Asset Allocation and International Investments Edited by
Greg N. Gregoriou
ASSET ALLOCATION AND INTERNATIONAL INVESTMENTS
Also edited by Greg N. Gregoriou ADVANCES IN RISK MANAGEMENT DIVERSIFICATION AND PORTFOLIO MANAGEMENT OF MUTUAL FUNDS PERFORMANCE OF MUTUAL FUNDS
Asset Allocation and International Investments
Edited by GREG N. GREGORIOU
Selection and editorial matter © Greg N. Gregoriou 2007 Individual chapters © contributors 2007 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2007 by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St. Martin’s Press, LLC and of Palgrave Macmillan Ltd. Macmillan® is a registered trademark in the United States, United Kingdom and other countries. Palgrave is a registered trademark in the European Union and other countries. ISBN-13: 978–0–230–01917–1 ISBN-10: 0–230–01917–X This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Asset allocation and international investments / edited by Gerg N. Gregoriou. p.cm. — (Finance and capital markets) Includes bibliographical references and index. ISBN 0–230–01917–X 1. Asset allocation. 2. Investments, Foreign. 3. Globalization—Economic aspects. 4. Portfolio management. I. Gregoriou, Greg N., 1956– II. Series. HG4529.5.A83 2006 332.67’3—dc22 2006045369 10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 09 08 07 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham and Eastbourne
To my mother Evangelia and in loving memory of my father Nicholas
This page intentionally left blank
Contents
Acknowledgments
xi
Notes on the Contributors
xii
Introduction
xvii
1 Time-Varying Downside Risk: An Application to the Art Market
1
Rachel Campbell and Roman Kräussl 1.1 Introduction 1.2 Art as an investment 1.3 Previous empirical studies 1.4 Empirical analysis 1.5 Data 1.6 Methodology 1.7 Results 1.8 Discussion 1.9 Conclusion
2 International Stock Portfolios and Optimal Currency Hedging with Regime Switching
1 3 4 5 6 9 10 11 13
16
Markus Leippold and Felix Morger 2.1 2.2 2.3 2.4 2.5
Introduction The model Estimation results Discussion Conclusion
16 18 21 26 39 vii
viii
CONTENTS
3 The Determinants of Domestic and Foreign Biases: An Empirical Study Fathi Abid and Slah Bahloul 3.1 Introduction 3.2 Theoretical framework of domestic and foreign biases 3.3 Data and preliminary statistics 3.4 The determinants of domestic and foreign biases 3.5 The empirical analysis 3.6 Additional tests 3.7 Conclusion
4 The Critical Line Algorithm for UPM–LPM Parametric General Asset Allocation Problem with Allocation Boundaries and Linear Constraints
42 42 44 46 56 67 71 74
80
Denisa Cumova, David Moreno and David Nawrocki 4.1 Introduction 4.2 The upside potential–downside risk portfolio model 4.3 An empirical example 4.4 Conclusion
5 Currency Crises, Contagion and Portfolio Selection
80 82 92 94
96
Arindam Bandopadhyaya and Sushmita Nagarajan 5.1 5.2 5.3 5.4 5.5
Introduction Stock market average rates of return and average volatility Stock market correlations Portfolio performance Conclusion
6 Bond and Stock Market Linkages: The Case of Mexico and Brazil
96 97 99 100 101
103
Arindam Bandopadhyaya 6.1 6.2 6.3 6.4
Introduction The estimation equations and data Results Conclusion
7 The Australian Stock Market: An Empirical Investigation
103 105 109 112
114
Adeline Chan and J. Wickramanayake 7.1 7.2 7.3
Introduction Existing evidence Hypothesis
114 115 118
CONTENTS
7.4 7.5 7.6
The data Data analysis and results Conclusion
ix
119 127 132
8 The Price of Efficiency – So, What Do You Think About Emerging Markets? 137 Zsolt Berényi 8.1 8.2 8.3 8.4 8.5
Introduction Higher moment performance analysis – the theory The efficiency gain/loss methodology Testing results Conclusion
9 Liquidity and Market Efficiency Before and After the Introduction of Electronic Trading at the Sydney Futures Exchange
137 138 140 143 149
151
Mark Burgess and J. Wickramanayake 9.1 Introduction 9.2 Review of the literature 9.3 Options data volume as a proxy for liquidity 9.4 Sample design 9.5 Analysis of results 9.6 Conclusion
10 How Does Systematic Risk Impact Stocks? A Study of the French Financial Market
151 152 154 159 165 178
183
Hayette Gatfaoui 10.1 10.2 10.3 10.4 10.5 10.6 10.7
Introduction Theoretical framework Empirical study The impact of systematic risk Further investigation Market benchmark comparison Conclusion
11 Matrix Elliptical Contoured Distributions versus a Stable Model: Application to Daily Stock Returns of Eight Stock Markets
183 185 187 190 195 201 209
214
Taras Bodnar and Wolfgang Schmid 11.1 Introduction 11.2 Small sample tests 11.3 Analysis of the power functions 11.4 Empirical study 11.5 Conclusion
214 216 221 222 224
x
CONTENTS
12 The Modified Sharpe Ratio Applied to Canadian Hedge Funds
228
Greg N. Gregoriou 12.1 12.2 12.3 12.4 12.5 Index
Introduction Literature review Data and methodology Empirical results Conclusion
228 229 230 231 233 235
Acknowledgments
I would like to thank Stephen Rutt, Publishing Director, and Alexandra Dawe, Assistant Editor, at Palgrave Macmillan for their suggestions, efficiency and helpful comments throughout the production process, as well as Keith Povey (with Elaine Towns) for copy-editing and editorial supervision of the highest order. In addition, I would like to thank the numerous anonymous referees in the US and Europe during the review and selection process of the articles proposed for this volume.
xi
Notes on the Contributors
The Editor Greg N. Gregoriou is Associate Professor of Finance and coordinator of faculty research in the School of Business and Economics at the State University of New York (Plattsburgh). He obtained his PhD (Finance) from the University of Quebec at Montreal and is the hedge fund editor for the peer-reviewed journal Derivatives Use, Trading and Regulation, published by Palgrave Macmillan, based in the UK. He has authored over fifty articles on hedge funds, and managed futures in various US and UK peer-reviewed publications, including Journal of Portfolio Management, Journal of Futures Markets, European Journal of Finance, Journal of Asset Management, European Journal of Operational Research and Annals of Operations Research. He has published four books with John Wiley and Sons Inc. and four with Elsevier.
The Contributors Fathi Abid is a Professor of Finance. He is Director of the research team MODESFI specializing in financial modeling and financial strategy. He lectures frequently on financial market theory and has taught investment and portfolio management at Tunisian and European universities. He has written and co-authored numerous articles in national and international scientific journals, books and proceedings. xii
NOTES ON THE CONTRIBUTORS
xiii
Arindam Bandopadhyaya is the Chairman and Associate Professor of Finance in the Accounting and Finance Department at UMass Boston, USA. He is also the Director of the College of Management’s Financial Services Forum. A recipient of the Dean’s Award for Distinguished Research, Dr Bandopadhyaya has published in journals such as the Journal of International Money and Finance, Journal of Empirical Finance, Journal of Banking and Finance and Review of Economics and Statistics. He has presented his work at national and international conferences such as those of the Financial Management Association, European Finance Association and European Economic Association. He has presented research reports of the Financial Services Form at the Boston Stock Exchange and the Federal Reserve Bank of Boston. Dr Bandopadhyaya teaches corporate finance, international finance and managerial economics. He has received teaching awards from the College of Management, including the Professor of the Year Award and the Betty Diener Award for Teaching Excellence. Slah Bahloul is an Assistant Professor of Finance at Higher School of Business Administration in Sfax, Tunisia. He is a research assistant in the MODESFI team and has taught international finance and financial decision-making. Zsolt Berényi holds an MSc in Economics from the University of Economic Sciences in Budapest, and a PhD in Finance from the University of Munich. His main interests lie in the risk and performance evaluation of alternative investments: hedge funds, CTAs and credit funds. After working for many years for the Deutsche Bank, HypoVereinsbank and KPMG at various locations throughout Europe, Zsolt now leads an independent consultancy in Budapest, Hungary. Taras Bodnar studied Mathematics at the Lviv National University, Ukraine from 1996 to 2001. He received a PhD in Economics in 2004 from the European University Viadrina, Frankfurt (Oder), Germany. Currently, he is a research assistant at the Department of Statistics, European University Viadrina. His fields of interest are quantitative methods in finance, nonstationary time series, elliptical distributions and econometric applications. Mark Burgess currently works in the financial services industry in Australia. He has a Bachelor of Business (Honors) degree from Monash University, Australia. Rachel Campbell completed her PhD on Risk Management in International Financial Markets at Erasmus University, Rotterdam, The Netherlands in 2001. She currently works at the University of Maastricht as an Assistant Professor of Finance. Her work has been published in a number of leading
xiv
NOTES ON THE CONTRIBUTORS
journals, including the Journal of International Money and Finance, Journal of Banking and Finance, Financial Analysts Journal, Journal of Portfolio Management, Journal of Risk, and Derivatives Weekly. She teaches with Euromoney Financial Training on Art Investment, and works as an Independent Economic Adviser for the Fine Art Fund in London, and for Fine Art Wealth Management, UK. Adeline Chan currently works in the financial services industry in Singapore. She has a Bachelor of Business (Honors) degree from Monash University, Australia. Denisa Cumova works in the fund management group at the Berenberg Bank in Hamburg, Germany. She received her PhD in Finance from the University of Technology, Chemnitz, Germany. Hayette Gatfaoui gained a PhD in “Default Risk Valuation of Financial Assets” University Paris 1 in 2000. He taught for five years at the University Paris 1 (Pantheon-Sorbonne) France, and is now an Associate Professor at Rouen Graduate School of Management, France. He is a specialist in applied mathematics (holding a Master’s degree in stochastic modeling for finance and economics). He is currently advising financial firms about risk measurement and risk management topics for asset management, and for credit risk management purposes. Dr Gatfaoui is also a referee for the International Journal of Theoretical and Applied Finance (IJTAF). His current research areas concern risk typology in financial markets, quantitative finance and risk analysis. Roman Kräussl obtained a first-class honors Master’s degree in Economics with a specialization in Financial Econometrics from the University of Bielefeld, Germany, in 1998. He completed his PhD on the Role of Credit Rating Agencies in International Financial Markets at Johann Wolfgang Goethe University, Frankfurtam Main, Germany, in 2002. As the Head of Quantitative Research at Cognitrend GmbH, he was closely involved with the financial industry. Currently he is Assistant Professor of Finance at Vrije Universiteit Amsterdam, The Netherlands and research fellow with the Centre for Financial Studies, Frankfurtam Main. Markus Leippold is Assistant Professor of Finance at the Swiss Banking Institute of the University of Zurich, Switzerland. Prior to moving back to academia he worked for Sungard, Trading and Risk Management Systems, and the Zurich Cantonal Bank. His main research interests are term structure modeling, asset pricing and risk management. He obtained his PhD in financial economics from the University of St. Gallen, Switzerland, in 1999. During his PhD studies, he was a research fellow at the Stern School of Business in New York. He has published in several journals, such as the
NOTES ON THE CONTRIBUTORS
xv
Journal of Financial and Quantitative Analysis, Journal of Economic Dynamics and Control, Journal of Banking and Finance, Review of Derivative Research, Journal of Risk, and Review of Finance. In 2003, he and his colleagues received an award from the German Finance Association for their paper on the equilibrium impacts of value-at-risk regulation, and an achievement award from RISK for their paper on operational risk. In 2004, their research paper on credit contagion won the STOXX Gold Award at the annual conference of the European Financial Management Association. David Moreno holds a PhD degree in Economics from the Universidad Carlos III, Madrid, Spain, and a BSc degree in Mathematics from the Universidad Complutense, Madrid. He is currently Assistant Professor of Financial Economics and Accounting at Universidad Pompeu Fabra, Barcelona and Co-Director of the Master’s Program in Finance. He has previously held teaching and research positions at the Financial Option Research Centre (Warwick Business School, UK), Universidad Carlos III de Madrid, and at the IESE Business School, Barcelona, Spain. His research interests focus on finance in continuous time, with special emphasis on derivatives markets, financial engineering applications, pricing of derivatives, empirical analysis of different pricing models, portfolio management and term structure models. His research has been published in a number of academic journals including Review of Derivatives Research and Journal of Futures Markets, as well as in professional volumes. He has presented his work at a number of international conferences and has given invited talks at many academic and nonacademic institutions. He is associate editor of Revista de Economía Financiera and a member of GARP (the Global Association of Risk Professionals). Felix Morger is a fourth-year PhD student at the Swiss Banking Institute of the University of Zurich, Switzerland. The main part of his thesis is concerned with the theoretical and empirical aspects of Bayesian learning models with Markov switching and their application to asset allocation. Prior to his PhD studies, he worked as a consultant in pension funds. Sushmita Nagarajan is a Senior Associate in the Structured Finance Group at Moody’s Investor Service, New York. Her areas of expertise are rating and monitoring various types of structured derivative products using Moody’s rating methodologies. She also provides quantitative analysis and research surrounding complex derivative products such as asset-backed commercial paper structures. Prior to joining Moody’s she was an intern at State Street Research and Management as a Fixed Income Research Analyst with emphasis on Collateralized Debt Obligations. She graduated summa-cumlaude with a MSc degree in Finance from Boston College, and has an MBA in Finance from Jawaharlal Nehru Technological University, India.
xvi
NOTES ON THE CONTRIBUTORS
David Nawrocki is the Katherine M. and Richard J. Salisbury Jr. Professor of Finance at Villanova University, Villanova, Pa., USA. He is a registered investment adviser and is the director of the Institute for Research in Advanced Financial Technology (IRAFT) at Villanova. Nawrocki’s research includes work on financial market theory, downside-risk measures, systems theory, portfolio theory, and business cycles. He received his PhD in Finance from the Pennsylvania State University, USA. Wolfgang Schmid is a Full Professor at the European University in Frankfurt (Oder), Germany. He received a PhD in Mathematics in 1984 at the University of Ulm, Germany. His fields of major statistical activities are quantitative methods in finance, statistical process control and econometric applications. J. Wickramanayake obtained his PhD in 1994 from La Trobe University, Australia. He completed his Master’s degree at Williams College, Williamstown, Ma., USA in 1982, and did postgraduate studies in the Netherlands in 1978. He has been a member of the Financial Services Institute of Australasia for over ten years. Dr Wickramanayake has more than twenty years’ experience as a financial analyst at a central bank. Currently, he teaches finance at both undergraduate and postgraduate levels at Monash University, Australia. Dr Wickramanayake’s research interests involve banking, financial markets, mergers/acquisitions, bankruptcy and business failures, fund management, superannuation and pension finance.
Introduction
Chapter 1 deals with the economic downturn during 2000 which left many investors with burnt fingers and weary of investing in equities. There has been a continued search for alternative asset classes to fulfill the need for preserving returns while not taking on too high a risk. One such innovative alternative is investing in art as an alternative to stocks, bonds and real estate. This chapter analyses in a detailed empirical study how the risk during the art market bubble increased dramatically before the collapse of the market in the early 1990s. Understanding how deviations from normality in the form of extreme market returns link to the creation of a bubble in asset prices is crucial to our understanding of risk-and-return relationships. Chapter 2 presents a model for strategic asset allocation and currency hedging for an international investor, where the returns on stock indices follow a Gaussian regime-switching model. The authors study a Bayesian investor, who has only partial information on the current regime switching model being active, but updates the investor’s beliefs over time. The results indicate that engaging in optimal currency hedging significantly improves the risk and return characteristics of the Bayesian investor. Chapter 3 describes an empirical study of the determinant factors of domestic and foreign home biases. Using the equity holdings of thirty countries, the authors find that a severe equity home bias exists for both developed and emerging markets. Stock market development, information costs and familiarity factors are found to contribute the most to explaining foreign bias, whereas investor’s behavior has a significant effect on domestic bias. Chapter 4 discusses how human beings have always engaged in different behavior above and below a target rate of return. As a result, reverse S-shaped utility functions have been utilized to describe this human investment behavior, ever since Friedman and Savage (1948) and Markowitz xvii
xviii
INTRODUCTION
(1952). Fishburn (1977) made this approach operational with the lower partial moment, LPM(a, t), model, which detailed risk-seeking and risk-averse behavior below a minimum target return. However, the Fishburn utility measures have attracted criticism, since they assume a linear utility (risk neutral) above the target return. Recently, the upper partial moment/lower partial moment (UPM/LPM) has been put forward as a solution to this problem. This chapter develops a UPM/LPM critical line algorithm that allows this model to be operational. Chapter 5 examines the characteristics of domestic and international portfolios from the perspective of a US investor in Asian emerging markets during a period where the economies have suffered a currency crisis. Among various portfolios constructed, a purely international portfolio posts superior performance compared to a purely domestic one or a combination of domestic and international portfolios in the post-crisis period. Chapter 6 investigates the Brady bond markets of the two largest LatinAmerican economies – Mexico and Brazil. Results indicate that, for the very near future, the yield in each market is determined primarily by past yields in the respective markets. However, over a longer-term horizon, the interrelationships between the bond markets and the stock markets of the two countries become increasingly important. Chapter 7 provides an evaluation and comparison between the explanatory power of the macroeconomic model of Chen et al. (1986) and the three-factor model of Fama and French (1993) in explaining the variation in returns in the Australian equity market for the decade of the 1990s. The empirical results show that firm attributes (Fama and French, 1993) alone are insufficient to explain returns and macroeconomic variables (Chen et al., 1986) can be combined in a better multifactor model to explain the variation in returns. Chapter 8 evaluates inter-market investment efficiency, which may be a complicated task, especially across investment forms with widely differing return characteristics. This chapter offers some new ideas on how to evaluate such investments, using the example of emerging markets. The authors show that replicating the expected return distribution using options, the efficiency of any investment portfolio – for example, not just “emerging market” or “equity” – can be assessed and compared. Chapter 9 examines whether the Sydney Futures Exchange (SFE) in Australia has benefited from the introduction of electronic trading on November 15, 1999. Empirical results in this study show that during the early stage, up to the beginning of August 2000 that the money SPI options were more liquid at times of high volatility after the automation of the SFE. However, the SPI futures were less liquid at times of medium to low market volatility after this event. The authors also found a cointegrating relationship between the Australian Stock Exchange (ASX) and the derivative market (SFE) before
INTRODUCTION
xix
and after the introduction of electronic trading supporting the semi-strong market efficiency hypothesis. Chapter 10 discusses how many researchers have focused on the common latent component underlying the evolution of stock returns. The authors propose to infer such an unobserved common component while employing the well known Black and Scholes (1973) option pricing formula. Their study is based on the assumption that any small stock market index is a distorted representative of such a latent component. Once this systematic risk factor is exhibited, the authors attempt to assess its impact on a basket of French stock returns. Chapter 11 explores the assumptions of independency and normality which are not appropriate in many situations of practical interest, especially for the data sets from emerging markets. The authors propose to make use of matrix elliptical distribution instead of the normal distribution. Empirically, they show that the assumptions of the elliptical symmetry cannot be rejected for daily returns. Chapter 12 applies the modified Sharpe ratio to a small sample of Canadian hedge funds. Many investors today use the traditional Sharpe ratio to measure risk-adjusted performance, but the proposed modified VaR Sharpe ratio is a superior and more precise method that can deal with the skewed/non-normal returns that hedge fund possess. The results show that the modified Sharpe ratio is more precise when examining non-normal returns.
This page intentionally left blank
CHAPTER 1
Time-Varying Downside Risk: An Application to the Art Market1 Rachel Campbell and Roman Kräussl
1.1 INTRODUCTION The economic downturn during 2000 left many investors with burnt fingers and weary of investing in equities. Since then, there has been a search for alternative asset classes to fulfill the need to preserve returns, while not involving too high a risk. Arising from the media’s continued concern about a potential bubble in the housing market, many investors are showing an increasing interest in alternative asset classes that are not so highly correlated with equities, and provide hedging potential as part of a diversified portfolio of investments. One such innovative alternative asset class to stocks, bonds and real estate is art, which is seen increasingly as not merely items with aesthetic value, but also as attractive investments with a potential capital gain. The planned launch of a Fund of Art Funds by ABN Amro in 2005, aiming to channel money into some existing (and some yet to be launched) independent art funds, serves to highlight this point. It is a well-known fact that investment in art is influenced strongly by income and other fundamental economic factors. The effect on the economy from a collapse in the art market depends on the contagious impact of the art market on the rest of the financial system, predominately through the banking system. Thus, what is the impact of a negative shock in the art market on the overall economy? We argue that the extent to which real effects are likely to occur from a bubble in the art market is likely to 1
2
T I M E -V A R Y I N G D O W N S I D E R I S K
be significantly less because of the type of investment that is made in the art market. There are two main reasons for this. First, as art is a luxury good, investors tend to invest money into the art market that would not necessarily be invested elsewhere in other asset classes beyond holding it as surplus cash. Second, the initial wealth levels of investors typically investing in art markets is higher, and therefore less at the mercy of the banking system, as the banks are unlikely to let such investors become insolvent. We argue that the likelihood of falling prices is only liable to affect the general economy to the extent that the losses made might reduce liquidity in financial markets. Even though booms in other markets, such as in real estate, may lead to a collapse in the initial market followed by a collapse in the banking sector, this is much less likely to be the case in the art market. Although the real effects from a collapse in the art market may be significantly less than in other financial markets, the development of bubbles in the art market is likely to be significantly greater. The rate at which prices in the art market are driven by taste and fashion, predominately via the media, is much greater than in other financial markets, where “value” is a greater function of market fundamentals. The development of a large bubble in the general price of all works of art was well documented in the early 1990s for most classes of art. Indeed, it would appear that there was a severe deviation away from the fundamental valuation of art pieces during this period. This provides an extremely interesting and unique data series with which to analyze the risk to the investor around the period of the bubble’s development. We focus on time-varying downside risk in relation to theory from behavioral finance. This, given our knowledge of the literature, is an area of research that has not been undertaken before. In this chapter we analyze the art market using a measure for time variance in the downside risk, which reflects “bubbliness” in the market. This estimate measures the changing probability of large movements occurring in the return distribution of the historical time series of art price data. Taking such an approach and using techniques developed in extreme value theory (EVT), we are able to provide some new insight into the creation and measurement of risk during times of the development of bubbles in financial markets. We focus on a particularly interesting case: the art market. This market is highly media- and taste-driven, is illiquid and lacks transparency, and thus offers an ideal application in which to observe downside risk with prices that may deviate significantly from fundamental values. This chapter is organized as follows. Section 1.2 briefly surveys the economic literature concerning art as an investment; we explore the financial aspects of art investing by emphasizing similarities and differences among financial assets. Section 1.3 discusses the data and the methodology, and presents the empirical results. Section 1.4 presents some behavioral
RACHEL CAMPBELL AND ROMAN KRÄUSSL
3
explanations for our results. Section 1.5 concludes and presents an outlook for future research.
1.2 ART AS AN INVESTMENT 1.2.1 The art market in general Financial assets tend to be very liquid, allowing for diversification benefits, and thus reduce risk. Additionally, they are relatively transparent. Most financial assets can be selected on the basis of a fairly small set of objective criteria. Fundamentals do exist and can be analyzed with standard finance tools. Such financial markets are characterized by a large number of individual buyers and sellers, transaction costs are low, and trades in perfectly (or nearly) identical assets are repeated millions of times daily in various exchanges. It goes without saying that, the first impression of the art markets is that they differ significantly from other types of financial markets. Most art markets would appear to be characterized by product heterogeneity, illiquidity, behavioral anomalies, market segmentation, information asymmetries, and almost monopolistic price setting. Moreover, there is no doubt that a substantial amount of the return from art investment is derived not from classical financial returns but rather from intrinsic aesthetic qualities through art as a consumption good. Art works are not liquid assets, and transaction costs are high. Short selling is not possible, and supply is rather inelastic in the short term. There are unavoidable delays between an owner’s decision to sell and the actual sale, since it takes about three to six months to “market a work” – that is, to have it accepted by the auction house, take photographs and print and distribute the catalogue, publish advertisements for the coming auction and so on. Investing in art typically requires substantial knowledge of art and the art market in general, and often a significant amount of capital to acquire a work of a well-known artist. Moreover, the art market is highly segmented and dominated by a few large auction houses. These auction houses, such as Sotheby’s and Christie’s, are used by a restricted number of buyers, mostly wealthy collectors, public museums or private foundations. Informational asymmetries are essential features of these markets. Furthermore, art sells only occasionally. Art objects are created by individuals. Accordingly, there is only a single, unique piece of original work available. This is an extreme case of a heterogeneous commodity. Therefore, financial risk in the art market is related to specific material risk factors associated with the unique physical nature of art works such as theft, fire, water damage, or the possible reattribution to another (less famous) artist. Moreover, the value of an art object is
4
T I M E -V A R Y I N G D O W N S I D E R I S K
determined by a complex and subjective set of beliefs about past, present and future prices. Art has little intrinsic value; its appeal is ultimately dependent on individual tastes and fashion, which can change over time. The future sales price of a piece of art depends on both the number of people who wish to buy the piece when it is put up for sale and the (available) wealth of the individuals or institutions who desire it at that time. The most distinctive difference between financial markets and the art market is that the individual investor’s expected return from investing in art consists not only of a rise in price. It also involves the psychic return from art works through their aesthetic qualities. Most empirical studies have been unable so far to quantify these psychic returns associated with art as a consumption good. Recognizing art as a consumption good helps in part to explain behavioral anomalies less well-known in modern financial markets.
1.3 PREVIOUS EMPIRICAL STUDIES In recent years, an extensive literature has arisen based on calculating the returns on art investments. Starting with Baumol (1986), these include, among others, empirical studies by Goetzmann (1993), Chanel (1995), Mei and Moses (2002) and Campbell (2005). Baumol (1986) and Goetzmann (1993) tend to concur that art is dominated as an investment product by stocks, bonds and real estate. Goetzmann (1993) finds a positive relationship between art investments and the stock market over shorter time periods. He argues that the high and significant positive correlation clearly makes art investment a poor instrument for the purposes of portfolio diversification. Goetzmann (1993) also finds evidence of a significant relationship between aggregate financial wealth and the demand for art. He concludes that this empirical finding is sufficient evidence that the demand for art increases with the wealth of art collectors since, in the twentieth century, art prices tended to follow stock market trends. Chanel (1995) follows this argumentation and concludes that financial markets react quickly to shocks in the economy. Profits generated on financial markets may be invested in art, so that developments in stock markets may be considered as leading indicators for returns in the art markets. Mei and Moses (2002) take a somewhat different view. They argue that a diversified portfolio of works of art play a more important role in portfolio diversification. They base their conclusions on their empirical finding that their art price index has lower volatility and a much lower correlation with other asset classes than was discovered in earlier research. Campbell (2005) focuses on the extent of downside risk, which is less for the art market during periods in which the stock market performs badly. This is highly likely to be driven by issues relating to theories from behavioral finance, caused not only by the low liquidity on the art market, but also to investors being
RACHEL CAMPBELL AND ROMAN KRÄUSSL
5
anxious not to sell off art works representing a symbol of their reputation and status during falling financial markets. Thus, maintaining art investment remains strong during periods of economic downturn. This helps to drive the hedging connotation of art in the portfolio. The cyclicality of the art and equity markets has been documented in a recent working paper by Bauer et al. (2005), who show that art investments perform well at times when other asset classes are performing badly. Despite this strength during downturns, bubbles are also evident in the art market. The famous bubble in the 1990s occurred because of the excessive demand for works of art by the Japanese. What happens to risk-and-return characteristics during these periods? Is risk time-varying during the expansion of the bubble? Could we have seen the extent to which a bubble was developing in the market? Before answering these questions, a number of preliminary queries need to be addressed on the definition and possible estimation of a bubble.
1.4 EMPIRICAL ANALYSIS 1.4.1 How to define a bubble in the art market? What is a bubble in financial markets? How do we define bubbles ex ante or even ex post? A financial market bubble may be defined loosely as a sharp increase in the price of an asset in a continuous process, with the initial rise generating investors’ expectations of further future increases and thereby attracting new buyers. These buyers are generally speculators, interested in profits from trading in the asset rather than the asset’s earning capacity. Such a definition implies that a high and increasing price is not justified and is fed by momentum investors who buy with the sole purpose of selling quickly to other investors at a higher price. In recent years, economists have tried to give additional substance to the definition of a financial market bubble by linking asset price movements to fundamentals. Fundamentals refer to those economic factors that together determine the price of any asset, such as cash flows and discount rates. For example, Stiglitz (1990, p. 13) defines a bubble in financial markets in the following way: “If the reason that the price is high today is only that the selling price will be high tomorrow – when fundamental factors do not seem to justify such a price – then a bubble exists”. In this context, Siegel (2003) argues that one cannot identify any asset price bubble immediately, because one has to wait a sufficient length of time to determine whether the previous asset prices can be justified by the asset’s subsequent cash flows. Unfortunately, it is not that easy to find an operational definition of a bubble in the art market. If a bubble is defined only as excess changes in prices that are not captured by underlying economic fundamentals, then
6
T I M E -V A R Y I N G D O W N S I D E R I S K
the extent to which bubbles occur in the art market is only by definition. Although some mass media treat the rapid rate of increase in some pieces of art as de facto evidence of a bubble in the art market, our understanding dictates that such rises alone are necessary but not sufficient evidence. Additional evidence is needed that relates current art prices to their fundamental determinants. It is vital to keep in mind that the only constraint limiting the price of a particular piece of art is the wealth of the agents willing to pay it. Economists have identified a number of transmission mechanisms where fluctuations in housing prices can have an effect on the overall economy. One potential effect of a severe home price decline could result from a consumption wealth effect. Although the magnitude of this effect remains controversial in some quarters, a number of empirical studies find significant wealth effects from real estate assets. For example, Case et al. (2001) show that if the magnitude of the wealth effect from housing is around 5 percent, then a severe decline would lead to reduction in consumption of roundabout US$150bn, which is about 2 percent of total personal consumption expenditures. Many analysts argue that the recent increase in home prices is symptomatic of a real-estate bubble that will burst eventually, just as the stock market bubble did in 2000. This would imply the erasing of a significant amount of household wealth. They add that such a decline of disposable income would have sharp adverse macroeconomic effects, as already indebted consumers reduce spending even further to improve their weakened financial situation. Despite the (technical and conceptual) difficulties of defining bubbles in art markets, we believe that the likelihood of falling prices will affect the general economy only to the extent that the losses made may reduce liquidity in financial markets.
1.5 DATA In order to look more specifically at both bull and bear markets, we use almost thirty years’ of monthly data, from January 1976 to December 2004, from the Art Market Research (AMR) database. AMR uses over 800 auction houses to collect sales data for hundreds of individual artists worldwide. This is the most comprehensive data set available for looking at performance during market extremes, since it is available as a monthly index. Indices are constructed for individual artists using the average prices of his or her paintings obtained in the market. A national index is constructed, comprised of a number of chosen artists for each country. Art indices for the USA and the UK art markets are used, covering a large number of artists over several movements and periods in the art scene, as well as a general index which covers the markets that dominate the global market for art.2 Log returns are
7
RACHEL CAMPBELL AND ROMAN KRÄUSSL
computed for the art indices over the period January 1976–December 2004 (see Table 1.1). For an art fund, the only dividend received is the pleasure of the painting for the portfolio manager, unless the art fund allows its investors to rent or borrow some of its paintings for display in their own homes, or the works of art are rented out to museums or art collectors on loan, thus providing an additional income stream. The indices do not cover sales by private dealers, or works of art that are bought in – that is, pieces put on the block but not selling; however, these represent a highly significant part of the global art market3 . Figure 1.1 displays the development of the average price indices
Table 1.1 Summary statistics: monthly log return data, January 1976– December 2004 ART 100 Annual average return
US 100
UK 100
5.27%
8.26%
5.12%
17.11%
15.86%
11.10%
Average
0.026
0.041
0.026
Standard deviation
0.121
0.112
0.078
−0.837
−0.817
−0.097
1.694
1.029
−1.083
Annual average standard deviation
Skewness Kurtosis
12,000 10,000
General US British
8000 6000 4000
0
1976 1977/03 1978/06 1979/09 1980/12 1982/03 1983/06 1984/09 1985/12 1987/03 1988/06 1989/09 1990/12 1992/03 1993/06 1994/09 1995/12 1997/03 1998/06 1999/09 2000/12 2002/03 2003/06 2004/09
2000
Figure 1.1 Art indices: International art performance, January 1976–December 2004 Note: The average price indices from AMR for the General Art Market (ART 100): the top US artists (US 100) and the top British artists (UK 100).
8
T I M E -V A R Y I N G D O W N S I D E R I S K
for the general art market, the top 100 US artists and the top 100 UK artists since the later 1970s. From Figure 1.1, it would appear that there was an enormous bubble in the art market during the early 1990s. This has been documented in the literature and is usually considered to have arisen from the effect of the expanding Japanese economy, with wealth flowing directly from this boom into the luxury art market. From the graph, it would appear that prices returned to their fundamental values in the two years following the bubble in 1990. Figure 1.1 also indicates a Japanese phenomenon after the collapse of the Japanese economy, with money flowing directly out of the art market during this period. Interestingly, the causality has been documented between these two markets, as well as between other equity markets and the art market, by Campbell (2005). The restrictive supply of art, recently cited as the reason for increasing prices, and hence returns being made in art investment, is also a driving factor behind the occurrence of bubbles in the art market. More recent developments using art as collateral for credit loans only serve to lengthen the extent and duration of the resulting price rises and the size of the financial bubble. This optimism is exacerbated by the tendency of investors and banks towards myopic disaster behavior. The highly leveraged positions of banks, holding collateral consisting of such opaque assets as real estate and artworks result in downside risk from the real estate and art markets being shifted to the banking sector. This effect is exaggerated by the feeling of wealth created by increases in property prices and art prices feeding on each other. This can have severe implications for the banking sector and the macroeconomy, so this is one reason to evaluate carefully how much of these assets act as collateral on the balance sheet. The extent to which a bubble is able to develop depends on the upward pressure on movements in prices, notably through such mechanisms as greater media coverage, so that a large response to large price changes occurs. The development of a bubble through the occurrence of large price movements should be reflected by greater conditional volatility in financial markets. Movements which occur with a greater probability than conditional volatility would suggest, can be accounted for by the use of a tail index. This is not a new methodology, but its application to the measurement of speculative asset-pricing bubbles, is, to our knowledge, new. Before we observe the relationship empirically between estimates for the probability of larger than conditionally normal movements in prices occurring during the period of development of a speculative bubble, we shall first outline the methodology for estimating a tail index. It is the conditional estimate of the tail index that we estimate using rolling observations, to see how the probability of larger than “normal” movements in prices change over the development and bursting of the bubble in the art market.
RACHEL CAMPBELL AND ROMAN KRÄUSSL
9
1.6 METHODOLOGY In order to analyze the extent to which the market moves away from fundamental values through larger than “normal” probabilities occurring in large movements of the return distribution, we shall apply below the Hill’s (1975) tail index estimator, which was further extended by Huisman et al. (1997). We use EVT to provide us with estimates of tail indices. EVT looks specifically at the distribution of the returns in the tails, and the tail fatness of the distribution is reflected by the tail index. This concept was first introduced by Hill (1975), and measures the speed with which the distribution’s tail approaches zero. The fatter the tail, the slower the speed and the lower the tail index given. An important feature about the tail index is that it equals the number of existing moments for the distribution. A tail index estimate equal to 2 therefore reveals that both the first and second moments exist, in that case the mean and the variance; however, higher moments will be infinite. By definition, the tail index for normal distribution equals infinity, since in that case, all moments exist. Since the number of degrees of freedom reflects the number of existing moments, the tail index can thus be used as a parameter for the number of degrees of freedom to parameterize the student-t distribution. To obtain tail index estimates, we use a modified version of the Hill estimator, developed by Huisman et al. (1997). Their estimator has been modified to account for the bias in the Hill estimator, with the additional advantage of producing almost unbiased estimates in relatively small samples. Specifying k as the number of tail observations, and ordering their absolute values as an increasing function of size, we obtain the tail estimator proposed by Hill. This is denoted by γ, which is the inverse of α: 1 ln(xn−j+1 ) − ln(xn−k ) k k
γ(k) =
(1.1)
j=1
Following the methodology of Huisman et al., we can use a modified version of the Hill estimator to correct for the bias in small samples. The bias in the Hill estimator stems from the fact that it is a function of the sample size. A bias-corrected tail index is therefore obtained by observing the bias of the Hill estimator as the number of tail observations increases up to κ, whereby κ is equal to half of the sample size γ(k) = β0 + β1 k + ε(k),
k = 1, . . . , κ
(1.2)
The optimal estimate for the tail index is the intercept β0 , while the α estimate is the inverse of this estimate. This is the estimate of the tail index that we use to estimate rolling estimations of the degree with which larger
10
T I M E -V A R Y I N G D O W N S I D E R I S K
Table 1.2 Alpha estimates for the All Art Index, January 1976–December 2004 Log returns Alpha
Gamma
SE
Kappa
All Art Index
Both Left Right
1 3.26994 3.18242 3.46838
0.305816 0.314226 0.288319
0.053088 0.081208 0.069766
1 3.26994 3.18242 3.46838
US Art Index
Both Left Right
2 3.87852 3.99025 3.13632
0.25783 0.250611 0.318845
0.044758 0.062053 0.079892
2 3.87852 3.99025 3.13632
UK Art Index
Both Left Right
3 9.60735 9.35147 10.4014
0.104087 0.106935 0.096141
0.018069 0.026795 0.023805
3 9.60735 9.35147 10.4014
Note: This table provides the alpha estimates using the Huisman et al. (1997) estimator for the All Art Index from Art Market Research, using monthly data.
than “conditionally normal” returns occur in the historical distribution of returns over time.
1.7 RESULTS Table 1.2 provides the alpha estimates using the Huisman et al.’s estimator over the period January 1976 to December 2004. We first look at the alpha estimates for the whole sample. We see that there is indeed deviation from the assumption of “normality”, since the alpha estimates are between 2 and 3. This would imply that the tail index is able to capture some of the additional movement occurring in returns beyond that of the assumption of normality, captured by volatility alone. There is a move away from fundamental distribution over time. In Figure 1.2, the inverse alpha estimates (gammas) using the previous eight years’ sample of monthly data are plotted next to the actual monthly returns. We see that, the more the returns fluctuate, the higher the inverse alpha estimate and the greater the movement away from fundamental values. Indeed, the correlation between volatility and alpha is −0.42, which is highly significant at the 95 percent confidence level. It has been shown that this measure increases during periods of instability.4 Therefore, we would expect that the use of the gamma estimate is a good indicator for a movement away from fundamental values, and the development of an asset pricing bubble.
11
RACHEL CAMPBELL AND ROMAN KRÄUSSL
General art index gamma Art returns General art index
/0 2
/0 2
04 20
/0 2
03 20
/0 2
02 20
/0 2
01 20
/0 2
00 20
/0 2
99 19
/0 2
98 19
97 19
96 19
95 19
94 19
93 19
91
92 19
19
90 19
89 19
87
88 19
19
02
86
19
19 85 /
84 19
⫺0.1
/0 2
1
/0 2
0
/0 2
2 /0 2
0.1
/0 2
3
/0 2
0.2
/0 2
4
/0 2
0.3
/0 2
5
/0 2
0.4
/0 2
6
Indices value
7
0.5
/0 2
Gamma estimates
0.6
0
Figure 1.2 Art returns and time-varying downside risk, January 1976–December 2004 Notes: Art Index. Returns and Gamma Estimates Monthly Data 96 Rolling Observation for Gamma Estimates for left tail of the Art Index using 96 observations to estimate the downside risk. Based on the data for the art market, we use the eight years’ monthly data available from 1976 to 1984 – a total of 96 observations – to calculate the conditional gamma estimate for the distribution. Obviously, the other moments of the distribution, the mean and the standard deviation are able to change conditionally over time, so that the gamma estimate is able to capture the extent of larger than conditionally normal movements occurring in the return distribution. The results are shown in Figure 1.2. There is an extremely high correlation between the bubble occurring in 1990 and the high values obtained from the tail index estimator over the period of the bubble. The gamma estimates converge to their average values after the bubble bursts in 1991, and maintain a value around the average value over the rest of the sample until the current period.
1.8 DISCUSSION It would appear that the phenomenon of the bubble developing in the art market may be captured through the use of the tail index estimator, which captures the probability of larger than conditionally normal movements in large returns occurring over the return distribution over time. The analysis so far has only been applied to the art market, but there is no reason why
12
T I M E -V A R Y I N G D O W N S I D E R I S K
the methodology may not be used for other financial markets in which it is thought that bubbles have occurred, or are thought to be present. Indeed, the use of a relatively small sample of observations to analyze the tail index estimator provides a robust estimator, which can be applied conditionally over the historical time series of returns. A further issue is that it is not possible to test strictly for efficiency in the art market. We have discussed so far possible reasons for inefficiencies of the art market – for example, information asymmetries. But there are good reasons why particular behavioral anomalies are even larger and more widespread in the art market compared to the financial markets. Many private collectors are not profit-oriented and are particularly prone to the behavioral anomalies that arise from leaving endowments, opportunity costs and sunk cost effects. Circumstantial evidence suggests that private collectors are strongly subject to the endowment effect, which implies that they value an art object owned to a greater extent than one not owned. The result is that people often demand much more to give up an object than they would be willing to pay to acquire it (see Thaler, 1980). This is what Samuelson and Zeckhauser (1988) call a status quo bias; that is, the preference for the current state that biases someone against both buying or selling an object. These anomalies are manifestations of an asymmetry of value that Tversky and Kahneman (1991) call “loss aversion”. Loss aversion means that the disutility of selling an object is greater than the utility associated with buying it. Loss aversion also explains why there is no market for renting art objects. Frey and Eichenberger (1995) argue that the consumption benefits of viewing art should be revealed in the rental fees for art objects. The consumer would pay a fee for enjoying art while being unaffected by price changes in the art market. The reason why such market-revealing pure psychic benefits from art do not exist must be sought in property rights and a corresponding ownership effect. While the decision to buy art might be based on financial calculations, the desire to possess a beautiful and internationally famous work that will impress friends and clients unquestionably adds to the attraction. The owner of a work of art has a monopoly over that specific object, while other assets may be held by many individuals. The major difference between investing in art and in common financial assets is that art is tangible and is associated with a given lifestyle. This implies that an art object yields additional benefits if it is owned and not just rented, because the art object’s aura is also appropriated (Benjamin, 1963). Apart from the endowment effect and its corresponding ownership effect, there is also the opportunity cost effect. This implies that many collectors isolate themselves from considering the returns of alternative uses for their investments. A third behavioral anomaly that plays a large part in the art market is the sunk cost effect. This describes the tendency to be excessively attached to activities (things) for which one has expended resources resulting
RACHEL CAMPBELL AND ROMAN KRÄUSSL
13
from past efforts at building up a (specific) art collection. Additionally, the self-deception theory suggests that the tendency to adjust attitudes to match past actions is a mechanism designed to persuade the individual that he or she is a skillful decision-maker. Are art investors reluctant to realize their losses? Or are investors extremely reluctant to realize their losses in art? Mental accounting is a kind of narrow framing that involves keeping track of gains and losses related to decisions in separate mental accounts. Thaler (1985) argues that individuals reexamine each account only intermittently when it is action-relevant. Mental accounting may explain the disposition effect (Shefrin and Statman, 1985) – that is, the excessive propensity to hold on to assets that have declined in value and to sell the winners. Such a mechanism may even be side-tracked when the individual avoids recognizing losses. Self-deception theory reinforces this argument, since a loss is an indicator of poor decision-making, and a self-deceiver maintains self-esteem by avoiding the recognition of this. Regret avoidance may also reflect a self-deception mechanism designed to protect self-esteem about poor decision ability. Kahneman et al. (1991) show that regret is stronger for individual decisions that involve action rather than passivity. This effect is also known as the “omission bias”. A bequest aspect is also highly relevant. Gifts from parents to their children, or inheritances of family members in the form of art objects are valued more highly by the owner than they would be purely for their monetary value. Frey and Eichenberger (1995) argue that, by selling the object, the owners are transferring with it part of their own “nature”.
1.9 CONCLUSION Using a unique set of data with which to observe and quantify the extent of a bubble in the art market, we have been able to gain a greater insight into the nature of bubbles with respect to the larger than “normal” movements that appear to occur during the build-up and breakdown of financial bubbles. More detailed analysis with regard to return distribution will no doubt enable a richer analysis of the make-up of the asset bubbles, and will be extremely interesting avenues for further research. By defining the degree of “bubbliness” in a market as the degree to which large movements are more likely to occur, the gamma of the distribution of historical returns can be estimated conditionally over time. We see that there is an extremely high correlation between the size of the gamma estimates and prices during the period of the bubble. The larger the gamma estimate, the greater the probability of more extreme movements in the return distribution. This should indeed be constant over time. However, we see that the correlation of the gamma estimates increases during the period of the bubble, and is thereafter fairly constant. We therefore premise that the bubble
14
T I M E -V A R Y I N G D O W N S I D E R I S K
can be defined ex post from a larger probability occurring in the tails of the distribution, observed conditionally over time – from rolling observations used to estimate the degree of “bubbliness” in the market. Although the results presented here are preliminary in nature, they provide an extremely innovative and interesting avenue for further research into the notion of bubbles in financial markets. The use of the art market, which represents a market in which deviations from fundamental values are much more likely, provides a particularly interesting market with which to observe such measures. There are many further areas that may need to be addressed before any definite conclusions can be drawn. For example, the use of this measure on alternative asset classes in which bubbles have been observed. The “dot.com” mania and real estate markets in particular. Although the results are in a preliminary form, they should help to generate further discussion and insight into the determination and measurement of bubbles in financial markets.
NOTES 1. All errors are the responsibility of the authors. Many thanks to participants at the conference on “Art: An Alternative Asset Class” at Sotheby’s, London, for their valuable comments. 2. Figures on the exact numbers of artists per index are available from the authors on request, or from AMR. 3. In a similar manner, the S&P 500 only represents a segment of the whole market for US equities. 4. See Pownall and Koedijk (1999) for an analysis of the Asian financial crises of 1997–8, with the use of the same methodology.
REFERENCES Bauer, R., Campbell, R. A. and Dil, N. (2005) “Art Diversification over the Business Cycle: The Case for the UK”, Working Paper, Maastricht University. Baumol, W. (1986) “Unnatural Value: or Art Investment as a Floating Crap Game”, American Economic Review, 76(2): 10–14. Benjamin, W. (1963) Das Kunstwerk im Zeitalter seiner technischen Reproduzierbarkeit, 4th edn (Frankfurt am Main: Edition Suhrkamp). Campbell, R. A. (2005) “Art as an Alternative Asset Class”, Working Paper, Maastricht University. Case, K. E., Quigley, J. M. and Shiller, R. J. (2001) “Comparing Wealth Effects: The Stock Market versus the Housing Market”, Advances in Macroeconomics, 5(1): 1–32. Chanel, O. (1995) “Is Art Market Behavior Predictable?”, European Economic Review, 39(3–4): 519–27. Frey, B. S. and Eichenberger, R. (1995) “On the Rate of Return in the Art Market: Survey and Evaluation”, European Economic Review, 39(3–4): 528–37. Goetzmann, W. N. (1993) “Accounting for Taste: Art and the Financial Markets over Three Centuries”, American Economic Review, 83(5): 1370–6.
RACHEL CAMPBELL AND ROMAN KRÄUSSL
15
Hill, B. (1975) “A Simple General Approach to Inference about the Tail of a Distribution”, Annals of Mathematical Statistics, 3(5): 1163–74. Huisman, R., Koedijk, K. G., Kool, C. and Palm, F. (1997) “Fat Tails in Small Samples”, Working Paper, Erasmus University, Rotterdam. Kahneman, D., Knetsch, J. and Thaler, R. (1991) “The Endowment Effect, Loss Aversion, and Status Quo Bias”, Journal of Economic Perspectives, 5(1): 193–206. Mei, J. and Moses, M. (2002) “Art as an Investment and the Underperformance of Masterpieces”, American Economic Review, 92(5): 1656–68. Pownall, R. A. and Koedijk, K. G. (1999) “Capturing Downside Risk in Financial Markets: the Case of the Asian Crisis”, Journal of International Money and Finance, 18(6): 853–70. Samuelson, W. and Zeckhauser, R. (1988) “Status Quo Bias in Decision Making”, Journal of Risk and Uncertainty, 1(1): 7–59. Shefrin, H. and Statman, M. (1985) “The Disposition to Sell Winners Too Early and Ride Losers Too Long: Theory and Evidence”, Journal of Finance, 40(3): 777–90. Siegel, J. J. (2003) “What Is an Asset Price Bubble? An Operational Definition”, European Financial Management, 9(1): 11–24. Stiglitz, J. E. (1990) “Symposium on Bubbles”, Journal of Economic Perspectives, 4(2): 13–18. Thaler, R. H. (1980) “Toward a Positive Theory of Consumer Choice”, Journal of Economic Behavior and Organization, 1(1): 39–60. Thaler, R. H. (1985) “Mental Accounting and Consumer Choice”, Marketing Science, 4(3): 199–214. Tversky, A. and Kahneman, D. (1991) “Loss Aversion in Riskless Choice: A ReferenceDependent Model”, Quarterly Journal of Economics, 106(4): 1039–61.
CHAPTER 2
International Stock Portfolios and Optimal Currency Hedging with Regime Switching Markus Leippold and Felix Morger
2.1 INTRODUCTION Despite the vast literature on optimal currency hedging, there still is considerable disagreement about how international investors should hedge their currency risk. One argument is that investors should fully hedge, since exchange-rate changes in excess of the forward discount rate average out. Therefore, hedging decreases the risk of foreign investment, but does not reduce its expected returns. In the words of Perold and Schulman (1988), currency hedging is a free lunch. However, there is a large branch of literature that does not agree with this viewpoint. As an early example, Froot (1993) argues that the free-lunch argument does not hold in the long run. If exchange rates and asset prices display mean reversion, the optimal hedging policy becomes time-varying. In particular, real exchange rates revert to their means according to the theory of purchasing power parity, and investors should maintain an unhedged foreign currency position. Therefore, for an investor with a long investment horizon, it becomes optimal not to hedge at all. Froot argues that real-exchange rates may deviate from their theoretical fair value over shorter horizons, and currency hedging in this context may become beneficial. As a compromise between these two extreme viewpoints, Black (1989) argues that, using Siegel’s paradox, there is a constant universal hedge 16
MARKUS LEIPPOLD AND FELIX MORGER
17
ratio between zero and one. However, Black has to impose some strong assumptions and because of the time-period sensitivity and significant variability and volatility of input parameters in the optimal hedge ratio, there is a significant dispersion in what constitutes the optimal constant hedge ratio. In contrast, the evidence of Glen and Jorion (1993), who analyze the performance of mean-variance efficient stock and bond portfolios from the G5 countries when hedging the associated currency risk with currency forwards, shows that there is a substantial improvement when using conditional time-varying hedging strategies. It is beyond the scope of this chapter to provide a full account of the existing literature on currency hedging, but we refer, for example, to the recent contribution by Dales and Meese (2001) for an overview. We note that most of the literature builds on simplifying assumptions on the dynamics of the underlying returns. Indeed, there is now ample empirical evidence against the normal distribution for return dynamics and a lot of statistical justification for so-called regime-switching models. For example, Turner et al. (1989), Garcia and Perron (1996), Gray (1996), Perez-Quiros and Timmermann (2000), Whitelaw (2000), Ang and Bekaert (2002a, 2002b), Ang and Chen (2002), Connolly, Stivers and Sun (2005) and Guidolin and Timmermann (2005a, 2006) report evidence of regimes in stock or bond returns. Therefore, in our study, we analyze the impact of such regime-switching models on optimal currency hedging. Closely related to our study are the works by Ang and Bekaert (2002a) and Guidolin and Timmermann (2005a). Both papers make use of regimeswitching models. Ang and Bekaert (2002) analyze the optimal investment strategy within a mean-variance framework. Concerning the modeling of regime switches, they do not consider a Bayesian updating rule to infer on state probabilities. Guidolin and Timmermann (2005b) assume preferences over the moments of wealth distribution. In addition, they explore the optimal asset allocation of an international portfolio with unhedged returns. They do not address the issue of optimal currency hedging. We use a more general CRRA (Constant Relative Risk Aversion) utility setting and we compare the non-Bayesian investor with the Bayesian one. We show that it really pays to go Bayes! Furthermore, we explicitly allow the investor to hedge his or her currency exposure. Whereas Guidolin and Timmermann (2005b) find that their model offers a rational explanation of the strong home bias observed in US investors’ asset allocation, our results contrast with their conclusion. While we find a slight decrease in foreign asset holdings for a strategy with unhedged returns, the strategy with optimal currency hedging substantially increases the exposure to foreign markets. Therefore, the home bias becomes even more puzzling. The plan of this chapter is as follows. In Section 2.2, we present the regimeswitching model and the optimization problem. Section 2.3 provides the estimation results for several model specifications. In Section 2.4, we provide
18
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
a discussion of several aspects of our results; in particular, we discuss the economic benefits of using regime-switching models and we analyze the optimal portfolio allocation. Section 2.5 concludes.
2.2 THE MODEL Regime-switching models consist of two generic processes, the state process st and the return process rt . The unobservable state process st determines which state is active at time t. For the state st , we assume a discrete firstorder Markov chain with S possible states or regimes. The constant transition probability for moving from state i to state j is denoted as pij = P{st+1 = j|st = i, st−1 = k, . . .} = P{st+1 = j|st = i},
for i, j = 1, . . ., S
and we collect all the pij ’s in the transition matrix P. For the N-dimensional return vector rt , we assume the state dependent dynamics drt = µ(st )dt + (st )dBt where Bt is a N × 1 dimensional Wiener process. Both the drift vector µ(st ) and the N × Ndimensional covariance matrix (st ) depend on the active regime. Therefore, the distribution of rt+1 conditional on the state st is a mixture of S normal distributions with probability density function f (rt+1 |st = i) =
S
pij f (rt+1 |st+1 = j)
j=1
We note that the regime-switching model defined above can account for skewed and fat-tailed returns. Furthermore, with pij > 1/S as a sufficient condition, we can also generate correlation breakdowns and volatility clusters. Both are often observed in the joint dynamics of international stock markets.
2.2.1 Portfolio selection with perfect knowledge of the active state We assume that the investor can invest in N assets, where the Nth asset is the risk-free asset. The investment horizon T is fixed. The investor has the possibility of rebalancing the asset allocation at the beginning of every period; for example, at times t = 0, . . ., T − 1. There are no transaction costs.
MARKUS LEIPPOLD AND FELIX MORGER
19
We further assume that the investor has a CRRA utility function defined over wealth, for example U(W) =
1 W 1−γ 1−γ
where we assume γ > 1 for the relative risk aversion coefficient. We start with the situation in which the investor has perfect knowledge of the active state. We denote by αt the vector of portfolio weights at time t. To maximize the investor’s terminal wealth, he/she has the following objective function: max E0 [U(WT )]
α0 ,...,αT−1
s.t.
Wt+1 = Wt (αTt exp (rt+1 ))
(2.1)
1 = αTt 1 where E0 [·] = E[·|F0 ]. To simplify notation, we write Wt+1 = Wt αTt exp (rt+1 ) = Wt Rt+1 (αt ) where Rt+1 is the portfolio’s gross return from time t to time t + 1. Using a dynamic programming approach, we can solve the optimization problem recursively. At each time step t, we have to maximize the indirect utility function J: J(W, r, s, θ, t) = max Et [Qt+1,T U(Wt+1 )] αt
given the parameter set θ = {µ(s), (s), P} and with the indirect utility Qt+1,T = Et+1 [(RT (α∗T−1 ) · . . . · Rt+2 (α∗t+1 ))1−γ ]
QT,T = 1
where α∗T−1 , . . .α∗t+1 are the optimal portfolio weights determined recursively. These optimal portfolio weights have to solve the corresponding first-order conditions (FOC) of the optimization problem in Equation (2.1). Given state st = i, we obtain the FOC of the investor’s allocation problem as .−γ
Et [Qt+1,T Rt+1 (αi,t )λt+1 |st = i] =
S
.−γ
pij Et [Qt+1,T Rt+1 (αi,t )λt+1 |st+1 = j] = 0
(2.2)
j=1
where αi,t = αt (st = i) and λi,t+1 is defined as the vector of excess returns over the risk-free rate, for example ⎛ 1 N )⎞ exp (ri,t+1 ) − exp (ri,t+1 ⎜ ⎟ .. λi,t+1 = ⎝ ⎠. . N−1 N ) exp (ri,t+1 ) − exp (ri,t+1
20
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
Whereas, for the case of one single regime with i.i.d. returns, the optimal asset allocation for a CRRA investor is constant over different time horizons (see, for example, Samuelson, 1969), the allocation in a multiple regime model depends on the prevailing regime st and on the investment horizon. This dependency arises because of the changing investment opportunity set induced by the regime switches.
2.2.2 Portfolio selection under hidden regime switches Since it is rather restrictive to assume that investors have perfect knowledge on the active state, we next assume that, at the moment they invest, they do not know which regime is active and they have to infer on the active regime using a specific updating procedure. For the updating procedure, we assume that the investor uses a Bayesian updating rule based on the filter developed by Hamilton (1989). To sketch out briefly the updating procedure, we denote the oneperiod-ahead forecast of investor beliefs as ut+1|t , for example, ut+1|t is the column vector of the probabilities P(st+1 = i|Ft ; θt ), and the set θt = {µ(s), (s), P} collects the time-t parameters of the regime-switching model. The optimal inference (posterior) and forecast (prior) on the active regime is found by iterating the equations uˆ t|t =
uˆ t|t−1 ◦ηt T 1 (uˆ t|t−1 ◦ηt )
(2.3)
and uˆ t+1|t = PT uˆ t|t
(2.4)
where ηt is a S × 1 vector of the multivariate normal densities determined by µ(st ) and (st ), 1 is the S × 1 unit vector, and by ‘◦’ we denote the elementby-element multiplication. Given a starting value u1|0 and a parameter set θ the algorithm defined by Equations (2.3) and (2.4) calculates for each time t the probability of a regime currently being active and also being active for the period ahead. To get parameter estimates, we maximize the likelihood function using a variate of the EM algorithm developed by Hamilton (1990). The log likelihood function L(θ) to be maximized is the sum of the denominator of Equation (2.3) over all t. We note that, with incomplete information on the current state, the indirect utility J is no longer a function of the active state st , but a function of the beliefs ut|t about the active state, where ut|t is the column vector of the probabilities P(st = i|Ft ; θt ). In particular, we have J(W, r, ut|t , θ, t) = maxαt Et (Qt+1,T U(X, Y))
MARKUS LEIPPOLD AND FELIX MORGER
21
We write the belief at time t + 1 given the filtration Ft+1 as vt+1|t+1 . For ease of notation, we drop the time indices from the beliefs ut|t and vt+1|t+1 whenever u and v are used as subscripts. Then, the FOC is given as .−γ
Et [Qt+1,T Rt+1 (αu,t )λt+1 |ut|t ] =
S i=1
ui,t|t
S j=1
1
.−γ
puv,j Et [Qv,t+1,T Rt+1 (αu,t )λt+1 |st+1 = j]dv = 0 (2.5)
pij 0
where puv,j = P{ut|t , vt+1|t+1 ; θt |st+1 = j}. By inspection, Equation (2.5) is a straightforward extension of the FOCs under full information on the active regime given in Equation (2.2). The expectation gives the FOCs for a given belief vt+1|t+1 , weights αu,t , and regime st+1 = j. The summation of the expectation over the different states j corresponds to the FOCs for the full information case. The summation over the beliefs ui,t|t and the integral over the combinations of beliefs vt+1|t+1 are because of the non-observability of the state process. With the state process being observable, ui,t|t equals 1 for the active regime and zero for all others. The integral over the combinations of beliefs vt+1|t+1 enters Equation (2.4), because the indirect utility Qv,t+1,T depends on the beliefs vt+1|t+1 , which are unknown at investment. The probability puv,j corresponds to the probability of the belief moving from ut|t to vt+1|t+1 , given that regime j is active at t+1. It assigns a weight to each expectation and its involved indirect utility. The probability puv,j can also be interpreted as the probability of occurrence of vt+1|t+1 given ut|t and regime st+1 = j. The optimal portfolio choice problem for a regime-switching model must be solved numerically. In the following empirical section, we apply a backward solution algorithm with a Monte Carlo simulation of size Z = 30,000. Furthermore, we have to discretize the state space of beliefs. For the model with two regimes, we have 3 grid points, and for the model with three regimes we have 6 grid points. With this parameterization the algorithm generates very accurate weights. Therefore, we do not consider more elaborated algorithms such as, for example, the algorithm proposed by Brandt et al. (2004) that is based on the Longstaff and Schwartz (2001) least-square Monte Carlo method.
2.3 ESTIMATION RESULTS 2.3.1 Data We take the perspective of a US investor, who allocates his/her wealth in risk-free assets and the US, UK and German stock markets. For the stock markets, we use MSCI (Morgan Stanby Capital International) country indices. We approximate the risk-free rate with the one-month Eurodollar rate. To
22
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
investigate the optimal currency hedging behavior of the investor, we let the investor take positions not only in the foreign market indices denominated in US dollars, but also in the foreign markets denominated in their local currency. The relative asset allocation between the unhedged and hedged indices determines the optimal hedging policy. The monthly data covers the period from December 1970 to June 2005, which results in 414 one-month excess returns over the risk-free rate. For the out-of-sample analysis, we use fifteen and a half years spanning the period January 1990 to June 2005. Table 2.1 presents the summary statistics for the excess returns. With the sole exception of the unhedged German index in the period prior to 1990, the index returns fail the Jarque–Bera test for normality, motivating the use of regime-switching models.
2.3.2 Specification test Using three model portfolios, we test for normality (Jarque–Bera – JB) and for the absence of serial correlation with three different likelihood ratio tests on the predictive density proposed by Berkowitz (2001). LR1;1 tests for serial correlation of lag one, LR2;1 for serial correlation of lag one and two, and LR2;2 for linear and squared serial correlation up to lag two. The latter is a test for omitted volatility dynamics. The results of the specification tests for the cases of two and three regimes are summarized in Table 2.2.1 We find that the Jarque–Bera statistic is reduced substantially by the use of regime-switching models. For example, we find that the Jarque–Bera statistics of the UK excess returns are above 2,000. In contrast, under the chosen specifications of the regime-switching models, the statistics for the UK are mainly below 20 during the period of the out-of-sample test. Hence, we are well advised to use a regime-switching model to account for the skewed and fat tails of stock market returns. In particular, for the subsequent analysis we shall use three regimes for both the unhedged and the optimal hedging strategy, and two regimes for the fully hedged strategy.
2.3.3 Parameter estimates as at June 2005 In this section, we present the parameter estimates for different model specifications. Table 2.3 presents the parameter estimates of the model specification with a single regime and unhedged indices. The mean excess returns are between 5 percent and 7.7 percent per year, and the volatilities are around 20 percent. The correlations are around 0.5. The highest correlation is between the UK and the US markets. All parameter estimates are significantly different from zero.
MARKUS LEIPPOLD AND FELIX MORGER
23
Table 2.1 Summary statistics, stock market returns, December 1970 – June 2005 GER, FH
GER, UH
UK, FH
UK, UH
Panel A: Summary statistics for June 2005 Moment statistics Mean 0.040 0.070
0.081
0.077
Std, dev. Skewness Kurtosis JB
US
0.050
0.198
0.216
0.208
0.230
0.154
−0.405
−0.214
1.367
1.337
−0.320
5.172
4.410
18.678
14.739
4.876
90.780
36.416
4319.407
2470.964
66.188
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
Correlation coefficients GER, FH 1.000 GER, UH
0.858
1.000
UK, FH
0.446
0.373
1.000
UK, UH
0.379
0.464
0.894
1.000
US
0.508
0.463
0.595
0.538
1.000
Panel B: Summary statistics for December 1989 Moment statistics Mean 0.035 0.088 0.108
0.093
0.032
Std, dev.
