Financial Market Analytics

FINANCIAL MARKET ANALYTICS JOHN L. TEALL

Q

Quorum Books Westport, Connecticut • London

Library of Congress Cataloging-in-Publication Data Teall, John L., 1958Financial market analytics / John L. Teall. p. cm. Includes bibliographical references and index. ISBN 1-56720-198-9 (alk. paper) 1. Investments—Mathematics. 2. Business mathematics. HG4515.3.T43 1999 332.6/0151—dc21 98-23975

I. Title.

British Library Cataloguing in Publication Data is available. Copyright © 1999 by John L. Teall All rights reserved. No portion of this book may be reproduced, by any process or technique, without the express written consent of the publisher. Library of Congress Catalog Card Number: 98-23975 ISBN: 1-56720-198-9 First published in 1999 Quorum Books, 88 Post Road West, Westport, CT 06881 An imprint of Greenwood Publishing Group, Inc. Printed in the United States of America

@r The paper used in this book complies with the Permanent Paper Standard issued by the National Information Standards Organization (Z39.48-1984). P In order to keep this title in print and available to the academic community, this edition was produced using digital reprint technology in a relatively short print run. This would not have been attainable using traditional methods. Although the cover has been changed from its original appearance, the text remains the same and all materials and methods used still conform to the highest book-making standards.

CONTENTS

Preface

ix

1 Introduction and Overview l.A Analytics and the Scientific Method in Finance l.B Financial Models l.C Empirical Studies l.D Research in Finance l.E Applications and Organization of this Book

1 1 3 4 5 13

2 Preliminary Analytical Concepts 2.A Time Value Mathematics 2.B Geometric Series and Expansions Application 2.1 Annuities and Perpetuities Application 2.2 Growth Models Application 2.3 Money and Income Multipliers 2.C Return Measurement 2.D Mean, Variance and Standard Deviation Application 2.4 Risk Measurement 2.E Comovement Statistics Application 2.5 Security Comovement 2.F Introduction to Simple OLS Regressions Application 2.6 Relative Risk Measurement Exercises

15 15 17 18 19 20 21 23 24 26 27 29 30 32

Contents

vi 3 Elementary Portfolio Mathematics

37 37 40 43 45

4 Matrix Mathematics

49 49 50 50 52 52 54 55 57 59 60 61 65 69 71 73 74 78

3.A Introduction to Portfolio Analysis 3.B Single Index Models 3.C Multi-Index Models Exercises

4.A Matrices, Vectors and Scalars Application 4.1 Portfolio Mathematics 4.B Addition, Subtraction and Transposes of Matrices 4.C Multiplication of Matrices Application 4.1 (continued) Portfolio Mathematics 4.D Inversion of Matrices 4.E Solving Systems of Equations Application 4.2 Coupon Bonds and Deriving Yield Curves Application 4.3 Arbitrage with Riskless Bonds Application 4.4 Fixed Income Portfolio Dedication 4.F Vectors, Vector Spaces and Spanning Application 4.5 The State Preference Model Application 4.6 Binomial Option Pricing Application 4.7 Put-Call Parity 4.G. Orthogonal Vectors Application 4.8 Arbitrage Pricing Theory Exercises

5

Differential Calculus 5.A Functions and Limits Application 5.1 The Natural Log 5.B Slopes, Derivatives, Maxima and Minima Application 5.2 Utility of Wealth 5.C Derivatives of Polynomials Application 5.3 Marginal Utility Application 5.4 The Baumol Cash Management Model Application 5.5 Duration Application 5.6 Bond Portfolio Immunization Application 5.7 Portfolio Risk and Diversification 5.D Partial Derivatives Application 5.8 Deriving the Simple OLS Regression Equation Application 5.9 Deriving Multiple Regression Coefficients 5.E The Chain Rule, Product Rule and Quotient Rule Application 5.10 Plotting the Capital Market Line 5.F Taylor Series Expansions Application 5.11 Convexity and Immunization Application 5.12 Risk Aversion Coefficients

83 83 84 84 87 89 91 91 94 97 97 99 99 101 102 104 112 113 115

Contents 5.G The Method of LaGrange Multipliers Application 5.13 Optimal Portfolio Selection Application 5.14 Plotting the Capital Market Line, Second Method Application 5.15 Deriving the Capital Asset Pricing Model Application 5.16 Constrained Utility Maximization Exercises Appendix 5.A Derivatives of Polynomials Appendix 5.B Rules for Finding Derivatives Appendix 5.C Portfolio Risk Minimization on a Spreadsheet

vu 116 118 119 122 124 127 131 132 133

6 Integral Calculus 6. A Antidifferentiation and the Indefinite Integral 6.B Definite Integrals and Areas Application 6.1 Cumulative Densities Application 6.2 Expected Value and Variance Application 6.3 Stochastic Dominance Application 6.4 Valuing Continuous Dividend Payments Application 6.5 Expected Option Values 6.C Differential Equations Application 6.6 Continuous Time Security Returns Exercises Appendix 6.A Rules for Finding Integrals

137 137 138 142 144 145 149 150 151 152 155 157

7 Introduction to Probability 7.A Random Variables and Probability Spaces 7.B Distributions and Moments 7.C Binomial Distributions Application 7.1 Estimating Probability of Option Exercise 7.D The Normal Distribution 7.E The Log-Normal Distribution Application 7.2 Common Stock Returns 7.F Conditional Probability Application 7.3 Option Pricing — Conditional Exercise Application 7.4 The Binomial Option Pricing Model Exercises

159 159 160 161 164 166 167 167 169 169 170 173

8 Statistics and Empirical Studies in Finance 8.A Introduction to Hypothesis Testing Application 8.1 Minimum Acceptable Returns 8.B Hypothesis Testing: Two Populations Application 8.2 Bank Ownership Structure 8.C Interpreting the Simple OLS Regression Application 8.3 The Capital Asset Pricing Model

175 175 176 179 179 181 184

Contents

vm Application 8.4 Analysis of Weak Form Efficiency Application 8.5 Portfolio Performance Evaluation 8.D Multiple OLS Regressions Application 8.6 Estimating the Yield Curve 8.E Event Studies Application 8.7 Analysis of Merger Returns 8.F Models with Binary Variables Exercises

189 191 193 198 199 201 208 211

9 Stochastic Processes 9.A Random Walks and Martingales 9.B Binomial Processes 9.C Brownian Motion, Weiner and Ito Processes 9.D Ito's Lemma Application 9.1 Geometric Weiner Processes Application 9.2 Option Prices — Estimating Exercise Probability Application 9.3 Option Prices — Estimating Expected Conditional Option Prices Application 9.4 Deriving the Black-Scholes Option Pricing Model Exercises

213 213 214 215 218 221

10 Numerical Methods 10.A Introduction 10.B The Binomial Method Application 10.1 The Binomial Option Pricing Model Application 10.2 American Put Option Valuation 10.C The Method of Bisection Application 10.3 Estimating Bond Yields Application 10.4 Estimating Implied Variances 10.D The Newton-Ralphson Method Application 10.4 (continued) Estimating Implied Variances Exercises

233 233 233 235 237 240 241 242 245 246 248

Appendix A Solutions to End-of-Chapter Exercises Appendix B Statistics Tables Appendix C Notation Definitions Glossary References Index

249 293 295 299 305 313

222 223 224 230

PREFACE

Evolution of highly sophisticated financial markets, innovation of specialized securities and increasingly intense competition among investors have driven the development and use of highly rigorous mathematical modeling techniques. The investment community has unleashed a plethora of complicated financial instruments, mathematical models and computer algorithms, often created by socalled rocket scientists. Practitioners and researchers have learned that mathematical models are crucial to financial decision making; yet the quantitative skills of practitioners and researchers are often "rusty." University students enrolled in finance courses often feel overwhelmed when their mathematical preparation is inadequate. They all too frequently suffer difficulty with mathematics to the point where they are unable to grasp even the intuition of financial techniques. Many books, manuals and instructors have responded by watering down quantitative content. Yet, mathematics is necessary to understand the implications, variations and limitations of financial techniques. The purpose of this book is to provide background reading in a variety of elementary mathematics topics used in financial analysis. It assumes that the reader has limited or no exposure to statistics, calculus and matrix mathematics. Broad coverage includes discussions related to portfolio management, derivatives valuation, corporate finance, fixed income analysis and other issues as well. This book's organization by quantitative topic differs from that of other financial mathematics books which tend to be organized by financial topic. Readers experiencing difficulty with quantitative technique typically need more review of mathematical technique to master financial technique. This book is intended to provide an informal introduction to a given mathematics topic which is then re-enforced through application to a variety of topics in finance. Coverage is broad, both in terms of coverage in mathematics and in finance. This heterogeneity of coverage has compelled separation and spreading of the

X

Preface

various finance topics throughout the book, but they are linked through Background Readings suggestions at the beginnings of most sections and applications. Exercises provided at the ends of chapters are intended to be completed with assistance of a basic calculator, though a computer-based spreadsheet may be helpful in some cases. This book is designed to provide prerequisite or parallel reading for other books such as Elton and Gruber [1995], Hull [1997], Copeland and Weston [1988], Alexander and Francis [1986], Cox and Rubinstein [1985] and Martin, Cox and MacMinn [1988]. The book may serve as a quantitative foundation for more advanced or specialized texts such as Campbell, Lo and MacKinlay [1997], Baxter and Rennie [1996] and Neftci [1996]. Another goal of this book is to enable readers to read less technical academic and professional articles. This book should serve several purposes in financial and academic communities: 1. As a reference book in the library of a finance practitioner likely to encounter problems of a quantitative nature. 2. To provide quantitative support to the researcher in finance without the mathematics skills necessary to master current finance methodology. Among the researchers who may benefit from this book are academics specializing in strategic management, accounting, business policy and certain fields of economics. In addition, financial engineers, systems analysts in financial services firms and policy makers may benefit from material presented in this book. 3. As a supplemental text for undergraduate and M.B.A. students likely to experience difficulties with the quantitative technique in finance courses. Material covered in this text will parallel coverage of several courses, including Financial Management, Principles of Investments, Portfolio Analysis, Options and Futures and Fixed Income Management. In some cases, this book may be appropriate as a supplement to doctoral students in finance. 4. As a primary text for a course such as "Applied Analytical Methods in Finance." 5. As a primary or secondary text for a prerequisite mathematics course offered by most M.B.A. programs covering linear mathematics, calculus and statistics. Most graduate business schools offer such courses to prepare students without adequate mathematics background for more quantitative aspects of their programs. 6. As a manual for a continuing education course intended to provide coverage of finance from an analytical perspective. In addition, participants in certain certification programs may benefit from this book. Objectives and organizations of individual chapters are described in Section 1 .E and in introductions to the chapters themselves.

Preface

XI

ACKNOWLEDGMENTS I am fortunate to have had a number of students, colleagues and friends assist and provide guidance in the preparation of this book. Steve Adams, Larry Bezviner, Michael Dang, Hyangbab Ku, Joe Mazzeo and Jay Pandit all contributed useful comments and corrections on earlier versions of the manuscript. My production editor, Deborah Whitford was most helpful in the preparation and editing of the manuscript as were Sang Kim, Daniel Terfossa and Philip Wong, who also furnished valuable assistance in the preparation tables and figures. My old friends Iftekhar Hasan and John Knopf provided encouragement and advice throughout the various stages of writing this book. Ed Downe and Ken Sutrick contributed most useful comments regarding specific sections of the manuscript. I am particularly grateful for the gracious and varied contributions of Peter Knopf and T. J. Wu. And most importantly, I never get anything accomplished without constant prodding and needling from Miriam Vasquez. I would like to attribute to my friends the various errors and shortcomings that will inevitably surface in this book, but I'm afraid that they are already on the verge of casting me off. Since I feel more secure in my relationships with my lovely Anne and my lovely Emily, I'll blame them.

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1 Introduction and Overview

l.A: ANALYTICS AND THE SCIENTIFIC METHOD IN FINANCE Traders, merchants, farmers and financiers have used mathematics to conduct business for many centuries. Throughout most of the past two millennia, commerce has been conducted with the use of simple arithmetic, integers and fractions. Western business accounts were maintained without the use of the number zero or decimals until after the twelfth century when the Hindu-Arabic numerical system was introduced to the West. Obviously, methods for performing routine computations have improved substantially over the years. The primary mechanical device for performing computations prior to the twelfth century was the abacus; even the simplest of arithmetic operations were cumbersome to perform with the Roman numerical system. For example, suppose that a merchant needed to borrow LXXXVIII denarii from one lender and XLIV from a second. What is the total sum to be borrowed? Next, compute the sum to be repaid, assuming a IX percent interest rate compounded monthly for VII years. Obviously, the Roman numerical system is less useful for performing routine arithmetic operations than for recording numbers of units. The thirteenth-century Italian mathematician, Leonardo Fibonacci (born Leonardo Pisano) introduced Arabic numerical notation to Europe in his book Liber Abaci. This treatise on arithmetic and algebra was enthusiastically received by his contemporaries, in part, because it contained a wealth of practical applications. Fibonacci discussed numerous applications of the Arabic numerical system to commerce, including interest calculations, weights and measures, exchange rates and bookkeeping. Use of the Hindu-Arabic numerical system and simple mathematics slowly worked its way into business and finance over the centuries. However, the ability to compile and manipulate data with simple arithmetic was insufficient to properly analyze most types of financial decisions. In addition, prior to 1950,

2

Chapter 1

financial studies had drawn little attention from the scholarly communities. Neither academic nor professional financial literature benefitted from rigorous scientific discipline. The literature tended to be primarily descriptive and anecdotal in nature, based largely on the experience and common sense of practitioners.1 Very few analytics beyond arithmetic and simple algebra were used in this literature.2 Furthermore, many of the earlier writings were inconsistent and contradictory with no methodology to resolve inconsistencies. A more axiomatic approach to financial research and technique was developed which worked its way into the financial literature in the 1950s and early 1960s. This approach was based on the methodology frequently used in the physical sciences. The scientific method applied to financial problem solving might be described as follows: 1. Observe, describe and measure financial phenomena. 2. Use previously obtained knowledge and experience to exclude all but those factors most relevant to the problem under consideration. 3. Describe, measure and model the causes, processes and implications of these financial phenomena. 4. Place the results of the model into some known law, framework or generalization and/or construct testable hypotheses or generalizations to explain the phenomena. This reasoning process of generalization from specific observations to form hypotheses or theories is called induction. 5. Observe and test these descriptions, measures, models, hypotheses and generalizations empirically. Derive and test predictions of models. 6. Revise and improve models to make better predictions. 7. Accept or continue to revise the models. This axiomatic approach to the study of finance, like the study of physical sciences, requires extensive use of mathematics. Mathematics provides us with a means of representing and simplifying complex financial systems in a concise and rigorous manner; it makes the study of finance far more exciting, enabling us to better understand investor motivations and behavior. Mathematics brings us closer to comprehending how and why investors behave in high-risk environments where they face stress and a variety of constraints. Many important developments in the financial industry owe their existence to the development of improved quantitative techniques. Financial engineering, techniques of option valuation, portfolio insurance, fixed income hedging strategies and index arbitrage are just a few of the modern financial developments which are highly dependent on mathematical technique. In fact, many of the quantitative developments in finance were initiated in the industry as analysts became more sophisticated in their pursuit of increased profits and improved risk management techniques. Financial analysis takes place in a highly competitive uncertain environment that includes many individuals and institutions. These players interact with one

Introduction and Overview

3

another over many periods of time which can be presumed to be infinitely divisible. While this environment is fascinating and exciting, its analysis requires application of many branches of mathematics, ranging from simple arithmetic, algebra and calculus to stochastic processes, numerical methods and probability theory. This book presents essential mathematical technique and its applications to financial analysis. l.B: FINANCIAL MODELS A model might be characterized as an artificial structure describing the relationships among variables or factors. Practically all of the methodology in this book is geared toward the development and implementation of financial models to solve financial problems. For example, the simple valuation models in Chapter 2 provide a rudimentary foundation for investment decision making, while the more sophisticated models in Chapter 9 describing stochastic processes provide an important tool to account for risk in decision making. The use of models is important in finance because "real world" conditions that underlie financial decisions are frequently extraordinarily complicated. Financial decision makers frequently use existing models or construct new ones that relate to the types of decisions they wish to make. Models proposing decisions which ought to be made are called normative models.3 The purpose of models is to simulate or behave like real financial situations. When constructing financial models, analysts exclude the "real world" conditions that seem to have little or no effect on the outcomes of their decisions, concentrating on those factors which are most relevant to their situations. In some instances, analysts may have to make unrealistic assumptions in order to simplify their models and make them easier to analyze. After simple models have been constructed with what may be unrealistic assumptions, they can be modified to match more closely "real world" situations. A good financial model is one which accounts for the major factors that will affect the financial decision (a good model is complete and accurate), is simple enough for its use to be practical (inexpensive to construct and easy to understand), and can be used to predict actual outcomes. A model is not likely to be of use if it is not able to project an outcome with an acceptable degree of accuracy. Completeness and simplicity may directly conflict with one another. The financial analyst must determine the appropriate trade-off between completeness and simplicity in the model he wishes to construct. This book emphasizes both theoretical and empirical models as well as the mathematics required to construct them. Theoretical models are constructed to simulate or explain phenomena; empirical models are intended to evaluate or measure relationships among "real world" factors. The financial analyst may construct and use a theoretical model to provide a framework for decision making and then use an empirical model to test the theory. Analysts also use

4

Chapter 1

empirical models to measure financial phenomena and to evaluate financial performance. In finance, mathematical models are usually the easiest to develop, manipulate and modify. These models are usually adaptable to computers and electronic spreadsheets. For example, the matrix-based models in Chapter 4 are easily accommodated by popular spreadsheet programs. Numerical techniques discussed in Chapter 10 are essentially models which are used to obtain numerical solutions for other models, usually with the aid of computer software. Mathematical models are obviously most useful for those comfortable with math; the primary purpose of this book is to provide a foundation for improving the quantitative preparation of the less mathematically oriented analyst. Other models used in finance include those based on graphs and those involving simulations. However, these models are often based on or closely related to mathematical models. Computers have played an important role in many types of financial analysis for a number of years. Since the early 1980s, computer-based spreadsheets have been used with increasing frequency, largely due to their ease of use. Among the better-known computer based spreadsheets are Lotus 123™, Excel™ and Quattro-Pro™. An electronic spreadsheet appears on the user's computer screen as a matrix or array of columns and rows where numbers, formulas or labels are entered. These entries may be related in a number of ways. Advantages of a computer-based spreadsheet over a paper-based spreadsheet include the speed and ease of computations and revisions offered by the computer. Spreadsheet use does not require mastery of a programming language; in fact, one can begin to use a spreadsheet within a few minutes after learning a small number of elementary commands and procedures. In addition, the models in this book are perfectly adaptable to more structured programs such as BASIC, FORTRAN, PASCAL, C+ +, and so on. l.C: EMPIRICAL STUDIES Empirical studies are intended to measure financial phenomena and performance and to test theories and models. Because financial studies usually concern measurements involving large numbers of firms or securities, empirical analysis makes extensive use of statistics. Financial analysts are fortunate in that they frequently have access to significant data resources. Securities markets record enormous quantities of prices and other trading statistics, firms create detailed accounting statements and various business and government agencies generate huge volumes of data pertaining to economics and commerce. Statisticians have developed highly sophisticated means of analyzing such data. Finance, unlike many of the social sciences, has not made extensive use of experimental methodologies. However, important contributions to the finance literature have been made by behavioral psychologists such as Daniel Kahnaman and Amos Tversky, who demonstrated that individuals in their decision making

Introduction and Overview

5

tend to over emphasize recent information and trends and under-emphasize prior information. Dale Griffin collaborated with Tversky on work arguing that experts tend to be more prone to overconfidence than novices while maintaining reputations for their expertise. They suggest that overconfident traders tend to be more aggressive in their trading strategies. Although use of experimental methodology is increasing in finance, the vast majority of empirical research in finance is highly dependent on statistical analysis of data. Many empirical tests are conducted for the purpose of testing theories and models. Scholars are concerned with testing the validity of their theories to explain the behavior and performance of financial markets. Practitioners in the financial industry benefit from testing their theories and models on historical or hypothetical data before actually investing money to implement them. Several important methodologies for empirical testing are presented in Chapter 8. This chapter discusses how financial theories can be tested based on either examination of the validity of underlying assumptions or the accuracy of predictions implied by the theories. l.D: RESEARCH IN FINANCE This section briefly reviews scholarly research in financial economics in order to provide readers lacking strong academic backgrounds in finance with resources that may prove useful for solving financial problems. It emphasizes the literature which either introduced or made extensive use of the quantitative concepts presented in this book. It is important to know how an existing body of literature can be used to solve financial problems. The reviews that follow also mention sections or applications in this book that discuss techniques related to the reviewed research. Early Research As discussed in Section l.A, financial literature prior to 1950 was primarily descriptive, anecdotal and prescriptive in nature. However, there were some important exceptions. One early exception was Daniel Bernoulli [1738], who wrote on diminishing marginal utility and risk aversion. At a meeting of mathematicians, he proposed a problem commonly referred to as the St. Petersburg Paradox. This problem was concerned with why gamblers would pay only a finite sum for a gamble with an infinite expected value. Louis Bachelier [1900] wrote his doctoral dissertation at the Sorbonne on the distribution of stock prices and option valuation. He provided a derivation for a probability density function which was later to be known as a Weiner process (Brownian motion process with drift; see Section 9.C). The option valuation model based on this process was quite similar to the better known and more recently developed Black-Scholes pricing model (Applications 9.3 and 9.4). His Brownian motion derivation predated the better publicized derivation of

6

Chapter 1

Brownian motion by Albert Einstein. Unfortunately, his research was ignored until the early 1950s when Leonard Savage and Paul Samuelson discussed the distribution of security prices and Case Sprenkle [1961] wrote his doctoral dissertation on option valuation. Irving Fisher [1896, 1907, 1930] wrote important treatises on the theory of interest rates and the internal rate of return (See Section 2.C). His 1896 paper set forth the Expectations Hypothesis of the term structure of interest rates (see Application 4.2 on deriving spot and forward rates). The well-known Fisher Separation Theorem demonstrates that the individual investment decision can be made independently of consumption preferences over time. This important theorem extends Application 5.16 on constrained utility maximization by introducing different asset investment classes. Alfred Cowles [1933] and Holbrook Working [1934] were statisticians who were concerned with capital markets efficiency, or more specifically, the random movement of stock prices. Their tests were somewhat similar to those described in Application 8.4 on weak form market efficiency. The axiomatic approach to financial research was in its infancy during the 1950s and early 1960s. Harry Markowitz [1952, 1959] is regarded as the originator of Modern Portfolio Theory. His research is the basis for Section 4. A and Application 5.7 regarding portfolio mathematics. Writing his doctoral dissertation in statistics, Markowitz described the impact on portfolio diversification of increasing the number of securities in a portfolio. His model also detailed the importance of selecting uncorrelated securities for portfolio management. Treynor [1961] used the results of Markowitz to value securities. Lintner [1956] and Gordon [1959] provided important research on corporate dividend policy and the valuation of corporate shares. Application 2.2 includes a derivation of the Gordon Stock Pricing Model. Modigliani and Miller [1958, 1961, 1963] were major innovators in corporate finance, particularly on issues related to dividend policy and capital structure. They were the first to offer a proof based on the Law of One Price, a concept used throughout this book, particularly in Chapters 4 and 9. Their papers demonstrated the irrelevance of corporate capital structure and dividend policies in perfect markets. Kenneth Arrow and Gerard Debreu [1954] and Debreu [1959] published in the economics literature a model for pricing commodities. This model was applied to the valuation of corporate assets and securities by Arrow [1964] and Hershleifer [1964] and [1965]. Application 4.5 on State Preference Theory is derived from technique presented in these papers. Tobin [1958] derived the Efficient Frontier and Capital Market Line based on the work of Markowitz. His model, which forms the basis of Applications 5.10 and 5.14, suggests that all investors in a market, no matter how differently they feel about risk, will hold the same stocks in the same proportions as long as they maintain identical expectations regarding the future. Investor portfolios will differ only in their relative proportions of stocks and bonds.

Introduction and Overview

7

Statisticians and econometricians have long been fascinated by the tremendous amounts of financial data made available by the various data services. For example, Kendall [1953], Muth [1961] and Eugene Fama [1965] all wrote on capital markets efficiency and the random nature of stock price returns. Samuelson [1965a] and Fama [1965] modeled asset price dynamics as a submartingale (See Section 9.A), where the best forecast for a future asset price is the current price with a "fair" return; price histories are otherwise irrelevant in forming forecasts. The Major Breakthroughs The major breakthroughs in financial research were in large part due to the more scientific approaches to financial analysis. As suggested on page 6, Markowitz provided the major breakthrough in portfolio analysis. The Capital Asset Pricing Model (CAPM) extended the work of Markowitz and Tobin to provide an important theory of capital markets equilibrium, enabling investors to value securities. The model states that security returns are linearly related to returns on the market and that firm specific risks do not affect security prices. Developed independently by Sharpe [1964], Lintner [1965] and Mossin [1966], the veracity of this model is still a hotly disputed issue in the financial literature. The derivation of Sharpe is the basis for Application 5.15, and Application 8.3 provides an example for its computation. The Black-Scholes Options Pricing Model is based on the construction of perfectly hedged portfolios (See Applications 4.6 and 9.4) and applied to the valuation of corporate securities. The perfect hedge and the equilibrium pricing framework are important features distinguishing their paper from earlier ones by Sprenkle [1961] and Samuelson [1965b]. The publication of the Black and Scholes paper coincided with the 1973 opening of the Chicago Board Options Exchange, the first and still largest stock options market. In addition, Black and Scholes applied their model to the valuation of risky debt and equity securities in the limited liability firm. A third model of equilibrium in financial markets, the Arbitrage Pricing Theory (APT), was provided by Stephen Ross [1976]. This model is based on a form of the Law of One Price, which, in general, states that investments generating identical cash flow structures should be valued identically. The APT provided a simple and more general theory of equilibrium in capital markets than the CAPM. The Arbitrage Pricing Theory is derived in Application 4.8. Utility Analysis and Risk Measurement Utility analysis is concerned with how people make and rank choices (See Applications 5.1 and 5.2). Bernoulli [1738] wrote the first paper on the relationships among diminishing marginal utility, risk aversion and expected value in an uncertain environment. John von Neumann and Oscar Morgenstem

8

Chapter 1

[1944] set forth axioms of utility analysis and maximization. John Pratt [1964], following Arrow [1964] and Markowitz [1952], discussed determination of risk premiums (See Application 5.12) based on utility functions of risk averse individuals. These concepts were significant in the development of the Efficient Frontier and the Capital Asset Pricing Model. Financial analysts have long realized that forecasting security returns is quite difficult. Furthermore, estimating security risk can also be time consuming and problematic. For example, analysts often use the volatility of historical returns as a surrogate for ex-ante risk as in Application 2.4. Two difficulties associated with the traditional sample estimator procedure, time required for computation and arbitrary selection of returns from which to compute volatilities, may be dealt with through the use of extreme value estimators (e.g., Parkinson [1980]). Latane and Rendleman [1976] suggest using volatilities implied by option pricing models, as in Application 10.4. Portfolio Analysis John Lintner [1965] performed the first empirical test of the Capital Asset Pricing Model using a two-stage regression. He rejected the CAPM based on his tests; however, his two-stage regression procedure was performed on individual stocks rather than portfolios. This enabled beta estimation errors to cloud his results. Black, Jensen and Scholes [1972] found evidence to support the CAPM based on their test of portfolios. Fama and MacBeth [1973] found that while the riskless rate and beta explained the structure of security returns, beta squared and unsystematic variances did not. This lended support to the validity of CAPM. In a widely quoted paper, Richard Roll [1977] presented an important criticism of the earlier CAPM tests. Essentially, he concluded that CAPM tests are flawed in that the market portfolio has not been properly specified. Market indices which have been used in tests are not identical to the actual market portfolio, and CAPM tests are very sensitive to the selected index. Furthermore, the linear relationship between security returns predicted by the CAPM must hold if the selected index is mean-variance efficient. Hence, according to Roll, the only valid CAPM test is whether the market portfolio is efficient, though performance of such a test is complicated due to the inability to properly specify the market. Several studies, including Chen, Roll and Ross [1986] of multi factor models and APT models have suggested that more than one index is needed to explain the correlation structure of security returns (see Section 3.C and Application 4.8). Fama and French [1992] found that stock betas did not explain long-term return relationships, although firm size and market-to-book ratios did. A number of more recent studies have been published and are underway (e.g., Kothari, Shanken and Sloan [1995] who found that the relationship between portfolio returns and beta is much stronger when annual returns rather than

Introduction and Overview

9

monthly returns are used). The APT equilibrium asset pricing model of Ross does not require assumptions as restrictive as the CAPM requires. The APT states that security returns will be linearly related to a series of factors, but does not state what those factors are. Roll and Ross [1980] and Chen, Roll and Ross [1986] use factor analysis in their tests, finding that APT was supported by their data. CAPM makes very restrictive assumptions regarding investor utility functions and/or security return distributions, while APT requires that security returns be linearly related to index values. The concept of stochastic dominance (Application 6.3) may be used without such restrictive assumptions by investors in choosing among portfolios (see Whitmore and Findlay [1978], Hadar and Russell [1969] and Meyer [1977]). First order stochastic dominance rules apply to all investors who prefer more wealth to less. Hanoch and Levy [1969] prove that second order stochastic dominance rules can be used by risk-averse investors who prefer more wealth to less. Fixed Income Analytics The term structure of interest rates is concerned with the relationship between fixed income instrument yields and their terms to maturity. Term structure models can be used to project future interest rates and to construct hedges when interest rates are ex-ante unknown. The Expectations Hypothesis regarding the term structure of interest rates (Fisher [1896], Lutz [1940] and Application 4.2) states that long-term interest rates are a geometric mean of current and projected short rates. This hypothesis is supplemented by the Liquidity Premium Hypothesis (Keynes [1936], Hicks [1946]) and the Market Segmentations Hypothesis (Walker [1954], Modigliani and Sutch [1966]). The Liquidity Premium Hypotheses state that long rates tend to exceed the geometric mean of current and projected short rates due to investor preferences to invest short term to avoid risk. The Market Segmentations Hypothesis states that long and short rates depend on supply and demand conditions for short term and long term debt. Macauley [1938] and Hicks [1946] developed the simple duration model measuring the sensitivity of a bond's price to interest rates. Bierwag [1977] discusses fixed income portfolio immunization techniques (see Applications 5.5, 5.6 and 5.11). Derivative Securities A derivative security may be defined simply as an instrument whose payoff or value is a function of that of another security, index or value. There exist a huge variety of derivative securities, including (but not limited to) options, futures contracts and swap contracts. Stock options are one of the more popular types of derivative securities. The model of Black and Scholes [1973] was

Chapter 1

10

unique in that it is based on the construction of perfectly hedged portfolios. The perfectly hedged portfolio should earn the riskless rate of return. Thus, unlike earlier models, that of Black and Scholes is an equilibrium asset pricing model. They also applied their model to the valuation of limited liability corporation debt and equity securities, realizing that the equity position in a limited liability stock firm is analogous to a call option to purchase the firm's assets. Later, this concept was applied to a variety of other types of assets and contracts. Black and Scholes [1972] performed the first empirical test of the BlackScholes model on over-the-counter dividend protected call options. Although their model seemed to work quite well, Black and Scholes suggested that a large fraction of deviation of actual options prices from formula values could be explained by transactions costs. Also, they found that the model overestimated values of calls on high risk stocks. Stoll [1969] found that the put-call parity relation (see Application 4.7) performed reasonably well. Smith [1976] and Whaley [1982] provide excellent reviews of the early empirical literature on option pricing. Essentially, they found that the Black-Scholes model, with various modifications, works quite well in determining option values. Cox, Ross and Rubinstein [1979] derive the Binomial Option Pricing Model, which has proven particularly useful in the valuation of American options (see Applications 4.6, 7.1, 7.4, 10.1 and 10.2). As the number of time periods in the lattice approaches infinity, the results of the Binomial Model approach those of the Black-Scholes Model. Forward markets provide for future transactions to buy or sell assets. Futures contracts might be regarded as standardized forward contracts, which are normally traded on exchanges and provide for margin requirements and marking to the market. Why do futures markets exist? Keynes [1923] and Hicks [1946] argue that producer risk aversity provides an incentive for producers to sell their products in advance in futures markets to avoid price uncertainty. Speculators may receive more favorable commodity prices in the process of resolving producer price uncertainty. The development of large numbers of other types of derivative contracts, including a variety of swaps and "exotic" options, has led to new bodies of literature. Creation of new contracts forms an important basis of what is often referred to as Financial Engineering. This area of finance deals largely with risk management. Useful review papers in this area include Smith, Smithson and Wilford [1990], Smith and Smithson [1990], Finnerty [1988] and Damodoran and Subrahmanyam [1992]. Capital Markets Efficiency An Efficient Capital Market is defined as a market where security prices reflect all available information. The level of efficiency existing in a market might be characterized as the speed in which security prices reflect information of a particular type. Fama [1970] classified these types of information and

Introduction and Overview

11

defined three types of market inefficiency. Weak form inefficiency exists when security prices do not reflect historical price information; that is, an investor can generate an abnormal profit by trading based on historical price information. Semistrong form inefficiencies exist when investors can generate abnormal returns based on any publicly available information. Strong form inefficiencies exist when any information, public or private can be used to generate abnormal trading profits. Granger [1968], in one of the earlier weak form efficiency tests, found a very weak relationship between historical and current prices — .057% of a given day's variation in the log of the price relative (Inil+RJ) is explained by the prior day's change in the log of the price relative. His methodology is somewhat similar to that discussed in Application 8.4. Numerous studies including Roll [1981], Keim [1983] and Reinganum [1981, 1983] have confirmed a January Effect. One explanation is year-end tax selling, when investors sell their "losers" at the end of the year to capture tax write-offs. Year-end tax selling bids down prices at the end of the year. They recover early the following year, most significantly during the first five days in January. There also exists substantial evidence that smaller firms outperform larger firms (Banz [1981], Barry and Brown [1984] and Reinganum [1983]). For example, if one were to rank all NYSE, AMEX and NASDAQ firms by size, one is likely to find that those firms which are ranked as smaller will outperform those which are ranked larger. This effect holds after adjusting for risk as measured by beta. There is evidence that these abnormally high returns are most pronounced in January. Basu [1977] and Fama and French [1992] find that firms with low price-toearnings (P/E) ratios outperform firms with higher price-to-earnings ratios. Fama and French find that the P/E ratio and firm size predict security returns significantly better than the Capital Asset Pricing Model. Semistrong form efficiency tests are concerned with whether security prices reflect all publicly available information. For example, how much time is required for a given type of information to be reflected in security prices? In one well-known test of market efficiency, Fama, Fisher, Jensen and Roll [1969] examined the effects of stock splits on stock prices. They argued that splits were related to more fundamental factors that affected prices. The importance of their paper stems from the development and use of the now standard event study methodology to test semistrong form efficiency. Brown and Warner [1980] compare and contrast various event study methodologies, including those discussed in Section 8.E. Selected Topics in Corporate Finance Generally, corporate finance is concerned with three types of decisions within the firm: the corporate investment decision, the corporate financing

12

Chapter 1

decision and the dividend decision (Van Home [1981]). The investment or capital budgeting decision concerns how the firm will use its financial resources. Dean [1951] recommended acceptance of capital budgeting projects whose internal rates of return (IRR) exceed market determined costs of capital. Lorie and Savage [1955] and Hershleifer [1958] criticized the IRR rules. They suggest the Net Present Value (NPV) rule as an alternative. Sections 2. A and 2.B in this book deals with these issues. Mason and Merton [1985] demonstrated how certain options embedded in capital budgeting projects can be evaluated. Major developments in cash management were contributed by Baumol [1952] (see Application 5.4) and Miller and Orr [1966]. These models generate cash balances under cases of certainty and uncertainty with respect to cash usage. Kim and Atkins [1978] modeled the accounts receivable decision as an investment, using the NPV technique. Beaver [1966] used a series of univariate tests of ratios to distinguish firms eventually filing for bankruptcy from those which did not. His tests were unable to make use of more than one ratio at a time. Altman [1968] extended this analysis, using the methodology of multi-discriminate analysis to forecast default on corporate debt issues. This paper had an enormous impact on the methods used by lending institutions and credit rating agencies in determining credit worthiness. However, the assumptions underlying the use of multi-discriminate analysis usually do not apply very closely. Furthermore, the numerical scoring system used by Altman has little intuitive meaning. Hence, other prediction models based on logit or probit analysis including Ohlson [1980] and Zavgren [1985] have been provided in the literature (See Section 8.F). John and John [1992] provided an extensive review of the literature in the more general area of financial distress and corporate restructuring. Research in the Profession Although most of the research in the financial profession is of a proprietary nature, there is much communication among researchers in the profession. For example, journals such as Risk are comprised of articles written by and for researchers in the derivatives and risk management professions. There are also substantial amounts of information exchanged between members of the profession and members of the academic communities. Among the journals and magazines specializing in applied financial research are The Financial Analysts Journal, Financial Management, The Journal of Investing, The Journal of Portfolio Management, The Review of Derivatives Research, The Journal of Applied Corporate Finance, The Midland Corporate Finance Journal, and The Journal of Financial Engineering. The journals Mathematical Finance and Applied Mathematical Finance specialize in applications of mathematics to finance. In addition, there are a number of useful books that focus on practitioner-oriented research, including Stern and Chew [1986] and Baxter and Rennie [1996],

Introduction and Overview

13

l.E: APPLICATIONS AND ORGANIZATION OF THIS BOOK The primary purpose of this text is to ensure that the reader obtains a reasonable degree of comfort and proficiency in applying mathematics to a variety of types of financial analysis. Chapter 2 provides a brief review of elementary mathematics of time value, return and risk. Specific applications often follow the description of the mathematical topic in this and in other chapters. A particularly large number of exercises is provided at the end of Chapter 2 for readers requiring substantial review. Chapter 3 discusses elementary portfolio return and risk measures along with a description of index models. The quantitative sophistication required for Chapters 2 and 3 does not extend beyond high school algebra. Chapter 4 delivers an introduction to matrix algebra along with a number of applications in finance. This chapter and Chapter 5 on differential calculus are probably the most important in the book. Chapter 5 also provides a large number of applications of differential calculus to finance. Chapter 6 discusses rudiments integral calculus, differential equations and a few simple applications. In large part, it is intended to set the stage for Chapters 7 and 9 on probability and stochastic processes. Chapters 7 and 8 provide a review of probability and statistics along with applications to analyzing risk, valuing securities and performing financial empirical studies. Event studies are emphasized in Chapter 8. Next, Chapter 9 discusses stochastic processes and continuous time mathematics. Particular emphasis is given to the analysis of options in both Chapters 9 and 10. Chapter 10 provides an introduction to some of the numerical methods most commonly used in finance. There are appendices at the ends of Chapters 5 and 6 and at the end of the text. Included in the endof-text appendices are detailed solutions to end-of-chapter exercises, statistics tables and a list of notation definitions. A glossary of terms follows the text appendices. This book is designed such that it will not be necessary for most readers to start at the beginning and read all the material prior to a given topic. Generally, reading the previous section (except for the first section in each chapter) will be sufficient background for the reader to comprehend any given section unless additional "Background Readings" are listed. Comprehending the section preceding an application should be sufficient to ensure understanding of that application, except, again, where additional "Background Readings" are listed. NOTES 1. For example, see Dewing [1920] which describes the life cycle of the firm. See also Weston [1981, 1994], Megginson [1996], Martin, Cox and MacMinn [1988], Elton and Gruber [1995] and Copeland and Weston [1988] who provide wide-ranging overviews of financial literature in the academic realm. Part of this literature review was based on these earlier reviews. 2. Exceptions to this are discussed in Section l.D.

