 81H3

From Appendix 8A, the n simultaneous equations used to solve Hi are R1  Rf D C H2 12 C H3 13 C    C Hn 1n R2  Rf D H1 12 C H2 22 C H3 23 C    C Hn 2n :: :

Using Cramer’s rule, H1 ; H2 , and H3 can be solved for as follows:

H1 12

Rm  Rf D H1 1n C H2 2n C H3 3n C    C Hn n2 (8.6) where:

H1 D

Rp Rf p2

.

The HS are proportional to the optimum portfolio weight Wi .i D 1; 2; : : : ; n/ by a constant factor K. To determine the optimum weight Wi ; Hi is first solved from the set of equations indicated in Equation (8.6). Having done so the Hi must be called to calculate Wi , as indicated in Equation (8.7). Wi D

Hi n P

ˇ ˇ ˇ 7 ˇ ˇ ˇ 4 ˇ ˇ ˇ ˇ 12

R1  Rf D H1 12 C H2 12 C H3 13 R3  Rf D H1 13 C H2 23 C H3 32 Example 8.1 provides further illustration. 2

Elton et al. (2006).

ˇ

49 12:6 ˇˇ

12:6 81

ˇ ˇ ˇ

D

33648:1  27:1177 6350:4 D D 3:97% 160030 160030 ˇ ˇ ˇ 64 7 28:8 ˇ ˇ ˇ ˇ ˇ ˇ 28 4 12:6 ˇ ˇ ˇ ˇ 28:8 12 81 ˇ

H2 D ˇ ˇ ˇ 64 28 28:8 ˇ ˇ ˇ ˇ ˇ ˇ 28 49 12:6 ˇ ˇ ˇ ˇ 28:8 12:6 81 ˇ D

.2540:2 C 9676:6 C 20736/  .9676:8 C 15876 C 3317:8/ 160030

D

32952:8  28870:6 D 2:55% 160030

H3 D

R2  Rf D H1 12 C H2 22 C H3 23

ˇ ˇ

28 28:8 ˇˇ

.4233:6 C 1451:5 C 27783/  .1111:3 C 9072 C 16934:4/ .10160:6 C 10160:6 C 254016/  .10160:6 C 63504 C 40642:6/

i D1

If there are only three securities, then Equation (8.6) reduces to:

ˇ

49 12:6 ˇˇ ˇ 12:6 81 ˇˇ

ˇ ˇ ˇ 64 ˇ ˇ ˇ 28 ˇ ˇ ˇ ˇ 28:8

(8.7)

Hi

ˇ ˇ

28 28:8 ˇˇ

D

Hi D kW i .i D 1; 2; : : : ; n/; and kD

(8.6b)

(8.6a)

ˇ ˇ ˇ 64 ˇ ˇ ˇ 28 ˇ ˇ ˇ ˇ 28:8 ˇ ˇ ˇ ˇ 64 ˇ ˇ 28 ˇ ˇ ˇ ˇ 28:8

ˇ ˇ

28 28:8 ˇˇ ˇ 49 12:6 ˇˇ ˇ ˇ ˇ ˇ ˇ 28:8 ˇˇ ˇ 12:6 ˇˇ ˇ 81 ˇˇ

12:6 81 28 49 12:6

.3225:6 C 2469:6 C 37632/  .3225:6 C 9480 C 9878:4/ 160030 20815:2 D D 13:01% 160030

D

128

C.-F. Lee et al.

Using Equation (8.7), W1 ; W2 , and W3 are obtained: W1 D

H1 3 P

D

Hi

3:97 3:97 D 3:97 C 2:55 C 13:01 18:53

The efficient frontier for this example is shown in the figure. A represents an efficient portfolio with Rp D 17:94 percent and p2 D 50:90 percent.

i D1

D 20:33% H2 2:55 W2 D 3 D P 19:53 Hi

8.3 Treynor-Measure Approach with Short Sales Allowed

i D1

Using the single-index market model discussed in previous chapter, Elton et al. (1976) define:

D 13:06% H3 13:01 W3 D 3 D P 19:53 Hi

p D

i D1

n P iD1

Wi2 ˇi2 m2

C

D 66:61%

n P n P iD1 j D1

Wi Wj ˇi ˇj m2

C

n P iD1

!1=2 2 Wi2 2i

j ¤i

Rp and p2 can be calculated by employing these weights:

Substituting of this value of p into Equation (8.3):

Rp D .15/ .0:2033/ C .12/ .0:1306/ C .20/ .0:0061/ D 3:049 C 1:5672 C 13:322

n P iD1

LD

D 17:9382%

  Wi Ri  Rf

p D

n P iD1

Wi2 ˇi2 m2

C

n n P P iD1 j D1

Wi Wj ˇi ˇj m2

C

n P iD1

!1=2 2 Wi2 2i

j ¤i

p2

D

3 X

Wi2 i2

i D1

C

3 3 X X

(8.8)

Wi Wj rij i j

i ¤j

i D1 j D1

D .0:2033/2 .64/ C .0:1306/2 .49/ C .0:6661/2 .81/

In order to find the set of Wi s that maximizes L, take the derivative of the above equation with respect to each Wi . Let

C2 .0:2033/ .0:1306/ .0:5/ .8/ .7/

Hi D

C2 .0:2033/ .0:6661/ .0:4/ .8/ .9/ C2 .0:2/ .0:1306/ .0:6661/ .7/ .9/ D 2:645 C 0:836 C 35:939 C 1:487 C 7:8 C 2:192 D 50:899%

Rp  Rf p2

! Wi

  since Rp  Rf =p2 is a constant factor for each security to be included in the portfolio. Hence, it can be cancelled by using a standard scaling method: Wi D

Efficient frontier for Example 8.1

Hi n P Hi

(8.9)

i D1

From Appendix 8B Hi can be obtained as follows: 1 n P .Rj Rf / 2  2 B m j D1 2i C ˇ 2

C B i B C i ˇ2 A  2 @ P j 2i 1 C m2 2 0

Hi D

  Ri  Rf 2 2i

j D1

(8.10)

ej

Equation (8.10) can be modified to: ˇi Hi D 2 2i

Ri  Rf  C ˇi

! (8.11)

8 Performance-Measure Approaches for Selecting Optimum Portfolios

 ˇ in which Ri  Rf ˇ ˇi is the Treynor performance measure. C  can be defined as m2 C D

i P .Ri Rf /ˇi j D1

1C

m2

2 2j

i P

ˇj2

j D1

2 2j

(8.12)

The Hi s must be calculated for all of the stocks in the portfolio. If Hi is a positive value, this indicates the stock will be held long, whereas a negative value indicates that the stock should be sold short. This method is called the Treynor measure approach. The argument will be clearer when the case of portfolio selection with short sales not allowed is discussed. To determine the optimum portfolio from the Hi s (such that 100% of funds are invested) the weights must be scaled. One method follows the standard definition of short sales, which presumes that a short sale of stock is a source of funds to the investor; it is called the standard method of short sales. This standard scaling method is indicated in Equation (8.9). In Equation (8.9), Hi can be positive or negative. This scaling factor includes a definition of short sales and the constraint: n X

jWi j D 1

i D1

(1) Security number

(2) Ri

(3) Ri  Rf

(4) “i

(5) 2 2i

1 2 3 4 5

15 13 10 9 7

10 8 5 4 2

1 2 1.43 1.33 1

30 50 20 10 30

From the information in the table, using Equations (8.10) and (8.11). Hi .i D 1; 2; : : : ; 5/ can be calculated: H1 D

1 30



 3:067 D 0:2311

13  5  3:067 D 0:0373 2



10  5 1:43 H3 D  3:067 D 0:0307 20 1:43



95 1:33 H4 D  3:067 D 0:0079 10 1:33

75 1 H5 D  3:067 D 0:0356 30 1

H2 D

2 50



15  5 1

According to Lintner’s method:

A second method (Lintner’s (1965) method of short sales) assumes that the proceeds of short sales are not available to the investor and that the investor must put up an amount of funds equal to the proceeds of the short sale. The additional amount of funds serves as collateral to protect against adverse price movements. Under these assumptions, the constraints on the Wi s can be expressed: n X

129

5 X

Now to scale the Hi values into an optimum portfolio, apply Equation (8.13): W1 D

jWi j D 1

W2 D

i D1

And the scaling factor is expressed as: Wi D

Hi n P jHi j

W3 D (8.13)

W4 D

i D1

W5 D

Example 8.2 provides further illustration. Example 8.2. The following example shows the differences in security weights in the optimal portfolio due to the differing short-sale assumptions. Data associated with regressions of the single-index model are presented in the table. The mean return, R, the excess return Ri Rf , the beta coef2 are presented ficient ˇi , and the variance of the error term 2i in columns 2–5.

jHi j D 0:3426

i D1

0:2311 D 0:6745 0:3426 0:0373 D 0:1089 0:3426 0:0307 D 0:0896 0:3426 0:0079 D 0:0231 0:3426 0:0356 D 0:1039 0:3426

Thus, the Lintner model states that an investor should invest 67.45% in security 1, 10.89% in security 2, and 8.96% in security 3. The investor should then sell short 2.31 and 10.39% of securities 4 and 5, respectively. If this same example using the standard defini 5 is scaled

P tion of short sales Hi , which provides funds to the i D1

investor:

130

C.-F. Lee et al. 5 X

where:

Hi D 0:2556

m2

i D1

W1 D W2 D W3 D W4 D W5 D

Ci

0:2311 D 0:9041 0:2556 0:0373 D 0:1459 0:2556 0:0307 D 0:1201 0:2556 0:0079 D 0:0309 0:2556 0:0356 D 0:1393 0:2556

i P

ˇj2

j D1

2 2j

(8.15)

Example 8.3 provides further illustration.

8.4 Treynor-Measure Approach with Short Sales Not Allowed Elton et al. (1976) also derive a Treynor-measure approach with short sales not allowed. From Appendix 8C, Equation (8.11) should be modified to: !

Ri  Rf  Ci C i ˇi

ˇj

If all securities have positive ˇi s, the following three-step procedure from Elton et al. can be used to choose securities to be included in the optimum portfolio.3 ı  1. Use the Treynor performance measure Ri  Rf ˇi to rank the securities in descending order. 2. Use Equation (8.15) to calculate Ci for all securities. 3. Include in ı the portfolio all securities for which  Ri  Rf ˇi is larger than Ci .

Example 8.3. The tape showing Center for Research in Security Prices was the source of 5 years of monthly return data, from January 2003 to December 2007, for the 30 stocks in the Dow Jones Industrial Averages. The value-weighted average of the NYSE index was used as the market while 3-month Treasury-bill rates were used as the risk-free rate. The single-index model was used with an ordinary leastsquares regression procedure to determine each stock’s beta. The following data were compiled for each stock. 1. 2. 3. 4. 5.

The mean monthly return Ri The mean excess return Ri  Rf The beta coefficient The variance of the residual errors  ı The Treynor performance measure Ri  Rf ˇi

All data are listed in the worksheet on next page, which lists the companies in descending order of the Treynor performance measure.4 To calculate the Ci as defined in Equation (8.15), cali h  . 2  . 2 i  P , culate Rj  Rf ˇj 2j ; Rj  Rf ˇj 2j

(8.14)

j D1

.

and

2 ˇj2 2j :

presented

ˇi 2i

j D1

Rj Rf 2 2j

1 C m2

Using this definition of short sales provides that the investor should invest 90.41% of his or her money in security 1, and so on. If all Wi s are added together, they equal 100%. This is true because the definition says that the funds received from selling short a security should be used to purchase more of the other securities. The difference between Lintner’s method and the standard method are due to the different definitions of short selling discussed earlier. The standard method assumes that the investor has the proceeds of the short sale, while Lintner’s method assumes that the short seller does not receive the proceeds and must provide funds as collateral. The method discussed in this section does not require the inputs of covariance among individual securities. Hence, it is a simpler method than that of the Sharpe performancemeasure method discussed in the previous section. The relative advantage of the Sharpe performance method over the Treynor performance method is exactly identical to the relative advantage of the Markowitz model over the Sharpe single-index model. In sum, the Treynor performance method for portfolio selection requires the information of both the risk-free rate and the market rates of return.

Hi D

D

i P

0:00207;

i P

. i P 2 ˇj2 2j

is

calculated

and

j D1

in the worksheet. Substituting m2 D h .

i  . 2 i P 2 Rj  Rf ˇj 2j , and ˇj2 2j

j D1

j D1

into Equation (8.15) produces Ci for every firm as listed in the last column in the worksheet. Using company HPQ as an example: 3 If the beta coefficient ˇi for i th security is positive, then the size of Hi depends on the sign of the term ı in parentheses. Therefore, if a security with a particular Ri  Rf ˇi is included in the optimum portfolio, all ı securities with a positive beta that have higher values of  R i  Rf ˇi must be included in the optimum portfolio. 4 This set of data has been analyzed in later chapters. The names of these 30 firms can be found in Table 19.1.

XOM MCD JNJ BA T CAT HPO CVX GE DIS KO JPM INTC AA UTX IBM AXP HD MRK VZ DD KFT GM MSFT C PG MMM WMT PFE BAC Rm Var.Rm / Rf

Ticker

Mean Ri 0:0185 0:0258 0:0043 0:0193 0:0115 0:0128 0:0202 0:0101 0:0088 0:0117 0:0080 0:0120 0:0121 0:0132 0:0077 0:0068 0:0076 0:0063 0:0038 0:0035 0:0038 0:0016 0:0004 0:0000 0:0010 0:0013 0:0017 0:0009 0:0035 0:0054 0:00887 0:002433333 0:000615571

0.0067 0.0073 0.0045 0.0080 0.0083 0.0113 0.0084 0.0102 0.0052 0.0067 0.0056 0.0070 0.0104 0.0098 0.0106 0.0066 0.0061 0.0083 0.0100 0.0066 0.0066 0.0064 0.0144 0.0114 0.0073 0.0099 0.0111 0.0059 0.0070 0.0095

Standard deviation i

Worksheet for Dow Jones industrial averages

R i  Rf 0:0160 0:0234 0:0018 0:0168 0:0091 0:0104 0:0178 0:0077 0:0064 0:0093 0:0056 0:0096 0:0096 0:0107 0:0053 0:0043 0:0051 0:0038 0:0014 0:0011 0:0014 0:0008 0:0028 0:0025 0:0035 0:0011 0:0042 0:0015 0:0059 0:0079

(1)

0.7905 1.3600 0.1178 1.2084 0.7139 0.8210 1.4747 0.7931 0.7268 1.0829 0.7845 1.3579 1.6463 1.9542 1.1547 1.1726 1.4069 1.2612 0.6459 0.8905 1.1670 0.7377 1.6717 1.0800 1.2265 0.3752 0.9803 0.2021 0.7079 0.7140

ˇi

(2)

0.00228 0.00204 0.00118 0.00299 0.00370 0.00720 0.00288 0.00587 0.00130 0.00197 0.00152 0.00183 0.00482 0.00341 0.00591 0.00177 0.00103 0.00319 0.00572 0.00217 0.00176 0.00213 0.01076 0.00706 0.00226 0.00574 0.00682 0.00206 0.00264 0.00506

2 2i

(3)

(5) .Ri Rf /ˇi ei2

5:5603 15:5812 0:1837 6:8054 1:7527 1:1; 863 9:0985 1:0377 3:5539 5:0882 2:8695 7:1293 3:2879 6:1519 1:0314 2:8744 6:9941 1:5228 0:1537 0:4538 0:9278 0:2825 0:4414 0:3761 1:8779 0:0729 0:6010 0:1498 1:5809 1:1104

(4) .Ri Rf / ˇi

0:0203 0:0172 0:0157 0:0139 0:0127 0:0127 0:0121 0:0097 0:0087 0:0086 0:0071 0:0071 0:0058 0:0055 0:0046 0:0037 0:0037 0:0031 0:0021 0:0012 0:0012 0:0011 0:0017 0:0023 0:0028 0:0030 0:0043 0:0076 0:0083 0:0110

5:5603 21:1416 21:3253 28:1307 29:8834 31:0697 40:1682 41:2059 44:7598 49:8480 52:7175 59:8467 63:1346 69:2865 70:3180 73:1924 80:1865 81:7093 81:8631 82:3169 83:2447 82:9622 82:5208 82:1447 80:2668 80:1939 79:5929 79:4430 77:8622 76:7518

(6) Cumulative sum of column (5) 273:92 905:36 11:73 488:69 137:56 93:62 754:92 107:18 406:76 593:91 403:88 1; 008:07 562:52 1; 120:50 225:46 776:21 1; 913:36 499:04 72:94 366:18 775:66 255:97 259:71 165:15 664:60 24:53 140:89 19:79 189:74 100:70

ˇi2 ei2

(7)

0.002929 0.007540 0.007574 0.008514 0.008683 0.008788 0.009362 0.009370 0.009317 0.009234 0.009086 0.008788 0.008563 0.008158 0.008065 0.007709 0.007029 0.006862 0.006833 0.006667 0.006344 0.006201 0.006051 0.005951 0.005548 0.005534 0.005439 0.005422 0.005246 0.005136 0.009370

Max C1

(9) C1

273:92 1; 179:29 1; 191:02 1; 679:71 1; 817:28 1; 910:90 2; 665:82 2; 773:00 3; 179:75 3; 773:66 4; 177:53 5; 185:61 5; 748:12 6; 868:63 7; 094:09 7; 870:31 9; 783:66 10; 282:71 1; 0355:65 1; 0721:83 1; 1497:49 1; 1753:45 1; 2013:16 1; 2178:30 1; 2842:91 1; 2867:43 1; 3008:32 13; 028:81 13; 217:85 13; 318:55

(8) Cumulative sum of column (7)

8 Performance-Measure Approaches for Selecting Optimum Portfolios 131

132

C.-F. Lee et al. Positive optimum weight for eight securities

 CHPQ D

Ticker

Hi

XOM MCD JNJ BA T CAT HPQ CVX Total C D

6.0189 6.4370 0.8050 2.1887 0.7820 0.4428 1.3769 0.0421 18.0935 0.009370

ˇi ="i2

.R i  Rf / ˇi

(A) optimum percentage

(B) mean

.A/  .B/

346.5158 665.6941 99.5753 404.4066 192.6984 114.0371 511.8963 135.1436

0.0203 0.0172 0.0157 0.0139 0.0127 0.0127 0.0121 0.0097

0.3327 0.3558 0.0445 0.1210 0.0432 0.0245 0.0761 0.0023 1.0000

0.0185 0.0258 0.0043 0.0193 0.0115 0.0128 0.0202 0.0101 Rp

0.00615 0.00919 0.00019 0.00233 0.00050 0.00031 0.00154 0.00002 0.02023

.0:0006155/ .40:6817114/ D 0:009362497 1 C .0:0006155/ .265:818464/

From Ci of the worksheet it is clear that there are ten securities included in the portfolio. The estimated ı 2thatshould beı ; Ri  Rf ˇi and Ci of these ten securities are ˇi 2i listed in the table at the top of this page. Substituting this information into Equation (8.14) produces Hi for all ten securities. Using security 5 as an example: HXOM D .346:5158/.0:0202988  0:009370285/ D 6:018948255 Using Equation (8.12) the optimum weights can be estimated for all ten securities, as indicated in the table. In other words, 33.27% of our fund should be invested in security XOM, 35.58% in security MCD, 4.45% in security JNJ, 12.10% in security BA, 4.32% in security T, 2.45% in security CAT, 7.61% in security HPQ, 0.23% in security CVX, based on the optimal weights, the average rate for the portfolio Rp is calculated as 2.02%, as presented in the last column of the table above.

8.5 Impact of Short Sales on Optimal-Weight Determination The Markowitz model of portfolio analysis. The Markowitz model requires a large number of inputs, as it is necessary to estimate the covariance between each pair of securities. Previously, the analysis was simplified by the assumption that security returns were related through a common response to some market index. This model, known as Sharpe’s

single-index model, greatly reduces the number of inputs necessary to analyze the risk and return characteristics of portfolios. In both the Markowitz and Sharpe models the analysis is facilitated by the presence of short selling. This chapter discusses a method proposed by Elton and Gruber for the selection of optimal portfolios. Their method involves ranking securities based on their excess return to beta ratio, and choosing all securities with a ratio greater than some particular cutoff level C  . It is interesting to note that while the presence of short selling facilitated the selection of the optimum portfolio in both the Markowitz and Sharpe models, it complicates the analysis when we use the Elton and Gruber approach. From Example 8.3, using the Dow Jones Industrial Average (DJIA), the absence of the short selling allowed formation of the optimal portfolio using only eight stocks.

8.6 Economic Rationale of the Treynor Performance-Measure Method Cheung and Kwan (1988) have derived an alternative simple rate of optimal portfolio selection in terms of the singleindex model. First, Cheung and Kwan relate Ci as defined in Equation (8.12) to the correlation coefficient between the portfolio rates of return with i securities Ri and market rates of return Rm .i /: Ci i D (8.16) m ‚i where: i D i m =i m I i m D covariance between Ri and Rm I ı  ‚i D Ri  Rf i ; the Sharpe performance

8 Performance-Measure Approaches for Selecting Optimum Portfolios

measure associatedwith the i th portfolioI and i and m D standard deviation for ith portfolio and market portfolio, respectively. Cheung and Kwan show that Pi and Ci display the same functional behavior for optimal portfolio selection. In other words, if portfolios are formed by adding securities successively from the highest rank to the lowest rank, the optimal portfolio is reached when the correlation of the portfolio returns and the index is at its maximum. Since the expected return on the index must be positive if the investor is to invest in stocks, an objective of the investor using the single-index model to establish the risk-return tradeoff is to pick securities that benefit the most from a market upswing. Hence, the role of index in the selection of securities for an optimum portfolio is demonstrated explicitly. Based on the single-index model and the risk decomposition discussed in previous chapters, the following relationships can be defined: 1: 2: 3:

i D

i m i m

i m D ˇi m2 2 i2 D ˇi2 m2 C 2i

(8.17)

From Equation (8.17), Cheung and Kwan define i in terms of ˇi2 ; m2 , and i . i D

q

ı

2 ˇi2 m2 i2 C 2i D

s 1

2 2i i2

2 in which 2i is the nonsystematic risk for the i th portfolio. They use both i and ‚i to select securities for an optimum portfolio, and they conclude that i can be used to replace ‚i in selecting securities for an optimum portfolio. Nevertheless, ‚i information is still needed to calculate the weights for each security. Hence, Cheung and Kwan’s i criteria is good only for understanding Elton and Gruber’s performance-measure method for portfolio selection.

8.7 Conclusion Following Elton et al. (1976) and Elton and Gruber (1987) we have discussed the performance-measure approaches to selecting optimal portfolios. We have shown that the performance-measure approaches for optimal portfolio selection are complementary to the Markowitz full variancecovariance method and the Sharpe index-model method.

133

These performance-measure approaches are thus worthwhile for students of finance to study following an investigation of the Markowitz variance-covariance method and Sharpe’s index approach.

References Alexander, G. J. and B. J. Resnick. 1985. “More on estimation risk and single rule for optimal portfolio selection.” Journal of Finance 40, 125–134. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Chen, S. N. and S. J. Brown. 1983. “Estimation risk and simple rules for optimal portfolio selection.” Journal of Finance 38, 1087–1093. Cheung, C. S. and C. C. Y. Kwan. 1988. “A note on simple criteria for optimal portfolio selection.” Journal of Finance 43, 241–245. Elton, E. J., M. J. Gruber, and M. W. Padberg. 1976. “Simple criteria for optimal portfolio selection.” Journal of Finance 31, 1341–1357. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Kwan, C. C. Y. 1984. “Portfolio analysis using single index, multiindex, and constant correlation models: a unified treatment.” Journal of Finance 39, 1469–1483. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance. Springer, New York. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Lintner, J. 1965. “The valuation of risk asset on the selection of risky investments in stock portfolio and capital budgets.” The Review of Economics and Statistics 57, 13–37. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, Hoboken, NJ. Markowitz, H. 1952. “Portfolio selection.” Journal of Finance 7, 77–91. Markowitz, H. 1959. Portfolio selection: efficient diversification of investments, Wiley, New York. Sharpe, W. F. 1963. “A simplified model for portfolio analysis.” Management Science 9, 277–293. Sharpe, W. F. 1964. “Capital asset prices: a theory of market equilibrium under conditions of risk.” Journal of Finance 19, 425–442. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Appendix 8A Derivation of Equation (8.6) The objective function L as defined in Equation (8.3) can be rewritten: # " n X   Max L D Wi Ri  Rf i D1

2 31=2 n n n X X X 4 Wi2 i2 C Wi Wj ij 5 i ¤j i D1

i D1 j D1

134

C.-F. Lee et al.

0

Then, following the product and chain rule, we have: d dL D dW i d Wi 2 4

n X

"

n X

  Wi Ri  Rf

Wi i2 C

n n X X

D4

n X

Wj ij 5

n n X X

i D1

n P

i ¤j kD

31=2

4

n X

i D1 j D1

Wi i2 C

n n X X

i D1

2 D4

n X

k

i D1

Wj ij 5 31=2

Wi2 i2 C

i D1

n n X X

42Wi i2 C 2

n X

i D1 j D1

Wi i2

C

n X

! Wi ij

  C Ri  Rf D 0

j ¤i

There is one equation like this for each value of i .

33=2

R1  Rf D H1 12 C H2 12 C    C Hn 1n

Wi Wj ij 5 3

R2  Rf D H1 21 C H2 22 C    C Hn 2n :: :

Wj ij 5

Rn  Rf D H1 n1 C H2 n2 C    C Hn n2

i D1 j D1

2

i ¤j Wi Wj ij

Ri  Rf D Hi 1i C H2 2i C    C Hi i2 C    C Hn ni

i D1 n X

n P n P

Define Hi D kW i , where the Wi are the fractions to invest in each security and the Hi are proportional to this fraction. Substituting Hi for kW i :

Wi Wj ij 5

Ri  Rf # " n X   1 Wi Ri  Rf 2

4

Wi2 i2 C

dL D .kW1 1i C kW2 2i C    C kWi i2 C    dW i CkWn ni / C .Ri  Rf / D0



2

i D1

Therefore:

i D1 j D1



  Wi Ri  Rf

i D1

i D1 j D1

Wi2 i2 C

(8A.2)

yields

31=2

n n X X

n P i D1

Wi Wj ij 5

" n # X   d

Wi Ri  Rf dW i i D1 " n # X   d C Wi Ri  Rf dW i i D1 2

Wj ij A

Defining

31=2

i D1 j D1

Wi2 i2 C

1

j D1

i D1

i D1

2

@Wi i2 C

#

n X

j D1

D0

.i D 1; 2; : : : ; n/

(8A.1) 2

Multiplying n P

Equation

by

4

n P

i D1

#1=2 Wi Wj ij

(8A.1)

Wi2 i2 C

n P

Appendix 8B Derivation of Equation (8.10)

i D1

Following the optimization procedure Equation (8A.2) in Appendix 8A:

; i ¤ j , and rearranging yields:

j D1

0 n B   P B Wi Ri  Rf B B i D1 Ri  Rf D B n n P n P BP 2 2 B Wi i C Wi Wj ij @ i D1 j D1 j D1 i ¤j

1 C C C C C C C A

n P

  dL D Ri  Rf  dW i 0

  Wi Ri  Rf p2

n X j D1

j ¤i

deriving

i D1

@Wi ˇi2 m2 C ˇi D0

for

1

2 A Wj ˇj m2 C Wi 2i

8 Performance-Measure Approaches for Selecting Optimum Portfolios

Let Hi D

h

Rp  Rf

.

i p2 Wi and, solving for any Hi ,

Ri  Rf Hi D  2 2i

ˇi m2

n P

Hj ˇj

j D1

(8B.1)

2 2i

135

Hi D

Hi ˇi D



Ri  Rf ˇi 2 2i

ˇi m2 

n P j D1 2 2i

d P d X

Hj ˇj2

j D1

Hj ˇj D

(8B.2)

Hj ˇj D

2 1 C 2i

2 2i

n P

ˇj2

j D1

2 2i

n X

Hj ˇj D

j D1

(8B.3) and let:

By substituting Equation (8B.3) into Equation (8B.1), Equation (8.10) is obtained.

Appendix 8C Derivation of Equation (8.15) This appendix discusses the use of performance measure to examine the optimal portfolio with short sales not allowed. Therefore, Equation (8B.1) from Appendix 8.B must be modified: n Ri  Rf ˇi m2 X  2 ˇj Hj C i Hi D 2 2i 2i j D1



Rj Rf 2 2j

ˇj (8C.3)

d P j D1

ˇj 2 2j

Notice since the set d contains all stocks with positive Hi :

n P .Rj Rf /ˇj j D1

j D1



1 C m2

j D1

Adding together the n equation of this form yields: n X

(8C.2)

Multiplying both sides by ˇj , summing over all stocks in d , and rearranging yields:

Multiplying both sides of the equation by ˇi : 

d Ri  Rf ˇi m2 X  Hj ˇj and i D 0 2 2 2i 2i j D1

(8C.1)

where Hi 0; i 0, and i Hi D 0 for all i . The justification of this equation can be found in Elton et al. (1976). Assuming all stocks that would be in an optimal portfolio (called d ) can be found, and then arranging these stocks as i D 1; 2; : : : ; d , for the subpopulation of stocks that make up the optimal portfolio:

Hj ˇj

j D1

d P

C D

d X

j D1 m2

Rj Rf 2 2j

ˇj

n P

ˇj2

j D1

2 2j

1 C m2

(8C.4)

Using Equation (8C.4), the following equation for Hi is obtained after substitution and rearranging from Equation (8C.1): ˇi Hi D 2 2i

Ri  Rf C ˇi

! C i

(8C.5)

Since i 0, the inclusion of i can only increase the value of Hi . Therefore, if Hi is positive with i D 0 can never make it zero and the security should be included. If Hi < 0 when i D 0, positive values of i can increase Hi . However, because the product of i and Hi must equal zero, as indicated in Equation (8C.1), positive values of i imply Hi D 0. Therefore, any security Hi < 0 when i D 0 must be rejected. Therefore, Equation (8.15) in the text can be used to estimate the optimal weight of a portfolio.

Chapter 9

The Creation and Control of Speculative Bubbles in a Laboratory Setting James S. Ang, Dean Diavatopoulos, and Thomas V. Schwarz

Abstract Persistent divergence of an asset price from its fundamental value has been a subject of much theoretical and empirical discussion. This paper takes an alternative approach of inquiry – that of using laboratory experiments – to study the creation and control of speculative bubbles. The following three factors are chosen for analysis: the compensation scheme of portfolio managers, wealth and supply constraints, and the relative risk aversion of traders. Under a short investment horizon induced by a tournament compensation scheme, speculative bubbles are observed in markets of speculative traders and in mixed markets of conservative and speculative traders. These results maintain with super-experienced traders who are aware of the presence of a bubble. A binding wealth constraint dampens the bubbles as does an increased supply of securities. These results are unchanged when traders risk their own money in lieu of initial endowments provided by the experimenter. Keywords Speculative bubbles r Experimental asset markets r Fundamental asset values r Tournament r Market efficiency r Behavioral finance

Speculative bubbles are induced in this study under a pictoria laboratory setting, where a New York Stock Exchange type of double oral auction market (without a specialist) involving many traders is modeled. Speculative bubbles occur when buyers are willing to bid higher and higher prices for an asset, which, in retrospect, is far in excess of its worth based on fundamentals. The bubbles ultimately burst and prices drop to a much lower level.1 The stock market crash in the U.S. in 1987, in Japan in 1991–1992, the dot com bubble in 2000, and the recent housing bubble in the U.S. are examples.2 In recent years, academicians and practitioners are slowly but grudgingly coming to the realization that the extant theories of stock market behavior (e.g., efficient market hypothesis and capital asset pricing theory), fail to explain the magnitude of fluctuations in the stock market. Not only have stock prices been found to fluctuate too much relative to fundamentals, but also there have been occurrences of speculative bubbles that could not be explained by arrival of new information. Several plausible explanations for bubbles are offered such as rational bubbles (Shiller 1988; West 1988), irrational bubbles (Ackert et al. 2002; Lei et al. 2001), judgment error (Ackert et al. 2006) and herding behavior (Froot et al. 1992).3

9.1 Introduction The purpose of this study is to investigate the formation of speculative bubbles in asset prices under a laboratory setting. Specifically, we investigate how to create, control, and dismantle bubbles, as well as the conditions in which bubbles may or may not arise. J.S. Ang () Department of Finance, Florida State University, Tallahassee, FL 32306, USA e-mail: [email protected] D. Diavatopoulos Villanova University, Villanova, PA 19085, USA e-mail: [email protected] T.V. Schwarz Grand Valley State University, Grand Rapids, MI 49504, USA e-mail: [email protected]

1 Stiglitz (1990), in his overview of a symposium on bubbles, defines the existence of bubbles to be: “if the reason that the price is high today is only because investors believe that the selling price will be high tomorrow – when ‘fundamental’ factors do not seem to justify such a price.” Similarly, he defines the breaking of a bubble as marked price declines that occur without any apparent new information. 2 Other notable example of bubbles include the Dutch tulip mania in the seventeenth century, the South Sea Islands Company bubbles (Voth and Temin 2003), John Low’s Mississippi Company Scheme Bubbles of the eighteenth century, the South Sea Islands Company bubbles, John Low’s Mississippi Company scheme bubbles of the eighteenth century, the U.S. stock market boom of the late 1920s, the Florida land price bubbles of the 1920s, the great bull market of the 1950s and 1960s, and the high-tech stock boom of the early 1980s and the boom and bust of the California and Massachusetts housing markets in recent years. However, due to the difficulties in specifying the fundamentals, there is still disagreements as to whether these cases could be explained by the fundamental example of Garber (1990) versus White (1990). 3 Outstanding surveys of this literature are provided by Porter and Smith (2003), Camerer (1989) and Sunder (1992).

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_9, 

137

138

While some work has been done to show that bubbles can be abated with experience (Dufwenberg et al. 2005), an understanding of the formation of speculative bubbles is still important to researchers for several reasons. First, bubbles could cause significant disruptions in the asset market, not only by creating a large redistribution of wealth among investors, but also by adversely affecting the supply of funds to the market as well as resource allocation among and within firms. Second, the identification of factors affecting the formation of bubbles is crucial in aiding regulators in designing policies to reduce the occurrence or magnitude of bubbles. In particular, if bubbles can be replicated in a laboratory setting, then various proposals to dampen bubbles could also be tested and compared for their effectiveness. Roll (1989) summarizes the difficulty with examining recent empirical results of the 1987 Crash in this regard. Third, an understanding of the dynamic process of bubble formation would contribute to our knowledge of how to model the behavior of asset prices (DeLong et al. 1989; Cutler et al. 1991). In spite of some interesting recent theoretical developments, empirical research on the existence of bubbles tends to be inconclusive and with low power; for example, Gurkaynak (2005), West (1988), Flood and Hodrick (1990). A major problem is the difficulty of specifying the fundamental value of an asset, as bubbles are defined as the price in excess of the fundamental values (Bierman 1995; Robin and Ruffieux 2001). Without being able to calculate the time series of the asset’s fundamental value, price movement could simply be caused by factors affecting the fundamental valuation of the asset; for example, change in risk aversion, arrival of new information, and so forth. And if the fundamental value could only be measured imperfectly using proxies such as past dividends, the imprecise estimates would, of course, reduce the power of any test. The experimental approach reduces this problem (Cason and Noussair 2001). By design, the value of the fundamentals can be specified in advance; hence there is no measurement problem. Any gross and persistent divergence of the asset price from the prespecified fundamental value can now be attributed to bubbles (Siegel 2003). In addition to reducing the identification/measurement problem, performing laboratory experiments to study asset bubbles has two other advantages. First, it allows different characteristics of the market institutions and participants to be introduced in a controlled manner. That is, relevant factors may be manipulated to create or discourage the formation of bubbles. This is an important feature because some of these factors may not be isolated in the real world for detailed study while other factors are simply proposals in the design of market institutions of the future. Second, by controlling the information available to market participants, we can control the role played by unrelated, or exogenous events (e.g., sunspots). Thus, the laboratory experiment approach to study asset market behavior

J.S. Ang et al.

complements the theory/model building process. The three types of variables chosen for analysis in this study are the following: 1. The compensation scheme of a portfolio manager. Allen and Gorton (1988) have argued that compensation schemes for portfolio managers may induce bubbles even in a finite horizon. Also, recent literature on tournaments (see James and Issac 2000; Ehrenberg and Bognanno 1990 and others) has shown that the level and structure of relative compensation influences participant behavior, while Hirota and Sunder (2005) have found that short horizons are important factors in the emergence of bubbles. Three types of compensation structure are used in these experiments: a linear compensation scheme based on portfolio performance, and two versions of compensation based on relative performance in a short-term horizon. 2. Wealth constraint (tight/loose); supply of securities. An infinite number of trades (e.g., overlapping generations and the availability of credit) is often cited as a prerequisite for bubbles. Ricke (2004) discusses how credit made available from margin could generate bubbles. High liquidity leads to bubbles in the work of Caginalp et al. (2001). Scheinkman and Xiong, (2003), Hong et al. (2006) analyze the effect of a short sales constraint on the formation of bubbles. Therefore, experimenting with wealth constraints may provide valuable insights into the effectiveness of certain policies (such as margin rule change, credit availability, and so forth) to control bubbles. 3. The type of investors in the market (speculative/conservative). This variable tests the Keynes-Hicks theory of speculation where differences in traders’ willingness to take risks is the foundation of speculative markets. Traders in the experiments are pre-tested for attitudes toward risk taking. Bubbles are observed under the following conditions: 1. A market of speculators with short-term investment horizon. 2. A market of mixed conservative and speculative traders with short-term investment horizon. 3. A market of mixed trader types with short-term investment horizon using their own money. On the other hand, bubbles are dampened under the following investment environments: 1. A market of conservative traders with a short-term horizon. 2. A market of mixed trader types with a long-term investment horizon. 3. A market of mixed trader types when the wealth of the traders, especially the bulls, is constrained. 4. A single-period trading environment.

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

The remaining part of this paper is organized into four sections. Section 9.2 presents the hypotheses to be tested by incorporating them into the experimental design, which is discussed in greater detail in Sect. 9.3. The results are reported in Sect. 9.4 with Sect. 9.5 summarizing and concluding the paper.

9.2 Bubbles in the Asset Markets The possibility of asset bubbles has long been recognized; however, more formal theoretical development is of relatively recent vintage. Harrison and Kreps (1978), for instance, suggest that in general the right to resell the asset makes traders willing to pay more for it than they would if obliged to hold it forever. Thus, market price could exceed fundamental value. Literature on rational bubbles emphasizes that once a bubble is started, it would be rational to price the bubble component even if it is expected to burst with positive probability. Brunnermeier and Nagel (2004) examine stock holdings of hedge funds during the recent NASDAQ tech bubble and find that the portfolios of these sophisticated investors were heavily tilted towards (overpriced) technology stocks. However, this does not seem to be the result of unawareness of the bubble on the part of hedge funds.4 At an individual stock level, hedge funds reduced their exposure before prices collapsed suggesting awareness and implicit pricing of the bubble component. On the other hand, whether bubbles can even get started has been questioned 64 Diba and Grossman (1987). Essentially, if there is a finite number of periods, starting from the next to the last period, the expectation that the bubble might end may be sufficient to keep it from ever starting. By the process of backward induction or an unraveling argument, bubbles will not exist. Moreover, if the number of trades is finite, withdrawal of early trades at a profit means the remaining traders would be at a negative sum game; that is, with finite trades, will there be a “greater fool” who gets stuck when the bubble bursts. Still, perturbing the model by adding uncertainties on the length of the horizon among traders or market size may preserve the possibility of asset bubbles. Smith et al. (1988) are among the first to investigate the incidence of bubbles. Their design was to give traders common beliefs (according to one of Tirole’s requirements) and long horizons of up to 15 trading periods. Bubbles are observed in 4

Griffin et al. examine the extant theoretical literature about bubbles that includes models where naive individuals cause excessive price movements and smart money trades against (and potentially eliminates) a bubble versus models where sophisticated investors follow market prices and help drive a bubble. In considering these competing views over the tech bubble period on NASDAQ they find evidence that supports the view that institutions contributed more than individuals to the spectacular NASDAQ rise and fall.

139

several of their experimental markets. It is unclear, however, what institutional setting, other than long trading periods, induces bubbles in their study. In a speculative market where bubbles could be present, speculative traders are more likely to purchase shares (and even more so, if bubbles are rationally priced) than riskaverse traders. Not only are they more willing to put a higher value on risky assets, they are also more likely to take the chance that they might not be able to sell out their inventory before the bubble bursts. Therefore, our first hypothesis is that bubbles are more likely to be formed in a market of risk taking traders (speculators). The compensation scheme could also affect the behavior of traders. For instance, Allen and Gorton (1988), and Allen and Gale (2000) show that an option-type compensation scheme for portfolio managers could induce speculative bubbles in asset prices. Portfolio managers are encouraged via incentive rewards to generate short-term trading gains even in a finite horizon world. The current practice of publishing and ranking the short-term investment performance of portfolio managers and the very substantial incentives to hedge fund managers’ performance that may be based on unrealized gains on illiquid assets could give rise to adverse incentives. Portfolio managers who are concerned about these rankings will either take on a riskier strategy for the possibility of outshining their peers or they will simply play it safe and follow the crowd. Both portfolio strategies could lead to the formation of bubbles. The play safe by “following the herd” strategy will cause asset prices to have a strong positive correlation in the short term. Portfolio managers would be buying when others are buying, thus creating an upward price trend, and selling when others were selling, thus bursting the bubble it created. On the other hand, pursuit of a risky strategy may be sufficient to create price leadership that is followed by others in the market. This would be more likely in an uncertain valuation environment. Voth and Temin (2003) suggest that riding the bubble may actually be a profitable strategy. Portfolio managers are subject to an occupational hazard: unless they produce winning results, they stand a good chance of being fired. On the other hand, star performers receive seven or even eight figure incomes as new cash flows into the funds they manage. This compensation system is similar to tournament models where participants are paid according to their relative performance among a group of peers rather than on their absolute performance. Tournament systems are likely to produce increased performance when (1) there is difficulty in monitoring the activities of the agent, (2), when the agent possesses valuable information (Baker 1992), and (3) when good performance measures are available (Baker 1992). All three of these conditions exist in the realm of professional money management, and therefore it is probable that a relative performance compensation system

140

will be effective in increasing manager performance.5 Thus, it is hypothesized that a tournament incentive scheme that encourages a short-term horizon for portfolio managers is more likely to create bubbles.6 Finally, an important policy question has been whether restricting the availability of credit or the supply of securities in a market (by raising the margin requirement or allowing short sales) could reduce or even eliminate the formation of speculative bubbles. (Ackert et al. 2006 find that price run ups and crashes are moderated when traders are allowed to short sell). Most countries, including the U.S. and Japan, have adjusted these conditions in the recent past through adjustments in the use of stock index futures and by easing credit conditions. These changes have tended to occur subsequent to large declines in the country’s equity markets. With limited or asymmetric ability to go short versus long, speculators on the long side have an advantage in acquiring funds for investment. Additionally, if the life of a bubble is uncertain and relatively long lasting, costly short sell will not be profitable even if the bubbles eventually burst. The usual experimental design often endows traders with a relatively large initial wealth such that the budget constraint is not binding. This experiment will test the effect of a tighter budget constraint by both reducing the initial endowment and increasing the supply of securities. It is hypothesized that a wealth constraint and/or relative increase in the supply of securities will reduce the incidence of bubbles. To summarize, the effect of three factors: attitude toward risk, investment horizon, and wealth constraint are examined as to their contribution to the creation and control of asset bubbles. They are tested by incorporating them into the experimental design of a laboratory setting described below, the importance of these factors in the formation and control of asset price bubble, singly and jointly, can now be formally examined.

9.3 Experimental Design The evolutionary nature of laboratory experimental research is such that the results of any study acts as a catalyst for new questions and therefore new experiments. As with Smith

5

Becker and Huselid (1992) and Ehrenberg and Bognanno (1990) have documented in field studies that such tournament compensation systems are effective in raising performance in professional golf and auto racing competitions. 6 It is possible that, if there is sufficient number of short horizon portfolio managers herding in the manner described by Froot et al. (1992), a bubble can start on basis of any information. Shleifer and Vishny (1990) also propose that the portfolio managers have short horizon; however, it is the risk of uncertain return from investing in the longer horizon that prevented disequilibrium to be arbitraged away.

J.S. Ang et al.

et al. (1988), we note that many of our latter experiments were directly motivated by the results obtained from our earlier ones. This progression of thought and analysis will be apparent in the later section on results. Herein, however, we present the method of our investigation in comprehensive form. The creation of “bubbles” within asset markets is examined under the control of three primary factors: (1) the degree of trader risk aversion, (2) trader investment horizon, and (3) available investment capital/supply of securities. Table 9.1 summarizes the design of 14 experiments used to investigate these factors on the presence of asset bubbles. Each of these experiments uses a common market mechanism that builds on the earlier work of Forsythe et al. (1982), Plott and Sunder (1982), and Ang and Schwarz (1985). These common features are summarized below.

9.3.1 General Market Design 1. A double-oral auction, similar to that used on the floor of major U.S. exchanges, is replicated. The recruited traders are physically present within a single room during the course of trading. These traders are independent and trade solely for their own account. There are no specialists or other privileged traders. 2. Only those shares of a single generic security are traded. The sole attribute of these shares is the payment of dividends at the end of each period. 3. Each market (experiment) has ten trading periods. These periods are further categorized into five trading years each of which consists of two contiguous trading periods (A and B). Endowments (discussed below) are reinitialized at the end of the second period of each year. Thus, the initial market represents a two-period model with each security entitled to two payoffs (dividends), one at the end of period A and the other at the end of period B.7 4. Each trading period last for 6 min, with opening, warning (at 5 and 51=2 min), and closing bells. Consequently, each experiment has a total of 60 (6 min 10 periods) trading minutes. During the 6-min periods, traders can observe the continually updated bid/ask and past transacted prices. 7

In the experiment, a trader has at least the following choices available: (a) Maintain the endowed position by not trading and receiving the stochastic payoffs at the end of each period; (b) Hold the securities through period A and sell in period B, in which case the investor will receive the first period dividend and the selling price; (c) Sell the initial holdings in period A to receive the sale price; (d) Buy additional shares in period A, receive dividends at the end of the period, and then sell the securities in period B; (e) Sell the securities in period A and then buy back securities in period B in order to receive the dividends; (f) Purchase a net amount of shares in both periods; (g) Purchase and sell shares within each period.

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

141

Table 9.1 Experimental Design This table categories five designs of fourteen experiments used to examine the impact of risk aversion, investment horizon, and credit/supply constraints (initial endowment) upon the formation and control of asset bubbles. Participant Groupsa Las Vegas

Design 1

Initial Endowmentb 2 securities 10,000 francs

Investment Horizonc Two period

Risk Aversiond Mixed

Las Vegas

2

2 securities 10,000 francs

ShortenedMixed

4m, 6m$

Las Vegas

3

5 securities 3,000 francs

ShortenedMixed

7x,8x,9t,10t

FSU1

4

2 securities 10,000 francs

Two period

FSU1

5

2 securities 10,000 francs

ShortenedSingle

Single Type

Experimentse 1,2,3,5$

11s, 13c 12sm, 14cm

Type

FSU2

2

2 securities 10,000 francs

ShortenedMixed

15, 16

FSU2

6

10 securities 1,000 francs

ShortenedMixed

17, 18

Albania

1

2 securities 10,000 francs

Two period

Albania

2

2 securities 10,000 francs

ShortenedMixed

24,25

Albania

6

10 securities 1,000 francs

ShortenedMixed

26

Albania

7

2 securities 5,000 francs

Single Period

Mixed

19

Albania

8

2 securities 5,000 francs

Single Period/Tournament

Mixed

20,21

Albania

9

20 securities 500 francs

Single Period/Tournament

Mixed

22

a

Mixed

23

The participant groups consist of the following: Las Vegas represents students from the University of Las Vegas at Nevada. FSU1 and FSU2 represent students from the Florida State University at two different time periods. Albania represents students from the University of Tirana in Albania.

b

The initial endowment refers to traders wealth position at the beginning of each trading year of an experiment. This endowment allows traders to sell (using provided securities) or buy (using francs, the currency used in these experiments). The additional securities and reduced currency endowments provided in Design 3 serves to better equate relative purchase and selling abilities. c Investment horizon refers to the horizon within which traders effectively operate. A two-period horizon refers to a market where period A securities are based on the dividends paid in both periods (A & B) of a trading year. In a shortened investment horizon, the trader is induced (via the tournament compensation schedule of Table 9.3) to operate with a horizon which is shorter than the two-period environment in which the securities will pay dividends. d Mixed risk aversion means that traders with various risk preferences were participants within the same market. Single type means that only speculative (s) or conservative (c) traders made up that market. The designations (s) and (c) appear next to experiments 11–14 in the last column. e Notation is as follows: $ represents a market where traders provided $20 of their own money to trade, the sum of which became the pool of money dispersed according to relative profit performance. m,x,t represents the number of traders receiving the tournament prize as outlined in Table 9.3. This tournament compensation was used to induce a shortened horizon market and was differentially paid to the top two (t) or the top six (x) traders. In experiments marked (m), the first three trading years paid a bonus to the top six traders followed by years where only the top two traders received bonuses.

9.3.2 Dividend Design 1. At the beginning of each year, each trader is endowed with trading capital and shares of the generic security. Each share pays dividends at the end of the first (A) and second (B) periods. The second period dividend is a liquidating

dividend. Reinitializing of position at the beginning of each year allows for replication of decision making in experimental markets.8 8 See Smith et al. (1988) for an example of when reinitialization is not used.

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Table 9.2 Dividend Design This table presents the cash flow payoffs which a single asset will provide to its owner. This payoff is different for Trader Types I, II, and III and therefore provides for different fundamental valuations. Rational Expectations Equilibrium are determined by the trader type with the highest valuation for that period.

I II III

Period A Dividend Statec G B 350 110 250 150 200 140

Expected Dividendd

Yearly Expected Dividende

230 200 170

Period B Dividend Statec G B 250 150 200 140 350 110

200 170 230

430 370 400

2

I II III

200 350 250

140 110 150

170 230 200

350 250 200

110 150 140

230 200 170

400 430 370

3

I II III

250 200 350

150 140 110

200 170 230

200 350 250

140 100 150

170 230 200

370 400 430

4

I II III

350 250 200

110 150 140

230 200 170

250 200 350

150 140 110

200 170 230

430 370 400

5

I II III

200 350 250

140 110 150

170 230 200

350 250 200

110 150 140

230 200 170

400 430 370

Yearsa

Trader Typeb

1

Expected Dividendd

a Each experiment is composed of five trading years, each of which contains two trading periods A and B. Ownership of an asset in period A entitles the bidder of both period A and Period B dividends (two-period valuation) whereas period B ownership merits only that period’s dividend (single-period valuation). b There are three trader types in each trading year with four traders in each category. These trader types only differ by the amount of dividend cashflows that the single traded asset will provide its holder. The four traders within each category are rotated within the other categories so as to maintain an uncertain valuation environment. c Dividend States refer to the stochastic payoff that will be provided to specific trader types given the occurrence of the G (Good) or B (Bad) state. The realization of the state of nature is determined at the end of each trading period by flipping a fair coin. d Given equal fifty percent probability of occurrence of G or B, the expected dividend is the simple average of period G and B payoffs. e The yearly expected dividend represents the summation of expected dividend for both periods A and B.  Signifies trader type with the highest expected value

2. The dividends to be paid at the end of each period are stochastic. Two equally likely dividend outcomes are possible, the good (G) state and the bad (B) state. The realized state is announced at the end of each 6-min trading period as determined by the flip of a coin by the experimenter. 3. The dollar amount of the dividends paid at the end of each period depends on the trader’s type and the realized state. The 12 traders who make up each market are classified into three types to allow for differences in induced values. The dividend payouts for these three trader types are summarized in Table 9.2. As an example, at the beginning of year 1, the four traders of type I are privately informed that for period A they will receive either 350 or 110 for the good (G) and bad (B) states, respectively, and for period B either 250 or 150 for the good and bad states, respectively. 4. The trader type with the highest expected dividend (0.5

Good Dividend C 0.5 Bad Dividend) is rotated each period so as to enhance trader uncertainty about equilibrium prices. Virtually all previous experimental studies have documented that given sufficient learning (through repeated trading) in a stationary dividend payout environment, prices will rather quickly approach the rational equilibrium level. This learning has two sources: (a) observation that one’s own payouts are not changing, and

(b) observation that market generated bids, offers, and transacted prices are not changing. Our expectation is that the greater the trader’s reliance on market generated (as opposed to prior dividend) information, the more likely bubbles are to occur due to bandwagon and other crowd psychologies. If, instead of bubbles, we should observe that prices converge to rational equilibrium prices (as in the constant dividend studies), then this would strengthen our knowledge concerning efficiency in these laboratory markets. This result would also suggest that trading methods based on historical prices alone would not have value.

9.3.3 Investment Horizon 1. Three types of investment horizon are provided within these experiments: a single-period, a two-period, and a shortened-horizon. Initially, at the beginning of each trading year, a trader is entitled to two stochastic dividends for each security held, one each at the end of periods A and B. Therefore, at the beginning of period A, a rational trader will value the security for both its period A and period B stochastic dividends. Hence, all A period

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

pricing should reflect a two-period investment horizon. Subsequent to the termination of period A trading and the announcement and payment of the period A dividend, period B trading proceeds. As the security is now only entitled to the B period dividend, a single-period investment horizon results for all B periods. Our hypothesis is that a shortened investment horizon increases the possibility of an asset pricing bubble. We test for this by creating a tournament compensation package in period A. The incentive for traders is to concentrate upon their single (A) period performance over the concerns of a rational twoperiod price. This incentive results in a shortened investment horizon.9 2. In this study, Initially, a trader’s dollar compensation is defined by the following: we alter the compensation structure to induce a change in the length of a trader’s investment horizon profit function: Pi D fŒdi;j;A  Xi;A C di;j;B  Xi;B C .Ri  Ci /

(9.1)

where Pi , dollar profit per trading year for trader i. It consists of dividend income and trading gains (losses) from both periods; f, the conversion rate of francs into dollars.10 ; di;j;t , the dividend paid in francs to trader i, given state j occurs in period t; j D G or B; t D A or B; Xi;t , the number of shares held by trader i at the end of period t; t D A or B; Ri , revenues in francs for trader i for all shares sold during periods A and B; Ci , costs in francs for trader i for all shares purchased during periods A and B. In order to induce pressure for a shortened investment horizon, an additional compensation package is introduced in Period A of some experiments as identified in Table 9.1. This tournament compensation system is based on the traders’ relative performance as measured by the Tournament Performance Index (TPI) below: TPIi D Ri  Ci C MXi;A

(9.2)

9 It is important to note that there is a difference between a one- and two-period horizon and a shortened-horizon. In a one-period model, only a single dividend is valued. In a two-period model, two dividends are valued. In our shortened-investment horizon, the trader is induced to operate within a horizon that is different from that of his operating environment. That is, within a two-period operating environment, the trader is given an incentive to operate with a shorter (possibly single) period horizon. This is quite different from a single-period model. This shortened horizon is a stronger test of market efficiency, in that the pressures are away from rather toward rational equilibrium prices, (as defined in Equation (9.4), subsequently). The methodology is meant to emulate modern portfolio managers operating in an environment of perpetual horizon stock securities yet receiving tournament incentives to outperform colleagues on a short-term basis. 10 Francs are the currency used within this study. They have been used successfully by Plott and Sunder (1982), Ang and Schwarz (1985), as well as others. Their primary benefit is to avoid the technical problem of dealing with small dollar amounts.

143

where Ri ; Ci , and Xi;A are as previously defined, and M represents the closing market value of the shares. This closing market value is taken to be the price of the second to last transacted price for that period. This procedure is introduced in order to reduce the possibility of manipulating market value by collaboration on a final transaction. It represents a simplified version of the priceaveraging process that takes place on most organized exchanges for the setting of opening and closing prices. 3. The tournament compensation system provides traders with an incentive to outperform each other in period A only. This incentive system increases the importance of single-period performance (in A) over two-period concerns; that is, it induces a shorter investment horizon in period A. A trader’s compensation is dependent upon his relative rank as summarized in Table 9.3. In Schedule Six, the top six (of twelve) traders are rewarded with francs ranging from 1,500 to 200. Schedule Two is an alternative schedule that is hypothesized to induce even greater competitive pressure as only the top two traders are compensated greatly.11 Ehrenberg and Bognanno (1990) and Table 9.3 Tournament Compensation Schedule This table presents the additional tournament compensation schedule provided to traders based on their relative profitability in period A of certain experiments (see shortened investment horizon listed in Table 1). Relative profitability is measured by: TPIi D Ri  Ci C M  Xi;A where TPIi is the tournament performance index for trader i, Ri is the revenues received from the sale of assets in period, Ci is the cost of assets purchased, M is the closing market value for the period, and Xi;A represents the end-of-period asset holdings. Together the index measures the total of realized and unrealized capital gains. The addition of the tournament compensation to period A provides an incentive for traders to prefer period A capital gains over equivalent period B dividends and thereby induces a shortened (from the two period model) investment horizon. Schedule Six (s)

Two (t)

Rank 1 2 3 4 5 6 7 to 12

Lowest

1

Highest

3000 francs

Lowest

1000 0

2 3 to 12

TPI Highest

Compensation 1500 francs 1000 700 400 200 200 0

a

Two compensation schedules are introduced. The first provides for those traders who do better than the average (i.e., the top six) receive the additional compensation list. In the second schedule, only the top two “superstars” are richly rewarded. The tournament literature (e.g. Baker 1991) suggests that tournament systems, and especially schedule two, provide effective incentive systems to increase performance. This design is meant to emulate the short-term performance pressures faced by professional money managers.

11

The compensation schemes depict the different ways portfolio managers are being rewarded: those who are above the average or beaten

144

Becker and Huselid (1992) find that the reward spread does cause increased performance incentives. Therefore, we expect Schedule Two to increase incentives for shortterm pricing behavior. Table 9.1 summarizes the experimental use of the performance reward schedule.

9.3.4 Risk Aversion 1. Prior to selection, each potential trader was given a lengthy questionnaire. Intermingled within this material were two psychological tests on risk taking: the Jackson Personality Inventory (1976) and the Jackson et al. (1972) tests.12 These two tests have been applied in laboratory (Ang and Schwarz 1985) and field studies (Durand et al. 2006) and are more practical to administer than the theoretical risk measures found in the economics literature.13 Those persons who score in the top 12, signifying the least risk averse, and the bottom 12, or the most risk averse, are invited to participate in the second stage of the experiment. 2. Traders for experiments 1–10 were students from the University of Las Vegas at Nevada and were recruited from a senior level options class. These students had all taken two statistics, a corporate finance, a valuation, a portfolio analysis, and an options course. They were well trained in arbitrage, present value, and expected value. From this pool of students, 12 were chosen to participate based on their attribute ranking in risk aversion. Participants were chosen so that a mix of risk aversion types were represented in the same market. Included were those the market (Schedule Six), and those who are the superstars (Schedule Two). 12 The authors are aware of the work of Holt and Laury (2002), which was not available at the time of this study. According to Holt and Laury their experiment shows that increases in the payoff level increase RRA. However, when estimating RRA, Holt and Laury assume that subject’s utilities depend only on payments in the experiment. They fail to account for the wealth subjects have from other sources (see Heinemann 2003). 13 The Jackson Personality Inventory is scientifically designed questionnaire for the purpose of measuring a variety of traits of interest in the study of personality. It was developed for use on populations of average or above average ability. Jackson states (1976), p.9, “It is particularly appropriate for use in schools, colleges, and universities as an aid to counseling, for personality research in a variety of settings, and in business and industry.” Of the 16 measurement scales of personality presented, one scale directly measures monetary risk-taking using a set 20 true and false questions. Mean and standard deviation measures for 2,000 male and 2,000 female college students are provided. Jackson et al. (1972) demonstrate four facets of risk taking: physical, monetary, social, and ethical. The authors’ questionnaires are situational in that the respondent is asked to choose the probability that would be necessary to induce the respondent to choose a risky over a certain outcome. Jackson (1977) presents high internal consistency correlation between the risk measurement techniques.

J.S. Ang et al.

who ranked at all levels of the scale, from high to low risk aversion. This was done so that differences in individual risk behavior could be tracked within an identical market environment. 3. Experiments 11–14 were conducted at Florida State University (FSU1) and as summarized in Table 9.1, these experiments were designed so that an experimental market consisted entirely of traders that were either relatively more risk averse (conservatives) or less risk averse (speculators). This experimental form allowed for evaluation of whether risk aversion is uniquely a necessary or sufficient condition for the presence of bubbles. 4. Experiments 15–26 were conducted at a later date at Florida State University (FSU2) and the University of Tirana in Albania. This was done to confirm the robustness of our results. We intentionally chose students from two different universities with different backgrounds to represent the two extremes in our test. Experiments 19–26 were administrated to subjects in Albania, who have a low degree of familiarity with capital markets while experiments 15–18 were administered to Florida State University students (FSU2) who had taken a financial engineering course and completed another more involved laboratory asset market experiment. Hence, we consider these FSU2 students to be super experienced relative to the students from Albania.

9.3.5 Validation Procedures The following procedures are incorporated into the experimental design to insure the reliability and external validity of the results: 1. To guard against the possibility that subjects’ experience with trading could change their attitudes toward risk taking, they were retested. Subsequent to the first four experiments, additional risk questionnaires were given to the participants. A Spearman rank correlation (with initial risk rankings) was 902 with a t-statistic of 6.61 indicating that there had been no significant change in the relative risk attributes of the traders. 2. Videos were used to verify recorded information, identify possible irregularities, and to train new subjects. 3. Subjects were given extensive training on the operation of the game; the main experiments were conducted on groups of experienced, if not super-experienced, traders. 4. Lengthy post experiment questionnaires were also given to the subjects. Among other things, these were used to verify that the traders considered their trading strategies taken at the time of trade to be rational.

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

9.4 Results and Analysis 9.4.1 Control Experiments For experiments 1–14, five experimental designs were used to test for the effects of risk aversion, investment horizon, and capital endowment on the presence of asset bubbles. These designs are summarized in Table 9.1. The first design consisting of experiments 1, 2, 3, and 5 were control markets where the two-period model was tested without extrane-

Fig. 9.1 Experiment #1

Fig. 9.2 Experiment #2

145

ous influence from the three treatment variables mentioned above. Figures 9.1–9.5 plot the series of resulting prices. Bid and ask prices are represented by a “C” symbol and are connected by a vertical solid line. Transacted prices are identified by a solid horizontal line connecting each trade. From earlier laboratory studies, we would expect prices to converge to rational expectation equilibrium levels after an initial period of learning. Two relevant concepts of equilibrium prices in these markets have been proposed (see Forsythe et al. 1982). The first is the Naive Equilibrium (NE) Price. The NE is the highest

146

J.S. Ang et al.

Fig. 9.3 Experiment #3

Fig. 9.4 Experiment #4

price any trader in the market is willing to pay based on his individual valuation of the expected dividends for the two periods or: NE D Max kŒE.DA;k / C E.DB;k /

(9.3)

where k classifies the trader type (1, 2, or 3) based on prior expected dividend valuations (see Table 9.2). E.DA;k / and E.DB;k / are the values of expected dividends in periods A and B to the k’th trader type.

The NE price is the market price that will prevail if the traders use only their private information to determine value. It is naïve in the sense that traders do not learn about the valuations of other traders from the market trading information. These traders also ignore the option value to trade (e.g., hold a security for a period and then sell it to another trader who would value it most in the remaining period). The second is the Perfect Foresight Equilibrium (PFE) price. It is equal to the highest total value that successive owners of the same share will pay; or, in the experiment, the

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

147

Fig. 9.5 Experiment #5

sum of the highest expected payoffs for periods A and B for all traders or: PFE D Maxk E.DA;k / C Max E.DB;k /

(9.4)

These prices represent two extreme benchmarks in the continuum of the value of capital market in discovering information through trading. The NE price gives no role to capital market in price discovery, while the PFE price assumes full discovery (i.e., trading in capital market can correctly identify the share’s highest value in each future holding period). They define, respectively, the lower and upper bounds of the share’s fundamental value. Thus with payoffs to traders and across holding periods under the control of the experimenters, we can now identify with certainty whether a stock is undervalued (when price is below NE), or overvalued (when price is above PFE), or is in a bubble (when price is grossly below NE or above PFE, as in a negative or positive bubble). If the experimental market captures a well functioning capital market, learning and repeated trails would cause prices to converge toward PFE. There are two properties in (Equations (9.3) and (9.4)) that are worth noting. First, NE and PFE prices are identical in a one period world when price determination is closer to a simple auction of a single period payoff. Second, when the payoff in the equations are dollars, as in cash dividends and capital gains or losses, NE and PFE give the risk neutral prices. In the absence of risk neutrality, a negative difference between observed prices and these prices may be interpreted as a risk premium. The results illustrated in Figs. 9.1–9.3 establish the validity of our experimental design as we are able to produce results similar to those obtained in previous experimental

studies. In particular, we are able to reproduce the result that prices converge to PFE with learning and repeated trials. These prices are plotted as a solid horizontal line and are greater than the NE prices.14 The inexperience of traders in experiment 1 is greatly reduced in experiments 2 and 3 as traders learn to cope with the large uncertainty in valuations (introduced by design). This pricing uncertainty increases the traders’ reliance on “market generated information” in order to determine valuation. A micro-analysis of traders’ accounts in experiment 2 shows that some traders became actively involved in arbitrating between the A and B periods of a trading year. As a consequence these prices tended toward their PFE equilibrium levels. Traders’ learning contributed to further pricing efficiencies in experiment 3. Some earlier “irrational” trades by selected individuals had resulted in substantial losses creating a “once-bitten” effect and more rational decisions were 14 Note that all odd numbered experiments used the dividend design in Table 9.2. In order to differentiate between (1) learning about a stationary environment and (2) learning efficient valuation within laboratory markets, we created nonstationarity in equilibrium prices across experiments. In particular, for all even numbered experiments, the dividend payoffs of Table 9.2 were simply cut in half so that rational equilibrium prices were also one half that of the odd numbered experiments. When this equilibrium dividend rotation is viewed in conjunction with the previously mentioned rotation of trader types, it becomes apparent that each individual trader was likely to view the environment (at least initially) as nonstationary. Consequently, any results that we show regarding equilibrium pricing and convergence would suggest that learning about valuation methods rather than a stationary environment creates rational valuation. That is, we are concerned about learning that takes place within the trader (how he values) not about the environment (stationary value). We are able to pursue this expanded question due to our debt to earlier authors who have already well established the presence of the latter.

148

followed subsequently. By the end of this experiment, prices in both the A and B periods were close to the PFE price.15 A final examination of the validity of the experimental design was performed by requiring each trader from Las Vegas experiments 5 and 6 to “invest” his own money ($20) into the markets. As a result, it was possible for traders to lose as well as to win. The results, illustrated in Fig. 9.5, show continued price convergence toward equilibrium levels.16 Of interest is the pattern of the bid-ask spread within a period. The data suggests that the primary resolution of uncertainty is obtained during the first transaction of a period. Subsequent trading tends to vary little from earlier levels with subsequently smaller bid-ask spread levels. We conclude the control section noting that the experimental design creates price-revealing trades that foster PFE equilibrium pricing. While consistent with earlier research, these results extend our knowledge into a much more uncertain (nonstationary) valuation environment more typical of real world asset markets. In addition, the validity of these results are not affected by whether or not a dollar investment is required from traders; trading behavior is similar under both environments.

9.4.2 The Formation of Bubbles With a well-functioning experimental design established, we now sequentially introduce our hypothesized treatment variables. In experiment 4, we introduce the shortened trading horizon with a tournament prize as described in the experimental design. At this point, we have an advantage over previous studies in that we were able to recruit the identical 12 traders back. This level of experience will lead to converging equilibrium prices as opposed to bubble formation.17 The effect of the tournament compensation is to shorten the traders’ investment horizon in period A from a PFE twoperiod model. By providing tournament payment based on period A relative ranking, there is an increased incentive to generate period A capital gains over equivalent period B div15

While period A prices exceeded the calculated PFE price of 460, this price is somewhat unknown to traders at this point. Prior trading results had created a history of B period prices averaging 320. Consequently, it was rational for a PFE trader to pay up to 550 (230 for A period plus 320 for B period sales price). The last trade in period 5A of 505 was well below that level. A more detailed presentation of the experimental results further reveals the rationality of these prices and is available from the authors upon request. 16 Again, period A prices seem to drift upward due to initial excess pricing in period B. 17 Our design is to eliminate the bubbles effect of miscalculation caused by inexperienced traders as suggested by White (1990) and King et al. (1990). It is more useful and realistic to study the formation and control of bubbles in markets of experienced traders.

J.S. Ang et al.

idends. The tournament compensation, while increasing the incentive to win, does not necessarily equate to higher equilibrium prices. The prize is paid to the largest (realized and unrealized) relative capital gains that can be achieved in either a bull or bear market. Extraordinary results are shown in Fig. 9.4 where five massive price bubbles are observed in each of the A periods. At this point, a new learning phase was initiated as traders competed strategically for the tournament prize. The dominant initial strategy centered on buying all available assets at increasing price levels thereby creating artificial price support for capital gains. While this often resulted in achieving the prize, it also meant dealing with an inventory of overvalued assets in period B. Some traders actually lost money for the year even though they obtained the prize. It is important to note that the bubbles did not discourage the traders from participating, and at least for awhile the number willing to participate actually increased. Examination of asset holdings reveals that there were four to five active prize seekers in later bubbles versus one to two initially. In addition, seven to nine traders continued to hold securities at the periods’ end rather than to sell out at extremely high bubble levels. The much higher increased tournament reward structure for “superstar” performers of periods 4 and 5 (see Schedule Two of Table 9.3) resulted in the largest bubbles (consistent with our predictions) and with the greatest variability in prices and bid-ask spreads. Again, the buying frenzy in period 5 was led by different traders than those in period 4. This continued rotation in trading leadership highlights that the results are not driven by a few misinformed traders. In fact, period B prices are very stable and efficiently priced. Furthermore, a trader questionnaire survey at the end of experiment 4 revealed that traders were fully cognizant of expected dividend value, yet they looked to both dividends and market generated information to determine value. Traders stated that they were influenced by the behavior of their peers and were motivated to earn as much as possible. Several traders noted that the introduction of the tournament compensation stimulated them to take on more risk. The net result of these effects was to create a herd or bandwagon effect centered on market-generated information. Despite the earlier findings of experiment 5, we tested the validity of these bubbles in an environment where traders used their own money rather than the experimenters.18 Would such wild speculation occur when a trader’s own money was at risk? Figure 9.6 clearly shows this answer to be yes. In all five A periods, average prices are more than twice the equilibrium value. As before, period B pricing is very efficient

18

To the authors’ knowledge, this is the first time traders in an experimental market of this type have used their own money to trade and still produced bubbles.

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

149

Fig. 9.6 Experiment #6

and stable. That is, although our traders engaged in bubble pricing, they arrived at it through rational means. The traders in these markets had now participated in six experiments, the most of any research to date. Yet, even in the presence of super-experienced traders we continue to find bubble formation. In addition, these traders were aware of the situation and made every opportunity to profit from the bubble.19

9.4.3 The Control of Bubbles It became readily apparent from the earlier experiments that restrictions on the supply side of the market were having an influence on market prices. Many traders found themselves bound in their actions by the institutional makeup of the experimental markets. Many of the traders suggested that they be allowed to short-sell in future experiments so as to implement sell strategies in overvalued markets. 19 For instance, new strategies were employed at various stages (which perpetuated continuing uncertainty in the markets). At one time, the market actually stood still for an extended period. Then traders began to liquidate at any price rather than to replicate their earlier strategy of waiting until late in the period to sell out at bubble prices. Other traders began to try and scalp the market by driving prices both up and down thereby generating capital gains in both price directions. Even others began to try and force losses on traders with large inventories and thereby improve their relative ranking. This was accomplished successfully in period 2A by selling at a loss (at a price below market prices) in order to create a low settle price, M (the second to last trade). Other attempts at this strategy followed in all remaining A periods. Nevertheless, bubbles persisted and many traders were frustrated in their inability to arbitrage them away.

As previously mentioned, the tournament compensation system does not alter PFE prices as the prize can be achieved in any type of market environment and with any type of price pattern. Given the results of our previous experiments as well as traders’ comments,20 it appeared that buyers (longs) had an advantage over sellers (shorts). Is it possible that the bubbles we observe were due to differential market position in addition to a shortened investment horizon? In order to answer this question, we conducted four more experiments that provided traders with initial endowments and better equated the position of buyers and sellers. Rather than being endowed with two securities and 10,000 francs of trading capital as before, each trader is initially endowed with five securities and 3,000 francs (see Table 9.3).21 The price patterns of experiments 7–10 are as startling as the dramatic bubbles earlier. We find that the market is immediately priced at a discount to PFE.22 This had never 20 Traders completed survey questionnaire at the completion of experiments 4, 6, and 10. 21 Given that experiment 6, period A prices averaged around 600, initial trading capital of 3,000 francs would provide buying power of roughly five securities. Consequently, the new buying power and selling power were a prior relatively equal. Even though period A prices turned out to be quite a bit lower in experiments 7–10, this did not create a great advantage to buyers since the supply of securities (5 traders 12 traders D 60) was relatively large for a 6-min trading period. As such, there was an ample supply of securities relative to buying power in order to drive prices down should traders turn bearish. 22 We are unable to recruit all 12 traders back for experiments 7–10 due to graduation, taking of jobs, etc. We were, however, able to retain 7 of the original 12 traders. These traders had now participated in six previous experiments. The five replacements were drawn from the original pool of subjects that had completed the risk attribute questionnaires. These new traders were chosen to replace the risk types that had

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happened in any of the tournament periods before. If this had been simply the result of learning, we would have expected a gradual decline from the lofty levels of experiments 4 and 6. Rather, we see an immediate discount price that generally remains at a discount throughout all four experiments.23 Overall, we consider this to be strong evidence that a necessary condition for the creation of the large bubbles of these markets is that the institutional environment be biased toward more purchasing ability relative to that of selling. In summary, experiments 7–10 highlight the importance of the supply of securities and the supply of investable funds that may be augmented by short selling. Bubbles observed in experiments 4 and 6 are immediately eliminated when the relative purchasing advantage of long traders is removed. Rational pricing reflecting a modest risk premium results even when traders are faced with a shortened investment horizon.

9.4.4 The Impact of Risk Aversion The results of previous experiments, especially 6, showed that trader risk aversion was an important factor in determining trader strategy and therefore price patterns. In general, it was found that speculative traders were more likely to seize

vacated, so that, in general, we maintained a wide dispersion of risk types within the market. In addition, some of these new traders had sat in as observers to previous experiments. Others viewed videos of the earlier experiments. All were instructed in the past experimental results and the various strategies previously used were explained. As such, we do not believe that this change is a critical factor in the continuation of our investigation. 23 An analysis of many of the last trades of period A for experiments 7–10 often shows either a sharp spike up or down. This illustrates that the traders had become very efficient (through learning) in their manipulation of closing prices. Given the large supply of securities available to squelch a price bubble, speculators were no longer singularly (due to large initial endowments of trading capital) able to create capital gains by driving market prices up. With this constraint, they quickly learned that all they needed to accomplish was to purchase the most securities at current prices and then drive the market up on the final few trades. This was often easily accomplished in that (1) only the second to last trade needed to be higher in line with the calculation rules of the TPI, and (2) as no surprise, there was always many traders who were willing to sell their securities at a price above the current level. The art to this strategy became a matter of timing; do not try to buy the market too early lest you run out of capital, and do not be too late lest you be unable to make the second to the last trade. There did not appear to be too much of a problem for buyers in accomplishing this in experiments 7 and 8; however, starting in experiment 9, some short traders, annoyed at bullish traders getting the tournament prize, began jockeying in these last seconds with the long traders to drive prices down. The results of such feuds appear in periods 3A, 4A, and 5A of experiment 9 and each A period of experiment 10. The winner of these duels increasingly became the trader who was best able to execute his trade. Eventually, trading activity become so frantic in the last 15 s of trading that the open outcry systems of double oral auction began to break down.

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upon the opportunity created by the introduction of uncertainty (via the tournament period) in search of capital gains. In contrast, the more conservative traders were likely to allow the speculators to act first by creating a positive price trend and would simply sell at inflated prices or they would allow speculators to first initiate the “burst” of the bubble and then follow in their footsteps. Consequently, the conservative traders were often those responsible for the perpetuation of a direction initially set by speculators. The purpose of experiments 11–14 were to further test these relationships.24 We chose at this time to create two separate trading groups according to risk aversion, each composed of 12 traders. These 24 traders were chosen from a pool of 70 students that completed the risk-ranking questionnaire described earlier. The 70 respondents were rank ordered from highest to lowest in risk aversion. The top 12 and bottom 12 students were chosen to participate in the experiments. This method allows us to obtain good separation according to risk aversion. Contrary to our previous experiments, these markets would be made up entirely of one risk aversion class. We label these two risk classes as speculators and conservatives. This is a relative nomenclature as all of these traders are considered to be risk averse, and we only presume to provide an ordinal measure of risk aversion Figs. 9.7–9.10. The design of these experiments follows that of experiments 1–6, as we wish to test for the presence of bubbles and the initial endowments of experiments 7–10 have already been shown to eliminate bubbles. All of these traders had previously participated in two experimental markets and therefore can be considered experienced. Nevertheless, we test for rationality of pricing in experiments 11 and 13 before introducing the shortened horizons in experiments 12 and 14. Figures 9.11–9.13 reveal that both markets are quite rational in that they charge a discount from PFE as a risk premium. As expected, the conservative traders of experiment 13 charge a larger risk premium than the speculative traders of experiment 11. This result provides strong evidence in support of our measure/separation of risk aversion. We also note that the speculative group exhibits prices above the PFE levels of period B. This is consistent with our earlier results where this was found in the single period case Fig. 9.14. Experiments 12 and 14 introduce the tournament compensation schedule to induce a shorter investment horizon. As expected, the speculative group seizes upon the opportunity and price bubbles are generated in the latter periods. Also to no surprise, the conservative group does not create

24

Experiments 11–14 were conducted at a second university and, therefore, the results provide information about the external validity of our experiments outside the setting of a single university.

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Fig. 9.7 Experiment #7

Fig. 9.8 Experiment #8

the pressure necessary to cause bubbles to form. As a result, we conclude that a necessary condition for asset bubbles is the presence of speculators.25

9.4.5 The Formation of Negative Bubbles We have just learned that the effect of the reduced investment horizon is to increase the incentive for short-term speculative

25

A detailed examination of individual trades reveals the speculative group of traders are found to be more innovative in designing new trading strategies both in the creating and bursting of bubbles. The finding is consistent with the observation made by Friedman (1992) in his review of a dozen NBER working papers on asset pricing. He finds these recent research results demonstrate that rational speculative behaviors such as

an attempt by investors to learn from other investors, to affect another’s opinion, or to simply engage in protective trading could in some context, such as imperfect information, magnify price fluctuations.

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Fig. 9.9 Experiment #9

Fig. 9.10 Experiment #10

gains and that speculative traders are those most eager to earn these profits. We now extend the research design to investigate the question of whether negative bubbles are also possible. We test this proposition by conducting four new experiments (labeled as experiments 15–18 in Table 9.1). We conduct experiments 15 and 16 as “controls” to replicate the positive bubble environment found in experiments 4, 6, and 12. Experiments 15 and 16 validate our previous results with a new set of experimental subjects while Figs. 9.15 and 9.16

plot the pattern of close prices relative to the equilibrium level (horizontal line). In both experiments large positive bubbles emerge in most trading years. We now pose the following question, “Would an environment opposite to that of Design 2 lead to negative bubbles?” We keep the structure of Design 2, but since it was the unequal endowment effect (more purchasing power versus selling pressure, under 2 securities, 10,000 francs) that created the ability to pursue profits in a positive bubble

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Fig. 9.11 Experiment #11

Fig. 9.12 Experiment #12

environment, we reverse the endowment effect in experiments 17 and 18 by providing 10 securities and 1,000 francs to each trader. This one change provides traders in experiments 17 and 18 with a much greater ability to buy relative to sell. The results plotted in Figs. 9.17 and 9.18 show a preponderance for negative bubbles. While the initial four years of experiment 17 show some learning adjustment to this new and difficult trading scheme, large price discounts emerge to the extent that period 5A’s closing price is insignificantly different than period 5B’s, which is a single period receiving only a single dividend. By experiment 18, each year shows

downward trending markets in each A period. The reader may notice that the positive bubbles seem to burst while the negative bubbles don’t. However, since our design did not allow more cash to be made available through borrowing or infusion, correction may not be observed in the short trading period.

9.4.6 Statistical Analysis Table 9.4 summarizes regression analyses of the impact of the variables just discussed upon the divergence of asset

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Fig. 9.13 Experiment #13

Fig. 9.14 Experiment #14

prices from their PFE levels in period A. In particular, we test the following relation: PL PFE D f .I; E; I E; T; I E T; S; I S; A; I A; $/ (9.5) where PL -PFE, the deviation from equilibrium for period A of each trading year where PL is the last trade of the period and PFE is the Perfect Foresight Equilibrium price; f , a linear additive model; I, a dummy variable representing the shortened Investment horizon according to Table 9.1. I D 1 for shortened horizon, and 0 otherwise (i.e., exper-

iments 4,610,12,14); E, a dummy variable representing the Endowment effect according to Table 9.1. E D 1 when 2 securities are issued, and 0 otherwise (i.e., experiments 1–6, 11–14); I E, an interaction dummy variable representing both a shortened investment horizon and two security endowment (i.e., experiments 4, 6, 12, 14); T, a dummy variable representing the Tournament effect according to Table 9.1. T D 1 when there is a tournament prize for two traders only, and 0 otherwise (i.e., experiments 4, 6, 12, 14 (years 4 and 5) and 9, 10); I E T, an interaction dummy variable representing a shortened investment horizon, a two security

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting Fig. 9.15 Experiment #15

Fig. 9.16 Experiment #16

Fig. 9.17 Experiment #17

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Fig. 9.18 Experiment #18

endowment, and a tournament effect (i.e., experiments 4, 6, 9, 10 (years 4 and 5)); S, a dummy variable representing the extent to which speculators participated in the experiments according to Table 9.1. S D 1 for experiments 11, 12 and 0 otherwise; I S, an interaction dummy variable representing the shortened investment horizon and a pure speculative trader market (i.e., experiment 12); A, the ratio of end-ofperiod asset inventory for speculative traders to total asset holdings. Speculative traders are those who scored in the top one-half of the risk measurement questionnaires; I A, an interaction variable for shortened investment horizon and ratio asset holdings for speculators (experiments 4, 6–10, 12, 14); $, a dummy variable representing experiments where traders risked their own money according to Table 9.1. $ D 1 when own money is used, and 0 otherwise (i.e., experiments 5 and 6). Due to their differential design, the results for experiments 1–10 appear separately in Panel A and those for experiments 11–14 in Panel B. Model 1 of Panel A tests the impact of (1) I D 1, a shortened horizon, (2) E D 1, a restricted endowment effect (wealth and supply effects), and (3) I D 1; E D 1, an interaction of a shortened horizon with restricted initial endowment. Given that the regression was run with no intercept, the coefficients represent estimates of each variable’s independent impact. The results suggest that neither a shortened investment horizon nor a biased endowment effect (advantage to “bulls” versus “bears”) is sufficient to induce bubble behavior. However, the interaction of these two variables is highly significant in explaining the bubble results of these experiments. That is, an environment that provides both the incentive and the ability to profit from a bubble will likely result in positive price divergence.

As hypothesized earlier, we test for the heightened effect of tournament incentives (i.e., T D 1) by examining the effect of “superstar” prizes paid to only the top two traders (as outlined in Tables 9.1 and 9.3). We also test for an interaction effect with a shortened horizon .I D 1/ and restricted endowment .E D 1/. Model 2 results are consistent with Model 1 in that a tournament effect is not sufficient in itself (t D 0:13 on T variable); however, in conjunction with a reduced horizon and restricted endowment, the tournament interacts to explain a significant part (t D 4:57 on I E T) of the bubbles in these experiments. In Model 3, we observe the impact of speculative traders vis à vis conservatives by introducing a measure of asset purchase activity. The end-of-period asset holdings for the speculative group (the top one half of traders in risk ratings) is compared to the total asset endowment for all traders. In the absence of any effect, assets should be evenly divided and this ratio, A, should be equal to 5. The results of Model 3 indicate that speculators independently do not impact the presence of a bubble (t D 0:01 for A); however, when speculators operate within a shortened horizon .I A/ they do significantly differentiate themselves from conservatives by buying more and contributing to positive price bubbles. Finally, the impact of the use of the trader’s own money is shown not to significantly alter the effects of the price bubbles (t D 0:54 for $). The R2 of 80 suggest that the vast majority of price deviation from PFE levels can be explained by investment horizon, endowment effects, and risk aversion. Panel B reports the results for experiments 11–14 where markets were either composed of all speculators (11, 12) or all conservatives (13, 14). Due to this makeup, variables A and I*A are not defined in these regressions, although S and I*S are substituted in their place and represent the speculative

represents the deviation from equilibrium for Period A of each trading year where PL is the last trade of the period and PFE is the Perfect Foresight Equilibrium price. a dummy variable representing the shortened Investment horizon according to Table 9.1. I D 1 for shortened horizon, and 0 otherwise (i.e., experiments 4,6–10,12,14) is a dummy variable representing the Endowment effect according to Table 9.1 E D 1 when 2 securities are issued, and 0 otherwise. (i.e., experiments 1–6, 11–14) is an interaction dummy variable representing both a shortened investment horizon and two security endowment (i.e., experiments 4,6,12,14) is a dummy variable representing the Tournament effect according to Table 9.1. T D 1 when there is a tournament prize for two traders only, and 0 otherwise (i.e., experiments 4,6,12,14 (years 4&5) and 9,10) is an interaction dummy variable representing a shortened investment horizon, a two security endowment, and a tournament effect (i.e., experiments 4,6 (years 4&5), and 9, 10). is a dummy variable representing experiments composed entirely of Speculators according to Table 9.1. S D 1 for experiments 11 & 12 and 0 otherwise. is an interaction dummy variable representing the shortened investment horizon and a pure speculative trader market (i.e., experiment 12) represents the ratio of end-of-period Asset inventory for speculative traders to total asset holdings. Speculative traders are those who scored in the top one-half of the risk measurement questionnaires. is an interaction variable for shortened investment horizon and ratio asset holdings for speculators (experiments 4,6–10,12,14) is a dummy variable representing experiments where traders risked their own money according to Table 9.1. $ D 1 when own money is used, and 0 otherwise (i.e., experiments 5 & 6)

b As reported in Table 9.1, experiments 1–10 consisted of traders with a wide range (mixed) of risk aversion. Experiments 11 & 12 were composed only of speculative traders with experiments 13 & 14 composed of conservatives. c NOINT means the regression was run by suppressing the intercept.    , , signify statistical significance levels at .10, .05, and .01, respectively.

I A $

I E T S I S A

PL -PFE I E I E T

Table 9.4 The Impact of Investment Horizon, Credit/Supply Constraints, Risk Aversion, and Other Variables This table shows the extent to which certain variables cause a deviation from perfect foresight equilibrium values.a The following regression is estimated separately for experiments 1–10 and 11–14 according to the experimental design of Table 1. PL -PFE D f(I, E, I E, T, I E T, S, I S, A, I A, $)  Model Intercept I E I E T I E T S I S A I A $R2 Panel A Experiments 1–10 .n D 50/b 1 NOINTc 45:1 .1:48/ 38:5 .1:26/ 512.3 (8.40) .67 .79 2 48:1 .1:43/ 9.6 (.23) 311.8 (6.28) 6.1 (0.13) 382.3 (4.57) 145.1 (1.80) 195.9 (2.23) 17.3 (0.38) 282.5 (3.08) 0.8 (0.01) 292.4 (1.77) 23:6 .0:54/ .80 3 178:1 .2:73/ Panel B Experiments 11–14 .n D 20/ 4 150:5 .11:17/ 109.1 (6.98) 89.9 (4.92) .75 61.1 (2.83) 67.0 (3.17) 45.8 (1.53) .83 5 139:0 .9:30/ 61.8 (2.71) a t-values in parentheses. Variables defined as follows:

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Table 9.5 Negative Bubble and Single Period Results This table shows the extent to which endowment in conjunction with other variables causes a deviation from perfect foresight equilibrium values.a The following regression is estimated separately for experiments 15–18, 19–22, and 23–26 according to the experimental design of Table 9.1.  PL -PFE D f .NI; E; I E; EN ; I EN / Model

Intercept

I E

NI

I EN

R2

201:3 .2:34/

.55

92:9 .6:89/

.57

208:0 .4:08/

.80

Panel A Experiments 15–18 .n D 20/ 6

NOINTb

45.0 (0.37)

400.6 (4.67)

Panel B Experiments 19–22 .n D 40/ 7

NOINTb

22:8.1:60/

22.5 (2.36)

Panel C Experiments 23–26 .n D 20/ 8 a

I E I E EN I EN 

174.5 (3.95)

t-values in parentheses. Variables defined as follows:

PL -PFE

b

130:0 .3:61/

represents the deviation from equilibrium for Period A of each trading year where PL is the last trade of the period and PFE is the Perfect Foresight Equilibrium price. is a dummy variable representing the shortened Investment horizon according to Table 9.1. I D 1 for shortened horizon, and 0 otherwise. is a dummy variable representing the Endowment effect according to Table 9.1. E D 1 when 2 securities are issued, and 0 otherwise. is an interaction dummy variable representing both a shortened investment horizon and two security endowment. is a dummy variable representing the sell side of the Endowment effect hypothesized to lead Negative bubbles. E D 1 when 10 securities are issued and 0 otherwise. is an interaction dummy variable representing both a shortened investment horizon and a ten security environment.

NOINT means the regression was run by suppressing the intercept. ;  ;  signify statistical significance levels at .10, .05, and .01, respectively.

markets (11 and 12) and the interaction of shortened horizon with a speculative market (12). In addition, a restricted endowment effect .E D 1/ is imposed for experiments 11–14 since experiments 7–10 clearly established their necessity in creating bubbles. Model 4 results highlight the significant positive effect of the combined shortened horizon/restricted endowment effect (t D 6:98 for I). More importantly, the speculative group statistically differs from conservatives with an additional mean price difference of 89.9 .t D 4:92/. Model 5 supports the results of experiments 1–10 in that: (1) a shortened investment horizon with restricted endowments leads to price bubbles .t D 2:71/ for I, (2) a heightened tournament incentive will heighten short-term horizons and lead to positive price effects (t D 2:83 for I T), and (3) speculators contribute to positive price bubbles in restricted endowment environments (t D 3:17 for S).26 The visual analysis of experiments 15–18 (negative bubble experiments) is confirmed by the regression results reported in Table 9.5. The variables are as defined earlier under equation 5 albeit the EN representing a dummy variable for the negative endowment effect. EN D 1 when the initial endowment equals 10 securities and 1,000 firms and 0 otherwise. In addition, since the shortened horizon variable I occurs for all years except 1A of each experiment, I and Furthermore, although insignificant, the p-value for I  S is equal to 14 suggesting that the speculative difference may be even greater under a shortened investment horizon.

26

E are highly correlated. The design is therefore set to only measure the interaction effects of a shortened horizon and endowment. The four periods (1A of each experiment) are the control periods where a shortened horizon is not present (dummy NI D 1 for not I). The parameter estimates of Model 6 show significant positive results for both positive and negative bubbles. The joint presence of a shortened horizon induced by a tournament payoff along with a buy side endowment (2 securities, 10,000 firms); that is, I E D 1, leads to an average increase of 400.6 francs in price levels. The single alteration of the endowment to sell side (10 securities, 1,000 francs) in the presence of a tournament leads to an average decrease in price of 201.3 francs. The estimate for NI reflects the insignificant impact of the control periods where the endowment effect is present but without the tournament payoff inducing a shortened horizon. So as in the earlier results, the combined effect of the incentive (i.e., the tournament) and the ability (i.e., the endowment) work to create both positive and negative price bubbles.

9.4.7 Further Tests To check the robustness of our results we conducted eight final experiments in a unique and different setting, the former communist country of Albania. Of its many unique

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

characteristics, one of the most important is its history of being the most isolated (politically and economically) country in Europe since World War II. Since democratic reforms began in 1991, a new business school was opened in the second largest city of Albania, Shkodra, where the third year students served as traders. Would the students whose country didn’t have a securities market or a history of free market trade show the same results as we had found at U.S. universities? While our previous experiments had the most experienced traders ever used in a study, these Albanian students may indeed represent the least experienced traders examined to date, which may be regarded as an extreme test of the validity of our results. Because of the newness of the trading experience for these students, a single period design was used in the first four experiments. For each experiment’s ten trading years (no period B), asset payoffs were for a single dividend payoff. The amounts used were the same as those of Table 9.2 so that equilibrium levels remained at 230 for each year. As shown in Table 9.1, Design 7 (experiment 19) consists of a single period security without a tournament effect. Design 8 (experiments 20 and 21) introduces the tournament payoff of Table 9.3 (Schedule Two) within the single period environment. This allows us to test for the presence of bubbles in the simpler pricing environment while also easing the learning experience of the Albanian students toward two-period tournament pricing. The pricing results for these three experiments can be seen in Figs. 9.19–9.21. Without the tournament in experiment 19, pricing is rational and typical showing a discount (risk pre-

Fig. 9.19 Experiment #19

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mium) of about 30 francs from the equilibrium level of 230. Near the end of experiment 20, the tournament effect appears to have created some price movement above equilibrium. This pressure continues into experiment 21 where prices trade at an average premium of 30 francs. Although these premiums do not constitute a bubble, it is clear they had a significant positive impact on pricing levels. Would this effect be eliminated (reversed) by changing the buy/sell pressure as was done earlier under Design 6 where negative bubbles were induced? Design 9 tests this proposition by changing the endowment from 2 securities and 5,000 francs to 20 securities and 500 francs. The results, reported in Fig. 9.22, show that even in these simple markets the endowment effect combined with tournament payoff leads to pricing away from equilibrium. These observations are confirmed by the regression results of Model 7 in Table 9.5 where buy side preference .I E D 1/ leads to significant increase in prices while sell side preference .I EN D 1/ leads to lower prices. The absence of a tournament payoff .NI D 1/ leads to insignificant price effects as investment horizon cannot be altered in a single period market. The Albanian students had now participated in four single period experiments and were ready to attempt two-period pricing. Experiment 23 was a simple two period pricing environment without any tournament payoff as in Design 1 (control). The plot of prices in Fig. 9.23 shows that the students initially struggled with two-period pricing since period A prices (two payoffs) differed little from period B prices (single payoff), although by the end of the experiment enough learning had developed.

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Fig. 9.20 Experiment #20

Fig. 9.21 Experiment #21

Experiments 24 and 25 introduce the shortened investment horizon (tournament effect) within the two-period framework as in Design 2 earlier. The price patterns in Figs. 9.24 and 9.25 show the creation of positive price bubbles to levels approaching 650 francs. Despite the histor-

ical background of this country and these students, they responded to market pressures in the same bubble-like manner. The last experiment, 26, alters the endowment to the sell side as before to see if negative bubbles can also be obtained. Price paths in Fig. 9.26 show a general downward trend of

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Fig. 9.22 Experiment #22

Fig. 9.23 Experiment #23

prices. The prices in period A show significant and growing discounts from the equilibrium levels of 460. These observations are confirmed by the regression results reported in Panel C of Table 9.5 with buy side endowment contributing 174.5 francs and sell side endowment reducing levels by 208.0 francs.

9.5 Conclusions The results of this study have a number of implications for real world markets. Experiments 1–6 seem to imply that within an environment that restricts selling pressures,

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Fig. 9.24 Experiment #24

Fig. 9.25 Experiment #25

a shortened investment horizon is sufficient to create asset bubbles. In application to the real world, short-term performance of traders, portfolio managers, and so forth could create pressures leading to price bubbles. Experiments 7–10 provide restrictions to the previous conclusion in that a shortened investment horizon creates bubble pressure only

when the market environment favors buyers over sellers. Unfortunately, most of our real world securities markets do have such a bias via restricted short sales, asymmetric leverage for longs versus shorts, restricted options and futures, and the like. Experiments 11–14 add to the puzzle by demonstrating the role of speculators within bubble formation. As a whole,

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Fig. 9.26 Experiment #26

the study suggests that necessary and sufficient conditions for the formation of asset bubbles are a shortened investment horizon, restricted selling activity relative to buyers, and the presence of speculators. We have also shown that repeated replication of these experiments under different settings still produce robust results. The first and third variables are a matter of fact within U.S. securities markets while restricted selling activity relative to buyers can take many forms. Either enhancing the buyer’s position or restricting the seller’s position is sufficient. Examples include increasing purchasing (speculative) ability through reduced stock margin levels, introduction of high leverage stock index futures, and in macroeconomic terms, a growing money supply or savings level. This latter variable may help explain the previous high levels of the Japanese equity market. The high level of Japanese savings creates very large endowments available for investment purchase. Given a limited supply of securities, our experimental markets show that these conditions will lead to a bubble. They also suggest that the bubble will burst when there is greater equating between the supply and demand. Recent changes in the Japanese institutional framework may, as predicted by this study, have lead to the bursting of that bubble. The primary prescription put forth for regulatory authorities to eliminate unnecessary market volatility resulting from asset bubbles is to create an institutional environment that does not restrict the transfer of information to the market. Structure the variables so that both bulls and bears have equal costs in executing their trades.

References Ackert, L., N. Charupat, B. Church and R. Deaves. 2002. “Bubbles in experimental asset markets: irrational exuberance no more.” Working paper 2002–2024. Ackert, L., N. Charupat, R. Deaves and B. Kluger. 2006. “The origins of bubbles in laboratory asset markets.” Working paper 2006–6, Federal Reserve Bank of Atlanta. Allen, F. and D. Gale. 2000. “Bubbles and crises.” Economic Journal 110, 236–255. Allen, F. and G. Gorton. 1988. “Rational finite bubbles,” in L. Rodney (Ed.). Working paper no. 41–88, White Center for Financial Research. Ang, J. and T. Schwarz. 1985. “Risk aversion and information structure: an experimental study of price variability in the securities market.” Journal of Finance 40, 924–844. Baker, G. P. 1992. “Incentive contracts and performance measurement.” Journal of Political Economy 100, 598–614. Becker, B. E. and M. A. Huselid. 1992. “The incentive effects of tournament compensation systems.” Administrative Science Quarterly 37, 336–350. Bierman, H., Jr. 1995. “Bubbles, theory and market timing.” Journal of Portfolio Management 22, 54–56. Brunnermeier, M. and S. Nagel. 2004. “Hedge funds and the technology bubble.” Journal of Finance 59, 2013–2040. Caginalp, G., D. Porter and V. Smith. 2001. “Financial bubbles: excess cash, momentum, and incomplete information.” The Journal of Psychology and Financial Markets 2(2), 80–99. Camerer, C. 1989. “Bubbles and fads in asset price: a review of theory and evidence.” Journal of Economic Surveys 3, 3–41. Cason, T. and C. Noussair. 2001. “The experimental study of market behavior,” Advances in Experimental Markets, Springer, New York, pp. 1–14. Cutler, D. M., J. M. Poterba and L. H. Summers. 1991. “Speculative dynamics.” Review of Economic Studies 58, 529–546.

164 DeLong, J. B., A. Schleifer, L. H. Summers and R. T. Waldmann. 1989. “The size and incidence of the losses from noise trading.” Journal of Finance 44, 681–696. Diba, B. T. and H. Grossman. 1987. “On the inception of rational bubbles.” Quarterly Journal of Economics 102, 197–700. Dufwenberg, M., T. Lindqvist and E. Moore. 2005. “Bubbles and experience: an experiment.” The American Economic Review 95, 1731–1737. Durand, R., R. Newby and J. Sanghani. 2006. “An intimate portrait of the individual investor.” Working paper. Ehrenberg, R. G. and M. L. Bognanno. 1990. “Do tournaments have incentive effects?” Journal of Political Economy 98, 1307–1324. Flood, R. P. and R. J. Hodrick. 1990. “On testing for speculative bubbles.” Journal of Economic Perspectives 4, 85–102. Forsythe, R., T. Palfrey and C. Plott. 1982. “Asset valuation in an experimental model.” Econometrica 50, 537–563. Friedman, B. M. 1992. “Monetary economics and financial markets: retrospect and prospect.” NBER Reporter, 1–5. Froot, K. A., D. S. Scharfstein and J. C. Stein. 1992. “Herd on the street: informational inefficiencies in a market with short-term speculation.” Journal of Finance 47, 1461–1484. Garber, P. 1990. “Famous first bubbles.” Journal of Economic Perspectives 4, 35–54. Gurkaynak, R. S. 2005. “Econometric tests of asset price bubbles: taking stock,” REDS Working paper No. 2005–2004. Harrison, M. and D. Kreps. 1978. “Speculative investor behavior in a stock market with heterogeneous expectations.” Quarterly Journal of Economics 92, 323–336. Hart, O. D. and D. M. Kreps. 1986. “Price destabilizing speculation.” Journal of Political Economy 94, 927–953. Heinemann, F. 2003. “Risk aversion and incentive effects: comment.” Working paper (Ludwig-Maximilians-Universitat Munchen). Hirota and Sunder. 2005. “Price bubbles sans dividend anchors: evidence from laboratory stock markets.” Working paper. Holt, C. A. and S. K. Laury. 2002. “Risk aversion and incentive effects.” American Economic Review 92, 1644–1655. Hong, H. G., J. A. Scheinkman and W. Xiong. 2006. “Asset float and speculative bubbles.” Journal of Finance 61, 1073–1117. Jackson, D. 1976. “Jackson personality inventory manual,” Research Psychologists Press, Goshen, NY. Jackson, D. 1977. “Reliability of the Jackson personality inventory.” Psychological Reports 40, 613–614. Jackson, D., D. Hourany and N. Vidmar. 1972. “A four dimensional interpretation of risk taking.” Journal of Personality 40, 433–501. James, D. and R. M. Isaac. 2000. “Asset markets: how they are affected by tournament incentives for individuals.” American Economic Review 90, 995–1004.