0.277
0.161 −0.228
Skewness Kurtosis JB
0.177
0.214
0.248
−0.180
0.009
1.459
1.338
5.138
3.716
16.296
12.305
5.545
42.827
4.440
1723.482
870.741
61.157
(0.000)
(0.109)
(0.000)
(0.000)
(0.000)
Correlation coefficients GER, FH 1.000 GER, UH
0.832
1.000
UK, FH
0.360
0.287
1.000
UK, UH
0.329
0.397
0.920
1.000
US
0.380
0.335
0.560
0.505
1.000
Notes: The table presents the summary statistics of the stock market returns in excess of the risk-free rate and includes the Jarque–Bera test. The market indices are MSCI country indices and the risk-free rate is the 1-month Eurodollar rate. The data are provided by Datastream. The statistics for Germany and the UK are given both as hedged (denoted as FH) and unhedged (denoted as UH) indices. Mean and standard deviation are annualized. Panel A gives the summary statistics for the entire sample. Panel B presents the numbers for the sub-sample from December 1970 until the start of the horse race (see section 2.2).
Table 2.4 presents the parameter estimates for a three-regime model specification with unhedged indices. Compared to the benchmark case in Table 2.3, the bear state exhibits negative drifts, slightly increased volatilities and correlations, and a relatively low persistence. Once the bear state is
24
Table 2.2 Specification tests, December 1989–December 2004 Test
1989.12
1992.6
1994.12
1997.6
1999.12
2002.6
2004.12
Panel A: Unhedged strategy 2 regimes JB 1* 2*
2**
2**
1*, 1**
1*, 1**
1**
LL-ratio
3*
1*
1*
2*
3*
1*
3*
3 regimes JB
1*
1**
1**
LL-ratio
2*
3*
3*
3*
3*
1*
2*,1**
Panel B: Fully hedged strategy 2 regimes JB 1*
1*
1*
1*
1**
2*
1*
2*
2*
3*
1*
3 regimes JB
1*
2*
1**
1**
LL-ratio
2*
2*
2*
1*, 1**
2*
3*
1*
Panel C: Optimally hedged strategy 2 regimes JB 2* 1* 2*
1*, 2**
3**
1*, 3**
5**
5*
4*, 1**
4*, 2**
2*, 4**
LL-ratio
LL-ratio
3*
4*
3 regimes JB
1**
1*
1*
4*
1*, 2**
LL-ratio
1*
4*, 1**
4*, 1**
4*, 1**
3*
2*
3*
Notes: The table presents an overview of the specification tests at intervals of 2.5 years. The entries in the table show the number of markets that failed the Jarque–Bera test and likelihood ratio test, respectively, at a significance level of 5% or 1%. Significance at the 5% level is denoted by * and at the 1% level by **. Panel A reports the results for the strategy without hedging, and Panel B results for the strategy with the hedged returns. Panel C presents the overview for the optimal hedging case.
Table 2.3 Parameters of the single regime strategy
Panel A: Drifts Mean excess return
GER
UK
USA
0.070 (0.008)
0.077 (0.008)
0.050 (0.006)
Panel B: Correlations and volatilities GER
0.216**
UK
0.464**
0.230**
US
0.463**
0.538**
0.154**
Notes: The table presents the parameter estimates for the international investor and single regime specification with the unhedged indices. Panel A gives the annualized mean excess returns. The values in brackets are the annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Significance at the 0.05 level is represented by ** of the corresponding covariances.
25
MARKUS LEIPPOLD AND FELIX MORGER
Table 2.4 Parameter estimates for the three regimes specification with unhedged indices Panel A: Drifts GER −0.123 (0.104)
UK −0.024 (0.310)
US −0.197 (0.143)
Low correlation state
0.123 (0.013)
0.104 (0.025)
0.106 (0.018)
High correlation state
0.035 (0.023)
0.056 (0.015)
0.031 (0.016)
Bear state
Panel B: Correlations and volatilities GER Bear state GER 0.241**
UK
UK
0.537**
0.507**
US
0.651**
0.609**
Low correlation state GER
0.200**
UK
0.361**
0.191**
US
0.172**
0.448**
High correlation state GER
0.235**
US
0.240**
0.136**
UK
0.850**
0.136**
US
0.825**
0.765**
0.144**
Panel C: Transition probabilities Bear state Bear state 0.842 (0.208)
Low corr. 0.026 (0.005)
High corr. 0.000 (0.086)
Low correlation state
0.158 (0.363)
0.934 (0.036)
0.076 (0.181)
High correlation state
0.000
0.041
0.924
Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the .05 level is represented by ** of the corresponding covariances.
active, the probability that it will remain active for a year is only 14 percent. The other two states are both growth states, but with significantly different correlation structures. Table 2.5 reports the results for the model specification with two regimes and fully hedged currency risk. There is a clear distinction between the two regimes. The first regime is a bear state with low drifts, especially for Germany with a drift of −22.3 percent, high volatilities and very low persistence. The probability that it switches to the bull state within two months is close to 50 percent. The second regime is a bull state with high excess returns ranging between 8 percent and 10 percent, low volatilities, and high
26
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
Table 2.5 Parameter estimates for the three regimes specification with fully hedged currency risk Panel A: Drifts Bear state
GER −0.223 (0.005)
UK 0.043 (0.013)
US −0.097 (0.007)
0.101 (0.006)
0.088 (0.010)
0.082 (0.010)
Bull state
Panel B: Correlations and volatilities GER Recession state GER 0.298**
UK
UK
0.431**
0.403**
US
0.521**
0.640**
Growth state GER
0.167**
UK
0.496**
0.140**
US
0.482**
0.584**
Panel C: Transition probabilities Bear state Bear state 0.721 (0.007) Bull state
0.280
US
0.237**
0.130**
Bull state 0.053 (0.081) 0.947
Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the .05 level is represented by ** of the corresponding covariances.
persistence. Similar to the bear state, the correlations are close to the sample average. Finally, Table 2.6 presents the parameter estimates for the model specification with three regimes with both hedged and unhedged indices. As for the unhedged strategy, the three regimes can be referred to as bear state, low correlation state, and high correlation state. The low correlation state can be interpreted as a transitory state.
2.4 DISCUSSION With the empirical specification at hand, we next discuss some of the implications of regime switching on asset allocation. With regard to the numerical calculation of the optimal portfolio strategies, we use a Monte Carlo simulation with sample size Z = 30,000. Furthermore, for all calculations, we suppose that the investor has a risk aversion parameter equal to γ = 5.
Table 2.6 Parameter estimates for the three regimes specification with hedged and unhedged indices Panel A. Drifts Bear Low High
GER, FH −0.234 (0.073) 0.061 (0.016) 0.110 (0.026)
Panel B: Correlations and volatilities GER, FH Bear state GER, FH 0.282** GER, UH 0.923** UK, FH 0.432** UK, UH 0.386** US 0.480**
GER, UH −0.186 (0.070) 0.094 (0.006) 0.125 (0.025)
UK, FH 0.037 (0.025) 0.091 (0.007) 0.073 (0.015)
UK, UH 0.072 (0.026) 0.063 (0.009) 0.110 (0.015)
US −0.133 (0.028) 0.072 (0.006) 0.076 (0.016)
GER, UH
UK, FH
UK, UH
US
0.283** 0.456** 0.453** 0.555**
0.467** 0.987** 0.643**
0.480** 0.615**
0.261**
Low correlation GER, FH GER, UH UK, FH UK, UH US
0.158** 0.790** 0.344** 0.301** 0.291**
0.195** 0.199** 0.408** 0.216**
0.162** 0.805** 0.517**
0.199** 0.443**
0.134**
High correlation GER, FH GER, UH UK, FH UK, UH US
0.233** 0.920** 0.844** 0.724** 0.853**
0.227** 0.809** 0.849** 0.835**
0.130** 0.860** 0.807**
0.130** 0.771**
0.137**
Low corr. 0.032 (0.075) 0.929 (0.063) 0.039
High corr. 0.000 (0.035) 0.080 (0.098) 0.920
Panel C: Transition probabilities Bear state Bear 0.783 (0.003) Low 0.217 (0.019) High 0.000
27
Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. FH is short for fully hedged and UH for unhedged. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the 0.05 level is represented by ** of the corresponding covariances.
28
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
2.4.1 Economic importance of regimes The economic importance of regimes is a very relevant issue for the appraisal of regime-switching strategies. We measure this “importance” as utility costs that an investor bears, when he/she gives up the optimal strategy and follows instead a suboptimal one. More precisely, we are interested in the monetary compensation c, also called certainty equivalent compensation, that makes an investor with time horizon T indifferent between the suboptimal weights α− and the optimal weights α∗ . Formally, we must solve E0 [U(WT (α∗ )|W0 = 1)] = E0 [U(WT (α− )|W0 = 1 + c)] for c. Using the fact that CRRA utility preferences are homogeneous in W0 , and since E0 [U(WT )|W0 = 1] =
Q0,T 1−γ
we obtain c=
Q∗0,T Q− 0,T
1/(1−γ) −1
We report the certainty-equivalent compensations in Table 2.7 as cents per invested dollar. In Table 2.7, the reported cost can be attributed directly to international investment and regime-switching (Panel A) and to regimeswitching only (Panel B), respectively. Panel C documents the economic importance of regimes in a purely international setting. Panel A of Table 2.7 displays the economic cost of investing in a domestic US portfolio while disregarding the possibility of international diversification. Relative to the international investment strategy with one regime and unhedged indices, the costs (in terms of invested dollars) are only 0.53 percent per year for all time horizons. Hence the strategy with a single regime does not seem to offer substantial benefits, at least when only a small number of (highly correlated) markets are involved (as is the case for our analysis). However, turning to the regime-switching strategies in Panel A, we see that the economic costs for the three regime-switching strategies may be as high as 2.7 percent per year. Consequently, international investment combined with regime-switching does pay off – even when the number of potential markets is low and their correlation high. Also, we note that ignoring regimes tends to increase the annualized economic costs as the time horizon increases. This finding contrasts with Guidolin and Timmermann (2004), who explore the required compensation for buy-and-hold strategies, and find decreasing costs with a longer time horizon. Their result is
Table 2.7 The economic importance of regime-switching strategies 6 months Total
1 year
Annualized
2 years Total
Annualized
5 years Total
Annualized
10 years Total
Annualized
Panel A: International strategies versus domestic US strategy, 1 reg. Internat., 1 reg. 0.26 0.53 0.53 1.06
0.53
2.66
0.53
5.39
0.53
UH, 3 reg.
2.51
14.14
2.68
31.37
2.77
1.01
2.02
2.32
5.09
FH, 2 reg.
0.71
1.42
1.50
3.10
1.54
8.06
1.56
16.88
1.57
OH, 3 reg.
1.03
2.06
2.20
4.57
2.26
11.90
2.27
25.24
2.28
Panel B: RS strategies versus international unhedged portfolio, 1 reg. UH, 3 reg. 0.74 1.49 1.78 3.94
1.95
10.95
2.10
23.98
2.17
FH, 2 reg.
0.45
0.90
0.98
2.05
1.02
5.34
1.05
11.07
1.06
OH, 3 reg.
0.61
1.23
1.36
2.87
1.42
7.44
1.44
15.46
1.45
Panel C: States of OH versus the OH bear state Low corr. 1.06 2.13
1.33
1.46
0.73
1.50
0.30
1.50
0.15
High corr.
1.97
3.97
2.90
3.75
1.86
4.32
0.85
4.35
0.43
Uncond. state
1.06
2.13
1.51
1.83
0.91
2.02
0.40
2.03
0.20
2005/6
1.89
3.82
2.80
3.62
1.80
4.17
0.82
4.19
0.41
29
Notes: Panels A and B report the economic cost of investing suboptimally. The suboptimal strategies have the same investment opportunities as the optimal strategies. The economic cost of these three panels are calculated with the indirect utilities, Q∗0,T of the unconditional state probabilities. Panel C documents the importance of regimes for the case of the optimal hedging strategy. It gives the cost of being in the bear regime compared to a situation with another, economically more promising state belief. In all panels, the economic costs are given as cents per invested dollar required, making an investor indifferent between the optimal and the suboptimal weights. UH denotes unhedged; FH fully hedged; and OH optimal hedging.
30
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
as a result of the inability of the investor to react to market changes. Panel B gives the required compensation for an international investor to ignore regime-switching. With a horizon of ten years, the costs of ignoring regimeswitching average almost 17 percent, or 1.6 percent per year. Comparing Panel B with the regime-switching strategies of Panel A, we see that the economic gains are about 30 percent (on average) lower in Panel B than in Panel A. In Panel A, only 30 percent of the certainty equivalents can be attributed to international diversification, and 70 percent result from to taking into account the possibility of regime-switches. In Panel C, we present the results on the economic significance of the regimes for the optimal hedging strategy. The panel gives the cost of being in the bear state compared to a situation with another, economically more promising state belief. These costs are high in the short term and, in contrast to the other panels, decrease with longer investment horizons. They are high in the short term because the regimes have very different estimated parameters, and therefore very different expected returns. They decrease because, in the long term, the probability for a switch to another regime increases and the initial regime becomes less influential. There are almost no additional costs related to the initial regime after the first five years. The costs reported in Table 2.7 are not only economically, but also statistically significant. To test for statistical significance, we recall that, given some regularity conditions for the likelihood function (see, for example, Poirier, 1995), the asymptotic distribution of the MLE of θ is A θˆ −−→ N(0, J(θˆ )−1 )
where J(θˆ ) is the matrix of second derivatives of the log likelihood function with respect to the estimated parameters and observed returns – for example J(θˆ ) = −
T ∂2 log (rt ) t=1
∂θˆ i ∂θˆ i
, i = 1, . . ., k, j = 1, . . ., k
(·) is the normal distribution, and k is the number of parameters to be estimated. Therefore, to derive confidence intervals we first simulate Q = 200 parameter sets θˆ q from N(θˆ , J(θˆ )−1 ). Then the economic costs are calculated for each set of parameters θˆ q . Figure 2.1 plots the confidence intervals of the required compensation when investing in the one-regime international strategy rather than the three-regime strategy with unhedged indices. The lower bound of the 95 percent confidence interval lies above zero. Therefore, the null hypothesis, that ignorance of regime-switching does not cause any utility losses, is
31
MARKUS LEIPPOLD AND FELIX MORGER
Required compensation
0.07 0.06 0.05
Estimate Median 67% Cl 95% Cl
0.04 0.03 0.02 0.01 0
6
12 Investment horizon
18
24
Figure 2.1 Confidence intervals of economic cost estimates Notes: The figure shows the bootstrapped confidence intervals of the compensation required for ignoring the unhedged regime-switching strategy with the parameters estimated as per 2005/6. The economic costs are calculated for the indirect utilities of the unconditional state probabilities. In the bootstrap procedure, 200 parameter sets are drawn from N(θˆ ,J(θˆ )−1 ).
clearly rejected. For a one-year horizon, the required compensation can be as high as 3.5 percent with a median compensation of more than 1 percent. With long horizons, the non-realized wealth related to ignorance of regimeswitching is substantial.
2.4.2 Strategies in competition: horse race Most often, a model performs very well in sample, but fails out of sample. In this section, we present an out-of-sample test (horse race) for our regimeswitching strategies. This horse race spans the period from January 1990 to June 2005, or 186 real data returns. At the beginning of each month, we calculate the optimal asset allocation for the different strategies, and at the end of each month, we calculate the strategies’ performance in terms of cumulated wealth. To calculate the optimal portfolio allocation at the beginning of each month, the model parameters and beliefs are estimated with the data available to the investor. So, for example, the optimal weights for January 1990 are calculated based on the excess returns from January 1971 to December 1989. The horse race is therefore a truly out-of-sample comparison of different strategies.2
32
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
4
3.5
International, 1 reg UH, 3 reg FH, 2 reg OH, 3 reg
Cumulative wealth
3
2.5
2
1.5
1 1990
1995
2000
2005
Investment date
Figure 2.2 Cumulative wealth of the strategies Notes: The figure plots the cumulative wealth during the horse race of the regime-switching strategies and a benchmark strategy. The horse race is out-of-sample and spans the period from 1990/1 to 2005/6. The optimal asset allocations are calculated for an investor with a one-month investment horizon. UH is short for unhedged, FH for fully hedged, and OH for optimally hedged. Cumulative wealth and Sharpe ratio
Figure 2.2 plots the evolution of cumulative wealth for the four strategies under consideration. International strategy with a single regime serves as benchmark for the regime-switching strategies. With the exception of the first two years, the cumulative wealth of all regime-switching strategies is always above the benchmark. By the end of June 2005, the optimal hedging strategy performs best, outperforming the single-regime benchmark strategy by 37 percent. We also observe that the optimal hedging strategy recovers fast from the Russian crises in autumn 1998, suffers during the bursting of the Internet bubble, but recovers very well with strong performance during 2004/5. In comparison, the unhedged regime-switching strategy outperforms the benchmark by only 18 percent. Towards the end of the bursting of the
MARKUS LEIPPOLD AND FELIX MORGER
33
Internet bubble, the investor allocates most of the capital into the risk-free asset, but misses the beginning of the recovery period. The strategy with the fully hedged currency risk outperforms the benchmark by only 8 percent. Up to January 2000, its performance is similar to the unhedged regime-switching strategy. Relative to the unhedged strategy, which bears all the currency risk, it outperforms in the period from January 2000 to the early 2002, since the value of the dollar relative to the euro and pound increased. On the other hand, it underperforms towards the end of the horse race because it cannot take advantage of the depreciation of the dollar from early 2002 to the end of 2004. In 2005, it again outperforms because of a rise in the dollar. Table 2.8 presents the descriptive statistics of the horse race for the whole period and for two subperiods, split at the peak of the Internet bubble in January 2000. In comparison to Figure 2.2, we include additional strategies. First, we include a purely domestic single-regime strategy. Second, for comparability with Ang and Bekaert (2004), we introduce a naïve investor, who uses a regime-switching model with three possible states, but does not follow a Bayesian updating rule. Instead, he or she treats the one state with the largest probability of being active as the active state, with a probability of one. Panel A and B report the summary statistics (annualized) for the different strategies and for market indices, respectively. The numbers for the subperiods confirm that the high total performance of the US market is because of its outperformance in the 1990s. The standard deviations of the returns are low for the single-regime strategies, high for the market indices, and intermediate for regime-switching. The single regime strategies exhibit large Sharpe ratios. The largest Sharpe ratio is generated by US domestic strategy, which can be attributed to the outperformance of the US market. The Sharpe ratios of the regime-switching strategies are lower than those for the singleregime strategies. This is because of the low variance of the single-regime strategies. Finally, the regime-switching strategy with three regimes and a naïve investor has a lower mean return and higher return volatility than the Bayesian investor with the unhedged regime-switching strategy. Clearly, it pays to go Bayes. These conclusions for the whole period still hold when looking at the two subperiods (Table 2.8). We end this section with a word of caution. We have to be careful when ranking the different strategies using the Sharpe ratio. The Sharpe ratio would be the optimal measure when investors are only concerned about mean and volatility (or if asset returns are normally distributed). However, since we assume a CRRA utility, the investor also has concerns about higher moments of the portfolio strategy. Therefore, for the ranking of the different strategies, a comparison based on the economic costs as in Section 2.4.1 would be appropriate.
34
Table 2.8 Descriptive statistics on the horse race 1990.1–2005.6 Mean
Standard deviation
1990.1–1999.12
2000.1–2005.6
Sharpe ratio
Mean
Standard deviation
Sharpe ratio
Mean
Standard deviation
Sharpe ratio
Panel A: Strategies Domestic, 1 reg.
6.65
6.22
1.07
10.70
5.56
1.92
−0.35
6.90
−0.05
International UH, 1 reg.
6.16
8.45
0.73
10.46
8.03
1.30
−1.22
8.86
−0.14
UH, “naïve”, 3 reg.
6.60
11.84
0.56
13.08
12.79
1.02
−4.25
9.12
−0.47
UH, 3 reg.
7.28
11.17
0.65
13.15
12.05
1.09
−2.62
8.76
−0.30
FH, 2 reg.
6.77
11.24
0.60
13.52
11.10
1.22
−4.48
10.85
−0.41
OH, 3 reg.
8.33
11.79
0.71
14.05
10.60
1.33
−1.35
13.38
−0.10
Panel B: Markets GER, FH
6.44
22.22
0.29
14.48
20.02
0.72
−6.75
25.53
−0.26
GER, UH
6.76
21.84
0.31
12.85
19.18
0.67
−3.50
25.95
−0.13
UK, FH
8.42
14.46
0.58
14.25
14.34
0.99
−1.44
14.39
−0.10
UK, UH
9.16
15.27
0.60
14.24
15.37
0.93
0.49
14.91
0.03
10.69
14.55
0.73
19.01
13.40
1.42
−2.98
15.87
−0.19
US
Notes: We report the descriptive statistics of the horse race for the whole period and two subperiods. The descriptive statistics consist of the returns’ mean and standard deviation, and the Sharpe ratio (SR). The mean and standard deviation are annualized. Panel A gives the descriptive statistics of the strategies, and Panel B those of the market indices. FH denotes fully hedged and UH unhedged.
35
MARKUS LEIPPOLD AND FELIX MORGER
GER, FH
1 Weight
Weight
1
0.5
0 1990
1995
2000
0.5
0 1990
2005
GER, UH
Weight
Weight
2005
1995
2000
0.5
0 1990
2005
US
1995
2000
2005
Risk free asset
1
1 Weight
Weight
2000
1
0.5
0.5
0 1990
1995
UK, UH
1
0 1990
UK, FH
1995
2000
Investment date
2005
0.5
0 1990
1995
2000
2005
Investment date
Figure 2.3 Asset allocation of the optimal hedging strategy Notes: The figure plots the asset allocation of the optimal hedging strategy. The weights are derived for an investor with a one month horizon. The investor has a risk aversion of γ = 5. Optimal weights
It is instructive to take a look at the behavior of the optimal portfolio strategy through time. Figure 2.3 plots the asset allocation of the optimal hedging strategy. We observe that the weights have sharp peaks and temporarily drop to zero. These sudden moves are mainly because of changes in the state probability estimates. As we can see in Figure 2.4, beliefs about the active state change frequently and provoke the large variation in portfolio weights. Looking at the unhedged and hedged indices of one specific country, the optimization usually puts a lot of weight into the index with the higher historical sample mean and ignores the other index. This result is most probably because of the high correlation between the hedged and unhedged indices. Taking the example of Germany, it is better to invest in the unhedged index, which offers the higher expected drift, and to forgo the additional, but
36
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
pr(St 1) Probability
1
0.5
0 1990
1995
2005
pr(St 2)
1 Probability
2000
0.5
0 1990
1995
2000
2005
2000
2005
pr(St 3) Probability
1
0.5
0 1990
1995 Investment date
Figure 2.4 Changing beliefs during the horse-race Notes: The figure displays the state probabilities of the unhedged strategy. The plots give the probability that a state is active at a certain investment date. The state beliefs are estimated together with the parameter estimates by the EM algorithm. small diversification offered by the hedged index. However, the exception to this rule is investment into the unhedged UK index starting from July 2002. Here, the optimization picks up the depreciation of the dollar, which makes the pound returns more valuable for the US investor. To take advantage of the depreciation, it reduces its holdings in the UK local return index to zero and shifts into the unhedged UK index. The impressive performance at the end of the horse race is the reward.
Optimal hedge ratio
Figure 2.5 plots the optimal hedge ratios for the two foreign markets. The figure confirms the observations made above. The optimal currency hedge
MARKUS LEIPPOLD AND FELIX MORGER
37
GER 1 Hedge ratio
0.8 0.6 0.4 0.2 0 1990
1995
2000
2005
2000
2005
UK
1 Hedge ratio
0.8 0.6 0.4 0.2 0 1990
1995 Investment date
Figure 2.5 Optimal hedge ratios, Germany and UK, during the horse-race Notes: The figure plots the optimal hedge ratio of the foreign markets, Germany and UK, during the horse race. The hedge ratio is calculated as the allocation to the local currency index divided by the total investment to the foreign market. The weights are derived for an investor with a one month horizon. The investor has a risk aversion of γ = 5. No dot is plotted at investment dates where the hedged and unhedged indexes of the foreign markets do not receive any weight at all. ratio is time-varying, and the optimal hedging policy depends on two factors. The first is the trade-off between the diversification offered by the index with the lower expected drift, and the difference in the drifts of the hedged and unhedged indices. The second factor concerns the ex-ante identification of moves in the exchange rates that motivate a preference for the index with the lower drift. The prerequisite for this are parameter estimates for the regime-switching model that allow the identification of favorable moves in the exchange rates. As we observe in Figure 2.5, our model generates the optimal hedge ratios for the German and UK markets that are generally either zero or one. In contrast, in an equity-only international CAPM with regime-switching betas, Ang and Bekaert (2002) find optimal hedge ratios of 53 percent for the high volatility regime and 40 percent for the low volatility regime.
38
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
International, 1 reg. UH,3 reg. OH,3 reg.
1
Weight
0.8 0.6 0.4 0.2 0 1990
1995
2000
2005
Investment date
Figure 2.6 Optimal foreign investments Notes: The figure plots the foreign investments of two different regime-switching strategies and one single-regime strategy. The foreign investments are calculated as the total wealth allocated to foreign markets. The weights are derived for an investor with a one-month horizon. UH = unhedged; OH = optimal hedging.
2.4.3 Optimal foreign investment Figure 2.6 compares the optimal foreign investments of the regime-switching strategies with the international unhedged strategy with a single regime. The international strategy with a single regime starts with an asset allocation of more than 43 percent to foreign markets and reduces this number steadily to less than 30 percent at the end of the horse race. The reduction is caused by increasing drift estimates for the US market. The average international investment is 34 percent. The unhedged regime-switching strategy invests almost the same fraction into the foreign markets, namely 30 percent on average. However, compared to the benchmark strategy, the foreign investment is very volatile over time, moving between 0 percent and 76 percent. Therefore, the introduction of regimes makes the foreign investments strongly time dependent. With the exception of very few data points, the foreign investment of the optimal hedging strategy lies above the benchmark strategy, moving between 4 percent and 100 percent. The average allocation to foreign markets is 63 percent. This is almost twice the percentage of the foreign holdings induced by the single regime strategy. Hence, the introduction of optimal currency hedging increases the optimal allocation to foreign markets compared to the single regime international strategy and, as a backdrop to these results, the home bias question becomes even more of a puzzle.
MARKUS LEIPPOLD AND FELIX MORGER
39
2.5 CONCLUSION We investigated different regime-switching models for international asset allocation. Taking the perspective of a US investor, we applied regime-switching models to the US, UK and German stock markets. We show that, with a correctly specified model, the economic gains from explicitly modeling regime switches are significant and substantial. The cost of ignoring international regime-switching without currency hedging is 2.8 percent per year for a domestic US investor with a ten-year horizon. Of these costs, 70 percent can be attributed to regime-switching and 30 percent to international diversification. Concerning the hedging behavior, the optimal hedging strategy depends strongly on the conditional expected returns of the hedged and unhedged market indices. Most of the time, currency risk is either completely hedged, if the hedged index has the higher expected return, or not hedged at all. The ultimate test of the quality of our model specifications is the outof-sample horse race, where for different strategies we compare the wealth generated during a period spanning fifteen and a half years, from January 1990 to June 2005. Regime-switching strategies generate higher returns than the international and domestic benchmark strategy with a single regime, but at the cost of higher volatilities. The optimal hedging strategy outperforms the international strategy with a single regime by 37 percent in total and 2 percent per year, respectively. During the horse race, the optimal hedging strategy outperforms the strategy without currency hedging by an annualized 1.1 percent and the strategy with a full hedge by an annualized 1.6 percent. Finally, analyzing the size of foreign investments, we find that the optimal foreign investment under regime-switching is highly time-varying. The average foreign investment for the strategy without currency hedging is 30 percent, which is just below the 34 percent of foreign investments generated by the international strategy with a single regime. However, with optimal currency hedging, the investments in foreign markets increase to an average of 63 percent. Hence, under optimal currency hedging, the home bias becomes even more severe.
NOTES 1. A detailed econometric analysis can be obtained from the authors (email:
[email protected]) 2. One could argue that our horse race is only pseudo out-of-sample, because the choice of the number of regimes is based on Table 12.2, which uses the whole sample. In real out-of-sample horse races, the specification test should be done at each investment date. This is certainly true. However, as Table 12.2 shows, the choice of regime would
40
INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING
be the same for each evaluated investment date. Because of this stability, it is legitimate to do the horse race without a specification test at each investment date. The horse race can be considered as a real out-of-sample test.
ACKNOWLEDGMENTS Part of this work was done when Markus Leippold was visiting professor at the Federal Reserve Bank of New York. The authors acknowledge the financial support of the Swiss National Science Foundation (NCCR FINRISK) and the University Research Priority Program “Finance and Financial Markets” at the University of Zurich.
REFERENCES Ang, A. and Bekaert, G. (2002a) “International Asset Allocation with Regime Shifts”, Review of Financial Studies, 15(4): 1137–87. Ang, A. and Bekaert, G. (2002b) “Regime Switches in Interest Rates”, Journal of Business and Economic Statistics, 20(2): 163–82. Ang, A. and Bekaert, G. (2004) “How Do Regimes Affect Asset Allocation?”, Financial Analysts Journal, 60(2): 86–99. Ang, A. and Chen, J. (2002) “Asymmetric Correlations of Equity Portfolios”, Journal of Financial Economics, 63(3): 443–94. Berkowitz, J. (2001) “Testing Density Forecasts with Applications to Risk Management”, Journal of Business and Economic Statistics, 19(4): 465–74. Black, F. (1989) “Universal Hedging: Optimizing Currency Risk and Reward in International Equity Portfolios”, Financial Analyst Journal, 45(4): 161–7. Brandt, M. W., Goyal, A., Santa-Clara, P. and Storud, J. (2004) “A Simulation Approach to Dynamic Portfolio Choice with an Application to Learning About Return Predictability”, Review of Financial Studies, 17(3): 831–73. Connolly, R., Stivers, C. and Sun, L. (2005) “Stock Market Uncertainty and the Stock–Bond Return Relation”, Journal of Financial and Quantitative Analysis, 40(1): 161–94. Dales, A. and Meese, R. (2001) “Strategic Currency Hedging”, Journal of Asset Management, 2(1): 9–21. Dempster, A. P., Laird, N. M. and Rubin, D. R. (1977) “Maximum Likelihood from Incomplete Data via the EM Algorithm”, Royal Statistical Society B, 39(1): 1–38. Froot, K. (1993) “Currency Hedging over Long Horizons”, NBER Working Paper No. W4355. Garcia, R. and Perron, P. (1996) “An Analysis of the Real Interest Rate under Regime Shifts”, The Review of Economics and Statistics, 78(1): 111–25. Glen, J. and Jorion, Ph. (1993) “Currency Hedging for International Portfolios”, Journal of Finance, 68(5): 1865–86. Gray, S. (1996) “Modeling the Conditional Distribution of Interest Rates as a RegimeSwitching Process”, Journal of Financial Economics, 42(1): 27–62. Guidolin, M. and Timmermann, A. (2004) “Strategic Asset Allocation and Consumption Decisions under Multivariate Regime Switching”, Working Paper, Federal Reserve Bank of St. Louis. Guidolin, M. and Timmermann, A. (2005a) “Economic Implications of Bull and Bear Regimes in UK Stock and Bond Returns”, Economic Journal, 115(500): 111–43.
MARKUS LEIPPOLD AND FELIX MORGER
41
Guidolin, M. and Timmermann, A. (2005b) “International Asset Allocation under RegimeSwitching, Skew and Kurtosis Preferences”, Working Paper, Federal Reserve Bank of St. Louis. Guidolin, M. and Timmermann, A. (2006) “An Econometric Model of Nonlinear Dynamics in the Joint Distribution of Stock and Bond Returns”, Journal of Applied Econometrics, forthcoming. Hamilton, J. D. (1989) “A New Approach to the Economic Analysis of Non-Stationary Time Series and the Business Cycle”, Econometrica, 57(2): 357–84. Hamilton, J. D. (1990) “Analysis of Time Series Subject to Changes in Regime”, Journal of Econometrics, 45(1): 39–70. Longstaff, F. A. and Schwartz, E. (2001) “Valuing American Options by Simulation: A Simple Least-Squares Approach”, Review of Financial Studies, 14(1): 113–47. Perez-Quiros, G. and Timmermann, A. (2000) “Firm Size and Cyclical Variations in Stock Returns”, Journal of Finance, 55(3): 1229–62. Perold, A. and Schulman, E. (1988) “The Free Lunch in Currency Hedging: Implications for Investment Policy and Performance Standards”, Financial Analyst Journal, 44(3): 45–50. Poirier, D. (1995) Intermediate Statistics and Econometrics: A Comparative Approach, Cambridge, Mass.: MIT Press. Samuelson, P. (1969) “Lifetime Portfolio Selection by Dynamic Stochastic Programming”, Review of Economics and Statistics, 51(3): 239–46. Turner, C., Startz, R. and Nelson, C. (1989) “A Markov Model of Heteroskedasticity, Risk, and Learning in the Stock Market”, Journal of Financial Economics, 25(1): 3–22. Whitelaw, R. (2000) “Stock Market Risk and Return: An Equilibrium Approach”, Review of Financial Studies, 13(3): 521–47.
CHAPTER 3
The Determinants of Domestic and Foreign Biases: An Empirical Study Fathi Abid and Slah Bahloul
3.1 INTRODUCTION The international capital asset pricing model (ICAPM), based on traditional portfolio theory developed by Sharpe (1964) and Lintner (1965), suggests that, to maximize risk-adjusted returns, investors should hold a world market portfolio of risky assets. However, domestic assets are heavily weighted in investors’ portfolios even after the relaxing of capital control after 1980. For example, in 1997, 89.9 percent of US investors’ equity portfolios were domestic equities, while the size of the USA in world market capitalization was about 48.3 percent (Ahearne et al., 2004). The wide disparity between actual and recommended international equity portfolio weights constitutes the equity home bias, one of the unresolved puzzles in international finance literature.1 Various attempts have been made to explain the home asset bias. First explanations have focused on the institutional factor. The existence of equity home bias may be related to barriers to capital flow (Black, 1974: Stulz, 1981), hedging possibilities against domestic risk (Glassman and Riddick, 1996), and information asymmetries (Ahearne et al., 2004). Dissatisfaction with 42
FATHI ABID AND SLAH BAHLOUL
43
institutional explanations has led some authors to consider explanations based on investor behavior: optimism of investors about their domestic markets (French and Poterba, 1991), unfamiliarity with foreign market (Huberman, 2001), and subjective competence in the home market (Kilka and Weber, 2000). Recent explanations consider the problem of corporate governance and investor protection explains the home asset bias. The presence of controlling shareholders and the lack of investor protection led to low investment rates in foreign markets (Dahlquist et al., 2003). Explanations for the home bias seem not to be explored sufficiently to provide convincing arguments. Most studies on home bias use a singlefactor model, but international investing behavior seems to be determined by many factors (Faruqee et al., 2004). Besides, most of the previous studies are from the perspective of developed countries, in particular US investors, and neglect emergent countries’ points of view. The purpose of this chapter is not to add a new explanation but rather to examine the determinants of home and foreign equity biases for the period 2001–02. We start by considering different groups of factors that might intervene to explain equity home bias. These factors are economic development, capital controls, stock market development, information costs, investor behavior, familiarity, investor protection, and other variables. Following Chan et al. (2005), work on the determinants of home bias using mutual fund investors for the period 1999–2000, this chapter applies similar methodology to study the cross-border behavior of investors from various countries, including both developed and emerging countries, for a recent period (2001–02) and different investment strategies. We use the Coordinated Portfolio Investment Survey (CPIS) dataset from the International Monetary Fund (IMF) that lists the aggregate stockholdings of both individual and institutional investors. The effects of two other causes (information costs and investors’ behavior) on the home bias will be analyzed in addition to those considered by Chan et al. (2005). As did Chan et al. (2005), we distinguish between the domestic and foreign components of the home bias. The domestic bias reflects the extent to which investors overweigh the local market in their holdings, while the foreign bias reflects the extent to which investors underweigh or overweigh foreign markets. We also use, as an additional test, a measure of the home bias defined by Ahearne et al. (2004). Using the two-year data on equity holdings of thirty countries, we find that equity home bias is a feature in both developed and emerging markets. All of the thirty countries show a domestic bias. The fraction of domestic assets held by local investors is much larger than the world-market capitalization weight of the country. However, domestic bias varies greatly across countries. Venezuela, for example, has the highest domestic bias. Investors from the USA, the European bloc and developed countries have the lowest domestic bias.
44
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
Results show that the impact on domestic and foreign biases is asymmetric. Stock market development and information costs affect domestic bias the most, whereas information costs and familiarity variables contribute significantly to explaining foreign bias. Investor behavior, in contrast, has a significant effect on domestic bias but not on foreign bias. Results indicate that factors such as economic development, capital control and investor protection have smaller effects on these biases. The remainder of the chapter is organized as follows. Section 3.2 provides a theoretical framework for the domestic and foreign biases. In section 3.3, we present the descriptive statistics of investors’ holdings, domestic bias and foreign bias. Section 3.4 discusses the different causes of home bias. Section 3.5 presents and interprets the results, and section 3.6 presents a number of additional tests.
3.2 THEORETICAL FRAMEWORK OF DOMESTIC AND FOREIGN BIASES Chan et al. (2005) have used the theoretical framework developed by Cooper and Kaplanis (1986) to analyze domestic and foreign biases. Cooper and Kaplanis’s model assumes that a representative investor in country i acts as an expected return maximizer for a given level of variance Max(wi R − wi ci )
(3.1)
subject to wi Vwi = v wi I = 1 where wi is a column vector, the jth element of which is wij ; wij is the proportion of individual i’s total wealth invested in risky securities of country j; R is a column vector of pre-tax expected returns; ci is a column vector, the jth element of which is cij ; cij is the deadweight cost to investor i of holding securities in country j; v is the given constant variance; V is the variance/covariance matrix of the gross (pre-cost, pre-tax) returns of the risky securities; and I is a unity column vector. The Lagrangean of the above maximization problem is L = (wi R − wi ci ) − (h/2)(wi Vwi − v) − ki (wi I − 1)
(3.2)
where h and ki are Lagrange multipliers. The derivation of objective function with respect to wi equal to zero lead to R − ci − hVwi − ki I = 0
(3.3)
FATHI ABID AND SLAH BAHLOUL
45
Therefore the optimal portfolio for investor i is wi = (V −1 /h)(R − ci − ki I)
(3.4)
where
ki = I V −1 R − I V −1 ci − h /I V −1 I
Given the individual portfolio holdings, the aggregation leads to the world capital market equilibrium. The clearing condition for the model is pi wi = w∗ (3.5) where pi is the proportion of world wealth owned by country i; w∗ is a column vector, the ith element of which is wi∗ ; and wi∗ is the proportion of the world market capitalization in country i’s market. Using Equations (3.4) and (3.5), and defining z as the global minimumvariance portfolio (V −1 I/(I V −1 I)), Cooper and Kaplanis have obtained hV(wi − w∗ ) = pi ci − ci − z pi ci − ci I (3.6) If deadweight costs are zero (cij are equal to zero for all i and j), the righthand side of Equation (3.6) is zero, and each investor holds the world market portfolio. If deadweight cost of any country/investor pair is equal to c, then the portfolio holdings of each investor will deviate from the world market portfolio. To examine the deviation, Chan et al. (2005) have considered the simple case when the covariance matrix, V, is diagonal with all variances equal to s2 . The deviation of the portfolio weight of investor i in country j from the world market portfolio is given by hs2 (wii − wi∗ ) = −cii + bi + ai − d, i = j 2
hs
(wij − wj∗ )
= −cij + bj + ai − d, i = j
(3.7) (3.8)
where a i = z ci bj = pk ckj pi ci d = z ai can be interpreted as the weighted average deadweight cost for investor i, bj as the weighted marginal deadweight cost for investors investing in country j, and d as the world weighted average marginal deadweight cost. Equation (3.7) measures the extent to which domestic asset holdings of
46
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
investor i deviate from those of the world market portfolio, whereas, Equation (3.8) measures the extent to which investors of country i holdings in foreign market j deviate from the world market portfolio. Similar to Chan et al. (2005), we refer to the former as domestic bias (DBIASi ) and the latter as foreign bias (FBIASij ). From Equation (3.7), investor i overweights domestic country (DBIASi > 0) if the deadweight cost for this investor investing in his/her own country i (cii ) is considerably less than the weighted average deadweight cost for world investors (bi ), or if the weighted average deadweight cost he/she faces (ai ) is large enough to discouraged him/her from investing in foreign markets. Equation (3.8) shows that country j asset holdings by investor i depend on the difference between the deadweight costs for investor i investing in country j (cij ) and the weighted average deadweight cost for world investors (bj ). If cij is greater than bj , investor i underweights country j (FBIASij < 0). Then, the more important the deadweight cost for investor i investing in country j, the greater is the foreign bias (a more negative FBIASij ).
3.3 DATA AND PRELIMINARY STATISTICS 3.3.1 Data sources The cross-border equity data is taken from a survey of international portfolio holdings coordinated by the IMF for seventy countries for the end of December in both 2001 and 2002.2 Data on explanatory variables are not available for all countries; we explore data for only thirty investing and forty-three receiving countries. The data of capital market capitalization is from the international federation of stock exchanges (FIBV).
3.3.2 Statistics on investor holdings For each of the thirty investing countries, we calculate the percentage allocation of local investors in forty-three countries as follows: Wij =
MVij 43
(3.9)
MVij
j=1
where Wij is the share of country j in investor holdings of investing country i; and MV ij is the market value of country j s asset holdings by investors of investing country i.
FATHI ABID AND SLAH BAHLOUL
47
The weight of country j in the world market portfolio is defined as the portfolio of the forty-three countries included in the sample MVj∗ Wj∗ = (3.10) 43 ∗ MVj j=1
Wj∗
where is the share of country j in the world market portfolio; and MVj∗ is the market capitalization of country j; We compute Wij and Wj∗ in 2001 and 2002 separately, and then take an average of the two values. Table 3.1 presents the distribution of the average equity allocations (in percentages) of thirty investing countries’ investors in forty-three national markets across the world. The table shows that domestic bias is revealed in all countries in the sample. Across all of the thirty countries, the shares of investors’ holdings in the domestic market are much more important than the world market capitalization weight of the country. Venezuela has the highest percentage of investors’ holdings of domestic equities (99.64 percent) and the lowest share in the world market portfolio (0.02 percent). Austria has the lowest percentage of domestic asset holdings (50.12 percent), though its share in the world market portfolio is 0.12 percent. The same table shows that investors do underweight foreign markets in their asset holdings. Generally, the share of foreign asset holdings is by far smaller than the shares of foreign country in the world market portfolio. Yet, there are some exceptions. For example, the German and Belgian investors hold a proportion of 4.37 percent and 7.17 percent respectively, of French assets, while the share of the French market in the world market portfolio is only 3 percent. This may provide preliminary evidence that geographical proximity plays an important role in determining the extent to which investors’ overweight foreign markets. Table 3.2 presents the average share of domestic assets held by investors from different blocs: European, American, Asia/Pacific and African. It shows that European investors have the smallest percentage of domestic asset holdings among the four blocs. From Table 3.3 it can be seen that investors from developed markets have a lower percentage of domestic asset holdings compared to investors from emerging markets.
3.3.3 Statistics on domestic and foreign biases Chan et al. (2005) have used the theoretical framework developed by Cooper and Kaplanis (1986) to compute domestic and foreign biases. The domestic bias for a specific country j (DBIASj ) refers to the deviation of the proportion of country j’s investors in the local market from its world market capitalization weight. Therefore, DBIASj is defined as the log ratio of the
48
Table 3.1 Equity allocation for thirty countries, 2001 and 2002, percentages Panel A: First 15 countries Country
% WMP Argentina Austria Australia Belgium Brazil Canada Chile
Czech Denmark Finland France Germany Greece Hong Italy Kong Republic
Argentina
0.10
76.67
0.00
0.00
0.00
0.06
0.00
0.00
0.00
0.23
0.00
0.00
0.00
0.00
Australia
1.51
0.00
83.39
0.31
0.07
0.00
0.27
0.00
0.00
0.17
0.04
0.07
0.08
0.00
0.00 0.06 0.12 0.22
Austria
0.12
0.00
0.01
50.12
0.04
0.00
0.02
0.00
0.57
0.13
0.01
0.02
0.13
0.00
0.00 0.08
Belgium
0.74
0.00
0.03
0.80
74.12
0.00
0.04
0.00
1.18
0.26
0.06
0.97
0.21
0.01
0.00 0.14
Brazil
0.61
0.18
0.03
0.05
0.06
99.16
0.09
0.05
0.00
0.13
0.00
0.05
0.02
0.00
0.00 0.23
Canada
2.53
0.00
0.18
0.32
0.09
0.00
75.16
0.01
0.01
0.21
0.06
0.14
0.08
0.00
0.23 0.12
Chile
0.21
0.02
0.00
0.00
0.00
0.00
0.00
97.29
0.00
0.01
0.00
0.00
0.00
0.00
0.00 0.01
China P.R.
2.00
0.00
0.01
0.04
0.01
0.00
0.01
0.00
0.00
0.31
0.00
0.03
0.01
0.00
1.15 0.02
Czech Republic
0.05
0.00
0.00
0.17
0.03
0.00
0.00
0.00
94.53
0.03
0.00
0.00
0.01
0.00
0.00 0.01
Denmark
0.32
0.00
0.05
0.09
0.09
0.00
0.08
0.00
0.00
62.55
0.41
0.06
0.06
0.00
0.00 0.04
Egypt
0.10
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00 0.00
Finland
0.65
0.09
0.08
0.92
0.63
0.00
0.20
0.00
0.06
1.01
79.46
0.66
1.24
0.02
0.01 0.48
France
3.00
0.05
0.65
3.40
7.17
0.03
1.03
0.01
0.39
2.49
1.65
73.87
4.37
0.11
0.09 2.65
Germany
3.44
0.07
0.42
11.58
1.76
0.02
0.59
0.04
0.75
1.54
1.00
3.01
73.56
0.09
0.06 2.19
Greece
0.30
0.00
0.01
0.08
0.03
0.00
0.01
0.00
0.00
0.03
0.01
0.01
0.02
98.71
0.00 0.03
Hong Kong
1.93
0.00
0.19
0.14
0.12
0.00
0.36
0.00
0.00
0.19
0.14
0.12
0.08
0.00
90.13
0.16
Hungary
1.02
0.00
0.00
0.81
0.01
0.00
0.00
0.00
0.11
0.06
0.00
0.01
0.01
0.01
0.00
0.02
India
0.47
0.00
0.02
0.01
0.01
0.00
0.02
0.00
0.00
0.06
0.00
0.01
0.02
0.00
0.00
0.04
Indonesia
0.11
0.00
0.00
0.02
0.01
0.00
0.01
0.07
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.03
Italy
1.99
0.15
0.20
0.86
0.96
0.01
0.35
0.00
0.16
1.04
0.43
1.67
1.36
0.02
0.01 77.53
Japan
8.61
0.00
1.06
1.81
0.68
0.00
1.68
0.00
0.00
1.93
0.72
1.08
0.76
0.01
0.44
1.63
Korea. Republic
0.82
0.00
0.05
0.15
0.05
0.00
0.25
0.00
0.00
0.49
0.04
0.09
0.11
0.01
0.31
0.21 0.03
Malaysia
0.27
0.00
0.01
0.03
0.01
0.00
0.01
0.00
0.00
0.07
0.00
0.02
0.01
0.00
0.14
Mexico
0.46
0.06
0.02
0.07
0.02
0.00
0.18
0.04
0.00
0.23
0.00
0.05
0.05
0.00
0.00
0.06
Netherlands
2.77
0.11
0.47
3.38
3.61
0.04
0.68
0.14
0.54
1.71
1.43
3.66
3.12
0.05
0.04
1.97
New Zealand
0.08
0.00
0.01
0.01
0.00
0.00
0.03
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.07
Norway
0.27
0.00
0.01
0.10
0.07
0.00
0.05
0.00
0.00
0.33
0.23
0.05
0.04
0.00
0.00
0.01
Pakistan
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
Philippines
0.08
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
Continued
49
50
Table 3.1 Continued Panel B: Second 15 countries Country
% WMP Argentina Austria Australia Belgium Brazil Canada Chile
Czech Denmark Finland France Germany Greece Hong Italy Kong Republic
Poland
0.11
0.00
0.00
0.20
0.02
0.00
0.00
0.00
0.05
0.01
0.00
0.00
0.01
0.00
0.00
0.02
Portugal
0.18
0.00
0.01
0.10
0.09
0.07
0.03
0.00
0.01
0.05
0.02
0.07
0.07
0.01
0.00
0.15
Russian Federation
0.41
0.00
0.01
0.24
0.03
0.00
0.02
0.00
0.04
0.10
0.09
0.02
0.08
0.01
0.00
0.05
Singapore
0.43
0.00
0.06
0.15
0.03
0.00
0.15
0.00
0.00
0.09
0.00
0.04
0.04
0.00
0.28
0.07
South Africa
0.66
0.00
0.05
0.06
0.05
0.00
0.06
0.00
0.00
0.06
0.00
0.04
0.04
0.00
0.00
0.04
Spain
1.85
1.08
0.14
0.84
1.11
0.20
0.34
0.00
0.04
0.73
0.29
1.93
1.23
0.02
0.01
0.74
Sweden
0.82
0.01
0.09
0.36
0.21
0.00
0.19
0.08
0.02
3.32
5.28
0.19
0.29
0.00
0.01
0.16
Switzerland
2.29
0.00
0.37
3.03
1.11
0.02
0.64
0.01
0.10
2.19
1.01
1.62
2.20
0.05
0.02
1.41
Taiwan
1.10
0.00
0.04
0.06
0.04
0.00
0.07
0.00
0.00
0.17
0.00
0.05
0.03
0.00
0.30
0.08
Thailand
0.16
0.00
0.01
0.06
0.06
0.00
0.02
0.00
0.00
0.07
0.00
0.02
0.03
0.00
0.15
0.03
Turkey
0.16
0.00
0.00
0.03
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.01
0.01
0.01
0.00
0.05
United Kingdom
7.79
0.17
1.57
5.61
2.96
0.06
2.74
0.04
0.30
5.98
3.20
3.95
4.25
0.37
4.73
2.80
49.44
21.32
10.74
13.99
4.61
0.30
14.58
2.20
1.11
11.96
4.43
6.39
6.35
0.46
1.74
6.36
0.02
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
United States Venezuela
Panel A: First 15 countries continued Country
% Japan Korea Malaysia Netherlands New Norway Portugal Singapore South Spain Sweden Switzerland UK WMP Zealand Africa
USA Venezuela
Argentina
0.10
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.04
0.00
0.02
0.02 0.00
0.00
Australia
1.51
0.16
0.00
0.05
0.34
5.28
0.84
0.00
0.77
0.06
0.00
0.25
0.13
0.76 0.28
0.00
Austria
0.12
0.00
0.00
0.00
0.06
0.00
0.09
0.09
0.00
0.00
0.00
0.06
0.13
0.03 0.01
0.00
Belgium
0.74
0.03
0.00
0.00
0.60
0.00
0.62
0.84
0.01
0.00
0.14
0.08
0.14
0.13 0.07
0.00
Brazil
0.61
0.01
0.00
0.00
0.06
0.00
0.05
0.26
0.00
0.00
0.10
0.03
0.04
0.15 0.14
0.00
Canada
2.53
0.14
0.01
0.01
0.27
0.40
0.43
0.02
0.15
0.02
0.01
0.22
0.37
0.21 0.59
0.00
Chile
0.21
0.00
0.00
0.00
0.05
0.00
0.00
0.00
0.00
0.00
0.06
0.00
0.00
0.02 0.01
0.00
China. P.R. 2.00
0.04
0.01
0.01
0.02
0.00
0.27
0.00
0.55
0.00
0.04
0.04
0.02
0.09 0.02
0.00
Czech Republic
0.05
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.01 0.00
0.00
Denmark
0.32
0.02
0.00
0.00
0.13
0.03
1.38
0.00
0.02
0.02
0.01
0.40
0.06
0.14 0.05
0.00 0.00
Egypt
0.10
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00 0.00
Finland
0.65
0.09
0.00
0.00
0.54
0.06
1.05
0.20
0.07
0.02
0.28
1.92
0.47
0.49 0.34
0.00
France
3.00
0.45
0.00
0.01
2.49
0.23
2.86
1.23
0.67
0.15
1.61
2.12
1.90
3.30 0.76
0.00
Germany
3.44
0.28
0.01
0.00
1.93
0.51
1.93
0.91
0.31
0.04
1.27
1.81
3.89
1.74 0.41
0.01 Continued
51
52
Table 3.1 Continued Panel B: Second 15 countries continued Country
% Japan Korea Malaysia Netherlands New Norway Portugal Singapore South Spain Sweden Switzerland UK USA Venezuela WMP Zealand Africa
Greece
0.30
0.01
0.00
0.00
0.04
0.00
0.08
0.00
0.00
0.01
0.01
0.02
0.03
0.05 0.02
0.00
Hong Kong
1.93
0.19
0.06
0.07
1.02
0.16
0.24
0.00
2.82
0.01
0.00
0.23
0.15
0.71 0.20
0.00
Hungary
1.02
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.01
0.03 0.01
0.00
India
0.47
0.00
0.00
0.01
0.12
0.00
0.00
0.03
0.25
0.00
0.05
0.01
0.01
0.10 0.06
0.00
Indonesia
0.11
0.00
0.01
0.02
0.01
0.00
0.00
0.00
0.86
0.00
0.00
0.00
0.01
0.04 0.02
0.00
Italy
1.99
0.16
0.00
0.00
1.00
0.08
0.85
0.76
0.06
0.03
0.83
0.57
0.57
0.91 0.24
0.00
Japan
8.61 90.51
0.10
0.01
1.43
2.23
4.05
0.14
1.56
0.14
0.50
1.97
1.14
2.87 1.28
0.00
Korea. Republic
0.82
0.02 99.31
0.01
0.16
0.09
0.36
0.00
1.07
0.00
0.00
0.17
0.14
0.46 0.24
0.00
Malaysia
0.27
0.02
0.05
99.23
0.05
0.00
0.01
0.00
4.40
0.00
0.00
0.04
0.02
0.09 0.02
0.00
Mexico
0.46
0.00
0.00
0.00
0.06
0.00
0.05
0.01
0.01
0.00
0.03
0.04
0.24
0.23 0.18
0.00
Netherlands 2.77
0.25
0.01
0.00
67.97
0.19
1.81
0.68
0.16
0.04
0.75
1.01
2.43
1.73 0.77
0.00
New Zealand
0.01
0.00
0.00
0.00
68.37
0.02
0.00
0.07
0.00
0.00
0.01
0.00
0.03 0.02
0.00
0.08
Norway
0.27
0.01
0.00
0.00
0.06
0.04
54.34
0.02
0.02
0.00
0.00
0.37
0.13
0.14 0.06
0.00
Pakistan
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00 0.00
0.00
Philippines
0.08
0.01
0.00
0.04
0.00
0.00
0.00
0.00
0.34
0.00
0.00
0.01
0.00
0.01 0.01
0.00
Poland
0.11
0.00
0.00
0.00
0.01
0.00
0.21
0.05
0.00
0.00
0.00
0.06
0.01
0.02 0.01
0.00
Portugal
0.18
0.01
0.00
0.00
0.06
0.00
0.11
87.49
0.01
0.00
0.26
0.03
0.03
0.13
0.02
0.00
Russian Federation
0.41
0.00
0.00
0.00
0.02
0.00
0.03
0.00
0.00
0.00
0.00
0.29
0.09
0.09
0.04
0.00
Singapore
0.43
0.04
0.00
0.38
0.11
0.07
0.20
0.00
74.08
0.01
0.00
0.09
0.04
0.29
0.17
0.00
South Africa
0.66
0.00
0.00
0.01
0.05
0.00
0.02
0.44
0.00
86.32
0.01
0.04
0.06
0.12
0.06
0.00
Spain
1.85
0.14
0.00
0.00
1.10
0.10
0.81
2.56
0.04
0.03
89.85
0.38
0.41
0.84
0.23
0.00
Sweden
0.82
0.07
0.00
0.00
0.34
0.13
2.95
0.06
0.06
0.04
0.05
64.65
0.36
0.66
0.13
0.01
Switzerland
2.29
0.32
0.00
0.00
1.60
0.19
1.81
0.27
0.26
0.10
0.34
1.83
75.75
1.55
0.58
0.00
Taiwan
1.10
0.02
0.00
0.01
0.09
0.00
0.28
0.00
1.03
0.00
0.00
0.12
0.08
0.35
0.14
0.00
Thailand
0.16
0.01
0.01
0.02
0.02
0.00
0.00
0.00
1.45
0.00
0.00
0.01
0.01
0.09
0.02
0.00
Turkey
0.16
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.02
0.00
0.04
0.11
0.01
0.00
United Kingdom
7.79
1.35
0.04
0.03
4.67
4.58
8.27
1.57
2.92
10.75
2.08
6.40
2.45
73.88
2.56
0.00
49.44
5.59
0.36
0.09
13.47
17.27
13.93
2.36
5.96
2.20
1.66
14.68
8.60
7.41
90.18
0.25
0.02
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
United States Venezuela
0.00 99.64
Notes: This table contains the distribution of 30 investing countries’ average investors’ allocations across 43 national markets for 2001 and 2002. The second column contains a country’s average stock market capitalization weight in the world market portfolio. Panel A – WMP: world market portfolio – Argentina, Australia, Austria, Belgium, Brazil, Canada, Chile, Czech Republic, Denmark, Finland, France, Germany, Greece, Hong Kong and Italy. Panel B – WMP: world market portfolio – Japan, Korea, Malaysia, Netherlands, New Zealand, Norway, Portugal Singapore, South Africa, Spain, Sweden, Switzerland, UK, USA, Venezuela.
53
54
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
Table 3.2 Average shares of domestic asset holdings by investors from different blocs American bloc
Asia/pacific bloc
African bloc
European bloc
89.68%
86.43%
86.32%
74.89%
Notes: American bloc: Argentina, Brazil, Canada, Chile, USA and Venezuela; Asia/Pacific: Australia, Korea, Hong Kong, Japan, Malaysia, New Zealand and Singapore; European bloc: Germany, Austria, Belgium, Denmark, Spain, Finland, France, Greece, Italy, Norway, Netherlands, Portugal, Czech Republic, UK, Sweden and Switzerland; African bloc: South Africa only.
Table 3.3 Shares of domestic asset holdings for developed and emerging countries Developed countries
Emerging countries
77.17%
93.24%
Notes: Developed countries: Germany, Australia, Austria, Belgium, Canada, Korea, Denmark, Spain, USA, Finland, France, Greece, Hong Kong, Italy, Japan, New Zealand, Norway, Netherlands, Portugal, UK, Singapore, Sweden, and Switzerland; Emerging countries: Argentina, Venezuela, Brazil, Chile, South Africa, Malaysia and Czech Republic.3
share of country j’s investors’ holdings in the domestic market (Wjj ) to the world market capitalization weight of country j (Wj∗ ): log
Wjj Wj∗
(3.11)
The foreign bias refers to the extent to which domestic investors underweight or overweight foreign markets in their asset holdings. The foreign bias (FBIASij ) is defined as log
Wij Wj∗
(3.12)
The average foreign bias of foreign investors in an investing country j is calculated by averaging FBIASj across all remaining countries. Table 3.4 exhibits the distribution of the domestic bias for domestic investors and the average foreign bias of foreign investors in an investing country, across thirty investing countries. Average values are computed for the sample periods 2001 and 2002. In general, we observe a significant cross-section variation in domestic bias and average foreign bias measures. The domestic bias fluctuates between 0.6 (USA) and 8.5 (Venezuela). The average foreign bias varies
FATHI ABID AND SLAH BAHLOUL
55
Table 3.4 Domestic bias of domestic investors and average foreign bias of foreign investors Country
Domestic bias
Average foreign bias
Argentina
6.71
−4.79
Australia
4.02
−3.82
Austria
6.07
−3.40
Belgium
4.61
−3.62
Brazil
5.09
−4.07
Canada
3.39
−4.05
Chile
6.15
−5.28
Czech Republic
7.59
−3.89
Denmark
5.27
−3.46
Finland
4.81
−2.24
France
3.21
−2.09
Germany
3.07
−2.44
Greece
5.80
−4.43
Hong Kong
3.85
−4.56
Italy
3.66
−3.05
Japan
2.35
−3.64
Korea
4.80
−3.99
Malaysia
6.14
−3.79
New Zealand
6.77
−4.19
Netherlands
3.20
−2.56
Norway
5.32
−3.58
Portugal
6.16
−2.95
Singapore
5.16
−3.82
South Africa
4.89
−4.71
Spain
3.88
−2.79
Sweden
4.37
−2.64
Switzerland
3.51
−2.69
UK
2.25
−1.97
USA
0.60
−2.64
Venezuela
8.50
−4.22
from −1.97 (UK) to −5.28 (Chile). The values of these measures are important relatively to those of Chan et al. (2005). We use aggregate cross-section data for institutional and individual investors. Chan et al. (2005) have used the data for mutual funds that are likely to invest more in foreign markets, so it seems acceptable to reach the end with different results.
56
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
Table 3.5 Domestic bias and average foreign bias Domestic bias
Average foreign bias
American bloc
5.07
−4.18
Asia/Pacific bloc
4.73
−3.97
European bloc
4.55
−2.99
African bloc
4.89
−4.71
Developed countries
4.21
−3.25
Developing countries
6.44
−4.39
The domestic and average foreign biases are calculated for investors from developed and developing countries as well as for investors from the American, European, Asia/Pacific and African blocs separately. Results are summarized in Table 3.5. The table shows that domestic bias and average foreign bias are less important for investors from the European bloc than for those from other regional blocs. Results corroborate once again the fact that European investors invest less in domestic assets. Developed countries also have a less important domestic bias and an average foreign bias compared to developing countries.
3.4 THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES There may be a large number of causes to explain home bias. We select several explanatory variables and regroup them into economic development, capital control, stock market development, information cost, familiarity, investor behavior, investor protection, and others. We calculate descriptive statistics of these variables for the years 2001 and 2002 separately. We find that an important number of these variables remain stable during this period. Consequently, we shall consider the descriptive statistics only for the year 2001, as shown in Table 3.6.
3.4.1 Economic development Chan et al. (2005) have found that the level of economic development of a country has a significant effect on the investment decisions of foreign investors. To study the impact of economic development on the home bias, we set a number of measures of economic development. We distinguish between gross domestic product (GDP) per capita in US dollars (GDPC); the real growth rate of gross domestic product (RGDP); the average of exports
Table 3.6 Summary statistics for the explanatory variables, 2001 Panel A: The first set of variables Country
Economic development GDP per capita US($)
Real GDP growth (%)
Trade volume (% of GDP)
Financial market development Foreign direct investment (% of GDP)
Transaction costs (basis points)
Stock market capitalization (% of GDP)
Capital control
Emerging market dummy
Capital flow Restrictions
Argentina
7,430
−4
17
0.01
70.8
0.12
1
5.8
Australia
18,995
4
35
0.01
51.0
1.02
0
6.1
Austria
23,603
1
77
0.03
45.6
0.13
0
8.1
Belgium
22,120
1
172
0.3
27.5
0.81
0
9.2
2,949
1
23
0.04
58.6
0.37
1
4.2
22,343
1
70
0.04
36.4
1.01
0
8.6
4,314
3
55
0.07
114.3
0.85
1
7.0
924
8
43
0.04
n.a.
0.45
1
2.7
5,593
3
122
0.10
72.9
0.16
1
7.0
29,713
1
61
0.06
41.3
0.53
0
9.0
1,511
4
17
0.01
n.a.