14

Chapter 1

3. Normative models, proposing what "ought to be," are distinguished from positive models which intend to describe "what is." Academicians are often most interested in positive models to describe various financial phenomena. Members of the profession are often interested in both types of models.

SUGGESTED READINGS Weston [1981, 1994], Megginson [1996], Martin, Cox and MacMinn [1988], Elton and Gruber [1995] and Copeland and Weston [1988] all provide wide-ranging overviews of financial literature in the academic realm. Bernstein [1992] provides a very readable discussion of financial literature in the academic realm and discusses many applications for practitioners. The focus of Bernstein's book is on academics who generated important discoveries in finance. Merton [1995] has prepared an overview of mathematics usage in finance along with a discussion of the vital role of mathematics in financial analysis.

2

Preliminary Analytical Concepts

2.A: TIME VALUE MATHEMATICS Interest is a charge imposed on borrowers by lenders for the use of the lenders' money. The interest cost is usually determined as a percentage of the principal (the sum borrowed). Interest is computed on a simple basis if it is paid only on the principal of the loan. Compound interest accrues on accumulated loan interest as well as on the principal. Thus, if a sum of money (X0) were borrowed at an annual interest rate (i) and repaid at the end of n years with accumulated interest, the total sum repaid (FVn or future value at the end of year n) is determined as follows: (2.1) Interest is computed on a compound basis when a borrower must pay interest on not only the loan principal, but on accumulated interest as well. If interest must accumulate for a full year before it is compounded, the future value of such a loan is determined as follows: (2.2) This compound interest formula can be derived easily from the simple interest formula by adding accumulated interest to principal at the end of each year to form the basis of the subsequent year's computations: (A) If interest is to be compounded m times per year (or once every fractional 1/m part of a year), the future value of the loan is determined as follows:

(2.3)

16

Chapter 2

Many continuous time financial models allow for continuous compounding of interest. As m approaches infinity (m-»oo), the future value of a loan or investment can be defined as follows: (2.4) where e is the natural log whose value can be approximated at 2.718. (See Application 5.1 for more details on this derivation.) Cash flows realized at the present time have a greater value to investors than cash flows realized later. The purpose of the present value concept is to provide a means of expressing the value of a future cash flow in terms of current cash flows. That is, the present value concept is used to determine how much an investor would pay now for the expectation of some cash flow CFn to be received at a later date: (2.5) where PV is the present value of a single cash flow to be received at time n and k is an investor-determined discount rate accounting for risk, inflation and the investor's time value of money. This present value formula is easily derived from the compound interest formula by noting that X0 (principal) and PV are analogous, as are FV and CFn. The present value function is merely the inverse of the future value function. The continuously compounded version of Equation 2.5 is PV = CFne"kn. This variation is generally used when the analyst does not wish to arbitrarily select a compounding interval. In addition, this continuously compounding variation enables the analyst to continuously adjust for interest or returns not withdrawn from the asset being evaluated. If an investor wishes to evaluate a series of cash flows, he needs only to discount each separately and then sum the present values of each of the cash flows: (2.6) Consider a cash flow series where the cash flows were expected to grow at a constant annual rate of g. The amount of the cash flow generated by that investment in year t (CFt) reflecting t—1 years of growth would be: (2.7) where CF! is the cash flow generated by the investment in year one.

Preliminary Analytical Concepts

17

2.B: GEOMETRIC SERIES AND EXPANSIONS A geometric expansion is an algebraic procedure used to simplify a geometric series. Suppose we wished to solve the following finite geometric series for S: (A) where c is a constant or parameter and x is called a quotient. If n is large, computations may be time consuming and repetitive. Simplifying the series may save substantial amounts of computation time. Essentially, the geometric expansion is a two-step process: 1. First, multiply both sides of the equation by the quotient: (B) 2. Second, to eliminate repetitive terms, subtract the above product from the original equation and simplify: (C)

(D) for x * 1. For example, if x were to equal (1 + i), the following two equations would be equal: (E)

(F)

Thus, for any geometric series where x ^ 1, the following summation formula holds: (2.8) Such geometric series and expansions are very useful in time value mathematics and problems involving series of probabilities.

18

Chapter 2

APPLICATION 2.1: ANNUITIES AND PERPETUITIES (Background reading: Section 2.A) An annuity is defined as a series of identical payments made at equal intervals. If payments are to be made into an interest bearing account, the future value of the account will be a function of interest accumulating on deposits as well as the deposits themselves. The future value annuity factor may be derived through the use of the geometric expansion. Consider the case where we wish to determine the future value of an account based on a payment of X made at the end of each year (t) for n years where the account pays an annual interest rate equal to i: (A) Thus, the payment made at the end of the first year accumulates interest for a total of ( n - 1 ) years, the payment at the end of the second year accumulates interest for ( n - 2 ) years and so on. The first step in the geometric expansion is to multiply both sides of Equation A by (1 +i): (B) Then we subtract Equation A from Equation B to obtain: (C) and rearrange to obtain: (D) which simplifies to: (2.9) A similar procedure is used to arrive at a formula for finding the present value of an annuity: (A) (B) (C) (D)

Preliminary Analytical Concepts

19

which simplifies to the following: (2.10) As the value of n approaches infinity in the annuity formula, the value of the right-hand-side term in the brackets l/(k(l +k)n) approaches zero. Thus, the present value of a perpetual annuity, or perpetuity is determined as follows: (2.11)

APPLICATION 2.2: GROWTH MODELS Suppose that we wished to value a cash flow series where the cash flow each year is expected to have grown at rate g over the prior year's cash flow. Thus, the cash flow in any year t (CFt) is C F ^ l +g). We can derive a present value growing annuity model as follows: (A)

(B)

(O (D)

(E)

(F) which simplifies to the following Present Value Growing Annuity formula: (2.12)

20

Chapter 2

When k> g, the Present Value Growing Annuity formula can be used to derive the Present Value Growing Perpetuity formula by allowing n to approach infinity: (2.13) When applied to stocks, this model is often referred to as the Gordon Stock Pricing Model. APPLICATION 2.3: MONEY AND INCOME MULTIPLIERS Suppose that the central bank of a country issues a fixed amount of currency K to the public and permits commercial banks to loan funds left by the public in the form of demand deposits of amount DD. The public obtains the currency and deposits it with the commercial banking system. Further, suppose that the central bank requires that commercial banks hold on reserve a proportion r of their demand deposits; that is, all commercial banks must leave with the central bank nonloanable reserves totaling r • DD. Whenever funds are loaned by a commercial bank, they are spent by the borrower. The borrower purchases goods from a seller; the seller then deposits its receipts into the commercial banking system, creating more funds available to loan. However, each deposit requires that the commercial bank increase its reserve left with the central bank by the proportion r: (A) Here, K is the currency originally issued by the central bank to the public and deposited in the commercial banking system. The amount rK is used to meet the reserve requirement while (l-r)K is loaned to the public, then redeposited into the commercial banking system. Of the (l-r)K redeposited into the banking system, (l-r)(l-r)K is available to loan after the reserve requirements are fulfilled on the second deposit. This process continues into perpetuity. Based on the currency issued by the central bank and its reserve requirement, what is the total money supply for this economy? We can determine total money supply through the following geometric expansion: (B) (C) (D) (2.14)

Preliminary Analytical Concepts

21

where we assume that K is positive and 0 < r < 1. Thus, the money multiplier here equals K/r. A central bank issuing $100 in currency with a reserve requirement equal to 10% will have a total money supply equal to $1,000. A similar sort of multiplier exists in the relationship between consumer autonomous consumption (consumption expenditures independent of income) and total income. Suppose that the following depicts the relationships among income Y, autonomous consumption TJ and income-dependent consumption cY: (A) If autonomous consumption were to increase by a given amount, this would increase income, resulting in an increase in income-dependent consumption. This would further increase income and consumption, and the process would replicate itself perpetually: (B) We can derive an income multiplier to determine the full amount of the change in income resulting from a change in autonomous consumption: (C) (D) (2.15) Thus, the income multiplier equals c/(l-c) = c/s where s represents the proportion of marginal income saved by individuals. 2.C: RETURN MEASUREMENT The purpose of measuring investment returns is simply to determine the economic efficiency of an investment. Thus, an investment's return expresses the cash flows generated by an initial cash outlay relative to the amount of that outlay. There exist a number of methods for determining the return of an investment. One can compute a holding period return on investment (ROIH) as follows: (2.16)

22

Chapter 2

where CFt represents the cash flow paid by the investment in time t and P0 is the initial investment outlay. One may standardize this holding period return by annualizing it as follows: (2.17) Although its concept is quite simple, this arithmetic mean return does not account for the timing of cash flows nor the compounding of investment profits. An alternative average return is the geometric mean return computed as follows: (2.18) where rt is the return on investment for a single period t. Another return measure which more appropriately accounts for the timing of investment cash flows is internal rate of return (IRR), which is that value for r which solves the following: (2.19) However, one should note that internal rate of return can be more difficult to compute than the other return measures. In addition, there may be multiple values for r (multiple IRRs) which satisfy equation 2.19 and no rule which consistently tells us which is appropriate. This may occur when there are negative cash flows following positive cash flows. One important variation of internal rate of return is the yield to maturity for a bond: (2.20) where P0 is the bond's purchase price, F its face value, y its yield to maturity (IRR) and INT its annual interest payments. One obtains yield to maturity by solving this equation for y. While this expression is appropriate for bonds making annual interest payments, the following can be used for bonds making semiannual interest payments (INT -r 2): (2.21)

Preliminary Analytical Concepts

23

2.D: MEAN, VARIANCE AND STANDARD DEVIATION The purpose of this and the following two sections is to introduce the reader to several important, though elementary concepts from probability and statistics. These concepts are applied to the measurement of risk in applications following these sections, then defined and discussed more rigorously in Chapters 6 and 7. Suppose we wish to describe or summarize the characteristics or distribution of a single population of values (or sample drawn from a population). Important characteristics include central location (measured by average, mean, median, expected value or mode), dispersion (measured by range, variance or standard deviation), asymmetry (measured by skewness) and clustering of data about the mean and extrema (measured by kurtosis). In many instances, we will be most interested in the typical value (if it exists) drawn from a population or sample; that is, we are interested in the "location" of the data set. Mean (often referred to as average) or expected values (sometimes referred to as weighted average) are frequently used as measures of location (or central tendency) because they account for all relevant data points and the frequency with which they occur. The arithmetic mean value of a population /i is computed by adding the values Xj associated with each observation i and dividing the result by the number of observations n in the population: (2.22) Future events whose actual outcomes are not certain may have associated with them numerous potential outcomes. Some of these potential outcomes may be more likely to be realized than others. The more likely outcomes are said to have higher probabilities associated with them. Probabilities are analogous to frequencies as a proportion of a population and may be measured as percentages summing to 100%. The expected value of a population E[x] is computed as a weighted average of the potential rates x h where probabilities Pj serve as weights:

(2-23) Other measures of location include median and mode. If we were to rank values in a data set from highest to lowest, that value with the middle rank would be regarded as the median value. Usually, ties are averaged. The value occurring with the highest frequency in a data set is referred to as the mode. Variance is a measure of the dispersion (variability and sometimes volatility or uncertainty) of values within a data set. In a finance setting, variance is also used as an indicator of risk. Variance is defined as the mean of squared deviations of actual data points from the mean or expected value of a data set.

24

Chapter 2

Deviations are squared to ensure that negative deviations do not cancel positive deviations, resulting in zero variances. High variances imply high dispersion of data. This indicates that certain or perhaps many data points are significantly different from mean or expected values. Population, sample and expected variances are computed as follows: (2.24)

(2.25)

(2.26) Standard deviation is simply the square root of variance. It is also used as a measure of dispersion, risk or uncertainty. Standard deviation is sometimes easier to interpret than variance because its value is expressed in terms of the same units as the data points themselves rather than their squared values. High standard deviations as high variances imply high dispersion of data. Standard deviations are computed as follows: (2.27)

(2.28)

(2.29)

APPLICATION 2.4: RISK MEASUREMENT When an individual orfirminvests, it subjects itself to uncertainty regarding the amounts and timing of future cash flows. Expected return is defined and used as a return forecast in this section. Expected return is expressed as a function of the investment's potential return outcomes and associated

Preliminary Analytical Concepts

25

probabilities. The riskiness of an investment is simply the potential for deviation from the investment's expected return. Thus, the risk of an investment is defined here as the uncertainty associated with returns on that investment. Expected return is defined mathematically as a function of returns Rj resulting from any one of n potential outcomes i with probability P^ (2.30) The statistical concept of variance is an indicator of uncertainty associated with the investment. It accounts for all potential outcomes and associated probabilities: (2.31) Unfortunately, in many real-world scenarios, it is very difficult to properly assign probabilities to potential outcomes. However, if we are able to claim that historical volatility indicates future variance (or, similarly, volatility or uncertainty is constant over time), we can use historical variance as our indicator of future uncertainty: (2.32) where "R" is the mean return over the n year sampling period. Standard deviation a is simply the square root of variance. It has the convenient property of being expressed in the same units of measurement as the mean. There are two primary difficulties associated with the traditional historical sample estimator procedure for variance, time required for computation and the arbitrary selection of returns from which to compute volatilities (returns based on prices from end of day, week, quarter, etc.). The time required for computation may be quite large when the sample selected must be large enough for statistical significance (60 monthly returns is a commonly used data set for variance computations). Extreme value indicators (based on security high and low prices) such as that derived by Parkinson [1980] can be very useful for reducing the amount of data required for statistically significant standard deviation estimates: (2.33) where HI designates the stock's high price for a given period and LO designates the low price over the same period. This estimation procedure is based on the assumption that underlying stock returns are log-normally distributed with zero drift and constant variance over time. Garman and Klass [1980] and Ball and

26

Chapter 2

Torous [1984] provide more efficient extreme value estimators using opening and closing prices while Rogers and Satchell [1991]) and Kunitomo [1992] provide drift adjusted (nonzero average return) models. A problem shared by both the traditional sample estimating procedures and the extreme value estimators is that they require the assumption of stable variance estimates over time; that is, historical variances equal future variances. A third procedure, first suggested by Latane and Rendleman [1976], is based on market prices of options, which may be used to imply variance or volatility estimates. For example, the Black-Scholes Option Pricing Model and its extensions provide an excellent means to estimate underlying stock variances if call prices are known. Essentially, this procedure determines market estimates for underlying stock variance based on known market prices for options on the underlying securities (see Chapter 10). Brenner and Subrahmanyam [1988] provide a simple formula to estimate an implied standard deviation (or variance) from the value c0 of a call option whose striking price equals the current market price S0 of the underlying asset:

(2.34) where T is the number of time periods prior to the expiration of the option. As the market price differs more from the option striking price, the estimation accuracy of this formula will worsen. 2.E: COMOVEMENT STATISTICS A joint probability distribution is concerned with probabilities associated with each possible combination of outcomes drawn from two sets of data. Covariance measures the mutual variability of outcomes selected from each set; that is, covariance measures the relationship between variability in one data set relative to variability in the second data set, where variables are selected one at a time from each data set and paired. If large values in one data set seem to be associated with large values in the second data set, covariance is positive; if large values in the first data set seem to be associated with small values in the second data set, covariance is negative. If data sets are unrelated, covariance is zero. Covariance between data set x and data set y may be measured as follows, depending on whether one is interested in covariance of a population, of a sample or expected covariance:

(2.35)

Preliminary Analytical Concepts

27

(2.36)

(2.37) The sign associated with covariance indicates whether the relationship associated with the data in the sets are direct (positive sign), inverse (negative sign) or independent (covariance is zero). The absolute value of covariance measures the strength of the relationship between the two data sets. However, the absolute value of covariance is more easily interpreted when it is expressed relative to the standard deviations of each of the two data sets. That is, when we divide covariance by the product of the standard deviations of each of the data sets, we obtain the coefficient of correlation pxy as follows:

(2.38)

A correlation coefficient equal to 1 indicates that the two data sets are perfectly positively correlated; that is, their changes are always in the same direction, by the same proportions, with 100% consistency. Correlation coefficients will always range between - 1 and +1. A correlation coefficient of - 1 indicates that the two data sets are perfectly inversely correlated; that is, their changes are always in the opposite direction, by the same proportions with 100% consistency. The closer a correlation coefficient is to —1 or +1, the stronger is the relationship between the two data sets. A correlation coefficient equal to zero implies independence (no relationship) between the two sets of data. The correlation coefficient may be squared to obtain the coefficient of determination (also referred to as r2 in some statistics texts and here as p2). The coefficient of determination is the proportion of variability in one data set that is explained by or associated with variability in the second data set. For example, p2 equal to .35 indicates that 35% of the variability in one data set is explained in a statistical sense by variability in the second data set. APPLICATION 2.5: SECURITY COMOVEMENT Standard deviation and variance provide us with measures of the absolute risk levels of securities; such absolute measures provide potential for deviation from the variable expected value. However, in many instances, it is useful to

28

Chapter 2

measure the risk of one security relative to the risk of another or relative to the market as a whole or to an index. The concept of covariance is integral to the development of relative risk measures. Covariance (COV [Rj,Rj] or au) provides us with a measure of the relationship between the returns of two securities. That is, given that two securities returns are likely to vary, covariance indicates whether they will vary in the same direction or in opposite directions. The likelihood that two securities will covary similarly (or, more accurately, the strength of the relationship between returns on two securities) is measured by Equation 2.39: (2.39) where Rki and Rjj are the return of stocks k and j if outcome i is realized and Pj is the probability of outcome i. E[Rk] and E[Rj] are simply the expected returns of securities k and j. The concept of covariance is also crucial to the • development of models of diversification and portfolio risk (see Chapter 3). Historical covariance can be used to measure security comovement or relative risk if one is willing to assume that historical comovement indicates future comovement: (2.40) The coefficient of correlation provides us with a means of standardizing the covariance between returns on two securities. For example, how large must covariance be to indicate a strong relationship between returns? Covariance will be smaller given low returns on the two securities than given high security returns. The coefficient of correlation pkj between returns on two securities will always fall between —1 and +1.1 If security returns are directly related, the correlation coefficient will be positive. If the two security returns always covary in the same direction by the same proportions, the coefficient of correlation will equal one. If the two security returns always covary in opposite directions by the same proportions, pk>j will equal negative one. The stronger the inverse relationship between returns on the two securities, the closer pkJ will be to negative one. If p kj equals zero, there is no relationship between returns on the two securities. The coefficient of correlation pkJ between returns is simply the covariance between returns on the two securities divided by the product of their standard deviations: (2.41)

Preliminary Analytical Concepts

29

2.F: INTRODUCTION TO SIMPLE OLS REGRESSIONS Regressions are used to determine relationships between a dependent variable and one or more independent variables. A simple regression is concerned with the relationship between a dependent variable and a single independent variable; a multiple regression is concerned with the relationship between a dependent variable and a series of independent variables. A linear regression is used to describe the relationship between the dependent and independent variable(s) to a linear function or line (or hyperplane in the case of a multiple regression). The simple Ordinary Least Squares regression (simple OLS) takes the following form: (2.42) The ordinary least squares regression coefficients a and b are derived by minimizing the variance of errors in fitting the curve (or m dimensional surface for multiple regressions involving m variables). Since the expected value of error terms equals zero, this derivation is identical to minimizing error terms squared (see the OLS derivation in Application 5.8). Regression coefficient bx is simply the covariance between y and x divided by the variance of x; bj and b0 are found as follows:

(2.43) (2.44) Appropriate use of the OLS requires the following assumptions: 1. Dependent variable values are distributed independently of one another. 2. The variance of x is approximately the same over all ranges for x. 3. The variance of error term values is approximately the same over all ranges of x. 4. The expected value of each disturbance or error term equals zero. Violations in these assumptions will weaken the validity of the results obtained from the regression and may necessitate either modifications to the OLS regression or different statistical testing techniques. The derivation of the OLS model are discussed in Sections 5.8 and 5.9 and numerous applications will be discussed in Chapters 3 and 8.

30

Chapter 2

APPLICATION 2.6: RELATIVE RISK MEASUREMENT A portfolio is simply a collection of investments. The market portfolio is the collection of all investments that are available to investors. That is, the market portfolio represents the combination or aggregation of all securities (or other assets) that are available for purchase. Investors may wish to consider the performance of this market portfolio to determine the performance of securities in general. Thus, the return on the market portfolio is representative of the return on the "typical" asset. An investor may wish to know the market portfolio return to determine the performance of a particular security or his entire investment portfolio relative to the performance of the market or a "typical" security. Determination of the return on the market portfolio requires the calculation of returns on all of the assets available to investors. Because there are hundreds of thousands of assets available to investors (including stocks, bonds, options, bank accounts, real estate, etc.), determining the exact return of the market portfolio may be impossible. Thus, investors generally make use of indices such as the Dow Jones Industrial Average or the Standard and Poor's 500 to gauge the performance of the market portfolio. These indices merely act as surrogates for the market portfolio; we assume that if the indices are increasing, then the market portfolio is performing well. For example, performance of the Dow Jones Industrials Average depends on the performance of the thirty stocks that comprise this index. Thus, if the Dow Jones market index is performing well, the thirty securities, on average are probably performing well. This strong performance may imply that the market portfolio is performing well. In any case, it is easier to measure the performance of thirty or five hundred stocks (for the Standard and Poor's 500) than it is to measure the performance of all of the securities that comprise the market portfolio. Beta measures the risk of a given security relative to the risk of the market portfolio of all investments. Beta is determined by Equations 2.45 and 2.46:

(2.45) (2.46) Beta may also be described as the slope of an Ordinary Least Squares regression line fit to data points comprising returns on Security i versus returns on some index such as one representing the market portfolio: (2.47) where c^ is the vertical intercept of this regression. Again, ft is computed based on Equations 2.45 and 2.46. The vertical intercept a{ of the regression line is

Preliminary Analytical Concepts

31

simply E[Rj J - ftE[IJ. The slope term ft may be interpreted as the change in the return of the security induced by a change in the index; ft is the risk of the asset relative to the risk of the market. The term e-ltl might be interpreted as the security's return associated with firm-specific factors and unrelated to It. The concepts of Beta, relative risk models and index models are discussed in much greater detail in Chapters 3, 5 and 8. NOTE 1. Many statistics textbooks use the notation (ri(j) to designate the correlation coefficient between variables (i) and (j)- Because the letter (r) is used in this text to designate return, we will use the lower case rho (py) to designate correlation coefficient. SUGGESTED READINGS Many of the topics in this chapter, including time value of money, return, risk and comovement statistics are discussed in Brealey and Myers [1996]. An even more elementary presentation of these topics is provided by Brealey, Myers and Marcus [1995]. The presentation in Brealey, Myers and Marcus is suitable as background reading for this chapter. Brown and Kritzman [1990] also discuss many of these topics, including the Parkinson extreme value variance estimator. Mayer, Duesenberry and Aliber [1987] and other texts in money and banking and in macroeconomics discuss money multipliers and their derivations. The textbook by Ben-Horim and Levy [1984] provides an excellent introductory presentation of statistics with numerous applications to finance.

32

Chapter 2

EXERCISES 2.1 The Doda Company has borrowed $10,500 at an annual interest rate of 9%. How much will be a single lump-sum repayment in eight years, including both principal and interest, if interest is computed on a simple basis; that is, what is the future value of this loan? 2.2 What would be the lump-sum loan repayment made by the Doda Company in Problem 2.1 if interest were compounded a. annually? b. semi-annually? c. monthly? d. daily? e. continuously? 2.3 Assume that you are advising a twenty-three-year-old client with respect to personal financial planning. Your client wishes to save, become a millionaire, and then retire. Your client intends to open and contribute to a tax deferred Individual Retirement Account each year until he retires with $1,000,000 in that account. a. If your client were to deposit $2,000 at the end of each year into his I.R.A., how many years must he wait until he retires with his $1,000,000? Assume that the account will pay interest at an annual rate of 10%, compounded annually. b. What would your answer to a. be if the interest rate were 12%? c. What would the client's annual payment have to be if he wished to retire at the age of forty with $1,000,000? Assume that the client will make deposits at the end of each year for 17 years at an annual interest rate of 10% and that his I.R.A. will be supplemented with another type of retirement account known as a 401(k), so that his total annual tax deferred deposits can exceed $2,000. d. What would your answer to c. be if your client were willing to wait until he is fifty to retire? e. What would your answer to d. be if your client were able to make deposits into an account paying interest at an annual rate of 12%? f. What would your answers to a., c. and d. be in the annual interest rate were only 4%? g. If the annual inflation rate for the next fifty years were expected to be 3%, what would be the purchase power of $1,000,000 in 17 years? In 27 years? h. What would be your answers to g. be if the inflation rate were expected to equal 9%? 2.4 The Starr Company has the opportunity to pay $10,000 for an investment paying $2,000 in each of the next nine years. Would this be a wise investment if the appropriate discount rate were a. 5%? b. 10%? c. 20%? 2.5 An investor has the opportunity to purchase for $4,900 an investment which will pay $1,000 at the end of six months, $1,100 at the end of one year, $1,210 at the end of

Preliminary Analytical Concepts

33

eighteen months, $1,331 at the end of two years, and $1,464.10 at the end of thirty months. Assuming that the investor discounts all of his cash flows at an annual rate of 20%, should he purchase this investment? Why or why not? 2.6 The Tray nor Company is selling preferred stock which is expected to pay a $50 annual dividend per share. What is the present value of dividends associated with each share of stock if the appropriate discount rate were 8% and its life expectancy were infinite? 2.7 The Lajoie Company is considering the purchase of a machine whose output will result in a $10,000 cash flow next year. This cash flow is projected to grow at the annual 10% rate of inflation over each of the next 10 years. What will be the cash flow generated by this machine in a. its second year of operation? b. its third year of operation? c. its fifth year of operation? d. its tenth year of operation? 2.8 What would be the present value of a fifty-year annuity whose first cash flow of $5,000 is paid in 10 years and whose final (fiftieth) cash flow is paid in 59 years? Assume that the appropriate discount rate is 12% for all cash flows. 2.9 An employee expects to make a deposit of $1,000 into his pension fund account in one year, with additional deposits to follow for a total of 40 years when he retires. The amount to be deposited in each year will be 5% larger than in the prior year (e.g., $1,050 deposited in the second year, $1,102.50 in the third year, etc.). Furthermore, the retirement account will accrue interest on accumulated deposits at an annual rate of 8%, compounded annually. What will be the terminal (future) value of the account at the end of the 40-year period? Show how to derive a computationally efficient expression to solve this problem. 2.10 The Chesbro Company is considering the purchase of an investment for $100,000 that is expected to pay off $50,000 in 2 years, $75,000 in 4 years and $75,000 in 6 years. In the third year, Chesbro must make an additional payment of $50,000 to sustain the investment. Calculate the following for the Chesbro investment: a. return on investment using an arithmetic mean return b. the investment internal rate of return c. describe any complications you encountered in part b 2.11 A $1,000 face value bond is currently selling at a premium for $1,200. The coupon rate of this bond is 12% and it matures in 3 years. Calculate the following for this bond assuming its interest payments are made annually: a. its annual interest payments b. its current yield c. its yield to maturity 2.12 Work through each of the calculations in Problem 2.11 assuming interest payments are made semiannually.

34

Chapter 2

2.13 The Galvin Company invested $100,000 into a small business 20 years ago. Its investment generated a cash flow equal to $3,000 in its first year of operation. Each subsequent year, the business generated a cash flow which was 10% larger than in the prior year; that is, the business generated a cash flow equal to $3,300 in the second year, $3,630 in the third year, and so on for 19 years after the first. The Galvin Company sold the business for $500,000 after its twentieth year of operation. What was the internal rate of return for this investment? 2.14 Plantaganet Products management is considering the investment in one of two projects available to the company. The returns on the two projects A and B are dependent on the sales outcome of the company. Plantaganet management has determined three potential sales outcomes for the company. The highest potential sales outcome for Plantaganet is outcome 1, or $800,000. If this sales outcome were realized, Project A would realize a return outcome of 30%; Project B would realize a return of 20%. If outcome 2 were realized, the company's sales level would be $500,000. In this case, project A would yield 15%, and Project B would yield 13%. The worst outcome, 3, will result in a sales level of $400,000, and return levels for Projects A and B of 1 % and 9%, respectively. If each sales outcome has an equal probability of occurring, determine the following for the Plantaganet Company: a. the probabilities of outcomes 1, 2 and 3 b. its expected sales level c. the variance associated with potential sales levels d. the expected return of Project A e. the variance of potential returns for Project A f. the expected return and variance for Project B g. standard deviations associated with company sales, returns on Project A and returns on Project B h. the covariance between company sales and returns on Project A i. the coefficient of correlation between company sales and returns on Project A j. the coefficient of correlation between company sales and returns on Project B k. the coefficient of determination between company sales and returns on Project B 2.15 Which of the projects in Problem 2.14 represents the better investment for Plantaganet Products? 2.16 Historical percentage returns for the Lancaster and York Companies are listed in the following chart along with percentage returns on the market portfolio: Year 1988 1989 1990 1991 1992

Lancaster York Market

4 7 11 4 5

19 4 -4 21 13

15 10 3 12 9

Calculate the following based on the preceding diagram: a. mean historical returns for the two companies and the market portfolio b. variances associated with Lancaster Company returns and York Company returns

Preliminary Analytical Concepts

35

as well as returns on the market portfolio c. the historical covariance and coefficient of correlation between returns of the two securities d. the historical covariance and coefficient of correlation between returns of the Lancaster Company and returns on the market portfolio e. the historical covariance and coefficient of correlation between returns of the York Company and returns on the market portfolio 2.17 Forecast the following for both the Lancaster and York Companies based on your calculations in Problem 2.16. a. variance and standard deviation of returns b. coefficient of correlation between each company's returns and returns on the market portfolio 2.18 Stock A will generate a return of 10% if and only if Stock B yields a return of 15%; Stock B will generate a return of 10% if and only if Stock A yields a return of 20%. There is a 50% probability that Stock A will generate a return of 10% and a 50% probability that it will yield 20%. a. What is the standard deviation of returns for Stock A? b. What is the covariance of returns between Stocks A and B? 2.19 Under what circumstances can the coefficient of determination between returns on two securities be negative? How would you interpret a negative coefficient of determination? If there are no circumstances where the coefficient of determination can be negative, describe why. 2.20 The following daily prices were collected for each of three stocks over a twelve day period. CORP. Z CORP. Y CORP. X DATE PRICE DATE PRICE PRICE DATE 60.375 1/09 20.000 1/09 50.125 1/09 20.000 1/10 50.125 1/10 60.500 1/10 60.250 1/11 1/11 20.125 50.250 1/11 1/12 1/12 1/12 60.125 20.250 50.250 60.000 20.375 1/13 1/13 1/13 50.375 1/14 60.125 1/14 20.375 50.250 1/14 62.625 1/15 1/15 21.375 52.250 1/15 60.750 1/16 1/16 21.250 52.375 1/16 1/17 1/17 60.750 21.375 52.250 1/17 60.875 1/18 21.500 52.375 1/18 1/18 60.875 52.500 1/19 1/19 21.375 1/19 60.875 1/20 1/20 21.500 52.375 1/20 Based on the data given above, calculate the following: a. returns for each day on each of the three stocks. There should be a total of eleven returns for each stock, beginning with the date 1/10 b. average daily returns for each of the three stocks c. daily return standard deviations for each of the three stocks

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3

Elementary Portfolio Mathematics

3.A: INTRODUCTION TO PORTFOLIO ANALYSIS (Background reading: Sections 2.C, 2.D and 2.E) A portfolio is simply a collection of investments held by an investor. It may be reasonable to be concerned with the performance of individual securities only to the extent that their performance affects overall portfolio performance. Thus, the performance of the portfolio is of primary importance. The return of an investor's portfolio is simply a weighted average of the returns of the individual securities that within it. The expected return of a portfolio may be calculated either as a function of potential portfolio returns and their associated probabilities (as computed in earlier sections) or as a simple weighted average of the expected individual security returns. Generally, the portfolio variance or standard deviation of returns will be less than a weighted average of the individual security variances or standard deviations. Portfolio Return The expected return of a portfolio may be calculated using Equation 3.1 where the subscript p designates the portfolio and the subscript j designates a particular outcome out of m potential outcomes: (3.1)

For many portfolio management applications, it is useful to express portfolio return as a function of the returns of the individual securities that comprise the portfolio: (3.2) The subscript i designates a particular security, and weights ws are the

38

Chapter 3

portfolio proportions. That is, a security weight w, specifies how much money is invested in security i relative to the total amount invested in the entire portfolio. For example, Wj is: $ invested in security i Total $ invested in portfolio p Thus, portfolio return is simply a weighted average of individual security returns. Portfolio Risk We can also define portfolio return variance as a function of potential portfolio returns and associated probabilities: .

(3-3)

m

a2p= S

(Rpj-E[Rp])2Pj

j = 1

It is important to note that the variance of portfolio returns usually is not simply a weighted average of individual security variances. In fact, in some instances, we can combine a series of highly risky assets into a relatively safe portfolio. The risk of a portfolio in terms of variance of returns can be determined by solving the following double summation: (3.1)

This expression can be rewritten as follows: (3.4a) When a portfolio is comprised of only two securities, its variance can be determined by Equation (3.5): (3.5) Larger portfolios require the use of Equation 3.4 or a variant of Equation 3.5 accounting for all products of security weights and standard deviations squared and all possible combinations of pairwise security covariances and weight products. For example, the three-security variation of Equation 3.5 is: (3.5a)

The implication of the covariance terms in Equations 3.4 through 3.5a is that security risk can be diversified away by combining individual securities into

Elementary Portfolio Mathematics

39

portfolios. Thus, the old stock market adage Don't put all your eggs in one basket really can be validated mathematically. Spreading investments across a variety of securities does result in portfolio risk that is lower than the weighted average risks of the individual securities. This diversification is most effective when the returns of the individual securities are at least somewhat unrelated; that is, lower covariances aU} result in lower portfolio risk. Similarly, the reduction of portfolio risk is dependent on the correlation coefficient of returns py between securities included in the portfolio. Since the covariance between security returns jy equals the product oxofix^ covariance will reflect the correlation coefficient. Thus, the lower the correlation coefficients between these securities, the lower is the resultant portfolio risk. In fact, as long as p^ is less than one, which, realistically is always the case between nonidentical securities, some reduction in risk can be realized from diversification. To derive the variance of portfolio p as a function of security variances, covariances and weights as in Equation 3.4, we begin with our standard variance expression as a function of m potential portfolio return outcomes j and associated probabilities as in Equation 3.3: (3.3) For the sake of simplicity, let the number of securities n in our portfolio equal two. From our portfolio return expression, we may compute portfolio variance as follows: (A) Next, we complete the square for Equation A and combine terms multiplied by the two weights to obtain: (B) Next, we bring the summation term inside the brackets: m

(O

40

Chapter 3

We complete our derivation by noting our definitions from Chapter 2 for variances and covariances as follows: (3.5) which is a special case of Equation 3.4. Similar derivations can be performed for portfolios comprising more than two securities. One of the most important problems in portfolio management concerns the selection of security weights that minimize portfolio risk at a given return level. For example, suppose that a security analyst has provided a portfolio manager with estimates concerning security expected returns, variances and covariances. The portfolio manager must determine how much to invest in each of the securities, subject to various constraints. This problem is dealt with extensively in Sections 5.7, 5.10 and 5.13 through 5.15. 3.B: SINGLE INDEX MODELS (Background reading: Sections 2.F and 3.A) Simple observation of security markets reveals a strong tendency for security returns to be affected by common factors, particularly the market portfolio. From a mathematical perspective, these factors represent a source of covariance or correlation between returns of pairs of securities. The single index model specifies a single source of covariance among security returns Rit, and denotes security returns as a linear function of this factor or index It: (3-6) where otx represents that portion of the return of security i which is constant and independent of the index It, ft represents the sensitivity of security i to index I and £it represents the portion of security i's return, which is security specific and unrelated to the index or to returns of other securities. The index models are simply regression models that presume that security returns are a linear function of one or more (in the case of multi-index models) indices. If index models can be used to generate security returns, then the process for obtaining security variances and covariances with respect to one another will be much simplified. The Single Index Model has several uses: 1. To reduce the number of inputs and computations required for portfolio analysis. In particular, the Single Index Model is useful for deriving forecasts for security and portfolio expected return, variance and covariance. 2. To build and apply equilibrium models (see Chapters 4 and 5) such as the Capital Asset Pricing Model and Arbitrage Pricing Theory. 3. To adjust for risk in event studies and back-testing programs (see Chapter 8).