J.S. Ang et al. King, R. R., V. L. Smith, A. Williams and M. Van Boening. 1990. “The robustness of bubbles and crashes,” Working paper, University of Washington, Seattle, WA. Lei, V., C. Noussair and C. Plott. 2001. “Nonspeculative bubbles in experimental asset markets: lack of common knowledge of rationality vs. actual irrationality.” Econometrica 69(4), 831–859. Plott, C. and S. Sunder. 1982. “Efficiency of experimental markets with insider information: an application of rational expectations models.” Journal of Political Economy 90, 663–698. Porter, D. P. and V. L. Smith. 2003. “Stock market bubbles in the laboratory.” Journal of Behavioral Finance 4(1), 7–20. Ricke, M. 2004. “What is the link between margin loans and stock market bubbles?” University of Muenster, Department of Banking No. 03–01. Robin, S. and B. Ruffieux. 2001. “Price bubbles in laboratory asset markets with constant fundamental values.” Experimental Economics 4, 87–105. Roll, R. 1989. “Price volatility, international market links, and their implications for regulatory policy.” Journal of Financial Services Research 3, 211–246. Scheinkman, J. A. and W. Xiong. 2003. “Overconfidence and speculative bubbles.” Journal of Political Economy 111, 1183–1219. Shiller, R. J. 1988. “Fashions, fads, and bubbles in financial markets,” in Knights, raiders, and targets, J. Coffee Jr., L. Lowenstein and S. Ross-Ackerman (Eds.). Oxford Press, New York. Shleifer, A. and R. W. Vishny. 1990. “Equilibrium short horizons of investors and firms.” American Economic Review 80, 148–153. Siegel, J. J. 2003. “What is an asset price bubble? an operational definition.” European Financial Management 9, 11–24. Smith, V., G. Suchanek and A. Williams. 1988. “Bubbles, crashes, and endogenous expectations in experimental spot asset markets.” Econometrica 56, 1119–1151. Stiglitz, J. 1990. “Symposium on bubbles.” Journal of Economic Perspective 4, 13–18. Sunder, S. 1992. “Experimental asset markets: a survey,” in Handbook of experimental economics, J. H. Kagel and A. E. Roth (Eds.). Princeton University Press, Princeton, NY. Voth, H.-J. and Temin, P. 2003. “Riding the South Sea Bubble,” MIT Economics, Working Paper No. 04–02. West, K. 1988. “Bubbles, fads, and stock price volatility tests: a partial evaluation.” Journal of Finance 43, 639–655. White, E. 1990. “The stock market boom and crash of 1929 revisited.” Journal of Economic Perspectives 4, 67–84.

Chapter 10

Portfolio Optimization Models and Mean–Variance Spanning Tests Wei-Peng Chen, Huimin Chung, Keng-Yu Ho, and Tsui-Ling Hsu

Abstract In this chapter we introduce the theory and the application of the computer program of modern portfolio theory. The notion of diversification is age-old: “don’t put your eggs in one basket,” obviously predates economic theory. However, a formal model showing how to make the most of the power of diversification was not devised until 1952, a feat for which Harry Markowitz eventually won the Nobel Prize in economics. Markowitz portfolio shows that as you add assets to an investment portfolio the total risk of that portfolio – as measured by the variance (or standard deviation) of total return – declines continuously, but the expected return of the portfolio is a weighted average of the expected returns of the individual assets. In other words, by investing in portfolios rather than in individual assets, investors could lower the total risk of investing without sacrificing return. In the second part we introduce the mean–variance spanning test that follows directly from the portfolio optimization problem. Keywords Minimum variance portfolio r Optimal risky portfolio r Capital allocation line r Mean–variance spanning tests

10.1 Introduction of Markowitz Portfolio-Selection Model Harry Markowitz (1952, 1959) developed his portfolioselection technique, which came to be called modern portfolio theory (MPT). Prior to Markowitz’s work, securityselection models focused primarily on the returns generated W.-P. Chen () Department of Finance, Shih Hsin University, Taipei, Taiwan e-mail: [email protected] H. Chung and T.-L. Hsu Graduate Institute of Finance, National Chiao Tung University, Hsinchu, Taiwan e-mail: [email protected]; [email protected] K.-Y. Ho Department of Finance, National Taiwan University, Taipei, Taiwan e-mail: [email protected]

by investment opportunities. Standard investment advice was to identify those securities that offered the best opportunities for gain with the least risk and then construct a portfolio from these. Following this advice, an investor might conclude that railroad stocks all offered good risk-reward characteristics and compile a portfolio entirely from these. The Markowitz theory retained the emphasis on return; but it elevated risk to a coequal level of importance, and the concept of portfolio risk was born. Whereas risk has been considered an important factor and variance an accepted way of measuring risk, Markowitz was the first to clearly and rigorously show how the variance of a portfolio can be reduced through the impact of diversification. He proposed that investors focus on selecting portfolios based on their overall risk-reward characteristics instead of merely compiling portfolios from securities that each individually have attractive risk-reward characteristics. A Markowitz portfolio model is one where no added diversification can lower the portfolio’s risk for a given return expectation (alternately, no additional expected return can be gained without increasing the risk of the portfolio). The Markowitz Efficient Frontier is the set of all portfolios of which expected returns reach the maximum given a certain level of risk. The Markowitz model is based on several assumptions concerning the behavior of investors and financial markets:

1. A probability distribution of possible returns over some holding period can be estimated by investors. 2. Investors have single-period utility functions in which they maximize utility within the framework of diminishing marginal utility of wealth. 3. Variability about the possible values of return is used by investors to measure risk. 4. Investors care only about the means and variance of the returns of their portfolios over a particular period. 5. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance.

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6. Return is desirable; risk is to be avoided.1 7. Financial markets are frictionless.

10.2 Measurement of Return and Risk Throughout this chapter, investors are assumed to measure the level of return by computing the expected value of the distribution, using the probability distribution of expected returns for a portfolio. Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation.

The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio combination of securities is equal to the weighted average covariance2 of the returns on its individual securities: n n X X     wi wj Cov ri ; rj Var rp D p2 D

(10.2)

i D1 j D1

Covariance can also be expressed in terms of the correlation coefficient as follows:   Cov ri ; rj D ij i j D ij

(10.3)

10.2.1 Return Given any set of risky assets and a set of weights that describe how the portfolio investment is split, the general formula of expected return for n assets is: E.rP / D

n X

wi E.ri /

where ij D correlation coefficient between the rates of return on security i; ri , and the rates of return on security j; rj I i , and j represent standard deviations of ri and rj respectively. Therefore: n n X   X wi wj ij i j Var rp D

(10.1)

i D1

where

n P

wi D 1:0; n D the number of securities, wi the

i D1

proportion of the funds invested in security i; ri ; rP the return on i th security and portfolio p; E. / the expectation of the variable in the parentheses. The return computation is nothing more than finding the weighted average return of the securities included in the portfolio.

1 Markowitz model assumes that investors are risk averse. This means that given two assets that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk. The exact tradeoff will differ by investor based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-return profile (i.e., if for that level of risk an alternative portfolio exists that has better expected returns). Using risk tolerance, we can simply classify investors into three types: risk-neutral, risk-averse, and risk-lover. Riskneutral investors do not require the risk premium for risk investments; they judge risky prospects solely by their expected rates of return. Riskaverse investors are willing to consider only risk-free or speculative prospects with positive premium; they make investment according the risk-return tradeoff. A risk-lover is willing to engage in fair games and gambles; this investor adjusts the expected return upward to take into account the “fun” of confronting the prospect’s risk.

(10.4)

i D1 j D1

Overall, the estimate of the mean return for each security is its average value in the sample period; the estimate of variance is the average value of the squared deviations around the sample average; the estimate of the covariance is the average value of the cross-product of deviations.

10.3 Efficient Portfolio Efficient portfolios may contain any number of asset combinations. We examine efficient asset allocation by using two risky assets for example. After we understand the properties of portfolios formed by mixing two risky assets, it will be easy to see how the portfolio of many risky assets might best be constructed.

2

High covariance indicates that an increase in one stock’s return is likely to correspond to an increase in the other. A low covariance means the return rates are relatively independent and a negative covariance means that an increase in one stock’s return is likely to correspond to a decrease in the other.

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10.3.1 Two-Risky-Assets Portfolio

E.rP / D wA E.rA / C wB E.rB /

asset B

r=0

r=1 0
r = −1 0

Standard Deviation

Fig. 10.1 Investment opportunity sets for asset A and asset B with various correlation coefficients

(10.5)

The variance of the rate of return on the two-asset portfolio is P2 D w2A A2 C w2B B2 C 2wA wB AB A B

r = −1

Expected Return

Because we now envision forming a portfolio from two risky assets, we need to understand how the uncertainties of asset returns interact. It turns out that the key determinant of portfolio risk is the extent to which the returns on the two assets tend to vary in tandem or in opposition. The degree to which a two-risky-assets portfolio reduces variance of returns depends on the degree of correlation between the returns of the securities. Suppose a proportion denoted by wA is invested in asset A, and the remainder 1  wA , denoted by wB , is invested in asset B. The expected rate of return on the portfolio is a weighted average of the expected returns on the component assets, with the same portfolio proportions as weights.

(10.6)

where AB is the correlation coefficient between the returns on asset A and asset B. If the correlation between the component assets is small or negative, this will reduce portfolio risk. First, assume that AB D 1:0, which would mean that assets A and B are perfectly positively correlated, the righthand side of Equation (10.6) is a perfect square and simplifies to p2 D w2A A2 C w2B B2 C 2wA wB A B D .wA A C wB B /2 or p D wA A C wB B Therefore, the portfolio standard deviation is a weighted average of the component security standard deviations only in the special case of perfect positive correlation. In this circumstance, there are no gains to be had diversification. Whatever the proportions of asset A and asset B, both the portfolio mean and the standard deviation are simple weighted averages. Figure 10.1 shows the opportunity set with perfect positive correlation – a straight line through the component assets. No portfolio can be discarded as inefficient in this case, and the choice among portfolios depends only on risk preference. Diversification in the case of perfect positive correlation is not effective. Perfect positive correlation is the only case in which there is no benefit from diversification. With any correlation coefficient less than 1:0 . < 1/, there will be a diversification effect, the portfolio standard deviation is less than the weighted average of the standard deviations of the component securities. Therefore, there are benefits to diversification whenever asset returns are less than perfectly correlated. Our

analysis has ranged from very attractive diversification benefits .AB < 0/ to no benefits at all AB D 1:0. Negative correlation between a pair of assets is also possible. Where negative correlation is present, there will be even greater diversification benefits. Again, let us start with an extreme. With perfect negative correlation, we substitute AB D 1:0 in Equation (10.6) and simplify it in the same way as with positive perfect correlation. Here, too, we can complete the square, this time, however, with different results. P2 D w2A A2 C w2B B2  2wA wB A B D .wA A  wB B /2 And, therefore, P D jwA A  wB B j

(10.7)

With perfect negative correlation, the benefits from diversification stretch to the limit. Equation (10.7) points to the proportions that will reduce the portfolio standard deviation all the way to zero. An investor can reduce portfolio risk simply by holding instruments that are not perfectly correlated. In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk.

10.3.2 The Concept of Markowitz Efficient Frontier Every possible asset combination can be plotted in riskreturn space, and the collection of all such possible portfolios defines a region in this space. The line along the upper edge of this region is known as the efficient frontier. Combinations

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Z

B’ Minimum Variance Portfolio (MVP)

Investment Opportunity Set

V

Efficient frontier of risky assets

Expected Return

Expected Return

A

Minimum variance portfolio

Individual assets

B 0

Standard Deviation

0

Standard Deviation

Fig. 10.2 Investment opportunity set for asset A and asset B

Fig. 10.3 The efficient frontier of risky assets and individual assets

along this line represent portfolios (explicitly excluding the risk-free alternative) for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolio lying on the efficient frontier represents the combination offering the best possible return. Mathematically, the efficient frontier is the set of mean–variance choices from the portfolio set where for a given variance no other portfolio offers a higher man return. Figure 10.2 shows investors the entire investment opportunity set, which is the set of all attainable combinations of risk and return offered by portfolios formed by asset A and asset B in differing proportions. The curve passing through A and B shows the risk-return combinations of all the portfolios that can be formed by combining those two assets. Investors desire portfolios that lie to the northwest in Fig. 10.2. These are portfolios with high expected returns (toward the north of the figure) and low volatility (to the west). The area within curve BVAZ is the feasible opportunity set representing all possible portfolio combinations. Portfolios that lie below the minimum-variance portfolio (point V ) on the figure can therefore be rejected out of hand as inefficient. The portfolios that lie on the frontier VA in Fig. 10.2 would not be likely candidates for investors to hold because they do not meet the criteria of maximizing expected return for a given level of risk or minimizing risk for a given level of return. This is easily seen by comparing the portfolio represented by points B and B 0 . Since investors always prefer more expected return than less for a given level of risk, B 0 is always better than B. Using similar reasoning, investors would always prefer B to V because it has both a higher return and a lower level of risk. In fact, the portfolio at point V is identified as the minimum-variance portfolio because no other portfolio exists that has a lower standard deviation. The curve VA represents all possible efficient portfolios and is the efficient frontier3, which represents the set of portfolios that

offers the highest possible expected rate of return for each level of portfolio standard deviation. Any portfolio on the downward sloping portion of the frontier curve is dominated by the portfolio that lies directly above it on the upward sloping portion of the frontier curve since that portfolio has a higher expected return and equal standard deviation. The best choice among the portfolios on the upward sloping portion of the frontier curve is not as obvious because in this region higher expected return is accompanied by higher risk. The best choice will depend on the investor’s willingness to trade off risk against expected return (Fig. 10.3).

3

The efficient frontier will be convex – this is because the riskreturn characteristics of a portfolio change in a non-linear fashion as

10.3.3 Short Selling Various constraints may preclude a particular investor from choosing portfolios on the efficient frontier, however. Short sale restrictions are only one possible constraint. Short selling is a usual regulated type of market transaction. It involves selling assets that are borrowed in expectation of a fall in the assets’ price. When and if the price declines, the investor buys an equivalent number of assets at the new lower price and returns to the lender the borrowed ones. Now, relaxing the assumption of no short selling, investors could sell the lowest-return asset B (here, we assume that E.rA / E.rB / and A B ). If the number of short sales is unrestricted, then by a continuous short selling of B and reinvesting in A the investor could generate an infinite expected return. The efficient frontier of unconstraint portfolio is shown in Fig. 10.4. The upper bound of the

its component weightings are changed. (As described above, portfolio risk is a function of the correlation of the component assets, and thus changes in a non-linear fashion as the weighting of component assets changes.) The efficient frontier is a parabola (hyperbola) when expected return is plotted against variance (standard deviation).

10 Portfolio Optimization Models and Mean–Variance Spanning Tests

Z’ Expected Return

A

without short sales with short sales

V P

0

Z

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Table 10.1 The minimum variance portfolio weight of two-assets portfolio without short selling The correlation of two assets Weight of Asset A Weight of Asset B B A  2B D1 wA D wB D A  B A  B B A  D 1 wA D wB D A C B A C B D0

wA D

B

B2 A2 C B2

wB D

A2  2B2 A2 C B2

Standard Deviation

Fig. 10.4 The efficient frontier of unrestricted/restricted portfolio

highest-return portfolio would no longer be A but infinity (shown by the arrow on the top of the efficient frontier). Likewise the investor could short sell the highest-return security A and reinvest the proceeds into the lowest-yield security B4 , thereby generating a return less than the return on the lowest-return assets. Given no restriction on the amount of short selling, an infinitely negative return can be achieved, thereby removing the lower bound of B on the efficient frontier. Hence, short selling generally will increase the range of alternative investments from the minimum-variance portfolio to plus or minus infinity.5 Relaxing the assumption of no short selling in this development of the efficient frontier involves a modification of the analysis of the efficient frontier of constraint (not allowed short sales). In the next section, we introduce the mathematical analysis of the efficient frontier with/without short selling constraints.

Mathematically, the portfolio selection problem can be formulated as a quadratic program. For two risky assets A and B, the portfolio consists of wA and wB , and the return of the portfolio is then rA wA C rB wB . The weights should be chosen so that the risk is minimized, that is Min P2 D w2A A2 C w2B B2 C 2wA wB AB A B wA

for each chosen return and subject to wA C wB D 1; wA 0; wB 0. The last two constraints simply imply that the assets cannot be in short positions. The minimum variance portfolio weights are shown in Table 10.1.6 Above, we simply use two-risky-assets portfolio to calculate the minimum variance portfolio weights. If we generalization to portfolios containing N assets, the minimum portfolio weights can then be obtained by minimizing the Lagrange function C for portfolio variance. Min p2 D

n n X X

wi wj ij i j

i D1 j D1

Subject to w1 C w2 C : : : C wN D 1

10.3.4 Calculating the Minimum Variance Portfolio In Markowitz portfolio model, we assume investors choose portfolios based on both expected return, E.rp /, and the standard deviation of return as a measure of its risk, p . Therefore, the portfolio selection problem can be expressed as maximizing the return with respect to the risk of the investment (or, alternatively, minimizing the risk with respect to a given return, hold the return constant and solve for the weighting factors that minimize the variance).

4

Rational investors will not short sell a high-return asset and buy a low-return asset. This case is just for extreme assumption. 5 Whether an investor engages in any of this short-selling activity depends on the investor’s own unique set of indifference curves.

C D

n n X X





wi wj Cov ri rj C 1 1 

i D1 j D1

n X

! Wi

(10.8)

i D1

in which 1 are the Lagrange multipliers, respectively, ij is the correlation coefficient between ri and rj , and other variables are as previously defined. By using this approach the minimum variance can be computed for any given level of expected portfolio return (subject to the other constraint that the weights sum to one). In practice, it is best to use a computer because of the explosive increase in the number of calculations as the number of securities considered grows. The efficient set that is gen6 The solution uses the minimization techniques of calculus. Write out the expression for portfolio variance from Equation (10.6), substitute 1  wA for wB , differentiate the result with respect to wA , set the deriva-

tive equal to zero, and solve for wA to obtain wA D wB D 1  wA .

B2 AB A B A2 CB2 2AB A B

and

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erated by the aforementioned approach (Equation (10.8)) is sometimes called the minimum-variance set because of the minimizing nature of the Lagrangian solution.

The Lagrangian function is C D

n n X X



One of the goals of portfolio analysis is minimizing the risk or variance of the portfolio. Previous sections introduced the calculation of minimum variance portfolio, and we minimize the variance of the portfolio subject to the portfolio weights’ summing to one. If we add a condition to Equation (10.8), which is subject to the portfolio attaining some target expected rate of return, we can get the optimal risky portfolio. Min p2 D

n n X X

Wi Wj ij i j

i D1 j D1

Subject to

n P

"

wi wj Cov ri rj C 1 E 

i D1 j D1

10.3.5 Calculating the Weights of Optimal Risky Portfolio



C2 1 

n X



n X

# wi E .ri /

i D1

! jwi j

(10.10)

i D1

If the restriction of no short selling is in minimization variance problem, it needs to add a third constraint: wi 0;

i D 1; : : : ; N

The addition of this non-negativity constraint precludes negative values for the weights (that is, no short selling). The problem now is a quadratic programming problem similar to the ones solved so far, except that the optimal portfolio may fall in an unfeasible region. In this circumstance, the next best optimal portfolio is selected that meets all of the constraints.

Wi E.Ri / D E  , where E  is the target ex-

i D1

n P

Wi D 1:0.

10.3.6 Finding the Efficient Frontier of Risky Assets The first constraint simply says that the expected return

pected return and

i D1

on the portfolio should equal the target return determined by the portfolio manager. The second constraint says that the weights of the securities invested in the portfolio must sum to one. The Lagrangian objective function can be written: C D

n n X X

  wi wj Cov ri rj

i D1 j D1

"



C1 E 

n X

# wi E.ri / C 2 1 

i D1

n X

! wi

(10.9)

i D1

Taking the partial derivatives of this equation with respect to each of the variables, w1 ; w2 ; : : : ; wN ; 1 , and 2 and setting the resulting equations equal to zero yields the minimization of risk subject to the Lagrangian constraints. Then, we can solve the weights and these weights represent an optimal risky portfolio by using matrix algebra. If there is no short selling constraint in the portfolio analn P wi D 1:0, should substitute to ysis, second constraint, n P

According to two-fund separation, the efficient frontier of risky assets can be formed by any two risky portfolios on the frontier. All portfolios on the mean–variance efficient frontier can be formed as a weighted average of any two portfolios or funds on the efficient frontier; this is called twofund separation. So, if we have any two points of the portfolio combinations, we can draw an entire efficient frontier of the risky assets. In previous sections, we have introduced the minimum variance portfolio and optimal risky portfolio; given the expected return, then, we can generate the entire efficient frontier by the separation property. Deriving the efficient frontier may be quite difficult conceptually, but computing and graphing it with any number of assets and any set of constraints is quite straightforward. Later, we will use Excel and MATLAB to generate the efficient frontier.

10.3.7 Finding the Optimal Risky Portfolio

i D1

jwi j D 1:0, where the absolute value of the weights jwi j

i D1

allows for a given wi to be negative (sold short) but maintains the requirement that all funds are invested or their sum equals one.

We already have an efficient frontier; however, how do we decide the best allocation of portfolio? One of the factors to consider when selecting the optimal portfolio for a particular investor is the degree of risk aversion. Investor’s willingness to trade off risk against expected return. This

10 Portfolio Optimization Models and Mean–Variance Spanning Tests

level of aversion to risk can be characterized by defining the investor’s indifference curve, consisting of the family of risk/return pairs defining the tradeoff between the expected return and the risk. It establishes the increment in return that a particular investor will require in order to make an increment in risk worthwhile. The optimal portfolio along the efficient frontier is not unique with this model and depends on the risk/return tradeoff utility function of each investor. We use the utility function that is commonly employed by financial theorists and the AIMR (Association of Investment Management and Research) where a portfolio with a given expected return E.rp / and standard deviation p is assigned the following utility function: U D E.rp /  0:005Ap2

(10.11)

where U is the utility value and A is an index of the investor’s risk aversion. The factor of 0.005 is a scaling convention that allows us to express the expected return and standard deviation in Equation (10.11) as percentages rather than decimals. We interpret this expression to say that the utility from a portfolio increases as the expected rate of return increases, and it decreases when the variance increases. The relative magnitude of these changes is governed by the coefficient of risk aversion, A. For risk-neutral investors, A D 0. Higher levels of risk aversion are reflected in larger values for A. Portfolio selection, then, is determined by plotting investors’ utility functions together with the efficient-frontier set of available investment opportunities. In Fig. 10.5, two sets of indifference curves labeled U1 and U2 are shown together with the efficient frontier. The U1 curve has a higher slope, indicating a greater level of risk aversion. The investor is indifferent to any combination of rp and p along a given curve. The U2 curve would be appropriate for a less riskaverse investor – that is, one who would be willing to accept relatively higher risk to obtain higher levels of return. The optimal portfolio would be the one that provides the highest utility – a point in the northwest direction (higher return and

U2

Expected Return

B

171

lower risk). This point will be at the tangent of a utility curve and the efficient frontier. Each investor is logically selecting the optimal portfolio given his or her risk-return preference, and neither is more correct than the other. In order to simplify the determination of optimal risky portfolio, we use the capital allocation line (CAL), which depicts all feasible risk-return combinations available from different asset allocation choices, to determine the optimal risky portfolio. To start, however, we will demonstrate the solution of the portfolio construction problem with only two risky assets (in our example, asset A and asset B) and a riskfree asset. In this case, we can derive an explicit formula for the weights of each asset in the optimal portfolio. This will make it easy to illustrate some of the general issues pertaining to portfolio optimization. The objective is to find the weights wA and wB that result in the highest slope of the CAL (i.e., the weights that result in the risky portfolio with the highest reward-to-variability ratio). Therefore, the objective is to maximize the slope of the CAL for any possible portfolio, P. The objective function is the slope that we have called CALS : The entire portfolio including risky and risk-free assets. CALS D

E.rp /  rf p

(10.12)

For the portfolio with two risky assets, the expected return and standard deviation of portfolio S are E.rP / D wA E.rA / C wB E.rB / 1=2  P D w2A A2 C w2B B2 C 2wA wB AB A B

(10.5) (10.13)

When we maximize the objection function, CALS , we have to satisfy the constraint that the portfolio weights sum to 1. Therefore, we solve a mathematical problem formally written as E.rp /  rf Max CALS D wi p P Subject to wi D 1. In this case of two risky assets, the solution for the weights of the optimal risky portfolio, S, can be shown to be as follows7 :

A

wA D

U1

ŒE.rA /rf B2 ŒE.rB /rf AB A B ŒE.rA /rf B2 CŒE.rB /rf A2 ŒE.rA /rf CE.rB /rf AB A B

wB D 1  wA Minimum variance portfolio

0

Efficient frontier of risky assets

Standard Deviation

Fig. 10.5 Indifference Curves and Efficient frontier

7

The solution procedure for two risky assets is as follows. Substitute for expected return from Equation (10.5) and for standard deviation from Equation (10.13). Substitute 1  wA for wB . Differentiate the resulting expression for CALs with respect to wA , set the derivative equal to zero, and solve for wB .

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CAL Expected Return

Then, we form an optimal complete portfolio8 given an optimal risky portfolio and the CAL generated by a combination of portfolio S and risk-free asset. We have constructed the optimal portfolio S, and then we can use the individual investor’s degree of risk aversion, A, to calculate the optimal proportion of complete portfolio to invest in the risky component. Assume that a risk-free rate is rf , and a risky portfolio with expected return E.rp / and standard deviation p will find that for any choice of y, the expected return of the complete portfolio is

Indifference Curve Opportunity Set of Risky Assets Optimal Risky Portfolio Optimal Complete Portfolio

rf 0

Standard Deviation

Fig. 10.6 Determination of the optimal portfolio

E.rC / D rf C yŒE.rp /  rf The variance of the overall portfolio is

3. Allocation funds between the risky portfolio and the riskfree asset:

C2 D y 2 p2 The investor attempts to maximum utility, U, by choosing the best allocation to the risky asset, y. To solve the utility maximization problem more generally, we write the problem as follows: Max U D E.rC /  0:005AC2 y

D rf C yŒE.rp /  rf  0:005Ay 2 p2 Setting the derivative of this expression to zero, we can solve for y yield the optimal position for risk-averse investors in the risky asset, y  , as follows: y D

E.rp /  rf 0:01Ap2

(10.14)

The solution shows that the optimal position in the risky asset is, as one would expect, inversely proportional to the level of risk aversion and the level of risk (measured by the variance) and directly proportional to the risk premium offered by the risky asset. Once we have reached this point, generalizing to the case of many risky assets is straightforward. Before we move on, let us briefly summarize the steps we followed to arrive at the complete portfolio (Fig. 10.6). 1. Specify the return characteristics of all securities (i.e., expected return, variance, and covariance). 2. Establish the risky portfolio: (a) Calculate the optimal risky portfolio S. (b) Calculate the properties of portfolio S using the weights determined in step and Equations (10.5) and (10.13). 8

The complete portfolio means that the entire portfolio including risky and risk-free assets.

(a) Calculate the fraction of the complete portfolio allocated to Portfolio S (the risky portfolio) and to riskfree asset (Equation (10.14)). (b) Calculate the share of the complete portfolio invested in each asset and in risk-free asset. For further information about modern portfolio theory, please refer to Lee et al. (1990), and Bodie et al. (2007, 2008).

10.4 Mean–Variance Spanning Test 10.4.1 Mean–Variance Spanning and Intersection Tests Huberman and Kandel (1987) first introduced a mean– variance spanning test. The method statistically tests whether adding a set of new assets can improve the investment opportunity set relative to a set of basis assets. In other words, mean–variance spanning tests enable us to analyze the effect on the mean–variance frontier of adding new assets to a set of benchmark assets. For ease of illustration, we define the union of both new assets and benchmark assets as augmented assets. If the mean–variance frontier of the benchmark portfolios and that of the augmented portfolios coincides, then there is spanning. In this case, investors do not benefit from adding new portfolios to their current portfolios. Whether an observed shift is statistically significant can be tested using regression-based mean–variance spanning tests. We briefly describe the approaches that we use to examine whether adding an IPO portfolio could significantly improve the investment opportunity set relative to a set of benchmark assets. For convenience, we follow the notations and treatment in Kan and Zhou (2008).

10 Portfolio Optimization Models and Mean–Variance Spanning Tests

We denote by K the set of benchmark portfolios with returns, R1t and by N the set of test assets with returns, R2t . Let the returns on N C K assets as Rt , we denote the   ex1 pected returns on N C K assets as  D EŒRt  2 and the variance-covariance matrix of N C K assets as V D   V11 V12 , where V is non-singular. We then esVarŒRt  V21 V22 timate the following model using ordinary least squares as:

173

LM D T

i D1

(10.15)

Equation (10.15) can be rewritten as R D XB C E in matrix O D notation with the estimators of B and ˙ being BO  Œ˛O ˇ P 1 0 1 0 0 O O .R  XB/. O Under the D T .R  XB/ .X X / .X R/ and P O0 normality assumption, we Phave t  N.0; / and vec.B /  0 0 1 N.vec.B /; .X X / ˝ /. Following Huberman and Kandel (1987), the null hypothesis of “spanning” is: H0S W ˛ D 0N ; ı D 1N  ˇ1K D 0N ;

(10.16)

where 0N is defined as a zero vector with N elements. If we fail to reject the null hypothesis, the benchmark assets then span the mean–variance frontier of the benchmark assets plus an IPO portfolio. In other words, failing to reject the null hypothesis implies that investors are not able to enlarge their investment opportunity set by adding an IPO portfolio. On the other hand, if the null hypothesis is rejected, adding an IPO portfolio does improve the investment opportunity  set. 0 We can rewrite this null hypothesis as ‚ D ˛ ı D     00N 1 00K O2N D AB  C , where A D and C D . 0 10K 10N O0 The distribution of the null P hypothesis is vec.‚ /  0 0 1 0 N.vec.‚ /; A.X X / A ˝ /. By defining " GO D TA.X X / A D 0

1

0

1 C O 01 VO111 O 1

O 01 VO111 1K

O 01 VO111 1K

10K VO111 1K

#

and 2 P1 1 X ˛O 0 O ˛O O O ‚ O0 D4 HO D ‚ P1 ˛O 0 O ıO

P1 3 ˛O 0 O ıO 5 P1 ıO0 O ıO

we then denote by 1 and by 2 the two eigenvalues of the matrix, HO GO 1 . The distributions of the asymptotic Wald, likelihood ratio, and Lagrange multiplier test statistics for the null hypothesis are: A

W D T .1 C 2 /  22N ; LR D T

2 X i D1

A

ln.1 C i /  22N ;





1 1



1

(10.19)

T KN N

 F2N;2.T KN / for N 2; (10.20)



1 T K 1 1  F2;.T K1/ for N D 1; (10.21) U 2

ˇ ˇ ˇ.ˇ ˇ ˇ ˇ ˇ where U D ˇGO ˇ ˇHO C GO ˇ. In general, the test for mean–variance spanning can be divided into two parts: (1) the spanning of the global minimum-variance (GMV) portfolio and (2) the spanning of the tangency portfolio. Therefore, we can rewrite all three asymptotic test statistics based on this geometric feature. For example, the Wald test can be rewritten as W DT

.O R1 /2 1 CT .O R /2

! 1 C OR .R1GMV /2 1 ; 1 C OR1 .R1GMV /2 (10.22)

where .O R1 /2 and .O R /2 are the global minimum-variance of the benchmark assets assets plus new as and benchmark  sets, respectively. OR1 R1 GMV is the slope of the asymptote of the frontier for the benchmark assets, and  mean–variance  OR R1 GMV is the slope of the tangency line of the mean– variance frontier for the benchmark portfolios plus new portfolios. The first term measures the change of the GMV portfolios due to the addition of a set of new assets. The second term measures whether there is an improvement of the squared tangency slope due to adding a set of new portfolios to the set of benchmark portfolios. Another relevant idea about testing the movement or change in the tangency portfolio is the mean–variance intersection test proposed in Huberman and Kandel (1987). If the mean–variance frontier of the benchmark assets and the mean–variance frontier of the augmented assets have only one point in common, this is known as intersection. Using Equation (10.15), the null hypothesis for intersection is: H0I W ˛  .1  ˇ1K / D 0;

(10.23)

where  is the risk-free rate. Following DeRoon and Nijman (2001), the test statistic for testing the intersection hypothesis can be rewritten in terms of the maximal Sharpe ratios as

(10.17) WI D T (10.18)

i A 2  2N : 1 C i

In addition, the exact finite sample distribution of the likelihood ratio test under the null, as Huberman and Kandel (1987) and Jobson and Korkie (1989) also show, is:

U2 R2t D ˛ C ˇ R1t C t ; t D 1; 2; : : : ; T:

2 X

! 1 C OR ./2 1 DT 1 C OR ./2 1

! OR ./2  OR1 ./2 ; 1 C OR1 ./2 (10.24)

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where OR1 ./ is the maximal Sharpe ratio attainable for the benchmark assets, and OR ./ is the maximal Sharpe ratio attainable for the augmented assets. We note that the Wald test statistic for intersection is chi-square distributed with degree of freedom N . The test statistic specified in Equation (10.24) indicates that the numerator is related to the difference in the squared maximal Sharpe ratios attainable for benchmark assets and augmented assets. The rejection of the null hypothesis of the intersection test implies that the mean–variance frontier of augmented assets does not have any point in common with the mean–variance frontier of benchmark assets based on the reference point of risk-free rate. Thus, the maximal Sharpe ratios are different between the augmented assets and benchmark assets.

10.4.2 Step-Down Tests for Mean–Variance Spanning

W2 D T

! cO C dO 1 C aO 1

1 ; 1 C aO cO1 C dO1

(10.28)

where aO 1 D O 01 VO111 O 1 ; bO1 D O 01 VO111 1K ; cO1 D 1k 0 VO111 1K O cO and dO are the analogues and dO1 D aO 1 cO1  bO12 . Here a; O b; O O of aO 1 ; b1 ; cO1 and d1 , based on benchmark assets plus new assets.

10.4.3 Mean–Variance Spanning Tests Under Non-normality and Heteroskedasticity The tests described so far assume that the returns are normally distributed and the error term in Equation (10.15) is homoskedastic. We can use a GMM Wald test to adjust for return non-normality and heteroskedasticity. The GMM Wald test is: O 0 /0 Œ.AT ˝IN /ST .AT 0 ˝IN / 1 vec.‚ O 0 / 22N; WaD T vec.‚ (10.29) A

Kan and Zhou (2008) report that the asymptotic tests have very good power in testing assets that can reduce the variance of the global minimum-variance (GMV) portfolio, but have little power against test assets that can only improve the tangency portfolio. They therefore suggest a step-down procedure that requires us to first test ˛ D 0N and then to test ı D 0N conditional on ˛ D 0N . If the rejection is due to the first test, we know that it is because the two tangency portfolios are very different. If the rejection is due to the second test, it is because the two GMV portfolios are very different. The first test .˛ D 0N / is straightforward. We simply ignore the null hypothesis of ı D 0N in Equation (10.16) and apply the same asymptotic Wald test procedure. Note that HO GO 1 then degenerates to a scalar and its eigenvalue, denoted as 3 , is actually itself. The second test (ı D 0N conditional on ˛ D 0N ) is a test of ı D 0N on estimating Equation (10.15) without an intercept. We follow the same asymptotic Wald test procedure detailed above. The matrix HO GO 1 in the second test is also a scalar. Thus, its eigenvalue, denoted as 4 , is itself. The step-down asymptotic Wald tests can be written as: A

W1 D T .3 /  2N ; A

W2 D T .4 /  2N :

(10.25) (10.26)

Equations (10.25) and (10.26) can also be rewritten using the similar notation of the finite sample step-down F tests as shown in Kan and Zhou (2008): W1 D T

aO  aO 1 1 C aO 1

;

(10.27)

where the moment condition is EŒg "t D E.X ˝ E/ # D 0 O 1 1 C aO 1 O 1 V11 00N.1CK/ ; ST D E Œgt 0 gt , and AT D . bO1 10K VO111 We can also conduct step-down GMM Wald tests to disentangle the two sources of the spanning test. The test for ˛ D 0N (denote as Wa1 ) is straightforward. We simply ignore the null hypothesis of ı D 0N in Equation (10.16) and use (10.29). However, we need to change AT to  Equation  1 C aO 1 01 VO111 due to the change in the null hypothesis. Testing ı D 0N conditional on ˛ D 0N (denote as Wa2 ) is actually a test of ı D 0N using Equation (10.15) without an intercept. (10.29) by chang  We can again use Equation 1 . Furthermore, new ST is ing AT to 10K VO11 C 1 01 different from ST of the joint test since X for the second step-down test has no intercept term. Finally, we note that both step-down GMM Wald test statistics are distributed as chi-square with N degrees of freedom. The details on how the change in matrix AT is made are shown as follows. Based on Kan and Zhou (2008), AT is the consistent estimate for AEŒXX 0 1 . For the first   stepdown test (testing ˛ D 0N ), A has to be changed as 1 0K . Therefore,   AT D AEŒXX 0 1 D 1 C aO 1 O 01 VO111 :

(10.30)

For the second step-down test (testing  0ı D 0N conditional on ˛ D 0N ), A has to be changed to 1K . Thus, i h AT D AEŒXX 0 1 D 10K .VO11 C 1 01 /1 ;

(10.31)

10 Portfolio Optimization Models and Mean–Variance Spanning Tests

where new X is different from X in Equation (10.30), since there is no intercept term in the regression equation estimated.

10.5 Alternative Computer Program to Calculate Efficient Frontier Several software packages can be used to generate the efficient frontier. In this section, we will demonstrate the method using Microsoft Excel and MATLAB.

10.5.1 Application: Microsoft Excel Excel is far from the best program for generating the efficient frontier and is limited in the number of assets it can handle, but working through a simple portfolio optimizer in Excel can illustrate concretely the nature of the calculations used in more sophisticated “black-box” programs. You will find that in Excel, the computation of the efficient frontier is fairly easy. Assume an American investor who forms a six-stockindex portfolio. The portfolio consists of six stock indexes: United States (S&P 500), United Kingdom (FTSE 100), Switzerland (Swiss Market Index, SMI), Singapore (Straits Times Index, STI), HongKong (Hang Seng Index, HSI), and Korea (Korea Composite Stock Price Index, KOSPI), with monthly price data from Jan. 1990 to Dec. 2006. He/she wants to know his/her optimal portfolio allocation. The Markowitz portfolio selection problem can be divided into three parts. First, we need to calculate the efficient frontier. Second, we need to choose the optimal risky portfolio given one’s capital allocation line (find the point at the tangent of a CAL and the efficient frontier). Finally, using the optimal complete portfolio allocate funds between the risky portfolio and the risk-free asset.

175

age returns, standard deviations, and the correlation matrix10 for the rates of return on the stock index. After we input Table 10.2A into our spreadsheet as shown, we create the covariance matrix in Table 10.2B using the relationship Cov.ri ; rj / D ij i j . Before computing the efficient frontier, we need to prepare the data to establish a benchmark against which to evaluate our efficient portfolios, so we can form a bordermultiplied covariance matrix. We use the target mean of 15% for example. To compute the target portfolio’s mean and variance, these weights are entered in the border column A47:B52 and border row B46:G46. We calculate the variance of this portfolio in cell B54 in Table 10.2D. The entry in B54 equals the sum of all elements in the border-multiplied covariance matrix where each element is first multiplied by the portfolio weights given in both the row and column borders. We also include two cells to compute the standard deviation and expected return of the target portfolio (formulas in cells B55 and B56).11 To compute points along the efficient frontier we use the Excel Solver in Table 10.2D (which you can find in the Tools menu).12 Once you bring up Solver, you are asked to enter the cell of the target (objective) function. In our application, the target is the variance of the portfolio, given in cell B54. Solver will minimize this target. You must then input the cell range of the decision variables (in this case, the portfolio weights, contained in cells A47:A52). Finally, you enter all necessary constraints into the Solver. For an unrestricted efficient frontier that allows short sales, there are two constraints: first, that the sum of the weights is 1.0 (cell A53 D 1), and second, that the portfolio’s expected return equals the target return 15% (cell B56 D 15). Once you have entered the two constraints you ask the Solver to find the optimal portfolio weights. The Solver beeps when it has found a solution and automatically alters the portfolio weight cells in row 46 and column A to show the makeup of the efficient portfolio.

10

10.5.1.1 Step One: Finding Efficient Frontier First, we need to calculate expected return, standard deviation, and covariance matrix. The expected return and standard deviation can be calculated by applying the Excel STDEV and AVERAGE functions to the historic monthly percentage returns data.9 Table 10.2A and B show aver-

9

The expected return and standard deviation of each index needs to be annualized.

The correlation matrix is calculated in Excel using the data analysis function that is found under the Tool Menu. Note that if Data Analysis does not appear on the Tool Menu you will need to select Add-in and add to the Menu. 11 Because the expected returns and the portfolio weights are represented by column vectors (denoted e and w, respectively, with row vector transposes e T and wT ), and the variance-covariance terms by matrix V , then the expressions can be written as simple matrix formulas. So, the calculation of expected return and variance of portfolio can use the Excel array functions. Matrix notation Excel formula: D SUMPRODUCT.w; e/ Portfolio return: wT e Portfolio variance: wT Vw D MMULT.TRANSPOSE.w/; MMULT.V; w// 12 If Solver does not show up under the Tools menu, you should select Add-Ins and then select Analysis. This should add Solver to the list of options in the Tools menu.

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Table 10.2 Performance of six stock indexes A

B

C

D

E

F

G

1 A. Annualized S tandard Deviation , Average Return of S &P 500, FTS E 100, S MI, S TI, HIS , KOS PI 1990-2006 2 3 4 5 6 7 8 9

Asset Data S&P 500 FTSE100 SMI STI HSI KOSPI

Avreage Ret. 9.57 6.75 11.48 8.76 15.98 7.91

Std. Dev. 13.90 14.01 17.07 24.09 26.25 32.20

(%)

FTSE100 0.7431 1 0.7434 0.5570 0.5559 0.4302

SMI 0.6643 0.7434 1 0.4832 0.4666 0.3301

STI 0.5692 0.5570 0.4832 1 0.7518 0.4236

HSI 0.5561 0.5559 0.4666 0.7518 1 0.3731

FTSE100 144.8102 196.4143 177.8828 188.0728 204.5563 194.1278

SMI 157.7058 177.8828 291.5349 198.7550 209.1882 181.5081

STI 190.6630 188.0728 198.7550 580.4614 475.5806 328.6319

HSI 203.0139 204.5563 209.1882 475.5806 689.3243 315.4615

10 B. Correlation Matrix 11 12 13 14 15 16 17 18

S&P 500 1 0.7431 0.6643 0.5692 0.5561 0.3747

S&P 500 FTSE100 SMI STI HSI KOSPI

KOSPI 0.3747 0.4302 0.3301 0.4236 0.3731 1

19 C. Covariance Matrix 20 21 22 23 24 25 26 27

S&P 500 193.3182 144.8102 157.7058 190.6630 203.0139 167.7584

S&P 500 FTSE100 SMI STI HSI KOSPI

KOSPI 167.7584 194.1278 181.5081 328.6319 315.4615 1036.9482

28 D. Boardered Covariance Matrix for Efficient Frontier with mean of 15% 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

(allow short selling up to 100% and limit the investment to 150%) Cell Formulas S&P 500 FTSE100 SMI STI Weights A32 A33 A34 A35 0.78 A33*B32*B21 A33*C32*C21 A33*D32*D21 A33*E32*E21 -0.54 A34*B32*B22 A34*C32*C22 A34*D32*D22 A34*E32*E22 0.60 A35*B32*B23 A35*C32*C23 A35*D32*D23 A35*E32*E23 -0.34 A36*B32*B24 A36*C32*C24 A36*D32*D24 A36*E32*E24 0.41 A37*B32*B25 A37*C32*C25 A37*D32*D25 A37*E32*E25 0.08 A38*B32*B26 A38*C32*C26 A38*D32*D26 A38*E32*E26 SUM(A33:A38) SUM(B33:B38) SUM(C33:C38) SUM(D33:D38) SUM(E31:E36) Portfolio variance SUM(B39:G39) Portfolio SD B40^0.5 Portfolio mean A33*B4+A34*B5+A35*B6+A36*B7+A37*B8+A38*B9

HSI A36 A33*F32*F21 A34*F32*F22 A35*F32*F23 A36*F32*F24 A37*F32*F25 A38*F32*F26 SUM(F33:F38)

KOSPI A37 A33*G32*G21 A34*G32*G22 A35*G32*G23 A36*G32*G24 A37*G32*G25 A38*G32*G26 SUM(G33:G38)

HSI 0.41 65.71 -45.19 51.39 -66.89 117.20 10.90 122.63

KOSPI 0.08 11.03 -8.71 9.06 -9.39 10.90 7.28 20.25

Results Weights 0.78 -0.54 0.60 -0.34 0.41 0.08 1.00 Portfolio variance Portfolio SD

S&P 500 0.78 119.12 -60.90 73.75 -51.05 65.71 11.03 147.42 292.92 17.11

FTSE100 -0.54 -60.90 56.37 -56.77 34.37 -45.19 -8.71 -72.65

It adjusts the entries in the border-multiplied covariance matrix to reflect the multiplication by these new weights, and it shows the mean and variance of this optimal portfolio-the minimum variance portfolio with mean return of 15%. These results are shown in Table 10.2D, cells B54:C56. You can find that they yield an expected return of 15% with a standard deviation of 17.11% (results in cells B56 and B55). To generate the entire efficient frontier, keep changing the required mean in the constraint (in cell B56), letting the Solver

SMI 0.60 73.75 -56.77 103.47 -40.39 51.39 9.06 132.04

STI -0.34 -51.05 34.37 -40.39 67.54 -66.89 -9.39 -56.77

work for you. If you record a sufficient number of points, you will be able to generate a graph with the quality of Fig. 10.7. If short selling is not allowed, you can compute the portfolio on efficient frontier with the short sales constraint applying Solver. As to the example described above, we need to impose the additional constraints that each weight (the elements in column A and row 46) must be nonnegative. Once they are entered, you repeat the variance-minimization exercise until you generate the entire restricted frontier. The outer

10 Portfolio Optimization Models and Mean–Variance Spanning Tests

177

Table 10.2 (continued) A

B

C

D

E

F

G

H

I

58 E. The Unrestricted Efficient Frontier 59

Mean

Std. Dev.

S&P 500

FTSE100

SMI

ST I

HSI

KOSPI

60

6.000

12.999

0.459

0.699

-0.111

0.061

-0.188

0.080

7.730

12.710

0.522

0.461

0.025

-0.016

-0.072

0.081

62

9.000

12.867

0.568

0.287

0.125

-0.073

0.012

0.081

63

11.000

13.716

0.640

0.013

0.282

-0.162

0.146

0.082

64

13.000

15.185

0.713

-0.261

0.439

-0.252

0.279

0.083

65

15.000

17.115

0.785

-0.536

0.596

-0.341

0.412

0.084

66

17.000

19.369

0.857

-0.810

0.753

-0.430

0.546

0.085

19.784

22.976

0.812

-1.000

0.940

-0.635

0.812

0.071

68

22.000

26.499

0.661

-1.000

1.063

-0.860

1.087

0.050

69

24.000

30.075

0.453

-1.000

1.191

-1.000

1.335

0.020

61

67

MVP

ORP

70 71 F. The Restricted Efficient Frontier 72

Mean

Std. Dev.

S&P 500

FTSE100

SMI

ST I

HSI

KOSPI

73

6.745

14.050

0.0000

1.0000

0.0000

0.0000

0.0000

0.0000

7.000

13.525

0.0586

0.8649

0.0000

0.0000

0.0000

0.0765

8.281

12.829

0.4797

0.4237

0.0190

0.0000

0.0000

0.0776

76

10.000

13.350

0.5922

0.0522

0.2402

0.0000

0.0384

0.0769

77

11.000

14.096

0.4743

0.0000

0.3262

0.0000

0.1411

0.0584

74 75

78

MVP

12.897

16.998

0.1292

0.0000

0.4839

0.0000

0.3775

0.0095

79

ORP

13.000

17.199

0.1104

0.0000

0.4924

0.0000

0.3904

0.0068

80

14.000

19.414

0.0000

0.0000

0.4396

0.0000

0.5604

0.0000

81

15.000

22.540

0.0000

0.0000

0.2176

0.0000

0.7824

0.0000

82

U=E (rp ) − 0.005As

83 Utility Function 84 Risk Aversion Index (A)

2 p

5

85

⇒ (1−y) rf + y* E ( rp )

86 G.Optimal Complete Risky Portfolio(Unrestricted Portfolio) Cell Formulas

87 88

Risk-free Asset

89 Return

SUMPRODUCT (B96:C96,B98:C98)

B69

90 Risk

0.00

C69

91 optimal position

1-C98

(C96-B96)/(0.01*B88*C97^2)

92

Optimal Overall Portfolio

Optimal Risky Portfolio

4.01

C98*C97

⇒ y *s p

(1-y,y)

93 95

Results

96

Risk-free Asset

Optimal Risky PortfolioOptimal Overall Portfolio

97 Return

4.01

19.8

13.44

98 Risk

0.00

23.0

13.73

99 optimal position

0.40

0.60

100

(1-y,y)

101 102 H. Optimal Complete Risky Portfolio (Restricted Portfolio) 103

Risk-free Asset

Optimal Risky Portfolio Optimal Overall Portfolio

104 Return

4.01

12.90

9.48

105 Risk

0.00

17.00

10.46

0.38

0.62

106 optimal position 107

(1-y,y)

108

frontier in Fig. 10.7 is drawn assuming that the investor may maintain negative portfolio weights, and the inside frontier obtained allowing short sales. Table 10.2E and F present a number of points on the two frontiers with and without short sales. You can see that the weights in restricted portfolios are never negative. The minimum variance portfolios in two frontiers are not the same. Before we move on, let us summarize the steps of using Solver to calculate the variance-minimization portfolio.

The steps with Solver are as follows: 1. Invoke Solver by choosing Tools then Options then Solver. 2. Specify in the Solver parameter Dialog Box: The Target cell to be optimized, specify max or min. 3. Choose Add to specify the constrains then OK. 4. Solve and get the results in the spreadsheet.

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Fig. 10.7 Efficient frontier of unrestricted and restricted portfolio Portfolio Return (%)

Portfolio Efficient Frontier

30 25 20 15 10 5 0

0

5

10

15

20

25

30

35

Portfolio Risk (%) Restricted

10.5.1.2 Step Two: Finding Optimal Risky Portfolio Now that we have the efficient frontier, we proceed to step two. In order to get the optimal risky portfolio, we should find the portfolio on the tangency point of capital allocation and efficient frontier. To do so, we can use the Solver to help us. First, you enter the of the target function, “maximum” the E.r /r reward-to-variability ratio ( pp f , we assume risk-free rate is 4.01%13 ) the slope of the CAL, input the cell range (the portfolio weights, contained in cells A47:A52), and other necessary constraints (such like the sum of the weights equal to one and others). Then ask the Solver to find the optimal portfolio weights. The results are shown in Table 10.2E and F. The optimal risky portfolio with short selling allowance has an expected return of 19.784% (in cell B67) with a standard deviation of 22.976% (in cell C67). The expected return and standard deviation of the restricted optimal risky portfolio are 12.897% (in cell B78) and 16.998% (in cell C78).

10.5.1.3 Step Three: Capital Allocation Decision A person’s allocation decision will be influenced by his or her degree of risk aversion. Now we have the optimal risky portfolio, and then we can use the concept of complete portfolio allocation funds between the risky portfolio and the risk-free asset. We use Equation (10.11) as our utility function and set the risk aversion equal to 5; the risk-free rate is 4.1%. First, we construct a complete portfolio with the risk-free asset and the optimal risky portfolio.

13

We use 3-month Treasury Bill interest rate as the risk-free rate; the average interest rate of a 3-month T-Bill is 4.09% from 1990/01 to 2006/12.

Unrestricted

S&P 500

FTSE100

SMI

STI

HSI

KOSPI

According to Equation (10.14), the optimal weight in E.rp /rf the risky portfolio is 0:01A and the optimal position 2 p

E.r /r

p f of the risk-free asset is 1  0:01A 2 . Then we can use p Equations (10.5) and (10.6) to calculate the expected return and standard deviation of the overall optimal portfolio. The results are shown in Table 10.2G and H. The optimal unrestricted (restricted) portfolio has a 13.44% (9.48%) expected return with 13.73% (10.46%) standard deviation. And the investor will invest 60% (62%) of portfolio value in risky portfolio and 40% (38%) in risk-free asset. To sum up the three steps, all the results are shown in Table 10.3.

10.5.2 Application: MATLAB By using the Solver, you can repeat the varianceminimization exercise until you generate the entire efficient frontier, just with a push of a button. The computation of the efficient frontier is fairly easy in Microsoft Excel. However, Excel is limited in the number of assets it can handle. If our portfolio consists of hundreds of asset, using Excel to deal with the Markowitz optimal problem will become more complicated. MATLAB can handle this problem. The Financial Toolbox within MATLAB provides a complete integrated computing environment for financial analysis and engineering. The toolbox has everything you need to perform mathematical and statistical analysis of financial data and display the results with presentation-quality graphics. You do not need to switch tools, convert files, or rewrite applications. Just write expressions the way you think of problems, MATLAB will do all that for you. You can quickly ask, visualize, and answer complicated questions. Financial Toolbox includes a set of portfolio optimization functions designed to find the portfolio that best meets investor requirements. Recall the steps when using Excel in

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Table 10.3 The results of optimization problem Unrestricted portfolio

Restricted portfolio

Minimum variance portfolio

Optimal risky portfolio

Optimal Overall portfolio

Minimum variance portfolio

Optimal risky portfolio

Optimal Overall portfolio

Portfolio return

7.73%

19.78%

13.44%

8.28%

12.89%

9.48%

Portfolio risk

12.71%

22.98%

13.73%

12.83%

16.99%

10.46%

Portfolio

previous section. In the portfolio optimization problem, we need to prepare several of the data for the computation preparation. First, we need the expected return, standard deviation, and covariance matrix. Then, we can calculate the optimal risky portfolio weights and know how allocation will be more efficient. Before a taking real portfolio selection example by using MATLAB, we introduce the several portfolio optimization functions that will be use in our portfolio selection analysis.

10.5.2.1 Mean–Variance Efficient Frontier Calling function frontcon returns the mean–variance efficient frontier for a given group of assets with user-specified asset constraints, covariance, and returns. The computation is based on sets of constraints representing the maximum and minimum weights for each asset and the maximum and minimum total weight for specified groups of assets. The efficient frontier computation functions require information about each asset in the portfolio. This data is entered into the function via two matrices: an expected return vector and a covariance matrix. Calling frontcon while specifying the output arguments returns the corresponding vectors and arrays representing the risk, return, and weights for each of the 10 points computed along the efficient frontier. As there are no constraints, you can call frontcon directly with the data you already have. If you call frontcon without specifying any output arguments, you get a graph representing the efficient frontier curve.

is a vector of the expected return of each portfolio, and PortWts is a matrix of weights allocated to each asset.14 The output data is represented row-wise, where each portfolio’s risk, rate of return, and associated weight is identified as corresponding rows in the vectors and matrix. ExpReturn and ExpCovariance are frontcon required information. NumPorts, PortReturn, AssetBounds, and GroupBounds all are optional information. ExpReturn specifies the expected return of each asset; ExpCovariance specifies the covariance of the asset returns. In NumPorts, you can define the number of portfolios you want to be generated along the efficient frontier. If NumPorts is empty (entered as []), frontcon computes 10 equally spaced points.15 PortReturn is vector of length equal to the number of portfolios containing the target return values on the frontier. If PortReturn is not entered or [], NumPorts equally spaced returns between the minimum and maximum possible values are used. AssetBounds is a matrix containing the lower and upper bounds on the weight allocated to each asset in the portfolio.16 Groups is number of groups matrix specifying asset groups or classes.17 GroupBounds matrix allows you to specify an upper and lower bound for each group. Each row in this matrix represents a group. The first column represents the minimum allocation, and the second column represents the maximum allocation to each group. Notice, the arguments in the parentheses are the input data and the items in the square bracket are the output returns of function. If you do not want the results to show up in output window, you may use the semicolon to suppress output from the evaluation of function. However, if you want to get all

Function: frontcon Syntax [PortRisk, PortReturn, PortWts] D frontcon(ExpReturn, ExpCovariance, NumPorts, PortReturn, AssetBounds, Groups, GroupBounds) PortRisk, PortReturn, and PortWts are the returns of frontcon, where PortRisk is a vector of the standard deviation of each portfolio, PortReturn

14

The total of all weights in a portfolio is 1. When entering a target rate of return (PortReturn), enter NumPorts as an empty matrix []. 16 The Default of lower asset bound is all 0s (no short-selling); default upper asset bound is all 1s (any asset may constitute the entire portfolio). 17 Groups.i; j/ D 1 (jth asset belongs in the ith group). Groups.i; j/ D 0 (jth asset not a member of the ith group). 15

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output results from the evaluation of function, you must add square brackets with output arguments when in expression feedback of functions in writing syntax. The names of input and output arguments can be user-specified. It is not necessary to be named as our example shows. You can name what you want, as long as the concept of data matches the function arguments required.

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tions. The number of columns of the matrix A, and the length of the vector Wts correspond to the number of assets. The number of rows of the matrix A, and the length of vector b correspond to the number of constraints. This method allows you to specify any number of linear inequalities to the function portopt.18 Function: portcons Syntax ConSet D portcons(varargin19)

10.5.2.2 Portfolios on Constrained Efficient Frontier If you want set some linear constraints when computing efficient frontier, you can call portopt function. The portopt computes portfolios along the efficient frontier for a given group of assets, based on a set of user-specified linear constraints. Typically, these constraints are generated using the constraint specification functions that we will describe in next section. Function: portopt

ConSet D portcons(‘ConstType’, Data1, : : :, DataN) creates a matrix ConSet, based on the constraint type ConstType, and the constraint parameters Data1, : : :, DataN. ConSet D portcons(‘ConstType1’, Data11, : : :, Data1N, ‘ConstType2’, Data21, : : :, Data2N, : : :) creates a matrix ConSet, based on the constraint types ConstTypeN, and the corresponding constraint parameters DataN1, : : :, DataNN.

Syntax [PortRisk, PortReturn, PortWts] D portopt(ExpReturn, ExpCovariance, NumPorts, PortReturn, ConSet)

10.5.2.4 Optimal Capital Allocation to Efficient Frontier Portfolios

Since the portopt function is able to cope with more general constraints than the frontcon function, the portopt function can be viewed as an advanced version of the frontcon function. The main difference between these two is ConSet; others are the same as described in frontcon function. ConSet, optional information, is a constraint matrix for a portfolio of asset investments, created using portcons. If not specified, a default is created. The syntax expressed above will return the mean–variance efficient frontier with user-specified covariance, returns, and asset constraints (ConSet). If portopt is invoked without output arguments, it returns a plot of the efficient frontier (Table 10.4).

A last function we describe here may be used to find an optimal portfolio. So far, we have deal with efficient portfolios, leaving the risk/return trade-off unresolved. We may resolve this tradeoff by linking mean–variance portfolio theory to the more general utility theory. Function portalloc computes the optimal risky portfolio on the efficient frontier, based on the risk-free rate, the borrowing rate, and the investor’s degree of risk aversion. It also generates the capital allocation line, which provides the optimal allocation of funds between the risky portfolio and the risk-free asset. In the Financial toolbox the function portalloc is provided, which yields the optimal portfolio assuming the quadratic utility function: U D E.rp /  0:005Ap2 , where the parameter A is linked to risk aversion; its default value is 3.

10.5.2.3 Portfolio Constraints While frontcon allows you to enter a fixed set of constraints related to minimum and maximum values for groups and individual assets, you often need to specify a larger and more general set of constraints when finding the optimal risky portfolio. The function portopt addresses this need, by accepting an arbitrary set of constraints as an input matrix. The auxiliary function portcons can be used to create the matrix of constraints, with each row representing an inequality. These inequalities are of the type A Wts0 b, where A is a matrix, b is a vector, and Wts is a row vector of asset alloca-

18

In reality, portcons is an entry point to a set of functions that generate matrices for specific types of constraints; portcons allows you to specify all the constraints data at once, while the specific portfolio constraint functions allow you to build the constraints incrementally. These constraint functions are pcpval, pcalims, pcglims, and pcgcomp. 19 varargin is variable length input argument list. The varargin statement is used only inside a function M-file to contain optional input arguments passed to the function. The varargin argument must be declared as the last input argument to a function, collecting all the inputs from that point onwards. In the declaration, varargin must be lowercase.

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Table 10.4 The detail information of portcons function (Source: MATLAB Financial toolbox) Constraint type Description Value Default

All allocations are 0; no short selling allowed. Combined value of portfolio allocations normalized to 1

NumAssets (required). Scalar representing number of assets in portfolio.

PortValue

Fix total value of portfolio to PVal.

PVal (required). Scalar representing total value of portfolio. NumAssets (required). Scalar representing number of assets in portfolio.

AssetLims

Minimum and maximum allocation per asset.

AssetMin (required). Scalar or vector of length NASSETS, specifying minimum allocation per asset. AssetMax (required). Scalar or vector of length NASSETS, specifying maximum allocation per asset. NumAssets (optional).

GroupLims

Minimum and maximum allocations to asset group.

Groups (required). NGROUPS-by-NASSETS matrix specifying which assets belong to each group. GroupMin (required). Scalar or a vector of length NGROUPS, specifying minimum combined allocations in each group. GroupMax (required). Scalar or a vector of length NGROUPS, specifying maximum combined allocations in each group.

GroupComparison

Group-to-group comparison constraints.

GroupA (required). NGROUPS-by-NASSETS matrix specifying first group in the comparison. AtoBmin (required). Scalar or vector of length NGROUPS specifying minimum ratios of allocations in GroupA to allocations in GroupB. AtoBmax (required). Scalar or vector of length NGROUPS specifying maximum ratios of allocations in GroupA to allocations in GroupB. GroupB (required). NGROUPS-by-NASSETS matrix specifying second group in the comparison.

Custom

Custom linear inequality constraints A PortWts’b.

Function: portalloc Syntax [RiskyRisk, RiskyReturn, RiskyWts, RiskyFraction, OverallRisk, OverallReturn] D portalloc(PortRisk, PortReturn, PortWts, RisklessRate, BorrowRate, RiskAversion) RiskyRisk is the standard deviation of the optimal risky portfolio; RiskyReturn is the expected return of the optimal risky portfolio; RiskyWts is a vector of weights allocated to the optimal risky portfolio; RiskyFraction is the fraction of the complete portfolio allocated to the risky portfolio; OverallRisk is the standard deviation of the

A (required). NCONSTRAINTS-by-NASSETS matrix, specifying weights for each asset in each inequality equation b (required). Vector of length NCONSTRAINTS specifying the right hand sides of the inequalities.

optimal overall portfolio; OverallReturn is the expected rate of return of the optimal overall portfolio. PortRisk is standard deviation of each risky asset efficient frontier portfolio; PortReturn is expected return of each risky asset efficient frontier portfolio; PortWts is weights allocated to each asset. RisklessRate is risk-free lending rate (for investing), a decimal number; BorrowRate is borrowing rate, which is larger than the riskless rate – a decimal number.20 RiskAversion is coefficient of investor’s degree of risk aversion (Default D 3). BorrowRate and RiskAversion are optional. As with frontcon, calling portopt while specifying output arguments returns the corresponding vectors and ar20

If borrowing is not desired, or not an option, set to NaN (default).

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rays representing the risk, return, and weights for each of the portfolios along the efficient frontier. Use them as the first three input arguments to the function portalloc. Calling portalloc while specifying the output arguments returns the variance (RiskyRisk), the expected return (RiskyReturn), and the weights (RiskyWts) allocated to the optimal risky portfolio. It also returns the fraction (RiskyFraction) of the complete portfolio allocated to the risky portfolio and the standard deviation (OverallRisk) and expected return (OverallReturn) of the optimal overall portfolio consisting of the risky portfolio and the risk-free asset. The overall portfolio combines investments in the risk-free asset and in the risky portfolio. The actual proportion assigned to each of these two investments is determined by the degree of risk aversion characterizing the investor. The value of RiskyFraction exceeds 1 (100%), implying that the risk tolerance specified allows borrowing money to invest in the risky portfolio, and that no money will be invested in the risk-free asset. This borrowed capital is added to the original capital available for investment. Calling portalloc without specifying any output arguments gives a graph displaying the critical points.

to create the matrix of constraints for short selling allowance and call portfopt compute constrained efficient frontier. Finally, use PortRisk, PortRet, PortWts input arguments to the function portalloc and find the optimal risky portfolio and the optimal allocation of funds between the risky portfolio and the risk-free asset. In this example the resulting optimal portfolio is formed with 88.92% in the risky portfolio and 11.08% in the risk-free asset. The optimal overall portfolio return is 17.03% with 20.76% standard deviation. The expected return of six-stockindex portfolio is 18.64%, and the portfolio risk is 23.35%. The best allocation proportion of each asset for the investor is to allocate 85.84% portfolio value in S&P 500 Index, 96.39% in Swiss Market Index (SMI), and 81.39% in Hang Seng Index (HSI), and short selling 100% portfolio value in FTSE 100 Index, 61.82% in Straits Times Index (STI), and 1.8% in Korea Composite Stock Price Index (KOPSI) (Figs. 10.8–10.10). For further information about computer program application of modern portfolio theory, please refer to Benninga (2008), Jackson and Staunton (2001), and Brandimarte (2006).

10.5.2.5 Six-Stock-Index Portfolio

10.6 Conclusion Following the previous example, assume that an investor forms a six-stock-index portfolio that is his/her optimal portfolio allocation. Assume that the risk free investment return is 4.1%, investor’s degree of risk aversion is 3, and the borrowing rate is 7%. We also allow short selling up to 100% of the portfolio value in each stock index, but limit the investment in any each stock index to 150% of the portfolio value. Now we use MATLAB to assist him/her. First, we calculate rate of return of the natural logarithm stock index price. Second, we use mean and cov functions to calculate the expected return of each stock index and covariance matrix of portfolio. Third, call portcons

In this chapter, we introduced the theory and the application of computer program of modern portfolio theory. In the first part, we illustrated the Markowitz portfolio model to show that investors could lower the total risk of investing without sacrificing return by investing in portfolios rather than in individual assets. The Markowitz portfolio selection problem involves three topics: (1) finding the efficient frontier, (2) constructing the capital asset line, and (3) determining the optimal portfolio. We also demonstrated the methods applied in these topics by using Microsoft Excel and MATLAB. In the second part, we introduced the mean–variance span-

clear all; clc; tic %portfolio.m load stock_index.txt; Portfolio = stock_index (:,1:6); [m n] = size(Portfolio); [AvrRet]= mean(Portfolio); [ExpRetVec]= AvrRet*12;

Fig. 10.8 MATLAB code for asset allocation

[CovMtx]= cov(Portfolio)*12;

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Fig. 10.8 (continued) % COMPUTE EFFICIENT FRONTIER (SHORT SELLING ALLOWED) NumPorts = n; AssetMin = [ -1 -1 -1 -1 -1 -1]; AssetMax = [ 1.5 1.5 1.5 1.5 1.5 1.5]; pval = 1; Constraint = portcons('PortValue',pval, NumPorts, 'AssetLims', AssetMin, AssetMax, NumPorts); [PortRisk,PortRet,PortWts] = portopt(ExpRetVec, CovMtx, 50, [], Constraint); [OptRisk,OptRet,OptWts]=portopt(ExpRetVec, CovMtx, NumPorts, [], Constraint); % OPTIMAL ASSET ALLOCATION RisklessRate = 0.041; BorrowRate = 0.07; RiskAversion = 3; [RiskyRisk, RiskyReturn, RiskyWts, RiskFraction, OverallRisk, OverallReturn] = portalloc (PortRisk, PortRet, PortWts, RisklessRate, BorrowRate, RiskAversion)

% PLOT THE RESULT

plot(PortRisk, PortRet, 'm-', ... 0, RisklessRate, 'k:square', RiskyRisk, RiskyReturn, 'k:diamond',... OverallRisk, OverallReturn, 'o',... [0; RiskyRisk], [RisklessRate; RiskyReturn],'r--') xlabel('Portfolio Risk'); ylabel('Portfolio Return '); title('Efficient Frontier'); legend('Efficient Frontier', 'Risk free asset', 'Risky Portfolio', 'Asset Allocation Point') grid on

RiskyRisk = 0.2335 RiskyReturn = 0.1864 RiskyWts = 0.8584 -1.0000 RiskFraction = 0.8892 OverallRisk = 0.2076 OverallReturn = 0.1703

0.9639

Fig. 10.9 MATLAB output of optimal allocation problem

-0.6182

0.8139

-0.0180

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Fig. 10.10 Optimal asset allocation

ning test that follows directly from the portfolio optimization problem. The mean–variance spanning tests enable us to analyze the effect on the mean–variance frontier of adding new assets to a set of benchmark assets. In conclusion, this chapter provides us with a complete overview and some techniques on portfolio analyses.

References Benninga, S. 2008. Financial modeling, 3rd ed., MIT, MA. Bodie, Z., A. Kane, and A. J. Marcus. 2007. Essentials of investment, 6th ed., McGraw-Hill/Inwin, New York. Bodie, Z., A. Kane, and A. J. Marcus. 2008. Investment, 7th ed., McGraw-Hill/Inwin, New York. Brandimarte, P. 2006. Numerical methods in finance and economics: a MATLAB-based introduction, 2nd ed., Wiley, New York.

DeRoon, F. A. and T. E. Nijman. 2001. “Testing for mean-variance spanning: a survey.” Journal of Empirical Finance 8, 111–155. Financial Toolbox 3. 2009. User’s Guide. The MathWorks, Inc. Huberman, G. and S. Kandel. 1987. “Mean-variance spanning.” Journal of Finance 42, 873–888. Jackson, M. and M. Staunton. 2001. Advanced modeling in finance using excel and VBA, Wiley, New York. Jobson, J. D. and B. Korkie. 1989. “A performance interpretation of multivariate tests of asset set intersection, spanning and meanvariance efficiency.” Journal of Financial and Quantitative Analysis 24, 185–204. Kan, R., and G. Zhou. 2008. “Tests of mean-variance spanning.” Working Paper, University of Toronto and Washington University. Lee, C. F., J. E. Finnerty, and D. H. Wort. 1990. Security analysis and portfolio management. Scott, Foresman/Little Brown, Glenview, IL. Markowitz, H. M. 1952. “Portfolio selection.” Journal of Finance 7, 77–91. Markowitz, H. M. 1959. Portfolio selection: efficient diversification of investments, Wiley, New York.

Chapter 11

Combining Fundamental Measures for Stock Selection Kenton K. Yee

Abstract Securities selection is the attempt to distinguish prospective winners from losers – conditional on beliefs and available information. This article surveys some relevant academic research on the subject, including work about the combining of forecasts (Operational Research Quarterly 20, 451–468, 1969), the Black-Litterman model (Journal of Fixed Income 1(2), 7–18, 1991; Financial Analysts Journal (September/October) 28–43, 1992), the combining of Bayesian priors and regression estimates (Journal of Finance 55(1), 179–223, 2000), model uncertainty and Bayesian model averaging (Statistical Science 14(4), 382–417, 1999; Review of Financial Studies 15(4), 1223–1249, 2002), the theory of competitive storage (Review of Economic Studies 59, 1–23, 1992), and the combination of valuation estimates (Review of Accounting Studies 12(2–3), 227–256, 2007). Despite its wide-ranging applicability, the Bayesian approach is not a license for data snooping. The second half of this article describes common pitfalls in fundamental analysis and comments on the role of theoretical guidance in mitigating these pitfalls. Keywords Value investing r Fundamental analysis r Noisy markets r Data snooping r Theory of storage

11.1 Introduction Even if one believes the market is efficient in the intermediate or long run, there is no reason to believe that instant market quotes reflect long-run value in the short run (Black 1986). On the contrary, market microstructure, liquidity, and behavioral phenomena suggest that at any given time prices may be noisy and returns skewed. Accordingly, investors, bankers, insiders, analysts, tax authorities, judges, and other capital market participants are constantly pressed to assess

K.K. Yee () Columbia Business School, New York, NY, USA e-mail: [email protected]

and reassess value not only to “beat the market” but also to set fair prices for contentious transactions. A large industry of institutional “tactical” asset managers exist to exploit temporary market mispricing and to push the limits of arbitrage. Obtaining value estimates or “scores” is the starting point for stock selection (Treynor and Black 1973; Klarman 1991; Penman 2001; Yee 2008b), value-based indexing in a noisy market (Chan and Lakonishok 2004; Arnott et al. 2005; Siegel 2006; Perold 2007), and beating benchmark indices through tactical asset allocation (Black and Litterman 1991, 1992; Lo 2007). Obtaining accurate value estimates is also useful at the firm level. Boards of directors and shareholders determine what prices to tender or accept for mergers and acquisitions (Yee 2005b). Tax authorities and other regulators must appraise estates, assess the fairness of transfer prices, and audit the value of employee stock option plans (ESOPs). When irreconcilable disputes arise, their assessments are adjudicated in court by a judge, who chooses a terminal appraisal after weighing conflicting testimony from adversarial experts. In principle, valuing a company is not rocket science. According to asset pricing theory, value is the sum of expected cash outflows suitably discounted. One way to estimate equity value is the discounted cash flow (DCF) method. DCF is implemented by forecasting expected cash inflows from operations, netting them against forecasted payments to creditors, and then discounting. The discount factor, which depends on interest rates, the equity risk premium, leverage, time, and possibly other factors, is estimated based on models such as CAPM or APT (Kaplan and Ruback 1995; Ang and Liu 2004). One may also transform the DCF formula into representations that take advantage of accounting information to predict cash flows and, accordingly, represent value (Feltham and Ohlson 1999; Yee 2005a; Ohlson and Gao 2006). In a certain world with perfect markets, DCF is cut and dry. But owing to uncertainty in prospective cash flows (which may be exacerbated by limited available information) and inherent shortcomings of CAPM and APT (which require estimating beta and forecasting values for the timevarying equity risk premium), DCF analysis inevitably leads

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to an imprecise answer. Indeed, DCF analysis – where subjectively shifting numbers around can lead to radically different answers – gives a new twist on the old adage that there are three types of lies: lies, damned lies, and equity valuation. In the face of such uncertainty and subjectivity, estimated value – like beauty – is in the eyes of the beholder even without further complications caused by heterogeneous beliefs or information asymmetry. A valuation result depends on the procedure one uses to obtain it. Along with DCF, the most popular procedures include asset-based valuation, the method of comparables, and ratio analysis. The method of comparables entails identifying a sample of peer companies and an accounting benchmark such as earnings, revenues, or forward earnings. Assuming the price-to-benchmark ratios of all peer companies are roughly equal, one infers the value of a company from the average or median price-tobenchmark ratio of its peers. Although the method of comparables is arguably the most popular valuation method, it is plagued with subjectivity in both the determination of pertinent comparable measures and the identification of a firm’s true peers (Bhojraj and Lee 2003). In practice, the outcome can be manipulated by including or excluding standouts or outliers from a candidate peer group. Another problem arises when modestly different procedures yield disparate answers when applied to the same company at the same time. No single procedure is conclusively most precise and accurate in all situations. Since no method universally dominates the others, it is not surprising that financial analysts frequently estimate DCF as well as method of comparable values for a more complete analysis. Accordingly, popular valuation textbooks – for example, Penman (2001) and Damodaran (2002) – promulgate multiple techniques and view them as being simultaneously and contextually informative, with no dominating method. When available, market value serves as yet another estimate of value. How should an analyst combine two or more noisy valuation estimates into a superior valuation estimate that makes optimal use of the given information? Although there is tremendous practitioner interest this question, the academic literature offers surprisingly little guidance on how to combine estimates (Yee 2004a, 2008a). When analysts do combine estimates, their techniques are usually ad hoc since there is no theoretical guiding framework. A common practice is to form a linear weighted average of the different valuations.1

Indeed, weighted average valuations have become industry practice in financial legal proceedings (Sharfman 2003) and IRS tax-related valuation disputes (Beatty et al. 1999). The Delaware Block Method is an important case in point. In the years following the Great Depression, the Delaware Court of Chancery, the leading corporate law jurisdiction in the United States, evolved a valuation procedure that, by the late 1970s, became known as the Delaware Block Method. Virtually all American courts employed the Delaware Block Method for appraising equity value in the 1970s and early 1980s. In Delaware, the Block Method was the legally prescribed valuation procedure for shareholder litigation up until 1983. In 1983, the Delaware Supreme Court updated the law to allow judges to consider “modern” methods like discounted cash flow. Subsequently, the Delaware courts currently use the Block Method, discounted cash flow, and hybrids of these methods (Yee 2002; Chen et al. 2007 and 2010). The Delaware Block Method is most easily characterized in algebraic terms, even though judges do not write down such explicit formulas in their legal options. The Block Method relies on three measures to estimate the fair value of equity:

1

2

The author’s equity-valuation students enrolled in an MBA elective at a leading business school choose to take weighted averages of their valuation estimates even though nothing in the class materials advises them to combine estimates. When asked why they do this, students report they learned to do this at work, other classes, or that weighing makes common sense. The students do not articulate how they determine their weights, and the weights vary greatly between students.

contemporaneous market price P net asset value2 of equity b 5-year trailing earnings average e capitalized by ®, a num-

ber estimated from price-to-earnings ratios of comparable firms. Then, the fair value V of equity at date t is estimated as the weighted average V D .1  b  e / P C b b  e ' e:

(11.1)

The weights .1  ›b  ›e /; ›b , and ›e – numbers between 0 and 1 – are adjudicated on a case-by-case basis. Earnings capitalization factor ¥ is also determined on a case-by-case basis and typically ranges between 5 and 15. Even if one accepts combining estimates by averaging them, as in the Delaware Block Method, how should the relative weights on P, b, and e be determined? Should more weight be given to market value than net asset value? Should net asset value receive more attention than earnings value? The simplest choice might be to take an equally weighted average, but even this parsimonious rule is ad hoc and unjustified. Net asset value is the liquidation value of assets minus the fair or settlement value of liabilities. Net asset value is not the same as accounting “book” value. Book value, because it derives from historical cost accounting, arcane accounting depreciation formulas, and accounting rules that forbid the recognition of valuable intangible assets, is known to be an unreliable (typically conservatively biased) estimate of net asset value.

11 Combining Fundamental Measures for Stock Selection

On this issue, the IRS valuation rules, honed by decades of spirited high stakes litigation and expensive expert testimony, do not lend insight. Revenue Ruling3 59–60 advises that “where market quotations are either lacking or too scarce to be recognized,” set 1  b  e D 0 and focus on earnings and dividend-paying capacity. For the comparables method, use the “market price of corporations engaged in the same or similar line of business.” Upon recognizing that “determination of the proper capitalization rate presents one of the most difficult problems,” the Ruling advises that “depending upon the circumstances in each case, certain factors may carry more weight than others: : :valuations cannot be made on the basis of a prescribed formula.” Intuition provides a bit more guidance. The intuitively sensible rule is to weigh the more credible estimates more and the less credible estimates less. For example, if a security is thinly traded, illiquidity may cause temporary price anomalies.4 Likewise, “animal spirits” may cause the price of hotly traded issues to swing unexplainably. In either extreme, price quotations are unreliable indicators of fundamental value and common sense says to weigh price quotations less in such cases. On the other hand, price should be given more weight when it is believed that market price is more efficient and has not been subject to animal spirits. Yee (2008a) proposes Bayesian “triangulation” as the appropriate conceptual framework for combining value estimates. Section 11.2 describes the Bayesian triangulation formula, which prescribes how to combine different valuation estimates and price into a superior valuation estimate. Section 11.3 illustrates how Bayesian triangulation can be used to back out how noisy Delaware shareholder litigation judges believe share prices are. Bayesian methods provide a coherent framework for integrating views of analysts and investors into empirical statistical models. Model conclusions blend the information content of observed data with subjective information input. A prominent example is the Black-Litterman (1991, 1992) model, which is reviewed in Sect. 11.4. Section 11.4 also describes other applications of the Bayesian framework in asset pricing research, including Pastor (2000), Cremers (2002), and the Bayesian combination of forecasts. Sections. 11.5–11.7 turn to the dangers of misusing the Bayesian framework as an excuse to justify data snooping. Section 11.5 recalls how data snooping misleads researchers 3 The Internal Revenue Code addresses the valuation of closely held securities in Section 2031(b). The standard of value is “fair market value,” the price at which willing buyers or sellers with reasonable knowledge of the facts are willing to transact. Revenue Ruling 59–60 (1959–1 C.B. 237) sets forth the IRS’s interpretation of IRC Section 2031(b). 4 For example, a large block trade of micro-cap shares may cause temporary price volatility. Non-synchronous trading may cause apparent excessive price stability, as seen in emerging equity markets.

187

into drawing false conclusions even within a seemingly rigorous Bayesian context. Using an example from commodities pricing, Sect. 11.6 illustrates how micro-economic theory can be used to lend credibility to empirical results. Section 11.7 describes data snooping pitfalls in financial statement analysis and deep-dive fundamental research. Section 11.8 concludes.

11.2 Bayesian Triangulation This section prescribes how to fold two different valuation estimates together with the market price to achieve a more precise valuation estimate. First, it introduces Theorem 11.1 and Equation (11.2), the formula prescribing how much to weigh the individual estimates for maximum accuracy of the triangulated valuation estimate. Then, it describes how to implement Equation (11.2) and give a recipe for estimating input parameters. To state Theorem 11.1, it is necessary to introduce notation. Assume the analyst has three distinct valuation estimates: market price P, an intrinsic valuation estimate VI , and a method of comparables estimate VC . In particular, VI might be a DCF or net asset valuation estimate, while VC might be an estimate based on peer-group forward price-to-earnings ratios or trailing price-to-book ratios. While raw valuation estimates are biased in practice (e.g., Lie and Lie 2002), assume that VI and VC are pre-processed to remove anticipated bias. (The theorem does not address the issue of how to remove systematic bias, whose removal is outside the scope of this article.) Let V denote what the analyst wants to estimate. While it is common to equate V with intrinsic value or the present value of expected dividends, one wants to avoid this trap since intrinsic value is not an observable. Rather, let us just say that V is the value one wants to estimate in a given contextual situation. From an active portfolio manager’s perspective, V represents the price target or “score” that determines a security’s “alpha” in the extended CAPM framework of Treynor and Black (1973; see also Grinold 1989). From a tax appraiser’s perspective, V represents the valuation that is most “fair.” From an empirical researcher’s perspective (see Sect. 11.4), V might identified as the future realization of market price at the investment horizon (say, next year’s price suitably discounted back to the current date). Then, from this perspective, the question is well defined: How does one use Bayesian triangulation to make the most accurate forecast of market price (or alpha) at the investment horizon? Theorem 11.1 states how to estimate V given inputs fP; VI ; VC ; ¢; ¢I ; ¢C ; ¡g where:

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¢ is the standard error of market price P, which is a noisy

measure of fundamental  value V. Specifically, V D P C e  where5 e  N 0;  2 . ¢I is the standard error of intrinsic valuation estimate VI , also a noisy of V. Assume VI D V C eI where   measure eI  N 0; I2 is uncorrelated with e. ¢C is the standard error of method of comparables estimate VC , also a noisy measure of V. In particular, VC D V C  eC where eC  N 0; C2 is uncorrelated with eI but is correlated with market noise so that corr .e; eC /   > 0. The amount e by which market price P differs from V is noise. Since market noise is unobservable, suppose that e is an unobservable mean-zero random error. Error e reflects Fischer Black’s (1986) assertion that “because there is so much noise in the world, certain things are essentially unobservable.” VI may represent either a DCF or liquidation valuation estimate of V. Assume its estimation error eI , is uncorrelated6 with market noise e. In contrast, because method of comparable estimates relies directly on market price, they are affected by temporary market movements. Hence, assume that eC , the method of comparables estimation error, is positively correlated with market noise e. The key difference between VI and VC is that corr .e; eI /  0 while corr .e; eC /   > 0. The central result is (Yee 2008a): Theorem 11.1. The most precise estimate of value V an analyst constructs from a linear combination of P, VI , and VC is _

V D .1  I  C /P C I VI C C VC ;

(11.2)

where kI D

.12 / 2 C2 .12 / 2 C2 C . 2 C C2 C 2C /I2

kC D

.CC /I2 .12 / 2 C2 C . 2 C C2 C

.1  I  C / D

2C /I2

.C C/I2 C : .12 / 2 C2 C . 2 C C2 C 2C /I2

Proof. See Appendix 11A.   Y N m;  2 means Y is normally distributed with mean m and standard deviation £. Note that we leave the possibility that the standard deviations may be heteroskedatic – nothing in our model requires them to be independent of V. 6 Accounting treatments like mark-to-market accounting or cookiejar accounting may correlate accounting numbers to market noise. If so, then DCF estimates that rely on accounting numbers to make their cash flow projections may correlate to market noise, so that corr.eI ; e/ ¤ 0. For this reason, the Appendix provides the generalization of Theorem 11.1 to the case when corr.eI ; e/ D I ¤ 0. 5

Appendix 11A also offers a generalization of Theorem 11.1 to the case when corr.eI ; e/ D I ¤ 0, when DCF error eI correlates to market noise e. In the interest of expository parsimony, assume that corr.eI ; e/ D 0 in the main text. Because the choice of weights ›I and ›C in Theorem 11.1 _ _ minimizes the Bayesian uncertainty of V , call V the _ Bayesian triangulation of P, VI , and VC . In particular, V is a more precise estimate than P, VI , or VC are individually. Bayesian triangulation enables analysts to aggregate the information content of the individual estimates P, VI , and _ VC into a more robust estimate, V , of fundamental value V. Bayesian triangulation combines three different estimates _ into a more precise valuation estimate, V . Equation (11.2) has the simple appealing features finan_ cial analysts should find reassuring. First, V is the most efficient unbiased linear estimator ofV conditional on P, VI , _ and VC . In this sense, V is the Bayesian rational estimate of fundamental value given P, VI , and VC . If a valuation estimate is exact, it deserves full weight and should be reflected in the expressions of Theorem 11.1. It is straightforward to verify that if a valuation estimate (P, VI , or VC / is exact, then the expressions in Theorem 11.1 assign it 100% weight and its counterparts zero weight: If I ! 0, then I D 1 and C D .1  I  C / D 0. If C ! 0, then C D 1 and I D .1  I  C / D 0. If  ! 0, then .1  I  C / D 1 and I D C D 0.

On the other hand, if a valuation estimate is infinitely imprecise (e.g., worthless), it deserves no weight and the formulas in Theorem 11.1 assign it zero weight: If I ! 1, then I D 0. If C ! 1, then C D 0. If  ! 1, then .1  I  C / D 0.

When a valuation estimate is worthless, the two remaining estimates have non-zero weights that depend on their relative precisions. If a valuation estimate is noisier, an analyst expects it to receive less weight and the competing estimates to obtain commensurately more weight. As depicted in Fig. 11.1, this is what happens. As the ratio of ¢C to ¢ increases, the method of comparables and intrinsic valuation estimates (¢C D ¢I in the Figure) become more imprecise relative to market price. Accordingly, when this happens in Fig. 11.1, intrinsic value weight ›I and the method of comparables weight ›C fall while the market value weight .1  ›I  ›C / increases. ›C exceeds ›I in Fig. 11.1 because the method of comparables estimate, by virtue of being correlated with market price, helps to cancel out market noise and so receives more weight than the uncorrelated intrinsic valuation estimate. Recall that ¡ is the correlation between market price noise and the method of comparables estimation error, VC .

11 Combining Fundamental Measures for Stock Selection Fig. 11.1 Triangulation weights ›C ; ›I , and 1  ›C  ›I as a function of z D ¢C =¢ assuming ¡ D 0:5 and ¢C D ¢I

189 Maximum Precision Weights

0.8 0.7

1-kI-kC

weight value

0.6 0.5 0.4 0.3

kC

0.2 kI 0.1 0 0

Assuming all firms have positive CAPM beta, the method of comparables error correlates to market noise because, by construction, the former relies on peers’ market prices to determine VC . However, the correlation is generally not exact for many reasons, one of which is that the method of comparables is subject to sources of error unrelated to the market price. Such errors stem from the use of imperfect peers, noisy benchmarks, or questionable benchmarks (e.g., manipulated earnings). Thus, financial analysts cannot expect that ¡ D 1 in realistic situations. However, for the purpose of shedding light on Equation (11.2), it is instructive to imagine what would happen in the ¡ D 1 limit. In this limit, algebra reveals that lim

!1  I D 0 and Equation (11.2) simplifies to _

0.5

C  P C  C VC . In other words, if the only V D  C C C source of error in the method of comparables were market noise, then there is no need to give weight to an independent DCF valuation estimate, VI . In this idealized world, V is perfectly estimated by weighing market price P and method of comparables estimate VC . In this limit the method of comparables error exactly cancels out market noise (perfect diversification). In real world applications, ¡ ¤ 1, the intrinsic value _ VI should always have positive weight, and V is an imperfect (though always maximum precision) estimate of V. In principle, one can apply Equation (11.2) by estimating the parameters ; C ; I , and  based on historical data and then evaluating the weights I and C based on those estimates. Alternatively, following Equation (11.2) it is possible

to run regressions like 1 D .1  I  C / C I VI=P C

C VC=P on appropriate historical data samples for the direct determination of I and C . Of course, back testing always presumes that the weights are constant over time and whether this is true must be decided on a case-by-case basis.

1

1.5 z

2

2.5

3

The main point here is conceptual rather than empirical. Theorem 11.1 provides Bayesian justification for the practice of combining individual valuation estimates by taking a linear weighted average. In such weighted averages, the more reliable estimates command more weight. Market price does not receive 100% weight unless price is perfectly efficient. Theorem 11.1 – like valuation generally – should be applied contextually. Theorem 11.1 directly applies only to a continuing firm that is not expected to undergo reorganization or liquidation in the foreseeable future. If a firm is near bankruptcy and liquidation, then liquidation value is the most relevant estimate of value. In this context, capitalized earnings, which is a measure of continuing value, does not deserve any weight; even if it might be accurate, it is accurate only as an estimate of continuing value, not of liquidation value. Similarly, if a firm is expected to be taken over in the near future, then the expected takeover price is the most relevant estimate of value independent of capitalized earnings or net asset value, which are not relevant in that context. Context matters (Yee 2008c; Chen et al. 2010).

11.3 Triangulation in Forensic Valuation Beating the market is one obvious motive for using triangulation. The triangulation formula may also be used to infer the implied imprecision of valuation estimates from the weights that practitioners assign those estimates. This section focuses on the second application in the context of shareholder appraisal litigation. Compared to other institutions that appraise equity value, the judicial system is unique because judges promulgate official opinions describing how they arrive at value

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assessments. Because judicial opinions are supposed to be neutral (unlike sell-side analyst reports), they offer unique first-person accounts of how experienced practitioners perform valuation. More importantly, judicial opinions indirectly reflect the views of the professional financial community because judges solicit testimony from competing appraisal experts before making an appraisal. Judges sometimes appoint a special master or neutral expert to provide an independent, unbiased assessment. Hence, judicial opinions typically benefit from input representing several diverse and sophisticated perspectives. In this light, judicial valuation is an indirect but articulate reflection of how valuation is practiced in the greater financial community. The Delaware Block valuation method, Equation (11.1), is the special case of Equation (11.2) when VI D b and VC D e. As described in the Introduction, the courts have not articulated a guiding principle for picking the weights b and e . Indeed, the articulated rule is that “No rule of thumb is applicable to weighting of asset value in an appraisal : : :; rather, the rule becomes one of entire fairness and sound reasoning in the application of traditional standard and settled Delaware law to the particular facts of each case.”7 As one would expect, judges, after hearing expert testimony, tend to assign more weight to net asset value in liquidation contexts, and more weight to capitalized earnings when valuing a continuing firm. Market price is assigned a weight based on availability and liquidity of the shares, which affect the reliability of price. See Seligman (1984) for vivid case-study descriptions of the facts underlying several sample cases. If one focuses on valuation outcomes rather than on the details of the legal and valuation process underneath, one readily observes the following trend: net asset value is usually weighted almost twice as much as market value or earnings value. As depicted in Table 11.1, a survey of a frequently cited sample of judicial appraisals finds that, on average, net asset weight ›b D 46% – almost twice the average value of the market value and earnings value weights. The weights vary tremendously from case-to-case; the standard deviation of ›b is 26%, more than half of the average value of ›b . The same holds for the standard deviations of the market value and earnings value weights.8 One explanation for the heavier average weight on net asset value in the tiny Table 11.1 sample is self-selection. It could be that liquidation or reorganization cases, where break-up or net asset value is more relevant, tend to be adjudicated in court more than going-concern cases. It might also be that cases litigated in court tend to involve compa-

nies that do not have liquid shares, or companies that do not have credible earnings numbers. Another explanation is simply that judges, or the experts who testify, trust net asset value more than price or capitalize earnings. For the sake of illustration, if one pretends that the latter explanation dominates, then it is possible to calculate the implied relative precisions that judges accord the individual value estimates by inverting the formulas in Theorem 11.1. If judges maximize precision, the average weights in Table 11.1 imply that, on average, judges view net asset value to be a substantially less noisy estimator of value than even market price. Inverting the relationship between I and C and ; I , and C yields

7

The literature on the use of Bayesian methods in asset pricing is extensive and range from Bayesian learning of fundamental attributes (e.g., Lewellen and Shanken 2002; Yee 2004b) to Bayesian judicial valuation in shareholder litigation (Yee 2008c). I shall not attempt to review this extensive literature

Bell vs. Kirby Lumber Corp., Del. Supr., 413 A.2d 137 (1980). For comparison, a random number homogeneously distributed between 0 and 1 has average value 50% and standard deviation 28.9%. Since the market value and earnings value weights have significantly smaller spread and their means differ from 50%, it does not appear these weights are homogeneously distributed random variables. 8

.1  KI  2KC / C C D 

and I D 

s 

p

.1  KI  2KC /2 2 C 4.1  KI  KC /KC 2KC

 .1  2 / C C  C  

:  1 C   I

Assuming that I  b  12 and C  e  14 yields q C D  and I D  1 2 . Hence, judges select weights as if they think that (1) market prices are as imprecise as those based on capitalized trailing earnings; and (2) net asset valuation q estimates are more precise than market noise (e.g., I 

 12 ). Again, the purpose of this back-of-the-envelope calculation is to illustrate how one might use historical data together with the framework offered by Theorem 11.1 to estimate the implied precisions of competing value estimates. A definitive calculation of implied precision requires a more comprehensive sample and controls for self-selection and other contextual biases. The next section turns to applications of Bayesian analysis in other areas of asset pricing, including how Pastor (2000), also reverse engineering a Bayesian weighted average formula, finds evidence that U.S. investors may exhibit home bias because they face less parameter uncertainty for U.S. than for foreign stock issues.

11.4 Bayesian Triangulation in Asset Pricing Settings

11 Combining Fundamental Measures for Stock Selection

Table 11.1 These cases are from Seligman (1984)

191

Case

Year

1  kb  ke

kb

ke

Levin vs. Midland-Ross In re Delaware Racing Ass’n Swanton vs. State Guarantee Corp. In re Olivetti Underwood Corp. Poole vs. N.V. Deli Maatschappij Brown vs. Hedahl’s-Q B & R Tome Land & Improvement Co. v. Silva Gibbons vs. Schenley Industries Santee Oil Co. vs. Cox In re Creole Petroleum Corp. In re Valuation of Common Stock of Libby Bell vs. Kirby Lumber Corp. Average Standard deviation

1963 1965 1965 1968 1968 1971 1972 1975 1975 1978 1979 1980

0:25 0:4 0:1 0:5 0:25 0:25 0:4 0:55 0:15 0 0:4 0 0:27 0:18

0:5 0:25 0:6 0:25 0:5 0:5 0:6 0 0:7 1 0:2 0:4 0:46 0:26

0:25 0:35 0:3 0:25 0:25 0:25 0 0:45 0:15 0 0:4 0:6 0:27 0:17

While Brown, Libby, Santee Oil, and Tome Land are not Delaware court cases, they all employ the Delaware Block Method.

here. Instead, this section briefly describes and compares three examples that illustrate how the spirit of Theorem 11.1 might play out in other asset pricing settings.

11.4.1 Black and Litterman: Combining Private Views with CAPM Black and Litterman (1991, 1992) and He and Litterman (1999) provide a way for users to integrate their personal “views” about securities into a portfolio selection framework. The idea is as follows. Suppose the actual excess returns of N assets are jointly normally distributed as r  MVN .; †/. Investors never directly observe the true expected excess returns . Instead, an investor’s prior is   MVN .…; †/, where … are estimates (based on, say, CAPM or APT) and  is positive number parameterizing the investor’s uncertainty in these estimates. Suppose by doing private research an investor develops additional private views about K  N special portfolios q, an N K matrix. That is, the investor believes that the K special portfolios have jointly normally distributed returns q 0   MVN .Q; /. Then Black and Litterman show that these additional portfolio views modify her priors on . Given these views, her posterior is that jq;Q;;  MVN . ; V /, where

1 h i . †/1 … C q1 Q   . †/1 C q1 q 0

1 : V  . †/1 C q1 q 0 The expression for  , like the expression for the combined value estimate in Theorem 11.1, is a weighted average – the weights on … and  sum to unity (in a matrix sense). An investor who trusts CAPM (or APT) has  ! 0,

which implies that her posterior is the same as her prior, that is,   MVN .…; †/. Likewise, an investor who does not hold private views about any special portfolios has   MVN .…; ; †/. At the other extreme, an investor who does not trust CAPM has  ! 1, which implies she relies exclusively on her K portfolio views to determine  1   q1 q 0 q1 Q. Most investors fall in between these extreme investors and combine information from special portfolios with their priors in accordance with Black and Litterman’s weighed average formula. Practitioners find the Black-Litterman model appealing for three reasons. The investment manager specifies personal bullish or

bearish views on the expected returns of as few or as many assets as desired. By suitably choosing the matrix elements of the view portfolios matrix q, an investor can impose either absolute views about the performance of single assets or relative views about the relative performance of pairs of assets. The Black-Litterman model avoids extreme corner solutions common in mechanical portfolio optimization with traditional short sales constraints.

11.4.2 Incorporating Bayesian Priors in Regression Estimates of Model Parameters The Black and Litterman model does not specify how to use historical information to improve the estimation of expected returns. Pastor and Stambaugh (1999) and Pastor (2000) describe how Bayesian priors may be used to improve the estimation of Treynor and Black (1973) model parameters ˛ and ˇ.

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K.K. Yee

In particular, suppose that an investor believes that expected excess returns (in excess of the risk-free rate) for a given firm are governed by the time series model rt D ˛ C ˇt C "t : An investor who believes in CAPM would believe that ˛ D 0 identically and that ˇ is the firm’s exposure to the market excess return. An investor with doubts about CAPM, market efficiency, or an investor who is benchmarking returns to something other than the market portfolio, however, would face uncertainty about the values of ˛ and ˇ. Assume that, although the investor does not know the firm’s ˛ and ˇ values precisely, her priors are that





˛ ˛ ;‰  MVN ˇ ˇ   and that "t  N 0;  2 is i.i.d. Pastor (2000, p. 216) shows that, given T periods of time series data, 0 0 1 1 r1 B :: B :: C R  @ : A and X  @ : rT

1 1 :: C ; : A

1 T

the investor’s Bayesian estimates for ˛ and ˇ are

˛O Bayes ˇOBayes



 1 D ‰ 1 C  2 X 0 X 

 ˛O OLS ˛ ‰ 1 ; C  2 X 0 X O ˇ ˇOLS



˛O OLS ˇOOLS



are the OLS estimates of ˛ and ˇ from the

˛ regression R D X C E, where E 0  .e1    eT /. Thus, ˇ the Bayesian estimates are a matrix weighted average of the prior means and the regression estimates. Pastor’s weighted average manifests properties analogous to the triangulation formula in Theorem 11.1. An investor who has more precise priors (e.g., smaller ‰ eigenvalues) will tend to place more weight on ˛ and ˇ. In contrast, an investor who has imprecise priors but less noisy time series data will put more weight on her regression estimates. Applying this model to data used for studying U.S. home bias in international asset allocation, Pastor was able to estimate an implied value for ‰11 , U.S. investors’ prior uncertainty for ˛. Pastor estimates that ‰11  1%, which suggests that U.S. investors may exhibit home bias because they do not face an exorbitant amount of parameter uncertainty from domestic issues. where

11.4.3 Bayesian Model Averaging Model-based Bayesian analysis is sensitive to model specification. Among others, Hoeting et al. (1999) and Cremers (2002) propose schemes based on Bayes’ theorem to take into account model selection uncertainty and adjust for potential data snooping biases that may arise if a researcher focuses on one particular model from a range of ex ante candidates after observing the data. In particular, Cremers’ approach is motivated by the disturbing fact that the evaluation of explanatory variables with regression t statistics typically assumes one specific model is correct (an implicit joint hypothesis), which overlooks the tremendous uncertainty the researcher has about the true model’s identity. Cremers (2002) simultaneously considers all linear (regression) models that can be constructed from a set of explanatory variables, assigns them each flat priors, and estimates their posterior probabilities from data. The grand model is the weighted average of individual candidate models. The posterior weight on each model is informed by regression results for each model. Cremers shows that specification of a prior for the inclusion of each candidate variable along with the expected R-square of the model regression is sufficient to compute the posterior weight for each model. Existence of a grand model also affects the interpretation of individual-model t statistics. Studies such as Cremers (2002) culminate from a rich stream of research on model and parameter uncertainty dating back to Zellner and Chetty (1965) and Klein and Bawa (1976) regarding how such uncertainty affects portfolios and the identification of efficient portfolios. A closely related recent study is Avramov (2002), which investigates the role of uncertainty about the returns-forecasting model on choosing optimal portfolios.