0.25
1
7.3
Finland
23,422
1
62
0.03
42.3
1.57
0
8.1
France
22,308
2
49
0.04
28.2
0.69
0
7.6
Germany
22,511
1
57
0.01
27.3
0.58
0
9.5
Greece
11,062
4
33
0.01
74.4
0.72
0
8.3
Hong Kong
24,213
0
241
0.15
53.2
3.11
0
9.6
5,088
4
124
0.05
103.8
0.20
1
8.8
India
463
5
19
0.01
44.4
0.23
1
2.0
Indonesia
676
3
62
−0.02
83.7
0.16
1
4.8
Brazil Canada Chile China Czech Republic Denmark Egypt
Hungary
57 Continued
Table 3.6 Continued Country
Italy Japan Korea, Republic of Malaysia Mexico Netherlands New Zealand Norway Pakistan Philippines Poland Portugal Russian Fed Singapore South Africa Spain Sweden Switzerland Taiwan Thailand Turkey UK USA Venezuela
58
Panel A: The first set of variables continued Economic development
Financial market development
Capital control
GDP per capita US($)
Real GDP growth (%)
Trade volume (% of GDP)
Foreign direct investment (% of GDP)
Transaction costs (basis points)
Stock market capitalization (% of GDP)
Emerging market dummy
Capital flow Restrictions
18,921 32,869 10,180 3,696 6,262 23,944 13,241 37,620 415 920 4,808 10,835 2,118 20,545 2,549 14,315 24,673 33,998 n.a. 1,888 2,119 24,211 35,118 5,123
2 0 4 0 0 1 3 2 3 3 4 2 5 −2 3 3 1 1 n.a. 2 −7 2 0 3
43 18 68 184 53 114 53 54 33 89 47 58 51 280 50 47 63 68 n.a. 110 50 42 19 36
0.01 0.00 0.01 0.01 0.04 0.13 0.04 0.01 0.01 0.01 0.03 0.05 0.01 0.13 0.06 0.05 0.06 0.04 n.a. 0.03 0.02 0.04 0.02 0.03
40.5 24.4 73.4 90.9 65.9 27.7 39.0 30.0 n.a. 113.2 n.a. 44.5 n.a. 74.0 88.5 39.4 29.3 41.5 59.7 87.4 45.3 46.6 28.5 102.7
0.48 0.54 0.40 1.35 0.20 1.81 0.35 0.41 0.08 0.29 0.14 0.42 0.25 1.36 1.29 0.80 1.08 2.15 n.a. 0.31 0.32 1.50 1.40 0.05
0 0 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1
8.7 8.4 n.a. 3.7 5.1 9.5 8.9 7.8 0.8 4.6 3.8 7.6 3.2 7.6 4.1 6.9 7.2 9.1 7.6 4.3 5.8 9.1 8.2 7.8
Panel B: The second set of variables Country
Capital control
Other variables
Investor protection
Intensity of capital control
Rule of law
Accounting
Minority
Expropriation
Argentina
0.05
5.35
45
4
5.91
6
0
Australia
0.00
10
75
4
9.27
10
Austria
0.00
10
54
2
9.69
9.5
Belgium
0.00
10
61
0
9.63
9.5
Brazil
0.05
Canada
0.00
6.32 10
Efficiency
Legal system dummy
Lag 2-year return
Return correlation (average)
2.5
0.017
1
1.1
0.136
0
−12.0
0.004
0
−16.6
−0.028
54
3
7.62
5.75
0
22.1
0.126
74
4
9.67
9.25
1
19.7
0.202
Chile
0.11
7.02
52
3
7.5
7.25
0
13.4
0.056
China
0.59
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
29.4
−0.037
Czech Republic
0.02
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
1.3
0.162
Denmark
0.00
10
62
3
9.67
0
12.1
0.090 −0.029
Egypt
0.29
Finland
0.00
France
0.00
Germany Greece
4.17
10
24
2
6.3
0
23.1
77
2
9.67
10
0
n.a.
8.98
69
2
9.65
8
0
0.00
9.23
62
1
9.9
9
0
0.00
6.18
55
1
7.12
7
0
n.a.
n.a.
Hong Kong
0.00
8.22
69
4
8.29
10
1
20.8
0.169
Hungary
0.03
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
India
0.59
4.17
57
2
7.75
8
1
20.0
0.144
Indonesia
0.14
3.98
65
2
7.16
2.5
0
14.0
0.056
10
6.5
n.a.
n.a.
5.9
0.051
1.5
0.145
59
Continued
Table 3.6 Continued Country
Italy Japan Korea, Republic of Malaysia Mexico Netherlands New Zealand Norway Pakistan Philippines Poland Portugal Russian Federation Singapore South Africa Spain Sweden Switzerland Taiwan Thailand Turkey UK USA Venezuela
60
Panel B: The second set of variables continued Capital control
Other variables
Investor protection
Intensity of capital control
Rule of law
Accounting
Minority
Expropriation
Efficiency
Legal system dummy
0.00 0.00 0.03 0.06 0.01 0.00 0.00 0.00 1.00 0.54 0.03 0.00 0.26 0.00 0.00 0.00 0.00 0.00 0.44 0.34 0.01 0.00 0.00 1.00
8.33 8.98 5.35 6.78 5.35 10 10 10 3.03 2.73 n.a. 7.8 n.a. 8.57 4.42 7.8 10 10 8.52 6.25 5.18 8.57 10 6.37
62 65 62 76 60 64 70 74 61 65 n.a. 36 n.a. 78 70 64 83 68 65 64 51 78 71 40
0 3 2 3 0 2 4 3 4 4 n.a. 2 n.a. 3 4 2 2 1 3 3 2 4 5 1
9.35 9.67 8.31 7.95 6.07 9.98 9.69 9.88 5.62 5.22 n.a. 8.9 n.a. 9.3 6.88 9.52 9.4 9.98 9.12 7.42 7 9.71 9.98 6.89
6.75 10 6 9 6 10 10 10 5 4.75 n.a. 5.5 n.a. 10 6 6.25 10 10 6.75 3.25 4 10 10 6.5
0 0 0 1 0 0 1 0 1 0 n.a. 0 n.a. 1 1 0 0 0 0 1 0 1 1 0
Lag 2-year return
−0.2 4.3 19.4 29.2 28.4 −2.9 n.a. −3.1 26.3 −23.9 n.a. −12.7 129.6 18.7 0.0 −13.3 18.5 −0.1 −9.4 −9.8 56.9 −4.2 4.8 43.8
Return correlation (average) 0.140 0.155 0.098 0.061 0.110 0.022 n.a. −0.062 0.084 0.066 n.a. 0.095 −0.003 0.153 n.a. 0.134 0.218 0.071 0.140 0.060 0.118 0.124 0.142 −0.002
Panel C: The third set of variables Country
Information costs
Investor behavior
Phone costs (By minute in $)
Investor behavior toward foreign market returns (basis points)
Distance (kilometers) (average)
Common language dummy (average)
Argentina
0.5743
−12
12,274
0.12
Australia
0.3881
−8
12,726
0.31
Austria
0.3323
−13
5,843
0.05
Belgium
0.3696
−28
5,970
0.12
Brazil
0.5861
41
11,417
0.02
Canada
0.2559
12
8,745
0.38
Chile
0.7472
−12
12,595
0.12
Czech Republic
0.6327
26
n.a.
0.00
Denmark
0.2706
−7
5,879
0.00
Finland
0.4442
N.A.
5,967
0.00
France
0.2997
−19
6,049
0.07
Germany
0.2417
−5
5 916
0.05
Greece
0.4996
N.A.
6 109
0.00
Hong Kong
0.2935
24
8 403
0.36
Italy
0.2991
Japan
0.5163
Korea, Republic of
0.3471
−2
Familiarity
6,044
0.00
12
8,809
0.05
20
8,324
0.05 Continued
61
62
Table 3.6 Continued Panel C: The third set of variables continued Country
Information costs
Investor behavior
Phone costs (by minute in $)
Investor behavior toward foreign market returns (basis points)
Familiarity Distance (kilometers) (average)
Common language dummy (average) 0.31
Malaysia
0.8926
5
8 846
Netherlands
0.3287
−19
5 956
0.05
New Zealand
0.2928
n.a.
13 996
0.31
Norway
0.3576
−25
5 997
0.00
Portugal
0.4850
−11
6 776
0.02
Singapore
0.3567
17
9 013
0.38
South Africa
0.3448
n.a.
9 536
0.31
Spain
0.3565
6 513
0.12
Sweden
0.2849
12
5 931
0.00
Switzerland
0.2556
−13
6 016
0.12
UK
0.3002
−9
6 093
0.31
USA
0.1560
−3
9 217
0.31
Venezuela
0.8008
−24
9 501
0.10
0.2
This table presents, summary statistics for each country, for eight groups of explanatory variables: (i) Economic development variables: GDP per capita, real GDP growth, trade volume (% of GDP) and foreign direct investment; (ii) Financial market development variables: transaction costs, stock market capitalization (% of GDP) and emerging market dummy variable; (iii) Capital control variables: capital flow restrictions and stock-holding restrictions; (iv) Investor protection variables: rule of law index, accounting standard index, minority investor protection index, risk of expropriation index and efficiency of judicial system index; (v) Other variables: past 2-year return and average return correlation; (vi) Information costs: average phone costs by minute; (vii) Investor behavior: average degree of pessimism toward foreign market return; (viii) Familiarity: average distance in kilometers and average common language dummy variable.4
FATHI ABID AND SLAH BAHLOUL
63
and imports scaled by GDP (TRADE); and foreign direct stock investment inward scaled by GDP (DI). All these variables are obtained from the World Development Indicators (WDI). Table 3.6 shows significant cross-sectional variation in the four measures of economic development. The country that has the highest value of GDPC is a developed country (Norway: US$37,620), while the most important value of RGDP is held by an emerging country (China: 8 percent). Belgium has the highest foreign direct stock investment inward relative to its GDP (DI), while the country that has the largest trade volume, as a percentage of GDP (TRADE) is Singapore.
3.4.2 Capital control Ahearne et al. (2004) suggest that, while capital controls have been greatly reduced in many countries, they still affect cross-border investment. In this chapter we measure capital control according to two parameters. The first is the intensity of capital control (RESTRICT) developed in Edison and Warnock (2003) and used by Ahearne et al. (2004). This measure is constructed by using International Finance Corporation (IFC) indexes. It equals one minus the ratio of the market capitalization of a country’s Investable (IFCI) and Global (IFCG) indexes.5 Restrictions vary greatly across developing countries. For industrial countries, the IFC does not publish investable indexes. We assume that, for these countries, Investable and Global indexes are identical. Table 3.6 shows that RESTRICT ranges between zero (in developed countries) to one (in Pakistan). The second variable (RFLOW) measures the restrictions of countries on capital flows. It is constructed by the Economic Freedom Network by assigning lower ratings to countries with more restrictions on foreign capital transactions and was used by Chan et al. (2005). Table 3.6 shows that RFLOW varies from 0.8 in Pakistan to 9.6 in Hong Kong.6 When a country imposes capital controls, this will stop, or at least discourage, foreign investors from holding stocks of companies in that country. When a country imposes higher capital control measures, the degree of foreign bias becomes higher (more negative FBIAS). Also, when a score on RFLOW is low (and the score on RESTRICT is important), domestic investors find it difficult to invest overseas, as it requires government approval. Then they will invest a large amount of their wealth in the domestic market and the domestic bias will consequently be important (more positive DBIAS).
3.4.3 Stock market development Chan et al. (2005) suggest that investors tend in general to invest more in developed stock markets. In fact, these markets present high liquidity and lower transaction costs. We measure the stock market development according to three distinct variables.
64
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
The first variable is the relative size of the stock market of each country, measured by the stock market capitalization as a percentage of the country’s GDP (SIZE). The value of SIZE varies from 0.05 percent in Venezuela to 3.11 percent in Hong Kong. The data on GDP are taken from the World Development Indicators (WDI) and data on market capitalization are from the International Federation of Stock Exchanges (FIBV). The second variable is the transaction costs associated with trading foreign securities (COST). The role of transaction costs in the explanation of the home bias has been neglected because of the existence of an important turnover rate of foreign assets compared to domestic assets (Tesar and Werner, 1995). However, Carmichael and Coen (2003) reveal, by using a simple OLG model of the world economy with transaction costs, that the introduction of very small transaction costs is sufficient to reproduce the large home bias observed in portfolios. We use the Elkins–McSherry Co. measure of transaction costs. The latter consists of three components: commissions, fees and market impact costs. This measure has been used by Ahearne et al. (2004) and Chan et al. (2005). Although the other explanatory variables are for the year 2001, we use the data for the year 1999.7 We assume that these transaction cost estimates do not change substantially, and we can utilize it in our analysis. We have transaction cost data for thirty-eight of the forty-three markets, ranging from 24.4 basis points for Japan to 114.3 for Chile. We also consider a dummy variable (DUMEMERG) that equals one for an emerging market, and zero otherwise. We expect that foreign investors will opt for investing in local markets which are large, developed and require low transaction costs. Foreign bias will be less important in these markets. The proportion of local asset holdings by domestic investors will be smaller and the domestic bias less important.
3.4.4 Information asymmetries Gehrig (1993) argues that one of the explanations for the home bias is that local investors spend too much on information about foreign markets. A high information cost discourages investors from investing abroad. Zhou (1998) shows that, with differential information, agents on average tilt their portfolio towards stocks about which they have better information. Portes and Rey (2005) use the volume of telephone call traffic as a proxy for information costs. To measure information costs, we shall consider the cost of international phone calls per minute from a country i to a country j.8 Table 3.6 shows that average values of information costs vary from US$0.1560 (USA) to US$0.8008 (Venezuela). High information costs between country i and country j make country i’s investors hold fewer assets in country j, so we expect an important FBIASij (more negative FBIAS).
FATHI ABID AND SLAH BAHLOUL
65
3.4.5 Investors’ behavior One explanation of the home bias is that investors in each country expect returns from their domestic equity market to be several hundred basis points higher than returns from foreign markets. French and Poterba (1991) show that investors may be relatively more optimistic about their home markets than are foreign investors. Strong and Xu (2003) find that fund mangers from the USA, UK, continental Europe and Japan show a significant relative optimism towards their equity markets. To measure an investor’s degree of optimism or pessimism towards a market, we use BEHAV, a variable that equals the difference between the market return implied by the actual portfolio holdings and the returns implied by an international value-weighted portfolio for each country. To determine this vector of return, we have used French and Poterba’s (1991) model: µ = λw∗
(3.13)
where µ is the vector of expected return; w∗ is optimal portfolio weights; is the covariance matrix; and λ is relative risk aversion. Average values of BEHAV vary from –28 basis points for Belgian investors to 41 basis points for Brazilian investors. In general, investors are pessimistic about foreign markets, which can help to explain why investors underweight foreign markets and overweight domestic markets.
3.4.6 Familiarity One explanation for the home bias is that investors may not be familiar with foreign markets. Huberman (2001) finds that “Shareholders of a Regional Bell Operating Company (RBOC) tend to live in the area which it serves, and an RBOC’s customers tend to hold its shares rather than other RBOCs’ equity”. Like Chan et al. (2005) and Sarkissan and Schill (2004), we use three proxies of familiarity9 variables for each pair of countries i and j. The first familiarity variable is common language. Data are obtained from the World Factbook 2001. For each pair of countries i (investing) and j, we construct a language dummy variable (DUMLANG) which equals one if i and j share the same language and zero otherwise. The second variable is geographical proximity (DISTANCE). Data are obtained from http://www.ksg.harvard.edu/people/sjwei. Average values of distance vary from 5,843 kilometers for Austria to 13,996 for New Zealand. The last variable is the amount of bilateral trades (TRADEB), with values ranging from 0 to 1. A value of 0.17 for TRADEB between the UK (investing) and the USA means that 17 percent of the total UK trade (imports
66
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
and exports) is with the USA. (Data are obtained from the United Nations Statistics Division Databases.) For the familiarity variables, we expect that investors from country i, who are more familiar with country j, through sharing a common language, being close to each other, or having larger bilateral trade volumes, will display less foreign bias (FBIASij ). Familiarity variables do affect domestic bias. Investors from a country that is isolated from the rest of the world will also hold a large proportion of domestic assets.
3.4.7 Investor protection La Porta et al. (1997) argue that capital markets are narrow in countries with poorer investor protection. Giannetti and Koskinen (2004) show that foreign investors are reluctant to invest in a country where expropriating minority shareholders is easy, while wealthy investors have an incentive to become controlling shareholders by investing a large proportion of their wealth in the stock market in a country with poor investor protection. Similarly to Chan et al. (2005), we use six measures of investor protection based on La Porta et al. (1997, 1998, 2000).10 The first measure is the rule of law index (LAW), elaborated by the International Country Risk Agency. It arranges countries on a scale ranging from zero to 10, with lower scores for countries with less respect for law and order. The index varies from 2.73 in the Philippines to 10 in twelve countries across the world. The second measure is the accounting standard index (ACC). This defines the amount and transparency of information available to investors. Table 3.6 shows that Sweden has the highest score (83) while the lowest is for Egypt (24). The third measure is the anti-director rights (MINORITY). It indicates the degree of protection for minority investors. Values vary from zero for Italy, Belgium and Mexico, to 5 for the USA. The fourth measure is the risk of expropriation index (EXPROP). This index is constructed by the International Country Risk Agency. It has a scale ranging from zero to 10, with lower scores for higher risk. This index varies from 5.22 for the Philippines to 9.98 for the USA, Switzerland and the Netherlands. The fifth measure is the efficiency of the judicial system (EFFICIENCY). This index is constructed by the Business International Corporation. Values range from 2.5 for Indonesia to 10 for fourteen developed countries. The sixth measure is a dummy variable that considers the type of legal system (DUMLEGAL). It equals 1 for common-law countries and zero otherwise. In fact, La Porta et al. (1997) have found that the French civil law countries have the weakest investor protection, particularly when compared to common-law countries.
FATHI ABID AND SLAH BAHLOUL
67
3.4.8 Other variables In addition to the above variables, we include several others that account for the home bias. The first variable is the two-year return (RET2). Bohn and Tesar (1996) find that US investors exhibit returns-chasing behavior, with a tendency to underweight countries whose stock markets have performed poorly. The second variable is the correlation between the returns of two countries (CORR). For each pair of countries, i and j, we compute the correlation coefficient using country returns in US dollars from 1999 to 2001. Data for indexes are from Yahoo Finance, and data for exchange rates are from the Oanda website. The Correlation coefficient is used as a proxy for the diversification potential between two countries.
3.5 THE EMPIRICAL ANALYSIS Following Chan et al.’s (2005) methodology, this section studies the causes of domestic and foreign biases. In all tests, we stack up all the observations on domestic-bias measure (DBIASj ) and regress them against each set of the explanatory variables; then we do the same for the foreign bias measures (FBIASij ). The dependent variables (DBIAS and FBIAS) are average values for the 2001–02 period, while, explanatory variables are for the year 2001.
3.5.1 Results concerning domestic bias The second column of Table 3.7 shows the predicted signs of the coefficients. The other columns contain estimates of explanatory variables for the eight categories separately and for all variables. Table 3.7 shows that stock market development variables have the most important explanatory power, with adjusted R2 of 58 percent, while familiarity variables exhibit a low adjusted R2 of 17 percent. For the market development variables, we find that DBIAS is negatively related to the size of the market (SIZE) and positively linked to transaction costs (COST). Large markets have a higher visibility and attract foreign investors, so domestic investors will invest less in large domestic markets. If transaction costs are very high, foreign investors will tend to invest less in local markets, and investments in domestic market by local investors will then be important. Information costs have a significant positive effect on DBIAS; the coefficient is 6.35 with a t-ratio of 4.67. If information costs in foreign market are very important, local investors opt for local markets, and then the domestic bias will be very important.
68
Table 3.7 Regression analysis of domestic bias Predicted sign
Economic development variables Coeff.
Constant
4.00
GDPC RGDP TRADE DI
− − − −
RFLOW RESTRICT
− +
SIZE COST DUMEMERG
− + +
INFOR
+
BEHAV
+
DUMLANG DIST
− +
LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL
− − − − − −
RET2 CORR
− +
Adjusted R2
−0.001 0.03 0.00 −0.97
Capital control
Stock market development
Information costs
Investor behavior
Familiarity
Investor protection
Other variables
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
8.43
7.10
5.16
−5.34
−1.82
2.10
3.47
3.82
9.03
−21.03
−2.10
15.87
6.30
5.36
10.51
28.71
1.83
0.00 −0.01 0.01 −4.63
−0.05 −0.06 1.12 −0.94
0.01 1.14
0.19 0.51
−0.77 2.40 −2.00
−1.97 2.50 −0.87
4.79
1.61
2.72
1.35
2.79 −3.27
0.52 −2.37
1.55 0.01 0.29 −1.94 −0.35 −1.62
2.37 0.23 1.09 −3.29 −0.80 −1.16
−0.02 1.14
−0.56 0.21
−4.15 0.16 0.71 −0.19 −0.34 4.17
−1.97 2.77 −1.20 2.94 −0.70
−3.62 3.55 −0.85 6.35
4.67 4.19
2.61 −5.14 2.94
−2.17 2.60 0.57 −0.03 −0.32 −1.48 −0.01 0.36
1.21 −0.89 −1.19 −2.74 −0.03 0.42 0.05 −11.55
0.35
All variables
0.26
0.58
0.42
0.19
0.17
0.38
0.23
2.17 −2.65
t-stat
0.86
Notes: DBIAS j : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flow restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: average common language dummy variables; DIST : average of log geographical distances; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: average of return correlations. * Bold numbers, indicate t-stat is significant.
FATHI ABID AND SLAH BAHLOUL
69
Concerning the investor-protection variables, only the index of risk expropriation is statistically significant at the 5 percent level with a coefficient reaching –1.48. Expropriation risk will have less of an impact on the decisions of local investors to invest in local markets than on foreign investors. As a result, when the expropriation risk is small (lager EXPROP), this will attract relatively more foreign investments to local markets. In this case, the domestic bias will be lower. Furthermore, the result of economic development variables shows that only GDP per capita (GDPC) has a significant negative coefficient. GDP per capita will reflect the development of a country and its financial markets. If GDPC is positive, a country will attract more foreign investors and domestic bias will consequently be less important. The two control variables (RFLOW) and (RESTRICT) are significant at the 5 percent level. The coefficient of RFLOW is positive. In fact, the lower the value of RFLOW, the more important are the restrictions facing the domestic investors in foreign markets and foreign investors in local markets. Then domestic investors will invest more in domestic markets and DBIAS will be important. The relation between DBIAS and RESTRICT is positive and this implies that if the restrictions on foreign investment are important, local investors will invest more in the domestic market. Investors’ behavior also has an impact on domestic bias. Its coefficient is 4.19 and the t-ratio 2.61. The more optimistic investors are about the local market, the more they invest in it; hence the importance of DBIAS. Common language (DUMLAN) and geographical proximity (DIST), which are proxies for familiarity, both have a significant impact on domestic bias. Countries that have the same language as a large number of countries in the world tend to have a smaller domestic bias. Countries that are farther away from the rest of the world have an important domestic bias. When, all explanatory variables are estimated jointly, results show significant coefficients generally for financial market development and for two variables of investor protection (LAW and EXPROP). Other variables, such as a common language, are not significant. Our findings corroborate those of Chan et al. (2005) especially regarding the importance of stock market development in the explanation of the domestic bias.
3.5.2 Results concerning foreign bias Table 3.8 shows regression results for the foreign bias measure, FBIASij . We introduce an additional independent variable, DBIASi , which controls the impact of domestic bias on foreign bias. If investors invest more in a domestic market, the proportion they could invest in other markets will be considerably lower. Then, if the domestic bias is important (DBIASi
Predicted sign
Economic development variables
Capital control
70
Table 3.8 Regression analysis of foreign bias Stock market development
Information costs
Investor behavior
Familiarity
Investor protection
Other variables
All variables
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff. t-stat
Coeff.
t-stat
−2.31 −0.51
−6.72 −12.2
1.01 −0.57
1.03 −13.0
−1.00 −0.18
−5.03 −4.06
−1.27 −0.52
−5.47 −10.9
8.46 −0.43
13.86 −10.6
−5.89 −0.55
−8.58 −12.8
−1.21 −5.05 −0.50 −10.7
15.32 −0.08
6.09 −1.48
0.00 0.05 0.00 −4.42
1.02 0.91 1.58 −2.12
−0.25 1.21
−2.57 2.21
0.30 −0.80 0.43
1.71 −1.87 0.77
Constant DBIAS
−
−2.15 −0.52
−8.36 −12.5
GDPC RGDP TRADE DI
+ + + +
0.001 −0.02 0.00 −0.99
9.12 −0.77 0.16 −0.62
RFLOW RESTRICT
+ −
SIZE COST DUMEMERG
+ − −
INFOR
−
BEHAV
+
DUMLANG TRADEB DIST
+ + −
LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL
+ + + + + +
RET2 CORR Adjusted R2
+ −
0.16 −1.31
4.23 −4.04 0.06 −0.42 −0.88
0.44 −1.55 −3.24 −4.86
−14.84 −2.83
−1.88 0.54 3.71 −1.21
2.89 2.80 −17.50
0.15
0.18
0.24
0.11
0.35
−9.96
−9.09
−4.78
0.86 4.26 −0.75 −0.54 −1.22 −15.98 0.04 0.03 −0.09 0.44 −0.11 −0.38
0.17
−4.92
0.20
0.40 4.03 −1.21 3.24 −1.91 −1.75 −0.02 −5.11 0.85 2.42 0.14
0.00 −0.01 −0.12 0.01 −0.21 0.59
−0.02 −0.80 −1.48 0.06 −3.13 2.21
−0.01 1.19 0.53
−1.49 2.70
Notes: FBIAS ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET 2: past 2-year return; CORR: return correlations between two countries; DBIAS i : control variable.
FATHI ABID AND SLAH BAHLOUL
71
important), foreign bias will be important (more negative FBIASij ). The coefficients of DBIASi are negative and significant for almost all specification models. Among the eight different categories of explanatory variables, familiarity has the greatest influence on foreign bias. All familiarity variables are statistically significant at the 5 percent level. Foreign investors tend to invest in a country that is geographically close to them, that enjoys a large bilateral trade volume, and with which they share a language. The more familiar investors are with a foreign market, the more they invest abroad and the less the foreign bias (FBIAS important). Moreover, investors are less willing to invest in a foreign market that require a high information cost. The coefficient of INFOR is –4.86 and the t-ratio equals –14.84. Results concerning the investor protection variables show that the two variables ACC and EXPROP have a significant effect on foreign asset holdings. Foreign investors prefer markets with more transparency of information (ACC important) and a low expropriation risk. The more investors’ rights are maintained, the smaller the foreign bias. Economic market development (DUMEMERG) and financial market development (GDPC) have a significant impact on foreign bias. Foreign investors tend to invest more in a developed market (DUMEMERG = 0) and in a country with an important GDP per capita. In other words, foreign investors prefer large and developed markets with high levels of transparency. Capital control variables have a strong impact on foreign bias. The coefficient on RFLOW and RESTRICT are, respectively, 0.16 with a t-ratio of 4.23 and –1.31 with a t-ratio of –4.04. This result suggests that a country with fewer restrictions on capital flows (RFLOW important) or/and on foreign asset holding (RESTRICT low) attract more foreign investors, then foreign bias in this country will be less important (FBIAS important). When we regress the foreign bias on all variables, some of the coefficients are no longer statistically significant at the conventional level. Information costs and familiarity variables (except TRADEB) remain statistically significant; yet investor protection variables are insignificant. These results corroborate those found by Chan et al. (2005), in particular regarding the importance effect of familiarity on the foreign bias.
3.6 ADDITIONAL TESTS 3.6.1 Results concerning home bias In addition to domestic and foreign bias measures, we have used another measure of home bias developed by Ahearne et al. (2004) to confirm the preceding results.
72
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
The home bias of country i’s investors against country j (HBIASij ) is share of country j in the portfolio of country i’s investors HBIASij = 1 − (3.14) share of country j in the world market portfolio Regression results of the home bias on eight categories of variables are presented in Table 3.9. Table 3.9 shows that familiarity variables are the most important determinants of home bias. In fact, they present the most important adjusted R2 (0.13). Coefficients are −0.21 for common language (DUMLANG), −1.63 for bilateral trade volume (TRADEB) and 0.31 for geographic proximity (DIST), with t-ratios, respectively, of −2.64, −2.84 and 10.52. Familiarity variables seem to contribute similarly in explaining the foreign bias. Among the remaining factors, only four variables are statistically significant at the conventional level. The first variable is information costs, with a coefficient of 0.52 and a t-ratio of 4.49. Then, if foreign investors are less informed about the local market, they will invest less and the home bias will be very important. The second variable is related to investor protection (ACC). Foreign investors prefer markets with more information transparency (ACC important). The third variable is the restriction on asset holdings by foreign investors (RESTRICT). If foreign investors have this constraint, they will invest less abroad, and home bias will be very important. The last variable is trade volume scaled by GDP (TRADE). Its coefficient is −0.02, with a t-ratio of −2.84. Foreign investors tend to hold assets in a country with very important trade volume scaled by GDP. When all the explanatory variables are estimated jointly, apart from correlation coefficient (CORR), only familiarity variables present a statistically significant coefficient that confirms the hypothesis. Results corroborate those of Ahearne et al. (2004), especially for the impact of information costs on home bias.
3.6.2 Domestic, foreign and home biases according to a world float portfolio Dahlquist et al. (2003) show that the prevalence of closely held firms in most countries helps to explain why these countries exhibit a home bias in equity holdings. Based on their estimates of the percentage of closely held market capitalization, we construct a world float portfolio with country weights based on the free-floating shares available to investors. We calculate the float adjusted domestic bias (DBIAS_FLOAT), foreign bias (FBIAS_FLOAT) and home bias (HBIAS_FLOAT).
Table 3.9 Regression analysis of home bias Predicted sign
Economic development variables Coeff.
Constant GDPC RGDP TRADE DI
− − − −
RFLOW RESTRICT
− +
SIZE COST DUMEMERG
− + +
INFOR
+
BEHAV
−
DUMLANG TRADEB DIST
− − +
LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL
− − − − − −
RET2 CORR
− +
Adjusted R2
Capital control
Stock market development
t-stat
Coeff.
t-stat
Coeff.
t-stat
0.90
13.28
0.71
6.39
0.95
2.28
−0.00 −0.00 −0.02 1.18
−1.75 −0.04 −2.84 1.83 0.00 0.26
Information costs
Investor behavior
Familiarity
Investor protection
Other variables
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
0.55
9.92
0.75
22.50
−1.86
−7.11
1.24
4.65
0.72
18.35
−2.03
−1.38
0.00 −0.01 0.01 1.91
0.93 −0.32 −1.41 1.57
0.09 0.10
1.61 0.31
−0.08 −0.07 −0.05
−0.82 −0.30 −0.17
0.09 2.04 −0.03 −0.06 0.10
−0.55 −0.51 0.81 0.52
4.49 −0.65
−1.06 −0.21 −1.63 0.31
−2.64 −2.84 10.52 −0.02 −0.01 0.04 0.00 0.01 0.03
−0.40 −2.70 1.24 0.04 0.22 0.38 0.00 −1.01
0.01
0.01
0.01
All variables
0.04
0.01
0.13
0.01
0.00
1.46 −0.10
0.25
1.15
−1.92
−1.73
−0.30 −2.64 0.34
−2.52 −3.24 7.65
−0.11 0.00 0.00 0.04 0.00 0.11
−1.58 −0.40 0.08 0.38 −0.03 0.71
0.00 0.61
−0.29 2.39
0.15
73
Notes: HBIAS ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index: ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET 2: past 2-year return; CORR: return correlations between two countries.
74
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
Results concerning the float-adjusted domestic bias
Table 3.10 presents the result of the DBIAS_FLOAT estimation. The results of the float-adjusted domestic bias are almost the same as those of the unadjusted domestic bias. Variables that have a significant impact on domestic bias also have a significant effect on DBIAS_FLOAT. Moreover, the predictive power is generally more important for DBIAS_FLOAT than for DBIAS. The results confirm those of Dahlquist et al. (2003) for the impact of corporate governance structure on the home asset bias.
Results concerning the float-adjusted foreign bias
Table 3.11 shows the result of the FBIAS_FLOAT estimation for the eight categories of explanatory variables. The foreign bias calculated on the basis of the world float portfolio seems to be influenced by the same characteristics as the foreign bias. Except bilateral trade volume (TRADEB), all significant variables for the foreign bias are statistically significant at the 5 percent level for the foreign bias float.
Results for the float-adjusted home bias
Results found for the float-adjusted home bias are qualitatively the same as those of the home bias. Information costs and familiarity variables remain statistically significant for the home bias float. See Table 3.12.
3.7 CONCLUSION This chapter presented an analytical study of the bilateral asset holdings of investors from thirty investing countries in forty-three receiving countries. Similarly to Chan et al. (2005), we distinguish between domestic bias (domestic investors overweighting the local markets) and foreign bias (foreign investors under-or overweighting the overseas markets). We find that home bias is a large phenomenon for both developed and developing nations. The results show that stock market development and information costs have an important impact on domestic bias, while information costs and familiarity variables have an important effect on foreign bias. Economic development, capital control and investor protection variables have only a small effect on theses biases. However, investor behavior has a significant impact only on the domestic bias. Additional tests show that, only information costs and familiarity variables have an important effect on home bias. Investment behavior appears to be determined by multiple-factor models.
Table 3.10 Regression analysis of domestic bias based on the world float portfolio Economic development variables Coeff. Constant GDPC RGDP TRADE DI
Capital control
Stock market development
Information costs
Investor behavior
Familiarity
Other variables
All variables
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
7.09
8.34
7.91
4.84
−5.68
−1.68
2.06
2.91
3.99
8.47
−26.95
−2.38
18.15
6.35
5.73
9.72
25.75
2.08
0.01 0.05 0.01 −0.86
−4.49 0.28 0.76 −0.15
0.00 0.04 0.02 −7.14
−0.15 0.23 2.16 −1.18
0.15 1.00
0.60 0.57
−1.04 2.29 −1.33
−3.32 3.03 −0.73
−0.39 4.56
RFLOW RESTRICT
−1.98 2.54 −1.37 3.15 −0.46
SIZE COST DUMEMERG
−3.59 3.30 −0.49 7.41
INFOR
4.67 5.12
BEHAV
2.86 −5.76 3.65
DUMLANG DIST
−2.15 2.85 0.62 −0.04 −0.39 −1.68 −0.01 0.60
LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL RET2 CORR Adjusted R2
Investor protection
0.35
0.24
0.59
0.42
0.22
0.20
0.40
1.16 −0.95 −1.28 −2.74 −0.03 0.63 0.05 −12.34 0.22
2.15 −2.45
t-stat
4.95
2.11
3.80
2.39
5.03 −3.13
1.20 −2.29
1.71 0.02 0.34 −2.02 −0.40 −2.40
3.34 0.46 1.64 −4.36 −1.17 −2.18
−0.03 2.09 0.93
−0.99 0.49
75
Notes: DBIAS_FLOAT j : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: average common language dummy variables; DIST : average of log geographical distances; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: average of return correlations.
Table 3.11 Regression analysis of foreign bias based on the world float portfolio 76
Economic development variables
Capital control
Stock market development
Information costs
Investor behavior
Familiarity
Investor protection
Other variables
All variables
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Constant DBIASFLOAT
−1.76 −0.43
−7.72 −13.6
−1.78 −0.42
−5.61 −13.5
0.46 −0.46
0.48 −14.0
−0.59 −0.22
−3.49 −6.6
−1.19 −0.43
−6.23 −11.8
8.12 −0.35
13.74 −11.6
−4.61 −0.44
−6.90 −13.8
−1.13 −0.42
−5.68 −11.7
13.34 −0.09
5.33 −2.1
GDPC RGDP TRADE DI
0.00 −0.03 0.00 −0.74
6.02 −0.89 0.47 −0.47
0.00 0.04 0.01 −4.45
0.54 0.76 2.66 −2.15
−0.20 1.29
−2.11 2.37
0.17 −0.76 0.93
0.95 −1.79 1.68
−4.78
−10.23
−9.16
−4.84
0.88 −0.95 −1.21
4.37 −0.68 −15.90
−0.01 −0.01 −0.02 0.23 −0.26 0.30
−0.04 −0.86 −0.25 1.23 −3.88 1.13
−0.01 1.19
−0.92 2.72
RFLOW RESTRICT
0.10 −1.16
2.66 −3.64 −0.08 −0.26 −0.69
SIZE COST DUMEMERG
−0.59 −0.98 −2.56 −4.16
INFOR
−13.13 −2.21
BEHAV
−1.50 0.51 2.41 −1.14
DUMLANG TRADEB DIST
2.82 1.85 −16.88 0.09 0.03 −0.10 0.31 −0.17 −0.19
LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL
−0.01 0.68
RET2 CORR Adjusted R2
0.96 3.75 −1.38 2.37 −2.99 −0.91
0.15
0.15
0.17
0.23
0.13
0.35
0.19
0.15
−3.77 1.99
0.53
Notes: FBIAS_FLOAT ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: return correlations between two countries; FLOAT _DBIAS j : control variable.
Table 3.12 Regression analysis of home bias based on the world float portfolio Economic development variables Coeff.
t-stat
Constant
0.78
7.02
GDPC RGDP TRADE DI
0.00 0.00 0.00 2.07
0.01 −0.04 −3.25 1.95
RFLOW RESTRICT
Capital control
Stock market development
Coeff.
t-stat
Coeff.
t-stat
0.42
2.26
1.21
1.76
0.02 0.42
Investor behavior
Familiarity
Investor protection
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
Coeff.
t-stat
0.37
4.01
0.62
10.92
−3.05
−7.06
1.04
2.39
0.66
INFOR
t-stat
Coeff.
t-stat
0.58
8.64
−3.14
−1.27
0.00 −0.03 −0.01 3.34
0.65 −0.55 −1.96 1.63
0.12 0.29
1.22 0.53
−0.04 −0.06 −0.40
−0.22 −0.14 −0.73
3.44 −0.93
−0.90 −0.35 −1.84 0.44
DUMLANG TRADEB DIST
−2.64 −1.97 8.98 −0.03 −0.01 0.06 0.01 0.04 −0.05
LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL
−0.51 −2.09 1.20 0.11 1.06 −0.37 0.00 0.11
RET2 CORR 0.01
0.01
0.01
0.02
0.01
0.10
0.01
All variables
Coeff.
0.20 −0.86 0.43
BEHAV
Adjusted R2
Other variables
0.99 1.96 0.02 −0.17 0.09
SIZE COST DUMEMERG
Information costs
0.00
0.98 0.46
0.28
0.78
−2.73
−1.45
−0.49 −3.70 0.51
−2.46 −2.69 6.78
−0.16 0.00 −0.02 0.00 0.03 0.15
−1.38 −0.21 −0.24 0.03 0.52 0.56
0.00 0.91
−0.37 2.12
0.12
77
Notes: HBIAS_FLOAT ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: return correlations between two countries.
78
THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES
Our results corroborate those of Chan (2005), namely about the effect of familiarity variables on domestic and foreign biases as well as the conclusion reached by Ahearne et al. (2004) as to the impact of information costs on the home bias.
NOTES 1. See Lewis (1999) and Karolyi and Stulz (2003) for a survey of literature on the home bias. 2. http://www.imf.org/external/np/sta/pi/datarsl.htm. 3. http://en.wikipedia.org/wki/developed_country. 4. For variable groups (vi), (vii) and (viii), average values are calculated for each investing country face other receiving countries. Average value of TRADEB (see page 000) is not presented with other familiarity variables because it is equal to 1/43 for each observation. 5. We thank Jack Glen for providing us with data. 6. Source: http://www.freetheword.com. 7. The data for the year 1999 are used by Bartram and Dufey (2001). 8. http://www.phone-rate-calculator.com. 9. In this chapter we use the term “familiarity” broadly to capture the effects of both asymmetric information and psychological factors. 10. Data are used by Chan et al. (2005).
REFERENCES Ahearne, A., Griever W. and Warnock F. (2004) “Information Costs and Home Bias: An Analysis of US Holding of Foreign Equities”, Journal of International Economics, 62(2): 313–36. Bartram, S. M. and Dufey, G. (2001) “International Portfolio Investment: Theory, Evidence and Institutional Framework”, Financial Market Institutions & Instruments, 10(3): 85–155. Black, F. (1974) “International Capital Market Equilibrium with Investment Barriers”, Journal of Financial Economics, 1(4): 337–52. Bohn, H. and Tesar, L. (1996) “US Equity Investment in Foreign Markets: Portfolio Rebalancing or Return Chasing”, American Economic Review, 86(2): 77–81. Carmichael, B. and Coen, A. (2003) “International Portfolio Choice in an Overlapping Generations Model with Transaction Costs”, Economic Letters, 80(2): 269–75. Central Intelligence agency (2001) The world factbook (Potomac Books). Chan, K., Covrig, V. and Ng, K. (2005) “What Determines the Domestic Bias and Foreign Bias? Evidence from Mutual Fund Equity Allocations Worldwide”, Journal of Finance, 60(3): 1495–534. Cooper, I. and Kaplanis, E. (1986) “Costs to Cross Border Investment and International Equity Market Equilibrium”, in J. Edwards, J. Franks, C. Mayer and S. Schaefer (eds) Recent Developments in Corporate Finance (Cambridge University Press). Dahlquist, M., Pinkowitz, L., Stulz, M. and Williamson, R. (2003) “Corporate Governance and the Home Bias”, Journal of Financial and Quantitative Analysis, 38(1): 87–110. Edison, H. and Warnock, F. (2003) “A Simple Measure of the Intensity of Capital Controls”, Journal of Empirical Finance, 10(1–2): 81–103. Faruqee, H., Li Sh. and Yan I. (2004) “The Determinants of International Portfolio Holdings and Home Bias”, Working Paper, WPL04/34.
FATHI ABID AND SLAH BAHLOUL
79
French, K. and Poterba, J. (1991) “Investor Diversification and International Equity Markets”, American Economic Review, 81(2): 222–26. Gehrig, T. (1993) “An Information Based Explanation of the Domestic Bias in International Equity Investment”, Scandinavian Journal of Economics, 95(1): 97–109. Giannetti, M. and Koskinen, Y. (2004) “Investor Protection and the Demand for Equity”, SSE/EFI Working Paper, Series in Economics and Finance, n.526. Glassman, D. A. and Riddick, L. A. (1996) “Why Empirical International Portfolio Models Fail: Evidence that Model Misspecification Creates Home Bias”, Journal of International Money and Finance, 15(2): 275–312. Huberman, G. (2001) “Familiarity Breeds Investment”, Review of Financial Studies, 14(3): 659–80. Karolyi, A. and Stulz, R. (2003) “Are Financial Assets Priced Locally or Globally?”, in G. Constantinides, M. Harris, and R. M. Stulz (eds), Handbook of the Economics of Finance, (Amsterdam: North-Holland). Kilka, M. and Weber, M. (2000) “Home Bias in International Stock Return Expectations”, Journal of Psychology and Financial Markets, 1(3–4): 176–92. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (1997) “Legal Determinants of External Finance”, Journal of Finance, 52(3): 1131–51. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (1998) “Law and Finance”, Journal of Political Economy, 106(6): 1113–55. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (2000) “Investor Protection and Corporate Governance”, Journal of Financial Economic, 58(1–2): 3–27. Lewis, K. (1999) “Trying to Explain Home Bias in Equities and Consumption”, Journal of Economic Literature, 37(2): 571–608. Lintner, J. (1965) “The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolio and Capital Budgets”, Review of Economics and Statistics, 47(1): 13–37. Portes, R. and Rey, H. (2005) “The Determinants of Cross-Border Equity Flows”, Journal of International Economics, 65(2): 269–96. Sarkissan, S. and Schill, M. (2004) “The Overseas Listing Decision: New Evidence of Proximity Preference”, Review of Financial Studies, 17(3): 769–809. Sharpe, W. F. (1964) “Capital Asset Prices: A Theory of Market Equilibrium under the Condition of Risk”, Journal of Finance, 19(3): 425–42. Strong, N. and Xu, X. (2003) “Understanding the Equity Home Bias: Evidence from Survey Data”, Review of Economics and Statistics, 85(2): 307–12. Stulz, R. M. (1981) “On the Effect of Barriers to International Investment”, Journal of Finance, 36(4): 923–34. Tesar, L. and Werner, I. (1995) “Home Bias and High Turnover”, Journal of International Money and Finance, 14(4): 467–92. Zhou, C. (1998) “Dynamic Portfolio Choice and Asset Pricing with Differential Information”, Journal of Economic Dynamic and Control, 22(7): 1027–51.
CHAPTER 4
The Critical Line Algorithm for UPM–LPM Parametric General Asset Allocation Problem with Allocation Boundaries and Linear Constraints Denisa Cumova, David Moreno and David Nawrocki
4.1 INTRODUCTION Assume that there are information costs and asymmetric information in the marketplace. These conditions have been associated with segmented markets, with investors having a preferred habitat. Therefore, investors will be searching for local minima and maxima in their preferred habitat rather than a global market optimization based on equilibrium market asset pricing. Investors will have unique utility functions depending on their preferred habitat, and attempt to maximize their utility through a localized solution. This idea is not new as it traces back to Simon (1955), who proposed satisficing investors who restrict their searches to localized searches with rationality bounded by the area of the search. Cyert and March (1963) followed with their behavioral theory which suggests that decision-makers 80
DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI
81
break complex problems into a sequence of simpler problems, which they solve sequentially using local rationality. This is a key component of the Kahneman and Tversky (1979) prospect theory, where investors set up multiple mental accounts and optimize each account. Under these conditions, the traditional Markowitz (1959) portfolio theory may be used to optimize within a preferred habitat or mental account. This leads to the question of whether the quadratic utility function of traditional mean-variance analysis is the appropriate utility model for portfolio analysis. There is much evidence indicating that investors are more sensitive to losses than gains.1 This introduces a discontinuous change in the shape of the investor’s utility function at some target return and plays a role in prospect theory, developed by Kahneman and Tversky (1979) and Tversky and Kahneman (1991). However, there is evidence that investors are not risk-averse throughout the range of returns, and will exhibit risk-seeking behavior in special situations. Friedman and Savage (1948) and Markowitz (1952) argue that willingness to purchase both insurance and lottery tickets implies reverse S-shaped (both concave and convex) utility functions. A reverse S-shaped utility function provides an explanation for investors engaging in risk-averse behavior for losses and risk-seeking behavior for gains. Fishburn (1977) proposed the lower partial moment (LPM) a, τ model to explain risk-seeking and risk-averse behavior below a target return (τ). Investor behavior is explained through a coefficient (a) as a < 1 is risk-seeking behavior and a > 1 is risk-averse behavior, thus the LPM (a, τ) model. The LPM (a, τ) model proved to be a very useful risk measure because of its flexibility in capturing investor behavior (Nawrocki, 1999). However, it was not immune to criticism. Kaplan and Siegel (1994a, 1994b) zeroed in on its characteristic of a linear utility function above the target return, which assumes that the investor is risk-neutral to all above-target returns. A recent paper by Post and van Vliet (2002) found evidence that, while investors are risk-averse to below-target returns, they are risk-seeking above the target return. In order to apply more realistic behavior to above-target returns, Sortino et al. (1999) proposed a performance measure, the upper partial moment/lower partial moment (UPM/LPM) ratio. Given the potential usefulness of the UPM/LPM model, we develop a critical line UPM/LPM portfolio optimization algorithm (CLA) with bounded investment constraints for investors to generate optimal solutions within separate mental accounts or preferred habitats. This algorithm is important as it allows us to study the behavior of UPM/LPM portfolios in greater detail. Section 4.2 of the chapter derives the UPM/LPM CLA and provides a proof that it is consistent with Kuhn–Tucker conditions. Section 4.3 presents a short empirical test to demonstrate that the resulting algorithm does work relative to the traditional mean-variance (EV) algorithm, and section 4.4 offers a summary and conclusions.
82
CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES
4.2 THE UPSIDE POTENTIAL–DOWNSIDE RISK PORTFOLIO MODEL The UPM is also known as the upside potential, and the LPM is a family of downside risk measures. Therefore, the upside potential–downside risk (UPM/LPM) model may be formulated as follows: Maximize E(UPMp ) =
K
pt [max{0, E(Rpt ) − τ}]c
t=1
Minimize E(LPMp ) =
K
(4.1)
pt [max{0, τ − E(Rpt )}]a
t=1
subject to n
wi = 1
(4.2)
i=1
Although Markowitz (1959) developed the CLA for mean-variance optimization, this algorithm is not exclusive to the mean/variance problem; it can also be constructed for other risk–return portfolio models. In the next section, we derive the CLA for general asset allocation problem with bounds and linear equality constraints for the UPM–LPM portfolio model. Let r = (r1 , r2 , . . . rn )T be a vector of asset returns and x = (x1 , x2 , . . . xn ) a related vector of investment. It is assumed that x varies in the compact and convex set S. General asset allocation problems in the UPM–LPM framework can be formulated as looking for a legitimate investment vector x = (x1 , x2 , . . . xn ) with minimal downside risk for a given portfolio upper partial moment (b0 ). The investment vector x is legitimate whenever it fulfills the constraints. Therefore, the general optimization problem may be stated as Select x = (x1 , x2 , . . . . xn ) for which min E(LPMp ) =
n n
xi xj CLPMij
i=1 j=1
=
n i=1
xi2 LPMi +
n i =j
xi xj CLPMij
(4.3)
DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI
83
subject to n n
E(UPMp ) =
xi xj CUPMij =
i=1 j=1
n i=1
xi2 UPMi +
a11 · x1 + a12 · x2 + · · · + a1n · xn = b1 a21 · x1 + a22 · x2 + · · · + a2n · xn .. .
n i=j
xi xj CUPMij
= b2 .. .
am1 · x1 + am2 · x2 + · · · + amn · xn = bm 0 ≤ x ≤ 1 ∀i = 1 . . . n where LPMi =
T
pt · [Max{0; (τ − rit )}]a
t=1
CLPMij =
T
pt · [Max{0; (τ − rit )}]a−1 (τ−rjt )
t=1
UPMi =
T
pt · [Max{0; (rit − τ)}]c
(4.4)
t=1
CUPMij =
T
pt · [Max{0, (rit − τ)}]c−1 (rjt − τ)
t=1
LPMi = CLPMij
for
∀i = j
UPMi = CUPMij
for
∀i = j
In matrix form, this optimization problem can be formulated as (a) min x
LPMp = xT · L · x
(b) UPMp = xT · U · x (c) A · x = b (d) 0 ≤ x with x = (x1 , x2 , . . . xn )T ⎞ ⎛ CLPM11 · · · CLPM1n ⎟ ⎜ .. .. .. L=⎝ ⎠ . . . CLPMn1 · · · CLPMnn
(4.5)
84
CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES
⎞ CUPM11 · · · CUPM1n ⎟ ⎜ .. .. .. U=⎝ ⎠ . . . CUPMn1 · · · CUPMnn ⎛
⎛
⎞ a11 · · · a1n ⎜ ⎟ A = ⎝ ... . . . ... ⎠ am1 · · · amn ⎛
⎞ b1 ⎜ ⎟ b = ⎝ ... ⎠ bm First, we have to prove whether this formulation is convex, as the Kuhn– Tucker conditions for finding a global optimum can only be applied in this case. The objective function E(LPMp ) is convex for all a ≥ 1 in bounded x ∈ S if and only if it is positive semi-definite for all bounded x ∈ S. The E(LPMp ) is a positive variable for all bounded x by a ≥ 1, which implies that E(LPMp ) (and the associated matrix L) is positive semi-definite. E(LPMp ) ≥ 02 for all bounded x by a ≥ 1 assures construction of LPM3 taking positive value or zero. Thus, we can state that the objective function E(LPMp ) is convex for all a ≥ 1 in bounded x ∈ S. With the exception of E(UPMp ) all constraints are linear, and hence convex. As the quadratic function E(UPMp ) ≥ 04 is bounded for all x by c ≥ 1, convexity is assured by the formulation of UPM taking positive values or zero. Therefore, E(UPMp ) (and the associated matrix U) is positive semi-definite, and therefore convex. Kuhn–Tucker conditions are based on Lagrangian multipliers. In this case, the Lagrangian function is5
1 1 T x · U · x − UPMp L = xT · L · x − λ · (A · X − b) − λu · 2 2 where λ = {λ1 , λ2 , . . . , λm } ∈ m and λu denote Lagrangian multipliers for constraints (Equations 4.5b and 4.5c). Using the matrix form, the Kuhn– Tucker conditions are now constructed.
4.2.1 Kuhn–Tucker conditions Equation (4.5) represents a convex quadratic minimization problem with convex constraints.6 The necessary and sufficient conditions for x to be a global optimum is that x fulfills the Kuhn–Tucker conditions.7
DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI
85
Let η represent the vector of partial derivatives of the Lagrangian L with respect to the n decision variables xi (0 ≤ i ≤ 1):
∂L ∂L ∂L , ,..., η = (η1 , η2 , . . . , ηn ) = ∂x1 ∂x2 ∂xn The Kuhn–Tucker conditions for (4.5) are formulated as (a) η = L · x − λ · A − λu · (U · x) ≥ 0 (b) x ≥ 0, λ ≥ 0 (c)
∀ η · x = 0 ⇔ (η > 0 ∧ x = 0)
1≤i≤n
and
(η = 0 ∧ x > 0)
(4.6)
(d) A · x = b (e) xT · U · x = UPMp Similar to Markowitz’s CLA condition (4.6c) implies that the partial derivative ηi of L with respect to xi equals zero if and only if xi is greater than zero – that is, if asset i is included in the base solution. This is the optimality condition. When xi equals zero, then the respective partial derivative ηi is positive. Equations (4.6a) and (4.6d) can be summarized, then Equation (4.6) can be rearranged in Equation (4.7), where it is assumed that all partial derivatives are equal to zero:
X U 0 X 0 L A · − λu · = (a) A 0 −λ 0 0 −λ b (b) x ≥ 0, λ ≥ 0 (c)
∀
1≤i≤n
η > 0 ⇔ x = 0 and η = 0 ⇔ x > 0
(4.7)
(d) η = L · x − λ · A − λu · (U · x) ≥ 0 (e) xT · U · x = UPMp Equation (4.7d) is added to ascertain that (4.6a) remains satisfied. Each xi > 0 represents a base or “IN” variable. Each xi = 0 is a non-base or “OUT” variable. The first-order condition (Lagrangean function equals zero) remains appropriate for the variables that are in the solution, as their bounds are not binding and hence could have been omitted entirely (at least for the risk tolerance being examined). Assets xi = 0 will not have any impact on portfolio expected LPM, and therefore, a particular IN-set in the L matrix can replace its ith row by identity vector ei , for every i, which is not in the IN-set. Vector ei has a 1 in its ith position and zero in other positions. Let this matrix be L. Let U be the U matrix with its ith row replaced by 0 vector, for every i,
86
CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES
which is not in the IN-set. Next, the A matrix is defined as A matrix with the ith row replaced by a 0 vector, for every i not in the IN-set. In vector 0 , the ith zero position should be replaced with a downside boundary b for x in the OUT-position. But this downside boundary is zero (xi = 0), so
0 the vector does not change. Thus, it holds true that b
0 x x U 0 L A − λ = · · u b −λ −λ 0 0 A 0
L A
A 0
− λu
L − λu · U A
A 0
U 0
0 0
0 x = · b −λ
(4.8)
0 x = · b −λ
For example, if we have three assets and the second is IN, the first and the third are OUT, and the A · X = b constraint is simply x1 + x2 + x3 = 1, we shall have ⎞ ⎡⎛ 1 0 0 0 ⎢⎜ CLPM21 CLPM22 CLPM23 1 ⎟ ⎟ ⎢⎜ ⎣⎝ 0 0 1 0⎠ 1 1 1 0 ⎛ ⎞⎤ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 x1 0 ⎜ CUPM21 CUPM22 CUPM23 0 ⎟⎥ ⎜ x2 ⎟ ⎜ 0 ⎟ ⎟⎥ · ⎜ ⎟=⎜ ⎟ − λu ⎜ ⎝ 0 0 0 0 ⎠⎦ ⎝ x3 ⎠ ⎝ 0 ⎠ 0 0 0 0 −λ 1 The matrix with the portfolio fractions emerges from
x −λ
=
L − λu · U A
A 0
−1
0 · b
Rearranging the inverse matrix gives
−1 0 0 A x = · −1 −1 −1 b −λ (A ) −(A ) · (L − λu · U) · (A ) After multiplying, we obtain
−1 A ·b x = −1 −1 −λ −(A ) · (L − λu · U) · (A ) · b
(4.9)
DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI
87
It would be useful to divide this equation with and without λu :
−1 0 x A ·b = + λu −1 −1 −1 −1 −λ (A ) · U · (A ) · b −(A ) · L · (A ) · b Then, we can simply write
x = + λu · θ −λ
(4.10)
In addition to Equation (4.10), Equation (4.7d) has to be satisfied, thus
x η = ((L − λu · U) A ) · ≥0 −λ
x Substituting matrix gives −λ η = ((L − λu · U) A ) ·
−(A
−1 )
· L · (A
−1
A
−1
·b
) · b + λu · (A
−1 )
· U · (A
−1
)·b
≥0
Again, it would be useful to divide this equation with and without λu , and then simplify the denotation η=L·A
−1
· b − λu · U · A
+ λu · A · (A
−1
−1
) · U · (A
· b − A · (A
−1
)≥0
−1
) · L · (A
−1
)·b (4.11)
η = ϑ + λu · ϑ ≥ 0 where Equation (4.8) forces η = 0 + λu · 0
for
i ∈ IN-set.
Equations (4.10) and (4.11) provide the Kuhn–Tucker conditions (6a, 6b, 6d) expressed as linear functions of λu in the same way as the CLA for M–V. These linear functions are known as “critical lines”. Thus it is easy to compute efficient segments and corner portfolios in similar way as with the CLA for M–V. Portfolios with the same structure are defined as portfolios with the same i-assets in the “IN”-set. As we change the value of the λu -parameter expressing risk tolerance, the portfolio structure does not change with the value of λu , but remains the same for a certain interval. Portfolios, where the portfolio structure changes, are called “corner” portfolios by Markowitz (1959). These corner portfolios divide the λu (efficient frontier) into piecewise intervals, for which piecewise critical lines will be calculated. Hence, the efficient frontier is said to be segmented.
88
CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES
4.2.2 Efficient segments on the efficient frontier Each efficient segment is determined by the interval λlow ≤ λE ≤ λhi , with which Equation (4.10) and (4.11) are satisfied. For any critical line, the interval boundaries are defined as λhi = min(λc , λd ) λlow = max(λa , λb , 0) where
λa =
βi >0
−∞
λb =
δi >0
λc =
for
i = 1, · · · , n
if υi ≤ 0
for
i = 1, · · · , n
for
i = 1, · · · , n
for
i = 1, · · · , n
max −i /θi βi <0
+∞
if θi ≤ 0
max −ϑi /υi −∞
λd =
max −i /θi
if θi ≥ 0
max −ϑi /υi δi <0
+∞
if υi ≥ 0
All points on the critical line which lie within the interval are efficient, for which holds8
x = + λu · θ −λ λlow ≤ λu ≤ λhi Additionally: (i) λlow < λhi ; (ii) there is no i such that θi = 0 and i < 0; and (iii) there is no i such that υi = 0 and ϑi < 0. Similarly, as in the CLA for M–V in Markowitz (1987, p. 159, fig. 7.1), the critical lines can also be illustrated as straight lines: ⎛ ⎞
x ⎝ −λ ⎠ = + θ λu (4.12) ϑ υ η
DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI
89
4.2.3 Adjacent efficient segments In this algorithm, adjacent segments of the efficient frontier for λu > 0 can be obtained in an analogous manner to the CLA for M–V. The procedure is explained again for critical lines related to the UPM–LPM model. Let denote l0 the starting critical line efficient on the interval λ0low ≤ λu ≤ λ0hi . Now, looking for the adjacent efficient segment defined by a new critical line, we can move in the direction of: (a) decreasing λu (b) increasing λu . In both cases, as λu changes, x can change status in one of the following three ways: (i) An xi on the downside boundary xi = 0 may go in the IN-set of the portfolio. In this case, the critical value of the λu -parameter is obtained when 0 = ηi = ϑi + υi · λu Therefore, the critical value of λu is λu = −
ϑ υ
(ii) Avariable in the IN-set may move to the OUT-set; that is, it will reach the downside boundary of zero. Then, the critical value of the λu -parameter is reached when xi = 0, or 0 = Y = + θ · λu For the critical value of λu we obtain λu = −
θ
(iii) The algorithm stops whenever λu = 0 or the minimum LPM portfolio is reached. If we decrease λu in the first case (a1), the lower bound λ1low of a new ϑ critical line l1 is determined by a λlow = − ; that is, the ηi ≥ 0 constraint. υ
90
CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES
; that is, by the θ xi ≥ 0 constraint. As λu has to be positive, the last lower bound can only be λu = 0. The upper bound of new critical line l1 is determined by λ0low ; that is, λ1hi = λ0low . For the adjacent efficient segment on the right side of the initial one, we have to move along the increasing λu (case b). The lower bound of new critical line l−1 equals the higher bound of the initial efficient segment; that is, ϑ −1 −1 λ0hi = λ−1 low . The higher bound λhi may be determined by (b1), then: λhi = − υ or by (b2), so that
In the second case (a2) λllow is determined by a λu = −
λhi = −
θ
In (b3), λhi = ∞, the portfolio with maximum E(UPM) portfolio is found. These relationships can be seen in Figure 4.1. In this manner, we obtain the interval λ1low ≤ λu ≤ λ1hi of a new critical line −1 1 l on the left side, and interval λ−1 low ≤ λu ≤ λhi on the right side of the initial efficient segment where all portfolios are efficient. For successive critical lines, l2 , l3 , l4 and so on., and l−2 , l−3 , l−4 and so on the above process has to be applied again. Obviously, the efficient segments are continuously linked, as in the CLA for M–V. The finiteness of this algorithm is guaranteed, because on the left side it is broken up by λu = 0 determining a portfolio with a minimum (LPM) downside risk, and on the right side a portfolio with a maximum (UPM) upside potential return with λu = ∞.9
LPM ∞ Min a LPM UPM
Feasible
Infeasible 0 UPM
Figure 4.1 Minimization formulation of the portfolio problem
DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI
91
The construction of the algorithm implies that when l0 satisfies conditions (i), (ii) and (iii) of the theorem for the efficient portfolio on the critical line, as does l1 (l−1 ).10 This results from the following: (i) If some i = κ on the critical line l0 xκ moves from the IN-set to the OUTset at λu = λ0low , in the new critical line l1 related ηκ will increase from zero at λu = λ0low as λu decreases. (ii) If for some i = κ on the critical line l0 ηκ moves from the IN-set to the OUT-set at λu = λ0low , in the new critical line l1 related xκ will increase from zero at λu = λ0low , as λu decreases. (iii) If some i = κ on the critical line l0 xκ moves from the IN-set to the OUTset at λu = λ0hi , in the new critical line l−1 related ηκ will increase from zero at λu = λ0hi as λu increases. (iv) If for some i = κ on the critical line l0 ηκ moves from the IN-set to the OUT-set at λu = λ0hi , in the new critical line l−1 related xκ will increase from zero at λu = λ0hi , as λu increases. The corner portfolios, which are determined by the values of λu where the adjacent critical lines intersect, could be used to determine other portfolios. The other portfolios are simply defined as convex combination of related corner portfolios in the neighbourhood. Then, if two corner portfolios in neighbourhood x and ‘x’ are defined by λu and λu , any portfolio xˆ determined by λu ≤ λˆ u ≤ λu could be obtained from xˆ = a · x + (1 − a) · ‘x’;
a ∈ [0; 1]
Proportion a determines λu = a · λu + (1 − a) · λu a ∈ [0; 1]. Therefore, the portfolio downside risk and upside potential can be obtained by the substitution of xˆ into the equation: LPMp = xˆ T · L · xˆ UPMp = xˆ T · U · xˆ This algorithm guarantees that the Kuhn–Tucker conditions in Equation (4.6) are satisfied, but Equation (4.6e) is not explicitly satisfied. However, it can be shown that λu and UPMp are in a positive relationship; that is, the increase of λu will lead to a increase in the portfolio’s expected upper partial moment UPMp . This implies that, for each UPMp = b0 , λu can be found such that Equation (4.6e) also holds.
92
CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES
Substituting the top n rows of vector
x −λ
into Equation (4.6e)
UPMp = xˆ T · U · xˆ provides UPMp = [ + λu · θ]T · U · [ + λu · θ] Thus, the value of λu solving this equation also satisfies Equation (4.6e).