Elementary Portfolio Mathematics

41

The Single Index Model is based on the following series of assumptions: 1. Security returns are linear in a common index as follows: 2. The parameters of the index model, a{ and ft, are computed through a linear regression procedure such that the risk premium is purely a function of the index, not security-specific risk. That is, E(eit) = 0. Furthermore, it is assumed that security specific risk is unrelated to the value of the index; that is, E(eit-It) = 0 = Cov(eit-It). 3. The index represents the only source of covariance between asset returns. That is, E(eit-ejt) = 0. Based on the Single Index Model, we may reflect the expected return of a security i or portfolio p as follows: (3.7)

(3.7a) where the parameters for the portfolio are simply a weighted average of the parameters for the individual securities. For sake of notational convenience, we use the expectations operator E[-] to replace the summation notation EJ = 1 [-]PJ; that is, for expected security return and variance, we have:

v (B) We can use Equations 3.6 and 3.7 and our standard definition for security variance to express security variance as a function of the index:

v We can complete the square of Equation A and write security variance as: (B) Because the covariance between the index and firm specific returns is assumed to be zero above (E[(eit-0)(I-E[I])] = 0), the cross product terms drop out: (C)

42

Chapter 3

Due to our definition of variance and the expected unsystematic risk premium (error) equalling zero, Equation C simplifies to:

(3.8) This expression has a particularly useful intuition: security variance is the sum of systematic or index induced variance $2{o\ and firm specific variance a2£i. Firm specific risk a2ei tends toward zero in a well-diversified portfolio, such that portfolio variance is expressed: (3.8a)

The Single Index Model can be used to substantially reduce the number of computations for covariances required for portfolio risk analysis. We see from Equation 4 that n2 covariance calculations are required to compute portfolio risk.1 For example, a ninety security portfolio will require 8100 covariance calculations. Thus, it is very useful to limit the number of calculations required for each covariance. Using the expectations operator notation, we can define covariance as follows: (3.9) Replacing Equations 3.6 and 3.7 into Equation 3.9, we have: (A) After performing multiplications within the brackets, and noting that ei and ej are

uncorrelated with the index such that the cross product terms drop out, Equation A simplifies to: (B) We bring the expectations operator inside the brackets to obtain: (C)

Since (e^) equals zero by our assumption above that the index captures all sources of covariance between pairs of securities, Equation C simplifies to: (3.10)

If our covariance calculations were to be based on 60 months of time series returns, we would compute a single beta value for each of n securities in a portfolio and a variance for the index itself. Thus, we could compute all (n2-n)-r-2 covariances from n betas and one variance. When n is large, the time to complete these computations will be substantially less than the time to compute (n 2 -n)/2 covariances from 60 months of raw returns data. In most cases, the single index model relies on an index representing market

Elementary Portfolio Mathematics

43

returns. The most frequently used index for academic studies is the S&P 500, but other indices such as those provided by the exchanges, Value Line and Russell may be used as well. Historical betas are most frequently estimated on the basis of covariances and variance drawn from sixty months of historical security returns. However, historical returns and their volatility are not necessarily the best indicators of future betas. Corporate circumstances change over time as does the market's evaluation of those circumstances. Furthermore, any historical beta estimate would be subject to sampling and measurement error. Blume [1975] has shown a tendency for betas to drift toward 1 over time. He proposed a correction for this tendency: (3.11)

where ft F is the forecasted beta for a future five-year period and ft H is the historical beta estimated using the procedure described above. The coefficients 70 and y] are determined by performing a regression of five-year betas against betas estimated over the immediately preceding five-year period. For example, the beta estimates ft F for the period 1955-1961 based on beta estimates for 1948-1954 ft H were obtained from adjustment coefficients 70 = .343 and y{ = .677. Note that the coefficients will normally sum to approximately one. Other adjustment procedures exist as well, including that proposed by Vasicek [1973]. Beaver, Kettler and Scholes [1970] and numerous papers authored by Barr Rosenberg, including Rosenberg and James [1976], have proposed estimating betas from firm fundamental factors including ratios. The advantage to this methodology is that the "fundamental beta" is not based on historical returns data but on current financial statement data supplemented with other current and relevant information. The fundamental beta forecast ft F is determined as a function of m firm fundamental factors Xj: (3.12)

The fundamental factors might include financial ratios such as debt-equity ratios, liquidity ratios and return measures. Other relevant factors might include firm size, sales growth rate, volatility of the industry and so forth. The coefficients are determined on the basis of a regression of historical betas on historical values for the various fundamental factors. 3.C: MULTI-INDEX MODELS (Background reading: Section 3.B) The Multi-Index Model enables the analyst to attribute multiple sources of covariance between security returns. The multi-index model can be used to estimate security returns, expected returns, variances and covariances as follows:

44

Chapter 3

(3.13) (3.14) (3.15) (3.16) Derivations for these measures are identical to those for the Single Index Model after adjusting the original statistical measures for the additional indices. One important problem from a practical perspective concerns how to obtain indices for the Multi-Index Model. Selection of these indices should be based on the sources of comovement among security returns. Potential indices might include market index returns, interest rates, commodity prices, financial ratios, firm size, and volatility of the industry. Any economic or fundamental factor might qualify as an index if it captures a significant portion of the comovement among security prices. NOTE 1. In sum, n2 covariances need to be computed for the standard portfolio variance model. However, this number of covariances can be reduced to (n2-n)/2 non-trivial covariances since n of the covariances will actually be variances (the covariance between any security i and itself is variance) and each aik will equal aki. By this formula, we can compute that a ninety security portfolio would require 4005 covariance calculations. SUGGESTED READINGS Elton and Gruber [1995] and Alexander and Francis [1986] both provide excellent detailed explanations of all of the topics covered in this chapter. In particular, see Chapters 4 through 7 in the Elton and Gruber text and Chapters 4 and 5 in the Alexander and Francis text. Both of these texts also cover many extensions to this important material. Brealey and Myers [1996] in Chapters 7 and 8 provide a readable introduction to portfolio return and risk. Other important topics in portfolio mathematics are covered in applications to matrix mathematics and calculus in Chapters 4 through 6 of this book. Topics related to OLS regressions are covered in Chapters 2, 4 and 8.

Elementary Portfolio Mathematics

45

EXERCISES 3.1 An investor is considering combining Douglas Company and Tilden Company common stock into a portfolio. Fifty percent of the dollar value of the portfolio will be invested in Douglas Company stock; 50% will be invested in Tilden Company stock. Douglas Company stock has an expected return of 6% and an expected standard deviation of returns of 9%. Tilden Company stock has an expected return of 20% and an expected standard deviation of 30%. The coefficient of correlation between returns of the two securities has been shown to be .4. Compute the following for the investor's portfolio: a. expected return b. expected variance c. expected standard deviation 3.2 Work through each of your calculations in Problem 1 again, assuming the following weights rather than those given originally: a. 100% Douglas Company stock; 0% Tilden Company stock b. 75% Douglas Company stock; 25% Tilden Company stock c. 25% Douglas Company stock; 75% Tilden Company stock d. 0% Douglas Company stock; 100% Tilden Company stock 3.3 How do expected portfolio return and risk levels change as the proportion invested in Tilden Company stock increases? Why? Prepare a graph with expected portfolio return on the vertical axis and portfolio standard deviation on the horizontal axis. Plot the expected returns and standard deviations for each of the portfolios whose weights are defined in Problems 1 and 2. Describe the slope of the curve connecting the points on your graph. 3.4 The common stocks of the Landon Company and the Burr Company are to be combined into a portfolio. The expected return and standard deviation levels associated with the Landon Company stock are 5% and 12%, respectively. The expected return and standard deviation levels for Burr Company stock are 10% and 20%. The portfolio weights will each be 50%. Find the expected return and standard deviation levels of this portfolio if the coefficient of correlation between returns of the two stocks is: a. 1 b. .5 c. 0 d. - . 5 e. - 1 3.5 Describe how the coefficient of correlation between returns of securities in a portfolio affects the return and risk levels of that portfolio. 3.6 An investor is considering combining securities A and B into an equally weighted portfolio. This investor has determined that there is a 20% chance that the economy will perform very well, resulting in a 30% return for security A and a 20% for security B. The investor estimates that there is a 50% chance that the economy will perform only adequately, resulting in 12% and 10% returns for securities A and B, respectively. The investor estimates a 30% probability that the economy will perform poorly, resulting in

Chapter 3

46

a - 9 % return for Security A and a 0% return for security B. These estimates are summarized as follows: Outcome Probability 1 .20 2 .50 3 .30

R^ .30 .12 -.09

R^ .20 .10 0

R^

a. What is the portfolio return for each of the potential outcomes? b. Based on each of the outcome probabilities and potential portfolio returns, what is the expected portfolio return? c. Based on each of the outcome probabilities and potential portfolio returns, what is the standard deviation associated with portfolio returns? d. What are the expected returns of each of the two securities? e. What are the standard deviation levels associated with returns on each of the two securities? f. What is the covariance between returns of the two securities? g. Based on your answers to part d in this problem, find the expected portfolio return. How does this answer compare to your answer in part b? h. Based on your answers to parts e and f, what is the expected deviation of portfolio returns? How does this answer compare to your answer in part c? 3.7 An investor has combined securities X, Y and Z into a portfolio. He has invested $1000 in security X, $2000 into security Y and $3000 into security Z. Security X has an expected return of 10%; security Y has an expected return of 15% and security Z has an expected return of 20%. The standard deviations associated with securities X, Y and Z are 12%, 18% and 24%, respectively. The coefficient of correlation between returns on securities X and Y is .8; the correlation coefficient between X and Z returns is .7; the correlation coefficient between Y and Z returns is .6. Find the expected return and standard deviation of the resultant portfolio. 3.8 An investor wishes to combine Stevenson Company stock and Smith Company stock into a riskless portfolio. The standard deviations associated with returns on these stocks are 10% and 18%, respectively. The coefficient of correlation between returns on these two stocks is - 1 . What must each of the portfolio weights be for the portfolio to be riskless? 3.9 Assume that the coefficient of correlation between returns on all securities equals zero in a given market. There are an infinite number of securities in this market, all of which have the same standard deviation of returns (assume that it is .5). What would be the portfolio return standard deviation if it included this infinite number of securities in equal investment amounts? Why? (Demonstrate your solution mathematically.) 3.10 The variance of returns on the market portfolio is .01 and the required or expected return for Portfolio A is .15. There is no riskless asset and the expected return on the market portfolio is .12. The variance of returns on Portfolio A is .02, and it is regarded as being sufficiently well diversified to have zero unsystematic variance. If Portfolio A currently has a market value of $2,000,000, what is its Beta?

Elementary Portfolio Mathematics

47

3.11 Briefly discuss the strengths and weaknesses of each of the following techniques as a means to estimate the anticipated covariance between returns of two securities: a. forecasted covariance as a function of potential return outcomes and their associated probabilities b. historical covariances c. single index betas d. multi-index betas e. fundamental betas 3.12 Under what circumstances does increasing the number of indices that I use in my index model improve my covariance estimates? Under what circumstances does decreasing the number of indices that I use in my index model improve my covariance estimates?

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4

Matrix Mathematics

4.A: MATRICES, VECTORS AND SCALARS A matrix is defined as an ordered rectangular array of numbers. A matrix enables one to represent a series of numbers as a single object, thereby providing for convenient systematic methods for completing repetitive computations. The following are examples of matrices:

The dimensions of a matrix are given by the ordered pair m x n, where m is the number of rows and n is the number of columns in the matrix. Thus, A is 3 x 2, B is 2 x 2, c is 2 x 1, and d is 1 x 1. Each number in a matrix is referred to as an element. The symbol a^ denotes the element in Row i and Column j of Matrix A, bl} denotes the element in Row i and Column j of Matrix B, and so on. Thus, a32 is 4 and c2l = 3. There are specific terms denoting special types of matrices. For example, a vector is a matrix with either only one row or one column. Thus, the dimensions of a vector are 1 x n or m x 1. Matrix c above is a column vector; a 1 x n matrix is a row vector. A scalar is a matrix with exactly one element. Matrix d is a scalar. A square matrix has the same number of rows and columns (m = n). Matrices B and d are square matrices. A symmetric matrix is a square matrix where ci(j equals cjti for all i and j ; that is, the i'th element in each row equals the j'th element in each column. Scalar d and matrices H, I, and J below are all symmetric matrices. A diagonal matrix is a symmetric matrix whose elements off the principal diagonal are zero, where the principle diagonal contains the series of elements where i = j . Scalar d and Matrices H, and I below are all diagonal matrices. An identity or unit matrix is a diagonal matrix

50

Chapter 4

consisting of ones along the principal diagonal. Both matrices H and I following are diagonal matrices; I is the 3 x 3 identity matrix:

APPLICATION 4.1: PORTFOLIO MATHEMATICS (Background reading: Sections 3.A and 4.A) Computing returns and variances for portfolios with large numbers of securities often involves large numbers of repetitive calculations. Use of matrices provides a means of organizing, systemizing and generally simplifying these series of calculations. Consider a portfolio comprised of three securities with the following characteristics and weights:

The following represent the returns and weights vectors for the three securities:

Bearing in mind that si,j = sj,i and that si2 = sij, we may represent a covariance

matrix for the securities as follows:

Note that each element Vy equals the covariance
Matrix Mathematics

51

For example:

Note that each of the matrices are of dimension 3 x 3 and that each of the elements in Matrix C is the sum of corresponding elements in Matrices A and B. The process of subtracting matrices is similar, where a^ - by = dy for A - B = D:

(4.2)a

For example:

The transpose A' of Matrix A is obtained by interchanging the rows and columns of Matrix A; that is, ay becomes a^. The following represent Matrix A and its transpose A' along with Vector x and its transpose x':

52

Chapter 4

4.C: MULTIPLICATION OF MATRICES (Background reading: Section 4.B) Two matrices A and B may be multiplied to obtain the product AB = C if the number of columns in the first Matrix A equal the number of rows B in the second.1 If Matrix A is of dimension m x n and Matrix B is of dimension n x q, the dimensions of the product Matrix C will be m x q. Each element c u of Matrix C is determined by the following sum: (4.3) For example, consider the following product:

Matrix C in the above is found as follows:

Notice that the number of columns (3) in Matrix A equals the number of rows in Matrix B. Also note that the number of rows in Matrix C equals the number of rows in Matrix A; the number of columns in C equals the number of columns in Matrix B. APPLICATION 4.1 (continued): PORTFOLIO MATHEMATICS In Section A, we represented the returns, weights, variances and covariances of securities in a portfolio with a series of appropriate matrices. We will now perform arithmetic operations on these matrices to determine the expected return and variance of the portfolio. First, we obtain the portfolio's expected return E[Rp] = w'r as follows:

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53

Note that we transposed the weights vector to make it conform for multiplication with the returns vector (the first matrix to multiply must have the same number of columns as the number of rows in the second matrix to multiply). Since our desired product is a single number (a 1 X 1 matrix), we want the first matrix to have one row and the second to have one column. This is why we premultiply by the transposed matrix. We find the variance of returns for this portfolio al = w'Vw as follows:

We obtain this product by multiplying from left to right (the commutative property does not hold for multiplication of matrices), starting with w'V:

= [.004+.004+.01 .001+.064+.02 .002+.016+.18] = [.018 .085 .198] We now multiply w'V by w to obtain the portfolio variance:

Note that the three matrices that we multiplied were of dimension 1 x 3, 3 x 3 and 3 x 1 . Our desired result is a single number, or a 1 x 1 matrix. Therefore, the first matrix in our product should have one row and the last matrix in our product should have one column. To ensure comformability for multiplication, the number of columns in each matrix should be the same as the number of rows in the following matrix. Confirm the following for our threesecurity portfolio based on the above weights and covariances:

54

Chapter 4

4.D: INVERSION OF MATRICES (Background reading: Section 4.C) An inverse Matrix A'1 exists for the square Matrix A if the product A"1 A or A A"1 equals the identity Matrix I. Consider the following product:

One means for finding the inverse Matrix A"1 for Matrix A is through the use of a process called the Gauss-Jordan Method. This method will be performed on Matrix A by first augmenting it with the identity matrix as follows:

(A) For the sake of convenience, call the above augmented Matrix A temporarily. Now, a series of row operations (addition, subtraction or multiplication of each element in a row) will be performed such that the identity matrix replaces the original Matrix A (on the left side). The right-side elements will comprise the inverse Matrix A"1. Thus, in our final augmented matrix, we will have ones along the principal diagonal on the left side and zeros elsewhere; the right side of the matrix will comprise the inverse of A. Allowable row operations include the following: 1. Multiply a given row by any constant. Each element in the row must be multiplied by the same constant. 2. Add a given row to any other row in the matrix. Each element in a row is added to the corresponding element in the same column of another row. 3. Subtract a given row from any other row in the matrix. Each element in a row is subtracted from the corresponding element in the same column of another row. 4. Any combination of the above. For example, a row may be multiplied by a constant before it is subtracted from another row. Our first row operation will serve to replace the upper left corner value with a one. We multiply Row 1 in A (Row 1 A) by .5 to obtain the following:

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55

where Row IB replaces Row 1 A. Now we obtain a zero in the lower left corner by multiplying Row 2 in A by 1/8 and subtracting the result from our new Row 1 to obtain Matrix B as follows: (B) Next, we obtain a 1 in the lower right corner of the left side of the matrix by multiplying Row 2B by 8/15:

We obtain a zero in the upper right corner of the left side matrix by multiplying Row 2 above by 2 and subtracting from Row 1 in B:

(C)

The left side of augmented Matrix C is the identity matrix; the right side of C is A 1 . Because matrices cannot be divided as numbers are in arithmetic, one performs an analogous operation by inverting the matrix intended to be the "divisor" and postmultiplying this inverse by the first matrix to obtain a quotient. Thus, instead of dividing A by B to obtain D, one inverts B and obtains D by the product AB"1 = D. This concept is extremely useful for many types of algebraic manipulations. 4.E: SOLVING SYSTEMS OF EQUATIONS (Background reading: Section 4.D) Matrices can be very useful in arranging systems of equations. Consider for example the following system of equations:

56

Chapter 4

This system of equations may be represented as follows:

We are not able to divide s by C to obtain x; instead, we invert C to obtain C"1 and multiply it by s to obtain x:

Therefore, to solve for Vector x, we first invert C by augmenting it with the Identity Matrix: (A)

(B)

(C)

Thus, we obtain Vector x with the following product:

(D)

Thus, we find that x{ = 1.8 and x2 = -1/3. Arbitrage is defined as the simultaneous purchase and sale of assets or portfolios yielding identical cash flows. Assets generating identical cash flows (certain or risky cash flows) should be worth the same amount. This is known as the Law of One Price. If assets generating identical cash flows sell at different prices, opportunities exist to create a profit by buying the cheaper asset and selling the more expensive asset. The ability to realize a profit from this type of transaction is known as an arbitrage opportunity. Solutions for multiple variables in systems of equations are most useful in the application of the Law of One Price and seeking arbitrage opportunity.

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APPLICATION 4.2: COUPON BONDS AND DERIVING YIELD CURVES (Background reading: Sections 2.C, 4.A and 4.E) The spot rate is the yield at present prevailing for zero coupon bonds of a given maturity. The t year spot rate is denoted here by y01, which represents the interest rate on a loan to be made at time zero and repaid in its entirety at time t. Spot rates may be estimated from bonds with known future cash flows and their current prices. We are able to obtain spot rates from yields implied from series of bonds when we assume that the Law of One Price holds. The yield curve represents yields or spot rates of bonds with varying terms to maturity. For example at a given point in time, the yield for one-year bonds may be 5% (y 0 , = .05), while the yield for five-year bonds may be 10% (y 05 = .10). This section is concerned with how interest rates or yields vary with maturities of bonds. The simplest bonds to work with from an arithmetic perspective are pure discount notes, notes which make no interest payments. Such notes make only one payment at one point in time — on the maturity date of the note. Determining the relationship between yield and term to maturity for these bonds is quite trivial. The return one obtains from a pure discount note is strictly a function of capital gains; that is, the difference between the face value of the note and its purchase price. Short-term U.S. Treasury Bills are an example of pure discount (or zero coupon) notes. Coupon bonds are somewhat more difficult to work with from an arithmetic perspective because they make payments to bondholders at a variety of different periods. A coupon bond may be treated as a portfolio of pure discount notes, with each coupon being treated as a separate note maturing on the date the coupon is paid. This slightly complicates the process for determining yields, but is necessary to avoid associating wrong yields with given time periods. Consider an example involving three bonds whose characteristics are given in Table 4.1. The three bonds are trading at known prices with a total of eight annual coupon payments among them (three for bonds A and B and 2 for bond C). Bond yields or spot rates must be determined simultaneously to avoid associating contradictory rates for the annual coupons on each of the three bills. Table 4.1 Coupon Bonds A, B and C BOND

CURRENT PRICE

FACE VALUE

COUPON RATE

YEARS TO MATURITY

A

962

1000

.10

3

B

1010.4

1000

.12

3

C

970

1000

.10

2

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58

Let Dt be the discount function for time t; that is, Dt = l/(l+y0Ay. Since y 0t is the spot rate or discount rate that equates the present value of a bond with its current price, the following equations may be solved for discount functions then spot rates:

This system of equations may be represented by the following system of matrices:

To solve this system we first invert Matrix CF, then use this inverse to premultiply Vector P0 to obtain Vector d:

D

Thus, we find from solving this system for Vector d that D, = .9, D2 = .8 and D3 = .72. Since Dt = 1/(1+y0t)', 1/D, = (l+y 0 t )', and y0il = 1/D1" - 1. Thus, spot rates are determined as follows:

Note that there exists a different spot rate (or discount rate) for each term

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59

to maturity; however, the spot rates for all cash flows generated by all bonds at a given period in time are the same. Thus, y0t will vary over terms to maturity, but will be the same for all of the bonds in a given time period. APPLICATION 4.3: ARBITRAGE WITH RISKLESS BONDS (Background reading: Application 4.2) The example provided in Application 4.2 above consists of three priced riskless bonds defining spot rates for all three relevant years. The cash flow structure of any three-year bond (for example, Bond D) added to the market can be replicated with some portfolio of bonds A, B, and C as long as its cash payments to investors are on the same dates as those made by at least one (in this example, two) of the three bonds A, B, and C. For example, assume that there now exists Bond D, a three-year, 11.5% coupon bond selling in this market for $990. This bond will make payments of $115 in years 1 and 2 in addition to a $1115 payment in year 3. A portfolio of bonds A, B and C can be comprised to generate the exact cash flow series. Thus, Bond D can be replicated by a portfolio of our first three bonds with the following weights: wA= .25, wB= .75 and w c =0, which are determined by the following system of equations or matrices:

To solve this system wefirstinvert Matrix CF, then use it to premultiply Vector cfD to obtain vector w:

Thus, we find from this system that wA = .25, wB = .75 and wc == 0. We determine the value of the portfolio replicating Bond D by weighting their current market prices: (.25 • $962) + (.75 • $1010.4) = $998.3. Based on the portfolio's price, the value of Bond D is $998.3, although its current market

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price is $990. Thus, one gains an arbitrage profit from the purchase of this bond for $990financedby the sale of the portfolio of Bonds A and B at a price of $998.3. Here, we simply swap a portfolio comprised of Bonds A and B for Bond D. Our cash flows in years 1,2 and 3 will be zero, although we receive a positive cash flow now of $8.3. This is a clear arbitrage profit. This arbitrage opportunity will persist until the value of the portfolio equals the value of Bond D. Thus, spot rates must be consistent for all bonds of the same risk class and maturity. APPLICATION 4.4: FIXED INCOME PORTFOLIO DEDICATION (Background reading: Application 4.3) Afixedincome fund is concerned with ensuring the provision of a relatively stable income over a given period of time. Typically, afixedincome fund must provide payments to its creditors, clients or owners for a given period. For example, a pension fund is often expected to make a series of fixed payments to pension fimd participants. Such funds must invest their assets to ensure that their liabilities are paid. In many cases, fixed income funds will purchase assets such that their cash flows exactly match the liability payments that they are required to make. This exact matching strategy is referred to as dedication and is intended to minimize the risk of the fund. The process of dedication is much the same as the arbitrage swaps discussed in Application 4.3 above; the fund manager merely determines the cash flows associated with his liability structure and replicates them with a series of default risk free bonds. For example, assume that a pension manager needs to make payments to pension plan participants of $1,500,000 in one year; $2,500,000 in two years; and $4,000,000 in three years. He wishes to match these cash flows with a portfolio of bonds E, F and G whose characteristics are given in Table 4.2. These three bonds must be used to match the cash flows associated with the fund's liability structure. For example, in year 1, Bond E will pay $1100, F will pay $120 and Table 4.2 Coupon Bonds E, F and G BOND

CURRENT PRICE

FACE VALUE

COUPON RATE

YEARS TO MATURITY

E

1010

1000

.10

1

F

1100

1000

.12

2

G

950

1000

.10

3

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Matrix Mathematics

G will pay $100. These payments must be combined to total $1,500,000. Cash flows must be matched in years 2 and 3 as well. Only one matching strategy exists for this scenario. The following system may be solved for b to determine exactly how many of each of the bonds are required to satisfy the fund's cash flow requirements: 1100 0 0

120

100

E

h

1120 100 0 CF

1,500,000"

b

1100

= 2,500,000 4,000,000

b

•

o. b

=

L

Inverting Matrix CF and multiplying by Vector L, we find that the purchase of 824.9704 Bonds E, 1907.467 Bonds F and 3636.363 Bonds G satisfy the manager's exact matching requirements. The fund's time zero payment for these bonds totals $6,385,979.9292. 4.F: VECTORS, VECTOR SPACES AND SPANNING (Background reading: Section 4.E) Earlier, we defined a vector as a matrix with either only one row or one column. A column Vector v comprising n real elements is said to be within the set Rn, which is regarded as the n-dimensional vector space. Rn is defined as the set of all vectors with n real valued elements or coordinates. The following represent vectors in three dimensional space:

Thus, Vectors a, b, and c are all elements of the three dimensional space R3. The n elements of a Vector v might be regarded as the coordinates of a point in n-dimensional space or an n-dimensional plane. All of the vectors falling within this plane are said to exist in set Rn. Two of the arithmetic operations applicable to vectors are vector addition and scalar multiplication. Linear combinations of given vectors are applications of these two operations to the vectors. If a vector in a given n-dimensional space can be expressed as a linear combination of a set of other vectors in the same space, we say that the given vector is linearly dependent on the other set of vectors. Similarly, if a set of vectors {x} can be multiplied by a series of scalars G: (where at least one of the scalars is nonzero) to obtain a vector of zeros, we say that linear dependence exists among the set of vectors {x}: (4.4)

62

Chapter 4

Thus if at least one of the scalars otx above has a nonzero value, linear dependence exists within the set of n vectors x. Linear independence within a set of vectors {x} exists where the set of scalars with each having a value of zero is the only set which can be used to multiply set of vectors {x} to obtain the n-dimensional zero vector. In this case, no vector in the set {x} can be defined as a linear combination of other vectors in the set. Linear dependence exists within vector sets A, B and C below because Equation (4.4) can be satisfied with scalars such that at least one a^O. Furthermore, within each set of vectors, any one vector may be described as a linear combination of the other two.

(A)

(B)

(C)

We determine that vector set A is linearly dependent by demonstrating that there exists a set of values for a satisfying:

It is obvious from the first and third equations that ^ equals zero. Using a{ equal to zero in the second equation, we find that any value for a2 will satisfy equation 2 as long as a 3 is equal but opposite in sign. Since at least one of the scalars a may be nonzero when satisfying the three equations, the set of three

63

Matrix Mathematics

vectors is linearly dependent. If three nonzero scalars cannot be found to satisfy the equations, linear independence exists within the set. Each of vector sets D, E and F below are linearly independent because the set of scalars satisfying Equation (4.4) for each set will all have zero values. For example, in vector set D, au a2 and a 3 must all equal zero for a vector of zeros to be a linear combination of the three vectors. Furthermore, no vector in set D can be defined as a linear combination of the other vectors in set D; no vector in set E can be defined as a linear combination of the other vectors in set E; and no vector in set F can be defined as a linear combination of the other vectors in setF.

(D)

(E)

1

0

0

0

1

0

0

0

1

1

1

0

1

0

1

0

1

1

"7 '

0

0

0

5

-5

0

0

3

(F)

Linearly independent vectors

Linearly independent vectors

Linearly independent vectors

A set of n vectors is said to span the n-dimensional vector space if that set of n vectors is linearly independent. A set of n vectors consisting of n elements each, which do not span a vector space, are not linearly independent. The following represent examples of sets of vectors which are not linearly independent:

(G)

(H)

9

0

0

4

3

1

6

9

3

7"

1

0

7

1

0

0

1

5

Linearly dependent vectors; 3-dimensional space is not spanned.

Linearly dependent vectors; 3-dimensional space is not spanned.

Chapter 4

64

Each of the above sets of vectors may be combined to equal a vector of zeros. Thus, any one vector in each set may be expressed as a linear combination of the other two vectors in the set. Neither of these sets of three vectors span the three-dimensional vector space. The following set of three vectors does span the three-dimensional space; none of these vectors can be expressed as a linear combination of the other two vectors:

(D

4 2 1 *i

2 1 0 X

2

6 0 2

3 dimensional space is spanned by these linearly independent vectors.

*3

Any n-dimensional vector can be expressed as a linear combination of a set of n other linearly independent vectors. That is, a set of vectors is a basis for a given vector space if any vector in the space can be expressed as a linear combination of the set of vectors forming the basis. The basis for Rn is said to span the n-dimensional space. For example, any three dimensional vector (such as [4,3,6]' which equals 4xt + 3x2 + 6x3) can be expressed as a linear combination of the three vectors in Set D above since the three vectors in set D are linearly independent. This set of three vectors is a basis for R3; these three vectors span R3. The same can be said for vector sets E and F; each of these sets represents a basis for and spans R3. In a sense, when one vector is linearly dependent on another n - 1 vectors, the information in the other n - 1 vectors can be used to replicate the information in the nth vector. In a financial sense where elements within a vector represent payoffs of a given security contingent on outcomes or associated with a points in time, the payoff structure of the nth security can be replicated with a portfolio comprising the other n - 1 securities on which its payoff vector is linearly dependent. When a set of n payoff vectors span the n potential outcome or time space, the payoff structure for any other security in the same outcome or time space can be replicated with the payoff vectors of the n securities spanning the payoff space. Securities whose payoff vectors can be replicated by portfolios of other securities must sell for the same price as those portfolios; otherwise, the Law of One Price is violated. Consider Application 4.3 above examining arbitrage opportunities in bond markets. For an arbitrage opportunity to exist, the payoff vector of a given bond must be replicated by a linear combination of payoff vectors for other bonds, yet sell for a price different from the portfolio which replicates it. That is, that bond's payoff vector must be linearly dependent on the payoff vectors of the n - 1 bonds in the portfolio; if the bond sells for a price other than that of the portfolio, an arbitrage opportunity exists. Application 4.4 concerning portfolio dedication also requires that the liability structure be dependent on the payoff vectors of bonds. In Application 4.2, spot rates are implied based on the

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assumption that arbitrage opportunities do not exist among the bonds used to construct the yield curve. If n spot rates are desired, a minimum of n bonds forming the basis for the n-dimensional time space are required. APPLICATION 4.5: THE STATE PREFERENCE MODEL (Background reading: Section 4.F) In some situations, investors may value securities by estimating expected cash flows and discounting them appropriately. This discounted expected cash flow model requires the estimation of expected cash flows based on knowledge of all possible cash flow outcomes which might be realized on the investments along with probabilities of each potential cash flow being realized. That is, expected cash flow is the sum of all potential cash flows weighted by their associated probabilities. The set of potential cash flows may be estimated easily enough; perhaps all values ranging between - oo and oo qualify as potential cash flows (though, of course, most of the potential cash flows might be regarded as being rather improbable). The greater difficulty with the discounted expected cash flow model is the estimation of probabilities to associate with potential cash flows. In many instances, these probabilities must be rather subjectively determined. The State Preference Model (sometimes referred to as the contingent claims analysis model) provides a means of valuing securities without the necessity of first estimating probabilities to associate with each potential cash flow. Essentially, the process used in this model requires first determining the number of potential relevant states of nature, finding prices of the same number of securities with known independent payoff structures (this determination is discussed later) and then valuing the securities of particular interest as functions of the known prices of securities which have already been valued. In terms of linear mathematics, this process requires first determining the vector space Rn, valuing n "control" securities with linearly independent payoff vectors, and pricing the payoff vectors of the securities of interest with linear combinations of prices from the "control" securities. We will make the following assumptions for our use of the State Preference Model: 1. There exist n potential states of nature in a one-time period framework (this could be extended to m time periods). Each security will have exactly one known payoff associated with each state of nature. 2. Each investor's utility or satisfaction is a function only of his level of wealth; the state of nature which is realized is important only to the extent that the investor's wealth is affected. 3. The n states of nature are mutually exclusive. 4. All relevant potential states of nature are part of the evaluation. 5. Capital markets are in equilibrium for all securities.

66

Chapter 4

6. There exist complete capital markets; that is, there exist n securities with known linearly independent payoff vectors and known values. These n securities form the basis for the n-dimensional outcome space. Suppose in a given economy with n potential outcomes there exists a security with a known payoff vector. For example, in a three-potential-state world, security x will pay $8 if outcome 1 is realized, $4 if outcome 2 is realized and $1 if outcome 3 is realized. The payoff vector for security x is given as follows:

Here, we have defined every potential payoff in this three state world, but have made no projections regarding probabilities to associate with the outcomes. What is the value of security x? The state preference model will be used to determine this value not based on discounted expected cash flows but based on the known values of other securities existing in this three-state world. The first step in the evaluation procedure is to decompose the security into an imaginary portfolio of pure securities. Define a pure security (also known as an elementary, primitive or Arrow-Debreu security) to be an investment which pays $1 if and only if a particular outcome or state of nature is realized and nothing otherwise. Thus, the payoff vector for a given pure security i in an npotential outcome economy will comprise n elements; 1 will be the i'th element and all other elements will be zero. The payoff vector for pure security 2 in a three-potential-outcome world is given as follows:

The vector name 2 is not to be confused with a numerical value of 2; it is merely the name of the vector. Pure security 2 will pay 1 if outcome 2 is realized; otherwise, it will pay zero. Payoff vectors for three pure securities will span the vector space for a three-outcome economy:

The three-dimensional vector space is spanned by three vectors if any vector in that space can be defined in terms of a linear combination of those three vectors. Thus, the payoff vector for any security existing in this three-outcome

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world may be represented as a function or, more specifically, a linear combination of the payoff vectors of the three pure securities spanning the threeelement vector space. For example, security x might be represented as a portfolio of 8 pure securities 1, 4 pure securities 2 and 1 pure securities 3: 8 4

1 = 8 •

1 X

0

0 + 4 •

0 = 8 •

1

1

0 + 1 •

0 + 4 •

2

0 1

+ 1 •

3

We are able to evaluate security x easily if we know values of the three pure securities. Suppose, the value of pure security 1 is given by Vx = .2 which suggests that an investor is willing to pay .2 for a security which pays 1 if and only if outcome one is realized. Furthermore, suppose that pure security 2 has a value of .4 and pure security 3 has a value of .3. The value of security x is determined as follows: Px = %?x + 4P2 + 1P3 = (8 • .2) + (4 • .4) + (1 • .3) = 1.6 + 1.6 + .3 = 3.5. An investor should be willing to pay 3.5 for security x. We have evaluated security x without estimating probabilities to associate with each outcome, without knowledge of investors risk preferences and without determining a discount rate. On the other hand, our evaluation is a function of value estimates of three imaginary or pure securities. Just how are we able to obtain estimates of values of securities which may not even exist in the "real world?" Here, we have to rely most heavily on the complete capital markets assumption number 6 from the list on pages 65-66 along with the concept of spanning and basis in n-dimensional space, the mathematical underpinnings of complete capital markets. A complete capital market exists in an n-potential-outcome economy where there are at least n securities with known, linearly independent payoff vectors. If the values of each of these n securities are known, then the value of any other security in this n-potential state world can be determined based on values of the original n securities. With n securities, we will be able to determine values of each of n pure securities. Since any other security can be expressed as a portfolio of these n pure securities, the value of that security must equal the value of the portfolio of its "component" pure securities. Suppose, for example, securities x and y exist in a two-potential-outcome world and have the following payoff vectors: x

8 4

y

2 10

We can demonstrate that these two payoff vectors are linearly independent. Thus, x and y span the two-dimensional vector space and there exist complete

Chapter 4

68

capital markets in this economy. This implies that we can decompose securities x and y into the following component pure securities:

where security x represents a portfolio comprising 8 of pure security 1 and 4 of pure security 2; security y represents a portfolio of 2 of pure security 1 and 10 of pure security 2. Suppose that security x has a market value of 5 and security y has a market value of 8. In the absence of arbitrage opportunities, a portfolio comprising 8 of pure security 1 and 4 of pure security 2 must also have a market value of 5 and a portfolio comprising 2 of pure security 1 and 10 of pure security 2 must have a market value of 8:

We may solve this system for Pj and P2 to obtain ?x = .25 and P2 = .75. Now, consider a third security z with the following payoff structure:

The price of market security z is determined from the prices of our two pure securities: Vz = (20 -.25) + (8 • .75) = 11 The State Preference Model developed here is based on a one-time-period framework with n-potential outcomes. One attractive feature of this model is that it can easily be extended to a multiple-time-period framework. That is, the payoff vectors can be expanded to include nt potential outcomes for each of T time periods t. Thus, the number of elements in a payoff vector in the expanded Time-State Preference Model is £[=1nt. A second attractive feature of the model is that it can be used to estimate "synthetic" or "risk-neutral" probabilities. Consider the example above consisting of securities x, y and z. The pure security prices estimated in this example were estimated to be ?x = .25 and P2 = .75. This suggests that investors would be willing to pay .25 for a pure security that pays 1 if and only if Outcome 1 is realized; they would be willing to pay .75 for a pure security that pays 1 if and only if Outcome 2 is realized. If we were willing to assume that investors are risk neutral, we may infer that they believe that Outcome 2 is more likely to be realized than Outcome 1 (in fact, three times more likely). In fact, since the pure security prices add to one (this will not be the case for

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69

investors who prefer money sooner rather than later), we may infer that investors associate probabilities of .25 to Outcome 1 and .75 to Outcome 2. These probabilities are referred to as synthetic probabilities because they are constructed from security prices based on the assumption that investors are risk neutral. Normally, we may obtain a synthetic probability pj as follows:

where Pj is the price of pure security i and Pj is the price of each of n pure securities j . APPLICATION 4.6: BINOMIAL OPTION PRICING (Background reading: Section 4.F and Application 4.5) Derivative securities are assets whose values are derived from the performance of other securities. Stock options are examples of derivative securities. One type of stock option is a call, which grants its owner the right (but not the obligation) to purchase shares of an underlying stock at a specified "exercise" price within a given time period (before the expiration date of the call). The value of the call at its expiration is a function of the value of its underlying stock. More specifically, the value of a call at its expiration is given by the maximum of either zero or the difference between the stock price at expiration and the exercise price of the call:

Thus, if the call is exercised at its expiration, its value is equal to the value of the underlying stock less the price at which it can be purchased due to the option. If the stock price is lower than the exercise price of the option at its expiration, the call is discarded; its value is zero. Consider a one-time-period, two-potential-outcome framework where there exists Company Q stock currently selling for $50 per share and a riskless $100 face-value T-bill currently selling for $90. Suppose Company Q faces uncertainty, such that it will pay its owner either $30 or $70 in one year. The T-bill will certainly pay its owner $100 in one year. Further assume that a call with an exercise price of $55 exists on one share of Q stock. This call will be worth either $0 or $15 when it expires, based on the value of the underlying stock.2 The payoff vectors for Stock q, the T-bill (b) and the call (c) are given as follows:

70

Chapter 4

The current prices of the stock and T-bill are known to be $50 and $90. Since their payoff vectors span the two-outcome space in this two-potential-outcome framework, they form complete capital markets and we can estimate pure security prices as follows:

We may solve this system for ?x and P2 to obtain ?x = .325 and P, = .575. The call value is determined:

Synthetic probabilities may be estimated as follows:

We may also use the concept of arbitrage to determine the value of the call. Again, since the payoff vectors of the stock and T-bill span the two outcome space, they form complete capital markets and the call can be valued based on the values of the stock and T-bill. In other words, a portfolio comprising the stock and T-bill can be formed to replicate the payoff structure of the call. Portfolio holdings are determined as follows:

We invert the payoffs matrix to obtain:

We find that #q = .375 and #b = -.1125. This implies that the payoff structure of a single call can be replicated with a portfolio comprising .375 shares of Q Company stock for a total of .375 • $50 = $18.75 and short-selling .1125 T-bills (in effect, borrowing .1125 • $90 = $10,125 at the T-bill rate). This portfolio requires a net investment of $18.75 - $10,125 = $8,625. Since the call on Q Company stock has the same payoff structure as this portfolio, its current value must be $8,625.