11.4.4 Bayesian Guidance for Improving Forecasting Suppose one uses historical data to evaluate the regression model Pt C1 D aP Pt C ab bt C aE Et C "t C1 ; where Pt C1 is the next-period market price (cum-dividend and suitably discounted), Pt is the current price, bt is the net asset value estimate, Et is the capitalized earnings estimate, and "t C1 is the error. Popular textbooks such as Damodaran (2002, Chap. 18) suggest regressions along similar lines. Then, this regression model is interpretable within the framework of Theorem 11.1 given Pt C1 represents the resolution

11 Combining Fundamental Measures for Stock Selection

of fair value at date t. From the perspective Equation (11.2), it is expected that aP C ab C aE D 1. (A deviation from aP C ab C aE D 1 is an indication that one or more of the value estimates fPt ; bt ; Et g is systematically biased over the sample.) Moreover, using the measured values of the regression coefficients, one could invert the weight formulas given in Theorem 11.1 to deduce the implied relative precisions of fPt ; bt ; Et g for predicting Pt C1 . The technique of using historical data to estimate parameter values in models like Equation (11.2) is, of course, not new. There is a long tradition of using historical and contemporaneous financial data to project whether a firm will become a winner or a loser (e.g., Altman 1968; Ohlson 1980; Fama and French 1992; Piotroski 2000; and Beneish et al. 2001). Using the Kalman filter, Yee (2004b) models how, if earnings reports provide noisy information about an unobservable fundamental attribute, then earnings realizations cause valuation coefficients to revise dynamically. Chan and Lakonishok (2004) and Arnott et al. (2005) estimate crosssectional and time-series models to predict future returns based on beginning-of-period weighted averages of bookto-price, earnings-to-price, cash-flow-to-price, and sales-toprice. The literature on using regressions to estimate weights for combining forecasts is even more developed. Originating with the seminal paper of Bates and Granger (1969), this literature has sprouted branches that explore behavioral as well as purely statistical aspects of how to optimally combine forecasts. Armstrong (1989), Clemen (1989), and Diebold and Lopez (1996) provide surveys. Borrowing generously from Diebold and Lopez, this section summarizes aspects of this literature that are relevant to combining value estimates. Combining methods fall roughly into two groups, variance-covariance methods and regression-based methods. Consider first the variance-covariance method, which is due to Bates and Granger (1969). Suppose one has two s steps i for i 2 f1; 2g. The composite ahead unbiased forecasts, yO;s forecast formed as the weighted average c 1 2 D ! yO;s C .1  !/ yO;s yO;s

is unbiased if, and only if, the weights sum to unity (as they do here). The combined forecast error will satisfy the same relation as the combined forecast; that is, "c;s D ! "1;s C .1  !/ "2;s 2 2 with a variance c2 D ! 2 11 C.1  !/2 22 C2! .1  !/ 12 2 2 where 11 and 22 are unconditional forecast error variances 2 is their covariance. The weight that minimizes the and 12 combined forecast error variance is

!D

2 11

2 22  12 : 2 C 22  212

193

The optimal weight is determined in terms of both the variances and covariances. Moreover, the forecasterror vari 2 2 . ; 22 ance from the optimal composite is less than min 11 While this suggests that combining forecasts only improves accuracy, there is a caveat. In practice, one must numerically estimate the unknown variances and covariances underlying the optimal combining weights. If one is averaging over L in-sample years, one estimates by replacing ! L P j "i;s ";s so that with O ij D L1  D1

!D

2 O 22  O 12 : 2 2 O 11 C O 22  2O 12

In the small samples typical for valuation studies, sampling error contaminates the combining weight estimates. Sampling error is exacerbated by co-linearity that normally exists among primary forecasts. Accordingly, combining weight estimates does not guarantee the reduction of out-of-sample forecast error. Fortunately, forecast combination seems to perform well in practice (Clemen 1989). Next, consider the time-series regression method of forecast combination. In this method, one regresses realizations y;s on previous forecasts of y;s to determine the optimal weights for combining current and future forecasts. In other words, one assumes that c 1 2 D ! yO;s C .1  !/ yO;s yO;s

and determines the optimal value of ! by back testing. Granger and Ramanathan (1984) show that the optimal weight has a regression interpretation as the coefficient vector of a linear projection of realizations y;s onto the forecasts subject to two constraints: the weights sum to unity, and no intercept is included. In practice, one simply runs the regression on available data without imposing these constraints. The regression method is widely used. There are many variations and extensions. Diebold and Lopez (1996) describe four examples in an excellent review article: timevarying combining weights, dynamic combining regressions, Bayesian shrinkage of combining weights toward equality, and nonlinear combining regressions. The remainder of this section summarizes the examples given by Diebold and Lopez and then warns against the overuse of combining regressions to over-fit in-sample data. For more details and a list of citations to the original literature, refer to the Diebold and Lopez article.

11.4.4.1 Time-Varying Combining Weights Granger and Newbold (1973) and Diebold and Pauly (1987) proposed time-varying combining weights, respectively, in

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the variance-covariance context and in the regression context. In the regression context, one may undertake weighted or rolling combining regressions, or estimate combining regressions with time varying parameters. Time-varying weights are attractive for a number of reasons. First, different learning speeds may lead to a particular forecast improving over time compared to others. If so, one wants to weight the improving forecast more. Second, the design of various forecasting models makes them better forecasting tools in some situations. In particular, a structural model with a well-developed wage-price sector may substantially outperform a simpler model when there is high inflation. In such times, the more sophisticated model should receive higher weight. Third, agents’ utilities or other properties may drift over time. 11.4.4.2 Dynamic Combining Regressions Serially correlated errors arise in combining regressions. Diebold (1988) considers the covariance stationary case and argues that serial correlation is likely to appear in unrestricted regression-based forecast combining regressions when the unconstrained coefficients do not automatically add up to unity. More generally, serial correlation may capture dynamical elements that are not captured by the explicit independent variables. Coulson and Robins (1993) point out that a combining regression with serially correlated disturbances is a special case of a combining regression that includes lagged dependent variables and lagged forecasts. 11.4.4.3 Bayesian Shrinkage of Combining Weights Toward Equality Fixed arithmetic averages of forecasts are often found to perform well even relative to theoretically optimal composites (Clemen 1989). This is because the imposition of a fixed-weights constraint eliminates variation in the estimated weights at the cost of possibly introducing bias with incorrectly selected weight. However, the evidence indicates that the benefits of imposing equal weights often exceed this cost. With this motivation, Clemen and Winkler (1986) propose Bayesian shrinkage techniques to allow for the incorporation of varying degrees of prior information in the estimation of combining weights. Least-squares weights and the prior weights correspond to the polar cases of the posterior-mean combining weights. The actual posterior mean combining weights are a matrix-weighted average of the two polar cases. They describe a procedure whereby the weights are encouraged, over time, to drift toward a central tendency (e.g., ! D 12 ). While the combining weights are coaxed toward the arithmetic mean, the data is still allowed some opportunity to reflect possibly contrary values.

K.K. Yee

11.4.4.4 Nonlinear Combining Regressions Theory does not specify that combining regressions should be linear. In fact, valuation theory suggests that a degree of nonlinearity is to be expected (Yee 2005a). As a practical matter, empiricists favor linear models (only) because of their simplicity, which is always a virtue. In combining regressions, one way to introduce nonlinearity is to allow the weights to depend on one or more hidden state variables that change dynamically over time. The value of the state variable must be inferred from past forecast errors or the value of other observables. Needless to say, the structural equations can become quite elaborate, the computation quite messy, and the results sensitive to numerical details. For these reasons, I do not recommend introducing nonlinearity or many variables without specific motivation from economic insight. Absent sound theoretical guidance, it is too easy for the “exploration” of complex econometric models to turn into fishing expeditions.

11.5 The Data Snooping Trap The Bayesian rubric does not offer a license to go on undisciplined data mining expeditions without guidance from economic insight and theory. Data snooping, even within a Bayesian rubric, is an unsatisfactory substitute for economic and factual insight if the goal is to achieve good out-of-sample performance. Data snooping is the practice of sifting through in-sample data to look for patterns and, then, over-eagerly interpreting an observed pattern to conclude that it is real if it passes common (but nonetheless inappropriate) statistical tests within the sample. The problem is that phantom patterns will exhibit chance statistical significance at the 1% level about 1% of the time. If a researcher tests 100 phantom patterns, there is a 63.4% .D 1  0:99100 / chance that one or more of them exhibit significance at the 1% level. If a researcher tests 500 phantom patterns, the probability of finding false positives jumps to 99.3% .D 1  0:99500 /. Thus, a data snooper who (either explicitly or implicitly) considers hundreds of potential patterns has almost a certain chance of uncovering one or more patterns that exhibit chance statistical significance. A creative and diligent researcher who is paid to keep on looking until she “finds something” is virtually certain to uncover phantom patterns. Although data snooping is dangerous in any field, data snooping is particularly treacherous in macroeconomics, finance, and accounting research, where data is plentiful and the use of statistics is liberal. With the advent of modern software, huge numbers of hypotheses about a single data set can be “tested” in a short time by algorithmically searching for variables combinations that exhibit correlation.

11 Combining Fundamental Measures for Stock Selection

A seemingly reasonable justification for the practice of data snooping is the argument that a researcher should let the data speak for itself. This rationale licenses the researcher to conjure up a broad set of hypotheses where one is bound to appear significant in sample. This is not to say that researchers should never listen to data. The point is that, upon hearing a pattern, a researcher cannot assess its significance based on the in-sample data set that spoke it because passing an in-sample test is tantamount to verifying a tautology. For example, in a room of 400 people, at least two are guaranteed to share a birthday. Let’s say Mary Jones and John Smith not only share a birthday but are the only two of the 400 to have birthdays on April 1st. A data-snooping researcher would seek to find something special in common between Mary and John, such as that they both play piano or fly airplanes. By rifling through a mental list consisting of hundreds or thousands of possible characteristics, the researcher will inevitably uncover some commonality between John and Mary. For instance, the researcher might find that John and Mary are the only two of the 400 who hold Columbia MBAs and work on Wall Street, facts exposed by exhaustively comparing everyone’s biographies. Then the snooper might propose the hypothesis that “being born on April 1st leads to a higher likelihood of getting a Columbia MBA and working on Wall Street.” With the benefit of hindsight, she can even conjure up plausible economic theories, such as “Kids with April 1st birthdates, being subject to more teasing in school, are more motivated to excel and pursue lucrative Wall Street positions in finance.” The in-sample data seemingly supports this position since not one of the other 398 randomly selected people in the sample had April 1st birthdays, attended Columbia Business School, or work on Wall Street. However, upon turning to out-of-sample tests, the researcher will fail to verify that Wall Streeters are more likely born on special days like April 1st than any other day. Why? Because the original hypothesis was uncovered by data snooping and inadequately tested using in-sample data. Numerous authors have pointed out the dangers of mining financial data. Lo and MacKinlay (1990) attempt to quantify the impact of data-snooping bias in cross-sectional tests of asset pricing models when firm characteristics used to sort stocks is correlated with the error in performance measures. More colorfully, Crack (1999) identified the following astrological anomaly by shifting through 20 years of CRSP daily data. Crack found that stock market volatilities are higher and mean returns lower on trading days surrounding new- and full-moons than surrounding other days. Why should we be skeptical of this anomaly? First, our current understanding of economics does not predict that the moon’s status should affect trading behavior. That it does would be a truly revolutionary idea – and revolutionary ideas require careful verification before acceptance. Second, Crack found that removing the trading days surrounding the

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October 1987 stock market crash, which happened right before a new moon, reduced the statistical significance of his market-volatility result (but not the mean-return result). “In the absence of any economic justification, and given my data snooping confessions,” Crack writes, “these results cannot be interpreted as anything other than an elegant and off-beat example of data snooping. Unfortunately, most data snooping in financial economics is more insidious – because plausible economics examples are typically involved.” (Crack 1999, p. 96) There is nothing wrong with sifting through lots of data per se. Problems arise only if one subsequently uses inappropriate statistical analysis to draw inferences from the exercise. In particular, if a researcher or his predecessors have considered more than one candidate hypothesis to explain patterns in the data, then the researcher must take into account the universe of hypotheses considered when evaluating each hypothesis. To this end, Sullivan et al. (1999) use bootstrapping analysis to adjust confidence intervals in hypothesis testing to correct for biases caused by data snooping. They show that 26 technical daily trading rules exhibiting an apparent ability to beat the market under traditional hypothesis testing lose that ability once the authors adjust their confidence intervals to take into account that prior academic and practitioner research had selectively identified these rules after examining a universe of over 7,000 closely related but equally ineffectual trading rules. Then, applying White’s (2000) “Reality Check” methodology to 100 years of daily DJIA pricing data, Sullivan et al. conclude that the snoopingadjusted p-value for even the best technical trading rule is over 30%; that is, they fail to reject the null hypothesis that even the best technical trading rule in a universe of thousands of rules is consistent with market efficiency.

11.6 Using Guidance from Theory to Mitigate Data Snooping The first way to mitigate data snooping pitfalls is to adjust hypothesis testing (using something like White’s Reality Check methodology) to take into account that multiple hypotheses have been implicitly or explicitly considered. The second way is to commit to a single hypothesis before looking for patterns in data. This is because, as Kahn et al. (1996) formalize in an insightful Bayesian model, new empirical evidence giving rise to a new hypothesis does not support that new hypothesis as strongly as empirical evidence that had been predicted by a pre-existing hypothesis supports that pre-existing hypothesis. In other words, pre-committing to a single hypothesis suppresses the ability consider multiple hypothesis and select a false one that happens to fit the data.

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For example, consider the following empirical exercise, which I carried out using data from the Commodity Research Bureau: 1. Take 15 monthly commodity cash price series spanning 1901–2007.9 2. Compute the trailing 3-month log return rlag3 for each commodity-month. 3. Pretend that these cash prices are tradable10 and implement a trend-following strategy indicated by the following commodity-level trading rule: buy bt rlag3 dollars if rlag3 > 0 sell st rlag3 dollars if rlag3 < 0; where bt ; st are the appropriate normalization weights so that the long and short commodity portfolios are of equal dollar value in each month t. 4. Compute the monthly returns of the resulting long-short commodity portfolio. 5. For comparison purposes, also track the monthly returns of a long-only equal-weighted portfolio of these 15 commodities. Call this portfolio the “index” portfolio. After computing the historical return series of the longshort and index portfolios, I sorted the months into quintiles based on the performance of the index portfolio. Figure 11.2 depicts average returns of these portfolios during the five monthly quintiles. As depicted, the long-short portfolio does well when the index portfolio does well AND when the index portfolio does 9

Most of my 15 series, including gold, began well after 1901. Only the cotton, corn, and wheat series started in January 1901. 10 While it is unrealistic to believe that investors can trade these commodities at the quoted historical cash prices, a parallel exercise (not reported here) using commodity futures prices would yield similar results.

poorly. While the long-short portfolio is positively correlated to the index in months of strongest index returns, it is negatively correlated to the index in the months of worst index returns. In other words, the long-short return distribution in Fig. 11.2 is U-shape. Is this U shape the result of data snooping or is it real economic phenomenon? The answer to this question is important because one would not expect data snooping results to persist into the future. On the other hand, if it is a real phenomenon, then understanding the causes would not only give us confidence to possibly exploit it – it would help predict analogous phenomena in other asset classes. There are many reasons that should raise red flags about the U shape. The accuracy and liquidity of historical price series dating back to the early twentieth century are always in doubt. Even if they are accurate, many of the reported prices might have been realized in thinly traded markets and, thus, not represent tradable market prices. In addition, survivorship bias is an issue given that the Commodity Research Bureau does not report time series of out-of-style commodities such as candle wax, which never became popularly traded. I will now argue that the U shape is real and not the result of data snooping (even though I examined several variations of trend trading strategies, all of which happily yielded a U shape). My argument is based on the fact that previously known characteristics of commodities predict the U-shape curve. Being implied by pre-existing theory rather than requiring the invention of new theory endows the U shape with more credibility in the sense of Kahn et al. (1996). Pre-existing theory implies the U shape in the following ways. First, it is well known (and re-confirmed in the first column of Table 11.2) that commodity-level returns are serially correlated. As Fung and Hsieh (2001) explain, if returns are serially correlated, trend followers are positioned properly to enjoy positive returns when prices move up and when they move down. More importantly, by weighing constituent

Trend following performs well on good and bad months. equal weighted

Fig. 11.2 The long-short trend following portfolio performs well on both the best and worst quintile months of the equal-weighted portfolio. As argued by Fung and Hsieh (2001), this is because trend following strategies yield return distributions similar to that of look-back straddles. Source: Author’s calculation using Commodity Research Bureau commodity pricing data

annualized average return

50% 40% 30% 20% 10% 0% 1

−10% −20% −30% −40%

2

long-short

11 Combining Fundamental Measures for Stock Selection

Table 11.2 Monthly serial correlations, annualized monthly returns, standard deviation, skewness, and kurtosis of 15 commodities based on monthly cash prices during 1901–2007 (Source: Commodities Research Bureau and author calculation)

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Serial correlation (%)

Annualized mean (%)

Annualized standard dev (%)

Annualized skewness (%)

Annualized kurtosis (%)

Butter Cotton Copper Cocoa Corn Eggs Gold Hogs Iron Live cattle Platinum Silver Soybeans Wheat Zinc

22:83 25:76 35:55 25:37 25:17 8:22 28:40 17:76 23:30 28:95 23:33 22:55 37:82 20:30 42:39

3:84 3:91 4:62 5:91 5:29 11:11 5:70 6:25 6:80 4:35 5:90 5:56 4:56 4:85 4:71

22:35 26:28 17:21 28:01 24:83 43:04 14:29 29:42 25:77 17:30 20:10 21:47 23:67 22:84 18:81

17:92 17:09 20:91 104:28 40:78 4:72 75:97 42:51 36:52 20:83 74:16 71:80 13:82 14:47 33:18

86:01 277:03 42:06 415:18 138:77 9:34 191:51 88:41 108:71 37:75 172:27 226:50 49:79 48:11 88:56

Median Mean Pooled

23:33 25:85 Inapplicable

5:56 5:56 5:57

21:47 23:69 24:73

36:52 36:99 31:60

88:56 132:00 137:38

The kurtosis reported here would be zero for a normal distribution. The “Median” and “Mean” rows report the median and mean of the commodity-level statistics. The “Pooled” row refer to the same statistic computed by pooling the returns of all commodity-months

commodities with their lag returns wr lag , trend followers enjoy their largest gains when the underlying assets make their most extreme repeated moves. As a result, trend following strategies yield portfolio return distributions that are empirically similar to the ex post payoff distribution of lookback straddles. Since trend following performs best when the investable universe has extreme performance in either direction, the long-short portfolio returns are U-shape. Thus, while its robustness is perhaps too appealing, the U shape in Fig. 11.2 is not surprising. The second reason I believe that the U shape is real is because the fact that inventories cannot fall below zero implies that commodity prices should be serially correlated according to micro-economic theory. According to the theory of competitive storage (e.g., Kaldor 1939; Deaton and Laroque 1992; Litzenberger and Rabinowitz 1995; and Gorton et al. 2007) commodity prices enjoy periodic spikes whenever unexpected demand shifts or production problems cause inventory shortages. Since it takes at least one successful production cycle to remedy a shortage, price hikes caused by shortages can persists for months until the market adjusts one way or another to the shortage. Deaton and Laroque (1992, Equation 10) showed that, if harvests are independent and identically distributed, then the expected commoditylevel returns depend nonlinearly on price:    p ; 1 : E ŒRt C1 / min pt p  is a constant that depends on many factors like the cost of storage. Hence, commodity prices deviate from the frictionless no-arbitrage relation obeyed by the prices of financial

assets. The returns of commodities in demand for direct consumption depend on current prices and are serially correlated. The third reason that the U shape is credible is because other predictions of the theory of competitive storage have found (partial) empirical support.11 For example, because inventory shortages periodically cause price hikes, the competitive storage predicts that the distributions of commodity returns should be positively skewed and leptokurtic. As indicated in last two columns of Table 11.2, most of our 15 commodity series exhibit positive skewness and positive kurtosis. Figure 11.3 depicts the skewed and leptokurtic return distributions of gold and cocoa.

11.7 Avoiding Data-Snooping Pitfalls in Financial Statement Analysis Like their colleagues who work with macroeconomic summary data sets, fundamental analysts – so-called for their bottoms-up, one-firm (or one-commodity) at a time approach to collecting and analyzing data – are subject to data snooping. This is because pertaining to each large public company (or commodity) is a rich history of financial statements and 11

Unfortunately, the theory of storage is still incomplete. As Deaton and Laroque (2003, p. 1) explain, the model, “although capable of introducing some autocorrelation into an otherwise i.i.d. process, appears to be incapable of generating the high degree of serial correlation of most commodity prices. : : : We are therefore left without a coherent explanation for the high degree of autocorrelation in commodity prices: : :”

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Cocoa's Monthly Cash-Price Return Distribution 1927-2007

Gold's Monthly Cash-Price Return Distribution 1941-2007 1000

1000

100

100

10

10

27%

24%

21%

18%

15%

9%

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0%

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1 −11%

1

Fig. 11.3 The monthly return distributions of the gold and cocoa on a log scale. As one can see, both distributions are positively skewed and leptokurtic. Source: Author’s calculation using Commodity Research Bureau commodity pricing data

anecdotes, that are rich with apparent trends and regularities that invite naïve extrapolation and data snooping. Within the financial statement analysis (FSA), data snooping leads to irrational behaviors like hindsight bias and the invention of faulty heuristics that cause the analyst to compound past mistakes by repeating them in the future. FSA is the process of transforming accounting information from quarterly and annual statements into forecasts of future cash flow and profit.12 In fundamental analysis, changing a few parameters generates a whole different inference, which leaves the door wide open to irrational behavior. Though fundamentalists’ “deep dive” research often adds value because damaging information may be buried in footnotes and managerial conference calls, too much data often obfuscates what’s important by burying it in a haystack. Too much data coupled with an inadequately principled perspective on FSA renders an analyst vulnerable to drawing irrational conclusions and issuing poor investment recommendations. Staring at a large amount of information tricks the human mind into seeing patterns, real or imagined, and drawing improper conclusions from those patterns. Even worst, an analyst might conjure up a bad heuristic that will guide her into making a similar mistake next time, thus compounding her initial mistake. In addition to improperly snooping accounting data, areas of FSA particularly prone to subjectivity include the scrubbing of accounting numbers to correct for untidy bookkeeping loopholes and earnings manipulation; the forecasting of cash flows; and the extrapolation of current growth into

12

Yee (2008b) deconstructs the round-trip process of fundamental analysis, stock selection, and profit taking. Klarman (1991) presents a hearty practitioner’s account.

the future when making valuation estimates. Scientifically accepted prescriptions do not exist for how to perform these everyday tasks. Analysts do not even agree on how to diagnose basic attributes like a firm’s earnings quality. While some analysts rate firms by assigning a numerical earnings quality score, many other analysts believe earnings quality is an ill-defined concept of dubious usefulness. Thus, FSA requires analysts to make a lot of subjective choices, each of which is another invitation to irrational behavior. Compounding the problem is the dizzying smorgasbord of competing valuation techniques spawned by the financial industry (as described in, for example, Damodaran 2002). It may never be determined which of these techniques are scientifically sound. Kaplan and Ruback (1995), Gilson et al. (2000), Lie and Lie (2002), Liu et al. (2001) and Yoo (2006) ran horse races between different valuation methods. Kaplan and Ruback estimated price-to-EBITDA and also DCF valuations for a sample of highly leveraged transactions (HLTs). For their sample of 51 HLTs between 1983 and 1989, they found price-to-EBITDA and DCF gave comparable levels of accuracy and precision for predicting the HLT transaction price. In particular, about half of their valuation estimates fell within 15% of the actual HLT transaction price. However, many will argue that the glass is half empty. Differences exceeding 15% in the other half of the Kaplan and Ruback sample demonstrates that price-to-EBITDA and DCF are inconsistent about half the time. Gilson et al. (2000) compared DCF to the method of comparables for companies emerging from bankruptcy. Using EBITDA multiples based on the industry median, they found about 21% of the valuation estimates fell within 15% of market values. The authors conjectured that Kaplan and Ruback’s HLT valuations were more precise than their bankruptcy ones due to greater information efficiency in the HLT setting. Using 1998 financial data from Compustat, Lie

11 Combining Fundamental Measures for Stock Selection

and Lie found that the method of comparables based on asset value yields more precise and less biased estimates than estimates based on sales and earnings multiples, especially for financial firms. Liu et al. (2001) and Yoo (2006) found that forward price-to-earnings ratios – when consensus analysts’ earnings forecasts are available – perform best in large sample studies. Consistent with Liu et al., and Yoo13 (whose studies are based on a much longer time period), Lie and Lie also conclude that forward price-earnings ratios are more accurate and precise estimators than trailing ones. However, Lie and Lie report that the accuracy and precision of their method of comparable estimates varied greatly according to company size, profitability, and the extent of intangible value. Such unresolved open-ended issues create uncertainty for fundamentalists and leave them subject to behavioral pitfalls, including overconfidence, or the belief that a favorable outcome is more likely than what will actually occur; over optimism, or overestimating the likelihood of good luck; and overreaction to recent events, such as an exciting earnings announcement or press release (Daniel et al. 2001). Other judgment biases common to fundamentalists include hindsight bias, or an over-reliance on anecdotal evidence; herding, or blindly following mass movements irrespective of sound judgment; and myopia, or a narrow fixation on recent, and perhaps exceptional, changes. These behavioral pitfalls influence what an analyst emphasizes or ignores, the frequency of forecast revisions, and the subsequent inferences. They lurk even if the analyst is free of incentive-driven biases or other sinister intentions – behavioral pitfalls hinder even an honest analyst attempting to make fair evaluations. A tenet of western civilization is Plato’s “rule of law” concept – the principle that government authorities can exercise power only in accordance with publicly disclosed laws adopted in accordance with established procedure. The rule of law constrains government authorities from behaving in capricious or self-serving ways contrary to the written law. (Plato’s Statesman, his Laws, and his student Aristotle’s Politics were among the first to enunciate the benefits of rule of law over “rule of men” almost 2,500 years ago.) The rule-of-law principle suggests that commitment to a disciplined FSA process would add value by constraining analysts’ discretion to interject in-the-moment psychology and emotions. Ideally, a conclusion from FSA should not depend on the personality or tastes of the analyst performing the exercise. Just as the rule of law disciplines police, a disciplined 13

Yoo (2006) empirically examined, in a large-scale study of Compustat firms, whether taking a linear combination of several univariate method of comparables estimates would achieve more precise valuation estimates than any comparables estimate alone. He concluded that the forward price-earnings ratio essentially beats the trailing ratios, or combination of trailing ratios, so much that combining it with other benchmark estimates does not help. Yoo did not consider DCF, liquidation value, or market price in his study.

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decision-making process adds value by forcing analysts to remove their personalities and psychology and conform to pre-agreed stock-selection criteria. I believe that fundamentalists would be wise to regularly step back from accounting minutiae and supplement their bottoms-up research with frameworks that instill discipline and consistency and prevent errors caused by psychological enticements to misinterpret details.

11.8 Conclusion Securities selection is the attempt to distinguish prospective winners from losers – conditional on beliefs and available information. This article surveys some relevant academic research on the subject, including work about the combining of forecasts (Bates and Granger 1969), the BlackLitterman model (1991, 1992), the combining of Bayesian priors and regression estimates (Pastor 2000), model uncertainty and Bayesian model averaging (Hoeting et al. 1999; Cremers 2002), the theory of competitive storage (Deaton and Laroque 1992), and the combination of valuation estimates (Yee 2007). Despite its wide-ranging applicability, the Bayesian approach is not a license for data snooping. The second half of this article describes common pitfalls in fundamental analysis and comments on the role of theoretical guidance in mitigating these pitfalls. Bayesian triangulation provides a conceptual framework that justifies two intuitive heuristics: 1. Estimate value as the weighted average of all available bona fide valuation estimates, including market price if available. 2. Assign greater weight to the more reliable estimates when some estimates are more reliable than are others. The triangulation formula, Equation (11.2), agrees with the Bayesian expectation value conditioned on the information set comprised of the individual valuation estimates. Applying standard Bayesian analysis, Equation (11.2) can be easily extended to provide a conceptual framework for folding in as many valuation estimates as desired. As Sect. 11.3 describes, given a particular choice of weights, one is able to reverse engineer Equation (11.2) to obtain the valuation noise implied by a given set of Bayesian weights. Reverse engineering reveals that shareholderlitigation judges believe market price is as noisy a valuation estimate as capitalized trailing earnings and significantly noisier than net asset value. This is interesting because, while conventional wisdom pre-dating Fischer Black has always believed that market prices and valuation estimates are noisy, it is also generally accepted that the amount of noise is an elusive quantity that cannot be observed. Inverting the

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Bayesian triangulation formula provides a way of estimating the implied noisiness of price and other valuation estimates from a credible sample of appropriate data. This exercise is suggested as an issue for future research. The Bayesian framework allows for uncertainty in its many forms, provides modeling flexibility, and permits investors and managers to insert subjective priors and views into a rational decision-making process. Despite its virtues, the Bayesian framework and the desire to optimize combining weights is not a license for data snooping. For fundamental analysts, financial statements and anecdotal information comprise an extensive set of data about each large public company (or commodity) that invites data snooping and armchair quarterbacking, which leads to hindsight bias. Thus, fundamental analysts would be wise to regularly step back from accounting minutiae and discipline their FSA with a less precision-oriented, more skeptical and theoretical perspective. In the investment business (to paraphrase Keynes) “it is better to be roughly right than precise-but-wrong” and data mining greatly enhances the likelihood of the latter. Therefore, fundamentalists who complement their bottoms-up research with top-down, quantitative economic discipline will achieve the best long-term investment results. If nothing else, quantified checklists, scorecards, and portfolio constraints can add value simply by disciplining analysts to resist overconfidence, hindsight bias, and psychological pitfalls to make emotionally detached assessments based on pre-agreed criteria. The emphasis should be on the evenness of the analytical process over time. Investing is a repetitive activity. Fortuitous outcomes arising from ex ante unsound decisions must not be celebrated since good luck is not reproducible. The key is commitment to a quantified process and being disciplined enough to stick with it in the long haul irrespective of short-term market swings. Acknowledgments The statements and opinions expressed in this article are those of the author as of the date of the article and do not necessarily represent the views of the Bank of New York Mellon, Mellon Capital Management, or any of their affiliates. This article does not offer investment advice.

References Altman, E. I. 1968. “Financial ratios, discriminant analysis and the prediction of corporate bankruptcy.” Journal of Finance 23, 589–699. Ang, A. and J. Liu. 2004. “How to discount cashflows with time-varying expected returns.” Journal of Finance 59(6), 2745–2783. Armstrong, J. 1989. “Combining forecasts: the end of the beginning or the beginning of the end?” International Journal of Forecasting 5(4), 585–588. Arnott, R., J. Hsu, and P. Moore. 2005. “Fundamental indexation.” Financial Analysts Journal 61(2), 83–99. Avramov, D. 2000. “Stock return predictability and model uncertainty.” Journal of Financial Economics 64, 423–458.

K.K. Yee Bates, J. and C. Granger. 1969. “The combination of forecasts.” Operational Research Quarterly 20, 451–468. Beatty, R., S. Riffe, and R. Thompson. 1999. “The method of comparables and tax court valuations of private firms: an empirical investigation.” Accounting Horizons 13(3), 177–199. Beneish, M., C. Lee, and R. Tarpley. 2001. “Contextual fundamental analysis through the prediction of extreme returns.” Review of Accounting Studies 6(2/3), 165–189. Bhojraj, S. and C. Lee. 2003. “Who is my peer? A valuation-based approach to the selection of comparable firms.” Journal of Accounting Research, 41(5), 745–774. Black, F. 1986. “Noise.” Journal of Finance, vol. 16(3), 529–542. Black, F. and R. Litterman. 1991. “Asset allocation: combining investor views with market equilibrium.” Journal of Fixed Income 1(2), 7–18. Black, F. and R. Litterman. 1992. “Global portfolio optimization.” Financial Analysts Journal (September/October) 28–43. Chan, L. and J. Lakonishok. 2004. “Value and growth investing: review and update.” Financial Analysts Journal 60(1), 71–87. Chen, F., K. Yee, and Y. Yoo. 2007. “Did adoption of forward-looking methods improve valuation accuracy in shareholder litigation?” Journal of Accounting, Auditing, and Finance 22(4), 573–598. Chen, F., K. Yee, and Y. Yoo. 2010. “Robustness of judicial decisions to valuation-method innovation: an exploratory empirical study.” Journal of Business, Accounting, and Finance, forthcoming. Clemen, R. 1989. “Combining forecasts: a review and annotated bibliography.” International Journal of Forecasting 5, 559–583. Clemen, R. and R. Winkler. 1986. “Combining economic forecasts.” Journal of Economic and Business Statistics 4, 39–46. Coulson, N. and R. Robins. 1993. “Forecast combination in a dynamic setting.” Journal of Forecasting 12, 63–67. Crack, T. 1999. “A classic case of “data snooping” for classroom discussion.” Journal of Financial Education 92–97. Cremers, K. 2002. “Stock return predictability: a Bayesian model selection perspective.” Review of Financial Studies 15(4), 1223–1249. Damodaran, A. 2002. Investment valuation: tools and techniques for determining the value of any asset, 2nd ed., Wiley, NY. Daniel, K., D. Hirshleifer, and A. Subrahmanyam. 2001. “Overconfidence, Arbitrage, and Equilibrium Asset Pricing.” Journal of Finance 56(3), 921–965. Deaton, A. and G. Laroque. 1992. “On the behavior of commodity prices.” Review of Economic Studies 59, 1–23. Deaton, A. and G. Laroque. 2003. “A model of commodity prices after Sir Arthur Lewis.” Journal of Development Economics 71, 289–310. Diebold, F. 1988. “Serial correlation and the combination of forecasts.” Journal of Business and Economic Statistics 6, 105–111. Diebold, F. and J. Lopez. 1996. “Forecast evaluation and combination,” in Handbook of statistics, G. S. Maddala and C. R. Rao (Eds.). North Holland, Amsterdam. Diebold, F. and P. Pauly. 1987. “Structural change and the combination of forecasts.” Journal of Forecasting 6, 21–40. Fama, E. and K. French. 1992. “The cross-section of expected stock returns.” Journal of Finance 47, 427–465. Feltham, G. and J. Ohlson. 1999. “Residual earnings valuation with risk and stochastic interest rates.” Accounting Review 74(2), 165–183. Fung, W. and D. Hsieh. 2001. “The risk in hedge fund strategies: theory and evidence from trend followers.” Review of Financial Studies 14, 313–341. Gilson, S., E. Hotchkiss, and R. Ruback. 2000. “Valuation of bankrupt firms.” Review of Financial Studies 13(1), 43–74. Gorton, G., F. Hayashi, and K. Rouwenhorst. 2007. The fundamentals of commodity futures, Yale working paper. Granger, C. and P. Newbold. 1973. “Some comments on the evaluation of economic forecasts.” Applied Economics 5, 35–47. Granger, C. and R. Ramanathan. 1984. “Improved methods of forecasting.” Journal of Forecasting 3, 197–204.

11 Combining Fundamental Measures for Stock Selection Grinold, R. 1989. “The fundamental law of active management.” Journal of Portfolio Management 15(3), 30–37. He, G. and R. Litterman. 1999. The Intuition Behind BlackLitterman Model Portfolios. Investment Management Division, Goldman Sachs. Hoeting, J., D. Madigan, A. Raftery, and C. Volinsky. 1999. “Bayesian model averaging: a tutorial.” Statistical Science 14(4), 382–417. Kahn, J., S. Landsburg, and A. Stockman. 1996. “The positive economics of methodology.” Journal of Economic Theory 68(1), 64–76. Kaldor, N. 1939. “Speculation and economic stability.” Review of Economic Studies 7, 1–27. Kaplan, S. and R. Ruback. 1995. “The valuation of cash flow forecasts: an empirical analysis.” Journal of Finance 50(4), 1059–1093. Klarman, S. 1991. Margin of safety: risk-averse value investing strategies for the thoughtful investor, HarperBusiness, New York, NY. Klein, R. and V. Bawa. 1976. “The effect of estimation risk on optimal portfolio choice.” Journal of Financial Economics 3, 215–231. Lewellen, J. and J. Shanken. 2002. “Learning, asset-pricing tests, and market efficiency.” Journal of Finance, 57(3), 1113–1145. Lie, E. and H. Lie. 2002. “Multiples used to estimate corporate value.” Financial Analysts Journal 58(2), 44–53. Litzenberger, R. and N. Rabinowitz. 1995. “Backwardation in oil futures markets: theory and empirical evidence.” Journal of Finance 50(5), 1517–1545. Liu, D., D. Nissim, and J. Thomas. 2001. “Equity valuation using multiples.” Journal of Accounting Research 40, 135–172. Lo, A. 2007. Where do alphas come from? A new measure of the value of active investment management, Working paper, Sloan School of Business. Lo, A. and A. MacKinlay. 1990. “Data snooping biases in tests of financial asset pricing models.” Review of Financial Studies 3, 431–468. Ohlson, J. 1980. “Financial ratios and the probabilistic prediction of bankruptcy.” Journal of Accounting Research 18(1), 109–131. Ohlson, J. and Z. Gao. 2006. “Earnings, earnings growth and value.” Foundations and Trends in Accounting 1(1), 1–70. Pastor, L. 2000. “Portfolio selection and asset pricing models.” Journal of Finance 55(1), 179–223. Pastor, L. and R. Stambaugh. 1999. “Cost of equity capital and model mispricing.” Journal of Finance 54(1), 67–121. Penman, S. 2001. Financial statement analysis and security valuation. McGraw-Hill, Boston. Perold, A. 2007. “Fundamentally flawed indexing.” Financial Analysts Journal 63(6), 31–37. Piotroski, J. 2000. “Value investing: the use of historical financial statement information to separate winners from losers.” Journal of Accounting Research 38 (Supplement), 1–41. Seligman, J. 1984. “Reappraising the appraisal remedy.” George Washington Law Review 52, 829. Sharfman, K. 2003. “Valuation averaging: a new procedure for resolving valuation disputes.” Minnesota Law Review 88(2), 357–383. Siegel, J. 2006. “The noisy market hypothesis.” Wall Street Journal June 14, 2006, A14. Sullivan, R., A. Timmermann, and H. White. 1999. “Data-snooping, technical trading rule performance, and the bootstrap.” Journal of Finance 54(5), 1647–1691. Treynor, J. and F. Black. 1973. “How to use security analysis to improve portfolio selection.” Journal of Business 46, 66–86. White, H. 2000. “Reality check for data snooping.” Econometrica 68(5), 1097–1126. Yee, K. 2002. “Judicial valuation and the rise of DCF.” Public Fund Digest 76, 76–84. Yee, K. 2004a. “Perspectives: combining value estimates to increase accuracy.” Financial Analysts Journal 60(4), 23–28. Yee, K. 2004b. “Forward versus trailing earnings in equity valuation.” Review of Accounting Studies 9(2–3), 301–329.

201 Yee, K. 2005a. “Aggregation, dividend irrelevancy, and earnings-value relations.” Contemporary Accounting Research 22(2), 453–480. Yee, K. 2005b. “Control premiums, minority discounts, and optimal judicial valuation.” Journal of Law and Economics 48(2), 517–548. Yee, K. 2007. “Using accounting information for consumption planning and equity valuation.” Review of Accounting Studies 12(2–3), 227–256. Yee, K. 2008a. “A Bayesian framework for combining value estimates.” Review of Quantitative Finance and Accounting 30(3), 339–354. Yee, K. 2008b. “Deep-value investing, fundamental risks, and the margin of safety.” Journal of Investing, 17(3), 35–46. Yee, K. 2008c. “Dueling experts and imperfect verification.” International Review of Law and Economics 28(4), 246–255. Yoo, Y. 2006. “The valuation accuracy of equity valuation using a combination of multiples.” Review of Accounting and Finance 5(2), 108–123. Zellner, A. and V. Chetty. 1965. “Prediction and decision problems in regression models from the Bayesian point of view.” Journal of American Statistical Association 60, 608–616.

Appendix 11A Proof of Theorem 11.1 This section establishes that the specific choice of weights _ ›I and ›C in Theorem 11.1 maximizes the precision of V _ or, equivalently, minimizes varŒV . To this end, consider an _ arbitrary linear weighing scheme V D P P C I VI C C VC , where ›P ; ›I , and ›C are yet to be determined real numbers. Plugging in V D P C e; VI D V C eI , and VC D V C eC and _ requiring EŒV D V immediately yields P D 1  I  C , _ which implies V D V C .1 C I C C /e C I eI C C eC . _ Computing the variance of this expression yields varŒV D .1  I  C /2  2 C I2 I2 C C2 C2  2.1  I  C /C C . _

The first order conditions for minimizing varŒV with respect to ›I and ›C may be written in matrix form as

 2 C I2 . C C / . C C /  2 C C2 C 2 C



I C



D

2 : . C C /

Inverting the matrix recovers the expressions for ›I and ›C _ given in the Theorem. These weights minimize varŒV , which _ 1 _ means they maximize varŒV , the precision of V . t u

11A.1 Generalization of Theorem 11.1 Theorem 11A.1. assumes that error eI from the intrinsic valuation estimate is uncorrelated to market noise e. If an analyst believes that corr.eI ; e/ D I ¤ 0, he or she should replace the weights in Theorem 11.1 with the following formulas: Œ.1  2 / C C . C C /I I  C .1  2 / 2 C2 C 2. C C /I  C I C Œ.1  I2 / 2 C 2 C C C2 I2 Œ.1  I2 / I C .I  C I /C  I kC D 2 2 2 .1   / C C 2. C C /I  C I C Œ.1  I2 / 2 C 2 C C C2 I2 kI D

202

K.K. Yee



and

 2 C I2 C 2I I . C C C I I / . C C C I I /  2 C C2 C 2C

.1  KI  KC / D

.1 

2 / 2 C2

Œ.C C  /I C I  C I C C 2. C C /I  C I C Œ.1  I2 / 2 C 2 C C C2 I2

Proof. Same logic as the proof of Theorem 11.1. When corr.eI ; e/ D I , the new first order optimization condition becomes

D



I C





. C I I / : . C C /

Inverting the matrix recovers the expressions for ›I and ›C . These weights maximize the precision of the triangulation _ estimate V . t u

Chapter 12

On Estimation Risk and Power Utility Portfolio Selection Robert R. Grauer and Frederick C. Shen

Abstract Previous studies show that combining a power utility portfolio selection model with the empirical probability assessment approach (EPAA) to estimate the joint return distribution frequently generates economically and statistically significant abnormal returns. In this paper, we examine additional ways of estimating joint return distributions that allow us to explore the stationary/nonstationary tradeoffs implicit in “expanding” versus “moving” window estimation methods; the benefits of alternative methods of allowing for the “memory loss” inherent in the moving-window EPAA; and the possibility that weighting more recent observations more heavily may improve investment performance. Keywords Estimation risk r Portfolio choice r Power utility In a strict sense, there wasn’t any risk——if the world had behaved as it did in the past. Merton Miller, discussing the fall of Long-Term Capital Management Quoted in Lowenstein (2000), page 61

12.1 Introduction Previous studies show that combining a power utility portfolio selection model with the empirical probability assessment approach (EPAA) to estimate the joint return distribution frequently generates economically and statistically significant abnormal returns. The results are perhaps surprising in light of the fact that the EPAA is based exclusively on historic return data in a 32-quarter “moving window”. In this paper, we examine eight additional ways of R.R. Grauer () Faculty of Business Administration, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6 e-mail: [email protected] F.C. Shen Head, Strategic and International, Group Risk Management, OCBC Bank, Singapore e-mail: [email protected]

estimating joint return distributions that allow us to explore the stationary/nonstationary tradeoffs implicit in “expanding” versus “moving” window estimation methods; the benefits of alternative methods of allowing for the “memory loss” inherent in the moving-window EPAA; and the possibility that weighting more recent observations more heavily may improve investment performance. Although the results show that the EPAA holds its own against the eight other estimation methods in industry-rotation and global asset-allocation settings, the rankings of the estimation methods differ (sometimes dramatically) across datasets, time periods, evaluation methods and investor risk aversion. The paper proceeds as follows. Section 12.2 reviews the literature. Section 12.3 outlines the basic multiperiod investment model and the base-case EPAA. Section 12.4 describes the data. Section 12.5 delineates eight additional ways of estimating the joint return distribution. Section 12.6 presents four methods used to evaluate the ex post performance of the power policies. Section 12.7 reports the results. Section 12.8 summarizes the paper and offers concluding comments. Section 12.9 is an addendum, which shows that combining a moving window approach (to estimate the means from regressions of returns on dividend yields and riskfree interest rates) with an expanding window (to estimate the joint return distribution) holds promise for generating higher portfolio returns while guarding against the possibility of losses in some long forgotten states of nature.

12.2 Literature Review Grauer and Hakansson (1982, 1985, 1986, 1987, 1995a) and Grauer et al. (1990) combine a discrete-time power utility portfolio selection model1 with the empirical probability assessment approach (EPAA) to examine domestic, global and industry rotation asset-allocation problems. The method often generates economically and statistically 1 See Mossin (1968), Hakansson (1971, 1974), Leland (1972), Ross (1974), and Huberman and Ross (1983).

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_12, 

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significant abnormal returns, which is surprising because the EPAA is based exclusively on historic return data in a 32quarter moving window. This raises the question of whether the results are primarily due to the following: the power utility formulation that underlies the multiperiod portfolio selection model, to the inputs of the model, or to the constraints. Grauer and Hakansson (1993) address the first possibility by comparing two approximation schemes for calculating the optimal portfolios in the multiperiod investment model. Specifically, the mean-variance and the quadratic utility approximations are compared to the more cumbersome power utility method. The mean-variance approximation works well for quarterly decisions, but not quite so well for annual decisions. On the other hand, when short sales are permitted, Grauer (2009) shows that the mean-variance model risks bankruptcy ex ante. In cases where the mean-variance model risks bankruptcy, the power utility model adopts radically different investment policies to preclude any ex ante probability of losing 100 percent (or more) of wealth in some state of nature. The second two possibilities may be considered under the general category of estimation risk. The primary thrust of the investigation of estimation error in the inputs has concentrated on alternative estimates of the means—and in the case of the mean-variance model on alternative estimates of the mean vector and covariance matrix. In a mean-variance framework, empirical evidence based on simulation analysis and out-of-sample performance suggests that correcting for estimation risk, particularly in the means, can improve investment performance substantially. See, for example, Jobson, Korkie and Ratti (1979), Jobson and Korkie (1980, 1981), and Jorion (1985, 1991). In a discrete-time multiperiod setting, the results are not as clear. Grauer and Hakansson (1995b, 2001) compare the investment policies and returns of three classes of estimators of the means— the historic or sample estimator; two forms of shrinkage estimators, the James-Stein and Bayes-Stein estimators; and a capital asset pricing model based estimator—with mixed results. Grauer (2008b) extends the analysis adding a dividend yield – riskfree rate estimator of the means.2 The results show that this estimator performs by far the best. In addition, Grauer and Hakansson (1998) employ an inflation adapter to adjust the means that is surprisingly successful in timing the market.

2 Solnik (1993) explores the use of conditioning variables to predict the means in a mean-variance portfolio selection framework. Merton (1971, 1973) points out that if returns are predictable, investors may wish to hedge against shifts in the opportunity set. Kandel and Stambaugh (1996), Brennan et al. (1997), and Barberis (2000), among others, employ a continuous-time framework to examine whether the optimal portfolio weights of an investor with a long horizon are different from those of an investor with a short horizon.

R.R. Grauer and F.C. Shen

Alternatively, it has been suggested that one might adjust for estimation risk by constraining portfolio weights. See, for example, Frost and Savarino (1988) and Grauer (2008c, 2009) in a mean-variance framework, Grauer and Shen (2000) in the context of the discrete-time multiperiod model, and Black and Litterman (1992) who argue against constraining the weights. Again, the evidence is mixed. Frost and Savarino find that constraining the weights improves performance. Grauer (2009) finds that the imposition of nonnegativity constraints improves performance dramatically. Grauer and Shen find that adding constraints beyond nonnegativity constraints does not lead to better performance, while Grauer (2008c) finds that adding (other) constraints beyond non-negativity constraints hinders performance in some periods and improves it in others. This paper investigates eight alternatives to the basecase EPAA. The paper has had an extremely long gestation period—its origins trace to the late 1980s, see Shen (1988). An appealing aspect of power utility functions is that they protect against any ex ante possibility of bankruptcy and in the case of the more risk-averse power utility functions against any large losses—but only in the states of nature captured in the moving-window based EPAA. We worried about what might happen if we ignored the possibility of some catastrophic event not captured in the moving window. Subsequent real-world events—the rise and fall of Long-Term Capital Management—showed how fundamental our concern is and how truly devastating it can be to base an investment strategy on a model that ignores even an infinitesimal possibility of disaster.3 The ability of the moving-window based EPAA to allow for nonstationary joint return distributions is a major strength. However, as noted, a potentially serious shortcoming is that the moving window is 32-quarters in length, which means for decisions made after say the mid- to late1940s (or the mid- to late-1990s) there is no memory of the losses suffered in the great depression (or in 1987).4 One way to alleviate the problem is to employ an “expanding window” that encompasses all-of-history, but assumes joint return distributions are stationary.5 The eight alternatives to the EPAA either: lengthen the estimation period; add “boom” and/or “disaster” states drawn from all-of-history

3 Lowenstein (2000) provides an extremely readable account of the rise and fall of Long-Term Capital. In addition to the quote by Merton Miller, he includes comments by Paul Samuelson, Eugene Fama, and William Sharpe, that question the idea that “real world” risk can ever be completely eliminated. 4 Results based on 40-quarter and from 8- to 15-year estimating periods have also been discussed. 5 Note that we say one way to alleviate the problem. It seems clear that no model can completely overcome the problem of a truly unforeseen event.

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to the moving window; or change the way probabilities of a state are estimated giving more weight to more recent observations. The alternatives allow us to explore the stationary-nonstationary tradeoffs implicit in the expanding versus moving window estimation methods; the benefits of alternative methods of allowing for the “memory loss” inherent in the moving-window EPAA; and the possibility that weighting more recent observations more heavily may improve investment performance.

12.3 The Multiperiod Investment Model The basic model is the same as the one employed in Grauer and Hakansson (1986) and the reader is therefore referred to that paper (specifically pages 288–291) for details. It is based on the pure reinvestment version of dynamic investment theory. At the beginning of each period t, the investor chooses a portfolio, xt , on the basis of some member,  , of the family of utility functions for returns r given by  max E xt

 X 1 1  .1 C rt .xt // D max t s .1 C rt .xt // x   t s (12.1)

subject to xi t 0; all xLt 0; xBt  0; X i

xi t C xLt C xBt D 1; X

mi t xi t  1;

(12.2) (12.3) (12.4)

i

Pr .1 C rt .xt / 0/ D 1;

(12.5)

where P d rts .xt / D xit rits C xLt rLt C xBt rBt is the ex ante return on i

the portfolio in period t if state s occurs,   1 D a parameter that remains fixed over time, xit D the amount invested in risky asset category i in period t as a fraction of own capital, xBt D the amount lent in period t as a fraction of own capital, xBt D the amount borrowed in period t as a fraction of own capital, xt D .x1t ; : : : ; xnt ; xLt ; xBt /, rit D the anticipated total return (dividend yield plus capital gains or losses) on asset category i in period t, rLt D the return on the riskfree asset in period t, d D the interest rate on borrowing at the time of the rBt decision at the beginning of period t, mit D the initial margin requirement for asset category i in period t expressed as a fraction,

ts D the probability of state s at the end of period t, in which case the random return rit will assume the value rits . Constraint (12.2) rules out short sales and ensures that lending (borrowing) is a positive (negative) fraction of capital. Constraint (12.3) is the budget constraint. Constraint (12.4) serves to limit borrowing (when desired) to the maximum permissible under the margin requirements that apply to the various asset categories. Constraint (12.5) rules out any ex ante probability of bankruptcy.6 Under the base-case scenario employed in most previous studies, the inputs to the model are derived from the empirical probability assessment approach (EPAA). Consider the EPAA and suppose quarterly revision is used. Then, at the beginning of quarter t, the portfolio problem (12.1)–(12.5) for that quarter uses the following inputs: the (observable) riskfree return for quarter t, the (observable) call money rate plus one percent at the beginning of quarter t, and the (observable) realized returns for the risky asset categories for the previous k quarters. Each joint realization in quarters tk through t1 is assigned a probability 1=k of occurring in quarter t. Thus, under the EPAA, estimates are obtained on a moving basis and used in raw form without adjustment of any kind. On the other hand, since the whole joint distribution is specified and used, there is no information loss—all moments and correlations are implicitly taken into account. It may be noted that the empirical distribution of the past k periods is optimal if the investor has no information about the form and parameters of the true distribution, but believes that this distribution went into effect k periods ago. With these inputs in place, the portfolio weights xt for the various asset categories and the proportion of assets borrowed are calculated by solving system (12.1)–(12.5) via nonlinear programming methods.7 At the end of quarter t, the realized returns on the risky assets are observed, along r (which may differ from with the realized borrowing rate rBt d 8 the decision borrowing rate rBt ). Then, using the weights selected at the beginning of the quarter, the realized return on the portfolio chosen for quarter t is recorded. The cycle is then repeated in all subsequent quarters.9

6

The solvency constraint (12.5) is not binding for the power functions, with  < 1, and discrete probability distributions with a finite number of outcomes because the marginal utility of zero wealth is infinite. Nonetheless, it is convenient to explicitly consider (12.5) so that the nonlinear programming algorithm used to solve the investment problems does not attempt to evaluate an infeasible policy as it searches for the optimum. 7 The nonlinear programming algorithm employed is described in Best (1975). 8 The realized borrowing rate is calculated as a monthly average. 9 Note that if k D 32 is under quarterly revision, then the first quarter for which a portfolio can be selected is b C 32, where b is the first quarter for which data is available.

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All reported returns are gross of transaction costs and taxes and assume that the investor in question had no influence on prices. There are several reasons for this approach. First, as in previous studies, we wish to keep the complications to a minimum. Second, the return series used as inputs and for comparisons also exclude transaction costs (for reinvestment of interest and dividends) and taxes. Third, many investors are tax-exempt and various techniques are available for keeping transaction costs low. Finally, since the proper treatment of these items is nontrivial, they are better left to a later study.

12.4 The Data The data used to estimate the probabilities of next period’s returns on risky assets, and to calculate each period’s realized returns on risky assets, come from several sources. The quarterly returns series for the U.S. asset categories (common stock, long-term corporate bonds, and long-term government bonds) are provided by Ibbotson Associates. The quarterly database on non-U.S. equity returns for 1960–99 covering seven countries (Canada, France, Germany, Japan, the Netherlands, Switzerland, and the United Kingdom) comes from Brinson Associates (1960–70) and from Ibbotson Associates (1970–99). All returns are expressed in U.S. dollars and represent total returns since both dividends (net of foreign taxes withheld) and capital appreciation (or depreciation) are taken into account. The returns from the U.S. value-weighted industry groups are constructed from the returns on individual NYSE firms contained in the Center for Research in Security Prices (CRSP) monthly returns database. The industry data in this paper updates the 1934–86 data in Grauer et al. (1990) through 1999. The firms are combined into 12 industry groups on the basis of the first two digits of the firms’ SIC codes, with value-weighted industry indices constructed from the same universe of firms. The riskfree asset is assumed to be 90-day U.S. Treasury bills maturing at the end of the quarter. We use the Survey of Current Business and the Wall Street Journal as sources. The borrowing rate is assumed to be the call money rate plus one percent for decision purposes (but not for rate of return calculations). The applicable beginning of period decision rate, d , is viewed as persisting throughout the period and thus as rBt riskfree. For 1934–76, the call money rates are obtained from the Survey of Current Business. For later periods, the Wall Street Journal is the source. Finally, margin requirements for stocks are obtained from the Federal Reserve Bulletin.10 10

There is no practical way to take maintenance margins into account in our programs. In any case, it is evident from the results that they would

R.R. Grauer and F.C. Shen

For comparative purposes, we calculate and include the returns on unlevered and levered equal- or value-weighted benchmark portfolios of the risky assets in the investment universe. Table 12.1 summarizes the compositions of these portfolios along with an enumeration of the asset categories included in the study.

12.5 Alternative Ways of Estimating the Joint Return Distribution Nine approaches to estimating the joint return distribution are described in this section. The results, reported in Section 12.7, determine whether one approach dominates the others empirically. The Empirical Probability Assessment Approach. The basecase EPAA is a moving window method for estimating next period’s joint return distribution where only the most recent k periods of jointly realized returns are used. As we move forward in time, the earliest observation is dropped while the most recent one is added. Thus, implicit in the approach is that the joint return distribution is stationary for the last k quarters. The interesting modeling question is: What is the optimal choice of the lag period k? Intuitively, the lag should be long enough to cover a business cycle and to insure that the number of states is greater than the number of asset categories.11 At the same time, the lag should be short enough to reflect up-to-date information and to allow for possible nonstationarities in the data. In this paper, we focus on the basecase 32-quarter estimating period that is employed in most of the previous multiperiod asset allocation papers. Under the EPAA in quarter t, each of the 32 most recent joint return realizations (states) is assumed to occur with a probability of 1/32. However, a possible shortcoming of the EPAA is that it only considers data within the moving window. Therefore, it may underestimate the probability of either “boom” or “disaster.” The “All-of-History” Approach. One way to overcome the possibility of underestimating either very good or very bad outcomes is to employ an expanding window that uses all past returns to estimate the (assumed stationary) joint return distribution. At the beginning of quarter t, returns from quarter 1 (the first available data) to quarter t1 are used. At the beginning of quarter tC1, returns from quarter 1 to quarter t are used to estimate the joint return distribution, and so on.

come into play only for the more risk-tolerant strategies (and even for them only occasionally) and the net effect would be relatively neutral. 11 If the number of assets were equal to the number of states, there would be a complete market. In the absence of constraints other than the budget constraint, the optimizer would then think it could achieve any arbitrary return distribution.

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Table 12.1 Asset category and fixed-weight portfolio symbols RL Riskfree Lending (90-day US Treasury Bills) GB Long-term US Government Bonds CB Long-term US Corporate Bonds CS US Common Stocks (S&P 500) B Borrowing FR French Equities GE German Equities JA Japanese Equities NE Dutch Equities CA Canadian Equities SI Swiss Equities UK British Equities EW Equal-weighted portfolio of risky assets in investment universe E5 50% in EW, 50% in RL E15 150% in EW, 50% in B E20 200% in EW, 100% in B

Each joint realization is given an equal probability 1=.t1/ of occurring in quarter t. Note that the probabilities become smaller as we move forward in time, unlike those in the moving window EPAA. For example, with 12 industries in the fourth quarter of 1999, each of 295 possible states is assumed to occur with equal probability. Comparisons of the returns generated from applying the all-of-history and EPAA allow us to explore the stationary-nonstationary tradeoffs implicit in the expanding versus moving window estimation methods. The “Sum-of-the-Digits” Approach. The polar opposite to the all-of-history approach focuses on the most up-to-date data, weighting the more recent observations in the k-quarter window of the EPAA more heavily than earlier observations. Probabilities are assigned by first letting KD

k X

i:

(12.6)

i D1

Then, at the beginning of quarter t, the joint realization of quarter t 1 is assigned probability k=K; the joint realization of quarter t  2 is assigned probability .k  1/=K, and so on, until the joint realization of quarter t k is assigned probability 1=K of occurrence. Thus, more weight is assigned to the recent observations using a linear decay function. The idea is borrowed from the exponentially weighted moving average model of univariate time series modeling, as well as from a common sense argument that the recent past may be a better guide to what might happen next period than is the more distant past. If k D 32, then K D ..33/.32//=2. Thus, in quarter t, the probability of the most recent joint return realization is 32=K and the probability of the joint realization from quarter t  k is 1=K. The “Disaster States” Approach. This approach combines the most recent joint return realizations in the EPAA moving window with the “worst” joint realizations that occurred

PETR FINA CDUR BASI FOOD CONS CAPG TRAN UTIL TEXT SERV LEIS

Petroleum Finance and Real Estate Consumer Durables Basic Industries Food and Tobacco Construction Capital Goods Transportation Utilities Textiles and Trade Services Leisure

VW V5 V15 V20

Value-weighted portfolio of risky assets in investment universe 50% in VW, 50% in RL 150% in VW, 50% in B 200% in VW, 100% in B

through all-of-history. Let k be the number of quarters used in the EPAA and n be the number of risky assets. At the beginning of quarter t, the worst state for each asset i (defined as its smallest return) from quarter 1 (the first observation) to quarter t1 (the most recent observation) is recorded. These n disaster states are then appended to the k most recent states of the EPAA to obtain n C k states as the estimate of the joint return distribution for quarter t. The probabilities of each state are determined by the following rule: the probability of each of the k most recent states is equal to 1=.k C j /, and the probability of each of the n disaster states is equal to j=..k C j /n/, where j is an integer such that 1 < j < n. Clearly, the disaster-states approach provides a more conservative estimate of the joint return distribution than the EPAA. Furthermore, letting j vary from 1 to n varies the degree of conservatism. (If j D n, equal probability is assigned to each state.) In this case, the interesting modeling question is: What is a reasonable value for the weighting factor j ? The tradeoff is to keep the investor aware of disaster states, but not to assign so much probability to them so as to completely drive the investor out of the equity markets. In the empirical work, we investigate two possibilities letting j D 1; 2. In the case of 12 industries and a 32-quarter estimation period, each of the 32 normal states (representing the joint return realizations on the 12 industries) is assigned a probability of either 1/33, if j D 1; or 1/34, if j D 2. Each of the 12 disaster states is assigned a probability of either 1/((33)12), if j D 1; or 2/((34)12), if j D 2. Thus, if j D 1 .j D 2/, 32/33 (32/34) of the total probability is assigned to the normal states and the remainder—either 1/33 or 2/34—to the 12 disaster states. The “Boom-Disaster States” Approach. This approach differs from the disaster states approach in that the “best” and “worst” outcomes for each asset category are recorded. Thus, at the beginning of quarter t, the best and worst state for each

208

asset i from quarter 1 to quarter t  1 are recorded. These 2n boom or disaster states are then appended to most recent k states of the EPAA to obtain 2n C k states as the estimate of the joint return distribution for quarter t. The amended rule for assigning the probabilities is: the probability of each of the k most recent states is equal to 1=.k C j /, and the probability of each of the 2n boom or disaster states is equal to j=..kCj /2n/, where j is an integer such that 1 < j < 2n. In the empirical work, we investigate three possibilities letting j D 1, 2 and 4. In the case of 12 industries and a 32-quarter estimation period, each of the 32 normal states is assigned a probability of either 1/33, if j D 1; or 1/36, if j D 4; while each of the 24 boom or disaster states is assigned a probability of either 1/((33)22), if j D 1; or 4/((34)24), if j D 4. Thus, if j D 1 .j D 4/, 32/33 (32/36) of the total probability is assigned to the normal states and the remainder—either 1/33 or 4/36—to the 24 boom or disaster states. The “All-of-History Sum-of-the-Digits” Approach. The final approach assigns sum-of-digits probabilities to each joint realization through all-of-history. In assigning the probabilities in quarter t, we first let the sum in (6) run from 1 to t  1 (instead of from 1 to k). Then, at the beginning of quarter t, the joint realization of quarter t  1 is assigned probability .t  1/=K; the joint realization of quarter t  2 is assigned probability .t  2/=K and so on, until the joint realization of quarter 1 is assigned probability 1=K. For example, with 12 industries in the fourth quarter of 1999, t D 296, the number of states equals 295, and K D ..296/.295//=2. Thus, the probability of the most recent joint return realization is 295=K and the probability of the joint realization from quarter 1 is 1=K.

12.6 Alternate Ways of Evaluating Investment Performance The expected utility-based objective function (1) of multiperiod portfolio theory has no direct tie to any of the commonly used statistical measures of investment performance. Therefore, we employ an eclectic mix of four methods for evaluating ex post performance: compound return versus standard deviation of return, Jensen’s (1968) alpha, Grinblatt and Titman’s (1993) portfolio change measure, and ex post certainty equivalent returns. Each of the methods has a certain intuitive appeal. On the other hand, each method embodies economic or statistical assumptions that may be at variance with: the investor’s objective function, the assumptions employed in estimating the joint return distributions, or the assumptions used in the other methods of measuring performance. Consequently, it is interesting to see whether the rankings of the different performance measures agree. Compound Return – Standard Deviation Pairs. Compound (or geometric mean) return is an intuitive, frequently

R.R. Grauer and F.C. Shen

used measure employed by both practitioners and academics in performance comparisons. It provides a perfect ranking of what a given initial amount would have grown to under reinvestment over the period under question. When it comes to measuring variability, standard deviation has few contenders. We make two comparisons. First, we compare the compound return - standard deviation pairs of the power policies generated from different ways of estimating the joint return distribution with those of the asset categories, and with those of the levered equal- and value-weighted benchmarks. Then, for each of the powers we compare the nine ways of estimating the return distributions ranking the estimation methods from high to low based on compound returns. Jensen’s Alpha. In academic circles, Jensen’s (1968) alpha is probably the most commonly employed performance measure. In this paper, for each portfolio j (power utility policy) we run the regression rjt  rLt D ˛j C ˇj .rmt  rLt / C ejt ;

(12.7)

where rjt is the quarterly rate of return on portfolio j; rmt is the quarterly rate of return on the CRSP value-weighted index, and rLt is the rate of return on three-month Treasury bills. The intercept ˛j is the measure of risk-adjusted investment performance. Positive (negative) values indicate superior (inferior) performance. The null hypothesis is that there is no superior investment performance and the alternative hypothesis is that there is. Thus, we report the results of onetailed t-tests. However, the test embodies both economic and statistical assumptions, and is not without its critics. In terms of economics, alpha—the intercept from Equation (12.7)—reduces the tradeoff between reward (ex post arithmetic average return) and risk (beta) envisioned in the Sharpe (1964) - Lintner (1965) capital asset pricing model into a single risk-adjusted rate of return that can be interpreted as the distance from the security market line. In statistical terms, both the alpha and beta in Equation (12.7) are assumed to be constant in the performance evaluation period. At a deeper level, Roll (1978) shows that the alphas and betas are a function of the benchmark portfolio (i.e., the proxy for the market portfolio). If the benchmark portfolio is exactly mean-variance efficient, there is no abnormal performance. If two benchmark portfolios are not exactly mean-variance efficient, they can, in principle, lead to completely different sets of “winners and losers”. See also Dybvig and Ross (1985), Elton et al. (1993), Grauer (1991), and Green (1986). The ambiguity this creates in a domestic market is bad enough. But with a universe that goes much beyond U.S. common stocks, the lack of a commonly accepted benchmark portfolio makes it difficult to justify employing the Jensen test. Thus, we limit our analysis of alphas to the industry dataset. As noted above, the assumptions embodied in the measures of performance may conflict with the assumptions

12 On Estimation Risk and Power Utility Portfolio Selection

embodied in the investors’ choices of optimal portfolios. For example, in this paper the investors are viewed as price-takers—no particular equilibrium model is assumed or relevant. Furthermore, the investors maximize expected utility employing power utility functions, whereas the Jensen measure is based on a model of market equilibrium where all investors are mean-variance decision makers. Grinblatt and Titman’s Portfolio Change Measure. In contrast to most other performance measures, Grinblatt and Titman’s (1993) Portfolio Change Measure (PCM) employs portfolio holdings as well as rates of return and does not require an external benchmark portfolio (i.e., a proxy for the market portfolio). In order to motivate the portfolio change measure, assume that uninformed investors perceive that the vector of expected returns is constant, while informed investors can predict whether expected returns vary over time. Informed investors can profit from changing expected returns by increasing (decreasing) their holdings of assets whose expected returns have increased (decreased). The holding of an asset that increases with an increase in its conditional expected rate of return will exhibit a positive unconditional covariance with the asset’s returns. The portfolio change measure is constructed from an aggregation of these covariances. For evaluation purposes, let X ri t .xi t  xi;t j /; PCM t D

209

as performance is measured. Thus, there we do not have to worry about possible gaming or window dressing problems that face researchers trying to gauge the performance of mutual funds. However, it is worth noting that the PCM imposes stronger assumptions of stationarity in returns than other measures of portfolio performance. Moreover, with the exception of the all-of-history approach, the strong stationarity assumption is not made in the other eight ways of estimating the joint return distributions. Ex Post Certainty Equivalent Returns. The final performance measure is based on knowledge of the investor’s utility function. At a point in time each investor, employing an estimate of the joint return distribution, maximizes the ex ante expected utility of returns by solving (12.1)–(12.5). For example, employing industry data, each investor makes a time series of 264 portfolio choices, beginning in the first quarter of 1934 and ending in the fourth quarter of 1999, one set for each of the nine ways of estimating the joint return distribution. Now, suppose the investor is asked to evaluate the ex post utility of each of the nine time series of portfolio returns. Define the ex post utility of the time series of (optimal ex ante) portfolio returns as EU D

T X 1 1 .1 C rt .xt // ; T  t D1

EU D

T X 1 In.1 C rt .xt //; T t D1

i

where rit is the quarterly rate of return on asset i at time t; xit and xi;t j are the holdings of asset i at time t and time t  j , respectively. This expression provides an estimate of the covariance between returns and weights at a point in time. Alternatively, it may be viewed as the return on a zero-weight portfolio. The portfolio change measure is an average of the PCM t s XX    ri t xi t  xi;t j =T ; (12.8) PCM D t

  1;

 ¤ 0;

 D 0;

(12.9)

where each portfolio return .1 C rt .xt // is assigned equal probability of occurrence. While Equation (12.9) gives the desired ranking of the nine time series of returns for each of the powers, the utility numbers are non-intuitive, particularly for the low powers. Therefore, the expected utility numbers from Equation (12.9) are transformed into certainty equivalent returns as follows

i 1

where T is the number of time-series observations. The portfolio change measure test itself is a simple t-test based on the time series of zero-weight portfolio returns; that is, p  t D PCM= .P CM / T ; where .PCM/ is the standard deviation of the time series of PCM t s. In their empirical analysis of mutual fund performance, Grinblatt and Titman work with two values of j that represented one- and four-quarter lags. We compute the PCM for the same two lags, but to save space we report results for the four-quarter lag only. The PCM seems particularly apropos in the present study because the portfolio weights are chosen according to a prespecified set of rules over the same quarterly time interval

1 C rCE D ŒEU  ;

  1;

1 C rCE D expŒEU ;

 D 0:

 ¤ 0; (12.10)

At the risk of oversimplifying, we draw the following analogy between the certainty equivalent return and Jensen’s alpha. Alpha reduces the reward (ex post average return) to risk (beta) tradeoff of the mean-variance capital asset pricing model into a single risk-adjusted rate of return. Similarly, we can think of the certainty equivalent return as reducing the reward (ex post average return) to risk (standard deviation) tradeoff of a specific power utility investor into a single risk-adjusted rate of return.12 12 The argument is simplified as power utility weights all moments of the distribution not just the mean and variance. We also note that Kandel

210

12.7 The Results We evaluate the ex post performance of ten power utility investment strategies when the joint return distributions are estimated nine different ways in industry-rotation and global asset-allocation settings. Four methods are employed in the evaluation: compound return – standard deviation pairs, Jensen’s alpha, Grinblatt and Titman’s portfolio change measure, and ex post certainty equivalent returns. Naturally, space limitations dictate that we can only report a subset of the results. To keep the figures relatively uncluttered we report compound return – standard deviation plots for seven of the ten utility functions focusing on four of the nine ways of estimating the joint return distribution—the base-case EPAA, the polar opposite all-of-history and sum-of-digits approaches, and the boom-disaster 1 approach. In the tables, we report on all nine estimation methods focusing on two power utility functions—the extremely risk-averse power 30 and the well-known much less risk-averse logarithmic utility function. The results in the tables are sorted from best to worst. In Table 12.2 the sort is by compound return, in Table 12.3 by alpha, in Table 12.4 by the portfolio change measure, and in Table 12.5 by certainty equivalent return. In addition, in each of the tables the results for the all-ofhistory, boom-disaster 1, EPAA, and sum-of-digits estimation approaches are presented in bold type so that the results may be more easily compared with the results in the figures. Compound Return – Standard Deviation Pairs. Fig. 12.1 plots the annual compound returns and the standard deviations13 of the realized returns for four sets of ten power utility strategies under quarterly revision for the 32-year period 1968–99 when the risky portion of the investment universe consists of 12 U.S. value-weighted industries. The first set of strategies (see dark circles) shows the returns generated from the base-case EPAA. The second set of strategies (see open circles) shows the returns from the all-of-history approach. The third set (see open squares) shows the returns generated from the sum-of-digits approach. The fourth set (see open triangles) shows the returns from the boom-disaster 1 approach. The figure also shows the compound returns and standard deviations of the returns of the up- and down-levered valueweighted benchmark portfolios (see open diamonds). With the exception of power 1, the EPAA strategies easily outperform the polar opposite sum-of-digits and all-of-history approaches. Moreover, the sum-of-digits approach outperforms the all-of-history approach. The boom-disaster 1 approach performs very well for high powers and not as well and Stambaugh (1996) and Grauer and Shen (2000) have employed certainty equivalent returns to measure investment performance. 13 For consistency with the geometric mean, the standard deviation is based on the log of one plus the rate of return. This quantity is very similar to the standard deviation of the rate of return for levels less than 25 percent.

R.R. Grauer and F.C. Shen

for the low powers. Allowing for the possibility of (boom and) disaster makes the more risk-averse investors even more conservative, penalizing their ex post performance—at least as measured in terms of ex post compound return. Finally, the active strategies perform well relative to the value-weighted benchmarks, except for the high power strategies based on the all-of-history approach.14 Figure 12.2 presents the compound return – standard deviation plot for the 32-year period 1968–99 when the risky portion of the investment universe is ten global asset categories. The difference between the sum-of-digits and all-ofhistory plots in Figs. 12.1 and 12.2 is striking. In the global universe, the all-of-history estimation approach clearly dominates the sum-of-digits method for the high (less risk-averse) powers—just the opposite of what occurs with the U.S. industry universe.15 As in the industry universe, the EPAA and boom-disaster 1 approaches closely shadow each other and, with the possible exception of the power 1 strategy, dominate the sum-of-digits and all-of-history approaches. Finally, the returns earned by the four estimation approaches do not so clearly dominate the levered equal-weighted benchmarks. Table 12.2 contains the compound return-standard deviation results for all nine estimation methods for the power 30 and the logarithmic utility function in the value-weighted industries and global universes. The table shows that for the power 30 there is little difference between the returns earned by the nine estimation methods. For the logarithmic utility function the differences are much more pronounced.16 Finally, the results reinforce the observed differences (see Figs. 12.1 and 12.2) in the rankings of the sum-of-digits and all-of-history estimation methods in the global and industry settings. Jensen’s Alpha. Table 12.3 shows the results for Jensen’s test for the power 30 and logarithmic utility functions in the value-weighted industries universes when the joint return distributions are estimated in nine ways in the 1934–99 and 1968–99 periods. We note three main points from the table. First, all but three of the policies exhibit positive abnormal returns and slightly more than one-third are statistically significant.17 Second, the EPAA and all-of-history estimation approaches consistently rank among the best and worst, respectively. Third, the rankings are highly correlated with the compound return rankings reported in Table 12.2. 14

Figure 12.1 also contains the compound return-standard deviation pairs for a combined approach discussed in Section 12.9. 15 We note, however, that the starting point for the all-of-history estimation period is 1926 in the industry dataset and 1960 in the global dataset. 16 The differences are even more pronounced in an equal-weighted industries universe, reported in the first version of the paper. 17 In the equal-weighted industries universe, all the power policies perform extremely well—especially in the 1968–90 period where the highly statistically significant abnormal returns range from approximately 45 to over 60 percent of excess returns.

16.57 16.47 16.37 15.12 14.65 14.42 13.47 13.42 10.77

6.94 6.72 5.25 5.05 4.97 4.86 4.86 4.66 4.53

31.22 31.85 30.14 26.36 28.00 35.05 21.36 25.24 19.54

7.60 7.36 3.76 3.52 3.50 3.43 3.29 3.30 3.29

Standard deviation

Boom Disaster 1 Boom Disaster 2 EPAA Disaster 1 Boom Disaster 4 Sum of Digits (SOD) Disaster 2 All of History SOD All of History

EPAA Sum of Digits (SOD) Boom Disaster 1 Boom Disaster 2 Disaster 1 All of History SOD Boom Disaster 4 Disaster 2 All of History

17.12 16.51 16.13 15.77 14.17 14.09 13.59 11.44 10.54

8.87 8.58 7.90 7.66 7.60 7.49 7.44 7.28 7.22

Compound return

30.02 29.72 28.86 21.32 27.76 32.51 13.35 23.62 17.50

3.80 4.33 2.99 2.81 2.76 2.70 2.65 2.65 2.55

Standard deviation

Disaster 1 All of History SOD Disaster 2 Boom Disaster 1 EPAA Boom Disaster 2 Boom Disaster 4 All of History Sum of Digits (SOD)

EPAA Boom Disaster 1 Sum of Digits (SOD) Boom Disaster 2 Disaster 1 Boom Disaster 4 Disaster 2 All of History SOD All of History

19.13 19.05 18.98 18.88 18.68 18.55 17.99 17.59 12.27

9.33 8.81 8.71 8.57 8.48 8.30 8.12 8.10 8.02

Compound return

43.44 47.63 44.24 42.34 42.35 42.75 43.74 45.06 42.31

3.92 3.50 4.33 3.37 3.29 3.22 3.09 3.16 2.94

Standard deviation

There are two investment universes: (1) 12 U.S. value-weighted industry groups and (2) eight country common stock indices, U.S. long-term government bonds, and U.S. long-term corporate bonds. The power portfolios are revised quarterly in the 1934–1999 and 1968–1999 periods. Borrowing is permitted. The compound return of each of the nine estimators is sorted from high to low. The benchmark empirical probability assessment approach, sum-of-digits, all-of-history, and boom-disaster 1 estimators are presented in bold type

Boom Disaster 1 EPAA Boom Disaster 2 Disaster 1 Boom Disaster 4 Sum of Digits (SOD) Disaster 2 All of History SOD All of History

Panel B: Power 0

Sum of Digits (SOD) EPAA Boom Disaster 1 Boom Disaster 2 Disaster 1 All of History SOD Boom Disaster 4 Disaster 2 All of History

Panel A: Power 30

Compound return

Table 12.2 Compound return and standard deviation of annual returns for power 30 and power 0 portfolios generated from nine estimators of the joint return distribution Value-weighted industries 1934–1999 Value-weighted industries 1968–1999 Global asset categories 1968–1999

12 On Estimation Risk and Power Utility Portfolio Selection 211

2.69 3.41 4.01 4.00 3.86 3.61 3.33 2.89 1.98

0.66 0.72 0.27 0.22 0.20 0.18 0.18 0.13 0.10 0.77 1.10 1.37 1.41 1.39 1.39 1.32 1.16 0.94

0.17 0.20 0.08 0.07 0.06 0.05 0.06 0.04 0.04 0.93 0.91 0.88 0.78 0.70 0.44 0.32 0.26 0.15

0.28 0.27 0.09 0.07 0.07 0.06 0.05 0.05 0.02 0.03 0.06 0.07 0.07 0.08 0.24 0.22 0.25 0.71

0.03 0.04 0.03 0.03 0.03 0.02 0.05 0.03 0.14 Disaster 1 Boom Disaster 1 EPAA Boom Disaster 2 Disaster 2 Sum of Digits (SOD) Boom Disaster 4 All of History SOD All of History

EPAA Sum of Digits (SOD) Boom Disaster 1 Disaster 1 Boom Disaster 2 Boom Disaster 4 Disaster 2 All of History SOD All of History 2.95 3.71 3.49 3.44 1.92 2.98 2.68 1.84 1.27

0.49 0.43 0.24 0.17 0.19 0.13 0.10 0.15 0.08 1.11 1.57 1.51 1.51 0.64 1.36 1.39 1.14 0.87

0.15 0.18 0.09 0.06 0.07 0.05 0.03 0.06 0.04

1.11 1.10 1.00 0.94 0.86 0.72 0.38 0:05 0.17

0.23 0.13 0.10 0.08 0.07 0.05 0.05 0.04 0.02

Alpha

0.10 0.10 0.15 0.12 0.08 0.24 0.28 0.53 0.65

0.03 0.22 0.04 0.04 0.05 0.06 0.04 0.12 0.19

1-tail p-value

The investment universe consists of 12 U.S. value-weighted industry groups. The power portfolios are revised quarterly in the 1934–1999 and 1968–1999 periods. Borrowing is permitted. Average arithmetic excess returns (raw returns less the riskfree lending rate) and alphas are measured in percent per quarter. The heteroskedasticity-consistent p-values measure the significance of the alphas relative to zero. The alpha of each of the nine estimators is sorted from high to low. The benchmark empirical probability assessment approach, sum-of-digits, all-of-history, and boom-disaster 1 estimators are presented in bold type

Disaster 2 Disaster 1 EPAA Boom Disaster 1 Boom Disaster 2 Sum of Digits (SOD) Boom Disaster 4 All of History SOD All of History

Panel B: Power 0

EPAA Sum of Digits (SOD) Boom Disaster 1 Boom Disaster 2 Disaster 1 Boom Disaster 4 All of History SOD Disaster 2 All of History

Panel A: Power 30

Table 12.3 Average excess returns, betas, and alphas for power 30 and power 0 portfolios generated from nine estimators of the joint return distribution Value-weighted industries 1934–1999 Value-weighted industries 1968–1999 Excess return Beta Alpha 1-tail p-value Excess return Beta

212 R.R. Grauer and F.C. Shen

1.53 1.33 1.10 1.03 0.97 0.96 0.66 0.45 0.33

0.15 0.14 0.09 0.07 0.07 0.05 0.04 0.04 0.02 0.00 0.00 0.00 0.01 0.00 0.03 0.01 0.07 0.03

0.02 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 EPAA Boom Disaster 1 Disaster 1 Disaster 2 Boom Disaster 2 Sum of Digits (SOD) All of History SOD All of History Boom Disaster 4

EPAA Boom Disaster 1 Sum of Digits (SOD) Boom Disaster 2 Disaster 1 Disaster 2 All of History SOD Boom Disaster 4 All of History 1.57 1.49 1.24 1.14 1.00 0.96 0.52 0.38 0.28

0.18 0.10 0.10 0.08 0.08 0.06 0.04 0.04 0.02 0.02 0.03 0.04 0.01 0.07 0.09 0.09 0.07 0.29

0.02 0.01 0.12 0.02 0.01 0.01 0.02 0.06 0.02

All of History All of History SOD Disaster 2 Disaster 1 EPAA Boom Disaster 1 Boom Disaster 2 Sum of Digits (SOD) Boom Disaster 4

EPAA Boom Disaster 1 All of History SOD Boom Disaster 2 Boom Disaster 4 Disaster 1 All of History Disaster 2 Sum of Digits (SOD)

1.05 0.98 0.90 0.61 0.52 0.51 0.47 0.47 0.43

0.13 0.11 0.09 0.09 0.09 0.09 0.08 0.08 0.02

0.01 0.01 0.05 0.16 0.22 0.21 0.22 0.30 0.23

0.02 0.02 0.01 0.02 0.02 0.02 0.00 0.03 0.43

1-tail p-value

There are two investment universes: (1) 12 U.S. value-weighted industry groups and (2) eight country common stock indices, U.S. long-term government bonds, and U.S. long-term corporate bonds. The power portfolios are revised quarterly in the 1934–1999 and 1968–1999 periods. Borrowing is permitted. The portfolio change measure (PCM), which employs a four-quarter lag, is measured in percent per quarter. The p-values measure the significance of the portfolio change measures relative to zero. The portfolio change measure of each of the nine estimators is sorted from high to low. The benchmark empirical probability assessment approach, sum-of-digits, all-of-history, and boom-disaster 1 estimators are presented in bold type

EPAA Boom Disaster 1 Disaster 1 Boom Disaster 2 Disaster 2 Sum of Digits (SOD) All of History SOD Boom Disaster 4 All of History

Panel B: Power 0

Sum of Digits (SOD) EPAA Boom Disaster 1 Boom Disaster 2 Disaster 1 Disaster 2 All of History SOD Boom Disaster 4 All of History

Table 12.4 Portfolio change measures for power 30 and power 0 portfolios generated from nine estimators of the joint return distribution Value-weighted industries 1934–1999 Value-weighted industries 1968–1999 Global asset categories 1968–1999 PCM 1-tail p-value PCM 1-tail p-value PCM Panel A: Power 30

12 On Estimation Risk and Power Utility Portfolio Selection 213

5.02 5.03 4.88 4.43 4.35 4.63 3.71 3.91 3.00

1.24 1.20 1.20 1.29 1.22 1.15 1.12 1.73 1.68 15.31 15.47 14.61 13.30 13.39 15.50 10.24 11.81 8.96

1.13 0.99 1.03 1.28 1.10 0.94 0.90 3.03 2.84 3.91 3.88 3.86 3.58 3.48 3.43 3.21 3.20 2.59

1.05 1.05 1.04 1.04 1.04 1.02 1.00 0.52 1.48 Boom Disaster 1 Boom Disaster 2 EPAA Disaster 1 Boom Disaster 4 Sum of Digits (SOD) Disaster 2 All of History SOD All of History

Boom Disaster 1 Boom Disaster 2 Disaster 1 Boom Disaster 4 All of History SOD Disaster 2 All of History EPAA Sum of Digits (SOD) 5.39 5.12 5.17 4.63 4.36 4.66 3.60 3.52 2.95

1.92 1.87 1.85 1.81 1.83 1.78 1.76 2.17 2.11 17.12 16.24 16.85 13.75 14.48 16.32 8.73 12.22 8.96

1.16 0.98 0.94 0.84 0.91 0.77 0.75 2.01 2.44 4.03 3.89 3.81 3.73 3.37 3.35 3.24 2.75 2.54

1.72 1.72 1.72 1.71 1.70 1.69 1.68 1.54 0.81

Disaster 1 All of History SOD Disaster 2 Boom Disaster 1 EPAA Boom Disaster 2 Boom Disaster 4 All of History Sum of Digits (SOD)

Boom Disaster 2 Boom Disaster 4 Disaster 1 All of History Boom Disaster 1 Disaster 2 All of History SOD EPAA Sum of Digits (SOD)

6.07 6.35 5.97 6.03 6.01 5.96 5.85 5.97 4.55

2.09 2.02 2.06 1.95 2.14 1.98 1.97 2.27 2.14

18.07 19.07 17.44 18.29 18.47 18.19 18.04 18.93 17.90

1.38 1.24 1.33 1.10 1.53 1.15 1.22 1.89 2.35

Standard deviation

4.47 4.46 4.44 4.42 4.37 4.35 4.22 4.14 2.94

1.78 1.78 1.78 1.77 1.77 1.77 1.74 1.70 0.74

Certainty equivalent

There are two investment universes: (1) 12 U.S. value-weighted industry groups and (2) eight country common stock indices, U.S. long-term government bonds, and U.S. long-term corporate bonds. The power portfolios are revised quarterly in the 1934–1999 and 1968–1999 periods. Borrowing is permitted. The arithmetic means, standard deviations, and certainty equivalent returns are measured in percent per quarter. The certainty equivalent return of each of the nine estimators is sorted from high to low. The benchmark empirical probability assessment approach, sum-of-digits, all-of-history, and boom-disaster 1 estimators are presented in bold type

Boom Disaster 1 EPAA Boom Disaster 2 Disaster 1 Boom Disaster 4 Sum of Digits (SOD) Disaster 2 All of History SOD All of History

Panel B: Power 0

Boom Disaster 2 Boom Disaster 4 All of History SOD Boom Disaster 1 Disaster 1 Disaster 2 All of History Sum of Digits (SOD) EPAA

Panel B: Power 30

Table 12.5 Certainty equivalent returns for power 30 and power 0 portfolios generated from nine estimators of the joint return distribution Value-weighted industries 1934–1999 Value-weighted industries 1968–1999 Global asset categories 1968–1999 Arithmetic Standard Certainty Arithmetic Standard Certainty Arithmetic mean deviation equivalent mean deviation equivalent mean

214 R.R. Grauer and F.C. Shen

12 On Estimation Risk and Power Utility Portfolio Selection 24 All of History 0

Sum of Digits

22

EPAA

Boom Disaster 1

0.5 −1

Combined Approach

20

1

Benchmarks

18

Compound Return

Fig. 12.1 Compound return versus standard deviation of ln.1 C r/ of annual portfolio returns for seven power portfolios in a U.S. industry universe. The investment universe consists of twelve U.S. value-weighted industry groups. The benchmark portfolios are the value-weighted CRSP index with differing degrees of leverage. The power portfolios are revised quarterly in the 1968–1999 period. Borrowing is permitted. The joint return distributions are estimated using the all-of-history, sum-of-digits, empirical probability assessment, boom-disaster 1, and combined approaches

215

0 −1

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.5

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1

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Standard Deviation

Grinblatt and Titman’s Portfolio Change Measure. Table 12.4 contains the results for Grinblatt and Titman’s portfolio change measure. Results for the value-weighted industries universe are shown in the 1934–99 and 1968–99 periods and for global asset universe in the 1968–99 period. In this case, we note four main points. First, all the policies exhibit positive abnormal performance. Second, the PCMs are highly statistically significant—all but nine of the 54 are statistically significant at the 10 percent level. Third, in the value-weighted industries universe, the PCMs are more statistically significant than the corresponding alphas in Table 12.3. Fourth, some of the PCM rankings are at complete odds with the compound returns and Jensen’s alpha rankings. For example, compound returns and Jensen’s alpha rank all-of-history ninth in seven of the eight cases reported in Tables 12.2 and 12.3. Yet, for the logarithmic investor in the global universe, compound returns ranks the all-of-history estimation method eighth while the portfolio change measure ranks it first. Ex Post Certainty Equivalent Returns. Table 12.5 shows quarterly arithmetic means, standard deviations, and certainty equivalent returns for power 30 and logarithmic utility in the value-weighted industries and the global uni-

verses. For the logarithmic investor, the certainty equivalent returns rankings are consistent with those of the other performance measures. For power 30, it is a completely different story. For this investor in the global setting, the EPAA ranks eighth by certainty equivalent returns as opposed to first by compound returns and the portfolio change measure. In the industry setting, the power 30 investor actually assigns negative certainty equivalent returns to the EPAA and sum-ofdigits estimation methods in 1934–99 period. Surprisingly, this can be traced to a single observation in the second quarter of 1962, where the EPAA and sum-of-digits estimation methods lose 16.4% and 13.9% respectively. By way of comparison, the third worst loss is 3.8% and the remaining six estimation methods lose less than 2.6%. These results go to the core of the estimation risk problem when the estimation method is based on a finite number of states of nature drawn from historical data. Given a long enough time series we will eventually find some state or quarter t where the ex ante joint return distribution drawn from the 32-quarter moving window will be characterized by either no losses or small losses. Yet, in the next quarter there will be a large loss. While all estimation methods are based on ex ante joint return distributions that contain estimation

216 22

−1

20

−1 0 0

18 0

−1

.5 .5

Compound Return

Fig. 12.2 Compound return versus standard deviation of ln.1 C r/ of annual portfolio returns for seven power portfolios in a global investment universe. The investment universe consists of eight country common stock indices, U.S. long-term government bonds, and U.S. long-term corporate bonds. The benchmark portfolios equal-weight the ten asset classes employing differing degrees of leverage. The power portfolios are revised quarterly in the 1968–1999 period. Borrowing is permitted. The joint return distributions are estimated using the all-of-history, sum-of-digits, empirical probability assessment, and boom-disaster 1 approaches

R.R. Grauer and F.C. Shen

16

−5 E20 −5

14

.5 E15 −1

EW −10 −5

12

−10

.5

−10

E5 −10

−30 −30

1

1

All of History

10

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−30

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RL

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20

30

40

50

60

Standard Deviation

error, the estimation methods that assign some probability to loss states outside the moving 32-quarter window will yield more conservative investment policies. As a consequence, when the large loss occurs in quarter t, the realized portfolio return may be (and in some cases is) smaller than the smallest ex ante portfolio return in any state. That is exactly what happens to the EPAA and sum-of-digits methods in 1962. In other scenarios or states, less risk-averse investment policies will yield larger gains than any gains seen in the ex ante joint return distributions. As the disutility of a loss outweighs the utility of an equal-size gain for all risk-averse investors, either the conservative or aggressive policies generated from different estimation methods could be preferred ex post depending on the degree of investor risk aversion. However, the extreme aversion to (unexpected) losses exhibited by the more risk-averse members of the power utility functions will almost certainly mean that the conservative investment policies of estimation methods, which account for the possibility of a large loss, will be preferred ex post.18

12.8 Conclusion Previous studies show that combining a power utility portfolio selection model with the empirical probability assessment approach to estimating the joint return distribution frequently generates economically and statistically significant abnormal returns. Given that the EPAA is based exclusively on historic return data in a 32-quarter moving window, a series of papers explore whether these somewhat surprising results—as well as those generated from mean-variance optimizers that employ historical data to generate the mean vector and covariance matrix—are attributable to the power utility formulation that underlies the multiperiod model, to the inputs to the model, or to the constraints. The latter two possibilities are usually discussed under the rubric of estimation risk. In turn, most of the analysis of the inputs focuses on alternate estimates of the means—or in the context of the mean-variance model in terms of the mean vector and covariance matrix. By way of contrast, this paper investigates eight alternatives to the base-case EPAA that either: lengthen the esti-

18

These results are consistent with Grauer and Shen (2000), who find that ex ante the policies of more risk-averse investors subject to fewer constraints are preferred to policies subject to more constraints.

Nonetheless the policies subject to more constraints are preferred ex post.

12 On Estimation Risk and Power Utility Portfolio Selection

mation period; add “boom” and/or “disaster” states drawn from all-of-history to the moving window; or change the way probabilities of a state are estimated giving more weight to more recent observations. The alternatives allow us to explore the stationary-nonstationary tradeoffs implicit in the expanding versus moving window estimation methods; the benefits of alternative methods of allowing for the “memory loss” inherent in the moving-window EPAA; and the possibility that weighting more recent observations more heavily may improve investment performance. Compound return-standard deviation pairs, Jensen’s alpha, Grinblatt and Titman’s portfolio change measure, and ex post certainty equivalent returns are employed in evaluating the ex post performance of ten power utility investment strategies generated from nine estimation methods of the joint return distributions in two investment universes. The results show that: the EPAA holds its own against other methods of estimating the joint return distribution; the all-ofhistory method does not perform very well; and the methods that add disaster or boom-disaster states to the EPAA moving window perform quite well. We find the latter result encouraging, as in our judgment, ignoring the possibility of disaster states in forming estimates of the joint return distribution may one day prove to be just as costly to the power utility strategies as the catastrophic events of August and September 1998 were to the arbitrage strategies of Long-Term Capital. However, while there is considerable agreement in the rankings of the estimation methods, the rankings differ (sometimes dramatically) across datasets, time periods, performance evaluation methods, and investor risk aversion. For the most part the evaluation measures rank the all-of-history estimation method at or near the bottom (i.e., ninth out of nine estimation methods). Yet, for the logarithmic investor in the global dataset, the portfolio change measure ranks the all-of-history estimation method first, and for the power 30 investor the certainty equivalent measure ranks the all-ofhistory estimation method fourth. For the power 30 investor in the industry dataset, compound return, Jensen’s alpha, and the portfolio change measure rank the EPAA and sum-ofdigits estimation method either first or second (or in one case first and third). Yet, the certainty equivalent measure ranks the EPAA and sum-of-digits method eighth or ninth. Although dramatic disagreement among the performance measures in certain instances is discouraging, it should not be surprising. The expected utility-based objective function of multiperiod portfolio theory has no direct tie to any of the commonly used measures of investment performance. And, while each of the performance measures has a certain intuitive appeal, each of them embodies economic or statistical assumptions that may be at variance with: the investor’s objective function, the assumptions employed in the different methods of estimating the joint return distri-

217

butions, or the assumptions used in the other methods of measuring performance. The results raise two fundamental issues. First, the performance measures may be particularly misleading if a catastrophic event does not show up in the sample. Second, even if a catastrophic event does occur, traditional performance measures may not be equipped to deal with it. For example, Grauer (2008a) finds that Jensen’s alpha and Grinblatt-Titman’s portfolio change measure rank a number of bankrupt mean-variance asset-allocation strategies ahead of perfect-foresight strategies that generate riches beyond one’s wildest dreams.

12.9 Addendum In light of the out-of-sample portfolio returns generated from models that forecast the means based on information variables in a moving window, see for example Grauer (2008b, c, 2009), we examine the effect of combining these movingwindow forecasts of the means with an expanding window approach that guards against the possibility of losses in some long forgotten states of nature. Dividend yields and riskfree interest rates are used to forecast the means. The following regression is run at each time t ri  D a0i C a1i dy 1 C a2i rL C ei  ; for all i and £; in the tk to t1 estimation period, where the i subscript denotes an industry, dy 1 is the annual dividend yield on the CRSP value-weighted index lagged one month so that it is observable at the beginning of quarter t; rLt is the (observable) beginning-of-quarter Treasury bill rate, and in this case k D 32. The one-period ahead forecast of the mean of industry i is rNDRit D aO 0i C aO 1i dyt 1 C aO 2i rLt ; where aO 0i ; aO 2i and aO 2i are the estimated coefficients and dyt 1 and rLt are observable at the beginning of period t C 1. That is, the quarterly variable dyt 1 is lagged one month and there is no need to lag rLt as it is observable at the beginning of the quarter. For each asset i , we add the difference .NrDRit  rNit /, where rNit is the historic mean from the expanding (all-of history) window, to each actual return on asset i in the expanding window estimation period. That is, in each estimation period, we replace the raw return series with the adjusted return series riA D ri  C .NrDRit  rNit /, for all i and . No adjustment is made to the EPAA variance-covariance structure or to the other moments. We limit reporting results to the compound returnstandard deviation pairs for the industry dataset in 1968–

218

1999. Figure 12.1 shows that the combination approach clearly dominates the other approaches. Acknowledgments We thank the Social Sciences Research Council of Canada and the Centre for Accounting Research and Education for financial support. The paper was presented at the University of British Columbia, the Northern Finance Association meetings in London Ontario, and the Pacific Northwest Finance Conference in Seattle. We thank C.F. Lee, William Ziemba, and seminar participants for valuable comments, and Reo Audette, Christopher Fong, Maciek Kon, Jaspreet Sahni, William Ting, and Jeffrey Yang for most capable assistance.

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219 Sharpe, W. F. 1964. “Capital asset prices: a theory of market equilibrium under conditions of risk.” Journal of Finance 19, 425–442. Shen, F. C. 1988. Industry rotation in the U.S. stock market: an application of multiperiod portfolio theory.” unpublished Ph.D. dissertation, Simon Fraser University, Canada. Solnik, B. 1993. “The performance of international asset allocation strategies using conditioning information.” Journal of Empirical Finance 1, 33–55.

Chapter 13

International Portfolio Management: Theory and Method Wan-Jiun Paul Chiou and Cheng-Few Lee

Abstract This paper investigates the impact of various investment constraints on the benefits and asset allocation of the international optimal portfolio for domestic investors in various countries. The empirical results indicate that local investors in less-developed countries, particularly in East Asia and Latin America, benefit more from global diversification. Although the global financial market is becoming more integrated, adding constraints reduces but does not completely eliminate the diversification benefits of international investment. The addition of portfolio bounds yields the following characteristics of asset allocation: a reduction in the temporal deviation of diversification benefits, a decrease in time-variation of components in optimal portfolio, and an expansion in the range of comprising assets. Our findings are useful for asset management professionals to determine target markets to promote the sales of national/international funds and to manage risk in global portfolios. Keywords International portfolio management r Investment constraints

13.1 Introduction In the past three decades, financial markets throughout the world have experienced evolutions that leave challenges and opportunities to international portfolio management: more openness to domestic and foreign investors, increasing integration with other countries, gradual liberalization from governmental control, and innovations of financial products and services. Specifically, as the investibility and the legal limitation to foreign investment vary across countries, it is W.-J.P. Chiou () Department of Finance, John L. Grove College of Business, Shippensburg University, Shippensburg, PA 17257, USA e-mail: [email protected] C.-F. Lee Department of Finance and Economics, Rutgers Business School, Rutgers University, New Brunswick, NJ 08854, USA

nature to question the asset allocation of the unrestricted efficient frontier suggested by Markowitz (1952).1 Furthermore, understanding changes in the benefits of international diversification strategies helps to structure dynamic portfolio rebalancing when the world financial market has been increasingly integrated. To ensure the achievability of strategies, the obstacle of the extreme allocations on the optimal international portfolio, such as negative and enormous weights, needs to be taken into account. This chapter explores the theory and method of global asset management by investigating the attributes of the diversifying portfolio. We also provide the evidence of significance to include various investment constraints by observing the change in weights and mean-variance efficiency. Previous studies investigate the strategies of international portfolio management with short-selling constraints (for a more detailed discussion, see Cosset and Suret 1995; De Roon et al. 2001; De Santis and Gerard 1997; Driessen and Laeven 2007; Fletcher and Marshall 2005; French and Poterba 1991; Harvey 1995; Li et al. 2003; Novomestky 1997; and Obstfeld 1994). Yet, the impact of other investment restrictions on the global diversification benefits remains unclear. There are three reasons for portfolio managers to consider the unattainability of the “corner solutions” on the efficient frontier. First, both profitability and liquidity of the diversifying portfolio should be taken into account when asset allocation in international markets is determined. The heavily positive/negative weights of investments in the minor markets recommended by less-constrained strategies may cause illiquidity of the portfolio. Second, the excessive foreign capital in- and out flows in minor markets tend to trigger instability in security prices. This may generate dramatic variation in returns, risks, and correlations. Finally, in many countries, particularly developing nations, governmental regulations prohibit foreign investors to short sell and/or

1

The investiblity of the stock market is indicated by the degrees with which foreign investors can trade as do local investors in the domestic markets and the liquidity of assets. See Bae et al. (2004).