4.3 AN EMPIRICAL EXAMPLE The database used in this study consists of monthly returns for a sample of forty stocks chosen randomly from the CRSP (Chicago Research Stock Prices) between January 1980 and December 2004. As noted in the introduction, the mean-variance framework will be correct if the stock returns are normally distributed and will not if they are non-normally distributed. We compute the Jarque–Bera test, and find that the null hypothesis of normality is rejected at the 5 percent level of significance in 87 percent of stocks. The median skewness is positive (0.41) and the median kurtosis (5.26) is also greater than 3, the kurtosis of a bell distribution. This basis analysis justifies
0.18 0.16
E-V UPM/LPM
UPM(a 2)
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.01
0.02
0.03
0.04
0.05 0.06 0.07 risk [LPM(c 2)]
0.08
0.09
0.1
0.11
Figure 4.2 This figure displays the efficient frontiers using Mean-Variance and UPM/LPM Models with 40 stocks
93
0.032 E-V UPM/LPM
0.03 0.028
UPM(a 0.5)
0.026 0.024 0.022 0.02 0.018 0.016 0.014 0.012 0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
risk [LPM(c 3)]
Figure 4.3 This figure displays the efficient frontiers using Mean-Variance and UPM/LPM Models with 40 stocks
0.25 E-V UPM/LPM
UPM(a 3)
0.2
0.15
0.1
0.05
0 0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
risk [LPM(c 5)]
Figure 4.4 This figure displays the efficient frontiers using the Mean-Variance and UPM/LPM Models with 40 stocks
94
CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES
the use of the LPM and UPM in portfolio optimization. The UPM/LPM formulation contained in Equation (4.6) and the traditional Markowitz EV was programmed into a critical line algorithm using the MATLAB program. The results for different values of the parameters c (in the UPM) and a (in the LPM) are provided in Figures 4.2 to 4.4. The UPM/LPM algorithm is compared to a Markowitz mean-variance (EV) algorithm. In both figures, the results are graphed on the UPM/LPM axis. Therefore, the UPM/LPM frontier should be expected to dominate the EV frontier. We observe that the bigger differences between EV and the UPM/LPM efficient frontiers occur for low levels of downside risk and differences are not found in the upper area of the figures. Moreover, we find higher differences when the degree of risk aversion used (parameter “a” in the LPM risk measure) increases. Thus, the biggest differences between the EV frontier and UPM/LPM occur in Figure 4.4. Nevertheless, the major issue is that the UPM/LPM frontier tracks alongside the EV frontier, demonstrating that the algorithm is working and can be used by practitioners to manage their portfolios according to different investor utility functions.
4.4 CONCLUSION The critical line algorithm is a very efficient and robust algorithm. In this chapter, the CLA was derived for a UPM/LPM portfolio formulation with investment constraints. The major assumption required to provide a convergent solution is to perform a linear transformation of the asymmetric UPM and LPM matrices into a symmetric matrix. Given this transformation, the Kuhn–Tucker conditions can be derived and the CLA developed for the UPM/LPM formulation. A usable UPM/LPM algorithm is valuable to portfolio managers, as they may be restricted to preferred habitats because of information costs and other institutional restrictions. In addition, different investors in various preferred habitats will exhibit different utility functions: in this case, a portfolio selection algorithm that can provide local optima while employing different investor utility functions may prove very useful.
NOTES 1. See Nawrocki (1999) for an overview of investor attitudes towards downside risk. 2. As matrix L is asymmetric, it needs to be transformed into a symmetrical matrix. Without losing its characteristics, it is possible to transform it by [LT + L]/2. We keep the same denotation. 3. Some anomalies with negative portfolio risk can occur.
DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI
95
4. As matrix U is asymmetrical, it has to be transformed into a symmetrical matrix in the same way as L. The denotation remains the same. 5. Results are slightly simpler when we write ½ x.CLPM.x and ½ x.CUPM.x rather than x.CLPM.x and x.CUPM.x. Of course, a point that minimizes x.CLPM.x also minimizes ½ x.CLPM.x. (The same reasoning holds true for CUPM.) 6. Another way for selecting an efficient portfolio would be by maximizing the expected chance potential E(UPMp ) for a given level of shortfall risk. However, this formulation does not fulfill necessary and sufficient conditions for Kuhn–Tucker conditions to solve the optimization for a global optimum. With a maximization problem, the objective function has to be concave, but E(UPMp ) is not. 7. The derivation of the Kuhn–Tucker conditions may be found in Intriligator (1971), pp. 22–36. 8. This proof is the same as in Markowitz (1987, p. 158). 9. This is the same proof as in: Markowitz (1987), p. 172. 10. This is the same proof as in: Markowitz (1987), p. 172.
REFERENCES Cyert, R. M. and March, J. G. (1963) The Behavioral Theory of the Firm, Englewood Cliffs, NJ: Prentice–Hall. Fishburn, P. C. (1977) “Mean-Risk Analysis with Risk Associated with Below-Target Returns,” American Economic Review, 67(2): 116–26. Friedman, M. and Savage, L. J. (1948) “The Utility Analysis of Choices Involving Risk”, Journal of Political Economy, 56: 279–304. Intriligator, M. D. (1971) Mathematical Optimization and Economic Theory, Englewood Cliffs, NJ: Prentice-Hall. Kahneman, D. and Tversky, A. (1979) “Prospect Theory: An Analysis of Decision Under Risk”, Econometrica, 47(2): 263–92. Kaplan, P. D. and Siegel, L. B. (1994a) “Portfolio Theory Is Alive and Well”, Journal of Investing, 3(3): 18–23. Kaplan, P. D. and Siegel, L. B. (1994b) “Portfolio Theory Is Still Alive and Well”, Journal of Investing, 3(3): 45–6. Markowitz, H. (1952) “The Utility of Wealth”, Journal of Political Economy, 60(2): 151–8. Markowitz, H. (1959) Portfolio Selection (1st edn) (New York: John Wiley). Markowitz, H. (1987) Mean-Variance Analysis in Portfolio Choice and Capital Markets (Cambridge, Mass.: Basil Blackwell). Nawrocki, D. (1999) “A Brief History of Downside Risk Measures”, Journal of Investing, 8(3): 9–25. Post, T. and Van Vliet, P. (2002) “Downside Risk and Upside Potential”, Working paper, Erasmus Research Institute of Management (ERIM). Simon, H. A. (1955) “A Behavioral Model of Rational Choice”, Quarterly Journal of Economics, 69: 99–118. Sortino, F., Van Der Meer, R. and Plantinga, A. (1999) “The Dutch Triangle”, Journal of Portfolio Management, 26(1): 50–8. Tversky, A. and Kahneman, D. (1991) “Loss Aversion in Riskless Choice: A ReferenceDependent Model”, Quarterly Journal of Economics, 106(4): 1039–62.
CHAPTER 5
Currency Crises, Contagion and Portfolio Selection Arindam Bandopadhyaya and Sushmita Nagarajan
5.1 INTRODUCTION Recent studies have shown evidence of asset market price correlation and contagion. For example, Karolyi and Stulz (1996) show evidence of co-movements of US and Japanese stock returns, and find large shocks to broad-based market indices positively affect both the magnitude and persistence of return correlations. Masson (1998) identifies several sources for contagion. In the so-called “monsoonal effect”, pressures common to affected assets are the source of the contagion. In the spillover effect, changes in fundamentals affecting one set of assets cause fundamental changes in other asset groups, leading to contagion. However, some observers (see, for example, Eichengreen and Mody, 1998) argue that these economic effects cannot fully explain contagion. In fact, a change in one set of asset prices may trigger changes elsewhere, for reasons unexplained by economic fundamentals, perhaps because there are shifts in the market’s attitude towards risk, leading to the notion of pure contagion. In another strand of the existing literature, authors examine the effect of currency crises on asset markets. For example, Barbone and Forni (1997) studied the effect of the Mexican crisis on the secondary market for Brady bonds, and found that it had a strong permanent effect on the risk assessment of Mexico as well as on countries with similar pre-crisis means and volatilities. Hartmann et al. (2001) and Erb and Harvey (1998) examine the 96
ARINDAM BANDOPADHYAYA AND SUSHMITA NAGARAJAN
97
correlation of returns in equity markets, contagion in currency markets, and the joint behavior of country risk measures during crisis periods. Research on currency crises as they relate to contagion has documented that there is an increase in the correlation of asset prices during periods of crisis. For example, Baig and Goldfajn (1998) find that there is an increase in the yield spreads of sovereign bonds during periods of crisis as compared to tranquil periods. Fleming et al. (1998) found strong volatility linkages between stock, bond and money markets, and they find an increase in the linkages after currency crises. In this chapter, we investigate the impact of a currency crisis (namely the Asian currency crisis of the late 1990s) on the returns and riskiness of the stock markets of the affected economies. We also examine the correlation of stock price returns in these markets, and observe how they changed during and after the currency crisis. We find that stock market returns declined during the crisis but recovered during the post-crisis period. The riskiness of the capital markets increased during the crisis and even though it declined after the crisis, it remains relatively high. There is an increase in the correlation in the stock price return during the crisis, consistent with results in the existing literature. Since the benefits of diversification across different assets are a function of the correlation of the returns of the assets in the portfolio, we investigate how the change in the correlation during and after the currency crisis affects the performance of various portfolios. We find that from the perspective of a US investor, a purely international portfolio posts superior performance compared to purely domestic, or a combination of domestic and international, portfolios in the post-crisis period. The rest of the chapter is organized as follows. Section 5.2 discusses the impact of the currency crisis on stock market rates of return and volatility. Section 5.3 examines how stock market correlations change after a crisis period. Section 5.4 examines the performance of domestic versus international portfolios before and after the currency crisis, and our conclusions are in section 5.5.
5.2 STOCK MARKET AVERAGE RATES OF RETURN AND AVERAGE VOLATILITY The study focuses on seven major East Asian economies: Japan, Hong Kong, Taiwan, Korea, Indonesia, Malaysia and the Philippines. The stock market in each of these economies is represented by their respective stock market index; Nikkei 225 (N225), Hang Seng Index (HIS), Taiwan Weighted Index (TWII), Seoul Composite (KS11), Jakarta Composite (JKSE), KLSE Composite (KLSE) and PSE Composite (PSI). The data pertaining to each of these markets is from Bloomberg’s comprehensive database. The database covers, on a monthly basis, the dollar price of the stock indices for each of the chosen
98
C U R R E N C Y C R I S E S, C O N T A G I O N A N D P O R T F O L I O S E L E C T I O N
emerging market economies. These prices are used to compute the monthly stock market return in each economy. The period examined is June 1995 to December 2001. We define the pre-crisis period as June 1995 to December 1996, the crisis period as January 1997 to May 1998, and the post-crisis period as June 1998 to December 2001. The return for each index that represents an emerging market, for both pre-crisis and post-crisis periods, is calculated from a simple arithmetical average of the monthly returns posted by each of the indices in the relevant period. For comparison purposes, the return on the S&P 500 Index during these different periods is also computed. To measure the riskiness of each market, the average standard deviation of returns over each sub-sample period is computed from the standard deviations of the monthly returns. These calculations are summarized in Table 5.1. In the pre-crisis period, average return ranged from 1.98 percent in the Hang Seng Index to −0.34 percent in the Nikkei 225. For all of the East Asian markets, there was a decline in the average return from the precrisis period to the crisis period. The Taiwan Weighted Index average return remained positive; the Jakarta Composite suffered the largest decline and posted the lowest average return at −8.47 percent. It is noteworthy that the average return in the S&P 500 Index increased during the crisis period compared to the pre-crisis average. The average returns increased after the crisis period in all the East Asian markets, with the exception of the Taiwan Weighted Index, which went into the negative range. The Nikkei 225 remained in the negative region in the post-crisis period, as did the PSE Composite. The Seoul Composite posted the highest average at a relatively robust
Table 5.1 Average return and average standard deviation in the East Asian stock markets, June 1995–December 2001, percentages Markets
N225
HIS
TWII
KS11
JKSE
KLSE
PSI
S&P 500
Pre-crisis period: June 1995–December 1996 Average return Average standard deviation
−0.34
1.98
0.96 −1.92
1.37
0.82
0.75
1.78
5.30
4.18
8.00
5.27
4.06
5.58
2.67
6.37
Crisis period: January 1997–May 1998 Average return Average standard deviation
−2.01 −1.63 7.44
12.33
0.06 −5.19 −8.47 −5.59 −4.26 10.39
18.42
21.97
19.24
10.96
2.39 4.21
Post-crisis period: June 1998–December 2001 Average return Average standard deviation
−0.52 7.44
0.96 −0.31 9.31
11.02
2.89
1.45
14.85
17.74
Note: See text for explanation of abbreviations of market names.
1.33 −1.10 12.42
13.25
0.26 5.42
ARINDAM BANDOPADHYAYA AND SUSHMITA NAGARAJAN
99
2.89 percent. In the meantime, the S&P 500 Index showed a significant decline in its average return. In the East Asian markets, pre-crisis average standard deviations ranged from 8.00 percent in the Taiwan Weighted Index to 4.06 percent in the KLSE Composite. Among all the markets, the S&P 500 index was the least volatile, with an average standard deviation of 2.67 percent. There was an increase in the average standard deviation in all the East Asian markets, most of them sharp. The Jakarta Composite became the most volatile, with an average standard deviation of 21.97 percent, and the Nikkei 225 demonstrating the lowest volatility. While remaining less volatile compared to its East Asian counterparts, the US market was not spared the volatility increase; average standard deviation in the S&P 500 index during the crisis period was 4.21 percent. The change in the volatility between the crisis period and the post crisis period is mixed. Volatility declined in the Hang Seng Index, Seoul Composite and Jakarta Index. In all the other markets, including the S&P 500 Index, the average standard deviation increased (with the exception of the Nikkei 225, where it stayed the same). In each market, post-crisis volatility was greater than the volatility experienced in the pre-crisis period.
5.3 STOCK MARKET CORRELATIONS In this section, we focus on the correlation of the rates of return across the various East Asian markets in the sample, and how the rates of return in the East Asian markets correlated with the US market. We also examine how these correlations were affected by the currency crisis. Using the monthly rates of returns that were computed in the previous section, we calculate the pairwise correlation across the different markets before and after the crisis. The pairwise correlation numbers are reported in Table 5.2. Most pre-crisis correlations were positive; the exceptions were the negative correlations between the Taiwan Weighted Index and the Hang Seng Index, and the PSI Composite and the Seoul Composite. The Jakarta Composite and the KLSE Composite had the highest positive correlation, and the Seoul Composite and the Hang Seng Index had a low positive correlation of 0.046. The Jakarta Composite was also highly correlated with the Hang Seng Index. The S&P 500 Index was the most correlated with the Hang Seng Index and least correlated with the Seoul Composite. Post-crisis correlations were all positive. Generally (in fact, there are only a few exceptions) there was an increase in the pairwise correlation across the markets after the currency crisis, which is interpreted by researchers as evidence of contagion. In some cases (for example, Hang Seng and Nikkei 225), the increase in the correlation is quite high. The PSE Composite and the Jakarta Composite became the two most highly correlated markets. The implication of the general increase in correlations after the crisis is obvious;
100
C U R R E N C Y C R I S E S, C O N T A G I O N A N D P O R T F O L I O S E L E C T I O N
Table 5.2 Return correlation between stock markets: pre and post currency crisis N225
HIS
TWII
KS11
JKSE
KLSE
PSI
S&P 500
Pre-crisis: June 1995–December 1996 NIKKEI 225
1
Hang Seng
0.153
Taiwan
0.266
−0.034
Seoul
0.297
0.046
0.247
1
Jakarta
0.353
0.672
0.248
0.159
1
KLSE
0.426
0.394
0.512
0.110
0.681
1
PSE
0.277
0.454
0.088
−0.253
0.663
0.522
1
S&P 500
0.482
0.654
0.160
0.143
0.574
0.414
0.326
1 1
1
Post-crisis: June 1998–December 2001 NIKKEI 225
1
Hang Seng
0.635
1
Taiwan
0.337
0.451
1
Seoul
0.661
0.625
0.468
1
Jakarta
0.412
0.495
0.316
0.571
1
KLSE
0.153
0.397
0.444
0.363
0.452
1
PSE
0.516
0.639
0.434
0.629
0.689
0.513
1
S&P 500
0.654
0.648
0.400
0.636
0.396
0.440
0.527
1
the diversification benefits of investing in the East Asian markets that are included in the sample declined after the precipitation of the crisis.
5.4 PORTFOLIO PERFORMANCE This section tests the performance of three hypothetically constructed portfolios in both the pre-crisis period and the post-crisis period. The portfolios were constructed under the assumption that a US investor seeks to invest only in the stock market indices of the East Asian markets and the domestic market (S&P 500). The portfolios have been classified as in Table 5.3. The returns, the standard deviations and the Sharpe ratios of each of these portfolios were calculated.1 The numbers indicate that Portfolio A posted a higher return relative to Portfolios B and C in the pre-crisis period. Not only did Portfolio A post the highest return, it also had the lowest standard deviation. As a result, this portfolio had the highest risk-adjusted return among the three portfolios, as measured by the Sharpe ratio. After the crisis, the returns of each portfolio declined, and the standard deviation of each
ARINDAM BANDOPADHYAYA AND SUSHMITA NAGARAJAN
101
Table 5.3 Classification of portfolios Portfolio
Type
Allocation
A
Purely domestic portfolio
100% investment in the S&P 500
B
Purely international portfolio
100% investment in an equally-weighted portfolio invested in the equity indices of the seven economies
C
Domestic + international portfolio
Equally weighted portfolio of portfolios A&B
portfolio increased. However, on a risk-adjusted basis, Portfolio B posted the best performance; in fact, this was the only portfolio that had a positive Sharpe ratio in the post-crisis period. These observations indicate that, before the currency crisis, US investors benefited more from domestic holdings rather than diversifying across the emerging market economies. With the restoration of these markets following the crisis, the demand for emerging market equities rose, leading to a higher return per unit of risk for a pure international portfolio. Shortly after these markets experienced a currency crisis, US investors would have in fact benefited by investing in a portfolio consisting of the East Asian market indices rather than concentrating investment in the domestic market.
5.5 CONCLUSION This chapter investigated the impact of the Asian currency crisis on the rates of return, volatilities, and the correlation of the rates of return of stock market indices of seven East Asian markets and the US market. Results indicate that average rates of returns declined during the crisis, but recovered in the postcrisis period. Stock market volatility increased during the crisis, and even though it declined in the post-crisis period, it remained high compared to pre-crisis levels. The East Asian markets were correlated with each other and with the US market before the crisis, and the correlations became tighter in the post-crisis period. Using three hypothetically constructed portfolios, we found that the US investor would have obtained the best risk-adjusted return by investing exclusively in the domestic market before the crisis. However, after the crisis, the best risk-adjusted return would have been found in an equally-weighted portfolio of the stock market indices of the seven East Asian economies.
NOTE 1. The 10-year US Treasury rate (www.finance.yahoo.com) for the relevant periods is used as the risk-free rate in computing the Sharpe ratios.
102
C U R R E N C Y C R I S E S, C O N T A G I O N A N D P O R T F O L I O S E L E C T I O N
REFERENCES Baig, T. and Goldfajn, I. (1998) “Financial Market Contagion in the Asian Crisis”, IMF Working Paper (98/155). Barbone, L. and Forni, L. (1997) “Are Markets Learning? Behavior in the Secondary Market for Brady Bonds”, World Bank Policy Research Working Paper. Eichengreen, B. and Mody, A. (1998) “Interest Rates in the North and Capital Flows to the South: Is There a Missing Link?”, International Finance, 1(1): 35–58. Erb, C. B. and Harvey, C. R. (1998) “Contagion and Risk”, Emerging Markets Quarterly, 2(2): 46–64. Fleming, J., Kirby, C. and Ostdiek, B. (1998) “Information and Volatility Linkages in the Stock, Bond and Money Markets”, Journal of Financial Economics, 49: 111–37. Hartmann, P., Straetmans, S. and de Vries, C. G. (2001) “Asset Market Linkages in Crisis Periods”, Tinbergen Institute Discussion Paper (TI-2001–071/2). Karolyi, G. A. and Stulz, R. M. (1996) “Why Do Markets Move Together? An Investigation of US–Japan Stock Market Return Comovements”, Journal of Finance, 51: 951–86. Masson, P. (1998) “Contagion: Monsoonal Effects, Spillovers and Jumps between Multiple Equilibria”, IMF Working Paper (98/142).
CHAPTER 6
Bond and Stock Market Linkages: The Case of Mexico and Brazil Arindam Bandopadhyaya
6.1 INTRODUCTION Research on emerging market bonds has been growing quickly (see, for example, Min, 1998).1 This can be attributed to two main factors. First, while emerging market bond capitalization is relatively small compared to the size of the fixed-income market, it has still attracted the attention of investors. There have been times (for example, in the summer of 1997) when the average performance of the Emerging Market Bond Index is better than that of the S&P 500, and is considerably better than the US high-yield index. Among the debt instruments in the emerging markets, Brady bonds are the most important. Overall, there are about US $200bn of Brady bonds outstanding. According to the Emerging Market Traders Association, the secondary market turnover for Brady bonds represents the majority of all trading in emerging market debt instruments. Second, the sovereign bond yield spreads over the yield of similar issues from the US Treasury has become a market-based measure of sovereign credit worthiness. It has been argued that the credit worthiness of economies as measured by agencies such as Institutional Investor, Moody’s and Standard and Poor’s are only ex post indicators and may not be useful measures for those who are more interested in the future performance of an economy. As a result, recent literature on sovereign credit worthiness has focused on market-determined 103
104
B O N D A N D S T O C K M A R K E T L I N K A G E S: M E X I C O A N D B R A Z I L
2500
2000 Venezuela 1500 Arg Brz 1000 Mex
Phi
1992/Q1 1992/Q2 1992/Q3 1992/Q4 1993/Q1 1993/Q2 1993/Q3 1993/Q4 1994/Q1 1994/Q2 1994/Q3 1994/Q4 1995/Q1 1995/Q2 1995/Q3 1995/Q4 1996/Q1 1996/Q2 1996/Q3 1996/Q4 1997/Q1 1997/Q2 1997/Q3 1997/Q4 1998/Q1
500
Figure 6.1 Brady bond-stripped yield spread for Argentina, Brazil, Mexico, the Philippines and Venezuela
indicators, such as the Brady bond stripped yield spread, as the purest form of market-based sovereign risk. It has been documented that market-based indicators of sovereign risk, including bond yield spreads, are highly correlated, with the correlation increasing in crisis periods (see, for example, Figure 6.1, in which the Brady bond-stripped yield spreads of Argentina, Brazil, Mexico, the Philippines and Venezuela appear). When there is an adjustment in the market’s perception of sovereign risk for one country because of country-specific events, it results in a revision of the market-based country-risk measure for other countries as well, even though there is no apparent reason for the revision. This phenomenon is known as contagion, and has been the focus of many studies (see, for example, Baig and Goldfajn, 1998, 2000). The literature on emerging bond markets has documented cross-sectional correlation across various bond markets in a number of ways.2 Izvorski (1998) computes the default probabilities implicit in the prices of Brady bonds for seven developing countries, and finds that there is high cross correlation in the estimated default probabilities, with most correlation coefficients in excess of 90 percent. Barbone and Forni (1997) studied the effect of the Mexican crisis on the secondary market for Brady bonds, and found that the crisis had a strong, permanent effect on the risk assessment of Mexico as well as on countries with similar pre-crisis means and volatilities.
ARINDAM BANDOPADHYAYA
105
A number of studies have focused on the association between bond and stock markets. Chordia et al. (2002) investigated the common determinants of bond and stock market liquidity. Fleming et al. (1998) examined and found strong volatility linkages between stock, bond and money markets, as well as an increase in the linkages since the crash of 1987. Kelly et al. (1998) focused on the bond-stock market relationship in emerging markets. This paper examines the Brady bond market of the two largest Latin– American economies – Mexico and Brazil – with the US stock market being a common exogenous variable to each market. Results indicate that the stripped yields of each market in the very near future are determined primarily by the past yields of their respective markets. However, over a longer-term horizon, the interrelationships between the bond markets and the stock markets of the two countries become important. Future yields in the Mexican bond market are affected by current returns in the Mexican stock market, and to some extent by yields in the Brazilian bond market. A significant portion of the future variation in the Brazilian bond market yield is explained by current variation of the yield in the Mexican bond market and returns in the Mexican stock market. The Brazilian stock market returns play a negligible role in both bond markets. The US equity markets, after controlling for the bond and stock markets in Mexico and Brazil, play an insignificant role in all the four markets studied. The rest of the chapter is organized as follows. Section 6.2 describes the data and the estimation equations. Section 6.3 discusses the results and the interpretation. Conclusions are in section 6.4.
6.2 THE ESTIMATION EQUATIONS AND DATA The primary variable of interest is the Brady bond stripped-yield spreads of Mexico and Brazil. Brady bonds are securities issued by a sovereign in exchange for sovereign debts to commercial banks as a part of debt renegotiations. These bonds are denominated in US dollars, and the principal and some of the interest is collateralized with US Treasury bonds. When evaluating a Brady bond, it is necessary to strip the principal and interest guarantees in order to extract the sovereign risk that is assessed by the investors on the issuing country. The Brady bond stripped-yield spread is the difference between the Brady bond stripped yield and the US Treasury bond yield with a similar maturity. Brady bond stripped yields are estimated in a vector autoregressive (VAR) framework. Specifically, the Brady bond yield of Mexico (BBYMEX) is postulated to be a function of its own past values, and the past values of the Brazilian Brady bond yield (BBYBRA), the past returns in the Mexican and the Brazilian stock markets (respectively, MEXRET and BRARET), and the past returns of the US equity market (NDXRET). Similarly, BBYBRA is a
106
B O N D A N D S T O C K M A R K E T L I N K A G E S: M E X I C O A N D B R A Z I L
function of its own past values and the past values of BBYMEX, BRARET, MEXRET and NDXRET. As part of the VAR system, the stock market returns in Mexico and Brazil are also postulated to be functions of their own past returns and the past values of the other endogenous variables in the system. The return of the US equity market is exogenous to the system. The VAR system takes the following form: BBYMEX t = BBYMEX t−i + BBYBRAt−i + MEXRET t−i + BRARET t−i + NDXRET t−i + c1 + ε1,t BBYBRAt = BBYBRAt−i + BBYMEX t−i + MEXRET t−i + BRARET t−i + NDXRET t−i + c2 + ε2,t MEXRET t = MEXRET t−i + BBYMEX t−i + BBYBRAt−i + BRARET t−i + NDXRET t−i + c3 + ε3,t BRARET t = BRARET t−i + BBYMEX t−i + BBYBRAt−i + MEXRET t−i + NDXRET t−i + c4 + ε4,t It should be noted that results in the estimation of a VAR system are sensitive to the ordering in which the variables occur. However, in this study, the results described in the next section are not qualitatively different if the ordering of the variables is different from the one that appears in the above system. The Brady bond stripped-yield spreads and the stock market returns using major market index of each economy are computed from April 1993 to
6000
Market level
4000
Mexico level Mexican returns
2000 0 ⫺2000 ⫺4000 ⫺6000
Time (04/93–03/98)
Figure 6.2 Mexican stock market levels and returns, April 1993–March 1998
ARINDAM BANDOPADHYAYA
107
March 1998. Stock market returns and levels in Mexico, Brazil and the USA appear in Figures 6.2, 6.3 and 6.4, respectively. The Mexican and Brazilian Brady bond stripped yield spreads appear in Figure 6.5. Descriptive statistics on the stock returns and bond yield spreads are in Table 6.1, and the correlation matrix on these variables is in Table 6.2.
15,000 Brazil level Brazilian returns
Market level
10,000
5000
0
⫺5000 ⫺10,000
Time (04/93–03/98)
Figure 6.3 Brazilian stock market levels and returns, April 1993 and March 1998
3000
Market level
2000
Nasdaq level NDX returns
1000 0 ⫺1000 ⫺2000 ⫺3000 Time (04/93–03/98)
Figure 6.4 US stock market levels and returns April 1993–March 1998
108
3000
Bond yield spread
2500
BBYMEX BBYBRA
2000 1500 1000 500 0
Time (04/93–03/98)
Figure 6.5 Mexican and Brazilian Brady bond yield spread, April 1993–March 1998
Table 6.1 Descriptive statistics on stock market returns and bond yield spreads Average MEXRET BRARET NDXRET
38.94005 200.583 38.70707
Standard deviation
High
Low
4207.979
−4801.37
9224.337
−5388.31
492.8731
2529.97
−2696.92
626.4084 1138.831
BBYMEX
666.3659
341.7765
2426.00
265.00
BBYBRA
757.5591
265.6788
1681.00
264.00
Table 6.2 Correlation matrix on stock market returns and bond yield spreads MEXRET
BRARET
NDXRET
BBYMEX
MEXRET BRARET
0.108534
NDXRET
0.072882
0.100373
BBYMEX
−0.009228
−0.097391
BBYBRA
−0.00361
−0.0225
0.018975 −0.00096
0.781324
ARINDAM BANDOPADHYAYA
110
MEXRET
100
BRARET
BBYMEX
109
BBYBRA
90 80 70 60 50 40 30 20 10 0 1
6
11
16
21
26
31 Period
36
41
46
51
56
61
Figure 6.6 Variance decomposition of Mexican stock market returns
6.3 RESULTS One of the primary uses of the VAR model described in Section 6.2 is to determine the variance decompositions of the variables of interest. The variance decomposition of the Mexican stock market return, the Brazilian stock market return, the stripped Mexican Brady bond yield spread, and the stripped Brazilian Brady bond yield spread appear in Figures 6.6, 6.7, 6.8 and 6.9, respectively. All four figures show that the US stock market plays a negligible role in each of these markets. From Figure 6.6, it is apparent that virtually all of the variation in the Mexican stock market return is explained by past returns in the Mexican stock market. The effect of past returns declines over time, but remains at about 84 percent for a lagged value of 60 days. Similarly, Figure 6.7 indicates that Brazilian stock market returns are explained almost solely by returns in that market in the past, with over 80 percent of the current variability being explained by a lagged value of 60 days. Figures 6.8 and 6.9 show that the Brady bond markets of both countries are related to one other, and that the Mexican stock market plays a significant role in the Brady bond yield spread of each market. Specifically, for the immediate future, the most important variables in both Brady bond markets are its own past yields. As one looks further into the future in the Mexican Brady bond market, the influence of the Mexican stock return becomes increasingly important. While 46 percent of the variability is still explained by a 60-day lagged value of the yield in this market, 41 percent of the current
110
B O N D A N D S T O C K M A R K E T L I N K A G E S: M E X I C O A N D B R A Z I L
110 100
MEXRET
BRARET
BBYMEX
26
36
BBYBRA
90 80 70 60 50 40 30 20 10 0 1
6
11
16
21
31 Period
41
46
51
56
61
Figure 6.7 Variance decomposition of Brazilian stock market returns
110 MEXRET
100
BRARET
BBYMEX
BBYBRA
90 80 70 60 50 40 30 20 10 0 1
6
11
16
21
26
31 36 Period
41
46
51
56
61
Figure 6.8 Variance decomposition of Mexican Brady bond stripped yield spread
variability is explained by a 60-day lagged value in the Mexican stock market. While the Brazilian markets increase in importance with the passage of time they remain relatively insignificant in the explanation of returns in the Mexican Brady bond market. The Brazilian Brady bond market is slightly
ARINDAM BANDOPADHYAYA
111
110 100
MEXRET
BRARET
11
21
BBYMEX
BBYBRA
90 80 70 60 50 40 30 20 10 0 1
6
16
26
31
36
41
46
51
56
61
Period
Figure 6.9 Variance decomposition of Brazilian Brady bond stripped yield spread more influential than the Brazilian stock market. A 60-day lagged Brazilian Brady bond yield explains about 7 percent of the current variability, while a 60-day lagged return in the Brazilian stock market explains only about 6 percent of current returns. The interdependence of the markets is most pronounced in the Brazilian Brady bond market. The most important variable in the short term in this market is also its own past yields. Sixty three percent of current variability is explained by the yield one day ago. But the Mexican Brady bond yields also have a significant role to play in the variation of current yields in this market. Thirty five percent of the current variability comes from the yield one day ago in the Mexican Brady bond market. What is quite noteworthy is that the effect of its own lagged values of yields in this market decreases dramatically. A 60-day lagged yield explains only 14 percent of current variability. The effect of the Mexican stock market becomes most pronounced in this market as time goes by. Over 42 percent of the current variability is related to Mexican stock market returns of 60 days ago. The Mexican Brady bond market maintains its strong influence in this market, increasing somewhat over time to explain 42 percent of the current variability with a 60-day lag. The results are of significant importance to practitioners in general, and to bond traders in particular. The results support the hypothesis that the two stock markets examined are quite independent of each other. However, the Mexican stock market plays a significant role in the Brady bond market of both of these countries. In the Mexican Brady bond market, the Mexican stock market plays a particularly important role, especially over relatively
112
B O N D A N D S T O C K M A R K E T L I N K A G E S: M E X I C O A N D B R A Z I L
longer periods of time. In the Brazilian Brady bond market, contemporaneous yields are primarily a function of past yields and past yields in the Mexican Brady bond market, but over a relatively longer period of time the Mexican bond and stock markets dominate the explanation of variations in this market.
6.4 CONCLUSION This chapter examined the relationship between the bond and stock markets of Mexico and Brazil, two of the main markets in Latin America. Results indicate that the stock markets of these two countries are independent of each other, with most of the variations in returns being explained by past returns of each respective market. The Mexican bond market is also independent of either the stock or the bond market in Brazil. However, the Mexican stock market affects yields in the Mexican bond market over a longer-term horizon. The Brazilian bond yields are closely tied to its own yields in the past. But perhaps the most prominent finding of the chapter is that the Brazilian bond yields are significantly affected by Mexican bond and stock market returns.
NOTES 1. Min (1998) finds that macroeconomic fundamentals, such as low domestic inflation rates, improved terms of trade and increased net foreign assets, are associated with lower yield spreads. Weak liquidity variables of a country, such as high debt-to-GDP ratio, low foreign reserves to GDP ratio, low export growth, high import growth and high debt service ratio, are associated with higher yields. 2. Contagion has been studied extensively in the context of stock and other asset markets (for example, see Karolyi and Stulz, 1996; Hartmann et al., 2001).
ACKNOWLEDGMENTS I thank James Grant and Anne Jones for valuable comments and suggestions. Any remaining errors are all mine.
REFERENCES Baig, T. and Goldfajn, I. (1998) “Financial Market Contagion in the Asian Crisis”, IMF Working paper (98/155). Baig, T. and Goldfajn, I. (2000) “The Russia Default and the Contagion to Brazil”, IMF Working paper (00/160). Barbone, L. and Forni, L. (1997) “Are Markets Learning? Behavior in the Secondary Market for Brady Bonds”, World Bank Policy Research Working paper.
ARINDAM BANDOPADHYAYA
113
Chordia, T., Sarkar, A. and Subrahmanayam, A. (2002) “Common Determinants of Bond and Stock Market Liquidity: The Impact of Financial Crisis, Monetary Policy, and Mutual Fund Flows”, Working paper, Kellogg School of Management, Northwestern University. Fleming, J., Kirby, C. and Ostdiek, B. (1998) “Information and Volatility Linkages in the Stock, Bond and Money Markets”, Journal of Financial Economics, 49: 111–37. Hartmann, P., Straetmans, S. and de Vries, C. G. (2001) “Asset Market Linkages in Crisis Periods”, Tinbergen Institute Discussion paper (TI-2001–071/2). Izvorski, I. (1998) “Brady Bonds and Default Probabilities”, IMF Working paper (98/16). Karolyi, G. A. and Stulz, R. M. (1996) “Why Do Markets Move Together? An Investigation of US–Japan Stock Market Return Comovements”, Journal of Finance, 51: 951–86. Kelly J. M., Martins, L. F. and Carlson J. H. (1998) “The Relationship between Bonds and Stocks in Emerging Markets”, Journal of Portfolio Management, 24(3): 110–22. Min, H. (1998) “Determinants of Emerging Market Bond Spread: Do Economic Fundamentals Matter?”, World Bank Policy Research Working paper.
CHAPTER 7
The Australian Stock Market: An Empirical Investigation Adeline Chan and J. Wickramanayake
7.1 INTRODUCTION For decades, both practitioners and academics have been searching for the ideal asset-pricing model. The birth of the capital asset pricing model (CAPM) as a single-factor model in the 1960s revolutionized the concept of asset pricing as it enabled the quantification of the risk–return relationship. For practitioners, asset-pricing models may be important to identify whether stocks are over- or undervalued, which could influence their trading decisions. Academics, however, are particularly interested in finding an accurate asset-pricing model to facilitate the testing of the efficient market hypothesis (EMH). Nevertheless, since the late 1970s, a number of studies have presented evidence that contradicts the CAPM. The apparent violations have spawned research into possible explanations. One of the identified sources of deviations from CAPM is that of missing risk factors and the misidentification of the market portfolio, as discussed in Roll (1977). Consequently, multifactor arbitrage pricing theory (APT) and, more generally, multifactor models were developed as superior alternatives to the CAPM. The idea of multifactor APT seems appealing as it not only allows multiple factors but also does not require the identification of the market portfolio (Roll and Ross, 1980). Studies on the multifactor models are often carried out by pre-specifying systematic macroeconomic variables (Chen et al., 1986; 114
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
115
Chen and Jordan, 1993; Clare and Thomas, 1994; Groenewold and Fraser, 1997). While there is consensus that APT is empirically sound, several studies have also documented the fact that APT is not an adequate pricing model (Reinganum, 1981). This is because it only considers systematic risk factors and does not take into account capital market anomalies, such as size and value factors (firm attributes), which are becoming increasingly important (Banz, 1981; Reinganum, 1981). The evidence of size and value factors has been documented strongly in the landmark study by Fama and French (1993), which attracted substantial academic interest. It was also found in subsequent studies that the overwhelming explanatory power of size and value factors has rendered macroeconomic variables redundant in explaining returns (He and Ng, 1994; Cochrane, 1999). Hence the controversy of whether macroeconomic variables have more explanatory power than the firm-attribute factors remains unresolved. Although the issues of multifactor models have been researched extensively in the USA, empirical research developing a model that captures both systematic risk and market anomalies is limited in Australia. Therefore, a study into this area will bridge the gap in the Australian empirical literature. A drawback of the traditional multifactor model is its failure in explaining capital market anomalies. Therefore, the main objective of this study is to see if a combination of both firm-attribute factors and macroeconomic variables can develop a parsimonious multifactor model to explain the returns in the Australian equity market for the period of 1990–99. The rest of the chapter is organized as follows: Section 7.2 reviews the literature that is pertinent to multifactor models and their underlying variables. Section 7.3 provides a discussion on the hypothesis tested in this study. Section 7.4 consists of a discussion on the data and methodology adopted here, and section 7.5 presents the results generated using the methodology outlined in section 7.4. The results are discussed and analyzed in relation to the hypotheses and findings of prior studies. Finally, section 7.6 comprises an overview of the study, a discussion of the major findings, and a conclusion.
7.2 EXISTING EVIDENCE One of the most important problems of modern financial economics is the quantification of the tradeoff between risk and expected return (MacKinlay, 1995). It was not until the development of the capital asset pricing model (CAPM) that economists were able to quantify any differences in returns. The CAPM, introduced by Sharpe (1964), Lintner (1965) and Mossin (1966) states that the required rate of return of any stock is equal to the risk-free rate of return plus a risk premium which reflects only non-diversifiable risk.
116
THE AUSTRALIAN STOCK MARKET
For decades, the CAPM was regarded as a good measure of risk and thus a good explanation of the fact that some assets earn higher returns than others. However, the set of assumptions underlying the CAPM was regarded as unrealistic and has been criticized strongly by Roll (1977), primarily because an efficient market portfolio cannot be measured and identified. As a result, the proxy for the market portfolio is not determined. Additionally, by using ex post data to calculate ex ante returns presents a problem of impracticality. Thus tests of its empirical validity cannot be constructed. Moreover, the CAPM typically displays/poor explanatory power as well as overestimating the risk-free rate and underestimating the risk premium (Groenewold and Fraser, 1997). Consequently, the risk represented by the CAPM’s beta will tend to produce inconsistent expected returns.1 Therefore, based on these findings, CAPM does not seem to provide a good description of stock returns. The multifactor arbitrage pricing theory (APT) formulated by Ross (1976, 1977) was put forward as a superior alternative to the CAPM. The appropriateness of the APT as an alternative to the CAPM lies with the fact that it agrees perfectly with what appears to be the intuition behind the CAPM. Since the APT applies only to subsets of the universe of assets, it eliminates the need to justify the choice of a market portfolio. This therefore rectifies the problem identified by Roll (1977). Also, in contrast with the CAPM, the APT is derived from a simple arbitrage argument for asset returns generated by a multifactor model. Its explanatory power is bound to be better, since it allows more than a single generating factor (for example, a market return index) to capture the systematic components of risk in the model. It has been documented that the APT possesses the potential to overcome several weaknesses of the CAPM (Chen, 1983; Groenewold and Fraser, 1997). Yet, despite its advantages, the APT is still far from eclipsing the CAPM. The earliest criticism made is that the APT only considers nondiversifiable risks,2 and this fact contradicts the growing evidence of market anomalies, which involves firm-specific attributes that are unsystematic. Friend et al. (1978) indicated that returns are related to both diversifiable and undiversifiable risk. Following that, Reinganum (1981) found that the multifactor APT cannot be justified as an adequate model of asset pricing because it did not consider the size effect. Hence it is not able to eliminate persistent abnormal returns. However, Chen (1983), using similar data to Reinganum (1981), argues that firm size do not contribute additional explanatory power to the APT. None the less, like the CAPM, the APT fails to explain market anomalies, which are pervasive in capital markets today. The other reason for its failure to replace the CAPM is that the APT is not capable of attaching proper economic meaning to the factors that appear in the pricing equation (Diacogiannis and Diamandis, 1997). Taking this
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
117
perspective, the set of Chen et al.’s (1986) macroeconomic variables has often been used in the application of the APT. A distinction is made here between the APT and multifactor models (MFM). More often than not, studies that involve macroeconomic and statistical factors are used in the context of the multifactor APT (for example, Connor and Korajczyk, 1993; Diacogiannis and Diamandis, 1997). It is important to recognize that, while the APT is a MFM, all MFMs are not necessarily APTs. Even though more complete MFMs have been developed with a theoretical underpinning found in the APT (Solnik, 1983), the assumption of no arbitrage is unimportant to multifactor models. Therefore, unlike the APT, the advantage of multifactor models is that they allow more flexibility since they are statistical models that simply quantify the relationship between independent factors and dependent stock returns. It is also not constrained by assumptions such as those underlying the APT and therefore, allows for the entry into the equation of any factors, including firm-specific attributes. In fact, with some blurring boundaries, as long as these common factors influence many stocks rather than being specific to a single stock, they can be introduced into the multifactor model (Solnik, 1983). It is probably the no-arbitrage assumption in the APT that differentiates it from multifactor models. Otherwise, they are prima facie essentially similar. There are primarily three types of multifactor models: statistical, macroeconomic and firm-attribute factor models (Connor, 1995). In general, statistical factor models often use a range of maximum-likelihood and principal component-based factor analysis procedures on observed security returns to identify unobserved pervasive factors in returns. The macroeconomic variables approach follows the intuition of a priori selecting systematic macroeconomic variables that might influence share prices. Firm-attribute factor models, on the other hand, rely on the empirical findings that company attributes such as firm size, dividend yield, book-to-market ratio and industry classifications explain a substantial proportion of common return (Connor, 1995). Fama and French (1993, 1996) proposed a three-factor model consistent with the multifactor APT to replace the single-factor CAPM. Their threefactor model includes factors related to size and book-to-market equity as risk factors. Although two bond factors, related to maturity and default risk, were also considered, it was shown that they cannot explain the crosssection of average returns. Fama and French (1996) found that over the period 1963–93 in the USA, the model explains patterns in returns observed when portfolios are formed on fundamentals-to-price ratios, and the reversal of long-term returns, as documented by DeBondt and Thaler (1985). However, one shortcoming of the model is that it fails to explain the continuation of short-term returns, documented by Jegadeesh and Titman (1993).
118
THE AUSTRALIAN STOCK MARKET
Therefore, the model needs to be further refined before it can be accepted as a replacement for the CAPM. There are some studies that consider a combination of macroeconomic variables and firm-attribute factors in developing a model to explain stock returns (Connor, 1995; Zhou, 1999; Liew and Vassalou, 2000). These studies all found mixed results as to whether the risk attributes in the firm-attribute factor model capture all the risk characteristics present in the macroeconomic factor betas, or that the macroeconomic variables still retain their explanatory power. As mentioned earlier, Faff (1988, 2001) used the statistical factor model in testing empirically issues concerning the multifactor APT in Australia. The results showed that over the period 1974–85, there was mixed support for the APT, having up to three unidentified factors priced. Groenewold and Fraser (1997) attempted to identify the factors in the multifactor APT following the macroeconomic variables constructed by Chen et al. (1986). The sample period of January 1980 to April 1994 was used, and it was found that only the inflation rate and rate of growth of M3 appeared to be prevalent in the ASX. Halliwell et al. (1999) replicated Fama and French’s (1993) study using Australian data. The results of the study suggested that the size effect and not the book-to-market effect provided considerable explanatory power over realised returns for the period 1980–91.
7.3 HYPOTHESIS The main research question here is whether the multifactor model, which combines both macroeconomic variables and firm-attribute factors, will provide a better explanation for returns. If the explanatory power of the macroeconomic variables is not subsumed by the firm-attribute factors, or vice versa, when combined, then the marriage of these factors is essential in building a parsimonious multifactor model. Hence, the underlying alternative hypothesis to be tested will be: HA : A combination of size and value factors with macroeconomic variables has the potential of providing a better model for explaining cross sectional variation in returns on stocks traded in the ASX.
The important point to note here is that the standard APT model does not consider market anomalies such as size and book-to-market values, because they are generally considered to be unsystematic firm-specific attributes and do not constitute valid APT factors. Among the number of financial variables that have been found to affect stock returns, for example, price–earnings ratios, cash-flow to price ratio and leverage, the mimicking risk factors of size (SMB) and book-to-market equity (HML) constructed by Fama and French (1993) are selected. These factors not only capture market anomalies but also
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
119
have the potential to be compatible with the APT model (Fama and French, 1993, 1996). Most studies on factor models involving firm attributes have presented evidence that the macroeconomic factor model pales in comparison when explaining returns (Connor, 1995; Cochrane, 1999), because these factors already contain the explanatory power of the macroeconomic factors and therefore it would be unnecessary to consider macroeconomic variables in the multifactor model (He and Ng, 1994). However, the literature of the APT suggests that macroeconomic variables may act as proxies for pervasive risk factors (Priestley, 1996; Kryzanowski and To, 1983). It was also shown in the literature review that studies in Australia have covered the statistical aspect (Faff, 1988), macroeconomic aspect (Groenewold and Fraser, 1997) and attribute factors aspect (Halliwell et al., 1999) of the multifactor model. By and large, few studies have considered combining both firm attributes, such as size and book-to-market factors, and macroeconomic factors in applying the multifactor APT. The macrofactor APT is often criticized for not being able to capture market anomalies (Reinganum, 1981), and there is increasing evidence that size and value factors do have strong explanatory power in determining returns. However, systematic macroeconomic factors inevitably affect returns. Therefore, in this study, it is proposed that these factors should complement each other in building a multifactor model for the purpose of explaining returns. By considering both economic conditions and capital market anomalies, a better model should be derived. This chapter examines the marriage of these factors, and seeks to estimate a parsimonious model for the purpose of explaining returns of stocks traded in the Australian Stock Exchange (ASX). Analogous to Chen et al. (1986) and Fama and French (1993), Fama and MacBeth (1973) regressions are run to estimate the model. The null hypothesis is postulated and tested against the above alternative hypothesis.
7.4 THE DATA 7.4.1 Explanatory returns This study uses a representative target population obtained by selecting directly from the original sample set that includes all publicly-listed companies on the ASX. Because of the stringent requirements imposed, the full set of all publicly-listed companies would provide better coverage and a larger sample size to filter from than the All Ordinaries Index. A point to note here is that the selection of firms based on the requirement of having to survive through the ten-year period might arouse concerns regarding survivorship bias. Nevertheless, this is common to all tests requiring
120
THE AUSTRALIAN STOCK MARKET
long-term data (Cheng, 1995) and thus, survivorship bias should not have any significant impact on the results. Information obtained from this database also includes market capitalization on a monthly basis. Returns are value-weighted in the spirit of minimizing variances, and value-weighting has been achieved by adjusting for capitalization changes. The sample period is from January 1990 to December 1999. A set of accounting data is obtained from both DataStream and IRESS. As only a limited set of accounting data (for example, book-tomarket value) is available on DataStream, the majority of the book-to-market values had to be calculated from IRESS, using the net-tangible asset (NTA) formula (see Table 7.1).3 Firms that have accounting data on DataStream and IRESS for the year t − 1 are matched with their monthly returns from January to December of year t. The accounting data employed are from 1989 to 1998, and an average sample size of 216 stocks is used in the monthly regressions.
Table 7.1 Sampling procedure Screening criteria
Number of firms remaining
Total observations (number of months)
Initial data set collected on all publicly listed stocks on the ASX between January 1990 and December 1999
1,451
120
Firms that:
1,114
120
Data collected from DataStream: firms that meet the 10-year requirement
337
120
Without either market or book-to-market value
249
120
Did not meet the 2-year requirement Did not survive for the full 10-year period or were delisted before portfolio formation
Firms with both market and book-to-market values
88
120
Data collected from IRESS: Book-to-market calculated from IRESS
140
120
Firms with negative book-to-market equity
12
120
Firms with both market and book-to-market values
128
120
Final sample of firms with market values and book-to-market values retrieved from DataStream and IRESS
216
120
Notes: The number of firms that satisfy the sample selection criteria are detailed in this table. The final row shows the resulting sample size for analysis during the period of January 1990 to December 1999. Source: DataStream International and IRESS.
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
121
The construction of the book-to-market (HML) and size (SMB) portfolios is consistent with Fama and French (1993). Using DataStream International and IRESS, portfolios of publicly-listed Australian stocks based on the ratio of book-to-market equity (BE/ME) and on market equity (ME) are formed. To be included in the analysis, firms have to be listed on DataStream International for at least two years prior to portfolio formation, and with their prices available in December t − 1. Portfolios are formed by following the sampling procedure listed in Table 7.1. Stocks in the final sample are then ranked according to their bookto-market ratio and size, in descending order. Size is represented by market value and is often regarded as a proxy for growth and profitability. For the size sort, all ASX stocks in the sample are sorted on size in January of each year from 1990 to 1999. The median sample size is then used to split the stocks into two groups, small and big (S and B), with equal numbers of companies in each group. Stocks for the book-to-market sort are sorted independently into three book-to-market equity (BE/ME) groups, based on the break points for the bottom 30 percent (low), middle 40 percent (medium) and top 30 percent (high). The decision to sort firms into three BE/ME and only two ME groups follows the evidence in Fama and French (1992) that book-to-market equity has a stronger role in explaining average returns than does size. Based on the independent sorts and ranking procedure in year t − 1, six portfolios are constructed (S/H, S/M, S/L, B/H, B/M, B/L) as in Table 7.2 from the intersections of the two ME and three BE/ME groups. For example, the S/L portfolio in Table 7.2 would contain stocks in both the small ME group and the low BE/ME group. Similarly, the B/L portfolio contains stocks in the big ME group as well as in the low BE/ME group. The portfolio construction procedure is detailed in Table 7.2.
Table 7.2 Portfolio construction procedure Market capitalisation/size
Book-to-market
Portfolio
Smallest 50%
Highest 30%
S/H (Small size firm with high BE/ME)
Medium 40%
S/M (Small size firm with medium BE/ME)
Lowest 30%
S/L (Small size firm with low BE/ME)
Highest 30%
B/H (Big size firm with high BE/ME)
Medium 40%
B/M (Big size firm with medium BE/ME)
Lowest 30%
B/L (Big size firm with low BE/ME)
Biggest 50%
Note: This table shows the six portfolios used in constructing the small-minus-big (SMB) and high BE/ME-minus low BE/ME (HML) indices.
122
THE AUSTRALIAN STOCK MARKET
Monthly value-weighted returns on these six portfolios are calculated from January of year t to December of year t, and portfolios are reformed annually in January of year t + 1. Returns are calculated in the beginning of January of year t to ensure that the most recent book equity for year t − 1 is known. Firms that have been sorted into these six portfolios in year t − 1 are then used to form mimicking portfolios that proxy for size and book to market factors in year t. The mimicking portfolio for size, SMB (small minus big), was meant to capture risk factors in returns related to size. SMB is the difference, each month, between the simple average of the returns on the three small stock portfolios (S/H, S/M, S/L) and the simple average of the returns on the three big stock portfolios (B/H, B/M, B/L). Hence, SMB is the difference between the returns on the small stock portfolio and the big stock portfolio with about the same weighted-average book-to-market equity. By taking differences, it should remove the influence of BE/ME, focusing on the different return behavior of stocks based solely on size. The book-to-market mimicking portfolio, HML, aimed to capture the risk factor in returns related to book-to-market equity. HML is the difference, each month, between the simple average of the returns on the two high BE/ME portfolios (S/H and B/H) and the average on the two low BE/ME portfolios (S/L and B/L). Again, the difference between the two returns should be largely free of the size factor, focusing instead on the different behaviors of high and low book-to-market companies.
7.4.2 Chen et al.’s (CRR) macroeconomic variables This study uses the CRR macroeconomic variables as proxies for the underlying economic risk factors affecting stock returns. These factors are shown in Table 7.3. Multicollinearity is always a possibility with multiple regressions. Therefore, correlation coefficients are calculated to assess the impact of this problem. Unlike Chen et al. (1986) where the strongest correlation was between URP and UTS, there appears to be no significant correlation existing between the variables. Hence the problem of multicollinearity should not
Table 7.3 Chen et al.’s (1986) set of macroeconomic variables (1) The unexpected change in the term structure – UTS (2) The unexpected change in risk premiums – URP (3) The change in expected inflation – DEI (4) The unexpected inflation rate – UI (5) The unexpected change in the growth rate in industrial production – MP
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
123
be of concern in this study. However, it can be observed from Table 7.4 that a notable correlation is between MP and UI, with the highest value, of 0.375.
7.4.3 The returns to be explained The portfolio return is defined as the weighted average of returns for all companies within the portfolio. Data from DataStream International was used for prices and monthly returns, adjusted for dividends and capitalization changes. The issue of thin trading may arise as far as shares are concerned. The two ways of dealing with a share that does not trade in a given month is either to use the last traded price from a previous month, or treat the price of that month as missing (Halliwell et al., 1999). This study uses the former method. The advantage of this approach is that it reduces the loss of return observations, but may induce serial correlations in returns.4 In the spirit of Fama and French (1993), excess returns on portfolios are generated on size and book-to-market equity as dependent variables in the time series regressions. Allocating stocks into five size and book-tomarket quintiles to form twenty-five portfolios would be ideal. However, this would produce average portfolios sizes of generally fewer than five stocks, inevitably leading to some portfolios having zero stocks. As a result, because of data limitations, only six portfolios were formed instead of the twenty-five proposed by Fama and French (1993).5 Accordingly, size is sorted independently and split into halves (50:50 split) while book-to-market takes a 30:40:30 split. Portfolios formed on size and BE/ME aim to determine if mimicking portfolios SMB and HML capture common factors in stock returns related to size and book-to-market equity. Since the selection of these break points is arbitrary, the choice of such a
Table 7.4 Correlation matrix for economic variables, January 1990– December 1999 Symbol
MP
UTS
DEI −0.03988
UI
RI
URP
MP
1.00000
0.219469
0.375158
0.243811
0.146363
UTS
0.219469
1.00000
0.120216
0.331228
0.011135
0.128267
0.120216
1.00000
0.301845
0.178308
0.173716
DEI
−0.03988
UI
0.375158
0.331228
0.301845
1.00000
0.281827
0.125792
RI
0.243811
0.011135
0.172308
0.281827
1.00000
0.049821
URP
0.146363
0.128267
0.173716
0.125792
0.049821
1.00000
Notes: MP = monthly growth rate in industrial production; UTS = unanticipated change in term structure; DEI = change in expected inflation; UI = unanticipated inflation; RI = return on value-weighted ASX All Ordinaries; URP = unanticipated change in risk premium.
124
THE AUSTRALIAN STOCK MARKET
division would not be unreasonable, as it would still achieve its original objective of providing a wide variation of returns. Size is proxied by market value of equities at the end of December in year t − 1. The book-to-market ratio is calculated at the end of December in year t − 1 by dividing the book value by market value at that date. Monthly value-weighted returns on these six portfolios are calculated from January of year t to December of year t, and portfolios are renewed in January of year t + 1. From the construction of these sets of six portfolios, value-weighted monthly returns on these portfolios from January to December of year t are calculated. Excess returns are taken as the returns on these portfolios less a proxy for the risk-free rate. The risk-free rate proxy is equivalent to the monthly return derived from the thirteen-week Treasury notes.6 The excess returns on these six portfolios for January 1990 to December 1999 will be used as the dependent variables for stocks in the cross-sectional regressions.
7.4.4 Portfolio characteristics To determine whether the sample is representative of the market as a whole, a number of characteristics of the sample were analyzed. Table 7.5 provides some indication of how many small and big firms were included in the sample. From Panel A of Table 7.5, of the firms sorted into the small-size portfolio, about 23 percent of the firms belonged to the highest BE/ME ratios and of the companies sorted into the big-size portfolio, about 18 percent belonged to the lowest BE/ME ratios. It can be seen from Panel B that the small sizehigh BE/ME ratio portfolio (for example, S/H portfolio), exhibit the highest average BE/ME ratio, of 3.736. This is consistent with the market in the USA, where the majority of small firms have the highest BE/ME ratio (Fama and French, 1993). An indication of the variation in firm size is provided in Panel C. It can be seen that the average smallest size is approximately US $128 m. This is considerably lower than the average size of the biggest firms, at US $1,768 m. This indicates that the sample contains both very small and very large firms, providing a good distribution of both. As for the average book-to-market ratios, it also spans a wide range of 0.25 to 3.74, indicating a relatively fair selection of book-to-market values.
7.4.5 Cross-sectional regressions The commonest test of the multifactor model is a two-stage test. The first stage involves the use of time series data to estimate a set of factor loadings
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
125
for each asset, and in the second stage the mean returns are regressed on the factor loadings in a cross-sectional regression. Following Fama and French (1992, 1993) and Chen et al. (1986), the Fama and MacBeth (1973) two-stage regression procedure is used to examine the relationship between individual
Table 7.5 Average characteristics of companies in each portfolio Year
N
Big High (B/H)
Small
Medium (B/M)
Low (B/L)
High (S/H)
Medium (S/M)
Low (S/L)
Panel A: Number of firms in the sample for each year 1990
216
14
51
43
51
36
21
1991
216
14
55
39
53
32
23
1992
216
15
58
35
50
28
30
1993
216
15
55
38
50
31
27
1994
216
25
52
31
40
34
34
1995
216
23
53
32
42
33
33
1996
216
17
58
33
47
28
33
1997
216
15
56
37
49
31
28
1998
216
10
55
43
55
31
22
1999
216
16
45
47
49
41
18
Average
16
Percentage
7.59
Year
54
38
49
33
27
24.91
17.50
22.50
15.05
12.45
Big High (B/H)
Medium (B/M)
Small Low (B/L)
High (S/H)
Medium (S/M)
Low (S/L)
Panel B: Weighted average book-to-market values 1990
1.497
0.712
0.299
2.367
0.770
0.279
1991
3.070
0.988
0.429
4.211
1.095
0.319
1992
2.179
0.664
0.275
2.333
0.704
0.227
1993
2.131
0.709
0.309
2.835
0.734
0.263
1994
0.864
0.459
0.182
2.112
0.427
0.170
1995
1.036
0.568
0.247
4.746
0.585
0.239
1996
0.977
0.593
0.230
3.468
0.570
0.248
1997
0.914
0.515
0.208
1.832
0.520
0.192
1998
7.304
0.608
0.256
6.112
0.672
0.220
1999
1.796
0.795
0.271
7.345
0.817
0.311
Average
2.177
0.661
0.271
3.736
0.689
0.247 Continued
126
THE AUSTRALIAN STOCK MARKET
Table 7.5 Continued Year
Big High (B/H)
Small
Medium (B/M)
Low (B/L)
High (S/H)
Medium (S/M)
Low (S/L)
Panel C: Weighted average size ($M) 1990
197
913
1,326
7
9
10
1991
46
625
1,398
4
6
7
1992
179
1,071
1,649
8
8
9
1993
387
951
1,628
8
9
8
1994
781
2,413
901
18
18
16
1995
620
2,068
1,177
14
18
16
1996
305
1,663
2,642
15
638
1,139
1997
371
2,623
1,719
21
1,172
28
1998
439
2,013
2,653
15
23
22
1999
882
2,021
2,591
13
25
23
Average
421
1,636
1,768
12
192
128
Notes: Panel A shows the actual and average number of companies of each portfolio each year; Panel B shows the average book-to-market ratios of each portfolio each year; Panel C shows average size (in millions) of the companies of each portfolio each year.
stock returns and the fundamental variables. Although this procedure is known to be prone to the errors-in-variables problem, securities are grouped into portfolios to reduce the noise in individual returns (Chen et al., 1986).7 Subsequently, separate cross-sectional regressions for each formation period from 1990 to 1999 is run, in which the dependent variable is the annual return for the Year + 1 on asset i and the independent variables are various characteristics of asset i. For each period t = 1, . . . , T the following regression is run: Rit = α0t +
K
λit βit + εit
for j = 1, 2, . . . , Nt
(7.1)
i=1
where Rit = the monthly required rate of return at time t; K = the number of explanatory variables; βit = the realization of explanatory variable i at time t; and Nt = the number of stocks in period t. The explanatory variables include macroeconomic variables such as the unexpected change in the term structure (UTS); unexpected change in risk premiums (URP); change in expected inflation (DEI); unexpected inflation rate (UI); unexpected change in the growth rate in industrial production (MP); and a market return index (RI) (Chen et al., 1986). The firm-attribute factors include the size factor (SMB) and book-to-market factor (HML) (Fama and French, 1993). In this study, about twelve firms, on average, per year
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
127
have negative book values during the formation periods and these are not included in the regressions. The coefficients of Equation (7.1), generated after the regressions, are the time series averages over the ten formation periods. Also, as some of the regressions display departures from homoskedasticity, all regressions are adjusted to be White-heteroskedasticity consistent (White, 1980). Therefore, the null hypothesis is that the time series average of the yearly regression slopes is zero, and the p-values of the estimates are computed for each year.