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Note that the present value of the T-bills equals B0 = 90 = 100 -i- (1.1111) = 100 -s- (1 +rf). We have structured the system of equations above in matrix format such that the following holds:

We solve this system simultaneously for #q and #b,findingthem to be .375 and -.1125, respectively. Since these ratios hold regardless of whether the stock price increases or decreases, The Law of One Price dictates that the following must hold:

where B0 and F are the current price and face value of the riskless bond. Similarly, a hedge portfolio with value B0 = VH can be constructed from one share of stock and #c calls. The value of this hedge portfolio is:

One interesting implication of this example is that when one can hedge options transactions against underlying stock and riskless assets, one can value the option as a function of the hedging proportions (#q and #b), the current value of the underlying stock and the riskless return. One need not use a riskadjusted return to value the option and one need not know the probabilities of stock price increases or decreases. This is the Cox-Ross Risk-neutrality Argument. This example represents the basis for mathematics underlying modern option pricing theory. Options are valued as a function of underlying stock and riskless assets which are combined into portfolios such that those portfolios have payoff structures identical to the options. The relation between the prices of the options and the portfolios which are constructed to replicate them is invariant with respect to their expected returns or discount rates used by investors. Hence, because it tends to be quite convenient, one can discount all cash flows generated by these securities at the riskless rate. APPLICATION 4.7: PUT-CALL PARITY (Background reading: Section 4.F and Application 4.6) A put is an option that grants its owner the right to sell the underlying stock at a specified exercise price on or before its expiration date. The owner of the option contract may exercise his right to buy or sell; however, he is not obligated to do so. In this application, we discuss a simple model concerning the

72

Chapter 4

relationship between put and call values. When this relationship holds, one is able to value a put based on the value of a call with identical terms. First, assume that there exists a European put (with a current value of p0) and a European call (with a value of C0) written on the same underlying stock which currently has a value equal to S0. Both options expire at time T and the riskless return rate is rf. Since the payoff function of the call at expiration is CT = MAX[S T -X, 0] and the payoff function for the put is pT = MAX[X-S T , 0], the following system describes the pricing of a put in terms of the underlying stock, exercise price of options and the call with the same terms as the put:

This put-call parity relation holds regardless of the number of potential outcomes in the state space. The securities need not span the outcome space for the put-call parity relation to hold. Consider the following numerical example where there are three potential stock prices, 80, 100 and 120 and a 105 exercise price for the options:

Because the put-call parity relation must hold at option expiry regardless of the underlying stock price, the following put-call parity relation must hold at time zero:

That is, a put is equivalent in value to a portfolio consisting of a short position in one share of stock underlying the put, an investment into a riskless asset certain to pay X at time T where X is die exercise price of options and T is the term to option expiry, and a long position in one call on the stock with the same expiration date and exercise price as the put. Suppose that the stock whose time T payoff function given above is currently selling for $102. Further assume that a one-year call (let T = 1) with

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an exercise price equal to $105 is currently selling for $8 and the current riskless return rate equals .10. A one-year put with an exercise price equal to $105 will be worth $1,007 according to the put-call parity relation. 4.G: ORTHOGONAL VECTORS (Background reading: Section 4.F) Two vectors x and y are said to be orthogonal when x'y = 0. Orthogonal vectors are linearly independent. Geometrically, the vectors x and y will represent perpendicular lines (lines placed at right angles with respect to one another). Symbolically, we write x 1 y to mean that Vector x is orthogonal to Vector y. One result from linear mathematics which is extremely useful in finance follows from two assumptions: 1. Assume that a set X of n - 1 vectors {xu x2, ..., x^} is orthogonal to another Vector w. That is, xj • w = 0, x2 • w = 0, and so on. 2. Furthermore, assume that because each of the n - 1 vectors of the X are orthogonal to w, an nth Vector xn must also be orthogonal to Vector w. Thus, x„ • w = 0. That is, assume that the n - 1 vectors from set X being orthogonal to w implies that Vector xn must also be orthogonal to Vector w. Result: Vector xn must be a linear combination of vectors {xt, x2, ..., xnA}. Thus, in order for the orthogonality between each vector in set X and Vector w to imply orthogonality between Vector xn and Vector w, Vector xn must be a linear combination of vectors {x^ x2, ..., xn.j}. In order for a set of n— 1 vectors to make this statement regarding Vector xn, xn must be a linear combination of the n - 1 vectors which imply that relationship between xn and w. Consider the following example where vectors a, b, c, d and w are given as follows:

Vectors a, b, and c are all orthogonal to Vector w as is Vector d. That is, premultiplication (or postmultiplication) of any one of vectors a, b, c or d by Vector w will result in a vector of zeros: w'a = 0, w'b = 0, w'c = 0 and w'd = 0. Furthermore, it is easy to demonstrate that Vector d is also a linear

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74

combination of vectors a, b, and c:3

Thus, if Vector d is a linear combination of vectors a, b, and c, it must be orthogonal to Vector w just as vectors a, b, and c are orthogonal to Vector w. Similarly, if Vector d must be orthogonal to Vector w whenever vectors a, b, and c are orthogonal to Vector w, Vector d must be a linear combination of vectors a, b, and c. The last statement provides that mathematical foundation to the Arbitrage Pricing Theory. APPLICATION 4.8: ARBITRAGE PRICING THEORY (Background reading: Section 4.G) Arbitrage Pricing Theory is a theory of capital markets equilibrium; that is, it describes the relationships among security prices and returns if the following assumptions hold: 1. Capital markets are competitive and frictionless. 2. Investors share homogeneous expectations. 3. Security returns are generated by the following factor model, where r is an n x 1 vector of security returns in an n security economy, j8 is an n x k matrix of returns in an economy where security prices are linearly related to k factors or indices, f is a k x 1 vector of factor values (a factor value is the difference between the actual index value and its expected value) prevailing during the time during which returns are measured, e is an n x 1 vector of residual return components for securities and Ij is the j'th index which generates security returns:

where:

R}s a random return for security i,

4. The number of securities, n, exceeds the number of factors, k. 5. Enough securities exist to diversify away unsystematic risk. Thus, w'e = 0. 6. Arbitrage opportunities cannot persist. Therefore, investors will not be able to form zero net investment, zero risk portfolios which earn a positive

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profit. The zero net investment, zero risk portfolio will have the following characteristics:

The sixth assumption implies that the following zero return condition must hold for any zero net investment, zero risk arbitrage portfolio:

where E[r] is the n x 1 security expected returns vector. Since there is no risk on this arbitrage portfolio, w'/Jf + w'e = 0, and the security expected returns vector is orthogonal to the weights vector:

The Equation a from Assumption 6 above states that the weights vector is orthogonal to the unit vector. The Equations b from Assumption 6 states that the weights vector is orthogonal to each of the beta coefficients vectors. Equations c from Assumption 6 states that the weights vector is orthogonal to the unique return components vector. Thus, these three sets of equations define the arbitrage portfolio — the zero net investment risk-free portfolio. The noarbitrage assumption states that the return on this arbitrage portfolio must equal zero. Because orthogonality in the first three sets of vectors implies orthogonality in the fourth set of vectors, the returns vector must be a linear

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76

combination of the unit, betas and epsilons vectors. Thus, the securities expected returns vector is orthogonal to the weights vector. Therefore, the securities expected return vector must be a linear combination of the unit, beta coefficients vector and unique risk components vector (though, the unique risk components has an expected value of zero). Thus: The APT Model

where the 5 values may be regarded as risk premia or coefficients relating the betas to expected returns. Thus, these 5 values may be interpreted as follows: the riskless return rate factor risk premia for i = 2 to k). Factor risk premia 5; may be described as the return on a portfolio with bj=0 for all indices except index i where bj = l. Consider the following example based on a two-index model which generates security returns in a particular economy:

Suppose that a given portfolio A has an expected return of .13, a beta with index 1 equal to 2 and has zero covariance to the second index. Portfolio B has an expected return of . 11, is uncorrelated with the first index and has a beta of 6 with the second index. Portfolio C has an expected return of .10 and has betas equal to one with each of the two indices. We may write the expected returns Vector E[r], the beta Matrix 0, the unit Vector i and the factor risk premia Vector 6 as follows:

The expected returns vector might be expressed as follows:

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We may solve the above for 50, fy and 52 for factor risk premia. Alternatively, his system may be expressed as a system of equations as follows:

The riskless return rate and two-factor risk premia are determined to be as follows: 50 = -05 = rf, d{ = .04, 52 = .01. Based on the above information, what would be the value of a security whose expected total expected payoff in one period is $100 assuming that its Beta (bx) with 8l is 2 and its Beta (b2) with 52 is .8? First, we find its expected return to be (.05 + 2 • .04 + .8 • .01) = .20. Thus, its current value is $100/1.2 = 83.333. NOTES 1. If two matrices may be multiplied, they are said to be conformable for multiplication. Any matrix may be multiplied by a scalar. One simply multiplies each of the elements of the matrix by the scalar to obtain the corresponding element of the product; that is, each element cM of C equals sa^ where C = sA. 2. The value of the call at expiration is the larger of either 0 or the difference between the value of its underlying security and the call exercise price. In this example, if the stock is worth $30, the call is worthless; if the stock is worth $70, the call is worth $15. 3. The coefficients .05, .02 and .04 may be found by forming a 5 x 3 Matrix X out of column vectors a,b and c, and solving the system XS = w for 5 where 6 is a 3 x 1 column of coefficients. Although no inverse to Matrix X exists, one can easily solve the system algebraically by bootstrapping or by reducing it to individual equations. SUGGESTED READINGS Elton and Gruber [1995] provide excellent readings pertaining to many of the applications in this chapter, including the estimation of yield curves, exact matching programs for bond portfolio management. In addition, Elton and Gruber discuss APT, put-call Parity and the Binomial Option Pricing Model. The appendix to Roll [1977] provides a comprehensive description of mean variance analysis, the Efficient Frontier, the Capital Asset Pricing Model and portfolio analysis in general with the use of matrix mathematics. However, this article may be regarded as being somewhat more advanced. Copeland and Weston [1988] and Martin, Cox and MacMinn [1988] discuss State Preference Theory and put-call parity as well as numerous other topics related to material presented in this chapter.

78

Chapter 4 EXERCISES

.1 Multiply the following matrices:

a.

b.

12

-2

1

3 4

2 _1 2

2

.04 .04i|

33.3333 -8.3333

,04 At

-8.3333

8.3333

.02' c.

.02 .16 .10]

.16 — .10

.02" d.

.16

[.02 .16 .10] =

.10 4.2 An investor has invested his capital into two funds, Fund I and Fund n. Each of these funds is comprised of three stocks, Stock 1, Stock 2, and Stock 3. The portfolio weights for each of the stocks in each of the funds and the stock returns are given in the following tables: w? Stock Return -Wlw0 1 .10 .40 .25 .35 Fund I 2 .18 .60 .20 Fundll .20 .26 3 a. Construct a single matrix of portfolio weights for the funds. Fund I will be represented in the first row and Fund II will be represented in the second row. b. Construct a column vector of stock returns. c. Multiply the weights matrix by the returns vector to obtain a column vector for returns on the two funds. d. Using matrix notation, demonstrate how one wouldfindthe return on the investor's overall portfolio if it were equally invested in the two funds. 4.3 A stock portfolio P is comprised of three stocks, A, B and C. The expected returns for the securities are .05 for Stock A, .08 for Stock B and .18 for Stock C. The variance of returns for Stock A is .01, .16 for Stock B and .25 for Stock C. The covariance between returns on Stocks A and B is .02, .04 between Stocks A and C and .10 between Stocks B and C. Stock A comprises 20% of the portfolio, B comprises 30% of the portfolio and C comprises the remaining 50%. a. Prepare an expected returns vector for the securities which comprise the portfolio. b. Noting that o^ = ajti, and that the return covariance between any security and itself

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is its variance, prepare a covariance matrix for the securities that comprise this portfolio. c. Prepare a weights vector for the portfolio. d. Using appropriate matrices, find the expected return of the portfolio. e. Using appropriate matrices, find the variance of returns for the portfolio. 4.4 Invert the following matrices:

a.

1 2

b.

3 4

d.

-2

.04 .04 .04 .16

33.33 -8.33

3 J.

c.

2

e.

-8.33 8.33

1

2 0

0

2 4

0

~2

4 8 20

4.5 Solve the following for x:

.04 .04 x1

.01

.04 .16 x2

.11

C

=

x

s

4.6 Solve the following for x: .08 .08 .1 1]

x

i

0

.08 .32 .2 1

x

2

0

x3

.15

.1

.2

0 0

1

1

0 0

C

1

x L 4

x

=

s

4.7 Assume that there are two three-year bonds with face values equaling $1000. The coupon rate of bond A is .05 and .08 for bond B. A third bond C also exists, with a maturity of two years. Bond C also has a face value of $1000; it has a coupon rate of 11%. The prices of the three bonds are $878.9172, $955.4787 and $1055.419, respectively. a. What are the spot rates implied by these bonds? b. Find a portfolio of bonds A, B and C which would replicate the cash flow structure of bond D, which has a face value of $1000, a maturity of three years and a coupon rate of 3%. 4.8 A life insurance company expects to make payments of $30,000,000 in one year; $15,000,000 in two years; $25,000,000 in three years; and $35,000,000 in four years to satisfy claims of policyholders. These anticipated cash flows are to be matched with a portfolio of the following $1000 face value bonds:

80

Chapter 4 BOND

CURRENT PRICE

COUPON RATE

YEARS TO MATURITY

1

1000

.10

1

2

980

.10

2

3

1000

.11

3

4

1000

.12

4

How many of each of the four bonds should the company purchase to exactly match its anticipated payments to policyholders? 4.9 Security A will pay 5 in outcome 1, 7 in outcome 2 and 9 in outcome 3. Security B will pay 2 in outcome 1, 4 in outcome 2 and 8 in outcome 3. Security C will pay 9 in outcome 1, 1 in outcome 2 and 3 in outcome 3. Both securities A and C currently sell for $5 and Security B currently sells for $3. What would be the value of Security D which will pay 1 in each of the 3 outcomes? 4.10 Suppose the Nixon Company needs to determine how much long-term capital it needs to raise to finance a $400,000 capital expenditure. It will raise $60,000 of this total through an increase in its working capital. An additional sum will be raised by retaining some of its earnings (retained earnings, RE). The long-term capital will be raised externally by selling bonds at an interest rate of 10%. The Nixon Company may use the following equation to determine its external financing needs (EFN, or in this example, how much money it will borrow): EFN = $400,000 - $60,000 - RE where RE represents its anticipated retained earnings. The Nixon Company's retained earnings (RE) are a function of its external financing needs (or, after accounting for taxes and dividends, the impact of interest payments of borrowed money on retained earnings). Suppose the Nixon Company is expected to generate $250,000 in operating income before interest and taxes, operates in a 20% income tax bracket and will retain 50% of its after-tax earnings. Given that the company will pay interest at a rate of 10% on its debt, retained earnings can be determined as follows: RE = ($250,000 - .10EFN)(1 - .20)(1 - .50) RE = $100,000 - .04EFN Thus, to compute EFN, these two equations need to be solved simultaneously for EFN and RE. We can use matrix notation to solve simultaneously for EFN and RE. What is EFN? That is, how much money must the Nixon Company raise by selling long-term debt? 4.11 Rollins Company stock currently sells for $12 per share and is expected to be worth either $10 or $16 in one year. The current riskless return rate is .125. What would be the value of a one-year call with an exercise price of $8?

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Matrix Mathematics

4.12 Harper Company stock currently sells for $12 per share and is expected to be worth either $10, $16 or $25 in one year. The current riskless return rate is .125. A oneyear call with an exercise price of $15 currently sells for $3. What would be the value of a one-year call with an exercise price of $9? 4.13 Which of the following vectors are orthogonal to a vector of ones?

1=

1 9 -6

2=

-6 10 -4 .

3=

-1 ' -4 5

4.14 Suppose that a given portfolio A has an expected return of .08, beta with index 1 equal to 1.5 and a beta of 2 with respect to the second index. Portfolio B has an expected return of .18, has a beta of 3 with respect to the first index and a beta of 2.5 with respect to the second index. Portfolio C has an expected return of .08 and has betas equal to one with each of the two indices. What are the three-index expected values implied by the portfolio expected returns?

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5

Differential Calculus

5.A: FUNCTIONS AND LIMITS A function is a rule which assigns to each number in a set a unique second number. Functions are frequently represented by equations, graphs and tables. The following example is a "generic" functional relationship in equation form: y = f(x), which reads y is a function of x. If y increases as x increases, we say that y is a direct or increasing function of x. The following are examples where y is an increasing function of x:

Functions (a), (b), and (c) are linear functions; graphs depicting the relationships between x and y would be linear. Equation (d) represents an exponential function. If y decreases as x increases, we say that y is a decreasing or inverse function of x. The following are examples where y is an decreasing function of x:

Assume that y = f(x). If x approaches (gets closer to) some value a (without actually equalling a) causing y to approach L, we say that the limit of

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Chapter 5

f (x) as x approaches a equals L. The limit is expressed as follows: (5.1) lim/(*) = L Again, Equation (5.1) reads: The limit of f (x) as x approaches a is L. The following are examples of limits:

Thus, the limit of function (a) is 0; the limit of function (b) is e; the limit of function (c) is ein and the limit of function (d) is 2. APPLICATION 5.1: THE NATURAL LOG One useful application of the limit is the derivation of the number e, defined as follows:

Thus as m approaches infinity, the value of function (1 + l/m)m approaches e, which is approximated at 2.71828. The number e has a variety of uses in finance, including discount and terminal value functions involving continuous compounding of interest (see Chapter 3). 5.B: SLOPES, DERIVATIVES, MAXIMA AND MINIMA (Background reading: Section 5.A) Assume that y is a function of x; that is, y = f (x). Presumably, a change in x may affect a change in y. For example, if y = 2x, a change in x by one will cause a change in y by 2. Therefore, in this example, the slope of the function is 2. The slope itself of this function does not change as x changes, this function is said to be linear; y is a linear function of x. The slope m of any line is defined as follows: (5.2) where xx - XQ = Ax do not equal zero. In our example above where y = 2x, if XQ = 5 and Xj = 11, then y0 = 10, y{ = 22, Ax = 6 and Ay = 12. Clearly, Ay -5- Ax = 2, the slope of the function. This slope or rate of change in y is constant with respect to x.

Differential Calculus

85

Figure 5.1 Changing Slope

Consider a second function: y = 7x4, which is plotted in Figure 5.1. It should be clear from Figure 5.1 that the slope of the function y changes with each change in x. Because the slope of function y changes as x changes, Equation (5.2) cannot be used to determine its slope at a given point, except where the change in x, Ax is approaching zero. Define h as Ax as Ax approaches zero. The derivative from calculus can be used to determine rates of change or slopes. For those functions whose slopes are constantly changing, the derivative is to find an instantaneous rate of change; that is, the change in y induced by the "tiniest" change in x. Assume that y is given as a function of variable x. If x were to increase by a small (infinitesimal — that is, approaching, though not quite equal to zero) amount h, by how much would y change? This rate of change is given by the derivative of y with respect to x, which is defined as follows: (5.3) dy If an infinitesimal value h (that is, a value approaching zero) were added to x, y would change by the derivative of y with respect to x multiplied by the change in x:

Consider a second function y = f (x). The derivative of this function with respect to x is itself a function: dy/dx = f'(x). Whenever this derivative is

86

Chapter 5

Figure 5.2 Concave Up Function

positive, an infinitesimal increase in x will lead to an increase in y. The slope of the curve representing the function is positive at points where the derivative is positive. The slope of the function represented by Figure 5.1 is positive at all points (since x>0 at all points in the figure). In Figure 5.2, the slope is positive to the right of the minimum point. However, in Figure 5.3, the slope is positive to the left of the maximum point. Whenever the derivative is negative, an infinitesimal increase in x will lead to an decrease in y. Such is the case to the left of the minimum point in Figure 5.2 and to the right of the maximum point Figure 5.3 Concave Down Function

Differential Calculus

87

in Figure 5.3. A derivative equal to zero implies that an infinitesimal change in x leads to no change in y. A zero derivative may suggest a minimum or maximum value for y, as is the case in Figures 5.2 and 5.3. In many instances, one may determine when a given function is maximized or minimized by determining when the derivative of the function has a value equal to zero. As we shall see later, this is the case for the functions represented in Figures 5.2 and 5.3. While the function representing the derivative (or, first derivative) indicates the slope of the original function, the function representing the derivative of the derivative (the second derivative) indicates the slope of the first derivative function and the concavity (change in the slope) of the original function. Notice that the slopes in Figures 5.1 through 5.3 change as x changes. The rate of change in a slope is determined by the second derivative: (5.4) Thus, the second derivative of a function is simply the derivative of the first derivative. The function represented in Figure 5.1 has a positive second derivative when x > 0 as suggested by the fact that it appears concave up, indicating that its slope is decreasing as x increases. Note, however, that its first derivative is always positive. The function represented in Figure 5.2 has a positive second derivative as suggested by the fact that it appears concave up, indicating that its slope is increasing as x increases. In fact, its slope is negative when x is small, is zero when the function is minimized and becomes positive when x rises above that level which minimizes y. The function represented in Figure 5.3 has a negative second derivative as suggested by the fact that it appears concave down, indicating that its slope is decreasing as x increases. In fact, its slope is positive when x is small, is zero when the function is maximized and becomes negative when x rises above that level which maximizes y-

APPLICATION 5.2: UTILITY OF WEALTH Let us assume that a given investor can associate some level of personal satisfaction (or, in economic terms, utility) with any given wealth level. In fact, it may be reasonable to assume that this level of utility increases as the investor's level of wealth increases; that is, an investor becomes happier as his level of wealth increases. Let us further assume that we can mathematically define the relationship between an investor's wealth and utility levels; that is, utility is a function of wealth (we will maintain the assumption throughout this chapter that W > 0):

88

Chapter 5

If we assume utility to be measurable, an example of such a utility function might be: An investor whose wealth is given by W will have a utility level equal to onehalf times the square root of his wealth level. Thus, an investor with this particular utility of wealth function whose wealth level is $10,000 would have a utility level of 50. If the investor's wealth level were to increase to $14,400, his utility level would increase to 60. Thus, this investor's utility level increases as he becomes wealthier. That is, there is a positive relationship between wealth and utility (the slope [f'(w)] of the utility function is positive): The slope of the utility of wealth function may also be described as the marginal utility of wealth function. Since the derivative of utility with respect to wealth is positive, the investor has positive marginal utility with respect to wealth. However, as the wealth level of the investor increases, further increases in wealth lead to progressively smaller increases in his utility level. Therefore, the investor experiences diminishing marginal utility with respect to wealth and the slope of his utility of wealth function becomes smaller: Thus, the second derivative of this investor's utility of wealth function is negative; its curve is concave down as in Figure 5.4. Figure 5.4 Utility of Wealth and Risk Aversion

Utility of wealth function for risk averse individual: f (W) > 0; f'(W) < 0 eg: Uv .5W

Differential Calculus

89

An individual experiences diminishing marginal utility with respect to wealth when an additional dollar is less important to him if he were wealthy than the same dollar would be if he were poor. For example, a gift of one dollar would probably provide more benefit to a homeless person than to a wealthy individual. Any individual who prefers more wealth to less will have an upward sloping utility of wealth function. His function will be concave down if he has diminishing marginal utility with respect to increases in wealth (see Figure 5.4). We demonstrate later that this type of individual is also risk averse. Most investors are probably of this type. An individual with constant marginal utility with respect to wealth will have a linear utility function and will be neutral towards risk. An individual with increasing marginal utility with respect to wealth will have a concave up utility function and will seek situations of risk. 5.C: DERIVATIVES OF POLYNOMIALS (Background reading: Section 5.B) One type of function which appears regularly in finance is the polynomial function. This type of function defines variable y in terms of a coefficient c (or series of coefficients Cj), variable x (or series of variables Xj) and an exponent n (or series of exponents nj). Strictly speaking, the exponents in a polynomial equation must be non-negative integers; however, the rules that we discuss here still apply when the exponents assume negative or non-integer values. Where there exists one coefficient, one variable and one exponent, the polynomial function is written as follows: (5.5) For example, let c = 7 and n = 4. Our polynomial function is written as follows: y = 7x4. The derivative of y with respect to x in Equation (5.5) is given by the following function:1 (5.6) Taking the derivative of y with respect to x in our example, we obtain: dy/dx = 7 . 4 . x3 = 28x3. Note that this derivative is always positive when x > 0; thus the slope of this curve is always positive when x > 0. Consider a second polynomial with more than one term (m terms total). In this second case, there will be one variable x, m coefficients (Cj) and m exponents (nj): (5.7)

Chapter 5

90 The derivative of such a function y with respect to x is given by:

(5.8) That is, simply take the derivative of each term in y with respect to x and sum these derivatives. Consider a second example, a second order (the largest exponent is 2) polynomial function given by: y = 5x2 - 3x + 2. The derivative of this function with respect to x is: dy/dx = lOx - 3. This function is plotted by Figure 5.2. This derivative is positive when x > .3, negative when x < .3 and zero when x = .3. Thus, when dy/dx > 0, y increases as x increases; when dy/dx < 0, y decreases as x increases, and when dy/dx = 0, y may be either minimized or maximized. The slopes in Figure 5.2 are consistent with these derivatives. Also notice that y is minimized when x = .3; at this point, dy/dx = 0. As suggested above, derivatives can often be used to find minimum and maximum values of functions. To find the minimum value of y in function y = 5x2 - 3x + 2, we set the first derivative of y with respect to x equal to zero and then solve for x. For our example, the minimum is found as follows: 10*-3 = 0 10* = 3 3 x = — 10 In order to ensure that we have found a minimum (rather than a maximum), we check the second derivative. The second derivative is found by taking the derivative of the first derivative. If the second derivative is greater than zero, we have a minimum value for y (the function is concave up). When the second derivative is less than zero, we have a maximum (the function is concave down). If the second derivative is zero, we have neither a minimum nor a maximum. The second derivative in the above example is given by: d2y/dx2 = 10, also written f"(x) = 10. Since the second derivative 10 is greater than zero, we have found a minimum value for y. In many cases, more than one "local" minimum or maximum value will exist. Consider a third example where our second order polynomial is given: y = -7x 2 + 4x + 5. The first derivative is: dy/dx = -14x + 4. Setting the first derivative equal to zero, we find our maximum as follows: -14JC + 4 =

-14* x

o

= -4 4 = — 14

Differential Calculus

91

We check second order conditions (the second derivative) to ensure that this is a maximum. The second derivative is: dVd*2 = ""14. Since —14 is less than zero, we have a maximum at 4/14. APPLICATION 5.3: MARGINAL UTILITY In the Utility of Wealth Application example, we defined a utility of wealth function for a particular individual as follows: U = .5 •

fW

We rewrite this utility function in polynomial form: _i

AW) = .5 • W2 Using the polynomial rule, we find the first derivative of the utility function as follows:

This derivative is positive, indicating that utility increases The second derivative of the utility function is found:

Thus, this utility function is concave down, indicating diminishing marginal utility with respect to wealth. Downward concavity in a utility of wealth function is consistent with risk aversion. This is because potential wealth increases associated with actuarialy fair gambles result in smaller utility changes than potential wealth decreases. The use of concavity in measuring and pricing risk aversion is discussed in Section 5.12. APPLICATION 5.4: THE BAUMOL CASH MANAGEMENT MODEL The Baumol Cash Management Model may be used by firms in the determination of optimal cash balances. This model assumes that one may associate two types of costs with obtaining and maintaining cash balances, brokerage costs and foregone interest costs. Brokerage Costs The model assumes that the firm's only source of cash is derived from the liquidation of marketable securities. Firms are able to generate interest or

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returns on its marketable securities. Thus, for example, if the firm realizes revenues from sales, these revenues are immediately deposited into marketable securities such that they produce interest for the firm. When the firm needs cash, it liquidates marketable securities. This sale of marketable securities will result in a brokerage commission B; the model assumes that this commission is fixed with respect to the level of securities sold. Thus, the firm will attempt to minimize its brokerage commissions by minimizing the number of times it must obtain cash by liquidating marketable securities. This suggests that the firm has an incentive to maintain high cash balances so that it does not incur excessive brokerage commissions by frequently having to order additional cash. Foregone Interest Costs This cash maintained by the firm generates no return. This provides an incentive for the firm to maintain as small a cash balance as is necessary to operate. Maintenance of a small cash balance implies that the firm maintains high levels of interest bearing marketable securities. The firm's problem is that it must minimize the sum of two costs, brokerage costs which are decreasing in the cash balance level and foregone interest costs which are increasing in the cash balance level. Brokerage, foregone interest and total costs are depicted in Figure 5.5 as a function of cash balance levels. Assume that the firm's cash usage is constant; that is, the firm uses the same level of cash each day. Further, for the sake of simplicity, assume that the firm Figure 5.5 Costs Associated with Cash Balances Costs

Total

Cost: $ minimum cost

Foregone Interest Cost i

Transactions Costs

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Figure 5.6 The Baumol Cash Management Model Cash Balance

C/2

Time permits its cash level to decline to zero before it replenishes its balances. Thus, the firm will start a given period with a cash balance of c and use this cash in equal amounts each day or period. When the cash balance has declined to zero, the firm will order additional cash by liquidating marketable securities, increasing its cash balance back to c. This process is depicted in Figure 5.6. Since cash usage is constant each period, the average cash balance equals the maximum balance, c, minus the minimum balance, 0, divided by 2. The firm's foregone interest or return cost is simply the average cash balance times the interest or return rate the firm could have earned had it otherwise invested in marketable securities: foregone interest cost - —

i = avg - i

where i is the interest rate foregone by the firm when maintaining cash balances. When the firm's cash balance declines to zero, it cannot operate unless it obtains additional cash. To obtain this cash, the firm engages its broker to sell marketable securities; the receipts of the sale are used to replenish cash balances and the process continues. Each time the broker is engaged to sell marketable securities enabling the firm to replenish its cash balances, a transactions cost is incurred. In the Baumol Model, any revenues received by the firm are immediately converted into return-bearing assets; thus, the firm must liquidate a portion of its return-bearing assets to obtain cash. When the firm sells marketable securities to raise cash, it will incur the fixed brokerage fee B for

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each transaction. The total number of transactions executed by the firm per time period is simply the total cash demand X over that period divided by the cash order quantity c. Therefore, the total transactions costs incurred by the firm over the relevant time period is determined: X transaction costs = — • B c The total costs $ associated with cash balances for the firm is simply the sum of its foregone interest and brokerage costs:

The firm's objective will be to select a cash order quantity c so as to minimize $, the total costs associated with cash balances. To minimize $, we need only to find the derivative of $ with respect to c, set it equal to zero and then solve for c. First, for the sake of simplicity, re-write $ in polynomial format:

Now we find the derivative of $ with respect to c and set it equal to zero:

We now solve algebraicly for c:

(5.9) Use of the Baumol Cash Management Model (Equation 5.9) will minimize the costs associated with cash balances under our assumptions. APPLICATION 5.5: DURATION (Background reading: Sections 2.A, 2.C and 5.C) Bonds and other debt instruments issued by the United States Treasury are generally regarded to be free of default risk and of relatively low liquidity risk.

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However, these bonds, particularly those with longer terms to maturity are subject to market value fluctuations after they are issued, primarily due to changes in interest rates offered on new issues. Generally, interest rate increases on new bond issues decrease values of bonds which are already outstanding; interest rate decreases on new bond issues increase values of bonds which are already outstanding. The duration model is intended to describe the proportional change in the value of a bond that is induced by a change in interest rates or yields of new issues. Many analysts use present value models to value treasury issues, frequently using yields to maturity of new treasury issues to value existing issues with comparable terms. It is important for analysts to know how changes in new-issue interest rates will affect values of bonds with which they are concerned. Bond duration measures the proportional sensitivity of a bond to changes in the market rate of interest. Consider a two-year 10% coupon treasury issue which is currently selling for $986.48. The yield to maturity y of this bond is 12%. Default risk and liquidity risk are assumed to be zero; interest rate risk will be of primary importance. Assume that this bond's yield or discount rate is the same as the market yields of comparable treasury issues (which might be expected in an efficient market) and that bonds of all terms to maturity have the same yield. Further assume that investors have valued the bond such that its market price equals its present value; that is, the discount rate k for the bond equals its yield to maturity y. If market interest rates and yields were rise for new treasury issues, then the yield of this bond would rise accordingly. However, since the contractual terms of the bond will not change, its market price must drop to accommodate a yield consistent with the market. Assume that the value of an n-year bond paying interest at a rate of c on face value F is determined by a present value model with the yield y of comparable issues serving as the discount rate k: (5.10) Assume that the terms of the bond contract, n, F and c are constant. Just what is the proportional change in the price of a bond induced by a proportional change in market interest rates (technically, a proportional change in [1+y])? This may be approximated by the bond's Macaulay Simple Duration Formula as follows: (5.11) Equation (5.11) provides a good approximation of the proportional change in the price of a bond in a market meeting the assumptions described above induced by an infinitesimal proportional change in (1 + y). To compute the bond's

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sensitivity, we first rewrite Equation (5.10) in polynomial form (to take derivatives later) and substitute y for k (since they are assumed to be equal): (5.12) First, find the derivative of PV with respect to (l+y):

(5.13) Equation (5.13) is rewritten:

(5.14) Since the market rate of interest is assumed to equal the bond yield to maturity, the bond's price will equal its present value. Next, multiply both sides of Equation (5.14) by (l+y)-s-P 0 to maintain consistency with Equation (5.11):

(5.15) Thus, duration is defined as the proportional price change of a bond induced by a infinitesimal proportional change in ( l + y ) or 1 plus the market rate of interest:

(5.16) Since the market rate of interest will likely determine the yield to maturity of any bond, the duration of the bond described above is determined as follows from Equation (5.16):

(5.17) Dur = (1"12) Gi!# <1+-12>2 = -1.87 This duration level of -1.87 suggests that the proportional decrease in the value of this bond will equal 1.87 times the proportional increase in market interest rates. This duration level also implies that this bond has exactly the same interest rate sensitivity as a pure discount bond (a bond making no coupon payments) which matures in 1.87 years.