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_13, 

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to hold more than a certain proportion of company shares.2 From legal and institutional aspects, a large percentage of foreign capital allocation to securities in those small financial markets is impractical. Accordingly, the results of this chapter are more realistic for asset management since illiquidity of portfolio and variation in return and volatility, as well legal limitations, are considered when the optimal portfolio strategies are formed. To appropriately estimate the effectiveness of international portfolio management, two straightforward measures, the increase in risk-adjusted premium by investing in the maximum Sharpe ratio (MSR) portfolio and the reduction in the volatility by investing the minimum variance portfolio (MVP) on the international efficient frontiers, are used. Additionally, this chapter extends the study in diversification benefits by including over-time comparison and cross-region analyses. An increase of correlations and reduction of returns in the global market cast a shadow of doubt on the effectiveness of international investment. An intertemporal analysis of the benefits of diversification is helpful to determine the impact of integration of the international financial market. Our empirical findings confirm the benefits of international portfolio management with various investment constraints. The cross-country comparison suggests that domestic investors in emerging markets, particularly in East Asia and Latin America, benefit more from international diversification. It is also found that the global diversification benefits vary as the world financial market has became more integrated. Although the lower and upper bounds of weights worsen the mean-variance efficiency of the optimal portfolio, they generate some desired attributes for asset management and therefore enhance the feasibility of asset allocation strategies. Specifically, the addition of overweighting investment constraints substantially reduces the time-variation in gains and weights of the global diversification. The expansion of coverage in the optimal portfolio makes the asset allocations more realistic. The chapter is organized as follows. Section 13.2 reviews development of global portfolio management. Section 13.3 reviews literature. The estimate of the benefits of international diversification is described in Sect. 13.4. Section 13.5 reports the empirical results of the benefits of international diversification. The over-time examination of components of the optimal portfolios is reported in Sect. 13.6. Conclusions are presented and relevant issues are discussed in Sect. 13.7.

2 For instance, the ownership of listed companies by foreigners cannot exceed the limit of 10% in Chile, 25% in South Korea, and 10% in Taiwan. In Brazil, international institutional investors cannot own more than 49% of voting share. In Switzerland, corporations are likely to issue international investors special shares that are traded at a premium over the same stocks available exclusively to Swiss nationals.

W.-J.P. Chiou and C.-F. Lee

13.2 Overview of International Portfolio Management International diversification has been a strong trend for investors in all countries after the mid-1990s. For European investors, such as ones in the Netherlands and the U.K., overseas portfolios play an important role in asset management. However, this is a relatively newer option in the portfolio for investors in the U.S. and most emerging markets. The enormous growth of financial markets outside North America after the mid-1980s is one of major reasons that trigger global diversification. Table 13.1 presents the average growth rates of market value in major 34 countries from 1993 to 2005 and the weights of world market capitalization as of the end of 2005. These sample countries are also grouped into seven regions: East Asia, North America, Latin America, North Europe, West/Central Europe, South Europe, and Oceania. The global value-weighted average annual growth rate is 10.5% during the period. The countries with the largest growth in market value were Finland, Turkey, and Greece, in which the increase rates are twice more than the world average. The variation in the growth of market value within each group of countries at a given developmental stage and in various areas is considerable. The fact that the equity markets in developing countries experienced a larger increase in value indicates the significance of international diversification for the U.S. investor. The magnitude of foreign markets also validates the trend toward international diversification. In the early 1970s, the New York Stock Exchange represented more than 60% of the world market value of less than $1 trillion in total. As shown in Figure 13.1, the size of the world market multiplied significantly in the next three decades and at the end of 2005, the stock market value in the above 34 countries was about 38 trillion. The shares among various regions varied drastically. The percentage of the U.S. equity market varies from more than 50% to less than 40% in the early 1990s and back to 43% by the end of 2005. The East Asia and Oceania region, which made up more than one-third of the market capitalization in the world in the early 1990s, reduced to about one-fifth at the end of 2005. Europe represents about one-third of the global market value. The stock markets in developed countries consistently represented more than 90% of the world capitalization. On the other hand, the capitalization of the emerging markets is small, and only Brazil, South Korea, and Taiwan were of a share greater than 1% in the world market as of the end of 2005. International diversification is inspired by two desired characteristics in portfolio management: a low global correlation to decrease the portfolio volatility and the expansion of investment targets to allow investors to allocate asset toward the markets with higher expected return. The above attributes eventually lead to a superior mean-variance

13 International Portfolio Management: Theory and Method

223

Table 13.1 Capitalization and performance of markets Panel A: Developed countries Sharpe ratio Country

Cap growth

Australia Austria Belgium Canada Switzerland Germany Denmark Spain Finland France U.K. Hong Kong Ireland Italy Japan Netherlands Norway New Zealand Singapore Sweden U.S.A. Average

14:81% 15:14% 12:72% 14:96% 13:09% 10:16% 14:28% 19:11% 24:37% 13:13% 9:60% 14:97% 16:12% 15:04% 5:36% 10:24% 20:00% 8:14% 13:62% 14:39% 10:75% 10:47%

Cap weight 2:08% 0:33% 0:78% 3:83% 2:41% 3:15% 0:48% 2:48% 0:54% 4:20% 7:89% 2:72% 0:29% 2:06% 11:80% 1:23% 0:49% 0:10% 0:66% 1:14% 43:88% 4:41%

Mean return

Region

0:113 0:087 0:081 0:107 0:124 0:096 0:137 0:105 0:114 0:111 0:093 0:125 0:099 0:077 0:009 0:117 0:116 0:067 0:084 0:136 0:114 0:097

Oceania C/W Europe C/W Europe N. America C/W Europe C/W Europe C/W Europe C/W Europe N. Europe C/W Europe C/W Europe E. Asia C/W Europe S. Europe E. Asia C/W Europe N. Europe Oceania E. Asia N. Europe N. America

Median 0:001 0:091 0:020 0:024 0:117 0:046 0:060 0:005 0:081 0:057 0:025 0:002 0:002 0:010 0:089 0:122 0:015 0:034 0:023 0:013 0:114 0:021

St. Dev. 0.069 0.134 0.127 0.093 0.151 0.111 0.093 0.116 0.162 0.091 0.133 0.115 0.140 0.083 0.060 0.158 0.090 0.136 0.107 0.128 0.171 0.079

Max 0:201 0:392 0:376 0:219 0:411 0:309 0:312 0:295 0:375 0:254 0:291 0:251 0:372 0:179 0:048 0:360 0:241 0:254 0:223 0:341 0:393 0:192

Time Dec-05 Dec-05 Aug-98 Sep-00 Mar-98 Jul-98 Apr-98 Aug-98 Jan-93 Jan-00 Mar-98 Sep-94 May-98 Apr-98 Jul-97 Jun-98 Jan-98 Nov-05 Feb-96 Mar-00 Jan-00 Apr-98

Min 0.122 0.172 0.196 0.169 0.197 0.199 0.142 0.145 0.261 0.139 0.293 0.145 0.200 0.163 0.240 0.213 0.183 0.249 0.190 0.139 0.147 -0.145

Time Oct-01 Jun-00 Apr-03 Aug-94 Apr-03 Apr-03 Apr-03 Aug-93 Apr-00 Apr-03 Apr-03 Oct-02 Mar-03 Apr-03 Sep-98 Apr-03 Oct-02 Dec-00 Sep-98 Oct-02 Apr-05 Apr-03

Panel B: Emerging markets Sharpe ratio Country

Cap growth

Cap weight

Mean return

Region

Median

St. dev.

Max

Month-Year

Min

Month-Year

Argentina Brazil Chile Greece Indonesia Korea, S. Malaysia Mexico Philippines Portugal Thailand Turkey Taiwan Average

7:48% 19:78% 12:48% 22:19% 15:84% 15:72% 5:37% 4:28% 7:62% 16:08% 6:11% 24:10% 12:74% 12:21%

0:12% 1:23% 0:35% 0:37% 0:21% 1:85% 0:47% 0:62% 0:10% 0:17% 0:32% 0:42% 1:23% 0:57%

0:185 0:193 0:182 0:133 0:082 0:077 0:064 0:223 0:029 0:045 0:060 0:129 0:062 0:108

L. America L. America L. America S. Europe E. Asia E. Asia E. Asia L. America E. Asia C/W Europe E. Asia S. Europe E. Asia

0:019 0:024 0:046 0:009 0:064 0:042 0:013 0:045 0:104 0:041 0:002 0:004 0:040 0:002

0.101 0.089 0.173 0.105 0.108 0.079 0.126 0.133 0.171 0.127 0.141 0.064 0.065 0.066

0:165 0:230 0:368 0:229 0:149 0:206 0:233 0:437 0:266 0:283 0:213 0:143 0:122 0:129

Mar-94 Jul-97 Nov-94 Oct-99 Dec-05 Dec-05 Jan-94 Feb-94 Oct-95 May-98 Jan-94 Jan-94 Sep-97 Feb-94

0.235 0.181 0.217 0.180 0.279 0.239 0.272 0.131 0.269 0.242 0.273 0.136 0.174 0.137

Jun-02 Oct-02 Oct-02 Jul-95 Oct-98 Jan-98 Nov-98 Feb-99 Nov-01 May-03 Sep-98 Feb-95 Oct-02 Oct-02

The growth rate of capitalization during the period of 1992:12–2005:12 and the world capitalization weight as of the end of 2005 in each country are reported. The mean of raw return for each market from 1988:01 to 2005:12 is annualized. The mean, median, standard deviation, maximum, and minimum of the time series of the monthly Sharpe ratio for each market are also presented. The averages for developed countries and emerging markets are calculated by their value-weighted portfolios.

efficiency of holding portfolio. However, several impediments to global diversification in recent years are discussed. The first criticism of global diversification is that consistently higher returns in domestic markets in certain countries challenge the need of overseas investment. It is questionable to claim greater expected return in the future by using the past superior performance. Table 13.1 summarizes the value-weighted average performance and the dynamics of the monthly Sharpe ratio for each market. The raw return in developing countries, on average, is higher than the stock markets in developed countries. However, the Sharpe ratios in

developed countries are higher than the ones in the emerging markets. The countries of the maximum mean-variance efficient domestic portfolio are the U.S., Switzerland, the Netherlands, and Finland. It is consistent with previous findings that the mean-variance efficiency of equity prices in emerging markets is lower due to higher volatility (Bekaert and Harvey 2003). Furthermore, the cross-area disparity of equity returns among emerging markets is considerable. Specifically, the stock return in Latin America outperformed the ones in other regions, while the equities in East Asian countries performed worse than the rest of the world.

224

W.-J.P. Chiou and C.-F. Lee Millions of U.S. Dollars 40,000,000

Oceania

35,000,000

30,000,000

North America

25,000,000

20,000,000

Latin America

15,000,000

East Asia 10,000,000

5,000,000

Europe

0 1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

Fig. 13.1 Stock market capitalization in the world. The equity market capitalization in different regions from 1992 to 2005 is demonstrated

Table 13.1 also indicates significant time-variation of riskadjusted performance in each stock market. Compared to the means of the Sharpe ratio, the time variation, which is measured by standard deviation, of the Sharpe ratio for each market is substantial. For the developed countries, the maximum domestic Sharpe ratio most likely occurred in 1998 and 2005, while for most emerging markets this occurred before 1998. The local investors in the majority of emerging markets generated the worst risk-adjusted performance in 1998 due to financial crises. On the other hand, investors in developed countries experienced the largest loss in 2002 and 2003 because of the evaporation of high-tech bubbles after 2000 and the economic recession worsened by the terrorist attack in 2001. Furthermore, there are nonsynchronous movements of mean-variance efficiency across countries, which imply the potential gains to domestic investors if they diversify their portfolios globally. For instance, there are ten markets of the best mean-variance efficiency and six markets of the worst in 1998. Similarly, four countries had the highest Sharpe ratio, and three countries had the lowest in 2000. This suggests that cross-market performances can differ drastically in the same year and that local investors may avoid losses in their home markets by allocating their funds optimally in other countries. Second, it has been observed that international correlations have trended aloft over the past decades. As shown in Table 13.2, the intertemporal comparison provides evidence of enhancement of correlation in the international financial market. The means of correlation of each market with all other countries in the second period is persistently greater than the ones in the first period. In the first period, some markets are negatively correlated or almost uncorrelated with certain areas. The enhancement of correlation is particularly

considerable for the emerging markets with the countries that are in the different regions. All means of correlation coefficient in the second period are increasing and positive. This finding is consistent with the enhancement of integration of the world financial market over the past two decades. Table 13.2 also shows the means of the correlation coefficients of each country with other countries grouped by regions in two sub-periods. The developed countries, in general, demonstrate higher correlations with other markets than the emerging markets do. Most countries also have the highest coefficients of correlation with the other countries in their home region and with the ones in North America. Although the magnitude of correlations increases over time, the phenomena that the emerging markets are less correlated with other countries can be constantly observed in the two sample periods. The fact that most markets are less correlated with the countries from other regions indicates possible diversification benefits from inter-continent investments. In addition, the stock markets in the rest of the world tend to have considerable co-movements with the U.S. markets. Third, the existence of home bias in international asset allocation suggests investors consider the barriers to overseas investments costly. The factors such familiarity with financial markets in other countries, political risk, market liquidity and efficiency, regulation on foreign ownership of domestic corporations, transaction costs, levies and taxes, and exchange rate risk, rationalize why an investor would want to overweigh his or her local portfolio compared with the weights of asset in the rest of the world. However, previous research suggests that the costs related to the previous variables are smaller than the improvement of mean-variance brought by international diversification. This implies that overseas asset allocation should not be avoided entirely.

Table 13.2 Coefficients of correlation among markets Panel A: Developed countries World average Latin America 88–96 97–05 88–96 97–05 Australia 0.28 0.55 0.14 0.58 Austria 0.31 0.43 0.07 0.39 Belgium 0.32 0.45 0.06 0.32 Canada 0.31 0.56 0.13 0.61 Switzerland 0.32 0.51 0.02 0.40 Germany 0.35 0.57 -0.01 0.52 Denmark 0.30 0.50 0.01 0.47 Spain 0.39 0.57 0.22 0.56 Finland 0.28 0.40 0.12 0.37 France 0.33 0.59 0.10 0.50 U.K. 0.38 0.56 0.06 0.51 Hong Kong 0.33 0.47 0.17 0.55 Ireland 0.37 0.48 0.09 0.43 Italy 0.28 0.49 0.05 0.46 Japan 0.27 0.38 0.06 0.34 Netherlands 0.40 0.58 0.02 0.49 Norway 0.34 0.54 0.14 0.58 New Zealand 0.25 0.48 0.06 0.44 Singapore 0.40 0.47 0.15 0.55 Sweden 0.38 0.55 0.13 0.52 U.S.A. 0.32 0.56 0.19 0.57 Panel B: Emerging markets World average Latin America 88–96 97–05 88–96 97–05 Argentina 0.06 0.36 0.18 0.57 Brazil 0.10 0.52 0.12 0.67 Chile 0.08 0.51 0.14 0.66 Greece 0.19 0.39 0.09 0.38 Indonesia 0.14 0.34 0.05 0.36 Korea, S. 0.15 0.38 0.06 0.33 Malaysia 0.33 0.34 0.10 0.38 Mexico 0.16 0.52 0.22 0.65 Philippines 0.27 0.37 0.15 0.41 Portugal 0.28 0.46 0.09 0.36 Thailand 0.28 0.41 0.17 0.42 Turkey 0.09 0.38 0.06 0.48 Taiwan 0.15 0.40 0.14 0.54 The averages of unconditional coefficients of correlation of East Asia 88–96 0.25 0.30 0.26 0.34 0.25 0.26 0.20 0.29 0.27 0.24 0.29 0.41 0.33 0.25 0.21 0.30 0.24 0.22 0.52 0.31 0.29 97–05 0.53 0.31 0.23 0.47 0.33 0.36 0.31 0.38 0.25 0.35 0.38 0.54 0.32 0.26 0.40 0.38 0.39 0.48 0.57 0.39 0.45

Central/West Europe 88–96 97–05 0.29 0.54 0.42 0.56 0.51 0.68 0.33 0.58 0.50 0.69 0.58 0.73 0.49 0.65 0.51 0.71 0.32 0.46 0.53 0.76 0.54 0.70 0.34 0.40 0.51 0.61 0.38 0.66 0.40 0.36 0.61 0.76 0.46 0.63 0.27 0.48 0.44 0.37 0.50 0.66 0.39 0.62

North Europe 88–96 97–05 0.43 0.57 0.34 0.39 0.35 0.47 0.39 0.67 0.38 0.55 0.42 0.69 0.45 0.59 0.55 0.63 0.55 0.55 0.35 0.73 0.54 0.63 0.35 0.43 0.50 0.51 0.40 0.59 0.38 0.40 0.51 0.65 0.55 0.54 0.39 0.47 0.45 0.42 0.57 0.65 0.40 0.65

South Europe 88–96 97–05 0.04 0.40 0.34 0.34 0.16 0.37 0.02 0.45 0.17 0.40 0.19 0.52 0.15 0.38 0.18 0.51 0.03 0.43 0.17 0.53 0.10 0.48 0.10 0.26 0.22 0.43 0.15 0.49 0.06 0.24 0.16 0.47 0.10 0.49 0.12 0.39 0.17 0.31 0.19 0.53 0.00 0.46

Oceania 88–96 0.66 0.22 0.15 0.42 0.27 0.22 0.19 0.42 0.38 0.23 0.45 0.30 0.34 0.19 0.26 0.39 0.38 0.66 0.37 0.47 0.29

97–05 0.70 0.52 0.42 0.63 0.52 0.54 0.46 0.58 0.41 0.55 0.57 0.55 0.53 0.46 0.52 0.55 0.61 0.70 0.60 0.55 0.54

North America East Asia Central/West Europe North Europe South Europe Oceania 88–96 97–05 88–96 97–05 88–96 97–05 88–96 97–05 88–96 97–05 88–96 97–05 0.16 0.42 0.05 0.33 0.02 0.31 0.02 0.36 0.14 0.36 0.17 0.37 0.08 0.63 0.08 0.42 0.08 0.56 0.22 0.53 0.16 0.41 0.13 0.54 0.14 0.61 0.14 0.50 0.04 0.46 0.08 0.52 0.01 0.48 0:03 0.56 0.06 0.40 0.11 0.23 0.29 0.50 0.18 0.45 0.35 0.33 0.15 0.38 0.20 0.35 0.20 0.48 0.11 0.26 0.14 0.25 0.13 0.19 0.20 0.42 0.19 0.42 0.22 0.45 0.13 0.32 0.23 0.39 0:07 0.27 0.16 0.53 0.41 0.36 0.46 0.49 0.33 0.26 0.37 0.26 0.16 0.20 0.30 0.36 0.25 0.69 0.21 0.48 0.12 0.48 0.20 0.55 0.00 0.45 0.14 0.57 0.34 0.43 0.39 0.51 0.24 0.29 0.20 0.26 0.16 0.20 0.29 0.48 0.20 0.50 0.18 0.26 0.40 0.64 0.32 0.57 0.41 0.40 0.28 0.42 0.36 0.46 0.41 0.57 0.26 0.31 0.21 0.31 0.21 0.18 0.20 0.62 0:05 0.52 0.11 0.25 0.11 0.38 0.03 0.51 0.35 0.33 0.01 0.41 0.12 0.49 0.25 0.47 0.13 0.32 0.14 0.36 0.05 0.28 0.06 0.45 each country with other countries of different regions during two periods, 1988:01–1996:12 and 1997:01–2005:12 are reported.

North America 88–96 97–05 0.43 0.66 0.17 0.41 0.41 0.48 0.65 0.79 0.39 0.58 0.33 0.71 0.27 0.65 0.40 0.66 0.33 0.60 0.40 0.72 0.51 0.70 0.49 0.57 0.40 0.58 0.29 0.56 0.26 0.50 0.51 0.68 0.41 0.66 0.28 0.51 0.48 0.57 0.44 0.71 0.65 0.79

13 International Portfolio Management: Theory and Method 225

226

13.3 Literature Review Previous empirical evidence suggests that investment constraints may not completely eliminate the benefits brought about by international diversification. Bekaert and Urias (1996), Chiou (2008), Chiou et al. (2009), De Roon et al. (2001), Driessen and Laeven (2007), Harvey (1995), Li et al. (2003), Pástor and Stambaugh (2000), and Wang (1998) confirm the benefits of international diversification even when short-sales are not allowed. Cosset and Suret (1995) find that including securities from high political-risk countries into the portfolio can increase mean-variance efficiency. Considering geographical restrictions, De Roon et al. (2001) find that investors in East Asia can still generate benefits from diversifying their portfolios by investing in other countries in Latin America. Green and Hollifield (1992) and Jagannathan and Ma (2003), on the other hand, investigate the impacts of imposing short-sales and upper-bound investment constraints on mean-variance efficiency and portfolio risk. Errunza et al. (1999) find that U.S. investors can utilize domestically traded American Depositary Receipts (ADRs) to duplicate the benefits of international diversification. The degree of international market integration is critical to the effectiveness of international diversification. Bekaert and Harvey (1995), Bekaert et al. (2005), De Jong and De Roon (2005), and Errunza et al. (1992) suggest that the world market is mildly segmented and that the degree of global market integration varies throughout time. The integration of international financial markets, in general, has gradually increased, but the emerging markets are still more segmented than those in developed countries. Bekaert et al. (2005) find that the correlations among individual stock price do not necessarily increase when the international financial market is more integrated, while the idiosyncratic risk does not show a particular time trend. Given the change in the world financial market, an intertemporal study in international diversification benefits would be useful. The institutional and cultural heterogeneities among countries are key factors determining nonsynchronous movement of security prices among markets. Beck et al. (2003) and Demirgüç-Kunt and Maksimovic (1998) suggest that the international differences of financial markets can be explained by natural endowments and legal tradition. La Porta et al. (1998, 2000), Djankov et al. (2003, 2008) show how law affects the protection to investors. Stulz and Williamson (2003) document that liberalization and development of financial markets relate to cultural background such as major religion and language. Bekaert and Harvey (2003) report the major characteristics of emerging markets and their chronological innovations. The differences in cultural background, natural endowments, institutional systems, and legal tradition deter integration of international financial markets so that investors may generate gains from overseas diversification.

W.-J.P. Chiou and C.-F. Lee

In sum, previous studies document the benefits of global diversification with constraints such as short-sales. The increase in correlations of international financial markets does not completely eliminate the benefits because of the dispersion of returns among countries. The difference of cultures, natural endowments, and institutional systems among markets cause the dispersion of returns among markets. The gap of understanding regarding the time-series of the benefits of constrained international diversifications needs to be filled.

13.4 Forming the Optimal Global Portfolio Suppose a representative investor desires to minimize the volatility of her portfolio, given the same return, by allocating funds among international markets. The expected returns in excess of the risk-free interest rate and the variancecovariance of N international asset returns can be expressed as a vector T D Œ1 ; 2 ; : : : ; N and a positive definite matrix V, respectively. Let  be the set of all real vectors of weighting wT such that wT 1 D w1 C w2 C : : : C wN D 1, where 1 is an N -vector of ones. We further assume that the best predictors of means, variances, and covariances are their historical averages. The investor thus follows the method of Markowitz (1952) to form the global efficient frontier. The problem of the optimal portfolio selection is then expressed as a Lagrange’s equation: minfw; ;g „ D

    1 T w VwC p  wT  C  1  wT 1 ; 2 (13.1)

where p denotes the expected return on the portfolio, and the shadow prices and  are two positive constants. The quadratic programming solution for asset spanning without investment constraints is:     wp D V1  C  V1 1 :

(13.2)

Various weighting constraints are considered. It is wellknown that short-selling is not allowed for foreign investors in many countries, particularly in less developed nations (see De Roon et al. 2001; Harvey 1995; Li et al. 2003; Pástor and Stambaugh 2000). Furthermore, the investor considers the liquidity of portfolio when global asset allocation is determined. To reflect the argument by Green and Hollifield (1992) and Jagannathan and Ma (2003), the relative magnitude of market value in each country is considered when the upper boundaries of weighting are characterized. The subset of portfolio weights PU with short-sale and over-weighting investment constraints .SS C OW.U // can be described as: PU D fwp 2  W 0  wi  U w.C ap/i  1; i D 1; 2 : : : ; N g; U > 1;

(13.3)

13 International Portfolio Management: Theory and Method

227

where w.Cap/i is the proportion of the market value of each country i in the world, and U is any real number. The restriction of weights is then incorporated in Lagrange’s Equation (13.1). Since the incentives of diversification are not only to seek higher yields but also to reduce volatility, the Maximum Sharpe ratio (MSR) represents the highest meanvariance efficiency that can be achieved by the international efficient frontier. Specifically, MSRJ D maxfwp g

 

12 ˇ

ˇ wTp  = wTp Vwp ˇwTp 2 PJ : (13.4)

For domestic investor in country i , the greatest increase in unit-risk return brought by global diversification is •i;J D MSRJ  SRi ;

(13.5)

1

where SRi D i =Vi 2 is the Sharpe ratio, and Vi is the variance of the domestic portfolio in country i . The other measure of benefits of diversification is the maximum reduction in volatility as a result of international diversification. Elton et al. (2007) suggest that investors may seek to minimize the variance of a portfolio because of the lack of predictability of expected returns. In this case, an investor may want to construct a minimum-variance portfolio (MVP). The weighting of the MVP can be characterized as: ˇ h io n ˇ wMVP D wp W minfwp g wTp V wp ˇwTp 2 PJ :

(13.6)

The maximum decline in volatility by diversifying internationally with various investment constraints is 1  ©i;J D 1  wTMVP;J VwMVP;J =Vi 2 :

(13.7)

In this chapter, the global efficient frontiers and the Sharpe ratio are estimated by using monthly returns in the previous 5 years. The time-series of ıi;J and "i;J with various investment constraints are generated by time-rolling efficient frontiers. The weights for the MSR portfolio and the MVP under various investment constraints in each month are also calculated.

13.5 The Benefits of International Diversification Around the World Figure 13.2 exhibits the efficient frontiers of the global portfolio at the end of each year from 1993 to 2005. Because of the time-variations in mean-variance efficiency and correlations among markets, the shape and size of efficient frontiers vary noticeably from 1993 to 2005. The movement of

efficient frontiers also does not follow a consistent direction or pattern. The efficient frontier first shifts to the northwest from 1993 to 1994 but then moves to the southeast (reduction in mean-variance efficiency) from 1994 to 1997. The positions of optimal portfolios hover around the same general area from 1998 to 1999. In 2000, the frontier moves northwest, consistent with the U.S. market dot-com boom. Then, from 2001 to 2003, the frontier moves noticeably to the southeast, seemingly reflective of the U.S. market crash. The period from 2002 to 2003 is by far the least mean-variance efficient of our sample period. Then the frontier moves northwest again in 2004 and 2005. The movement of the efficient frontiers seems to reflect the world markets and business cycles. Figure 13.3 graphically demonstrates the time-series of potential diversification benefits to the domestic investors. Graph A shows that the average improvement of meanvariance efficiency to the local investor was at its peak from 1993 to 1994 when only short-sales constraint is considered. On the other hand, when the limit of over-weighting investment is taken into account, ı is at its maximum during 1996 and 2000. In Graph B, " with various restrictions was slightly lower from 1996 to 1998 and from 2003 to 2005. The intertemporal change of the benefit of risk reduction is not as considerable as that of the one of mean-variance efficiency improvement. The cross-strategy comparison shows that the benefits under various constraints are not proportional. In Graph A, for instance, the divergence of the Sharpe ratio benefits among various trading limits is significant when the Sharpe ratio benefit of the no-short-selling portfolio is high. A similar pattern can be found in Graph B. This suggests that the effectiveness of the less restrictive diversifying strategies is more sensitive to the variants of asset returns in the menu. Consideration of these constraints eliminates uncertainty in portfolio performance since some portion of investment shifts to the second-best alternatives, which generally are the national indices representing larger markets with high mean-variance efficiency. Table 13.3 reports means of ı and " of each country over the sample period. The gains of international diversification to local investors in emerging markets, both the improvement of mean-variance efficiency and the decrease in volatility, are greater than the ones in developed countries. This result holds for the various scenarios of weighting constraints. The diversification benefits to the investors in emerging markets are eroded considerably by restrictive upper bounds, especially ı. The domestic investors in developed countries still can increase mean-variance efficiency of their portfolios by approximately 0.136 and reduce volatility by 2.5% via the global diversification with constraints of SS C OW.3/. Table 13.3 also shows that the comparative advantages vary from region to region. The local investors in East Asia,

228

W.-J.P. Chiou and C.-F. Lee

Fig. 13.2 Mean-variance efficient frontiers: 1993–2005. This figure shows the short-sale constrained international efficient frontiers at the beginning of each year from 1993 to 2005. The efficient frontiers are constructed by the monthly returns in previous 5 years

Monthly Expected Return 0.05 0.04 94

0.03

93 01

00

0.02

98

96

99 97

05

0.01 0.00 0.00

03

02

04 95

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

−0.01 Monthly St. Dev.

Fig. 13.3 The benefits of Panel A: δ international diversification. The time-series of the means of the benefits of international diversification for all sample countries are presented. The increase in the risk-adjusted premium and the decrease in volatility of domestic portfolio by international diversification are •i;J D MSRJ  SRi and  1 ©i;J D 1  wTMVP;J VwMVP;J =Vi 2 , respectively

Panel B: ε

on average, enjoy superior increments in the risk-adjusted premium and the greatest shrinkages in volatility from global diversification. For home investors in Latin America, the ben-

efits of incorporating overseas stocks in their portfolios primarily are to reduce risk. The benefits for investors in Europe and North America are relatively trivial compared to the ones

13 International Portfolio Management: Theory and Method

Table 13.3 The benefits of international diversification to domestic investors

229

Panel A: Developed countries SS C OW(10)

SS C OW(5)

SS C OW(3)

ıi;J

"i;J

ıi;J

"i;J

ıi;J

"i;J

ıi;J

"i;J

0:3472 0:4358 0:2563 0:2954 0:1908 0:2588 0:2221 0:2403 0:1875 0:2776 0:2938 0:2726 0:2639 0:3044 0:4308 0:1968 0:3253 0:3248 0:3175 0:2028 0:1617 0:2765

2:03% 2:27% 1:49% 2:28% 1:90% 2:77% 2:09% 3:21% 6:04% 2:19% 0:85% 5:21% 2:38% 3:75% 3:05% 1:51% 3:55% 3:52% 3:69% 4:47% 1:01% 2:82%

0:2475 0:2693 0:1937 0:2614 0:0953 0:1803 0:1609 0:1575 0:1217 0:1645 0:1813 0:1450 0:1950 0:2413 0:3196 0:1307 0:2500 0:2549 0:1434 0:1520 0:0995 0:1888

1:94% 2:18% 1:42% 2:13% 1:82% 2:68% 1:91% 3:08% 5:98% 2:15% 0:60% 5:16% 2:22% 3:63% 2:97% 1:42% 3:49% 3:45% 3:47% 4:29% 0:81% 2:70%

0:2196 0:2506 0:1584 0:2444 0:0603 0:1375 0:1187 0:1467 0:0965 0:1156 0:1487 0:1094 0:1553 0:2063 0:2835 0:0975 0:1966 0:2167 0:1073 0:1188 0:0707 0:1552

1:78% 2:07% 1:31% 2:08% 1:76% 2:52% 1:80% 3:04% 5:93% 2:06% 0:51% 5:14% 2:12% 3:54% 2:87% 1:33% 3:39% 3:34% 3:42% 4:18% 0:76% 2:62%

0:1993 0:2351 0:1318 0:2293 0:0395 0:1206 0:0956 0:1387 0:0862 0:0837 0:1311 0:0876 0:1316 0:1806 0:2669 0:0783 0:1810 0:1974 0:0859 0:1007 0:0474 0:1356

1:64% 1:94% 1:18% 1:99% 1:67% 2:33% 1:67% 2:89% 5:84% 1:95% 0:39% 5:11% 1:95% 3:39% 2:75% 1:25% 3:24% 3:21% 3:36% 4:09% 0:63% 2:50%

SS C OW(10)

SS C OW(5)

SS C OW(3)

"i;J

ıi;J

ıi;J

ıi;J

SS Australia Austria Belgium Canada Switzerland Germany Denmark Spain Finland France U.K. Hong Kong Ireland Italy Japan Netherlands Norway New Zealand Singapore Sweden U.S.A. Average

Panel B: Emerging markets SS ıi;J Argentina Brazil Chile Greece Indonesia Korea, S. Malaysia Mexico Philippines Portugal Thailand Turkey Taiwan Average

0:3163 0:2756 0:2344 0:3178 0:4113 0:3924 0:3113 0:2054 0:3538 0:2951 0:3116 0:3426 0:3754 0:3187

9:32% 9:64% 4:40% 5:63% 11:18% 7:14% 4:93% 6:82% 6:59% 2:96% 8:46% 14:80% 6:89% 7:60%

0:1914 0:1778 0:1511 0:1884 0:2806 0:2927 0:1750 0:0725 0:2136 0:2185 0:1881 0:2240 0:2739 0:2037

"i;J 9:11% 9:51% 4:21% 5:54% 11:00% 6:89% 4:86% 6:75% 6:52% 2:86% 8:26% 14:61% 6:71% 7:45%

0:1814 0:1508 0:1195 0:1532 0:2534 0:2469 0:1452 0:0687 0:1796 0:1926 0:1688 0:1676 0:2386 0:1743

"i;J 8:99% 9:42% 4:10% 5:44% 10:92% 6:78% 4:80% 6:58% 6:46% 2:74% 8:13% 14:54% 6:66% 7:35%

0:1699 0:1307 0:1025 0:1223 0:2396 0:2286 0:1285 0:0617 0:1570 0:1799 0:1517 0:1515 0:2175 0:1570

"i;J 8:88% 9:28% 3:97% 5:31% 10:83% 6:65% 4:73% 6:39% 6:39% 2:61% 8:02% 14:43% 6:49% 7:23%

The medians of benefits of international diversification with short-sales (SS) and various sets of short-sale plus over-weighting (SS C OW(U )) investment constraints to domestic investor in different countries are reported. The increase in the risk-adjusted premium and the decrease in volatility of domestic portfolio by 1  international diversification are •i;J D MSRJ  SRi and ©i;J D 1  wTMVP;J VwMVP;J =Vi 2 , respectively.

in the rest of world. The relations of the more constrained diversification benefits among regions are similar to the results of the less constrained portfolios. In the long term, the international diversification benefits are time-varying and do not fall significantly. The more restrictive upper bounds of portfolio weights shrink the benefits of international diversification and their temporal variation. The optimal global diversifying strategies indeed generate benefits to the local investors, even though the short-sale and over-weighting constraints are increasingly restrictive. In general, local investors in emerging markets, particularly in East Asia and Latin America, benefit more from international diversification. This result holds for global portfolios with various investment restrictions.

13.6 The Optimal Portfolio Components Table 13.4 shows the average weight of each country for the maximum Sharpe Ratio (MSR) portfolio and the MVP over the period. The comparison of components in the optimal portfolios under various investment constraints indicates the infeasibility of less restrictive diversifying strategies. For the portfolio with short-sale restrictions, weighting in emerging markets is disproportionate to its distribution in the world capital market value. On average, the investors who wish to maximize portfolio mean-variance efficiency should place 31.41% of wealth in emerging markets, which represent merely 7.4% of total market value of all countries at the end of 2005. Among them, although the Chilean equity

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W.-J.P. Chiou and C.-F. Lee

Table 13.4 Means of portfolio weights Panel A: Developed countries MSR Portfolio Australia Austria Belgium Canada Switzerland Germany Denmark Spain Finland France U.K. Hong Kong Ireland Italy Japan Netherlands Norway New Zealand Singapore Sweden U.S.A. Sum

SS 0.0010 0.1196 0.0256 0.0040 0.1120 0.0002 0.0278 0.0011 0.1640 0.0017 0.0043 0.0009 0.0139 0.0007 0.0000 0.0282 0.0000 0.0014 0.0000 0.0132 0.1663 0.6859

SS C OW(10) 0.0404 0.0025 0.0026 0.0493 0.1282 0.0034 0.0137 0.0221 0.0353 0.0667 0.0073 0.0590 0.0049 0.0125 0.0014 0.0191 0.0059 0.0021 0.0104 0.0095 0.3830 0.8792

MVP SS C OW(5) 0.0211 0.0018 0.0017 0.0324 0.0742 0.0128 0.0098 0.0244 0.0188 0.0814 0.0254 0.0426 0.0031 0.0193 0.0086 0.0119 0.0033 0.0015 0.0084 0.0107 0.5085 0.9218

SS C OW(3) 0.0131 0.0013 0.0012 0.0194 0.0476 0.0144 0.0059 0.0182 0.0118 0.0794 0.0421 0.0308 0.0021 0.0162 0.0217 0.0077 0.0020 0.0010 0.0061 0.0090 0.5963 0.9475

SS 0.0180 0.0530 0.1197 0.0496 0.0441 0.0072 0.0264 0.0000 0.0000 0.0000 0.2426 0.0000 0.0067 0.0059 0.0585 0.0353 0.0000 0.0062 0.0000 0.0000 0.1845 0.8577

SS C OW(10) 0.0351 0.0105 0.0093 0.0578 0.0690 0.0273 0.0265 0.0000 0.0000 0.0075 0.3317 0.0001 0.0051 0.0128 0.0661 0.0119 0.0013 0.0020 0.0022 0.0000 0.2477 0.9239

SS C OW(5) 0.0503 0.0061 0.0049 0.0361 0.0888 0.0258 0.0145 0.0000 0.0010 0.0153 0.2435 0.0016 0.0055 0.0283 0.0855 0.0059 0.0007 0.0013 0.0016 0.0000 0.3273 0.9438

SS C OW(3) 0.0356 0.0043 0.0033 0.0235 0.0658 0.0233 0.0094 0.0000 0.0015 0.0297 0.1553 0.0049 0.0045 0.0427 0.1078 0.0066 0.0005 0.0011 0.0011 0.0000 0.4374 0.9584

Panel B: Emerging markets MSR Portfolio Argentina Brazil Chile Greece Indonesia Korea, S. Malaysia Mexico Philippines Portugal Thailand Turkey Taiwan Sum

MVP

SS

SS C OW(10)

SS C OW(5)

SS C OW(3)

SS

SS C OW(10)

SS C OW(5)

SS C OW(3)

0.0011 0.0055 0.1049 0.0065 0.0044 0.0600 0.0372 0.0640 0.0241 0.0000 0.0010 0.0049 0.0006 0.3141

0.0019 0.0158 0.0112 0.0060 0.0034 0.0241 0.0232 0.0216 0.0030 0.0003 0.0071 0.0026 0.0006 0.1208

0.0010 0.0138 0.0058 0.0039 0.0020 0.0148 0.0145 0.0122 0.0015 0.0008 0.0045 0.0018 0.0016 0.0782

0.0006 0.0093 0.0035 0.0029 0.0013 0.0095 0.0099 0.0074 0.0009 0.0008 0.0029 0.0013 0.0023 0.0525

0.0026 0.0004 0.0290 0.0008 0.0017 0.0249 0.0347 0.0031 0.0071 0.0216 0.0000 0.0067 0.0096 0.1423

0.0015 0.0008 0.0094 0.0061 0.0016 0.0256 0.0120 0.0015 0.0014 0.0044 0.0001 0.0042 0.0077 0.0761

0.0010 0.0006 0.0054 0.0062 0.0015 0.0190 0.0072 0.0007 0.0008 0.0029 0.0003 0.0023 0.0084 0.0562

0.0006 0.0005 0.0040 0.0062 0.0010 0.0120 0.0054 0.0005 0.0006 0.0020 0.0005 0.0015 0.0069 0.0416

The average weights of the MSR portfolio and the MVP with various investment constraints for each country over the sample period are reported.

market represented only about 0.35% of world capitalization, the average weight is 10.49%. The extreme weights also occur in Austria, Finland, Ireland, South Korea, Mexico, and the Philippines. As the overweighting investment constraints are included and become more restrictive, the percentage of the portfolios on the assets in developing countries decreases. An overwhelming amount of investments also can be found in small-cap areas, such as Latin America, Northern Europe, Southern Europe, and Oceania when only shortselling is considered. On the other hand, the weights for the areas of large market value, such as North America

and Central/Western Europe are zero for a number of periods. The more restrictive OW constraints “enforce” portfolio weighting to distribute to other second-best mean-varianceefficient markets of large capitalization. The weightings of those more constrained portfolios are more balanced than the ones of the short-sale-forbidden portfolios. The percentage of the non-zero-weight month in the testing period is reported in Table 13.5. On the whole, assets in the developed countries were more likely to be selected in the optimal portfolios with restrictive upper bounds. When only short-sale was not allowed, three national

13 International Portfolio Management: Theory and Method

231

Table 13.5 Percentage of non-zero portfolio weight Panel A: Developed countries MSR portfolio

MVP SS C OW(5)

Australia Austria Belgium Canada Switzerland Germany Denmark Spain Finland France U.K. Hong Kong Ireland Italy Japan Netherlands Norway New Zealand Singapore Sweden U.S.A. Developed Countries

SS 2:56 14:74 7:05 4:49 42:95 0:64 17:31 1:92 51:28 1:92 5:77 5:13 10:26 0:64 0:00 25:00 0:64 2:56 0:00 8:97 43:59 95:51

SS C OW(10) 24:36 16:67 23:08 21:15 60:26 7:05 43:59 18:59 58:97 19:87 10:26 35:26 18:59 12:82 2:56 35:90 19:23 21:79 25:64 14:74 72:44 100:00

(%) 25:00 24:36 29:49 25:00 64:10 17:95 59:62 26:92 60:90 31:41 23:72 42:31 23:08 24:36 4:49 44:23 21:15 31:41 37:18 28:85 82:05 100:00

SS C OW(5) SS C OW(3) 25:64 29:49 33:33 25:00 67:95 23:08 57:69 32:05 63:46 44:87 33:97 51:28 26:28 28:21 8:33 46:15 21:79 35:26 44:87 39:74 85:26 100:00

SS SS C OW(10) 43:59 58:33 46:79 71:15 73:72 78:21 29:49 30:77 36:54 71:15 11:54 25:00 58:33 81:41 0:00 0:00 0:00 0:64 0:00 13:46 56:41 64:10 0:00 2:56 17:31 21:15 23:08 50:00 70:51 78:21 21:15 21:15 0:00 4:49 19:23 20:51 0:00 6:41 0:00 0:64 67:95 75:00 100:00 100:00

(%) 64:74 82:05 82:05 30:77 89:74 29:49 83:33 0:00 8:33 22:44 65:38 8:97 43:59 80:77 84:62 21:15 5:77 26:92 7:05 0:00 100:00 100:00

SS C OW(3) 73:08 96:15 94:87 30:77 92:31 35:90 91:67 0:64 8:33 39:10 67:31 20:51 56:41 91:03 92:95 41:67 6:41 36:54 9:62 0:64 100:00 100:00

Panel B: Emerging markets MSR portfolio

MVP SS C OW(5)

SS Argentina Brazil Chile Greece Indonesia Korea, S. Malaysia Mexico Philippines Portugal Thailand Turkey Taiwan Emerging Markets

5:77 18:59 30:13 13:46 7:69 13:46 14:10 25:64 30:77 0:00 1:28 12:18 1:92 67:31

SS C OW(10) 25:64 43:59 51:28 24:36 25:64 28:21 45:51 49:36 36:54 3:85 35:90 19:23 2:56 86:54

SS C OW(5)

(%)

SS C OW(3)

26:28 54:49 51:92 26:92 30:13 35:26 54:49 52:56 36:54 16:03 43:59 24:36 7:05 91:03

28:85 55:77 52:56 32:69 31:41 34:62 59:62 53:21 37:82 26:28 47:44 28:85 12:18 92:95

SS 22:44 5:13 44:87 7:69 14:10 44:87 23:08 8:97 13:46 32:05 0:00 28:21 33:97 98:08

SS C OW(10) 22:44 8:33 47:44 36:54 18:59 43:59 25:64 7:69 16:67 46:15 1:92 31:41 39:74 93:59

(%) 28:85 7:05 50:00 51:28 22:44 43:59 30:77 9:62 19:23 58:33 4:49 31:41 51:92 94:23

SS C OW(3) 32:69 5:77 60:26 73:72 25:00 44:23 39:10 8:33 23:08 69:23 8:97 33:97 51:92 99:36

This table reports the percentages of the months of non-zero portfolio weight for the MSR portfolio and the MVP with various investment constraints for each country during the sample period.

indices (Japan, Singapore, and Portugal) were never selected in the MSR portfolio, and eight national indices (Spain, Finland, France, Hong Kong, Norway, Singapore, Sweden, and Thailand) were never selected in the MVP. Furthermore, only five markets (Switzerland, Finland, U.S.A., Chile, and Portugal) were included in more than 30% of the sample period. As the overweighting investments are increasingly constrained, the securities in any group of countries were more frequently included in the MSR portfolio as well as the

MVP. In the most restrictive case, securities in 22 markets were included in the MSR portfolio and 23 markets in the MVP for more than 30% of the sample period, respectively. Figure 13.4 shows the numbers of national indices selected in the optimal portfolios over the sample period. This cross-strategy comparison supports the essentialness of the upper bounds of portfolio weighting. The timeseries average of market indices for the no-short-selling MSR portfolio is 4.2 and implies, overall, that more than

232

W.-J.P. Chiou and C.-F. Lee

Panel A: MSR Portfolio

Panel B: MVP

Fig. 13.4 Number of selected national indices in the optimal portfolios

80% of the international portfolios are redundant in the same month. The inclusion of overweighting constraints expands the coverage of the optimal portfolios to 9:6 ŒSS C OW.10/ ; 11:8 ŒSS C OW.5/ , and 13:3 ŒSS C OW.3/ . Although a certain portion of Sharpe ratio benefits are lost due to compulsory diversifications, those strategies also increase the invariance of weighting and benefits, while at the same time expanding the assets chosen in the optimal portfolios. As the upper bounds are increasingly constrained, the variations in the elements of the optimal portfolio diminish. A similar conclusion can also be found in the weighting of the MVP.

13.7 Conclusion This chapter adds to the current literature by investigating the impact of weighting bounds on the benefits and asset allocation of the globally diversified portfolio. The empirical results suggest that boundaries of weighting, such as shortsale and overweighting, decrease but not completely eliminate the benefits of global investment. Domestic investors in emerging markets, particularly in East Asia and Latin America, benefit more from international diversification. This finding holds even though the global financial market has become increasingly integrated. In addition, more restrictive

13 International Portfolio Management: Theory and Method

investment constraints enhance the feasibility of the optimal strategies. Specifically, some appealing traits for asset management transpire: a reduction in the temporal deviation of diversification benefits, a decrease in time-variation of components in optimal portfolio, and an expansion in the range of comprising assets. The findings to the issues discussed in this chapter provide useful insights for international asset management professionals. Future research of the benefits of global diversification strategies may apply dynamic asset pricing theory and take into account the demands of hedging market exposure and exchange rate. Chang et al. (2005) examine the demands of market risk hedge and currency exposure hedge in international asset pricing. Chan et al. (1999) and Fleming et al. (2001) confirm the economic value in modeling a variancecovariance matrix when the efficient frontier is formed. Bekaert and Harvey (1995) and Bekaert et al. (2005) document the time-variation of the integration of international financial markets. The purpose of this chapter, however, is to investigate the time-varying international diversification benefits and their changes caused by various investment constraints over the long term. Due to the time-variation in expected return, volatility, and correlation among the international assets, application of conditional or dynamic asset pricing models to parameterize the optimal international diversification is helpful to professionals of asset management.

References Bae, K., K. Chan, and A. Ng. 2004. “Investibility and return volatility.” Journal of Financial Economics 71, 239–263. Beck, T., A. Demirgüç-Kunt, and R. Levine. 2003. “Law, endowments, and finance.” Journal of Financial Economics 70, 137–181. Bekaert, G. and C. R. Harvey. 1995. “Time-varying world market integration.” Journal of Finance 50, 403–444. Bekaert, G. and C. R. Harvey. 2003. “Emerging markets finance.” Journal of Empirical Finance 10, 3–56. Bekaert, G., C. Harvey, and A. Ng. 2005. “Market integration and contagion.” Journal of Business 78, 39–70. Bekaert, G. and M. Urias. 1996. “Diversification, integration and emerging market closed-end funds.” Journal of Finance 51, 835–869. Bekaert, G., R. Hodrick, and X. Zhang. 2005. International stock return comovements, NBER working paper, Columbia Business School, Columbia. Chan, L. K., J. Karceski, and J. Lakonishok. 1999. “On portfolio optimization: forecasting covariances and choosing the risk model.” Review of Financial Studies 12, 937–974. Chang, J., V. Errunza, K. Hogan, and M. Hung. 2005. “An intertemporal international asset pricing model: theory and empirical evidence.” European Financial Management 11, 218–233. Chiou, W. 2008. “Who benefits more from international diversification?” Journal of International Financial Markets, Institutions, and Money 18, 466–482.

233 Chiou, W., A. C. Lee, and C. Chang. 2009. “Do investors still benefit from international diversification with investment constraints?” Quarterly Review of Economics and Finance 49(2), 448–483. Cosset, J. and J. Suret. 1995. “Political risk and the benefit of international portfolio diversification.” Journal of International Business Studies 26, 3013–18. De Jong, F. and F. A. De Roon. 2005. “Time-varying market integration and expected returns in emerging markets.” Journal of Financial Economics 78, 583–613. Demirgüç-Kunt, A. and V. Maksimovic. 1998. “Law, finance, and firm growth.” Journal of Finance 53, 2107–2137. De Roon, F. A., T. E. Nijman, and B. J. Werker. 2001. “Testing for mean-variance spanning with short sales constraints and transaction costs: the case of emerging markets.” Journal of Finance 56, 721–742. De Santis, G. and B. Gerard. 1997. “International asset pricing and portfolio diversification with time-varying risk.” Journal of Finance 52, 1881–1912. Djankov, S., R. La Porta, F. Lopez-de-Silanes, and A. Shleifer. 2003. “Courts.” Quarterly Journal of Economics 118, 453–518. Djankov, S., R. La Porta, F. Lopez-de-Silanes, and A. Shleifer. 2008. “The law and economics of self-dealing.” Journal of Financial Economics 88, 430–465. Driessen, J. and L. Laeven. 2007. “International portfolio diversification benefits: cross-country evidence from a local perspective.” Journal of Banking and Finance 31, 1693–1712. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2007. Modern portfolio theory and investment analysis. Wiley, NJ. Errunza, V., K. Hogan, and M. Hung. 1999. “Can the gains from international diversification be achieved without trading abroad?” Journal of Finance 54, 2075–2107. Errunza, V., E. Losq, and P. Padmanabhan. 1992. “Tests of integration, mild segmentation and segmentation hypotheses.” Journal of Banking and Finance 16, 949–972. Fleming, J., C. Kirby, and B. Ostdiek. 2001. “The economic value of volatility timing.” Journal of Finance 56, 329–352. Fletcher, J. and A. Marshall. 2005. “An empirical examination of the benefits of international diversification.” Journal of International Financial Markets, Institutions and Money 15, 455–468. French, K. R. and J. M. Poterba. 1991. “Investor diversification and international equity markets.” American Economic Review 81, 222–226. Green, R. C. and B. Hollifield. 1992. “When will mean-variance efficient portfolios be well diversified?” Journal of Finance 47, 1785–1809. Harvey, C. 1995. “Predictable risk and returns in emerging markets.” Review of Financial Studies 8, 773–816. Jagannathan, R. and T. S. Ma. 2003. “Risk reduction in large portfolios: why imposing the wrong constraints helps.” Journal of Finance 58, 1651–1683. La Porta, R., F. Lopez-de-Silanes, A. Shleifer, and R. Vishny. 1998. “Law and finance.” Journal of Political Economy 106, 1113–1155. La Porta, R., F. Lopez-de-Silanes, A. Shleifer, and R. Vishny. 2000. “Investor protection and corporate governance.” Journal of Financial Economics 58, 3–27. Li, K., A. Sarkar, and Z. Wang. 2003. “Diversification benefits of emerging markets subject to portfolio constraints.” Journal of Empirical Finance 10, 57–80. Markowitz, H. 1952. “Portfolio selection.” Journal of Finance 7, 77–91. Novomestky, F. 1997. “A dynamic, globally diversified, index neutral synthetic asset allocation strategy.” Management Science 43, 998–1016.

234 Obstfeld, M. 1994. “Risk-taking, global diversification, and growth.” American Economic Review 84, 1310–1329. Pástor, L. and R. Stambaugh. 2000. “Comparing asset pricing models: an investment perspective.” Journal of Financial Economics 56, 335–381.

W.-J.P. Chiou and C.-F. Lee Stulz, R. M. and R. Williamson. 2003. “Culture, openness, and finance.” Journal of Financial Economics 70, 313–349. Wang, Z. 1998. “Efficiency loss and constraints on portfolio holdings.” Journal of Financial Economics 48, 359–375.

Chapter 14

The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model: A Case of World Equity Fund Market Chin W. Yang, Ken Hung, and Jing Cui

Abstract Due to limited numbers of reliable international equity funds, the Markowitz investment model is ideal in constructing an international portfolio. Overinvestment in one or several fast-growing markets can be disastrous as political instability and exchange rate fluctuations reign supreme. We apply the Le Châtelier principle to the international equity fund market with a set of upper limits. Tracing out a set of efficient frontiers, we inspect the shifting phenomenon in the mean–variance space. The optimum investment policy can be easily implemented and risk minimized. Keywords Markowitz quadratic programming model r Le Châtelier Principle r International equity fund and added constraints

14.1 Introduction As the world is bubbling in the cauldron of globalization, investment in foreign countries ought to be considered as part of an optimum portfolio. Needless to say, returns from such investment can be astronomically high as was evidenced by several international equity funds in the period of the Asian flu. On the flop side, however, it may become disastrous as political system and exchange rate markets undergo structural changes. The last decade has witnessed a substantial increase in international investments partly due to “hot money” from OPEC, the People’s Republic of China, and various types of quantum funds, many of which are from the U.S. From the report of Morgan Stanley Capital International Perspectives, North America accounted for 51.6% of the

K. Hung () Texas A&M International University, Laredo, TX, USA e-mail: [email protected] C.W. Yang and J. Cui Clarion University of Pennsylvania, Clarion, PA, USA

world equity market. Most recently, however, the European Union has begun to catch up with the U.S. Along with booming Asian economies, Russia, India and Brazil are making headway into the world economic stage. In addition, the Japanese economy has finally escaped from its decade-long depression. Obviously, the opportunities in terms of gaining security values are much greater in the presence of international equity markets. Unlike the high correlation between domestic stock markets (e.g., 0.95 between the New York Stock Exchange and the S&P index of 425 large stocks), it is rather low between international markets. For example, correlations between the stock indexes of the U.S. and Australia, Belgium, Germany, Hong Kong, Italy, Japan and Switzerland were found to be 0.505, 0.504, 0.489, 0.491, 0.301, 0.348 and 0.523, respectively (Elton and Gruber 2007) In effect, the average correlation between national stock indexes is 0.54. The same can be said of bonds and Treasury bills. The relatively low correlation across nations offers an excellent playground for international diversification, which will no doubt reduce the portfolio risk appreciably. An example was given by Elton and Gruber (2007) that 26% of the world portfolio, excluding the U.S. market, in combination with 74% of the U.S. portfolio reduced total minimum risk by 3.7% compared with the risk if investment is made exclusively in the U.S. market. Furthermore, it was found that a modest amount of international diversification can lower risk even in the presence of the large standard deviation in returns. Solnik found that 15 of 17 countries had higher returns on stock indexes than those on the US equity index for the period of 1971– 1985. In the light of the growing trend of globalization and expanding GDP, a full-fledged Markowitz rather than a world CAPM is appropriate to arrive at an optimum international portfolio. It is of greater importance to study this topic for it involves significantly more dollar amounts (sometime in billions of dollars). The next section presents data and the Markowitz quadratic programming model. Sect. 14.3 introduces the Le Châtelier principle in terms of maximum investment percentage on each fund. Sect. 14.4 discusses the empirical results. A conclusion is given in Sect. 14.5.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_14, 

235

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C.W. Yang et al.

14.2 Data and Methodology Monthly price data are used from March 2006 to March 2007 for ten international equity markets or 130 index prices. Ust 1 / , we obtain 120 index returns from ing return rt D .PtPP t 1 which 10 expected return ri .I D 1; 2 : : : 10/ and 45 return covariances ¢ij and 10 return variances are obtained. In addition, we assume future index return will not deviate from expected returns. In his pioneering work, Markowitz (1952, 1956, 1959, and 1991) proposed the following convex quadratic programming investment model. Min xi ; xj v D

X

xi2 ii C

i 2I

XX

xi xj ij

(14.1)

i 2I j 2J

subject to X

ri xi k

(14.2)

xi D 1

(14.3)

xi 0 8i 2 I

(14.4)

i 2I

X i 2I

where xi D proportion of investment in national equity fund i ii D variance of rate of return of equity fund i ij D covariance of rate of return of equities i and j ri D expected rate of return of equity fund i k D target rate of return of the portfolio i and j are sets of positive integers Subscripts i D 1; 2; 3 : : : : : : 10 denote ten equity funds: Fidelity European Growth Fund, ED America Fund, Australia Fund, Asean Fund, Japan Fund, Latin America fund, Nordic Fund, Iberia Fund, Korea Fund, United Kingdom Fund, respectively. Note that exchange rate is not considered here, as some studies suggest the exchange rate market is fairly independent of the domestic equity market (Elton and Gruber 2007, pp. 266–267). Moreover, rates of returns on equities are percentages, which are measurement free regardless of which local currencies are used. Markowitz model is the very foundation of modern investment with quite a few software available to solve Equations (14.1) through (14.4). A standard quadratic programming does not take advantage of the special feature in the Markowitz model and as such the dimensionality problem sets limits to solving a large-scale portfolio. One way to circumvent this limitation is to reduce the Markowitz model to linear programming framework via the Kuhn-Tucker conditions. The famed duality theorem that the objective function of the primal (maximization) problem is bounded by that of the dual (minimization) problem lends support to solving

quadratic programming as linear programming with ease. We employ LINDO (Linear, Interactive, and Discrete Optimizer) developed by LINDO systems Inc. (2000). The solution in test runs is identical to that solved by other nonlinear programming software, such as LINGO. We present the results in Section 14.4.

14.3 The Le Châtelier Principle in the Markowitz Investment Model When additional constraints are added to a closed system, the response of variables to such perturbation become more “rigid” and may very well increase the total investment risk, since decision variables have fewer choices. If the original solution set remains unchanged as additional constraints are added, it is a case of envelope theorem in differential calculus (Samuelson 1960 and 1972; Silberberg 1971 and 1974; Labys and Yang 1996; Silberberg and Suen 2000). In economics terms, it is often said that long-run demand is more elastic than that of a short run. In reality, however, the decision variables are expected to change as constraints are added. For instance, optimum investment proportions are bound to vary as the Investment Company Act is enacted. In particular, the result by Loviscek and Yang (1997) indicates the loss in efficiency is 1–2 percentage points due to the Investment Company Act, which translates into millions of dollars loss in daily return. The guidelines of the Central Securities Administrators Council declare that no more than 5% of a mutual fund’s assets may be invested in the securities of that issuers. This 5% rule extends to investment in companies in business less than 3 years, warrants options, and futures. It is to be pointed out that the Investment Company Act applies to domestic mutual funds. In the case of world equity markets, the magnitudes are greater and one percentage point may translate into billions of dollars. To evaluate the theoretical impact in terms of the Le Châtelier principle, we formulate the Lagrange Equation from (14.1) to (14.4).

X X xi (14.5) LDvC k ri xij C  1  Where  and  are Lagrange multiplies indicating change in the risk in response to change in right hand side constraints. In particular  assumes important economic meaning: change in total portfolio risk in response to an infinitesimally small change in k while all other decision variables adjust to their new equilibrium levels, that is,  D dv=dk. Hence, the Lagrange multiplier is of pivotal importance in determining the shape of the efficient frontier in the model. In the unconstrained Markowitz model the optimum investment proportion in stock (or equity fund) i is a free variable between 0 and 1. In the international equity

14 The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model

market, however, diversification is more important owing to the fact that (1) correlations are low and (2) political turmoil can jeopardize the portfolio greatly if one put most of his/her eggs in one basket. As we impose the maximum investment proportion on each security from 99 to 1%, the solution to the portfolio selection model becomes more restricted (i.e., the values of optimum investment proportion are bounded within a narrower range as the constraint is tightened). Such as impact on the objective function v is equivalent to the following: as the system is gradually constrained, the limited freedom of optimum X gives rise to a higher and higher risk level as k is increased. This is to say, if parameter k is increased gradually, the Le Châtelier principle implies that in the original Markowitz minimization system, isorisk contour has the smallest curvature to accommodate the most efficient adjustment mechanism as shown below: ˇ 2 ˇ ˇ 2  ˇ ˇ 2  ˇ ˇ @ v ˇ ˇ@ v ˇ ˇ@ v ˇ ˇ ˇ ˇ ˇ ˇ ˇ (14.6) ˇ @k 2 ˇ  ˇ @k 2 ˇ  ˇ @k 2 ˇ where v and v are the objective function (total portfolio risk) corresponding to the additional constrains of xi  s and xi  s for all i and s > s represent different investment proportions allowed under V and V and abs denotes absolute value. Using the envelope theorem (Dixit 1990), it follows immediately that @ fL.xi ; k/ D v.xi .k//g d fL.xi .k/; k/ D v.xi .k//g D dk @k D  jxi D xi .k/ (14.7) as such, Equation (14.6) can be mode to ensure the following inequalities: ˇ ˇ ˇ  ˇ ˇ  ˇ ˇ @ ˇ ˇ @ ˇ ˇ @ ˇ ˇ ˇˇ ˇ ˇ ˇ (14.8) ˇ @k ˇ ˇ @k ˇ  ˇ @k ˇ Table 14.1 The Markowitz investment model without upper limits Expected rate 11% 12% 13% 14% of return Objective function value

237

Well-known in the convex analysis, the Lagrange multiplier œ is the reciprocal of the slope of the efficient frontier curve frequently drawn in investment textbooks. Hence, in the mean–covariance space the original Markowitz efficient frontier has the steepest slope for a given set of xi ’s. Care must be exercised that the efficient frontier curve of the Markowitz minimization system has a vertical segment corresponding to a range of low ks and a constant v. Only within this range do the values of optimum xs remain unchanged under various degrees of maximum limit imposed. Within this range constraint Equation (14.2) is not active. That is the Lagrange multiplier is zero. As a result, equality relation holds for Equation (14.8). Outside this range, the slopes of the efficient frontier curve are different given to the result of Equation (14.8).

14.4 An Application of the Le Châtelier Principle in the World Equity Market Equations (14.1) through (14.4) comprise the Markowitz model without upper limits imposed on stocks. We calibrate the target rate (annualized) k to 34% from 11% with an increment of 1%. At 11%, X2 (ED America Fund) and X3 (Australia Fund) split the portfolio. As k increase X3 dominates while X2 is given less and less weight, which leads to the emergence of X8 (Iberia Fund). At 30%, only X4 (Asean Fund) and X8 (Iberia Fund) remain in the portfolio (Table 14.1). The efficient frontier leftmost curve in Fig 14.1 takes the usual shape in the mean–variance space. As we mandate a 35% maximum limit on each equity fund – the least binding constrain – X1(Fidelity European Growth Fund) X2, X3, X8 and X10 (United Kingdom Fund) take over the portfolio from k D 10% to 15% before X4 drops out 15%

16%

17%

18%

19%

6:78  102 6:82  102 6:95  102 7:16  102 7:39  102 7:64  102 7:91  102 8:20  102 8:58  102

X1 (Fidelity 0 European growth fund)

0

0

0

0

0

0

0

0

X2 (ED America fund)

0.5

0.400806

0.300101

0.219762

0.162278

0.104793

0.047309

0

0

X3 (Australia 0.5 fund)

0.599194

0.699899

0.75939

0.772629

0.785867

0.799106

0.791753

0.699589

X4 (Asean fund)

0

0

0

0

0

0

0

0

0.01296

X5 (Japan fund)

0

0

0

0

0

0

0

0

0 (continued)

238

Table 14.1 (continued) X6 (Latin 0 America fund) X7 (Nordic 0 fund) X8 (Iberia 0 fund) X9 (Korea 0 fund) X10 (United 0 Kingdom fund) Expected rate 20% of return Objective 9:05  102 function value X1 (Fidelity 0 European growth fund) 0 X2 (ED America fund) X3 (Australia 0.634087 fund) X4 (Asean 0.057536 fund) X5 (Japan 0 fund) X6 (Latin 0 America fund) X7 (Nordic 0 fund) X8 (Iberia 0.308377 fund) X9 (Korea 0 fund) X10 (United 0 Kingdom fund)

C.W. Yang et al.

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.020848

0.065094

0.109339

0.153585

0.208247

0.287451

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

21%

22%

23%

24%

25%

26%

27%

28%

9:59  102 0.101866

0.108515

0.115818

0.123775

0.132387

0.141654

0.151575

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.568585

0.503083

0.437581

0.372078

0.306576

0.241074

0.175572

0.110069

0.102111

0.146687

0.191262

0.235838

0.280414

0.324989

0.369565

0.41414

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.329304

0.35023

0.371157

0.392084

0.41301

0.433937

0.454864

0.47579

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

50% 45% 40% 35% Expected return

15% 20% 25% 30% 35% 100% SML

30% 25% 20% 15% 10% 5% 0% 0

0.05

0.1

0.15

0.2 Risk

Fig. 14.1 Efficient frontiers with various upper bounds

0.25

0.3

0.35

Table 14.2 Solutions of the Markowitz model with 35% upper limit Expected rate 9% 10% 11% 12% of return 7:43  102 7:58  102 7:73  102 Objective 7:33  102 function value 0.085217 0.133884 0.110655 0.087426 X1 (Fidelity European growth fund) X2 (ED 0.35 0.35 0.35 0.35 America fund) X3 (Australia 0.35 0.35 0.35 0.35 fund) 0 0 0 0 X4 (Asean fund) X5 (Japan 0 0 0 0 fund) X6 (Latin 0 0 0 0 America fund) X7 (Nordic 0 0 0 0 fund) X8 (Iberia 0 0.029984 0.079497 0.12901 fund) X9 (Korea 0 0 0 0 fund) 0.214783 0.136133 0.109848 0.083564 X10 (United Kingdom fund) Expected rate 20% 21% 22% 23% of return 14% 8:07  102

0.040788

0.35

0.35 0.000716 0 0

0 0.227105 0 0.031391

25%

13% 7:90  102

0.064197

0.35

0.35 0 0 0

0 0.178523 0 0.05728

24%

26%

0.008995

0

0.267461

0

0

0

0.007761

0.35

0.35

0.015784

8:26  102

15%

27%

0

0

0.247432

0

0

0

0.058159

0.35

0.344409

0

8:48  102

16%

28%

0

0

0.289142

0

0

0

0.064676

0.35

0.296182

0

8:80  102

17%

29%

0

0

0.330852

0

0

0

0.071193

0.35

0.247955

0

9:13  102

18%

(continued)

0

0

0.35

0

0

0

0.093635

0.35

0.206365

0

9:48  102

19%

14 The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model 239

Table 14.2 (continued) 9:87  102 Objective function value 0 X1 (Fidelity European growth fund) X2 (ED 0.170407 America fund) X3 (Australia 0.35 fund) X4 (Asean 0.129594 fund) X5 (Japan 0 fund) X6 (Latin 0 America fund) X7 (Nordic 0 fund) X8 (Iberia 0.35 fund) X9 (Korea 0 fund) X10 (United 0 Kingdom fund) 0.107476

0

0.09849

0.35 0.20151 0 0

0 0.35 0 0

0.102903

0

0.134448

0.35 0.165552 0 0

0 0.35 0 0

0

0

0.35

0

0

0

0.237468

0.35

0.062532

0

0.112404

0

0

0.35

0

0

0

0.273427

0.35

0.026573

0

0.117688

0

0

0.35

0

0

0

0.314597

0.335403

0

0

0.123998

0

0

0.35

0

0.043638

0

0.35

0.256362

0

0

0.139536

0

0

0.35

0

0.162544

0

0.35

0.137456

0

0

0.170337

0

0

0.35

0

0.28145

0

0.35

0.01855

0

0

0.206101

0

0

0.35

0

0.3

0

0.35

0

0

0

0.228593

240 C.W. Yang et al.

Table 14.3 Solutions of the Markowitz model with 30% upper limit Expected rate of return 8% 9% 10% 7:64  102 7:78  102 Objective function value 7:60  102 X1 (Fidelity European 0.133913 0.192955 0.169726 growth fund) X2 (ED America fund) 0.3 0.3 0.3 X3 (Australia fund) 0.3 0.3 0.3 X4 (Asean fund) 0 0 0 X5 (Japan fund) 0 0 0 X6 (Latin America 0 0 0 fund) X7 (Nordic fund) 0 0 0 X8 (Iberia fund) 0 0.007437 0.05695 X9 (Korea fund) 0 0 0 X10 (United Kingdom 0.266087 0.199608 0.173324 fund) Expected rate of return 18% 19% 20% 9:71  102 0.101255 Objective function value 9:33  102 X1 (Fidelity European 0 0 0 growth fund) X2 (ED America fund) 0.289177 0.253218 0.21726 X3 (Australia fund) 0.3 0.3 0.3 X4 (Asean fund) 0.110823 0.146782 0.18274 X5 (Japan fund) 0 0 0 X6 (Latin America 0 0 0 fund) X7 (Nordic fund) 0 0 0 X8 (Iberia fund) 0.3 0.3 0.3 X9 (Korea fund) 0 0 0 X10 (United Kingdom 0 0 0 fund) 12% 8:09  102 0.123268 0.3 0.3 0 0 0 0 0.155977 0 0.120755 22% 0.110586 0 0.145344 0.3 0.254656 0 0 0 0.3 0 0

11% 7:93  102 0.146497 0.3 0.3 0 0 0 0 0.106463 0 0.14704 21% 0.105743 0 0.181302 0.3 0.218698 0 0 0 0.3 0 0

0 0.3 0 0

0.109385 0.3 0.290615 0 0

23% 0.115785 0

0 0.200288 0 0.09668

0.3 0.3 0.004001 0 0

13% 8:26  102 0.099031

0 0.3 0 0

0.059706 0.3 0.3 0 0.040294

24% 0.126251 0

0 0.240643 0 0.074284

0.3 0.3 0.011046 0 0

14% 8:44  102 0.074027

0 0.3 0 0

0.00518 0.3 0.3 0 0.09482

25% 0.139566 0

0 0.280998 0 0.051888

0.3 0.3 0.018091 0 0

15% 8:63  102 0.049023

0 0.3 0 0

0 0.19239 0.3 0 0.20761

26% 0.169119 0

0 0.3 0 0.033198

0.3 0.3 0.040426 0 0

16% 8:83  102 0.026376

0 0.3 0.1 0

0 0 0.3 0 0.3

27% 0.232735

0 0.3 0 0.017805

0.3 0.3 0.076368 0 0

17% 9:05  102 0.005827

14 The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model 241

Table 14.4 Solutions of the Markowitz model with 25% upper limit Expected rate of return 8% 9% 10% 7:94  102 8:18  102 Objective function value 7:94  102 X1 (Fidelity European 0.25 0.25 0.205568 growth fund) X2 (ED America fund) 0.25 0.25 0.25 X3 (Australia fund) 0.25 0.25 0.25 X4 (Asean fund) 0 0 0 X5 (Japan fund) 0 0 0 X6 (Latin America 0 0 0 fund) X7 (Nordic fund) 0 0 0 X8 (Iberia fund) 0 0 0.083917 X9 (Korea fund) 0 0 0 X10 (United Kingdom 0.25 0.25 0.210515 fund) Expected rate of return 18% 19% 20% 0.10134 0.104717 Objective function value 9:82  102 X1 (Fidelity European 0.02553 0.004981 0 growth fund) X2 (ED America fund) 0.25 0.25 0.25 X3 (Australia fund) 0.25 0.25 0.25 X4 (Asean fund) 0.168138 0.20408 0.237302 X5 (Japan fund) 0 0 0 X6 (Latin America 0 0 0 fund) X7 (Nordic fund) 0 0 0 X8 (Iberia fund) 0.25 0.25 0.25 X9 (Korea fund) 0 0 0 X10 (United Kingdom 0.056332 0.040939 0.012698 fund) 12% 8:50  102 0.157274 0.25 0.25 0.007287 0 0 0 0.173471 0 0.161969 22% 0.124825 0 0.16235 0.25 0.25 0 0.08765 0 0.25 0 0

11% 8:33  102 0.182278 0.25 0.25 0.000242 0 0 0 0.133115 0 0.184365 21% 0.112819 0 0.216876 0.25 0.25 0 0.033124 0 0.25 0 0

0 0.25 0 0

0.107825 0.25 0.25 0 0.142175

23% 0.138059 0

0 0.213826 0 0.139573

0.25 0.25 0.014331 0 0

13% 8:67  102 0.13227

0 0.25 0 0

0.053299 0.25 0.25 0 0.196701

24% 0.152521 0

0 0.25 0 0.117903

0.25 0.25 0.02437 0 0

14% 8:85  102 0.107728

0 0.25 0.01243 0

0 0.23757 0.25 0 0.25

25% 0.170996 0

0 0.25 0 0.10251

0.25 0.25 0.060312 0 0

15% 9:06  102 0.087178

0 0.25 0.25 0

0 0 0.25 0 0.25

26% 0.318692 0

0 0.25 0 0.087117

0.25 0.25 0.096254 0 0

16% 9:28  102 0.066629

0 0.25 0 0.071725

0.25 0.25 0.132196 0 0

17% 9:54  102 0.04608

242 C.W. Yang et al.

Table 14.5 Solutions of the Markowitz model with 20% upper limit Expected rate of return 11% Objective function value 8:91  102 X1 (Fidelity European growth fund) 0.2 X2 (ED America fund) 0.2 X3 (Australia fund) 0.2 X4 (Asean fund) 0 X5 (Japan fund) 0 X6 (Latin America fund) 0 X7 (Nordic fund) 0 X8 (Iberia fund) 0.2 X9 (Korea fund) 0 X10 (United Kingdom fund) 0.2 Expected rate of return 18% Objective function value 0.105156 X1 (Fidelity European growth fund) 0.199003 X2 (ED America fund) 0.2 X3 (Australia fund) 0.2 X4 (Asean fund) 0.2 X5 (Japan fund) 0 X6 (Latin America fund) 0.000997 X7 (Nordic fund) 0 X8 (Iberia fund) 0.2 X9 (Korea fund) 0 X10 (United Kingdom fund) 0 12% 8:94  102 0.196097 0.2 0.2 0.015936 0 0 0 0.187968 0 0.2 19% 0.115271 0.136659 0.2 0.2 0.2 0 0.063341 0 0.2 0 0

13% 9:12  102 0.16853 0.2 0.2 0.044255 0 0 0 0.2 0 0.187215 20% 0.126117 0.074314 0.2 0.2 0.2 0 0.125686 0 0.2 0 0

14% 9:33  102 0.14798 0.2 0.2 0.080197 0 0 0 0.2 0 0.171822 21% 0.137694 0.01197 0.2 0.2 0.2 0 0.18803 0 0.2 0 0

15% 9:56  102 0.127431 0.2 0.2 0.116139 0 0 0 0.2 0 0.156429 22% 0.157136 0 0.131176 0.2 0.2 0 0.2 0 0.2 0.068824 0

16% 9:83  102 0.106882 0.2 0.2 0.152081 0 0 0 0.2 0 0.141037 23% 0.183105 0 0.044055 0.2 0.2 0 0.2 0.003234 0.2 0.152711 0

17% 0.101129 0.086332 0.2 0.2 0.188023 0 0 0 0.2 0 0.125644 24% 0.244151 0 0 0.2 0.2 0 0.2 0 0.2 0.2 0

14 The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model 243

Table 14.6 Solutions of the Markowitz model with 15% upper limit Expected rate of return 14% 15% Objective function value 0.106891 0.107619 X1 (Fidelity European growth fund) 0.15 0.15 X2 (ED America fund) 0.15 0.15 X3 (Australia fund) 0.15 0.15 X4 (Asean fund) 0.15 0.15 X5 (Japan fund) 0.1 0.000542 X6 (Latin America fund) 0 0 X7 (Nordic fund) 0 0.099458 X8 (Iberia fund) 0.15 0.15 X9 (Korea fund) 0 0 X10 (United Kingdom fund) 0.15 0.15 16% 0.114122 0.15 0.15 0.15 0.15 0 0.07293 0.02707 0.15 0 0.15

17% 0.12277 0.15 0.15 0.15 0.15 0 0.129408 0 0.15 0 0.120592

18% 0.134269 0.15 0.15 0.15 0.15 0 0.15 0.035453 0.15 0.019024 0.045523

19% 0.150141 0.142108 0.15 0.15 0.15 0 0.15 0 0.15 0.107892 0

20% 0.175912 0.056087 0.043913 0.15 0.15 0 0.15 0.15 0.15 0.15 0

21% 0.195432 0.1 0 0.15 0.15 0 0.15 0.15 0.15 0.15 0

244 C.W. Yang et al.

14 The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model

(Table 14.2). Beyond it, X2, X3, X4 and X8 stay positive till k D 24%. At k D 26% and on X3, X4, X6 (Latin America Fund) and X8 prevail the solution set. The efficient frontier curve is the second leftmost as shown in Fig 14.1, indicating the second most efficient risk – return lotus in the market. As we lower the maximum limit on each equity fund to 30%, again X1, X2, X3, X8 and X10 enter the Markowitz portfolio base with X1, X2, X3, and X8 dominating other funds till k D 22% (Table 14.3). At k D 27%; X4 D 0:3, X6 (Latin America Fund) D 0:3; X8 D 0:3, and X9 (Korea Fund) D 0:1 constitute the solution set. The efficient frontier is next right to that with maximum investment limit set at 35%. When the maximum investment limit is lowered to 25%, the diversification across world equity market is obvious (Table 14.4); X1, X2, X3, X4, X8 and X10 become positive at k D 11% to k D 20%. At k D 21%, X10 is replaced by X6 till k D 24%. At the highest k D 26%, the international equity fund portfolio is equally shared by X4, X6, X8 and X9. Again, the efficient frontier has shifted a bit to the right, a less efficient position. Table 14.5 reports the results when the maximum investment amount is set at 20%. This constraint is clearly so overbearing that six of ten equity funds are positive at k D 12%, through k D 17%. The capital market manifests itself in buying majority of equity funds. As a consequence, the efficient frontier is second to the right most curve. Table 14.6 presents the last simulation in this paper: an imposition of a 15% maximum investment limit on each equity fund. This is the most binding constraint and as such we have eight positive investments at k D 15% and 16%. At its extreme, an equity fund manager is expected to buy every equity fund except X5 at k D 18%. The solution at k D 21%, the optimum portfolio consists of X1 D 0:1; X3 D X4 D X6 D X7 D X8 D X9 D 0:15. The over diversification is so pronounced that its efficient frontiers appears to the right most signaling the least efficient locus in the mean–variance space.

14.5 Conclusion The Le Châtelier principle developed in thermodynamics can be easily applied to international equity fund market in which billions of dollars can change hand at the drop of a hat. We need to point out that considering international capital investment is tantamount to enlarging opportunity set greatly especially in the wake of growing globalization coupled with prospering world economy. In applying the Le Châtelier principle, there is no compelling reason to fix the optimum portfolio solution at a fixed set. As such, optimum portfolio investment amounts (proportions) are most likely to change and lead one to a suboptimality. The essence of the Le Châtelier principle is that over diversification has its

245

own price, an exorbitant one, due to its astronomically high value of volume transacted. The forced equilibrium is far from being an efficient equilibrium as the efficient frontiers keep shifting to the right as the maximum investment proportion is tightened gradually. Is cross-border capital investment worthwhile? On the positive side, the enlarged opportunity will definitely moves solution set to the northwest corner of the mean–variance space, which corresponds to higher indifference curve. On the negative side, over diversification in the international equity market has its expensive cost: shifting efficient frontier to the southeast corner of the mean–variance space. If foreign governments view these capital investments negatively, a stiff tax or transaction cost would slow down such capital movement. Notwithstanding these negative considerations, investment in international equity market is on the rise and will become even more popular in the future.

References Dixit, A. K. 1990. Optimization in economic theory, 2nd ed., Oxford University Press, Oxford. Elton, E. J. and M. J. Gruber. 2007. Modern portfolio theory and investment analysis 7th ed., Wiley, New York. Labys, W. C. and C.-W. Yang. 1996. Le Chatelier principle and the flow sensitivity of spatial commodity models. in Recent advances in spatial equilibrium modeling: methodology and applications, van de Bergh, J. C. J. M., P. Nijkamp and P. Rietveld (Eds.). Springer, Berlin, pp. 96–110. Loviscek, A. L. and C.-W. Yang. 1997. “Assessing the impact of the investment company act’s diversification rule: a portfolio approach.” Midwest Review of Finance and Insurance 11(1), 57–66. Markowitz, H. M. 1952. “Portfolio selection.” The Journal of Finance VII(1), 77–91. Markowitz, H. M. 1956 “The optimization of a quadratic function subject to linear constraints.” Naval Research Logistics Quarterly 3, 111–133. Markowitz, H. 1959. Portfolio selection: Efficient Diversification of Investment, John Wiley and Sons, New York. Markowitz, H. M. 1991. “Foundation of portfolio theory.” Journal of Finance XLVI(2), 469–477. Samuelson, P. A. 1948. Foundations of Economic Analysis, Cambridge, MA: Harvard University Press. Samuelson, P. A. 1960. An extension of the Le Chatelier principle. Econometrica pp. 368–379. Samuelson, P. A. 1972. “Maximum principles in analytical economics.” American Economic Review 62(3), 249–262. Schrage, L. 2000. Linear, integer and quadratic programming with LINDO, Palo Alto, CA: The Scientific Press. Silberberg, E. 1971. “The Le Chatelier principle as a corollary to a generalized envelope theorem.” Journal of Economic Theory 3, 146– 155. Silberberg, E. 1974. “A revision of comparative statics methodology in economics, or how to do economics on the back of an envelope.” Journal of Economic Theory 7, 159–172. Silberberg, E. and W. Suen. 2000. The structure of economics: a mathematical analysis, McGraw-Hill, New York. Value Line, Mutual fund survey New York: Value Line, 19. Yang, C.W., K. Hung, and J. A. Fox. 2006. “The Le Châtelier principle of the capital market equilibrium.” in Encyclopedia of Finance, C. F. Lee and A. C. Lee (Eds.). Springer, New York, pp. 724–728.

Chapter 15

Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints ´ Darinka Dentcheva and Andrzej Ruszczynski

Abstract We consider the problem of constructing a portfolio of finitely many assets whose return rates are described by a discrete joint distribution. We present a new approach to portfolio selection based on stochastic dominance. The portfolio return rate in the new model is required to stochastically dominate a random benchmark. We formulate optimality conditions and duality relations for these models and construct equivalent optimization models with utility functions. Two different formulations of the stochastic dominance constraint: primal and inverse, lead to two dual problems that involve von Neuman–Morgenstern utility functions for the primal formulation and rank dependent (or dual) utility functions for the inverse formulation. The utility functions play the roles of Lagrange multipliers associated with the dominance constraints. In this way our model provides a link between the expected utility theory and the rank dependent utility theory. We also compare our approach to models using value at risk and conditional value at risk constraints. A numerical example illustrates the new approach. Keywords Portfolio optimization r Stochastic dominance r Stochastic order r Risk r Expected utility r Duality r Rank dependent utility r Yaari’s dual utility r Value at risk r Conditional value at risk

15.1 Introduction The problem of optimal portfolio selection is subject of major theoretical and computational studies in finance. A fundamental issue while dealing with uncertain outcomes is a theoretically sound approach to their comparison. The theory of stochastic orders plays a fundamental role in economics (see Mosler and Scarsini 1991; Whitmore and

Findlay 1978). These are relations that induce partial order in the space of real random variables in the following way. A random variable R dominates the random variable Y if EŒu.R/ EŒu.Y / for all functions u./ from certain set of functions, called the generator of the order. The concept of stochastic dominance is very popular and widely used in economics and finance because of its relation to models of risk-averse preferences (Fishburn 1964). It originated from the theory of majorization (Hardy et al. 1934) for the discrete case, and was later extended to general distributions (Quirk and Saposnik 1962; Hadar and Russell 1969; Rotschild and Stiglitz 1969). Stochastic dominance of second order is defined by the set of nondecreasing concave functions: a random variable R dominates another random variable Y in the second order if EŒu.R/ EŒu.Y / for all nondecreasing concave functions u./ for which these expected values are finite. Thus, no risk-averse decision maker will prefer a portfolio with return rate Y over a portfolio with return rate R. A popular approach is the utility optimization approach. Von Neumann and Morgenstern (1944) developed the expected utility theory: for every rational decision maker there exists a utility function u./ such that the decision maker prefers outcome R over outcome Y if and only if EŒu.R/ > EŒu.Y / . This approach can be implemented also very efficiently; however, it is almost impossible to elicit the utility function of a decision maker explicitly. More difficulties arise when a group of decision makers with different utility functions have to reach a consensus. Recently, the dual utility theory (or rank dependent expected utility theory) has attracted much attention in economics. This approach was first presented by Quiggin (1982) and later rediscovered in a special case by Yaari (1987). From a different system of axioms than those of von Neumann and Morgenstern, one derives that every decision maker has a certain rank dependent utility function w W Œ0; 1 ! R. Then a nonnegative outcome R is preferred over a nonnegative outcome Y , if and only if

D. Dentcheva Stevens Institute of Technology, Hoboken, NJ, USA A. Ruszczy´nski () Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

Z1

Z1 w.p/dF .1/ .RI p/ 

 0

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_15, 

w.p/dF.1/ .Y I p/; 0

(15.1) 247

´ D. Dentcheva and A. Ruszczynski

248

where, F.1/ .RI / is the inverse distribution function of R. For a comprehensive treatment of the rank dependent utility theory, we refer to Quiggin (1993), and for its application in actuarial mathematics, see Wang et al. 1997; Wang and Yong 1998. Another classical approach, pioneered by Markowitz (1952, 1959, 1987), is the mean-risk approach, which compares the portfolios with respect to two characteristics. One is the expected return rate (the mean) and another one is the risk, which is given by some scalar measure of the uncertainty of the portfolio return rate. The mean-risk approach recommends the selection of Pareto-efficient portfolios with respect to these two criteria. In a mean-risk portfolio model we combine these criteria by specifying some parameter as a tradeoff between them. As a parametric optimization problem the mean-risk model can be solved numerically very efficiently, which makes this approach very attractive (Konno and Yamazaki 1991; Ruszczy´nski and Vanderbei 2003). In this paper we formulate a model for risk-averse portfolio optimization and demonstrate its relation to the expected utility approach and to rank dependent utility approach. We optimize the portfolio performance under an additional constraint that the portfolio return rate stochastically dominates a benchmark return rate, for example, the return rate of an index. The model is based on the publications of Dentcheva and Ruszczy´nski (2003a, b, c; 2004a, b) where a new model of risk-averse optimization has been introduced. This approach has a fundamental advantage over mean-risk models and utility function models. All data for our model are readily available. In mean-risk models the choice of the risk measure has an arbitrary character, and it is difficult to argue for one measure against another. Similarly, optimization of expected utility requires the form of the utility function to be specified. Our analysis, departing from the benchmark outcome, generates implied utility function of the decision maker. It is implicitly defined by the benchmark used, and by the problem under consideration. We provide two problem formulations in which the stochastic dominance has a primal or inverse form: a Lorenz curve. The primal form has a dual problem in terms of expected utility functions, and the inverse form has a dual problem in terms of rank dependent utility functions. In this way our model provides a link between this two competing economic approaches. Duality relations with coherent measures of risk are explored in Dentcheva and Ruszczy´nski (2008).

15.2 The Portfolio Problem Let R1 ; R2 ; : : : ; Rn be random return rates of assets 1; 2; : : : ; n. We assume that EŒjRj j < 1 for all j D 1; : : : ; n.

Our aim is to invest our capital in these assets in order to obtain some desirable characteristics of the total return rate on the investment. Denoting by x1 ; x2 ; : : : ; xn the fractions of the initial capital invested in assets 1; 2; : : : ; n we can easily derive the formula for the total return rate: R.x/ D R1 x1 C R2 x2 C : : : C Rn xn :

(15.2)

Clearly, the set of possible asset allocations can be defined as follows: X D fx 2 Rn W x1 Cx2 C: : :Cxn D 1; xj 0; j D 1; 2; : : : ; ng: In some applications one may introduce the possibility of short positions (i.e., allow some xj ’s to become negative). Other restrictions may limit the exposure to particular assets or their groups, by imposing upper bounds on the xj ’s or on their partial sums. One can also limit the absolute differences between the xj ’s and some reference investments xN j , which may represent the existing portfolio, and so on. Our analysis does not depend on the detailed way this set is defined; we only use the fact that it is a convex polyhedron. All modifications discussed above define some convex polyhedral feasible sets and are, therefore, covered by our approach. The main difficulty in formulating a meaningful portfolio optimization problem is the definition of the preference structure among feasible portfolios. If we use only the mean return rate EŒR.x/ , then the resulting optimization problem has a trivial and meaningless solution: invest everything in assets that have the maximum expected return rate. For these reasons, the practice of portfolio optimization usually resorts to two approaches. In the first approach we associate with portfolio x some dispersion measure .R.x// representing the variability of the return rate R.x/. In the classical Markowitz model the function .R.x// is the variance of the return rate, .R.x// D VŒR.x/ ; but many other measures are possible here as well. The mean-risk portfolio optimization problem is formulated as follows: max EŒR.x/  .R.x//: x2X

(15.3)

Here,  is a nonnegative parameter representing our desirable exchange rate of mean for risk. If  D 0, the risk has no value and the problem reduces to the problem of maximizing the mean. If  > 0 we look for a compromise between the mean and the risk. Alternatively, one can minimize the risk function .x/, while fixing the expected return rate EŒR.x/ at some value m, and consider a family of problems

15 Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints

parametrized by m. The reader is referred to the book by Elton et al. (2006) for the modern perspective on mean-risk analysis in portfolio theory. The general question of constructing mean-risk models that are in harmony with the stochastic dominance relations has been the subject of the analysis of the recent papers by Ogryczak and Ruszczy´nski (1999; 2001; 2002). We have identified there several primal risk measures, most notably central semideviations, and dual risk measures, based on the Lorenz curve, which are consistent with the stochastic dominance relations. The second approach is to select a certain utility function u W R ! R and to formulate the following optimization problem (15.4) max EŒu.R.x// : x2X

It is usually required that the function u./ is concave and nondecreasing, thus representing preferences of a risk-averse decision maker (Fishburn 1964; 1970). Recently, a dual (rank dependent) utility model attracts much attention. It is based on distorting the cumulative probability distribution of the random variable R.x/ rather than applying a nonlinear function u./ to the realizations of R.x/. The corresponding problem has the following form Z1 F.1/ .R.x/; p/dw.p/:

max x2X

(15.5)

249

functions. For a real random variable V , its first performance function is defined as the right-continuous cumulative distribution function of V : F1 .V I / D PfV  g

for  2 R:

A random return V is said (Lehmann 1955; Quirk and Saposnik 1962) to stochastically dominate another random return S in the first order, denoted V .1/ S , if F1 .V I /  F1 .S I /

for all  2 R:

We can say that V is “stochastically larger” than S , because it takes values lower than  with smaller (or equal) probabilities than S , no matter what the target  is. The second performance function F2 is given by areas below the distribution function F , Z F2 .V I / D

F1 .V I / d 

for  2 R;

1

and defines the weak relation of the second order stochastic dominance (SSD). That is, random return V stochastically dominates S in the second order, denoted V .2/ S , if F2 .V I /  F2 .S I /

for all  2 R:

0

Here F.1/ .R.x/; p/ is the p-quantile of the random variable R.x/, and w./ is the rank dependent utility function, which distorts the probability distribution. We discuss this in Sect. 15.3.2. The challenge in both utility approaches is to select the appropriate utility function or rank dependent utility function that represent our preferences and whose application leads to nontrivial and meaningful solutions of Equation (15.4) or (15.5). In this paper we propose an alternative approach: introducing a comparison to a benchmark return rate into our optimization problem. The comparison is based on the stochastic dominance relation. More specifically, we consider only portfolios whose return rates stochastically dominates a certain benchmark return rate.

15.3 Stochastic Dominance 15.3.1 Direct Forms In the stochastic dominance approach, random return rates are compared by a point-wise comparison of some performance functions constructed from their distribution

(see Hadar and Russell 1969; Rotschild and Stiglitz 1969). We can express the function F2 .V I / as the expected shortfall (see, for example, Levy 2006; Ogryczak and Ruszczy´nski 1999): for each target value  we have F2 .V I / D EŒ.  V /C ;

(15.6)

where .  V /C D max.  V; 0/. The function F.2/ .V I / is continuous, convex, nonnegative and nondecreasing. It is well defined for all random variables V with finite expected value. Due to this representation, the second order stochastic dominance relation V .2/ S can be equivalently characterized by the system of inequalities on the expected shortfall below any target : EŒ.  V /C  EŒ.  S /C

for all  2 R:

(15.7)

Also, we obtain an equivalent characterization in terms of the expected utility theory of von Neumann and Morgenstern (see, for example, Hanoch and Levy 1969; Levy 2006; Müller and Stoyan 2002): For any two random variables V; S the relation V .1/ S holds true if and only if for all nondecreasing functions u./ defined on R we have EŒu.V / EŒu.S / :

(15.8)

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any two random variables V; S with finite expectations, the relation V .2/ S holds true if and only if Equation (15.8) is satisfied for all nondecreasing concave functions u./.

For

In the context of portfolio optimization, we consider stochastic dominance relations between random return rates defined by Equation (15.2). Thus, we say that portfolio x dominates portfolio y in the first order, if F1 .R.x/I /  F1 .R.y/I /

for all  2 R:

This is illustrated in Fig. 15.1. Similarly, we say that x dominates y in the second order .R.x/ .2/ R.y//, if F2 .R.x/I /  F2 .R.y/I /

for all  2 R:

The second order relation is illustrated in Fig. 15.2. Recall that the individual return rates Rj have finite expected values and thus the function F2 .R.x/I / is well defined.

Let us consider the inverse model of stochastic dominance, frequently referred to as Lorenz dominance. For a real random variable V (for example, a random return rate) we define the left-continuous inverse of the cumulative distribution function F1 .V I / as follows: F.1/ .V I p/ D inf f W F1 .V I / pg

for

0 < p < 1:

Given p 2 .0; 1/, the number q D q.V I p/ is called a p-quantile of the random variable V if PfV < qg  p  PfV  qg: For p 2 .0; 1/ the set of p-quantiles is a closed interval and F.1/ .V I p/ represents its left end. Directly from the definition of the first order dominance we see that V .1/ S , F.1/ .V I p/ F.1/ .S I p/ for all 0 < p < 1: (15.9) The first order dominance constraint can be interpreted as a continuum of probabilistic (chance) constraints, studied in stochastic optimization (see, Dentcheva 2005; Prékopa 2003). Our analysis uses the absolute Lorenz function F.2/ .V I / W Œ0; 1 ! R, introduced in (Lorenz 1905). It is defined as the cumulative quantile:

1

F1(R(x);h)

F1(R(y);h)

15.3.2 Inverse Forms

0.5

Zp F.2/ .V I p/ D

F.1/ .V I t/dt

for

0 < p  1;

0

(15.10) h

0

Fig. 15.1 First order stochastic dominance R.x/ .1/ R.y/

F.2/ .V I 0/ D 0: Similarly to F2 .V I /, the function F.2/ .V I / is well defined for any random variable V , which has a finite expected value. We notice that Z1 F.2/ .V I 1/ D

F2(R(y);h)

F.1/ .V I t/dt D EŒV : 0

F2(R(x);h)

h

Fig. 15.2 Second order dominance R.x/ .2/ R.y/

By construction, the Lorenz function is convex. Lorenz functions are commonly used for inequality ordering of positive random variables, relative to their (positive) expectations (see Gastwirth 1971; Muliere and Scarsini 1989). Such a Lorenz function, p 7! F.2/ .V I p/=EŒV , is convex and nondecreasing. The absolute Lorenz function, however, is not monotone, when negative outcomes occur.

15 Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints

251

It is well known (see, for example, Ogryczak and Ruszczy´nski 2002) that we may fully characterize the second order dominance relation by using the function F.2/ .V I /: V .2/ S , F.2/ .V I p/ F.2/ .S I p/ for all 0  p  1: (15.11)

F(–2)(R(x);p)

F(–2)(R(y);p) This characterization of stochastic dominance by Lorenz p functions is widely used in economics and statistics. 1 We now provide an equivalent characterization by rank 0 0.5 dependent utility functions. It is analogous to the characteriFig. 15.4 Second order dominance R.x/ .2/ R.y/ in the inverse form zation by expected utility functions. Dentcheva and Ruszczy´nski (2006b) provide the following characterization.

For any two random variables V; S the relation V .1/ S

holds true if and only if for all nondecreasing functions w./ defined on [0,1] we have Z1

Z1 F.1/ .V I p/dw.p/

0

F.1/ .S I p/dw.p/: 0

(15.12)

Similarly, we say that x dominates y in the second order .R.x/ .2/ R.y//, if F.2/ .R.x/I p/ F.2/ .R.y/I p/

f or al l p 2 Œ0; 1 : (15.13)

Recall that the individual return rates Rj have finite expected values and thus the function F.2/ .R.x/I / is well defined. The second order relation is illustrated in Fig. 15.4.

For any two random variables V; S with finite expec-

tations, the relation V .2/ S holds true if and only if Equation (15.12) is satisfied for all nondecreasing concave functions w./. The functions w./ appearing in this characterization are rank dependent (dual) utility functions. In the context of portfolio optimization, we consider stochastic dominance relations between random return rates defined by Equation (15.2). Thus, we say that portfolio x dominates portfolio y in the first order, if F.1/ .R.x/I p/ F.1/ .R.y/I p/

for all p 2 .0; 1/:

15.3.3 Relations to Value at Risk and Conditional Value at Risk There are fundamental relations between the concepts of Value at Risk .VaR/ and Conditional Value at Risk .CVaR/ and the stochastic dominance constraints. The VaR constraint in the portfolio context is formulated as follows. We define the loss rate L.x/ D R.x/. We specify the maximum fraction !p of the initial capital allowed for risk exposure at risk level p 2 .0; 1/, and we require that PŒL.x/  !p 1  p:

This is illustrated in Fig. 15.3.

Denoting by VaRp .L.x// the left .1  p/-quantile of the random variable L.x/, we can equivalently formulate the VaR constraint as VaRp .L.x//  !p : The first order stochastic dominance relation between two portfolios is equivalent to the continuum of VaR constraints. Portfolio x dominates portfolio y in the first order, if

F(–1)(R(x);p) F(–1)(R(y);p)

VaRp .L.x//  VaRp .L.y// for all p 2 .0; 1/: p

0

0.5

1

Fig. 15.3 First order stochastic dominance R.x/ .1/ R.y/ in the inverse form

The CVaR at level p, roughly speaking, has the following form CVaRp .L.x// D EŒL.x/jL.x/ VaRp .L.x// :

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This formula is precise if VaRp .L.x// is not an atom of the distribution of L.x/. More precisely we express it as follows: 1 CVaRp .L.x// D p

Zp

15.4.1 Direct Formulation

VaRt .L.x//dt: 0

We note that 1 CVaRp .L.x// D  F.2/ .R.x/; p/: p

(15.14)

Another description uses extremal properties of quantiles and equivalently represents CVaR as follows (Rockafellar and Uryasev 2000): (15.15)

A CVaR constraint on the portfolio x can be formulated as follows: (15.16) CVaRp .L.x//  !p : Using Equations (15.14) and (15.13) we conclude that the second order stochastic dominance relation for two portfolios x and y is equivalent to the continuum of CVaR constraints: R.x/ .2/ R.y/ , CVaRp .L.x//  CVaRp .L.y// for all p 2 .0; 1 : (15.17) Assume that we compare the performance of a portfolio x with a random benchmark Y (for example, an index return rate or another portfolio return rate) requiring R.x/ .2/ Y . Then the fraction !p of the initial capital allowed for risk exposure at level p is given by the benchmark Y : !p D CVaRp .Y /;

p 2 .0; 1 :

Assume that Y has a discrete distribution with realizations yi ; i D 1; : : : ; m. Then relation Equation (15.7) is equivalent to EŒ.yi  R.x//C  EŒ.yi  Y /C ;

i D 1; : : : ; m: (15.18)

This result does not imply that the continuum of CVaR constraints Equation (15.17) can be replaced by finitely many constraints of form CVaRpi .R.x// CVaRpi .Y /;

The starting point for our model is the assumption that a benchmark random return rate Y having a finite expected value is available. It may have the form of Y D R.z/, for some benchmark portfolio z. It may be an index or our current portfolio. Our intention is to have the return rate of the new portfolio, R.x/, preferable over Y . Therefore, we introduce the following extension of the optimization problem Equation (15.3): max EŒR.x/  .R.x//



 1 CVaRp .L.x// D inf EŒ.  R.x//C   :  p

15.4 The Dominance-Constrained Portfolio Problem

i D 1; : : : ; m;

with some fixed probabilities pi ; i D 1; : : : ; m. The reason is that we do not know at which probability levels the CVaR constraints have to be imposed.