7.5 DATA ANALYSIS AND RESULTS Table 7.6 reports any correlations between the variables, and Table 7.7 presents the results of the regressions. The correlation matrix of Table 7.6 is computed for the period of January 1990 to December 1999. From the correlation matrix above, it is observed that the correlation between all the variables is low. It is interesting to note that the highest correlation exists between SMB and DEI, with a value of 0.45, although neither variable seems to share a common underlying series. Table 7.7 reports the results of combining the five variables of Chen et al. (1986) and the two main factors of Fama and French (1993). The question of whether the market return index should be included in the regression was raised. While Chen et al. (1986) and Chan et al. (1985) included both the value-weighted and equally-weighted NYSE indices, Chen et al. (1986) argues that they are unlikely to reflect information actually available in the market. This is because macroeconomic time series are generally smoothed, averaged and substantially revised at subsequent dates. Hence portfolio returns will only reveal a weak statistical relation with Chen et al.’s (1986) economic innovations. In contrast, the market return index in both the mean variance theory and empirical evidence of the CAPM show up strongly (Clare and Thomas, 1994). Nevertheless, following Clare and Thomas (1994), the inclusion of the market index effectively addresses two issues. First, at a practical level, the market may contain information that is not captured by the set of variables used in this study. Therefore, if the market return index appears to be significant, this indicates a misspecification of the model in this study. Second, it can also be seen as a direct test for the multifactor model against the CAPM. If we conclude that the return on the market index is the only systematic component of risk where none of the other factors are priced, then CAPM can be accepted in place of the multifactor model. It is important to note that MP and URP remain statistically significant at the 5 percent level, and the explanatory power of SMB and HML is not affected by the inclusion of the macroeconomic variables. SMB and HML
128
Table 7.6 Correlation matrix for economic variables and firm-attribute factors, January 1990–December 1999 Symbol MP UTS DEI
MP
UTS
1.00000
0.219469
0.219469
1.00000 0.120216
−0.03988
DEI −0.03988
UI
RI
URP
SMB
HML
0.375158
0.243811
0.146363
0.280345
−0.08152
0.120216
0.331228
0.011135
0.128267
0.200902
−0.04021
1.00000
0.301845
0.178308
0.173716
0.044575
1.00000
0.281827
0.125792
0.134956
−0.12138
0.008734
UI
0.375158
0.331228
0.301845
RI
0.243811
0.011135
0.172308
0.281827
1.00000
0.049821
0.113015
−0.07116
URP
0.146363
0.128267
0.173716
0.125792
0.049821
1.00000
0.049423
−0.02097
1.00000
−0.00393
SMB HML
0.280345 −0.08152
0.200902 −0.04021
0.44575 0.008734
0.134956 −0.12138
0.113015 −0.07116
0.049423 −0.02097
−0.003926
1.00000
Notes: MP = monthly growth rate in industrial production; UTS = unanticipated change in term structure; DEI = change in expected inflation; UI = unanticipated inflation; RI = return on value weighted ASX All Ordinaries; URP = unanticipated change in risk premium; SMB (Small minus Big) = the difference each month between the simple average of the returns on the three stock portfolios (S/L, S/M, S/H) and the simple average of the returns on the three big stock portfolios (B/L, B/M, B/H); HML (high minus low) = the difference each month between the simple average of the returns on the two high BE/ME portfolios (S/H and B/H) and the simple average of the returns on the two low BE/ME portfolios (B/L and S/L).
Table 7.7 Regression of returns on Chen et al.’s (1986) macroeconomic variables and Fama and French (1993) factors, January 1990–December 1999 Rit = αit + β1it MP1t + β2it UTS2t + β3it DEI3t + β4it UI4t + β5it URP5t + β6it RI6t + β7it SMB7t + β8it HML8t + εit α it (a)
MP (b)
UTS (c)
DEI (d)
UI (e)
URP (f)
RI (g)
SMB (h)
HML (i)
Adjusted F Stats Durbin LM(1) LM(12) Prob Watson (m) (n) R2 (j) (F Stats) (DW) (k) (l)
(1) S/H
−0.021 2.670 (0.325) (0.004)
0.001 0.008 (0.863) (0.439)
−0.003 (0.434)
−2.631 0.000 (0.002) (0.378)
0.356 (0.000)
0.372 (0.000)
0.623
25.614 (0.000)
2.180
0.296
0.251
(2) S/M
−0.004 2.318 (0.835) (0.001)
0.001 0.003 (0.808) (0.791)
−0.009 (0.460)
−1.824 0.000 (0.010) (0.447)
0.181 (0.000)
−0.057 (0.365)
0.393
7.154 (0.000)
1.915
0.661
0.499
(3) S/L
−0.019 1.982 (0.378) (0.014)
0.004 0.005 (0.223) (0.616)
−0.005 (0.202)
−3.605 0.000 (0.000) (0.445)
0.331 (0.000)
−0.389 (0.000)
0.653
28.967 (0.000)
2.158
0.372
0.898
(4) B/H
−0.016 1.764 (0.462) (0.032)
0.003 0.004 (0.401) (0.699)
−0.006 (0.056)
−4.058 0.000 (0.000) (0.382)
−0.056 (0.112)
0.117 (0.025)
0.265
3.941 (0.000)
2.135
0.429
0.927
(5) B/M
−0.011 2.762 (0.483) (0.000)
0.003 0.006 (0.211) (0.550)
−0.006 (0.099)
−2.915 0.000 (0.000) (0.506)
−0.042 (0.198)
−0.068 (0.204)
0.291
4.518 (0.000)
2.044
0.795
0.414
(6) B/L
−0.017 2.448 (0.350) (0.001)
−0.001 0.007 (0.831) (0.463)
−0.005 (0.222)
−2.052 0.000 (0.001) (0.240)
−0.033 (0.459)
−0.122 (0.137)
0.249
3.604 (0.001)
2.243
0.172
0.131
129
Notes: p-values are in parentheses. All regressions are White heteroskedasticity consistent. S/H = small size firm with high BE/ME; S/M = small size firm with medium BE/ME; S/L = small size firm with low BE/ME; B/H = big size firm with high BE/ME; B/M = big size firm with medium BE/ME; B/L = big size firm with low BE/ME; MP = monthly growth rate in industrial production; UTS = unanticipated change in term structure; DEI = change in expected inflation; UI = unanticipated inflation; RI = return on value weighted ASX All Ordinaries; URP = unanticipated change in risk premium; SMB (small minus big) = the difference each month between the simple average of the returns on the three stock portfolios (S/L, S/M, S/H) and the simple average of the returns on the three big stock portfolios (B/L, B/M, B/H); HML (high minus low) = the difference each month between the simple average of the returns on the two high BE/ME portfolios (S/H and B/H) and the simple average of the returns on the two low BE/ME portfolios (B/L and S/L).
130
THE AUSTRALIAN STOCK MARKET
are statistically significant in the small-size portfolios (S/H, S/M, S/L); however, this does not explain any returns in the big stock portfolios (B/H, B/M, B/L). It appears that returns in the big stock portfolio (B/H, B/M, B/L) can only be explained by macroeconomic factors. Again, the three macroeconomic factors that are significantly pervasive across the big stock portfolios are MP, UI and URP. It may be the case that large firms in Australia are more diversified in their operations compared to small-sized firms and therefore are probably immune to firm-attribute factors such as size and book-to-market ratios. However, they still cannot protect themselves from nondiversifiable macroeconomic factors. It is not surprising that, with the combination of these factors, at least four factors and up to five factors could explain the returns of the stocks listed in the ASX. An attempt was made to construct a parsimonious model via the process of eliminating variables. The two variables that have been considered for exclusion are UTS and DEI, since they are the least significant in the regressions. Based on the adjusted R2 values, it appears that the model that excluded UTS provided the strongest explanatory power among the three models considered.8 The final parsimonious model, together with a set of diagnostics, is shown in Table 7.8. Since a two-stage ordinary least squares (OLS) regression was used, it is important to ensure that residuals are not autocorrelated. In general, there is no evidence of first-order residual autocorrelation as shown by the Durbin–Watson (DW) test (Durbin and Watson, 1950, 1951), and Lagrange multiplier (LM1 and LM12) test statistics (Godfrey, 1978) show that there is no first and twelfth-order residual autocorrelation. For the DW and LM1 tests, the null hypothesis to be tested is that residuals of the model are not first-order serially correlated against the two-sided alternative hypothesis. This is not surprising; it is expected that autocorrelation would not be present as first differences have already been taken for the variables. Table 7.8 also displays a number of diagnostics measures. It can be seen that the F-statistics indicate statistically significant regressions for each of the six cross-sectional regressions and the adjusted R2 s range from 0.256 for the portfolio B/L to 0.699 for the portfolio S/L. It can be seen that the intercept term is insignificant for all six cases, with the values ranging from −0.018 to 0.020. A well-specified asset-pricing model produces intercepts that are indistinguishable from zero (Merton, 1973). Therefore these results support the contention that the multifactor model containing both Chen et al. (1986) variables and Fama and French (1993) factors does indeed provide better explanatory power on equity returns. Additionally, the statistically insignificant market-return index indicates that the model is not misspecified, and the market does not contain missing priced factors (Clare and Thomas, 1994). The findings here differ in comparison to prior studies. He and Ng (1994) investigated whether size and book-to-market values of equity are proxying for macroeconomic risks found in Chen et al. (1986). It was found that,
Table 7.8 Regression of returns on the parsimonious model using Chen et al.’s (1986) macroeconomic variables and Fama and French (1993) factors, January 1990–December 1999 Rit = αit + β1it MP1t + β2it DEI2t + β3it UI3t + β4it URP4t + β5it RI5t + β6it SMB6t + β7it HML7t + εit α it (a)
MP (b)
DEI (c)
UI (d)
URP (e)
RI (f)
SMB (g)
HML (h)
Adjusted R2 (i)
Fstats Prob (F Stats) (j)
DW (k)
LM(1) (l)
LM(12) (m)
(1)
S/H
−0.020 (0.347)
2.683 (0.001)
0.008 (0.465)
−0.003 (0.478)
−2.620 (0.000)
0.000 (0.374)
0.357 (0.000)
0.372 (0.000)
0.627
29.525 (0.000)
2.179
0.298
0.298
(2)
S/M
−0.003 (0.869)
2.332 (0.001)
0.003 (0.785)
−0.009 (0.014)
−1.813 (0.010)
0.000 (0.474)
0.182 (0.000)
−0.057 (0.363)
0.399
8.241 (0.000)
1.917
0.666
0.502
(3)
S/L
−0.014 (0.378)
2.062 (0.011)
0.005 (0.582)
−0.004 (0.288)
−3.540 (0.000)
0.000 (0.509)
0.336 (0.000)
−0.390 (0.000)
0.699
32.856 (0.000)
2.142
0.421
0.956
(4)
B/H
−0.013 (0.540)
1.817 (0.002)
0.004 (0.678)
−0.006 (0.068)
−3.030 (0.000)
0.000 (0.413)
−0.052 (0.134)
0.116 (0.025)
0.268
4.439 (0.000)
2.130
0.444
0.964
(5)
B/M
−0.007 (0.674)
2.830 (0.000)
0.006 (0.524)
−0.005 (0.141)
−2.861 (0.000)
0.000 (0.594)
−0.037 (0.235)
−0.068 (0.200)
0.290
4.982 (0.000)
2.035
0.842
0.410
(6)
B/L
−0.018 (0.304)
2.434 (0.001)
0.007 (0.464)
−0.005 (0.182)
−2.096 (0.001)
0.000 (0.223)
−0.034 (0.437)
−0.122 (0.136)
0.256
4.148 (0.000)
2.243
0.170
0.127
131
Notes: p-values are in parentheses. All regressions are White-heteroskedasticity consistent. S/H = small size firm with high BE/ME; S/M = small size firm with medium BE/ME; S/L = small size firm with low BE/ME; B/H = big size firm with high BE/ME; B/M = big size firm with medium BE/ME; B/L = big size firm with low BE/ME; MP = monthly growth rate in industrial production; DEI = change in expected inflation; UI = unanticipated inflation; RI = return on value weighted ASX All Ordinaries; URP = unanticipated change in risk premium; SMB (small minus big) = the difference each month between the simple average of the returns on the three stock portfolios (S/L, S/M, S/H) and the simple average of the returns on the three big stock portfolios (B/L, B/M, B/H); HML (high minus low) = the difference each month between the simple average of the returns on the two high BE/ME portfolios (S/H and B/H) and the simple average of the returns on the two low BE/ME portfolios (B/L and S/L).
132
THE AUSTRALIAN STOCK MARKET
when combined, the introduction of size into the regressions eliminates the explanatory power not only of risk premium (URP) but also of term structure (UTS). In contrast, when book-to-market is considered in place of size, the average slopes on term structure and risk premium remain significant. It was then concluded that the size factor subsumes macroeconomic risk factors, and that macroeconomic factors do not explain the book-to-market effect (He and Ng, 1994). Fama and French (1993) found that in the fivefactor model, the term structure series and default premium did not explain returns in the presence of firm attribute factors. Although the results found in this study are inconsistent with those of previous studies, the explanatory power of both size and value factors is not disregarded. Nevertheless, it is apparent from Table 7.7 that only the macroeconomic variables were capable of explaining returns for the big stock portfolios (B/H, B/M, B/L), and size and book-to-market factors did not subsume the macroeconomic risk factors. Instead, as can be seen in Table 7.8, these two competing sets of variables complement each other in providing a more comprehensive explanation for returns. This supports the hypothesis in this study that a combination of macroeconomic variables and firm-attribute factors do provide better explanatory power than when they are used separately.
7.6 CONCLUSION In this study, the explanatory power of size and book-to-market equity, together with the Chen et al. (1986) macroeconomic variables, were examined. Using a multifactor asset-pricing framework, it was found that Fama and French’s (1993) size and book-to-market factors do not subsume CRR’s (1986) macroeconomic variables. When all these factors are considered together, only the variable UI (unanticipated inflation) seemed marginally to have lost its explanatory power. Overall, in the presence of other macroeconomic variables, the explanatory power of the model improved. Furthermore, macroeconomic variables are the only variables capable of explaining returns for the big stock portfolios. Therefore, these two sets of variables complement each other and can be combined to offer better explanatory power. Thus the model captures both systematic risk factors and market anomalies affecting shares traded in the ASX. As a result, the empirical findings support the alternative hypothesis that a combination of both firm-attribute factors and macroeconomic variables would provide a better model. Furthermore, the results imply that the firm-attribute factors and macroeconomic variables do not capture similar risk characteristics important for pricing stocks.
APPENDIX Table 7.A1 Regression of returns on Chen et al.’s (1986) macroeconomic variables and Fama and French (1993) factors, excluding factor DEI, January 1990–December 1999 Rit = αit + β1it MP1t + β2it UTS2t + β3it UI3t + β4it URP4t + β5t RI5jt + β6it SMB6t + β7it HML7t + εit α it (a)
MP (b)
UTS (c)
UI (d)
URP (e)
RI (f)
SMB (g)
HML (h)
Adjusted R2 (i)
Fstats Prob (F Stats) (j)
DW (k)
LM(1) (l)
LM(12) (m)
(1)
S/H
−0.025 (0.288)
2.517 (0.002)
0.001 (0.835)
−0.002 (0.593)
−2.524 (0.000)
0.000 (0.342)
0.357 (0.000)
0.374 (0.002)
0.625
29.292 (0.000)
2.165
0.338
0.286
(2)
S/M
−0.005 (0.777)
2.269 (0.015)
0.001 (0.797)
−0.009 (0.025)
−1.790 (0.012)
0.000 (0.433)
0.181 (0.000)
−0.056 (0.366)
0.299
8.234 (0.000)
1.908
0.632
0.207
(3)
S/L
−0.022 (0.313)
1.888 (0.022)
0.004 (0.213)
−0.004 (0.259)
−3.539 (0.000)
0.000 (0.410)
0.331 (0.000)
−0.388 (0.000)
0.656
33.286 (0.000)
2.138
0.435
0.953
(4)
B/H
−0.018 (0.415)
1.691 (0.044)
0.003 (0.393)
−0.006 (0.074)
−3.022 (0.000)
0.000 (0.359)
−0.056 (0.112)
0.117 (0.024)
0.171
4.514 (0.000)
2.116
0.495
0.946
(5)
B/M
−0.014 (0.379)
2.658 (0.000)
0.003 (0.195)
−0.005 (0.143)
−2.843 (0.000)
0.000 (0.455)
−0.041 (0.217)
−0.066 (0.222)
0.195
5.126 (0.000)
2.033
0.843
0.490
(6)
B/L
−0.020 (0.269)
2.322 (0.002)
−0.001 (0.854)
−0.004 (0.290)
−1.998 (0.001)
0.000 (0.208)
−0.032 (0.483)
−0.121 (0.153)
0.152
4.056 (0.001)
2.220
0.218
0.182
133
Notes: p-values are in parentheses. All regressions are White-heteroskedasticity consistent. S/H = small size firm with high BE/ME; S/M = small size firm with medium BE/ME; S/L = small size firm with low BE/ME; B/H = big size firm with high BE/ME; B/M = big size firm with medium BE/ME; B/L = big size firm with low BE/ME; MP = monthly growth rate in industrial production; UTS = unanticipated change in term structure; UI = unanticipated inflation; RI = return on value weighted ASX All Ordinaries; URP = unanticipated change in risk premium; SMB (small minus big) = the difference each month between the simple average of the returns on the three stock portfolios (S/L, S/M, S/H) and the simple average of the returns on the three big stock portfolios (B/L, B/M, B/H); HML (high minus low) = the difference each month between the simple average of the returns on the two high BE/ME portfolios (S/H and B/H) and the simple average of the returns on the two low BE/ME portfolios (B/L and S/L).
134
THE AUSTRALIAN STOCK MARKET
NOTES 1. Returns to high beta stocks will tend to be overestimated and low beta stocks to be underestimated. 2. This is consistent with the CAPM, in which only undiversifiable risk is rewarded. Diversifiable risk is believed to be eliminated via portfolio formation. 3. Net-tangible assets is total shareholder’s funds, net of preference capital, minority interests and intangible assets. 4. However, an LM test conducted on the dependent variables of the returns showed no sign of serial correlations. Therefore, by using the last trading price, it did not affect the returns. See Appendix, Table 7.A1. 5. Fama and French (1993) used quintile break points for both size and book-to-market equity. The 25 size-BE/ME portfolios are formed as intersections of the 5 size and 5 BE/ME groups. In addition, Fama and French (1993) had a sample size containing approximately 4,800 stocks from NYSE, Amex and NASDAQ, compared to 216 stocks used in this study. 6. Other Australian studies (Groenewold and Fraser, 1997; Halliwell et al., 1999) have also used the thirteen-week Treasury notes as a proxy for the risk-free rate. 7. An effort has been made to construct portfolios in the spirit of Fama and French (1993) so as to spread expected returns over a wide range in order to improve the discriminatory power of the cross-sectional regression tests. 8. The regression of the model that excluded DEI can be found in the Appendix, Table 7.A1.
ACKNOWLEDGMENT The authors would like to thank Mr Shrimal Perera for assistance in this research.
REFERENCES Banz, R. W. (1981) “The Relationship between Returns and Market Values of Common Stocks”, Journal of Financial Economics, 9(1): 3–18. Chamberlain, G. and Rothschild, M. (1983) “Arbitrage, Factor Structure and Mean Variance Analysis on Large Asset Markets”, Econometrica, 51(5): 1281–304. Chan, K. C., Chen, N. F. and Hsieh, D. A. (1985) “An Exploratory Investigation of the Firm Size Effect”, Journal of Financial Economics, 14(3): 451–71. Chen, N. F. (1983) “Some Empirical Tests of the Theory of Arbitrage Pricing”, Journal of Finance, 38(5): 1393–414. Chen, S. J. and Jordan, B. D. (1993) “Some Empirical Tests in the Arbitrage Pricing Theory: Macrovariables vs. Derived Factors”, Journal of Banking and Finance, 17(1): 65–90. Chen, N. F., Roll, R. and Ross, S. A. (1986) “Economic Forces and the Stock Market”, Journal of Business, 59(3): 383–403. Cheng, C. S. (1995) “The UK Stock Market and Economic Factors: A New Approach”, Journal of Business Finance and Accounting, 22(1): 129–42. Clare, A. D. and Thomas, S. H. (1994) “Macroeconomic Factors, the APT and the UK Stock Market”, Journal of Business Finance and Accounting, 21(3): 309–29. Cochrane, J. H. (1996) “A Cross-Sectional Test of an Investment-Based Asset Pricing Model”, Journal of Political Economy, 104(3): 572–621. Cochrane, J. H. (1999) “New Facts in Finance”, Economic Perspectives, 23(3): 36–58.
A D E L I N E C H A N A N D J. W I C K R A M A N A Y A K E
135
Connor, G. (1995) “The Three Types of Factor Models”, Financial Analysts Journal, 51(1): 42–7. Connor, G. and Korajczyk, R. (1988) “Risk and Return in an Equilibrium APT: Application of a New Methodology”, Journal of Financial Economics, 21(2): 255–89. Connor, G. and Korajczk, R. (1993) “A Test for the Number of Factors in an Approximate Factor Model”, Journal of Finance, 48(4): 1263–91. Daniel, K. and Titman, S. (1997) “Evidence on the Characteristics of Cross-Sectional Variation in Stock Returns”, Journal of Finance, 52(1): 1–33. DeBondt, W. F. M. and Thaler, R. H. (1985) “Does the Stock Market Overreact?”, Journal of Finance, 50(3): 557–81. Durbin, J. and Watson, G. S. (1951) “Testing for Serial Correlation in Least Squares Regression II”. Biometrika, 38(1/2): 159–78. Durbin, J. and Watson, G. S. (1950) “Testing for Serial Correlation in Least Squares Regression I”. Biometrika, 37(3/4): 409–28. Diacogiannis, G. P. (1986) “Arbitrage Pricing Model: A Critical Examination of its Empirical Applicability for the London Stock Exchange”, Journal of Business Finance and Accounting, 13(4): 489–504. Diacogiannis, G. P. and Diamandis, P. (1997) “Multi-Factor Risk–Return Relationships”, Journal of Business Finance and Accounting, 24(3–4): 559–71. Faff, R. (2001) “An Examination of the Fama and French Three-Factor Model Using Commercially Available Factors”, Australian Journal of Management, 26(1): 1–18. Faff, R. W. (1988) “An Empirical Test of the Arbitrage Pricing Theory on the Australian Stock Returns 1974–85”, Accounting and Finance, 28(2): 23–43. Faff, R. W. (1993) “Empirical Investigations into Asset Pricing in the Australian Equity Market”, Ph.D. manuscript, Monash University, Australia. Fama, E. F. (1991) “Efficient Capital Markets: II”, Journal of Finance, 46(5): 1575–617. Fama, E. and French, K. R. (1993) “Common Risk Factors in the Returns on Stocks and Bonds”, Journal of Financial Economics, 33(1): 3–56. Fama, E. F. and French, K. R. (1995) “Size and Book-to-Market Factors in Earnings and Returns”, Journal of Finance, 50(1): 131–55. Fama, E. F. and French, K. R. (1996) “Multifactor Explanations of Asset Pricing Anomalies”, Journal of Finance, 51(1): 55–83. Fama, E. F and MacBeth, J. D. (1973) “Risk, Return and Equilibrium: Some Empirical Tests”, Journal of Political Economy, 81(3): 607–36. Fama, E. F. and French, K. R. (1997) “The Cross-section of Expected Stock Returns”, Journal of Finance, 47(2): 427–65. Friend, I., Westerfield, R. and Granito, M. (1978) “New Evidence on the Capital Asset Pricing Model”, Journal of Finance, 33(3): 903–17. Godfrey, L. G. (1978) “Testing for Higher Order Serial Correlation in Regression Equations when Regressors Include Lagged Dependent Variables”, Econometrica, 46(4): 1303–13. Groenewold, N. and Fraser, P. (1997) “Share Prices and Macroeconomic Factors”, Journal of Business Finance and Accounting, 24(9–10): 1367–83. Halliwell, J., Heaney, R. and Sawicki, J. (1999) “Size and Book to Market Effects in Australian Share Markets: A Time Series Analysis”, Accounting Research Journal, 12(1): 122–37. He, J. and Ng, L. K. (1994) “Economic Forces, Fundamental Variables, and Equity Returns”, Journal of Business, 67(4): 599–609. Jagannathan, R. and Wang, Z. (1996) “The Conditional CAPM and the Cross Section of Expected Returns”, Journal of Finance, 51(1): 3–53. Jegadeesh, N. and Titman, S. (1993) “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency”, Journal of Finance, 48(1): 65–91.
136
THE AUSTRALIAN STOCK MARKET
Jorion, P. (1991) “The Pricing of Exchange Rate Risk in the Stock Market”, Journal of Financial and Quantitative Analysis, 26(3): 363–76. Kramer, C. (1994) “Macroeconomic Seasonality and the January Effect”, Journal of Finance, 49(5): 1883–91. Kryzanowski, L. and To, M. C. (1983) “General Factor Models and the Structure of Security Returns”, Journal of Financial and Quantitative Analysis, 18(1): 31–52. Lewellen, J. (1999) “The Time-Series Relations Among Expected Return, Risk and Bookto-Market”, Journal of Financial Economics, 54(1): 5–43. Liew, J. and Vassalou, M. (2000) “Can Book-to-Market, Size and Momentum Be Risk Factors That Predict Economic Growth”, Journal of Financial Economics, 57(2): 221–45. 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. MacKinlay, C. (1995) “Multifactor Models Do Not Explain Deviations from the CAPM”, Journal of Financial Economics, 38(1): 3–28. MacKinlay, C. (1997) “Event Studies in Economics and Finance”, Journal of Economic Literature, 35(1): 13–39. McElroy, M. B. and Burmeister, E. (1988) “Arbitrage Pricing Theory as a Restricted Nonlinear Multivariate Regression Model”, Journal of Business and Economic Statistics, 6(1): 29–42. Merton, R. C. (1973) “An Intertemporal Capital Asset Pricing Model”, Econometrica, 41(5): 867–87. Mossin, J. (1966) “Equilibrium in a Capital Asset Market”, Econometrica, 34(4): 768–83. Pindyck, R. S. and Rubinfeld, D. L. (1998) Econometric Models and Economic Forecasts, (Singapore: Irwin/McGraw-Hill). Priestley, R. (1996) “The Arbitrage Pricing Theory, Macroeconomic and Financial Factors, and the Expectations Generating Processes”, Journal of Banking and Finance, 20(5): 869–90. Reinganum, M. R. (1981) ‘The Arbitrage Pricing Theory: Some Empirical Results’, Journal of Finance, 36(2): 313–21. Reyfman, A. (1997) “Labour Market Risk and Expected Asset Returns”, Unpublished Ph.D. thesis, University of Chicago, USA. Roll, R. (1977) “A Critique of the Asset Pricing Theory Tests”, Journal of Financial Economics, 4(2): 129–76. Roll, R. and Ross, S. A. (1980) “An Empirical Investigation of the Arbitrage Pricing Theory”, Journal of Finance, 35(5): 1073–102. Ross, S. A. (1976) “The Arbitrage Theory of Capital Asset Pricing”, Journal of Economic Theory, 13(3): 341–80. Ross, S. A. (1977) “Return, Risk and Arbitrage”, in I. Friend and J. Bricksler (eds), Risk and Return in Finance, vol. 1 (Cambridge, Mass.: Cambridge University Press). Sawyer, K. R. (1998) “Testing the Bounding Conditions of Arbitrage Pricing”, Working paper, University of Melbourne, Australia. Sharpe, W. F. (1964) “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk”, Journal of Finance, 19(3): 425–42. Sinclair, N. A. (1982) “An Empirical Test of the Arbitrage Pricing Theory”, Unpublished Ph.D. thesis, University of New South Wales, Australia. Solnik, B. (1983) “International Arbitrage Pricing Theory”, Journal of Finance, 38(2): 449–57. Thaler, R. H. (1999) “The End of Behavioural Finance”, Financial Analysts Journal, 55(1): 12–17. White, H. (1980) “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity”, Econometrica, 48(4): 817–38. Zhou, G. (1999) “Security Factors as Linear Combinations of Economic Variables”, Journal of Financial Markets, 2(2): 403–32.
CHAPTER 8
The Price of Efficiency – So, What Do You Think About Emerging Markets? Zsolt Berényi
8.1 INTRODUCTION With the appearance of assets reaching beyond the risk–return characteristics of conventional assets, performance measurement have also had to answer new questions. Option-like structures, dynamic strategies, leverage, the introduction of asset classes such as hedge funds, CTAs, credit funds and so on, produced novel shapes of risk–return. In answering these questions, on the one hand, the argument can still be used that traditional performance measurement tools – such as the Sharpe ratio – may provide a useful means for such cases as well. This may be true under specific circumstances; however, it is also likely that for certain investment products and markets, this line of reasoning could have serious flaws. This may be the case for unsettled markets, as well as for assets with inherent non-normality in returns, thin liquidity and accompanying sensitivity to adverse market movements. Such as, for example, investment funds on emerging markets. In this chapter, I aim at evaluating the performance of investment funds from the Central and Eastern European regions. I believe that these gradually maturing emerging markets are a good example with which to 137
138
THE PRICE OF EFFICIENCY
demonstrate differences in (and the usefulness of) conventional and new methods for performance evaluation. That is, I intend to demonstrate that, if decision-makers rely on some statistical properties of past returns, it is advisable to consider (a) higher moments of the return distribution in question; and (b) some sort of replicating strategy for performance assessment. In this concept, portfolio efficiency will be quantified based on a replication of the risk-return shape. The efficiency gain/loss is simply the difference between expected returns of the particular fund and that of the individually constructed benchmark. In addition to this, I also use a so-called doublelognormal (DLN) framework for estimating underlying prices to be used during the fund replication process. The combination of the efficiency gain/loss approach and the DLN framework is believed to deliver a more complete methodology for taking the special characteristics of such investments into consideration.
8.2 HIGHER MOMENT PERFORMANCE ANALYSIS – THE THEORY In this section, we provide an overview of the theoretical background used during the analysis. It contains two basic building blocks, from which a third follows. First, it is understood that higher moments are to be considered in order to get a more complete view of performance. Second, we give an explanation for using the replication principle. These two principles can be then combined into the efficiency gain/loss measure, the basis of the current investigation.
8.2.1 The role of higher moments It is widely accepted that return non-normality may cause traditional performance measures – especially the CAPM-based ones – to produce controversial results. From this background, a series of research projects have been carried out, and several alternative performance measurement techniques proposed. The acceptance of these methods is, however, at best, disputed. For most practical applications, the Sharpe ratio still plays, in spite of its shortcomings, a leading role. One possible way to account for return non-normality (and other return irregularities) is the inclusion of higher moments in the analysis. In this chapter, we take this route, and examine the usefulness of the highermoment-based efficiency gain/loss methodology when dealing with emerging market investments. This methodology involves the use of optioned portfolios, in order to create individual benchmarks for each asset by
ZSOLT BERÉNYI
139
replicating certain characteristics (for example, higher moments) of the original return distribution.
8.2.2 The rationale Investment funds are sold on the basis of investors’ choice. When making their decision, investors also consider the track record of the fund’s management. This is so since, first, return distribution data from the past is believed to give an indication of the manager’s skill and experience; and second, statistical measures such as variance, and especially higher moments, are often treated as being stable enough to make reasonable predictions as to future return distribution (dice do have a memory, after all?). In this respect, past and future views are linked together. That is, investors are concerned about past returns, since some properties of the past return distribution serve as a basis for making investment decisions for the future. This remains the same in cases when assets exhibit return profiles different from traditional ones, and is certainly also true for emerging market investment funds. However, the latter case requires slightly different methods to deal with the perspective of unsettled markets, lower liquidity, dynamic strategies or scarce information.
8.2.3 Portfolio replication Portfolio replication may essentially serve two purposes. First, it is possible to create synthetic benchmark assets for performance assessment purposes. Second, it is also possible to define efficient portfolios for investment purposes (Kat and Palaro, 2006). In this chapter, we concentrate on the first area only and refer to portfolio replication as constructing individual benchmark assets for each fund in question for performance evaluation purposes. The rationale behind replication can be explained by referring to Dybvig’s (1988a) and (1988b) payoff distribution pricing model. Dybvig proposed a pricing framework for assets with arbitrary return distributions, in which agents minimize the cost of any one-period return distribution, regardless of the factors that drive state probabilities. He labeled the price of the minimum cost portfolio for any return distribution as the ‘distributional price’, to distinguish it from the normal asset price. Building on Dybvig’s work, we can assert that return distributions can, and should, be compared with each other directly. This theory, of course, should also work for any synthetically created return distribution as well. For example, it is well known that by using options return distributions can be shaped arbitrarily. Consequently, given an efficient underlying (such as an equity index), by using options, it is also
140
THE PRICE OF EFFICIENCY
possible to create optioned portfolios ‘mimicking’ other portfolios in risk characteristics. Dynamic strategies can essentially deliver the same results. That is, optioned portfolios can be used for comparison (performance evaluation) purposes, regardless of the underlying – even between asset classes, in fact. Indeed, the use of optioned benchmark portfolios for such purposes is not a new idea. For example, Glosten and Jagannathan (1994) proposed the use of options in order to re-create contingent claims for mutual fund performance evaluation. Dynamic strategies can also be used to create (“mimic”) any particular risk profile, as proposed by Amin and Kat (2003) to evaluate hedge fund returns in using path-independent dynamic strategies. It is important to note that, for most practical applications, portfolio replication does not mean replicating the full return distribution of the asset in question. This would certainly be possible by using, for example, factor models to replicate monthly returns; if factor returns and factor sensitivities of a particular asset are known, this methodology will deliver nice results. But in fact, this is seldom the case – especially not, for example, for hedge funds or emerging market investments. Hence, a full portfolio replication is not feasible, but neither is it necessary. In practice, instead of aiming at a full replication of return series, it is sufficient to reproduce certain characteristics of the particular return distribution. For example, we can create a distribution with a similar set of higher central moments. We do so by first choosing an asset with sufficient liquidity and traded derivatives (such as futures or options). Next, this underlying is combined with derivatives and the risk-free asset, in order to mimic the individual risk–return shapes. Note that, based on earlier research (for example, Berényi, 2003) it is recommended to use three moments: the variance, skewness and kurtosis, as these are believed to give a thorough view of the risk perceived by investors. Combining the above effect of options and the payoff distributional pricing theory, synthetic optioned portfolios can be used to generate both artificial benchmarks and real portfolios if current investment opportunities are not attractive enough.
8.3 THE EFFICIENCY GAIN/LOSS METHODOLOGY 8.3.1 Definition The efficiency gain/loss methodology denotes, in essence, a comparison of expected returns between two assets with a similar risk profile (see, for example, Amin and Kat, 2003; Berényi, 2003). For each risk profile, we can create an individual benchmark asset with an equivalent risk exposure. Consequently, the efficiency gain/loss measure is understood as the difference in
ZSOLT BERÉNYI
141
expected returns of the asset in question and that of its replicating benchmark counterpart. The risk exposure taken, as mentioned above, is defined by the higher central moments of the return distribution. That is, we build on the assumption that those three moments will cover a considerably large amount (if not all) of the risk perceived by the investors. It is important to bear in mind that we rely essentially on moments of past return distributions, or, put another way, on the assumption that past return distribution can be extrapolated meaningfully into the future. The interpretation of the efficiency gain/loss measure is a straightforward one. The expected return of the benchmark asset is the alternative return one may achieve when holding the risk exposure constant. Since the alternative optioned return may be realized freely without restrictions, it embodies the minimum acceptable return for a given risk profile. In other words, if a fund’s risk profile can be replicated with a higher expected return, then it cannot be efficient, and vice versa: if it is efficient, then it cannot be replicated.
8.3.2 The benchmark Basically, there are two possible ways for generating benchmark assets. The first methodology relies on using dynamic replication strategies, by trading futures on traditional assets (for example, equity index futures, but bond, currency and interest rate futures can also be used). The second methodology makes use of options for the same purpose. For both methodologies, though, we require at least one efficient underlying with traded futures and/or options. For simplicity, we shall use options. Stock markets usually encompass a substantial number of investors generating liquidity for these instruments and guaranteeing an efficient price formation for return non-normality also on emerging markets. For the construction of the benchmark asset, we apply nonlinear optimization to reproduce the higher moment return characteristics of the given fund. We use the most liquid equity index underlying denominated in the same currency as the fund in question, combined with options and the – market-specific – risk-free asset. The nonlinear optimization algorithm for constructing benchmark assets can be formulated as follows: for any investment fund l, construct a benchmark asset L from the assets i (that is, the underlying asset, the risk-free asset and the options) and with return rL , by maximizing the expected benchmark return: Maximize E(rL ) =
i
xi E(ri )
(8.1)
142
THE PRICE OF EFFICIENCY
σL2 = σl2 for the variance (target rate),
subject to
s3L = s3l (target rate skewness), i
kL3 = kl3 (target rate kurtosis) and
(8.2)
xi = 1 as budget constraints,
where E(ri ) is the expected return on asset i; xi is the investment in asset i (for example, the underlying asset, the risk free asset, as well as the options); σL , sL , kL are the standard deviation, skewness and kurtosis for the benchmark asset L; and σl , sl , kl , are respective moments for the fund l. It is worth noting that the magnitude of the individual benchmark return also gives an indication of the overall level of risk taken. The reason for this follows from the nature of optioned markets, where different positions with identical expected returns (identical price) can be substituted for each other; therefore they necessarily have a similar perceived risk level.
8.3.3 The underlying The efficiency gain/loss analysis can certainly be completed simply on the basis of historical return distributions. However, given the degree and characteristics of the variability of historical returns on such markets, instead of using historical returns with a Monte Carlo simulation, we propose using the future implied distribution. The implied probability distribution can be obtained from the information embodied in option prices. That is, since option prices contain information on investors’ future anticipations of the underlying asset’s future price distribution, this methodology can deliver a far more reliable estimate for the future return process than can historical returns. The information contained in option prices may be extracted by assuming a specific underlying process and running a nonlinear optimization by minimizing a specific target function (for example, minimizing the distance between observed and calculated option prices). The underlying distribution is often assumed to follow a lognormal distribution. However, for unsettled markets, bimodal distribution (for example, election effect) or periods of irregular market behavior (crises), it may be necessary to use a double lognormal setting – for example, combining two lognormal distributions in order to calculate the underlying distribution that best fits the observed option prices. The mixture of two lognormals does a reasonable job in, for example, periods of high negative skewness, or high kurtosis.
ZSOLT BERÉNYI
143
Throughout the estimation procedure, the DLN distribution function for futures prices of the underlying, g[…] is given by the following equation (see, for example, Melick and Thomas, 1997): g[f0 ] = wg1 [f0 ] + (1 − w)g2 [f0 ]
(8.3)
where gi [f0 ] =
√
1 2πσi · f0
· exp
ln(f0 ) − µi σi
2 2
(8.4)
with w being the weight of he first lognormal distribution. The parameters of the above equations are estimated by minimizing the sum of squared errors over all option strikes and prices.
8.4 TESTING RESULTS 8.4.1 Data for the analysis During our analysis we concentrate on Hungarian investment funds because, for this market, option prices and also relatively long data series are easy to obtain. Investment fund daily returns from August 1996 to December 2005 were used. The data set contains 144 investment funds, all denominated in Hungarian forint (HUF). For the sake of simplicity, missing returns were substituted by the risk-free rate. For demonstrating the usefulness of the efficiency gain/loss measure to assess performance consistently even across different asset classes, the data set contains equity investment funds, mixed bond–equity funds, bonds, real estate, money market investment funds, and guaranteed funds. It must be noted that, if holding to the maturity, the return on guaranteed funds certainly follows a pattern very unlike other investment forms. However, for a relatively short time period, they display a risk–return shape comparable to normal investment funds. From daily returns, we generate semi-ex ante annual returns by bootstrapping, drawing 250 samples from the set of daily data, and repeating the procedure 1,000 times. This methodology is analogous to the methodology applied, for example, by Ederington (1995). For the underlying asset, daily index data from the Budapest Stock Exchange (BUX) was used. Options on the index have a maturity of three months. For the risk-free rate, the Budapest Interbank Offered Rate (BUBOR)
144
THE PRICE OF EFFICIENCY
0.0014 0.0012
DLN SLN
Probability
0.0010 0.0008 0.0006 0.0004 0.0002 0.0000 8,000
13,000 18,000
23,000
28,000 33,000
38,000 43,000
End-of-period value
Figure 8.1 SLN and DLN distributions
was chosen. As daily (overnight) returns were not always available, the onemonth interest rates were used as a proxy for the overnight interest data. In the course of the analysis, returns net of the corresponding risk-free rates were applied.
8.4.2 Return distribution of the benchmark asset To calculate the underlying index distribution on the basis of observed option prices, a nonlinear optimization was run to obtain the parameter of the DLN that fits the observed option prices; that is, the squared error between observed and calculated option prices was minimized. We calculated the DLN distribution using the pins obtained for the underlying by historical simulation. By doing this, correlations between assets and synthetic portfolios can be computed more easily. The results from both the SLN and DLN calculations can be seen in Figure 8.1. Both the SLN and the DLN distributions display a very good fit to the observed option prices. Moreover, most remarkably, the calculated distributions for the end-of-period future underlying price do not seem to differ from each other significantly. That is, either the implied distribution is clearly a lognormal one, or it is an obvious sign for market participants calculating option prices based directly on the Black–Scholes formula.
ZSOLT BERÉNYI
145
8.4.3 Basic performance characteristics First, the normality of the investment fund returns were examined with the Jarque–Bera test statistic. For our sample, it has been found that the normality assumption cannot be rejected in about 24 percent of the cases at 95 percent confidence; that is, more than three-fourths of the cases exhibit considerable return non-normality – this should in fact not be a surprise, and it is in line with the preliminary expectations regarding emerging market investments. It is also worth noting that a huge majority (about 91 percent) of the funds exhibit negative skewness. Since our analysis covers various asset classes with completely different strategies and characteristics, the risk levels taken cannot be compared directly. Instead, the risk exposures can be analyzed using risk-adjusted performance measures such the efficiency gain/loss measure.
8.4.4 Higher moment performance characteristics In order to assess the value created by the fund manager, as discussed above, the higher-moment-based efficiency gain/loss methodology was used. That is, for every fund, a replicating portfolio was constructed, comprising an underlying index, options and risk-free assets. It is worth noting again that replicating funds have the greatest expected return for a given set of variance, skewness and kurtosis. The calculation of the expected return on the replicating portfolio will certainly be influenced by the maximum allowable weights for each security/option. It implies that, if no short-selling is possible, the replicating returns will be lower than (or at best equal to) the case with fewer constraints. During the analysis, replicating returns were calculated both when short selling was allowed, and where only long positions could be held with no additional leverage. The latter could be, theoretically, applicable for markets with specific constraints, and for private investors. Figure 8.2 contains the results of the first calculation with short selling allowed. (Note that all the Figures below contain annualized data, sorted by magnitude, for easier understanding and interpretation.) The outline shows a completely normal investment set: the majority of the funds show considerable negative efficiency, but it is also possible to find investments with superior performance; that is, there are some funds whose performance cannot simply be replicated. Figure 8.3 shows a comparison between an unconstrained case and a case where short selling is not allowed. As expected, there is a clear difference to be observed between the short selling and no short selling cases. Nevertheless, this difference is not substantial: the Spearman rank correlation between the two rankings is quite high, with a value above 0.93. It can
146
THE PRICE OF EFFICIENCY
10.0%
Efficiency gain/loss (Percent)
5.0%
0.0%
⫺5.0%
⫺10.0%
⫺15.0%
1
6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146
Investment funds
Figure 8.2 Efficiency gain/loss for the CEE investment funds investigated
10.0% Short selling Not short selling
Efficiency gain/loss (Percent)
5.0%
0.0%
⫺5.0%
⫺10.0%
⫺15.0%
1
8
15 22
29 36 43 50 57 64 71 78
85 92 99 106 113 120 127 134 141
Investment funds
Figure 8.3 Efficiency gain/loss for CEE investment funds: a comparison of replicating strategies also be seen that a large part of the funds still shows a considerable negative performance in both cases. However, it is also observable that in the no-short-selling case, the ratio of funds with inferior performance is much lower.
ZSOLT BERÉNYI
15.0%
147
5 Efficiency gain/loss
12.0%
4
Efficiency gain/loss (Percent)
Sharpe 9.0%
3
6.0%
2
3.0%
1
0.0%
0
⫺3.0%
⫺1
⫺6.0%
⫺2
⫺9.0%
⫺3
⫺12.0%
⫺4
⫺15.0%
1
8
15
22 29 36 43 50 57 64 71 78
85 92 99 106 113 120 127 134 141
⫺5
Investment funds
Figure 8.4 Efficiency gain/loss measure and the Sharpe ratio for CEE investment funds
Furthermore, in the constrained case, there are several assets offering a performance not significantly different from their replicating counterpart (that is, neither superior nor inferior). This implies that, for private investors with limited market access, investment funds may offer a useful low-cost alternative in quite a few cases. It is important to note that there are cases in which the efficiency gain/loss is to a great extent higher in the no-short-selling case. This indicates the presence of performance attributes that can be achieved with short selling only – such attributes are, for example, large negative skewness. This coincides with our findings regarding the number of funds with negatively skewed returns. Next, we also had a look at the rankings produced by the Sharpe ratio. Interestingly, this measure also exhibits a relatively high rank correlation with the unconstrained as well as with the constrained efficiency gain/loss: in both cases, the value of the Spearman rank correlation index is above 0.8 (for the constrained case slightly below 0.9), which indicates a reasonably close relationship. These findings can be seen in the Figure 8.4, which shows a relatively close co-movement of the rankings produced by the Sharpe ratio and the unconstrained efficiency gain/loss ratio.
148
THE PRICE OF EFFICIENCY
Table 8.1 Investment fund strategies (peer groups) Strategy
No. of funds in peer group
Equity only
6
Bond long
17
Bond short
11
Mixed equity
28
Mixed bond
7
Mixed balanced
8
Money market
23
Real estate
14
Others (for example, guaranteed) Sum total
30 144
8.4.5 Peer group analysis After analyzing the properties of the investment funds investigated, it was decided to extend the analysis to gain a broader view of investment funds in different asset classes. From this background, for each strategy, a peer group index was built, delivering the average return of all investment funds (with naïve diversification). The role of peer groups is twofold. First, a very basic examination of peer group returns is performed. Second, we use peer group returns, together with their replicating counterparts, to calculate correlations with an investor’s hypothetical portfolio. The peer groups contain, by definition, investment funds following the same strategy. Table 8.1 contains the identified peer-group building strategies. As expected, money market, short bond, but also real estate returns show (a) a low volatility of returns within the peer group; and (b) relatively low overall levels of risk taken, so that these groups are relatively homogeneous. However, it is also true for every peer group that, because of the averaging effect, variance, skewness and kurtosis of the peer group returns are considerably lower than among the individual peer group members. This implies also lower replicating returns at the peer group level because of the lower risk exposure. The efficiency gain/loss, however, still produces satisfactory returns in such cases. It is certainly true that the higher the overall risk level, the higher the possible replicating return. To the correlation analysis, the peer group average returns were compared to a hypothetical portfolio, containing 65 percent shares, 25 percent bonds and 10 percent in money market instruments, using continuous rebalancing in order to keep asset shares constant.
ZSOLT BERÉNYI
149
Table 8.2 Correlation between an investor’s portfolio and peer groups/replicating assets Correlation with the investor’s portfolio Strategy
Peer groups
Replicating asset for the peer group
Equity only
0.127
−0.834
Bond long
0.571
0.104
Bond short
0.097
0.068
Mixed equity
0.003
0.239
Mixed bond
0.149
0.334
Mixed balanced
0.258
0.388
−0.037
0.576
Real estate
0.004
0.332
Others (guaranteed)
0.006
0.267
Money market
The aim of the action is to compare correlation coefficients between the investor’s existing (hypothetical) fund and both the investment funds in question, as well as their respective replicating counterparts. The reason behind this is that while we do not intend to replicate correlations, it might be interesting to know whether we have to consider this alternative as well (see Kat and Palaro, 2006). In Table 8.2, we give an overview of the correlations between the investor’s hypothetical portfolio and the peer groups and their replicating counterparts, respectively. It can be seen that the correlation coefficients of the replicating peer group benchmark asset are usually higher than those of the peer groups. This is very probably attributable to the fact that replicating portfolios build on the equity index and its derivatives. Thus, if the investor’s portfolio contains a considerable amount of equity investments, we have to accept a higher correlation. This is the bad news. However, the good news is that it is also possible to create assets with considerable negative correlation as well, if that is what we are aiming at.
8.5 CONCLUSION In this chapter, we investigated the performance of different investment funds on an emerging market with the efficiency gain/loss methodology. Our findings can be summarized as follows. We found that this methodology can be extended meaningfully to assess the value created by fund managers, also beyond the traditional markets most investors focus on, and even across asset classes. Emerging markets
150
THE PRICE OF EFFICIENCY
also have their own stars, among a series of inferior investments, which we believe can be identified using the efficiency gain/loss methodology. In addition, we found that, for investors with limited market access, several investment funds offer a good alternative. Next, we found that, while using the DLN settings may be useful under certain circumstances, it is clearly not necessary in the market investigated. That is, the probability distribution implied in option prices can be described fairly well by a single lognormal function. We also found that the rank correlation of the efficiency gain/loss and the Sharpe ratio is quite high. Thus, the Sharpe ratio may again provide a lowcost alternative to the higher moment measures for this particular market and time period. However, it is not robust enough for an ex ante analysis, since fund managers can outsmart it by using options. Finally, we also found that replicating portfolios, being built on an equity index and options, usually exhibit a higher correlation with the investor’s hypothetical portfolio – at least if one does not account for correlation when generating replicating portfolios. However, the good news is that by using options it is also possible to create assets with negative correlation if the analysis is extended in this direction.
REFERENCES Amin, G. and Kat, H. (2003) “Hedge Fund Performance 1990–2000: Do the Money Machines Really Add Value?”, Journal of Financial and Quantitative Analysis, 38(22): 251–74. Berényi, Z. (2003) Risk and Performance Evaluation with Skewness and Kurtosis for Conventional and Alternative Investments (Frankfurt: Peter Lang). Dybvig, P. H. (1988a) “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market”, Review of Financial Studies, 1(1): 67–88. Dybvig, P. H. (1988b) “Distributional Analysis of Portfolio Choice”, Journal of Business, 61(3): 369–93. Glosten, L. R. and Jagannathan, R. (1994) “A Contingent Claim Approach to Performance Evaluation”, Journal of Empirical Finance, 1(2): 133–60. Ederington, L. H. (1995) “Mean-Variance as an Approximation to Expected Utility Maximization: Semi Ex-Ante Results”, in M. Hirschey and M. W. Marr (eds), Advances in Financial Economics (London: JAI Press), (1) 81–98. Kat, H. M. and Palaro, H. P. (2006) “Who Needs Hedge Funds? A Copula-Based Approach to Hedge Fund Return Replication”, Working paper, Cass Business School, City University, London. Melick, W. R. and Thomas, C. P. (1997) “Recovering an Asset’s Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis”, Journal of Financial and Quantitative Analysis, 32(1): 91–115.
CHAPTER 9
Liquidity and Market Efficiency Before and After the Introduction of Electronic Trading at the Sydney Futures Exchange Mark Burgess and J. Wickramanayake
9.1 INTRODUCTION On October 28, 1999, the Floor Members of the Sydney Futures Exchange (SFE) voted to transfer SFE’s floor-based futures and options contracts to an electronic trading system. On March 4, 1999, the SPI ceased floor-trading and became screen-traded. Later, on Friday, November 12, 1999 the remainder of the floor-trading ceased, and on Monday, November 15, 1999 the Sydney Futures Exchange (SFE) became a completely automated system, with all trading conducted by SYCOM IV. The Australian Stock Exchange (ASX) became completely automated in October 1990 with its electronic trading system, SEATS (ASX website, 2000). Now that the spot (ASX) and derivative (SFE) markets have similar trading structures, a unique opportunity is provided to examine empirically 151
152
LIQUIDITY AND MARKET EFFICIENCY
whether automation has influenced the derivative markets’ operational, informational and market efficiency in comparison to the ASX. Completely automated trading structures have been introduced at various futures exchanges around the world. Despite the efficiency gains that accompany such automation, there is a reluctance to move away from open outcry trading system, citing early evidence that such exchanges were more liquid than electronic exchanges. However, more recent studies have suggested that electronic trading is superior to open outcry in many respects, including liquidity. This study provides evidence as to whether the claims by Sydney Futures Exchange on the advantages of the new structure are in fact true. The major advantage perceived is that liquidity and the price discovery process of the market will be enhanced because of the lower costs of trading, faster trade execution, cleaner information dissemination, and greater transparency with respect to prices and quotes. The outline of the chapter is as follows. Section 9.2 provides a selected literature review while identifying the four research propositions. Sample design, model and statistical procedures are given in Section 9.3, while empirical results are analyzed in Section 9.4. Concluding remarks are in Section 9.5.
9.2 REVIEW OF THE LITERATURE 9.2.1 Dynamics of a changing market structure Since the late 1980s, rapid advancements in information technology have transformed the financial markets. The improved technology has not only enhanced the computing and modeling skills used by professional investors in financial markets, but has also been a catalyst for development and change in the structure of the market’s themselves. Electronic screens in securities exchanges are replacing the traditional open outcry floor-trading structures with an automated trading system (Kofman and Moser, 1997). This change reflects the demand by market participants, based on issues centered on increasing operational efficiency1 while maintaining the market’s ability for equal or greater informational efficiency.2 An automated system claims to have benefits for market participants through lower infrastructure costs, improved price transparency, superior platforms and listing for new products, global distribution, and improvements in speed of trade execution. With the increasing implementation of automated trading structures, the important review by Massimb and Phelps (1994) raised a number of contentious issues regarding the benefits and shortcomings of electronic trading systems. At face value, the issues relating to automation appear straightforward. However, according to Massimb and Phelps (1994) automating an
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
153
existing trading-floor structure can raise doubts about its possible benefits. They have raised the issue that the dynamics of the open outcry system is not effortlessly captured or supported by the electronic trading structure, and that the introduction of electronic trading forces a trade-off between the advantages of both systems. Massimb and Phelps (1994, p. 39) suggest that: Exchanges, regulators and investors evaluating the relative merits of open-outcry and electronic matching should note, however, that the trading environment and the trade matching algorithms embedded in electronic matching fail to capture those features of open outcry that account for its success and liquidity.
Automation does offer and create operational efficiencies (see Ates and Wang, 2005), though it can be traded off against the liquidity from a trading floor system involving open outcry. The liquidity depletion is derived from the apparent impact of trading transparency that the “local traders” require to fulfill their perceived role in the market, which is to provide liquidity.
9.2.2 Liquidity The changeover from an open outcry structure to an electronic trading structure has in the past altered the liquidity of futures markets. A study by Massimb and Phelps (1994), among others, argues that the open outcry structure is more liquid. The empirical results presented by Shyy and Lee (1996) also show higher liquidity in an open outcry structure. However, the study by Pirrong (1996) supports the view that automation increases liquidity within the market. Technology is progressing rapidly, and the markets are becoming more complex. Most securities markets have begun to shift from the traditional trading floors to automated trading systems to cope with the increase in volume and to lower transaction costs, with the added bonuses of allegedly reduced human error and providing greater liquidity (Massimb and Phelps, 1994). Recently Bortoli et al. (2004) have shown that lower brokerage commissions have resulted from electronic trading in Sydney Futures Exchange. The successful transition to an automated trading system must carry across the key dynamics and attributes from the open outcry trading system (Massimb and Phelps, 1994). When considered along with the benefits of lower infrastructure costs and more efficient transaction pricing, automation seems very appealing. However, Pirrong (1996) attributes the success of a trading structure to the market’s ability to generate and maintain liquidity.
154
LIQUIDITY AND MARKET EFFICIENCY
9.2.3 Assumptions under the mixture of distributions hypothesis The study by Frino et al. (2004) found that the liquidity of automated systems decreased during times of high volatility, compared to the open-floor trading system. The mixture of distributions hypothesis (MDH) developed by Clark (1973), Epps and Epps (1976) and Tauchen and Pitts (1983) provides a theoretical framework to explain that returns and trading volume are driven by the same underlying latent news arrival, or information flow. The arrival of unexpected good news results in a price increase, and the arrival of bad news results in a price decrease. Both of these events are accompanied by above-average trading activity in the market as it adjusts to the new equilibrium. Accordingly, the absolute price change and trading volume should be positively correlated. Copeland (1976) states that the trading volume can be used as a proxy for information arrival. A theoretical explanation of the sequential information flow between stock and option markets was provided by Copeland. The price–volume relationship was also found to have significant implications for the entire futures market, as the absolute change in price is reflected by an increase in trading volume (Karpoff, 1987). Therefore, days with high volumes suggest high market volatility, and days with a low volume imply small market volatility. So, if the market index and market index futures are essentially reacting to the same information, as explained by the sequential flow of information (Copeland, 1976), the markets should both adjust to new information, and have a corresponding effect on trading volume levels (Cornell, 1981; Martell and Wolf, 1985). Prior research by Brailsford (1996) investigated the empirical relationship between trading volume, returns and volatility on the All Ordinaries Index on the Australian Stock Exchange (ASX) from April 24, 1989 to December 31, 1994. Brailsford (1996) found support for the mixture of distributions hypothesis by explaining how returns are generated, and implications of the MDH for inferring return behavior (non-normality in returns through the arrival of information) from trading volume.
9.3 OPTIONS DATA VOLUME AS A PROXY FOR LIQUIDITY The following discussion on options data attempts to establish a link with the MDH approach and further justify the utilization of the liquidity ratio analysis. Options data can be incorporated into the study of liquidity by using a technique developed by Rubinstein (1994), in that the strike prices of the options contracts can be grouped into portfolios. The portfolios are ranked in terms of moneyness that in turn can be used to assess the liquidity of the option contract traded. Moneyness is a term used to describe the relative closeness of the strike price to the stock price. The advantage of
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
155
using this technique is that it allows a simplification of the variable to be tested in the data set, to provide a more in-depth analysis. This approach will be followed in this study. Each portfolio will be sorted by the strike price:stock price ratio to determine moneyness, and then further sorted by the option contracts based on time to expiration. This approach will provide an insight into volume liquidity in the options market since the automation of the SFE. Anthony (1988) investigates the causal effects of the sequential flow of information between the stock market and options market, using their trade volumes based on the assumptions of the mixture of distributions hypothesis discussed earlier. Daily volume data of twenty-five companys’ call options for the period January 1982 through June 1983 listed on the CBOE (Chicago Board Options Exchange) were used. Anthony’s study is useful, as it suggests that liquidity is generated by information arrival and thus can be identified through volume changes. The informational effects can be held constant, as it is incorporated into the prices of both markets, and therefore ratio analysis, as discussed earlier, can provide an insight into the effects of the automation of the SFE. Jarnecic (1999) also investigates the lead–lag relationship of intraday data between trading volumes of stocks and stock options of the ASX and ASX options (note that these ASX options are not traded at the SFE). The results indicate that any lead relationship was found to have been eliminated when frequent trading occurred in both markets. Jarnecic’s study appears to highlight the overstating of the lead–lag relationship through the examination of intraday data.3 The use of daily data as in this investigation will not be prone to such results. In this study, liquidity ratios will be constructed from the volume data of the All Ordinaries Index, and the share price index (SPI) futures and option contracts. These ratios will be used to investigate the liquidity of the SFE both before and after automation. The mixture of distributions hypothesis will be used as a theoretical explanation to categorize the ratios into volume groups based on their size. Thus each group corresponds to a level of market volatility, and can therefore be used to analyze whether the futures and options contracts increase or decrease in liquidity in periods of high and low volatility. It is likely that there will be no change in the liquidity ratios. This is based on the assumption that the automation transition in the SFE has been conducted carefully to maintain the existing dynamics of the old floor structure and enhance them with the benefits of automation. It should be noted that Australian stock and futures markets are small when compared to larger international markets that have substantially greater trading volumes. Based on the above discussion, the following research proposition is drawn: Research Proposition 1: The liquidity and liquidity ratio will change after the SFE became an electronic trading system.
156
LIQUIDITY AND MARKET EFFICIENCY
9.3.1 Price discovery and operational efficiency of a market structure Price discovery is the differential reaction of the different markets to new information, and the rate at which the new information is incorporated into price. Fama’s (1970) semi-strong form4 of the “efficient market hypothesis” (EMH) states that the price discovery process must be very small, to prevent earning abnormal returns. Therefore the market structure is important, as it must optimize the price discovery process between markets to provide no significant excess returns to market participants (O’Hara, 1995). Market efficiency5 also depends on the notion of operational efficiency: this is the market’s ability to provide liquidity and speed of execution, and to minimize transactional costs. The concepts of price discovery and operational efficiency should complement each other according to Freund et al. (1997). Their study on the Toronto Stock Exchange (TSE) measured the speed of transmission of daily and monthly returns data of twenty-five stocks for the period January 1976 to March 1981, the period during which the TSE introduced the new electronic trading system. This change in trading system allowed Freund et al. (1997) to determine any change in market efficiency resulting from a change in operational structure efficiency. In a recent study on France, Germany, South Korea and the UK, Copeland et al. (2004) rejected the random walk hypothesis for both open outcry and electronic trading systems. Their results suggest that there has been no increase in efficiency as a result of the introduction of electronic trading.
Cointegration
As stated earlier, the objective of the present study is to investigate the market efficiency of the Sydney Futures Exchange (SFE) both before and after automation. In order to provide an insight on the pricing of information with regard to the futures and spot market, and to determine the existence of any change in the relationship between the two markets as a result of automation, cointegration methodology can be used. The cointegration approach can be used to investigate empirically the relationship between the spot price and futures price from their corresponding markets, to investigate the predictive price effect of the quoted futures contract on the underlying asset price effect. Mananyi and Struthers (1997) examined the semi-strong form of the efficient market hypothesis (EMH) of the monthly spot and futures cocoa prices traded on the London Futures and Options Exchange from January 1985 to December 1991. Using cointegration methodology, the study found that behavior of cocoa prices was inconsistent with the semi-strong form of EMH. It could not find evidence that the spot and futures prices were cointegrated
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
157
in a long-term equilibrium relationship in the cocoa markets. Mananyi and Struthers (1997) attributed these inconsistent findings to irrationality associated with the complex institutional arrangements characterizing the type of futures traded. Kempf and Korn (1998) examined the intraday minute-by-minute integration between the German stock index DAX and DAX futures from June 1995 to March 1996. The importance of their study lies in the fact that the DAX was being traded simultaneously in both the open outcry trading system and the screen-traded system. The investigation allowed the examination of the relationship between spot and futures markets for different spot-trading systems. It was found that the markets are more closely integrated when both the DAX Index and DAX futures are screen-traded; however, the performance of both systems at times of high and low trading activity does not seem to affect either system. Kempf and Korn (1998) attribute the higher degree of integration to the delayed information flow and order execution lags on the floor-traded markets. Daily price data were used by Ackert and Racine (1999) to examine whether equity spot and future markets were cointegrated. For this they used data on the Standard and Poor’s S&P 500 index and associated futures contract closing prices from January 4, 1988 to June 30, 1995. It was concluded that Standard and Poor’s S&P 500 index and futures price and interest rate are cointegrated, which is consistent with the view that no-arbitrage assumption is reasonable in equity markets. Groenewold (1997) reports the results of various tests of the EMH using daily data (as used in this study) of the Australian Statex Actuaries Price Index, Statex Actuaries Accumulation Index, and the All Ordinaries Price and Accumulation Indexes, against New Zealand’s NZSE-40 Index, and NZ Gross Index for the period 1975 through 1992. The study tests semi-strong form efficiency using Johansen cointegration tests; and the two market’s lead–lag relationship by testing with Granger causality. The cointegration and lead–lag methodology (as used in this study) found that indexes across regions were not cointegrated, and not Granger-caused over the period examined. Turkington and Walsh (1999) used intraday frequency data for the period January through December 1995 to examine the Australian All Ordinaries index and the SPI (Share Price Index) futures, to investigate the impact of market structure on trading through cointegration and causality. They found that, over the sample period, the market was cointegrated and that causality was bi-directional. The previous Australian studies are important to this investigation, for two reasons. First, both studies (those of Groenewold, and Turkington and Walsh) used similar methodologies that provided an insight to the expected results of the pre-automation period examined. Second, the present study plans to examine whether a change in the trading structure will change the
158
LIQUIDITY AND MARKET EFFICIENCY
cointegrating and causal relationships found in the previous studies. Based on the literature reviewed here, the following research propositions can be formulated: Research Proposition 2: The share price index (SPI) is cointegrated with the All Ordinaries Index six months before the introduction of electronic screen trading at the Sydney Futures Exchange. Research Proposition 3: The share price index (SPI) is cointegrated with the All Ordinaries Index six months after the introduction of electronic screen trading at the Sydney Futures Exchange.
The lead–lag relationship
Lead–lag is a term used to describe the relationship between two markets. The relationship shows the flow of information being simultaneously and unbiasedly incorporated into each market’s prices (Fama, 1970). If the price in each market incorporates the new information at different intervals of time, the market that does so in the quickest way will appear to lead the other. A substantially debated stream of finance literature stems from the issue of price discovery, and that markets are not semi-strong form efficient, as they do not simultaneously incorporate new public information into their prices (Grünbichler et al., 1994). As a result, it is possible for one trading market structure to discover prices more rapidly than another type. Does the open outcry floor-trading structure or electronic screen-trading structure provide a faster price discovery, or create a lead–lag relationship? There is significant subsequent research investigating lead–lag relationships between asset and derivative markets. Much of the research by Grünbichler et al. (1994), Kofman and Moser (1997) and Turkington and Walsh (1999) have all examined stock indexes and the stock index futures from various markets from around the world, to determine whether operational efficiency has improved as a result of a structural change in the market. The data used by Brooks et al. (1999) were daily returns for the FTSE 100 index and Index futures contracts in the UK for the period January 1985 to December 1993, and the S&P 500 Index and Index futures contracts in the USA for the period January 1983 to December 1993. They selected the daily data set as it would not be likely to overstate the lead–lag relationships that occur with intraday data. The results showed that, contrary to the use of traditional intraday data and methodology, the periods where the futures market leads the cash market are few and far between. Any lead– lag relationship detected will not last long. They conclude that the results are consistent with the prediction of the standard cost-of-carry model6 and market efficiency.