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Application of the Simple Macaulay Duration model does require several important assumptions. First, it is assumed that yields are invariant with respect to maturities of bonds; that is, the yield curve is flat. Furthermore, it is assumed that investors' projected reinvestment rates are identical to the bond yields to maturity. Any change in interest rates will be infinitesimal and will also be invariant with respect to time. The accuracy of this model will depend on the extent to which these assumptions hold. APPLICATION 5.6: BOND PORTFOLIO IMMUNIZATION (Background reading: Application 5.5) Earlier, we discussed bond portfolio dedication, which is concerned with matching terminal cash flows or values of bond portfolios with required payouts associated with liabilities. This process assumes that no transactions will take place within the portfolio and that cash flows associated with liabilities will remain as originally anticipated. Clearly, these assumptions will not hold for many institutions. Alternatively, one may hedge fixed income portfolio risk by using immunization strategies, which are concerned with matching the present values of asset portfolios with the present values of cash flows associated with future liabilities. More specifically, immunization strategies are primarily concerned with matching asset durations with liability durations. If asset and liability durations are matched, it is expected that the net fund value (equity or surplus) will not be affected by a shift in interest rates; asset and liability changes offset each other. Again, this simple immunization strategy is dependent on the following: 1. 2. 3. 4.

Changes in (1 + y) are infinitesimal. The yield curve is flat (yields do not vary over terms to maturity). Yield curve shifts are parallel. Only interest rate risk is significant.

APPLICATION 5.7: PORTFOLIO RISK AND DIVERSIFICATION (Background reading: Section 3.A) Here, we shall demonstrate that the risk of a portfolio decreases as the number of securities in the portfolio increases. First, in Section 3. A, we defined portfolio variance as follows:

(A)

For sake of simplicity, we shall assume the following:

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1. The portfolio with which we will work is equally weighted; that is, Wj = Wj = 1/n for each security weight. 2. All securities have the same variance, ax. 3. Each security has the covariance with every other security. This covariance will be equal to the average covariance between pairs of securities. 4. The average security variance in an equally weighted portfolio is larger than the average covariance with other securities. This condition must hold in all cases unless all security returns are perfectly correlated. In the case where all security returns are perfectly correlated, return variances will equal return covariances. These assumptions permit us to rewrite Equation A as follows:

First, note that E-^lAOa- is the mean of the security variances. Also, note that there are n terms related to security variances and n(n-1) covariance terms. The average covariance is written:

Since the average covariance term will be added (n-1) times, we now write portfolio variance as follows:

To demonstrate that portfolio variance decreases as n increases, we merely demonstrate that the derivative of Q\ with respect to n is negative:

which will be true whenever the average security variance exceeds the average covariance between different securities. This will be the case whenever the correlation coefficient between security returns is less than one. Note from Equation D that as the number of securities in the portfolio approaches infinity, the portfolio's risk approaches the average covariance between securities. Thus, only covariance risk is significant for large, welldiversified portfolios. If security returns are entirely independent (ai} = 0), portfolio risk approaches zero as the number of securities included in the portfolio tends toward infinity.

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5.D: PARTIAL DERIVATIVES (Background reading: Section 5.C) A multivariate function exists when a dependent variable y is a function of a series of independent variables (e.g, y = f[xltx2]). The partial derivative (e.g, Sy/Sxj) of a function is concerned with the rate of change in the dependent variable induced by change in one of its independent variables, while holding other variables constant. The following represents a function and its relevant partial derivatives:

Many finance applications are concerned with changes in the dependent variable induced by simultaneous changes in the independent variables. One may use the total derivative dy to indicate changes in the dependent variable y induced by changes in one or more of n independent variables XJ:

In the example above, the total derivative would be determined as follows:

APPLICATION 5.8: DERIVING THE SIMPLE OLS REGRESSION EQUATION (Background reading: Sections 2.F, 3.B and 4.E) A simple ordinary least squares regression is used to fit a line to a series of data points. In this section, we refer to a as the fitted value for the vertical intercept of the regression line; B is the fitted value for the slope of this line. One attempts to maintain small vertical distances (errors: €;) of predicted dependent variable values (Yj) on the line from actual data point values (Yj). The following equations represent the line fit by the regression and the actual or empirical values of the dependent variable: (5.18)

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Assume that each of the data points expressed in terms of Y and X are known. We define our line in terms of regression coefficients a and B. Our objective when obtaining regression coefficients is to minimize the sum of error terms squared. First, we solve Equation (5.18) for e{i which is then squared and summed to obtain:

(5.19) We rewrite Equation (5.19) as follows: (5.20) Note that if we were to attempt to minimize the sum of error terms themselves, we would find that our forecasted values Y{ equal infinity for all i; this causes each of our error terms to equal minus infinity. We minimize squared error terms so that we do not fit a line such that all of the error terms equal minus infinity; we want error terms to be close to zero. Squaring the error terms results in positive values which we will want to minimize. Also, we show later that the normal distribution is defined in terms of squared variable values, adding to the attractiveness of working with least squares. Minimizing the squared error terms is analogous to fitting the line as close as possible to the data points. To minimize the sum of error terms squared, we find partial derivatives with respect to a and B and set them equal to zero: (5.21.a)

(5.21.b) Next, we cancel 2's to simplify and solve equation (5.21.a) for EY, and equation (5.21.b) for EXjYj to obtain our set of normal equations: (5.22)

To finally obtain our regression coefficients, we solve the above normal equations simultaneously for a and B. We can show algebraicly that solving the

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above system is identical to solving the following for regression coefficients a and B:

(5.23)

APPLICATION 5.9: DERIVING MULTIPLE REGRESSION COEFFICIENTS (Background reading: Sections 2.F, 3.C, 4.E and Application 8) The multiple ordinary least squares regression expresses a single dependent variable y in terms of a series of independent variables and their coefficients:

The regression coefficients are derived in much the same manner as the regression coefficients in the simple regression. Squared error terms are summed and minimized by setting partial derivatives with respect to coefficients equal to zero. This set of equations, after slight rearrangement, is referred to as normal equations. We continue by solving the system of normal equations simultaneously for the coefficients. The following represents the system of normal equations:

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The above system of equations may be represented in matrix form. One may solve for a and bx to Bn as follows:

This system may be rewritten as follows: where m is the number of independent variables and n is the number of observed data points and matrices are defined as follows:

6

This is the classic Ordinary Least Squares Regression Model.

5.E: THE CHAIN RULE, PRODUCT RULE AND QUOTIENT RULE (Background reading: Section 5.D) The functions presented in Section 5.C are all given in polynomial form. Many functions are not or cannot be presented in this manner. The chain rule may be used to find derivatives for some of these functions. For example, consider the following function: Although the this function can be reduced to polynomial form (y = 245x2 + 21 Ox + 45; its derivative using the polynomial rule is 490x + 210), we will apply the chain rule to find its derivative. The first step in applying the chain rule here is to define a function u such that u = (3 + 7x), we may define y as

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y = 5u2. Furthermore, we note that du/dx = 7 and dy/du = lOu. The chain rule is quite simple, though it does have powerful implications: (5.24)

That is, if y can be written as a function of function u, which itself is a function x, then the derivative of y with respect to x equals the derivative of y with respect to function u multiplied by the derivative of function u with respect to x. In our example, we find the derivative of y with respect to x as follows:

Another useful tool from calculus is the product rule, which may be applied to a function such as y = (3x + 7)(2x + 5). Define a function u as (3x + 7) and a function v as (2x + 5). The product rule, defined as follows may be applied to find the derivative of function y where function y equals function u times function v (y = u • v): (5.25) Thus, the derivative of y with respect to x may be found as follows:

Another useful tool from calculus is the quotient rule, which may be applied to a function such as y = (3x + 7)/(2x + 5). Again, define a function u (the numerator) as (3x + 7) and a function v (the denominator) as (2x + 5). The quotient rule, defined as follows may be applied to find the derivative of function y: (5.25) The quotient rule states that the derivative of y with respect to x equals the derivative of the numerator with respect to x times the denominator minus the derivative of the denominator with respect to x times the numerator all divided

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by the denominator squared. Thus, the derivative of y with respect to x is found as follows:

APPLICATION 5.10: PLOTTING THE CAPITAL MARKET LINE (Background reading: Sections 4.A and 4.D) The Efficient Frontier represents the risk-return combinations (in standard deviation-expected-return space) of the most efficient portfolios available to an investor. That is, the Efficient Frontier contains the risk-return coordinates of each portfolio which minimizes risk given a portfolio return level (alternatively, the Efficient Frontier contains the risk-return coordinates of each portfolio which maximizes return given a risk level. If a riskless asset is part of the investor's opportunity set, the Efficient Frontier is given by a line which has a vertical intercept equal to the riskless return rate and extends through the risk-return coordinates of the market portfolio. In this section, we present a graphical derivation of the Capital Market Line from the Efficient Frontier. Based on this intuitive derivation, we discuss an analytical procedure for its derivation using a simple example. Consider, for example, a market where the average coefficient of correlation between returns on securities is .4. (This is not a particularly unrealistic assumption.) For the sake of simplicity, assume that there exist in this market five securities, (A) through (E), whose return-risk combinations are given in Figure 5.7. First, combine securities (A) and (B) into a portfolio. Return and risk combinations of the resultant portfolio will fall somewhere on the curve extending between the two securities, depending on their relative weights (see Figure 5.8). Similarly, securities (B) and (C) can be combined into portfolios as can securities (C) and (D), and (D) and (E) (see Figure 5.7). We have constructed a series of curves representing risk-return combinations of an infinite number of two security portfolios. These resultant portfolios themselves can be combined into additional portfolios. For example, consider portfolios (AB) and (BC) in Figure 5.9. These portfolios can be combined into further portfolios as can portfolios (BC) and (CD) as well as (CD) and (DE). The resultant portfolios can all be combined into additional portfolios (Figure 5.9). Notice that as the portfolios become more diversified, they become more efficient. Thus, the curves representing the risk-return combinations of these portfolios fall further to the northeast on the risk-return space. However, the benefits of this diversification must reach a limit. This is because the portfolios that are being combined are more correlated than the individual securities that they contain. On the curve indicating this limit, further diversification cannot result in more efficient portfolios. The upward sloping portion of this curve is

Differential Calculus

Figure 5.7 Portfolio Possibilities Frontier: Two Securities

Figure 5.8 Portfolio Possibilities: Pairwise Combinations

105

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Figure 5.9 The Efficient Frontier Efficient Frontier/

FeasibleRegion

called the Efficient Frontier. The most efficient portfolios of risky assets will have risk-return combinations falling on the Efficient Frontier. Suppose that there exists a riskless asset whose variance or standard deviation of expected returns is zero. Thus, an investor purchasing such an asset will certainly receive the return he originally expected. Though this asset is riskless, the investor will require a return, compensating him for inflation and his time value of money. This risk-free rate of return (rf) can be approximated with the short-term treasury bill rate. The risk-free asset can be combined with any portfolio of risky assets. Such a portfolio will have a risk-return combination which is simply a weighted average of the risky portfolio's and the risk-free asset's risk-return combinations. For example, consider a portfolio of risky assets with expected return and standard deviation levels of 10% and 20% and a risk-free asset with an expected return of 5% (see Figure 5.10). If the portfolio and the risk-free asset were combined into a new portfolio with equal weights (wf = wm = .5), the resultant portfolio would have expected return and standard deviation levels of 7.5% and 10%:

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Differential Calculus Figure 5.10 Combining the Portfolio of Risky Assets with the Riskless Asset E[RP] 20 Borrowing Portfolios

10

wf = 0

075

w f = .5

wf = 1 .10

.20

.60

^P

Notice that the correlation coefficient between returns on any risky asset and the risk-free asset must be zero. Thus, both portfolio expected returns and portfolio standard deviations will be a linear combination of the individual security returns and standard deviations only when (p^ = 1 or, as in this case, when a risk-free asset is combined with a risky investment. If an investor has the opportunity to borrow money at the risk-free rate of return (rf), he has the opportunity to create a negative weight (wf) for the risk-free asset. For example, if an investor had an initial wealth level of $1000, but wished to invest $3000 in a risky asset with an expected return of 10%, he could borrow $2000 at the risk-free rate of 5% if the lender were certain the investor would fulfill his debt obligation. Since the investor is borrowing money rather than lending (buying treasury bills is, in effect, lending the government money), the weight associated with the risk-free asset is negative. Because the total sum invested in the risky asset is three times as great as the investor's initial wealth level, (wA) is equal to 3. The investor's expected portfolio return level is 20%, higher than the return of either of the assets comprising the portfolio: E[Rp] = (-2 • .05) + (3 • .10) = .20 Notice that the sum borrowed is twice as great as the investor's initial wealth level, thus (wf) is equal to -2. The standard deviation of returns on the portfolio is .6:

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Figure 5.11 Combinations of Risky Portfolios with Riskless Assets

Notice that the portfolio standard deviation is higher than the standard deviations of either of the assets comprising the portfolio. Therefore, borrowing money (creating leverage) permits the investor to increase his expected returns; however, he must also face additional risk. Consider an investor who has the opportunity to invest in a combination of a risk-free asset and one of several risky portfolios (A) through (E) depicted in Figure 5.11. Which of these five portfolios is the best to combine with the risk-free asset? Notice that the portfolios with risk-return combinations on the line connecting the risk-free asset and portfolio (C) dominate all other portfolios available to the investor. Thus, any portfolio whose risk-return combination falls on lines extending through portfolios (A), (B), (D), and (E) will be dominated by some portfolio whose risk-return combination is depicted on the line extending through portfolio (C). This line has a steeper slope than all other lines between the risk-free asset and risky portfolios. The investor's objective is to choose that portfolio of risky assets enabling him to maximize the slope of this line; that is, the investor should pick that portfolio with the largest possible (0p), where (9p) is defined by Equation (5.27):

(5.27)

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Therefore, the investor should invest in some combination of portfolio (C) and the risk-free asset. If the curve connecting portfolios (A) through (E) were the Efficient Frontier, then portfolio (C) would be referred to as the market portfolio. This is because every risk averse investor in the market should select this portfolio of risky assets to combine with the riskless asset. Notice that the line extending through portfolio (C) is tangent to the curve at point (C). The best portfolio of risky assets to combine with the risk-free security lies on the Efficient Frontier, tangent to the line extending from the risk-free security. This line is referred to as the Capital Market Line. Notice that portfolios on the Capital Market Line dominate all portfolios on the Efficient Frontier. If a risk-free security exists, the Capital Market Line represents risk-return combinations of the best portfolios of securities available to investors. Thus, an investor's risk-return combinations are constrained by the Capital Market Line. The most efficient portfolio on the Efficient Frontier to combine with the riskless asset is referred to as the market portfolio (depicted by [M] in Figures 5.11 and 5.12). Thus, the market portfolio lies at a point of tangency between the Efficient Frontier and the Capital Market Line. All investors should hold portfolios of risky assets whose weights are identical to those of the market portfolio. The Capital Market Line combines the market portfolio with the riskless asset. This line can be divided into two parts: the lending portion and the borrowing portion. If an investor invests at point (M) on the Capital Market Figure 5.12 The Efficient Frontier and Capital Market Line Capital Market Line Efficient Frontier Market Portfolio

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Line, all of his money is invested in the market portfolio. If he invests to the left of (M), his portfolio is a lending portfolio. That is, he has purchased treasury bills, in effect, lending the government money, and invested the remainder of his funds in the market portfolio. If he invests to the right of point (M), he has a borrowing portfolio. In this case, he has invested all of his funds in the market portfolio and borrowed additional money at the risk-free rate to invest in the market portfolio. Consider a second simple example where there exist two risky securities 1 and 2 in the stock market of Noplacia. A particular investor in this market has projected the following characteristics for these stocks along with a riskless treasury bill:

There also exists a riskless treasury instrument (bill) available for investors of Noplacia. The expected return or implied interest rate on this bill is 8%. Given this interest rate and the above stock projections, determine: 1. the stock weightings for the optimal portfolio of risky securities for this investor 2. the expected return of his portfolio of stocks 3. the risk of his stock portfolio, as measured by standard deviation 4. the characteristics of the Capital Market Line faced by this investor; that is, the equation for the Capital Market Line Since our objective is to select a portfolio of the two risky assets such that the slope of the Capital Market Line is maximized, we will select stock portfolio weights such that 0 P is maximized. To accomplish this, we will find partial derivatives of 0 p with respect to weights of each of the two stocks, set the partial derivatives equal to zero and solve for the weight values Wj and w2. First, we write 0 p for the simple two stock portfolio as follows:

Next, we use the quotient rule to find the derivative of 9 P with respect to Wj and w2:

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However, we need to use the chain rule to find the derivative of the denominator with respect to Wj and w2:

Notice from equation (A) that the derivative of (E[Rp]-rf) with respect to v/x equals (E[R,]-rf). Next, we substitute our results of equation set (C) into equation set (B) making use of the far right hand side of equation (A):

Because the derivatives from Equation Set D are both set equal to zero, we may multiply the numerator by af and maintain the equality. Next, we rewrite Equation Set D as follows:

(E)

To continue the process of simplification, define the variable z, to be Wj(E[Rp]rf)/ffp and rewrite equation set (D) as follows:

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Substituting appropriate values from our example, we find:

which yields:

Solving the above simultaneously yields zx = 1.174603 and Zj = .698412. Since E[Rp], rf and crp are the same for both zx and z^ portfolio weights vtx and w2 will be linearly related to their z values. Thus, the portfolio weights are determined as follows:

The return and risk levels of the portfolio (m) of risky stocks are simply:

Thus the answers to the four problems proposed earlier are as follows:

4. The equation for the Capital Market Line is given as follows:

B

This process is easily expanded to include as many securities as may exist in the market. Matrix mathematics such as that presented in Chapter 4 would simplify computations. The Efficient Frontier can be plotted by varying the riskless rate; an additional "market portfolio" is obtained each time a new riskless return is used in the computations. 5.F: TAYLOR SERIES EXPANSIONS (Background reading: Section 5.C) Taylor series expansions are frequently used in finance to evaluate a function f(xx) at a point \x which is different from a second point XQ at which

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f(xo) has already been evaluated. That is, the Taylor series may be used to approximate a rate of change in f(x) induced by a change in x. The derivative was used earlier to determine a rate of change in f(x), however, it is normally accurate only for infinitesimal changes in x. The Taylor series approximation may be used for finite changes in x. The Taylor series is defined as follows for a function f(x) which is differentiable n times:

(5.28)

For example, consider the function y = 4x3. Let x0 = 5, such that we have f(x0) = 500, f (xo) = 12xo2 = 300, f"(x0) = 24XQ = 120, f'"(x0) = 24, and all higher order derivatives equal to zero. Now, suppose we wish to increase x by Ax = 8 to X! 13. The Taylor series expansion may be used to evaluate Xj as follows:

In this example, our approximation for f(xx) was exact because f(x) was differentiable only three times (n=3) and our approximation used all three derivatives. In many cases, we will be able to obtain reasonable, though not precise approximations with fewer than n derivatives. Generally, Taylor series approximations will improve as order of the approximating equation increases (as n increases). If the equation is differentiable only n times, the Taylor series approximation of the nth order will be precise. APPLICATION 5.11: CONVEXITY AND IMMUNIZATION (Background reading: Application 5.5) Earlier, we used duration to determine the approximate change in a bond's value induced by a change in interest rates (1 + y). However, the accuracy of the duration model is reduced by finite changes in interest rates, as we might reasonably expect. Duration may be regarded as a first order approximation (it only uses the first derivative) of the change in the value of a bond induced by a change in interest rates. Convexity is determined by the second derivative of the bond's value with respect to (1 4- y). The first derivative of the bond's price with respect to (1 + y) is given:

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(5.29) We find the second derivative by determining the derivative of the first derivative as follows:

(5.30)

Convexity is merely the second derivative of P0 with respect to (1 + y) divided by P0. The first two derivatives may be used in a Taylor series to approximate new bond prices induced by changes in interest rates as follows:

(5.31)

Consider a five-year ten percent $1000-face-value coupon bond currently selling at par (face value). We may compute the present yield to maturity of this bond as y0 = .10. The first derivative of the bond's value with respect to (1 +y) at y0 = .10 is found from Equation (5.29) to be 3790.786769 (duration is 3790.786769 - 1.1 -H 1000 = 4.169865446); the second derivative is found from Equation (5.30) to be 19,368.34238 (convexity is 19,368.34238 -*• 1000 = 19.36834238). If bond yields were to drop from .10 to .08, the actual value of this bond would increase to 1079.8542, as determined from a standard present value model. If we were to use the duration model (first-order approximation from the Taylor expansion, based only on the first derivative), we estimate that the value of the bond increases to 1075.815735. If we use the convexity model second-order approximation from Equation (5.30), we estimate that the value of the bond increases to 1079.689403. The second-order approximation may also be written as: (5.32)

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Note that this second estimate with the second-order approximation generates a revised bond value which is significantly closer to the bond's actual value as measured by the present value model. Therefore, the duration (Application 5.5 above) and immunization (Application 5.6 above) models are substantially improved by the second order approximations of bond prices (the convexity model). The fund manager wishing to hedge portfolio risk should not simply match durations (first derivatives) of assets and liabilities, he should also match their convexities (second derivatives). APPLICATION 5.12: RISK AVERSION COEFFICIENTS (Background reading: Application 5.3) This application is concerned with the measurement of investor risk aversion. Since one might expect that an investor is likely to prefer certainty to uncertainty, one might expect that he would require a premium to accept a risk of a given level (or pay a premium to eliminate a given risk). The higher the premium that an investor would require to accept a given risk, the more risk averse we can infer that he is. Assume that the investor selects his investment so as to maximize the expected utility level which he associates with his level of wealth W. Also, assume that his wealth level is subject to some uncertainty represented by 1 whose expected value is zero. Thus, 2 represents an actuarialy fair gamble or random number which can assume any value, but has an expected value of zero. Assume that the investor would be willing to pay a premium ir to eliminate this risk. Our problem here is to determine the maximum premium that he would be willing to pay; we will use the level of this premium to measure the investor's level of risk aversity. First, we note that the maximum premium that the investor is willing to pay would be that which equates the utility associated with his current uncertain level of wealth with the level of utility he would realize if he "bought insurance" and eliminated his risk: (5.33) Thus, expected utility is currently a function of the current level of wealth and the gamble; if the gamble is eliminated, utility will be a function of the current wealth level minus the insurance premium. Our problem is to solve this equality for 7r. We solve by performing a Taylor series expansion around both sides of the equality:

Since E[Z] = 0, <% = E[22] and E[2]U'(W) may be dropped from the equality. Following convention in the theoretical financial literature, we will approximate by dropping all of the left-hand side higher order terms not explicitly stated in the above equality. This convention is quite reasonable if we are willing to

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assume that the risk 2 is normally distributed since E[23] = E[Z4] = ... = 0 . We will also approximate the right-hand side of the utility function above by dropping all terms and derivatives of higher order than one to obtain:

Now, we solve for the risk premium as follows: (5.34) When used in this context, 7r is referred to as the Absolute Risk Aversion Coefficient, which indicates an investor's aversion to a given risk o\y based on his utility of wealth function U(w) and his current level of wealth.

5.G: THE METHOD OF LAGRANGE MULTIPLIERS (Background reading: Sections 4.E and 5.C) Earlier, we discussed the application of derivatives to problems requiring determination of minimums or maximums. In many realistic cases, optimization problems will impose constraints which must be considered. For example, portfolio managers may wish to define portfolio weights so as to minimize portfolio risk; however, they may be required to generate certain minimum income levels for their clients. Thus, risk minimization is subject to satisfying a return constraint. In a microeconomics framework, a consumer may wish to maximize his utility function subject to a budget constraint. The method of LaGrange multipliers is intended to enable optimization subject to constraints. This method merely revises the original function to be optimized by adding one term for each constraint representing the product a function of the constraint and a LaGrange multiplier X. The method of LaGrange multipliers will frequently work for nonlinear functions with more than one independent variable. For example, suppose we wish to minimize the function y = 3x2 - 12x - 4z subject to the constraint that 2x + 5z = 20. This problem may be written as follows:

which reads: Objective function: Minimize y = 3x2 - 12x - 4z subject to 2x + 5z = 20. The LaGrange function is constructed as follows:

Notice that we have added a second function to the original function to be minimized. This second function is the product of a single LaGrange multiplier

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X and the constraint, written such that its numerical value will be zero. The LaGrange multiplier may be interpreted as the change in y that will be induced by a change in the constraint (in this case, the 20); that is, X is a sensitivity variable. If the constraint is not binding, X will equal zero; if the constraint is binding in this example, 2x + 5z will equal 20. Therefore, in either case, the numerical value of the function that we have added to our original function to be optimized will be zero. The numerical value of our original function will be unchanged, although its derivatives will be affected by the constraint. We continue our problem by setting equal to zero partial derivatives of function L with respect to each of our variables x, z and X (i.e., finding firstorder conditions):

This system is written and solved in matrix format as follows:

The inverse for the coefficients matrix is found, and we solve for security weights and our LaGrange multiplier as follows:

We find that x = 52/30, z = 248/75 and X = -4/5. Our minimum value for y = -25.0133 and our constraint is binding since X is nonzero; 2x + 5z = 20. The term X may be regarded as a sensitivity coefficient. It indicates the change in y that would result from a relaxation in the constraint. For example, increasing the constraint by 1 would change our objective variable y by -4/5; decreasing our constraint by 1 would increase y by .8. If we were to relax our constraint by allowing 2x + 5z to be as small as 19, we would be able to change our values for x and z, increasing y to -24.2133. In many instances, the

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LaGrange multiplier will only indicate approximate changes in the objective variable induced by changes in the constraint. APPLICATION 5.13: OPTIMAL PORTFOLIO SELECTION (Background reading: Section 4.A) An investor has the opportunity to comprise a portfolio of three assets with the following expected return and standard deviation levels: Asset A B C

E[R] .10 .20 .30

o .20 .30 .40

The investor's objective is to construct a portfolio which enables him to minimize his risk level such that his expected portfolio return is at least 15%. He wishes to invest all $100,000 of his cash into this portfolio, hence, the weights of his portfolio should sum to 1. Our problem is to determine how much of his wealth the investor should place into each of the three securities. First, we compute relevant variances and covariances. Our objective function and constraints are given as follows:

The LaGrange function is constructed as follows:

Our first-order conditions are given by the following:

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119

This system is written and solved in matrix format as follows:

We now invert Matrix C then solve for Vector x. We find the following weights: wA = .625, wB = .25 and wc = .125; our LaGrange multipliers are: \x = .225 and X2 = .0525. The expected return and standard deviation of portfolio returns are .15 and .207665, respectively. The portfolio variance is .043125. Since the first LaGrange multiplier is .225, an increase by .01 in the return constraint would be expected to change portfolio weights and lead to an increase in portfolio variance of approximately .00225. We would find by actually inserting .16 into the return constraint of the LaGrange function that portfolio variance increases to .045642. APPLICATION 5.14: PLOTTING THE CAPITAL MARKET LINE, SECOND METHOD (Background reading: Applications 5.10 and 5.13) The Capital Market Line represents the return-risk combinations of the set of most efficient portfolios of risky and risk-free assets. With a known riskless rate of return, one can derive the Capital Market Line by simply determining the return-risk combination of a single efficient portfolio including risky assets. One

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can use the LaGrange function procedure described in Application 13 to determine the weights of an efficient portfolio with a target return rp > rf: (5.35) Consider an example from Application 5.10 example where there exist two risky securities A and B in the stock market of Noplacia. A particular investor in this market has projected the following characteristics for these stocks:

There also exists a riskless treasury instrument (bill) available for investors of Noplacia. The expected return or implied interest rate on this bill is 8%. Given this interest rate and the above stock projections, we can determine with the LaGrange optimization procedure: 1. 2. 3. 4.

the the the the

stock weightings for the market portfolio of risky securities expected return of the market portfolio of stocks risk of the market portfolio as measured by standard deviation equation for the Capital Market Line

Any mean-variance efficient (optimal) portfolio with some fraction invested in the riskless asset will have standard deviation and expected return coordinates lying on the Capital Market Line. Suppose, for example, that we set our target return for a particular mean-variance efficient portfolio Q equal to 10%. Security weights for this optimal portfolio are determined by the same LaGrange optimization procedure described in Application 5.13 above. Our objective function and constraints are given as follows:

The LaGrange function is constructed as follows:

Our first order conditions are given by the following:

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121

Solution of the above set of equations simultaneously reveals that wA = .201, wB = .120, wf = .679, \x = .342 and X2 = -.027. The standard deviation of returns for this portfolio Q is computed to be .0586 and its expected return equals . 10. With these two coordinates, we are able to plot this optimal portfolio on the expected return-standard deviation space; that is, we have the coordinates for one point on the Capital Market Line. A second point on this line is known as well — the point represented by the risk-free asset with an expected return equal to .08. Thus, the vertical intercept for the Capital Market Line equals .08. The slope of the Capital Market Line can be determined from any optimal portfolio (such as portfolio Q) and the riskless asset: (5.36) Based on the return-risk characteristics of optimal portfolio Q, we find that our equation for the Capital Market Line is as follows: (5.37) Obtaining the weights of the market portfolio from portfolio Q is straightforward. First, we note that portfolio Q is a simple combination of the riskless asset and the market portfolio. We also note that the market portfolio is comprised of risky assets only; the weight associated with the riskless asset is zero. Since proportion .679 of portfolio Q is the riskless asset, 1 - .679 or proportion .321 of Portfolio Q is comprised of risky assets. The weights of assets A and B in the risky component of portfolio Q are determined as follows:

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Since this mean variance efficient portfolio does not contain the riskless asset, its weights must be identical to those of the market portfolio. Since the weights of Securities A and B in the market portfolio are .627 and .373, the expected return and standard deviation levels of the market portfolio are determined:

The following represents the equation for the Capital Market Line:

(5.38) Note that the equation for the Capital Market Line is the same when it is based on the market portfolio as when it is based on efficient portfolio Q. The Efficient Frontier can be plotted by varying the riskless rate; an additional "market portfolio" is obtained each time a new riskless return is used in the computations. APPLICATION 5.15: DERIVING THE CAPITAL ASSET PRICING MODEL (Background reading: Application 5.14) The Capital Asset Model is derived from the same methodology as the Capital Market Line. The Capital Asset Pricing Model (CAPM) is a model of market equilibrium; that is, the CAPM provides a securities pricing model when supply equals demand for traded assets. The CAPM is concerned with the pricing relationships among securities and the relationships among security and market portfolio returns. The first step in deriving the CAPM is to perform the LaGrange optimization procedure just as in Application 5.14. We shall include in the investment opportunity set the riskless asset since the CAPM is based on the assumption that it is available. The general form of the LaGrange function is as follows:

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123

(5.35) Our first order conditions are given as follows:

Note that the covariances of the riskless asset with respect to any other asset equals zero. Our next step in deriving the CAPM is to subtract the derivative L of with respect to wf from the derivative of L with respect to the weight of any other single asset i to obtain:

This equation can be rewritten in a more simplified form: (5.39) We shall define the market portfolio as that portfolio of n risky assets which can be most efficiently combined with the riskless asset paying return rf. Note that by definition of aUp we can obtain the following simplification that is key to the development of the CAPM:

(5.40)

Thus, we can write the following: (5.41)

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This relationship holds for all securities and all portfolios, including the market portfolio:

This relationship allows us to write \x as follows:

Next, we substitute for A, in equation (5.41): (5.42) Cancel the 2s, rearrange terms, and we obtain the Capital Asset Pricing Model: (5.43) E[R] = r Thus, under assumptions consistent with the Capital Asset Pricing Model, security returns are linearly related to the riskless rate and returns on the market portfolio. APPLICATION 5.16: CONSTRAINED UTILITY MAXIMIZATION (Background reading: Application 5.3) In Applications 5.2 and 5.3, we discussed utility as a function of wealth. Here, we discuss utility as a function of consumption in a two-time-period framework. An investor with an initial wealth level equal to W0 plans to consume his wealth over two time periods. That wealth which is not consumed in the first time period is invested at an interest rate of r and the total, including interest, is consumed in the second period. For example, suppose that the individual's utility is described as a function of consumption in Time zero and Time one as follows:

The constraints on the individual's consumption are given:

Our problem is to find this individual's optimal intertemporal consumption (consumption over time) bundle, subject to his wealth constraint and prevailing interest rate; that is:

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125

First, we will set up an appropriate LaGrange function:

where X is the single LaGrange multiplier. Next, we will find first-order conditions:

Rewrite first-order conditions as follows:

We solve the system simultaneously by multiplying the first condition by .90909 and then subtracting the second condition from the result. Nothing is changed in the third condition:

Then we multiply the first equation in the above pair by 55 and subtract the second:

Therefore, C, = 8.040935 and C0 = 12.69005, resulting in a utility level of 17.82747. NOTE 1. This rule is derived in Appendix 5.A. SUGGESTED READINGS The presentation of calculus in this chapter was very informal, lacking the theoremproof orientation of a more rigorous calculus text such as Salas and Hille [1978]. This

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more rigorous presentation of calculus would be most useful for more advanced studies in financial mathematics. A number of finance texts provide a variety of readings on applications discussed in this chapter. For example, Elton and Gruber [1995] provide excellent readings pertaining to construction of the Efficient Frontier, duration, immunization and derivations of the Capital Asset Pricing Model. Copeland and Weston [1988] discuss constrained utility maximization and present an alternative derivation of the Capital Asset Pricing Model. Roll [1977] provides an insightful derivation of the Capital Asset Pricing Model using LaGrange optimization and matrix mathematics. Brealey and Myers [1996] discuss the Baumol Cash Management Model and constrained utility maximization.

Differential Calculus

127 EXERCISES

5.1 Find derivatives of y with respect to x for each of the following:

5.2 Find second derivatives of y with respect to x for each function in Problem 1. 5.3 Identify those functions which have finite minimum values for y. For these functions, what values for x minimize y?

5.4 Identify those functions in Problem 3 which have finite maximum values for y. For these functions, what values for x maximize y? 5.5 a. b. c. d.

Find the duration of the following pure discount bonds: A $1000 face value bond maturing in one year currently selling for $900 A $1000 face value bond maturing in two years currently selling for $800 A $2000 face value bond maturing in three years currently selling for $1400 A portfolio consisting of one of each of the three bonds listed in parts a, b and c of this problem

5.6 What is the relationship between the maturity of a pure discount bond and its duration? 5.7 Find the duration of each of the following $1000 face value coupon bonds assuming coupon payments are made annually: a. 3-year 10% bond currently selling for $900 b. 3-year 12% bond currently selling for $900 c. 4-year 10% bond currently selling for $900 d. 3-year 10% bond currently selling for $800 5.8 Based on duration computations, what would happen to the prices of each of the bonds in Question 7 if market interest rates (1 +r) were to decrease by 10%? 5.9 What is the duration of a portfolio consisting of one of each of the bonds listed in problem 5.7? 5.10 Consider each of the following functions:

For each of the functions a through f above, find partial derivatives for y with respect to x. Then find partial derivatives for y with respect to z for each function.

Chapter 5

128 5.11 Find derivatives for y with respect to x for each of the following:

5.12 Investors have the opportunity to invest in any combination of the securities given in the table below:

Find the slope of the Capital Market Line. 5.13 Investors have the opportunity to invest in varying combinations of riskless treasury bills and the market portfolio. Investors' investment portfolios will have expected returns equal to [Rp] and standard deviations of returns equal to ap. Let w m be the proportion of a particular investor's wealth invested in the market portfolio. Obviously, the investor's proportional investment in the riskless asset is w f = ( l - w m ) . Prove (or derive) the following:

Note: If you successfully complete parts a and b, you have derived the equation for the Capital Market Line (where the market portfolio characteristics are known). Now, complete part c: c. What happens to the slope of the Capital Market Line as each investor's level of risk aversity increases? 5.14 How would you expect transactions costs to affect borrowing rates of interest? How would lending rates of interest be affected? How would the Capital Market Line be affected by transactions costs? 5.15 A securities analyst has recommended the purchase of two stocks, A and B to include in the portfolio for one of your clients. The analyst has forecasted returns and risk levels as measured by standard deviation of returns and covariances as follows: Expected Standard Security Return Deviation A .08 .30 B .12 .60

COV(A,B) = 0

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129

Your client can borrow money at a rate of 6%, lend money at 4% and has $30,000 to invest. Your client, while not particularly risk averse, wishes to minimize the risk of his portfolio given that his expected return is at least 18%. How much money should he borrow or lend? How much should he invest in each of the two stocks? 5.16 Define an investor's utility (U) as the following function of his wealth level (w): U = lOOOw - .Olw2. This investor currently has $10,000. Answer the following: a. What is his current utility level? b. Find the utility level he would associate with 12,000. c. Use a Taylor series second order approximation to estimate the investor's utility level after his wealth level is increased by $2,000 from its current level of $10,000. 5.17 Find durations and convexities for each of the following bonds: a. A 10% five-year bond selling for $1079.8542 yielding 8% b. A 12% five-year bond selling for $1000 yielding 12% 5.18 For each of the bonds listed in Problem 5.17 above, complete the following assuming all interest rates (yields) change to 10%: a. Use the duration (first order) approximation models to estimate bond value changes induced by changes in interest rates (yields) to 10%. b. Use the convexity (second order) approximation models to estimate bond value changes induced by changes in interest rates (yields) to 10%. c. Find the present values of each of the bonds after yields (discount rates) change to 10%. 5.19 Our objective is to find that value for x which enables us to maximize the function y = 15x2 - 3x subject to the constraint that .5x = 100. Set up and solve a LaGrange function for this problem. 5.20 Solve the following: MAX Y = 5 + 3x + 10X2 s.t: 5x = 10. 5.21 An investor has the opportunity to comprise a portfolio of two assets with the following expected return and standard deviation levels: Asset A B

ErRI a .10 .20 .20 .40 PAB = -5

Determine the following: a. Optimal portfolio weights given each of the following expected return constraints: i. E(Rp) = .15 ii. E(Rp) = .12 hi. E(Rp) = .18 b. Optimal portfolio weights given each of the same expected return constraints in part (a) above, securities A and B from above and assuming the existence of a riskless asset with a 9% expected return.