(15.19)

subject to R.x/ .2/ Y;

(15.20)

x 2 X:

(15.21)

Similarly to Equation (15.3), we optimize a mean-risk objective function, but we introduce a constraint that the portfolio return dominates a benchmark. Even when  D 0 and we maximize just the expected value of the return rate, our model will still lead to nontrivial solutions, due to the presence of the dominance constraint Equation (15.20). To increase flexibility of model Equations (15.19)– (15.21), we may also allow a uniform shift of R.x/ by a constant c, as in the following model: max EŒR.x/  .R.x//  ıc subject to R.x/ C c .2/ Y; x 2 X: Here ı > 0 can be interpreted a cost of the shift c. Observe that the shift c may also become negative, in which case we are rewarded for uniformity of dominating Y . The shift c may be interpreted as an additional cash added to the return, and ı is the interest to be paid when the loan is paid back. To simplify the derivations, from now on we focus on the simplest formulation of the dominance-constrained problem: max EŒR.x/

(15.22)

subject to R.x/ .2/ Y;

(15.23)

x 2 X:

(15.24)

15 Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints

We can observe the first advantage of our problem formulation: all data in it are readily available. Moreover, the set defined by Equation (15.23) is convex (Dentcheva and Ruszczy´nski 2003c; 2004a, c). Let us assume now that Y has a discrete distribution with realizations yi attained with probabilities i ; i D 1; : : : ; m. We also assume that the return rates have a discrete joint distribution with realizations rjt ; t D 1; : : : ; T; j D 1; : : : ; n, attained with probabilities pt ; t D 1; 2; : : : ; T . Then the formulation of the stochastic dominance relation Equation (15.23) resp. Equation (15.18) simplifies even further. Introducing variables sit representing the shortfall of R.x/ below yi in realization t; i D 1; : : : ; m; t D 1; : : : ; T , we can formulate problem Equations (15.22)–(15.24) as follows: max

T X

pt

i D1

n X

xj rjt

(15.25)

j D1

subject to n X

xj rjt C sit yi ;

i D 1; : : : ; m;

t D 1; : : : ; T;

j D1

(15.26) T X

pt Sit 

t D1

m X

k .yi  yk /C ;

253

If the set X is a convex polyhedron, problem Equations (15.25)–(15.29) becomes a large scale linear programming problem. It may be solved by general-purpose linear programming solvers. However, the size of the problem increases dramatically with the number of assets n, their return realizations T , and benchmark realizations m, which makes it impractical for even moderate dimensions (in thousands). For the purpose of solving these problems, we developed a specialized decomposition method presented in Dentcheva and Ruszczy´nski (2006a).

15.4.2 Inverse Formulation Assume that the return rates have a joint discrete distribution realizations rjt ; t D 1; : : : ; T and j D 1; : : : ; n, attained with probabilities pt ; t D 1; 2; : : : ; T . Moreover, we assume that all probabilities pt are equal, that is, pt D 1=T; t D 1; : : : ; T . This is the case of empirical distributions. Correspondingly, we assume that Y has a discrete distribution with m D T equally probable realizations yt ; t D 1; : : : ; T . We use the symbol RŒt .x/ to denote the ordered realizations of R.x/; that is, RŒ1 .x/  RŒ2 .x/  : : :  RŒT .x/:

i D 1; : : : ; m; (15.27)

lD1

sit 0;

i D 1; : : : ; m;

t D 1; : : : ; T:

(15.28)

x 2 X:

(15.29)

Indeed, or every feasible point x of (15.22)–(15.24), setting 0 sit D max@0; yi 

n X

1 xj rjt A ;

i D 1; : : : ; m;

j D1

t D 1; : : : ; T; we obtain a feasible pair .x; s/ for Equations (15.26)– (15.29). Conversely, for any feasible pair .x; s/ for Equations (15.26)–(15.29), inequalities Equations (15.26) and (15.28) imply that n X

sit max.0; yi 

xj rjt /;

i D 1; : : : ; m;

t D 1; : : : ; T:

j D1

Taking the expected value of both sides and using Equation (15.27) we obtain F2 .R.x/I yi /  F2 .Y I yi /;

Since R.x/ has a discrete distribution, the functions F2 .R.x/I / and F.2/ .R.x/I / are piecewise linear. Owing to the fact that all probabilities pt are equal, the break points of F.2/ .R.x/I / occur at t=T , for t D 0; 1; : : : ; m. The same applies to F.2/ .Y I /. It follows from Equation (15.13) that the stochastic dominance constraint Equation (15.23) can be equivalently expressed as



t t F.2/ Y I ; F.2/ R.x/I T T

Note that F.2/ .R.x/I 0/ D F.2/ .Y I 0/ D 0. We have

t 1 X t D RŒk .x/; F.2/ R.x/I T T

t D 1; : : : ; T:

kD1

Therefore problem Equations (15.22)–(15.24) can be written with an equivalent inverse form of the dominance constraint: max EŒR.x/ subject to t X

i D 1; : : : ; m:

Therefore, problem Equations (15.22)–(15.24) is equivalent to problem Equations (15.25)–(15.29).

t D 1; : : : ; T:

RŒk .x/

kD1

x 2 X:

t X

yŒk ;

(15.30) t D 1; : : : ; T;

(15.31)

kD1

(15.32)

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It was shown in (Ogryczak and Ruszczy´nski 2002) that the P function x 7! tkD1 RŒk .x/ is concave and positively homogeneous. It is also polyhedral. Therefore, Equation (15.31) are convex polyhedral constraints. If the set X is a convex polyhedron, problem Equations (15.30)–(15.32) has an equivalent linear programming formulation. All these transformations are possible due to the crucial assumption that the probabilities of all elementary events are equal. If they are not equal, the break points of the function F.2/ .R.x/I / depend on x, and therefore inequality Equation (15.13) cannot be reduced to finitely many convex inequalities. This is in contrast to the primal formulation, where the discreteness of Y alone was sufficient to reduce the stochastic dominance constraint to finitely many convex inequalities. We have to observe that the quantile formulation Equation (15.31) of stochastic dominance constraints is more involved than the primal formulation, and requires more sophisticated computational methods. Using Equation (15.31) directly would require employing nonsmooth optimization methods to solve problem Equations (15.30)–(15.32). Equivalent formulation with linear constraints has very many constraints, because of the large number of pieces of the function x 7! Pt nski (2010) dekD1 RŒk .x/. Still, Dentcheva and Ruszczy´ veloped a highly efficient cutting plane method, which significantly outperforms direct approaches.

It will play for problem Equations (15.22)–(15.24) a similar role to that of a Lagrangian. It is well defined, because for every u 2 U and every x 2 Rn the expected value EŒu.R.x// exists and is finite. The following theorem has been proved in a more general version in (Dentcheva and Ruszczy´nski 2003c). Theorem 15.1. If xO is an optimal solution of Equations (15.22)–(15.24) then there exists a function uO 2 U such that L.x; O uO / D max L.x; uO /

(15.34)

EŒOu.R.x// O D EŒOu.Y / :

(15.35)

x2X

Conversely, if for some function uO 2 U an optimal solution xO of Equation (15.34) satisfies Equations (15.23) and (15.35), then xO is an optimal solution of Equations (15.22)–(15.24). We can also develop duality relations for our problem. With the function Equation (15.33) we can associate the dual function D.u/ D max L.x; u/: x2X

We are allowed to write the maximization operation here, because the set X is compact and L.; u/ is continuous. The dual problem has the following form min D.u/: u2E

The set U is a closed convex cone and D./ is a convex function, so Equation (15.36) is a convex optimization problem.

15.5 Optimality and Duality 15.5.1 Primal Form From now on we assume that the probability distributions of the return rates are discrete with finitely many realizations realizations rjt ; t D 1; : : : ; T; j D 1; : : : ; n, attained with probabilities pt ; t D 1; 2; : : : ; T . We also assume that there are finitely many ordered realizations of the benchmark outcome Y W y1 < y2 <    < ym . The probabilities of these realizations are denoted by i ; i D 1; : : : ; m. We also assume that the set X is compact. We define the set U of functions u W R ! R satisfying the following conditions: u./ is concave and nondecreasing u./ is piecewise linear with break points yi ; i D 1; : : : ; m u.t/ D 0 for all t ym

It is evident that U is a convex cone. Let us define the function L W Rn U ! R as follows L.x; u/ D EŒR.x/ C u.R.x//  u.Y / :

(15.36)

(15.33)

Theorem 15.2. Assume that Equations (15.22)–(15.24) has an optimal solution. Then problem Equation (15.36) has an optimal solution and the optimal values of both problems coincide. Furthermore, the set of optimal solutions of Equation (15.36) is the set of functions uO 2 U satisfying Equations (15.34)–(15.35) for an optimal solution xO of Equations (15.22)–(15.24). Note that all constraints of our problem are linear or convex polyhedral, and therefore we do not need any constraint qualification conditions here. The “Lagrange multiplier” u is directly related to the expected utility theory of von Neumann and Morgenstern. We have established earlier that the second order stochastic dominance relation is equivalent to Equation (15.8) for all utility functions in U. Our result shows that one of them, uO ./, assumes the role of a Lagrange multiplier associated with Equation (15.23). A point xO is a solution to Equations (15.22)–(15.24) if there exists a utility function

15 Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints

uO ./ such that xO maximizes over X the objective function EŒR.x/ augmented with this dual utility. We see that the optimization problem in Equation (15.34) is equivalent to max EŒv.R.x// ; x2X

255

Conversely, if for some function wO 2 W we find an optimal solution xO of Equation (15.39) that satisfies Equations (15.31) and (15.40), then xO is an optimal solution of Equations (15.30)–(15.32).

(15.37)

where v./ D  C u./. At the optimal solution the function vO ./ D  C uO ./ is the implied utility function. It attaches higher penalty to smaller realizations of R.x/ (bigger realizations of L.x/). By maximizing L.R.x/; u/ we look for x such that the left tail of the distribution of R.x/ is thin. It is important to stress that the optimal function uO ./ is piecewise linear, with break points at the realizations y1 ; : : : ; ym of the benchmark Y . Therefore, the dual problem has also an equivalent linear programming formulation.

We can also develop a duality theory based on Lagrangian Equation (15.38). For every function w 2 W the problem max ˆ.x; w/ x2X

is a Lagrangian relaxation of problem Equations (15.30)– (15.32). Its optimal value, ‰.w/, is always greater than or equal to the optimal value of Equations (15.30)–(15.32). We define the dual problem as min ‰.w/:

(15.42)

w2W

15.5.2 Inverse Form In addition to the assumption that all involved distributions are discrete, we also assume that all probabilities pt are equal, and that m D T . We introduce the set W of concave nondecreasing functions w W Œ0; 1 ! R. It is evident that W is a convex cone. Recall the identity Z1 EŒR.x/ D

F.1/ .R.x/I p/dp: 0

Let us define the function ˆ W X W ! R, as follows Z1

Z1 F.1/ .R.x/I p/dp C

ˆ.x; w/ D 0

F.1/ .R.x/I p/d w.p/ 0

Z1 

F.1/ .Y I p/d w.p/:

(15.38)

0

It plays a role similar to that of a Lagrangian of Equations (15.30)–(15.32). Theorem 15.3. If xO is an optimal solution of Equations (15.30)–(15.32) then there exists a function w O 2W such that ˆ.x; O w/ O D max ˆ.x; w/ O x2X

Z1

The set W is a closed convex cone and ‰./ is a convex function, so problem Equation (15.42) is a convex optimization problem. Duality relations in convex programming yield the following result. Theorem 15.4. Assume that problem Equations (15.30)– (15.32) has an optimal solution. Then problem Equation (15.42) has an optimal solution and the optimal values of both problems coincide. Furthermore, the set of optimal solutions of Equation (15.42) is the set of functions wO 2 W satisfying Equations (15.39)–(15.40) for an optimal solution xO of Equations (15.30)–(15.32). The “Lagrange multiplier” w in this case is related to rank dependent expected utility theory. We have established earlier that the second order stochastic dominance relation is equivalent to Equation (15.12) for all dual utility functions in W. Our result shows that one of them, w./, O assumes the role of a Lagrange multiplier associated with Equation (15.31). A point xO is a solution to Equations (15.30)–(15.32) if there exists a dual utility function w./ O such that xO maximizes over X the objective function EŒR.x/ augmented with this dual utility. We can transform the Lagrangian Equation (15.38) in the following way: Z1

Z1 F.1/ .R.x/I p/dp C

ˆ.X; w/ D 0

F.1/ .R.x/I p/d w.p/ 0

Z1 

Z1 F.1/ .R.x/I O p/d w.p/ O D

0

(15.39)

Z1 F.1/.YI p/dw.p/ D F.1/ .R.x/I p/d v.p/

0

 (15.40)

0

Z1

F.1/ .Y I p/d w.p/: O 0

(15.41)

F.1/ .Y I p/d w.p/; 0

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where v.p/ D p Cw.p/. At the optimal solution the function vO .p/ D p C w.p/ O is the quantile utility function implied R1 by the benchmark Y . Since F.1/ .Y I p/dw.p/ is fixed, 0

the problem at the right hand side of Equation (15.39) becomes a problem of maximizing the implied rank dependent expected utility in X . It attaches higher weights to quantiles corresponding to smaller probabilities p. By maximizing ˆ.R.x/; w/ we look for x such that the left tail of the distribution of R.x/ is thin. Similarly to von Neumann–Morgenstern utility function, it is very difficult to elicit the dual utility function in advance. Our model derives it from a random benchmark. The optimal function w./ O is piecewise linear, with break points at Tt ; t D 1; : : : ; T . Therefore, the dual problem has also an equivalent linear programming formulation. This property, however, is conditioned on the assumption of equal probabilities.

15.6 Numerical Illustration We have tested our approach on a basket of 719 real-world assets, using 616 possible realizations of their joint return rates (Ruszczy´nski and Vanderbei 2003). Historical data on weekly return rates in the 12 years from spring 1990 to spring 2002 were used as equally likely realizations. Implied utility functions corresponding to dominance constraints for four benchmark portfolios.

We have used four benchmark return rates Y . Each of them was constructed as a return rate of a certain index composed of our assets. As we actually know the past return rates, for the purpose of comparison we have selected equally weighted indexes composed of the N assets having the highest average return rates in this period. Benchmark 1 corresponds to N D 26, Benchmark 2 corresponds to N D 54, Benchmark 3 corresponds to N D 82, and Benchmark 4 corresponds to N D 200. Our problem was to maximize the expected return rate, under the condition that the return rate of the benchmark portfolio is dominated. Since the benchmark point was a return rate of a portfolio composed from the same basket, we have m D T D 616 in this case. We have solved the problem by our method of minimizing the dual problem that was presented in Dentcheva and Ruszczy´nski (2006a). The implied utility functions from Equation (15.37) obtained by solving the optimization problem Equation (15.34) in the optimality conditions are illustrated in Fig. 15.5. We see that for Benchmark Portfolio 1, which contains only a small number of fast-growing assets, the utility function is linear on almost the entire range of return rates. Only very negative return rates are penalized. A different situation occurs when the benchmark portfolio contains more assets and is therefore more diversified and less risky. In order to dominate such a benchmark, we have to use a utility function which introduces penalty for a broader range of return rates and is steeper. For the broadly based index in Benchmark Portfolio 4, the optimal utility function is smoother and is nonlinear even for positive return rates. It

Return Rate −0.03

−0.03

−0.02

−0.02

−0.01

0.00

0.00

0.01

0.01 0.000

−0.005

Fig. 15.5 Implied utility functions

Utility

Utility Function for Benchmark 1 Utility Function for Benchmark 2 Utility Function for Benchmark 3 Utility Function for Benchmark 4

−0.010

15 Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints

is worth mentioning that all these utility functions, although nondecreasing and concave, have rather complicated shapes. It would be a very hard task to determine in advance the utility function that should be used to obtain a solution dominating our benchmark portfolio. Obviously, the shape of the utility function is determined by the benchmark within the context of the optimization problem considered. If we change the optimization problem, the utility function will change. Finally, we may remark that our model Equations (15.22)–(15.24) can be used for testing the statistical hypothesis that the return rate Y of the benchmark portfolio is nondominated.

15.7 Conclusions We presented a new approach to portfolio selection based on stochastic dominance. The portfolio return rate in the new model is required to stochastically dominate a random benchmark, for example, the return rate of an index. We formulated optimality conditions and duality relations for these models and constructed equivalent optimization models with utility functions. Two different formulations of the stochastic dominance constraint: primal and inverse, lead to two dual problems that involve von Neuman–Morgenstern utility functions for the primal formulation and rank dependent (or dual) utility functions for the inverse formulation. The utility functions play the roles of Lagrange multipliers associated with the dominance constraints. In this way our model provides a link between the expected utility theory and the rank dependent utility theory. A numerical example illustrates the new approach and demonstrates the efficacy of the method. Future challenges are extensions of the approach to multivariate and multistage outcomes and benchmarks.

References Dentcheva, D. 2005. Optimization models with probabilistic constraints. in Probabilistic and randomized methods for design under uncertainty, G. Calafiore and F. Dabbene (Eds.). Springer, London, pp. 49–97. Dentcheva, D. and A. Ruszczy´nski. 2003a. “Optimization under linear stochastic dominance.” Comptes Rendus de l’Academie Bulgare des Sciences 56(6), 6–11. Dentcheva, D. and A. Ruszczy´nski. 2003b. “Optimization under nonlinear stochastic dominance.” Comptes Rendus de l’Academie Bulgare des Sciences 56(7), 19–25. Dentcheva, D. and A. Ruszczy´nski. 2003c. “Optimization with stochastic dominance constraints.” SIAM Journal on Optimization 14, 548–566. Dentcheva, D. and A. Ruszczy´nski. 2004a. “Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints.” Mathematical Programming 99, 329–350.

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Dentcheva, D. and A. Ruszczy´nski. 2004b. “Convexification of stochastic ordering constraints.” Comptes Rendus de l’Academie Bulgare des Sciences 57(3), 5–10. Dentcheva, D. and A. Ruszczy´nski. 2004c. “Semi-infinite probabilistic optimization: first order stochastic dominance constraints.” Optimization 53, 583–601. Dentcheva, D. and A. Ruszczy´nski. 2006a. “Portfolio optimization with stochastic dominance constraints.” Journal of Banking and Finance 30/2, 433–451. Dentcheva, D. and A. Ruszczy´nski. 2006b. “Inverse stochastic dominance constraints and rank dependent expected utility theory.” Mathematical Programming 108, 297–311. Dentcheva, D. and A. Ruszczy´nski. 2008. “Duality between coherent risk measures and stochastic dominance constraints in risk-averse optimization.” Pacific Journal of Optimization 4, 433–446. Dentcheva, D. and A. Ruszczy´nski. 2010. Inverse cutting plane methods for optimization problems with second order stochastic dominance constraints, Optimization in press. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, Wiley, New York. Fishburn, P. C.1964. Decision and value theory, Wiley, New York. Fishburn, P.C. 1970. Utility theory for decision making, Wiley, New York. Gastwirth, J. L. 1971. “A general definition of the Lorenz curve.” Econometrica 39, 1037–1039. Hadar, J. and W. Russell. 1969. “Rules for ordering uncertain prospects.” The American Economic Review 59, 25–34. Hanoch, G. and H. Levy, 1969. “The efficiency analysis of choices involving risk.” Review of Economic Studies 36, 335–346. Hardy, G. H., J. E. Littlewood and G. Polya. 1934. Inequalities, Cambridge University Press, Cambridge, MA. Konno, H. and H. Yamazaki. 1991. “Mean-absolute deviation portfolio optimization model and its application to Tokyo stock market.” Management Science 37, 519–531. Lehmann, E. 1955. “Ordered families of distributions.” Annals of Mathematical Statistics 26, 399–419. Levy, H. 2006. Stochastic dominance: investment decision making under uncertainty, 2nd ed., Springer, New York. Lorenz, M. O. 1905. “Methods of measuring concentration of wealth.” Journal of the American Statistical Association 9, 209–219. Markowitz, H. M. 1952. “Portfolio selection.” Journal of Finance 7, 77–91. Markowitz, H. M. 1959. Portfolio selection, Wiley, New York. Markowitz, H. M. 1987. Mean-variance analysis in portfolio choice and capital markets, Blackwell, Oxford. Mosler, K. and M. Scarsini (Eds.). 1991. Stochastic orders and decision under risk, Institute of Mathematical Statistics, Hayward, California. Muliere, P. and M. Scarsini, 1989. “A note on stochastic dominance and inequality measures.” Journal of Economic Theory 49, 314–323. Müller, A. and D. Stoyan. 2002. Comparison methods for stochastic models and risks, Wiley, Chichester. Ogryczak, W. and A. Ruszczy´nski. 1999. “From stochastic dominance to mean-risk models: semideviations as risk measures.” European Journal of Operational Research 116, 33–50. Ogryczak, W. and A. Ruszczy´nski. 2001. “On consistency of stochastic dominance and mean-semideviation models.” Mathematical Programming 89, 217–232. Ogryczak, W. and A. Ruszczy´nski. 2002. “Dual stochastic dominance and related mean-risk models.” SIAM Journal on Optimization 13, 60–78. Prékopa, A. 2003. Probabilistic programming. in Stochastic Programming, Ruszczy´nski, A. and A. Shapiro. (Eds.). Elsevier, Amsterdam, pp. 257–352. Quiggin, J. 1982. “A theory of anticipated utility.” Journal of Economic Behavior and Organization 3, 225–243.

258 Quiggin, J. 1993. Generalized expected utility theory – the rankdependent expected utility model, Kluwer, Dordrecht. Quirk, J.P and R. Saposnik. 1962. “Admissibility and measurable utility functions.” Review of Economic Studies 29, 140–146. Rockafellar, R. T. and S. Uryasev. 2000. “Optimization of conditional value-at-risk.” Journal of Risk 2, 21–41. Rothschild, M. and J. E. Stiglitz. 1969. “Increasing risk: I. A definition.” Journal of Economic Theory 2, 225–243. Ruszczy´nski, A. and R.J. Vanderbei. 2003. “Frontiers of stochastically nondominated portfolios.” Econometrica 71, 1287–1297. von Neumann, J. and O. Morgenstern. 1944. Theory of games and economic behavior, Princeton University Press, Princeton.

´ D. Dentcheva and A. Ruszczynski Wang, S. S., V. R. Yong, and H.H. Panjer. 1997. “Axiomatic characterization of insurance prices.” Insurance Mathematics and Economics 21, 173–183. Wang, S. S. and V. R. Yong. 1998. “Ordering risks: expected utility versus Yaari’s dual theory of risk.” Insurance Mathematics and Economics 22, 145–161. Whitmore, G. A. and M. C. Findlay. (Eds.). 1978. Stochastic dominance: an approach to decision-making under risk, D.C.Heath, Lexington, MA. Yaari, M. E. 1987. “The dual theory of choice under risk.” Econometrica 55, 95–115.

Chapter 16

Portfolio Analysis Jack Clark Francis

Abstract In 1952, Harry M. Markowitz published a seminal paper about analyzing portfolios. In 1990, he was awarded the Nobel Prize for his portfolio theory. Markowitz portfolio analysis delineates a set of highly desirable investment portfolios. These optimal portfolios have the maximum return at each plausible level of risk, computed iteratively over a range of different risk levels. Conversely, Markowitz portfolio analysis can find the same set of optimal investments by delineating portfolios that have the minimum risk over a range of different rates of return. The set of all Markowitz optimal portfolios is called the efficient frontier. Portfolio analysis analyzes rate of return statistics, risk statistics (standard deviations), and correlations from a list of candidate investments (stocks, bonds, and so on) to determine which investments, and in what proportions (weights), enter into every efficient portfolio. Further analysis of Markowitz’s portfolio theory reveals interesting asset pricing implications. Keywords Portfolio r Rate of return r Portfolio analysis r Standard deviation rCorrelation rWeight rVariance-covariance matrix r Risk minimization r Return maximization r Efficient portfolios r Efficient frontier

The objective of portfolio analysis is to determine a set of “efficient portfolios.” Figure 16.1 is a two-dimensional graph in risk-return space that depicts the set of efficient portfolios as a curve between points E to F. This curve is called the efficient frontier; it is a set of optimum investment portfolios generated by analyzing an underlying group of individual assets. The assets that are being analyzed can be thought of as being individual issues of stocks and bonds. These individual assets are represented by the dots lying below the efficient frontier in Fig. 16.1.

16.2 Inputs for Portfolio Analysis Portfolio analysis requires certain data as inputs. If n individual assets (stocks and bonds) are the potential investments, the inputs to portfolio analysis are as follows: 1. n expected returns. 2. n variances of returns. 3. .n2  n/=2 covariances, or correlations. For instance, to generate a three-security portfolio, the analysis requires the statistics listed in Table 16.1.

16.1 Introduction 16.3 The Security Analyst’s Job Portfolio analysis is a mathematical procedure that determines a selection of optimum portfolios for investors to consider. The procedure was first made public in 1952 by Harry Markowitz.1 The theory caused a scientific revolution in finance. Before Markowitz’s scientific procedure investment counselors passed off “common sense guidelines” to their clients as valuable expert advice.

The security analyst needs to analyze “one period” rates of return defined in Equation (16.1). For example, consider James’s purchase of a stock for $54 per share. He sold that stock 1 year later for $64 to realize a capital gain of $10 and a cash dividend of 80 cents per share. Jim’s total income from the stock is $10.80 and his rate of return is 20%.

J.C. Francis () Baruch College, New York, NY, USA e-mail: [email protected] 1

The analysis was originally presented in an article: Markowitz (1952). Markowitz expanded his presentation in a book, Markowitz (1959).

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_16, 

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16.4 Four Assumptions Underlying Portfolio Analysis Portfolio analysis is all based on four behavioral assumptions.

Fig. 16.1 Investment opportunities have an efficient frontier that is convex

Table 16.1 Input statistics needed for a 3-asset portfolio Expected Variances of Asset returns returns Covariances E.r1 / E.r2 / E.r3 /

1 2 3

12 D Var.r1 / 22 D Var.r2 / 32 D Var.r3 /

¢1;2 D Cov.r1 ; r2 / ¢2;3 D Cov.r2 ; r3 / ¢1;3 D Cov.r1 ; r3 /

0

1 One-year @ rate of return; A D $10:80 Income $54 Invested r D 20%   .Pr ice change; $10/ C .Cash dividend; $0:80/ D .Purchase price; $54/ (16.1) The length of the one-time period can vary within wide limits. It definitely may not be a short-run period or a very long-run period. But between, say, one month and 10 years, the portfolio manager can select any planning horizon that fits the investor’s needs. The security analyst should formulate estimates of the probability distribution of returns for each security that is an investment candidate. These probability distributions may be estimated using historical data. If so, the historical distributions may need to be adjusted to reflect anticipated factors that were not present historically. Figure 16.2 and Table 16.2 provide two examples of probability distributions of rates of return for the same share of stock. The security analyst should also estimate covariances (or correlation coefficients) between all securities under consideration. Given these pieces of information, the input data, portfolio analysis can determine the optimal portfolio for any investor described by the assumptions listed in the next section.

1. All investors endeavor to maximize a one-period expected utility function that exhibits diminishing marginal utility of returns (or wealth). This is equivalent to saying investors have positive but diminishing marginal utility of wealth (or returns). It means the investors prefer more wealth over less wealth, and they are risk-averse. 2. Investors’ risk estimates are proportional to the variability of the one-period rates of returns, as measured by the variance or standard deviation.2 3. Investors are willing to base their decisions solely in terms of expected return and risk. That is, their utility (U) is a function of variability of return .¢/ and expected return [E(r)]. Symbolically, U D fŒ¢; E.r/ .3 4. For any given level of risk, investors prefer higher returns to lower returns. Symbolically, •U=•E.r/ > 0. Conversely, for any given level of rate of return, investors prefer less risk over more risk. Symbolically, •U=•¢ < 0. These assumptions will be maintained throughout this discussion of portfolio analysis. Essentially, portfolio analysis is based on the premise that the most desirable assets are those that have either of the following two characteristics. 1. The investment has the minimum expected risk at any given expected rate of return. Or, conversely, 2. The investment has the maximum expected rate of return at any given level of expected risk. Assets described by either of the two preceding sentences are usually portfolios rather than individual assets. These assets are called efficient portfolios whether they contain one or many assets. Efficient portfolios can be derived by a process we will call “Markowitz diversification.”

16.5 Different Approaches to Diversification Many people define diversification as “not putting all your eggs in one basket” or “spreading your risks.” These popular definitions of diversification seem naïve once you consider the covariances (or correlations) between securities. “Markowitz diversification” is the particular form of diversification activity implied by modern portfolio analysis; it is 2

Problems can arise from using the variance to measure risk. See Szego (2002). 3 The mean and variance may not always provide the optimal selection. See Malevergne and Sornette (2005).

16 Portfolio Analysis

261

Fig. 16.2 Finite probability distribution of returns for a stock with three possible outcomes

Table 16.2 Probability distribution of returns for a $100 stock State of nature Event’s Stock’s Cash probability ending price dividend Column A B C Booming economy Slow growth Recession

0:3 D 30% 0:4 D 40% 0:3 D 30% 1:0 D 100%

$130.00 $110.00 $90.00

superior to naive diversification. Markowitz diversification involves combining investments with less-than-perfect positive correlation in order to reduce risk in the portfolio without unnecessarily sacrificing any of the portfolio’s potential return. In general, the lower the correlation of the assets in a portfolio, the less risky the portfolio will be. This is true regardless of how risky the assets in the portfolio are when analyzed in isolation. Markowitz explained that “in trying to make variance [of returns] small it is not enough to invest in many securities. It is necessary to avoid investing in securities with high covariances among themselves.”4

16.6 A Portfolio’s Expected Return Formula Consider the two individual securities in Table 16.3. Suppose that, say, 71.1% of the portfolio’s funds are invested in asset A and the other 28.9% in asset B. Symbolically, xA D 71:1% and xB D 28:9%, where the weight assigned to each asset is denoted xi for asset i. Throughout portfolio analysis the weights of the candidate assets must sum to one (or 100%), as shown in Equation (16.2). xA C xB D 1:0 D 100%

4

Markowitz (1952).

(16.2)

0 0 0

Single-period rate of return D

Products of A and D sum to E(r) E

0:3 D 30% 0:1 D 10% 0:1 D 10%

0:3  0:3 D 0:09 0:4  0:1 D 0:04 0:3  0:1 D 0:03 E.r/ D 0:16 D 16%

Table 16.3 Statistics describing two assets Investment Expected return (%) Standard deviation (%) A B

7 20

14.86 36.6

The formula for the weighted average return from this portfolio is below. E.rp / D xA E.rA / C xB E.rB /

(16.3)

Equation (16.3a) and (16.3) can be equivalently rewritten in terms of one weight since xB D 1:0  xA . E.rp / D xA E.rA / C .1  xA /E.rB / D xA E.rA / C xB E.rB / D xA .7%/ C xB .20%/ D .:711/.7%/ C .:289/.20%/ D .0:711/.0:07/ C .0:289/.0:2/ D 10:75% (16.3a) Equation (16.3a) concludes that the portfolio with 71.1% of its funds invested in asset A and the other 28.9% in asset B has an expected rate of return of 10.75%.

16.7 The Quadratic Risk Formula for a Portfolio A portfolio’s total risk is measured by either the variance or its square root, the standard deviation, of its rates of return.

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The variance of returns from a portfolio made up of n assets is defined in Equation (16.4). Var.rp / D

n n X X

xi xj ij

(16.4)

i D1 j D1

Equation (16.4) defines the variance-covariance matrix for a portfolio of n assets. The variance-covariance formula is expanded below to show how the variances and covariances of every asset determine the portfolio’s risk. Col. 1 VAR.rp / D Cx1 x1 ¢11 Cx2 x1 ¢21 Cx3 x1 ¢31 : : :. Cxn x1 ¢n1

Col. 2 Cx1 x2 ¢12 Cx2 x2 ¢22 Cx3 x2 ¢32 : : :. Cxn x2 ¢n2

Col. 3 Cx1 x3 ¢13 Cx2 x3 ¢23 Cx3 x3 ¢33 : : :. Cxn x3 ¢n3

Col. n  1 : : :. : : :. : : :. : : :. : : :.

Col. n Cx1 xN ¢1N Cx2 xN ¢2N Cx3 xN ¢3N : : :. Cxn xn ¢nn

Row 1 Row 2 Row 3 Row n  1 Row n

where p D standard deviation of portfolio’s rates of return, i2 and ii D two equivalent conventions for variance of returns from asset i, and, ij and cov.ri ; rj / D equivalent representations for covariance of returns between assets i and j Since x1 x2 ¢12 C x2 x1 ¢21 D 2x2 x1 ¢21 and x2 x2 ¢22 D x2 2 ¢2 2 , we can abbreviate the matrix above by rewriting it as the equivalent matrix below. Col. 1 VAR.rp / D Cx1 x1 ¢11 C2x2 x1 ¢21 C2x3 x1 ¢31 : : :. C2xn x1 ¢n1

Col. 2

Col. 3

Cxn xn ¢nn

Row 1 Row 2 Row 3 Row n  1 Row n

In general, a variance-covariance matrix can be described by the following characteristics. (1) Since it is an n by n matrix, it is a square (not rectangular) matrix. (2) All the diagonal boxes (on the diagonal from the upper left corner to the lower right corner) contain variances, since Cov.ri ; ri / D ¢ii D ¢i 2 D VAR.ri /. (3) Since ¢ij D ¢ji , or equivalently, Cov.rj ; ri / D Cov.ri ; rj /, it is a symmetric matrix in which the boxes above the diagonal are the mirror image of the boxes below the diagonal. The variance-covariance matrix can be equivalently rewritten as follows: VAR.rp / D

i D1

xi2 i2 C

n X n X

The covariance, denoted ¢ij , is related to the correlation coefficient as shown in Equation (16.5). ij D i j ij

(16.5)

where ij is the correlation coefficient between variables i and j . Note that the covariance of the random variable i with itself equals the variance, ii D i i ii D Var.ri /. Also observe that Equation (16.5) can be solved for ij to obtain a definition of the correlation coefficient. The covariance measures how two variables covary. If two assets are positively correlated their covariance will also be positive. For example, almost all NYSE-listed common stocks have positive covariances with each other. If two variables are independent, their covariance is zero. For example, the covariance of the stock market and sun spot activity is zero. If two variables move inversely, their covariance is negative.

16.9 Portfolio Analysis of a Two-Asset Portfolio

Col. n  1 Col. n

Cx2 x2 ¢22 C2x3 x2 ¢32 Cx3 x3 ¢33 : : :. : : :. : : :. C2xn x2 ¢n2 C2xn x3 ¢n3 : : :.

n X

16.8 The Covariance Between Returns from Two Assets

Reconsider the two individual assets denoted A and B in Table 16.3. Equation (16.3a) shows how to compute the expected return of a portfolio that has 71.1% of its funds invested in asset A and the other 28.9% in asset B. This section shows how to reduce the n n variance-covariance matrix, Equation (16.4), into a 2 2 matrix that is appropriate to analyze the risk of a portfolio that contains assets A and B. Equation (16.6) is a simple case of Equation (16.4); it defines the standard deviation of returns for a two-asset portfolio of common stocks, denoted A and B.5 p D

q

xA2 A2 C xB2 B2 C 2xA xB AB

(16.6)

Substituting Equation (16.5) into Equation (16.6) yields Equation (16.7). Equation (16.7) shows how the correlation between assets A and B affects the portfolio’s risk. p D

q

xA2 A2 C xB2 B2 C 2xA xB A B AB

(16.7)

xi xj ij

i D1 j D1 i ¤j

0

1 1 0 1 0 Sum of the Sum of the n The 2 0 D @ variances on A C @ .n  n/=2 A D @ portfolio s A total risk the diagonal covariances (16.4a)

5

The risk formula for a two-asset portfolio is derived below: (EDITOR: Please get this footnote all together on one page.) rp D x1 r1 C x2 r2 by definition E.rp / D x1 E.r1 / C x2 E.r2 /

16 Portfolio Analysis

Substituting in the risk statistics from Table 16.3 into Equation (16.7) yields: q x 2 .:1486/2 C xB2 .:366/2 C 2xA xB .:1486/.:366/AB q A D xA2 .0:04/ C xB2 .0:16/ C 0:16xA xB AB (16.7a) Figures 16.3a–d are graphs in risk-return space of assets A and B and the portfolios that can be formed from them at three different values for their correlation coefficient, AB D 1; 0, and C1. These four figures were prepared by plotting the risk and return of the various portfolios composed of A and B for all the positive weights that sum to unity (that is, xA C xB D 1) and for three different correlation coefficients, AB D 1; 0, and C1. To understand Fig. 16.3, it is informative to verify a few points in the figures by substituting some appropriate numbers into Equations (16.3a) and (16.7a) and calculating the portfolio’s expected return and risk. For example, Equation (16.3a) shows that the portfolio that has xA D 0:711 and xB D 0:289 has an expected return of 10.75%, regardless of what value the correlation coefficient assumes. However, the risk of the portfolio with xA D 0:711 and xB D 0:289 varies directly with the correlation coefficient, AB , as shown in Equation (16.7b). p D

p D

q

x 2  2 C xB2 B2 C 2xA xB A B AB s A A .0:711/2 .0:1486/2 C .0:289/2 .0:366/2 D C2.0:711/.0:289/.:1486/.0:366/AB p D 0:0111 C :0113 C :0224AB (16.7b)

16.9.1 Perfectly Positively Correlated Returns, Fig. 16.3a Portfolio analysis of the two-asset portfolio illustrated in Fig. 16.3a shows that when the correlation coefficient between the rates of return from assets A and B is at

These two equations are used in the following derivation.  2 Var.rp / D E rp  E.rp / the definition of Var.rp / D E fx1 r1 C x2 r2  Œx1 E.r1 / C x2 E.r2 / g2 by substitution for rp D E fx1 Œr1  E.r1 / C x2 Œr2  E.r2 / g2 collecting like terms n

263

its maximum value of positive unity, a linear risk-return relationship results. This straight line between assets A and B in risk-return space is derived by first setting AB to positive unity in Equation (16.7b). Next the values of the two assets’ weights are varied from zero to one (that is, 0 < x < C1) in an inverse manner so that they always sum to positive unity (namely, xA C xB D 1). Finally, the infinite number of pairs of positive values for xA and xB that sum to one are substituted, one pair at a time, into the portfolio risk formula, Equation (16.7b), and the portfolio return formula, Equation (16.3). The infinite number of risk and return statistics for the two-asset portfolio thus derived trace out the straight line in Fig. 16.3a when AB D C1:0.

16.9.2 Uncorrelated Assets, Fig. 16.3b If the rates of return from assets A and B are zerocorrelated, substantial risk reduction benefits can be obtained from diversifying between them. This beneficial risk reduction can be seen by noting what happens to portfolio risk Equation (16.7b) when AB equals zero. The last quantity on the right-hand side of Equation (16.7b) becomes zero when AB D 0. Clearly, this reduces the portfolio’s risk level below what it was when this correlation was a larger value (for instance, when AB D C1). The results of the uncorrelated returns are illustrated in Fig. 16.3b. The portfolio’s expected return is unaffected by changing the correlation between assets – this is because AB is not a variable in the portfolio return Equation (16.3). All differences between the portfolios generated when AB D C1 and the portfolios generated when AB D 0 are risk differences stemming from Equation (16.7b). Figure 16.3b shows that the portfolios with AB D 0 have less risk at every level of expected return than the same portfolios in Fig. 16.3a with AB D C1. The substantial risk reductions available by diversifying across uncorrelated assets are readily available to all investors. Empirical research has shown that common stock price indices, bond price indices, and commodity price indices all tend to be uncorrelated. Thus, an investor who diversifies across different market assets can expect to benefit from the diversifying between uncorrelated assets as indicated in Fig. 16.3b.

D E x12 Œr1  E.r1 / 2 C x22 Œr2  E.r2 / 2 o C2x1 x2 Œr1  E.r1 / Œr2  E.r2 /

16.9.3 Perfectly Negatively Correlated Returns, Fig. 16.3c

D x12 E Œr1  E.r1 / 2 C x22 E Œr2  E.r2 / 2 C2x1 x2 E fŒr1  E.r1 / Œr2  E.r2 / g D x12 Var.r1 / C x22 Var.r2 / C 2x1 x2 Cov.r1 ; r2 / restating definitions

The lowest possible value for a correlation coefficient is negative unity. When the correlation coefficient in portfolio risk

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Fig. 16.3 Portfolio analysis for two assets in risk-return space. (a) The portfolios that can be constructed from A and B when they are perfectly positively correlated lie along a straight line. (b) When A and B are uncorrelated, the portfolio possibilities lie along a curve. (c) If A and B are perfectly negatively correlated, the portfolios that can be created include the riskless portfolio Z. (d) The correlation between the assets determines which efficient frontier is possible

Equation (16.7b) reaches negative unity, the last term on the right-hand side of the equation assumes its maximum negative value for any given pair of values for xA and xB . In fact, when AB D 1, the portfolio’s risk can be reduced to zero risk for one particular set of portfolio weights. For example, for the portfolio made of A and B, Fig. 16.3c shows that when xA D 0:711; xB D 0:289, and AB D 1:0, the portfolio’s risk diminishes to zero. For all other weights the portfolio’s risk is above zero. But, relative to all similar portfolios, portfolio risk is always at its lowest possible level when AB D 1. If it seems dubious that two perfectly negatively correlated risky assets like A and B can be combined to form a riskless portfolio like the one at xA D 0:711 and xB D 0:289 in Fig. 16.3c, consider an example. In this example we assume two assets are perfectly negatively correlated, so their prices and returns move inversely. As a result of these inverse movements, whatever losses one asset experiences are offset by gains from the other asset. This inverse interaction can leave the portfolio with zero variability of return for the correct values for xA and xB . Thus, a riskless portfolio can be created from risky components.

16.9.4 Portfolio Analysis Using Markowitz Diversification, Fig. 16.3d Figure 16.3d summarizes the three illustrated examples from Figs. 16.3a–c. Summarizing, Fig. 16.3a shows that at point P the weights are xA D 0:711 and xB D 0:289, and for a correlation of AB D C1, the portfolio’s total risk is p D 21:2 percent. Figure 16.3b shows that if AB D 0, then p D 15 percent at point W . And, p Fig. 16.3c illustrates the case when AB D 1, then p D 0 D 0 at point Z. Figure 16.3d contains the three points (P; W , and Z, respectively) with E.r/ D 10:75% from Figs. 16.3a–c. Markowitz diversification can lower risk to zero if the securities analyst can find securities whose rates of return have very low enough correlations. Unfortunately, there are not many securities that have correlations that are very low. The typical pair of NYSE-listed stocks has returns that are correlated about  D 0:5. Using Markowitz diversification requires a data bank of financial statistics for many securities, a computer, and some econometric work.

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16.10 Mathematical Portfolio Analysis A solution technique for portfolio analysis that uses differential calculus and linear algebra is explained in this section. Calculus can be used to find the minimum risk portfolio for any given expected return E  . Mathematically, the problem involves finding the minimum portfolio variance.6 That is, Minimize Var.rp / D

n n X X

xi xj ij

(16.8)

i D1 j D1

subject to two Lagrangian constraints. The first constraint requires that the desired expected return E  be achieved. This is equivalent to requiring that the following equation cannot be violated. n X xi E.ri /  E  D 0 (16.9) i D1

The second constraint requires that the weights sum to unity. This constraint is equivalent to requiring that Equation (16.10) is true. n X

xi  1 D 0

(16.10)

i D1

C2

n X

!

i D1

xi  1

(16.11)

i D1

The minimum risk portfolio is found by setting dz=dxi D 0 for i D 1; : : : ; n and dz=dj D 0 for j D 1, 2 and solving the resulting system of equations for the xi ’s. The number of assets analyzed, n, can be any positive integer.

16.11 Calculus Minimization of Risk: A Three-Security Portfolio For a three-security portfolio, the objective function to be minimized is shown below.

6

dz dx1 dz dx2 dz dx3 dz d1 dz d2

For more information about mathematical portfolio analysis, see Francis and Alexander (1986).

D 2x1 11 C 2x2 12 C 2x3 13 C 1 E1 C 2 D 0 D 2x2 22 C 2x1 12 C 2x3 23 C 1 E2 C 2 D 0 D 2x3 33 C 2x1 13 C 2x2 23 C 1 E3 C 2 D 0 D x1 E1 C x2 E2 C x3 E3  E  D 0 D x1 C x2 C x3  1 D 0

(16.13) All the equations in system Equation (16.13) are linear, since the weights (xi ’s) are the variables and they are all of degree one. As a result, the system of equations may be solved as a system of simultaneous linear equations. The matrix representation of this system of linear equations is shown below as matrix Equation (16.14). 3 32 3 2 0 x1 1 6 7 6 7 17 7 6x2 7 6 0 7 6 6 7 7 1 7 6x3 7 D 6 0 7 7 0 5 42 5 4 1 5 E 0 1 (16.14) This system may be solved several different ways. With matrix notation, the inverse of the coefficient matrix, denoted C 1 , may be used to find the solution (weight) vector x follows: Cx D k C 1 Cx D C 1 k (16.15) Ix D C 1 k x D C 1 k 2

Combining these three quantities above yields the Lagrangian objective function of the risk minimization problem with a desired return constraint: ! n n X n X X  xi xj ij C 1 xi E.r1 /  E zD i D1 j D1

z D x12 11 C x22 22 C x32 33 C 2x1 x2 12 C 2x1 x3 13 C2x2 x3 23 C 1 .x1 E1 C x2 E2 C x3 E3  E  / C2 .x1 C x2 C x3  1/ (16.12) Setting the partial derivatives of z with respect to all variables equal to zero yields equation system Equation (16.13) below.

211 6 221 6 6 231 6 4 1 E1

212 222 232 1 E2

213 223 233 1 E3

E1 E2 E3 0 0

The solution will give the n weights in terms of E  . If n D 3 we have: x1 D a1 C d1 E  x2 D a2 C d2 E  (16.16) x3 D a3 C d3 E  where the ai and di are constants. For any desired E  the equations give the weights of the minimum risk portfolio. These are the weights of a portfolio in the efficient frontier. By varying E  the weights may be generated for the entire efficient frontier. The risk, Var.Vp /, of the efficient portfolios may be calculated, and the efficient frontier may be graphed. As a numerical example, the data from the three-security portfolio problem in Table 16.4 is solved to obtain the following coefficients matrix.

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Table 16.4 Statistical inputs for portfolio analysis of three common stocks Asset E.ri / Var.ri / D ¢ ii Cov.ri ; rj / D ¢ ij E.r1 / D 5% D 0:05 E.r2 / D 7% D 0:07 E.r3 / D 30% D 0:3

1 2 3

2

211 6221 6 6231 6 41 E1

212 222 232 1 E2

213 223 233 1 E3

E1 E2 E3 0 0

¢11 D 0:1 ¢22 D 0:4 ¢33 D 0:7

3 2 2.:1/ 1 6 2.  :1/ 17 7 6 6 17 7 D 6 2.0/ 41 5 0 0

:05

¢12 D 0:1 ¢13 D 0:0 ¢23 D 0:3

2.  :1/ 2.:4/ 2.:3/ 1 :07

3 2.0/ :05 1 2.:3/ :07 1 7 7 2.:7/ :3 1 7 7 1 0 05 :3

0 0 (16.17)

Multiplying the inverse of this coefficients matrix by the constants vector k yields the weights vector .C 1 k D x/ as shown below.7 C 1 k D W 2 :677 :736 6 :736 :800 6 :064 6 :059 4 1:433 2:790 :789 :447

:059 :064 :005 4:223 :236

:789 :447 :236 :522 :095

3 2 3 32 1:433 x1 0 2:790 7 6 0 7 6 x2 7 7 6 7 76 4:223 7 6 0 7 D 6 x3 7 15:869 5 4 1 5 4 2 5 1 :552 E

date investments provides a practical approach to portfolio analysis. Markowitz portfolio analysis can consider both the risk and return of dozens or hundreds of different securities simultaneously – the number is limited only by the size of the computer and the number of securities for which the risk and return input statistics are available. No human mind can simultaneously evaluate numerous different investment opportunities and balance the risks and returns of them all off against one another to find efficient portfolios that dominate all other investment opportunities. Markowitz portfolio analysis is essentially a mathematics problem requiring that many different equations be solved simultaneously. The computations can be done on a large scale by using a computer program that does “quadratic programming.” Quadratic programming (QP) minimizes the portfolio’s risk (a quadratic equation) at each level of average return for the portfolio.9 Markowitz portfolio theory has remained undiminished for over half a century. Commercially available raw data and new computers make it easier and cheaper to employ Markowitz’s QP every year. Now is an excellent time to utilize Markowitz’s Nobel Prize winning theory to manage your portfolio.

(16.18)

Evaluating the weights vector yields the system of equations below. x1 D :789  1:433E  x2 D :447  2:790E  x3 D :236 C 4:223E  1 D :522  15:869E  2 D :095 C :522E 

(16.19)

The weights in the three equations sum to unity (that is, x1 C x2 C x3 D 1 for any given value of E  ). And, they are all linear functions of E  , and, they all represent the weights of the three securities in the efficient portfolio at the point where E.rp / D E  . Varying E  generates the weights of all the efficient portfolios.8

References Francis, J. C. and G. Alexander. 1986. Portfolio analysis, 3rd ed. Prentice-Hall, Englewood cliffs, NJ. Francis, J. C. and R. Ibbotson. 2002. Investments: a global perspective, Prentice-Hall, Upper Saddle River, New Jersey, ISBN: 0-13890740-4. Holden, C. W. 2005. Excel modeling in investments, 2nd ed. PrenticeHall, Upper Saddle Ridge, New Jersey, ISBN: 0-13-142426-2. Holden C. W. 2009. Holden, excel modeling and estimation, 3rd ed. Prentice-Hall, Upper Saddle Ridge, New Jersey. Markowitz, H. 1952. “Portfolio selection.” Journal of Finance 7(1), 77–91. Markowitz, H. 1959. Portfolio selection. Cowles Foundation monograph, 16 Wiley, New York. Malevergne, Y. and D. Sornette. 2005. “Higher-moment portfolio theory; capitalizing on behavioral anomalies of stock markets.” Journal of Portfolio Management 31, 49–55. Stevens, G. V. G. 1998. “On the inverse of the covariance matrix in portfolio analysis.” Journal of Finance, October, 53(5), 1821–1827. Szego, G. 2002. “Measures of risk.” Journal of Banking and Finance 26, 1253–1272.

16.12 Conclusion In conclusion, it is helpful to note that using a computer to apply Markowitz diversification to a collection of candi7 Portfolio solutions depend on the covariance matrix. See Stevens, G. V. G. “On the inverse of the covariance matrix in portfolio analysis.” Journal of Finance, October 1998, 53(5), 1821–1827. 8 Additional details and further implications have been published by Francis and Ibbotson (2002).

9 Computer programs to perform portfolio analysis and other forms of investment analysis are available. For an Excel template that you can construct yourself, see Chapter 9 of Craig W. Holden, Excel Modeling In Investments, Second Edition, Prentice-Hall, Upper Saddle Ridge, New Jersey, 2005, ISBN: 0-13-142426-2. Or, see Chapters 5 and 6 of Holden’s Third Edition book entitled Excel Modeling and Estimation, 2009.

Chapter 17

Portfolio Theory, CAPM and Performance Measures Luis Ferruz, Fernando Gómez-Bezares, and María Vargas

Abstract This chapter is focused on the “Portfolio Theory” created by Markowitz. This theory has the objective of finding the optimum portfolio for investors; that is, that which gives tangency between an indifference curve and the efficient frontier. In this chapter, the mathematics of this model is developed. The CAPM, based on this theory, gives the expected return on an asset depending on the systematic risk of the asset. This model detects underpriced and overpriced assets. The critics expressed against the model and their application possibilities are also analyzed. Finally, the chapter centers on performance measures related to portfolio theory (classic indices, derivative indices and new approaches) and on the performance persistence phenomenon employing the aforementioned indices, including an empirical example. Keywords CAPM r Performance measures r Penalized internal rate of return index (PIRR) r Penalized internal rate of return for beta index (PIRR for Beta) r Portfolio theory

17.1 Portfolio Theory and CAPM: Foundations and Current Application 17.1.1 Introduction Portfolio theory has become highly developed and has strong theoretical support, making it essential in the toolbox of any financial expert. In 1990, three American economists shared the Nobel Prize for Economics: Harry Markowitz, Merton Miller, and

William Sharpe. Markowitz is considered the creator of portfolio theory. He began this field of research with his work of 1952, making major additions with his works of 1959 and 1987. Sharpe attempted to simplify the model of his mentor, Markowitz, with his work of 1963, based on which he created the capital asset pricing model (CAPM) in 1964. Miller has also worked on portfolio topics. Another Nobel Prize winner for Economics, James Tobin, contributed to the advance of portfolio theory with his work in 1958. Merton,1 Black and Fama are others who have made great advances in the field. A portfolio is a combination of assets, such as a group of stocks that are diversified to some extent. A portfolio is characterized by the return it generates over a given holding period. Portfolio theory describes the process by which investors seek the best possible portfolio in terms of the tradeoff of risk for return. In this context, risk refers to the situation where the investor is uncertain about which outcomes may eventuate, but can assess the probability with which each outcome may arise. In a risky environment, the choice of an optimal portfolio will depend on the distribution of returns, as well as the preferences of the investor – his or her appetite for risk. It is clear that all decision makers have their own personal preferences, but it is possible to posit some generalizations. First, let us suppose that all decision makers prefer more wealth to less and, second, that decision makers are risk averse. We define risk as the variability of the results of an activity; this variability can be measured by standard deviation or by variance. If we assume, as is customary, the marginal utility of an individual as a decreasing function of wealth, then the least risky activity is preferred, given the same average results. Although the risk aversion behavior of individuals has been questioned, the literature is based on the assumption

L. Ferruz () and M. Vargas University of Zaragoza, Zaragoza, Spain e-mail: [email protected] F. Gómez-Bezares University of Deusto, Bilbao, Spain

1

Merton also won a Nobel Prize in 1997 along with Scholes.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_17, 

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that risk must be rewarded to be accepted by the individual. This reward is made by awarding “prizes” for risky activity with a higher expected payoff. Therefore, the return premium represents the reward for risk.

17.1.2 The Mean-variance Model Decision makers will be interested in knowing the probability distribution for the return on their investment. If we assume that this probability distribution is normal, it will be defined by two parameters: the mathematical expectation (mean), and the standard deviation (or its square, the variance). Therefore, investors can be said to make decisions based on mean and variance, hence the “meanvariance” model. Asset i will produce, in period t, a return ri , that is a random variable with average i and standard deviation ¢i . Consequently, each asset will be a point on the ¢ map. The map, shown in Fig. 17.1, represents the group of possible investments (portfolios) from which the investor will be able to choose. Investors must then choose among the various points on this map of possible opportunities. Each individual has a particular preference, which constitutes the individual’s system of indifference curves. An indifference curve is that place of the points on the map that yield equal utility for the individual. In Fig. 17.2, a system of indifference curves typical for an individual who is averse to risk is shown. Characteristically, the indifference curves are positive-sloped and convex, since the higher the deviation in results, the higher is the mean demanded by individuals to maintain themselves at the same level of indifference. On the other hand, the furthest the curve is from the horizontal axis, the greater the level of utility that it represents.

17.1.3 The Efficient Frontier A portfolio is efficient when it yields a higher average return for a given risk, and a lower risk for a determined average return. Such portfolios make up the efficient frontier, which is a subgroup of the minimum-variance frontier, within which are located those portfolios that have a lower risk for a determined average return. We now show the process for obtaining the minimumvariance frontier, following the work of Merton (1972).2 We start with a portfolio made up of n securities (all risky and such that none can be obtained by the linear combination of any others) with ri returns and in wi weightings: R0 D .r1 ; r2 ; : : : ; rn / W0 D .w1 ; w2 ; : : : ; wn / We also adopt the following nomenclature: U is a vector of ones, so W0 U D 1 (the proportions must add up to 1) P D R0 W is the return on the portfolio and E.P/ D E.R0 /W, the average return † is the variance and covariance matrix of returns on n securities VAR.P/ D W0 †W, the variance of the portfolio COV.R; P/ D †W, the co-variance vector of the returns on the securities and the portfolio. Determination of the minimum-variance frontier results in minimization of the variance for each value of the average return; that is, variance is minimized by assuming a return expectation E : MIN W W0 †W Subject to W E.P/ D E.R0 /W D E W0 U D 1:

μ

μ

Possible Opportunities

σ σ Fig. 17.1 Figure 17.1 represents the group of possible investments (portfolios) from which the investor will be able to choose

Fig. 17.2 Figure 17.2 shows a system of indifference curves typical for an individual who is averse to risk

2

Adapted by Gómez-Bezares (2006).

17 Portfolio Theory, CAPM and Performance Measures

μ

m =

A + C

269

D ×s C

m =

μ A C

D(C s 2 −1)

+

C

Z A C

r0 m =

1

A C

2

−

D(Cs −1)

σ

C

σ

C

m =

A C

−

D C

Fig. 17.4 Figure 17.4 shows an efficient frontier with a risk-free asset

×s

values of œ1 we obtain the sub-group called the efficient frontier, that, as well as having minimum variance for a given expectation, has maximum expectation for a given variance.

Fig. 17.3 Figure 17.3 shows that the frontier is a hyperbola on the mean-deviation map

Finding the minimum conditioned by Lagrange, we obtain W vector, which provides the proportions, depending on the portfolio expectation, in which appear the securities of the portfolios that compose the frontier: W D

.C E.P /  A/

P1

E.R/ C .B  A E.P // D

P1

U

;

(17.1)

P P where: A D U 0 1 E.R/I B D E.R0 / 1 E.R/I C D P U 0 1 U ; and D D BC  A2 . It can be demonstrated that the frontier is a hyperbola on the mean-deviation map (see Fig. 17.3), and a parabola on the mean-variance map (Merton 1972)3 and Gómez-Bezares (2006). Furthermore, we can rewrite W D E.P/G C H, where G and H are vectors, in such a way that, given that each wi is linearly related with E(P) and knowing the W values of two portfolios that are on the frontier, we obtain G and H and, therefore, we obtain the Ws of all portfolios that make up the frontier. If we designate œ1 the Lagrange multiplier corresponding to the first condition, whose formula is œ1 D 2ŒC E.p/  A =D, it can be demonstrated that œ1 represents the slope – measured by its inverse – of the minimum-variance frontier on the mean-variance map. Depending on how œ1 varies, we obtain a new portfolio of minimum variance. For positive 3

See also Roll (1977).

17.1.4 Efficient Frontier with a Risk-free Asset It is possible to construct portfolios as a combination of risky and risk-free assets. On the mean-deviation map, a straight line represents the new portfolios. The highest straight line will be the tangent to the previous efficient frontier on point Z; thus, a new efficient frontier will appear (Fig. 17.4). On the left of Z, we have portfolios composed of a proportion of risk-free assets, and another of portfolio Z; on the right, we invest more than 100% in portfolio Z, accepting debt at the risk-free rate. We designate W as the vector of the weightings of investments in risky assets, and .1  W0 U/ as the proportion invested in the risk-free asset, with r0 return. In this case, we also minimize variance for a value of the E mean: MIN W W0 †W Subject to:E.P/ D E.R0 /W C .1  W0 U/ r0 D E Resolving the minimum conditioned by Lagrange, we obtain the vector W: P .E.P /  r0 / 1 .E.R/  U r0 / D .E.P /  r0 / M; W D B  2Ar0 C C r02 (17.2) designating M the corresponding vector. œ continues to be a linear function of E (P): D

2 ŒE.P /  r0 B  2Ar0 C C r02

(17.3)

270

in such a way that the various W vectors are proportional to the M vector. This leads to Tobin’s Separation Theorem (1958): When there is a risk-free asset (that can be sold short) on the frontier, all W vectors (that contain risky assets) will have the same proportions of assets, with only the proportion between these and the risk-free asset varying.

17.1.5 Efficient Frontier with Inequality Restrictions Until now the only restriction considered has been that the totality of the proportions invested in securities must add up to the unit. Other equality restrictions could be added and it would still hold that on the frontier, each portfolio is a linear combination of any two portfolios, and that the W values are a linear function of E (P), and therefore of œ1 . However, the problem becomes more complex when we add inequality restrictions (standard problem) to the basic problem, which contains only equality restrictions. In the standard problem, there are restrictions such as: ai  wi  bi , so that the relationships stop being linear for any value of œ1 , but are linear within each interval. For example, it could be that up to a value of œ1 D x no inequality restriction appears; that is, there is a basic sub-problem. If, at this point, any restriction stops, it becomes an equality up to œ1 D y, meaning that between x and y there is a new basic sub-problem (here the evolution will be linear). Therefore, a standard problem is a group of basic sub-problems. Points x and y are called cornerpoints, meaning that each portfolio on the frontier is a linear combination of two adjacent corner portfolios. In this way, if there is no risk-free asset, between each two of the adjacent corner portfolios, we will have a piece of hyperbola, but the complete frontier will not yet be a hyperbola (see Fig. 17.1). If securities cannot be emitted (sold short) except for the risk-free asset, whose short sale means getting into debt, but securities can be bought, the restrictions – that will only affect the risky assets – will take the form 0  wi  1. In this case, since we are able to put ourselves into debt in the risk-free asset, the upper line that is the efficient frontier will be unlimited towards the right. Therefore, the efficient part of the frontier will be a straight line that responds to a basic sub-problem within a standard problem.

17.1.6 Application to the Market Continuing with the case of inequality restrictions (supposing that there are limitations on short-selling, as in Fig. 17.1),

L. Ferruz et al.

μ R*

r0

σ Fig. 17.5 Figure 17.5 represents an optimum portfolio considering that the risk-free asset can be emitted without restriction

individuals will search for their optimum portfolio, which will be that portfolio that reaches the highest possible indifference curve, that is, that in which tangency is produced between one of the indifference curves and the efficient frontier. The problem here is that we do not know where the market portfolio will be located (the sum of all individuals’ portfolios), nor whether it will be efficient. However, if we assume a risk-free asset that can be emitted without restriction, then all individuals will invest a part of their wealth in the risk-free asset – positively or negatively – and another part in the R portfolio, which is the market portfolio. This is because all individuals will invest the risk-based portion of their wealth in the proportions of R (see Fig. 17.5). Of course, we assume uniform expectations (agreement) among investors. It is also assumed that there is equilibrium between supply and demand and a uniform interest rate for everyone, in terms of lending and borrowing. In this case, the efficient frontier is the so-called Capital Market Line, CML.

17.1.7 The Capital Asset Pricing Model The capital asset pricing model (CAPM), or Sharpe-Lintner model, stands out among asset pricing models. This model reflects how the expected return on an asset is a function of the expected returns on the market and the risk-free asset and of the relevant (systematic) risk of that asset. If this model is complied with, at equilibrium, all assets adjust their values so as to offer the return that corresponds to them, thereby allowing investors to determine whether an asset is undervalued or overvalued. This model was created in the pioneering work of Sharpe (1961, 1964), Lintner (1965), among others, and has been adapted over time to reflect different aspects of the real financial world.

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271

Although the starting hypotheses of the CAPM place us in markets with perfect competition, which creates the potential for lack of realism, the conclusions of the model adapt well to reality.4 In addition, recent technological advances, new products, and greater knowledge of the markets make markets in general seem ever more like the ideal markets supported by the CAPM. Portfolio theory is the fundamental theoretical base of the CAPM. This model accepts that there is a risk-free asset with r0 return (that can be emitted), and claims that the securities that make up a portfolio on the efficient frontier, with P return, must comply with the following equation E.R/ D r0 U C ŒE.P /  r0

COV.R; P / : VAR.P /

(17.4)

According to this model, the expected return on a security will depend on the portfolio P and the return on the risk-free asset; P represents any portfolio that is located on the efficient frontier. Furthermore, if we suppose there is agreement, all individuals will make the same forecast of the portfolio map, and if the market has reached equilibrium, the market portfolio R will be efficient. We return to this concept later.

17.1.8 The Market Model Sharpe (1963) attempts to reach a simplification of the Markowitz model by proposing the market model. The simplifying hypothesis of this model posits that the relations between returns on the various assets stem solely from the relation between returns on all assets and that of a market index. It also assumes the existence of a linear relation between the return on each asset and that of the index. Using the market portfolio as a proxy for the market index, we have the model Rit D ’i C “i Rt C ©it ;

(17.5)

where Rit is the return on the asset i in period t, and it is usually considered that ©it is an error term such that:

Hence, we obtain ¢ 2 .Ri / D “2i ¢ 2 .R / C ¢ 2 .©i /

(17.6)

in such a way that the first member of this equation represents the risk of the asset measured with its variance, and the second has two components: the first measures the systematic risk and the second the diversifiable risk. We commented above that in this model the only relationship between the returns on the assets is the shared relationship with the return on the index, therefore, COV.©it ©jt / D 0 if i ¤ j, but this condition is difficult to accept since the application of factorial analysis demonstrates that there are sectorial factors, which corroborates the idea that, aside from the correlation through the market, there are other sources of correlation. It is precisely based on said condition of zero covariance that it is demonstrated that diversifiable risk really does disappear in the portfolio; however, fortunately, even though there are sector effects, a careful diversification enables us to maintain the distinction between systematic and diversifiable risk. Systematic risk derives from the general economy and affects all assets to some extent, and diversifiable, or specific risk, is a risk that affects one asset in particular. If an asset has a high diversifiable risk, it will be easy to find other assets that do not correlate with it, forming a diversified portfolio with much less risk. Systematic risk is measured through the first component of the second member of the formula (17.6), but given that market variance is equal for all the assets, we can measure this risk using the beta parameter. This parameter is the regression slope between the asset return and that of the market (Security Characteristic Line, SCL). We will call a security an “aggressive security”if it increases market variations .“ > 1/ and “defensive” if it decreases them .“ < 1/, as shown in Fig. 17.6. The market portfolio has a slope equal to the unit, while the risk-free asset has a null slope. There are also assets with negative slopes that allow the total risk of a well-diversified portfolio to be reduced. SCL (aggressive) Ri

E.©it / D 0 VAR.©it / D ¢ 2 .©i / for all t’s; .homoscedasticity/ COV.©it ©it’ / D 0 for all t’s; t’ .no autocorrelation/ COV.©it; Rt / D 0 for all t’s:

SCL (defensive)

R* 4

Although we will discuss this matter later.

Fig. 17.6 Figure 17.6 shows an aggressive and a defensive security

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17.1.9 Relation Between the Market Model and CAPM The relevant risk from a security is its systematic risk, since it is this risk that each security introduces to the market portfolio, which by definition does not have diversifiable risk. On the other hand, the systematic risk of a portfolio is the linear combination of the systematic risks of the securities that comprise it. Securities have differing levels of systematic risk, depending on how defensive or aggressive they are, and must yield returns accordingly. The CAPM confirms that securities must produce returns that are dependent entirely on their systematic risk. If we consider formula (17.4) of the CAPM and substitute P for R (market portfolio) (since this is the only efficient portfolio made up solely of risky assets), we obtain  COV.R; R /  E.R/ D r0 U C E.R /  r0 VAR.R /

(17.7)

and, given that the regression betas between the asset return / , we arrive at and that of the market index are ˇ D COV.R;R VAR.R / the usual equation from the CAPM   E.R/ D r0 U C E.R /  r0 ˇ;

(17.8)

where “ is the vector that contains the betas. Considering CAPM formula (17.8), we note that assets must yield returns, depending on their systematic risk or beta as follows: their expected return is equal to that on the risk-free asset plus a premium for risk, which is the product of multiplying the premium that the market receives by the quantity of systematic risk accepted. Expected returns on assets, at equilibrium, must be adapted to the previous equation, leading to the Security Market Line, SML (Fig. 17.7). Securities located above the SML will be undervalued, as they produce greater returns than expected and, as they are much in demand, their prices will increase until they reach the SML; those securities located under the SML will be overvalued and will reach the SML by a process inverse to that of those located above the SML. E(Ri )

17.1.10 The Pricing Model The CAPM is an asset pricing model because it considers that assets are undervalued (overvalued) when their expected return is above (below) that demanded by the model. However, the constraint that a security must produce returns that depend on its systematic risk goes against the idea of business diversification; that is, the investment is not valued by the risk that it brings to the company, rather by the risk brought to the shareholder. However, and contrary to what is proposed by the CAPM, which does not consider diversifiable risk, there are investments that are difficult to diversify, as is the case for owners of family companies who invest the majority of their savings in their businesses. Also, the fact that few securities are listed on the stock market hinders diversification. These are only two reasons among many of why risk diversification is complex, and why diversifiable risk can be important. There are other occasions when it is necessary to pay attention to total risk and not just systematic risk – for example, bankruptcy risk, which is dependent on total risk. All in all, the CAPM produces several interesting results that cannot be ignored. From the CAPM, we deduce an important maxim: a firm must do only what the shareholder cannot do individually; that is, the firm should diversify to seek synergies; however, to diversify diversifiable risk, the shareholder alone is sufficient (this is true, above all, in large companies).

17.1.11 Contrasts and Controversy Regarding the CAPM

SML

E(R* )

r0 VAR(R*) β=1

Therefore, intelligent investors will eliminate diversifiable risk from their portfolios because it is not rewarded; they will retain only the systematic risk and will be rewarded as a result. The extent of their aversion to risk will lead them to accept a certain quantity of risk along the CML. On the CML are located the efficient portfolios, which do not have diversifiable risk, only systematic – given that the market portfolio has systematic risk only, and that all portfolios on the CML have one portion of the market portfolio and a portion of the risk-free asset.

COV(Ri,R*) BETAi

Fig. 17.7 Figure 17.7 shows the Security Market Line

There are numerous studies that criticize the starting hypotheses of the CAPM and that analyze the effects of their relaxation. In general, we point out two major levels of criticism: for professionals of the financial world, the CAPM is too theoretical, and for the academic world, the model does not adequately reflect a reality that is, in fact, much more

17 Portfolio Theory, CAPM and Performance Measures

complex. However, the most important thing is to determine whether the CAPM can adapt to reality – that is, whether it is sufficiently robust. At the beginning of the 1970s, the CAPM was viewed optimistically. By the late 1970s, the first criticisms, which grew in the 1980s, arose, and by the 1990s there was a great deal of controversy associated with the CAPM. To determine whether the CAPM actually works, it seems logical to study whether, using past data, the predicted relationship between an asset’s return and its systematic risk is borne out; there are diverse methodologies that enable this task. Equation (17.8) represents an ex ante model that has the drawback that expectations cannot be observed; but if we employ rational expectations, we may test it based on past data (ex post model). To do this, a regression is applied between average returns on assets and their betas (empirical market line) Rj D y0 C y1 ˇj C uj ; (17.9) where y0 represents the risk-free interest rate, and y1 the market premium for risk. The first contrasts are favorable to the CAPM. In this way for example, Black et al. (1972) achieve a good linear fit between average returns and betas with a positive y1 , although y0 does not represent the risk-free interest rate. Fama and MacBeth (1973) also obtain results consistent with a linear model, in which beta is the only risk measure and a positive risk premium is observed. In general, at this time, a consensus held that the market correctly prices securities and that the CAPM is a good pricing model. However, the first critics were not long in arriving. Roll (1977) holds that the only way to test the CAPM is to check whether the market portfolio is efficient ex post; but a problem with the CAPM is that the real market portfolio cannot be observed, as there are no portfolios that include all the assets of the economy. Another criticism of the CAPM is from Van Horne (1989), who makes clear that betas vary considerably according to the index employed as a proxy for the real market portfolio. However, the work that generated the most controversy about the CAPM was that of Fama and French (1992), which shows a small power of betas to explain average returns, and which emphasizes the existence of other variables that do explain them, such as size and book-to-market ratio. Nevertheless, Kothari et al. (1992) examine Fama and French (1992) and find that the tests used are not powerful enough. They also observe, using a different source, that the relationship between average returns and the book-to-market ratio is weak, although they do not make the same observation regarding size. They also find that annual betas make the relationship with returns clearer. That is, in short, they consider that beta continues to be a useful instrument to explain average returns.

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Roll and Ross (1994) also analyze the work of Fama and French (1992) and point out that a positive and precise relationship between expected returns and betas will be found if the market index used to find the betas is in the efficient frontier. The problem is that the market portfolio cannot be observed, meaning that proxies must be found, and if the proxy used is not efficient, it should not surprise us that the average returns/betas relationship is not found. Therefore, these authors conclude that although the CAPM cannot be rejected, it is overly dependent on the proxy chosen for the market portfolio, which detracts from the CAPM’s robustness. Later, Fama and French (1993) proposed a multifactor model with five factors to explain return on stocks and bonds (market, size, and book-to-market for stocks, along with two specific factors for bonds). In this way, they reach an approach strongly related to the APT. Fama and French (1996a) defend their model, insisting that beta, in itself, cannot explain expected returns. Fama and French (1996b) maintain that average returns are related to several variables, which, given that they are not explained by the CAPM, are called anomalies. However, these anomalies disappear when a model with three factors is constructed. However, the controversy continued. MacKinlay (1995) contends that the multi-factor models do not resolve all deviations of the CAPM. Kim (1995) believes that the problems with using betas to explain expected returns stem from error problems in the variables. Jagannathan and Wang (1996) think that the problem lies in the fact that empirical studies assume that betas remain constant, and they propose a conditional CAPM that is criticized by authors such as Ghysels (1998), and Lewellen and Nagel (2006), who believe that the conditional CAPM does not explain the pricing errors of the traditional model, since the variation in betas and the risk premia would have to be very large to explain such major anomalies in pricing as momentum or value premium. More recently, Daniel et al. (2001), using the Daniel and Titman test (1997), rejected the model of Fama and French (1993) when evaluating whether the value premium found in the average returns on the largest Japanese stocks represents a compensation for the risk taken. This work was probably undertaken as a way to counter criticism from Davis et al. (2000), who state that a three-factor model better explains the asset premium than does the Daniel and Titman characteristics model (1997). Many other authors have joined the controversy about the CAPM, for example, Chan and Lakonishok (1993), Grundy and Malkiel (1996), Malkiel and Xu (1997), and Bruner et al. (1998), among others. Among the most recent studies on portfolio theory, that of Dimson and Mussavian (1999) stands out. These authors conducted a review of financial asset pricing theory and point out that derivative pricing is the latest trend. Another notable

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study is Jegadeesh and Titman (2001) finding support for the hypothesis of behavioral models, which states that the returns on momentum strategies stem from subsequent reactions to the formation of these strategies. Lastly, Liu (2006) demonstrates that liquidity is an important source of risk, and that liquidity is not considered by the CAPM or by the model of Fama and French. This author believes that a two-factor model (liquidity and market factors) better explains stock returns and justifies the effect of the book-to-market ratio, which cannot be explained by the three-factor model of Fama and French. In general, there is consensus that investors seek more return when a greater risk is taken, but it is not so clear how they determine either the risk or the reward demanded. Several asset pricing models have attempted to answer these questions; among them, the CAPM is the most often employed model by both financial experts and academics, despite the difficulties inherent in its empirical contrast. The CAPM is not a complete model; rather, it is a simplification of reality. But its competitors do not achieve superior results, and they are more complex. Furthermore, the models that are created from empirical results and have little theoretical basis, such as that of Fama and French, are susceptible to data mining bias, as Black (1993) shows, making models with rigorous theoretical foundation, such as the CAPM, preferable. Additionally, and despite all the criticism that it has been subjected to since its appearance, we are able to confirm that the CAPM is very efficacious within companies when it is used to determine investment and finance policies. It is also useful in the markets, as it helps to determine the return that should be demanded from an asset, as well as the level of risk that is warranted. CAPM also enables us to evaluate fund managers, comparing the returns they achieve with the risk they bear.

According to the classic financial literature, we have two alternative methods to measure performance in portfolio management: On the one hand, we may estimate average past returns on portfolios during a set time horizon and, later, adjust said returns by the behavior of the reference market in which such portfolios invest, and by a representative measure of the risk borne by such portfolios (if the total risk is considered, this is usually measured through variability in the portfolio return, that is, through the standard deviation5 of the series of returns; however, if only the systematic risk is considered, it will be measured through the beta coefficient as explained in the previous section of this chapter). The premise here is that we, as individuals, are averse to risk, and consequently demand a higher expected return for bearing a higher risk; it is precisely the performance measures that, when relating average return to its risk, inform us as to how the analyzed portfolio is rewarding us in terms of return received for the accepted risk. On the other hand, portfolio management performance can also be measured using a history-based composition of portfolios, real or estimated, and by developing evaluation methodologies for them. The contributions made by Sharpe (1992), Grinblatt and Titman (1993), and Daniel et al. (1997) are important. However, this work focuses on the first alternative for the measurement of performance. In this section we first examine the classic performance measures usually employed to study stock market portfolios. We will see that the differences between the various measures are attributable to the relevant risk, as well as to the way in which the method of beating the market is measured. Later, we examine other measures that attempt to solve some problems present in the classic measures and, finally, we touch on the latest performance measures that enable better adjustment to the present reality.

17.2 Performance Measures Related to Portfolio Theory and the CAPM: Classic Indices, Derivative Indices, and New Approaches

17.2.2 Classic Performance Measures 17.2.2.1 The Sharpe Ratio (1966)   r0 ; (17.10)  where  is the average return on a portfolio or asset; r0 is the risk-free asset return, and ¢ is the standard deviation of the return on the portfolio or asset. SD

17.2.1 Introduction When evaluating the behavior of a security, portfolio, or investment fund, there are three characteristics that should be considered: return, risk, and liquidity. This last item can be assumed in markets that are sufficiently liquid. It is the return–risk relationship that must be analyzed, which leads to the issue of performance measures.

5

However, when there is a problem of asymmetry in the distribution of returns, the standard deviation is not a suitable risk measure, as Sortino and Price (1994), Eftekhari et al. (2000), and Pedersen and Satchell (2002) indicate; alternative measures must be considered, such as the standard semi-deviation or the Gini index, among others.

17 Portfolio Theory, CAPM and Performance Measures

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The Sharpe ratio calculates the return premium obtained by the asset or portfolio per unit of total risk accepted. Given that this index responds to total risk, it should be used to evaluate portfolios with a tendency to eliminate diversifiable risk. However, the indices of Treynor and Jensen, as we will see below, are based on the systematic risk, meaning that they should be used to evaluate portfolios that are not very diversified, or individual securities.

17.2.2.2 The Treynor Ratio (1965)

T D

  r0 ; ˇ

(17.11)

where “ is the systematic risk of the portfolio or asset. This index reflects the return premium obtained by the asset or portfolio per unit of systematic risk borne.

17.2.2.3 The Jensen Index (1968, 1969) 



J D .  r0 /    r0 ˇ; 

the aim of replicating the risk of the reference portfolio, according to the equation M2 D

 



 

 

(17.12)

where  is the average return obtained by the market portfolio. Usually, the market portfolio is approached using a benchmark or a stock market index. This index determines the difference between the excess return on the asset or portfolio over the risk-free asset, and the excess that it should have obtained according to the CAPM. Therefore, an asset or portfolio will be said to have obtained a good performance result when it obtains a positive Jensen index value. If the Sharpe or Treynor ratios are used as the performance measures to determine whether a portfolio or asset has obtained a good result, it will be necessary to compare this ratio with that obtained by the market portfolio. Also, this comparison will allow us to discover whether active management is superior to passive management. Furthermore, regardless of which performance index we use, the portfolio or asset that obtains the highest value in such index will always be considered superior.

17.2.3 M2 and M2 for Beta Another indicator of performance based on total risk is Modigliani and Modigliani (1997) M2 measure. This measure compares portfolio returns, adjusted for risk, with those in the appointed reference portfolio. It is calculated as that portfolio return that has been leveraged up or down with

  1 r0 ;

(17.13)

where ¢  is the standard deviation of the return on the benchmark, and r0 is the interest rate representing the associated cost of the quantity borrowed, or the interest rate paid for the money loaned. Therefore, M2 is a measure directly comparable with market average return,  , meaning that the evaluation of management will be positive when M2 is higher than  . Also, it can be proven that M2 generates the same rankings as the Sharpe ratio. Later, Modigliani (1997) proposed a new version of M2 that allows the use of systematic risk, rather than total risk: M2 for beta, which generates the same ranking as the Treynor ratio. The idea behind this measure is that we borrow or lend such that the portfolio has the same beta as the market, later comparing the resulting portfolios returns: M2 for beta D r0 C





.  r0 / ˇ

(17.14)

The interest in the use of M2 and M2 for beta measures, which technically coincide with the Sharpe and Treynor ratios, respectively, lies in the fact that they are measured as returns, which is much easier to understand by common investors than the Sharpe and Treynor risk premiums. In addition, Modigliani (1997) criticizes the Jensen index, claiming that differences in return cannot be compared when the risks are very different.

17.2.4 Information Ratio and Tracking Error

IR D

   TRACKINGERROR

(17.15)

Tracking error is defined as the difference between the return obtained within a period by a portfolio and that obtained by the benchmark. The variability of the tracking error is the deviation of said differences .¢TRACKING-ERROR /. The information ratio was developed by Sharpe (1994) and, assuming a normal distribution of probability, it reveals the probability that the tracking error will be negative; that is, the probability that the portfolio will yield lower returns than the benchmark. This means that the ratio is very useful for evaluating the manager’s ability to beat the market. However, Modigliani and Modigliani (1997) do not consider this a good performance measure as it does not consider total risk.

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17.2.5 PIRR Index

mi

The PIRR index came into being as an attempt to resolve the problem of the Sharpe ratio’s expression as a quotient. Indeed, if we assume that the distribution of returns on an asset is normal .; ¢/, which is the norm, the numerical value of the Sharpe ratio results from standardizing r0 , meaning that the application of this ratio to the ranking of funds depending on their performance, necessitates classifying them according to the probability that, within any subperiod of time, their returns will be lower than r0 . This means that this is not a good criterion for classification, because it is not logical that fund X be preferable to fund Y merely because the return on X has a lower probability of falling below the free risk rate. In fact, Y could have an average return much higher than that for X. All of this reasoning led Gómez-Bezares (1993) to question the risk quotient penalty that the Sharpe ratio imposes, and to study a possible linear penalty; this led him to opt for the penalty of the NPV,6 ultimately leading to the PPV7 PPV D NPV  tNPV ;

(17.16)

where t is the penalty parameter that will vary according to the risk aversion of the decision maker, and the NPV has been calculated at the risk-free interest rate, as we will later penalize it by subtracting t standard deviations. The PPV could be interpreted as the certainty equivalent of a risky investment with a determined average and standard deviation of the NPV.8 In accordance with this idea, Gómez-Bezares et al. (2004) penalize the IRR,9 giving rise to the PIRR.10 PIRR D IRR  tIRR

(17.17)

We may interpret the PIRR as the certainty equivalent return on a risky asset with a mean and a standard deviation of IRR; that is, a performance measure adjusted by risk. Returning to the previous nomenclature we have PIRR D   t:

(17.18)

The next step consists of assigning a value to t. To do this, we assume an efficient market in which r0 and  (affected by ¢  ) are considered to be equivalent (in the sense that they

D

mD

R*

m* PIRRD

mC

C

PIRRC r0

aC

a*

aD

s* = s D

sC

si

Fig. 17.8 Figure 17.8 shows a comparison between the S and PIRR indices

are similarly desirable for the group of investors): r0 D   t  . Calculating t, we have t D .  r0 / =  , which we observe is equal to the Sharpe ratio for the market portfolio .S /. Therefore, we reformulate PIRR PIRR D   S  :

(17.19)

In Fig. 17.8, we compare the S and PIRR indices. We represent the C and D portfolios and the market portfolio, and observe that the Sharpe ratio coincides with the tangent of the angle that forms the straight line that links r0 with one of the portfolios with the horizontal line corresponding to r0 ; therefore, it is deduced that the Sharpe ratio classifies assets based on a system of indifference straight lines that bundle in such a way that the asset located on the highest line is the best one. However, we can see that PIRR is equivalent to drawing straight lines parallel to the line that joins r0 and the market portfolio (PIRR is the point where each line crosses the vertical axis). Thus, we have in this case, a system of parallel indifference lines. Finally, it is easy to observe that S and PIRR lead to different classifications of portfolios: SC > SD > S I PIRRD > PIRRC > r0 (observe that PIRR D r0 ).

6

Net present value. Penalized present value. 8 Two other characteristics of this system are: if we assume that NPV follows a normal distribution, it is easy to understand t as the standardized value of the PPV; and the system is equivalent to using parallel indifference lines. 9 Internal rate of return. 10 Penalized internal rate of return. 7

17.2.6 PIRR Index for Beta The Treynor ratio (1965) poses similar problems to the Sharpe ratio; therefore, we will also impose a linear penalty of the risk, in this case

17 Portfolio Theory, CAPM and Performance Measures

PIRR for Beta D   t0 “;

277

(17.20)

where t0 is the penalty parameter, and we again assume that the certainty equivalent of the market portfolio is the riskfree rate: r0 D   t 0 ˇ  D   t 0 (given that “ D 1); calculating the value, we have: t 0 D   r0 , therefore PIRR for Beta D   .  r0 /“:

17.3 Empirical Analysis: Performance Rankings and Performance Persistence 17.3.1 Performance Rankings In this section, we put into practice what was discussed in the previous section. To do this, we use an example: We take weekly data from the returns on a sample of 198 equity investment funds over 6 years. We also take a stock market index to represent the market portfolio and the risk-free asset is approached by the one-month Treasury bill.11 With these data, we calculate the Sharpe and Treynor ratios for each year, as well as the Jensen and PIRR12 indices. In Table 17.1, we show the percentage of funds that outperform the market portfolio based on each of the performance measures considered, and for each of the 6 years analyzed.13 From Table 17.1 it is evident that, based on the S and PIRR indices, exactly the same percentage of investment funds outperform the stock market index. This result is not

The data have been taken from Vargas’ PhD Dissertation (2006). The remainder of the performance measures shown in the previous section are not applied in this example because, as has been stated previously, they give classifications of assets equivalent to those given by one of the measures considered in this example (except IR). 13 We do not show an individual analysis of the results obtained for each fund, for each performance measure, and for each period because of our purpose of abbreviating. 12

S PIRR J T

45% 45% 39% 40%

44% 44% 56% 55%

67% 67% 79% 79%

40% 40% 51% 51%

27% 27% 30% 30%

46% 46% 49% 49%

(17.21)

This system can be justified, as can the previous one, by parallel indifference straight lines. It is also apparent that this is equivalent to the Jensen index (1968, 1969). There are other performance proxies: conditional performance analysis, which considers return and risk expectations to be time-varying; style analysis, which attributes portfolio performance to a group of characteristics that define them; market timing analysis, which breaks down performance into two parts: (1) that which can be attributed to the manager’s ability to select securities (stock picking) and (2) that which can be attributed to the manager’s ability to anticipate the movements of the market (market timing); among others. However, this chapter does not focus on these analyses.

11

Table 17.1 Table 17.1 reports the percentage of funds that outperform the market portfolio based on different performance measures Year 1 Year 2 Year 3 Year 4 Year 5 Year 6

surprising because, as shown in Fig. 17.8, when an asset beats the market based on the Sharpe ratio, it also beats it based on the PIRR index. We observe very similar results between the J and T14 indices. In general, the results obtained are similar based on the four performance measures applied, which leads us to analyze the level of correlation that exists between the classifications of the funds, depending on their performance, derived from the application of the four measures. To do this, we first create a ranking of funds based on each measure, then we apply the Spearman’s rho and Kendall’s tau-b correlation coefficients to evaluate the level of similarity between these classifications.

17.3.1.1 Spearman’s Rho Coefficient P 6 di2 rs D 1  ; N.N 2  1/

(17.22)

where N is the number of funds (198, in this case), and di is the difference between the place occupied by fund i in two classifications. The Spearman’s rho correlation coefficient agrees with Pearson’s coefficient, but the former is an adaptation of the latter for ordinal data.

17.3.1.2 Kendall’s Tau-b Coefficient

D

2.P  Q/ ; N.N  1/

(17.23)

where P is the number of agreements between two classifications, and Q is the number of disagreements. This coefficient takes values between 1 and 1.1  £  1/, and has the disadvantage that, in the case of a draw, the coefficient cannot reach the values 1 and 1, meaning that in such a situation, it would be necessary 14 These should also coincide exactly, but they do not. However, in our example, the first and second years do not coincide and this is due to the special characteristics of one of the analyzed funds that give a negative beta during these 2 years.

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Table 17.2 Table 17.2 shows the results for the two correlation coefficients applied on the classifications of the funds considering different performance measures15

L. Ferruz et al.

J-PIRR

J-S

S-PIRR

S-T

PIRR-T

J-T

1st year

Kendall Spearman

0.946 0.994

0.835 0.960

0.862 0.969

0.883 0.926

0.790 0.900

0.796 0.898

2nd year

Kendall Spearman

0.830 0.926

0.754 0.890

0.849 0.958

0.858 0.952

0.728 0.892

0.795 0.890

3rd year

Kendall Spearman

0.793 0.874

0.702 0.826

0.832 0.946

0.754 0.815

0.648 0.771

0.801 0.918

4th year

Kendall Spearman

0.873 0.941

0.789 0.912

0.879 0.972

0.833 0.907

0.809 0.917

0.869 0.971

5th year

Kendall Spearman

0.986 0.999

0.816 0.945

0.821 0.947

0.977 0.997

0.814 0.945

0.813 0.943

6th year

Kendall Spearman

0.946 0.991

0.860 0.970

0.882 0.978

0.957 0.995

0.860 0.972

0.868 0.971

15

All of the coefficients in the table are significant at the 1% level.

to correct the expression of the coefficient. However, in the example that we have given, there are no draws. In Table 17.2, we show the results of the application of the two correlation coefficients on the classifications of the funds derived from the application of the four performance measures analyzed in the example. Table 17.2 shows a strong similarity between the order of preference determined from the different performance measures, as the coefficients obtained are very close to the unit and are significant at the 1% level. We can therefore see that the risk linear penalty achieves a methodological improvement of the Sharpe ratio that, in any case, generally maintains the rankings of the funds.

17.3.2.1 Malkiel’s z-statistic (1995) p Z D .Y  np/ = np .1  p/;

(17.24)

where Z represents a statistic that, approximately, follows the normal distribution (0,1); Y represents the number of winning portfolios in two consecutive periods; n equals WW C WL. This test shows the proportion of WW compared to WW C WL in such a way that p represents the probability that a winning portfolio over a period will continue to be a winner in the following period. We assign the value of 0.5 to p. If Z > 1:96, we reject the null hypothesis of nonpersistence at a significance level of 5%.

17.3.2 Performance Persistence

17.3.2.2 Brown and Goetzmann’s Statistic (1995)

In this section, we analyze the possible existence of persistence in the performance results achieved by our sample. To do this, we compare the performance achieved by the funds in two consecutive periods. We apply two methodologies. First is the non-parametric methodology, based on contingency tables and several statistics. It consists of comparing the performance rankings in two consecutive periods, distinguishing, in both periods, two portfolio subgroups (“winners” and “losers”) based on the criterion of the median: a fund is a “winner” if its performance is above the median, and a “loser” if it is below. Therefore, we classify the funds as WW if they are winners over two consecutive periods, and WL if they are winners in one period and then losers in a later period, and so on. In addition, the statistical tests of Malkiel (1995), Brown and Goetzmann (1995), and Kahn and Rudd (1995) are applied to determine the significance of the persistence.

ln.OR/ ; (17.25) ln.OR/ where OR is Brown and Goetzmann’s Odd’s ratio: WWLL OR D and where ln.OR/ D WLLW , q 1= 1 1 1 W W C =LL C =W L C =LW . Again, a value of Z > 1:96 would confirm a trend toward persistence in performance at the 5% level. ZD

17.3.2.3 Kahn and Rudd’s Statistic (1995) 2 n  n X X Oij  Eij  D ; Eij i D1 j D1 2

(17.26)

where Oij is the real frequency of the i-th row and the j-th column; Eij is the expected frequency of the i-th row and the j-th column.

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279

In the case of a 2 2 contingency table, the distribution shows one degree of freedom. A priori, the four expected frequencies would present the same figure (total number of funds, divided by four), meaning that we could reformulate the ¦2 statistic 2 D

.LL  N =4/2 .LW  N =4/2 .W W  N =4/2 C C N =4 N =4 N =4 C

.W L  N =4/2 ; N =4

(17.27)

where N is the total number of portfolios evaluated. If the chisquared has a critical value above 3.84, it would be indicative of performance persistence at a significance level of 5%. Following the above example, Tables 17.3 and 17.4 show the performance persistence results – performance is measured by the Sharpe ratio16 – obtained from application of the non-parametric methodology; in Table 17.3, we show the contingency table, and in Table 17.4, the results of the statistics.

16 We do not show persistence results obtained from applying other performance measures, due to space considerations and given that the aim of this example is not to analyze a sample, but rather to explain how the different methodologies of persistence analysis work.

Table 17.3 Table 17.3 reports the contingency table for the Sharpe ratio

Table 17.4 Table 17.4 shows the results of the statistics that check the persistence in performance measured by Sharpe ratio

WW LL LW WL No. funds

Z-Malkiel P OR Z- B & G P ¦2 - K & R P  

.

Table 17.5 Table 17.5 reports the results of performance persistence with the parametric methodology when performance is measured with the PIRR index

Pp.t C1/ D ˛ C ˇ Pp.t / C "p ;

(17.28)

where Pp.tC1/ and Pp.t/ represent the performance of portfolio p in the periods t C 1 and t, respectively. Positive beta values, together with a significant t statistic, would confirm the existence of performance persistence, while a negative estimation of said coefficient would indicate the existence of the opposite behavior. Continuing with the above example, in Table 17.5 we show the results of performance persistence–performance

Years 1, 2

Years 2, 3

Years 3, 4

Years 4, 5

Years 5, 6

Total

31 31 15 15 92

34 36 18 20 108

35 36 20 21 112

47 47 33 33 160

51 51 48 48 198

198 201 134 137 670

Years 1, 2

Years 2, 3

Years 3, 4

Years 4, 5

Years 5, 6

Total

2.3591 0.0183 4.2711 3.2641 0.0011 11.1304 0.0008

1.9052 0.0568 3.4000 3.0335 0.0024 9.6296 0.0019

1.8708 0.0614 3.0000 2.7998 0.0051 8.0714 0.0045

1.5652 0.1175 2.0285 2.2021 0.0277 4.9000 0.0269

0.3015 0.7630 1.1289 0.4263 0.6699 0.1818 0.6698

3.3328 0.0009 2.1679 4.9146 0.0000 24.5075 0.0000

/ indicates statistical significance at a level of 5% (1%).

No. Funds ’ T’ “ T“ R2 

We can observe from Tables 17.3 and 17.4 the presence of the phenomenon of performance persistence. In fact, based on Table 17.3, we can confirm that the number of funds that repeat as W or L in two consecutive years is greater than the number of funds that change their category (399 vs. 271). Furthermore, Table 17.4 shows the presence of the phenomenon of persistence given that statistics significant at the level of 1% are obtained for the whole sample. The other methodology that allows us to analyze the presence of the phenomenon of performance persistence is the parametric methodology, based on regression analysis in such a way that it determines, by means of ex post values, whether the relationship between performance in a certain period and the corresponding performance of an earlier period is statistically significant. To do this, the following regression is applied

Years 1, 2

Years 2, 3

Years 3, 4

Years 4, 5

Years 5, 6

92 0.000117 0.654 0.840 5.450 0.248

108 0.001 11.912 0.063 1.036 0.010

112 0.000377 2:743 0.602 6.098 0.253

160 0.000096 1.329 0.246 3.725 0.081

198 0.000382 4.171 0.449 4.385 0.089

indicates statistical significance at the 1% level.

280

being measured by the PIRR index – obtained through the application of the parametric methodology. This table confirms the presence of the phenomenon of performance persistence, except between the second and third years (non-significant beta).

17.4 Summary and Conclusions “Portfolio theory,” created by Markowitz, has reached a high level of development thanks to the improvement work carried out on it by several authors. This theory operates in a risk-based environment (the future return on an asset is known in terms of probability) and has the objective of determining the optimum portfolio for investors. To do this, it places all assets on the   ¢ map (mean and standard deviation of the random return variable, respectively), and to choose among them, it considers the investor’s preferences, represented by a system of indifference curves. Therefore, the optimum portfolio is that which gives tangency between an indifference curve and the efficient frontier (a group of portfolios with maximum return and minimum risk). Furthermore, “portfolio theory” is used as a theoretical foundation for an asset pricing model: the CAPM. This model, created by Sharpe, considers that the expected return on an asset depends on the market expected return and on the risk-free asset return, as well as on the systematic risk (non-diversifiable) of the asset. At equilibrium, according to CAPM, all assets adjust their values to offer the return that corresponds to them, meaning that it is easy to detect if an asset is underpriced or overpriced. A method of simplification of the relationships between securities also comes from Sharpe, who created the “market model,” according to which the relationships between the returns on assets are solely due to their common relationship with the market index return. This model allows us to distinguish between systematic and diversifiable risk (the latter risk is not remunerated, as it can be eliminated by the investor). The CAPM has been the object of various critiques, especially regarding its starting hypotheses that are alleged to distance this model from reality. However, it has been shown that its conclusions, to a sufficiently significant extent, are fulfilled. Moreover, CAPM has not been overcome by other more complex pricing models, and its usefulness for firms and in markets has been proven. The second part of this chapter refers to performance measures related to portfolio theory; therefore, classic indices are shown along with other, newer indices that attempt to overcome certain limitations present in the classic indices.

L. Ferruz et al.

Finally, we show an example in which we analyze the performance of a sample of equity investment funds using several performance measures. We contrast the similarity between the rankings of the funds (depending on their performance) that are obtained from several measures and, finally, we analyze the possible presence of the phenomenon of performance persistence throughout the period analyzed. Acknowledgement This chapter follows Gómez-Bezares (2006). The authors express their gratitude to Professor Stephen Brown for his helpful comments, which have greatly contributed to the improvement of this chapter.

References Black, F. 1993. “Beta and return.” Journal of Portfolio Management 20, 8–18. Black, F., M. C. Jensen and M. Scholes. 1972. “The capital asset pricing model: some empirical tests.” in Studies in the theory of capital markets, Jensen (Ed.). Praeger, New York. Brown, S. J. and W. N. Goetzmann. 1995. “Performance persistence.” Journal of Finance 50, 679–698. Bruner, R. F., K. M. Eades, R. S. Harris and R. C. Higgins. 1998. “Best practices in estimating the cost of capital: survey and synthesis.” Financial Practice and Education 8(1), 13–28. Chan, L. K. C. and J. Lakonishok. 1993. “Are the reports of beta’s death premature?” Journal of Portfolio Management, summer 19, 51–62. Daniel, K., M. Grinblatt, S. Titman and R. Wermers. 1997. “Measuring mutual fund performance with characteristic-based benchmarks.” Journal of Finance 52, 1035–1058. Daniel, K. and S. Titman. 1997. “Evidence on the characteristics of cross-sectional variation in stock returns.” Journal of Finance 52, 1–33. Daniel, K., S. Titman and K. C. J. Wei. 2001. “Explaining the crosssection of stock returns in Japan: factors or characteristics?” Journal of Finance 56, 743–766. Davis, J. L., E. G. Fama and K. R. French. 2000. “Characteristics, covariances and average returns: 1929 to 1997.” Journal of Finance 55(1), 389–406. Dimson, E. and M. Mussavian. 1999. “Three centuries of asset pricing.” Journal of Banking and Finance 23(12), 1745–1769. Eftekhari, B., C. S. Pedersen and S. E. Satchell. 2000. “On the volatility of measures of financial risk: an investigation using returns from European markets.” The European Journal of Finance 6, 18–38. Fama, E. F. and K. R. French. 1992. “The cross-section of expected stock returns.” Journal of Finance 47(2), 427–465. Fama, E. F. and K. R. French. 1993. “Common risk factors in the returns on stocks and bonds.” Journal of Financial Economics 33, 3–56. Fama, E. F. and K. R. French. 1996a. “The CAPM is wanted, dead or alive.” Journal of Finance 51(5), 1947–1958. Fama, E. F. and K. R. French. 1996b. “Multifactor explanations of asset pricing anomalies.” Journal of Finance 51(1), 55–84. Fama, E. F. and J. D. MacBeth. 1973. “Risk, return and equilibrium: empirical tests.” Journal of Political Economy 81(3), 607–636. Ghysels, E. 1998. “On stable factor structures in the pricing of risk: Do time-varying betas help or hurt?” Journal of Finance 53, 549–573. Gómez-Bezares, F. 1993. “Penalized present value: net present value penalization with normal and beta distributions.” in: Capital budgeting under uncertainty, Aggarwal (Ed.). Prentice-Hall, Englewood Cliffs, New Jersey, pp 91–102.

17 Portfolio Theory, CAPM and Performance Measures Gómez-Bezares, F. 2006. Gestión de carteras, Desclée de Brouwer, Bilbao. Gómez-Bezares, F., J. A. Madariaga and J. Santibánez. 2004. “Performance ajustada al riesgo.” Análisis Financiero 93, 6–17. Grinblatt, M. and S. Titman. 1993. “Performance measurement without benchmarks: an examination of mutual fund returns.” Journal of Business 66(1), 47–68. Grundy, K. and B. G. Malkiel. 1996. “Reports of beta’s death have been greatly exaggerated.” Journal of Portfolio Management 22(3), 36–44. Jagannathan, R. and Z. Wang. 1996. “The conditional CAPM and the cross-section of expected returns.” Journal of Finance 51(1), 3–53. Jegadeesh, N. and S. Titman. 2001. “Profitability of momentum strategies: an evaluation of alternative explanations.” Journal of Finance 56, 699–720 Jensen, M. C. 1968. “The performance of mutual funds in the period 1945–1964.” Journal of Finance 39, 389–416. Jensen, M. C. 1969. “Risk, the pricing of capital assets, and the evaluation of investment portfolios.” Journal of Business 42(2), 167–247. Kahn, R. N. and A. Rudd. 1995. “Does historical performance predict future performance?” Financial Analysts Journal 51, 43–52. Kim, D. 1995. “The errors in the variables problem in the cross-section of expected stock returns.” Journal of Finance 50(5) 1605–1634. Kothari, S. P., J. Shanken and R. G. Sloan. 1992. Another look at the cross-section of expected stock returns, Working Paper, Bradley Policy Research Center, University of Rochester, New York. Lewellen, J. and S. Nagel. 2006. “The conditional CAPM does not explain asset-pricing anomalies.” Journal of Financial Economics 82, 289–314. Lintner, J. 1965. “The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets.” Review of Economics and Statistics 47(1), 13–37. Liu, W. 2006. “A liquidity-augmented capital asset pricing model.” Journal of Financial Economics 82, 631–671. MacKinlay, A. C. 1995. “Multifactor models do not explain deviations from the CAPM.” Journal of Financial Economics 38, 3–28. Malkiel, B. G. 1995. “Returns from investing in equity mutual funds 1971 to 1991.” Journal of Finance 50, 549–572. Malkiel, B. G. and Y. Xu. 1997. “Risk and return revisited.” Journal of Portfolio Management, 23, 9–14. Markowitz, H. 1952. “Portfolio selection.” Journal of Finance 7(1), 77–91. Markowitz, H. 1959. Portfolio selection: efficient diversification of investments, Wiley, New York.