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
159
The present study on trading structures in the SFE will use an approach similar to that in Brooks et al. (1999), to determine the lead–lag relationship of the S&P/ASX 200. Based on their model, data on the All Ordinaries Index and the share price index futures (SPI) in the SFE will be used to identify cross-correlations and bi-correlations. For this purpose, daily data will be used. It is likely that there will be no change in the lead–lag relationship between the ASX and the SFE as the new system was implemented gradually over several months (March to November 1999). This leads to the development of the next research proposition: Research Proposition 4: The electronic market lead–lag relationship will remain unchanged after the change from the trading floor system to the electronic trading system.
9.4 SAMPLE DESIGN 9.4.1 Data sources The data for the period from January 2, 1998 to August 1, 2000, as required for the study, has been collected from two sources. The event date is defined as March 3, 1999. The event date is the day on which the SPI transferred from being floor-traded to being an automated electronic system. This event date should not to be confused with the full automation of the system that occurred on November 15, 1999. The Sydney Futures Exchange data for the Share Price Index consisting of futures and call option contracts was downloaded from the Sydney Future Exchanges website (http://www.sfe.com.au). The call option data has been used since call options in general are traditionally more frequently traded and have a greater depth of liquidity when compared to put options. The historical daily data files downloaded from the SFE website are daily date-labelled open, high, low, close prices, expiry, strike price and volume for night and day with settlement, open interest, risk and volatility for the SPI futures and options contracts. The Australian Stock Exchange data was collected from the DataStream database that contained all the relevant data required for this research. The information required for the All Ordinaries Index was daily-dated open, high, low, close price and volume. The secondary data used in the investigation, such as ratios, explained later in the chapter, were calculated from the primary data sources as discussed above. As option data is utilized in the testing of Hypotheses 1 and 2, it has to be arranged properly to achieve the best results. As option data have a large number of option contracts with differing strike prices traded at any one time, the contracts must be organized into moneyness portfolios. As shown
160
LIQUIDITY AND MARKET EFFICIENCY
Table 9.1 Portfolio moneyness Portfolios
Ratio
Striking price range FOTM
OTM
ATM
ITM
FITM
>9%
9% to 3%
3% to −3%
−3% to −9%
<−9%
Source: Rubinstein (1994).
in Table 9.1, the contracts are organized into a ratio corresponding to the underlying spot index. This indicates the moneyness of the contract, and allows the strike prices of contracts to be simplified into five basic moneyness portfolios (Rubinstein, 1994). These are “Far out of the money” (FOTM); “Out of the money” (OTM); “At the money” (ATM); “In the money” (ITM); and “Far in the money” (FITM). The option moneyness portfolios have classified the strike prices of the option contracts into groups relative to the spot index at that point in time. The option contracts have different dates to expiration. These have also been isolated into groups of expiration. Only the call option contracts with expirations of one month, two months and three months will be investigated in this study. This is because option contracts with dates to expiration outside this period (three months) were found to show highly inconsistent behavior.
9.4.2 Methodology The methodological approach is aimed at two aspects of the automation of the Sydney Futures Exchange. The first aspect is the analysis of the market’s liquidity by analysis of the volume. The second aspect, as discussed earlier, is the price discovery process utilizing cointegration and causality theories in examining these relationships between the Australian Stock Exchange and the Sydney Futures Exchange. The following section provides a discussion on variable specification, model and statistical procedures.
9.4.3 Model and statistical procedures Liquidity
The core of this research centres on determining the effect of the structural change in trading at the SFE, and whether there has also been a change in liquidity. Measurements of the pre-automation group against the postautomation group can provide an indication as to the liquidity effects over these two periods. To test the liquidity of the market, the call option data will be based on moneyness portfolios, as discussed earlier.
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
161
Data organization
As mentioned earlier, the data will be organized into pre-automation and post-automation groups as in Freund et al.’s (1997) study, as it will provide sub-groups to compare with one other. Within these two groups, the data will be further classified into portfolios of moneyness and expiration (Rubinstein, 1994). The option moneyness portfolios will be examined in full, with not only descriptive statistics, but also by investigating the mean contract size as per sample and per days traded. Analysis of variance will be carried out to test the difference of means through the moneyness portfolios, along with the analysis under the assumptions of the MDH. The mixture of distribution hypothesis (MDH) provides a theoretical assumption to explain why an absolute change in price occurs with the increase in volume, as both are related to the amount of information in the market. Therefore, if there is a large amount of information in the market (highly volatile), the price will accordingly fluctuate unbiasedly, reflecting the true value, and therefore it requires a proportional amount of trading volume to move the market to reach this equilibrium (Richardson and Smith, 1994). The data will therefore, categorize the days by volume size so as to represent market volatility. Thus days of high market volatility will be shown to have large volume size within the group, and the small-volume days will represent low market volatility. This will allow an investigation of liquidity change to be measured at different stages of market volatility, to determine whether the SFE has benefited from the automation of the open outcry trading floor to become an electronic, screen-traded system.
Descriptive statistics
The comparison of the SPI futures contract and All Ordinaries Index for the daily volume will be based initially on an analysis of the descriptive statistics. This analysis will provide an insight into the nature of the All Ordinaries Index and futures contract variables. The ratio tests (described in detail below) will provide a more coherent and appropriate analysis of the liquidity change in the index and futures index across the sub-periods. Any liquidity change found from the ratio investigation will be considered more influential than simply relying on the descriptive statistics results. Further insight will be provided by the analysis of variance (ANOVA) tests performed. MDH will be used (as discussed above) as a theoretical assumption to investigate differing market volatilities, by groups of volume size. This provides a unique methodology to analyze the liquidity of the market when it is highly volatile (large-volume groups), and at low market volatility (small-volume groups).
162
LIQUIDITY AND MARKET EFFICIENCY
Ratio analysis
Ratio analysis is the second aspect to the investigation of liquidity change at the Sydney Futures Exchange. This is done by statistical analysis of a combination of ratios between the call option and futures index (daily and nightly) volumes, holding expiration to maturities constant against the each other and against the daily index volumes. The method of calculation of the liquidity ratios is given in Appendix on page 179. A ratio of the futures volume to the spot volume will be calculated for two periods (pre-automation and post-automation) to determine a simple liquidity ratio. If the ratio increases or decreases after automation, this can be seen as signal of a change in overall liquidity resulting from automation at the SFE. The ratios will then be subjected to ANOVA tests to compare the means, and determine whether there has been any significant results from the cross evaluation of the two periods. ANOVA is a statistical method for determining the existence of differences among several population means. Using an ANOVA test assumes independent random sampling for each of the r populations, and that the r populations are normally distributed, with means µi that may or may not be equal, but with equal variance σ 2 . The null hypothesis for the ANOVA test is that the mean of population one is equal to the mean of population two, with the alternate hypothesis stating that the population mean one does not equal population mean two.
Regression analysis
Regression analysis aims to complete the investigation of liquidity to provide a concise statement concluding the advantages sought or lost in the introduction of electronic trading at the SFE. In regression analysis, the dependent variable is the SPI futures to All Ordinaries Index ratio. The independent variables used are the open interest, settlement price of the futures market, the “at the money” (ATM) ratio, and the ATM volume. Dummy variables will be included to represent the pre- and post-periods, along with dummy variables for the mixture of distribution groups representing descending volume size as a proxies of different levels of market volatility. This regression equation follows: y = α + (β1 In (OI)) + (β2 In (SETT)) + (β3 Dummy1 Pre_Post) + (β4 Dummy2 VOL_1) + (β5 Dummy3 VOL_2) + (β6 Dummy4 VOL_3) + (β7 Dummy5 VOL_4) + (β8 Dummy6 VOL_5) + et
(9.1)
where y = share price index/All Ordinaries Index; OI = open interest; SETT = settlement price of the share price index; Dummy1 Pre_Post = dummy variable for the pre-automation and post-automation period (0 if it is
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
163
the pre-automation period, and 1 if it is the post-automation period); Dummy2 VOL_1 = dummy variable for the largest volume group (0 if it is group volume 1, and 0 otherwise); Dummy3 VOL_2 = dummy variable for the second largest volume group (0 if it is group volume 2, and 0 otherwise). Dummy4 VOL_3 = dummy variable for the third largest volume group (0 if it is group volume 3, and 0 otherwise). Dummy5 VOL_4 = dummy variable for the fourth largest volume group (0 if it is group volume 4, and 0 otherwise). Dummy6 VOL_5 = dummy variable for the fifth largest volume group (0 if it is group volume 5, and 0 otherwise); and et = random error term. These results, found from the regression analysis, can be compared with the ANOVA results. This concludes the methodology used in this study to investigate the liquidity of the SFE before and after the introduction of electronic trading. The following section explains the methodologies used to investigate the price discovery process of the Sydney Futures Exchange.
9.4.4 Cointegration Unit root tests
Following the theory of non-stationary time series developed by Engle and Granger (1987), cointegration of a linear combination of two or more series can occur. Two unit root tests will be conducted in this study, to determine whether the time series is stationary or non-stationary. The first test is the Augmented Dickey–Fuller (ADF) test; with the second test being the Phillip– Perron (PP) test. The null hypothesis being tested is that there is a unit root in the variable (non-stationary), and alternate hypothesis stating that there is no unit root (stationary). Initially, all the variables are to be categorized into pre-automation, postautomation and the total period, as was done in the study by Freund et al. (1997). The ADF and PP tests will then be performed on the sub-groups. However, to increase the precision of testing, the log of the variable will also be taken. Analyzing the results for each period and then cross-analyzing them against the results for the other periods will determine if the variables are non-stationary. The variables that are found to be integrated of order one, I (1) in all three test groups will be further tested for cointegration and causality. Johansen cointegration test
Engle and Granger (1987) showed that a linear combination of two or more non-stationary series might be stationary. For this study, the Johansen cointegration test will be performed to identify the number of cointegrated
164
LIQUIDITY AND MARKET EFFICIENCY
relationships (vectors) in the equations. The stationary linear combination is called the cointegrating equation, and may be interpreted as a long-run equilibrium relationship between the variables. In the procedure developed by Johansen and Juselius (1990), the likelihood ratio is compared with the critical value at the given level of significance. This technique will be used as it is relatively more powerful than the two-step procedure proposed by Engle and Granger (Taylor and Sarno, 1997). Following Johansen and Juselius (1990) the Vector Autoregressive (VAR) system can be written as Yt = ξ + 1 Yt−1 + 2 Yt−2 + · · · · · + k − 1 Yt−k+1 + k Yt − k + εt
(9.2)
where Yt is a k × 1 vector of variables, are k × k coefficient matrices, ξ is k × 1 vector of constants and εt is a vector of disturbance or error terms. The long-run static equilibrium corresponding to Equation (9.2) is Y = 0
(9.3)
where the long-run coefficients matrix is defined by 1 − 1 2 − · · · · · − k =
(9.4)
Equity and futures prices cannot be cointegrated unless the time series is non-stationary. The unit root tests discussed in the previous section will uncover the properties of the time series. The Johansen procedure tests the restrictions imposed by cointegration on the unrestricted Vector Auto Regression (VAR) involving the series. If we cannot reject the hypothesis that the number of cointegrating equations is zero, the series is not cointegrated. To allow for the possibility of changes over time in the cointegrating relationship over the two sub-periods, the Johansen (1991) framework can be used. The first sub-group is, as mentioned earlier, pre-automation, and the second sub-group post-automation. This sub-period approach to sub-grouping was also performed in the study by Ackert and Racine (1999). In conclusion, Johansen’s procedure will be used to verify the existence of cointegration, and examine the number of cointegration vectors and the coefficients of the long-run equilibrium relationship. Lead–lag relationship (causality)
Figure 9.1 shows the information flow through the ASX and SFE. The Granger Causality (Granger, 1969, 1981) test in this study aims to provide and establish the relationship of price discovery from the asset market to the futures market on a daily basis. The implementation of this methodology will, we hope, determine whether the structural alignment of the SFE systems has had any effect on the price discovery process.
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
Night SPI
Day AOI open
Day AOI close
Close
165
Night SPI Open
Day SPI open
Day SPI close
Figure 9.1 Information flow in market prices overtime through the ASX and SFE Note: AOI – All Ordinaries Index; SPI – Share Price index. The Granger (1969, 1981) causality test will be run on the two sub-periods (pre- and post-automation). This will be done to establish whether any causality of variables is present. This result will then be contrasted with the post-automation results to establish whether the price discovery process has been maintained, or increased/decreased the informational flow between markets, to lead one price against another.
9.5 ANALYSIS OF RESULTS 9.5.1 Descriptive statistics The discussion begins with the presentation of the descriptive statistics as shown in Tables 9.2 and 9.3. The objective of this section is to provide an insight into the nature of the volume variables through the presentation of their statistical characteristics. These results are presented for the preautomation period (represented by “PRE” in all the tables) from January 2, 1998 to March 3, 1999. The post-automation period (represented by “POST” in all the tables) from March 3, 1999 to August 1, 2000. Table 9.2 reports the volume liquidity variables, while Table 9.3 presents the descriptive statistics of the volume liquidity ratio variables. As shown in Table 9.2, the post-automation statistics depict greater volume averages (means), along with greater standard deviations and trading ranges than those of the pre-automation period. Both the skewness and the kurtosis of the post-automation period are higher when compared with those of the pre-automation period. Table 9.3 presents the ratios of the variables included in Table 9.2. They aim to isolate the relative change of one volume in relation to the other volume, allowing for the comparison of trading activity of each market (futures market and share market). The volume liquidity variables from Table 9.2 are used in the calculation of the liquidity ratios in Table 9.3.
166
Table 9.2 Descriptive statistics for the volume liquidity variables Statistics PRE POST on liquidity ATM SPI ATM SPI
PRE AOI
POST AOI
PRE SPI Futures
POST SPI Futures
Mean
208.46
332.95
211070.20
236418.19
10390.62
10671.72
Standard error
22.26
35.19
3243.51
3710.86
214.77
195.55
Median
65.0
59.5
206381.0
224115.0
9967.0
9963.5
Standard deviation
323.38
535.94
54948.61
67411.17
3663.63
3552.38
Sample variance
104576
287228
Kurtosis
4.73
7.08
2.15
4.10
1.23
Skewness
2.16
2.50
0.74
1.33
0.80
1.23
Range
1764
2894
449689
555846
25096
23852
3019349809 4544266018 13422165 12619411 2.71
Minimum
1
1
31339
67891
807
4332
Maximum
1765
2895
481028
623737
25903
28184
Notes: Descriptive statistics are for the volume variables stated above: share price index (SPI); All Ordinaries Index (AOI); “at the money” option contracts (ATM).
Table 9.3 Descriptive statistics for the liquidity ratios Statistics on liquidity ratios
PRE ATM/SPI Futures
POST ATM/SPI Futures
PRE ATM/AOI
POST ATM/AOI
PRE SPI Futures/ AOI
POST SPI Futures/ AOI
Mean
0.464
0.473
0.348
0.355
0.751
0.747
Standard error
0.015
0.015
0.011
0.012
0.002
0.001
Median
0.500
0.488
0.375
0.374
0.752
0.746
Standard deviation
0.203
0.221
0.151
0.168
0.028
0.026
Sample variance
0.041
0.049
0.023
0.028
0.001
0.001
Kurtosis
−0.932
−1.295
−0.927
−1.300
0.473
2.351
Skewness
−0.263
−0.190
−0.268
−0.164
−0.320
−0.497
Range
0.790
0.774
0.568
0.578
0.179
0.210
Minimum
0.071
0.073
0.055
0.055
0.647
0.606
Maximum
0.861
0.848
0.623
0.633
0.825
0.816
Notes: The natural logarithms of all the liquidity ratios were taken to provide statistics. Variable definitions are the same as given in footnotes to Table 9.2.
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
167
In Table 9.3, the post-automation mean values and standard deviations reveal that “at the money” (ATM) option contract volume ratio (columns 2 through 5) has increased in proportion to the change in the other volume denominator (the SPI futures and the All Ordinaries Index – AOI). However, the post-automation means and standard deviations infer the opposite about the SPI futures, in that volume ratio has marginally decreased relative to the pre-automation volume ratio (last 2 columns of Table 9.3). The ATM ratios have seen an increase in kurtosis and a reduced skewness in the post-automation results (columns 2 through 5). The SPI futures/AOI ratio (columns 5 and 7) have increased in kurtosis, skewness and range after automation. The next section will endeavor to examine whether liquidity in the Sydney Futures Exchange has changed. Overall, Tables 9.2 and 9.3 suggest that the market seems to have been more active in the post-automation period, based on trading volumes and volume ratios. This could be because of an increase in the level of trading related to new information entering the market, such as the information relating to the technology stock crash in early 2000.
9.5.2 ANOVA results The examination of the pre-automation and post-automation periods involves the use of ANOVA tests. ANOVA is a statistical test to determine the existence of differences among sample means. An ANOVA test assumes that independent random sampling for each of the r sample, with the r sample in the study being normally distributed, with means µi that may or may not be equal, but with equal variance σ 2 . Thus the null hypothesis test of an analysis of variance states that the mean of sample one is equal to the mean of sample two. The alternate hypothesis states that the mean of sample one does not equal the mean of sample two. The pre-automation mean will be ANOVA-tested to determine if it is equal to the post-automation mean. This analysis will reveal any significant change in the volume variables over the two periods surrounding the automation of the Sydney Futures Exchange. The ANOVA results are presented in two panels in Table 9.4. Panel A of Table 9.4 is the option moneyness portfolios.7 Panel C of Table 9.4 presents the ANOVA test results for the liquidity ratios given earlier in Table 9.3. Under the assumption of the mixture of distribution hypothesis (MDH),8 Table 9.4 can be used to examine the change in liquidity relating to the change in market volatility. The MDH approach aims to examine whether the post-automation period mean daily volume has significantly increased relative to the pre-automation mean. The results can then be compared to determine the level of trading after automation to a particular level of market volatility. The following section provides an analysis of the option contract statistics and ANOVA tests on the above lines.
168
LIQUIDITY AND MARKET EFFICIENCY
Table 9.4 ANOVA tests Sample size
Panel A: Option moneyness Pre ATM (1) Post ATM (1)
Mean
ANOVA: Single Factor ( p-value)
507 706
86.3432 109.5864
0.04061∗∗
Pre OTM (2) Post OTM (2)
50 70
48.2800 272.1143
0.00140∗∗∗
Pre FITM (3) Post FITM (3)
108 126
224.3796 125.5794
0.03522∗∗
Pre ITM (3) Post ITM (3)
338 494
122.2781 180.9919
0.01092∗∗
Pre ATM (3) Post ATM (3)
209 402
119.6459 166.7463
0.07833∗
Pre ATM (1a) Post ATM (1a)
211 232
208.4597 332.9483
0.00363∗∗∗
Panel B: Liquidity of AOI Pre AOI Post AOI
287 330
211070.2021 236418.1939
Panel C: Liquidity ratios Pre ATM/AOI Post ATM/AOI
209 231
0.0011 0.0014
0.0990∗
Pre Futures/AOI Post Futures/AOI
287 330
0.0508 0.0464
0.000884∗∗∗
Pre Log Futures/AOI Post Log Futures/AOI
287 330
0.7513 0.7468
0.0381∗∗
5.24017E-07∗∗∗
Notes: (1) denotes contractual expiration of 1 month: (1a) denotes contractual expiration of 1 month (refined data set for the use for ration analysis); (2) denotes contractual expiration of 2 months; (3) denotes contractual expiration of 3 months; ATM = at the money; OTM = out of the money; ITM = in the money; FITM = far in the money; AOI = All Ordinaries Index; ∗∗∗ Denotes rejection at the 1% significance level; ∗∗ Denotes rejection at the 5% level; ∗ Denotes rejection at the 10% level.
Option contracts
In Table 9.4 ANOVA test results are presented. The null hypothesis is that the pre-automation variable mean is equal to the post-automation variable mean. The results presented are those that rejected the null hypothesis. Table 9.4 (Panel A) reveals that mean values of the post-automation option moneyness portfolio are greater than the mean values of post-automation option moneyness, apart from pre-automation and post-automation FITM9 (three months to expiry portfolios). In general, the results show the option contracts that are “at the money” (ATM) or close to the money (“in the money” and “out of the money”) have mean values that reject the null
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
169
hypothesis. Even the option contract with the decreased mean values in the post-automation period, the “far in the money” (FITM with three months to expiry), is found to be significant at the 5 percent level. This may have occurred because of a shift in market sentiment about likely events in three months ahead. However, the closer to the money contracts (ITM and ATM) with three months to expiry also have mean values that have increased significantly. Overall, these result show that there has been a liquidity change in the SFE in accordance with Research Proposition 1, with the evidence showing an increase in volume trading in the post-automation period. In relation to the All Ordinaries Index (AOI), the null hypothesis is rejected, as shown in Table 9.4 (Panel B). The AOI may have had a significant mean value increase as the market had more trading volume generally in the post-automation period (such as the trade activity surrounding the technology crash in April 2000). Table 9.4 (Panel C) also presents significant results for the liquidity ratio tests. The ANOVA results are consistent with the earlier findings using descriptive statistics. The null hypothesis is rejected to find a statistically significant increase in the AOI volume ratio means in the post-automation period. The “at the money” option contract with one month to expiry relative to the market index (ATM/AOI) also rejects the null hypothesis to find that the post-automation mean value increases. Further consistency with the descriptive statistics is shown with the significant SPI futures to AOI liquidity (log futures/AOI) ratio mean value decreasing in the post-automation period in a statistically significant manner. Overall, the results given in Table 9.4 seem to imply that option trading appears to have become increasingly accepted, compared to the rate of application of the other derivatives product (futures contract) examined. The evidence supporting this notion comes from the liquidity ratio ANOVA results (Table 9.4, Panel C) for the increase in the “at the money” options result, when contrasted to the decrease in the futures contract (results show an inverse relationship). Based on the above empirical evidence, there is clear support for Research Proposition 1, since there has been a significant change in liquidity and liquidity ratios. The next section will further examine whether the change in liquidity and liquidity ratios has been a result of the electronic trading system, or whether the market has been more volatile in the post-automation period, resulting in higher volumes traded.
ANOVA ratio results ranked to market volatility
This section attempts to provide a better insight to the previous ANOVA tests. The aim is to determine whether the increase in volume is related to automation in the SFE, or to a more volatile market. This will be done by an
170
LIQUIDITY AND MARKET EFFICIENCY
examination of the pre-automation and post-automation liquidity ratio variables in the form of volume groups, under the theoretical assumptions that underlie the mixture of distributions hypothesis (MDH). This will enable a more concise analysis of the liquidity ratios, as the liquidity performance can be measured at different market volatilities. MDH is used to explain why the relationship of the absolute change in price is positively correlated with trading volume. In simple terms, it takes a proportional amount of volume to move the market price to reflect the new information adequately (Harris, 1982; Tauchen and Pitts, 1983). Therefore, volume size can be used as a proxy for market volatility. Volume groups have been generated to control for the impact of informational content for the market. They can be categorized by ranking the volume based on size. Volume groups will have equal sample sizes. The first groups will contain the days of the largest volumes. This is because a large number of trades for the days were required to establish equilibrium within the market. The following groups will then be nominated in descending volume size to a group, with volume group 5 containing the smallest volume sample. When assessing the groups for the ratio analysis, the denominator in the ratio will be ranked. The volume groups in the liquidity ratio will be assessed on the volume size of the denominator. Such groups will therefore be representative of the information arrival in the market, and the necessary amount of volume to change the price to reflect the new information. This methodology enables information to be held constant, as a level of market volatility, and thus allow the corresponding volume group pre-automation and post-automation mean values to be ANOVA tested. This will determine the level of liquidity in the market at differing levels of market volatility. Hence it is possible to find out whether there has been a change of liquidity since the introduction of electronic trading at the SFE. The significant results are found in Table 9.5. Results of ANOVA tests found only four of the five volume groups rejected the null hypothesis (the mean values of the pre-automation variable equals the mean value of the post-automation variables). Statistically significant findings in Table 9.5 seem to add further weight to the previous ANOVA test results presented in Table 9.4. The null hypothesis for the mean values being equal to one another have been rejected. As shown in Table 9.5, the post-automation large-volume groups (groups 1 and 2) “at the money” (ATM) liquidity ratio mean values increase relative to the pre-automation volume means. This result suggests that option contracts in the SFE are more heavily traded on days of high market volatility at the Australian Stock Exchange (ASX). The SPI futures liquidity ratio is significant, as the null hypothesis is rejected to reveal that a decrease in the medium-to-small-volume group (groups 3 and 4) mean values for the post-automation period.
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
171
Table 9.5 ANOVA results for liquidity ratio groups Volume group
ANOVA summary for corresponding volume group Group variable
Group sample size
Average
p-value
3
PRE Night Futures/AOI POST Night Futures/AOI
42 42
0.00595 0.00461
0.03380∗∗
1
PRE ATM/Futures POST ATM/Futures
42 42
0.01110 0.02696
0.01427∗∗∗
2
PRE ATM/Futures POST ATM/Futures
42 42
0.01614 0.03715
0.01829∗∗∗
2
PRE Futures/AOI POST Futures/AOI
42 42
0.04979 0.04266
0.05886∗
3
PRE Futures/AOI POST Futures/AOI
42 42
0.05291 0.04726
0.06909∗
4
PRE Futures/AOI POST Futures/AOI
42 42
0.05426 0.04849
0.052025∗
2
PRE ATM/AOI POST ATM/AOI
42 42
0.000652 0.001297
0.0954∗
Notes: ∗∗∗ Denotes rejection at the 1% significance level; ∗∗ Denotes rejection at the 5% significance level; ∗ Denotes rejection at the 10% significance level; AOI = All Ordinaries Index; ATM = at the money.
In summary, based on the results presented in Table 9.5, it can be suggested tentatively that the option contracts appear to have been embraced by market participants on days of high market volatility, as an apparent substitute for futures trading. This is shown in the significant decrease in the futures volumes in the post-automation period for days with mediumto-small market volatility (groups 3 and 4 in Table 9.5). Overall, the results seem to show that the market participants have a shifting preference in their choice of derivative products after automation. The option market has seemingly become more liquid on days with high volatility (groups 1 and 2 in Table 9.5), while futures trading has become less liquid on days of medium volatility, relative to the reaction of the All Ordinaries Index. As shown above, the ANOVA results in Table 9.5 also support Research Proposition 1. The results from the ANOVA tests clearly show that there has been a change in liquidity at the SFE since the start of electronic trading. This change in liquidity appears to be independent of the more volatile market, as determined by the volume groups. To give further strength to the acceptance of Research Proposition 1, the next section will provide a definitive conclusion about the change in liquidity by regression analysis. It aims to provide further evidence on the automation process of the SFE, and determine whether the change in liquidity is a result of volatile market conditions.
172
LIQUIDITY AND MARKET EFFICIENCY
Ratio analysis
The final set of tests on the examination of liquidity at the Sydney Futures Exchange is based on regression analysis. The tests aim to provide additional evidence regarding the presence of liquidity change after the introduction of electronic trading at the SFE. A series of regressions have been run using the futures-to-AOI volume (SPI futures/AOI) ratio as the dependent variable (see Table 9.6). Independent variables used in the regressions are open interest, futures settlement price, “at the money” option portfolio volume, and the “at the money” volume portfolio as a ratio of the All Ordinaries Index. As shown in Table 9.6, the regression analysis shows that dummy variables representing the volumes of the pre-automation and post-automation periods are statistically significant, in line with the volume groups shown in Table 9.5. The regression diagnostics given in Table 9.6 are satisfactory. The pre-automation to post-automation dummy (pre–post) coefficient is also significant. This indicates a change in the SPI futures volume relative to the change in volume in the spot index (AOI) after the automation of the SFE. The significant results are supported by the market volatility volume groups being related to an increase in the option portfolios. The smaller volume group (group 5) has been found to be significant and negatively related to the dependent variable. This appears to demonstrate that there has been a change in futures volume trading. The “at the money” (ATM) contracts, as a ratio of the All Ordinaries Index (ATM–AOI in Table 9.6) is significant in determining the dependent variable. In relation to Research Proposition 1, it has been demonstrated clearly from the use of dummy variables that there has been a change in liquidity and liquidity ratios since the automation of the SFE. The results of the regression analysis agree with the above findings in the descriptive statistics and ANOVA analysis to support Research Proposition 1. The results discussed above lead the study to conclude that since the automation of the Sydney Futures Exchange, liquidity has decreased in the SPI futures market and increased in the SPI call options market relative to the derivative product traded. There appears to have been a substitution effect in the futures market for “at the money” option contracts. This effect has increased at times of high volatility of new information entering the market, when the option liquidity has increased significantly, whereas the futures liquidity has declined. This can be seen in the overall lower trading volumes in the post-automation period for medium to smaller volatile days of market participants reacting to news announcements. Therefore it seems to be the case that the options market has become more liquid since the automation of the SFE, while the futures market is not as liquid as it was before automation.
Table 9.6 Summary of regression analysis on liquidity Variable
SPI Futures/AOI
SPI Futures/AOI
SPI Futures/AOI
SPI Futures/AOI
SPI Futures/AOI
SPI Futures/AOI
SPI Futures/AOI
SPI Futures/AOI
SPI Futures/AOI 0.0380(0.0000)
C
0.0372(0.4981)
0.03599(0.5115)
0.0335(0.5411)
0.02457(0.6525)
0.0048(0.9314)
0.0037(0.9462)
0.0014(0.9796)
−0.0077(.8890)
log(Open interest)
−.0002(0.9433)
−6.97E-05(0.9817)
−6.97E-05(0.9817)
−6.97E-05(0.9817)
0.0015(0.6336)
0.0016(0.6029)
0.0016(0.6029)
0.001598(0.6029)
–
0.0011(0.6859) 0.6335(0.0378)∗∗
0.0012(0.6581) 0.6400(0.0357)∗∗
0.0012(0.6581) 0.6400(0.0357)∗∗
0.0012(0.6581) 0.6400(0.0357)∗∗
0.0028(0.2892)
0.0029(0.2743)
0.0029(0.2743)
0.002907(0.2743)
–
–
–
–
–
–
log(Settlement) ATM–AOI log(ATM) Pre–Post VOL_1 VOL_2 VOL_3
0.0004(0.2090) −0.0003(0.2180) −0.0003(0.2180) −0.000346(0.218) – −0.0049(0.0001)∗∗∗ −0.0049(0.0002)∗∗∗ −0.0049(0.0002)∗∗∗ −0.0049(0.0002)∗∗∗ −0.0051(0.0001)∗∗∗ −0.0051(0.0001)∗∗∗ −0.0051(0.0001)∗∗∗ −0.00508(0.0001)∗∗∗ −0.0045(0.0003)∗∗∗ 0.0195(0.0000)∗∗∗ 0.0182(0.0000)∗∗∗ 0.0207(0.0000)∗∗∗ 0.0297(0.0000)∗∗∗ 0.0197(0.0000)∗∗∗ 0.0185(0.0000)∗∗∗ 0.0208(0.0000)∗∗∗ 0.029971(0.0000)∗∗∗ 0.0306(0.0000)∗∗∗ 0.0118(0.0000)∗∗∗ 0.0106(0.0000)∗∗∗ 0.0130(0.0000)∗∗∗ 0.0220(0.0000)∗∗∗ 0.0117(0.0000)∗∗∗ 0.0105(0.0000)∗∗∗ 0.0128(0.0000)∗∗∗ 0.02197(0.0000)∗∗∗ 0.0222(0.0000)∗∗∗ – – 0.0023(0.2359) 0.01144(0.0000 0.0119(0.0000)∗∗∗ – – 0.0024(0.2105) 0.0114(0.0000) –
–
–
–
−0.0089(0.0000)∗∗∗ – – −0.0024(0.2105) – −0.0102(0.0000)∗∗∗ −0.114(0.0000)∗∗∗ −0.0089(0.0000)∗∗∗ – −0.0103(0.0000)∗∗∗
VOL_4 VOL_5 R2
−0.0023(0.2359) −0.0114(0.0000)
– −0.0091(0.0000)∗∗∗
0.009122(.0000)∗∗∗
0.0095(0.0000)∗∗∗
–
–
0.4404
0.4336
0.4415
0.4437
0.4437
0.4437
0.4385
0.4404
0.4404
ADJ. R 2
0.4318
0.4326
0.4326
0.4326
0.4287
0.4293
0.4293
0.4292
0.4266
SE of regression
0.0123
0.0123
0.0123
0.0123
0.0124
0.0124
0.0124
0.01236
0.0124
Sum squared resid
0.0612
0.0609
0.0609
0.0609
0.0615
0.0613
0.0613
0.061298
0.0621
1227.786
1228.588
1228.588
1228.588
1223.189
1223.908
1223.908
1223.908
1224.8819
log likelihood Durbin–Watson stat
1.6169
1.6149
1.6149
1.6149
1.6159
1.6153
1.6153
1.61525
1.6043
Mean dependent variable
0.0501
0.0501
0.0501
0.0501
0.0501
0.0501
0.0501
0.050096
0.0501
SD dependent variable
0.0163
0.0163
0.0163
0.0163
0.0164
0.0164
0.0164
0.016366
0.0163
Akaike info criterion
−5.9357
−5.9347
−5.9347
−5.9347
−5.9278
−5.9264
−5.9264
−5.92638
−5.9313
Schwarz
−5.857
−5.8467
−5.8467
−5.8467
−5.8494
−5.8382
−5.8382
−5.8382
−5.8726
F-stat
45.5127
−40.0769
−40.0769
−40.0762
−44.841
39.4519
39.4519
39.452
−66.9981
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Prob (F-stat) Notes:
∗
0.10 significance level;
∗∗
0.05 significance level;
∗∗∗
0.01 significance level.
174
LIQUIDITY AND MARKET EFFICIENCY
9.5.3 Price discovery analysis Time series properties
Before testing for cointegration, the order of integration of the individual time series needs to be determined. Tests for unit roots and determining order of integration involve the use of Dickey–Fuller (ADF) and Phillip– Perron (PP) tests. The procedure for testing for stationarity properties of the variables was discussed earlier. In each case, the null hypothesis is that the variable under investigation has a unit root (non-stationary on the level) while the alternative hypothesis is that the variable does not have a unit root (stationary on the level). In Table 9.7, the results of the ADF and PP tests show that variables cannot reject the null hypothesis of a unit root. This conclusion is reached after comparing the t-statistics of the level variables with the critical values from the ADF and PP unit root tests provided by MacKinnon (1991) at the corresponding significance level. Recall, from the previous discussion, that if the t-statistic is greater than the critical value in absolute terms the null hypothesis is rejected; thus the conclusion that the series is stationary on the first difference, or I (1). The results shown in Table 9.7 are consistent with findings by Ackert and Racine (1999) and Groenewold (1997), that price quotes for the index futures (Day SPI Close) and the index (AOI Close) are integrated to the order of one. Furthermore, Table 9.7 provides additional evidence to support Ackert and Racine’s (1999) and Groenewold’s (1997) findings in that both the day and night open and close prices for the index futures and index have non-stationary time series properties for daily data.
Table 9.7 Time series properties of the variables Variable
Pre-automation
Post-automation
Property
ADF∗
PP∗∗
ADF∗
PP∗∗
Night SPI Open
−1.866245
−2.13309
−2.495074
−2.804573
I (1)∗∗∗
Night SPI Close
−2.021747
−2.346416
−2.518351
−2.82888
I (1)∗∗∗
Day SPI Open
−1.915622
−2.203659
−2.61018
−2.198428
I (1)∗∗∗
Day SPI Close
−1.876280
−2.043389
−2.531016
−2.792202
I (1)∗∗∗
Day AOI Open
−1.71164
−1.809892
−2.552641
−2.817274
I (1)∗∗∗
Day AOI Close
−1.743018
−1.829806
−2.512903
−2.780272
I (1)∗∗∗
Notes: ∗ Critical values for ADF range from −3.1359 to −3.1368 at 10%, from −3.4260 to −3.4278 at 5%, and −3.9919 to −3.9947; ∗∗ Critical values for PP are ranging from −3.1359 to −3.1364 at 10%, from −3.4268 to −3.4275, at 5%, and −3.9917 to −3.9930; ∗∗∗ denotes variable is non-stationary as it is integrated at level one.
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
175
As shown in Table 9.7, all the variables are integrated as of order one. The automation of the SFE does not seem to have changed the non-stationary property of the daily and night open and close relationship held between the index prices. Similar results were found by Ackert and Racine (1999) and Groenewold (1997). As the time series for the pre-automation and postautomation have confirmed that they are integrated as of order one, the series can now be tested for cointegration.
Johansen cointegration tests
Johansen’s likelihood ratio tests can be applied to determine the value of r, the number of cointegrated relationships between the variables in Table 9.7. Following the procedure developed by Johansen and Juselius (1990) the number of cointegrated relationships (vectors) in the equation (shown in Tables 9.8 and 9.9) is determined by comparing the likelihood ratio to the critical value at the given level of significance. Tables 9.8 and 9.9 show that the null hypothesis of no cointegrating vectors can be rejected at 1 percent levels of significance for the variables tested in the pre- and post-automation periods. The above results support the cointegration hypothesis incorporated in Research Propositions 2 and 3. The results seen in Tables 9.8 and 9.9 show that all variables are cointegrated. The conclusion is that the SPI futures contract is cointegrated with the All Ordinaries Index both before and after the introduction of electronic trading at the SFE. Findings in this study are consistent with the findings of other studies (Kempf and Korn, 1998; Ackert and Racine, 1999; Turkington and Walsh,
Table 9.8 Johansen cointegration test log results for the pre-automation period Automation Variables
H0
H1
Eigenvalues
Likelihood ratio
SPI Night Close, AOI Day Open
r=0 r≤1
r>0 r>1
0.133768 0.018052
56.15137∗∗ 6.321137
AOI Day Close, SPI Night Open
r=0 r≤1
r>0 r>1
0.15986 0.017473
66.55955∗∗ 6.116847
SPI Day Open, AOI Day Open
r=0 r≤1
r>0 r>1
0.149069 0.017637
62.18897∗∗ 6.174793
AOI Day Close, SPI Day Close
r=0 r≤1
r>0 r>1
0.162063 0.017576
67.50692∗∗ 6.153183
AOI Day Close, SPI Day Open
r=0 r≤1
r>0 r>1
0.132571 0.01729
55.40282∗∗ 6.051996
Note: ∗∗ denotes rejection at 1% significant level.
176
LIQUIDITY AND MARKET EFFICIENCY
Table 9.9 Johansen cointegration test log results for the post-automation period Post-automation variables
H0
H1
Eigenvalues
Likelihood ratio
SPI Night Close, AOI Day Open
r=0 r≤1
r>0 r>1
0.171061 0.008075
66.54364** 2.756496
AOI Day Close, SPI Night Open
r=0 r≤1
r>0 r>1
0.102473 0.010521
40.47313** 3.60673
SPI Day Open, AOI Day Open
r=0 r≤1
r>0 r>1
0.177129 0.09194
69.62966** 3.149613
AOI Day Close, SPI Day Close
r=0 r≤1
r>0 r>1
0.076209 0.007512
29.6022** 2.571375
AOI Day Close, SPI Day Open
r=0 r≤1
r>0 r>1
0.087967 0.00663
33.56882** 2.261839
Note: ** denotes rejection at 1% significant level.
1999) that the behavior of index prices is consistent with the semi-strong form of the efficient market hypothesis. The results shown in Tables 9.8 and 9.9 provide evidence that the spot and futures prices are in a cointegrated equilibrium relationship. However, no previous empirical evidence has been provided to support the findings in this study for night open and close prices for a cointegrated relationship. Overall, results using a cointegration approach provide evidence supportive of a semi-strong form of the efficient market hypothesis.
Granger causality tests
As discussed earlier, the price discovery relationship is centered around the lead–lag behavior of the ASX All Ordinaries Index and the SFE SPI futures contracts. The causal relationship was examined using the Granger causality test to detect the direction of information flow as reflected in price change. The results for the Granger causality performed on the day-traded indexes are seen in Panel A in Table 9.10, and the night-traded indexes (SPI) are shown in Panel B. The causal relationships are tested on the open and close prices for the SPI futures and the All Ordinaries Index. As the unit root test results shown in Table 9.3 show that the variables are integrated of order one, I (1), the first differenced variables are used in the causality tests. It is observed from Panel A in Table 9.10 that the day traded All Ordinaries open value (day AOI open value) does cause the share price index (day open value), while the share price index (day close value) does cause the All Ordinaries Index (day close value). Causality seems to run both ways
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
177
Table 9.10 Granger causality test results Null Hypothesis
Panel A: Day traded series SPI Day Open, SPI day open does not Granger AOI Open cause AOI open AOI open does not Granger cause SPI day open AOI Close, SPI Day Close
AOI close does not Granger cause SPI day close SPI day close does not Granger cause index close
Panel B: Night traded series AOI Close, AOI close does not Granger SPI Night Open cause SPI night open SPI night open does not Granger cause AOI close SPI Night Close, SPI night close does not Granger AOI Open cause AOI open AOI open does not Granger cause SPI night close AOI Close, AOI close does not Granger SPI Night Close cause SPI night close SPI night close does not Granger cause AOI close
Pre-automation Post-automation F-stat F-stat 44.6516∗∗∗ 0.13576
4.30455∗∗∗ 0.36373
1.94637 0.37811
2.48783∗ 1.58893
1402.57∗∗∗ 0.72121
0.59129 59.8435∗∗∗
33.2951∗∗∗ 0.82011
123.388*** 1.09953
5.46812∗∗∗ 0.38735
22.0058∗∗∗ 1.99863
Note: ∗∗∗ (∗ ) denotes significance at the 1% (10%) level.
(bi-directional causality) from/to the All Ordinaries Index (AOI) to/from share price index futures in the day trades. The night trades of series are shown in Panel B of Table 9.10. The share price index (SPI) night close causes the All Ordinaries Index day close value in the post-automation period, whereas the All Ordinaries Index (AOI) day close causes the share price index (SPI) night close in the pre-automation period. Thus causality seems to run both ways (bi-directional causality) from/to the All Ordinaries Index (AOI) to/from night traded share price index futures in the day trades in the night traded period. Comparing the pre-automation and the post-automation periods, the results suggest that, since automation, the structural alignment between the night traded futures and the day traded futures seems to have created a synergy from the 24-hour trading of the SPI futures contract. This can be seen in Table 9.10 (Panel B), with the shift to the SPI night open causing the AOI close in the “post-” period. The results broadly show bi-directional results over the 24-hour period before and after the introduction of electronic trading at the SFE. Thus Research Proposition 4 is supported in that
178
LIQUIDITY AND MARKET EFFICIENCY
the lead–lag relationship remains unchanged (still bi-directional) after the start of electronic trading at the SFE.
9.6 CONCLUSION This study examined whether the Sydney Futures Exchange has benefited from the introduction of electronic trading. Under the first research proposition, empirical tests were carried out on the liquidity of the Sydney Futures Exchange with the analysis of the “at the money” (ATM) SPI call options, and SPI futures contracts. This provided an insight to the change of liquidity between the two derivative markets. By classifying the option contracts by date to expiry and the closeness of the strike price to the spot price, the ATM volume was analyzed along with the SPI futures against the All Ordinaries Index (AOI). The tests were run by classifying the volumes into size groups, following the assumptions of the mixture of distributions hypotheses so as to provide relative levels of market volatility against which to compare the liquidity ratios. The results show that the “at the money” SPI options were more liquid in times of high volatility after the SFE became automated. The SPI futures are less liquid in times of medium to low market volatility. This overall result supports Research Proposition 1, that the SFE’s liquidity has changed with the introduction of electronic trading. Therefore it can be concluded that the liquidity of the Sydney Futures Exchange seems to have increased the operational efficiency within the SPI call options market, while there seems to have been a decline in the operational efficiency of SPI futures market. The importance of the analysis of liquidity in this study is that it was able to account for times of high volatility, such as the technology crash at the beginning of 2000. This was shown clearly by segmenting the option and futures market responses to the differing levels of market volatilities. The examination of the price discovery process was incorporated into the last three research propositions. Research Propositions 2 and 3 were used to test semi-strong form market efficiency. Under this assumption, the trading prices in the Australian Stock Exchange (ASX) and Sydney Futures Exchange (SFE) should have a long-run cointegrating relationship. The results confirm the belief that ASX and the SFE are semi-strong efficient. The existence of cointegration between the two markets before and after the introduction of electronic trading supported the semi-strong market efficiency. The presence of a bi-directional lead–lag relationship between the SPI futures price and the All Ordinaries Index price before and after the introduction of electronic trading supported the fourth research proposition. The automation of the SFE did alter the price discovery process. First, it appeared to synergize the night traded market. This is most probably a
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
179
result from the SFE day and night structural alignment, of both instruments being traded on the same system. However, market leads and lags were bi-directional both before and after automation. This suggests that the electronic trading structure does not greatly enhance the price discovery price of the SFE. If it did, this would be observed in the SPI futures leading the AOI. Therefore, it can be concluded that a change in the liquidity is evident in the SPI futures and SPI call option contracts, but the price discovery process does not appear to have been enhanced by the automation of the Sydney Futures Exchange in the early stages up to August 2000.
APPENDIX Ratio variables Ratio_A =
Day_ATM_Volume Day_Index_Volume
Ratio_C =
Day_Futures_Volume Day_Index_Volume
Ratio_D =
Night_Futures_Volume Day_Index_Volume
Ratio_E =
Day_ATM_Volume Day_Futures_Volume
Day
= Traded during the day trading session
Night = Traded during the night trading session Futures = Share price index futures contract ATM
= “At the money” SPI call option contracts
Index = All Ordinaries Index
NOTES 1. Massimb and Phelps (1994) defined operational efficiency as the market’s ability to lower the costs of trading, and its execution speed of orders between buyer and seller. See Ates and Wang (2005) for the latest evidence on operational efficiency in the US futures market. 2. Informational efficiency means that all traders have equal access to all public information, and that the information is quickly reflected in trading prices (Tsang, 1999). 3. Overstatement of the lead–lag relationship will be discussed later in the chapter.
180
LIQUIDITY AND MARKET EFFICIENCY
4. Fama (1970) has identified three levels of market efficiency. The semi-strong form of the “efficient market hypothesis” states that prices reflect all publicly available information. 5. Market efficiency is discussed in detail in the following section. 6. Cost-of-carry is the cost involved in storing an asset and the interest lost on funds tied up therein. 7. A moneyness portfolio denotes the categorization of strike prices relative to the spot price, so as to create portfolios reflecting the moneyness at that point in time (Rubinstein, 1994). 8. A brief discussion on the mixture of distribution hypothesis was given earlier. 9. Variables definitions are given in the footnotes to Table 9.4.
REFERENCES Ackert, L. F. and Racine, M. D. (1999) “Stochastic Trends and Cointegration in the Market for Equities”, Journal of Economics and Business, 51(2): 133–43. Anthony, J. H. (1988) “The Interrelation of Stock and Option Market Trading Volume Data”, Journal of Finance, 43(4): 949–64. Ates, A. and Wang, G. H. K. (2005) “Information Transmission in Electronic versus OpenOutcry Trading Systems: An Analysis of U.S. Equity Index Futures Markets”, Journal of Futures Markets, July, 25(7): 679–715. Australian Stock Exchange (2000) website address: www.asx.com.au/B1400.htm (accessed April 24, 2000). Bortoli, L., Gareth, A. and Jarnecic, E. (2004) “Differences in the Cost of Trade Execution Services on Floor-Based and Electronic Futures Markets”, Journal of Financial Services Research, August, 26(1): 73–87. Brailsford, T. (1996) “The Empirical Relationship between Trading Volume, Returns and Volatility”, Accounting and Finance, 36(1): 89–111. Brenner, K. and Kroner, K. (1995) “Arbitrage, Cointegration, and Testing the Unbiasedness Hypothesis in Financial Markets”, Journal of Financial and Quantitative Analysis, 30(1): 23–42. Brooks, C., Garrett, I. and Hinich, M. J. (1999) “An Alternative Approach to Investigating Lead–Lag Relationships between Stock Index and Stock Index Futures Markets”, Applied Financial Economics, 9(6): 605–13. Clark, P. K. (1973) “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices”, Econometrica, 41(1): 135–55. Copeland, L., Lam, K. and Jones, S.-A. (1976) “The Index Futures Markets: Is Screen Trading more Efficient?”, Journal of Futures Markets, 24(4): 337–57. Copeland, T. E. (1976) “A Model of Asset Trading under the Assumption of Sequential Informational Arrival”, Journal of Finance, 31(5): 1149–68. Cornell, B. (1981) “The Relationship between Volume and Price Variability in Futures Markets”, Journal of Futures Markets, 1(2): 303–16. Dickey, D. A. and Fuller, W. (1979) “Distribution of the Estimators for Auto Regression Time Series with a Unit Root”, Journal of the American Statistical Association, 74(2): 427–31. Domowitz, G. (1993) “A Taxonomy of Automated Trade Execution Systems”, Journal of International Money and Finance, 12(3): 607–31. Engle, R. F. and Granger, C. W. J. (1987) “Co-integration and Error Correction: Representation, Estimation, and Testing”, Econometrica, 55(2): 251–76.
M A R K B U R G E S S A N D J. W I C K R A M A N A Y A K E
181
Epps, T. W. and Epps, M. L. (1976) “The Stochastic Dependence of Security Price Change and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis”, Econometrica, 44(2): 305–21. Fama, E. F. (1970) “Efficient Capital Markets: A Review of Theory and Empirical Work”, Journal of Finance, 25(2): 383–417. Fama, E. F. (1991) “Efficient Capital Markets”, Journal of Finance, 46(5): 1575–617. Franke, G. and Hess, D. (1995) “Anonymous Electronic Trading Versus Floor Trading”, Working Paper, Series II, no. 285, Universitat Konstanz. Freund, W. C., Larrain, M. and Pagano, M. S. (1997) “Market Efficiency Before and After the Introduction of Electronic Trading at the Toronto Stock Exchange”, Review of Financial Economics, 6(1): 29–56. Frino, A., Bortoli, L., and Jarnecic, E. (2004), “Differences in the Cost of Trade Execution Serouson Floor-based and Electronic Future, Markets”, Journal of Financial Services Research, 26(1): 73–87. Frino, A. I. and Jarnecic, E. (2000) “An Empirical Analysis of the Supply of Liquidity by Locals in Futures Markets: Evidence from the Sydney Futures Exchange”, Pacific Basin Finance Journal, 8(3–4): 443–56. Frino, A., McInish, T. and Toner, M. (1998) “The Liquidity of Automated Exchanges: New Evidence from the German Bund Futures”, Journal of International Financial Markets, Institutions and Money, 8(3–4): 225–41. Granger, C. W. J. (1969) “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods”, Econometrica, 37(3): 424–38. Granger, C. W. J. (1981) “Some Properties of Time Series Data and Their Use in Econometric Model Specification”, Journal of Econometrics, 16(1): 121–30. Groenewold, N. (1997) “Share Market Efficiency: Tests Using Daily Data for Australia and New Zealand”, Applied Financial Economics, 7(6): 645–57. Grünbichler, A., Longstaff, F. A. and Schwartz, E. S. (1994) “Electronic Screen Trading and the Transmission of Information: An Empirical Examination”, Journal of Financial Intermediation, 3(2): 166–87. Harris, L. (1982) “Transaction Data Tests of the Mixture of Distributions Hypothesis”, Journal of Financial and Quantitative Analysis, 22(2): 127–41. Hemler, M. L. and Longstaff. F. (1991) “General Equilibrium Stock Index Futures Prices: Theory and Empirical Evidence”, Journal of Financial and Quantitative Analysis, 26(3): 287–308. Hinich, M. (1996) “Testing for Dependence in the Input to a Linear Time Series Model”, Journal of Nonparametric Statistics, 6(3): 205–21. Hurst, H. E. (1951) “The Long-term Storage Capacity of Reservoirs”, Transactions of the American Society of Civil Engineers, 116(3): 143–52. Jarnecic, E. (1999) “Trading Volume Relations between the ASX and ASX Options Market: Implication of Microstructure”, Australian Journal of Management, 24(1): 77–91. Johansen, S. (1991) “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Autoregressive Models”, Econometrica, 59(6): 1551–80. Johansen, S. (1995) “Likelihood-based Inference in Cointegrated Vectors Autoregressive Models”, (Oxford University Press). Johansen, S. and Juselius, K. (1990) “Maximum Likelihood Estimation and Inference on Cointegration – with Application to the Demand for Money”, Oxford Bulletin of Economics and Statistics, 52(2): 169–210. Karpoff, J. M. (1987) “The Relation between Price Change and Trading Volume: A Survey”, Journal of Financial and Quantitative Analysis, 22(1): 109–26. Kempf, A. and Korn, O. (1998) “Trading System and Market Integration”, Journal of Financial Intermediation, 7(3): 220–39.
182
LIQUIDITY AND MARKET EFFICIENCY
Kofman, P. and Moser, J. (1997) “Spread Information Flows and Transparency Across Trading Systems”, Applied Financial Economics, 7(3): 281–94. MacKinnon, J. G. (1991) “Critical Values for Cointegration Tests”, in R. F. Engle, and C. W. J. Granger (eds), Modelling Long-run Economic Relationships (London: Oxford Publishing Company). Mananyi, A. and Struthers, J. J. (1997) “Cocoa Market Efficiency: A Cointegration Approach”, Journal of Economic Studies, 24(3): 141–51. Manaster, S. and Mann, S. C. (1998) “Life in the Pits: Competitive Market Making and Inventory Control”, Review of Financial Studies, 36(1): 51–64. Martell, T. F. and Wolf, A. S. (1985) “Determinants of Trading Volume in Futures Markets”, Working paper, Centre for the Study of Futures Markets, Columbia Business School, USA. Massimb, M. and Phelps, B. (1994) “Electronic Trading, Market Structure and Liquidity”, Financial Analysts Journal, 50(1): 39–50. O’Connor, S. M. (1993) “The Development of Financial Derivatives Markets: The Canadian Experience”, Technical Report, No. 62 (Ottawa: Bank of Canada). O’Hara, M. (1995) Market Microstructure Theory (Cambridge, Basil Blackwell). Perron, P. (1990) “Testing for a Unit Root in a Time Series with a Changing Mean”, Journal of Business and Economic Statistics, 8(1): 153–62. Peters, E. (1992) “R/S Analysis Using Logarithmic Returns: A Technical Note”, Financial Analysts Journal, 48(6): 81–2. Phillips, C. C. and Perron, P. (1988) “Testing for a Unit Root in Time Series Regression”, Biometrika, 8(2): 153–62. Pirrong, C. (1996) “Market Liquidity and Depth on Computerized and Open Outcry Trading Systems: A Comparison of DTB and LIFFE Bund Contracts”, Journal of Futures Markets, 16(5): 519–43. Ragunathan, V. and Peker, A. (1997) “Price Variability, Trading Volume and Market Depth: Evidence from the Australian Futures Market”, Applied Financial Economics, 7(5): 447–54. Richardson, M. and Smith, T. (1994) “A Direct Test of the Mixture of Distributions Hypothesis: Measuring the Daily Flow of Information”, Journal of Financial and Quantitative Analysis, 29(1): 101–16. Rubinstein, M. (1994) “Implied Binomial Trees”, Journal of Finance, 49(3): 771–818. Shyy, G. and Lee, J. (1996) “Price Transmission and Information Asymmetry in Bund Futures Markets: LIFFE vs. DTB”, Journal of Futures Markets, 15(1): 437–55. Sydney Futures Exchange (SFE) (1999) Full Electronic Trading: A Guide for Customers (Sydney, Australia: SFE). Tauchen, G. and Pitts, M. (1983) “The Price Variability–Volume Relationship on Speculative Markets”, Econometrica, 51(2): 485–505. Taylor, M. P. and Sarno, L. (1997) “Capital Flows to Developing Countries: Log and Short-term Determinants”, World Bank Economic Review, 11(3): 183–212. Tsang, R. (1999) “Open Outcry and Electronic Trading in Futures Exchanges”, Bank of Canada Review, Spring, 21–39. Turkington, J. and Walsh, D. (1999) “Price Discovery and Causality in the Australian Share Price Index Futures Market”, Australian Journal of Management, 24(2): 97–113.
C H A P T E R 10
How Does Systematic Risk Impact Stocks? A Study of the French Financial Market Hayette Gatfaoui
10.1 INTRODUCTION Systematic risk is known to affect the market prices of traded financial assets (Stulz, 1999a, 1999b, 1999c). Indeed, the capital asset pricing model (CAPM) theory argues that each financial asset bears an undiversifiable risk known as systematic or market risk, as introduced by Sharpe (1963, 1964, 1970) and Treynor (1961) among others.1 Such a risk can be estimated through a well-diversified portfolio so far as this portfolio presents as low as possible an idiosyncratic risk (French and Poterba, 1991). Recent literature focuses mainly on a sound assessment of the influence of systematic risk on financial assets, along with the beta coefficient in a CAPM framework. Koutmos and Knif (2002) estimate the influence of systematic risk while employing timevarying distributions (for example, conditional distributions depending on past innovations). Using market stock indices of the financial markets under consideration, they find that financial assets’ betas are stationary meanreverting processes with an average degree of persistence equal to four days. Gençay, Selçuk and Whitcher (2003) use wavelet techniques to assess the influence of systematic risk on any asset, or equivalently to compute its beta 183
184
A STUDY OF THE FRENCH FINANCIAL MARKET
in a CAPM model. These authors use the S&P 500 index as a systematic risk benchmark. Therefore, common practice resorts to available stock indices as proxies for a well-diversified market portfolio, and pays little attention to the sound assessment of systematic risk itself.2 However, a study by Campbell et al. (2001) shows that the number of stocks in such an index has to be high enough to offset idiosyncratic risk. They find that the number of assets required to create a well-diversified portfolio has grown over time. Therefore, using market indices with an insufficient number of stocks (for example, small stock indices) may be an inaccurate and even wrong benchmark for systematic risk. Indeed, a market stock index represents a sub-set of the whole range of financial assets that should enter the composition of an actual market portfolio, according to the critique of Roll (1977). This author underlines that the actual market portfolio is non-identifiable, since a market portfolio should be composed of stocks, bonds, real estate and human capital assets, among others. However, Campbell et al. (2001) show that market volatility (which is that part of the global volatility related to market factor, and specifically their market factor proxy) tends to drive global volatility. Therefore, in this chapter we address the question of how to find a proxy for the market factor, such as the systematic risk factor, in markets where only small stock indices are available, and where options on such indices are traded. It is a hard task, since the undiversifiable risk is not directly observable and can only be estimated. Hence, lacking a portfolio diversified enough to represent the market factor accurately, we attempt to infer the fair level of market risk factor from only observed available stock indices and related European call prices. The chapter is organized as follows. Section 10.2 introduces the assumptions and theoretical framework aimed at finding a proxy for the systematic risk factor. Section 10.3 employs an empirical application of such a framework, focusing on the French financial market and its CAC40 stock index. Section 10.4 studies the impact of the implied market factor on a pool of French stocks. The impact of systematic risk is analyzed through a twostep methodology, namely a correlation study and a Granger causality test. For further investigation, section 10.5 attempts to test for a non-linear relationship between both prices and returns of the implied market factor, and French financial assets. This study is realized in two stages: a linear regression analysis and a volatility analysis. The linear regression analysis considers first simple regressions of returns, and then Jensen-type (1968, 1969) regressions. The volatility study considers weekly rolling volatilities of asset returns. Section 10.6 attempts to draw some conclusions while comparing our implied market factor with other available market stock indices. A two-step study is undertaken, considering first the explanatory power corresponding to each stock index. The empirical weekly forecasting performance underlying each available market proxy is then assessed, employing the average absolute relative error as a performance measure. Finally, the
HAYETTE GATFAOUI
185
study ends with concluding remarks and suggestions for future research in section 10.7.
10.2 THEORETICAL FRAMEWORK In this section, we introduce our assumptions and the related theoretical framework allowing the induction of the market factor.
10.2.1 Valuation setting We assume that any small stock index is a non-perfect proxy of the systematic risk factor. Specifically, we suppose that a small stock index is a disturbed observation of the market factor.
Assumptions. Any small stock index It , at current time t, depends on market factor Xt such that It = t Xt
(10.1)
where t represents a (strictly) positive determinist scale factor that is time∗ . Moreover, is a continuous and derivable varying and bounded on R+ t function of time. We assume implicity that any small stock index is diversified so as to exhibit a sufficiently low level of idiosyncratic risk. Therefore, the scale factor encompasses this. This parameter is not purely, or mainly, driven by an idiosyncratic component. Hence the scale parameter can encompass many effects/factors such as liquidity phenomena, and short-term shocks resulting from some announcement effects or specific events occurring in the financial market. Further, all the assumptions of the Black and Scholes (1973) option valuation framework are supposed to hold. To sum up, trading is continuous; there are no dividend payments, no transaction costs and no taxes. Moreover, there is no arbitrage opportunity and a constant spot risk-free interest rate r prevails in the complete market.3 We also assume that the market fact tor follows a geometric Brownian motion such as dX Xt = µ dt + σ dWt where t is the current date; µ and σ are constant drift and volatility parameters of the systematic factor’s instantaneous rate of return;4 Wt is a standard Brownian motion under the historical probability.
Dynamic of the stock index. Applying Ito’s lemma in the risk-neutral universe and on-time sub-set [t, T], the stock index dynamic writes under risk neutral probability σ2 1 ∂t d ln (It ) = + r− dt + σ dWt∗ (10.2) t ∂t 2
186
A STUDY OF THE FRENCH FINANCIAL MARKET
σ2 ∗ ∗ T (T − t) + σ W which rewrites IT = It exp r − T−t , where (Wt ) is a 2 t standard Brownian motion. Of course, we could estimate t and T while building a well-diversified portfolio. Such a portfolio should be a good proxy of market factor so that the market is complete. However, along with Roll’s critique and Campbell et al. (2001), we address the question of how to proxy the market factor from a small-stock index, which is an imperfect proxy of market factor. Hence, we consider the prices of options on a small index. Indeed, observed index prices and call market prices will give information about both scale and market factors.
10.2.2 Option pricing We introduce a call pricing formula for European calls on the small-stock index I.
Call’s dynamic in a no-dividend framework. We consider a European call on stock index I whose strike price and expiring date are, respectively, K and T. At maturity, such a call is valued C(T, IT ) = max(0, IT − K) = (IT − K)+ . Like Black and Scholes (1973), we apply the no-opportunity arbitrage valuation principle, which states that the current value of any contingent claim is equal to the discount expected value of its future cash flows under risk neutral probability. Then, our European call Q Q price writes C(t, It ) = Et e−r(T−t) (IT − K)+ , where Et [.] is the expectation operator under risk neutral probability Q, conditional on the information set Ft = σ{Ws ,0 ≤ s ≤ t} available at current date t. Therefore, from Equations (10.1) and (10.2) of the stock index, the pricing formula for a European call on stock index I at current date t reads: C(t, It ) ≡ C(T − t, K, It , r, t , T , σ) =
where N(.)
T It N(d1 ) − K e−r(T−t) N(d2 ) t (10.3)
is the cumulative distribution function of the stan
I σ2 ln T + ln Kt + r + 2 (T − t) √ t √ ; d = d − σ T −t= dard normal law; d1 = 2 1 σ T −t 2 I σ ln T + ln Kt + r − 2 (T − t) t √ . If we assume that the small-stock index is a σ T −t
perfect proxy of market factor, we get the classical Black and Scholes (1973) option pricing formula, since we have t = T = 1 for each date t < T. Therefore, introducing a disturbance in our modifies the classical Black setting
T and Scholes formula through ratio t . However, assuming a Black and Scholes (1973) setting to value a call on a stock index is inappropriate in so
HAYETTE GATFAOUI
187
far as the no-dividend assumption is unrealistic. That is why we adapt the previous formula to account for a stock index comprising dividend-paying equities.