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5.22 Securities A, B and C have expected standard deviations of returns equal to 0, .40 and .80, respectively. Securities A, B and C have expected returns equal to .05, .15 and .25, respectively. The covariance between returns on B and C is 0. What are the security weights of the optimal portfolio with an expected return of .2?

Differential Calculus

131 APPENDIX 5.A DERIVATIVES OF POLYNOMIALS

The derivative of the polynomial y = cxn with respect to x is determined by:

(A.l)

This equality results from the Binomial Theorem; the term (,) reads n choose one. Generally, the function (•) can be used to determine the number of ways a sample of size j can be taken from a population of size n. Its value is determined as follows: (A.2) For example, © reads 5 choose 2 and has a value equal to 5! -r[21(5-2)1] = 120^[2(3-2)] = 10. Thus, there are 10 combinations of 2 outcomes from a sample of 5. To simplify the right-hand side of Equation A.l, we first note that the cxn terms cancel out. Next, we note that h is divided into each of the remaining terms, leaving us with: (A.3) However, since h is approaching zero, all terms multiplied by h or h raised to any positive integer power will approach zero. This leaves us with: (A.4)

APPENDIX 5.B RULES FOR FINDING DERIVATIVES Function f(x) = c f(x) = ex f(x) = cxn f(x) = g(x)+h(x) f(x) = g(x)-h(x) f(x) = g(x)^h(x) f(x) = g(h(x)) f(x) = ln(x) f(x) = ex f(x) = e*x) f(x) = cx

Derivative f (x) = 0 f (x) = c f (x) = cnx11-1 f (x) = g'(x)+h'(x) f(x) = g'(x)-h(x)+h'(x)-g(x) f (x) = [g'(x)-h(x)-h'(x)-g(x)]/[h(x)]2 f (x) = g'(h(x))4i'(x) f (x) = 1/x f (x) = ex f (x) = g'(x)-e*(x) f (x) = cxln(c)

Example f(x) = 7 f(x) = 7x f(x) = 7x3 f(x) = 7x 3 +5x f(x) = (2+7x)(3x 4 +llx) f(x) = (2+7x)-K3x 4 +llx) f(x) = (10+4x 2 ) 7 f(x) = ln(x) f(x) = ex f(x) = e 5 x f(x) = 5X

Derivative f (x) = 0 f (x) = 7 f (x) = 21x2 f (x) = 21x 2 +5 f(x) = 7(3x 4 +llx)+(12x 3 +ll)-7 f(x) = [7(3x 4 +llx)-7(12x 3 +ll)]/(3x 4 +llx) 2 f(x) = 74-2(10+4x 2 ) 6 f (x) = 1/x f (x) = ex f (x) = .5e 5x f (x) = 5xln(5)

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APPENDIX 5.C PORTFOLIO RISK MINIMIZATION ON A SPREADSHEET The system which follows may be used to solve for n portfolio weights such that the variance of returns for that portfolio is minimized and the portfolio target expected return is achieved. This system essentially results from the series of equations resulting from partial derivatives of a LaGrange Function intended to minimize portfolio risk subject to a return and a weights constraint (see Application 5.13).

This system may be applied regardless of the size of the portfolio. Note that each of the first n elements along the principal diagonal of the coefficients matrix are security variances and that the coefficients matrix is symmetric (Row i equals Column j). Also note that the second column from the right (and the second row from the bottom) consists of the negative of security expected returns and two zeros. The right-most column (and bottom-most row) consists of negative ones and two zeros. One may use the above system on a spreadsheet to solve for the system of n weights and 2 LaGrange multipliers. The table following represents a Lotus 123™ spreadsheet printout of a system used to solve for optimal weights in a two security portfolio. The target return for this portfolio is .07 and the expected security returns are .05 for security A and .15 for security B. Security return-standard deviations are expected to be .1 for A and .5 for B. The covariance between returns on A and B is expected to be -.025. The spreadsheet which we may construct for the weights is given by the table on the next page. The left part of the table represents numerical values displayed by the spreadsheet; the right part represents actual cell entries. Rows 1 - 2 are numerical inputs for the file from the problem to be solve; Rows 4 - 7 are the rows of the coefficients matrix to be inverted. One inverts this coefficients matrix by invoking the following command (in Lotus 123™):[ /dmia4..d7a9 {ENTER} This "/" in this routine invokes the 123 menu, "d" invokes the data menu, "m" invokes the matrix menu, "i" commands the spreadsheet to invert a matrix, "a4..d7" defines the matrix to be inverted and "a9" specifies the upper left-hand corner of the block of cells where the inverse matrix will be placed. Hitting the {ENTER} key completes the series of entries. Thus, Cells a9 through dl2 represent the inverse of the coefficients matrix. Cells e9 through el2 represent the solutions vector. We must multiply the solutions vector by the inverse of the coefficients matrix as follows: /dmm a9..dl2 f9..fl2 al4 {ENTER}

PORTFOLIO OPTIMIZATION PROBLEM (Spreadsheet routine) Problem: Minimize portfolio variance given the following inputs: a A = .1 aB « .5

A

1 .1 2 -.025 3 4 .02 5 -.05 6 -.05 7 -1 8 9 0 10 0 11 10 12 -1.5

13

14 .8 15 .2 16 .54 17 -.021

18 19 20

^A.B

B -.025 .5 -.05 .5 -.15 -1 0 0 -10 .5

= -.1325

c -.05 -.15 0 0

E[RA3 - .05 E[RJ - .15

D .05 .15

E .07

Iw, - 1 A

B

+bi

^A.B

0 0 -.07 -1

2*bl 2*b2~2

-d2 -fl

a9..dl2 is inverted matrix al4 al5 al6 a!7

is is is is

C

W

A B

W

*\

x\

D E[RA3 E[RB]

OB

2*ar2 2*a2 -dl -fl

-1 -1 0 0

10 -1.5 .5 -10 -62 3.8 3.8 -.245

F 1

rp « .07

-dl -d2 0 0

E rp

F 1

-fl -fl 0 0

-fl

0 0 -el

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Differential Calculus

Again, "/" invokes the 123 menu, "d" invokes the data menu, "m" invokes the matrix menu, "m" commands the spreadsheet to multiply two matrices, "a9..dl2" defines the first matrix to be multiplied, f9..fl2 defines the second matrix to be multiplied, and "al4" specifies the upper left hand corner of the block of cells where the product of the two matrices will be placed. Problems involving more than two securities will simply require the insertion of additional rows and columns into the coefficients in the spreadsheet file. Note that each new column will read down the same way as each new row reads across. The number of rows and columns in any coefficients matrix for this type of problem will be (n+2). Suppose we wished to invert the coefficients matrix A4:D7 described above using an Excel™ spreadsheet. To invert this matrix, first point to the square block of ceils in which the inverse matrix will be placed. In this example, the block in which the inverse matrix will be placed is A9:D12. Next, enter the matrix inverse formula into the upper left corner cell (in this case A9) of that block following the example below: =MINVERSE(A4:D7) where the formula is entered by simultaneously pressing the {CTRL-SHIFT-ENTER} keys. This inverse matrix will be placed in block A9:D12. To post-multiply this range by a second array, say in block F9.F12, first point to the block in which the solutions vector will be placed. In this example, the block would be A14:A17. Next, enter the following formula into the upper left corner cell of this block: =MMULT(A9:D12,F9:F12) Again, enter by simultaneously pressing {CTRL-SHIFT-ENTER} keys. NOTE 1. In Quattro-Pro™, the entry sequence is /tami a4..d7 a9 {ENTER}. In Excel™, one can either use the formula wizard, use the Lotus 123™ sequence or use the sequence described a little later here.

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6 Integral Calculus

6.A: ANTIDIFFERENTIATION AND THE INDEFINITE INTEGRAL (Background Reading: Section 5.C) The two most important concepts in calculus are the derivative and the integral. A geometric interpretation of the derivative is the slope of a curve given by function f(x); the analogous geometric interpretation of the integral is the area under a curve represented by the function f(x). Integrals are most useful for finding areas under curves, and for finding expected values and variances based on continuous distributions. As the D operator is used for summing countable numbers of objects, integrals are used for performing summations of uncountably infinite objects. One may find the integral of a function using the process of antidifferentiation which is the inverse process of differentiation. If F(x) is a function of x whose derivative equals f(x), then F(x) is said to be the antiderivative or integral of f(x), written as follows: (6.1)

The integral sign J is used to denote the antiderivative of the integrand f(x); the indefinite integral is denoted by j f(x)dx. We infer the following from Equation (6.1): (6.2) Consider the following function: f(x) = 3x2. This function is the derivative of what function? That is, what is the antiderivative of f(x)? F(x) = x3 + k is the antiderivative of f(x) where k is simply a real valued constant since:

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Thus, the derivative of the function F(x) is our original function f(x); F(x) is the antiderivative of f(x). The constant of integration k must be included in the antiderivative. Thus, all of the following could be antiderivatives of 3x2: F(x) = x3 + 2, F(x) = x3 + 600 and F(x) = x3 + 3.5. It is important for the antiderivative computation to be able to accommodate any of these possible constant values k. The following are a few of the rules which apply to the computation of indefinite integrals (where k is a real valued constant): (6.3) (6.4) (6.5) (6.6) (6.7) Other rules are provided in Appendix 6.A. 6.B: DEFINITE INTEGRALS AND AREAS (Background Reading: Section 6.A) Consider a function f(x) = lOx - x2. Suppose that we wish to find the area under a curve represented by this function over the range from x = 0 to x = l . The lower limit of integration is 0; the upper limit of integration is 1. We will first show how to find the area under a curve by demonstrating a method similar to one first proposed by Archimedes. In a sense, we will divide the area under the curve into a number of rectangles (see Figure 6.1). Data for Figure 6.1 are given in Table 6.1. We will reduce sizes of these rectangles and allow their numbers to approach infinity (n -* oo). Next, we will find the area of each rectangle. Each of these rectangles, which are numbered sequentially, will have a width of x r x M = 1/n -* 0 and a height of f(x*) where x* is some value between Xj and x^ (for sake of simplicity here, assume x* = Xj). Since the number of these rectangles under the curve will approach infinity, we have the width of each of these rectangles to approach (though not quite equal) zero. The area of each of these rectangles (where the product is non-negative) is simply the product of its height and width: (6.8)

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Integral Calculus

Figure 6.1 Finding the Area Under a Curve Using the Method of Archimedes

When Xj - Xj.j = . 1, the sum of the areas of the 10 rectangles equals 5.115. As the number of rectangles approaches infinity, and their widths approach zero, the sum of their areas will approach 4 2/3.

Table 6.1 The Area Under the Curve Represented by y = lOx - x2 Data Point i 1 2 3 4 5 6 7 8 9 10

YiXi 0.99 1.96 2.91 3.84 4.75 5.64 6.51 7.36 8.19 9.00

0.1 0.2 0.3 0.4 0.5 0.6 0.6 0.8 0.9 1.0

Xi - Xj.,

yt • (X| - x M )

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.099 0.196 0.291 0.384 0.475 0.564 0.651 0.736 0.819 0.900

E[y, • (x, - x,,)] = 5.115

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Thus, the area of a region extending from x=a to x=b under a curve can be found with the use of the definite integral over the interval from x=a to x=b as follows:

(6.9) The width of each rectangle equals (Xj-x^) = (b-a)/n -* 0 and the height of each rectangle equals f(x*). Thus, we begin to find the area under the curve in the example presented above as follows: (A) Since b - a equals 1, each Xj - xlA will equal 1/n and we obtain: (B) Next, we separate terms, note that our initial Xj value equals 0 and that each x} value equals i/n (since our units of increase are 1/n and i represents the number of increases accounted for at some point i in the summation). This enables us to obtain:

(C)

Next, we simplify and replace our summations by using standard results of series Ei/n2 and EiVn3:1

(D)

As n approaches oo, it is easy to see that the area under the curve extending from x = 0 to x = l approaches 5 - 1/3 = 4%.

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141

In summary, this process of determining the area under a curve within a defined region involved the summation of areas of an infinite number of rectangles of infinitesimal width which lie within this area. We have properly calculated the area under our curve, however, this process proposed by Archimedes is quite time-consuming. Another, more elegant, method makes use of the Fundamental Theorem of Integral Calculus, based on a brilliant insight by Isaac Barrow. This theorem is stated as follows: Iff(x) is a continuous function within the range from x=a tox-b, F(x) is the antiderivative off(x), the following must hold:

and

(6.10) Thus, we may use the Fundamental Theorem of Integral Calculus to find the area under the function f(x) = lOx - x2 by using antiderivatives as follows:

(E)

Notice that the constants of integration k canceled out. Essentially, we found the antiderivative of our function at a (or 0), then subtracted this antiderivative from the antiderivative of our function at b (or 1). Consider a second function: y = -7x2 + 4x + 5 represented by Figure 6.2. Suppose we wished to find the area between this curve and the horizontal axis within the range from x = 0 to x = 5. Again, we may use the Fundamental Theorem of Integral Calculus to find the area under the curve by using antiderivatives as follows:

(A)

Thus, the area under this curve in the range from x = 0 to x = 5 equals -216%. Actually, the area between this curve and the horizontal axis net of the area under the curve but above the horizontal axis within this range equals 216%.

Chapter 6

142 Figure 6.2 The Area Between the Curve and Horizontal Axis

APPLICATION 6.1: CUMULATIVE DENSITIES (Background Reading: Section 2.D) A continuous probability distribution P(x) may be used to determine the probability that a randomly distributed variable will fall within a given range or below a given value. Among the continuous probability distributions used by statisticians are the normal distribution, the uniform distribution and the gamma distribution. A probability density function is a theoretical model for a frequency distribution. The (density at x*)-dx equals the probability that a continuous random variable x lies between x* and x* + dx where dx -* 0.2 Thus, in a sense, the density function may be used to determine the probability p(Xj) that a continuous random variable Xj will be exactly equal to a constant x*. However, it is important to note that because the continuous random variable Xj may assume any one of an infinity of potential values, the probability that it assumes any particular exact value x* approaches zero. In any case, a density function p(x) will be found from a differentiable distribution function P(x) as follows: (6.11) Hence, the distribution function P(x) may be found from the density function as follows: (6.12)

P(x) = fp(x)dx

We will invent our own density function for this application which will be particularly simple to integrate. From this density function, we can obtain a distribution function. Suppose that the potential or random return ri for a given stock is expected to range from 0 to 10%. Further suppose that potential returns track some random continuously distributed variable xx ranging from 0 to 1; in

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143

fact, returns will be expressed as rx = f(Xj) = .lOxj, or 10% of the value of this randomly distributed variable. Assume that a density function (p(xj) = P'(Xj)} for the randomly distributed variable xx is given by the following: p(x) = 6(x — x2) for 0 < x < 1 and 0 elsewhere. This density function p(x) is analogous to a frequency distribution for x. The indefinite integral of this density function for x, the distribution function, is determined as follows:3,4

(A) We can use a definite integral to determine the probability that the random variable xx is less than some constant x*; this probability will be the same as for rx being less than .lOxj. The distribution function is simply the cumulative density function. For example, we determine the probability that xx will be less than .5 and that r} will be less than .05 as follows:

(B) Note that the lower limit of integration is 0 because the density function is nonzero only over the interval from zero to one. Thus, there is a fifty percent probability that xx will be less than .5 and that vx will be less than .05. We can also use definite integrals to determine the probability that the random variable will fall within a specified range. For example, we can integrate the density function p(x) over the interval from .2 to .5 to determine the probability that Xj will fall between .2 and .5:

(C) The probability that the return will range between .02 and .05 is also equal to .396. Similarly, we can determine that the probability that x will fall between .4 and .6 (and the probability that r will fall between .04 and .06) to be .296:

(D)

Chapter 6

144 APPLICATION 6.2: EXPECTED VALUE AND VARIANCE (Background Reading: Application 6.1)

In this application, we will continue to use the density function from Application 6.1: p(x) = 6(x - x2) for 0 < x < 1 and 0 elsewhere. We will evaluate integrals of this density function to generate an expected value and a standard deviation for our randomly distributed variable. To find the expected return, use the density function to weight each random return r^ (A) for 0 < x < 1 and 0 elsewhere.5 The indefinite integral of this density function for r, the distribution function, is determined as follows: (B) The expected value of this random variable r is determined as follows:

(C)

Thus, the expected return for this security equals .05. The variance of returns may be determined using the following: (D) The variance of returns from our distribution is determined as follows:

(E) i

Thus, the variance of returns in this distribution is .0005 and the standard deviation of returns is .0223606.

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145

APPLICATION 6.3: STOCHASTIC DOMINANCE (Background Reading: Applications 5.12 and 6.1) Many types of portfolio selection models make assumptions regarding either the form of probability distribution of returns faced by investors or about the form of investor utility of wealth functions. For example, the Capital Asset Pricing Model assumes either that security returns are normally distributed or that investors have quadratic utility functions. In reality, measurement of investor utility functions is, at best, extremely difficult. Determining the actual probability distribution of security returns is usually either difficult or impossible. Thus, portfolio selection may be aided by a set of rules which does not rely on determination of the exact return distribution and requires only the most essential information regarding investor preferences. The concept of stochastic dominance is such an example. It does not rely excessively on the exact form of investor utility functions and it does not necessarily require that return distribution functions be fully specified. Thus, stochastic dominance may be a useful portfolio selection tool when we are able to make only the barest of assumptions or observations regarding utility and probability functions. In portfolio analysis, a portfolio is considered dominant if it is not dominated by any other portfolio. One portfolio is considered to dominate a second portfolio if, from a given perspective or based on specific criteria, its performance is at least as good as the second portfolio under all circumstances (or states of nature) and superior under at least one circumstance. For example, first order stochastic dominance exists where one security has at least as high a payoff under each potential state of nature and a higher under at least one state. Table 6.2 lists three orders of stochastic dominance and the circumstances under which each might be used as a portfolio selection rule. In Table 6.2, U(w) designates the utility of wealth function, and A(w) represents the absolute risk aversion coefficient defined as follows:6 (6.13)

Table 6.2 Orders of Stochastic Dominance

Order of Stochastic Dominance First order Second order Third order

Used by Investors When More is preferred to less: U'(w)>0 Safety is preferred to risk: U"(w)<0 Investors have decreasing absolute risk aversion: A'(w) = {[U"(w)] - [U'(w)]}2 - {U'"(w) + U'(w)} < 0

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146

It is quite reasonable to assume that investors prefer more to less. Thus, whenever one asset exhibits first order stochastic dominance (defined below) over a second asset, the first asset will be preferred. Whenever an investor is risk averse and prefers more to less, an asset which exhibits second order stochastic dominance over a second will be preferred. Similarly, whenever an investor has decreasing risk aversion with respect to wealth, he is risk averse and he prefers more to less, an asset which exhibits third order stochastic dominance over an alternative asset will be preferred. Suppose that there exist two assets f and g whose payoffs f(x) and g(x) are dependent on some ordered random variable x such that f (x)>0 and g'(x)>0. Thus, as the value of random variable x increases, the payoffs on securities f and g increase. We will not specify the exact characteristics of individual investor utility functions; we state only that investors will prefer a higher payoff to a lower payoff. The probability distribution functions P£x) or Pg(x) can be used to represent the probability that security payoffs x will be less than or equal to some constant x*. Define the following probability distribution functions for payoffs on securities f and g: (6.14)

(6.15) Security f is said to exhibit first, second or third order stochastic dominance over security g if the appropriate conditions from Table 6.3 hold. First order stochastic dominance by security f over security g implies that for each potential security payoff x*, the probability that security g has a smaller payoff pg(x < x*) than x* exceeds (or equals with at least one instance exceeding) Table 6.3 Stochastic Dominance Conditions Order of Stochastic Dominance First order Second order Third order

Conditions Pf(x) £ Pg(x) for all x Pf(x) < Pg(x) for some x U'(w) > 0 f i.P/xJdx £ J *wPg(x)dx for all x { "^P^xjdx < J ^oePg(x)dx for some x U'(w) > 0 ; U"(w) < 0 { *«,( J !^C0Pf(x)dx)dx <; J *„( { !00Pg(x)dx)dxV x I -oo I ^Pf(x)dxdx < J ;„( J !wPg(x)dx)dx for some x U'(w) > 0 ; U"(w) < 0 ; A'(w) < 0

Integral Calculus

147

the probability that security f will have a smaller payoff than x*. Thus, for each state of nature x with density (probability) p(x), security f has at least as high a payoff as does g (and in least one state, a higher payoff). The probability that security g has a payoff lower than some specified amount exceeds (or equals with at least one instance exceeding) the probability that f will have a payoff lower than that amount. Investors preferring more to less will favor the security which exhibits first order stochastic dominance over another. Thus, when U'(w)>0, security f is preferred to security g. Second order stochastic dominance is concerned with the dispersion of payoffs. Second order stochastic dominance exists when the cumulative distribution function (which is the cumulative-cumulative density function) for security f never exceeds the cumulative distribution function for security g. In other terms, the cumulative distribution function that g has a payoff lower than some specified amount exceeds the cumulative distribution that f will have a payoff lower than that amount. Although this connection might be somewhat confusing, second order stochastic dominance essentially implies that if the probability of payoffs for security g at the lower end of the potential range are exceeded by the probability of payoffs at the higher end of the range for f, then f exhibits second order stochastic dominance over g. Risk averse investors prefer securities which exhibit second order stochastic dominance. Consider an example where a risk averse investor who prefers more wealth to less has the opportunity to invest in a security f whose future value is a function of a randomly distributed variable x. The density function for f is given by the following: pf(x) = 6(x - x2) for 0 ^ x < 1 and 0 elsewhere. The security f will have a payoff equal to f(x) with probability equal to Pf(x). In addition, the investor has the opportunity to purchase a second security g whose density function is given by pg(x) = 12(x2 - x3) for 0 <> x <> 1 and 0 elsewhere. Density functions for the payoffs for securities f and g are given in the upper graph in Figure 6.3. Security g will have a payoff equal to g(x) with probability equal to pg(x). For the sake of simplicity, we shall assume that f(x) = g(x) = x. If the investor is to choose one of the two securities based on first order stochastic dominance criteria, he will first determine cumulative densities (ignoring constants of integration) as follows: (A)

(B) These integrals are plotted in the lower graph in Figure 6.3. Notice that Pf(x) < Pg(x) for all x. This means that security f has a higher probability of a smaller payoff than g at every potential payoff x. F£x) ^ Pg(x) for all x; therefore, security g will be preferred to security f. First, note that pf = pg = 0 when x ^ 0 and when x > 1. Also note that Pf = Pg = 1 when x = 1.

148

Chapter 6

Figure 6.3 Stochastic Dominance

However, we can demonstrate with algebra that when 0 < x < 1, Pf > Pg. Thus, the probability that the payoff on security f is less than any constant in the range (0,1) is never less (though it may be greater) than the probability that the payoff on security g will be less than that constant. Thus, security g is preferred to security f. We can also demonstrate that j Pf ^ J Pg for all x; therefore, security g exhibits second order stochastic dominance over security f. We integrate Equations (A) and (B) above to obtain:

(C) (D)

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149

Again, note that J Pf = J Pg = 0 when x ^ 0 and { Pf = J Pg = 1 when x > 1. However, we can demonstrate with algebra that when 0 < x < 1, J Pf > j Pg. Thus, the risk averse investor will prefer security g to security f, even if first order stochastic dominance did not exist. Whenever first order stochastic dominance exists, investors preferring more to less will choose the asset which exhibits first order dominance. Risk averse investors who prefer more to less will always prefer an asset which exhibits second order stochastic dominance over an alternative asset, regardless of whether first order stochastic dominance exists. APPLICATION 6.4: VALUING CONTINUOUS DIVIDEND PAYMENTS (Background Reading: Section 2.A) An index fund holds securities in such quantities so as to match the return structure of a particular market index. For example, one could create a fund of securities intended to match the S&P 500 Index. Such a fund might consist of 500 stocks weighted such that the fund's returns and dividend yields replicate those of the S&P 500 Index. The index fund with 500 stocks is likely to receive frequent dividend payments. With many of its 500 stocks paying dividends on a quarterly basis, the fund is likely to receive dividends almost on a daily (or continuous) basis. Hence, one may model the dividend receipt structure of such a fund as though dividends represent continuous payments. Suppose that this fund were to receive dividends on a continuous basis at a rate of $1000 per year starting at time t=0. Assume that these dividends will be discounted at a continuously compounded rate of k=5% per year. The present value of all dividends received in any interval [t,t4-dt] is determined as follows: (6.16) PV[t,t+dt] = PV[0>t+dt] - PV[0>t] where PV[t,t+dt] equals the present value of dividends received during the interval [t,t+dt]. The amount of dividend payment to be received at any infinitesimal time interval dt equals f(t)dt = lOOOdt. The present value of this sum equals f(t)ektdt = 1000e05tdt. Thus, the present value of dividends received over the infinitesimal interval dt is: (A)

PV[t,t+dt] = PV[0,t+dt] - PV[0,t] = Me'^dt = lOOOe"dt To find the present value of a sum received over a finite interval beginning with t=0, one may apply the definite integral as follows: (6.17) Thus, in our numerical example, one may find the present value of dividends to be paid to the fund from time 0 for five years as follows:

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Chapter 6

(B)

Another useful application of this methodology is to the valuation of index contracts which are not dividend protected. For example, suppose that one can take a position on a contract to purchase a fund at time T. Any dividends to be received by the fund will be paid immediately to current fund investors as they are received. These dividends to investors represent "dividend leakage" from the fund. The current value of the fund to investors would reflect any dividends expected to be received prior to time T; however, the purchaser of the fund at time T would not receive dividends paid prior to time T. Thus, the fund purchaser should subtract from his valuation of the fund the present value of any dividends to be paid by the fund prior to time T. If the fund described above is currently worth $20,000 to current investors, the investor with the contract to purchase the fund in five years may value the fund at $20,000 - $4,423.98 = $15,576.02 based on the present value of dividends that he will not receive. APPLICATION 6.5: EXPECTED OPTION VALUES (Background Reading: Application 4.6) A European call option grants its owner the right to purchase stock for a specified exercise price X on the expiration date of the contract in n periods. If the price of the underlying stock does not exceed the exercise price of the option, the owner of the call disposes of the option contract and it expires worthless. Thus, the option is exercised only if S n >X, where its expiration value would be Sn-X. One might estimate the expected future value of the cash flow associated with the call by: (6.18) where p(Sn) is the probability that a given value for Sn will be realized when the option expires. In certain instances, it may be easier to estimate the value of the call with the following two-step procedure: 1. Estimate the probability that the stock price at time n will exceed the exercise price X of the call; that is, estimate the probability that the call will be exercised and 2. Estimate the expected value of the call given that the stock's price exceeds the exercise price of the call. The product of the two estimates given above results in the expected future value of the call:

Integral Calculus

151

(6.19) The probability that the call will be exercised is given by: (6.20) The expected value of the call given that it is exercised is given by: (6.21) where p(x) equals the probability that Sn = x. Thus, the expected value of the call is simply the product of Equations (6.20) and (6.21), which is identical to Equation (6.18): (6.22) 6.C: DIFFERENTIAL EQUATIONS A differential equation concerns the unknown function for which derivatives exist. A differential equation can be used to describe how a system evolves over an interval of time. Consider the following example of a simple differential equation involving dependent variable y and independent variable x: (6.23)

^ =x dx The solution to a differential equation is a function which, when substituted for the dependent variable, satisfies the equality. In a sense, the differential equation describes the direction or change in a system; the solution provides the path of the system. The following solves the differential equation (6.23): (6.24) Thus, we verify the solution to differential equation (6.23) by noting that it represents the derivative of y with respect to x in its solution equation (6.24). The order of a differential equation is the order of its highest derivative and the degree of a differential equation is the power to which its highest order derivative is raised. Equation (6.23) is a first order differential equation of the first degree and the following is an example of a second order differential equation of the second degree: (6.25)

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Chapter 6

A differential equation is said to be differentiable if it can be rewritten in the form g(y)dy = f(x)dx. A separable differential equation written in this form can be solved by the following: (6.26) The following is an example of a separable differential equation:

(A) $ - £ To solve this equation, we first separate the variables as follows: (B) Next we integrate both sides: (C) We solve for y as follows:

(D) Redefining k our constant to be 3(k2-k!), we find the general solution for our differential equation (A) to be: (E)

A particular solution results when k assumes a specific value. APPLICATION 6.6: CONTINUOUS TIME SECURITY RETURNS It is often convenient to model security price changes dSt on a continuous basis using differential equations. The following is an example of separable differential equation used to model price changes for a security: (6.27) where the drift term, /x, represents the rate of return for the security over an infinitesimal period. The solution to this differential equation gives the state of the system at a point in time. Differential equation (6.27) can be solved by the following:

(6.28)

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153

From equation (6.6) in Section 6.A, and because j /idt = /* J dt = /it, we obtain the following when St > 0: (6.29) which can be rewritten as: (6.30) (6.31) Equation (6.31) represents a general solution to our differential equation (6.27). If we define ek to be S0, a particular solution to equation (6.27) would be: (6.32) Differential equations such as (6.27) are very useful in the modeling of security prices and are adaptable to the modeling of stochastic (random) return processes. This topic is discussed in detail in Section 9.C. Suppose that a security with value St in time t generating returns on a continuous basis were to double in value after five years. Further suppose that its value after eight years were $1000. What would have been the initial value of this security S0? First, Equation (6.27) can be used to model the return generating process:

The solution to this equation is given by Equation (6.32): Thus, /t = .2 • ln(2) = .1386294. Thus, we can solve for the security's initial value as follows: S0 = $1000e'81386294 = $329.87698 Consider a second example involving a money market mutual fund which collects $100,000,000 in interest per year from its bond investments. Then, the interest is paid to the fund's investors in equal amounts during each interval of time (day or smaller time period) during the year such that they can be modeled as being continuous. If interest payments are to be discounted at an annual rate of five percent, what would be the present value of the interest payment stream over a one year period? To solve this, first note that the amount of interest payment to be received by investors at any infinitesimal time interval dt equals f(t)dt = $100,000,000dt. The present value of this sum equals f(t)ektdt =

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Chapter 6

$100,000,000e*05ldt. To find the present value of a sum received over a finite interval beginning with t = 0, one may apply the definite integral as follows:

Filling in values, we obtain:

NOTES 1. These may be simplified as follows:

These results are well known and may be verified by induction. 2. p(x) is a continuous version of P} which was used for the probability associated with a particular outcome i drawn from a discrete set of potential outcomes. See Section D in Chapter 2. 3. We will assume that k = 0 here. 4. Note the similarity in the discrete expected value expression EXjPjfromChapter 2, Section D and the continuous version { f(x)p(x). 5. f(x)p(x) is analogous to R ^ used earlier in Chapter 2, Section 2.D and Application 2.4 to designate return outcome i weighted by probability outcome i. 6. An investor is said to exhibit decreasing absolute risk aversion (A'(w)<0) if he accepts more portfolio risk (in absolute or dollar terms) as his wealth increases. See Application 5.12 for a more detailed explanation. SUGGESTED READINGS Neftci [1996] furnishes a review of calculus, including applications of integral calculus to finance. The presentation of Neftci provides a good foundation to securities valuation in a stochastic environment. Elton and Gruber [1995] discuss Stochastic Dominance as do Copeland and Weston [1988]] and Martin, Cox and MacMinn [1988], who also provide a review of integral calculus. Salas and Hille [1978] use a more rigorous theorem-proof orientation towards integral calculus in their text.

155

Integral Calculus

EXERCISES 6.1 a. b. c. d. e. f. g. h.

Integrate each of the following functions over x; f(x) = 0 f(x) = 7 f(x) = 21x2 f(x) = 21x2 + 5 f(x) = ex f(x) = .5e 5x f(x) = 5xln(5) f(x) = 1/x

6.2 Solve each of the following definite integrals: a. b. c. 6.3 Assume that the density function {p(xj) = P'(Xj)} for a randomly distributed variable x is given by the following: p(x) = 3x2 for 0 < x < 1 and 0 elsewhere. a. Find the distribution function P(x) for p(x). b. Find the expected value of x in the range .5 to 1. c. Find the expected value of x in the range 0 to .5. d. Find the expected value of x in the range 0 to 1. e. Find the variance of x in the range 0 to 1. 6.4 Suppose that the terminal cash flow for a given investment is equally likely to assume any value in the range (0, 100) and will not assume any value outside of this range. That is, a uniform density function is associated with the random investment cash flow. More specifically, the density function for the random variable is given by p(x) = .01 for 0 =£ x < 100 and 0 elsewhere. a. Find the distribution function P(x) for p(x). b. Find the expected value of x in the range 50 to 100. c. Find the expected value of x in the range 0 to 50. d. Find the expected value of x in the range 0 to 100. e. Find the variance of x in the range 0 to 100. f. What would be the expected future cash flow (contingent on its exercise) of a call option written on this investment if its exercise price were $50? That is, what is the expected cash flow of the option conditional on its exercise? g. What would be the expected cash flow of a call option written on this investment if its exercise price were $50? That is, what is the unconditional expected cash flow associated with this option?

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6.5 Assume that the density function pf for a randomly distributed variable {pf(x) = Pf(x)} is given by the following: pf(x) = 3x2 for 0 < x < 1 and 0 elsewhere. A second density function pg for a randomly distributed variable {pg(x) = Pg(x)} is given by the following: pg(x) = (2x 3 +x) for 0 < x < 1 and 0 elsewhere. a. Find Pf(x) and Pg(x). b. Demonstrate whether there exist conditions of First Order Stochastic Dominance. c. Demonstrate whether there exist conditions of Second Order Stochastic Dominance. 6.6 Suppose a muuial fund collects $2,000,000 in dividends per year from its various securities. The dividends are paid in equal amounts each interval (day or smaller time period) during the year such that they can be modeled as being continuous. If dividends are to be discounted at an annual rate of ten percent, what would be the present value of the dividend stream over the next twenty years? 6.7 Suppose that one applies the discount function PV = CFte * to a given cash flow CFt. Further assume that this cash flow has an initial value equal to CF0 and will grow continuously at an annual rate of g. Thus, the present value of any given cash flow is given by PV = CFoe^e"*. Derive a formula for valuing a growing annuity PVGA where cash flows are growing continuously, paid continuously and discounted at a continuously compounded rate. 6.8 A retiree with $1,000,000 invested at an annual interest rate of 5% will withdraw $100,000 per year for living expenses. Assume that the retiree withdrawals will be made continuously throughout the year. a. Let PV designate the present value of the account, FVt designate the value of the account at time t, PMT designate the payment made from the account during each year and let i designate the interest rate. Write an appropriate differential equation describing the rate of change in the retiree's account during a given time period. b. Solve the differential equation to find the balance of the account at any time T. c. Based on the solution to the relevant differential equation and the numbers given in the example, how much money would the retiree have in his account after 7 years? d. If the retiree continues to spend $100,000 per year, at what point (how many years) will he run out of money?

Integra] Calculus APPENDIX 6.A: RULES FOR FINDING INTEGRALS

The following integrals have no closed form solutions:

Polynomial functions are often used to estimate such integrals.

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7

Introduction to Probability

7.A: RANDOM VARIABLES AND PROBABILITY SPACES (Background Reading: Section 2.D) Because financial decision making is so much a function of uncertainty, probability theory is one of the most essential tools of financial model building. Probability theory is also the foundation for statistical theory on which empirical estimation is based. Furthermore, probability theory is also the foundation for stochastic processes, the random movement of variables over time. Foundations for probability theory include set theory from mathematics, though it is not necessary to fully understand set theory to make certain applications of probability to finance. This chapter is intended to equip the reader with a sampling of elementary definitions and tools from probability theory, along with applications to finance. Probability Spaces This section is intended to formalize and explain some definitions that are frequently used in probability theory. However, the remainder of this section is not essential for understanding subsequent sections of this chapter, though it may be helpful for grasping certain sections in Chapter 9. Understanding this section may be useful if the reader wishes to study a more formal presentation of probability theory in another book. In particular, familiarity with the more formal notation here will facilitate decomposition of martingales and changes of measure, which are important topics in books and articles pertaining to the mathematics of derivatives. A probability space consists of three elements: a sample space, events and associated probabilities. The sample space is a set Q of all potential outcomes or elementary states of the world co. An event 0 is a subset of elementary states

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160

(JJ taken from Q and is an element of the set of events $. For example, a specific event u which will result in some stock price, is the set {o)n, w21} and is itself an element of <J>. Elements cou and co21 are taken from the set of all potential outcomes Q; set Q represents all potential outcomes for factors that investors use to value the stock. Set $ may be characterized as the power set for fi from which specific events are drawn.1 Random variable X is simply a mapping from set $ of events. In our example, the random variable X simply represents the stock price resulting from a specific event comprised of outcomes co. If an event, represented by a number, varies in repeated samplings, it is referred to as a random variable. We define more formally a random variable X as follows: A random variable defined on a probability space (Q,$,P) is a function X:Q-+R satisfying: This set is an event in . It is customary to define a distribution function for X in the following manner:

Thus, a subset of elementary states of the world o) may drawn from the nonempty set fl of all potential states of the world; this unique subset of states comprises a particular event . The probability measure, P, is a mapping from $ to [0,1]. The anticipated relative frequency with which a given random variable occurs in a long series of trials might be regarded as the probability associated with the random variable. The probability of any particular event occurring is between zero and 1 (0 < P[] ^ 1); the probability of the empty event occurring equals zero (P[0] = 0]), and the probability that at least one of the potential events will occur equals 1 (P[fl] = j w(=ndP[
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each and every value of X*. A random variable that may assume any value within an interval is said to be continuous. The distribution function for a continuous random variable y, denoted F(y*) is defined:

The derivative of F(y*) with respect to y* is referred to as the density function for the random variable y. The density for a continuous random variable y represents the probability that the variable assumes a value between y* and y*+dy where dy -* 0. The first moment of a random variable y is its expected value while the variance of a random variable is the second moment about its mean fi. More generally, the ith moment of a random variable about its mean is defined to be E[y - fi]1. Skewness (distribution asymmetries if non-zero: (E[y - /i]3)) and kurtosis (heaviness of tails implying higher probability of extreme observations if positive: (E[y - /i]4)) are the third and fourth moments about the mean of a distribution. One particularly useful class of distributions is the stable class. Suppose that there exists a series of random variables y which are drawn from independent but identical distributions. If these distributions are stable, then they are invariant under addition. This means that the class of the distribution of the sum of random variables is the same as that for each of the identically distributed random variables. For example, if stock returns over time are independently and identically distributed, then their sums should be of the same distribution class if their distributions are in the stable class. This concept is particularly important for portfolio mathematics and the study of return generating processes. If a series of variables are drawn from independent and identical distributions (IID) of the stable class with a mean /A, and those distributions have a finite variance a2, then the Central Limit Theorem can be invoked. This powerful theorem states that the distribution of the sum (and mean) of those random variables will converge to a normal distribution as the number of variables summed approaches infinity. In many cases, the convergence of a particular stable distribution will be quite rapid. For example, many distributions will converge to approximately normal within only ten to thirty trials. 7.C: BINOMIAL DISTRIBUTIONS Consider a sampling of n identical trials, in which each trial can result in one of two potential outcomes. One such sampling might be based on a stock whose price can either rise or fall in each of n sequential transactions. Suppose that the probability associated with a stock price increase in each transaction is given to be p, implying a probability of (1-p) for a price decline in each transaction. Further assume that each transaction is independent and that prior transaction price changes have no effect on subsequent transaction price changes.