281 Markowitz, H. 1987. Mean-variance analysis in portfolio choice and capital markets, Basil Blackwell, Oxford. Merton, R. C. 1972. “An analytic derivation of the efficient portfolio frontier.” Journal of Financial and Quantitative Analysis 7, 1851–1872. Modigliani L. 1997. “Don’t pick your managers by their alphas,” U.S. Investment Research, U.S. Strategy, Morgan Stanley Dean Witter, June. Modigliani, F. and L. Modigliani. 1997. “Risk-adjusted performance.” Journal of Portfolio Management 23(2), 45–54. Pedersen, C. S. and S. E. Satchell. 2002. “On the foundations of performance measures under asymmetric returns.” Quantitative Finance 3, 217–223. Roll, R. 1977. “A critique of the asset pricing theory’s tests.” Journal of Financial Economics 4(2) 129–176. Roll, R. and S. A. Ross. 1994. “On the cross-sectional regression between expected returns and betas.” Journal of Finance 49, pp. 101–121. Sharpe, W. F. 1961. “Portfolio analysis based on a simplified model of the relationships among securities.” Ph.D. Dissertation, University of California, Los Angeles. Sharpe, W. F. 1963. “A simplified model for portfolio analysis.” Management Science 9(2), 277–293. Sharpe, W. F. 1964. “Capital asset prices: a theory of market equilibrium under conditions of risk.” Journal of Finance 19(3), 425–442. Sharpe, W. F. 1966. “Mutual fund performance.” Journal of Business 39, 119–138. Sharpe, W. F. 1992. “Asset allocation: management style and performance measurement.” Journal of Portfolio Management 18, 7–19. Sharpe, W. F. 1994. “The Sharpe ratio.” Journal of Portfolio Management 21, 49–58. Sortino, F. and L. Price. 1994. “Performance measurement in a downside risk framework.” Journal of Investing 3, 59–65. Tobin, J. 1958. “Liquidity preference as behavior towards risk.” The Review of Economic Studies 67, 65–86. Treynor, J. L. 1965. “How to rate management of investment funds.” Harvard Business Review 43, 63–75. Van Horne, J. C. 1989. Financial management and policy, PrenticeHall, Englewood Cliffs, New Jersey. Vargas, M. 2006. “Fondos de inversión españoles: análisis empírico de eficiencia y persistencia en la gestión.” PhD Dissertation, University of Zaragoza, Spain.

Chapter 18

Intertemporal Equilibrium Models, Portfolio Theory and the Capital Asset Pricing Model Stephen J. Brown

Abstract Intertemporal equilibrium models of the kind discussed in (Asset pricing, Princeton University Press, Princeton, 2001) have become the standard paradigm in most advanced asset pricing courses. The purpose of this chapter is to explain the relationship between this paradigm and the portfolio theory paradigm common in most of the prior asset pricing literature. We show that these paradigms are merely different ways of looking at the same economic phenomena, and that insights can be gained from each approach. Keywords CAPM r Hansen Jagannathan bounds r Intertemporal equilibrium r Portfolio theory r Sharpe ratio r Stochastic discount factor

18.1 Introduction John Cochrane’s recent textbook (Cochrane 2001) offers a compelling paradigm for asset pricing, starting with a consumption-based intertemporal equilibrium model, bypassing the normal development via single period portfolio theory that leads to the traditional Capital Asset Pricing Model (CAPM). This is a very elegant and persuasive treatment that generalizes CAPM to a multiperiod economy, and has direct implications for derivative pricing. Finally, and perhaps most importantly, well trained economists find this approach provides an intuitive and persuasive access to the central results of the financial economics literature. As a result, this text is required reading in most advanced level asset pricing courses. While this approach to the asset pricing literature is the very model of clarity and simplicity, the text does not discuss the connection to the earlier portfolio-theory based literature. As Cochrane notes “I skip very lightly over many

parts of asset pricing theory that have faded from current applications. Some examples are portfolio separation theorems, properties of various distributions, or asymptotic APT. While portfolio theory is still interesting and useful, it is no longer a cornerstone of pricing. Rather than use portfolio theory to find a demand curve for assets, which intersected with a supply curve gives prices, we now go to prices directly. One can then find optimal portfolios, but it is a side issue for the asset pricing question” (p. xvi). While this statement is correct, experience has shown that it gives the student the false but nevertheless comforting perception that the text subsumes all relevant developments in financial economics, and the necessary corollary that the entire literature of financial economics can be safely ignored. This is just as well, because the language and terminology of the text are so much at variance with the established portfolio-theory based literature that most beginning students find this earlier literature very difficult to understand. Our purpose is to show how the intertemporal equilibrium asset pricing model relates to the established portfolio-theory literature. An interesting byproduct of this reconciliation is the important role of the maximized Sharpe ratio. It was this insight that led in the first instance to the development of the CAPM in Sharpe (1964).

18.2 Intertemporal Equilibrium Models Hansen and Jagannathan (1991) have developed a very simple and elegant approach to developing an intertemporal equilibrium model that can relate to observed asset returns. This approach builds on earlier work by Breeden (1979) and Rubinstein (1976). The investor’s problem is to allocate wealth in order to maximize the utility of consuming both now and in the future. In other words, the investor is faced with an intertemporal choice problem of the form:

S.J. Brown () NYU Stern School of Business, New York University, New York, NY, USA e-mail: [email protected]

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_18, 

2 Max Et 4

1 X

3 ı j U.ct Cj /5

j D0

283

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S.J. Brown

where ct Cj represents future consumption, and ı is a subjective discount factor applied to future consumption. For a given budget constraint, the first order conditions for this problem imply that   U 0 .ct / D ı j Et .1 C Ri;t Cj /U 0 .ct Cj / for all assets i and periods j into the future.1 Dividing through by the marginal utility of consumption today, we have the important result that   1 D Et .1 C Ri;t Cj /mt;j U 0 .c

a vector of factors some combination of which can proxy for consumption growth. A third idea originally due to Hansen and Jagannathan (1991) is that since the stochastic discount factor prices all financial claims, we might be just as well off inferring the stochastic discount factor from the observed set of asset returns. This important insight allows us to interpret the stochastic discount factor in terms of the mean-variance efficient portfolio, an interpretation that yields the standard CAPM as a direct implication. Intertemporal equilibrium imposes an important constraint on the cross-section of expected returns on different securities. Starting from the basic formula that holds for all securities i :

/

where mt;j D ı j U 0 .ct Cj is the intertemporal marginal rate t/ of substitution. It also has the interesting interpretation of being a stochastic discount factor (sometimes also referred to as a pricing kernel) because it takes an asset with uncertain per dollar future payoff $.1 C Ri;t Cj / back to the present to be valued at one dollar. If there is a riskless asset in this econ  .1 C RF;t Cj / mt;j D , then 1 D E omy with return R F;t Cj t     .1 C RF;t Cj / Et mt;j or Et mt;j D .1CR1F;t Cj / , so that the expected value of the stochastic discount factor is equal to the discount factor used when the future payment is in fact without any risk. For the subsequent discussion we will drop the time subscripts.

18.3 Relationship to Observed Security Returns Cochrane (2001) argues that this intertemporal equilibrium model provides a straightforward way to value all financial claims. Security prices are determined so that the expected value of the growth of a dollar invested discounted by the stochastic discount factor equals the value of a dollar today. This is a direct implication of the equation 1 D  Et .1 C Ri;t Cj / mt;j . The difficulty is however that the stochastic discount factor m is not observable. There are three general approaches to this problem. The first is to specify m directly through assumptions made about utility and using measures of consumption. The chief difficulty associated with this approach is obtaining accurate and timely measures of aggregate consumption c. An alternative approach is to use

1 D E Œ .1 C Ri /m D E Œ .1 C i C . Ri  i /m D .1 C i / EŒm C E Œ. Ri  i /m 1 If there is a risk-free rate EŒm D 1CR , this implies the asF set pricing relation i  RF D .1 C RF / E Œ.Ri  i /m , which says that the risk premium is negatively proportional to the covariance between the asset return and the stochastic discount factor. In difficult economic times, consumption is depressed and the intertemporal rate of substitution m is high. A negative covariance between asset returns and m is therefore a source of risk to investors. As we mention above, we cannot directly observe m. However, since the same m prices all assets in the economy we should be able to infer it from asset prices and returns. Suppose there are N assets in the economy and we infer (the unobserved) m by a hypothetical regression on the set of returns in the economy, so that

m D m C

N X

.Rj  j / j

j D1

where the j are the hypothetical regression coefficients. The asset pricing relationship can then be written: i  RF D .1 C RF / E Œ. Ri  i /m D .1 " C RF / m E Œ. Ri  i / #.1 C RF / N P .Rj  j / j E . Ri  i / j D1

D

N P

E. Ri  i / .Rj  j / Zj

j D1 1 Cochrane [2001] along with many economists define return as the growth in value of one dollar invested, or the gross return 1 C Ri;tCj where Ri;tCj is the “net return” or total return (income plus capital gain) in period t C j for security i . This leads to some confusion, as Cochrane uses the symbol R to denote gross return, whereas much of the finance literature uses the same symbol to signify net return. We use the net return convention most commonly employed in the financial economics literature.

where the coefficients Zj D .1 C RF /j . This equation must hold for all assets, so we have the system of equations 1  RF D Z1 12 C Z2 12 C Z3 13 C : : : C ZN 1N 2  RF D Z1 21 C Z2 22 C Z3 23 C : : : C ZN 2N

18 Intertemporal Equilibrium Models, Portfolio Theory and the Capital Asset Pricing Model

3  RF D Z1 31 C Z2 32 C Z3 32 C : : : C ZN 3N :: : N  RF D Z1 N1 C Z2 N 2 C Z3 N 3 C : : : C ZN N2 which the reader will recognize as Equation (9.1) used in Elton et al. (2007) to identify the mean variance efficient portfolio with riskless lending and borrowing.2 From this fact we conclude immediately that the hypothetical regression coefficients j are proportional to mean variance efficient portfolio weights, and hence that the best estimate of the stochastic discount factor m can be written as a linear function of the return on a mean variance efficient portfolio, m D a C b RMV .

18.4 Intertemporal Equilibrium and the Capital Asset Pricing Model If we further identify the mean variance efficient portfolio MV as the market portfolio, then the asset pricing relation above immediately implies the standard CAPM. To see this, note that using the returns-based proxy m for the discount factor m   i  RF D .1 C RF / E . Ri  i /m and substituting in for m we find that expected excess return is given by

285

There is an important intuition here that establishes that the particular mean variance efficient portfolio is the market portfolio. From the interpretation of the portfolio as resulting from regressing the (unknown) stochastic discount factor m on the set of observed security returns, the variability ex2 D plained by the observed return portfolio, m2  D b 2 MV

.MV RF /2 2 , is maximized. This implies that the Sharpe ratio .1CRF /2 MV  RF given as MVMV D .1 C RF /m D m is also maximized, m which further establishes that RMV is the return on the market

portfolio. This last equation is very important because it establishes a nexus between the financial markets on the one hand and consumer preferences on the other. The fact that we can represent the maximized Sharpe ratio as the ratio of the standard deviation of the stochastic discount factor to its mean presents many financial economists with a serious problem. The average risk premium of the market measured in units of risk is far too high to be explained by any consumptionbased representation of the stochastic discount factor. This is referred to as the “equity premium puzzle.” As Cochrane (2001) p.24 notes, the Sharpe ratio measured in real (not nominal) terms has been about 0.5 on the basis of the past 50 years of data for the United States. Assuming the time  separable power utility function, the ratio m is approxim mately the risk-aversion coefficient times the standard deviation of the log of consumption. Since the rate of change in consumption is considerably less than the variance of market returns, this implies that investors are very risk averse, with a coefficient of risk aversion at least 50. A degree of risk aversion this large is difficult to motivate.

i  RF D .1 C RF / a E Œ. Ri  i / b .1 C RF / E Œ. Ri  i /RM V : The first term is zero by definition. The expectation in the second term is the covariance between the return on security i and the return on the mean variance efficient portfolio, which is beta times the variance of this portfolio return: 2 i  RF D b .1 C RF / M V ˇi

which must hold for all assets, including the mean variance efficient portfolio with beta equal to one, so that b D MV RF and the CAPM follows immediately,  .1CR / 2 F

MV

i  RF D ˇi .M V  RF /:

2 Note that this system of equations corresponds to the first order conditions   Rf D †z for a maximum value of the Sharpe ratio p  D .x 0 Rf /= x 0 †x, where x is the vector of portfolio weights, † is the covariance matrix of returns, and z is proportional to the portfolio weight vector x that maximizes the Sharpe ratio.

18.5 Hansen Jagannathan Bounds The relationship between the Sharpe ratio and the moments of the stochastic discount factor gives rise to another fascinating insight due to Hansen and Jagannathan (1991). Suppose we represent the stochastic discount factor m D a C b RMV . We have shown that this choice of m prices all assets, E Œ.1 C R/m D 1. Consider another discount factor m D a C b RMV C ", where " is uncorrelated with returns R on every single asset in the economy, and has zero expectation. Then this discount factor will also price all assets E Œ.1 C R/m D 1. This means that there are many possible discount factors, with m D m and m2 D m2  C "2 > m2  . However the choice of m D m minimizes the volatility of the stochastic discount factor and is most preferred by investors. Hence, we have the interesting  RF F R bounds: mm m D MVMV MV , which are referred m to as the “Hansen-Jagannathan bounds.” The inequality on the left hand side refers to the fact that consumption risk will increase if there is a source of risk " that cannot be hedged

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by the set of assets represented by returns R. Imperfections in the capital markets make the world a riskier place than it need to be. The inequality on the right hand side reflects the reality that limits to diversification through short sale restrictions or other factors limit the opportunities available to investors. Another useful interpretation of these contrasting inequalities is that the problem of choosing a returns-based discount function by minimizing the volatility of the discount factor m is equivalent to determining a portfolio that maximizes the portfolio Sharpe ratio.

18.6 Are Stochastic Discount Factors Positive? Unfortunately, this interpretation of the stochastic discount factor leads to a distinctly unattractive implication. Since MV RF 1 1 D a C bMV , a D 1CR C .1CR MV m D 1CR / 2 f f F

 D b .1 C RF / LPMMV;c ˇMV;c;i

where LPM MV;c D EŒ.RMV  c/2 jRMV < c is referred to as the lower partial moment of the return on the mean variance efficient portfolio, a measure of downside risk, and where  D E Œ.Ri  i /.RMV  c/ =LPM MV;c is referred to ˇMV;c;i as the lower partial moment beta, the contribution of security i to the downside risk of the market. As before, we have immediately a linear asset pricing model similar to the standard  , CAPM, except that the lower partial moment beta, ˇMV;c;i replaces the more familiar beta, ˇMV . This generalized asset pricing model was first derived by Bawa and Lindenberg (1997) who showed that it corresponded with an equilibrium model where agents have utility functions for wealth displaying diminishing absolute risk aversion. They further show that if returns are multivariate Normal or Student-t,  risk measure collapses to ˇMV and the standard the ˇMV;c;i CAPM result follows.

MV

MV RF 1 and therefore m D 1CR  .1CR .RMV  MV /, so that 2 f F /MV the implied discount rate is negative whenever the market 2

return exceeds its mean by an amount equal to MVMV RF , the standard deviation of the market return divided by the Sharpe ratio. Using the example given above, if the Sharpe ratio is about 0.5, the discount factor will be negative whenever the market return is two standard deviations above its mean. It is intuitive that the discount factor should fall as market return increases; after all we are assuming that consumers have diminishing marginal utility. It is not intuitive that the discount factor should be negative, and in fact it is easy to show that there are arbitrage opportunities that arise when the representative investor is willing to throw his or her money away in these circumstances. As a practical matter, the empirical representation of the discount factor in terms of the return on the market portfolio is rarely negative, and so a constraint on the discount factor to ensure that it is nonnegative should not lead to any major implications for asset prices. Hansen and Jagannathan suggest such a restriction but find it makes little difference in the case of equity markets. This assertion however depends strongly on the assumption that market returns are distributed normally. If returns are positively skewed, then the positive discount factor restriction may have a greater impact. It is worth pursuing a little the implications of a positive discount factor restriction. We can represent this positive factor as mC D a C b.RMV  c/ , where .RMV  c/ is the payoff of a put on the market index with exercise price 1 C c. As before, we have   i  RF D .1 C RF / E . Ri  i /mC   D .1 C RF / a E . Ri  Ri / b .1 C RF / E Œ. Ri  i /.RM V  c/

18.7 Conclusion A straightforward maximization of expected utility subject to a budget constraint leads to the CAPM via the maximized Sharpe ratio. This standard portfolio theory optimization problem has a dual representation involving shadow prices. This essential duality is illustrated by the Hansen Jagannathan bounds, which show that the maximized Sharpe ratio is a lower bound on the ability of investors to hedge consumption risk. Many economists find this representation to be more intuitive, although the pricing implications are the same. However, it is difficult and in many cases confusing to switch from one way of considering the problem to another. The purpose of this chapter is to clarify the relationship between the two approaches and to show that they are just two ways of looking at the same problem. Careful consideration suggests that the intertemporal equilibrium model cannot really avoid problems associated with the properties of various distributions, or issues that gave rise to consideration of an asymptotic APT. The analysis assumes that we can observe the mean-variance efficient portfolio or proxy for it on the basis of a meanvariance analysis based on the sample covariance matrix of returns. Indeed, whether a given portfolio is mean-variance efficient is a central question of modern portfolio theory.3 The intertemporal model additionally assumes that security returns are expressed in real terms, when observed returns are generally expressed in nominal terms, necessitating a separate adjustment for price level changes. As we see, the need to generalize the CAPM to account for the properties 3

See for example Gibbons et al. (1989).

18 Intertemporal Equilibrium Models, Portfolio Theory and the Capital Asset Pricing Model

of various distributions has an interesting parallel in the need to constrain the stochastic discount factor to be positive. In short, there is much to be gained from considering the asset pricing problem both from a stochastic discount factor and from a portfolio theory framework.

References Bawa, V. and E. Lindenberg. 1977. “Capital market equilibrium in a mean-lower partial moment framework.” Journal of Financial Economics 5, 189–200. Breeden, D. 1979. “An intertemporal asset pricing model with stochastic consumption and investment opportunities.” Journal of Financial Economics 7, 265–296.

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Cochrane, J. 2001. Asset pricing, Princeton University Press, Princeton. Elton, E., M. Gruber, S. Brown and W. Goetzmann. 2007. Modern portfolio theory and investment analysis, Wiley, Hoboken. Gibbons, M., S. Ross and J. Shanken. 1989. “A test of the efficiency of a given portfolio.” Econometrica 57, 1121–1152. Hansen, L. P. and R. Jagannathan. 1991. “Implications of security market data for models of dynamic economies.” Journal of Political Economy 99, 225–262. Rubinstein, M. 1976. “The valuation of uncertain income streams and the pricing of options.” Bell Journal of Economics 7, 407–425. Sharpe, W. 1964. “Capital asset prices: a theory of market equilibrium under conditions of risk.” Journal of Finance 19(3) 425–442.

Chapter 19

Persistence, Predictability, and Portfolio Planning Michael J. Brennan and Yihong Xia

Abstract We use a model of stock price behavior in which the expected rate of return on stocks follows an OrnsteinUhlenbeck process to show that levels of return predictability that cause large variation in valuation ratios and offer significant benefits to dynamic portfolio strategies are hard to detect or measure by standard regression techniques, and that the R2 from standard short run predictive regressions carry little information about either long run predictability or the value of dynamic portfolio strategies. We propose a new approach to portfolio planning that uses forward-looking estimates of long run expected rates of return from dividend discount models. We show how such long run expected rates of return can be used to estimate the instantaneous expected rate of return under the assumption that the latter follows an Ornstein-Uhlenbeck process. Simulation results using four different estimates of long run rates of return on U.S. common stocks suggest that this approach may be valuable for long horizon investors. Keywords Portfolio planning Dividend discount model

r

Return predictability

r

19.1 Introduction Hakansson (1970) and Merton (1971) extended the principles of dynamic portfolio optimization to situations of stochastic (time-varying) investment opportunities over 30 years ago, but interest in this topic languished until the 1990s under the soothing influence of the efficient markets paradigm and the belief that the investment opportunity set, if not exactly constant, was sufficiently close to constant that myopic portfolio strategies that ignored time variation in investment opportunities could be safely employed. M.J. Brennan () University of California, Los Angeles, CA, USA e-mail: [email protected] Y. Xia Wharton School, Philadelphia, PA, USA

However, the debate about excess stock price volatility,1 the evidence of mean reversion in stock prices,2 and of the predictive power of instruments such as the dividend yield, the book-to market ratio, the term spread and the short term interest rate,3 have revived interest in dynamic portfolio theory in recent years. This interest has been further stimulated by the behavior of stock prices in the late 1990s, which drew attention to the implications of valuation ratios such as the market dividend yield and the book to market ratio for expected future returns of equity securities,4 and therefore for portfolio strategies. The dynamic portfolio models that have been developed and calibrated to U.S. stock returns by Brennan et al. (1997), Barberis (2000), Campbell and Viceira (2001), and Xia (2001), among others, suggest that time variation in investment opportunities may have significant effects on optimal portfolio strategies for long-lived investors, and that failure to take account of time variation in expected returns may carry significant costs. The analysis of Kandel and Stambaugh (1996) suggests that short horizon investors also can benefit from predictability, even if it is highly uncertain. These models all rely on statistical relations between (excess) returns on stocks and (usually one of) a set of predictor variables or instruments such as those mentioned above. However, doubt has been cast on the usefulness of dynamic portfolio models by the weakness and instability of the estimated relations between stock returns and the instruments. It has also been suggested that in sample “predictability” may be an illusion, the result of uncertainty and learning about the parameters of the underlying stochastic process for dividends.5 Consistent with this suggestion, Goetzmann and Jorion (1993) argue that long run predictive regressions lack

1

See, for example, Shiller (1981), LeRoy and Porter (1981), Kleidon (1986), Marsh and Merton (1986). 2 See, for example, Fama and French (1988b). 3 See, for example, Keim and Stambaugh (1986), Fama and French (1988a) and so forth. 4 See, for example, Lewellen (2004). 5 See Brav and Heaton (2002) and Lewellen and Shanken (2002).

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_19, 

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power in small samples,6 while Bossaerts and Hillion (1999) find no evidence of out-of-sample excess return predictability at the monthly frequency in 14 countries using the standard instruments such as lagged returns, interest rates, and dividend yields. Goyal and Welch (2003) demonstrate that the dividend yield has no out of sample predictive power for annual stock returns during the period 1926–2002. They report that the dividend yield “is primarily forecasting future market returns over horizons greater than 10 years. It does not forecast market returns (or dividend growth) over horizons less than 5 years.” In a more recent paper, Goyal and Welch (2004) show similar results for a wide range of variables7 and conclude that “for all practical purposes, the equity premium has not been predictable, and any belief about whether the stock market is now too high or too low has to be based on a theoretical prior.” These results of Goyal and Welch seem to be consistent with the finding of Philips et al. (1996) that investment managers who followed Tactical Asset Allocation strategies during the period 1988–1994 failed to outperform static strategies. On the other hand, there is continuing evidence of predictability in equity returns, especially over long horizons. Fama and French (1988b), Poterba and Summers (1988), Lo and MacKinlay (1988), Cochrane (1999), and Campbell and Yogo (2003) all find evidence of temporary components in stock prices that becomes stronger at long horizons. Informal evidence of long run predictability is further apparent from the long swings and upward trend in valuation ratios during the twenteeth century – swings that do not appear to have been mirrored in changing dividend growth rates.8 The apparent paradox of strong evidence of long run predictability and only weak, if any, evidence of out-of-sample short run predictability is resolved by recognizing the importance of the time series behavior of the predictor variable that determines the time variation in the conditional mean return. This time series behavior has implications both for the estimation of the relation and for the economic importance of a given short run relation. In this paper we demonstrate, first, that levels of return predictability that are of first order economic significance for long-term investors are likely to be hard to detect from short run returns using the standard econometric methods that have been employed, even though they imply major deviations of valuation ratios from their long run “equilibrium” levels. In 6

Campbell (2001) shows that long-horizon regression tests have serious size distortions when asymptotic critical values are used, but some versions of such tests have power advantages remaining after size is corrected. 7 Campbell and Thompson (2004) show that the findings of Goyal and Welch (2004) are no longer true if sensible restrictions are imposed on the signs of predictive coefficients and return forecasts. 8 Campbell and Shiller (1998, 2001) and Fama and French (2002), among others, find that dividend yields have little or no forecast power for subsequent dividend growth in the years since 1870.

M.J. Brennan and Y. Xia

showing that (in-sample) predictability may be hard to find when it exists, our paper complements the analyses of Goyal and Welch (2003) and Bossaerts and Hillion (1999), which show that predictability may appear to exist for certain instruments even when no true (out-of-sample) predictability exists. Second, we show how long run measures of expected return that are anchored in fundamentals and are derived by comparing current stock prices with expected future dividends may be used to construct portfolio policies for long run investors. We provide time series estimates of instantaneous expected returns derived from these Dividend Discount Model measures of long run expected returns, and provide evidence that these measures would have been useful to long horizon investors.

19.2 Detecting and Exploiting Predictability In this section we use the simplest model of stock return predictability, in which the proportional drift of the stock price follows a standard Ornstein-Uhlenbeck process, to analyze two issues. The first is the likelihood of being able to detect return predictability using standard approaches if the stock price drift is perfectly observable. The second issue that we address is the value of following an optimal strategy that takes account of the time variation in the drift parameter, relative to the value of an unconditional strategy that treats the drift as constant. We conclude that levels of timevariation in expected returns that are very important for investors with long horizons (20 years or more) may well be undetectable using the standard statistical approaches and, even if detectable, are likely to be estimated with considerable imprecision.

19.2.1 Implications of a Model of Stock Price Behavior Our basic framework relies on the following continuous time model of the dynamics of the stock price with dividends reinvested, which we denote by P : dP D .˛ C ˇ/dt C P dzP ; P d D .  /dt C  dz ;

(19.1) (19.2)

where dzP and dz are correlated standard Brownian motions with the correlation P . In this formulation, t is to be thought of as a perfect signal of the drift of the asset price process, ˛ Cˇt ; however, for the most part we shall assume that ˛ D 0, and ˇ D 1; then  can be interpreted as the drift of the stock price.

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The discrete time equivalent of the Ornstein-Uhlenbeck process (19.2) is the AR(1) process: t   D e  .t   / C t :

(19.3)

 

where is the time interval between observations of . Let R.t; t C / denote the cumulative return on the stock over the interval Œt; t C  , then

t





The conditional variance of the -period return, R.t; t C /, which we denote by V ./  Var.R.t; t C /jt /, depends on the volatility of the return process P , the volatility of the signal  , the horizon , and the mean reversion parameter , and can be written as: 

1  e  1  e  C 2  2    2ˇP  P 1e  : C    2

 C P2  (19.5)

Since the variance of s conditional on t .s t/, 2

 2 .s jt / D 2 .1  e 2.st / /, is decreasing in , one might conjecture that the conditional variance of the cumulative return V ./ would also be decreasing in the mean reversion intensity . The following expression for the derivative of the variance with respect to  shows that this conjecture is true only if the correlation between innovations in P and innovations in  is positive. When the correlation is negative (as we shall typically assume it is), the sign of the derivative is indeterminate:  2 ˇ 2 1  e 2 1  e  @V D  3 2  4 C @    C

 1 C e  



if P 0 if P < 0

(19.6)

where

t

ˇ 2 2

 2ˇP P 1  e  e   C 2   <0 indeterminate

R.t; t C / D a C b t C ";

1 1e D ˛ C ˇ  P2  C ˇ.t  / 2  Z t C ˇ C .1  e .t C s/ /dz .s/  t Z t C C P dzP .s/: (19.4)

V ./ D



To analyze the role of t as a predictor of the cumulative return, R.t; t C /, rewrite Equation (19.4) as

R.t; t C /  ln P .t C /  ln P .t/

Z t C 1 ˛ C ˇ.s/  P2 ds D 2 t Z t C C P dzP ;

2e2  1 C e 2 2

2e  2 C 2e  

1 1  e  ; a  ˛ C ˇ  P2   ˇ 2 

(19.7)

1  e  ; (19.8)  Z t C Z ˇ t C " .1  e .t C s/ /dz .s/ C P dzP .s/;  t t b  ˇ

and " is independent of t . Note first that b increases with  as long as ˇ > 0 and that b < ˇ 8 > 0. In addition, b ! ˇ as the horizon goes to infinity, so that an innovation in  has a permanent effect on the stock price. As  ! 0; b ! 0 and b = !ˇ. Second b is decreasing in the intensity of the mean reversion parameter . As  ! 1 the process for  tends to a constant; deviations of  from  are transient, and as a result, b ! 0. On the other hand, as  ! 0, the process for  approaches a random walk, and b ! ˇ , or b = !ˇ so that an increase in  increases the expected rate of return in all future periods by the same amount. Thus, as has been pointed out by Cochrane (1999) and others, the degree of persistence in  has major implications for the predictive role of  in long horizon regressions, and two signals that differ in their persistence parameter  may have the same predictive importance for returns at one horizon, but quite different importance at a different horizon. This suggests, as we shall see, that it may be dangerous to attempt to assess the relevance of predictive variables by examining their power at a single horizon. A simple measure of the predictive power of a signal t is the R2 from the regression (19.6) at different return intervals . The unconditional distribution of the predictive variable, 2

t , is normal with mean  and variance 2 , and the unconditional variance of R.t; t C / is simply b2 Var.t / C Var."/. The theoretical R2 can be expressed explicitly in terms of the model parameters as:

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R2

b2 Var.t / ; b2 Var.t / C Var."/

D   1e  2 

C

2 

h    2 1e C

 1e 2 

1e 2 2

The expression for R2 depends on four parameters: the cumulative return interval , the mean reversion intensity of P the signal , the noise-to-signal ratio ˇ , and the correlation  between the innovations to the return and to the signal P . While all four parameters are important in determining the magnitude of the theoretical regression R2 , the dependence of R2 on  has received special attention in the literature. For example, Fama and Bliss (1987), Cochrane (1997, 1999), and Campbell (2001) have all explored the relation between R2 and  empirically finding that R2 normally increases with the return interval . While Campbell et al. (1997) derive an approximate relation between a two-period and a one-period R2 in a discrete model, our simple continuous time setup enables us to examine the relation between R2 and all relevant parameters, including , in closed form. To analyze formally the relation between R2 and , differentiate R2 with respect to : 2 ˇ 2 1e

@.R2 /  2 2 D  2 2 @ b Var.t / C Var."/ " 2 ˇ 2

.2e C e 2  1/ 2 CP2 .2e C e   1/  2P  ˇP  2 .2e C Ce  1/ ;  The expression outside the bracket is always positive but the term inside the bracket may be positive or negative depending on the parameter values. In general, R2 first increases and then decreases with the return interval , and the suffi2 > 0 is that  <  c where  c is the cient condition for @R @ solution to the nonlinear equation 2e C e 2  1 D 0. The turning point at which R2 starts to decrease with  depends on all four parameter values. It may also be verified that R2 ! 0 as  ! 0. Therefore, a variable that perfectly predicts the drift of the stock price will appear to have no predictive power if the predictive regressions are run using sufficiently short horizon returns. How short is “sufficiently” short depends on parameter values. In addition to the hump-shaped relation between R2 2 2 @R are also indeterminate. On and , the signs @R P @ and @

@R2 @P 

ˇ

the other hand, < 0, cet. par. the theoretical R2 is always decreasing in the correlation coefficient P and attains

i

C 2

P ˇ

2

 P  C 4P ˇ 

1e  

:

(19.9)

its highest value when P D  1. Since equation (19.9) is complicated, we shall explore the parameter sensitivities of the R2 at different horizons numerically by considering several reasonable scenarios.

19.2.2 Detecting Predictive Power in Finite Samples For our numerical analysis we choose the following “global” parameters. p We set the volatility of the annual cumulative return V .1/ D 0:14, which is approximately the annual volatility of real return on the S&P 500 index for the period 1950–2003. We set ˛ D 0 and ˇ D 1 so that  is the drift of the return process. With this in mind, we set the interest rate r D 3% and the long run mean  D 9%, which implies a mean equity premium of 6%. Finally, the volatility of the stationary distribution of  is defined by  v  p2 . We set v at 4%:9 this means that there is a 68% chance that the equity premium at any moment will lay between 4% and 8%, and a 6.6% probability that it will be negative.10 This seems to us to represent a substantial amount of variation in the equity premium that we should have a reasonable chance of detecting. Within these global parameters we consider nine scenarios, the product of three values of the mean reversion parameter, ›, and three values of the correlation between innovations in  and in returns, P . The three values of › are 0.02, 0.10, and 0.50, which correspond to half-lives for innovations in  of 34.7, 6.9, and 1.4 years and therefore would seem to span the economically interesting range. The values of P are 0:90; 0:5, and 0.0. As mentioned above, we restrict our attention to nonpositive correlations because it is to be expected that, as innovations in expected returns are equivalent to innovations in discount rates, they should be negatively related to contemporaneous stock returns;11 Xia (2001) reports a correlation of 0:93 between monthly innovations in the dividend yield and stock returns for the period January 9

We shall find that this corresponds to the empirical estimates of this parameter reported below. 10 Boudoukh et al. (1993) report evidence that the ex ante risk premium is negative in some states of the world. 11 It is possible for innovations in discount rates to be positively correlated with current returns if the innovations in discount rates are highly correlated with innovations in cash flow news.

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1950 to December 1997, which is similar to Barberis (2000) who also uses the dividend yield as a predictor of stock returns.

Table 19.1 reports various statistics for our nine different scenarios. The first four lines of the table report the parameters describing each scenario. The volatility of the drift,  ,

Table 19.1 Stock return model parameters and parameter estimates Scenarios (i) (ii) (iii) (iv) (v) › 0.02 0.02 0.02 0.10 0.10  0.008 0.008 0.008 0.018 0.018 P  0.00 0:50 0:90 0.00 0:50 P 0.140 0.142 0.144 0.140 0.144

(vi) 0.10 0.018 0:90 0.148

(vii) 0.50 0.040 0.00 0.139

(viii) 0.50 0.040 0:50 0.147

(ix) 0.50 0.040 0:90 0.155

Panel A: Variance ratios 1 month(s) 1.00 6 1.00 1 year(s) 1.00 2 1.00 3 1.01 4 1.02 5 1.02

1.03 1.01 1.00 0.98 0.95 0.93 0.91

1.05 1.03 1.00 0.95 0.91 0.86 0.82

1.00 1.00 1.00 1.01 1.03 1.06 1.09

1.05 1.03 1.00 0.95 0.92 0.89 0.87

1.10 1.05 1.00 0.90 0.82 0.75 0.69

0.98 0.99 1.00 1.04 1.07 1.11 1.13

1.10 1.05 1.00 0.94 0.91 0.89 0.88

1.20 1.10 1.00 0.86 0.77 0.70 0.66

Panel B: 1-month R2 Asymptotic 0.7% 25% 0.1% Median 0.2% Mean 0.3% 75% 0.5%

0.7% 0.2% 0.4% 0.5% 0.7%

0.6% 0.5% 0.7% 0.7% 1.0%

0.7% 0.2% 0.5% 0.6% 0.9%

0.6% 0.4% 0.6% 0.7% 1.0%

0.6% 0.6% 0.8% 0.9% 1.1%

0.7% 0.3% 0.6% 0.7% 1.0%

0.6% 0.3% 0.6% 0.7% 1.1%

0.5% 0.4% 0.7% 0.7% 1.0%

Panel C: 12-month R2 Asymptotic 7.4% 25% 0.6% Median 2.3% Mean 3.0% 75% 5.8%

7.4% 2.3% 5.0% 6.9% 8.5%

7.4% 5.4% 7.9% 8.4% 11.0%

6.9% 2.0% 5.1% 6.6% 9.6%

6.9% 3.7% 6.8% 7.7% 10.8%

6.9% 6.4% 9.0% 9.5% 12.1%

4.8% 1.7% 4.6% 5.7% 8.4%

4.8% 2.4% 5.2% 6.4% 9.2%

4.8% 3.3% 6.0% 6.8% 9.5%

Panel D: 1-month bias-corrected regression coefficient b Theoretical b 0.083 0.083 (2.39) (2.36) 25% 0.038 0.041 (0.58) (0.65) Median 0.082 0.082 (1.27) (1.26) Mean 0.082 0.094 (1.31) (1.26) 75% 0.127 0.133 (2.08) (1.87)

0.083 (2.33) 0.037 (0.65) 0.081 (1.22) 0.101 (1.21) 0.146 (1.79)

0.083 (2.38) 0.058 (1.32) 0.085 (2.05) 0.085 (2.05) 0.112 (2.80)

0.083 (2.32) 0.053 (1.28) 0.081 (1.88) 0.086 (1.88) 0.113 (2.48)

0.083 (2.27) 0.051 (1.29) 0.080 (1.81) 0.089 (1.81) 0.117 (2.30)

0.082 (2.36) 0.057 (1.56) 0.081 (2.29) 0.081 (2.29) 0.104 (2.96)

0.082 (2.23) 0.056 (1.49) 0.080 (2.10) 0.082 (2.11) 0.107 (2.76)

0.082 (2.14) 0.054 (1.44) 0.078 (1.94) 0.082 (1.97) 0.107 (2.52)

Panel E: 12-month bias-corrected regression coefficient b 25% 0.453 0.491 (0.56) (0.65) Median 0.987 0.980 (1.28) (1.22) Mean 0.987 1.113 (1.30) (1.25) 75% 1.549 1.587 (2.04) (1.87)

0.419 (0.62) 0.933 (1.18) 1.170 (1.19) 1.717 (1.78)

0.654 (1.22) 0.974 (1.93) 0.971 (1.96) 1.314 (2.73)

0.595 (1.20) 0.924 (1.78) 0.963 (1.81) 1.283 (2.42)

0.575 (1.22) 0.910 (1.74) 1.005 (1.77) 1.337 (2.29)

0.495 (1.11) 0.778 (1.83) 0.787 (1.83) 1.059 (2.51)

0.481 (1.09) 0.761 (1.74) 0.791 (1.78) 1.080 (2.41)

0.464 (1.08) 0.741 (1.66) 0.784 (1.69) 1.070 (2.26)

Historical

1.00 1.04 0.88 0.88 0.86

Panel A reports theoretical variance ratios based on Equation (19.5) for the returns generated under each of the scenarios. The historical data in this panel is from Table 19.1 of Poterba and Summers (1988), where the data is the real returns from 1871 to 1985 and it is scaled so that the 1-year ratio is 1.00. Panels B-C of this table report the distributions of statistics from ordinary least squares predictive regressions with non-overlapping observations of the form R.t; t C  / D a C b t C " for  D 1; 12 months. The distributions are estimated from 2000 simulations of the return process given by Equations (19.3) and (19.4) for 840 months using the parameter values for each of 9 scenarios Panels D-E report the bias-corrected regression coefficients and their t -ratios using the Amihud and Hurvich (2004) approach where both the point estimate and the standard errors are adjusted. The Stambaugh (1999) correction yields almost the same point p estimates and is omitted from the table. The t -ratios are reported in parentheses. The exogenous parameters are v D 4%;  D 9%; r D 3%; V .1/ D 14%; ˛ D 0, and ˇ D 1

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is set to be consistent with the exogenously chosen values of p v and › using the formula  D v 2. The volatility of the instantaneous return P is chosen to be consistent with p V .1/ D 0:14 and the values of the other parameters using the quadratic Equation (19.5). Panel A reports the variance ratios implied by the nine different scenarios. The variance ratio for m months is defined by the expression: VRm D

m // 12 Var.R.t; t C 12 ; m Var.R.t; t C 1//

  m is the cumulative m-month return, and where R t; t C 12 the variances for the returns are calculated using expression (19.5). Under a pure random walk the variance ratio will be equal to unity for all values of m. Mean reversion or negative autocorrelation in stock prices, such as would be introduced by changing discount rates, will cause the variance ratios to decline with the return calculation interval. The right hand column of the table reports the variance ratios calculated by Poterba and Summers (1988) for real U.S. equity returns in the period 1871–1985. An interesting feature of the PoterbaSummers ratios is that they increase for the 2-year calculation period, which implies small positive autocorrelation at this horizon. Comparing the variance ratios under the nine scenarios with those reported by Poterba and Summers, we see that scenarios (v) and (viii) ( D 0:05 and › D 0:10, 0.50) are the most consistent with the empirical data. Panels B-E present, for each scenario, means and quantiles of the distribution of statistics for the regression (19.6) for 1-month and 1-year forecast periods. The theoretical (asymptotic) R2 and predictive coefficient b , calculated using definition (19.9) and (19.8), are reported as benchmarks. The means and quantiles of the distributions of the regression R2 and estimated values of predictive coefficient, b are calculated from Monte Carlo simulations of returns. Each simulation is for 70 years (840 months), roughly corresponding to the length of data available on the CRSP tape. Each scenario is simulated 2000 times, and the predictive regression (19.6) is run for 1-month and 1-year forecast horizons. The 1 year regressions use nonoverlapping observations and the ordinary least squares estimates of the slope coefficient and its tstatistic are adjusted for bias using the Amihud and Hurvich (2004) approach. The Stambaugh (1999) bias-adjusted slope coefficients are indistinguishable from those obtained under the Amihud and Hurvich (2004) adjustment. The theoretical R2 for the 1-month regressions are between 0.5% and 0.7%, and for all scenarios the average simulated R2 lies between 0.3% and 0.9%. These values compare with those reported by Bossaerts and Hillion (1999) for a shorter sample period of between 2% and 9.8% using a combination of popular predictors. At the annual forecast horizon the theoretical R2 rises to between 4.8% and 7.4%, while

the median of the simulations yield estimates between 3.0% and 9.5%, which compares with the R2 of 17% reported by Cochrane (1999) where overlapping observations of pricedividend ratio are used to predict annual excess stock returns. The 25th percentile of the R2 distribution is around 3% points below the mean, so the Goyal and Welch (2003) estimate of 5.83% for the period 1926–1990 is well within the range of estimates that are consistent with our scenarios. Figure 19.1 plots the theoretical R2 as a function of the return interval, , for the nine different scenarios. Quite different patterns arise from the different sets of parameter values. Three features stand out: First, the generally humped shape of the graphs that is consistent with the theoretical analysis in the previous subsection; second, the small differences between the R2 at the 1-year return interval for the different scenarios as compared with the large differences in R2 for longer intervals; and third, the importance of P , as well as the persistence parameter , in determining the long horizon R2 . When  D 0:10, for example, the R2 at the 20-year interval ranges from 15.9% when P D 0 to 46.1% when P D 0:9. Figures 19.2a and b plot the 10- and 20-year theoretical R2 against the 1-year theoretical R2 . The 1-year R2 is independent of the correlation P by construction, since we set p V .1/ exogenously at 14%. The same R2 at the 1-year horizon can be associated with quite different R2 at long horizons: when  D 0:1, for example, the values of the 1-year R2 are all about 6.9%, but the R2 at the 20-year horizon for P D 0:0 is 15.9% vs. 46.1% for P D 0:9. Thus, modest variation in the 1-year R2 between 6.5% and 7.5% is associated with enormous variation in values of R2 at the 10- and 20-year intervals – from under 20% to over 70% in the case of the 20-year R2 . Figures 19.3a and b show the separate influences of  and P on the long horizon R2 across the nine scenarios. It is clear from these figures that a high value of  (0.5) greatly attenuates the long-horizon predictability of the return. On the other hand, the predictability of long run stock returns when expected returns are highly persistent depends strongly on the degree of negative correlation between the drift innovations and the stock return. For example, Scenario (i), which has a highly persistent expected return . D 0:02/ but zero correlation .P D 0:0/, yields less return predictability at the 10- and 20-year horizons than does Scenario (vi), which has less persistent expected returns . D 0:10/ but a highly negative correlation .P D 0:9/. These examples show that, for a given unconditional volatility of the instantaneous expected return, the long run predictability of stock returns depends crucially on the correlation between innovations in expected returns and the realized stock return. This is not surprising since, in the absence of underlying cash flow uncertainty, the correlation would be perfect and long run returns would be perfectly predictable. Thus, it is dangerous

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Fig. 19.1 Theoretical R2 as a function of the horizon. The figure plots the theoretical regression R2 as a function of the investment horizon  , where  varies from 0.08 (1 month) to 20 years for the nine scenarios reported in Table 19.1. The exogenous parameters are v D 4%;  D 9%; r D p 3%; V .1/ D 14%; ˛ D 0, and ˇD1

to infer anything about the long run predictability of stock returns from the R2 of 1-year regressions, particularly when the sampling variability of the R2 is taken into account. For example, Table 19.1 Panel B shows that under Scenario (ii) there is a 50% chance of estimating an R2 below 5% in an annual regression using 70 years data, although, as shown in Table 19.3a below, the theoretical R2 for 20-year returns is almost 57%. Panels D and E report statistics for the bias-adjusted coefficients of t and their associated t-ratios from predictive regressions for 1-month and 1-year returns using the simulated data. For most of the scenarios, the median t-statistic is less than 2, and the predictive relation between the signal  and the stock return would not be identified as significant in roughly two-thirds of the cases.12 The shorter data samples that are typically used in empirical studies to identify potential signals of the stock price drift make it even less likely that a significant relation will be found, and if the drift is not a perfect linear function of the signal as we have assumed, then the likelihood of identifying a signifi-

cant relation is further reduced. Similar results are obtained for the annual forecast period. Now, however, it is noticeable that both the theoretical and the estimated value of b decline for large  that reduces the persistence of the signal. The evidence in Table 19.1 shows that for the stock price processes that we have considered it is likely to be difficult to identify a significant relation between even a perfect signal of the expected return on stocks and the realized returns; even if a statistically significant relation is found, the standard error of the coefficient will be large, which makes the use of the estimate for portfolio planning purposes problematic.13 Thus, the failure of Goyal and Welch (2004) to find any ex-post predictive power in the dividend yield for excess returns on common stocks is not surprising even if the equity premium has a standard deviation as high as 4% and the dividend yield is a perfect signal of the expected return. Throughout this section we have assumed that the standard deviation in the equity premium is 4%. While this seems to us to be quite large, we have found that it is broadly consistent with the levels of stock return predictability and negative autocorrelation

12

13 See Xia (2001) for an analysis of the effects of parameter uncertainty on optimal portfolio planning.

Of course, if the coefficient is not corrected for bias, the t -statistic is somewhat higher.

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Fig. 19.2 Theoretical 10- and 20-year R2 vs. 1-year R2 . The figure plots the theoretical regression R2 at long horizon (10 and 20 years) as a function of 1-year R2 for the nine scenarios reported in Table 19.1. The exogenous parameters are v D 4%;  D 9%; r D p 3%; V .1/ D 14%; ˛ D 0, and ˇD1

in stock returns that previous researchers have found. We consider next whether this level of return predictability is of economic significance for variation in the level of stock prices.

19.3 Stock Price Variation and Variation in the Expected Returns To assess the implications for the level of stock prices of time variation in expected returns of the magnitude discussed above, we employ a simple valuation model in which the instantaneous expected rate of return on the stock, ,14 follows the O-U process described by Equation (19.2). Since stock prices vary with dividend growth expectations as well as expected returns, we assume that the expected growth rate in

dividends is constant in order to isolate the effects of expected returns on stock prices. Thus, the dividend on the stock is assumed to follow a geometric Brownian Motion with constant expected growth rate, g: dD D gdt C D dzD : D

Let V .D; /  Dv ./ denote the value of the stock, which is homogeneous of degree one in the level of the dividend.15 Ito’s Lemma implies that the process for capital gains plus dividends may be written as:  00  Ddt C dV 1 v 2 v0 1 D  C Œ.  / C D C g C dt V 2 v v v CD dzD C 15

14

Note that we are assuming here that ˛ D 0 and ˇ D 1.

(19.10)

v0  dz ; v

(19.11)

Note that P is used to denote the stock price with dividends reinvested and V to denote the ex-dividend value.

19 Persistence, Predictability, and Portfolio Planning

297

Fig. 19.3 Theoretical 10- and 20-Year R2 as a function of  and P  . The figure plots the theoretical regression R2 at long horizon (10 and 20 years) as a function of the mean reverting parameter , and the correlation between the innovations of the process of the instantaneous return and its drift P  . The exogenous parameters are v D 4%;  D 9%; r D p 3%; V .1/ D 14%; ˛ D 0, and ˇD1

Table 19.3a Long run return predictability and the value of dynamic vs. unconditional strategies Horizon Scenarios (i) (ii) (iii) (iv) (v) (vi) › 0:02 0:02 0:02 0:1 0:1 0:1  0:008 0:008 0:008 0:018 0:018 0:018 P  0 0:5 0:9 0 0:5 0:9 P 0:14 0:142 0:144 0:14 0:144 0:148

(vii) 0:5 0:04 0 0:139

(viii) 0:5 0:04 0:5 0:147

(ix) 0:5 0:04 0:9 0:155

1 year

R2 t D 0:05 t D 0:09 t D 0:13

7:40% 1:01 1 1:01

7:40% 1:01 1 1:01

7:40% 1:01 1 1:01

6:90% 1:01 1 1:01

6:90% 1:01 1 1:01

6:90% 1:01 1 1:01

4:80% 1:01 1 1:01

4:80% 1:01 1 1:01

4:80% 1:01 1 1:01

5 years

R2 t D 0:05 t D 0:09 t D 0:13

26:50% 1:04 1 1:04

28:80% 1:03 1 1:06

31:00% 1:03 1:01 1:08

18:80% 1:05 1:02 1:03

22:40% 1:03 1:02 1:06

26:80% 1:02 1:03 1:11

4:60% 1:05 1:03 1:04

5:90% 1:04 1:04 1:06

7:80% 1:04 1:05 1:09

10 years

R2 t D 0:05 t D 0:09 t D 0:13

38:00% 1:1 1:02 1:06

44:10% 1:06 1:02 1:14

50:70% 1:04 1:04 1:25

20:40% 1:11 1:05 1:06

27:80% 1:07 1:06 1:14

39:80% 1:07 1:13 1:35

2:60% 1:09 1:08 1:08

3:60% 1:09 1:09 1:11

5:60% 1:13 1:15 1:2

20 years

R2 t D 0:05 t D 0:09 t D 0:13

45:60% 1:27 1:07 1:09

56:80% 1:14 1:07 1:32

71:10% 1:09 1:25 2:16

15:90% 1:3 1:19 1:16

24:60% 1:17 1:16 1:27

46:10% 1:29 1:57 2:25

1:30% 1:2 1:18 1:18

1:90% 1:19 1:2 1:22

3:30% 1:36 1:39 1:46

40 years

R2 t D 0:05 t D 0:09 t D 0:13

43:70% 2:17 1:5 1:29

58:00% 1:46 1:27 1:58

79:30% 1:42 2:43 9:02

8:80% 1:9 1:7 1:62

15:20% 1:43 1:42 1:56

38:30% 2:32 3:25 5:47

0:60% 1:43 1:41 1:41

1:00% 1:45 1:45 1:48

1:80% 2 2:05 2:14

This table reports the theoretical values under different scenarios of R2 from regressions of long at the beginning of the period. It also reports the ratios of the certainty equivalent wealth for an certainty equivalent wealth under an unconditional strategy for different horizons and initial values p ters are v D 4%;  D 9%; r D 3%; V .1/ D 14%; ˛ D 0, and ˇ D 1. The risk

where D   D D is the covariance between the dividend growth and the expected return. Equating the drift term in (19.11) to the expected rate of return  yields the following ordinary differential equation for the price-dividend ratio, v: 1 00 2 v  Cv0 Œ./CD C.g /vC1 D 0: 2 2

(19.12)

run returns on the value of  optimal dynamic strategy to the of t . The exogenous parameaversion parameter is  D 5

and we impose a lower reflecting boundary on  at  such that v0 . / D 0.16 Equation (19.11) is a quasi-structural model of return predictability that corresponds to the purely statistical model (19.1–19.2). In both models the expected return is an exogenously specified O-U process, but (19.11) shows how the realized stock return depends on the innovations in the expected return and in the dividend. The instantaneous



< . The boundary with the growth condition: g C 22  D  condition for v./ for high values of  is lim!1 v./ D 0;

See Feller (1951). In our simulations below, we set  D 2:5%, which is about three standard deviations away from the long run  D 9%. 16

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M.J. Brennan and Y. Xia

volatility of the stock price, V , is: P2

D

V2



D2

C

v0  v

2

v0 C 2 D ; v

(19.13)

so that the stochastic expected return  drives a wedge between V and D . If v0 is zero (i.e.,  is a constant), then V D D . Note that the assumption that V (and therefore P ) is constant is made for simplicity, which is obviously 0 inconsistent with the above relation (19.13) in which vv is stochastic since it is a function of . In order to explore the sensitivity of the price-dividend ratio, v, to the expected rate of return , when the expected rate of return follows the stochastic process (19.2), the partial differential equation was solved numerically for the values of  and P that describe the scenarios used in Table 19.1. The real expected dividend growth rate, g, and the volatility of the real dividend growth rate, D , are set to 1.85% and 12.02% respectively, which are the sample mean and volatility of the historical annual real dividend growth rate in Shiller’s data for the period from 1872 to 2001.17 This leaves one parameter, D , of the system (19.10 and 19.2) to be determined endogenously. The covariance between the total return pro0 cess and d is V  D  D C vv 2 , and there is an approximate equality between P and V : P  P  V  D  D C

v0 2  : v 

(19.14)

19.4 Economic Significance of Predictability

This is combined with (19.13) to determine D as: 2 D 1

P2 2 .1  P /: D2

10.8%, across the six available scenarios. The price-dividend ratios in the case of negative D tend to be greater, but show broadly similar patterns to those in Panel A, except for scenarios (ii) and (iii) where the growth condition failed. The price-dividend ratio is very sensitive to the variation in , the sensitivity being determined almost entirely by the persistence parameter . When  D 0:10, an decrease in  from one standard deviation above its mean to one standard deviation below its mean increases the price-dividend ratio by 66–80%. The variation is much greater when  D 0:02, and much less when  D 0:50. However, the value of P has very little effect on the sensitivity of the price-dividend ratio to , although, as we have seen, this parameter is a major determinant of the predictability of long run returns. The results in Tables 19.1 and 19.2 show that it is possible for changes in the expected rate of return, , to have very large effects on stock prices without those changes being easily detected by standard statistical approaches. For example, Table 19.1 shows that in Scenario (v), in which  D 0:10, the median bias-adjusted t-ratio on the predictor variable in regressions of 1-year (1-month) returns on  is only 4.78 (4.88) when 70 years of data are available. Yet Table 19.2 shows that in this same scenario a change in  from one standard deviation below the mean to one standard deviation above the mean can increase the dividend yield by 74% without any change in dividend growth rate expectations.

(19.15)

Conditional on P > D , there exists no solution for D when P D 0 (Scenarios (i), (iv), and (ix)), but there are two equal solutions of opposite sign for D in the other scenarios. In this case, Equation (19.15) together with (19.14) implies that v0 < 0, which is intuitive because a higher discount rate  leads to a smaller price-dividend ratio. Panels A and B of Table 19.2 report, for the scenarios considered in Table 19.1 for which a feasible solution exists, the price-dividend ratios and the corresponding dividend yields for D > 0, while Panels C and D report these variables for D < 0. The price-dividend ratio in the case of positive D ranges from a low of 9.25 and a high of 30.12, with the corresponding dividend yield ranging from 3.3% to

Under the scenarios that we have described above, an investor faces a situation in which there is little short run predictability in stock returns, but there may be very significant long run predictability, depending on the persistence of the equity premium and the correlation of innovations in the premium with stock returns. Under these circumstances it is natural to ask whether there are likely to be significant costs for an investor who ignores time variation in expected returns. We answer this question by comparing the certainty equivalent wealth of an investor under an “unconditional” strategy with the certainty equivalent wealth under the “optimal” strategy. The investor is assumed to have an iso-elastic utility function defined over end-of-period wealth at the investment horizon T :18 max E0 Œu.WT / ; x

17 The P/D ratio and dividend data are from Robert Shiller’s web page http://www.econ.yale.edu/shiller/data.htm. Fama and French (2002) report that the mean dividend growth rate for the S&P 500 was 2.74% for the period 1872 to 1950 and was 1.05% (1.3% from Shiller’s data) for the period 1951 to 2000, which contrasts with the real earnings growth rate of 2.82% over the last half century.

18 Brennan and Xia (2002) and Wachter (2002) show that the determination of an optimal consumption and investment program for an investor with time-additive utility can be represented as a time integral of optimal utility of terminal wealth problems.

19 Persistence, Predictability, and Portfolio Planning

299

Table 19.2 Stock return models and valuation ratios Scenarios (i) (ii) › 0.02 0:02  0.008 0:008 P  0.00 0:50 P 0.140 0:142

(iii) 0:02 0:008 0:90 0:144

(iv) 0.10 0.018 0.00 0.14

(v) 0:10 0:018 0:50 0:144

(vi) 0:10 0:018 0:90 0:148

(vii) 0.50 0.040 0.00 0.139

(viii) 0:50 0:040 0:50 0:147

(ix) 0:50 0:040 0:90 0:155

Panel A: Price-dividend ratios .D > 0/ D n.a. t D .  v / D 5% n.a. t D  D 9% n.a. t D  C v D 13% n.a.

0:421 30:12 14:55 9:66

0:887 25:47 13:42 9:25

n.a. n.a. n.a. n.a.

0:387 19:47 14:37 11:43

0:879 17:67 13:20 10:60

n.a. n.a. n.a. n.a.

0:324 15:17 13:92 12:98

0:864 14:28 13:11 12:23

Panel B: Dividend yields .D > 0/ t D .  v / D 5% n.a. t D  D 9% n.a. t D  C v D 13% n.a.

3:3% 6:9% 10:4%

3:9% 7:5% 10:8%

n.a. n.a. n.a.

5:1% 7:0% 8:8%

5:7% 7:6% 9:4%

n.a. n.a. n.a.

6:6% 7:2% 7:7%

7:0% 7:6% 8:2%

Panel C: Price-dividend ratios .D < 0/ D n.a. 0:421 t D .  v / D 5% n.a. 57:80 t D  D 9% n.a. 20:26 t D  C v D 13% n.a. 11:27

0:887 191:45 46:56 17:52

n.a. n.a. n.a. n.a.

0:387 23:50 17:03 13:30

0:879 27:38 19:59 15:10

n.a. n.a. n.a. n.a.

0:324 16:41 15:05 14:03

0:864 17:62 16:15 15:05

0:5% 2:1% 5:7%

n.a. n.a. n.a.

4:3% 5:9% 7:5%

3:7% 5:1% 6:6%

n.a. n.a. n.a.

6:1% 6:6% 7:1%

5:7% 6:2% 6:6%

Panel D: Dividend yields .D < 0/ t D .  v / D 5% n.a. t D  D 9% n.a. t D  C v D 13% n.a.

1:7% 4:9% 8:9%

This table reports valuation ratios for a security whose dividend follows: dD D gdt C D dzD D and the expected rate of return on the security follows an Ornstein-Uhlenbeck process: d D .  /dt C  dz The real dividend growth rate g D 1:85% and the volatility D D 12:02% is from the sample mean and volatility of historical data of real annual dividend growth from 1872 to 2001, provided by Robert Shiller. The correlation, D is derived endogenously across the nine scenarios considered in Table 19.1, and “n.a.” means that no solution exists for D in that scenario. The growth condition failed in scenario (ii) and (iii) p under D < 0 The other exogenous parameters are v D 4%;  D 9%; r D 3%; V .1/ D 14%; ˛ D 0, and ˇ D 1

where u.WT / D

8 < :

1

e

T

e T

WT 1 ln WT

 > 1;  D 1:

The dynamic portfolio choice problem is subject to the dynamic budget constraint: dW D Œx.˛ C ˇ  r/ C r dt C xP dzP ; W with x defined as the proportion of wealth invested in the single risky asset whose stochastic process was given in Equations (19.1–19.2). The risk free interest rate, r, is assumed to be constant for simplicity.19 The “unconditional” strategy, which is based on the assumption that the equity premium is constant, allocates a constant fraction of wealth 19 For models of dynamic investment strategies with stochastic interest rates see Campbell and Viceira (2001) and Brennan and Xia (2002).

to the risky asset, but the allocation to the risky asset under the “optimal” strategy depends on t as shown below. Denoting the equity premium by yt  ˛ C ˇt  r, the investor’s optimal dynamic policy20 is to invest a fraction of his wealth, xt , in the risky asset, where: xt D

P  yt C ŒB./ C C./yt ; P P2

(19.16)

and   T  t is the time remaining to the investment horizon. The investor’s indirect utility, J.W; ; /  maxxt E0 Œu.WT / , is given by: J.W; ; / De

t

  1 W 1 2 exp A./ C B./yt C C./yt 1 2

20 This was first derived by Kim and Omberg (1996); it is discussed in more detail in Xia (2001).

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M.J. Brennan and Y. Xia

 e t

W 1 o ' .; t/; 1

(19.17)

where A./; B./ and C./ are defined in Appendix 19A. The optimal equity allocation, xt , is a time-dependent function of the equity premium y. Xia (2001) shows that the allocation in general is a non-monotonic function of the investment horizon, and that if the return process is negatively correlated with the process for , i.e., P < 0, which we have argued is likely to be case, then the optimal equity allocation x  most likely increases with the investment horizon . Let x u denote the constant equity allocation under the unconditional strategy that treats the equity premium, y  ˛ C ˇ  r, as a constant. Then xu 

y : P2

(19.18)

Although the optimal unconditional strategy is based on an equity premium that is assumed to be constant, y, the evolution of the investor’s wealth depends on the true dynamics of the equity premium   r. As shown in Appendix 19B, the indirect utility function for the unconditional strategy under the true dynamics of the equity premium is given by:

Fig. 19.4 The certainty equivalent wealth ratio CEWR as a function of  and P  . The figure plots the certainty equivalent wealth ratio .CEWRou ; CEWRom ; CEWRob / between the optimal dynamic strategy and the unconditional, the myopic, and the buy-and-hold strategies at the 20-year-horizon for t D 5%; 9%; 13% as a function of the mean reverting parameter , and the correlation between the innovations of the process of the instantaneous return and its drift P  . The exogenous parameters are v D 4%;  D 9%; r D p 3%; V .1/ D 14%; ’ D 0, and ˇD1

J u .W; ; / D e t  e t

W 1 exp fD./ C E./T g 1

W 1 u ' .; t/; 1

(19.19)

with D./ and E./ defined in Appendix 19B. The Certainty Equivalent Wealth under the optimal strategy, CEW o .; /, is defined as the amount of wealth to be received at the horizon with certainty that would give the investor the same expected utility as he receives under the optimal strategy with an initial $1 of wealth to invest: 1

CEW o .; /  Œ' o .; t/ 1 :

(19.20)

The Certainty Equivalent Wealth for the unconditional strategy, CEW u .; /, is defined analogously. We shall analyze the ratio of the certainty equivalents under the two strategies, CEW Rou , which is defined by CEW Rou .; / 

CEW o .; / : CEW u .; /

The ratio, CEW R, is a measure of the value of taking account of variation in the equity premium. Table 19.3a and Fig. 19.4a–c reports values of CEW Rou for different horizons across the nine different scenarios for

19 Persistence, Predictability, and Portfolio Planning

the stochastic stock price processes, and for different initial values of t , when the coefficient of relative risk aversion,  , is equal to 5. We include scenarios (i), (iv), and (vii) in the table for completeness, although the pattern of declining variance ratios generated by these scenarios is inconsistent with the empirical evidence, and the zero value of D contrasts with the negative value we expect to be associated with discount rate effects. Therefore, we restrict our discussion to the other six scenarios. When t D , the optimal strategy offers no measurable advantage over a 1-year horizon in any of the scenarios; a gain of up to 5% over a 5-year horizon; a gain of between 2% and 15% over 10 years, depending on the scenario, and a gain of 7–57% over 20 years, again depending on the scenario. Under Scenario (v) the gain is 2% for a 5-year horizon, 6% for a 10-year horizon, and 16% for a 20-year horizon. Focusing on the 20-year horizon, the gain from the optimal strategy when t is one standard deviation above its mean may be substantial, particularly when P D 0:9 (scenarios (iii) and (vi), and (ix)) and, for any given scenario, the increase in the CEW R is greater for a one standard deviation increase in t than is the reduction for a half standard deviation decrease in t . Note that, while the R2 of the predictive regressions was determined primarily by the value of , the influence of P on CEW R is much greater; for example, comparing scenarios (iv) and (vi) at the 20-year horizon when t D , the advantage of the optimal strategy increases from 19% to 57% as P decreases from zero to 0:90. This is because a large negative correlation between stock returns and the expected rate of return reduces the volatility of long horizon returns; this effect is taken into account by the optimal strategy that invests more than the unconditional strategy in stocks for  >  when the horizon is long. The benefit of the optimal strategy is a non-monotone function of the persistence parameter , tending to be greatest for  D 0:10 at the longer horizons. The short run predictability of returns as measured by the 1-year R2 is not, however, necessarily associated with higher certainty equivalent for the optimal strategy relative to the unconditional strategy. This is illustrated in Fig. 19.3, where the certainty equivalent ratio under the optimal and the unconditional strategies for a 20-year horizon, CEW Rou , is plotted against the 1-year R2 for the nine scenarios. Interestingly, Table 19.3a shows that there is also no clear relation between the advantage of the optimal strategy and the long run predictability of returns (20-year regression R2 ) when  D : the correlation between the 20-year CEW R across scenarios and the 20-year R2 is 0:05. Less surprisingly, since  carries more information about future investment opportunities when the R2 is high, the 20-year CEW R has a correlation of 0.48 with the 20-year R2 when  is one standard deviation below its mean and a correlation of 0.57 when  is one standard deviation above its mean.

301

Finally, it is interesting to note that there is also no relation between CEW Rou and the monthly or annual out-of-sample predictability as measured by the root mean squared error (RMSE).21 In summary, the gains to the optimal strategy can be very large for long horizon investors. For scenarios (v) and (vi), which correspond closely to the behavior of the expected rate of return that we shall extract from forecasts of long run rates of return, the gains run from 16% to 125% over a 20-year horizon, depending on the initial value of . The gains of the optimal strategy come from both market timing and hedging. Market timing is simply the variation in the equity allocation with the equity premium. Hedging is the additional allocation to equities that results from the negative correlation between innovations to rates of return and returns on the equity security, which implicitly recognizes that equities are not so risky in the long run.22 The benefits of market timing, but not of hedging, are captured by the myopic rule, x m : xm 

˛ C ˇt  r yt D : 2 P P2

(19.21)

The certainty equivalent wealth associated with the myopic strategy is calculated by evaluating the expected utility associated with the myopic strategy, and the details are given in Appendix 19C. Table 19.3b reports the certainty equivalent wealth ratios between the optimal and the myopic strategies CEW Rom . The results show that the hedging gains offered by the optimal strategy but not by the myopic strategy, are zero when P D 0 and of the order of 6% at the 20-year horizon when P D 0:50, but are as high as 38–66% when P D 0:90. For comparison, the certainty equivalent wealth associated with the optimal buy-and-hold strategy, x b , which is described in Appendix 19D, was calculated numerically. The initial equity allocation of the buy-and-hold strategy depends on the value of t , but subsequent changes in the allocation are determined entirely by the realized asset returns. The certainty equivalent wealth ratios between the optimal and the buy-and-hold strategies CEW Rob , reported in Table 19.3c and Fig. 19.4g–i, show that the buy-and-hold strategy is

21 The out of sample RMSE is calculated from the simulated data used for Table 19.1; the predictive relation is estimated from the first 65 years of data, and the out-of-sample RMSE is calculated from the differences between the predicted and the realized returns over the following five years. The results, which are not reported here, are available on request. Goyal and Welch (2003, 2004) use the out-of-sample RMSE as a measure of the value of a predictive instrument. 22 Stambaugh (1999) and Barberis (2000) compare myopic and buyand-hold strategies.

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Table 19.3b Long run return predictability and the value of dynamic vs. myopic strategies Horizon

(v)

(vi)

(vii)

(viii)

(ix)

›  P  P

Scenarios

(i) 0:02 0:008 0:00 0:140

(ii) 0:02 0:008 0:50 0:142

(iii) 0:02 0:008 0:90 0:144

(iv) 0:10 0:018 0:00 0:140

0:10 0:018 0:50 0:144

0:10 0:018 0:90 0:148

0:50 0:040 0:00 0:139

0:50 0:040 0:50 0:147

0:50 0:040 0:90 0:155

1 year

R2 t D 0:05 t D 0:09 t D 0:13

7:4% 1:00 1:00 1:00

7:4% 1:00 1:00 1:00

7:4% 1:00 1:00 1:00

6:9% 1:00 1:00 1:00

6:9% 1:00 1:00 1:00

6:9% 1:00 1:00 1:00

4:8% 1:00 1:00 1:00

4:8% 1:00 1:00 1:00

4:8% 1:00 1:00 1:00

5 years

R2 t D 0:05 t D 0:09 t D 0:13

26:5% 1:00 1:00 1:00

28:8% 1:00 1:00 1:00

31:0% 1:00 1:00 1:00

18:8% 1:00 1:00 1:00

22:4% 1:00 1:00 1:00

26:8% 1:00 1:01 1:02

4:6% 1:00 1:00 1:00

5:9% 1:00 1:00 1:00

7:8% 1:01 1:01 1:02

10 years

R2 t D 0:05 t D 0:09 t D 0:13

38:0% 1:00 1:00 1:00

44:1% 1:00 1:00 1:01

50:7% 1:00 1:02 1:04

20:4% 1:00 1:00 1:00

27:8% 1:00 1:01 1:02

39:8% 1:02 1:06 1:12

2:6% 1:00 1:00 1:00

3:6% 1:01 1:01 1:01

5:6% 1:04 1:05 1:06

20 years

R2 t D 0:05 t D 0:09 t D 0:13

45:6% 1:00 1:00 1:00

56:8% 1:01 1:02 1:05

71:1% 1:04 1:16 1:41

15:9% 1:00 1:00 1:00

24:6% 1:02 1:04 1:06

46:1% 1:20 1:38 1:66

1:3% 1:00 1:00 1:00

1:9% 1:02 1:02 1:02

3:3% 1:12 1:13 1:14

40 years

R2 t D 0:05 t D 0:09 t D 0:13

43:7% 1:00 1:00 1:00

58:0% 1:04 1:09 1:20

79:3% 1:35 2:32 5:56

8:8% 1:00 1:00 1:00

15:2% 1:08 1:11 1:14

38:3% 2:10 2:76 3:86

0:6% 1:00 1:00 1:00

1:0% 1:05 1:05 1:05

1:8% 1:31 1:33 1:34

This table reports the theoretical values under different scenarios from regressions of R2 long run returns on the value of  at the beginning of the period. It also reports the ratios of the certainty equivalent wealth for an optimal dynamic strategy to the certainty equivalent p wealth under a myopic strategy for different horizons and initial values of t . The exogenous parameters are v D 4%;  D 9%; r D 3%; V .1/ D 14%; ˛ D 0, and ˇ D 1. The risk aversion parameter is  D 5

Table 19.3c Long run return predictability and the value of dynamic vs. buy-and-hold strategies Horizon

(v)

(vi)

(vii)

(viii)

(ix)

›  P  P

Scenarios

(i) 0:02 0:008 0:00 0:140

(ii) 0:02 0:008 0:50 0:142

(iii) 0:02 0:008 0:90 0:144

(iv) 0:10 0:018 0:00 0:140

0:10 0:018 0:50 0:144

0:10 0:018 0:90 0:148

0:50 0:040 0:00 0:139

0:50 0:040 0:50 0:147

0:50 0:040 0:90 0:155

1 year

R2 t D 0:05 t D 0:09 t D 0:13

7:4% 1:00 1:01 1:02

7:4% 1:00 1:01 1:02

7:4% 1:00 1:01 1:02

6:9% 1:00 1:01 1:02

6:9% 1:00 1:01 1:02

6:9% 1:00 1:02 1:03

4:8% 1:01 1:01 1:03

4:8% 1:01 1:01 1:03

4:8% 1:01 1:02 1:03

5 years

R2 t D 0:05 t D 0:09 t D 0:13

26:5% 1:02 1:07 1:14

28:8% 1:02 1:08 1:15

31:0% 1:01 1:09 1:16

18:8% 1:04 1:08 1:13

22:4% 1:04 1:10 1:16

26:8% 1:05 1:12 1:18

4:6% 1:08 1:10 1:12

5:9% 1:09 1:12 1:15

7:8% 1:10 1:14 1:17

10 years

R2 t D 0:05 t D 0:09 t D 0:13

38:0% 1:04 1:16 1:29

44:1% 1:05 1:19 1:35

50:7% 1:05 1:24 1:43

20:4% 1:10 1:17 1:26

27:8% 1:13 1:23 1:36

39:8% 1:18 1:36 1:53

2:6% 1:19 1:21 1:24

3:6% 1:24 1:27 1:30

5:6% 1:32 1:36 1:42

20 years

R2 t D 0:05 t D 0:09 t D 0:13

45:6% 1:11 1:31 1:56

56:8% 1:15 1:47 1:88

71:1% 1:21 1:83 2:57

15:9% 1:24 1:34 1:48

24:6% 1:38 1:56 1:81

46:1% 1:76 2:27 2:92

1:3% 1:44 1:46 1:51

1:9% 1:59 1:63 1:68

3:3% 1:90 1:96 2:05

40 years

R2 t D 0:05 t D 0:09 t D 0:13

43:7% 1:27 1:55 2:00

58:0% 1:47 2:12 3:28

79:3% 2:11 5:24 15:66

8:8% 1:58 1:70 1:92

15:2% 2:09 2:40 2:91

38:3% 4:81 6:73 10:28

0:6% 2:11 2:15 2:22

1:0% 2:65 2:71 2:81

1:8% 3:97 4:07 4:25

This table reports the theoretical values under different scenarios of R2 from regressions of long run returns on the value of  at the beginning of the period. It also reports the ratios of the certainty equivalent wealth for an optimal dynamic strategy to the certainty equivalent wealth under an optimal buy-and-hold strategy for different horizons and initial values of t . The exogenous parameters are v D 4%;  D 9%; p r D 3%; V .1/ D 14%; ˛ D 0, and ˇ D 1. The risk aversion parameter is  D 5

19 Persistence, Predictability, and Portfolio Planning

303

Fig. 19.5 The certainty equivalent wealth ratio under the optimal and the unconditional strategies for a 20-year horizon vs. the 1-year predictive regression R2 . The figure plots the certainty equivalent wealth ratio (CEWRou ; CEWRom ; CEWRob ) between the optimal and the unconditional for a 20-year horizon investor CEWRou , as a function of 1 year for the nine scenarios reported in Table 19.1. The initial value of  is set at respectively, 5%, 9% and 13%. The exogenous parameters are v D 4%;  D 9%; r D p 3%; V .1/ D 14%; ˛ D 0, and ˇD1

extremely inefficient.23 At the 20-year horizon the gains ofthe optimal strategy when  D 0:1 are of the order of 38– 81% when D D 0:50 and 76–192% when D D 0:90 Fig. 19.5. To this point, we have shown that time-variation in expected stock returns, which may be very difficult to detect by standard regression methods, may nevertheless imply both significant variation in stock prices that is unrelated to changes in cash flow expectations, and substantial potential gains to the use of dynamic portfolio strategies for long horizon investors. However, the gains that we have calculated assume that it is possible to observe the instantaneous expected return on the stock and, as we have seen, regression estimates of the relation between expected returns and even perfect instruments of it are likely to be very imprecise. Therefore, 23

This is in contrast to the findings of Brennan and Torous (1999) who show that a buy-and-hold strategy performs well relative to a rebalancing strategy when the investment opportunity set is treated as constant.

in the next section, we explore the use of estimates of long run expected returns derived from a dividend discount model (DDM) as inputs to dynamic portfolio models.

19.5 Forecasts of Equity Returns Regressions that attempt to predict stock returns from instruments such as the dividend yield or the interest rate lack strong theoretical restrictions on the regression coefficients and, as we have seen in Sect. 19.2, the data are likely to yield very imprecise estimates of the coefficients even when the instruments are perfect. It is not surprising therefore that such regressions have essentially no out-of-sample predictive power. An alternative to the simple regression approach is to estimate the expected return on stocks by comparing the current level of the stock market with forecasts of future

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dividends on the market portfolio – the Dividend Discount Model (DDM) approach. The advantage of this approach, which has long been employed in “Tactical Asset Allocation” models, is that the expected return is estimated directly and that there is no need to estimate a regression coefficient relating the stock (excess) return to the predictor instruments. The offsetting disadvantage is that the rate of return estimated from the DDM is a long run internal rate of return and there is no reason to believe that this will be equal to the instantaneous expected rate of return even if the dividend forecasts are unbiased. Therefore, it is necessary to develop a model to derive the expected instantaneous rate of return from the long run expected rate of return.

19.5.1 Models and Estimation Procedure We employ two models to convert the estimated DDM long run expected rate of return into an estimate of t , the instantaneous expected rate of return. The first model assumes that the dividend growth rate g is a known constant. The second model assumes that the growth rate follows an OrnsteinUhlenbeck process.24 Both methods assume that the instantaneous expected rate of return follows an O-U process as in Equation (19.2). Our basic input data are direct estimates of the DDM long run “expected rate of return,” t ,25 as defined by the discounted cash flow model: Pt D

1 X Et ŒDt C  D1

1 C t

;

(19.22)

where Pt is the level of the stock price index at time t, and Et ŒDt C is the dividend on the index expected at time t to be paid at time t C .

Note that Equation (19.23) rests on the assumption that the dividend expectations in (19.22) form a geometric series. It was shown in Sect. 19.3 that if the expected rate of return, , follows an O-U process and the dividend growth rate is constant, then the price-dividend ratio v./ also satisfies the ordinary differential equation (ODE) (pde). Since v is a monotonic function of , at each point in time there exists a t whose implied v from the ODE is equal to the pricedividend ratio associated with t from Equation (19.23) for a given value of g. This implies a (nonlinear) one-to-one mapping between t and t for a given set of parameters for the stochastic process for ;   .; v ; ; D /. As a result, these parameters and the time series of t can be estimated by an iterative process: starting with an initial value of  D  0 ; t is calculated from the time series of t ; then  is estimated from the time series of t , and a new t series is re-calculated from t ; the process continues until convergence is achieved. We denote the resulting Model 1 estimates by 1 and 1 . This iterative procedure is essentially a non-linear Kalman filter in which the latent variable t is a nonlinear function of the observable variable t . The transition equation for t is the discrete-time equivalent of the O-U process: t C t D .1  e  t / C e  t t C "

(19.24)

and the observation equation (which contains no observation error) is: t D f .t / D g C

D D g C .v.t ; //1 : P

The mapping of v implied by the DDM to the v implied by the ODE is equivalent to solving the nonlinear function f numerically.

19.5.1.2 Model 2: Stochastic g 19.5.1.1 Model 1: Constant g When the dividend growth rate, g, is constant, there is a oneto-one correspondence between t and the price-dividend ratio, P =D, which is given by the (discrete time version of the) Gordon growth model: vt 

24

Pt 1Cg ; D Dt t  g

(19.23)

Brennan and Xia (2001) assume a similar model for the dividend growth rate. 25 There is no assurance that t , the solution to Equation (19.22), will be equal to the expected rate of return except when all future dividends are known and discount rates are constant.

To allow for the possibility that dividend growth rate expectations are stochastic, the instantaneous dividend growth rate is assumed to follow an Ornstein-Uhlenbeck process: dg D g .g  g/dt C g dzg :

(19.25)

Two steps are then required to derive  from : first the instantaneous dividend growth rate, g, and the parameters of its stochastic process, g  .g ; g ; g/, are estimated and then, given the time series of gt and g , the time series of t and the parameters of its stochastic process,  are estimated.

19 Persistence, Predictability, and Portfolio Planning

305

With stochastic dividend growth, the DCF valuation formula t can be written as: X …s EŒ.1 C gt Ci /jgt Pt i D1 D ; Dt .1 C kt /s sD1 1

(19.26)

For a given set of parameters of the dividend growth process g ; EŒ.1Cgt Ci /jgt is a known linear function of gt . That is, Equation (19.26) is used to solve for the time series gt , given g , the observed series of price-dividend ratios P =D, and the DDM expected rates of return t . The estimated time series gt is then used to estimate g , and theprocedure  iterates until there is a consistent set of gt and g ; gt ; t . This procedure can again be interpreted as a Kalman filter in which g is the latent variable with the transition equation gt C t D .1  e g t /g C e g t gt C "; while the observation equation relating the observable variables  and P =D to the unobservable variable g is given implicitly by (19.26).   Conditional on gt ; t from the first step, t is determined step. The price-dividend ratio

in the second  v  v ; gI  ; g now satisfies the following two-statevariable partial differential equation: 0D

1 2 1  vgg C g  g vg C 2 v 2 g 2 CŒg .g  g/ C g D gD vg C Œ .  / C D D v C Œg   v C 1:

(19.27)

Equation (19.27) depends on the set of parameters g , which was estimated in the first step, and the unknown  , which is to be estimated together with t in the second step. For a given value of   , the PDE (19.27) is solved numerically

for v ; gI  ; g . Given gt from the first step, a mapping between t and t ; t D f .t ; gt /, is defined by setting v ; gt I  ; g equal to the observed price dividend ratio P/D. The time series of t estimates are then used to estimate a new  and the process is iterated until convergence. This can be interpreted as another Kalman filter in which the transition equation of the latent variable t is given by (19.24) while the nonlinear observation equation is: P =D D v.; gt I  ; g /: The resulting estimates from Model 2 are denoted by 2 and 2 .

19.5.2 Estimates Quarterly data on real dividends and the price-dividend ratio for the S&P 500 index are obtained for the sample period 1950.1 to 2002.2. Four different DDM discount rate series, t , are used for illustrative purpose. The first two series are estimates of the real long run expected rate of return on equities that would have been assessed by investors at each date. The Arnott and Bernstein (2002) (A&B) series,  1 , is constructed by adding to the current dividend yield an estimate of the expected long run real growth rate in dividends that in turn is equal to a forecast of real GNP per capita growth less a “dilution factor” – these are estimated using an average of the experience over the previous 40 years and the experience since the series began in year 1810. The Ilmanen (2003) (IL) series,  2 , is constructed by adding together estimates of the “dividend yield” and the long run growth rate of dividends. The former is calculated from a smoothed earnings yield multiplied by 59%. The latter is an average of 2% and the past 10, 20, 30, 40 and 50 years’ geometric average real growth rate in corporate earnings. Both the A&B and the IL series were constructed around 2002 for research purposes and use data or parameter values that may not have been available to market participants at the date of the forecast. However, the next two series,  3 and  4 , were constructed in real time for use by investment professionals in asset allocation strategies. The series  3 is provided by Wilshire Associates (WA), an investment management consulting firm, and the series  4 is from Barclays Global Investors (BGI), a global investment management firm. They are both in nominal terms and are constructed by first aggregating I/B/E/S consensus estimates of growth rates in earnings per share out for 5 years for individual firms in the S&P 500 index and then letting the growth rates converge linearly to the economy-wide average growth rate over the next 10 years. The resulting growth rates are then used to project current dividends and to calculate an implied DDM rate of return from the current level of the S&P 500 index. These two nominal series are obtained from real-time forecasts of fundamentals that do not use any future information. Table 19.4 reports the estimated parameters of the  process for each series under the two assumptions about dividend growth. Several features stand out in the real series. First, the estimates of  ;  , and  are largely unaffected by whether the expected dividend growth rate is assumed to be constant or not. Second, the estimates of both  and  are somewhat higher for the Ilmanen series than for the A&B series; the effect of these differences on v , the standard deviation of the stationary distribution of , are largely offsetting

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M.J. Brennan and Y. Xia

Table 19.4 Parameter estimates of the  series Parameters    Scenario (vi) Table 19.2 0.100 0:0180 0:090 Real models A&B 1;1 0.085 0:0173 0:047 A&B 1;2 0.083 0:0172 0:045 IL 2;1 0.122 0:0196 0:066 IL 2;2 0.115 0:0224 0:066 BGI 3;1 BGI 3;2 WA 4;1 WA 4;2

Nominal models 0.091 0:0239 0.085 0:0214 0.122 0:0336 0.095 0:0266

0:133 0:113 0:137 0:111

v

P

D

D

0:0400

0:900

0:1200

0:879

0:0419 0:0423 0:0397 0:0467

0:977 0:981 0:884 0:885

0:0852 0:0776 0:0852 0:0822

0:126 0:106 0:117 0:066

0:0560 0:0519 0:0680 0:0611

0:812 0:657 0:682 0:714

0:0859 0:2076 0:0859 0:2202

0:234 0:249 0:095 0:209

g

g

g

Dg

0:103

0:0090

0:328

0:413

0:025

0:0034

0:379

0:055

0:209

0:0294

0:378

0:326

0:220

0:0339

0:402

0:417

This table reports the parameter estimates associated with the instantaneous expected return, , series, which are derived from the long run discount rate ›. The two real › series are calculated in Arnott and Bernstein (2002) (A&B) and Ilmanen (2002) (IL), while the two nominal › series are provided by Barclays Global Investors (BGI) and Wilshire Associates (WA). In each › series, we derive  under two cases. In case I, the dividend growth rate g is assumed to be a constant. In case II, the dividend growth rate g is assumed to follow a mean-reverting process. When the real › from Ilmanen or A&B is used as the long run discount rate, g is set to 0.86% in the first case, while g is set to 0.86% in the second case. When the nominal › is used, g is set to 4.82% in the first case, while g is set to 4.82% in the second case

and the estimates are very close to the value of 4% that we have assumed in our simulations in Sects. 19.2 and 19.3. Third, the estimates of the correlation P are all between 0:884 and 0:981; this is what we should expect since  is a proxy for the discount rate. Finally, most of the parameters are quite close to the values that were assumed or derived for Scenario (vi) in Table 19.2 (which assumes a constant expected dividend growth rate). The exception is D , which is much higher in Scenario (vi). We suspect that this is because D was set equal to 12.02% in Table 19.2 while its estimates is only 8.52% in Table 19.4.26 The estimates of the parameters of the nominal  process are broadly similar to those estimated for the real models except that in both cases the Model 2D is over 20%, but this is offset by the much higher level of mean reversion of the dividend growth rate g ; in addition, D is positive in both nominal models, while it is negative in the two real models. Figure 19.6 shows that when the A&B series  1 is used as the input, the estimates of  from the two models, 1;1 and 1;2 , are virtually coincident, suggesting that there is little advantage in allowing for a stochastic dividend growth rate. Note that both estimates of the instantaneous expected rate of return, , are much more volatile than the underlying DDM  series from which they are derived: the maximum value of 1;2 is over 16% in June 1982 and the minimum is around 3% in March 2000; in contrast, the maximum and minimum values of the DDM , which occur in the same 2 months, were respectively 7.9% and 2.1%. 26

As noted above, 12.02% is the sample volatility of annual real dividend growth over the period 1872–2001, while 8.52% is the estimate of D in Equation (19.25) derived from quarterly data for the period 1950.1 to 2002.2 using the algorithm described under Model 2 above.

Figure 19.7 plots the corresponding  estimates from the Ilmanen series  2 . As in the previous case, the 2;1 and 2;2 series track each other closely except around the end of 1999 when the decline in the Model 2 estimate, 2;2 , is much more dramatic than that in 2;1 : 2;2 reaches a minimum of 3:2% just before the end of the bull market in December 2000. For comparison, Figure 19.8 plots the  i and the i;2 .i D 1; 2/ for both the A&B and the Ilmanen series. While the two series of i;2 .i D 1; 2/ generally track each other quite well, there are periods of significant difference. In the 1950s the Ilmanen estimate exceeds the A&B estimate by up to 3.8%. Significant differences of the opposite sign occur during the period 1983–1996; in September 1987, the Ilmanen estimate was less than 1% while the A&B estimate was over 6.5%. Figures 19.9 and 19.10 plot the quarterly nominal estimates of  i ; i;1 and i;2 .i D 3; 4/ from BGI and WA for the shorter periods for which these series are available. These last two series differ from the previous two in three significant ways. First, they are truly ex-ante; second, they are nominal expected rates of return; third, they are based on analysts’ forecasts of earnings growth rates that are known to be upward biased. Not surprisingly, these nominal  series lie everywhere above the two real  series. Nevertheless, they show a similar pattern of increase during the 1970s, the BGI series  3 (the WA series  4 ) reaching a peak of 19.1% (18.8%) in the third quarter of 1981 (second quarter 1982). After the peak, there is a prolonged decline in both series. Although for both series, i;1 and i;2 .i D 3; 4/ move largely in parallel, i;1 is everywhere above i;2 .i D 3; 4/ and the difference between them is much larger than that observed for the A&B and the IL real series. Figure 19.11 shows  i and the corresponding instantaneous expected return i;2 .i D 3; 4/ together. The two i;2

19 Persistence, Predictability, and Portfolio Planning Fig. 19.6 Estimates of the real 1;1 and 1;2 series (1950.1 to 2002.2). The figure plots the long run expected real return ›1 from Arnott and Bernstein (2002) (A&B) together with the estimated series of 1;1 and 1;2 , corresponding to the two cases of constant g and mean-reverting g

Fig. 19.7 Estimates of the real 2;1 and 2;2 series (1950.1 to 2002.2). The figure plots the long run expected real return ›2 from Ilmanen (2003) (IL) together with the estimated series of 2;1 and 2;2 , corresponding to the two cases of constant g and mean-reverting g

307

308 Fig. 19.8 Estimates of the real 1;2 and 2;2 Series (1950.1 to 2002.2). The figure plots the long run expected real return ›1 and ›2 together with their corresponding 1;2 and 2;2 series under the assumption of mean-reverting g

Fig. 19.9 Estimates of the nominal 3;1 and 3;2 Series (1972.4 to 2002.1). The figure plots the long run expected real return ›3 from Barclays Global Investors (NGI) together with the estimated series of 3;1 and 3;2 , corresponding to the two cases of constant g and mean-reverting g

M.J. Brennan and Y. Xia

19 Persistence, Predictability, and Portfolio Planning Fig. 19.10 Estimates of the nominal 4;1 and 4;2 series (1973.1 to 2004.1). The figure plots the long run expected real return ›4 from Wilshire Associate (WA) together with the estimated series of 4;1 and 4;2 , corresponding to the two cases of constant g and mean-reverting g

Fig. 19.11 Estimates of the nominal 3;2 and 4;2 series (1973.1 to 2002.2). The figure plots the long run expected real return ›3 and ›4 together with their corresponding 3;2 and 4;2 series under the assumption of mean-reverting g

309

310

series track each other closely except for the period 1980– 1982 as well as 1999–2000, when the WA estimate 4;2 exceeds the BGI estimate 3;2 by 1–3%, and the period 2001–2002 when the WA estimate 4;2 is below the BGI estimate 3;2 by 2–5%.

19.5.3 Return Prediction We have already shown that even when return predictability is of great economic importance, the statistical evidence of predictability may be weak and hard to detect using standard statistical methods. Therefore, even if the  series contain valuable information for portfolio planning, the statistical evidence of their predictive power may be weak. Panel A of Table 19.5 reports the results of regressions of quarterly real returns on the S&P 500 index on values of i;2 .i D 1; 2/ derived from the A&B and IL real DDM series, while Panel B reports the results of regressing quarterly nominal returns on the estimates of the nominal i;2 .i D 3; 4/ derived from the BGI and WA nominal DDM series. We report results for the whole sample period from 1950.2 to 2002.2, and also the two approximate halves of the period, omitting the influential 1974.3 quarter when the real return on the S&P 500 was approximately 28%. For the two real  series, the effect of omitting this quarter is to raise both the estimated coefficient towards its theoretical value of unity and to raise the regression R2 . While the regression coefficients are not significantly different from either zero or their theoretical value of unity, the point estimates are close to unity. The explanatory power of the predictive variable in both models is considerably greater in the first half of the sample period where the regression R2 is around 4–5% and the predictive coefficient is statistically significant at the 5% level. In all other periods, however, regression R2 ’s are all below 2% and none of the predictive coefficients is significant at the 5% level. Panel B reports the corresponding results for nominal returns using the predictors derived from the BGI and WA nominal DDM models. These models have greater predictive power than the two real models for the relevant sample periods, with R2 at around 2–3%, despite the fact that they were made in real time and contain no “look-ahead” bias. The estimates of the coefficients of the predicted return are little affected by the omission of 1974.3 and, as in the two previous cases, are close to the theoretical value of unity but are not significantly different from zero either.

M.J. Brennan and Y. Xia

The evidence from Table 19.5 is thus broadly consistent with the earlier observation that the predictive coefficient estimates, while close to their theoretical values, are associated with large estimation error and that the weak statistical evidence of predictive power of  series at a quarterly horizon does not provide much information on their economic importance to investors with horizons of 20 years or longer.

19.5.4 Historical Simulations In this section we report the results of simulating the optimal and unconditional policies using each of four  series, A&B 1;1 , IL 2;1 , BGI 3;2 , and WA 4;2 , for a long horizon investor with a relative risk aversion coefficient,  , of 5 when the equity allocation is constrained to lie between 0 and 100%. Allocations to stocks were revised quarterly, and borrowing and short sale constraints were imposed. Under the unconditional constrained strategy, the fraction of the portfolio that is allocated to the risky asset is determined from Equation (19.18) subject to the constraint that 0  x u  1, and is constant over time. Under the constrained optimal strategy, the fraction of wealth allocated to the risky asset is determined by solving an optimal problem whose value function depends on ; it is described in Appendix 19A. The return on the market portfolio is taken as the historical real (nominal) return on the S&P 500 index for each quarter and the riskless interest rate is taken as the realized real (nominal) return on a 30-day Treasury from CRSP for the real 1;1 and 2;1 (the nominal 3;2 and 4;2 ). The instantaneous return volatility, P D 15:7% (17.1%), is determined from equation (var) by setting V .0:25/ equal to the sample volatility, 7.74% (8.40%), of quarterly S&P 500 index real (nominal) return, when either 1;1 or 2;1 (3;2 or 4;2 ) is used. The same value of P is used in both the optimal and the unconditional policies. The other parameters for the optimal policies are taken from the appropriate line of Table 19.4, and the constant unconditional equity premium  for the unconditional policy is calculated from the corresponding  from Table 19.4.27 For both the optimal and the 27

The unconditional equity premium  for the unconditional strategy was calculated in three different ways: (1) using the same  as that used in calculating the optimal strategy; (2) setting  to the sample mean of the S&P 500 Index excess return during the whole sample period of 1929 to 2002; and (3) setting  to the gradually updated sample mean excess return with the initial value calculated from 1929 to 1949 (or 1972 for the BGI or WA series). The first approach ensures that differences between the wealth realized under the optimal and the unconditional strategies are not caused by different assumptions about the

19 Persistence, Predictability, and Portfolio Planning

311

Table 19.5 Quarterly return prediction A. Real return predictive regressions

1

Sample period 1950.2–2002.2

Obs. 209

2

1950.2–1974.2, 1974.4–2000.2

208

3

1950.2–1974.2

97

4

1974.4–2002.2

111

1;2 as Predictor a0 a1 0.005 0.874 (0.46) (1.74) [0.43] [1.60]

R2 (%) 1.43

2;2 as Predictor a0 a1 0.009 0.701 (0.86) (1.53) [0.79] [1.27]

R2 (%) 1.12

0.005 (0.43) [0.38]

0.981 (2.02) [1.83]

1.94

0.008 (0.85) [0.75]

0.800 (1.81) [1.50]

1.57

0:019 (0.92) [1.04]

2.157 (2.08) [2.45]

4.37

0:020 (1.09) [1.06]

1.961 (2.42) [2.21]

5.79

0.011 (0.82) [1.07]

0.708 (1.22) [1.86]

1.35

0.017 (1.40) [2.36]

0.494 (0.88) [1.80]

0.70

B. Nominal return predictive regressions

1

Sample period 1973.2–2002.2

Obs. 117

2

1973.2–1974.2, 1974.4–2000.2

116

3;2 as Predictor a0 a1 0.003 1.026 (0.20) (1.86) [0.18] [1.69] 0.008 0.953 (0.48) (1.80) [0.41] [1.60]

2

R (%) 2.89

2.75

4;2 as Predictor a0 a1 0.006 0.924 (0.37) (1.80) [0.34] [1.66] 0.010 0.852 (0.67) (1.73) [0.54] [1.42]

R2 (%) 2.74

2.57

This table reports the results of regressing real and nominal quarterly returns of the S&P 500 stock index on forecasts of the return at the beginning of the quarter calculated from the estimated value of i;2 .i D 1; 2; 3; 4/ and the estimated parameters of the joint stochastic process using equation   1:0  e =4 R.t; t C 0:25/ D a0 C a1 i;2 t C "t ; i D 1; 2; 3; 4  where R.t; t C 0:25/ is the one quarter real return on the S&P 500 stock index in Panel A and is the corresponding nominal return in Panel B. In Panel A, the real return on the S&P 500 index is regressed on the estimated real A&B and IL i;2 .i D 1; 2/ series. In Panel B, nominal returns on the index is regressed on the estimated nominal BGI and WA i;2 .i D 3; 4/ series. The OLS t -ratios are reported in parenthesis and Newey-West adjusted t -ratios are in brackets

unconditional strategies under the real  (nominal ), the risk free rate  is set at a constant 1.1% (4.1%), which is the sample mean of the realized real (nominal) return of the 30-day Treasury bill rate from 1950 to 2002. Figures 19.12 and 19.13 summarize the results of the simulations under the real A&B 1;1 and IL 2;1 . For each figure the investor is assumed to be concerned with maximizing the expected utility of wealth on the last date included in the figure. For example, Fig. 19.12a describes the evolution of wealth of an investor who starts investing at the end of the first quarter (or equivalently the beginning of the second quarter) of 1950 with a horizon of the end of the first quarter of 1970; thus his initial horizon is 20 years and deunconditional equity premium. Unconditional strategies based on (2) and (3) have similar realized wealth as that based on (1) except for the period 1950–1970, during which unconditional strategies based on (2) and (3) significantly outperform that based on (1) but still underperform the optimal strategy. Since the relative performance of the optimal strategy is consistent across the three unconditional strategies, we only report results of the unconditional strategy based on (1) and omit those based on (2) and (3) for brevity.

creases each period. Investment decisions are assumed to be made at the beginning of each quarter based on the current value of . The figure shows that an investor, who would have followed the optimal strategy based on 1;1 over this period, would have vastly outperformed his unconditional counterpart. Much the same pattern is visible in Fig. 19.13a, which is based on 2;1 . Figures 19.12b and 19.13b show that the optimal strategies continue to outperform the unconditional strategies but by a smaller margin over the subsequent 20-year period 1970–1990. Both the 1;1 and 2;1 based strategies lose more in the oil-price related bear market of 1974 on account of their more aggressive stock positions but more than make up for this by the end of the period. A 12year horizon investor over the period 1990–2002 does better (about as well) under the optimal strategies than (as) under the unconditional strategy using 1;1 .2;1 / series. Finally, the simulations for an investor with a 52-year horizon starting in 1950 show that the “optimal” investor ends up well ahead. Since the i;1 and i;2 .i D 1; 2/ series are trivially different we do not report the results for the 2 series.

312 Fig. 19.12 Cumulative real wealth under the optimal and unconditional strategies for a long horizon. Investor (1;1 as the predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real A&B 1;1 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1950.06 with a terminal date in 1970.03. In panel b, the investment horizon is also 20 years and the investor starts investing in 1970.06 with a terminal date in 1990.03. In panel c, the investment horizon is 13 years and the investor starts investing in 1990.06 with a terminal date in 2002.06. In panel d, the investment horizon is 53 years and the investor starts investing in 1950.06 with a terminal date in 2002.06 Fig. 19.13 Cumulative real wealth under the optimal and unconditional strategies for a long horizon. Investor (2;1 as the predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real IL 2;1 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1950.06 with a terminal date in 1970.03. In panel b, the investment horizon is also 20 years and the investor starts investing in 1970.06 with a terminal date in 1990.03. In panel c, the investment horizon is 13 years and the investor starts investing in 1990.06 with a terminal date in 2002.06. In panel d, the investment horizon is 53 years and the investor starts investing in 1950.06 with a terminal date in 2002.06

M.J. Brennan and Y. Xia

19 Persistence, Predictability, and Portfolio Planning Fig. 19.14 Cumulative nominal wealth under the optimal and unconditional strategies for a long horizon. Investor (3;2 as the Predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real BGI 3;2 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1973.03 with a terminal date in 1992.12. In panel b, the investment horizon is around 10 years and the investors start investing in 1993.03 with a terminal date in 2002.03. In panel c, the investment horizon is about 30 years and the investor starts investing in 1973.03 with a terminal date in 2002.03

Fig. 19.15 Cumulative nominal wealth under the optimal and unconditional strategies for a long horizon. Investor (4;2 as the predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real WA 4;2 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1973.06 with a terminal date in 1993.02. In panel b, the investment horizon is around 10 years and the investors start investing in 1993.06 with a terminal date in 2002.06. In panel c, the investment horizon is about 30 years and the investor starts investing in 1973.06 with a terminal date in 2002.06

313

314

Both the A&B and Ilmanen DDM series are constructed by projecting historical growth rates of aggregate series such as profits or GDP. In contrast, the nominal BGI and WA DDM series constructed by investment professionals relies on bottom-up forecasts of individual firm growth rates. Figures 19.14 and 19.15 compare the (nominal) wealth outcomes from the unconditional vs. the optimal strategy derived from BGI 3;2 and WA 4;2 series for the 20-year period 1973–1993 and the 11-year period 1993–2003. In the first period, the wealth outcome under the optimal strategy is 53–67% higher than that under the unconditional strategy: it is noteworthy that these strategies do not suffer the large losses of the A&B and Ilmanen-based strategies during 1974. For the 11-year investment period 1993–2003, the optimal strategies based on both 3;2 and 4;2 series underperform the unconditional ones by 10–20%. It is interesting to note that the unconditional strategy out performs the optimal strategies during the bubble period at the end of the 1990s, but that it substantially under performs as the bubble collapses during the period 2000–2003: it is precisely in such circumstances that we would expect that a dynamic strategy that takes account of the long run expected rate of return implicit in asset prices to do well. Figures 19.14c and 19.15c compare the outcome of the unconditional strategies with those of the optimal ones that start with a 30-year horizon in 1973. Over this longer horizon, the optimal strategies show a consistent advantage. While we should be careful from inferring too much from these historical simulations that represent only a single sample path of stock prices for a single level of risk aversion, it is encouraging that the optimal dynamic strategies tend to outperform naive unconditional strategies, even when they are based on real time data.28

19.6 Conclusion There is considerable disagreement about whether or not stock returns are time-varying and predictable. On the one hand, there is ample evidence of in-sample return predictability from regressions of stock returns on instruments such as the dividend yield and short-term interest rate, and from variance ratio tests. On the other hand, it has been argued 28

The instantaneous expected returns estimated from the DDM long run expected returns depend on parameters of the expected return  and the dividend growth g processes,  ;  , and g etc. These parameters were estimated using data from the whole sample period so that our estimates of instantaneous expected returns, even when they are based on real time DDM estimates, rely on future data. For the A&B and IL series, we also estimated  for the period of 1950–2002 by first estimating the parameters using data only from 1900 to 1949, and the superior performance of the optimal strategy remains unchanged. We do not have long enough sample for the BGI and WA series, which are only available starting from 1973, to carry out this robustness check.