Call’s dynamic in a dividend framework. Since most of the stocks that constitute financial indices pay dividends, we assume that index I pays a dividend at a continuous annualized rate q (see Merton, 1973; Black, 1975). Therefore, under the Black and Scholes’ world and dividend-paying assumptions, the current price of underlying It has to be replaced with It e−q(T−t) . Then, adjusting the European call pricing formula in Equation (10.3) to become a dividend-paying framework, a European call on a dividend-paying stock index I is valued as C(T − t, K, It , r, t , T , σ) =
T It e−q(T−t) N(d1 ) − K e−r(T−t) N(d2 ) t (10.4)
where N (.) is the cumulative distribution function of the standard
I e−q(T − t) σ2 ln ( T ) + ln t K + r + 2 (T − t) √ t √ ; d2 = d1 − σ T − t = normal law; d1 = σ T − t −q(T−t) I e σ2 ln T + ln t K + r − 2 (T − t) t √ . In European call formula Equations σ T −t
(10.3) or (10.4), all parameters are known except the scale parameter at instants t and T (for example, t and T ), and volatility parameter σ. Therefore we shall use our knowledge about observed index prices and market prices of European index calls to extract information about the scale parameter and volatility parameter σ. Such a process will give information about the market factor itself.
10.3 EMPIRICAL STUDY We apply our European call pricing here to the French stock market and its CAC40 stock index.
10.3.1 Data We use Bloomberg daily closing data from January 2, 2002 to March 19, 2002, a total of 55 observations by series. We observe one-month r1M , twomonth r2M and three-month r3M risk-free interest rates, and consider market prices of the CAC40 French stock index. This index is composed of the forty most liquid and representative stocks listed on the French financial market, and pays a continuous annualized dividend rate q. The CAC40 INDEX is a weighted stock index whose weights are proportional to each of its forty
188
A STUDY OF THE FRENCH FINANCIAL MARKET
Table 10.1 Index information Index
q(%)
nb K
Spread
CAC40
2.2650
3
4238.99–4682.79
Table 10.2 Features of CAC40 INDEX calls Call name
Strike price (€ )
CAC 3/02 C4000
4,000
CAC 3/02 C4500
4,500
CAC 3/02 C5000
5,000
stocks’ capitalization. We also obtain closing prices of three European calls on CAC40 while considering option contracts of the continuous listing class. These calls are traded on the French options market called MONEP (Marché des Options Négociables de Paris). Let q, nbK and spread be, respectively, the dividend rate, the number of different strike prices of CAC40 INDEX calls, and the variation bounds of the index value (that is, lowest–highest in euros) over the studied time period (see Table 10.1). European calls on the CAC40 INDEX, maturing on March 27, 2002, exhibit the features shown in Table 10.2. Over our time horizon, time to maturity of calls falls from 84 calendar days to 8 calendar days (6 working days). Part of these data will help us to compute the risk-free interest rate, which must be defined. Given our European call pricing formula, we compute the risk-free rate as a function of time to maturity. We choose a quadratic interpolation method to infer our short-term risk-free rate from the one-, two- and three-month term risk free rates. Let r(t, T) be the risk free rate at current time t for time horizon T. This rate is then described by relation r(t, T) = a(T − t)2 + b(T − t) + c with a = 72[r1M (t) − 2r2M (t) + r3M (t)], b = 12[r2M (t) − r1M (t) − (a/48)] = −30r1M (t) + 48r2M (t) − 18r3M (t) and c = r1M (t) − (a/144) − (b/12) = 3r1M (t) − 3r2M (t) + r3M (t). This method gives a good risk-free rate proxy, given that European calls’ time to maturity (for example, (T − t)) is, at most, three calendar months.
HAYETTE GATFAOUI
189
This proxy is employed to infer the fair value of the market risk factor from the CAC40 INDEX and related European call prices. Incidentally, the lack of control of the CAC40 INDEX weights introduces size effects into this index, among others. Therefore, an important role is played by some specific effects that are peculiar to any given highly-capitalized firm belonging to the CAC40 INDEX. In this light, the market benchmark role of CAC40 is strongly compromised. Dow Jones STOXX market indices bypass such a bias by applying some weight constraint when a given stock’s weight exceeds some specific threshold among the indices under consideration (high free-floating market capitalization).
10.3.2 Induction of systematic risk We explain how to estimate the level of market factor from market prices of a small stock index and the closing prices of European calls on such an index. From Equation (10.4), the estimation of the market factor’s level requires the estimation of the scale parameter at instants t and T (that is, t and T ), and volatility parameter σ (the volatility of the market factor’s instantaneous rate of return). As we observe market prices of the CAC40 INDEX (the small-stock index) and closing prices of related European calls, one solution consists of inverting Equation (10.4) relative to the scale parameter at times t and T, and the volatility parameter. We estimate these implied parameters while minimizing the sum of squared valuation errors at each fixed date t as ⎧ ⎫ nbK ⎨
2 ⎬ Min CObs T − t, Kj , It − C T − t, Kj , It , r, t , T , σ ⎭ t , T , σ ⎩ j=1
where Kj ∈ {4000, 4500, 5000}, and CObs (T − t, Kj , It ) are the European call’s market price. We solve this non-linear minimization problem numerically with a quasi-Newton method, and a Davidon–Fletcher–Powell type of algorithm. First, we get T = 2.3050 and XT = 2033.8482. Second, results allow plotting the implied values of t and σ against time to maturity. Whaley, (1982) argues that valuation errors do not necessarily depend on options’ moneyness, hence we draw plots according to time rather than moneyness. The implied volatility parameter σ is time-varying with a quadratic trend (a “smirk” type trend). Moreover, implied time series t and σ exhibit the statistical profiles shown in Table 10.3. We then observe a non-normal behavior for t and σ, namely leptokurtic distributions. Specifically, the volatility of the systematic risk factor should be modeled by a non-normal stochastic process or time-varying series. This stylized fact is known as the Black and Scholes volatility bias characterizing non-normal observed market asset returns. Knowing the market trend, we can now characterize the impact of systematic risk on the French financial market.
190
A STUDY OF THE FRENCH FINANCIAL MARKET
0.2350
2.305 2.3
0.2200
2.295
0.2150
2.29
0.2100
2.285
0.2050
2.28
0.2000
2.275
0.1950
2.27
0.1900
2.265
0.1850
2.26
84 80 78 76 72 70 66 64 62 58 56 52 50 48 44 42 38 36 34 30 28 24 22 20 16 14 10 8
Volatility level
0.2250
2.31 Implied volatility Implied lambda
Scale factor’s level
0.2300
Time to maturity (days)
Figure 10.1 Daily implied scale factor and market factor volatility
Table 10.3 Descriptive statistics t Mean
2.2881
σ 0.2069
Xt 1952.2699
Standard deviation
0.0086
0.0069
52.0441
Skewness
0.2144
1.1208
−0.1669
−1.0728
2.9231
−0.8446
3.0589
31.0976
1.8901
Excess Kurtosis Jarque-Bera Statistic
10.4 THE IMPACT OF SYSTEMATIC RISK Given our market factor’s estimation, we try to quantify its impact on prices of French stocks. Our primary econometric study is composed of a correlation study and a Granger causality test.
10.4.1 Correlation We study correlations between implied market factor and, on the one hand, French stock indices (CAC40, SBF120 and SBF250), and on the other ten French stocks: Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi (see Table 10.4). Most of commonly used descriptive statistics are valid only under the strong assumption of an elliptical distribution. When this is not the case, statistics
HAYETTE GATFAOUI
191
Table 10.4 Correlation of assets with the implied market factor Asset return
Correlation coefficient
Asset return
Correlation coefficient
SBF120
0.9959
Valéo
0.5288
SBF250
0.9952
Société Générale
0.7078
CAC40
0.9967
L’Oréal
0.6775
Air Liquide
0.4667
Renault
0.5365
Danone
0.2002
Schneider
0.4736
Vivendi
0.7982
Thomson
0.5329
Totalfina Elf
0.6569
are false. Indeed, this point fits some of the current questions considered by the Basel Committee. Szego (2002) and Artzner et al. (1999, 2000), highlight the coherency problem of risk measures such as linear correlation or covariance. Such risk measures are valid only for, at least, stationary distributions when not elliptical. Specifically, leptokurtic distributions violate one main property ensuring risk measures’ coherency, namely the sub-additivity principle. Following this concern, we compute correlations between the return of the implied market factor and returns of French stocks. Returns of both series are stationary over the time period studied. We then study the link between evolutions of both the systematic risk factor’s return and French asset returns. The average correlation of our three stock indices is 0.9959. The implied market factor is highly correlated with stocks, whose correlation coefficients range from 0.2002 for Danone to 0.7982 for Vivendi. In the rest of the chapter, we study the dependency between systematic risk and French stocks.
10.4.2 Causality Any causality study needs a vector autoregressive (VAR) specification as a starting point. We first introduce our VAR specification and then apply a Granger causality test. VAR specification
We look for a link between the implied market risk’s return RX and French stock or index returns RS . Hence, we consider VAR representations linking RX to RS with S ∈ {SBF120,5 SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi}. A VAR model allows us to test for a statistical relation between variables. Moreover, any VAR process parameters have to be estimated for
192
A STUDY OF THE FRENCH FINANCIAL MARKET
stationary time series such as our asset returns. The related bidimensional VAR with p lags, called VAR(p), writes Yt = A0 + A1 Yt−1 + A2 Yt−2 + · · · + Ap Yt−p + εt where Yt = [RXt RSt ] is the vector of variables; A0 = [a01 a02 ] is a vector of j
constant parameters; Ap = [aip ] 1 ≤ i,j ≤ 2 is the coefficient matrix for lag p; and εt = [ε1t ε2t ] is the vector of innovations that is assumed to follow a normal law. In practice, disturbances may be correlated contemporaneously with each other, without being correlated with, on the one hand, their own lagged values, and on the other, all the lagged values of the variables. When disturbances (εt ) are correlated, the variation of one error component has an impact on the other components – variables have a synchronous influence on each other. A causality analysis allows then to study the kind of influence variables have on each other. Moreover, the optimal lag is determined while minimizing Akaike and Schwarz information criteria. We investigate optimal lags of one to five days while looking for a weekly influence at most, as compared to the four days of persistence documented by Koutmos and Knif (2002) for beta estimates. The maximum likelihood method then gives an optimal lag p of one. Such a first-order relationship between asset returns may result from asynchronous trading in the financial markets or asset prices’ speed of adjustment to new private/public information. Indeed, large firms’ asset prices integrate information more easily and quickly than small ones’ asset prices, since large firms’ assets are usually more liquid: as large firms’ assets are usually traded more frequently than small firms’ assets, their prices adjust more quickly to the arrival of new information. Moreover, McKenzie and Faff (2003) show that trading volumes and market returns determine time-varying autocorrelations of asset returns. This setting leads to the results shown in Table 10.5. a01 a111 a211 In each column, the coefficients of returns are displayed as 0 1 a2 a21 a221 with their related Student statistics between brackets under each coefficient. Moreover, the R2 statistic related to the estimation of each univariate relation is displayed in percent as: [R2 (RXt ) R2 (RSt )] . Recall that we have the next VAR(1) bivariate specification:
RXt RS t
=
a01 a02
+
a111 a211 a121 a221
RXt−1 RSt−1
+
ε1t ε2t
(10.5)
Our VAR(1) specification does not exhibit any influence between the implied market factor’s return and returns of French indices. As a rough guide, we also compute the statistics and coefficients related to our ten stocks’ VAR(1) specification (see Table 10.6).
HAYETTE GATFAOUI
193
Table 10.5 VAR results for stock indices Index return CAC40
SBF120
SBF250
R 2 (%)
Coefficients −0.0546
−1.1382
1.0206
(−0.3488)
(−0.6794)
(0.6428)
1.2644
−0.0219
−0.6581
0.5539
0.7644
(−0.1327)
(−0.3721)
(0.3305)
−0.0487
−0.4722
0.4102
(−0.2888)
(−0.3079)
(0.2666)
0.5899 0.4677
−0.0053
−0.4560
0.4037
(−0.0317)
(−0.2995)
(0.2643)
−0.0609
−0.5750
0.5389
(−0.3473)
(−0.4061)
(0.3619)
0.7086
−0.0030
−0.5750
0.4852
0.4455
(−0.0178)
(−0.3738)
(0.3441)
At the 5 percent level of Student test, Air Liquide and Renault stocks impact implied a systematic risk factor, while Société Générale stock influences implied a systematic risk factor at the 10 percent level. We further investigate these results through a causality test. Granger causality test
A natural application of VAR modeling is a causality test. Granger (1969) defines causality as follows: RXt is said to be the cause of RSt when taking into account the information set associated to RXt helps to improve predictions of RSt . Analyzing causality of RXt towards RSt is equivalent to realizing a test with constraints on the coefficients of RXt in its VAR representation (Equation (10.5)) (a restricted VAR specification for RXt , also known as RVAR). Specifically, consider assumption H0 : a121 = a221 = 0. If we accept H0 , then RXt does not cause RSt . To test assumption H0 , we compare the unrestricted VAR (for example, UVAR, in Equation 10.5) with the VAR specification restricted to H0 (RVAR). The related test statistic is the likelihood ratio L = (n − c) ln{|RVAR |/|UVAR |} where n is the number of observations; c is the numberof estimated coefficients in each univariate relation of the UVAR model; RVAR , UVAR are the covariance matrices of restricted and unrestricted VAR models, respectively; |A| represents the determinant of matrix A. In this case, L is assumed to follow a chi-square law with two degrees of freedom (for example, χ2 (2)). Therefore, we reject H0 assumption for a given test level α if L is greater than the critical value of the χ2 (2) law 2 for level α (for example, L > χcritical (2); see Hamilton, 1994).
194
Table 10.6 VAR results for stocks Stock return Air Liquide
Danone
Vivendi
Totalfina Elf
Valéo
R 2 (%)
Coefficients −0.0588 (−0.4002) 0.1104 (0.5906)
−0.2090 (−1.4016) −0.2699 (−1.4223)
0.2525 (2.0454) 0.0547 (0.3481)
8.1354 4.1368
−0.0352 (−0.2366) 0.0329 (0.2584)
−1.1071 (−0.7840) 0.1050 (0.8981)
0.2532 (1.5238) −0.1050 (−0.7379)
4.8662 2.2082
−0.0078 (−0.0477) −0.5975 (−1.5866)
−0.1317 (−0.5784) 0.0990 (0.1881)
0.0367 (0.3665) 0.0317 (0.1371)
0.7152 0.5276
−0.0326 (−0.2106) 0.1860 (1.1717)
−0.0787 (−0.4343) −0.0572 (−0.3077)
0.0208 (0.1160) −0.1560 (−0.8388)
0.4753 3.9099
−0.0281 (−0.1817) 0.2474 (1.0045)
−0.0611 (−0.3786) 0.0482 (0.1875)
−0.0047 (−0.0455) 0.0080 (0.0485)
0.4527 0.1308
Stock return Société Générale
L’Oréal
Renault
Schneider
Thomson
R 2 (%)
Coefficients −0.0808 (−0.5361) 0.1979 (0.7984)
−0.3013 (−1.5920) 0.1241 (0.3985)
0.2120 (1.7529) −0.0068 (−0.0341)
6.2121 0.5678
−0.0032 (−0.0209) 0.1839 (1.0659)
0.0924 (0.5057) 0.0766 (0.3671)
−0.1809 (−1.2728) −0.5131 (−3.1637)
3.5726 24.0139
−0.1423 (−0.9186) 0.4518 (1.6360)
−0.2426 (−1.5658) 0.0960 (0.3477)
0.1884 (2.1443) 0.1453 (0.9278)
8.8326 3.5828
−0.0233 (−0.1521) 0.1484 (0.5015)
−0.0384 (−0.2468) 0.4611 (1.5319)
−0.0278 (−0.3571) −0.1657 (−1.1000)
0.7017 4.8022
−0.0295 (−0.1934) 0.0393 (0.1082)
−0.0671 (−0.4149) 0.0559 (0.1451)
0.0017 (0.0250) −0.0438 (−0.2698)
0.4498 0.1454
HAYETTE GATFAOUI
195
Table 10.7 Granger statistics for indices and stocks Asset return
L
Probability
CAC40
0.1385 0.4132
0.7114 0.5233
SBF120
0.0897 0.0711
SBF250
Stock return
L
Probability
Valéo
0.0352 0.0021
0.8520 0.9639
0.7658 0.7909
Société Générale
0.1588 3.0727
0.6920 0.0858
0.1397 0.1310
0.7101 0.7190
L’Oréal
0.1348 1.6199
0.7151 0.2090
Air Liquide
2.0229 4.1838
0.1612 0.0461
Renault
0.1209 4.5982
0.7296 0.0369
Danone
0.8066 2.3218
0.3734 0.1339
Schneider
2.3468 0.1275
0.1318 0.7226
Vivendi
0.0354 0.1343
0.8515 0.7156
Thomson
0.0211 0.0006
0.8852 0.9801
Totalfina Elf
0.0947 0.0135
0.7596 0.9081
Note: bold indicates a chi-squares results.
Studying relationships between an implied systematic risk factor’s return and French stock returns, we tested two assumptions, namely: “H0 : RXt does not Granger cause RSt ” and “H0∗ : RSt does not Granger cause RXt ”, and obtained the results shown in Table 10.7. For each asset, the first and second lines correspond to the results of H0 and H0∗ assumptions, respectively. At the 15 percent level, Air Liquide and Renault returns cause the implied market factor’s return (RXt ). Enlarging our test level to 40 percent, Société Générale also causes the implied market factor’s return (RXt ). Our study therefore shows a smaller impact of the implied market factor on French assets than was expected. Our results’ weakness may come from the small sample size used. For further investigation, we look for contemporaneous links between variables without lag consideration. Specifically, we test for a non-linear influence of the implied market factor’s price on the prices of French stocks and indices.
10.5 FURTHER INVESTIGATION We attempt to exhibit non-linear dependence and “quadratic” causality between implied market factor and French stocks. Non-linearity is captured through the study of returns. We proceed in two steps: a regression analysis of asset returns and a volatility analysis of these daily returns.
196
A STUDY OF THE FRENCH FINANCIAL MARKET
10.5.1 Simple regression Focusing on a non-linear link between the price of the implied market factor and the price of an asset is equivalent to regressing this asset return on the return of the implied market factor. Specifically, we look for the following kind of relationship β
S t = αS X t S
(10.6)
where αS and βS are constant terms, and St ∈ {SBF120, SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo Moreover, we make the approximation that Vivendi}.
t−1 t RSt = StS−S for each time t ranging from 2 to 55, and rewrite ≈ ln SSt−1 t−1 Equation (10.6) as a logarithm variation between times (t − 1) and t
RSt = βS RXt
(10.7)
for t ∈ {2, … , 55}. Consequently, the non-linear link between Xt and St is equivalent to a linear regression of St return (RSt ) on Xt return (RXt ). Such a study is practical, given that returns are stationary variables here. Moreover, our methodology consists of applying a single-index model that translates into a one-factor model that is close to CAPM (see Tables 10.8 and 10.9)6 . Regressions of French asset returns on the return of the implied market factor are all significant at the 1 percent level, apart from Danone stock’s
Table 10.8 Regression results for stock indices β
Student t
R 2 (%)
CAC40
1.0514
86.3097
99.2930
SBF120
0.9925
73.1893
99.0179
SBF250
0.9460
64.8178
98.7496
Index return
Table 10.9 Regression results for stocks β
Student t
R 2 (%)
Air Liquide
0.5650
3.8326
21.2243
Danone
0.1645
1.4883
Vivendi
1.8302
9.0541
Totalfina Elf
0.6694
6.2260
Valéo
0.8384
4.4587
Stock return
β
Student t
R 2 (%)
Société Générale
1.1088
7.1866
48.4465
3.9693
L’Oréal
0.8737
6.6437
44.7481
58.9820
Renault
0.9527
4.3560
18.8525
40.7725
Schneider
0.9480
3.9014
21.6706
25.0821
Thomson
1.2628
4.5825
28.2329
Stock return
HAYETTE GATFAOUI
197
regression. Among available French stock indices, the highest explanatory power is reached for CAC40 (for example, R2 (CAC40) = 99.2930%), whereas the highest explanatory power among French stocks is reached for Vivendi stock (for example, R2 (ex) = 58.8820%). Therefore, the implied market factor has an important influence, in terms of explaining daily returns, on all our financial assets apart from Danone. Such a pattern indicates that the residual risk factor (idiosyncratic risk factor) additional to the systematic risk factor explains the main part of Danone stock’s evolution. We also tested for the assumption “H0 : βS = 1” in Equation (10.7) for stocks. We found that βS has a significant unit value only for Valéo, Société Générale, L’Oréal, Renault Schneider and Thomson stocks. Therefore, these six assets are driven purely by market trends as represented by the implied market factor. Moreover, Air Liquide’s, Danone’s, and Totalfina Elf’s, stock returns absorb the influence of the implied market factor’s return, whereas Vivendi’s stock return amplifies such an impact. Finally, our ten stocks are globally market-driven, since their returns exhibit a positive link with that of the implied market factor. Such a finding is coherent with the work of Campbell et al. (2001). Brailsford and Faff (1997) found poor support for CAPM when studying Australian daily stock returns. Moreover, in a daily stock return setting, Koutmos and Knif (2002) show that the simple regression model works well for systematic risk measurement purposes (estimating the beta coefficient). However, a dynamic model with time-varying parameters is better for forecast purpose (forecasting efficient conditional beta estimates). Our main goal is to assess the impact of systematic risk on French stocks rather than value the validity and performance of CAPM (that is, to assess the mean-variance efficiency of our market proxy). Explanations about validity and performance of CAPM are proposed by Roll (1977) and Campbell et al. (1997), among others. Jagannathan and Wang (1996) also propose a good performance study. However, given the closeness of our single-index model to CAPM, we further investigate some linear dependency between returns of our implied market factor and French stocks. For this purpose, we employ the one-factor model of Jensen (1968, 1969) for each time t ranging from 2 to 55, namely RSt − r1M (t) = αS + βS (RXt − r1M (t)) + εt , where RXt is the return of the implied market factor X at time t; RSt is the return of stock S at time t; r1M (t) is the one-month French risk-free rate; βS is the sensitivity of stock S to the implied market factor X; αS is a constant term of regression; εt is a random normal error with zero expectation and constant variance; RSt − r1M (t) and RXt − r1M (t) are, respectively, stock S and implied market factor X market-risk premia. Jensen’s methodology allows the assessment of a risk-adjusted performance, the relevant risk measure being the beta of Sharpe (1963). The alpha coefficient of the previous regression is known as Jensen’s alpha and represents the abnormal return or excess return of a given stock relative to its CAPM return if this model were valid. In fact, alpha is the
198
A STUDY OF THE FRENCH FINANCIAL MARKET
Table 10.10 Jensen’s regression results for stocks Stock return
α
Student t α
β
Student t β
R 2 (%)
CAC40
0.1945
4.6472
1.0511
88.2375
99.3366
SBF120
0.0191
0.4367
0.9922
79.8204
99.1905
SBF250
−0.1257
−2.7697
0.9457
73.1643
99.0379
Air Liquide
−1.3580
−2.6126
0.5640
3.8116
21.8380
Danone
−2.7691
−7.0750
0.1663
1.4924
4.1072
Vivendi
−0.2698
−0.5857
0.8737
6.6611
46.0417
Totalfina Elf
0.4514
0.6223
0.9511
4.6061
28.9774
Valéo
0.0167
0.0195
0.9475
3.8880
22.5227
Société Générale
0.5767
1.0732
1.1062
7.2304
50.1336
L’Oréal
0.9755
1.0011
1.2599
4.5414
28.3987
Renault
−0.9362
−2.5121
0.6680
6.2962
43.2574
Schneider
−0.2530
−0.3877
0.8357
4.4978
28.0081
Thomson
2.2339
3.3201
1.8333
9.5707
63.7881
non-equilibrium return that the stock brings in over the studied time horizon (see Table 10.10). Jensen’s alpha is significant for SBF250, CAC40, Air Liquide, Danone, Renault and Thomson stock returns. And the alpha is negative (that is, an abnormal return leading to a loss in value for an investment in the considered stock) for SBF250, Air Liquide, Danone, Vivendi, Renault and Schneider returns. Moreover, the beta coefficient is significant for all indices and stocks apart from Danone’s stock return, whose beta is close to zero. Hence, Danone’s stock evolution is uncorrelated or extremely low-correlated with the market, which means that this stock is low or not sensitive to market evolution. Put differently, Danone and Thomson stocks exhibit the lowest and highest beta coefficients (systematic risk), respectively, whereas beta estimates of CAC40, Société Générale, L’Oréal and Thomson returns lie above unity (for example, amplify the market effect). Finally, the explanatory power of our regressions is globally good in so far as CAC40 and Thomson stocks exhibit the highest explanatory power among indices and stocks, respectively (R2 (CAC40) = 99.3366% and R2 (Thomson) = 63.7881%). In contrast, Danone stock exhibits the lowest explanatory power (for example, R2 (Danone) = 4.1072%). Consequently, the implied market factor generally has a strong impact and influence in explaining stock return evolutions. Such an influence is nevertheless insufficient to explain the whole evolution of assets given both the limited explanatory power of regressions and the significance of Jensen’s alpha. Such a pattern can be explained by firm-specific features (for example, size effect) that are left aside while considering only systematic risk’s impact on French stocks (see Fama and French, 1992, 1993; Berk, 1995). This point is emphasized with
HAYETTE GATFAOUI
199
Danone stock (Danone) whose evolution is not explained by the overall latent systematic risk factor prevailing in the French market. However, stronger evidence concerning the influence of the implied market factor on French stocks can be found while further investigating non-linear dependency between asset returns.
10.5.2 Volatility impact Investigating non-linear relationships between returns, we study the influence of the implied market factor on the volatility of our French assets. Indeed, linear causality analysis is unable to account for non-linear dependency between financial assets (for example, implied market factor and French stocks). Non-linear phenomena describing both financial markets and the underlying dynamics of the various assets composing such markets have been widely documented in the financial literature. Mele (1998) explains different kinds of non-linear dynamics, volatility and equilibrium that may describe a financial market. The simple existence of conditional heteroskedasticity in asset prices already describes some non-linear patterns in financial markets (Gourieroux and Jasiak, 2001). Put differently, exhibiting links between asset volatilities is a means of accounting for the non-linear features and effects that prevail between assets in markets. To this end, testing for a quadratic dependency between returns, we consider the weekly rolling volatilities of assets. As one calendar week represents five working days (a financial week), the weekly rolling volatility of return RSt at date t is written as σ(RSt ) = t t 1 2 with R = 1 (R − R ) (RSi ) for t ∈ {6, . . . , 55}. We analyze the S S S i t t 5 5 i=t−4
i=t−4
impact of the volatility of the implied market factor while considering the following first differences regressions: (10.8) σ RSt = aS σ RXt + ηt where ∀t ∈ {7, . . . , 55}, ∀ Xt , σ(RXt ) = σ(RXt ) − σ(RXt−1 ); aS is a constant coefficient; ηt is a “normal” disturbance; St ∈ {SBF120, SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi}. Results for first difference regressions (Equation (10.8)) of the weekly rolling volatilities of French assets on the weekly rolling volatility of the implied market factor are listed in Tables 10.11 and 10.12. Volatility regressions in Equation (10.8) are significant at a 1 percent level for CAC40, Vivendi, Totalfina Elf, Valéo, Société Générale, L’Oréal, Renault, SBF120 and SBF250. Among French indices, SBF250 presents the highest explanatory power (R2 (SBF250) = 97.1677%) whereas Vivendi exhibits the highest explanatory power (R2 (Vivendi) = 52.1625%) among stocks. Results
200
A STUDY OF THE FRENCH FINANCIAL MARKET
Table 10.11 Volatility regression results for stock indices a
Student t
R 2 (%)
CAC40
1.0329
40.2860
97.1143
SBF120
0.9614
39.3774
96.9823
SBF250
0.9349
40.6838
97.1677
Index return
Table 10.12 Volatility regression results for stocks Stock return
a
Student t R 2 (%)
Stock return
a
Student t R 2 (%)
2.2470 Société Générale
0.8547
2.8667 14.4792
0.2888
0.2015 L’Oréal
0.4608
4.0223 23.6650
7.2466
52.1625 Renault
0.8075
2.7771 13.8271
0.3720
2.8797
14.7152 Schneider
−0.0938 −0.3519
0.3325
0.6732
2.7991
14.0162 Thomson
−0.5603 −1.3889
3.7211
Air Liquide
0.1634
1.1522
Danone
0.0402
Vivendi
1.7898
Totalfina Elf Valéo
suggest that the implied market factor has a strong influence on the weekly rolling volatilities of CAC40, L’Oréal, Renault, SBF120, SBF250, Société Générale, Totalfina Elf, Valéo, and finally Vivendi assets. However, from the explanatory power of regressions, the implied market factor fails to explain the whole evolution of assets. As shown by Campbell et al. (2001) and Goyal and Santa-Clara (2003), idiosyncratic risk should be the additional factor explaining the part of stock returns that is unexplained by systematic risk factor. Results require a global remark drawing a comparison between the global market information embedded in both the CAC40 stock index and its filtered counterpart, as represented by implied market factor X. The results we get while using the CAC40 stock index as a market proxy instead of an implied market factor give preliminary insights. Indeed, as the correlation between returns of the CAC40 stock index and implied market factor is 99.6666 percent, why should the CAC40 stock index not be the market? Given our results, the average (that is, arithmetic mean over time horizon) price of the CAC40 Index is 2.2881 times the average price of the implied market factor, with an average scale factor of 2.2883. Stated differently, the CAC40 INDEX average return is 3.8164 times the implied market factor’s average return. However, we also notice that the average ratio of the CAC40 INDEX return to the implied market factor’s return is 0.9636. These preliminary statistics suggest that CAC40 is a good market proxy if we assume that the implied market factor is the actual market portfolio. However, we cannot draw such
HAYETTE GATFAOUI
201
a conclusion when considering the CAC40 INDEX as a market proxy. Moreover, being interested only in a view of the level of market return, CAC40, as well as the SBF120 and SBF250 indices, seem to be a convenient approximation of market return (as represented by implied market factor). Recall that both SBF120 and SBF250 indices are correlated almost as highly with the implied market factor as the CAC40 stock index. Nevertheless, for forecast purpose, this viewpoint changes greatly. Such results are introduced and summarized in the next section, which makes a comparison between our four different benchmarks.
10.6 MARKET BENCHMARK COMPARISON To answer the question about choosing between implied market factor and the CAC40 INDEX, we compare the results we get when employing successively Xt , CAC40, SBF250 and SBF120 indices as market benchmarks. First, we summarize the results obtained for our four different benchmarks, then we make a study of forecasting performance.
10.6.1 Basic empirical study We have estimated the three previous types of regressions that we called simple, Jensen and volatility regressions. We summarize the results in this section, displaying only relevant results to save space. We use two criteria to discriminate between market benchmarks. First, we consider the explanatory power of related regressions, and second, how close beta estimates of such regressions lie relative to the beta estimates we get for implied market factor X. In this way, we observe the impact of the bias, which comes from the fact that studied stocks are part of our three French stock indices. Recall that the Granger causality test allows for the classifying of benchmark-based relations with decreasing value of significance as Xt , SBF250, SBF120 and the CAC40 INDEX. We display the results for the explanatory power of our three types of regressions in Table 10.13. The first, second and third lines of each asset correspond to the simple, Jensen and volatility regressions, respectively. As regards simple regressions, SBF250-based regressions exhibit the highest explanatory power for 50 percent of stocks (Valéo, Société Générale, Schneider, Renault and Thomson returns) whereas CAC40-based regressions exhibit the highest explanatory power for 30 percent of stocks (Air Liquide, Totalfina Elf and L’Oréal returns). In a less powerful way, SBF120-based regressions exhibit the highest explanatory power for 10 percent of stocks (Danone returns) analogously to Xt -based regressions (Vivendi returns). As regards Jensen-type regressions, considering the proportion of stocks
202
Table 10.13 Explanatory power of regressions (percentages) Stock
Xt
CAC40
SBF120
SBF250
Stock
21.2243 21.8380 2.2470
21.4424 21.9272 4.0148
20.1345 20.5040 3.1089
20.0694 20.3780 2.7512
Danone
3.9693 4.1072 0.2015
4.1214 4.2446 0.0354
4.1579 4.2681 0.1733
Vivendi
58.9820 46.0416 52.1625
57.6816 46.8405 47.8086
Totalfina Elf
40.7725 28.9774 14.7152
Valéo
25.0821 22.5227 14.0162
Air Liquide
Xt
CAC40
SBF120
SBF250
Société Générale
48.4465 50.1336 14.4792
50.4244 51.7562 17.1930
50.2269 51.2400 15.8954
50.4256 51.2706 16.1946
4.0683 4.1740 0.0841
L’Oréal
44.7481 28.3987 23.6650
45.8329 27.9946 26.9866
44.8268 29.3573 25.4735
44.0374 29.5735 23.1302
56.7106 45.6010 47.8346
56.4111 44.7000 49.3285
Renault
18.8525 43.2574 13.8271
18.8192 44.9391 14.9337
18.9169 42.8588 14.1422
19.4062 42.1550 13.6872
42.8574 28.2652 17.3273
41.1395 27.7130 15.9644
40.6282 27.8279 14.6540
Schneider
21.6706 28.0081 0.3325
22.2030 28.9954 0.3286
23.7321 30.7258 0.1534
24.1688 31.3020 0.0724
26.4305 22.8926 14.3045
28.5368 24.2689 14.3560
29.3201 24.6283 15.0222
Thomson
28.2329 63.7881 3.7211
27.9002 63.2092 3.4569
29.3254 63.0331 2.6751
29.5660 63.2194 3.1387
HAYETTE GATFAOUI
203
that are best explained (highest explanatory power) through our four market benchmarks leads to classifying benchmarks with a decreasing proportion’s of value as CAC40 (40 percent of stocks, namely Air Liquide, Vivendi, Société Générale and Renault returns); SBF250 (30 percent of stocks, namely Valéo, L’Oréal and Schneider returns); Xt (20 percent of stocks, namely Totalfina Elf, and Thomson returns); and SBF120 (10 percent of stocks, namely Danone returns). As regards the explanatory power of volatility-based regressions, comparing the proportion of best-explained stocks through our different market proxies allows us to order results with decreasing proportion’s of value as CAC40 (50 percent of stocks, namely Air Liquide, Totalfina Elf, Société Générale, L’Oréal and Renault returns); Xt (40 percent of stocks, namely Danone, Vivendi and Thomson returns); SBF250 (for example, 10 percent of stocks, namely Valéo’s return); and SBF120. However, explanatory power-based results are probably upward-biased because of the weights of studied stocks that belong to available French stock indices (for example, important free-floating market capitalization weights in indices). To bypass such a bias, we consider how close the beta estimates of our different benchmark-based regressions are to the beta estimates of Xt -based regressions. For an overview, we display related beta estimates in Table 10.14. For each stock return, the first, second and third levels refer to simple, Jensen-type and volatility regressions, respectively. As regards simple regressions, SBF120-based regressions exhibit the closest estimates for Danone, Vivendi, Totalfina Elf, Société Générale, L’Oréal, Renault and Thomson stocks (70 percent of stocks) while CAC40- and SBF250-based regressions exhibit the closest estimates for Valéo and Schneider stocks (20 percent of stocks), and Air Liquide stock (10 percent of stocks), respectively. As regards Jensen-type regressions, SBF120-based regressions exhibit the closest beta estimates for Danone, Vivendi, Valéo, Totalfina Elf, Société Générale, L’Oréal, Renault and Thomson stocks (80 percent of stocks) while SBF250- and CAC40-based regressions exhibit the closest estimates for Air Liquide (10 percent of stocks) and Schneider stocks (10 percent of stocks), respectively. As regards volatility regressions, CAC40-based regressions exhibit the closest beta estimates for Danone, Totalfina Elf, Valéo, Société Générale, L’Oréal, Renault and Schneider stocks (70 percent of stocks) while SBF250- and SBF120-based regressions exhibit the closest estimates for Air Liquide and Thomson stocks (20 percent of stocks), and Vivendi-stock (10 percent of stock), respectively. Hence, given the closeness to the beta estimates of Xt -based regressions, the SBF120 index seems to be the best proxy for implied market factor for both simple and Jensen-type regressions (best for 80 percent, on average). Therefore, assuming that Xt is the actual market portfolio and given that the correlation between returns of Xt and SBF120 is 0.9959, employing SBF120 as a market proxy follows the findings of both Kandel and Staumbaugh (1987) and Shanken (1987). But differently, the CAC40 INDEX seems to be the best proxy of the implied market factor for
204
Table 10.14 Beta estimates for three types of regressions (1) Stock
Xt
CAC40
SBF120
SBF250
Stock
Air Liquide
0.5650 0.5640 0.1634*
0.5381 0.5359 0.2005*
0.5521 0.5486 0.1922*
0.5756 0.5734 0.1876*
Société Générale
Danone
0.1645*
0.1588*
0.1688*
0.1749*
L’Oréal
0.1663* 0.0402*
0.1603* 0.0448*
0.1702* 0.0605*
0.1764* 0.0543*
Vivendi
1.8302 0.8737 1.7898
1.7167 0.8356 1.6351
1.8017 0.8728 1.7560
1.8831 0.9059 1.8353
Renault
Valéo
0.6694 0.9511 0.3720
0.6495 0.8907 0.3851
0.6739 0.9337 0.3969
0.7020 0.9808 0.3914
Totalfina Elf
0.8384 0.9475 0.6732
0.8135 0.9058 0.6489
0.8907 0.9873 0.6979
0.9448 1.0427 0.7348
Note: *Non-significant estimates at a 5% test level.
Xt
CAC40
SBF120
SBF250
1.1088 1.1062 0.8547
1.0714 1.0658 0.8879
1.1312 1.1227 0.9170
1.1875 1.1773 0.9526
0.8737
0.8378
0.8767
0.9107
1.2599 0.4608
1.1861 0.4672
1.2858 0.4883
1.3530 0.4808
0.9527 0.6680 0.8075
0.9024 0.6457 0.8006
0.9562 0.6675 0.8365
1.0102 0.6940 0.8471
Schneider
0.9480 0.8357 −0.0938*
0.9091 0.8063 −0.0892*
0.9930 0.8786 −0.0610*
1.0497 0.9297 −0.0391*
Thomson
1.2628 1.8333 −0.5603*
1.1898 1.7305 −0.5161*
1.2901 1.8294 −0.4903*
1.3572 1.9207 −0.5445*
HAYETTE GATFAOUI
205
volatility regressions. We investigate such preliminary results through a performance study.
10.6.2 Forecasting performance We attempt to discriminate between our four market proxies while considering their weekly forecasting performance in our three types of regressions. For this purpose, we first estimate regressions on the initial time horizon less one week of data (five observations). Then, we forecast corresponding returns or volatilities on the remaining week. Finally, we assess the related performance while computing the related average absolute relative error. Hence, we can assess the realized forecasting error relative to the actual level of return or volatility. Simple, Jensen-type and volatility regressions are successively estimated on t ∈ {2, … , 50}, {2, … , 50} and {7, … , 50} time horizons, respectively. We drop the last week of data for t ∈ {51, … , 55}. We display in Table 10.15 the beta estimates for our three types of regressions while employing successively our four market proxies. For each stock return, the first, second and third lines refer to simple, Jensen-type and volatility regressions, respectively. With regard to the closeness of our beta estimates to Xt -based beta estimates, a similar conclusion to that in the previous sub-section applies. On average (for 75 percent of stocks), the SBF120 index represents the best proxy of implied market factor for both simple and Jensen-type regressions whereas the CAC40 INDEX is the best proxy of Xt for volatility regressions. For the second part of our study, we first use previous regression estimates to forecast related returns and volatilities in the last week of our initial time horizon. Then, to assess the weekly forecasting performance of our benchmark-based regressions, we compute the corresponding average relative absolute error (average normalized absolute error). Such a performance measure allows us to highlight the percentage of forecasting errors relative to the actual level of both returns and volatilities under consideration. For this purpose, we compute respective forecasting errors eS of stock S during the last week of data as 55 ∗ ˆ St R − R 1 S eS = t ∗ RS 5 t t=51 ˆ St − rˆ1M (t) 55 R∗ − r1M (t) − R S 1 t eS = ∗ 5 RSt − r1M (t) t=51 and
55 ∗ R σ − σ ˆ RSt 1 S t eS = ∗ 5 σ RSt t=51
206
Table 10.15 Beta estimates for three types of regressions (2) Xt
CAC40
SBF120
SBF250
Stock
Air Liquide
0.5789 0.5808
0.5520 0.5522
0.5664 0.5654
0.5911 0.5895
Société Générale
0.1397*
0.1748*
0.1740*
0.1714*
Danone
0.1644
0.1582
0.1630*
0.1672*
Stock
Vivendi
Valéo
Totalfina Elf
L’Oréal
Xt
CAC40
SBF120
SBF250
1.1122 1.1170
1.0748 1.0750
1.1314 1.1282
1.1878 1.1827
0.8820
0.9159
0.9537
0.9977
0.8533
0.8187
0.8569
0.8910
0.1654
0.1596
0.1650*
0.1696*
1.2302
1.1586
1.2588
1.3232
0.0200
0.0288
0.0465*
0.0363*
0.4747
0.4778
0.5062
0.5036
1.8708
1.7585
1.8422
1.9279
0.9135
0.8667
0.9235
0.9751
0.8577
0.8201
0.8565
0.8898
0.6808
0.6582
0.6842
0.7133
1.8490
1.6884
1.8192
1.9103
0.7195
0.7232
0.7699
0.7714
Renault
0.6756
0.6574
0.6868
0.7178
0.9949
0.9546
1.0407
1.0998
0.9291
0.8714
0.9200
0.9667
0.8648
0.8373
0.9086
0.9622
0.3930
0.4014
0.4228
0.4228
−0.1106*
−0.1105*
−0.1098*
−0.0954*
0.8592
0.8370
0.9125
0.9686
1.2321
1.1606
1.2607
1.3247
1.0017
0.9567
1.0395
1.0969
1.8570
1.7554
1.8495
1.9424
0.7090
0.6761
0.7209
0.7621
−0.6688*
−0.6184*
−0.6215*
−0.6960*
Note: *Non-significant estimates at a 5% test level.
Schneider
Thomson
HAYETTE GATFAOUI
207
for simple, Jensen-type and volatility regressions. RS∗t , RS∗t − r1M (t) and σ ∗ (RSt ) are, respectively, observed market values of stock return, market risk premium and the first difference of weekly rolling volatility of stock St , while Rˆ St , (Rˆ St − rˆ1M (t)) and σ(R ˆ St ) are corresponding respective regression estimates of asset return, market risk premium and first-order weekly rolling volatility of asset St . We display our results in Table 10.16, where the first, second and third lines of each stock refer to simple, Jensen-type and volatility regressions, respectively. With regard to simple regressions, we classify benchmark-based regressions as an increasing function of the average relative absolute error, and get Xt implied factor, SBF250, SBF120 and CAC40 indexes. Indeed, Xt -based regressions exhibit the lowest absolute errors for Air Liquide, Totalfina Elf, Renault, and Schneider returns (40 percent of the stocks) while SBF250 and SBF120-based regressions exhibit the lowest ones for Danone, Vivendi and Thomson returns (30 percent of stocks), and Valéo, Société Générale, and L’Oréal returns (30 percent of stocks), respectively. Moreover, the mean of average absolute relative errors over all stocks for SBF250-based regressions is 130.8780 percent, and lies below the mean of SBF120-based regressions at is 131.2630 percent. On average, for 50 percent of cases, Xt average relative absolute errors are below the observed absolute errors of other stock indexbased simple regressions. With regard to Jensen-type regressions, ordering benchmark-based regressions with increasing value of average relative absolute errors results in Xt implied factor, SBF250, SBF120 and CAC40 indexes. In particular, Xt -based regressions exhibit the lowest average relative absolute errors for Totalfina Elf, Valéo, L’Oréal, Renault and Schneider returns (50 percent of the stocks), while SBF250 and SBF120-based regressions exhibit the lowest errors for Air Liquide, Danone, Société Générale and Thomson returns (40 percent of stocks), and Vivendi returns (10 percent of stocks), respectively. On average for 63.3334 percent of cases, Xt -based regressions exhibit average relative absolute errors that lie below the ones observed for the other stock index-based Jensen regressions. With regard to volatility regressions, ordering benchmark-based regressions with increasing value of average relative absolute errors results in Xt implied factor, SBF250, SBF120 and CAC40 indexes. Specifically, Xt -based regressions exhibit the lowest average relative absolute errors for Vivendi, Totalfina Elf, Société Générale and Renault returns (40 percent of stocks) while SBF250-based regressions exhibit the lowest for Air Liquide, Valéo and Schneider returns (30 percent of stocks). CAC40-based regressions exhibit lowest average relative absolute errors for L’Oréal and Thomson returns (20 percent of stocks), whereas SBF120-based regressions exhibit the lowest ones for Danone returns (10 percent of stocks). On average, for 50 percent of cases, Xt -based average relative absolute errors are lower than other benchmark-based average relative absolute errors.
208
Table 10.16 Average absolute relative errors for three types of regressions (percentages) Stock
Xt
CAC40
SBF120
SBF250
Air Liquide
190.2920 34.1998 93.6811
198.2960 34.4080 91.3792
199.7410 34.5058 89.5556
202.1020 34.1037 89.0935
Danone
108.7540 40.9120 100.5850
108.2980 40.8938 100.3970
106.8060 40.5194 97.5889
Vivendi
198.2090 21.3727 170.7980
203.5010 21.4338 181.0560
98.5802 75.7842 98.5722
Valéo
Mean error
Totalfina Elf
Stock
Xt
CAC40
SBF120
SBF250
Société Générale
156.1130 35.9794 156.3100
156.9040 34.9911 163.5170
145.9650 32.6395 165.2350
147.3510 32.2085 174.7310
105.9940 40.3935 97.6414
L’Oréal
58.8281 206.2060 228.1930
58.7250 208.0390 206.3310
56.9955 209.0130 242.5520
58.6216 206.6140 268.5740
179.0770 20.8868 285.0580
167.8610 21.3040 324.9240
Renault
118.5050 24.4820 76.1971
119.2220 24.7314 77.8722
123.4980 26.0944 101.8760
123.9320 26.5305 104.7220
99.4735 77.2388 103.1750
105.3530 78.5040 154.4290
107.1970 78.5563 187.6690
Schneider
178.6000 117.4490 134.1100
180.7920 119.7430 132.5430
200.0930 118.7390 127.7270
204.9850 119.3130 126.0830
119.8940 33.9410 124.9710
121.7060 34.1834 138.8180
118.5630 34.1897 122.2820
118.5890 33.9615 121.3980
Thomson
87.5991 93.7890 235.0740
84.4210 96.5114 91.5840
76.5346 90.8615 113.5140
72.1501 90.3767 164.9720
131.5370 68.4115 141.8490
133.1340 69.2174 128.6670
131.2630 68.5953 149.9820
130.8780 68.3362 165.9810
HAYETTE GATFAOUI
209
The extremely good performance exhibited by the explanatory powers of some index-based regressions probably results from an upward bias. Such a bias comes from the non-negligible weights assigned to studied stocks from which French market indices are composed. Also, beta estimate-based analysis indicates globally that, among observed market proxies, SBF120based regressions exhibit the closest beta estimates relative to the ones we get for Xt -based regressions. On the other hand, CAC40-based regressions are far behind SBF120-based ones when considering linear relations between asset returns. Hence, in the prospect of an assessment of the impact of systematic risk on French stocks, the CAC40 INDEX represents a much less accurate approximation of market return than the implied market factor (fewer significant relations between returns). Finally, our forecasting performance study based on average relative absolute errors suggests that the Xt benchmark (implied market factor) is, on average, a more powerful proxy for systematic risk factor than our three French stock indices. Specifically, our implied market factor has a far more powerful forecasting performance than the CAC40 stock index. Such a viewpoint is sustained by the results of Jensen-type regressions when we employ the CAC40 INDEX as a market proxy (rather than implied market factor). Regarding the negative estimated abnormal returns (negative alpha coefficients): these have a higher absolute value for the CAC40 benchmark than for the implied market factor. Hence, the CAC40 benchmark leads to higher abnormal losses. On the other hand, regarding positive estimated alpha coefficients, abnormal returns brought in by French stocks are lower for the CAC40 benchmark than for the implied market factor. Moreover, there is a difference of sign for Jensen’s alpha of the stock Valéo. In conclusion, assuming that implied market factor X is an accurate proxy of the actual market portfolio, the CAC40 INDEX leads to an underestimation of positive abnormal returns and an overestimation of losses or negative abnormal returns brought in by French stocks. Consequently, CAC40 stock index does not represent the market, though it has been employed to infer the level of implied market risk factor (through some non-linear filtering methodology).
10.7 CONCLUSION Considering the wide literature about systematic risk initiated by Sharpe (1963) and the debate initiated by the famous critique by Roll (1977), we addressed the problem of finding a good market risk proxy when considering a small stock index with traded options on such an index. We proceeded in five steps: a theoretical framework; an empirical application of this setting; two empirical studies assessing impact of the implied systematic risk
210
A STUDY OF THE FRENCH FINANCIAL MARKET
on French financial assets; and a critical study (forecasting performance analysis of available market benchmarks). First, our theoretical setting assumed that the small stock index was a disturbed observation of the actual market factor. This stock index depends on the market factor through a scale parameter, which is a continuous function of time. We further assumed that the market factor follows a geometric Brownian motion. Then, we induced an analytical formula pricing European calls on the stock index. All the parameters of our closed form formula are known apart from the scale parameter at times t and T, and the volatility of the market factor. Second, inverting our European call pricing formula given observed market prices of European index calls, we calculated the values of the scale factor at dates t and T, and the volatility parameter. These estimates allowed the computation of the implied market factor’s level from stock index prices. We applied this empirical study to the French financial market, and considered its CAC40 stock index. Results showed that the implied volatility parameter is time-varying, and the distributions of both volatility and market factor are leptokurtic. Third, we attempted to assess the implied market factor’s impact on a basket of French stocks and indices. We studied correlations between the implied market factor’s return and French asset returns. Results are poor in so far as our VAR study and the Granger causality test only show the strong influence of Air Liquide and Renault daily stock returns on the implied market factor’s return. Fourth, we investigated a non-linear relationship between French asset prices and the level of implied market factor. This led to the study of linear regressions of French asset returns on the implied market factor’s return. Our linear framework assumes that residual risk, which we assimilated to idiosyncratic risk, is normally distributed, with zero mean and constant variance. Results obtained are fruitful in that the implied market factor’s return appears to have a strong influence on French asset returns, apart from on Danone stock. Indeed, regressions exhibit high explanatory power. Further, we also estimated first differences regressions of French assets’ weekly rolling volatilities on the weekly rolling volatility of implied market factor. The implied market factor exhibits a strong link with CAC40, L’Oréal, Renault, SBF120, SBF250, Société Générale, Totalfina Elf, Valéo and Vivendi assets. However, it fails in explaining the whole evolution of assets, probably because idiosyncratic risk plays an important role. Indeed, such a risk factor can explain that part of assets’ evolution which remains unexplained by systematic risk factors, as suggested by Campbell et al. (2001) and Goyal and Santa-Clara (2003). Finally, we attempted to discriminate between implied market factor and French stock indices as a market proxy. Specifically, our forecasting performance study examines the highest correlation observed between both the CAC40 INDEX and implied market factor returns. Namely, the CAC40 INDEX can be a useful proxy to estimate current level of market return, whereas implied market factor is a more convincing market benchmark for
HAYETTE GATFAOUI
211
forecast purpose and performance measure (systematic risk level), along with the CAPM setting. Suggested improvements are, first, the lengthening of the time period. A larger sample could give stronger and more significant results. Second, building a diversified portfolio (which replicates market factors accurately) would give a systematic risk benchmark to be compared with implied market factor. Prior to this, we should address what the optimal number of stocks and optimal composition of a well-diversified portfolio should be to achieve a sound and standardized assessment of systematic risk. Third, as in CAPM theory, firm-specific risk or unsystematic risk was not considered to explain realized stock returns. Our single factor framework ignores the part of any return’s global variance that is a result of firm-specific patterns. However, firm-specific factors are important explanatory variables for asset returns, as shown by Fama and French (1992, 1993) and Berk (1995). Hence, given Roll’s critique and advice in favor of multi-index models, future research should apply at least a two-factor model accounting for both systematic and idiosyncratic risk factors.
NOTES 1. Improved versions of CAPM are also given by Mossin (1966), Lintner (1965, 1969) and Black (1972). Dynamic versions of CAPM are also proposed, along with intertemporal models like that in Merton (1974). 2. Milevsky (2002) studies two dimensions of diversification, namely the number of stocks in a portfolio, and the time horizon for investment. The author discusses the benefits of the number of stocks diversification versus time horizon diversification. 3. Under completeness, financial asset prices can be reached (for example, each market variable is observable or has a proxy). 4. Drift and volatility parameters must satisfy the Lipschitz conditions, which ensure the existence and uniqueness of the solution to the stochastic differential equation satisfied by the market factor’s dynamic (given a starting value). 5. The SBF120 index is a weighted stock index composed of the forty values of CAC40 INDEX and another eighty most liquid French stocks. The SBF250 index is a weighted index composed of the 120 stocks of the SBF120, and 130 stocks selected for their importance and sector representativity. There is no traded option on such indices in the MONEP. 6. We also performed regressions with a constant term. Unfortunately, the constant coefficient does not generally appear to be significant, and its estimated value is very different from the levels of the one-, two- or three-month French risk-free rates observed in the market. We notice that constant term of regression =(1 – β) rf 55 1 for rf ∈ {¯r1M , r¯2M , r¯3M } with r¯iM = 54 riM (t) whatever i = 1, 2, 3. t=2
ACKNOWLEDGMENTS I would like to thank participants and referees at the MODSIM conference (Townsville, Australia, July 2003) and Professor Peter Verhoeven (Curtin University of Technology)
212
A STUDY OF THE FRENCH FINANCIAL MARKET
for their helpful comments. I am also grateful to Professor Pascal St-Amour (HEC Montreal) and attendees of the EFMA annual meeting (Basle, Switzerland, July 2004) for their interesting remarks and suggestions. Finally, I thank participants at the 17th IAE National Days (Lyon, France, September 2004) and Deloitte Risk Management Conference (Antwerp, Belgium, May 2005) for their interesting comments. The usual disclaimer applies.
REFERENCES Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (1999) “Coherent Measures of Risk”, Mathematical Finance, 9(3): 203–28. Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (2000) “Risk Management and Capital Allocation with Coherent Measures of Risk”, Working paper, ETH Zentrum. Berk, J. B. (1995) “A Critique of Size-Related Anomalies”, Review of Financial Studies, 8(2): 275–86. Black, F. (1972) “Capital Market Equilibrium with Restricted Borrowing”, Journal of Business, 45(3): 444–55. Black, F. (1975) “Fact and Fantasy in the Use of Options”, Financial Analysts Journal, 31(July/August): 36–72. Black, F. and Scholes, M. (1973) “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 81(1): 637–54. Brailsford, T. J. and Faff, R. W. (1997) “Testing the Conditional CAPM and the Effect of Intervaling: A Note”, Pacific-Basin Finance Journal, 5(5): 527–37. Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997) The Econometrics of Financial Markets (Princeton, NJ: Princeton University Press). Campbell, J. Y., Lettau, M., Malkiel, B. G. and Xu, Y. (2001) “Have Individual Stocks become More Volatile? An Empirical Exploration of Idiosyncratic Risk”, Journal of Finance, 56(1): 1–43. Fama, E. and French, K. (1992) “The Cross-Section of Expected Stock Returns”, Journal of Finance, 47(2): 427–65. Fama, E. and French, K. (1993) “Common Risk Factors in the Returns on Stocks and Bonds”, Journal of Financial Economics, 33: 3–56. French, K. R. and Poterba, J. M. (1991) “International Diversification and International Equity Markets, American Economic Review, 81(2): 222–6. Gençay, R., Selçuk, F. and Whitcher, B. (2003) “Systematic Risk and Timescales”, Quantitative Finance, 3, April: 108–16. Gourieroux, C. and Jasiak, J. (2001) Financial Econometrics: Problems, Models, and Methods (Princeton, NJ: Princeton University Press). Goyal, A. and Santa-Clara, P. P. (2003) “Idiosyncratic Risk Matters!”, Journal of Finance, 58(3): 975–1007. Granger, C. W. J. (1969) “Investigating Causal Relations by Econometric Models and Cross Spectral Methods”, Econometrica, 37(3): 424–38. Hamilton, J. D. (1994) Time Series Analysis (Princeton, NJ: Princeton University Press). Jagannathan, R. and Wang, Z. (1996) “The Conditional CAPM and the Cross-Section of Expected Returns”, Federal Reserve Bank of Minneapolis, Research Department Staff Report 208. Jensen, C. M. (1968) “The Performance of Mutual Funds in the Period 1945−1964”, Journal of Finance, 23(2): 389–415. Jensen, C. M. (1969) “Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios”, Journal of Business, 42(2): 167–247.
HAYETTE GATFAOUI
213
Kandel, S. and Staumbaugh, R. (1987) “On Correlations and Inferences about MeanVariance Efficiency”, Journal of Financial Economics, 18(1): 61–90. Koutmos, G. and Knif, J. (2002) “Estimating Systematic Risk Using Time-Varying Distributions”, European Financial Management, 8(1): 59–73. Lintner, J. (1965) “The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets”, Review of Economics and Statistics, 47(1): 13–37. Lintner, J. (1969) “The Aggregation of Investor’s Diverse Judgments and Preferences in Purely Competitive Security Markets”, Journal of Financial and Quantitative Analysis, 4(4): 347–400. McKenzie, M. D. and Faff, R. W. (2003) “The Determinants of Conditional Autocorrelations in Stock Returns”, Journal of Financial Research, 26(2): 259–74. Mele, A. (1998) Dynamiques Non Linéaires, Volatilité et Equilibre, Collection Approfondissement de la Connaissance Economique, Economica. Merton, R. C. (1973) “The Theory of Rational Option Pricing”, Bell Journal of Economics & Management Science, 4(1): 141–83. Merton, R. C. (1974) “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates”, Journal of Finance, 29(2): 449–70. Milevsky, M. A. (2002/03) “Space-Time Diversification: Which Dimension is Better?”, Journal of Risk, 5(2) Winter: 45-71. Mossin, J. (1966) “Equilibrium in a Capital Asset Market”, Econometrica, 34: 768–83. Rogers, L. C. G. and Williams, D. (1994a) Diffusions, Markov Processes and Martingales: Foundations, vol. 1 (Cambridge University Press). Rogers, L. C. G. and Williams, D. (1994b) Diffusions, Markov Processes and Martingales: It¯o Calculus, vol. 2 (Cambridge University Press). Roll, R. (1977) “A Critique of the Asset Pricing Theory’s Tests”, Journal of Financial Economics, 4(2): 129–76. Shanken, J. (1987) “Multivariate Proxies and Asset Pricing Relations: Living with the Roll Critique”, Journal of Financial Economics, 18(1): 91–110. Sharpe, W. F. (1963) “A Simplified Model for Portfolio Analysis”, Management Science, 9: 499–510. Sharpe, W. F. (1964) “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk”, Journal of Finance, 19(3): 425–42. Sharpe, W. F. (1970) Portfolio Theory and Capital Markets (New York: McGraw-Hill). Stulz, R. M. (1999a) “International Portfolio Flows and Security Markets”, In M. Feldstein, (ed.), International Capital Flows (Chicago: University of Chicago Press). Stulz, R. M. (1999b) “Globalization, Corporate Finance, and the Cost of Capital”, Journal of Applied Corporate Finance, 12: 8–25. Stulz, R. M. (1999c) “Globalization of Equity Markets and the Cost of Capital”, Working paper, Dice Center, Ohio State University. Szego, G. (2002) “Measures of Risk”, Journal of Banking and Finance, 26(7): 1253–72. Treynor, J. (1961) “Toward a Theory of the Market Value of Risky Assets”, unpublished manuscript. Whaley, R. (1982) “Valuation of American Call Options on Dividend-paying Stocks”, Journal of Financial Economics, 10(1): 29–58.
C H A P T E R 11
Matrix Elliptical Contoured Distributions versus a Stable Model: Application to Daily Stock Returns of Eight Stock Markets Taras Bodnar and Wolfgang Schmid
11.1 INTRODUCTION The assumptions of independency and normality are not appropriate in many situations of practical interest, especially in modeling financial data from emerging markets. It was pointed out in numerous studies that daily financial data is heavily tail distributed (Blattberg and Gonedes, 1974; Fama, 1976; Engle, 1982; Bollerslev, 1986; Nelson, 1991; Rachev and Mittnik, 2000). These studies proposed to pick up the assumptions of t-distribution, symmetric stable distribution, or the autoregressive conditional heteroskedasticity (ARCH) process instead of normality. In this chapter, the much weaker assumption of matrix ellipticalcontoured distribution is imposed on the asset returns. This family covers a wide range of distributions – for example, the matrix normal distribution, the matrix mixture of normal distribution, the matrix t-distribution and 214
TARAS BODNAR AND WOLFGANG SCHMID
215
the matrix symmetric stable distribution. Elliptical distributions, whose contours of equal densities have the same elliptical shape as the normal distribution, provide an attractive alternative to the multivariate stable law. These distributions have been already discussed in financial literature (Chamberlain, 1983; Owen and Rabinovitch, 1983; Zhou, 1993; Berk, 1997; Bodnar and Schmid, 2003, 2004). For instance, Owen and Rabinovitch (1983) showed that Tobin’s separation theorem, Bawa’s rules of ordering certain prospects can be extended to elliptically contoured distributions. While Chamberlain (1983) showed that elliptical distributions imply mean– variance utility functions, Berk (1997) argued that one of the necessary conditions for the capital asset pricing model (CAPM) is an elliptical distribution for the asset returns. Furthermore, Zhou (1993) extended findings of Gibbons et al. (1989) by applying their test for the validity of the CAPM to elliptically distributed returns. The first paper dealing with the application of matrix elliptically contoured distributions in finance, however, seems to be Bodnar and Schmid (2003). They introduced a test for the global minimum variance. It is analyzed whether the lowest risk is larger than a given benchmark value or not. The aim of the present study is to derive the statistical procedures for testing the elliptical symmetry of multivariate sample, for example, matrix ellipticity. While there are several procedures for testing the multivariate elliptical and spherical symmetry under independency assumptions (Beran, 1979; Baringhaus, 1991; Fang et al., 1993; Heathcote, et al., 1995; Koltchinskii and Li, 1998; Manzotti et al., 2002; Zhu and Neuhaus, 2003), the matrix elliptical symmetry was not treated in literature up to now. Furthermore, only the limiting distributions of above-mentioned statistics were derived in the studies. Conversely, our approaches are based on the small sample tests. Empirically we show that daily returns of seven developed countries follow a matrix elliptical distribution. This result is in line with the findings of Andersen et al. (2001), who, using the distributional properties of realized volatility, argue that daily returns can be well approximated by the mixture of normal distributions (the partial case of elliptical family). The remainder of the chapter is organized as follows. The main results are presented in section 11.2. Under the null hypothesis of matrix elliptical symmetry, the finite sample distributions of the proposed statistics are derived in all cases when the type of elliptical symmetry, location vector and scale matrix are known or unknown. The test’s powers are considered in section 11.3. In section 11.4 we implement our findings empirically by considering daily returns of seven developed stock markets. We show that the null hypothesis of the matrix ellipticity cannot be rejected for this data. Furthermore, as the test power for the symmetric stable distribution is always very high, a practitioner should be very careful with modeling daily financial data by using the stable law. Final remarks and conclusions are presented in Section 11.5. All proofs are given in the Appendix.