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Assign a random variable y to denote the number of price increases in a given series of n transactions. Suppose that the stock will experience n=4 transactions over a given period and that we wish to know the probability that the stock price will have increased in exactly y* of those four transactions. First, we may determine the number of potential orderings of price increases (+) and decreases (-) as is indicated in Table 7.1. For example, the first transaction price may result in an increase or a decrease which will be followed by a decrease or increase, and so on. Thus, five transactions may be ordered sixteen ways with respect to price increases and decreases. If each transaction is equally likely to result in a price increase or decrease, then each ordering has equal probability (p = Vi). In this case, the probability that any particular ordering will be realized is given as follows: (7.1) If the probability of a price increase differs from the probability of a decrease, the probability associated with some orderings will be higher than for other orderings. For example, if the probability of a price increase equals .6, then the probability of a particular ordering with y* increases is determined as follows: (7-2)

For example, the probability of three price increases followed by a single decrease (+ + +-) equals .0864: When p>.5, orderings with more price increases are more probable than orderings with fewer price increases. For example, the probability of a single price increase followed by three decreases (+—) equals .0384. The number of orderings or ways that y* increases can be achieved in n transactions is determined as follows:

(7.3) For example, the number of ways that three price increases can be achieved in four transactions equals 4:

This result can be verified by counting the number of appropriate entries in Table 7.1. The binomial probability distribution for y will determine the probability of y* price increases in n transactions:

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Table 7.1 Orderings in 5 Transactions +-++ +-++--+ +---

++++ +++++-+ ++--

--++ --+---+ ----

-+++ -++-+-+ -+--

(7.4) Thus, from the orderings in Table 7.1, the probability that exactly y*=3 price increases will result from n = 4 transactions where p = . 6 is determined as follows:

One can determine a cumulative binomial probability distribution by summing the distribution function as follows: (7.5) The cumulative probability distribution provides the probability that the number of increases y is no greater than Y*. For example, the probability that the number of price increases in our example does not exceed 3 is determined as follows:

Similarly, the probability that four consecutive price increases are realized in the above system of transactions equals .1296. The expected value and standard deviation of a binomial random variable (Y) are: (7.6)

E(Y) = np

(7.7) where the random variable Y assumes a value of 1 with probability p and 0 with probability (1-p) in each of n trials.

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APPLICATION 7.1: ESTIMATING PROBABILITY OF OPTION EXERCISE (Background Reading: Application 6.5) A European call option is exercised only if Sn>X, where its expiration value would be Sn-X. The Binomial Option Pricing Model is based on the assumption that the underlying stock follows a binomial return generating process. This means that for any single period during the life of the option, the stock's value will change by one of two potential values. For example, the stock's value will be either u times its current value or d times its current value. Thus, if the stock is currently worth 100, u equals 1.2 and d equals .8, the stock's value in the forthcoming period will be either 120 (if outcome u is realized) or 80 (if outcome d is realized). Suppose there exists a European call trading on this particular stock during this n time period framework with an exercise price X = 90. Suppose the call expires at the end of one period when the stock value is either 120 or 80. Thus, if the stock were to increase to 120, the call would be worth 30 (cu = uS r X = 120-90 = 30), since one could exercise the call by paying 90 for a stock which is worth 120. If the stock value were to decrease to 80, the value of the call would be zero (cd = 0) since no investor would wish to exercise by paying 90 for shares which are worth only 80. Thus, the value of the call is given by: (7.8) If the term to expiration of the call were n> 1 periods, the value of the underlying stock at the expiration of the call would be S0uYdnY, depending on how many times Y the stock experienced an increase in price. The expiration value of the option would equal S0uYdnY - X. For example, if n were to equal 4, the value of the underlying stock with the initial price of $100 at the end of four periods would be S4 = $100-1.2Y-.84Y depending on how many times the stock price increased. The expiration value of the option would equal c4 = $100-1.2Y-.84Y - $90. Therefore, if the stock were to increase in three out of the four time periods, the expiration value of the option would be c4 = $100-1.23-.843 - $90 = $48.24. More generally, the expiration value of the call would be given as follows: (7.9) To determine the expected value of the option, one would use the binomial distribution function described in Section 7.C. First suppose that the probability that the stock will increase in each of the n time periods equals .75. Thus, 1-p, or the probability that the stock decreases in any given period equals .25. In this example, the stock must experience two or more increases in order for option holders to be willing to exercise their calls.3 The probability that the call option will be exercised is determined as follows:

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Thus, there is approximately a ninety five percent chance that the call will be exercised. See Application 7.4 in this chapter to determine the expected value of the call given that it is exercised. More generally, the first step in estimating the probability that the call will be exercised is to determine the minimum number of price increases j needed for Sn to exceed X: (7.10) We solve this inequality for the minimum non-negative integer value for j such that ujdnjS0 > X. Take logs of both sides to obtain: (7.11) Next divide both sides by log(u/d) and simplify. Thus, we shall define a to be the smallest non-negative integer for j where Sn>X:

(7.12) The call option is exercised whenever j > a. The probability that this will occur is given by the following binomial distribution: (7.13) where p is the probability that an increase in the stock's price will occur in a given trial. It is assumed that p does not vary over trials. The next step in obtaining a model for valuing call options is to determine the expected value of the call given that the stock value exceeds the exercise price of the option. Application 7.3 provides general details on conditioned expected values and the binomial option pricing model is derived in Application 9.1. One difficulty in applying the binomial model to options is estimating probabilities for potential stock prices. Application 4.5 discusses the general

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computation of synthetic probabilities, and Section 10.B discusses probability estimation in the binomial framework. 7.D: THE NORMAL DISTRIBUTION The binomial distribution is in the class of stable distributions. If a finite variance is associated with a particular binomial distribution, then, as implied by the Central Limit Theorem, as the number of trials (n) drawn from a binomial distribution approach infinity (n-*oo), the binomial distribution approaches the normal distribution. The normal density function is defined as follows: (7.13) The area under the normal density function (represented by the normal curve) within a specified range (a,b) is found by evaluating the following definite integral:4 (7.15) Similarly, the Standard Normal Distribution for N(Y) is found by evaluating the following integral: (7.16) Unfortunately, no closed-form solution exists for either of the above integrals; they can only be represented as integrals. Thus, in practice, the Cumulative Standard Normal Density for N( Y) is frequently approximated with a polynomial function created to represent the cumulative density such as the following: (7.17) where:

k = 1/(1+aY) b, = 0.319381530 b3 = 1.781477937 b5 = 1.330274429

a = 0.2316419 b2 = -0.356563782 b4 = -1.821255978

One estimates a cumulative density for y by inserting the appropriate value for Y in the above polynomial function. Suppose, for example, that one were working with a normally distributed sample of random values with an expected value equal to zero and a standard deviation equal to one. What would be the

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probability that the probability of drawing a random number from this sample with a value less than .8? One would determine this probability by simply substituting .8 for Y in the above equations to obtain N(Y) = .7881. Alternatively, one could use the z-table at the end of this book to find the zvalue associated with a random variable .80 standard deviations greater than the mean. The value corresponding to .80 is .2881; one adds .50 to this value to obtain z=.7881. Thus, slightly over 78% of the area under the normal curve lies to the left of .8 standard deviations to the right of the mean. Notice that the normal distribution has several distinguishing characteristics. It is continuous (any fractional value may be selected), symmetric (skewness = E[y - fi]3 = 0), and ranges from -oo and 4- oo. The normal curve (representing the normal distribution) is bell shaped, meaning that a greater concentration of observations will be clustered about the mean than in the tails of the distribution such that its kurtosis (E[y - fi]4) equals 3. 7.E: THE LOG-NORMAL DISTRIBUTION Suppose that we use the following to forecast the price St+dt of a given security at time t-f-dt based on its current price St: (7.18) where a is the asset's instantaneous rate of return. The increment in time, dt, is assumed to be approaching zero. This return might be written as follows: (7.19) If this instantaneous return a is distributed normally with mean and variance fi and a2, then the return r is said to be distributed log-normally:

(7.20)

APPLICATION 7.2: COMMON STOCK RETURNS Suppose that the log of price relatives (instantaneous returns) for a stock a is normally distributed with an expected value over one year equal to 10% and a variance equal to .16. That is, a ~ N(/x,a2) with fi = .10 and a2 = .16. Assume that the current price of the stock is $100. The expected arithmetic return r and variance for the stock will be computed from Equations (7.20) as follows:

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The expected price of the stock in one year equals Sx = $119.7. Consider the short-term returns rM>t of a non-dividend-paying stock which are compounded over time. Further assume that the returns are independent and identically distributed (IID) with a finite variance. That is, assume that the distribution of returns is stable. The long-term return of the stock can be computed with the following: (7.21) By taking logs of both sides, this can be rewritten as follows:

We can apply the central limit theorem to the above random short-term returns with a stable distribution to note that the long-term distribution approaches normality in the limit. Thus, since the logs of (l4-r 0T ) are approximately normally distributed, long-term returns are approximately log-normally distributed. One useful feature of the log-normal distribution to describe stock returns is that the minimum value the ln(l4-rt.lt) can assume is minus infinity. This value is consistent with a value of -100% for return rt_lt and is also consistent with the limited liability feature of equity. Shareholders can lose a maximum of 100% of what they have invested in the firm. Thus, if the firm should go bankrupt in any time period t, its return for that period would be -100%. The natural log of (l4-rt.lt) for that period would be -oo and both the short-term return rt.u and the long-term return r0T would be -100% as shareholders would lose all of their investment. The log-normal distribution has several desirable properties for use in describing the distribution of security returns. First, it permits a security's price to increase at an increasing rate when its compound growth rate is constant. Thus, the log-normal distribution is able to appropriately reflect compound growth rates. Second, the log-normal distribution, while allowing the instantaneous rate ln(l 4-rM t) to range from -oo to 4- oo, the finite return rt_u can only range from -100% to oo. These ranges are consistent with the limited liability feature of equity. Third, the log-normal distribution is right skewed, which is consistent with the empirical evidence on the distribution of stock returns.

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7.F: CONDITIONAL PROBABILITY Conditional probability indicates the likelihood of a particular event A given the occurrence of another event B. The probability of event A given B is defined as follows: (7.22) The above statement reads: The probability of A given B equals the probability of both A and B divided by the probability of B. The probability of both A and B is normally determined as follows: P[A fl B] = P[A]P[B\A] = P[B]P[A\B] Thus, the probability of both A and B equals the probability of A times the probability of B given A, which equals the probability of B times the probability of A given B. Two events are deemed independent when occurrence of A has no impact on the probability of B: (7.23)

p[A fl B] = P[A]P[B]

Suppose that a random variable has a continuous distribution and that V is a function of x. The conditional expectation of V given a particular information set might be written as follows: (7.24)

E[V\I) = f"Vf(V\DdV

Conditional probability can also be used to determine the expected value of a variable V given its value exceeds some constant X: (7.25)

E[V\V>X\ =

rv(x)P(x)dx

Jx

fcp(x)dx

This concept is most useful for option pricing because the expected value of an option equals its expected value contingent on the underlying asset's price falling within a specified range. APPLICATION 7.3: OPTION PRICING — CONDITIONAL EXERCISE (Background Reading: Application 6.5) In Application 6.5, we saw that the expected future value of a call was simply the product of the probability that the call will be exercised multiplied by the expected value of the call given that it is exercised. The product of the two estimates given above results in the expected future value of the call: (7.26)

E[cn] = P[S>X]-E[(Sn-X)\S>X)

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The probability of exercise and expected value conditional on exercise (from Equation 7.25) are determined as follows: (7.27)

(7.28) The product of Equations (7.27) and (7.28) yields: (7-29) To value the call, one needs to determine the appropriate distribution of stock returns to calculate the appropriate function for p(S„). In the discrete outcome setting, these values are given as follows: (7.30)

(7.31)

(7.32)

APPLICATION 7.4: THE BINOMIAL OPTION PRICING MODEL (Background Reading: Application 7.1) In Application 7.1, we estimated the probability that an option would be exercised based on the assumption that the underlying stock returns follow a binomial distribution. Here, we estimate the expected value of the call. When the call is exercised, its value is ujdnjS0 - X. Noting that the binomial distribution is discrete rather than continuous (hence summation rather than integral notation), wefindthe expected value of the call when exercised with the following:

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(7.33)

where p(j), the probability that the stock will experience exactly j price increases in n trials, is given by: (7.34) Thus, the unconditioned expected future value of the call option is given by: (7.35) We discount E[cn] at the per-jump rate rf (where there are n jumps) to obtain the present value of the call: (l'5®)

This is the binomial option pricing model.5 NOTES 1. This statement is true when fi has a finite number of elements; otherwise, represents the set of all events for which a probability can be computed. Note that the power set $ of fi is the set of all subsets of fi. That is, fi represents the set of all combinations of outcomes for all combinations of factors that investors use to value stocks. Iffiis comprised of n < oo elements, its power set * has 2n elements including the empty set. 2. Understanding Section 7.A and Application 6.1 may be helpful, though is not crucial to understanding this section. 3. If the stock experiences only one price increase, its expiration date price will equal $1(XM.21\83=$61.44, not enough to warrant option exercise. If the stock experiences two price increases, its expiration date price will be $92.16. 4. Section 6.B and Application 6.1 may be useful for this section. 5. In some instances, the continuous-time discount function e"m will be used to replace the discrete discount function (1 + rf)n.

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SUGGESTED READINGS Baxter and Rennie [1996], Hull [1997] and Neftci [1996] all provide good informal introductions of probability theory and applications to the pricing of derivative securities. Perhaps the most widely read classic on probability theory is An Introduction to Probability Theory and its Applications by William Feller [1968]. Obviously, the book by Feller provides a much more comprehensive discussion of probability theory than this chapter. A more accessible text for many readers would be Hastings [1997]. The Binomial Option Pricing Model was originally derived by Cox, Ross and Rubinstein [1979] and is carefully described in Cox and Rubinstein [1985] and Elton and Gruber [1995].

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7.1 Sixty percent of all stocks on a given exchange increased over the period of one year. Based on a particular investor's portfolio of five stocks, find the following: a. The probability that no stocks increased in value b. The probability that exactly 1 stock increased in value c. The probability that exactly 2 stocks increased in value d. The probability that exactly 3 stocks increased in value e. The probability that exactly 4 stocks increased in value f. The probability that exactly 5 stocks increased in value 7.2 Forty percent of all stocks on a given exchange increased over the period of one year. Based on a particular investor's portfolio of five stocks, find the following: a. The probability that no stocks increased in value b. The probability that at least 1 stock increased in value c. The probability that at least 2 stocks increased in value d. The probability that at least 3 stocks increased in value e. The probability that at least 4 stocks increased in value f. The probability that exactly 5 stocks increased in value 7.3 Examination of trade-by-trade data for a given stock reveals that the stock has a 51 % probability of increasing by $.0625 on any given transaction and a 49% probability of decreasing by $.0625 on any given transaction. The stock has a current market value equal to $100 and is expected to trade ten times per day starting today. a. What is the probability that the stock's price will exceed $99.99 at the end of today? b. What is the probability that the stock's price will exceed $100.50 at the end of today? c. What is the probability that the stock's price will exceed $110 at the end of 10 days? 7.4 Examination of price data for a given stock reveals that the stock has a 41% probability of increasing by 42% in any given six-month period and a 58% probability of decreasing by 30% during any six month period. The stock has a current market value equal to $60; a one-year call option exists on the stock with an exercise price equal to $50. Assume an annual riskless return rate equal to 5%. a. What is the probability that the stock's price will exceed $100 at the end of six months? b. What is the probability that the stock's price will exceed $100 at the end of one year? (One year represents two six-month intervals.) c. What are the two potential stock prices at the end of the first six-month period? What are the probabilities associated with each of these prices? d. What are the three potential stock prices at the end of the second six-month period? What are the probabilities associated with each of these prices? e. What are the three potential call option values at the end of one year? That is, what would be the value of the call option conditional on each of the three potential stock prices being realized? f. If the stock price increases during the first six-month period, two potential stock prices are possible at the end of one year. What are these two prices? What are the potential call values? Based on these two potential prices and associated

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probabilities, what would be the value of the call option if the stock price increases during the first six-month period? g. If the stock price decreases during the first six-month period, two potential stock prices are possible at the end of one year. What are these two prices? What are the potential call values? Based on these two potential prices and associated probabilities, what would be the value of the call option if the stock price decreases during die first six-month period? h. Based on the two potential call option values estimated in parts f and g and their associated probabilities, what is the current value of the call? 7.5 Consider a one time period, two potential outcome framework where there exists Company Q stock currently selling for $50 per share and a riskless $100 face value TBill currently selling for $90. Suppose Company Q faces uncertainty, such that it will pay its owner either $30 or $70 in one year. Further assume that a call with an exercise price of $55 exists on one share of Q stock. a. What are the two potential values the call might have at its expiration? b. What is the riskless rate of return for this example? Remember, the treasury bill pays $100 and currently sells for $90. c. What is the hedge ratio for this call option? That is, how many shares of stock should be shorted for each call option purchased in order to maintain a perfectly hedged portfolio? (See also Application 4.6 for information on hedge portfolio construction.) d. What is the current value of this option? 7.6 Suppose that the log of price relatives (instantaneous returns) for a stock a is normally distributed with an expected value equal to 6% per annum and a variance equal to .08 per annum. What are the expected arithmetic return r and the variance for the stock? 7.7 The following table lists closing prices for a stock over a six-day period.

a. b. c. d.

Day

Price

Price Relative

Log of Price Relative

0

100

NA

NA

1

104

2

102

3

104

4

102

5

106

From this table, compute the following: Daily price relatives (St/St.!) Logs of daily price relatives ln^/S^) Mean of daily logarithmic relatives \i Variance of daily logarithmic relatives a1

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Statistics and Empirical Studies in Finance

8.A: INTRODUCTION TO HYPOTHESIS TESTING (Background Reading: Sections 2.D, 2.E and 2.F) In Section l.A, we discussed the process of induction to form testable hypotheses or theories from specific observations. These hypotheses or theories are useful if they provide a means to make meaningful predictions. Normally, testing a theory involves the collection of additional observations to determine whether they support the theory's predictions. If the additional observations do not confirm the predictions, then one has grounds for rejecting the theory. The observations collected to test a theory are frequently represented by numbers or data. Statistics may be defined as a branch of mathematics concerning the collection, organization, interpretation and presentation of numerical facts and data. Inferential statistics pertains to how one forms conclusions beyond observed sample data. In most cases, statistical inference concerns the generalization of sample results to a population. In many instances, one may make use of statistical inference to test a hypothesis. By convention, a hypotheses test usually involves formulation of a null (or maintained) hypothesis along with a competing alternative (research or challenging) hypothesis. The null hypothesis H0 usually is the claim that the population parameter equals some "maintained" value (note that null frequently connotates no difference, no impact or nothing). The null hypothesis normally includes an equality sign or either < or ^ signs. The alternative hypothesis HA is the claim that the population parameter differs from the maintained value. The alternative hypothesis normally includes a strict inequality sign. Such tests are usually structured in a conservative manner such that the burden of proof is on the alternative hypothesis. One supports the research or alternative hypothesis by demonstrating the null hypothesis to be false (rejecting the null hypothesis). One rejects the null hypothesis only when the probability of its being true is

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sufficiently low (the conventional probability, known as a level of significance, is .05 or .01). In some instances, the appropriate level of significance for a hypothesis test can be based on the relative costs of rejecting the null hypothesis when it is true or accepting it when it is false. One might list the steps of a typical statistical hypothesis test as follows: 1. 2. 3. 4.

Define the null hypothesis, HQ. Define the alternative hypothesis, HA. Determine a level of significance, a, for the test. Determine the decision rule or test statistic along with acceptance or rejection regions or critical value based on a. 5. Perform computations. 6. Form conclusions.

The decision rule or test statistic is a given function of a measurement drawn from the sample on which the statistical decision will be based. The rejection region consists of those values of the test statistic which will lead to rejection of the null hypothesis. The critical value marks the boundary between the acceptance and rejection regions. An experiment involving a given sample drawn from a population has some probability of resulting in an erroneous conclusion. Thus, one's hypothesis test may lead to an incorrect acceptance of the null hypothesis (Type I error) or an incorrect rejection of the null hypothesis (Type 11 error). The power of a test refers to the probability of not committing a Type II error. This is equivalent to the probability of accepting the alternative hypothesis when it is correct. A test is considered to be superior when its power is higher. Statistics are most useful for empirical studies in finance. This chapter provides, at best, a very superficial overview of a few of the applications of statistical methodology in finance. The reader is advised to consult a more comprehensive text (such as those listed in Suggested Readings at the end of this chapter) for a more detailed presentation of statistical methodology and its applications to financial problems. APPLICATION 8.1: MINIMUM ACCEPTABLE RETURNS Consider an example involving a bank which invests in various mutual funds for its trust clients. Suppose that this bank adopts a rule that no fund can be purchased unless it is almost certain that the fund's return will not be negative over a one-month period. The bank uses a computer-based analytical model for predicting expected fund returns and standard deviations. Obviously, no fund can guarantee a positive return; thus, the bank will consider a given fund only if the probability is sufficiently high that its return will not be negative. For any given fund i under consideration, our null hypothesis will be that the fund return will

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equal or exceed zero. Our alternative hypothesis is that the fund's return will be negative. We express our hypotheses more formally as follows: H0: ^ > 0 HA: R, < 0 In this example, we shall assume that we are willing to reject the null hypothesis if we are 95% certain that our randomly selected return will be non-negative. Thus, we will test at the 5% level of significance. Suppose, for example, that the bank is considering the purchase of a fund whose monthly return is approximately normally distributed with an expected (or mean) return of .01 and a standard deviation of .02. We may use the normal curve to find the probability that the stock's return in a given month will be nonnegative (Pr[Rj > 0]). First, we will define a normal deviate (our test statistic) for data point i (in this case, Xj 0) as follows: (8.1)

Determining the probability that the stock's return is non-negative is identical to determining the probability that the normal deviate for the stock's returns exceeds .5; Pr[Rj ^ 0] = Prfo ^ .5]. Our next step is to find the value on a z-table corresponding to .5 (or use the appropriate polynomial given in Section 7.D). This z-value will be compared to the critical value (l-o:=.95) separating our acceptance and rejection regions. By matching the appropriate row (0.5) and column (0.00) on the z-table, we find this corresponding value to be .1915. We then find the probability that R| > 0 and z} > .5 (remember, these probabilities are identical) by adding 50% (the area to the right of the mean on the normal curve) to .1915. This results in a z-value equal to .6915. Thus the probability that Zj > .5 or that the stock return will exceed 0 is found to be .6915. This zvalue is less than our critical value of .95; hence, we fail to reject the null hypothesis that the return on the fund will exceed zero. Even though our z-value suggests that the return is more likely to be positive than negative, the conservatism of our test does not lead us to conclude that the return will be positive. In summary, the steps of our test were organized as follows: 1. Define the null hypothesis HQ.- R; > 0. 2. Define the alternative hypothesis HA: Rx < 0.

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3. Determine a level of significance a for the test: .05 is our level of significance; thus we must be 95% certain that our null hypothesis is incorrect before we reject it. 4. Determine the decision rule or test statistic:

along with acceptance or rejection regions or critical values. Our critical value = .95. 5. Perform computations: z-value = .6915. 6. Draw conclusions: We reject the null hypothesis that the fund's returns are non-negative because our z-value is less than our critical value. Next, we shall make a slight revision to our example from above. Suppose we examined 120 months of returns on the fund, finding that the monthly returns were approximately normally distributed, with a mean value of .01 and a standard deviation of .02. We are interested in selecting one return data point at random from our sample. Suppose we hypothesize that the return which we randomly select will be positive; that is, our alternative hypothesis (HA) is that Rt > 0. We then construct a null hypothesis (Ho) that the return will be less than or equal to zero. We express our hypotheses more formally as follows:

Can we say with a reasonable degree of certainty that our randomly selected return will not be less than zero? If so, we will reject our null hypothesis. When we wish to test a statistical hypothesis, we generally construct a test of its "negative." This negative usually is represented by the null hypothesis. If we cannot reject the null hypothesis, we must reject the alternative hypothesis. Suppose we are willing to reject the null hypothesis if we are not 95% certain that our randomly selected return will be less than or equal to zero. Using the z-table, we find that the probability .95 corresponds to a critical value (or z score) of 1.645 (deduct .50 from .95 to obtain .45, then look up the value for .45). Since our null hypothesis here is concerned with a standard (Rj = 0) less than the mean (R. = .01), we reject our null hypothesis if -1.645 is less than the normal deviate for Rt = 0:

We fail to reject our null hypothesis that Rj < 0. Thus we reject our alternative hypothesis that RQ > 0 at the 95% level of significance.

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8.B: HYPOTHESIS TESTING: TWO POPULATIONS Application 8.1 concerned hypothesis testing with respect to the mean of a single population. Here, we are concerned with comparing two means, fix and fi2 for populations 1 and 2 with standard deviations ax and a2. We shall assume that our samples are independent and drawn from populations whose data are normally distributed. Our test will be based on samples of sizes nx and n2. The samples will have means and variances equal to X! and X 2 and S2X and S2, respectively. We will base our testing methodologies on test statistics and distributions somewhat different from those used in Application 8.1. Suppose that we wanted to test whether the means of two populations were different based on samples drawn from those populations. Our hypotheses and test statistics might be as follows:

(8.2)

where S, and S2 are the sample variances. If we are testing whether Xx and X 2 are equal, then our hypothesized difference in means Xx - X 2 = 0 is used for computing our test statistic. Our test statistic assumes that our data follows a student-1 distribution. A variety of other types of tests involving samples from two populations can be constructed as well. For example, tests can be developed to determine whether variances differ, other tests can be based on samples with matched pairs of observations, and so on. A statistics or econometrics text, such as those discussed at the end of this chapter, can be consulted to provide additional testing methodologies. APPLICATION 8.2: BANK OWNERSHIP STRUCTURE Some empirical evidence exists that financial services firms with substantial shareholdings by institutions are less likely to fail than firms with smaller levels of institutional shareholders. Some financial observers argue that institutional investors are more likely to hold stock in firms which are less likely to fail. Other observers have suggested that institutional shareholders pressure firm managers to perform well and maintain solvency. Data in Table 8.1 represent the proportions of institutional shareholdings taken from two samples of commercial banks. Each proportion represents the percentage of the bank's outstanding shares held by investment institutions. The first sample are from 20

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Chapter 8

Table 8.1 Institutional Ownership for Banks Solvent Banks

Failed Banks

Bank

%Owned by Institutions

Bank

%Owned by Institutions

A B C D E F G H I J K L M N 0 P Q R S T

31.40 14.60 5.10 8.10 5.30 1.90 2.40 1.90 18.80 11.40 14.30 13.70 24.80 18.70 36.60 11.00 17.10 6.60 30.10 12.10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

24.30 9.80 57.60 22.70 7.30 36.50 46.50 26.90 25.70 16.00 12.40 10.90 23.90 25.40 10.70 26.10 8.40 30.30 25.50 25.40

banks which remained solvent and the second from 20 banks which failed. Our problem here is to test the hypothesis that institutions which failed had smaller proportions of institutional shareholders (µi) than those firms which remained

solvent (fi2). More formally, our null and alternative hypotheses are given as follows:

Our two samples of data are drawn from populations whose means, fix and /i2, and standard deviations, ax and a2, are unknown. Suppose we wish to test our hypothesis at the 1% level of significance. Twenty data points are drawn from each of the two samples; hence, there are 38 degrees of freedom for this test. The appropriate critical value for our test is determined as follows:

Statistics and Empirical Studies in Finance

181

If the test statistic is less than -2.423 taken from the relevant t-table, we can reject our null hypothesis that fix > fi2. The appropriate test statistic is from Equation 8.2. Our preliminary computations from the Table 8.1 are as follows:

Thus, our test statistic is computed as follows:

Since -2.55484 < -2.423, we can reject at the 1 % level of significance the null hypothesis that institutional shareholder proportions at failed banks is greater than or equal to proportions at solvent banks. This supports the contention that institutional shareholders hold greater percentages of shares of solvent banks than of failed banks. 8.C: INTERPRETING THE SIMPLE OLS REGRESSION (Background Reading: Sections 2.F, 3.B and Application 5.8)1 Section 2.F described the Ordinary Least Squares Regression (OLS) as a means to determine relationships between a dependent variable and one or more independent variables. Section 2.F also reviewed basic assumptions underlying the OLS methodology. There are numerous types of regressions depicting different types of relationships among variables. Table 8.2 provides details on some of these different types of regressions. A simple regression is concerned with the relationship between a dependent variable and a single independent variable. Regression coefficients b0 and bx represent the vertical intercept and the slope in the statistical linear relationship between the dependent variable yx and the independent variable xx. Thus the vertical intercept b0 represents the regression's forecasted value for yj when Xj equals zero and the slope of the regression bx represents the change in yx (the value forecast by the regression for yt) induced by a change in Xj. The error term ex represents the vertical distance between the value yx forecasted by the regression based on its true value y-^ that is, ^j = ^ - &. The OLS regression minimizes the sum or average of these error terms squared. The size of the sum of the squared errors (often called SSE or, when divided by (n-2), the variance of errors o1) will be used to measure the predictive strength of the regression equation. A regression with smaller error terms or smaller oj is likely to be a better predictor, all else held constant.

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Table 8.2 Classes of Ordinary Least Squares Regressions By Number of Variables

By Shape of Curve

Simple

Linear

Multiple

Linear

Simple

Curvilinear*

Simple

Loglinear*

Multiple

Curvilinear*

Multiple

Loglinear*

Simple

Nonlinear**

Multiple

Nonlinear**

Example

* Curvi-iinear regressions may be transformed into linear regressions. In these examples, the transformation is completed by finding the log of both sides, while ignoring the error term, since its expected value is zero. ** Non-linear regressions cannot be transformed into linear regressions.

Once we have determined the statistical relationship between yj and Xj based on our OLS, our next problem is to measure the strength of the relationship, or its significance. One of the more useful indicators of the strength of the regression is the coefficient of determination or p2 discussed in Section 2.E. The coefficient of determination (often referred to as r-square) represents the proportion of variation of variable y that is explained by its regression on x. It is determined as follows:

(8.3)

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183

This coefficient of determination may also be expressed as either of the following: (8.4)

(8.5)

The sum H(yx - "y")2 represents total variation in y; the sum Ee2 represents the variation in y not explained by the regression on x. Assume that there exists for a population a true OLS regression equation yj = j30 + JSJXJ + 6j representing the relationship between yx and xx> without measurement or sampling error. However, we propose the regression yx = b0 + bjXj + €j, whose ability to represent the true relationship between yj and xx is a function of our ability to measure and sample properly. Our sampling coefficients b0 and bx are merely estimates for the true coefficients /?0 and ft and they may vary from sample to sample. It is useful to know the significance of each of these sampling coefficients in explaining the relationship between yj and x,. Our estimate bx for the slope coefficient /31 may vary from regression to regression, depending on how our sample varies. Our estimates for bx will follow a t-distribution if our sample of yj's is large or normally distributed; if our sample is sufficiently large, our estimates for bx may be characterized as normally distributed. One potential test of the significance of our coefficient estimate bx is structured as follows:

Our null hypothesis is that y is unrelated or inversely related to x; our alternative hypothesis is that y is directly related to x. The first step in our test is to compute the standard error s e ^ ) of our estimate for bx as follows:

(8.6)

The standard error for bj is, in a sense, an indicator of our level of uncertainty regarding our estimate for b,. The numerator within the radical indicates the

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variability unexplained by the regression; the denominator indicates total variability. Our next step is to find the test statistic for bx. This is analogous to standardizing or finding the normal deviate in our earlier hypothesis tests: (8.7) We next compare this test statistic to a critical value from a table representing the t-distribution or representing the z distribution. The process for determining the statistical significance of the vertical intercept b 0 is quite similar to that for determining the statistical significance for bx. We first designate appropriate hypotheses, such as those which follow:

The primary difference in the process is in determining se(b0):

(8.8)

Next, we find our t-statistic as follows:

(8.9) We then compare our t-statistic to the appropriate critical value just as we did when testing the significance of the slope coefficient. This particular test involves two tails, since our alternative hypothesis is a strict inequality. Be certain to make appropriate adjustments to the critical value (for example, divide G: by 2 for two tailed tests) when making comparisons. APPLICATION 8.3: THE CAPITAL ASSET PRICING MODEL (Background Reading: Section 3.C) Historical returns for Holmes Company stock, along with those of the market portfolio and Treasury Bill (T-Bill) rates rf are summarized in Table 8.3. This table also computes risk premiums on Holmes Company stock and the market portfolio. We may determine the relationships between risk premiums of the Holmes Company shares and risk premiums on the market portfolio by the use of an

Statistics and Empirical Studies in Finance

185

Table 8.3 Holmes and Market Returns Year

Holmes

rH-rf

Market

r«-rf

T-Bill

1986

12%

.06

10%

.04

6%

1987

18%

.12

14%

.08

6%

1988

7%

.01

6%

0

6%

1989

3%

-.03

2%

-.04

6%

1990

10%

.04

8%

.02

6%

ordinary least squares regression. Similarly, if we were to regress risk premiums of the stock (rH t - rf) against risk premiums of the market (rm t - rf), we would be able to construct a Characteristic Line and find a Capital Asset Pricing Model Beta (see Figure 8.1). Our regression equation is given as follows: (rH.t" rf,t)

==

<*H + bH(rmtt - rffl) 4- eH,t

where aH designates the vertical intercept for the regression and bH is the regression slope. The beta from the Capital Asset Pricing Model is frequently used as a measure of a stocks risk relative to the risk of the market portfolio. Figure 8.1 Characteristic Line for Security A U.14 «

0.120.1 0.080.060.040.02~ -.04 ^ ^ s ^

^

-0.02« -0.04 -

.02

.04

.08

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Chapter 8

We compute the stock beta through the following steps: a. Calculate the variance of risk premiums for market portfolio:

b.

Calculate the covariance between risk premiums on the stock and risk premiums on the market portfolio:

J

[(.06-.04) • (.04-.02) + (.12-.04) • (.08-.02) + (.01-.04) • (0-.02) + (-.03-.04) • (-.04-.02) + (.04-.04) • (.02-.02)] + (5-1) (.02) - (.02) + (.08) • (.06) + (-.03) - (-.02) + (-.07) - (-.06) + 0 4 = .0025

c.