M.J. Brennan and Y. Xia

that in-sample predictive regressions perform poorly out of sample, so that the evidence of time variation should be discounted. In this paper, we have shown that time variation in expected returns that implies both large variation in stock market valuation ratios and substantial gains to long-term dynamic investment strategies is likely to be hard to detect by standard statistical methods. As a result, weak statistical evidence for return predictability does not in itself imply that return predictability is economically insignificant. In view of the difficulty of estimating expected returns from regressions of realized returns on instruments such as the interest rate or dividend yield, we suggest that it is likely to be more productive to estimate the expected long run rate of return by comparing the current level of stock prices with forecasts of expected future dividends in the dividend discount model (DDM) paradigm. This forwardlooking approach has the advantage that it does not rely on hard-to-estimate regression coefficients from past data. The disadvantage is that the rate of return that emerges from the dividend discount rate model is a long run expected rate of return. In order to use the DDM expected rate of return estimate in the dynamic portfolio planning, we show how the instantaneous expected rates of return can be estimated from the DDM long run expected rate of return under the assumption that the instantaneous expected rate of return follows an Ornstein-Uhlenbeck process; the technique is also extended to a setting in which the expected growth rate of dividends, instead of being constant, also follows an Ornstein-Uhlenbeck process. Time series of expected rates of return are estimated for four time series of DDM expected rates of return. Two of them are historical “back-casts,” and two are real time estimates provided by investment professionals. Simulations using realized S&P 500 index returns and the 30-day Treasury bill rates suggest that there may be significant benefits from the use of expected instantaneous rates of return derived from dividend discount model expected returns, even when the expected returns are estimated in real time. In this paper, however, we have examined the benefit of the optimal market timing strategy without taking account of misspecification of, or errors in estimating, the stochastic process for the expected instantaneous rate of return. Determining the sensitivity of DDM-based dynamic portfolio strategies to errors in specifying and estimating the stochastic process for the instantaneous expected return is a task for future work.

References Amihud, Y. and C.M. Hurvich. 2004. “Predictive regressions: a reduced bias estimation method”. Journal of Financial and Quantitative Analysis 39, 813–41.

19 Persistence, Predictability, and Portfolio Planning Arnott, R. D. and P. L. Bernstein. 2002. “What risk premium is normal?” Financial Analysts Journal 58, 64–85. Barberis, N. 2000. “Investing for the long run when returns are predictable.” Journal of Finance 55, 225–264. Bossaerts, P. and P. Hillion. 1999. “Implementing statistical criteria to select return forecasting models: what do we learn?” Review of Financial Studies 12, 405–428. Boudoukh, J., M. Richardson and T. Smith. 1993. “Is the ex ante risk premium always positive? – A new approach to testing conditional asset pricing models.” Journal of Financial Economics 34, 387–408. Brav, A. and J. B. Heaton. 2002. “Competing theories of financial anomalies.” Review of Financial Studies 15, 575–606. Brennan, M. J., E. S. Schwartz and R. Lagnado. 1997. “Strategic asset allocation.” Journal of Economic Dynamics and Control 21, 1377–1403. Brennan, M. J. and W. N. Torous. 1999. “Individual decision making and investor welfare.” Economic Notes 28, 119–143. Brennan, M. J. and Y. Xia. 2001. “Stock price volatility and the equity premium.” Journal of Monetary Economics 47, 249–283. Brennan, M. J. and Y. Xia. 2001. “Assessing asset pricing anomalies.” Review of Financial Studies 14, 905–942. Brennan, M. J. and Y. Xia. 2002. “Dynamic asset allocation under inflation.” The Journal of Finance 57, 1201–1238. Campbell, J. Y. 2001. “Why long horizons? A study of power against persistent alternative.” Journal of Empirical Finance 8, 459–491. Campbell, J. Y., A. W. Lo and A. Craig MacKinlay. 1997. The econometrics of financial markets, Princeton University Press, Princeton, New Jersey. Campbell, J. Y. and R. J. Shiller. 1998. “Valuation ratios and the long run stock market outlook.” Journal of Portfolio Management 24, 11–26. Campbell, J. Y. and R. J. Shiller. 2001. “Valuation ratios and the long run stock market outlook: an update.” in Advances in behavioral finance, vol II, N. Barberis and R. Thaler (Eds.). Russell Sage Foundation, 2004. Campbell, J. Y. and S. B. Thompson. 2004. “Predicting the equity premium out of sample: can anything beat the historical average?” working paper, Harvard University. Campbell, J. Y. and L. Viceira. 2001. “Who should buy long term bonds?” American Economic Review 91, 99–127. Campbell, J. Y. and M. Yogo. 2003. “Efficient test of stock return predictability.” working paper, Harvard University. Cochrane, J. H. 1997. “Where is the market going? Uncertain facts and novel theories.” Economic Perspectives XXI(6) (Federal Reserve Bank of Chicago), 3–37. Cochrane, J. H. 1999. “New facts in finance.” Economic Perspectives XXIII(3) (Federal Reserve Bank of Chicago), 36–58. Fama, E. F. and R. R. Bliss. 1987. “The information in long-maturity forward rates.” American Economic Review 77, 680–692. Fama, E. F. and K. R. French. 1988a. “Dividend yields and expected stock returns.” Journal of Financial Economics 23, 3–25. Fama, E. F. and K. R. French. 1988b. “Permanent and temporary components of stock prices.” Journal of Political Economy 96, 246–273. Fama, E. F. and K. R. French. 2002. “The equity premium.” Journal of Finance 57, 637–659. Feller, W. 1951. “An Introduction to Probability Theory and its Applications”, John Wiley, New York. Goetzmann, W. N. and P. Jorion. 1993. “Testing the predictive power of dividend yields.” The Journal of Finance 48, 663–679. Goyal, A. and I. Welch. 2003. “Predicting the equity premium with dividend ratios.” Management Science 49, 639–654. Goyal, A. and I. Welch. 2004. “A comprehensive look at the empirical performance of equity premium prediction.” Working paper, Yale University.

315 Hakansson, N. H. 1970. “Optimal investment and consumption strategies under risk for a class of utility functions.” Econometrica 38, 587–607. Ilmanen, A. 2003. “Expected returns on stocks and bonds.” Journal of Portfolio Management Winter 2003, 7–27. Kandel, S. and R. F. Stambaugh. 1996. “On the predictability of stock returns: an asset-allocation perspective.” The Journal of Finance 51, 385–424. Keim, D. B. and R. F. Stambaugh. 1986. “Predicting returns in the stock and bond markets.” Journal of Financial Economics 17, 357–390. Kim, T. S. and E. Omberg. 1996. “Dynamic nonmyopic portfolio behavior.” Review of Financial Studies 9, 141–161. Kleidon, A. W. 1986. “Variance bounds tests and stock price valuation models.” Journal of Political Economy 94, 953–1001. LeRoy, S. and R. Porter. 1981. “The present value relation: tests Based on Variance Bounds.” Econometrica 49, 555–574. Lewellen, J. W. 2004. “Predicting returns using financial ratios.” Journal of Financial Economics, 74, 209–235. Lewellen, J. W. and J. A. Shanken. 2002. “Learning, asset-pricing tests, and market efficiency.” The Journal of Finance 57, 1113–1145. Lo, A. W. and A. Craig MacKinlay. 1988. “Stock market prices do not follow random walks: evidence from a simple specification test.” Review of Financial Studies 1, 41–66. Marsh, T. A. and R. C. Merton. 1986. “Dividend variability and variance bounds tests for the rationality of stock market prices.” American Economic Review 76, 483–498. Merton, R. C. 1971. “Optimum consumption and portfolio rules in a continuous-time model.” Journal of Economic Theory 3, 373–413. Philips, T. K., G. T. Rogers and R. E. Capaldi. 1996. “Tactical asset allocation: 1977–1994.” Journal of Portfolio Management 23, 57–64. Poterba, J.M. and L.H. Summers. 1988. “Mean reversion in stock prices: evidence and implications”, Journal of Financial Economics 22, 27–59. Shiller, R. 1981. “Do stock prices move too much to be justified by subsequent changes in dividends?” American Economic Review 71, 421–436. Stambaugh, R. F. 1999. “Predictive regression.” Journal of Financial Economics 54, 375–421. Wachter, J. 2002. “Portfolio and consumption decisions under meanreverting returns: an exact solution for complete markets.” Journal of Financial and Quantitative Analysis 37, 63–91. Xia, Y. 2001. “Learning about predictability: the effect of parameter uncertainty on dynamic asset allocation.” The Journal of Finance 56, 205–246.

Appendix 19A The Optimal Strategy The investor is assumed to have an iso-elastic utility function defined over end-of-period wealth at the investment horizon T : 8 3 2 < T WT1r  > 15 max E0 4u.WT / D e ; 1r x  D1 : e T ln WT subject to the following dynamic budget constraint: dW D Œx.˛ C ˇ  r/ C r dt C xP dzP ; W

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where x is defined as the proportion of wealth invested in the single risky asset whose stochastic process was given in Equations (19.1–19.2). The risk free interest rate, r, is assumed to be constant for simplicity. Under the iso-elastic utility function, the indirect utility function, J.W; ; t/  maxxt Et Œu.WT / , is homogeneous in W : W 1r J.W; ; t/ D e t T .; t/; 1r where satisfies the following Bellman equation:  0 D max x

  1 2  ' C .  / C .1   /x P ' 2  C .1   /Œx.˛ C ˇ  r/ C r 

1   .1   /x 2 P2 Ct 2



a3 D

C./ D

2a3 .1  e  / ; .  a2 / C . C a2 /e 

B./ D

4a3 .˛ C ˇ  r/.1  e  2 /2 ; Œ.  a2 / C . C a2 /e 

P  yt C ŒB./ C C./yt ; P P2

and the function .; t/ under the unconstrained optimal strategy is reduced to: 1

.; t/ D exp A./ C B./yt C C./yt2 2

(19A.5) 

" A./ D a3

.ˇS /2 2 2 .˛ C ˇ  r/2 C   a2 2

(19A.6) !

# C r.1   /   

(19A.1)

where,  D @2 ;  D @ , and t D @ are the partial @ @t derivatives of with respect to  or t. Denote the equity premium by yt  ˛ C ˇt  r and the remaining investment horizon by   T  t, then the investor’s unconstrained optimal dynamic policy xt is given by:



(19A.4)

then q D a23 4a1 a3 > 0 for all  1, which is the condition for the well-behaved normal case. In this paper, we focus on dynamic strategies under  > 1, so A./; B./, and C./ are given by the normal solution:

C

4a3 .˛ C ˇ  r/Œ.2a2 C /e   4a2 e  3 Œ.  a2 / C . C a2 /e 

C

ˇ ˇ 2a3 .ˇs /2 ˇˇ .  a2 / C . C a2 /e  ˇˇ ln ˇ ˇ; 2 2  a22

@2

xt D

1 ; P2

 2

C 2a2  

(19A.7)

p where  D q. If the optimal allocation x  is constrained to be between zero and one (i.e., no borrowing or short-sale is allowed), then no closed-form solution is available. Both the constrained optimal policy x  and the associated function are solved numerically from the Bellman Equation (19A.1) subject to the constraint 0  x   1.



o

where A./; B./, and C./ are solutions to a system of three ordinary differential equations with boundary conditions of A./ D 0; B./ D 0; and C./ D 0 at  D 0: The details of the equations are contained in Kim and Omberg (1996) for the general HARA utility and in Xia (2001) for the CRRA utility. In particular, let

2  1 2 P ; 1C (19A.2) a1 D ˇ 

.1   /ˇ P a2 D 2  ; (19A.3) P

Appendix 19B The Unconditional Strategy If the investor believes that the equity premium is constant, then the investment opportunity set is constant from the investor’s perspective. The unconditional strategy, xu D

y ˛ C ˇ  r D ; 2 P P2

is based on the long run mean of the equity premium. However, the wealth process of the investor evolves according to the true dynamics of the equity premium given by Equations (19.1–19.2): dW D Œx u .˛ C ˇ  r/ C r dt C x u P dzP ; W

(19B.1)

19 Persistence, Predictability, and Portfolio Planning

317

The indirect utility function J u .W; ; / is then given by:   W 1r W 1r J u .W; ; t/ D Et e T T D e T T Et 1r 1r  

  WT exp .1   / ln Wt  

 1r WT T WT exp Et .1   / ln De 1r Wt 

 WT C Vart .1   / ln : Wt Solving for ln WWTt and its first two conditional moments from Equation (19B.1) gives the following result: J.W; ; t/ D e t

Wt1r D. /CE. /t e ; 1r

(19B.2)

where "

C.1   /

2 "

2ˇP C .1   / 

.1   /x u ˇ  C .1   / x u ˇP C.1   / 



 J .W; ; t/ D Et e m

!#

x u ˇ2 2

 D e T

WT1r m

.; t/ 1r (19C.2)

where WT is determined via the wealth dynamics (19C.1). Similar to the indirect utility function under the optimal strategy, we conjecture that the function m is of the form m . /CB m . /yC 1 C m . /y 2 2

m D eA

:

(19C.3)

where Am ./; B m ./, and C m ./ are solutions to a system of ordinary differential equations (ODE) similar to the case of the optimal strategy: dCm D a1m .C m /2 C a2m C m C a3m ; d

(19C.4)

with the boundary conditions:

1  e 

Am ./ D 0; B m ./ D 0; and C m ./ D 0 at  D 0: This system of ODEs has exactly the same form of solutions as that under the optimal strategy, but with slightly different coefficients, .a1m ; a2m ; a3m /. In particular, 

a1m

Appendix 19C The Myopic Strategy

D

ˇ 2 2

¤

a1 ; a2m

D a2 and a3m D

If the investor has a short investment horizon, then it is optimal to adopt the myopic market-timing strategy, ˛ C ˇt  r ; P2

The wealth process of the investor evolves according to the following dynamics: dW D Œx m .˛ C ˇt  r/ C r dt Cx m P dzP ; W

WT1r 1r

 1  dAm D a1m C m C .B m /2 C .˛ C ˇ  r/B m d 2 C.1   /r  ; (19C.6)



xu2 ˇ 2 2 1  e 2 1  ; (19B.3)  C .1   /2 2 2 2 1  e  E./ D .1   /x u ˇ : (19B.4) 

xm;t D

T

dBm 1 D a1m B m C m C a2m B m C .˛ C ˇ  r/C m ; d 2 (19C.5)

1 D./ D .1   / r C .x u /2 P2 2 ˇ 2 2

and the investor’s indirect expected utility function is given by

(19C.1)

.1   /ˇ P D2  P



1 D a3 P2

where a2 and a3 are given in (19A.2–19A.3) in Appendix 19A. Therefore, the solution to Am ./; B m ./, and C m ./ has exactly the same expression as that given by equations (19A.5–19A.7) in Appendix 19A except that a1 is replaced by a1m and  is replaced by m 

q a22  4a1m a3 :

318

M.J. Brennan and Y. Xia

Appendix 19D The Optimal Buy-and-Hold Strategy The dynamics of the stock price (with dividends reinvested) given in Equations (19.1–19.2) implies that the stock price at the time T conditional on information at t is:

 1 (19D.1) PT D Pt exp ˛ C ˇ  r  P2 .T  t/ 2 Cˇ.t  / Z

T

Cˇ t

1  e .T t / 

1  e .T s/ dz .s/ C P 

Z

T t

 dzP .s/ :

The optimal buy-and-hold strategy, x b , which is the proportion of current wealth Wt invested in the stock, then solves the following optimization problem by normalizing Wt to one dollar: 2

1 3 x b PPTt C .1  x b /e r.T t / 6 7 max Et 4 5 : (19D.2) b 1   x The strategy x b , the indirect utility J b , and the certainty equivalent wealth CEW b are all solved numerically.

Chapter 20

Portfolio Insurance Strategies: Review of Theory and Empirical Studies Lan-chih Ho, John Cadle, and Michael Theobald

Abstract A portfolio insurance strategy is a dynamic hedging process that provides the investor with the potential to limit downside risk while allowing participation on the upside so as to maximize the terminal value of a portfolio over a given investment horizon. First, this paper introduces the basic concepts and payoffs of a portfolio insurance strategy. Second, it describes the theory of alternative portfolio insurance strategies. Third, it empirically compares the performances of various portfolio insurance strategies during different markets and time periods. Fourth, it summaries the recent market developments of portfolio insurance strategies, especially in terms of the variations of features in CPPI investments. Finally, it addresses the impacts of these strategies on financial market stability. Keywords CPPI r OBPI r Risk-based portfolio insurance

20.1 Introduction Portfolio insurance refers to any strategy that protects the value of a portfolio of risky assets. The risky assets can be stocks, bonds, currencies, or even alternative assets, such as commodities, real assets, hedge funds, credits and so forth. If the value of the risky asset declines, the insurance or hedge will increase in value to help offset the decline in price of the hedged risky assets. If the price of the risky asset increases, the increase of the insured portfolio will be less than the increase in the risky asset itself but will nevertheless still increase.

L.-c. Ho () Department of Foreign Exchange, Central Bank of the Republic of China (Taiwan), 2, Roosevelt Road, Sec. 1, 10066 Taipei, Taiwan, ROC e-mail: [email protected] J. Cadle and M. Theobald Accounting and Finance Subject Group, University of Birmingham, Birmingham, UK

Table 20.1 illustrates how portfolio insurance works. In this example, the underlying risky asset is purchased for $95 and $5 is spent on portfolio insurance. The minimum amount that the insured investor can realize is $95, but the uninsured portfolio can fall in value to a low of $75 if the market falls. If the value of the risky asset increases, the value of the insured portfolio will increase, but at a smaller rate. Figure 20.1 illustrates the profit and loss of the insured and uninsured portfolio. Portfolio insurance allows market participants to alter the return distribution to fit investors’ needs and preferences for risk. Figure 20.2 shows the effect of insurance on the expected returns of a portfolio. Notice that the uninsured portfolio has greater upside potential as well as greater downside risk, whereas the insured portfolio limits the downside loss to the cost of the hedge. The upside potential of the insured portfolio is always below that of the uninsured portfolio. The cost of the insurance is the lower return for the insured portfolio should prices increase. While some investors would prefer the greater upside potential that the uninsured portfolio offers, risk-averse investors would prefer the limited-risk characteristics that the hedged portfolio offers.

20.2 Theory of Alternative Portfolio Insurance Strategies This section introduces the mechanisms behind various portfolio insurance strategies. These strategies share a common characteristic, a convex payoff function, which implies a “buy high and sell low” rule for the risky asset. If the market level declines an investor sells a fraction of the risky asset and buys the riskless asset; if the market level rises an investor switches from the riskless to the risky asset and rallies with the market. For illustrative purposes, we construct a portfolio that comprises a risky currency, the euro (EUR), and a risk-free currency, the US dollar (USD) overnight deposit account. For comparison purposes, we assume that the

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_20, 

319

320

Table 20.1 Mechanics of portfolio insurance: an example

L.-c. Ho et al.

Initial investment Cost of portfolio insurance

$100 $5

Amount of investment going toward securities

$95

Amount invested D 100 Value of portfolio at year end ($)

Return on uninsured portfolio (%)

Return on insured portfolio ($)

Net return on insured portfolio (%)

75

25

95

5

80

20

95

5

85

15

95

5

90

10

95

5

95

5

95

5

100

0

95

5

105

5

100

0

110

10

105

5

115

15

110

11

120

20

115

15

125

25

120

20

130

30

125

25

Fig. 20.1 Gains and losses of insured and uninsured portfolios: an example

initial value of EUR in the portfolio is 25%, short positions in EUR are not allowed, but that an investor can borrow via USD to buy EUR.

20.2.1 Portfolio Insurance with Synthetic Put (Option-Based Portfolio Insurance) The most popular portfolio insurance strategy is the synthetic put approach of Rubinstein and Leland (1981), also

referred to as an option-based portfolio insurance (OBPI) strategy. If an investor holds a risky asset and buys one atthe-money put option on that asset, the value of the resultant portfolio will not be less than the exercise price net of the premium at expiration. Thus, the investor effectively hedges the portfolio against downside risk. However, the implied volatility is usually higher than the historical volatility in option markets, which indicates that an actual option is more expensive than a synthetic one. Therefore, a synthetic option is often used as an alternative hedging vehicle in practice.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

321

Fig. 20.2 Expected returns on insured and uninsured portfolios

In a currency portfolio, a synthetic put can be created by dynamically rebalancing the portfolio between the risky (foreign) and riskless (domestic) currencies according to the delta of a put option, D D e rf T ŒN.d1 /  1 :

(20.1)

Specifically, the investment proportion of the risky currency in the portfolio value can be expressed as: WSynP ut D

Se rf T N.d1 / S C D S D ; V S CP

(20.2)

where P is the premium of a European put option on foreign currency from Garman-Kohlhagen (1983), P D Xe rT N.d2 /  Se rf T N.d1 /;

determined by the changes of delta. Table 20.2 illustrates the adjustment process. One important parameter in the synthetic put approach is the volatility. Rendleman and O’Brien (1990), Do (2002), Do and Faff (2004), and Bertrand and Prigent (2005) use different measures of volatility, such as implied, historical and Leland’s (1985) adjusted historical volatility, and discuss the effects of volatility estimation on the synthetic put strategy. Additionally, in order to reduce transaction costs, some revisions on the rebalancing discipline are adopted, such as the time discipline with weekly/monthly rebalancing, the price discipline with adjustments according to changes in the value of the risky asset itself, or the volume discipline corresponding to changes in the number of shares that should be held.

(20.3)

p ln.S=X /C.rrf C 2 =2/T p and d2 D d1   T ; S is and d1 D  T the spot exchange rate, r is the domestic risk-free interest rate, rf is the foreign risk-free interest rate, ¢ is the volatility of spot exchange rate return, X is the exercise price, T is the time to expiration, and N. / is the cumulative standard normal distribution function. Assume that the initial portfolio contains 1,000,000 units of EUR. In order to make the initial value of the EUR position in the portfolio equal to 25%, an investor needs to create a synthetic put with a delta of 0:75 and sell 750,000 units of the EUR. The dollar income from the EUR sold is kept in an USD cash account and earns the overnight deposit rate. As the EUR depreciates, the synthetic put moves further in-the-money and an investor sells more EUR; conversely, an investor withdraws money from the USD cash account to buy EURs as the EUR appreciates. The daily adjustment is

20.2.2 Constant Proportion Portfolio Insurance Another simplified approach, not involving complex “Greeks,” is the constant proportion portfolio insurance (CPPI) strategy developed by Black and Jones (1987) and Perold (1986), and extended to fixed-income instruments by Hakanoglu et al. (1989). In CPPI, the exposure to the risky asset, E, is always kept at the cushion times a multiplier, m. The cushion, C , is the difference between the portfolio value, V , and a protected floor value, FL, and the multiplier is constant throughout the investment period: C D V  FL;

(20.4)

E D m C:

(20.5)

2007/1/1 2007/1/2 2007/1/3

1.3201 1.3272 1.3170

(1) 0.750000 0.729593 0.764881

(2) 750,000 20,407 35,288

(3) $330,025 $358,884 $309,651

.4/ D .3/=.9/ 25.0 27.1 23.5

Table 20.2 Daily adjustment process of portfolio insurance with synthetic put EUR asset EUR amt. USD Synthetic put EURUSD Delta EUR asset (%) sold ./ value (5) $990,075 $27,084 $46,474

$ Income of EUR sold

USD Cum. Interest income income .6/ D .5/C.6/C.7/ (7) $990,075 $145 $963,137 $142 $1,009,753 $148

.9/ D .3/ C .6/ .10/ D d.9/=.9/ $1,320,100 $1,322,020 0.15 $1,319,404 0.20

.8/ D .7/=.9/ 75.0 72.9 76.5

Portfolio return (%)

Portfolio USD value

USD asset (%)

322 L.-c. Ho et al.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

The investment proportion of the risky currency in the portfolio value can be expressed as: WCPPI D

E : V

323

20.2.3 Portfolio Insurance with Downside Risk Control (Risk-Based Portfolio Insurance)

(20.6)

In a portfolio of securities, the floor is usually set as the discount price of a zero-coupon bond, which approaches the par value at the end of the investment horizon. In a currency portfolio, the floor is set as the forward rate, with a maturity equal to the investment horizon, that appreciates with time; that is, Ft D F0  exp.r0  t/, where F0 D S0  exp..r0  rf;0 / T /. Note that when the domestic riskfree rate is higher than the foreign risk-free rate, the forward rate is at a premium, which would result in short-selling of the risky asset at the beginning of the investment horizon; that is, E D m C D m .V  FL/ D m Units .S  F / < 0. In order to make the initial value of the EUR position in the portfolio equal to 25%, a discount factor (DF) is needed, and the multiplier1 also needs to be trimmed at the beginning of investment horizon. Thus, the synthetic floor is set as: Ft D DF  F0  exp.r0  t/:

(20.7)

Table 20.3 illustrates the adjustment process. As EUR appreciates, the portfolio value increases more than the floor level increases, the cushion increases, and investors buy EUR, vice versa. There are various market developments in CPPI mechanism with additional features, such as constraints on the investment level, constraints on leverage, variable floors, and variable multipliers and so forth, which will be introduced in later section.

The idea of risk management has become widespread and has been applied to the asset allocation problem. By dynamically controlling for downside risk while maximizing the expected return of a portfolio, the concept of the modern riskbased asset allocation process is analogous to that of the classic portfolio insurance strategy. Zhao and Ziemba (2000) are the first to explore the portfolio payoffs between the two approaches. There are basically two popular measures of downside risk, one is the Value-at-Risk (VaR) and the other more coherent one is the Expected Shortfall (ES). Accordingly, there are two risk-based portfolio insurance (RBPI) strategies. VaR-based portfolio insurance is a strategy that permanently controls the shortfall risk of the portfolio. The allocation to the risky asset is adjusted each day so that the shortfall probability – the probability of realizing a portfolio return .RP / that falls below a pre-specified threshold .R/ at the end of the investment period – does not exceed a target value, ˛ (say 5%): Prob.RP < R/  ˛: (20.8) To operationalize the shortfall risk, Herold et al. (2005) use the lower partial moment of order zero, which is equivalent to VaR, and assume a normally distributed portfolio returns: LPM 0 .RP / D N

  u ; 

RP D W  RA C .1  W / RF ;

(20.9) (20.10)

where u and  are the mean and the volatility of portfolio returns, respectively, and  is the required portfolio return. RA and RF are the return of the risky and the riskless asset, respectively. Controlling for a fixed shortfall probability, the dynamic allocation process is solved for the weight of the risky asset so that the portfolio return is always higher than the required return (i.e., VaR): 1

The magnitude of the multiplier depends on the discount factor. The lower the discount factor, the lower the multiplier in order to keep the initial exposure fixed at 25%; that is, E D m .V  FL/.

NVaR˛;t .RP / D ut .RP /  N 1 .˛/t .RP /:

(20.11)

Table 20.3 Daily adjustment process of constant proportion portfolio insurance Floor discount factor .DF/ D 0:94; multiplier .m/ D 5:24592 CPPI

2007/1/1 2007/1/2 2007/1/3

EURUSD

1.3201 1.3272 1.3170

Portfolio USD value

Floor level

EUR asset USD value

EUR amount

(1)

(2)

.3/ D ..1/–.2// m

(4)

$1,320,100 $1,322,020 $1,319,562

1.2572 1.2574 1.2577

$330,025 $338,752 $324,510

250,000 255,238 246,401

EUR asset (%) .5/ D .3/=.1/ 25.0 25.6 24.6

USD asset USD value

USD asset (%)

Portfolio return(%)

.6/ D .1/–.3/

.7/ D .6/=.1/

.8/ D d.1/=.1/

$990,075 $963,137 $1,009,753

75.0 74.4 75.4

0.15 0.19

324

L.-c. Ho et al.

Table 20.4 illustrates the daily adjustment process. Since the VaR of a portfolio return does not change dramatically each day during a certain period of time, the frequency of adjustment is much less than that in traditional portfolio insurance strategies. In comparison with the traditional portfolio insurance strategy, Herold et al. (2005) show that the VaR-based model can be regarded as a generalized version of CPPI with a dynamic and time-varying implied multiplier. The risky currency investment proportion is then given by: WVaR D

 ; Z˛  

(20.12)

where Z˛ is the ˛-quantile of the standard normal distribution and the process drift is assumed to be equal to zero at the daily differencing interval. The implied multiplier of the VaR model is obtained by: W V C =V V 1 1 E D  D D ; mimpl D D C C Z˛   C Z˛   VaR .RP / (20.13) where  D C=V is expressed in percentage terms. Thus, the inverse of the CPPI multiplier can be interpreted as the maximum loss or worst case return that is allowed to occur over the next period. Bertrand and Prigent (2001) propose that the upper bound on the multiplier can be determined by the extreme value theory (EVT). ES-based portfolio insurance is a strategy that permanently controls the expected shortfall of the portfolio. The expected shortfall is defined as the expected value of the loss of a portfolio in a certain percentage of worst cases within a given holding period. The allocation to the risky asset is adjusted each day so that the expected shortfall does not exceed a target value. Following the idea of Herold et al. (2005), Hamidi et al. (2007) propose a conditional CPPI multiplier that links the traditional CPPI strategy with the RBPI strategy. The conditional CPPI multiplier is determined by keeping the risk exposure of a portfolio constant, and the risk exposure can be defined either by the VaR or by the ES: mcond D

1 1 or D VaR˛ .RP / ES˛ .RP /

where HVaR˛;t .RP / D Percentile.RPt 260 ; : : : ; RPt I 5%/  R, among which HVaR˛;t .RP / is the historical VaR of portfolio returns at date t at the 1  ’% confidence level, RPt 260 ; : : : ; RPt indicates the daily portfolio return series given a certain weight of the risky asset during the past year, and EŒ is the expectation operator. Table 20.5 illustrates the daily adjustment process. Since the ES of a portfolio return does not change dramatically each day during a certain period of time, the frequency of adjustment is also much less than that in the traditional portfolio insurance strategies.

(20.14)

By assuming a normally distributed portfolio return, the expected shortfall and the VaR are scalar multiples of the standard deviation. There will be no differences across portfolio strategies employing variance, VaR and ES risk measures. Therefore, Ho et al. (2008) use the historical distribution of portfolio returns to operate the ES-based portfolio insurance strategy: HES˛;t .RP / D EŒRP jRP < H VaR˛;t .RP /  R; (20.15)

20.3 Empirical Comparison of Alternative Portfolio Insurance Strategies In the late 1980s and early 1990s, much of the research in this area focused on simulated comparisons of the performances among alternative portfolio insurance strategies. In the 2000s, most studies use empirical data to analyze the circumstances where one strategy would outperform the others based on some modifications on certain portfolio insurance mechanisms, such as volatility input, rebalancing intervals and so forth. The performances are typically evaluated by floor protection – whether they achieve the desired floor at the end of the investment horizon, the cost of insurance – the opportunity cost of forfeiting upward markets, and the Sharpe ratio. Listed below are some of the studies that have appeared in academic journals.

20.3.1 Zhu and Kavee (1988) Zhu and Kavee evaluate and compare the performances of the two traditional portfolio strategies, the synthetic put approach of Rubinstein and Leland (1981) and the constant proportion approach of Black and Jones (1987). They employ a Monte Carlo simulation methodology, assuming lognormally distributed daily returns with an annual mean return of 15% and paired with different values for market volatility, in order to discover whether these strategies can really guarantee the floor return and how much investors have to pay for the protection. Both strategies are able to reshape the return distribution so as to reduce downside risk and retain a certain part of the upside gains. However, they demonstrate that a certain degree of protection can be achieved at a considerable cost. There are two types of costs in implementing a portfolio insurance strategy. The first is the explicit cost, that is, the transactions costs. The other is the implicit cost, which is the average return forgone in exchange for protection against

EURUSD

1.3201 1.3272 1.3170

Control VaR

2007/1/1 2007/1/2 2007/1/3

EUR adj. rule (2) 25.0 25.0 25.0

Portfolio USD value

(1) $1,320,100 $1,322,020 $1,319,617

EUR asset USD value .4/ D .3/ S $330,025 $331,800 $329,250

EUR amount .3/ D .2/ IniAmt 250,000 250,000 250,000

Table 20.4 Daily adjustment process of VaR-based portfolio insurance ’ D 5.%/

.5/ D .4/=.1/ 25.0 25.1 25.0

EUR asset (%)

$0 $0

.6/ D d.3/ S

USD deposit adjust .C/

.7/ D .6/ C .7/ C .8/ $990,075 $990,220 $990,367

Cumulated USD amount

(8) $145.2 $146.3 $145.3

Interest income

.9/ D .7/=.1/ 75.0 74.9 75.0

USD asset (%)

0.15 –0.18

.10/ D d.1/=.1/

Portfolio return (%)

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies 325

EURUSD

1.3201 1.3272 1.3170

Control ES

2007/1/1 2007/1/2 2007/1/3

EUR adj. rule (2) 25.0 25.0 25.2

Portfolio USD value

(1) $1,320,100 $1,322,020 $1,319,617

EUR asset USD value .4/ D .3/ S $330,025 $331,800 $331,374

EUR amount .3/ D .2/ IniAmt 250,000 250,000 251,613

Table 20.5 Daily adjustment process of ES-based portfolio insurance ’ D 5.%/

.5/ D .4/=.1/ 25.0 25.1 25.1

EUR asset (%)

$0 $2;124

.6/ D d.3/ S

USD deposit adjust .C/

.7/ D .6/ C .7/ C .8/ $990,075 $990,220 $988,242

Cumulated USD amount

(8) $145.2 $146.3 $145.0

Interest income

.9/ D .7/=.1/ 75.0 74.9 74.9

USD asset (%)

0.15 –0.18

.10/ D d.1/=.1/

Portfolio return (%)

326 L.-c. Ho et al.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

327

Using simulated stocks and bills prices, Perold and Sharpe examine and compare how the four dynamic asset allocation strategies – namely, the buy-and-hold, constant mix, CPPI, and OBPI – perform in bull, bear, and flat markets and in volatile and not-so-volatile markets. CPPI and OBPI strategies sell stocks as the market falls and buy stocks as the market rises. This dynamic allocation rule represents the purchase of portfolio insurance and has a convex payoff function, which results in a better downside protection and a better upside potential than a buy-andhold strategy. However, they do worse in relatively trendless, volatile markets. Conversely, a constant mix strategy – holding a constant fraction of wealth in stocks – buys stocks as the market falls and sells them as it rises. This rebalancing rule effectively represents the sale of portfolio insurance and has a concave payoff function, which leads to less downside protection than, and not as much upside as, a buy-and-hold strategy. However, it does best in relatively trendless and volatile markets. Perold and Sharpe suggest that no one particular type of dynamic strategy is best in all situations. Financial analysts can help investors understand the implications of various strategies, but they cannot, and should not, choose a strategy without a substantial understanding of an investor’s circumstances and desires.

where St is the price of the risky asset at date t; k0 the strike price, r0 the annual continuously compounded riskless rate of interest, ¢0 the ex ante volatility parameter at the beginning of the insurance period, and T the maturity of the option. Assume that at the beginning of the insurance period, one manager predicts a high-volatile market and a 20% annualized volatility, while a second manager believes a lowvolatile market and a 10% volatility. The delta for the first manager is higher, which means he would buy more insurance and allocate less to a position in the risky asset. Assume that the ex-post volatility turns out to be 10%. The second manager will have made the proper allocation between risky and riskless assets. In contrast, the high-estimate manager will have invested too little in the risky asset. This misallocation would lose the opportunity of participating in the price appreciation in a strong market. Thus, a manager who underestimates volatility will typically end up buying less insurance than is necessary to ensure a given return, while a manager who overestimates volatility will buy more insurance than is necessary and forgo gains. As for the issue of portfolio rebalancing, Rendleman and O’Brien examine adjustment frequencies by time intervals; namely, daily, weekly, monthly, and bi-monthly. The effect (error) is measured as the difference of the horizon insurance values between noncontinuous and continuous trading. They suggest that weekly rebalancing produces an amount of error that appears to be tolerable by most portfolio managers. More importantly, they address the biggest potential risk of implementing portfolio insurance strategies – the gap risk, by simulating the performance of the OBPI strategy over the period of the October 1987 market crash. They indicate that most insured portfolios would have fallen short of their promised values because the managers would not be able to adjust the portfolio in time before a big drop in the market. The gap risk will be discussed further in the financial stability section.

20.3.3 Rendleman and O’Brien (1990)

20.3.4 Loria et al. (1991)

Rendleman and O’Brien address the issue that the misestimation of volatility input can have a significant impact on the final payoffs of a portfolio using a synthetic put strategy. In an OBPI strategy, the daily portfolio adjustments depend on the delta of the put option on the risky asset. In the original Black and Scholes (1973) valuation equation, the delta at each day is a function of the following variables:

Loria, Pham, and Sim simulate the performance of a synthetic put strategy using futures contracts based on the Australian All Ordinaries Index for the period April 1984–March 1989. Their study contains 20 consecutive nonoverlapping 3-month insurance periods whose expiration dates coincide with the expiration dates of each SPI futures contract traded on the Sydney Futures Exchange. Four implementation scenarios are examined: a zero floor versus 5% floor that correspond to a portfolio return of 0% and 5% annually,

the downside risk. When the market becomes volatile, the protection error of the synthetic put approach increases, and the transaction costs may be unbearable. On the other hand, while the constant proportion approach may have lower transactions costs, its implicit cost may still be substantial.

20.3.2 Perold and Sharpe (1988)

Dt D f .St ; kN0 ; rN 0 ; 0 ; T /;

(20.16)

328

respectively; and a realized volatility versus Leland’s (1985) modified volatility. They report that there is no perfect guarantee of loss prevention under any scenario. Even in the scenario with a 5% floor and modified volatility, 2 out of 20 contracts do not meet the desired floor. In addition, the OBPI strategy is most effective under severe market under severe market conditions. In other periods characterized by insignificant market declines, the value of the insured portfolio is below that of the market portfolio. Loria, Pham, and Sim suggest futures mispricing may be one potential culprit for this outcome.

L.-c. Ho et al.

have a realized volatility of less than 15%; violation of either one or both of these conditions is regarded as indicating a turbulent period. All portfolio insurance strategies achieve 100% floor protection during tranquil periods whereas the futures-based OBPI approach records the highest portfolio return. During turbulent times, futures-based portfolio insurance continues to perform quite well. The 1987 stock market crash makes the assessment difficult because of trading halts. However, assuming the futures continued to trade during that crisis, from the algorithm’s perspective, the futures-based CPPI maintains a positive return, while the OBPI results in a negative return.

20.3.5 Do and Faff (2004) 20.3.6 Cesari and Cremonini (2003) This empirical paper is an extension of Loria et al. (1991). Do and Faff examine two approaches (the OBPI and CPPI) and conduct simulations across two implementation strategies (via Australia stock index and bills, and via SPI futures and stock index). Furthermore, they consider the use of implied volatility as an input into the model, and the use of a zero floor versus 5% floor. The dataset consists of 59 nonoverlapping 3-month insurance periods, which span from October 1987 to December 2002. Thus, their key contributions relative to Loria et al. (1991) are the examination of a futures-based CPPI, a fine-tuned algorithm that allows for dividend payments, the consideration of ex ante volatility information, and more up to date data. In terms of floor protection, the futures-based portfolio insurance implementation generally dominates its index-andbill rival in both floor specifications, which reflects the low transaction costs in the futures market. Furthermore, the perfect floor protection is possible when implied volatility is used rather than using ex-post volatilities. From the cost of insurance perspective, the futures-based strategy generally induces a lower cost of insurance than its index-and-bill rival in both floor specifications, which reflects the same reason as above. However, the cost of insurance is higher when implied volatility is used compared to ex-post volatilities. The possible explanation is that the implied volatility is often higher than the same period ex-post volatility, which results in overhedging. As for the performances between the OBPI and CPPI approaches, there is no strong evidence to distinguish between them. Within the futures-based implementation, the synthetic put appears to dominate the CPPI with respect to floor protection, while the latter appears to slightly outperform in terms of upside participation. They also examine whether portfolio insurance strategies work under stress conditions by assessing the strategies’ effectiveness during tranquil and turbulent periods. Tranquil periods are defined as ones that return more than 4% and

This study is an extensive comparison of a wide variety of traditional portfolio insurance strategies. There are basically five dynamic asset allocation strategies: (1) buy-and-hold (BH); (2) constant mix (CM); (3) constant proportion (without and with the lock-in of profits, CP and CPL); (4) the option-based approach (with three variations, BCDT, NL, PS); and (5) technical strategy (with two kinds of stop-loss mechanism, MA and MA2). Therefore, nine strategies in total are considered. For each strategy, eight measures for risk, return and risk-adjusted performance are calculated; namely, mean return, standard deviation, asymmetry, kurtosis, downside deviation, Sharpe ratio, Sortino ratio, and return at risk. The strategies are then compared in different market situations (bear, no-trend, bull markets) and with different market volatility periods (low, medium and high periods), taking into account transaction costs and discrete rebalancing of portfolios. The three market situations are defined accordingly if market average returns fall into the three ranges: .30%; 5%/; .5%; C5%/; .C5%; C30%/, and the three ranges for the volatile periods are (10%, 15%), (15%, 25%), (25%, 30%). Transaction costs are treated in two ways: a proportional cost to the value traded and a correction to the Leland’s (1985) option volatility. Two main rebalancing disciplines are used: a time discipline with weekly adjustment, and a price discipline with adjustment only when prices are increased/decreased by 2.5% with respect to the previous rebalance time. Monte Carlo simulations of MSCI World, North America, Europe, and Pacific stock market returns show that no strategy is dominant in all market situations. However, in bear and no-trend markets, CP and CPL strategies appear to be the best choice. In a bull market or in no-trend but with high volatility market, the CM strategy is preferable. If the market phase is unknown, CP, CPL, and BCDT strategies are recommended.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

In addition, these results are independent of the volatility level and the risk-adjusted performance measure adopted.

20.3.7 Herold et al. (2005) By constructing a fixed income portfolio, in which the risky asset is the JPMorgan government bond index and the riskless asset is cash (1-month yield), Herold, Maurer, and Purschaker compare the hedging performances between the traditional CPPI strategy and the risk-based (specifically, VaR-based) strategy with a 1-year investment horizon that begins at each year from 1987 to 2003. CPPI avoids losses in the bear years of 1994 and 1999. The mean return is inferior to (about 40 base points below) that of the risk-based strategy. CPPI also produces a higher turnover.

20.3.8 Hamidi et al. (2007) Although Hamidi, Jurczenko, and Maillet propose a conditional CPPI multiplier determined either by VaR or by ES, only a VaR-based measure is studied in this empirical work, and the ES measure is absent for future research. The data set contains 29 years of daily returns of the Dow Jones Index, from January 2, 1987 to May 20, 2005, 4,641 observations in total. They use a rolling window of 3,033 returns to estimate the VaR, and there are 1,608 VaRs in the out-of-sample period. They resort to eight methods of VaR calculation: one non-parametric method using the historical simulation approach; three parametric methods based on distributional assumptions: namely, the normal VaR, the RiskMetrics VaR based on the normal distribution, and the GARCH VaR based on the Student-t distribution; four semi-parametric methods using quantile regression to estimate the conditional autoregressive VaR (CAViaR): namely, the symmetric Absolute Value CAViaR, the Asymmetric Slope CAViaR, the IGARCH(1,1) CAViaR, and the Adaptive CAViaR. According to the 1,608 back-testing results, the Asymmetric Slope CAViaR is the best model to fit the data. After having calculated the VaR values, the conditional multipliers can be determined. The estimations spread between 1.5 and 6, which are compatible with multiple values used by practitioners in the market (between 3 and 8). Using the time-varying CPPI multipliers estimated by different methods, and a “multi-start” analysis – the fixed 1-year investment horizon beginning at every day of the out-of-sample period – they find that the final returns of these insured portfolios are not significantly different.

329

20.3.9 Ho et al. (2008) This empirical study presents a complete structure of comparing traditional portfolio insurance strategies (OBPI, CPPI) with modern risk-based portfolio insurance strategies (VaR-, ES-based RBPI). By constructing a currency portfolio, in which the risky asset is the Australian dollar and the riskless asset is a US dollar overnight deposit, Ho, Cadle, and Theobald compare the dynamic hedging performances between the traditional and the modern strategies with a 1-year investment horizon that begins at each year from 2001 to 2007. When implementing the OBPI strategy, the delta is calculated based on the modified Black-Scholes formula, Dt D f .St ; kN0 ; rt ; t ; T /. That is, daily annualized historical volatilities are used as inputs in the put-replication process instead of the constant ex ante or the implied volatility in the original model to mitigate the volatility misestimation problem. Besides, the latter two parameters would make the portfolio daily returns more volatile. The interest rates are also updated daily for the same reason. When CPPI is implemented, the possible upper bounds of the multiplier are examined via the EVT, which ranges from 4 to 6 corresponding to different confidence levels. When risk-based approaches are employed, both a historical distribution and a normal distribution with exponentially weighted volatility of risky asset returns are assumed. A daily rebalancing principle is adopted in their research without any modification to show the original results of hedging. The performances are evaluated from six differing perspectives. In terms of the Sharpe ratio and the volatility of portfolio returns, the CPPI is the best performer, while the VaR based upon the normal distribution is the worst. From the perspective that the return distribution of the hedged portfolio is shifted to the right and in terms of both the average and the cumulative portfolio returns across years, the ES-based strategy using the historical distribution ranks first. Moreover, the ES-based strategy results in a lower turnover within the investment horizon, thereby saving transaction costs.

20.4 Recent Market Developments 20.4.1 Type of Risky Asset Over the past few years, portfolio insurance reemerged due to low structuring and trading costs. According to Pain and Rand (2008), OBPI investments have not been popular, which reflects the difficulty in explaining options to investors. However, the CPPI strategy is much more prevalent, stemming from a broadening in the asset classes

L.-c. Ho et al.

B/W in Print

330

Fig. 20.3 Structured note issuance by type of risky asset

for which investors need principal protection. Many of principal-protected structured notes are not only designed to link the performance of traditional assets, such as stocks, currencies, interest rates and bonds, but also to invest in alternative risky assets, such as hedge funds, fund of hedge funds, credit derivatives, commodities, real estates, private equities, and any combination of the above assets.

Fig. 20.4 Structured note issuance on alternative assets

institutional and retail clients, such as pension funds, insurance companies, private banks who purchase products for onward sale to their clients, and high net wealth individuals. Among which, the institutional investors are especially important in continental Europe.

20.4.4 Modified CPPI Mechanisms 20.4.2 Market Size Figure 20.3 shows the evolution of shares of structured note issuance by type of risky asset. The share of notes linked to equity and alternative assets increased significantly in 2002–2007. In 2007, the shares of currencies and equities are both about 40%, alternative assets occupy more than 10%, and bonds have the least share. Figure 20.4 illustrates the evolution of issuance of structured notes linked to alternative assets. The total issuance has grown five times, from about US $5 billion in 2002 to more than US $45 billion in 2007.

20.4.3 Market Participants According to Pain and Rand (2008), portfolio insurance products are more prevalent in Europe than in the United States. The key issuers (sellers) are typically large investment banks that can provide the necessary structuring and marketing expertise. The main investors include both

Since the inception of portfolio insurance strategies, CPPI investments have evolved to incorporate various different features. Of particular importance are the key components in the design of CPPI investments (i.e., the floor, the multiplier, and the exposure to risky assets).

20.4.4.1 Variations in Floor Ratchet floor: The performance of CPPI investments is price dependent; any gains at a particular point in time may still be lost if the underlying asset price subsequently falls. To address this, products with a “ratchet” mechanism have been introduced to lock in a proportion of upside performance. More specifically, the floor can be set as a percentage of the historical maximum net asset value of the portfolio. That is, the floor jumps up with the portfolio value in order to reduce the risky asset allocation when the market peaks. The time-invariant portfolio protection (TIPP) strategy proposed by Estep and Kritzman (1988) reflects this variation.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

Margin floor: The converse to the above case is when the market falls substantially for the cushion to approach zero, the exposure to the risky asset also approaches zero. Under a conventional CPPI, this leads to “cash-out” where the portfolio is fully invested in the risk-free asset, with no possibility of regaining any exposure. One method to counter this is to adjust the starting floor value, by augmenting this by a margin, which can later be used to help regain some exposure after a near cash-out situation. Straight-line floor: The floor in a conventional CPPI is a zero-coupon bond, which is sensitive to the level of interest rates. When interest rates fall, the floor would rise and the allocation switches away from the risky asset. This in turn would limit the subsequent potential upside from the CPPI. However, this sensitivity can be removed by allowing the floor to vary linearly with time.

331

offers the flexibility to build in additional features, such as zero-coupon, fixed or contingent coupons, and any variations mentioned above. Swaps: An alternative idea in the structured products area could be a collateralized swap. The client has a leveraged exposure to the risky asset and receives a leveraged return on the performance of the asset with no upfront payment required. In return, the client will accrue a financing fee on the leverage amount, payable periodically or at maturity. The swap normally requires the client to have a pre-agreed credit line with the issuer or pledge shares or cash to the issuer as collateral.

20.5 Implications for Financial Market Stability

20.4.4.2 Variations in Multiplier

20.5.1 Amplification of Market Price Movements Rather than having a constant multiplier in a conventional CPPI, dynamic portfolio insurance (DPI) allows for the multiplier to vary over time in relation to the volatility of the risky asset and reflects investors’ appetite for risk. In fact, DPI is a cushion management technique that quantifies the level of risk born by the investment in the risky asset. There is often a maximum level for the multiplier, which can be determined by stress testing on the returns of the risky asset using EVT. The risk-based portfolio insurance strategies in the earlier section reflect this feature. 20.4.4.3 Variations in the Exposure to Risky Assets Constraints on the investment level: Another method to avoid the “cash-out” situation when a market falls is to incorporate a minimum level of investment in the risky asset. Conversely, to avoid unbounded investment in the risky asset as a market rallies, a maximum investment level can also be imposed. Constraints on leverage: Exposure to the risky asset of more than the initial available funds can be achieved by allowing borrowing. There may be constraints on how much can be borrowed. Volatility caps: Some CPPI products include mechanisms that allow the percentage exposure to the risky asset to be reduced if its realized volatility exceeds a certain level.

The portfolio insurance strategies mentioned in this chapter share a common “buy high and sell low” rule for the risky asset. When the underlying asset markets move in one particular direction, either trending up or down, these actions could conceivably have feedback effects in markets that amplify price developments. If the underlying asset markets are deep and liquid, the feedback effects may be limited. However, if the underlying asset markets are less liquid (for example, CPPI products linked to hedge funds on credit derivatives), even small actions may reinforce market prices. Moreover, some underlying assets that appear less correlated in normal conditions may become more correlated in stress situations when CPPI products all look to reduce risky positions simultaneously. Taken together the stress situations and the inherently less liquid nature of certain underlying asset markets, the feedback would have a large impact. Fortunately, according to Pain and Rand (2008), market contacts reveal that the amount of portfolio insurance-related dynamic hedging flows still remain modest relative to the size of the corresponding underlying asset markets, even in those new asset classes. Moreover, market contacts did not perceive portfolio insurance as a significant factor during the 2007 financial market turmoil whereas it was perceived as the driving force during the 1987 stock market crash.

20.4.5 Structured Products 20.5.2 Gap Risk Structured notes: Many of the issuances of principalprotected products are designed as medium-term structured notes, with maturities ranging from 5 to 15 years. The model

The biggest potential risk of implementing a portfolio insurance strategy is that the payoffs are particularly

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vulnerable to sudden huge drops in the risky asset prices before the portfolio can be rebalanced. In such cases, the value of the insured portfolio would fall below the floor; however, a hard guarantee, such as 100% principal protection, to the investors must be fulfilled at the end of investment horizon. Therefore, the issuers of CPPI products would suffer a gap risk. In particular, if the issuers issue several CPPI products written on different underlying assets, and those seemingly uncorrelated assets suddenly become much more correlated in stressed conditions (such as the credit crunch led by the subprime mortgage crisis since September 2007), then the scale of the gap risk may be very much underestimated. The unexpected huge losses may jeopardize the issuers and the stability of financial markets. In theory, the issuer of CPPI products can hedge such exposure to gap risk by options. The issuer needs to model the likely worst-case move in the risky asset price before the next rebalancing opportunity and build the cost of this implicit option into the premiums and fees charged to the investors or provided by capital. But, in practice, the pricing of such options can be quite complex, because the issuer does not exactly know the underlying asset price processes and their correlations in stressed markets. Furthermore, it is difficult for the issuers to find available options through which to hedge their exposures. Pain and Rand (2008) report that some issuers of CPPI products create securities that package up the gap risk and sell these to investors, including private banks and funds. But these structured derivatives are not popular due to the complicated nature of risk. Other issuers seek to limit their exposure to gap risk by other imprecise hedging vehicles or so-called proxy hedging. But, this in turn, exposes the issuers to the basis risk.

20.6 Conclusion A portfolio insurance strategy is a dynamic hedging process that provides the investor with the potential to limit downside risk while allowing participation on the upside so as to maximize the terminal value of a portfolio over a given investment horizon. This paper looks at the basic concept of portfolio insurance, the alternative dynamic hedging strategies available to the portfolio manger, the empirical comparison of the performances of various portfolio insurance strategies, the recent market development of portfolio insurance techniques, and finally, the impact of portfolio insurance on financial market stability.

L.-c. Ho et al.

References Bertrand, P. and J. L. Prigent. 2001. Portfolio insurance: the extreme value approach to the CPPI method, Working paper, Thema University of Cergy. Bertrand, P. and J. L. Prigent. 2005. “Portfolio insurance strategies: OBPI versus CPPI.” Finance 26(1), 5–32. Black, F. and R. Jones. 1987. “Simplifying portfolio insurance.” Journal of Portfolio Management 14, 48–51. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–659. Cesari, R. and D. Cremonini. 2003. “Benchmarking, portfolio insurance and technical analysis: a Monte Carlo comparison of dynamic strategies of asset allocation.” Journal of Economic Dynamics and Control 27, 987–1011. Do, B. H. 2002. “Relative performance of dynamic portfolio insurance strategies: Australian evidence.” Accounting and Finance 42, 279–296. Do, B. H. and R. W. Faff. 2004. “Do futures-based strategies enhance dynamic portfolio insurance?” The Journal of Futures Markets 24(6), 591–608. Estep, T. and M. Kritzman. 1988. “TIPP: Insurance without complexity.” Journal of Portfolio Management 14(4), 38–42. Garman, M. B. and S. W. Kohlhagen. 1983. “Foreign currency option values.” Journal of International Money and Finance 2, 231–237. Hakanoglu, E., R. Kopprasch, and E. Roman. 1989. “Constant proportion portfolio insurance for fixed-income investment: a useful variation on CPPI.” Journal of Portfolio Management 15(4), 58–66. Hamidi, B., E. Jurczenko, and B. Maillet. 2007. An extended expected CAViaR approach for CPPI. Variances, Working paper, University of Paris-1. Herold, U., R. Maurer, and N. Purschaker. 2005. “Total return fixedincome portfolio management: a risk-based dynamic strategy.” Journal of Portfolio Management 31, 32–43. Ho, L. C., J. Cadle and M. Theobald. 2008. An analysis of risk-based asset allocation and portfolio insurance strategies, Working paper, Central Bank of the Republic of China (Taiwan). Leland, H. 1985. “Option pricing and replication with transaction costs.” Journal of Finance 40, 1283–1301. Loria, S., T. M. Pham, and A. B. Sim. 1991. “The performance of a stock index futures-based portfolio insurance scheme: Australian evidence.” Review of Futures Markets 10(3), 438–457. Pain, D. and J. Rand. 2008. “Recent developments in portfolio insurance.” Bank of England Quarterly Bulletin Q1, 37–46. Perold, A. R. 1986. Constant proportion portfolio insurance, Working paper, Harvard Business School. Perold, A. R. and W. F. Sharpe. 1988. “Dynamic strategies for asset allocation.” Financial Analysts Journal 44(1), 16–27. Rendleman, R. J. Jr. and T. J. O’Brien. 1990. “The effects of volatility misestimation on option-replication portfolio insurance.” Financial Analysts Journal 46(3), 61–70. Rubinstein, M. and H. E. Leland. 1981. “Replicating options with positions in stock and cash.” Financial Analysts Journal 37, 63–72. Zhao, Y. and W. T. Ziemba. 2000. “A dynamic asset allocation model with downside risk control.” Journal of Risk 3(1), 91–113. Zhu, Y. and R. C. Kavee. 1988. “Performance of portfolio insurance strategies.” Journal of Portfolio Management 14(3), 48–54.

Chapter 21

Security Market Microstructure: The Analysis of a Non-Frictionless Market Reto Francioni, Sonali Hazarika, Martin Reck, and Robert A. Schwartz

Abstract The Capital Asset Pricing Model describes a frictionless world characterized by infinite liquidity. In contrast, trading in an actual marketplace is replete with costs, blockages, and other impediments. Equity market microstructure focuses on how orders are handled and turned into trades in the non-frictionless environment. For over three decades, the literature has grown while, concurrently, trading systems around the world have been reengineered. After depicting the frictionless CAPM, we consider the development of microstructure analysis, concentrating on issues germane to market architecture. We then consider the design of one facility, Deutsche Börse’s electronic platform, Xetra. Important insights were gained from the microstructure literature during Xetra’s planning period (1994–1997), and Xetra’s implementation marked a huge step forward for Germany’s equity markets. Nevertheless, academic research and the design of a real world marketplace remain works in progress. Keywords Market microstructure r CAPM r Market makers r Liquidity r Price volatility r Price discovery r Trading costs r Electronic markets

21.1 Introduction Security market microstructure addresses issues that involve the implementation of portfolio (investment) decisions in a marketplace. Implementation entails the placement and handling of orders in a securities market, and their translation into trades and transaction prices. The process 

This chapter includes material from Francioni et al. (2008) and from Schwartz (1988), which was reprinted in Schwartz and Francioni (2004).

R.A. Schwartz and S. Hazarika () Zicklin School of Business, Baruch College, CUNY, New York, NY, USA e-mail: [email protected] R. Francioni and M. Reck Deutsche Börse, Frankfurt, Germany

links fundamental information concerning equity valuation (which is of primary concern to portfolio managers) to prices and transaction volumes that are realized in the marketplace. The quality of the link depends on the rules, procedures, and facilities of a securities market, and on the broader regulatory and competitive environment within which the market operates. Widespread interest on the part of the securities industry, government, and academia is testimony to the importance of market microstructure analysis. The subject addresses issues that concern investors, broker/dealer intermediaries, market regulators, exchanges, and other trading venues as well as the broad economy. Interest in microstructure has increased sharply over the past three and a half decades, spurred in particular by three events: the U.S. Securities and Exchange Commission’s (SEC) Institutional Investor Report (1971), the passage by the U.S. Congress of the Securities Acts Amendments of 1975, and the sharp stock market drop on October 19, 1987. Further, the advent of computer-driven trading in recent years has enabled researchers to capture electronically the full record of all trades and quotes, and this has provided empirical researchers with far richer data (referred to as “high frequency data”) for analyzing trading and price setting. Over the years, microstructure analysis has expanded and, concomitantly, exchange structure has strengthened. We consider both of these developments in this chapter. First, we set forth the major challenges that the microstructure literature addresses. Second, we consider the properties of a frictionless trading environment. Third, we present a broad view of the direction in which microstructure analysis has been and is evolving. Fourth, we turn to one application – the design of an actual marketplace: Deutsche Börse’s electronic trading system, Xetra. The German market was the last of the major European bourses to introduce an electronic trading platform, and it is state of the arts, which makes Deutsche Börse a particularly interesting case in point. Fifth, in the concluding section, we consider the bumpy and hazardous road that takes us from theory to the development of an actual marketplace.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_21, 

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21.2 Microstructure’s Challenge Microstructure analysis has four broad applications. First (and this is a key focus of the chapter), it gives guidance to market structure development. The link with market structure is straightforward: the critical factor that drives microstructure analysis is friction in the marketplace (i.e., the explicit and implicit costs of implementing portfolio decisions), and trading costs dependent on the architecture of the marketplace, which determines how orders are handled and turned into trades. The flipside of friction is illiquidity, and a primary function of a market center is to amass liquidity. Microstructure’s second application is to facilitate the development of trading strategies and algorithms for asset managers and broker/dealer intermediaries. The importance of this application is evident in the current development of computer-driven algorithmic trading. Algorithms can be finetuned to take account of, for example, the probability of a limit order executing, time of day effects such as market openings and closings, the search for liquidity in a fragmented environment, and the choice of a trading modality (e.g., a continuous limit order book market, a quote-driven dealer market, a periodic call auction, a block trading facility, or hybrid combinations of the above). The third application of microstructure analysis concerns tests of market efficiency. In the 1970s, at a time when the subject was first emerging, the Efficient Markets Hypothesis (EMH) was widely accepted by financial economists as a cornerstone of modern portfolio theory, and it continues to receive broad academic support today. The hypothesis addresses informational as distinct from operational efficiency (the latter refers to the containment of transaction costs by superior market design). According to the EMH, a market is informationally efficient if no participant is able to achieve excess risk adjusted returns by trading on currently available information. Many of the EMH tests have considered one major part of the information set – market information (e.g., recent quotes, trading volume, and transaction prices). If prices properly reflect all known information, then (in a frictionless market at least) they must change randomly over time; hence the term “random walk.” Earlier studies, based on daily data, generally supported the random walk hypothesis. However, with the advent of high frequency data, the footprints of complex correlation patterns have been detected. This observation, along with superior knowledge of the impact of trading costs on returns behavior, is casting a new light on market efficiency. Whether inefficiency is thought of in operational or informational terms, the EMH is not as stellar as it once was. In its fourth application, microstructure analysis sheds light on how new information is incorporated into security prices. In a zero cost, frictionless environment, share values would be continuously and instantaneously updated with the release of new information. In actual markets, however, information must be received and assessed, traders’ orders

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must be placed and processed, and executions must be delivered and accounts cleared and settled. Costs, both explicit (e.g., commissions) and implicit (e.g., market impact), are incurred throughout this chain of events. Highlighted in much microstructure literature are the costs that some participants incur when, in an asymmetric information environment, other participants receive information first and trade on it to the disadvantage of the uninformed. Asymmetric information is not the only reality, however. In light of the size, complexity, and imprecision of much publicly available information, one might expect that investors in possession of the same (large) information set will form different expectations about future risk and return configurations. This situation is referred to as “divergent expectations.”1 Asymmetric information and divergent expectations together reflect a rich set of forces that impact the dynamic behavior of security prices. This overview of microstructure’s four broad applications underscores that trading frictions are the subject’s raison d’être. Participant orders cannot be translated into trades at zero cost (markets are not perfectly liquid), and trades typically are not made at market clearing (i.e., equilibrium) prices. Trading decision rules (algorithms) are needed because the costs of implementing portfolio decisions can sharply lower portfolio performance. In fact, much algorithmic trading is designed to control trading costs, rather than to exploit profitable trading opportunities. Today, trading is recognized as an activity that is both distinct from investing and equivalently professional. Market structure is of concern to the buyside desks precisely because markets are not perfectly liquid, and neither are they perfectly efficient, either informationally or operationally. Consequently, better market structure can deliver superior portfolio performance for participants. What is the economic service, one might ask, that an equities market provides? The fuzzy link that connects information and prices in the non-frictionless environment underscores two major market functions – price discovery and quantity discovery. Price discovery refers to participants collectively searching for equilibrium prices. Quantity discovery refers to the difficulty that participants who would be willing to trade with each other actually have finding each other and trading when markets are fragmented. This difficulty is accentuated because some participants (primarily institutional investors) do not immediately reveal the total size of their orders (doing so would unduly drive up their market impact costs). Market structure affects both the accuracy of price discovery and the completeness of quantity discovery. The link between market structure and price discovery depends on the environment within which participants are operating. At one end of the spectrum, investors can be equally 1

For a recent discussion, see Davis et al. (2007).

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

informed and form homogeneous expectations based on the information they all possess. At the other end, they can be differentially informed and form divergent expectations with regard to commonly shared information. When investors who share common information all agree on share values (i.e., have homogeneous expectations), prices can be “discovered” in the upstairs offices of research analysts. When investors are not equally informed, and when they form different expectations based on common information, prices must be discovered in the marketplace. In this second environment, the economic service provided by an exchange is clear – it “produces the price.” Regarding quantity discovery, handling the orders of large institutional customers is a challenge. It is not at all uncommon for an institution to want to buy or to sell, for instance, 500,000 shares of a company that has an average daily trading volume of 300,000 shares. Executing an order of this size can easily drive prices away from the trader before the job has been completed. The adverse price move is a market impact cost. Institutions attempt to control their market impact costs by trading patiently and, as much as possible, invisibly. Good market structure can help. To this end, a number of alternative trading systems (ATSs) have been formed in recent years, and dark (i.e., non-transparent) liquidity pools have emerged. With prices discovered in the marketplace, participants employ trading strategies when they come to the market to implement their portfolio decisions. Participants with differential information that will soon become public determine how best to meter their orders into the market so as to move prices to new levels with minimal speed. Additional questions that any trader might ask include: “If I trade now, at the current moment, how will the price that I will receive compare with the average price that shares are trading at today?” “Is price currently at a sustainable, validated level, or is it likely to move higher or lower in the coming hours, minutes, or even seconds?” “Would I do better to be patient and place a limit order, or submit a market order and get the job done right away?” “Should I attempt to trade now in the continuous market, or wait for a closing call?” The orders that a set of participants reveal to the market depend on how questions such as these are answered, and prices that are set and trading volumes that are realized depend on the orders that are revealed. The categories of trading costs that receive the most attention on the part of exchanges, regulators, and academicians are generally those that are the most straightforward to measure: commissions and bid-ask spreads. Increasingly precise measures of market impact are also becoming available, and this cost too is being widely taken into account. On the other hand, the opportunity cost of a missed trade, being far more difficult to quantify, is often overlooked. Also more challenging is quantifying a cost that has received little formal attention: realizing executions at poorly discovered

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prices. The problem, of course, is that equilibrium values are not observable and appropriate benchmark values are not easily defined.

21.3 The Perfectly Liquid Environment of CAPM Peter Bernstein’s (2007) piece in the Journal of Portfolio Management has the intriguing title, “The Surprising Bond Between CAPM and the Meaning of Liquidity.” In it he wrote, “The more liquid an asset, the greater the dominance of systematic risk over stock specific risk.” We build on this insight in this section. In so doing, we formalize the fact that the Capital Asset Pricing Model (CAPM) describes an extreme case, a totally frictionless world where liquidity is infinite and systematic risk has complete dominance over stock specific risk. The analysis provides a good platform from which to launch a discussion of market microstructure, the study of a non-frictionless environment. CAPM models the prices of the individual equity shares that, in aggregate, comprise the market portfolio. Following standard methodology, we start our analysis of the frictionless environment by taking the market portfolio to be one single asset (e.g., an all encompassing exchange traded fund). We consider the demand of an agent to hold shares of this one risky asset when the only alternative is the riskless asset. We show that an individual agent’s demand curve to hold shares of the risky asset is downward sloping, and then use this curve to re-derive certain key CAPM equations to show that the associated demand to hold shares of each individual equity issue in that portfolio is infinitely elastic, and that therefore, the market for the individual shares is infinitely liquid. In the CAPM world, each individual equity issue in the market portfolio has an intrinsic value that is given by the parameter that locates the height (on the price vector) of that infinitely elastic demand. In the section that follows, we turn to the non-frictionless environment of microstructure analysis where individual stock demand curves are downward sloping, the liquidity of individual shares is, therefore, finite, and individual shares do not have intrinsic values. To obtain the representative investor’s demand curve to hold shares of the risky market portfolio, first we state the agent’s utility (of wealth) function. The demand curve to hold shares of the market portfolio may then be obtained directly from the utility function. The derivation follows Ho et al. (1985). We make the following assumptions: The investor’s portfolio comprises a risk-free asset and

one risky asset (shares of the market portfolio). Share price and share holdings are continuous variables.

336 Short selling is unrestricted. The existence of a brief trading period, T0 to T1 , which is

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of P1 , the expected utility of end of period wealth, written as a function of P and Q, given N0 and C0 , would be

followed by a single investment period, T1 to T2 . All transactions made during the trading period are settled

at point in time T1 . The investor seeks a portfolio at the beginning of the investment period (at time T1 ) that will maximize the expected utility of wealth to be realized at the end of the investment period (at time T2 ). Investor expectations with respect to the share price at the end of the investment period (at time T2 ) are exogenously determined (expectations are independent of the current price of shares). Investors are risk averse. The following variables are used: C0 D holdings of the risk-free asset at the beginning of the trading period .T0 / C1 D holdings of the risk-free asset at the beginning of the investment period .T1 / N0 D number of shares of the market portfolio held at the beginning of the trading period .T0 / N1 D number of shares of the market portfolio held at the beginning of the investment period .T1 / R0  1 D risk-free rate of interest over the trading period R1  1 D risk-free rate of interest over the investment period P1 D price at which shares of the market portfolio are purchased or sold during the trading period P2 D price at which shares of the market portfolio can be sold at the end of the investment period .T2 / rm D P2 =P1  1 D return on the market portfolio Q D number of shares traded by the investor at the beginning of the investment period .T1 /I Q > 0 indicates a purchase; Q < 0 indicates a sale

21.3.1 The Expected Utility of End of Period Wealth The participant starts the investment period with C1 dollars of the risk-free asset and N1 shares of the market portfolio (the risky asset). Therefore, wealth at T2 is given by C1 R1 CN1 P2 . As of T1 , this wealth is uncertain because P2 is uncertain. As of T1 , the expected utility of end of period wealth can be written as (21.1) EU.C1 R1 C N1 P2 / The investor starts the trading period with C0 dollars of the risk-free asset and N0 shares of the risky asset. If during the trading period the decision maker were to exchange holdings of the risk-free asset for Q shares of the risky asset at a price

h.P1 ; QjN0 ; C0 / D EU Œ.C0 R0  QP1 /R1 C .N0 C Q/P2 (21.2) where C0 R0  QP1 D C1 and N0 C Q D N1 . Equation (21.2) can be rewritten as h.P1 ; QjN0; C0 / D c C gQ.a  bQ  P1 /

where

(21.3)

c D U.W/ N20 U 0 .W /=R1 g D U0 .W/R1 a D ŒE.P2 /  2 N0 =R1 b D =R1 1 D  ŒU00 .W/=U0 .W/ Var.P/ 2

The step from Equation (21.2) to Equation (21.3) involves a Taylor expansion of the investor’s utility around the expected value of wealth if the investor does not trade.2 The procedure is a convenient way of introducing the variance term into the utility function.3

21.3.2 The Reservation Demand Curve Equation (21.3) can be further assessed with the use of risk aversion and risk premium measures that are defined in Appendix A. Specifically, using Equation (21.3), we now obtain both a reservation price demand curve and an ordinary demand curve to hold shares of the risky asset. We consider the reservation demand curve first. The reservation price for a purchase or a sale is the maximum price the decision maker would be willing to pay to buy a given number of shares .Q > 0/, or the minimum price the decision maker would be willing to receive to sell a given number of shares .Q < 0/ when the only alternative is not to trade at all. Equation (21.3) shows that, if no trade is made (that is, if Q D 0), the decision maker’s expected utility is equal to c. The reservation price for any value of Q is the price that equates the expected utility Œh.Pl ; QjN0 ; C0 / if 2 For a discussion of the Taylor procedure see, for example, R. G. D. Allen, Mathematical Analysis of Economists, London, England: Macmillan, 1960. 3 Two further assumptions are required to obtain Equation (21.3): (1) the third derivative of utility with respect to wealth is small enough to ignore; and (21.2) the squared deviation of the expected rate of return on the risky asset from the risk-free rate is small enough to ignore.

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

the trade is made, with the expected utility (c) if no trade is made. Thus, the reservation price for any value of Q is given by (21.4) h.P R ; QjN0 ; C0 / D c where PR is the reservation price associated with the trade of Q shares. Given Equation (21.3), for Equation (21.4) to be satisfied, we must have a  bQ  P1 D 0. Hence the reservation price demand curve is P R D a  bQ

(21.5)

21.3.3 The Ordinary Demand Curve Using Equation (21.3), we can also obtain the ordinary demand curve. At any value of P1 , the decision maker selects the value of Q that maximizes expected utility. Hence, the ordinary price demand curve is given by @h 0 .P ; QjN0 ; C0 / D 0 @Q

We thus see that the investor achieves an optimal holding of the risky asset by obtaining the number of shares that equates the marginal risk premium with the market price of risk.

21.3.5 The Investor’s Optimal Point on the Capital Market Line The demand model can be used to assess the investor’s optimal point on the capital market line. Let rp be the return on the combined portfolio (N1 shares of the market portfolio and C1 dollars of the risk-free asset). From Appendix Equation (21.7) we have D rp

D

D

The derivation in this section through Sect. 21.3.6 follows Schwartz (1988). When the investor has traded the optimal number of shares of the market portfolio at the market determined price per share, his or her risk premium can be related to the market price of risk. Assessing, the ordinary demand curve at P0 D P1 gives P1 D

E.P2 / 2 N1  R1 R1

(21.8)

Multiplying by R1 =P1 , rearranging, and recognizing that ŒE.P2 /=Pl  1 D E.rm / and R1  1 D rf , we get 2 N1 D E.rm /  rf P1

(21.9)

M % D E.rm /  rf

(21.10)

Therefore, we have

where M % is the marginal risk premium (see Appendix 21A). Note that the right-hand side is the price of risk.

2

2 1 W RA Var.rp / 2 N1

(21.11)

Because ¢p D .NP=W/¢m , we have Var .rp / D ¢p .NP=W/¢m and can write Equation (21.11) as

(21.7)

21.3.4 The Risk Premium and the Market Price of Risk

W N1

which, using RA D U00 .W/=U0 .W/, the measure of absolute risk aversion, can be written as

(21.6)

where P0 is the “ordinary” price associated with the trade of Q shares. Differentiating h in Equation (21.3) with respect to Q, setting the derivative equal to zero, and rearranging gives P 0 D a  2bQ

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1 RA p 2



PW N1

m

(21.12)

Substituting (21.12) into (21.9) and simplifying gives RR P D

E.rm /  rf m

(21.13)

where RR .D WRA / is the measure of relative risk aversion. Equation (21.13) shows that for the investor to hold an optimal combined portfolio, the market price of risk per standard deviation of the market portfolio must be equal to the investor’s coefficient of relative risk aversion times the standard deviation of the combined portfolio’s return. Letting w D N1 P1 =W, substituting w¢m ; D ¢p into Equation (21.13), and rearranging gives wD

E.rm /  rf Var.rm /RR

(21.14)

Equation (21.14) shows that the percentage of wealth that the risk averse participant invests in the market portfolio is positively related to the expected return E.rm /, and negatively related to rf ; Var.rm /, and RR . Investors all face the same values of E.rm /; rf and Var.rm /, but differ according to their degree of risk aversion. More risk averse investors (larger RR ) have smaller optimal values of w and hence are more

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apt to lend at the risk-free rate (which implies w < 1); less risk averse investors (smaller RR ) have larger optimal values of w and hence are more likely to borrow at the risk-free rate (which implies w > 1). The right-hand side of Equation (21.13) is the market price of risk per standard deviation of the market portfolio. The total compensation for risk taking is the price of risk times the standard deviation that the investor accepts (here, the standard deviation of the combined portfolio). Multiplying both sides of Equation (21.13) by p , we obtain 

E.rm /  rf RR Var.rp / D m

 p

(21.15)

Adding rf to both sides of Equation (21.15) gives the investor’s total compensation for waiting and for risk taking: 

E.rm /  rf E.rp / D rf C RR Var.rp / D rf C m

 p (21.16)

Equation (21.16) shows that the location of the investor’s optimal point on CAPM’s capital market line depends on his or her measure of relative risk aversion .RR /.

21.3.6 The ith Risky Asset’s Point on the Security Market Line We now assess the demand model to show the location of an ith risky asset on the security market line. In so doing, we establish that the demand for the ith risky asset is infinitely elastic. Equation (21.10) shows that the marginal risk premium for each investor, as a percentage of P1 , will equal E.rm /  rf . Therefore, for each investor, RA Var.P2 /N1 D E.rm /  rf P1

(21.17)

It follows from Equation (21.17) that investors with lower values of RA hold a larger number of shares, such that the product RA N1 is the same for all investors. Because rm D .P2 =P1 /  1; Var.rm / D Var.P2 /=P1 2 . Substituting Var.rm /P1 2 D Var.P2 / into Equation (21.17) and simplifying gives RA Var.rm /P1 N1 D E.rm /  rf

(21.18)

Using P1 N1 D wW we obtain wRR Var.rm / D E.rm /  rf

(21.19)

Equation (21.19) can be interpreted as an equilibrium condition for each investor. Because wRR D RA N1 P1 , and given that the product RA N1 is constant across investors, RR w is constant across all investors. [It is also clear from Equation (21.19) that the product wRR must be constant across all investors, because E.rm /; rf , and Var.rm / are the same for all.] The equilibrium condition for each investor with respect to the market portfolio implies an equilibrium condition for each investor with respect to any ith risky asset in the market portfolio. The CAPM shows that the relevant measure of risk for the ith risky asset is “i D ¢im =Var.rm/. Therefore, writing Var.rm / D ¢im =“i , substituting into Equation (21.19), and multiplying both sides by “i we get wRR m D ˇi ŒE.rm /  rf

(21.20)

Adding rf to both sides of Equation (21.20) gives CAPM’s security market line, rf C wRR i m D rf C ˇi ŒE.rm /  rf D E.ri /

(21.21)

Where E.ri / is the expected return on the ith stock in the market portfolio. Equation (21.21), assessed at w D 1, shows that the expected return for the ith risky asset depends on its covariance with the market return, and on the measure of relative risk aversion for an investor whose optimal combined portfolio contains the market portfolio only. The equation also shows that the ith risky asset’s specific location on the security market line depends on the covariance of the asset’s return with the return on the market portfolio, and hence that its expected return depends only on ˇi , its systematic risk. It follows from the above discussion that the demand to hold shares of the market portfolio is downward sloping, while the demand for each individual stock in the market portfolio is infinitely elastic. The reason is that perfect substitutes do not exist for the aggregate portfolio, but they do exist for the individual stocks. Only one factor characterizes any ith stock – ˇi , its covariance with the market. But the covariance for any stock can be duplicated exactly by an appropriate combination of two or more other stocks, and all holdings that have the same covariance must yield the same expected return. If they were to yield different expected returns, an unlimited number of shares of the higher yielding position would be bought, and an unlimited number of shares of the lower yielding position would be sold short until, with costless trading, the buying and selling pressures bring the two prices into exact equality. Unlimited buying (selling) at any price lower (higher) than the beta appropriate, CAPM price manifests an infinitely elastic demand to hold shares. That is, at an infinitesimally higher price no shares will be held, and at an infinitesimally lower price demand will be unlimited.

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

Bernstein’s (2007) two insights immediately follow: a stock’s systematic risk totally dominates its specific risk, and the market for each ith stock is infinitely liquid at the price which translates into E.ri /, its systematic risk appropriate return. As we turn to the non-frictionless market, the infinitely liquid, infinitely elastic property of CAPM is a good point of departure from the frictionless world. A common denominator in many microstructure analyses is that the demand to hold shares of individual stocks is downward sloping (which means that shares do not have intrinsic values). Market makers post bid and ask quotes that, when raised, result in more public sales to the market maker and, when lowered, result in more public purchases from the market maker. Bid and ask quotes can be distributed over multiple price points in competitive dealer markets as well as on public limit order books. Trading is not costless. Both explicit costs (e.g., commissions and taxes) and implicit costs (e.g., market impact costs) are incurred. Information is complex and imprecise, and thus investors commonly disagree about its interpretation. Arbitrage is not costless, and perfect substitutes for individual issues do not exist. Share values depend not only on the calculations of systematic risk in the upstairs markets, but also on how orders interact in the marketplace. As a consequence of all of this, trades that are made and the transaction prices that they are made at also depend on the structure of a marketplace. Microstructure analyses address these realities that CAPM does not comprehend.

21.4 What Microstructure Analysis Has to Offer: Personal Reflections In this section we review the development of microstructure analysis. Our objective is not to provide a comprehensive survey of the literature, but to highlight some of the important themes that have given guidance to market structure development. More detailed information can be obtained from Cohen et al. (1979) who have provided an early survey of the field; from O’Hara (1997) who discusses important theoretical microstructure models; from Madhavan (2000); Biais et al. (2005); Parlour and Seppi (2008) who have provided more recent surveys; and from Hasbrouck (2007) who deals with empirical microstructure research and research methodology. We first focus on the early literature, next turn to more recent developments, and finally present our thoughts concerning an important direction in which future microstructure research ought to head.

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21.4.1 The Early Focus The first contributions to the new field in financial economics that came to be called “microstructure” were made by a couple of people who participated in the SEC’s Institutional Investor Report (1971). A handful of others independently started to focus on microstructure topics in the early 1970s. Eventually a few of the early researchers came to recognize the commonality of their interests and, applying the title of Garman’s (1976) well-known paper, “Market Microstructure,” they gave the field its name. Much of the early literature focused on dealers and exchange specialists. These market makers were viewed as the suppliers of immediacy to investors, and the spread was considered the price they charge for providing this service in an environment where order arrival is non-synchronous. Of key importance was the relationship between spreads and the costs of market making. The earlier market maker studies were in large part motivated by a desire to determine whether or not these intermediaries were realizing monopoly profits and, if so, whether or not their profits were attributable to market making being a natural monopoly. Spreads that are greater than the costs of market making would be taken as an indication of monopoly power on the part of the dealers, and spreads that were negatively related to trading volumes would indicate economies of scale in market making, which could imply a natural monopoly (Stigler 1964). Spreads were indeed found to decrease with transactions volume, but reasons other than market making being a natural monopoly were advanced (Smidt 1971; Tinic 1972). The general picture that emerged was that the trading costs incurred by investors could be lowered by strengthening competition between market maker intermediaries. In particular, competition in the NYSE market was deemed inadequate, as specialists and the Exchange itself were viewed as having monopoly positions: each stock was assigned to just one specialist; the NYSE’s order consolidation rule (Rule 390) precluded in-house executions by requiring that exchange members send their orders for NYSE-listed securities to an exchange; and commissions were fixed and unjustifiably high (Tinic and West 1980).4 Not surprisingly, the focus on the market maker firms led several researchers to model market maker pricing decisions (i.e., the setting of their bid and ask quotes). These 4

Another major issue addressed by the microstructure literature at that time was the impact of information on trading volume and price (Copeland 1976; Beja and Hakansson 1977; Beja and Goldman 1980).

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included Bagehot (1971); Stoll (1978); Amihud and Mendelson (1980); Ho and Stoll (1980, 1981, 1983); Mildenstein and Schleef (1983). With one exception (Bagehot 1971), the early formulations dealt with inventory considerations. A market maker firm holding an undesirably long position would lower the quotes (i.e., lower the offer so as to sell more shares, and reduce the bid so as to discourage others from selling shares to it). Reciprocally, a market maker who was short would raise the quotes. This response on the part of the public (buy more shares when the market maker’s offer is lower, and sell more shares when the market maker’s bid is higher) is evidence that the public’s demand to hold shares of any specific stock was taken to be downward sloping. A variety of mathematical tools were used to solve for optimal market maker quotes. These models also gave further insight into the cost components of the market maker’s spread (Stoll 1989). While insightful, the early inventory-based pricing models suffered from some shortcomings. First, the early formulations for the most part assumed monopoly market makers, although some of these models were applied to markets such as the New York Stock Exchange where exchange specialists were in fact competing with other floor brokers and customer limit orders (Demsetz 1968). The application of theory further suffered from the reality that the price of immediacy for an investor is not the spread of an individual market maker, or even an average market maker spread, but the inside spread (i.e., the lowest ask across all market makers minus the highest bid).5 It is important to note that dealer spreads could individually remain relatively invariant with respect to transaction volume while the inside spread fell appreciably. A further shortcoming of most of these earlier models is that they did not take account of a major cost incurred by market makers: the losses generated by trading with better informed investors. Recognition of this reality (which is also outside the scope of the frictionless world of CAPM) led to a development that did much to establish microstructure as an important new field in financial economics – the introduction of market maker models that were based, not on inventory management, but on controlling the cost incurred when some investors are in possession of information that the market maker and other investors have not yet received. Bagehot (1971) was the first to embark on this line of thought. He was later followed by, among others, Gloston and Milgrom (1985) and Kyle (1985). With information asymmetries, the market maker always looses when trading with a better informed participant. For microstructure theorists at the time, this meant that, for the dealer market not to fail, some investors must trade for

5

For further discussion, see Cohen et al. (1979).

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reasons that are not related to information.6 Liquidity considerations (i.e., an investor’s personal cash flow needs) was one such motive for public buying and selling. A third participant type was also introduced along with the liquidity traders – noise traders (participants who trade on price moves as if they contain information when in fact they do not). This trio of informed traders, liquidity traders, and noise traders was used to show how markets could function and, in so doing, enable new information to be incorporated into security prices (Grossman and Stiglitz 1980; Milgrom and Stokey 1982; Kyle 1985; Glosten and Milgrom 1985; Copeland and Galai 1983; Easley and O’Hara 1987, 1991, 1992). At this stage in its early development, the microstructure pricing models were predominantly market maker models. One exception should be noted, however: a National Book System proposed by Mendelson et al. (1979) contained a comprehensive description of an order driven automated trading system that provided guidance for designing the first exchange-based electronic trading systems. For a more recent discussion of automated trading systems, see Domowitz and Steil (1999). Most equity markets around the globe are now order driven, limit order book markets that might include market makers in a hybrid structure (as does the NYSE), but are not basically quote driven (i.e., dealer) markets (as was the old Nasdaq and London Stock Exchange). The limit order book markets are driven by the orders placed by the investors themselves, not by market maker intermediaries.

21.4.2 The Current Focus Over the years, microstructure analysis has grown extensively on both the theoretical and empirical fronts. Concomitantly, the securities markets themselves have evolved, becoming ever more technologically developed, more global in outreach, but also more fragmented between different trading facilities. One important new direction microstructure research has taken is to further model the order driven market, an environment where natural buyers and sellers provide immediacy to each other because some, who are patient, are willing to post limit orders while others, who demand immediacy, choose to submit market orders that execute against the posted limit orders. Understanding the costs of, and motives for, placing limit orders as distinct from market orders was called for.

6

A market supported by informational trading only can indeed function if agents trade with each other because their expectations are divergent. When the information that triggers trading is common knowledge, the condition may be thought of as one where agents are agreeing to disagree.

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

With limit orders, the very existence of the bid-ask spread has to be explained. That is, with a sufficiently large number of participants placing priced orders, one might expect that orders would be posted at virtually every available price point in the neighborhood of equilibrium, and that the spread would disappear. Cohen et al. made this point in their review paper (Cohen et al. 1979), and they analyzed the existence of the spread in Cohen et al. (1981).7 They further wrote, “With regard to modeling the market spread, we suggest that a straightforward aggregation from individual spreads is not possible in a system where there is no clear distinction between demanders and suppliers of immediacy, and where traders meet in a dynamic, interactive environment that incorporates the impact of investor order placement strategies.” Strategic order placement clearly required further analysis. The task, however, was not simple. Some of the first papers in this area assumed, as is true for a dealer market, that limit order and market order participants are two separate, exogenously fixed groups that are separated by a firewall (Glosten 1994). This assumption, while simplifying mathematical modeling, unfortunately distills out much of the richness of an order driven market. More recent models have eliminated the firewall (Handa and Schwartz 1996a; Foucault 1999; Parlour 1998; Handa et al. 2003; Faucault et al. 2005; Goettler et al. 2005). With the choice between limit order and market order endogenous, for any market to function, participants must divide naturally into four groups that reflect two dichotomies (one between buyers and sellers and the other between limit order and market order placers), not the standard two (buyers and sellers). With order type selection endogenous in the order driven market, the balance between immediacy demanders and immediacy suppliers becomes a second equilibrium that must be understood. That is, one needs to recognize the conditions under which some participants will choose to be liquidity demanders (place market orders) while others choose to be liquidity suppliers (place limit orders). If a reasonable balance is not achieved between these two groups, the order driven market will fail (as indeed it does for thinner, small cap stocks). Increasingly, these issues have been handled, and some sophisticated limit order models have been developed.8 Microstructure analysis of trading systems has expanded to include periodic call auctions.9 The economics of a call auction are quite different from those of continuous trading and, consequently, so too are the order placement strategies

7

Cohen et al. (1981) describe the tradeoff between execution probability and price improvement in the optimal choice between limit and market orders. 8 See Bach and Baruch (2007) for a recent discussion and further references. 9 See Economides and Schwartz (1995) for a description of alternative call market structures.

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that participants should employ when they approach a call market. Call auctions do not, by their very nature, supply immediacy. Rather, orders that are entered during a call’s bookbuilding phase are held for a periodic crossing at a single clearing price at the (generally predetermined) time of the market call. Consequently, buy and sell orders submitted to a call do not execute when they arrive even if they match or cross in price (matching and crossing orders execute immediately in a continuous trading environment). This being the case, limit and market orders have a different meaning in a call: limit orders do not supply immediacy to market orders, and market orders are simply extremely aggressively priced limit orders (i.e., a market order to sell in a call effectively has a limit price of zero, and a market order to buy effectively has a limit price of infinity). Today, virtually all modern, electronic exchanges open and close their continuous markets with call auctions. Consequently, participants face further decisions when operating in a call plus continuous, hybrid market: how to submit an order to a call auction that is followed by continuous trading (e.g., an opening call), and how to submit an order to a continuous trading environment that is followed by a call auction (e.g., a closing call). Taking these tactical decisions into account is part of the complexity of microstructure analysis. Technological developments have simultaneously enabled new trading venues to emerge (which can fragment markets) while providing connectivity between them (which can consolidate markets). Concurrently, regulatory initiatives have been motivated by the desire to intensify inter-market competition. Questions can be raised, however, concerning fragmentation of the order flow. The conventional wisdom has been that the consolidation of order flow improves liquidity, and exposing each order to all other displayed orders gives investors the best prices for their trades. Consolidating trading in a single market provides incentives to liquidity suppliers to compete aggressively for market orders by revealing their trading interest, and by being the first to establish a more favorable price (if time is used as a secondary priority rule). On the other hand, arguments in favor of trading on multiple markets include the benefits of inter-market competition, and the fact that traders with disparate motives for trading may want different marketplaces to trade in (i.e., the “one size does not fit all” argument). And so, different markets develop to serve diverse investor needs (such as achieving a faster execution versus obtaining a better price). One growing need among large institutional investors, the ability to trade large orders with minimal market impact, has led to the advent of dark pool, block trading facilities such as Liquidnet, Pipeline, and ITG’s Posit that aid in quantity discovery. This development in the industry has spawned a related line of research on off-exchange and upstairs trading (Seppi 1990; Grossman 1992; Keim and Madhavan 1996; Madhavan and Cheng 1997).

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A spectrum of market quality issues have been of long and continuing importance to microstructure researchers. These include market transparency,10 both pre- and posttrade, (Porter and Weaver 1998), the accentuation of intraday price volatility, and correlation patterns that have been observed in high frequency data (Engle and Granger 1987). Other important issues include price clustering and tick sizes (Harris 1991, 1994; Angel 1997). Applications such as transaction cost analysis (TCA) and algorithmic trading have received increasing attention (Domowitz et al. 2001). The relative performance of floor based versus electronic trading is another important issue (Domowitz and Steil 1999). A major line of empirical research was pioneered by Hasbrouck 1993 who decomposes transaction prices into two components: a random walk component and a stationary component. The random-walk component is identified with an efficient price that the market is trying to discover. The stationary component is viewed as microstructure noise. Microstructure noise is commonly explained by features such as the bid-ask spread, market impact, and the discreteness of the pricing grid. The noise component has also been attributable to price discovery itself being a dynamic process (Menkveld et al. 2007 and Paroush et al. 2008).11 Numerous empirical studies have focused on two of the world’s premier markets, the New York Stock Exchange and NASDAQ (Hasbrouck 1991, 1995; Hasbrouck and Sofianos 1993; Christie and Schultz 1994; Christie et al. 1994; Bessembinder and Kaufman 1997a, b; Bessembinder 1999, 2003; Barclay et al. 2003, among others). Many other studies have considered European markets, Asian markets, and other markets around the world (e.g., Biais et al. 1995; Sandas 2001; Ozenbas et al. 2002).12 Across all of these markets, structural and performance differences have been noted, but also major similarities have been observed. It is apparent that, despite the influence of historic and cultural considerations, trader behavior and market performance around the globe depend largely on microstructure realities. Alternatively stated, trading rooms and markets around the world bear striking resemblances to each another. Another recent line of research has considered how search costs affect bid-ask spreads in financial markets. To this end, Duffie et al. (2005) present a dynamic model of market makers under the assumption of no inventory risk and information that is symmetrically distributed. They show that sophisticated investors who have better search and bargaining

10

Trading systems differ in their degree of transparency. Pagano and Röell (1996) investigate whether greater transparency enhances market liquidity by reducing the opportunities for taking advantage of uninformed participants. 11 Also see Hasbrouck (1995), Harvey and Huang (1991) and Jones et al. (1994). Further references are provided by Menkveld et al. (2007). 12 Also see Bessler (2006) for discussion and further references.

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abilities face tighter bid-ask spreads. This is in contrast to traditional information-based models that imply that spreads are wider for more sophisticated (i.e., better informed) investors. As we have noted, unlike in the frictionless market arena of CAPM, amassing liquidity is a primary function of a marketplace and market structure features are generally designed with liquidity implications in mind. Asset managers also take liquidity into account, along with the two other standard variables of modern portfolio theory, risk and return. Difficulties in defining, measuring, and modeling liquidity are formidable, however, and the literature that deals with it directly is relatively sparse (Bernstein 1987). Nevertheless, liquidity considerations have permeated the microstructure literature, both explicitly and implicitly.13 Looking back over the development of microstructure analysis, two observations stand out. First, microstructure studies have in multiple ways given direction to market structure development. Second, to a remarkable extent, the various theoretical microstructure models that are center stage today, and many empirical analyses that are based upon them, share a common structural framework – the asymmetric information paradigm. This consistency is desirable in that it implies that the field has grown by accretion rather than by replacement. Consequently, new insights are more apt to refine than to contradict old conclusions. Consistency, however, is not desirable if the common structural framework becomes overly rigid and restrictive, and if it yields incomplete and/or misleading answers to questions involving trader behavior, market structure, and regulatory policy. At times, a literature starts to advance along new fronts. We consider this possibility next for the microstructure literature.

21.4.3 Future Directions As we have noted, the current focus in the literature is on asymmetric information-based models, which are characterized as follows. Trading is driven by informational change, liquidity needs, and noise trading. The information motive for trading is the first mover of the three (liquidity and noise trading are required so that a market will not fail). Further, order arrival in the continuous environment is generally taken to be asynchronous. For a continuous trading regime to function with asynchronous order arrival, the presence of a limit order book and/or a market maker intermediary is required.

13

For further discussion and references regarding liquidity see Amihud and Mendelson (1986), Chordia et al. (2000, 2008), Hasbrouck and Seppi (2001), Amihud (2002) and Pastor and Stambaugh (2003).

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

Information trading is of keen interest because it represents the process by which new information is reflected in share values. In the standard asymmetric information models, it is assumed that all participants in possession of the same information form equivalent expectations concerning future risk and return configurations. When information changes, however, participants may not all receive the news at the same time; some receive it before others, a reality that, at any point in time, can divide traders into two groups – the informed and the uninformed. Informed participants will never trade with each other; consequently, liquidity and noise traders must be present for a market to function. As noted, asymmetry of information, for the most part, lies at the heart of the standard microstructure models of today. The homogeneous expectations assumption has been tempered of late. As a further departure from the infinitely liquid, zero cost environment of CAPM, it is being recognized that some participants produce “private information” (namely, that they further process information so as to gain insights that are not immediately available to others). Whether participant expectations differ because of the actual production of private information, or simply because different people interpret the same information or news announcement differently, the expectations of a group of investors can be divergent. Also at the heart of the asymmetric information models is the presumption that a stock has a fundamental value that bears a unique relationship, not to trader activity in the marketplace, but to the fundamental information that informed traders possess. The process of information being fully reflected in prices under asymmetric information is the act of informed and uninformed agents trading with each other until any discrepancy between a market price and a fundamental value is eliminated. The process can be viewed as arbitrage. In the earlier dealer models, the market maker was assumed to know a stock’s fundamental value. In later models, informed traders but not the market maker knows the fundamental values (Kyle 1985). Especially in the later models, price discovery is not instantaneous; rather, it is a protracted process that depends on the individual strategies employed by the informed and uninformed agents. In recent years, an alternative paradigm has been emerging: a divergent expectations environment (Miller 1977). While institutionally realistic, this paradigm has met with considerable academic resistance. For one thing, homogeneous expectations environments are far easier to deal with mathematically and homogeneity has, in many applications, proven to be a useful modeling assumption. The assumption has also been retained for another reason. As an attribute of individual rationality, it is presumed that intelligent agents facing the same information and applying the same

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(correct) analytic techniques will reach the same conclusions and, therefore, will have homogeneous expectations. Fundamental information, however, is enormous in scope. It is complex and imprecise, and our tools for analyzing it are relatively crude. In the presence of fuzzy information, expectations can be divergent. Allowing for divergent expectations opens another path for microstructure analysis, and it introduces new questions concerning agent behavior, market structure, and regulatory policy. Moreover, a further element can enter the analysis in a divergent expectations environment: along with forming their own opinions, agents may also respond to the opinions of others; that is, exhibit adaptive valuation behavior (Paroush et al. 2008; Davis et al. 2007).14 Just how agents communicate with each other and respond to each others’ opinions is a subject for ongoing research. The topic also opens another interface with behavioral finance. Price discovery acquires a different meaning in a divergent expectations environment, and this has important implications for market structure. When asymmetric information characterizes a community of investors, the strategic behavior of informed agents can affect the path that price takes when news moves a share value from one equilibrium to another, but the new equilibrium is path independent. With divergent expectations, the new equilibrium is path dependent – it depends on how the opinions of a diverse set of agents are integrated (Paroush et al. 2008). Alternatively stated, with divergent expectations, price discovery is a coordination process and, as such, is directly effected by market structure. In the standard asymmetric information environment, the key dichotomy is between informed and uninformed participants. But a second dichotomy also exists – one that separates large institutional customers from small retail customers. One might expect that the informed investor set would largely comprise the institutional customers. After all, the institutions are professional; they can afford to continuously monitor information and respond to news, and their very size (all else constant) reduces their per share cost of doing so. With divergent expectations, however, there is no presumption that institutional customers can, because of their size, consistently evaluate shares more accurately. On the contrary, institutions commonly disagree with each other and, as a consequence, commonly trade with each other. In the divergent expectations environment, institutional investors do not necessarily have an advantage over retail customers as fundamental analysts. In fact, their size makes 14 Adaptive valuation behavior refers to individual agents becoming more bullish (bearish) when learning of the relatively bullish (bearish) attitudes of others.

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trading more difficult and they incur higher transaction costs. So what accounts for their popularity? The value added by the mutual funds, pension funds, and so forth comes largely from their ability to facilitate diversification. Further, they can bring a systematic, professional, and disciplined approach to portfolio management (Davis et al. 2007).

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Big Board from a floor-based “slow” market into a hybrid that includes a “fast market” electronic venue. As of this writing, the floor-based component of the NYSE’s hybrid has been markedly reduced in importance. Several specialist firms have ceased operations, other floor brokers have departed, and the trading room areas have collapsed from five to two.

21.5 From Theory to Application 21.5.2 Regulatory Initiatives Microstructure analysis is inherently involved with analyzing the detailed functioning of a marketplace. The literature has a strong theoretical component and, to a large extent, is structured to yield insights into the effect of market design (structure and regulation) on market performance. Hopefully, theory can provide a broad roadmap for real world market architects to follow. In this section we provide a broad overview of major technology and regulatory changes that have taken place in the United States and Europe.15

21.5.1 Technological Developments Two exogenous forces have driven market structure change: technology and regulation. Regarding technology, the first big step was taken in 1971 in the U.S. when the National Association of Securities Dealers (NASD) introduced an electronic automated quotation (AQ) display system called NASDAQ. The Toronto Stock Exchange was the first exchange to introduce an electronic order driven platform – its Computer Assisted Trading System (CATS) in 1977. Following in Toronto’s footsteps, London instituted SEAQ in 1986, Paris rolled out its Cotation Assistée en Continu (CAC) in 1986, and Deutsche Börse’s Xetra came to life in 1997. Also in 1997, the London Stock Exchange introduced its Stock Exchange Trading System (SETS) limit order platform. By the end of the twentieth century most of the exchanges in Europe had converted to electronic limit order book platforms. Change came more slowly in the U.S. Instinet introduced an electronic platform in 1969. Nearly 30 years later, Instinet became known as an Electronic Communications Network (ECN). In short order, a slew of other ECNs emerged, led most prominently by Archipelago and Island. In 2002, NASDAQ implemented its own electronic platform, which, at the time, was called “SuperMontage.” Most recently, in the spring of 2006, the newly privatized NYSE Group initiated its Hybrid Market, a facility that has transformed the 15

Further discussion of market structure development is provided by Harris (2003).