216
MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS
11.2 SMALL SAMPLE TESTS Before presenting the main results of the study, we introduce the family of the matrix elliptical contoured distributions and briefly discuss their main properties. Following Gupta and Varga (1993) a random matrix X of dimensions k × n is said to have a matrix variate elliptically contoured distribution if its characteristic function has the form
(T) = exp(tr(iT M)) (tr(T T)) where T and M are k × n matrices, is a k × k positive semidefinite matrix, and : [0, ∞) → R. The symbol tr stands for the trace of a matrix. This family of distributions we denote by Ek,n (M, , ). If X = (X1 , … ,Xn ) ∼ Ek,n (M, , ) and if its second moments exist then it holds with M = (µ1 , … ,µn ) that the random vectors X1 , … , Xn are uncorrelated and Xi ∼ Ek,1 (µi , , ) (see Gupta and Varga, 1993, theorem 2.4.1, corollaries 2.4.1.1 and 2.4.1.2, theorem 2.3.2). Thus the columns of X follow a vector elliptically contoured distribution. It holds that E(X) = M and = Cov(Xi ) = −2 (0) (see Fang and Zhang, 1990, theorem 2.6.5). Note that the columns are independent if X follows a matrix variate normal distribution, for example, if is taken as being equal to exp(−x/2) (Gupta and Varga, 1993, theorem 2.1.5). Assuming X to be absolutely continuous, it follows that X ∼ Ek,n (M, , ) if and only if the density of X has the form f (X) = det()−n/2 h(tr((X − M) −1 (X − M)), where h and determine each other for specified k and n (see Gupta and Varga, 1993, theorem 2.2.1). The matrix elliptically contoured distributions possess several desirable properties that have been observed for financial assets. It presents an extension of the assumption of an independent normal sample. First, the returns must not be independent, and, second, they may have heavy tails. The aim of our study is to compare the ability of the matrix elliptical and symmetric stable distributions to explain the stochastic behavior of daily asset returns. For these purposes, we derive statistical procedures for testing the matrix elliptical symmetry of a multivariate sample. The cases of known and unknown nature of elliptical symmetry, scale parameters and location vector are considered. Let us denote the set of distribution functions with the known location vector, the known scale matrix, and the known characteristic function by µ,, = {F : F ∈ Ek,n (µ × 1, , )}
(11.1)
If some parameters are not precisely known we put points on the corresponding places in Equation (11.1). For example, when the location vector µ is unknown, we put .,, = {F : F ∈ Ek,n (µ × 1, , ), µ ∈ Rk }
(11.2)
TARAS BODNAR AND WOLFGANG SCHMID
217
In a similar way, the testing hypotheses are denoted. When the location vector is unknown we obtain H0,.,, : X ∼ F ∈ .,,
against H1,.,, : X ∼ F ∈ .,,
(11.3)
11.2.1 Known type of elliptical symmetry (known characteristic function) In this section we derive several procedures to test the ellipticity of the sample with the precise definition of the type of elliptical symmetry, for example, to test if X ∼ Ek,n (µ × 1 , , ) with the known characteristic function . First, we consider the case of the known scale matrix and unknown location vector µ. The test hypothesis is H0,.,, : X ∼ F ∈ .,,
against H1,.,, : X ∼ F ∈ .,, .
(11.4)
Let us denote the following random variable ˆ τ τ Q1 = τ τ
(11.5)
where τ is a nonzero vector of constants and is estimated by ˆ =
n 1 1 1 (Xt − X)(Xt − X) = X(I − 11 )X n−1 n−1 n
(11.6)
t=1
The stochastic representation of the random matrix X is essential for deriving the distribution of Q1 . Let be positive definite. It holds that X ∼ Ek,n (M, , ) if and only if X has the same distribution as M + R 1/2 U, where U is a k × n random matrix and vec (U ) is uniformly distributed on the unit sphere in Rkn , R is a non-negative random variable, and R and U are independent (see Gupta and Varga, 1993, theorem 2.5.2). The expression M + R 1/2 U is a stochastic representation of X, that is, it holds that X ≈ M + R 1/2 U. The symbol A ≈ B says that the two random variables A and B have the same distribution. The variable R is called the generating variable of X. The distribution of R2 is equal to the distribution of ni=1 (Xi − µi ) −1 (Xi − µi ). If X is absolutely continuous, then R is also absolutely continuous and its density is fR (r) =
2πnk/2 nk−1 r h(r2 ) nk 2
(11.7)
for r ≥ 0 (see Gupta and Varga, 1993, theorem 2.5.5). Note that for the matrix 2 ∼ χ2 . The index N refers to the normal variate normal distribution RN nk distribution.
218
MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS
Lemma 1. Let X = (X1 . . . Xn ) ∼ Ek,n (M,,) and n > k. Let be positive definite and suppose that X is absolutely continuous. Then it holds that (n − 1) Q1 has a stochastic representation R2 b, for example, (n − 1)Q1 ≈ R2 b with R being the generating variable of X, nk−n+1 b ∼ B( n−1 ) (Beta distribution), and the random variables R and b are independent. 2 2 ,
The proof of the lemma is given in the Appendix. From the result of Lemma 1, the moment sequence of random variable Q is calculated mi = E(Qi ) =
k E(R∗2i ) (i + n−1 2 ) ( 2 ) k (n − 1)i ( n−1 2 ) (i + 2 )
(11.8)
where R∗ is the generating variable of X1 . Finally, from Lemma 1 we obtain Theorem 1 Let X = (X1 , . . . , Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that under the null hypothesis H0,.,, the test statistic T1 = (n−1)Q1 has the same distribution as R˜ 1 , where R˜ 1 is the generating variable of En−1,1 (.,., ).
In case of the known scale matrix and the known location vector µ we consider the following random variable Q2 =
ˆ ˜ τ τ τ τ
with
n ˆ˜ = 1 (X − µ)(X − µ) t t n
(11.9)
t=1
The test hypothesis is given by H0,µ,, : X ∼ F ∈ µ,,
against
H1,µ,, : X ∼ F ∈ µ,,
ˆ˜ has a Wishart distriFrom the result of Lemma 1 and the fact that n bution with n degrees of freedom and the parameter matrix , namely, ˆ˜ ∼ W (n, ), it follows n k Theorem 2 Let X = (X, . . . Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that under the null hypothesis H0,µ,, the test statistic T2 = nQ2 has the same distribution as R˜ 2 , where R˜ 2 is the generating variable of En,1 (.,.,).
11.2.2 Unknown type of elliptical symmetry (unknown characteristic function) In contrast to the distributional properties of the proposed statistics in section 11.2.1, the test statistics presented in this section are distributional free within the class of matrix elliptical contoured distributions. Their distributions, in general, are presented by central F-distributions with some degrees of freedom. These results make them available to be applied without specifying the concrete type of elliptical symmetry. Again, the cases with known
TARAS BODNAR AND WOLFGANG SCHMID
219
and unknown location vectors are treated separately. We omit the moments requirements imposed on the asset returns. The proposed approaches can be applied to rather heavy-tailed distributions, even which do not possess the first and higher moments. Let X = (X(1) , X(2) ), where X(1) = (X1 , . . . , Xn1 ) and X(2) = (X1+n1 , . . . , Xn ) with n2 = n − n1 . First, we treat the case of unknown location vector µ. When the scale matrix is unknown one has to estimate it by previous observations. Then based on the first n1 observations we estimate ˆ 1) = (n
n1 1 1 1 (Xt − X¯ (1) )(Xt − X¯ (1) ) = X(1) (I − 11 )X(1) n1 − 1 n1 − 1 n1 t=1
Using the rest of the n2 observations we obtain ˆ 2) = (n
1 n2 − 1
n
(Xt − X¯ (2) )(Xt − X¯ (2) ) =
t=n1 +1
1 1 X(2) (I − 11 )X(2) n2 − 1 n2
Finally, to test the null hypothesis H0,.,.,. : X ∼ F ∈ .,.,.
against
H1,.,.,. : X ∼ F ∈ .,.,.
(11.10)
we use the result of the following theorem Theorem 3 Let X = (X1 . . . Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that ˆ 2 )τ/τ (n ˆ 1 )τ has a central under the null hypothesis H0 ,.,.,. the test statistic T3 = τ (n F-distribution with n2 − 1 and n1 − 1 degrees of freedom.
The proof of the theorem is given in the Appendix. If the location vector µ is known, a similar statistic as above is considered. However, the estimators of the scale parameters are given by ˆ˜ (n 1) =
n1 1 (Xt − µ)(Xt − µ) n1 − 1
ˆ˜ and (n 2)
t=1
n 1 = (Xt − µ)(Xt − µ) n2 − 1 t=n1 +1
correspondingly. From Muirhead (1982, theorem 3.2.8) and Fang and Zhang (1990, theorem 5.1.1) we obtain Let X = (X1 , . . . , Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that ˆ ˆ˜ ˜ 2 )τ/τ (n under the null hypothesis H0,µ,.,. the test statistic T4 = τ (n 1 )τ has a central F-distribution with n2 and n1 degrees of freedom. Theorem 4
Note that the distributions of T1 , T2 , T3 and T4 statistics do not depend on τ.
220
MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS
11.2.3 Further statistics If the type of ellipticity is unknown, additionally, the following four statistics are considered: −1 ˆ τ −1 τ n − k ˆ −1 ˆ )−1 − −1 T5 = (LR()L (τ τ) −1 p τ τ ˆ τ τ −1 ˆ τ −1 τ × − −1 τ τ ˆ −1 τ τ ˆ˜ −1 τ −1 τ n − k ˆ −1 ˆ˜ )−1 ˜ τ) − −1 (τ (LR()L T6 = ˆ˜ −1 τ p τ τ τ ˆ˜ −1 τ −1 τ × − −1 ˆ −1 τ τ ˜ τ τ −1 ˆ (n2 )τ n − k ˆ −1 T7 = (τ (n2 )τ) (n2 ) − p ˆ −1 τ τ −1 ˆ (n2 )τ −1 ˆ 2 ))L ) × (LR((n − ˆ −1 (n2 )τ τ
ˆ −1 (n1 )τ
ˆ −1 (n1 )τ τ
ˆ −1 (n1 )τ
ˆ −1 (n1 )τ τ
ˆ˜ −1 (n )τ ˆ˜ −1 (n )τ n − k ˆ −1 2 1 ˜ (n2 )τ) T8 = − (τ ˆ −1 ˆ −1 p ˜ ˜ τ (n2 )τ τ (n1 )τ ˆ˜ −1 × (LR((n 2 ))L )
ˆ˜ −1 (n )τ ˆ˜ −1 (n )τ 2 1 − ˆ −1 ˆ −1 ˜ (n2 )τ ˜ (n1 )τ τ τ
where R(A) = A−1 − A−1 ττ A−1 /τ A−1 τ. The statistic T5 is used to test the null hypothesis H0,.,,. : X ∼ .,,. , while the statistic T6 corresponds to the hypothesis H0,µ,,. : X ∼ µ,,. , the statistic T7 to H0,.,.,. : X ∼ .,.,. , and the statistic T8 to H0,µ,.,. : X ∼ µ,.,. . L is a p × k matrix of constants, p ≤ k − 1, such that (L,τ) is of full rank p + 1. The distributions of these statistics have already been derived in Bodnar (2004) and Bodnar and Schmid (2004). They do not depend on the type of elliptical symmetry within the class of matrix elliptical contoured. The critical values of the T7 - and T8 statistics can be obtained by numerical integration from the mathematical software package Mathematica. They are twice as large as the corresponding ones of central F-distributions (T5 - and T6 statistics).
Power
TARAS BODNAR AND WOLFGANG SCHMID
0.40 0.39 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13
221
T4 T8
0
1
2
3
4
5
6
7
8
9
10
Degrees of freedom
Figure 11.1 Power functions of the T4 and T8 tests for testing sample elliptical symmetry against independent multivariate t-distribution with different degrees of freedom
11.3 ANALYSIS OF THE POWER FUNCTIONS In this section we deal with the power functions of the proposed tests. Note that the rejection of the null hypothesis may be caused by a change in the covariance matrix or misspecification of the underlying distribution. As the first case is not of interest to us, we fix the covariance matrix of the process. Then the rejection of the null hypothesis is the result of an incorrect specification. Since a huge number of alternative hypotheses can be modeled, we do not consider all of them. Furthermore, because of the analytical difficulties of deriving the distributional function under the alternative hypothesis H1 to calculate the power of the test, we apply a Monte Carlo study. Several situations are modeled by drawing a sample of an independent multivariate t-distribution in the first case and an independent symmetric multivariate stable distribution in the second one. The location vector is chosen to be 0, and the scale matrix is identical. This choice is not restrictive, as neither of the T4 and T8 statistics depend on τ, and T8 is independent of L. In all cases, 104 seven-dimensional vectors of the corresponding distributions are drawn. The procedures for generating a multivariate symmetric stable distribution and a multivariate t-distribution are discussed in the Appendix. The powers of the T4 and T8 tests are shown in Figure 11.1 for multivariate t-distributions with different degrees of freedom. The powers of both tests decrease as the degrees of freedom increase. It is not surprising that the t-distribution converges to the normal when the degree of freedom tends
Power
222
MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS
0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30 0.28
T8
T4
0.6
0.8
1.0
1.2 1.4 Stability index
1.6
1.8
2.0
Figure 11.2 Power function of the T4 and T8 tests for testing sample elliptical symmetry against independent symmetrical multivariate stable distribution with different stability indices to infinity. Finally, the power functions of the T4 and T8 statistics are almost the same for the different degrees of freedom. Figure 11.2 contains the power functions of the T4 and T8 tests for independent symmetric stable distributions with different stability indices. The power of a T8 test is always higher than the power of T4 . When the stability index is around one, the power is over 0.6. For larger values of stability indices this probability decreases. It is around 35 percent when the stability index values are around 1.65, the recommended value for describing daily data (Blattberg and Gonedes, 1974). We make use of these results in the next section, when an empirical example of the daily returns of seven developed stock markets is discussed.
11.4 EMPIRICAL STUDY In this section, the results of the empirical study are presented. Because the location parameter, the scale matrix and the type of elliptical symmetry are usually unknown in a practical situation, we make use of the T8 statistic for testing the null hypothesis that daily returns follow a matrix ellipticalcontoured distribution. It is seen how the finite sample properties of these statistics can be used. We consider the daily price data from Morgan Stanley Capital International for the equity markets of seven developed countries (France, Germany, Italy, Japan, Spain, the UK and the USA) for the period January 1, 1994 to December 31, 2000. We group our data set by half-year
TARAS BODNAR AND WOLFGANG SCHMID
223
Table 11.1 The 5% and 10% critical values for the T8 statistic depending on the sample sizes n1 , n2 (k = 1) 0.05
α n1 \n2
63
0.1
0.05
63
64
0.1 64
0.05
0.1
65
65
63
8.195
5.695
7.695
5.495
7.265
5.287
64
8.815
5.927
8.190
5.691
7.689
5.498
65
9.581
6.192
8.792
5.927
8.176
5.684
Table 11.2 Value of the T8 statistic for different linear restrictions Year\Test
T 8, l 1
T 8, l 2
T 8, l 3
T 8, l 5
T 8, l 6
1994, I
2.608
1.283
5.175
1994, II
0.106
3.781
0.271
0.840
6.629
0.418
5.597
0.008
1995, I
2.616
1.702
0.168
1995, II
3.869
0.128
2.482
4.285
0.123
0.650
0.014
2.981
5.399
0.0
5.077
1996, I
3.552
3.827
1996, II
2.883
5.263
4.098
2.190
0.137
0.667
0.305
8.092
1997, I
0.001
3.569
4.247
0.025
0.170
0.761
1.146
1997, II
0.084
1.034
1998, I
0.270
1.808
0.255
1998, II
0.075
0.899
1999, I
0.108
1999, II
2.756
2000, I 2000, II
0.599
T 8, l 4 11.50
13.79 3.209
0.809 0.005
0.017
3.436
2.668
0.159
1.726
4.466
2.224
5.387
0.786
3.592
0.152
0.021
3.736
0.027
0.206
2.605
1.968
1.149
1.629
0.136
1.743
0.734
0.518
0.019
0.637
0.142
2.176
0.675
2.753
8.155
2.568
3.298
0.314
25.63
20.73
4.815 11.93
T 8, l 7
11.57
0.673 4.682
11.18
data. Furthermore, each group is additionally partitioned into two subsamples of three-month data sets. The first sub-sample is used to calculate ˆ 1 ) and the second for (n ˆ 2 ). (n In Table 11.1, the 5 percent and 10 percent critical values of the T8 statistics for different sample sizes, n1 , n2 ∈ 63,64,65 are presented. These change significantly with a change of sample size. However, the critical values are almost the same on the diagonals that are parallel to the main diagonal of the table. The values of the T8 statistic are presented in Table 11.2. They are calculated for different linear restrictions, for example, l1 = (1, 0, 0, 0, 0, 0, 0), . . . , l7 = (0, 0, 0, 0, 0, 0, 1). The values of the statistics that are greater than the 5 percent critical value are indicated in bold type. The null hypothesis of matrix elliptical symmetry is rejected in null cases out of fourteen for the T8
224
MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS
statistic with the l1 and l2 linear restrictions. In other cases there are only two rejections per a column. In general we observe seven rejections out of ninety-eight for the 5 percent level of significance, and ten rejections out of ninety-eight for the 10 percent level. Keeping in mind that the rejection of the null hypothesis can also be effected by changes in the covariance structure of the stock returns process, we are unable to reject the matrix ellipticity of the considered returns. From the other side, using the results of section 11.3 (the test power is very high for the multivariate symmetric stable distribution), one should be very careful with the assumption that daily stock returns follow a multivariate stable law.
11.5 CONCLUSION In this chapter, several statistics for testing the null hypothesis of a matrix elliptical-contoured distribution are proposed. The finite sample properties are derived in all cases of known and unknown types of elliptical symmetry, scale matrix and location vector. The T1 - and T2 statistics do not possess the invariance property with respect to matrix elliptical-contoured distributions and their null distributions are specified by the corresponding generating variables. From the other side, the statistics from T3 to T8 are distributionally free. Their stochastic properties, apart from T7 and T8 , are based on the central F-distribution with some degrees of freedom. The control limits of the T7 - and T8 statistics can be obtained by numerical calculations in the software package Mathematica. We applied the T8 statistic in a situation of practical interest by considering the daily stock returns of seven developed stock markets. The null hypothesis of the matrix elliptical symmetry is rejected ten times out of ninety-eight for the 10 percent level of significance. In addition, the rejection of the null hypothesis may be caused by changes in the covariance structure of the underlying distribution. Keeping everything together, we conclude that the results of the empirical study provide support to model the daily data by matrix elliptical-countered distributions. They are in line with the suggestions of Andersen et al. (2001) and Andersen et al. (2004), who argued that daily returns normalized by the realized volatility can be well approximated by normal distribution. Furthermore, researchers should be very careful with the application of the multivariate symmetric stable law in modeling daily data. Instead, the assumption of a matrix elliptical-contoured distribution should be maintained.
APPENDIX In this section, the proofs of Lemma 1 and Theorem 3 are given. Furthermore, we deal with the problem of generating independent multivariate t- and symmetrical stable distributed random vectors.
TARAS BODNAR AND WOLFGANG SCHMID
225
We say that a characteristic function belongs to the class (U) if it can be equal to another characteristic function in the neighborhood of zero without being identical to it. Correspondingly, a characteristic function belongs to the class (U) if it does not belong to (U). We denote a set of orthogonal matrices of the order k by O(k).
Proof of Lemma 1 We have it that Q ≈ R2 τ
(1/2 U(I − 11 /n)U 1/2 )τ τ τ
= R2 Q∗ . A similar presentation is obtained
2 Q . The index N is used to when X is matrix normally distributed; for example, QN ≈ RN ∗ 2 ∼ χ2 , R2 Q ∼ χ2 , it follows from Fang and Zhang indicate the normal case. Because RN N ∗ n−1 nk 1 nk − n + 1 2 Q ≈ R2 b . Furthermore, we ) exists such that RN (1990, p. 59) that b∗ ∼ B( n − ∗ N ∗ 2 , 2 2 > 0) = 1 and P(b > 0) = 1. have it that P(RN ∗ ˆ is positive definite with From the assumption of the lemma, it follows that probability 1 (see Muirhead, 1982, theorem 3.1.4). Hence, Q is greater than 0 with probability 1, and therefore, P(Q∗ > 0) = 1. From the above consideration, it follows that the density of ln b∗ is
fln b∗ (t) =
n−1
et 2 (1 − et ) 0,
nk−n+1 −1 2
,
if if
t≤0 t>0
∞ Hence, for any positive r it holds 0 ert fln b∗ (t)dt = 0 < ∞. Using the results of Fang and Zhang (1990), it follows that characteristic function φln b∗ ∈ (U). Thus, from the property of the operation ≈ (see Fang and Zhang, 1990, p. 38) it follows that Q∗ ≈ b∗ . As a 1 nk − n + 1 result we obtain Q∗ (R2 ) ≈ R2 Q∗ ≈ R2 b∗ , where b∗ ∼ B( n − ) and R2 and b∗ are 2 , 2 independently distributed.
Proof of Theorem 3 Let us consider T3 =
ˆ 2 )τ ˆ 2 )τ τ τ τ (n τ (n = ˆ 1 )τ ˆ 1 )τ τ τ τ (n τ (n
The rest of the proof follows from Muirhead (1982, theorem 3.2.8) and Fang and Zhang (1990, theorem 5.1.1).
Drawing samples from multivariate t- and symmetric stable distributions The way of generating samples of independent multivariate t-distributions and symmetric stable distributions follow immediately from their stochastic representations. From Fang et al. (1990, p. 85) we obtain that the k-dimensional t-distributed random vector Y is equal to Y = Z/
χ n
(11.11)
226
MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS
where Z has a k variate normal distribution, χ has a chi-squared distribution with n degrees of freedom, and Z and χ are independently distributed. The stochastic representation of the k-dimensional stable random vector with the index of stability equal to α, 0 < α < 2 is given by S=
√ AZ
(11.12)
where Z has a k variate normal distribution, A has an univariate α/2-stable distribution with skewness parameter equal to 1, the location parameter 0, and the scale parameter (cos(πα/4))2/α , and Z and A are independently distributed (see Samorodnitsky and Taqqu, 1994, p. 77). Following Kantner (1975), the stochastic representation of A is
A = cos (πα/4)
1/α
sin ((1 − α/2)θ) sin (αθ/2)α/(2−α) sin (θ)2/(2−α) W
(2−α)/α (11.13)
where θ is uniform on (0,π),W has a standard exponential distribution, and θ and W are independently distributed.
REFERENCES Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2004) “Parametric and Nonparametric Measurements of Volatility”, in Y. Aït-Sahalia and L. P. Hansen (eds), Handbook of Financial Econometrics, (Amsterdam: North-Holland). Andersen, T. G., Bollerslev, T., Diebold, F. X. and Ebens, H. (2001) “The Distribution of Realized Exchange Rate Volatility”, Journal of the American Statistical Association, 96(453): 42–55. Baringhaus, L. (1991) “Testing for Spherical Symmetry of a Multivariate Distribution”, The Annals of Statistics, 19(2): 899–917. Beran, R. (1979) “Testing Ellipsoidal Symmetry of a Multivariate Density”, The Annals of Statistics, 7(1): 150–62. Berk, J. B. (1997) “Necessary Conditions for the CAPM”, Journal of Economic Theory, 73(1): 245–57. Blattberg, R. C. and Gonedes, N. J. (1974) “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices”, The Journal of Business, 47(2): 244–80. Bodnar, T. (2004) “Optimal Portfolios in an Elliptical Model – Statistical Analysis and Tests for Efficiency”, Ph.D. thesis, Europa University Viadrina, Frankfurt (Oder), Germany. Bodnar, T. and Schmid, W. (2003) “The Distribution of the Global Minimum Variance Estimator in Elliptical Models”, EUV Working paper 22. Bodnar, T. and Schmid, W. (2004) “A Test for the Weights of the Global Minimum Variance Portfolio in an Elliptical Model”, EUV Working paper 2. Bollerslev, T. (1986) “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, 31(3): 307–27. Chamberlain, G. A. (1983) “A Characterization of the Distributions that Imply MeanVariance Utility Functions”, Journal of Economic Theory, 29(1): 185–201. Engle, R. F. (1982) “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation”, Econometrica, 50(4): 987–1008. Fang, K. T. and Zhang, Y. T. (1990) Generalized Multivariate Analysis (Berlin: Springer-Verlag Beijing: Science Press). Fang, K. T., Kotz, S. and Ng, K. W. (1990) Symmetric Multivariate and Related Distributions (London: Chapman & Hall).
TARAS BODNAR AND WOLFGANG SCHMID
227
Fang, K. T., Zhu, L. X. and Bentler, P. M. (1993) “A Necessary Test of Goodness of Fit for Sphericity”, Journal of Multivariate Analysis, 45(1): 34–55. Fama, E. F. (1965) “The Behavior of Stock Market Prices”, Journal of Business, 38(1): 34–105. Fama, E. F. (1976) Foundations of Finance (New York: Basic Books). Gibbons, M. R., Ross, S. A. and Shanken, J. (1989) “A Test of the Efficiency of a Given Portfolio”, Econometrica, 57(5): 1121–52. Gupta, A. K. and Varga, T. (1993) Elliptically Contoured Models in Statistics (Dondrecht: Kluwer Academic). de Haan, L. and Rachev, S. T. (1989) “Estimates of the Rate of Convergence for Max-Stable Processes”, The Annals of Probability, 17(2): 651–77. Heathcote, C. R., Cheng, B. and Rachev, S. T. (1995) “Testing Multivariate Symmetry”, Journal of Multivariate Analysis, 54(2): 91–112. Kantner, M. (1975) “Stable Densities under Change of Scale and Total Variation Inequalities”, The Annals of Probability, 3(4): 697–707. Koltchinskii, V. I. and Li, L. (1998) “Testing for Spherical Symmetry of a Multivariate Distribution”, Journal of Multivariate Analysis, 65(2): 228–44. Manzotti, A., Perez, F. J. and Quiroz, A. J. (2002) “A Statistic for Testing the Null Hypothesis of Elliptical Symmetry”, Journal of Multivariate Analysis, 81(2): 274–85. Muirhead, R. J. (1982) Aspects of Multivariate Statistical Theory (New York: John Wiley). Nelson, D. (1991) “Conditional Heteroscedasticity in Stock Returns: A New Approach”, Econometrica, 59(2): 347–70. Owen, J. and Rabinovitch, R. (1983) “On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice”, The Journal of Finance, 38(3): 745–52. Rachev, S. T. and Mittnik, S. (2000) Stable Paretian Models in Finance. (New York: John Wiley). Samorodnitsky, G. and Taqqu, M. S. (1994) Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance (New York/London: Chapman & Hall). Tu, J. and Zhou, G. (2004) “Data-generating Process Uncertainty: What Difference Does It Make in Portfolio Decisions?”, Journal of Financial Economics, 72(2): 385–421. Zhou, G. (1993) “Asset-pricing Tests under Alternative Distributions”, The Journal of Finance, 48(5): 1927–42. Zhu, L. X. and Neuhaus, G. (2003) “Conditional Tests for Elliptical Symmetry”, Journal of Multivariate Analysis, 84(2): 284–98.
C H A P T E R 12
The Modified Sharpe Ratio Applied to Canadian Hedge Funds Greg N. Gregoriou
12.1 INTRODUCTION The assessment of portfolio performance is fundamental for both investors and fund managers, and this applies also to Canadian hedge funds. Traditional portfolio measures present some limitations when applied to hedge funds. For example, the Sharpe ratio uses the excess reward per unit of risk as a measure of performance, with risk represented by the standard deviation. The mean-variance approach to the portfolio selection problem developed by Markowitz (1952) has frequently been the subject of undue criticism because of its utilization of variance as a measure of risk exposure when examining the non-normal returns of hedge funds. The value-at-risk (VaR) measure for financial risk has recently grown to be accepted as a traditional measure in investment firms, large banks and pension funds. As a result of the recurring frequency of down-markets since the collapse of Long-Term Capital Management (LTCM) in August 1998, VaR has played a paramount role as a risk management tool, and is considered to be a mainstream technique for estimating and conveying the exposure a hedge fund has to market risk. With the wide acceptance of VaR, and specifically, of modified VaR as a relevant risk-management tool, a more suitable portfolio performance measure for hedge funds can be formulated in terms of a modified Sharpe ratio.1 228
GREG N. GREGORIOU
229
Using the traditional Sharpe ratio to rank hedge funds will underestimate the tail risk, and then overestimate performance. Therefore, the further the distribution is from the normal, the greater the risk of underestimation. In this chapter, we rank nine funds according to the Sharpe and modified Sharpe ratios. Our results indicate that the modified Sharpe is lower and more accurate when examining non-normal returns.
12.2 LITERATURE REVIEW Many hedge funds produce statistical reports for clients using the traditional Sharpe ratio, which can be misleading because funds will have a tendency to look better in terms of risk-adjusted returns. The drawback of using a traditional Sharpe ratio is that no distinction is made between upside and downside risk, but rather a fund is penalized for upside risk as much as downside risk and does not differentiate irregular losses compared to repeated losses. VaR has emerged progressively in the finance literature as a prevailing measure of risk. However, its simple version also presents some limitations because of the skewed returns hedge funds possess. Methods of measuring VaR, such as the delta-normal method developed by Jorion (2000), are straightforward and simple to apply. However, the formula has its drawbacks, since the assumptions of normality of the distributions are violated largely because of the use of short-selling and derivatives strategies, such as futures and options, frequently used by hedge funds. Current methods have been proposed to properly assess the VaR for nonnormal returns as developed by Rockafellar and Uryasev (2001) using a conditional VaR for general loss distributions, while Agarwal and Naik (2004) construct a mean conditional VaR demonstrating that mean-variance analysis also underestimates tail risk. Furthermore, Favre and Galeano (2002) also developed a technique to properly assess funds with non-normal distributions. The authors demonstrate that the modified VaR (MvaR) can significantly improve the accuracy of the traditional VaR. The difference between the modified VaR and the traditional VaR is that the latter considers only mean and standard deviation, while the former takes into account higher moments such as skewness and kurtosis. In addition, it is possible to reduce the probability of large negative returns by at least 15 percent (Favre and Singer, 2002). The modified VaR allows us to define a modified Sharpe ratio, which is more suitable for hedge funds. For example, when two portfolios have the same mean and standard deviation, in essence they may be different because of extreme losses. Therefore an advantage exists when using the modified VaR measure and modified Sharpe ratio.
230
THE MODIFIED SHARPE RATIO AND CANADIAN HEDGE FUNDS
12.3 DATA AND METHODOLOGY The dataset we use contains hedge fund returns for fifty funds in Canada. However, the majority of the funds commenced operations in 2001 and have been discarded because of the small number of data points available at the time of writing. Only nine live Canadian hedge funds reporting monthly performance figures spanning the period January 1998–December 2002 have been investigated. We obtain data from Beck and Nagy (2003). This period contains the extreme market event of August 1998 as well as the September 11, 2001 terrorist attacks. We use the Extreme Metrics software and assume a risk-free rate of 0 percent to compute the results, using a 95 percent VaR probability. This means that the investor is able to borrow and reinvest in the market portfolio at zero cost in order to move along the capital market line. This assumption simplifies the ranking of assets, especially when some of them have an average return below the risk-free rate, which yields a negative Sharpe or modified Sharpe ratio. The difference between the traditional and modified Sharpe ratios is that, in the latter, the standard deviation is replaced by the modified VaR (at 95 percent) in the denominator. The traditional Sharpe ratio is generally defined as the excess return per unit of standard deviation, as represented by the following equation: Sharpe ratio =
R p − RF σ
(12.1)
where RP = return of the portfolio; RF = risk-free rate; and σ = standard deviation. Since Equation (12.1) presents several limitations for non-normal distribution, a modified Sharpe ratio can be defined in term of modified VaR, as follows: Modified Sharpe ratio =
Rp − RF , MVaR
(12.2)
with 1 1 MVaR = W[µ − {zc + (zc2 − 1)S + (zc3 − 3zc )K 6 24 1 − (2zc3 − 5zc )S2 }σ] 36
(12.3)
where RP = return of the portfolio; RF = risk-free rate; σ = standard deviation; Zc = is the critical value for probability (1 − α) − 1.96 for a 95 percent probability; S = skewness; and K = excess kurtosis. The detailed derivation of the formula for modified VaR is beyond the scope of this chapter. Readers are guided to Favre and Galeano (2002) for a more detailed explanation.
GREG N. GREGORIOU
231
12.4 EMPIRICAL RESULTS 12.4.1 Descriptive statistics Table 12.1 displays monthly statistics on mean return, standard deviation, skewness, excess kurtosis, normal and modified VaR, Jarque–Bera statistic and compounded returns of the hedge funds during the examination period. The average of the compounded returns and mean monthly returns are greatest in the highest group and least in the lowest group – an expected finding. In addition, we find that positive skewness is more pronounced in the lowest group, yielding more positive monthly returns, whereas the top group has the least average positive skewness. A likely explanation is that smaller hedge funds can better control skewness in negative extreme market events, and on average will have less negative monthly returns. The lowest group (see Table 12.1, Panel C) has the highest volatility and lowest returns, which could be attributed to hedge funds taking on more risk to achieve greater returns while increasing assets under management.
12.4.2 Performance discussion Market risk and performance results are also presented in Table 12.1. First we observe that the middle group has, in absolute value, the lowest normal and modified VaR, so is less exposed to extreme market losses. Furthermore, we find that the non-normality when skewness and kurtosis are considered simultaneously using the Jarque–Bera tend to be the largest for small hedge funds. With regard to performance, we notice that the lowest group has the lowest traditional Sharpe and modified Sharpe ratios. It appears that medium-sized hedge funds can do a better job in controlling riskadjusted performance than either small or large funds. Since medium-sized hedge funds receive a greater inflow of money than small funds, they can alter their allocation more frequently. However, there exists a huge difference of assets under management between large and medium funds. When receiving a vast inflow of capital, large hedge funds could be overwhelmed and might experience trouble in producing superior risk-adjusted returns than medium-sized hedge funds. Capacity constraints may exist, since the Toronto Stock Exchange is relatively small compared to the US markets, and trading securities may further restrict large Canadian hedge funds, thus making trading sporadic, especially when leverage and short-selling is involved. Smaller hedge funds with fewer assets might have no choice but to hold their portfolio for a long period of time, irrespective of changing economic conditions.
Fund name
232
Table 12.1 Descriptive statistics of Canadian hedge funds, 1998–2002 Assets Mean Std. dev. Skewness Excess Modified Normal Traditional Modified Jarque–Bera Compound (millions (%) (%) kurtosis VaR VaR Sharpe sharpe statistic return $) 99% 99% ratio ratio (%)
Panel A: Sub-sample 1: Top 3 funds Arrow Clocktower
325
1.4
3.5
0.2
0.00
−6.3
−6.8
0.18
0.16
Goodwood Fund
200
1.6
4.6
0.4
0.5
BPI Global Opportunities
195
1.5
5.9
0.9
1.1
−8.2
−9.1
0.15
−8.3
−12.3
0.14
Average
240
1.5
4.67
0.5
0.53
−7.60
−9.4
−3.9
0.26
127.73
0.13
1.92
137.07
0.09
10.88
119.97
0.16
0.13
4.35
128.26
−4.6
0.14
0.12
0.97
65.10
Panel B: Sub-sample 2: Middle 3 funds −0.4
Horizons Mondiale
125
0.9
2.3
0.2
Horizons Univest 2
107
0.9
0.7
0.2
1.2
−0.8
−0.9
0.75
0.66
3.74
74.44
82
1.8
5.7
0.00
3.4
−11.5
−16.2
0.13
0.09
28.57
162.52
104.67
1.2
2.9
0.13
1.4
5.4
7.2
0.34
0.29
11.09
100.69
0.10
0.07
24.79
112.03
Vertex Average
Panel C: Sub-sample 3: Bottom 3 funds Friedberg TT Equity Hedge
6
1.6
8.0
1.1
2.3
−11.9
−17.1
Horizons Strategic
3
1.8
7.0
3.2
18.4
−0.1
−14.5
0.10
0.09
948.02
77.22
Hillsdale Market Neutral ($US)
2
0.2
4.2
0.4
1.8
−9.6
−9.7
−0.02
−0.02
10.18
6.18
Average
3.67
1.2
6.4
1.57
7.5
−7.2
−13.76
0.06
0.05
327.66
65.14
GREG N. GREGORIOU
233
When we compare the results between the traditional and the modified Sharpe ratios, we find that the traditional Sharpe ratio is higher, confirming that tail risk is underestimated.
12.5 CONCLUSION It is of critical importance to understand that complications will arise when a traditional measure of risk-adjusted performance, such as the traditional Sharpe ratio, is used to investigate fat tails and non-normal returns of hedge funds. Institutional investors must use the modified Sharpe ratio to measure the risk-adjusted returns correctly; and the modified VaR is recommended to measure extreme negative returns because the normal VaR only considers the first two moments of a distribution, namely mean and standard deviation. The modified VaR, however, takes into consideration the third and fourth moments of a distribution – skewness and kurtosis. Using both the modified Sharpe and modified VaR will enable investors to obtain a more accurate picture without any bias. Furthermore, the modified VaR is lower than the normal VaR because of negative skewness in hedge fund returns and the small excess positive kurtosis. The statistics we have presented can be applied to all hedge fund and commodity trading adviser (CTA) classifications to evaluate non-normal returns. We believe many institutional investors wanting to add hedge funds and funds of hedge funds to traditional stock and bond portfolios must request additional and more appropriate statistics such as the modified Sharpe ratio in analyzing the returns of hedge funds.
NOTE 1. The standard VaR, which assumes normality and uses the traditional standard deviation measure, looks only at the tail of the distribution of extreme events. This is common when examining mutual funds, but when applying this technique to funds of hedge funds, difficulties arise because of the non-normality of returns (Favre and Galeano, 2002). The modified VaR takes into consideration the mean, standard deviation, skewness and kurtosis to evaluate correctly the risk-adjusted returns of funds of hedge funds. Computing the risk of a traditional investment portfolio consisting of 50% stocks and 50% bonds with the traditional standard deviation measure could underestimate the risk by as much as 35% (Favre and Singer, 2002).
REFERENCES Agarwal, V. and Naik, N. (2004) “Risks and Portfolio Decisions Involving Hedge Funds”, Review of Financial Studies. 17(1): 63–98. Beck, P. and Nagy, M. (2003) Hedge Funds for Canadians (Toronto: John Wiley).
234
THE MODIFIED SHARPE RATIO AND CANADIAN HEDGE FUNDS
Favre, L. and Galeano, J. A. (2002) “Mean-Modified Value-at-Risk with Hedge Funds”, Journal of Alternative Investments, 5(2): 21–5. Favre, L. and Singer, A. (2002) “The Difficulties in Measuring the Benefits of Hedge Funds”, Journal of Alternative Investments, 5(1): 31–42. Jorion, P. (2000) Value at Risk (New York: McGraw-Hill). Markowitz, H. (1952) “Portfolio Selection”, Journal of Finance, 77(1): 77–91. Rockafellar, R. T. and Uryasev, S. (2001) “Conditional Value-at-Risk for General Loss Distributions”, Research Report, ISE Dept, University of Florida.
Index
ABN Amro 1 accounting standard index 59–60, 66, 67–74, 75–7 Ackert, L.F. 157, 164, 174, 175 African bloc 47, 54, 56 Agarwal, V. 229 Ahearne, A. 42, 43, 63, 64, 71–2, 78 Air Liquide 190–209, 210 alpha art market 9–10 Jensen’s alpha 197–8 American bloc 47, 54, 56 Amin, G. 140 analysis of variance (ANOVA) 161, 162, 167–73 Andersen, T.G. 215, 224 Ang, A. 17, 33, 37 Anthony, J.H. 155 anti-director rights 59–60, 66, 67–74, 75–7 arbitrage pricing theory (APT) 114–15, 116–17 Argentina 104 art 1–15 art market 3–4; defining a bubble in 5–6 data 6–8 empirical studies 4–5 methodology 9–10 results 10–11 Art Market Research (AMR) database 6 Artzner, P. 191 Asia/Pacific bloc 47, 54, 56 Asian currency crisis 96–102 at the money (ATM) option contracts 160, 167, 168–9, 172 Ates, A. 153
auction houses 3 Augmented Dickey–Fuller (ADF) test 163, 174 Australian Statex Actuaries Price Index 157 Australian Stock Exchange (ASX) 151–2 All Ordinaries Index 154, 157, 158, 159, 161, 165–7, 168, 169, 175, 176–8 lead–lag relationship with SFE 155, 158–9, 164–5, 176–8 multifactor model 114–36; cross-sectional regressions 124–7; data 119–27; data analysis and results 127–32; parsimonious model 119, 130–2; portfolio characteristics 124; returns to be explained 123–4 Austria 47 automation 151–82 analysis of results 165–78; ANOVA results 167–73; descriptive statistics 165–7; price discovery analysis 174–8 sample design 159–65; cointegration 163–5; data sources 159–60; methodology 160; model and statistical procedures 160–3 Baig, T. 97, 104 Barbone, L. 96, 104 Bauer, R. 5 Baumol, W. 4 bear state 23–6, 27 Beck, P. 230 behavioral anomalies
12–13 235
236
INDEX
behavioral theory 80–1 Bekaert, G. 17, 33, 37 Belgium 47, 63 benchmarks benchmark assets in efficiency gain/loss methodology 140–1; return distribution 144 market benchmarks for French stocks 201–9; forecasting performance 205–9 Benjamin, W. 12 bequests 13 Berk, J.B. 211, 215 Berkowitz, J. 22 beta 197–8, 203–5, 206 bilateral trades 65–6, 67–74, 75–7 Black, F. 17, 185, 186 Black and Scholes volatility bias 189 Blattberg, R.C. 222 blocs 47, 54, 56 Bodnar, T. 215, 220 Bohn, H. 67 bond market–stock market linkages 103–15 book-to-market value 118–19, 121–2, 123–4, 125, 126–32, 133 Bortoli, L. 153 bounded rationality 80 Brady bonds 103–13 stripped-yield spreads 104, 105, 107, 108, 109–11 Brailsford, T. 154, 197 Brandt, M.W. 21 Brazilian bond market–stock market linkages 103–13 Brooks, C. 158, 159 Brownian motion 185–6 bubbles 11–14 art market 1–2, 5, 8, 10–11; defining a bubble in the art market 5–6 Internet 32–3 real estate 6 “bubbliness”, degree of 13–14 CAC40 index 187–209, 210–11 call pricing formula 186–7 Campbell, J.Y. 184, 186, 197, 200, 210 Campbell, R.A. 4, 8 Canadian hedge funds 228–34 capital asset pricing model (CAPM) 114, 115–16, 183, 197, 215 international (ICAPM) 42 capital controls 43, 44, 57–8, 59–60, 63, 67–74, 75–7
capital flow restrictions 57–8, 63, 67–74, 75–7 Carmichael, B. 64 Case, K.E. 6 causality analysis impact of systematic risk on French stocks 191–5; Granger causality test 193–5; VAR specification 191–3 lead–lag relationship 164–5, 176–8 Central and Eastern European investment funds 143–50 certainty-equivalent compensation 28–30 Chamberlain, G.A. 215 Chan, K. 43, 44, 45, 46, 47, 55, 56, 63, 64, 65, 67, 69, 74, 78, 127 Chanel, O. 4 Chen, J. 17 Chen, N.F. 116, 117, 118, 119, 125, 126, 127, 130, 132 Cheng, C.S. 119 China 63 Chordia, T. 105 Clare, A.D. 127, 130 Clark, P.K. 154 cocoa prices 156–7 Coen, A. 64 cointegration 156–8, 163–5, 175–8 Johansen test 163–4, 175–6 lead–lag relationship 155, 158–9, 164–5, 176–8 unit root tests 163, 174 collateral, art as 8 compensation, certainty-equivalent 28–31 Connor, G. 117 contagion 104 currency crises and portfolio selection 96–102 Cooper, I. 44–5, 47 Coordinated Portfolio Investment Survey (CPIS) dataset 43 Copeland, L. 154, 156 corner portfolios 87, 91 correlation studies correlation coefficient and home bias 67–74, 75–7 impact of systematic risk on French stocks 190–1 peer group analysis 148–9 stock market returns 97, 99–100, 101 cost-of-carry 158, 180 creditworthiness 103–4
INDEX
critical line UPM/LPM portfolio optimization algorithm (CLA) 80–95 derivation 82–4 efficient segments on the efficient frontier 88; adjacent efficient segments 89–92 empirical example 92–4 Kuhn–Tucker conditions 84–7 cross-sectional regressions 124–32 CRR (Chen, Roll and Ross) macroeconomic variables 122–3, 126, 127–32, 133 cumulative wealth 32–4 currency crises 96–102, 104 portfolio performance 100–1 stock market average rates of return and average volatility 97–9, 101 stock market correlations 97, 99–100, 101 currency hedging 16–41 economic importance of regimes 28–31 estimation results for regime-switching models 21–6, 27 optimal foreign investment 38, 39 out-of-sample test for regime-switching strategies 31–7, 39 Cyert, R.M. 80 Dahlquist, M. 43, 72 daily financial data 214 matrix elliptical contoured distributions 222–4 Dales, A. 17 Danone 190–209, 210 DAX 157 DeBondt, W.F.M. 117 default probabilities 104 depreciation of the dollar 33, 36 descriptive statistics automation of SFE 161, 165–7 Canadian hedge funds 231, 232 developed countries 47, 54, 56 Diacogiannis, G.P. 116 Diamandis, P. 116 disposition effect 13 distributional price 139 dividend-paying framework 187 dollar, depreciation of the 33, 36 domestic bias 42–79 data sources 46
237
determinants of 56–67; capital control 57–8, 59–60, 63; economic development 56–63; familiarity 61–2, 65–6; information costs 61–2, 64; investor protection 59–60, 66; investors’ behavior 61–2, 65; other variables 67; stock market development 57–8, 63–4 results of empirical analysis 67–9 statistics on 47–56 statistics on investor holdings 46–53, 54 theoretical framework 44–6 world float portfolio 72–4, 75 double-lognormal (DLN) framework 138, 142–3, 144, 150 Dow Jones STOXX market indices 189 downside risk 229 time-varying 1–15 Durbin–Watson (DW) test 130, 131 Dybvig, P.H. 139 East Asian stock markets 96–102 economic development 43, 44, 56–63, 67–77 economic importance of regimes 28–31 Ederington, L.H. 143 Edison, H. 63 efficiency gain/loss 138, 140–3, 149–50 benchmark 141–2 definition 140–1 higher moment performance characteristics 145–7 underlying 142–3 efficient frontiers 87–94 efficient segments on 88; adjacent efficient segments 89–92 mean-variance and UPM/LPM models 92–4 efficient market hypothesis (EMH) 114 semi-strong form 156, 175–6, 180 Eichenberger, R. 12, 13 Eichengreen, B. 96 electronic trading see automation elliptical distributions see matrix elliptical contoured distributions emerging markets Brady bonds see Brady bonds domestic and foreign biases 47, 54, 56, 64, 67–77 performance evaluation 137–50
238
INDEX
endowment effect 12 Engle, R.F. 163 Epps, M.L. 154 Epps, T.W. 154 Erb, C.B. 96–7 European bloc 47, 54, 56 expected inflation, change in 122–3, 126, 127–32, 133 expropriation, risk of 59–60, 66, 67–77 extreme value theory (EVT) 9 Faff, R.W. 118, 119, 192, 197 Fama, E.F. 115, 117–19, 121, 123, 125, 126, 127, 130, 132, 156, 158, 180, 211 familiarity 43, 44, 61–2, 65–78 Fang, K.T. 219, 225 far in the money (FITM) option contracts 160, 168–9 Faruquee, H. 43 Favre, L. 229, 230 firm-attribute factors 211 multifactor models 117, 118–19; ASX 121–32, 133 Fishburn, P.C. 81 Fleming, J. 97, 105 foreign bias 42–79 data sources 46 determinants of 56–67; capital control 57–60, 63; economic development 56–63; familiarity 61–2, 65–6; information costs 61–2, 64; investor protection 59–60, 66; investors’ behavior 61–2, 65; other variables 67; stock market development 57–8, 63–4 results of empirical analysis 69–71 statistics on 47–56 statistics on investor holdings 46–53, 54 theoretical framework 44–6 world float portfolio 72–4, 76 foreign direct investment (FDI) 56–63, 67–77 Forni, L. 96, 104 France 47 impact of systematic risk on stocks in French financial market 183–213 Fraser, P. 116, 118, 119 French, K.R. 43, 65, 115, 117–19, 121, 123, 125, 126, 127, 130, 132, 183, 211 Freund, W.C. 156, 161, 163 Frey, B.S. 12, 13 Friedman, M. 81
Friend, I. 116 Frino, A. 154 Froot, K. 16 Fund of Art Funds 1 futures see Sydney Futures Exchange (SFE) gamma estimates 9, 10–11, 13 Galeano, J.A. 229, 230 Garcia, R. 17 GDP per capita 56–63, 67–77 Gehrig, T. 64 Gençay, R. 183–4 geographical proximity 47, 61–2, 65, 67–74, 75–7 Germany 47 currency hedging and regime switching 21–6, 27, 33–6; optimal hedge ratio 36–7 DAX 157 Giannetti, M. 66 Gibbons, M.R. 215 Glassman, D.A. 42 Glen, J. 17 Glosten, L.R. 140 Goetzmann, W.N. 4 Goldfajn, I. 97, 104 Gonedes, N.J. 222 Gourieroux, C. 199 Goyal, A. 200, 210 Granger, C.W.J. 163 Granger causality test automation of SFE and lead–lag relationship 164–5, 176–8 systematic risk and French stocks 193–5 Gray, S. 17 Groenewold, N. 116, 118, 119, 157, 174, 175 Grünbichler, A. 158 Guidolin, M. 17, 28 Gupta, A.K. 216, 217 Halliwell, J. 118, 119, 123 Hamilton, J.D. 20 Hartmann, P. 96–7 Harvey, C.R. 96–7 He, J. 119, 131–2 hedge funds, Canadian 228–34 hedging, currency see currency hedging; optimal currency hedging hidden regime switches 20–1 high correlation state 25–6, 27
INDEX
higher moment performance analysis 138–40, 145–7 portfolio replication 139–40 rationale 139 role of higher moments 138–9 Hill, B. 9 home bias 42–79 Ahearne measure 71–2, 73 causes 56–67 data and preliminary statistics 44–56 empirical analysis 67–71 theoretical framework of domestic and foreign biases 44–6 world float portfolio 72–4, 77 Hong Kong 97–100 horse race (out-of-sample test) 31–7, 39 house prices 6 Huberman, G. 43, 65 Huisman, R. 9 Hungarian investment funds 143–50 in the money (ITM) option contracts 160, 168–9 Indonesia 97–100 industrial production growth rate, unexpected change in 122–3, 126, 127–32, 133 inflation change in expected 122–3, 126, 127–32, 133 unexpected inflation rate 122–3, 126, 127–32, 133 information costs 43, 44, 61–2, 64, 67–78 information flow 154, 155, 164–5 informational efficiency 152, 179 intensity of capital control 63, 67–74, 75–7 interest rates risk-free 188–9 unexpected change in term structure 122–3, 126, 127–32, 133 international capital asset pricing model (ICAPM) 42 International Finance Corporation (IFC) 63 Internet bubble 32–3 investor behavior domestic and foreign biases 43, 44, 61–2, 65, 67–77 UPM/LPM critical line algorithm 80–95
investor protection 67–77 Izvorski, I. 104
239
43, 44, 59–60, 66,
Jagannathan, R. 140, 197 Japan 5, 8, 97–100 Jarnecic, E. 155 Jarque–Bera test statistic 22, 24, 145, 231, 232 Jasiak, J. 199 Jegadeesh, N. 117 Jensen, C.M. 184, 197 Jensen-type regressions 197–8, 201–3, 204, 207, 208 Johansen, S. 164, 175 Johansen cointegration test 163–4, 175–6 Jorion, P. 17, 229 judicial system efficiency 59–60, 66, 67–77 Juselius, K. 164, 175 Kahneman, D. 12, 13, 81 Kantner, M. 226 Kaplan, P.D. 81 Kaplanis, E. 44–5, 47 Karolyi, G.A. 96 Karpoff, J.M. 154 Kat, H.M. 139, 140 Kelly, J.M. 105 Kempf, A. 157 Kilka, M. 43 Knif, J. 183, 192, 197 known characteristic function 216, 217–18 known location vector 216, 218, 219 Kofman, P. 152, 158 Korea 97–100 Korn, O. 157 Koskinen, Y. 66 Koutmos, G. 183, 192, 197 Kuhn–Tucker conditions 84–7 kurtosis 229, 231, 232, 233 La Porta, R. 66 Lagrange multiplier tests 130, 131 language, common 61–2, 65, 67–77 law, rule of 59–60, 66, 67–77 lead–lag relationship 155, 158–9, 164–5, 176–8 Lee, J. 153 legal system, type of 66, 67–77 leptokurtic distributions 189, 191
240
INDEX
likelihood ratio tests 22, 24 Lintner, J. 42, 115 liquidity 151–82 options data volume as a proxy for 154–9; price discovery and operational efficiency of a market structure 156–9 liquidity ratios 162, 165–7, 168, 169 and market volatility 169–71 London Futures and Options Exchange 156–7 Long-Term Capital Management (LTCM) 228 Longstaff, F.A. 21 L’Oréal 190–209 loss aversion 12 low correlation state 25–6, 27 lower partial moment (LPM) model 81 see also upside potential–downside risk portfolio model MacBeth, J.D. 119, 125 MacKinlay, C. 115 MacKinnon, J.G. 174 macroeconomic variables 117, 118–19 multifactor model for ASX 122–3, 126, 127–32, 133 Malaysia 97–100 Mananyi, A. 156–7 March, J.G. 80 market efficiency 156 market factor 185–7, 189–90 impact of systematic risk on French stocks 190–211 market return index 126, 127–32, 133 market structure 151–82 dynamics of a changing market structure 152–3 price discovery and operational efficiency of 156–9 Markowitz, H. 81, 82, 87, 88, 228 Massimb, M. 152–3 Masson, P. 96 matrix elliptical contoured distributions 214–27 analysis of the power functions 221–2 empirical study 222–4 small sample tests 216–20; further statistics 220; known type of elliptical symmetry 217–18; unknown type of elliptical symmetry 218–19
McKenzie, M.D. 192 mean–variance analysis 81, 228 critical line UPM/LMP model and 92–4 Meese, R. 17 Mei, J. 4 Mele, A. 199 Merton, R.C. 130 Mexico bond market–stock market linkages 103–13 currency crisis 96, 104 Min, H. 103, 112 mixture of distributions (MDH) hypothesis 154, 161, 170 modified Sharpe ratio 228–34 modified VaR (MvaR) 228, 229, 230, 231, 232, 233 Mody, A. 96 MONEP (Marché des Options Négociables de Paris) 188 moneyness portfolios 154–5, 159–60, 161, 167, 168–9 monsoonal effect 96 Moser, J. 152, 158 Moses, M. 4 Mossin, J. 115 Muirhead, R.J. 219, 225 multifactor arbitrage pricing theory (APT) 114–17 multifactor models (MFM) 114–36 data 119–27; cross–sectional regressions 124–7; CRR macroeconomic variables 122–3; explanatory returns 119–22; portfolio characteristics 124; returns to be explained 123–4 data analysis and results 127–32 existing evidence 115–18 parsimonious model for ASX 119, 130–2 multivariate t-distributions 221–2, 225–6 Nagy, M. 230 Naik, N. 229 Nawrocki, D. 81 New Zealand Gross Index 157 Ng, L.K. 119, 131–2 no-opportunity arbitrage valuation principle 186 non-linearity 195–201 Norway 63 NZSE-40 Index 157
INDEX
omission bias 13 open outcry 152, 153 operational efficiency 152, 153, 179 of a market structure 156–9 opportunity cost effect 12 optimal currency hedging 16–41 certainty equivalent compensation 28–31 estimation results for regime-switching models 21–6, 27 horse race for regime-switching strategies 31–7, 39 optimal foreign investment 38, 39 optimal hedge ratios 36–7 optimal weights 35–6 options benchmark assets 141–2; return distribution 144 option moneyness portfolios 154–5, 159–60, 161, 167, 168–9 options data volume as a proxy for liquidity 154–9 portfolio replication 139–40 pricing 186–7; dividend framework 187; no-dividend framework 186–7 out-of-sample test (horse race) 31–7, 39 out of the money (OTM) option contracts 160, 168–9 Owen, J. 215 ownership effect 12 Pacific/Asia bloc 47, 54, 56 Palaro, H.P. 139 parsimonious multifactor model 119, 130–2 payoff distribution pricing model 139 peer group analysis 148–9 Perez-Quiros, G. 17 perfect knowledge 18–20 performance Canadian hedge funds 231–3 forecasting 205–9 performance evaluation 137–50 efficiency gain/loss methodology 140–3 higher moment performance analysis 138–40 testing results 143–9; basic performance characteristics 145; data for the analysis 143–4; higher moment performance
241
characteristics 145–7; peer group analysis 148–9; return distribution of the benchmark asset 144 Perold, A. 16 Perron, P. 17 Phelps, B. 152–3 Philippines 97–100, 104 Phillip–Perron (PP) test 163, 174 phone call costs 61–2, 64, 67–77 Pirrong, C. 153 Pitts, M. 154 Portes, R. 64 portfolio replication 139–40, 145–7, 150 Post, T. 81 Poterba, J. 43, 65, 183 power functions 221–2 price art price indices 6–8 and trading volume 154 price discovery analysis 156–9, 174–9 prospect theory 81 Rabinovitch, R. 215 Racine, M.D. 157, 164, 174, 175 ratio analysis 162, 172–3, 179 real-estate bubble 6 real GDP growth rate 56–63, 67–77 regime switching 16–41 economic importance of regimes 28–31 estimation results 21–6, 27; data 21–2, 23; parameter estimates 22–6; specification test 22, 24 implications on asset allocation 26–38 model 18–21; portfolio selection under hidden regime switches 20–1; portfolio selection with perfect knowledge of the active state 18–20 optimal foreign investment 38, 39 strategies in competition 31–7, 39; cumulative wealth and Sharpe ratio 32–4; optimal hedge ratio 36–7; optimal weights 35–6 regression analysis cross-sectional regressions 124–32 impact of systematic risk on French stocks 196–209 liquidity and automation of SFE 162–3, 172–3 Reinganum, M.R. 115, 116, 119
242
INDEX
Renault 190–209, 210 replication, portfolio 139–40, 145–7, 150 return correlations 96 East Asian economies 97, 99–100, 101 returns Australian stock market 123–4, 127–32 bond and stock market linkages 105–12 East Asian stock markets average rates of return 97, 97–9, 101 return distribution of benchmark asset for Hungarian investment funds 144 reverse S-shaped utility functions 81 Rey, H. 64 Richardson, M. 161 Riddick, L.A. 42 risk downside see downside risk upside 229 upside potential–downside risk portfolio model 80–95 risk aversion 81, 92, 93, 94 risk-free interest rate 188–9 risk premiums, unexpected change in 122–3, 126, 127–32, 133 risk-seeking behavior 81 Rockafellar, R.T. 229 Roll, R. 114, 116, 184, 197, 209 Ross, S.A. 114, 116 Rubinstein, M. 154, 160, 161 rule of law 59–60, 66, 67–74, 75–7 S&P 500 index 157 Samorodnitsky, G. 226 Samuelson, W. 12 Santa-Clara, P.P. 200, 210 Sarkisson, S. 65 Sarno, L. 164 Savage, L.J. 81 SBF120 index 190–209, 210 SBF250 index 190–209, 210 scale factor 185–7, 189–90 Schill, M. 65 Schmid, W. 215, 220 Schneider 190–209 Scholes, M. 185, 186 Schulman, E. 16 Schwartz, E. 21 Selçuk, F. 183–4 self-deception theory 13
Sharpe, W.F. 42, 115, 183, 197, 209 Sharpe ratio 138, 228, 229, 230, 231, 232, 233 CEE investment funds 147, 150 modified 228–34 regime switching and optimal currency hedging 32–4 Shefrin, H. 13 short selling 145–7 Shyy, G. 153 Siegel, J.J. 5 Siegel, L.B. 81 Simon, H.A. 80 simple regression analysis 196–9, 201, 202, 203, 204, 206, 207, 208 Singapore 63 Singer, A. 229 size 118–19, 121–33 skewness 229, 231, 232, 233 Smith, T. 161 Société Générale 190–209, 210 Solnik, B. 117 Sortino, F. 81 spillover effect 96 SPI futures/All Ordinaries Index ratio 162–3, 172–3 Statex Actuaries Accumulation Index 157 statistical multifactor models 117 Statman, M. 13 status quo bias 12 Stiglitz, J.E. 5 stock index dynamic 185–6 stock market capitalization 57–8, 64, 67–77 stock market development 43, 44, 57–8, 63–4, 67–77 stock markets linkages to bond markets 103–13 currency crises, contagion and portfolio selection 96–102 Strong, N. 65 Struthers, J.J. 156–7 Stulz, R.M. 96, 183 sunk cost effect 12–13 Sydney Futures Exchange (SFE) 151–82 lead–lag relationship with ASX 155, 158–9, 164–5, 176–8 Share Price Index (SPI) 157, 158, 159, 161, 165–7, 175, 176–8 SPI futures/All Ordinaries Index ratio 162–3, 172–3
INDEX
symmetric stable distributions 221, 222, 225–6 systematic risk 183–213 empirical study 187–90; data 187–9; induction of systematic risk 189–90 impact 190–5; causality 191–3; correlation 190–1; Granger causality test 193–5 market benchmark comparison 201–9; basic empirical study 201–5; forecasting performance 205–9 non-linearity 195–201; simple regression analysis of asset returns 196–9; volatility impact 199–201 theoretical framework 185–7; option pricing 186–7; valuation setting 185–6 Szego, G. 191 tail index estimator 8, 9–12 Taiwan 97–100 Taqqu, M.S. 226 Tauchen, G. 154 Taylor, M.P. 164 term structure, unexpected change in 122–3, 126, 127–32, 133 Tesar, L. 64, 67 Thaler, R.H. 12, 13, 117 Thomas, S.H. 127, 130 Thomson 190–209 time series properties 174–5 time-varying downside risk 1–15 Timmerman, A. 17, 28 Titman, S. 117 Toronto Stock Exchange (TSE) 156, 231 Totalfina Elf 190–209, 210 trade, scaled by GDP 56–63, 67–77 trading volume 154, 155, 161, 165–7 market volatility and 169–71 transaction costs 64, 67–77 Treynor, J. 183 Turkington, J. 157, 158 Turner, C. 17 Tversky, A. 12, 81 two-year return 67–77 underlying distribution 142–3 unexpected inflation rate 122–3, 126, 127–32, 133 uniqueness of art works 3
243
unit root tests 163, 174 United Kingdom art market 6–8, 10 currency hedging and regime switching 21–6, 27, 33–6; optimal hedge ratio 36–7 United States art market 6–8, 10 regime switching and currency hedging 21–7, 33–6 stock market levels and returns 107 unknown characteristic function 218–19 unknown location vector 216–19 upper partial moment/lower partial moment (UPM/LPM) ratio 81 see also upside potential–downside risk portfolio model upside potential–downside risk portfolio model 80–95 efficient segments on the efficient frontier 88; adjacent efficient segments 89–92 empirical example 92–4 Kuhn–Tucker conditions 84–7 upside risk 229 upward bias 209 Uryasev, S. 229 Valéo 190–209, 210 valuation 185–6 value, art and 3–4 value-at-risk (VaR) 228, 229, 231, 232, 233 modified (MvaR) 228, 229, 230, 231, 232, 233 Van Vliet, P. 81 Varga, T. 216, 217 variance decompositions 109, 110, 111 vector autoregressive (VAR) models 164 Brady bonds 105–12 systematic risk and French stocks 191–3, 194 Venezuela 43, 47, 104 Vivendi 190–209, 210 volatility Asian currency crisis 97–9, 101 automation of SFE 161, 169–71 Black and Scholes volatility bias 189
244
INDEX
volatility continued impact of systematic risk on French stocks 199–205, 207, 208 volatility parameter 185, 189–90
Whaley, R. 189 Whitcher, B. 183-4 White, H. 127 world float portfolio
Walsh, D. 157, 158 Wang, G.H.K. 153 Wang, Z. 197 Warnock, F. 63 Weber, M. 43 Werner, I. 64
Xu, X.
65
Zeckhauser, R. 12 Zhang, Y.T. 219, 225 Zhou, C. 64 Zhou, G. 215
72–7