Major regulatory initiatives have played an important role in jump-starting these market structure changes. The 1975 Congressional Securities Acts Amendments was the first sizable regulatory foray into market structure development. The Amendments precluded the fixing of commission rates and mandated the development of a National Market System (NMS). In 1997, the U.S. Securities and Exchange Commission instituted its new Order Handling Rules (OHRs), which require that market makers holding customer limit orders display those orders in their quotes, and that dealers at least match any quotes that they themselves display on an ECN (either by bettering the quotes that they offer customers or by posting their superior quotes in NASDAQ’s SuperMontage). Following the OHRs, four other regulatory initiatives were introduced in the U.S. in relatively fast succession. In 1998, the SEC release Regulation of Exchanges and Alternative Trading Systems (RegATS), set forth rules that allow ATSs the alternative of registering as national exchanges or as broker-dealers. In 2000, the NYSE, under pressure from the SEC, rescinded its order consolidation rule (Rule 390). In 2001, the U.S. markets completed the transition from fractional to decimal pricing, which resulted in the minimum tick size decreasing from 1/16 of a dollar or 6.25 cents (it had earlier been 1/8 of a dollar or 12.5 cents) to one cent. In 2005, the SEC adopted Regulation NMS, the key provision of which is that better-priced limit orders cannot be traded through (the trade-through rule was fully implemented in 2007). On the eastern side of the Atlantic, the first major regulatory initiative was taken in 1993 when the Investment Services Directive opened the door for cross-border trading by introducing the single European passport. As discussed in Schwartz and Francioni (2004), “Passporting defines a system of mutual acceptance of other EU countries’ rules without truly harmonizing all of the details of the various rules.” Major regulatory change has come again to the European arena in the form of the Markets in Financial Instruments Directive (MiFID). Key provisions in MiFID include a best execution requirement (echoes of the 1975 U.S. Securities Acts Amendments), a quote disclosure requirement for upstairs broker/dealers (echoes of the U.S. Order Handling Rules),

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

and the disallowance of order focusing rules (echoes of the U.S. SEC pressuring the withdrawal of NYSE Rule 390). A major regulatory difference is that, no trade-through rule has been imposed on the European markets (unlike under the U.S. SEC’s Reg NMS).

21.6 Deutsche Börse: The Emergence of a Modern, Electronic Market We turn in this section to the designing of an actual marketplace. Our focus is on Deutsche Börse: it is the dominant stock exchange in Germany, the last of the major European Borses to go electronic, and its technology is state of the art. Important insights were gained from the microstructure literature during Xetra’s planning period and the system’s implementation has marked a huge step forward for Germany’s equity markets. But our roadmap, which is undoubtedly incomplete today, was even more limited in the 1994–1997 years when Xetra was being designed. And, there is always the danger that the cartographer whose map is being used has some misconceptions (e.g., believes in the existence of the Northwest Passage).

21.6.1 The German Equities Market in the Mid-1990s As recently as the mid-1990s, the German market had major structural defects that would undermine its competitiveness in the European arena. In recognition of this, Deutsche Börse, the newly founded exchange operator of the Frankfurter Wertpapierbörse (FWB), became the leading force for change.16 In the mid-1990s, Frankfurt’s trading floor was the major marketplace for German stocks, but the German market was badly fragmented. Kursmaklers, the equivalent of specialists, concentrated much of the liquidity in their order books. A primitive (by today’s standards) electronic trading system, IBIS (which was owned by FWB), operated in parallel with the floor trading. IBIS’s central component was an open limit order book that had hit and take functionality, but did not match orders automatically. The electronic system captured about 40% of the trading volume in the 30 large-cap DAX stocks, but no link existed between IBIS and the floor. Seven other floor-based regional exchanges were also operating in Germany with technical infrastructures that were similar to 16

FWB also owned the futures and options exchange Deutsche Termine Börse. After the 1997 merger with SOFFEX, DTB became Eurex.

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those in Frankfurt. In total, the regionals at that time were attracting roughly 10% of German exchange-based trading volume. Moreover, off-board trading has been (and still is) prevalent in Germany (Davis et al. 2006). Transparency for floor trading (pre-trade transparency in particular) was low. Quotes were not distributed publicly (they were available on the floor only). Price priority between different trading venues was not enforced and orders executed in one market commonly traded through orders waiting to be executed in another market. Market manipulation and other abuses of power and position were believed to be rife on the old Frankfurt floor. Given the appreciable market fragmentation, poor transparency, imperfect inter-market linkages, and dubious floor behavior, transaction costs were high. Changes, both structural and regulatory, were called for. The result was the development of Xetra, an electronic orderdriven trading system that comprises two principal modalities – a continuous order book platform and periodic singleprice call auctions.17

21.6.2 Designing a New Trading System Xetra’s development started in 1994, and the system was launched in 1997.18 Strong external forces also motivated this reengineering of Deutsche Börse’s market structure: regulatory reform, soaring trading volumes, pan-European harmonization of the exchange industry, vibrant cross-border competition for order flow, and the rising concerns of market participants about the future performance of Germany’s financial markets. Through Xetra’s design stage, microstructure theory, even as it existed at the time, was an indispensable guide. This new field in financial economics, with its origin in issues concerning the competitive and architectural structure of an equity market, should have been able to give guidance to the development of an actual marketplace such as Xetra. To an extent, it has fulfilled its promise. The literature gave Deutsche Börse a broad roadmap, and it has highlighted underlying relationships and other important considerations that a market architect should be aware of. Building the Xetra model involved specifying principles that the new market should implement, and the system’s functionality also had to be defined. Most importantly, the new market system was to provide equal and decentralized access to all of its participants. Further, the system’s functionality and the market information delivered to users (both pre- and post-trade) were to be the same for all traders.

17 18

For further discussion and descriptions, see Francioni et al. (2008). Appendix B provides details of Xetra’s design.

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A trader’s location should not matter. With this in mind, Deutshe Börse’s fundamental architectural decision was to structure a hybrid market that included two major modalities – a continuous electronic order driven platform, and periodic call auctions that were used primarily for market openings and closings.19 An absolutely critical attribute of an order driven trading system is its ability, vis-à-vis its competitors, to win the battle for liquidity. Regarding this matter, the earlier microstructure literature has given some guidance, but liquidity is a complex attribute to deal with. As it is not easy to define and measure, liquidity has been very difficult to model and assess. However, as noted above, the measurement and analysis of liquidity are currently attracting considerably more attention in the microstructure literature. Price discovery and transparency are two other issues for which the microstructure literature has provided valuable guidance. The architects at Deutsche Börse recognized that price discovery is a primary function of a market center, and their major reason for introducing the call auctions was to sharpen its accuracy, particularly at market openings and closings. Understanding that transparency is important while recognizing that it should not be excessive, the decision was made to disclose only the indicative clearing price (not the full book of orders) in the pre-call, bookbuilding period. Microstructure literature has given insights into the operations of the public limit order book for continuous trading. At the time, recognition was also emerging of periodic call auctions, a modality that was clearly differentiated from, but could effectively be used with, the continuous market. With regard to continuous trading, microstructure analyses of the use of limit and market orders and of the interaction between these two order types proved to be most valuable. However, a deeper understanding of the economics of an order driven market now exists than was the case in the 1994– 1997 period when Xetra was being designed. Another important contribution of microstructure theory has been the classification of traders according to their needs for immediacy and their propensities to be either givers or takers of liquidity. The differentiation between informed and uniformed traders also proved to be valuable, particularly with respect to the market maker role that has been incorporated into Xetra. Specifically, market makers, referred to as “designated sponsors,” were included to bolster liquidity provision for smaller cap stocks. A balance had to be achieved between the obligations imposed on the designated sponsors and the privileges granted to them. To accomplish this, information had to be assessed concerning the role of dealers in general (e.g., NASDAQ-Type market makers) and special19

Interestingly, the microstructure literature on call auctions was relatively sparse at that time. For an early discussion, see Handa and Schwartz (1996b).

R. Francioni et al.

ists in particular (e.g., NYSE-Type specialists). That balance defined the designated sponsors’ role in Xetra, and secured their willingness to accept it. Market microstructure insights also yielded the understanding needed to transform the specialist role into the newly designed designated sponsor role. Interestingly, the specialist role in the NYSE has now been changed to better harmonize with the Exchange’s electronic platform, and these intermediaries are now referred to as “designated market makers.” But designing an automated trading system is indeed a complex task, and the gap between theory and implementation is both large and intricate. Trading decisions can be made in a large variety of ways that run the gamut from humans interacting directly with humans without computers, to humans trading via electronic order handling and execution systems, to computers making trading decisions that are sent electronically to a computerized market (e.g., computerdriven algorithmic trading). Since the mid-1990s, market structure development has involved mainly the design of an electronic trading facility. Deutsche Börse took account of the fact that automation impacts both the way in which trading decisions are made, and the process by which prices are determined and trades executed in a market center. An electronic market requires the specification of an array of critical features (e.g., the trading modalities employed, rules of price and quantity determination, and basic features such as order types and trading parameters). With an electronic market, the software that implements a desired market structure must be specified on a level of detail that far exceeds what is required for human intermediated trading. For instance, a human agent (specialist) has historically handled price determination at NYSE openings. This function is performed with reference to various rules, but the specialist is also free to exercise reasonable judgment. Further, human-to-human interactions can evolve naturally as problems, opportunities, and new competitive pressures arise. In contrast, with a fully electronic opening, every possible condition that can occur must be recognized and a rule for dealing with it specified, and electronic interaction can be changed only by rewriting the code that specifies with stepby-step precision just how orders are handled and turned into trades and transaction prices. How does one achieve the precise specifications that a computerized trading system must have? In 1994, the market architects at Deutsche Börse could study the operations of other electronic platforms (e.g., CATS in Toronto and CAC in Paris). Doing so was helpful but of limited value given that Deutsche Börse was looking to develop a distinctive system. When moving into new territory, market structure development is a venture. How does one know in advance whether or not it will work? How can one determine whether or not the new system will be viable from a business perspective?

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

Nevertheless, design decisions have to be made, technical requirements must be specified, and the system must be built. The decisions involved represent huge financial bets on whether or not a new market structure will attract sufficient liquidity. Prototyping a new market in the design phase helps the assessment process, but doing so was considerably more difficult in 1994 than it is today with the advent of superior information technology and testing capabilities. In 1994, the architects were forced to rely more on their own educated judgment and on any insights they might gain from microstructure research. Those who are involved in the design of an actual market realize that the devil is in the details. Consider, for instance, the specification of a call auction. A call has excellent theoretical properties, but how should an actual auction be designed? It is straightforward to say that the market clearing price in a call auction should be the value that maximizes the number of shares that trade. But what should the specific rule be for selecting the clearing price if two prices both result in the same maximum trade size? Additionally, how transparent should the book be in the pre-call, order entry period? Are further design features needed to counter the possibility of gaming? And so on. Other considerations that for the most part are outside the scope of the microstructure literature also came into play during the design of Xetra. Information technology issues such as scalability, open architecture, and system reliability are of critical importance. So too are procedures for post-trade clearing and settlement. One of the final steps in the structural design of the new German market was the introduction in 2003 of a central counterparty (with a CCP, counterparty risk management was centralized and trading became fully anonymous, both pre- and post-trade). Electronic trading is also a prerequisite for highly efficient straight-through processing (STP involves all stages of a trade’s life cycle). Information technology has further facilitated the timely capture of market data (all trades, quotes, market index values, etc.) and has expedited its delivery to users. With regard to these diverse applications, Deutsche Börse has achieved a closer integration between trading on Xetra and the broader market infrastructure.

21.7 Conclusion: The Roadmap and the Road A market architect must have a roadmap that, broadly speaking, says where one ought to head and roughly how to get there. To this end, the microstructure literature has added clarity, articulation, and intellectual support. Briefly stated, the objective is to reduce trading frictions (costs), sharpen price discovery, and facilitate quantity discovery. The means of achieving this broad objective involve the amassing of

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liquidity. This is done through the appropriate use of limit order books for both continuous and call auction trading and, where appropriate, the inclusion of broker/dealer intermediaries. Further insights are gained from microstructure’s indepth analyses of trading motives (new information, liquidity needs, and technical trading signals). The literature has also provided guidance with regard to issues such as transparency and the consolidation (fragmentation) of order flow. But theory, even if it does provide a good roadmap, can take one only so far. The closer one gets to the design of an actual system, the more apparent the complexities of trading and trading systems become. The road actually traveled is indeed bumpy and hazardous. System designers know that “the devil is in the details.” They have to grapple with issues ranging from scalability, reliability, and other IT requirements, to business considerations concerning the ultimate profitability of a trading venue. The market architects at Deutsche Börse recognized these issues and their new system, Xetra, has marked a huge step forward for the German equity market. Today, important problems persist with regard to market design in Germany (and in all other markets around the world). Two fundamental questions concerning market architecture that have yet to be adequately answered: (1) What is the best way to deal with large, institutional orders? (2) How is liquidity creation best handled for mid-cap and smallcap stocks? At the same time, important microstructure topics continue to emerge at the academic research desks. Are there limits beyond which microstructure theory cannot provide guidance? Are there limits to the level of efficiency that a real world market can ever achieve? Undoubtedly, both answers are “yes” but, without question, neither of these limits has as of yet been reached. Quite clearly, microstructure research and the design of an actual marketplace remain works in progress.

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348 Beja, A. and N. H. Hakansson. 1977. “Dynamic market processes and the rewards to up-to-date information.” Journal of Finance 32, 291–304. Bernstein, P. 1987. “Liquidity, stock markets and market makers.” Financial Management 16, 54–62. Bernstein, P. 2007. “The surprising bond between CAPM and the meaning of liquidity.” Journal of Portfolio Management 34, 11–11. Bessembinder, H. 1999. “Trade execution costs on Nasdaq and the NYSE: a post-reform comparison.” The Journal of Financial and Quantitative Analysis 34, 387–407. Bessembinder, H. 2003. “Quote-based competition and trade execution costs in NYSE-listed stocks.” Journal of Financial Economics 70, 385–422. Bessembinder, H. and H. Kaufman. 1997a. “A comparison of trade execution costs for NYSE and Nasdaq-listed stocks.” The Journal of Financial and Quantitative Analysis 32, 287–310. Bessembinder, H. and H. Kaufman. 1997b. “A cross-exchange comparison of execution costs and information flow for NYSE-listed stocks.” Journal of Financial Economics 46, 293–319. Bessler, W. (Ed.). 2006. Bösen, Banken und Kapitalmärkte, Duncker & Humblot, Berlin. Biais, B., P. Hillion, and C. Spatt. 1995. “An empirical analysis of the limit order book and the order flow in the paris bourse.” Journal of Finance 50, 1655–1689. Biais, B., L. Glosten, and C. Spatt. 2005. “Market microstructure: a survey of microfoundations, empirical results, and policy implications.” Journal of Financial Markets 8, 217–264. Chordia, T., R. Roll, and A. Subrahmanyam. 2000. “Commonality in liquidity.” Journal of Financial Economics 56, 3–28. Chordia, T., R. Roll, and A. Subrahmanyam. 2008. “Liquidity and market efficiency.” Journal of Financial Economics 87, 249–268. Christie, W. and P. Schultz. 1994. “Why do NASDAQ market makers avoid odd-eighth quotes?” Journal of Finance 49, 1813–1840. Christie, W., J. Harris, and P. Schultz. 1994. “Why did NASDAQ market makers stop avoiding odd-eighth quotes?” Journal of Finance 49, 1841–1860. Cohen, K., S. Maier, R. Schwartz, and D. Whitcomb. 1979. “Market makers and the market spread: a review of recent literature.” The Journal of Financial and Quantitative Analysis 14, 813–835. Cohen, K., S. Maier, R. Schwartz, and D. Whitcomb. 1981. “Transaction costs, order placement strategy, and existence of the bid-ask spread.” The Journal of Political Economy 89, 287–305. Copeland, T. 1976. “A model of asset trading under the assumption of sequential information arrival.” Journal of Finance 31, 1149–1168. Copeland, T. and D. Galai. 1983. “Information effects on the bid-ask spread.” Journal of Finance 38, 1457–1469. Davis, P., M. Pagano, and R. Schwartz. 2006. “Life after the big board goes electronic.” Financial Analysts Journal 62(5), 14–20. Davis, P., M. Pagano, and R. Schwartz. 2007. “Divergent expectation.” Journal of Portfolio Management 34(1), 84–95. Demsetz, H. 1968. “The cost of transacting.” The Quarterly Journal of Economics 82(1), 33–53. Domowitz, I. and B. Steil. 1999. “Automation, trading costs, and the structure of the securities trading industry.” Brookings-Wharton Papers on Financial Services, 33–92. Domowitz, I., J. Glen, and A. Madhavan. 2001. “Liquidity, volatility, and equity trading costs across countries and over time.” International Finance 4, 221–256. Duffie, D., L. Pedersen, and N. Garleanu. 2007. “Valuation in over-thecounter markets.” Review of Financial Studies 20, 1865–1900. Economides, N. and R. Schwartz. 1995. “Electronic call market trading.” Journal of Portfolio Management 21, 10–18. Easley, D. and M. O’Hara. 1987. “Price, trade size, and information in securities markets.” Journal of Financial Economics 19, 69–90. Easley, D. and M. O’Hara. 1991. “Order form and information in securities markets.” Journal of Finance 46, 905–928.

R. Francioni et al. Easley, D. and M. O’Hara. 1992. “Time and the process of security price adjustment.” Journal of Finance 47, 577–606. Engle, R. and C. Granger. 1987. “Co-integration and error correction: representation, estimation and testing.” Econometrica 55, 251–276. Foucault, T. 1999. “Order flow composition and trading costs in a dynamic order driven market.” Journal of Financial Markets 2, 99–134. Foucault, T., O. Kaden, and E. Kandel. 2005. “The limit order book as a market for liquidity.” Review of Financial Studies 18, 1171–1217. Francioni, R., S. Hazarika, M. Reck and R. A. Schwartz. 2008. “Equity market microstructure: taking stock of what we know.” Journal of Portfolio Management 35, 57–71. Garman, M. 1976. “Market microstructure.” Journal of Financial Economics 3, 33–53. Glosten, L. 1994. “Is the electronic open limit order book inevitable?” Journal of Finance 49, 1127–1161. Glosten, L. and P. Milgrom. 1985. “Bid, ask, and transaction prices in a specialist market with heterogeneously informed agents.” Journal of Financial Economics 14, 71–100. Goettler, R., C. Parlour, and U. Rajan. 2005. “Equilibrium in a dynamic limit order market.” Journal of Finance 60, 2149–2192. Grossman, S. 1992. “The information role of upstairs and downstairs markets.” Journal of Business 65, 509–529. Grossman, S. and J. Stiglitz. 1980. “On the impossibility of informationally efficient markets.” American Economic Review 70(3), 393–408. Handa, P. and R. Schwartz. 1996a. “Limit order trading.” Journal of Finance 51, 1835–1861. Handa, P. and R. Schwartz. 1996b. “How best to supply liquidity to a securities market.” Journal of Portfolio Management 22, 44–51. Handa, P., R. Schwartz, and A. Tiwari. 2003. “Quote setting and price formation in an order driven market.” Journal of Financial Markets 6, 461–489. Harris, L. 1991. “Stock price clustering and discreteness.” Review of Financial Studies 4, 389–415. Harris, L. 1994. “Minimum price variations, discrete bid-ask spreads, and quotation sizes.” Review of Financial Studies 7, 149–178. Harris, L. 2003. Trading and exchanges: market microstructure for practitioners, Oxford University Press, New York. Harvey, C. and R. Huang. 1991. “Volatility in the foreign currency futures market.” Review of Financial Studies 4, 543–569. Hasbrouck, J. 1991. “Measuring the information content of stock trades.” Journal of Finance 46, 179–207. Hasbrouck, J. 1993. “Assessing the quality of a security market: a new approach to transaction-cost measurement.” Review of Financial Studies 6(1), 191–212. Hasbrouck, J. 1995. “One security, many markets: determining the contribution to price discovery.” Journal of Finance 50, 1175–1199. Hasbrouck, J. 2007. Empirical market microstructure, Oxford University Press, New York. Hasbrouck, J. and D. Seppi. 2001. “Common factors in prices, order flows and liquidity.” Journal of Financial Economics 59, 383–411. Hasbrouck, J. and G. Sofianos. 1993. “The trades of market makers: an empirical analysis of NYSE specialists.” Journal of Finance 48, 1565–1593. Ho, T. and H. Stoll. 1980. “On dealer markets under competition.” Journal of Finance 35, 259–267. Ho, T. and H. Stoll. 1981. “Optimal dealer pricing under transactions and return uncertainty.” Journal of Financial Economics 9, 47–73. Ho, T. and H. Stoll. 1983. “The dynamics of dealer markets under competition.” Journal of Finance 38, 1053–1074. Ho, T. R. Schwartz, and D. Whitcomb. 1985. “The trading decision and market clearing under transaction price uncertainty.” Journal of Finance 40, 21–42.

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market Jones, C., G. Kaul, and M. Lipson. 1994. “Information, trading and volatility.” Journal of Financial Economics 36, 127–154. Keim, D., and A. Madhavan. 1996. “The upstairs market for large-block transactions: analysis and measurement of price effects.” Review of Financial Studies 9, 1–36. Kyle, A. 1985. “Continuous auctions and insider trading.” Econometrica 53, 1315–1335. Madhavan, A. 2000. “Market microstructure.” Journal of Financial Markets 3, 205–258. Madhavan, A. and M. Cheng. 1997. “In search of liquidity: an analysis of upstairs and downstairs trades.” Review of Financial Studies 10, 175–204. Mendelson, M., J. Peake, and T. Williams. 1979. “Toward a modern exchange: the Peake-Mendelson-Williams proposal for an electronically assisted auction market,” in Impending changes for securities markets: what role for the exchange?, E. Bloch and R. Schwartz (Eds.). JAI Press, Greenwich, CT. Menkveld, A., S. Koopman, and A. Lucas. 2007. “Modelling aroundthe-clock price discovery for cross-listed stocks using state space methods.” Journal of Business and Economic Statistics 25(2), 213–225. Mildenstein, E., and H. Schleef. 1983. “The optimal pricing policy of a monopolistic marketmaker in the equity market.” Journal of Finance 38, 218–231. Milgrom, P. and N. Stokey. 1982. “Information, trade and common knowledge.” Journal of Economic Theory 26(1), 17–27. Miller, E. 1977. “Risk, uncertainty and divergence of opinion.” Journal of Finance 32, 1151–1168. O’Hara, M. 1997. Market microstructure theory, Basil Blackwell, Cambridge, MA. Ozenbas, D., R. Schwartz, and R. Wood. 2002. “Volatility in U.S. and European equity markets: an assessment of market quality.” International Finance 5(3), 437–461. Pagano, M. and A. Röell. 1996. “Transparency and liquidity: a comparison of auction and dealer markets with informed trading.” Journal of Finance 51, 579–611. Parlour, C. 1998. “Price dynamics in limit order markets.” Review of Financial Studies 11, 789–816. Parlour, C. and D. Seppi. 2008. “Limit order markets: a survey,” Handbook of financial intermediation & banking, A. W. A. Boot and A. V. Thakor (Eds.), Elesevier, Amsterdam. Paroush, J., R. Schwartz, and A. Wolf. 2008. The dynamic process of price discovery in an equity market, Working paper, Baruch College, CUNY. Pástor, L. and R. Stambaugh. 2003. “Liquidity risk and expected stock returns.” Journal of Political Economy 113, 642–685. Porter, D. and D. Weaver. 1998. “Post-trade transparency on Nasdaq’s national market system.” Journal of Financial Economics 50(2), 231–252. Sandas, P. 2001. “Adverse selection and competitive market making: empirical evidence from a limit order market.” Review of Financial Studies 14, 705–734. Schwartz, R. 1991. Reshaping the equity markets: a guide for the 1990s, Harper Business, New York. Schwartz, R. and R. Francioni. 2004. Equity markets in action, Wiley, New York. Seppi, D. 1990. “Equilibrium block trading and asymmetric information.” Journal of Finance 45, 73–94. Smidt, S. 1971. “Which road to an efficient stock market: free competition or regulated monopoly?” Financial Analyst Journal, 27, 18–20, 64–69. Stigler, G. 1964. “Public regulation of the securities markets.” Journal of Business 37, 117–142. Stoll, H. 1978. “The supply of dealer services in securities markets.” Journal of Finance 33, 1133–1151.

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Appendix 21A Risk Aversion and Risk Premium Measures Our analysis of the perfectly liquid CAPM environment makes reference to two measures of risk aversion and to several dimensions of a risk premium. We provide details concerning both of these in this appendix.

21A.1 Risk Aversion We use two risk aversion measures: (1) RA D U00 .W/=U0 .W/ is a measure of absolute risk aversion, and (2) RR D WRA is a measure of relative risk aversion. Because U00 < 0 for a risk averse decision maker, RA ; RR > 0 for risk aversion. Larger values of RA and RR indicate higher degrees of risk aversion. RA is a measure of absolute risk aversion because it reflects the decision maker’s reaction to uncertainty in relation to the absolute (dollar) gains/losses in an uncertain situation. RR is a measure of relative risk aversion because it reflects the decision maker’s reaction to uncertainty in relation to the percentage gains/losses in an uncertain situation.20

21A.2 Risk Premiums A risk premium is the minimum dollar compensation a decision maker requires to hold a risky asset in place of an alternative that involves no risk. Specifically, a decision maker would be indifferent between a riskless investment with a certain return of D dollars, and a risky investment with an expected dollar return of E(Z) equal to D plus the investor’s risk premium. In general, the investor’s risk premium depends upon his or her utility function and initial wealth, and upon the distribution of Z. Pi . / in Equation (21.3) is a risk premium: equals onehalf of RA (the measure of the investor’s absolute risk aversion) times Var.P2 /, which measures the absolute (dollar) risk attributable to holding one share of the market portfolio. The uncertainty associated with holding N shares of the risky 20

For further discussion, see J. Pratt, “Risk Aversion in the Small and the Large,” Econometrica, January 1964.

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asset is Var.NP2 / D N2 Var.P2 /; thus the total risk premium for holding N shares is T D N12

(21A.1)

Dividing Equation (21A.1) by N1 .D N0 C Q/ gives the risk premium per share (the average risk premium): A D N1

(21A.2)

Differentiating Equation (21A.1) with respect to N1 gives the risk premium for a marginal share (the marginal risk premium): (21A.3) m D 2 N1 Dividing Equation (21A.3) by P1 expresses the marginal risk premium as a percentage of current price: M % D

M 2 N1 D P1 P1

(21A.4)

The return on the combined portfolio of N1 shares of the market portfolio and C1 dollars of the risk-free asset is rP D

P2 1 P1



P1 N1 W





P1 N1 C 1 rf W

(21A.5)

and the variance of the return on the combined portfolio is  Var

P2 P1



P1 N1 W



D

N1 W

2 Var.P2 /

(21A.6)

Thus the investor’s risk premium associated with the uncertain return realized from the combined portfolio is rp D

N1 W

2

(21A.7)

Appendix 21B Designing Xetra This appendix provides further detail on the development and design of Deutsche Börse’s electronic trading platform, Xetra. The first steps in designing Xetra involved specifying principles that the new market should implement, and defining the system’s functionality. This was done by Deutsche Börse working together with key market participants. Most importantly, the new market system was to provide equal and decentralized access to all its participants. Further, the system’s functionality and the market information delivered to users (whether pre- or post-trade) were to be the same for all traders. A trader’s location should not matter. Equity trading in the German market has been and continues to be order driven. This was true both for IBIS

and for floor trading that was managed by a Kursmakler specialist acting in the capacity of auctioneer, broker, and dealer. It was clear from the beginning that Xetra should run an open limit order book (open in the sense that aggregated order volume is displayed at all price points in the order book). Additionally, order matching was automated and trader anonymity ensured. Core features of an electronic trading system are determined by the market structure that it implements. The structure defines how orders are handled and translated into trades and transaction prices. Xetra’s market model comprises diverse sub-models, each with a single trading modality, or a combination of multiple modalities (i.e., it is a hybrid). Most importantly, Xetra implements both continuous trading and periodic call auction trading. This differentiation is required to cope with liquidity differences among stocks, and different liquidity needs among users depending on the size of their orders and motives for trading. The market for all stocks opens and closes with a call auction, while less liquid stocks trade in multiple call auctions per day. Once the building-blocks were defined (i.e., continuous trading and call auctions), and their combinations specified, the next design step was to detail the specific features of each of the modalities. Those features are either static (i.e., represent basic structures such as the order book), or dynamic (i.e., define processes and behavior such as order matching). The next two sections of this appendix consider the systems design in more detail for continuous trading and periodic call auction trading, respectively.

21B.1 Continuous Trading By the mid-1990s, order books for continuous trading with price and time priorities had been implemented around the globe. In designing Xetra, Deutsche Börse’s market architects could refer to a wide range of existing examples, and to a broad microstructure literature. Once the eligible order types were identified, the center piece of the development was the definition of the detailed rules of price-time matching. The complexity of this definition was broken down into a finite set of individual cases that involved various order book situations combined with various incoming orders, for which the trading outcome was to be defined by a rule. All rules collectively described the dynamics of order matching. A major challenge in designing continuous trading involves the measures that should be taken to provide an orderly market in periods of sharply elevated price volatility. To deal with this, the concept of a “price corridor” was formulated. Diverse corridors around historical prices were defined that set the benchmark for an “orderly” price for the next

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

trade. If a price occurred that lay outside its corridor, trading was to be halted (briefly) with the entire order book transported into a call auction. The purpose of the call was to allow the market to consolidate in both space and time. Trading in the continuous market was resumed upon completion of the call. Lastly, all trading parameters for the continuous platform had to be determined. This included specifying tick-sizes, breadth of the price corridors, and durations and timings. Together, this provided a comprehensive overview of the “steering wheels” for the newly designed market.

21B.2 Call Auction Trading The purpose of Xetra’s call auctions is threefold: (1) to open and close continuous trading, (2) to trade less liquid stocks in multiple calls per day with no continuous trading offered, and (3) to stabilize the market in times of large price moves. Despite those multiple purposes, a single design was defined for the auctions. Additionally, certain key consistencies between continuous trading and the call had to be achieved. For example, both limit and market orders that could be submitted to continuous trading were allowed entry into the call order book. This seemingly simple requirement was difficult to implement because it expanded the universe of possible order book configurations (and therefore necessitated more complex matching rules). Additional procedures for setting the clearing price were also required to guard against erroneous pricing that could be caused by market orders overpowering an insufficient number of limit orders. As with continuous trading, price and time priority execution rules were stipulated. Most crucial was the degree of transparency that the calls would offer. Sufficient information about the order book had to be delivered for market participants to have relevant price and quantity information concerning actual market situations, but detailed information was suppressed to inhibit excessive information leakage and gaming. The pre-call information now available in Xetra is the highest bid and the lowest offer posted in the call when these orders do not cross, or the indicative call auction price that is calculated when the order book is crossed. In other words, the full order book content is not visible – pre-call, the Xetra screen displays only the potential outcome of the call at each point in time. When Xetra was under development, call auction trading at pre-specified times was managed on the floor by specialists who were responsible for price determination, timing,

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and the provision of dealer liquidity. The challenge was to reengineer the call so that it could be run by a computer, not by a human intermediary. The issue that Deutche Börse was facing was also grappled with by market microstructure academicians and other market architects. Substantial external guidance was received in the planning process. In particular, important inputs were obtained concerning the optimal degree of transparency for the call’s anti-gaming measures. The availability at the time of a variety of different call auction designs (both used and proposed) enabled Xetra’s calls to be designed relatively quickly.

21B.3 Electronic Trading for Less Liquid Stocks Kursmaklers (specialists) on the Frankfurt floor (both today and in the past) provide immediate liquidity at times when external liquidity is insufficient. The desire was strongly expressed, with two provisos, for a market maker to be incorporated into Xetra’s order driven model for less liquid stocks. The two provisos were (1) market participants must all have equal access to information, and (2) equal access to functionality must be maintained at a maximum level. Consequently, any changes that would favor the dealers were kept to a minimum. The dealers were referred to as “designated sponsors.” Like market makers in general, the designated sponsors were given both privileges and obligations. The primary obligation is that, on request of other market participants, the designated sponsor must provide quotes for a minimum volume and maximum spread in a stock during continuous trading. Additionally, multiple designated sponsors were included, so that they might compete with each other. Concurrently, the fulfillment of each sponsor’s obligation is measured, and the results published. The designated sponsors’ primary privilege is that they can see the identity of the quote requesters in an environment that otherwise ensures complete anonymity. Further, a sponsor balances the order book in all call auctions for the stocks that it is registered in. This gives the designated sponsors a last mover advantage (the freedom to trade against any imbalance that might exist at the market clearing price). With this privilege, a designated sponsor can influence the clearing price so as to execute orders that otherwise would not have transacted in that call. Lastly, the designated sponsors, depending on their measured performance, receive fee reductions.

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21B.4 Xetra’s Implementation and the Migration of Liquidity to Xetra Since 1997 Xetra went operational in fall 1997. At the beginning, the new system attracted roughly 60% of trading in the most liquid segment of the market, the 30 DAX stocks. Trading on Xetra for mid-cap stocks was not as successful – market share for this segment was about 20%, as the less liquid stocks largely continued at that time to trade on the floor. But the 1997 launch was just the start of a sequence of releases that have continued through the current time. One more recent innovation was the “continuous call auction.” With this facility, calls are not held at pre-specified

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times but are triggered by the occurrence of a “critical” liquidity situation. The continuous call comprises a dealerauctioneer who is responsible for providing a base level of liquidity in each call, as well as controlling its timing. Additionally, Xetra allows internalization of trading by member firms. Consequently, Xetra, which originally started as an exchange trading system, now also serves as the technical platform for OTC trading. Major innovations have benefited a broad range of cap sizes and, across the board, floor trading has continued to decline. Xetra has now been rolled out to 260 member firms in Europe, and its market share currently stands at 95% of all on-exchange trading in Germany today.

Chapter 22

Options Strategies and Their Applications Cheng Few Lee, John Lee, and Wei-Kang Shih

Abstract In this chapter we introduce different types of options and their characteristics. Then, we develop put-call parity theorems for European, American, and futures options. Finally, we discuss option strategies and their investment applications. Keywords Call option r Put option r American option r European option r Exercise price r Time to maturity r Futures option r Long straddle r Short straddle r Long vertical spread r Short vertical spread r Calendar spread r Protective put r Covered call r Collar

22.1 Introduction The use of stock options for risk reduction and return enhancement has expanded at an astounding pace over the last 20 years. Among the causes of this growth, two are most significant. First, the establishment of the Chicago Board Option Exchange (CBOE) in 1973 brought about the liquidity necessary for successful option trading, through public listing and standardization of option contracts. The second stimulus emanated from academia. In the same year that the CBOE was established, Professors Fischer Black and Myron Scholes published a paper in which they derived a revolutionary option-pricing model. The power of their model to predict an option’s fair price has since made it the industry standard. The development of option-valuation theory shed new light on the valuation process. Previous pricing models such as CAPM were based on very stringent assumptions, such as there being an identifiable and measurable market portfolio, as well as various imputed investor attributes, such

C.F. Lee and W.-K. Shih () Rutgers University, Newark, NJ, USA e-mail: [email protected] J. Lee Center for PBBEF Research, New York, NY, USA

as quadratic utility functions. Furthermore, previous theory priced only market risk since investors were assumed to hold well-diversified portfolios. The strength of the Black-Scholes and subsequent option-pricing models is that they rely on far fewer assumptions. In addition, the option-valuation models price total risk and do not require any assumptions concerning the direction of the underlying securities price. The growing popularity of the option concept is evidenced by its application to the valuation of a wide array of other financial instruments (such as common stock and bonds) as well as more abstract assets including leases and real estate agreements. This chapter aims to establish a basic knowledge of options and the markets in which they are traded. It begins with the most common types of options, calls, and puts, explaining their general characteristics and discussing the institutions where they are traded. In addition, the concepts relevant to the new types of options on indexes and futures are introduced. The next focus is the basic pricing relationship between puts and calls, known as put-call parity. The final study concerns how options can be used as investment tools. The chapter on option valuation that follows utilizes all these essential concepts to afford a deeper conceptual understanding of valuation theory.

22.2 The Option Market and Related Definitions This section discusses option-market and related definitions of options, which are needed to understand option valuations and option strategies.

22.2.1 What Is an Option? An option is a contract conveying the right to buy or sell a designated security at a stipulated price. The contract

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normally expires at a predetermined time. The most important element of an option contract is that there is no obligation placed upon the purchaser: it is an “option.” This attribute of an option contract distinguishes it from other financial contracts. For instance, while the holder of an option has the opportunity to let his or her claim expires unused if so desired, futures and forward contracts obligate their parties to fulfill certain conditions.

22.2.2 Types of Options and Their Characteristics A call option gives its owner the right to buy the underlying asset while a put option conveys to its holder the right to sell the underlying asset. An option is specified by five essential parts: 1. 2. 3. 4. 5.

The type (call or put) The underlying asset The exercise price The expiration date The option price

While the most common type of underlying asset for an option is an individual stock, other underlying assets for options exist as well. These include futures contracts, foreign currencies, stock indexes, and U.S. debt instruments. In the case of common stock options (on which this discussion is exclusively centered), the specified quantity to which the option buyer is entitled to buy or sell is one hundred shares of the stock per option. The exercise price (also called the strike price) is the price stated in the option contract at which the call (put) owner can buy (sell) the underlying asset up to the expiration date, the final calendar date on which the option can be traded. Options on common stocks have expiration dates 3 months apart in one of three fixed cycles. 1. January/April/July/October 2. February/May/August/November 3. March/June/September/December The normal expiration date is the third Saturday of the month. (The third Friday is the last trading date for the option.) As an example, an option referred to as an “ABC June 25 call” is an option to buy one hundred shares of the underlying ABC stock at $25 per share, up to its expiration date in June. Option prices are quoted on a per-share basis. Thus, a stock option that is quoted at $5 would cost $500 ($5 100 shares), plus commission and a nominal SEC fee. A common distinction among options pertains to when they can be exercised. Exercising an option is the process of

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carrying out the right to buy or sell the underlying asset at the stated price. American options allow the exercise of this right at any time from when the option is purchased up to the expiration date. On the other hand, European options allow their holder the right of exercise only on the expiration date itself. The distinction between an American and European option has nothing to do with the location at which they are traded. Both types are currently bought and sold in the United States. There are distinctions in their pricing and in the possibility of exercising them prior to expiration. Finally, when discussing options, the two parties to the contract are characterized by whether they have bought or sold the contract. The party buying the option contract (call or put) is the option buyer (or holder), while the party selling the option is the option seller (or writer). If the writer of an option does not own the underlying asset, he or she is said to write a naked option. Table 22.1 shows a listing of publicly traded options for Johnson & Johnson at September 21, 2007.

22.2.3 Relationships Between the Option Price and the Underlying Asset Price A call (put) option is said to be in the money if the underlying asset is selling above (below) the exercise price of the option. An at-the-money call (put) is one whose exercise price is equal to the current price of the underlying asset. A call (put) option is out of the money if the underlying asset is selling below (above) the exercise price of the option. Suppose ABC stock is selling at $30 per share. An ABC June 25 call option is in the money .$30  25 > 0/, while an ABC June 35 call option is out of the money .$30  35 < 0/. Of course, the expiration dates could be any month without changing the option’s standing as in, at, or out of the money. The relationship between the price of an option and the price of the underlying asset indicates both the amount of intrinsic value and time value inherent in the option’s price, as shown in Equation (22.1): Intrinsic Value D Underlying Asset Price Option Exercise Price

(22.1)

For a call (put) option that is in the value (underlying asset price > exercise price), its intrinsic value is positive. And for at-the-money and out-of-the-money options the intrinsic value is zero. An option’s time value is the amount by which the option’s premium (or market price) exceeds its intrinsic value. For a call or put option: Time Value D Option Premium  Intrinsic Value

(22.2)

22 Options Strategies and Their Applications

Table 22.1 Options quotes for Johnson & Johnson at 09/21/2007

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Stock price at 9/21/2007 D $65.12 Call options expiring Fri Jan 18, 2008 Strike

Symbol

Last

Bid

Ask

Vol

Open int

40 45 50 55 60 65 70 75 80 85 90

JNJAH.X JNJAI.X JNJAJ.X JNJAK.X JNJAL.X JNJAM.X JNJAN.X JNJAO.X JNJAP.X JNJAQ.X JNJAR.X

24 20 15:5 11:1 6:4 2:65 0:55 0:1 0:05 0:05 0:05

25.5 20.5 15.7 10.9 6.4 2.65 0.55 0.05 N/A N/A N/A

25:6 20:7 16 11:1 6:5 2:7 0:6 0:1 0:05 0:05 0:05

10 3 11 33 275 1;544 845 2 10 0 0

5;427 2;788 8;700 10;327 32;782 70;426 48;582 13;629 4;497 3;275 3;626

Put options expiring Fri Jan18, 2008 Strike Symbol Last 40 JNJMH.X 0:05 45 JNJMI.X 0:1 50 JNJMJ.X 0:12 55 JNJMK.X 0:25 60 JNJML.X 0:7 65 JNJMM.X 1:8 70 JNJMN.X 5 75 JNJMO.X 13:3

Bid N/A 0.05 0.1 0.25 0.65 1.85 4.9 9.7

Ask 0:05 0:1 0:15 0:3 0:7 1:95 5 9:9

Vol

Open int 1;370 5;002 14;004 31;122 69;168 46;774 1; 582 20

where intrinsic value is the maximum of zero or stock price minus exercise price. Thus an option premium or market price is composed of two components, intrinsic value and time value. In-the-money options are usually the most expensive because of their large intrinsic-value component. An option with an at-the-money exercise price will have only time value inherent in its market price. Deep out-of-the-money options have zero intrinsic value and little time value and consequently are the least expensive. Deep in-the-money cases also have little time value, and time value is the greatest for at-the-money options. In addition, time value (as its name implies) is positively related to the amount of time the option has to expiration. The theoretical valuation of options focuses on determining the relevant variables that affect the time-value portion of an option premium and the derivation of their relationship in option pricing. In general, the call price should be equal to or exceed the intrinsic value: C Max.S  E; 0/ where: C D the value of the call option; S D the current stock price; and E D the exercise price. Figure 22.1 illustrates the relationship between an option’s time value and its exercise price. When the exercise

0 3 1 99 227 30 20 0

Fig. 22.1 The relationship between an option’s exercise price and its time value

price is zero, the time value of an option is zero. Although this relationship is described quite well in general by Fig. 22.1, the exact relationship is somewhat ambiguous. Moreover, the identification of options with a mispriced time-value portion in their total premium motivates interest in a theoretical pricing model. One more aspect of time value that is very important is the change in the amount of time value an option has as its duration shortens. As previously mentioned, options with a

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Exercise price X January 15 $5.50 20 25

April Max.20  15; 0/ C1:25 D $6:25 Max.20  20; 0/ Max.20  20; 0/ C1:00 D $1:00 C2:00 D $2:00 Max.20  25; 0/ Max.20  25; 0/ C5:0 D $0:50 C1:25 D $1:25

July Max.20  15; 0/ C3:50 D $8:50 Max.20  20; 0/ C5:00 D $5:00 Max.20  25; 0/ C3:50 D $3:50

Other values are shown in the table above.

22.2.3.1 Additional Definitions and Distinguishing Features Fig. 22.2 The relationship between time value and time to maturity for a near-to-the-money option (assuming a constant price for the underlying asset)

longer time to maturity and those near to the money have the largest time-value components. Assuming that a particular option remains near to the money as its time to maturity diminishes, the rate of decrease in its time value, or what is termed the effect of time decay, is of interest. How time decay affects an option’s premium is an important question for the valuation of options and the application of option strategies. To best see an answer to this question refer to Fig. 22.2. In general, the value of call options with the same exercise price increases as time to expiration increases: C .S1 ; E1 ; T1 /  C .S1 ; E1 ; T2 / where T1  T2 and T1 and T2 are the time to expiration. Note that for the simple case in Fig. 22.2 the effect of time decay is smooth up until the last month before expiration, when the time value of an option begins to decay very rapidly. This effect is made clearer if we refer to Chap. 23, in which the components of the option-pricing model was examined. Example 22.4 shows the effect of time decay. Example 22.1. It is January 1, the price of the underlying ABC stock is $20 per share, and the time premiums are shown in the following table. What is the value for the various call options? Exercise price X 15 20 25

January $0:50 1:00 0:50

April $1:25 2:00 1:25

July $3:50 5:00 3:50

Solution 22.1. Call premium D Intrinsic value C Time premium C15;Jan D Max.20  15; 0/ C 0:50 D $5:50 per share or $550 for 1 contract

Options may be specified in terms of their classes and series. A class of options refers to all call and put options on the same underlying asset. For example, all AT&T call and put options at various exercise prices and expiration months form one class. A series is a subset of a class and consists of all contracts of the same class (such as AT&T) having the same expiration date and exercise price. When an investor either buys or sells an option (that is, is long or short) as the initial transaction, the option exchange adds this opening transaction to what is termed the open interest for an option series. Essentially, open interest represents the number of contracts outstanding at a particular point in time. If the investor reverses the initial position with a closing transaction (that is, sells the option if he or she originally bought it or vice versa) then the open interest for the particular option series is reduced by one). While open interest is more of a static variable, indicating the number of outstanding contracts at one point in time, volume represents a dynamic characteristic. More specifically, volume indicates the number of times a particular option is bought and sold during a particular trading day. Volume and open interest are measures of an option’s liquidity, the ease with which the option can be bought and sold in large quantities. The larger the volume and/or open interest, the more liquid the option. Again, an option holder who invokes the right to buy or sell is exercising the option. Whenever a holder exercises an option, a writer is assigned the obligation to fulfill the terms of the option contract by the exchange on which the option is traded. If a call holder exercises the right to buy, a call writer is assigned the obligation to sell. Similarly, when a put holder exercises the right to sell, a put writer is assigned the obligation to buy. The seller or writer of a call option must deliver 100 shares of the underlying stock at the specified exercise price when the option is exercised. The writer of a put option must purchase 100 shares of the underlying stock when the put option is exercised. The writer of either option receives the premium or price of the option for this legal obligation. The

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maximum loss an option buyer can experience is limited to the price of the option. However, the maximum loss from writing a naked call is unlimited; the maximum loss possible from writing a naked put is the exercise price less the original price of that put. To guarantee that the option writer can meet these obligations, the exchange clearinghouse requires margin deposits. The payment of cash dividends affects both the price of the underlying stock and the value of an option on the stock. Normally, no adjustment is made in the terms of the option when a cash dividend is paid. However, strike price or number of shares may be adjusted if the underlying stock realizes a stock dividend or stock split. For example, an option on XYZ Corporation with an exercise price of $100 would be adjusted if XYZ Corporation stock split two for one. The adjustment in this case would be a change in the exercise price from $100 to $50, and the number of contracts would be doubled. In the case of a noninteger split (such as three for two), the adjustment is made to the exercise price and the number of shares covered by the option contracts. For example, if XYZ Corporation had an exercise price of $100 per share and had a three-for-two split, the option would have the exercise price adjusted to $662=3 , and the number of shares would be increased to 150. Notice that the old exercise value of the option, $10,000 ($100 100 shares) is maintained by the adjustment ($662=3 150 shares).

22.2.4 Types of Underlying Assets Although most people would identify common stocks as the underlying asset for an option, a variety of other assets and financial instruments can assume the same function. In fact, options on agricultural commodities were introduced by traders in the U.S. as early as the mid-l800s. After a number of scandals, agricultural commodity options were banned by the government. They were later reintroduced under tighter regulations and in a more standardized tradable form. Today, futures options on such agricultural commodities as corn, soybeans, wheat, cotton, sugar, live cattle, and live hogs are actively traded on a number of exchanges. The biggest success for options has been realized on financial futures. Options on the S&P 500 index futures contracts, NYSE index futures, foreign-currency futures, 30-year U.S. Treasury bond futures, and gold futures have all realized extraordinary growth since their initial offerings back in 1982. Options on futures are very similar to options on the actual asset, except that the futures options give their holders the right (not the obligation) to buy or sell predetermined quantities of specified futures contracts at a fixed price within a predetermined period. Options on the actual asset have arisen in another form as well. While a number of options have existed for various

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stock-index futures contracts, options now also exist on the stock index itself. Because of the complexity of having to provide all the stocks in an index at the spot price should a call holder exercise his or her buy right, options on stock indexes are always settled in cash. That is, should a call holder exercise his or her right to buy because of a large increase in the underlying index, that holder would be accommodated by a cash amount equal to the profit on his contract, or the current value of the option’s premium. Although the options on the S&P 100 stock index at the Chicago Board Options Exchange (CBOE) are the most popular among traders, numerous index options are now traded as well. These include options on the S&P 500 index, the S&P OTC 250 index, the NYSE composite and AMEX indexes (computer technology, oil and gas, and airline), the Philadelphia Exchange indexes (gold/silver), the Value Line index, and the NASDAQ 100 index.

22.2.5 Institutional Characteristics Probably two of the most important underlying factors leading to the success of options have been the standardization of contracts through the establishment of option exchanges and the trading anonymity brought about by the Option Clearing Corporations and clearinghouses of the major futures exchanges. An important element for option trading is the interchangeability of contracts. Exchange contracts are not matched between individuals. Instead, when an investor or trader enters into an option contract, the Option Clearing Corporation (or clearinghouse for the particular futures exchange) takes the opposite side of the transaction. So rather than having to contact a particular option writer to terminate an option position, a buyer can simply sell it back to the exchange at the current market clearing price. This type of anonymity among option-market participants is what permits an active secondary market to operate. The sources of futures options traded on the various futures exchanges mentioned earlier are determined by the open-auction bidding, probably the purest form of laissezfaire price determination that can be seen today. With the open-auction-bidding price mechanism there are no market makers, only a large octagonal pit filled with traders bidding among themselves to buy and sell contracts. While some traders buy and sell only for themselves, many of the participants are brokers representing large investment firms. Different sides of the pit usually represent traders who are dealing in particular expiration months. As brokers and other pit participants make trades they mark down what they bought or sold, how much, at what price, and from whom. These cards are then collected by members of the exchange

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who record the trades and post the new prices. The prices are displayed on “scoreboards” surrounding the pit. While stock options and options on commodities and indexes are traded in a similar fashion, one major difference prevails – the presence of market makers. Market makers are individuals who typically trade one type of option for their own account and are responsible for ensuring that a market always exists for their particular contract. In addition, some option exchanges utilize board brokers as well. These individuals are charged with the maintenance of the book of limit orders (orders from outside investors that are to be executed at particular prices or when the market goes up or down by a prespecified amount). Essentially, market makers and board brokers on the options exchanges share the duties performed by the specialists on the major stock exchanges. Although stocks can be bought with as little as 50% margin, no margin is allowed for buying options – the cost of the contract must be fully paid. Because options offer a high degree of leverage on the underlying asset, additional leveraging through margins is considered by regulators to be excessive. However, if more than one option contract is entered into at the same time – for instance, selling and buying two different calls – then, of course, a lower cost is incurred, since the cost of one is partially (or wholly) offset by the sale of the other.

22.3 Put-Call Parity This section addresses a most important concept, called put-call parity (for option valuation). The discussion includes European options, American options, and future options.

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Theorem 22.1. Put-Call Parity for European Options with No Dividends. Ct;T D Pt;T C St  EBt;T where: Ct;T D value of a European call option at time t that matures at time T .T > t/; Pt;T D value of a European put option at time t, that matures at time T ; St D value of the underlying stock (asset) to both the call and put options at time t; E D exercise price for both the call and put options; Bt;T D price at time t of a default-free bond that pays $1 with certainty at time T (if it is assumed that this risk-free rate of interest is the same for all maturities and equal to r – in essence a flat-term structure – then Bt;T D e r.T t / , under continuous compounding), or Bt;T D 1= .1 C r/T t for discrete compounding. Equation (22.3) uses the following principle. If the options are neither dominant nor dominated securities, and if the borrowing and lending rates are equal, then the return patterns of a European call and a portfolio composed of a European put, a pure discount bond with a face value equal to the options exercise price E, and the underlying stock (or asset) are the same.1 In understanding why the put-call parity theorem holds, and to support the theorem, two additional properties of option pricing must be provided: Property 22.1. At maturity (time T ) the call option is worth the greater of ST  E dollars or zero dollars: CT D Max.0; ST  E/

22.3.1 European Options As an initial step to examining the pricing formulas for options, it is essential to discuss the relationships between the prices of put and call options on the same underlying asset. Such relationships among put and call prices are referred to as the put-call parity theorems. Stoll (1969) was the first to introduce the concept of put-call parity. Dealing strictly with European options he showed that the value of a call option would equal the value of a portfolio composed of a long put option, its underlying stock, and a short discounted exercise price. Before stating the basic put-call parity theorem as originally devised by Stoll, it must be assumed that the markets for options, bonds, and stocks (or any other underlying asset we choose) are frictionless.

(22.3)

(22.4)

As an example, suppose that the call option has an exercise price of $30. At maturity, if the stock’s (asset’s) price is $25, then the value of the call is the maximum of (0, 25–30) or (0, 5), which of course is zero. If an option sells for less than (ST  E) its intrinsic value, an arbitrage opportunity will exist. Investors would buy the option and short sell the stock, forcing the mispricing to correct itself. Consequently, this first property implies that a call option’s value is always greater than zero. An equivalent property and argument exist for the value of a put option as well.

1

Any security x is dominant over any security y if the rate of return on x is equal to or greater than that of y for all states of nature and is strictly greater for at least one state. For an expanded discussion of this subject, see Merton (1973) and Smith (1976).

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Property 22.2. At maturity, the value of a put option is the greater of (E  ST ) dollars or zero dollars: PT D Max.0; E  ST /

P0;1yr D $5 C $110

Example 22.2. A call option with 1 year to maturity and exercise price of $110 is selling for $5. Assuming discrete compounding, a risk-free rate of 10%, and a current stock price of $100, what is the value of a European put option with a strike price of $110 and 1-year maturity?

.1:1/1

 $100

P0;1yr D $5

(22.5)

Using the same line of reasoning and argument as for the call option, the second property also implies that the value of a put option is never less than zero. Table 22.2 provides proof of this first put-call parity theorem. Suppose at time t, two portfolios are formed: portfolio B is just a long call option on a stock with price St an exercise price of E, and a maturity date at T . Portfolio A consists of purchasing one hundred shares of the underlying stock (since stock options represent one hundred shares), purchasing (going long) one put option on the same stock with exercise price E and maturity date T , and borrowing at the risk-free rate an amount equal to the present value of the exercise price or EBt;T with face value of E. (This portion of the portfolio finances the put, call, and stock position.) At maturity date T , the call option (portfolio B) has value only if ST > E, which is in accordance with Property 1. For portfolio A, under all these conditions the stock price and maturing loan values are the same, whereas the put option has value only if E > ST . Under all three possible outcomes for the stock price ST , it can be seen that the values of portfolios A and B are equal. Proof has been established for the first put-call parity theorem. Example 22.2 provides further illustration.



1

22.3.2 American Options Of course, this first put-call parity theorem holds only under the most basic conditions (that is, no early exercise and no dividends). Jarrow and Rudd (1983) give an extensive coverage of the effects of more complicated conditions on put-call parity. These authors demonstrate that the effect of known dividends is simply to reduce, by the discounted value (to time t) of the dividends, the amount of the underlying stock purchased. In considering stochastic dividends, the exactness of this pricing relationship breaks down and depends on the degree of certainty that can be maintained about the range of future dividends. Put-call parity for American options is also derived under various dividend conditions. Jarrow and Rudd demonstrate that as a result of the American option’s early exercise feature, strict pricing relationships give way to boundary conditions dependent on the size and certainty of future dividends, as well as the level of interest rates and the size of the exercise price. To summarize, they state that for sufficiently high interest rates and/or exercise prices it may be optimal to exercise the put prior to maturity (with or without dividends). So the basic put-call parity for an American option with no dividends and constant interest rates is described by the following theorem. Theorem 22.2. Put-Call Parity for an American Option with No Dividends Pt;T C S  EBt;T > Ct;T > Pt;T C St  E

(22.6)

Solution 22.2. Increasing the generality of conditions results in increasing boundaries for the equilibrium relationship between put and

Pt;T D Ct;T C EBt;T  St Table 22.2 Put-call parity for a European option with no dividends

Time T (maturity) ST > E

ST D E

ST < E

1. Buy 100 shares of the stock .St /

ST

ST

ST

2. Buy a put (Pt , maturity at T with exercise price E)

0

0

E  ST

3. Borrow EBt;T dollars

E .ST E/

E 0

E 0

.ST  E/

0

0

Time t strategy Portfolio A

Portfolio A value at time T Portfolio B 1. Buy a call (Ct maturing at T with exercise price E)

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call options. The beauty of these arguments stems from the fact that they require only that investors prefer more wealth to less. If more stringent assumptions are made, then the bounds can be made tighter. For an extensive derivation and explanation of these theorems see Jarrow and Rudd (1983). Example 22.3 provides further illustration. Example 22.3. A put option with 1 year to maturity and an exercise price of $90 is selling for $15; the stock price is $100. Assuming discrete compounding and a risk-free rate of 10%, what are the boundaries for the price of an American call option? Solution 22.3. Pt;T C S  EBt;T > Ct;T > Pt;T C St  E

1 > Ct;T > $15 C $100  $90 $15 C $100  $90 .1:1/1 $33:18 > Ct;1yr > $25

Futures Options As a final demonstration of put-call parity the analysis is extended to the case where the underlying asset is a futures contract. The topic of futures contracts and their valuation will be more fully examined. Nevertheless, this chapter takes time to apply put-call parity when the options are on a futures contract because of the growing popularity and importance of such futures options. A futures contract as described in as a contract in which the party entering into the contract is obligated to buy or sell the underlying asset at the maturity date for some stipulated price. While the difference between European and American options still remains, the complexity of dividends can be ignored since futures contracts do not

pay dividends. Put-call parity for a European futures option (when interest rates are constant) is as follows: Theorem 22.3. Put-Call Parity for a European Futures Option. Ct;T D Pt;T C Bt;T .Ft;T  E/

(22.7)

where Ft;T is the price at time t for a futures contract maturing at time T (which is the underlying asset to both the call and put options). Option pricing Properties 1 and 2 for call and put options apply in an equivalent sense to futures options as well. However, to understand this relationship as stated in Equation (22.7) it must be assumed that the cost of a futures contract is zero. While a certain margin requirement is required, the majority of this assurance deposit can be in the form of interest-bearing securities. Hence as an approximation a zero cost for the futures contract is not unrealistic. Again, the easiest way to prove this relationship is to follow the same path of analysis used in proving Theorem 1. Table 22.3 indicates that the argument for this theorem’s proof is similar, with only a few notable exceptions. The value of the futures contract at time T (maturity) is equal to the difference between the price of the contract at time T and the price at which it was bought, or FTT Ft;T . This is an outcome of the fixed duration of a futures contract as opposed to the perpetual duration of common stock. Second, because no money is required to enter into the futures contract, the exercise price is reduced by the current futures price and the total is lent at the risk-free rate. (Actually this amount is either lent or borrowed depending on the relationship between Ft;T and E at time t. If Ft;T  E < 0, then this amount will actually be borrowed at the risk-free rate.) Why are there options on spot assets as well as options on futures contracts for the spot assets? After all, at expiration the basis of a futures contract goes to zero and futures prices

Table 22.3 Put-call parity for a European futures option

Time T (maturity) FTT > E

FTT D E

1. Buy a futures contract .Ft;T /

FTT  Ft;T

FTT  Ft;T

FTT  Ft;T

2. Buy a put (Pt;T on Ft;T with exercise price E and maturity T )

0

0

E  FTT

3. Lend Bt;T .FtT  E/ dollars

Ft;T E .FTT E/

Ft;T E 0

Ft;T E 0

FTT  E

0

0

Time t strategy

FTT < E

Portfolio A

Portfolio A’s value at time T Portfolio B 1. Buy a call (Ct;T on Ft;T with exercise price E and maturity T )

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equal spot prices; thus, options in the spot and options on the future are related to the same futures value, and their current values must be identical. Yet a look at the markets shows that options on spot assets and options on futures for the same assets sell at different prices. One explanation for this is that investors who purchase options on spot must pay a large sum of money when they exercise their options, whereas investors who exercise an option on a future need only pay enough to meet the initial margin for the futures contract. Therefore, if the exercise of the option is important to an investor, that investor would prefer options on futures rather than options on spot and would be willing to pay a premium for the option on the future, whereas the investor who has no desire to exercise the option (remember, the investor can always sell it to somebody else to realize a profit) is not willing to pay for this advantage and so finds the option on spot more attractive.

22.3.3 Market Application Put options were not listed on the CBOE until June 1977. Before that time, brokers satisfied their clients’ demands for put option risk-return characteristics by a direct application of put-call parity. By combining call options and the underlying security, brokers could construct a synthetic put. To illustrate, the put-call parity theorem is used when a futures contract is the underlying asset. Furthermore, to simulate the option broker’s circumstances on July 1, 1984, the equation is merely rearranged to yield the put’s “synthetic” value: Pt;T D Ct;T  Bt;T Ft;T C Bt;T E

(22.8)

So instead of a futures contract being purchased, it is sold. Assume the following values and use the S&P 500 index futures as the underlying asset. Ct;T D $ 3:35; Ft;T D 154:85 (September contract); E D 155:00; and Bt;T D 0:9770 (current price of a risk-free bond that pays $1 when the option and futures contract expire, average of bid and ask prices for T -bills from The Wall Street Journal). According to Equation (22.8), the put’s price should equal the theorem price: Pt;T D $3:497. The actual put price on this day (July 1, 1984) with the same exercise price and expiration month was Pt;T D $3:50. With repeated comparisons of the theorem using actual prices, it becomes clear that put-call parity is a powerful equilibrium mechanism in the market.

22.4 Risk-Return Characteristics of Options One of the most attractive features of options is the myriad ways in which they can be employed to achieve a particular combination of risk and return. Whether through a straight option position in combination with the underlying asset or some portfolio of securities, options offer an innovative and relatively low-cost mechanism for altering and enhancing the risk-return tradeoff. In order to better grasp these potential applications this section analyzes call and put options individually and in combination, relative to their potential profit and loss and the effects of time and market sentiment.

22.4.1 Long Call The purchase of a call option is the simplest and most familiar type of option position. The allure of calls is that they provide the investor a great deal of leverage. Potentially, large percentage profits can be realized from only a modest price rise in the underlying asset. In fact, the potential profit from buying a call is unlimited. Moreover, the option purchaser has the right but no obligation to exercise the contract. Therefore, should the price of the underlying asset decline over the life of the call, the purchaser need only let the contract expire worthless. Consequently, the risk of a long call position is limited. Figure 22.3 illustrates the profit profile of a long call position. The following summarizes the basic risk-return features for a long-call position. Profit potential: unlimited Loss potential: limited (to cost of option) Effect of time decay: negative (decrease option’s value) Market expectation: bullish As the profit profile indicates, the time value of a long call declines over time. Consequently, an option is a wasting asset. If the underlying asset’s price does not move above the exercise price of the option E by its expiration date T , the buyer of the call will lose the value of his initial investment (the option premium). Consequently, the longer an investor holds a call, the more time value the option loses, thereby, reducing the price of the option. This leads to another important point – taking on an option position. As with any other investment vehicle, the purchaser of a call expresses an opinion about the market for the underlying asset. Whereas an investor can essentially express one of three different sentiments (bullish, neutral, or bearish) about future market conditions, the long call is strictly a bullish position. That is, the call buyer only wins if the underlying asset rises in price. However, depending on the exercise price of the call, the

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Fig. 22.3 Profit profile for a long call

Fig. 22.4 Profit profile for a short call

buyer can express differing degrees of bullishness. For instance, since out-of-the-money calls are the cheapest, a large price increase in the underlying asset will make these calls the biggest percentage gainers in value. So an investor who is extremely bullish would probably go with an out-of-themoney call, since its intrinsic value is small and its value will increase along with a large increase in the market.

22.4.2 Short Call Selling a call (writing it) has risk-reward characteristics, which are the inverse of the long call. However, one major distinction arises when writing calls (or puts) rather than buying them. That is, the writer can either own the underlying asset upon which he or she is selling the option (a covered write), or simply sell the option without owning the asset (a naked write). The difference between the two is of consid-

erable consequence to the amount of risk and return taken on by the seller. Let us first examine the profit profile and related attributes of the naked short call, displayed in Fig. 22.4. When the writer of a call does not own the underlying asset, his or her potential loss is unlimited. Why? Because if the price of the underlying asset increases, the value of the call also increases for the buyer. The seller of a call is obliged to provide a designated quantity of the underlying asset at some prespecified price (the exercise price) at any time up to the maturity date of the option. So if the asset starts rising dramatically in price and the call buyer exercises his or her right, the naked-call writer must go into the market to buy the underlying asset at whatever the market price. The naked-call writer suffers the loss of buying the asset at a price S and selling it at a price E when S > E (less the original premium collected). When common stock is the underlying asset, there is no limit to how high its price could go. Thus, the naked-call writer’s risk is unlimited as well. Of course, the naked-call writer could have reversed position by buying

22 Options Strategies and Their Applications

back the original option he sold – that is, zeroing out the position – however, this also done at a loss. The following summarizes the basic risk-return features for a naked shortcall position. Profit potential: limited (to option premium) Loss potential: unlimited Effect of time decay: positive (makes buyer’s position less valuable) Market expectation: bearish to neutral The naked short-call position is obviously a bearish position. If the underlying asset’s price moves down, the call writer keeps the entire premium received for selling this call, since the call buyer’s position becomes worthless. Once again, the naked-call writer can express the degree of bearishness by the exercise price at which he or she sells the call. By selling an in-the-money call, the writer stands to collect a higher option premium. Conversely, selling an out-of-themoney call conveys only a mildly bearish to neutral expectation. If the underlying asset’s price stays where it is, the value of the buyer’s position, which is solely time value, will decay to zero; and the call writer will collect the entire premium (though a substantially smaller premium than for an in-the-money call). While the passing of time has a negative effect on the value of a call option for the buyer, it has a positive effect for the seller. One aspect of an option’s time value is that in the last month before the option expires, its time value decays most rapidly. Why? Time value is related to the probability that the underlying asset’s price will move up or down enough to make an option position increase in value. This probability declines at an accelerating (exponential) rate as the option approaches its maturity date. The consideration of time value, then, is a major element when investing in or hedging with options. Unless an investor is extremely bullish,

Fig. 22.5 Profit profile for a covered short call

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it would probably be unwise to take a long position in a call in its last month before maturity. Conversely, the last month of an option’s life is a preferred time to sell since its time value can more easily and quickly be collected. Now consider the other type of short-call position, covered-call writing. Because the seller of the call owns the underlying asset in this case, the risk is truncated. The purpose of writing a call on the underlying asset when it is owned is twofold. First, by writing a call option, one always decreases the risk of owning the asset. Second, writing a call can increase the overall realized return on the asset. The profit profile for a covered short call (or a covered write) in Fig. 22.5 provides further illustration. The following summarizes the basic risk-return features for the covered short-call position. Profit potential: limited (exercise price  asset price C call premium) Loss potential: limited (asset price  call premium) Effect of time decay: positive Market expectation: neutral to mildly bullish By owning the underlying asset, the covered-call writer’s loss on the asset for a price decline is decreased by the original amount of the premium collected for selling the option. The total loss on the position is limited to the extent that the asset is one of limited liability, such as a stock, and cannot fall below zero. The maximum profit on the combined asset and option position is higher than if the option was written alone, but lower than simply owning the asset with no short call written on it. Once the asset increases in price by a significant amount the call buyer will very likely exercise the right to purchase the asset at the prespecified exercise price. Thus, covered-call writing is a tool or strategy for enhancing an asset’s realized return while lowering its risk in a sideways market.

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22.4.3 Long Put Again, the put option conveys to its purchasers the right to sell a given quantity of some asset at a prespecified price on or before its expiration date. Similar to a long call, a long put is also a highly leveraged position, but the purchaser of the put makes money on the investment only when the price of the underlying asset declines. While a call buyer has unlimited profit potential, a put buyer has limited profit potential since the price of the underlying asset can never drop below zero. Yet like the long-call position, the put buyer can never lose more than the initial investment (the option’s premium). The profit profile for a long put is shown in Fig. 22.6. The following summarizes the basic risk-return features for the profit profile of a long-put position. Profit potential: limited (asset price must be greater than zero) Loss potential: limited (to cost of put) Effect of time decay: negative or positive Market expectation: bearish An interesting pricing ambiguity for this bearish investment is how the put’s price is affected by the time decay. With the long call there is a clear-cut relation – that is, the effect of the time decay is to diminish the value of the call. The relationship is not so clear with the long put. Although at certain prices for the underlying asset the value of the long-put position decreases with time, there exist lower asset prices for which its value will increase with time. It is the put’s ambiguous relationship with time that makes its correct price difficult to ascertain. (This topic has been explored in Chap. 5.) One uniquely attractive attribute of the long put is its negative relationship with the underlying asset. In terms of the capital asset pricing model, it has a negative beta (though usually numerically larger than that of the underlying asset, due to the leverage affect). Therefore, the long put is an ideal hedging instrument for the holder of the underlying asset who wants to protect against a price decline. If the investor is

Fig. 22.6 Profit profile for a long put

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wrong and the price of the asset moves up instead, the profit from the asset’s price increase is only moderately diminished by the cost of the put. (See Chap. 23 for more on hedging and related concepts.)

22.4.4 Short Put As was true for the short-call position, put writing can be covered or uncovered (naked). The risk-return features of the uncovered (naked) short put are discussed first. For taking on the obligation to buy the underlying asset at the exercise price, the put writer receives a premium. The maximum profit for the uncovered-put-writer is this premium, which is initially received. Figure 22.7 provides further illustration. While the loss potential is limited for the uncoveredput-writer, it is nonetheless still very large. Thus, someone neutral on the direction of the market would sell out-of-themoney (lower exercise price) puts. A more bullish sentiment would suggest that at-the-money options be sold. The investor who is convinced the market will go up should maximize return by selling a put with a larger premium. As with the long put, the time-decay effect is ambiguous and depends on the price of the underlying asset. The following summarizes the basic risk-return features for the profit profile of an uncovered short-put position. Profit potential: limited (to put premium) Loss potential: limited (asset price must be greater than zero) Effect of time decay: positive or negative Market expectation: neutral to bullish Referring again to Fig. 22.5 for the combined short-call and long-asset position, notice the striking resemblance of its profit profile at expiration to that for the uncovered short put. This relationship can be seen mathematically by using

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Fig. 22.7 Profit profile for an uncovered short call

Fig. 22.8 Profit profile for a long straddle

put-call parity. That is, the synthetic put price PT D E C CT ST , or at expiration the value of the put should equal the exercise price of the call option plus the call option’s value minus the value at time T of the underlying asset. Buying (writing) a call and selling (buying) the underlying asset (or vice versa) allows an investor to achieve essentially the same risk-return combination as would be received from a long put (short put). This combination of two assets to equal the risk and return of a third is referred to as a synthetic asset (or synthetic option in this case). Synthesizing two financial instruments to resemble a third is an arbitrage process and is a central concept of finance theory. Now a look at covered short puts is in order to round out the basics of option strategies. For margin purposes and in a theoretical sense, selling a put against a short-asset position would be the sale of a covered put. However, this sort of position has a limited profit potential if the underlying asset is anywhere below the exercise price of the put at expiration. This position also has unlimited upside risk, because the short position in the asset will accrue losses while the profit from the put sale is limited. Essentially, this position is equivalent to the uncovered or naked short call, except that the latter has less expensive transaction costs. Moreover, because the time value for put options is generally less than that of calls, it will be advantageous to short the call.

Strictly speaking, a short put is covered only if the investor also owns a corresponding put with exercise price equal to or greater than that of the written put. Such a position, called a spread, is discussed later in this chapter.

22.4.5 Long Straddle A straddle is a simultaneous position in both a call and a put on the same underlying asset. A long straddle involves purchasing both the call and the put. By combining these two seemingly opposing options an investor can get the best risk-return combination that each offers. The profit profile for a long straddle in Fig. 22.8 illustrates the nature of this synthetic asset. The following summarizes the basic risk-return features for the profit profile of a long-straddle position. Profit potential: unlimited on upside, limited on downside Loss potential: limited (to cost of call and put premiums) Effect of time decay: negative Market sentiment: bullish or bearish The long straddle’s profit profile makes clear that its riskreward picture is simply that of the long call overlapped by the long put, with each horizontal segment truncated

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(represented by the horizontal dashed lines on the bottom). An investor will profit on this type of position as long as the price of the underlying asset moves sufficiently up or down to more than cover the original cost of the option premiums. Thus, a long straddle is an effective strategy for someone expecting the volatility of the underlying asset to increase in the future. In the same light, the investor who buys a straddle expects the underlying asset’s volatility of price to be greater than that imputed in the option price. Since time decay is working against the value of this position, it might be unwise to purchase a straddle composed of a call and put in their last month to maturity when their time decay is greatest. It would be possible to reduce the cost of the straddle by purchasing a high-exercise-price call and a low- exercise put (out-of-the-money options); however, the necessary up or down movement in the asset’s price in order to profit is larger. Example 22.4 provides further illustration. Example 22.4. Situation: An investor thinks the stock market is going to break sharply up or down but is not sure which way. However, the investor is confident that market volatility will increase in the near future. To express his position the investor puts on a long straddle using options on the S&P 500 index, buying both at-the-money call and put options on the September contract. The current September S&P 500 futures contract price is 155.00. Assume the position is held to expiration. Transaction: 1. Buy 1 September 155 call at $2.00 ( 500 per contract) 2. Buy 1 September 155 put at $2.00 Net initial investment (position value) Results: 1. If futures price D 150.00: (a) 1 September call expires at $0 (b) 1 September put expires at $5.00 (c) Less initial cost of put Ending position value (net profit) 2. If futures price D 155.00: (a) 1 September call expires at $0 (b) 1 September put expires at $0 Ending position value (net loss) 3. If futures price D 160.00: (a) 1 September call expires at $5.00 (b) 1 September call expires at $0 (c) Less initial cost of put Ending position value (net profit)

Maximum loss potential: $2,000, the initial investment. Breakeven points: 151.00 and 159.00, for the September S&P 500 futures contract.2 Effect of time decay: negative, as evidenced by the loss incurred, with no change in futures price (result 2).

22.4.6 Short Straddle For the most part, the short straddle implies the opposite riskreturn characteristics of the long straddle. A short straddle is a simultaneous position in both a short call and a short put on the same underlying asset. Contrary to the long-straddle position, selling a straddle can be an effective strategy when an investor expects little or no movement in the price of the underlying asset. A similar interpretation of its use would be that the investor expects the future volatility of the underlying asset’s price that is currently impounded in the option premiums to decline. Moreover, since the time decay is a positive effect for the value of this position, one appropriate time to set a short straddle might be in the last month to expiration for the combined call and put. Figure 22.9 shows the short straddle’s profit profile, and Example 22.5 provides further illustration. The following summarizes the basic risk-return features for the profit profile of a short-straddle position. Profit potential: limited (to call and put premiums) Loss potential: unlimited on upside, limited on downside Effect of time decay: positive Market expectation: neutral

($1,000) ($1,000) ($2,000)

($1,000) $2,500 ($1,000) $500 ($1,000) ($1,000) $2,000 $2,500 ($1,000) ($1,000) $500

Summary: Maximum profit potential: unlimited. If the market had contributed to move below 150.00 or above 160.00, the position would have continued to increase in value.

Example 22.5. Situation: An investor thinks the market is overestimating price volatility at the moment and that prices are going to remain stable for some time. To express his opinion, the investor sells a straddle consisting of at-the-money call and put options on the September S&P 500 futures contract, for which the current price is 155.00. Assume the position is held to expiration. Transaction: 1. Sell 1 September 155 call at $2.00 ( 500 per contract) 2. Sell 1 September 155 put at $2.00 Net initial inflow (position value)

2

$1,000 $1,000 $2,000

Breakeven points for the straddle are calculated as follows: Upside BEP D Exercise price C Initial net investment (in points) 159:00 D 155:00 C 4:00 Downside BEP D Exercise price  Initial net investment (in points) 159:00 D 155:00 C 4:00 151:00 D 155:00  4:00

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Fig. 22.9 Profit profile for a short straddle

Results: 1. If futures price D 150.00: (a) 1 September 155 call expires at 0 (b) I September 155 put expires at $5.00 (c) Plus initial inflow from sale of put Ending position value (net loss) 2. If futures price D 155.00: (a) 1 September 155 call expires at 0 (b) I September 155 put expires at 0 Ending position value (net profit) 3. If futures price D 160.00: (a) 1 September 155 call expires at $5.00 (b) I September put expires at 0 (c) Plus initial inflow from sale of call Ending position value (net loss)

$1,000 ($2,500) $1,000 ($500) $1,000 $1,000 $2,000 ($2,500) $1,000 $1,000 ($500)

Summary: Maximum profit potential: $2,000, result 2, where futures price does not move. Maximum loss potential: unlimited. If futures price had continued up over 160.00 or down below 145.00, this position would have kept losing money. Breakeven points: 151.00 and 159.00, an eight-point range for profitability of the position.3 Effect of time decay: positive, as evidenced by result 2.

22.4.7 Long Vertical (Bull) Spread When dealing strictly in options, a spread is a combination of any two or more of the same type of options (two calls or two puts, for instance) on the same underlying asset. A vertical spread specifies that the options have the same 3

Breakeven points for the short straddle are calculated in the same manner as for the long straddle: exercise price plus initial prices of options.

maturity month. Finally, a long vertical spread designates a position for which one has bought a low-exercise-price call (or a low-exercise-price put) and sold a high-exercise-price call (or a high-exercise-price put) that both mature in the same month. A long vertical spread is also known as a bull spread because of the bullish market expectation of the investor who enters into it. Actually, the long vertical spread (or bull spread) is not a strongly bullish position, because the investor limits the profit potential in selling the highexercise-price call (or high-exercise-price put). Rather, this is a popular position when it is expected that the market will more likely go up than down. Therefore, the bull spread conveys a bit of uncertainty about future market conditions. Of course, the higher the exercise price at which the call is sold, the more bullish the position. An examination of the profit profile for the long vertical spread (see Fig. 22.10) can tell more about its risk-return attributes. The following summarizes the basic risk-return features for the profit profile of a long-vertical-spread position. Profit potential: limited (up to the higher exercise price) Loss potential: limited (down to the lower exercise price). Effect of time decay: mixed Market expectation: cautiously bullish Although profit is limited by the shorted call on the upside, the loss potential is also truncated at the lower exercise price by the same short call. There are other reasons for this being a mildly bullish strategy. The effect of time decay is ambiguous up to the expiration or liquidation of the position. That is, if the asset price, St , is near the exercise price of the higher-price option EH , then the position acts more like a long call and time-decay effect is negative. Conversely, if St is near the exercise price of the lower-price option EL , then the bull spread acts more like a short call and the time-decay effect is neutral. Consequently, unless an investor is more than mildly bullish, it would probably be unwise to put on a bull spread with the low exercise price call near the current price of the

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Fig. 22.10 Profit profile for a long vertical spread

asset while both options are in their last month to expiration. Example 22.6 provides further illustration. Example 22.6. Situation: An investor is moderately bullish on the West German mark. He would like to be long but wants to reduce the cost and risk of this position in case he is wrong. To express his opinion, the investor puts on a long vertical spread by buying a lower-exercise-price call and selling a higher-exercise-price call with the same month to expiration. Assume the position is held to expiration. Transaction: 1. Buy 1 September 0.37 call at ($ 587.50) $0.0047 ( 125000 per contract) 2. Sell 1 September 0.38 call at $ 1 62.50 $0.0013 Net initial investment ($ 425.00) (position value) Results: 1. If futures price D 0.37: (a) 1 September 0.37 call expires at $0 (b) 1 September 0.38 call expires at $0 Ending position value (net loss) 2. If futures price D 0.3800: (a) 1 September 0.37 call expires at $0.01 (b) 1 September 0.38 call expires at $0 Less initial cost of 0.37 call Ending position value (net profit) 3. If futures price D 0.39: (a) 1 September 0.37 call expires at $0.02 (b) 1 September 0.38 call expires at $0.01 Less initial premium of 0.37 call Plus initial premium of 0.38 call Ending position value (net profit)

($587.50) $162.50 ($425.00) $1,250.00 $162.50 ($587.50) $825.00

Summary: Maximum profit potential: $825.00, result 2. Maximum loss potential: $425.00, result 1. Breakeven point: 0.3734.4 Effect of time decay: mixed. positive if price is at high end of range and negative if at low end.

22.4.8 Short Vertical (Bear) Spread The short vertical spread is simply the reverse of the corresponding long position. That is, an investor buys a highexercise-price call (or put) and sells a low-exercise-price call (or put), both having the same time to expiration left. As the more common name for this type of option position is bear spread, it is easy to infer the type of market sentiment consistent with this position. The profit profile for the short vertical spread is shown in Fig. 22.11. As the profit profile indicates, this strategy is profitable as long as the underlying asset moves down in price. Profit is limited to a price decline in the asset down to the lower exercise price, while risk is limited on the upside by the long-call position. From the time-decay effects shown, a mildly bearish investor might consider using options in the last month to expiration with the EL option near the money. The following summarizes the basic risk-return features for the profit profile of a short-vertical-spread position. Profit potential: limited (down to EL ) Loss potential: limited (up to EH ) Effect of time decay: mixed (opposite to that of long vertical spread) Market sentiment: mildly bearish

$2,500.00 ($1,250.00) ($587.50) $162.50 $825.00

4

Breakeven point for the long vertical spread is computed at a lower exercise price plus price of long call minus price of short call .0:3734 D 0:3700 C 0:0047  0:0013/.

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Fig. 22.11 Profit profile for a short vertical spread

Fig. 22.12 Profit profile for a neutral calendar spread

22.4.9 Calendar (Time) Spreads A calendar spread (also called a time or horizontal spread) consists of the sale of one option and the simultaneous purchase of another option with the same exercise price but a longer term to maturity. The objective of the calendar spread is to capture the faster erosion in the time-premium portion of the shorted nearer-term-to-maturity option. By taking a position in two of the same type options (two calls or two puts), both with the same exercise price, the investor utilizing this strategy expresses a neutral opinion on the market. In other words, the investor is interested in selling time rather than predicting the price direction of the underlying asset. Thus, a calendar spread might be considered appropriate for a sideways-moving or quiet market. However, if the underlying asset’s price moves significantly up or down, the calendar spread will lose part of its original value. Figure 22.12 displays the calendar spread’s profit profile and related riskreturn attributes.

The profit profile shows that this strategy will make money for a rather narrow range of price movement in the underlying asset. While similar in nature to the short straddle (both are neutral strategies), the calendar spread is more conservative. The reason? It has both a lower profit potential and lower (limited) risk than the short straddle. The lower potential profit is the result of only benefiting from the time decay in one option premium instead of two (the call and the put) for the short straddle. Moreover, taking opposite positions in the same type of option at the same exercise price adds a loss limit on each side against adverse price moves. The following summarizes the basic risk-return features for the profit profile of a neutral calendar-spread position. Profit potential: limited Loss potential: limited (to original cost of position) Effect of time decay: positive. (Option sold loses value faster than option bought.) Market sentiment: neutral

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The calendar spread does not have to be neutral in sentiment. By diagonalizing this spread it is possible to express an opinion on the market. For instance, by selling a near-term higher-exercise-price option and purchasing a longer-term lower-exercise-price option, the investor is being bullish in position. Such a position is thus referred to as a bullish calendar spread. Why is it bullish? Remember that with the neutral calendar spread we are concerned solely with benefiting from the faster time decay in the premium of the shorted near-term option. Any significant movement in price upwards, for instance, would have not been profitable because it would have slowed the time decay and increased the intrinsic value of the shorted near-term option. In fact, we would eventually lose money because the difference in premiums between the near-term and the longer-term options (the spread) would narrow as the underlying asset’s price increased. However, the bullish calendar spread is much like a long vertical (or bull) spread in that a modest increase in price for the asset up to the higher exercise price will be profitable. At the same time, though, the bullish calendar spread also reaps some of the benefits from the greater time decay in the nearer-term option’s premium. While this strategy might sound superior to the straight bull spread, it really depends on market conditions. With a bullish calendar spread, its gain from time decay will probably not be as great as that from a neutral calendar spread, nor will its bullish nature be as profitable as a straight bull spread in the event of a modest price increase for the underlying asset. The real world application will be discussed in the next section.

22.5 Examples of Alternative Option Strategies In this section we are considering several other option strategies and using the options of Constellation Energy Group, Inc. (CEG) in the following examples. Below is the information published on July 13, 2007 for all options that will expire in October 2007. CEG stock closed at $94.21 on July 13, 2007 (Table 22.4).

22.5.1 Protective Put Assume that an investor wants to invest in the CEG stock on July 13, 2007, but does not desire to bear any potential loss for prices below $95. The investor can purchase CEG stock and at the same time buy the put option CEGVS.X with a strike price of $95. Let S0 ; ST , and X

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Table 22.4 Call and put option quotes for CEG at 07/13/2007 Call option expiring close Fri Oct 19, 2007 Strike

Symbol

Bid

Ask

70 75 80 85 90 95 100 105 110 115

CEGJN.X CEGJO.X CEGJP.X CEGJQ.X CEGJR.X CEGJS.X CEGJT.X CEGJA.X CEGJB.X CEGJC.X

23:5 19 14:6 11:4 8 5 2:85 1:5 0:65 0:2

25:5 20:9 16:4 12:2 8:5 5:4 3:2 1:75 0:9 0:45

Bid 0:15 0:4 0:9 1:6 2:95 4:9 7:7

Ask 0:35 0:65 1:15 1:85 3:4 5:5 8:6

Put option expiring close Fri Oct 19, 2007 Strike Symbol 70 CEGVN.X 75 CEGVO.X 80 CEGVP.X 85 CEGVQ.X 90 CEGVR.X 95 CEGVS.X 100 CEGVT.X

denote the stock purchase price, future stock price at the expiration time T, and the strike price, respectively. Given S0 D $94:21, X D $95, and the premium for the put option $5.5, Table 22.5 shows the values for Protective Put at different stock prices at time T. The profit profile of the Protective Put position is constructed in Fig. 22.13.

22.5.2 Covered Call This strategy involves investing in a stock and selling a call option on the stock at the same time. The value at the expiration of the call will be the stock value minus the value of the call. The call is “covered” because the potential obligation of delivering the stock is covered by the stock held in the portfolio. In essence, the sale of the call sold the claim to any stock value above the strike price in return for the initial premium. Suppose a manager of a stock fund holds 1,000 shares of CEG stock on July 13, 2007 and she plans to sell the CEG stock if its price hits $100. Then she can write 1,000 share of the call option CEGJB.X with a strike price of $100 to establish the position. She shorts the call and collects premiums. Given that S0 D $94:21; X D $100, and the premium for the call option $2.85, Table 22.6 shows the values for Covered Call at different stock prices at time T. The profit profile of the Covered Call position is constructed in Fig. 22.14.

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Table 22.5 Value of protective put position at option expiration

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Long a put at strike price

$95:00

Premium

Buy one share of stock

$5:50

Price

Stock Price

Payoff

Profit

Payoff

Profit

Payoff

Profit

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$24:21 $19:21 $14:21 $9:21 $4:21 $0:79 $5:79 $10:79 $15:79 $20:79 $25:79

$25:00 $20:00 $15:00 $10:00 $5:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00

$19:50 $14:50 $9:50 $4:50 $0:50 $5:50 $5:50 $5:50 $5:50 $5:50 $5:50

$95:00 $95:00 $95:00 $95:00 $95:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$4:71 $4:71 $4:71 $4:71 $4:71 $4:71 $0:29 $5:29 $10:29 $15:29 $20:29

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

Long put .X D $95/

$94:21

One share of stock

Fig. 22.13 Profit profile for protective put

Protective put value

Protective Put : Profit $30 $20

Profit

$10

One Share of Stock

$0 Long Put (X=$95)

−$10 −$20

Protective Put Value

−$30 $70

$75

$80

$85

$90

$95

$100 $105 $110 $115 $120

Stock Price

Table 22.6 Value of covered call position at option expiration

Write a call at strike price Buy one share of stock Stock

One share of stock

Price

Payoff

Profit

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$24:21 $19:21 $14:21 $9:21 $4:21 $0:79 $5:79 $10:79 $15:79 $20:79 $25:79

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

Premium Price

$100.00 Written call .X D $100/ Payoff $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $5:00 $10:00 $15:00 $20:00

$2:85 $94:21

Covered call Profit

Payoff

Profit

$2:85 $2:85 $2:85 $2:85 $2:85 $2:85 $2:85 $2:15 $7:15 $12:15 $17:15

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $100:00 $100:00 $100:00 $100:00

$21:36 $16:36 $11:36 $6:36 $1:36 $3:64 $8:64 $8:64 $8:64 $8:64 $8:64

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Fig. 22.14 Profit profile for covered call

Covered Call : Profit $30

Profit

$20 One Share of Stock

$10

Written Call (X=$100)

$0 -$10

Covered Call

-$20 -$30 $70

$75

$80

$85

$90

$95 $100 $105 $110 $115 $120

Stock Price

Table 22.7 Value of collar position at option expiration Long a put at strike price $85:00 Write a call at strike price

Premium

$105:00

$1:85

Premium

Buy one share of stock

$1:50

Price

$94:21

Stock

One share of stock

Long put .X D $85/

Write call .X D $105/

Collar value

Price

Payoff

Profit

Payoff

Profit

Payoff

Payoff

Profit

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$24:21 $19:21 $14:21 $9:21 $4:21 $0:79 $5:79 $10:79 $15:79 $20:79 $25:79

$15:00 $10:00 $5:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00

$13:15 $8:15 $3:15 $1:85 $1:85 $1:85 $1:85 $1:85 $1:85 $1:85 $1:85

$85:00 $85:00 $85:00 $85:00 $90:00 $95:00 $100:00 $105:00 $105:00 $105:00 $105:00

$9:56 $9:56 $9:56 $9:56 $4:56 $0:44 $5:44 $10:44 $10:44 $10:44 $10:44

Profit

$0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $5:00 $10:00 $15:00

Fig. 22.15 Profit profile for collar

$1:50 $1:50 $1:50 $1:50 $1:50 $1:50 $1:50 $1:50 $3:50 $8:50 $13:50

Collar : Profit

Profit

$30 $20

One Share of Stock

$10

Long put (X=$85)

$0

Write Call (X=$105) Collar Value

−$10 −$20 −$30 $70

$75

$80

$85

$90

$95

$100

$105

$110

$115

$120

Stock Price

22.5.3 Collar A collar combines a protective put and a short call option to bracket the value of a portfolio between two bounds. For example, an investor holds the CEG stock selling at $94.21. Buying a protective put using the put option CEGVQ.X with an exercise price of $85 places a lower bound of $85 on the

value of the portfolio. At the same time, the investor can write a call option CEGJA.X with an exercise price of $105. The call and the put sell at $1.50 and $1.85, respectively, making the net outlay for the two options to be only $0.35. Table 22.7 shows the values of the Collar position at different stock prices at time T. The profit profile of the Collar position is shown in Fig. 22.15.

22 Options Strategies and Their Applications

22.6 Conclusion This chapter has introduced some of the essential differences between the two most basic kinds of options: calls and puts. A delineation was made of the relationship between the option’s price or premium and that of the underlying asset. The option’s value was shown to be composed of intrinsic value, or the underlying asset price less the exercise price, and time value. Moreover, it was demonstrated that for an option the time value decays over time, particularly in the last month to maturity. Index and futures options were studied to introduce these important financial instruments. Put-call parity theorems were developed for European, American, and futures options in order to show the basic valuation relationship between the underlying asset and its call and put options. Finally, investment application of options and related combinations were discussed, along with relevant risk-return characteristics.

References Amram, M. and N. Kulatilaka. 2001. Real options, Oxford University Press, USA. Ball, C. and W. Torous. 1983. “Bond prices dynamics and options.” Journal of Financial and Quantitative Analysis 18, 517–532. Bhattacharya, M. 1980. “Empirical properties of the Black-Scholes formula under ideal conditions.” Journal of Financial and Quantitative Analysis 15, 1081–1106. Black, F. 1972. “Capital market equilibrium with restricted borrowing.” Journal of Business 45, 444–445. Black, F. 1985. “Fact and fantasy in the use of options.” Financial Analysts Journal 31, 36–72. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 31, 637–654. Bodhurta, J. and G. Courtadon. 1986. “Efficiency tests of the foreign currency options market.” Journal of Finance 41, 151–162. Bookstaber, R. M. 1981. Option pricing and strategies in investing, Addison-Wesley Publishing Company, Reading, MA. Bookstaber, R. M. and R. Clarke. 1983. Option strategies for institutional investment management, Addison-Wesley Publishing Company, Reading, MA. Brennan, M. and E. Schwartz. 1977. “The valuation of American put options.” Journal of Finance 32, 449–462. Cox, J. C. and M. Rubinstein. 1979. “Option pricing: a simplified approach.” Journal of Financial Economics 8, 229–263. Cox, J. C. and M. Rubinstein. 1985. Option markets, Prentice-Hall, Englewood Cliffs, NJ. Eckardt, W. and S. Williams. 1984. “The complete options indexes.” Financial Analysts Journal 40, 48–57. Ervine, J. and A. Rudd. 1985. “Index options: the early evidence.” Journal of Finance 40, 743–756.

375 Finnerty, J. 1978. “The Chicago board options exchange and market efficiency.” Journal of Financial and Quantitative Analysis 13, 28–38. Galai, D. and R. W. Masulis. 1976. “The option pricing model and the risk factor of stock.” Journal of Financial Economics 3, 53–81. Galai, D., R. W. Masulis, R. Geske, and S. Givots. 1988. Option markets, Addison-Wesley Publishing Company, Reading, MA. Gastineau, G. 1979. The stock options manual, McGraw-Hill, New York. Geske, R. and K. Shastri. 1985. “Valuation by approximation: a comparison of alternative option valuation techniques.” Journal of Financial and Quantitative Analysis 20, 45–72. Hull, J. 2005. Options, futures, and other derivatives, 6th Edition, Prentice Hall, Upper Saddle River, NJ. Jarrow, R. A. and A. Rudd. 1983. Option pricing, Richard D. Irwin, Homewood, IL. Jarrow, R. A. and S. Turnbull. 1999. Derivatives securities, 2nd Edition, South-Western College Pub, Cincinnati, OH. Lee, C. F. and A. C. Lee 2006. Encyclopedia of finance, Springer, New York. Liaw, K. T. and R. L. Moy. 2000. The Irwin guide to stocks, bonds, futures, and options, McGraw-Hill Companies, New York. Macbeth, J. and L. Merville. 1979. “An empirical examination of the Black-Scholes call option pricing model.” Journal of Finance 34, 1173–1186. McDonald, R. L. 2005. Derivatives markets, 2nd Edition, Addison Wesley, Boston, MA. Merton, R. 1973. “Theory of rational option pricing.” Bell Journal of Economics and Management Science 4, 141–183. Rendleman, R. J. Jr. and B. J. Barter. 1979. “Two-state option pricing.” Journal of Finance 34, 1093–1110. Ritchken, P. 1987. Options: theory, strategy and applications, Scott, Foresman. Rubinstein, M. and H. Leland. 1981. “Replicating options with positions in stock and cash.” Financial Analysts Journal 37, 63–75. Rubinstein, M., H. Leland, and J. Cox. 1985. Option markets, PrenticeHall, Englewood Cliffs, NJ. Sears, S. and G. Trennepohl. 1982. “Measuring portfolio risk in options.” Journal of Financial and Quantitative Analysis 17, 391–410. Smith, C. 1976. “Option pricing: a review.” Journal of Financial Economics 3, 3–51. Stoll, H. 1969. “The relationships between put and call option prices.” Journal of Finance 24, 801–824. Summa, J. F. and J. W. Lubow. 2001. Options on futures, John Wiley & Sons, New York. Trennepohl, G. 1981. “A comparison of listed option premium and Black-Scholes model prices: 1973–1979.” Journal of Financial Research 4, 11–20. Weinstein, M. 1983. “Bond systematic risk and the options pricing model.” Journal of Finance 38, 1415–1430. Welch, W. 1982. Strategies for put and call option trading, Winthrop, Cambridge, MA. Whaley, R. 1982. “Valuation of American call options on dividend paying stocks: empirical tests.” Journal of Financial Economics 10, 29–58. Zhang, P. G. 1998. Exotic options: a guide to second generation options, 2nd Edition, World Scientific Pub Co Inc, Singapore.

Chapter 23

Option Pricing Theory and Firm Valuation Cheng Few Lee, Joseph E. Finnerty, and Wei-Kang Shih

Abstract In this chapter, we introduce the basic concepts of call and put options. Second, we discuss the Black-Scholes option pricing model and its application. Third, we discuss how to apply the option pricing theory in capital structure issue. Finally, the warrant, one type of equity options, is discussed in detail. Keywords Warrants r Executive stock option r Publicly traded option r Call option r Put option r American option r European option r Exercise price r Time to maturity r Black-Scholes option pricing model r Capital structure

23.1 Introduction We discuss the emergence of options and option pricing, including several types of options and how their value is determined. We begin looking at the basic concepts of options in Sect. 23.2, then go on to discuss factors that affect the value of options in Sect. 23.3. Hedging, hedge ratio, and option valuation are discussed in Sect. 23.4. Section 23.5 discusses how option pricing theory is used to investigate the capital structure question. We close the chapter with a look at the type of option called the warrant in Sect. 23.6 and a summary of this chapter is discussed in Sect. 23.7.

23.2 Basic Concepts of Options In general, there are three types of equity options: (1) warrants, (2) executive stock options, and (3) publicly traded options. A warrant is a financial instrument issued by a corporation that gives the purchaser the right to buy a fixed number of shares at a set price for a specific period. There C.F. Lee and W.-K. Shih () Rutgers University, Newark, NJ, USA e-mail: [email protected] J.E. Finnerty University of Illinois at Urbana-Champaign, Champaign, IL, USA

are two major differences between a warrant and a publicly traded option. The first is that the maturity of the warrant is normally less than 9 months. The second difference is that the warrant is an agreement between the corporation and the warrant’s buyer. This means that if the warrant’s owner decides to exercise his right and purchase stock, the corporation issues new shares and receives the cash from the sale of those shares. The publicly traded option is an agreement between two individuals who have no relationship with the corporation whose shares are being optioned. When the publicly traded option is exercised, money is exchanged for shares between individuals and the corporation receives no funds, only a new owner. Executive stock options are a means of compensation for corporate employees. For services rendered, the manager or the employee has the right to buy a specific number of shares for a set price during a given period. Unlike warrants and publicly traded options, executive stock options cannot be traded. The option’s owner has only two choices: exercise the option or let it expire. Like a warrant, should the owner decide to exercise the option, the corporation receives money and issues new shares. The use of executive stock options for management compensation raises an interesting agency question. The firm’s managers may make investment and financing decisions that increase the firm’s risk in order to increase the value of their stock options. Such action could have a detrimental effect on the bondholders and other creditors of the firm. Thus, we will see that the value of an option is directly related to the variability or riskiness of the underlying asset, which in this case is the firm. Publicly traded options are probably the most widely known of the three types of equity option instruments. An important date in the history of these options is 1973, when the Chicago Board Options Exchange was founded. Although it was possible to trade options over the counter before that time, trading volume was relatively low. This date marks the beginning of a phenomenal growth in the popularity of options as a financial instrument. Indeed, in terms of the value of securities traded, the Chicago Board Options Exchange is running neck and neck with the New York and Tokyo stock

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exchanges as the world’s largest securities market. There are now five options-trading centers in the United States – the Chicago Board Option Exchange, the American Stock Exchange, the Philadelphia Stock Exchange, the Pacific Stock Exchange, and the New York Stock Exchange – and there is a steady stream of proposals for new listings on these exchanges. The share volume of trading and the general acceptance of these financial instruments renders options valuation an important subject for study. However, even more significant for our purposes, the theory of options valuation has important applications in financial management and in the valuation of other financial instruments. Black and Scholes (1973) point out those corporate securities can be viewed as options or contingent claims on firm value. In viewing a firm’s securities as options, we need to evaluate the interdependencies of bonds and stocks. As we have seen, a bond’s price is the present value of future interest and principal payments, and a stock’s price is determined by future dividends. The contingent-claims approach to security valuation differs in that it considers the valuation of all of the firm’s classes of securities simultaneously. Bonds and stocks are valued in terms on the value placed on the firm’s assets; that is, their value is contingent on the firm’s assets or investments. Banz and Miller (1978) developed an approach to making capital budgeting decisions based on a contingent claims framework. They created a method for calculating the NPV of an investment based on estimates derived from using an option valuation model. In this chapter, we consider the question of option valuation and how options are used to make financial management decisions. However, before doing so, we must become acquainted with some of the terminology of options trading. There are two basic types of options: puts and calls. A call gives the holder the right to buy a particular number of shares of a designated common stock at a specified price, called the exercise price (or strike price), on or before a given date, known as the expiration date. Hence, the owner of shares of common stock can create an option and sell it in the options market, thereby increasing the return or income on his or her stock investment. Such an option specifies both an exercise price and an expiration date. On the Chicago Board of Options Exchange, options are typically created for 3, 6, or 9 months. The actual expiration date is the third Saturday of the month of expiration. A more venturesome investor may create an option in this fashion without owning any of underlying stock. This is called naked option writing and can be very risky, especially if the value of the underlying asset has a high degree of variability. A put gives the holder the right to sell a certain number of shares of common stock at a set price on or before the expiration date of the option. In purchasing a put, the owner of shares has bought the right to sell these shares by the expira-

C.F. Lee et al.

tion date at the exercise price. As with calls, the creator can own the underlying shares (covered writing) or not (naked writing). The owner of a put or call is not obligated to carry out the specified transaction, but has the option of doing so. If the transaction is carried out, it is said to have been exercised. For example, if you hold a call option on a stock that is currently trading at a price higher than the exercise price, because the stock could be immediately resold at a profit. (You also could sell the option or hold i