Calculate beta for the stock by dividing the covariance between the stock and market risk premiums by the variance of premiums on the market portfolio:

This beta implies that one might expect that a 1 % increase in the market's return will lead to a 1.25% increase in the return of Holmes Company stock. However, this interpretation does omit consideration of nonmarket influences or firmspecific influences on its return. In any case, Holmes Company stock is regarded as having 1.25 times the level of systematic risk of the market or average security. The Characteristic Line for the Holmes Company will have a vertical intercept aH = .015 and a slope bH = 1.25. Normally, if we believe that the Capital Asset Pricing Model holds, we believe that a = 0 for all securities. If a does not equal zero, we might conclude either that Capital Asset Pricing Model did not hold for the testing period, or that our measuring or sampling techniques were insufficient or inappropriate (note that we only have five data points in this example). Suppose that we merely wish to test the hypothesis that our beta is greater than zero. Our null hypothesis and alternative hypotheses are given as follows:

Statistics and Empirical Studies in Finance

187

Assume we wish to test at a 95% level of confidence; that is, we wish to conclude that there is at least a 95 % probability that we are correct if we reject our null hypothesis. We perform a one-tailed test of our null hypothesis with 3 (n-2) degrees of freedom assuming that /3H follows a student-t distribution. Using the data for the Holmes Company, determine the statistical significance of the Holmes Company stock beta. We need to compute a standard error so that we may compute a test statistic. Our first step is to compute Eeand divide by its number of degrees of freedom, n-2, as illustrated by Table 8.4. Now that we have computed the standard error for the estimate of /3H = .064549, we may compute the test statistic for /JH using Equation 8.7:

Table 8.4 Holmes Stock Beta Significance a+0H(rm-rf)

€,

e]

.050

.065

-.005

.000025

.08

.100

.115

.005

.000025

.01

0

0

.015

-.005

.000025

.015

-.03

-.04

-.050

-.035

.005

.000025

.015

.04

.02

.025

.04

0

0

SSE =

.000100

Year

ff

rH-rf

fm-ff

1986

.015

.04

1987

.015

.06 .12

1988

.015

1989 1990

A\ w.Sauaired error = —

ftifrm-rf)

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Chapter 8

Since we may assume that /3H follows a student t distribution, may compare our t-statistic to a critical value found in the t-distribution table, given the appropriate level of confidence and degrees of freedom. In this case, the significance of our test is a = 1 - .95 = .05 and the number of degrees of freedom k equals n - 2 = 3. Our critical value for the test is found:

Since 19.36492 > 2.353, we reject the null hypothesis in favor of the alternative hypothesis. Therefore, it appears that market risk premiums are significant in explaining risk premiums of Holmes Company stock. The process for determining the statistical significance of the vertical intercept aH is quite similar to that for determining the statistical significance for bH. The primary difference in the process is in determining se(o:):

Suppose we are interested in testing the following at the 95% level of confidence:

Given our a value of .015 and our standard error of .002886, we find our tstatistic using Equation 8.9:

Next, we find our critical value tl~J2 = t 3 ^. Note that a is divided by 2 since we are performing a two tailed test. Our critical value is 3.182. Since our t statistic exceeds this critical value, we are unable to reject our null hypothesis; we cannot conclude that aH = 0. Therefore, based on our data, we must reject the applicability of the Capital Asset Pricing Model. We should stress at this point that our data set is too small to reach meaningful conclusions about the Capital Asset Pricing Model. Finally, we may determine the coefficient of determination (r-square) from Equations 8.3, 8.4 or 8.5 above. Based on the coefficient of determination of .992 determined from Equation 8.4, we conclude that 99.2% of all of the variation in risk premiums for Holmes Company stock can be explained by risk

189

Statistics and Empirical Studies in Finance

premiums on the market portfolio. A typical reporting format with t-statistics in parentheses for an OLS regression such as that we have just completed follows: .015

yi =

+

( 5.19615 )

1.25 • xt

r-squared = .992

( 19.3649 )

SSE = .001 d.f. = 3

APPLICATION 8.4: ANALYSIS OF WEAK FORM EFFICIENCY Weak form market efficiency tests are concerned with whether historical price sequences may be used to indicate future price levels. Markets are said to be weak form efficient if security prices fully reflect their history such that knowledge of security price histories cannot be used to generate abnormally high returns.2 Technical stock analysis is concerned with the examination of historical price sequences for particular patterns or sequences which would signal the profitability of buying, selling or holding securities. Thus, tests of weak form market efficiency are, in effect, tests of the usefulness of technical analysis in stock selection. There are an infinite number of potential types of price sequences which may be of interest to a technical analyst. Suppose, for example, an analyst were interested in whether there existed a relationship between the daily return on a security and its return on the prior day. Consider from Table 8.5 the sequence of daily closing prices for a given stock Q, from which we compute returns. The Table 8.5

Return Sequences Date

t

Pricet

Return

Return^

02/01/94

1

49

02/02/94

2

50

.020408

02/03/94

3

51

.020000

.020408

02/04/94

4

52

.019607

.020000

02/05/94

5

55

.057692

.019607

02/06/94

6

57

.036363

.057692

02/07/94

7

58

.017543

.036363

02/08/94

8

59

.017241

.017543

02/09/94

9

58

-.016949

.017241

02/10/94

10

55

-.051724

-.016949

02/11/94

11

53

-.036363

-.051724

02/12/94

12

52

-.018867

-.036363

Chapter 8

190

prices given in Table 8.5 assume that the stock traded each day, including weekends. Although there are many ways to determine the nature of the relationship between the return on a security and its prior day return, we will examine whether there exists a linear relationship based on a simple ordinary least squares regression of the form: ft = a + bit.x. We report regression results as follows:

Note that t-statistics are given in parentheses. Computations to obtain the tstatistics given above are described in Table 8.6. Given our standard error Table 8.6 Significance of Return Correlations t

Return,

Return,.!

f

^

cj

3 4 5 6 7 8 9 10 11 12

.020000

.020408

013735

.00626

.000039239

.019607

.020000

013420

.00618

.000038278

.057692 .036363

.019607 .057692

013118 042510

.04457 -.00614

.001986848 .000037788

.017543

.036363

026049

-.00850

.000072353

.017241

.017543

011525

.00571

-.016949 -.051724

.017241 -.016949

011291

-.02824

.000032673 .000797554

015095

-.03662

.001341651

-.036363

-.051724

041934

.00557

.000031029

-.018867

-.036363

030079

.01121

.000125692

SSE =

.004503110

Statistics and Empirical Studies in Finance

191

estimates se(a) = .007760684 and se(b) = .05606069533, we find our tstatistics to be t(a) = -.25959 and t(b) to be 3.2593. Suppose our two tests were expressed formally as follows:

Since we will assume that a and b follow student t distributions, we may compare our t-statistic to critical values found in the t-distribution table, given the appropriate level of confidence and degrees of freedom. We are performing two tail tests for significance. Suppose that we wish to test for significance at a level of .95 (.025 on each tail of the distribution). In this case, the significance of our test is a = .025 and the number of degrees of freedom k equals 10-2 = 8. Our critical value for the test is found: C2 = t8025 = 2.306. While we fail to reject our first null hypothesis that a = 0, we reject our second null hypothesis that b = 0. Thus, our test does provide evidence that the returns for this security are linearly related to returns on prior days. If we felt that we could generalize these results to stocks in general (note that our sample consisted of only one stock over a very limited time span), we may use this relationship as the basis for a trading rule. However, before implementing such a trading rule, we should ensure that our result cannot be explained by factors that investors price (such as very high risk or inflation). In some cases, corrections can be made by subtracting out various types of systematic risk premiums from returns and performing regressions on abnormal return components. Also, one should be particularly careful to ensure that apparent weak form inefficiencies are not due to actual market inefficiencies such as the inability to trade, high transactions costs or high taxes. Such inefficiencies can easily consume all of the apparent profits from a trading rule. APPLICATION 8.5: PORTFOLIO PERFORMANCE EVALUATION Proper benchmarks for comparison must be established when evaluating portfolio returns. Ideally, a benchmark should make allowances for portfolio risk. Here, we will use the Jensen alpha measure to evaluate the performance of a portfolio:

Presumably, a positive alpha (a) which is statistically significant will indicate that a portfolio outperforms the market on a risk-adjusted basis. Consider Table 8.7, which records returns over a twenty-year period for a portfolio (p). The

192

Chapter 8

Table 8.7 Portfolio p Performance Year

^

Rm

1975

0.14

1976

rf

Rp-rf

0.05

0.03

0.11

0.02

0.11

0.03

0.02

0.09

0.01

1977

0.04

-0.01

0.02

0.02

-0.03

1978

0.16

0.11

0.03

0.13

0.08

1979

0.03

-0.12

0.02

0.01

-0.14

1980

0.14

0.09

0.03

0.11

0.06

1981

0.26

0.13

0.04

0.22

0.09

1982

0.26

0.18

0.05

0.21

0.13

1983

0.13

0.04

0.05

0.08

-0.01

1984

-0.08

-0.11

0.04

-0.12

-0.15

1985

0.11

0.07

0.05

0.06

0.02

1986

0.22

0.17

0.06

0.16

0.11

Rnf r f

1987

0.22

0.16

0.07

0.15

0.09

1988

-0.01

-0.05

0.06

-0.07

-0.11

1989

0.04

-0.08

0.05

-0.01

-0.13

1990

0.28

0.21

0.06

0.22

0.15

1991

0.22

0.11

0.07

0.15

0.04

1992

0.21

0.11

0.08

0.13

0.03

1993

-0.04

-0.11

0.07

-0.11

-0.18

1994

0.18

0.12

0.05

0.13

0.07

table also records hypothetical and the market and riskless return rates. To compute the Jensen alpha measure, we first convert returns to risk premiums by subtracting riskless rates from returns on the portfolio and on the market. We then run a regression of portfolio risk premiums against market risk premiums to obtain the following:

First, we can conclude that the portfolio beta (.966203) is statistically significant at the 1% level. But more importantly, we can conclude that Jensen's alpha is statistically significant at the 5% level. Although we have concluded statistical

Statistics and Empirical Studies in Finance

193

significance for alpha, we should question the relevance of data as old as twenty years to evaluation of current fund management. 8.D: MULTIPLE OLS REGRESSIONS (Background Reading: Sections 2.F, 3.C and Application 5.9) A multiple OLS regression is concerned with the relationship between a dependent variable and a series of m independent variables. The regression equation produces a (m+1) dimensional surface which, where m ^ 3 can usually be plotted on a graph. The following represents the relationship between yi and xjfi based on a multiple linear regression involving m independent variables: (8.10) Coefficient b0 is the vertical intercept, just as in the simple OLS model. The m coefficients bx to bm are slope coefficients; each coefficient b} represents the change in yi induced by a change in variable Xjti holding all other variables constant. Suppose that an investor wished to test the theory that stock returns for Company Y are related to market returns and industry returns. Table 8.8 lists historical returns for Stock Y, the market and the industry. What is the relationship between Stock Y returns and those of the market and the industry? The following equation is intended to describe this relationship:

We obtain regression coefficients b0, bXi and b2 by running a multiple regression of RY>t on Rm>, and R, t. Mathematical procedures for determining regression coefficients are given in Application 5.9. We find the summary statistics and regression coefficients for our example as follows:

After computing regression coefficients, we determine the significance of our results. Since the objective of the multiple regression is to minimize the sum of error terms squared (SSE = De2, or variance of error terms of), we can base many of our tests of significance on the standard error of the estimate:

Chapter 8

194

Table 8.8 Returns for Stock Y, the Market and the Industry Year

Stock Y Return

Market Return

Industry Return

1980

.15

.10

.40

1981

.25

.10

-.02

1982

.50

.25

.50

1983

.35

.25

.50

1984

-.18

-.03

-.40

1985

-.30

.08

-.50

1986

.40

.30

.60

1987

-.17

-.05

-.23

1988

-.35

-.25

-.40

1989

.35

.15

.65

n = 10 m = 2 df= 7

There are n data points. There are two independent variables. There are 7 degrees of freedom: n-m-1 = 10-2-1.

b0 = -.00928 bi = .66608 b2 = .44850

The vertical intercept of the regression line The slope along the RM axis holding R, constant The slope along the Rj axis holding RM constant

(8.11)

Thus, in our example, we would use our regression coefficients to forecast a value for each year's return for Stock Y, based on the regression coefficients and actual values for market and industry returns. Table 8.9 summarizes these computations leading to a standard error of the estimate equal to .109275727. We will use this standard error estimate later to indicate the goodness of fit of the regression plane. Earlier in this chapter, we used the coefficient of determination (often referred to as r-square or p2) to measure the goodness of fit of a simple OLS regression. The coefficient of determination measures the proportion of variability in the dependent variable that may be explained by variability in the

Statistics and Empirical Studies in Finance

195

Table 8.9 Computation of Standard Error Year

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989

Stock Return

.15 .25 .50 .35 -.18 -.30

.40 -.17 -.35

.35

+ + + + + + + + + +

-b0

-bx • Market Return

.00928 - .666 • .10 .00928 - .666 • .10 .00928 - .666 • .25 .00928 - .666 • .25 .00928 - .666 -.03 .00928 - .666 • .08 .00928 - .666 • .30 .00928 - .666 -.05 .00928 - .666 -.25 .00928 - .666 • .15

-b2• Industry = Return

- .448 • .40 - .448 -.02 - .448 .50 - .448 .50 - .448 •-.40 - .448 -.50 - .448 .60 - .448 •-.23 - .448 •-.40 - .448 .65

= = = = = = = = = =

*t

-.08652 .20164 .11878 -.03122 .02846 -.12000 -.05932 -.02438 .00498 -.03182

e]

.00748571 .04065868 .01410868 .00097468 .00080997 .01440000 .00351886 .00059438 .00002480 .00101251

SSE = .0835830

independent variables. The multiple regression coefficient of determination (multiple R-square) is determined in much the same way as that of the simple OLS regression: (8.4)

2 _ y

Total Variation in Y Explained by the Regression Total Variation in y

Our coefficient of determination computations for the multiple regression will be based on the following: (8.12) For our example above, we compute the multiple coefficient of determination as follows:

196

Chapter 8

Thus, 90.8% of the variability in returns for security Y can be explained by variability in market and industry returns. The multiple correlation coefficient pY = (.908) 5 = .953. The analyst should be aware that adding independent variables to a multiple regression will improve its statistical fit. For example, in the extreme case, if the number of independent variables in a regression equals the number of data points, the regression will explain 100% of the variability of the dependent variable. However, this seemingly perfect ability to explain will probably hold only within the sample being tested and will not likely reflect the strength of the actual relationship between the dependent variable and the set of independent variables. Including too many independent variables will appear to strengthen the statistical relationship among variables (as indicated by an increasing p2) within the sample while actually weakening the regression's ability to predict outside of the sample. One can safeguard against this by making use of an adjusted rsquare, computed as follows: (8.13)

Based on the adjusted r-square, approximately 80% of the variation in returns can be explained by variability in market and industry returns. Note that as the number of data points n in the sample increases, adjusted r-square increases. As the number of independent variables m in the regression increases, adjusted rsquare decreases. Our test of the significance of the simple OLS regression was whether the explanatory variable had no effect on the dependent variable; that is, our null hypothesis b = 0 was tested against the alternative b =£ 0. In the multiple regression, we are concerned with whether the dependent variable is independent of all of our explanatory variables. Thus, we test the following null hypothesis against its alternative:

HA: The regression coefficients do not all equal zero. In the simple regression, we used the t-statistic to test our null hypothesis since we only dealt with a single explanatory variable. In the case of the multiple regression, we use the F-statistic:

Statistics and Empirical Studies in Finance

197

(8.14)

This F-statistic might be interpreted as the proportion of variation explained by the regression to the variance not explained by the regression. In our example, the F-statistic is computed as follows:

Our critical value for this F-statistic at the 99% confidence level is 9.55, given 2 (m) and 7 (10-2-1) degrees of freedom. Our actual F-statistic of 34.9 far exceeds this critical value. Therefore, we may state with 99% confidence that our market and industry variables are statistically related to returns on security Y. Our next problem is to determine the statistical significance of each of our individual regression coefficients. First, we compute standard errors for each coefficient. For our example, they are found to be:3

The standard errors for the regression coefficients are indicators of our level of uncertainty regarding our estimates for b0, bx and b2. Our next step is to find the test statistic for the regression coefficients:

We then can compare these test statistics to critical values from a table representing the t-distribution or representing the z distribution. We find, given 7 degrees of freedom, that b0 is not statistically significant at the 80% level (critical value is 1.415); bx is significant at the 80% level (the critical level at a 90% confidence level would be 1.895), and that b2 is significant at the 99% level (the critical level is 3.499.) If our level of significance for tests is .05, we would reject only the null hypothesis stating that l^ = 0. We would fail to reject our other two null hypotheses. However, based on our earlier F-test results, we would not reject the null hypothesis that the combination of market and industry returns have no impact on the stock returns in our example.

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198 APPLICATION 8.6: ESTIMATING THE YIELD CURVE (Background Reading: Application 4.2)

In Application 4.2, we were concerned with the yield curve describing the relationships among spot and forward rates over different intervals of time. We defined the following discount function Dt:

where yt is the spot rate which varied over time. Our solution technique for the different discount functions Dt and yields yt required that we analyze a series of bonds maturing and making coupon payments on specific dates. In particular, our solution technique required that we have at least one bond for each yield we wished to estimate and that bonds make payments on identical dates. In reality, we may have difficulty finding bonds which make payments on common dates; furthermore, the bonds which we select may not be priced consistently. Our solution technique would not imply a spot rate for any date that would not be consistent with at least one bond payment. Here, we will consider an alternative technique for mapping out a yield curve. Suppose that a fixed income manager believes that the following equation describes the relationship between bond discount functions and time (t): Dt = a + bxt + b2t2 + e, where a, bx and b2 are multiple OLS regression coefficients. We can use the multiple regression technique to determine spot rates from the data in Table 8.10 derived from zero coupon bonds. Based on a two-independent-variable OLS model, what would this fund manager predict the for the yield for a 2.5 year bond? Note that none of the bonds mature or make a coupon payment in exactly 2.5 years, so that we cannot compute D 25 using the solution technique from Application 4.2. Thus, to estimate the 2.5 year yield, we shall perform an OLS regression of Dt on t and t2. The first step in our computations is to calculate each value for Dt from yt. We find that Dx = .917431, D2 = .82946, D3 = .741162, D4 = .6635 and D5 = .590785. We regress Dt against t and t2 to obtain the following regression equation and t-statistics: Dt = 1.014818 - .09956* - .002939*2 (552.74) (-33.1) (5.99)

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Table 8.10 Bond Yields and Maturity Data t2

Bond

Yield

t

A B

.060 .082

1 2

4

C

.100

3

9

D

.114

4

16

E

.125

5

25

1

Inserting t = 2.5 into this equation, we find that D 25 = .784287. This leads to a yield y 25 solution of .102072. Our standard error estimates for a, bx and b2 are, respectively, .001836, .003 and .000491. Thus, based on resulting tstatistics, our estimates for a, bx and b 2 are statistically significant at the .01 level. 8.E: EVENT STUDIES An event study is concerned with the impact of a given firm-specific corporate event on the prices of the company's securities. For example, an event study might be conducted for the purpose of determining the impact of corporate earnings announcements on the stock price of the company. Event studies have been performed on announcements of corporate events such as dividends, earnings, takeovers, insider transactions, managerial changes, and so on. Event studies are used to measure market efficiency and to determine the impact of a given event on security prices. A number of studies have suggested a relatively high degree of efficiency in capital markets. If this suggestion is true, then one would expect that security prices would continuously reflect all or nearly all available information. Suppose that security prices are a function of all available information, and new information occurs randomly (otherwise, it would not be new information). In this case, one would expect that security prices would fluctuate randomly as randomly generated news is impounded in security prices. Thus, the "purchase or sale of any security at the prevailing market price represents a zero NPV transaction."4 In a perfectly efficient market, any piece of new relevant information would be immediately reflected in security prices. One should be able to determine the relevance of a given type of information by examining the impact of its occurrence on security prices. Thus, non-random performance of security prices immediately after a given event suggests that news of the event has a significant effect on security values. The degree of efficiency in a market

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to a given type of information may be reflected in the speed that the market reacts to the new information. At any given point in time, security prices might be affected by a large number of randomly generated pieces of new information or events. An event study is concerned with the impact of a specific type of new information on a security's price. Given that more than one piece of news may be affecting the security's price at any given point in time, one will probably need to study more than one firm to determine how the given type of information will affect securities. Thus, a population or sampling of firms experiencing the given event will be gathered; the impact of the event on each of the firms' securities will be studied simultaneously. Thus, the first step in conducting an event study is to gather an appropriate sample of firms experiencing the event. The impact of the event on security prices is typically measured as a function of the amount of time which elapses between event occurrence and stock price change. In a relatively efficient market, one might expect that the effect of the event on security prices will occur very quickly after the first investors learn of the event. Event studies are usually based on daily, hourly or even trade-to-trade stock price fluctuations. However, we frequently are forced to study only daily security price reactions since more frequent data is not readily available. Additionally, if markets are relatively efficient, one should obtain security price information as soon as possible after the event is known; although, determining when the information is known may be problematic. For example, analysts are often able to predict, with a reasonably high degree of accuracy, firm earnings and, on the basis of their predictions, trade securities. The impact of corporate earnings changes may be realized in security prices long before earnings reports are officially released. Therefore, one may need to study the impact of a given event, news item or announcement by considering security price reactions even before the event occurs. One should also take care in deciding on the precise nature of the event. For example, the event itself may actually be more important than its announcement. A dividend announcement may be of much greater interest than actual payment of the dividend; thus, many studies of the reaction of share prices to dividends actually use the dividend announcement date as the event date. Event studies typically standardize security price reactions by measuring the timing of security price reactions relative to the date of the event. For example, suppose that Company X announced its earnings on January 15 and Company Y announced its earnings on February 15. Let the base period time (t=0) for Company X be January 15 and the base period time (t=0) for Company Y be February 15. January 16 and February 16 (one day after the events) will be denoted as (t= 1) for the respective companies. Thus, the timing of the corporate events are standardized and we are able to measure average security price reactions 1,2, and so forth days after (and before) the event occurs.

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Although stock return generating processes may be modeled as random walks if capital markets are efficient, one might expect a general drift in returns; that is, one might expect that investors will earn a "normal" return on their securities. Thus, excess or abnormal return randomness is observed when markets are efficient, except for a very short period after relevant new information is available. The abnormal return in a given period for security i, 6jt, is the difference between its actual or ex-post return Rxi and its expected, normal or ex-ante return E[RM]: eM = Ritl-E[RitJ. To measure the impact of an event on security returns, one must have a consistent means of measuring normal returns. Brown and Warner [1980], in their classic study of event study methodologies, suggest three models of normal returns: 1. Mean Adjusted Returns. The normal return for a security equals a constant Kj. Typically, the mean return for the security over a sampling of time periods outside of the testing period serves as the constant Kj. The expected return for the security is assumed to be constant over time, though ex-ante returns will vary among securities. Thus, the abnormal return for the security is found: elit = Rit - Kj. 2. Market Adjusted Returns. The normal return for a security at a given point in time equals the market return for that period. The expected returns for all securities are assumed to be the same during a given period, though they vary over time. Abnormal returns are found: eM = RM - R mt . 3. Market and Risk Adjusted Returns. Here, normal returns are assumed to be generated by a single index model. Typically, security returns are linearly related to market returns through stock betas. These risk-adjusted returns vary across securities and over time. Abnormal returns may be determined: 6M = Rit - ft(Rm,t-rfit). One may test the significance of an event by averaging the abnormal performance for the sampling of securities around the event dates. If abnormal returns are not statistically significantly different from zero during the relevant testing period, one may conclude that the test did not provide evidence indicating event significance. If either no abnormal performance is detected around the event date or abnormal performance rapidly disappears, we have evidence of market efficiency with respect to that type of information. APPLICATION 8.7: ANALYSIS OF MERGER RETURNS Takeover attempts have affected many firms in recent years. Recent price reactions following takeover announcements suggest that targets of takeover attempts experience significant positive abnormal returns. Suppose we wish to test the following hypotheses based on an event study:

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1. Takeover targets experience positive stock price reactions to takeover announcements. 2. Markets react efficiently to takeover announcements. Our first step in this event study is to locate an appropriate sampling of companies to study. Suppose, we wish to base our study on the following three targets of takeover attempts: Company X: Merger announcement date Jan. 15, 1998 Company Y: Merger announcement date Feb. 15, 1998 Company Z: Merger announcement date Apr. 10, 1998 We will establish an 11-day testing period around the event dates.5 Table 8.11 provides our three target firm stock prices during 12-day periods around merger announcement dates. We standardize event dates (merger announcements occur on the seventh date, standardized at day 0) and compute returns for each stock during each of the days in the testing period, as in Table 8.12. Table 8.11 Target Company Stock Prices Company X Date Price

Company Y Date Price

Company Z Date Price

1/09

50.125

2/09

20.000

4/04

60.375

1/10

50.125

2/10

20.000

4/05

60.500

1/11

50.250

2/11

20.125

4/06

60.250

1/12

50.250

2/12

20.250

4/07

60.125

1/13

50.375

2/13

20.375

4/08

60.000

1/14

50.250

2/14

20.375

4/09

60.125

1/15

52.250

2/15

21.375

4/10

60.625

1/16

52.375

2/16

21.250

4/11

60.750

1/17

52.250

2/17

21.375

4/12

60.750

1/18

52.375

2/18

21.500

4/13

60.875

1/19

52.500

2/19

21.375

4/14

60.875

1/20

52.375

2/20

21.500

4/15

60.875

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Table 8.12 Target Company Stock Returns Day

Company X Return

Company Y Return

Company Z Return

-6

NA

NA

NA

-5

0

0

.00207

-4

.00249

.00625

-.00413

-3

0

.00621

-.00207

-2

.00248

.00617

-.00207

-1

-.00248

0

.00208

0

.03980

.04907

.00831

1

.00239

-.00584

.00206

2

-.00238

.00588

0

3

.00239

.00584

.00205

4

.00238

-.00581

0

5

-.00238

.00584

0

The next step in this study is to determine normal returns for each of the securities for each date. We could use any of the three adjustment methods discussed above (with more information), though for reasons of computational simplicity, we have decided to use the Mean Adjusted Return method. We may compute mean daily returns for each security for a period outside of our testing period. Suppose we compute average daily returns and standard deviations for each of the stocks for 180 day periods prior to the testing periods (data not given). Assume that we have found normal or expected daily returns along with standard deviations as follows: Stock X Y Z

Normal Return .000465 .000520 .000082

Standard Deviation .00415 .00637 .00220

Next, we compute periodic residuals for each stock during each date in the testing period along with the average residual over the sample for each date as in Table 8.13.

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Table 8.13 Target Firm Stock Residuals

Day -5 -4 -3 -2 -1 0 1 2 3 4 5

Co. X

Co. Y

Co. Z

Average

-.000465

-.000520

.001988

.000333

.002028

.005729

-.004214

.001181

-.000465

.005690

-.002156

.001022

.002021

.005652

-.002161

.001837

-.002947

-.000520

.002001

-.000488

.039335

.048559

.008233

.032042

.001926

-.006368

.001979

-.000820

-.002852

.005361

-.000082

.000809

.001926

.005327

.001975

.003076

.001920

-.006334

-.000082

-.001498

-.002846

.005327

-.000082

.000799

One of our objectives is to determine whether any daily residual is statistically significantly different from zero. Following standard hypotheses testing techniques presented earlier in this chapter, standard deviations for each of the average daily residuals are computed along with normal deviates ([et - 0] -r a€i) as in Table 8.14. Our test for each daily average residual (ARJ is structured more formally as follows: H0: ARt < 0

HA: ARt > 0

We shall assume the residuals follow a t-distribution and we will perform a onetailed test with a 95% level of significance. Given 1 = 3 - 2 degrees of freedom, the critical value for each test will be 6.314. Based on our computations above, we find that none of the residual t-statistics (normal deviates) exceed 6.314. Thus, we may not conclude with a 95% level of confidence that any residual differs from zero. Based on the confines of the test that we established here, we may not conclude that markets appear inefficient with respect to merger announcements. Perhaps, in part due to our small sample with such a small number of degrees of freedom, we cannot conclude that merger announcements has any effect on security returns. Note that this example was structured so as to facilitate computations; it is unlikely that a realistic test would be structured with a sample set of only three firms.

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Table 8.14 Target Firm Average Residuals and Standard Deviations Day

Average

Standard

Normal

-5

.000333

.001169

.28545

-4

.001181

.004103

.28780

-3

.001022

.003372

.30326

-2

.001837

.003192

.57561

-1

-.000488

.002020

-.24197

0

.032042

.017251

1.85739

1

-.000820

.003922

-.20921

2

.000809

.003412

.23711

3

.003076

.001591

1.93273

4

-.001498

.003515

-.42623

5

.000799

.003394

.23550

The tests performed above were concerned with whether merger announcements significantly affected stock prices in any given date around the time of the announcement. We found no significant effect for any single day returns. In some other instances, we may find that while no effect is found on the residual for any particular date, the effect might be realized over a period of days. Perhaps, we may even wish to broaden our test to determine whether some of the effect might be realized over a period of time before the date of the announcement. We may wish to compute cumulative average residuals to determine cumulative effects over time: (8.15) Cumulative average residuals are found in Table 8.15. Cumulative average residuals may also be found by summing individual firm residuals and dividing by the number of firms (in this example, 3) as in Table 8.16.6 Next, we test for statistical significance of cumulative average residuals by computing standard deviations of the cumulative residuals of the firms for each day and computing normal deviates. For example, the sample standard deviation of cumulative residuals for day -5 is computed based on the following:

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Table 8.15 Target Firm Cumulative Average Residuals

Day

Average Residual

Cumulative Average Residual

-5 -4 -3 -2 -1 0 1 2 3 4 5

.000333

.000333

.001181

.001514

.001022

.002537

.001837

.004375

-.000488

.003886

.032042

.035929

-.000820

.035108

.000809

.035917

.003076

.038993

-.001498

.037495

.000799

.038294

The normal deviate for a given date is simply the cumulative average residual for that date divided by the standard deviation applicable to that date. Daily standard deviations of cumulative residuals along with their normal deviates are given in Table 8.17. Note that normal deviates do not increase significantly enough on the merger announcement date such that it will exceed the critical value of 6.314. Thus, if our hypotheses concerning each date t in our testing period were given as follows:

we would not be able to reject the null hypothesis that CARt < with 95% confidence for any date.

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Table 8.16 Target Firm Cumulative Average Residuals Day

Cumulative Residual, X

Cumulative Residual, Y

Cumulative Residual, Z

Cumulative Average Residual

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5

NA

NA

NA

NA

-.00046

-.00052

.00198

.000333

.00156

.00520

-.00226

.001514

.00109

.01089

-.00438

.002537

.00311

.01655

-.00654

.004375

.00017

.01603

-.00454

.003886

.03950

.06459

.00369

.035929

.04143

.05822

.00567

.035108

.03858

.06358

.00558

.035917

.04050

.06891

.00756

.038993

.04245

.06257

.00748

.037495

.03958

.06790

.00739

.038294

Table 8.17 Target Firm Normalized Cumulative Average Residuals

Day

Standard Deviation

Normal Deviate

-5 -4 -3 -2 -1 0 1 2 3 4 5

.001432 .003717 .007742

.23307 .40749 .32775

.011599

.37720

.010778

.36056

.012468

3.77089

.021347

1.66432

.023212

1.56549 1.60622

.024538 .022204 .024190

1.70760 1.60047

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8.F: MODELS WITH BINARY VARIABLES OLS Regression models discussed earlier in this chapter assumed that both independent and dependent variables were continuous rather than discrete. Furthermore, we assumed that the dependent variables in our regressions were to be normally distributed. Here, we wish to consider scenarios where noncontinuous or qualitative variables such as bankrupt or solvent, state of incorporation, Over the Counter (OTC) or New York Stock Exchange (NYSE) markets, and so on, might be included in a regression analysis. We will examine so-called qualitative response models. Consider a scenario where we wished to determine whether the length of time a security has been publicly traded affects its return. Suppose we wished to control our study for stock risk, for whether a given stock was listed on the NYSE (versus traded over the counter) and for whether the shares are of a domestic or foreign firm. Since NYSE listing and foreign versus domestic domicile are not quantitative variables, we can establish dummy variables (also known as indicator variables which binary or 0, 1 variables) for these qualitative variables as follows:

Where bx represents the risk (beta) of Stock i, Tx represents the length of time that the stock has been publicly traded, our regression model can be structured as follows:

A statistically significant positive (33 value implies that a stock is expected to earn a higher return if it is listed on the New York Stock Exchange and a positive /34 value implies that the stock return is expected to be higher if is issued by a domestic firm. For a second example, suppose that one wants to perform a study of interest rates offered by U.S. banks controlling for the banks state of domicile. The states could be arranged alphabetically, starting with Alabama. However, one should not include in the study just one dummy variable with fifty possible values because any single number (such as a mean) based on the numbers drawn from several states would be meaningless. For example, the average between Alabama and Maine is meaningless. Instead, one should establish fifty (0,1) dummy variables, one for each state. If a bank is located in a given state, the dummy variable for that state for that bank might be one; all other state dummy variables would then be zero. However, one should be aware of the problems discussed earlier associated with using too many independent variables. Regression models can also be constructed with binary dependent variables. Suppose we wished to identify managerial characteristics that seemed related to corporate takeover defenses. We might project that the following managerial

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209

characteristics affected the probability that a tender offer would be contested by an incumbent management team: 1. 2. 3. 4.

Level of CEO compensation X u AgeofCEOX 2 i Proportion of the target firm's shares held by the CEO X3tl Proportion of the bidding firm's shares held by the CEO X* -

We can establish a binary independent variable TCj which assumes a value of 1 if the target firm's CEO contests a takeover attempt and a value of zero otherwise. Thus, our regression model might be constructed as follows:

A model of this type is referred to as a linear probability model. This model does have an important drawback in that its error will display heteroscedasticity since they will obviously be related to the independent variables since Y or TQ must equal zero or one. Furthermore, if we could constrain each value for TQ to fall within the range {0,1}, we may be able to infer a probability from Y or TQ; that is, TQ may represent the probability that the CEO contests a takeover. The linear probability model fails to do this because Y can assume any value, including those which cannot represent a probability. To allow the dependent variable to imply a probability in a binary model, we could use a logit model, which is based on the cumulative logistic probability function given as follows: (8.16) Pi can be interpreted as the probability that the binary dependent variable Yj equals 1. In our example regarding managers contesting takeovers, this can be rewritten as follows: (8.17)

Pj here can be interpreted as the probability that the CEO contests the takeover attempt. Now, solve for the two equations above for xp or /J + j33X3>i + j34X4 j to determine how to use the OLS or the linear probability model presented above to determine probabilities P: (8.18)

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Chapter 8

Thus, one can use the linear probability model to estimate beta coefficients and values for ln[Pj/(l-Pj)]. The actual process for obtaining the logit regression coefficients is beyond the scope of this book, though many good econometrics texts will outline the computational process.7 Other qualitative response models with binary dependent variables include the multi-discriminate analysis procedure, the probit model and the tobit model. NOTES 1. Application 5.8 may be useful for understanding this section, but it is not crucial. 2. An abnormally high return exceeds a normal return required to compensate for time value of money and risk. 3. The computational procedure is not described here. Consult a statistics text for details. 4. See Brown and Warner [1980]. 5. Actual event studies typically use 30, 45 or 60 day testing periods with many more than 3 firms. The sample used here is small to simplify the computational process. 6. Some differences in the cumulative average residuals from the Table 8.15 will result due to rounding. 7. See for example, Greene [1993]. SUGGESTED READINGS This chapter was intended only to provide a very superficial overview of a few of the many empirical testing methodologies used in finance. The textbook by Ben-Horim and Levy [1984] provides an excellent introductory presentation of statistics with numerous applications to finance. The econometrics text by Greene [1993] discusses many important testing methodologies, providing derivations and applications. An excellent book on econometrics and empirical studies in finance is The Econometrics of Financial Markets by Campbell, Lo and MacKinlay [1997]. This book provides details on how econometrics has been used in many important financial studies.

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EXERCISES 8.1 Given the following data for GNP and sales, use a simple OLS regression to forecast 1996 sales for Smedley Company: X GNP ($000) 18,000 32,000 47,000 72,000 61,000 80,000 90,000

Y $ Sales 10,000 12,000 16,000 23,000 19,000 22,000

Year 1990 1991 1992 1993 1994 1995 1996

8.2 The following represents sales levels for company Y and GNP levels for years 1989 to 1996: Year 1989 1990 1991 1992 1993 1994 1995 1996

Y Sales GNP (000s) ($ billions) 400 20 30 450 50 500 80 600 100 700 150 800 250 1000 9 1200

Based on this limited data set, forecast a sales level for 1996. 8.3 Historical returns for Holmes Company stock, Warren Company stock and the market portfolio along with Treasury Bill (T-Bill) rates are summarized in the following chart: Year 1986 1987 1988 1989 1990

Holmes Co. 12% 18% 7% 3% 10%

Warren Co. 4% 20% 2% -3% 9%

Market 10% 14% 6% 2% 8%

T-Bill 6% 6% 6% 6% 6%

a. Calculate return standard deviations for each of the stocks and the market portfolio. b. Calculate correlation coefficients between returns on each of the stocks and returns on the market portfolio. c. Prepare graphs for each of the stocks with axes (Rh - Rft) and (Rmt - Rft), where Rit is the historical return in year (t) for stock (i) ; Rmt and Rft are historical market and

212

Chapter 8

risk-free returns in time (t). The axes that you will label are for a Characteristic Line. Plot Characteristic Lines for each of the two stocks, d. Calculate Betas for each of the stocks. How do your Betas compare to die slopes of the stock Characteristic Lines? 8.4 Based on the following data, would you conclude that the price of Canseco Company stock (ct) affected by the number of employees (yt) that the company hires? Year c1 y1

1 2 3 4 5 6 7 8 9 10

325 350 335 364 355 385 375 405 401 438 433 473 466 512 492 547 537 590 576 630

8.5 The following table lists five pure discount bonds along with their yields and terms to maturity. BOND YIELD t A 1 .060 B .082 2 C .100 3 D .114 4 E .125 5 Based on a multiple regression model, with t and t2 as independent variables, what would you predict for the yield of a 4.5-year bond?

9

Stochastic Processes

9.A: RANDOM WALKS AND MARTINGALES (Background Reading: Section 7.A) Stochastic processes generate outcomes which are influenced by random effects over time. In Section 7.A, we defined a random variable as a function x:fl-*R where the triple (w,4>,P) is a probability space satisfying: V X e R , the set (co e Q: x (co) <. X\ This set is an event in $ such that P[x*X] = PftaeQ: x(u)±X}]=Fx(X) Next, we will consider sequences of random variables. For example, let X!,x2,...,xn be random variables defined on the same probability space (Q,,P). The joint cumulative distribution function (cdf) F is:

The random variables x are independent if:

A stochastic process is a sequence of random tvariables defined on xa common probability space (fi,4>,P) and indexed by time t. The values of xt(oj) define the sample path of the process leading to state wEl]. The terms x(w,t), xt(w) and x(t) are synonymous. A discrete time process is defined for a finite or countable set of time periods. This is distinguished from a continuous time process which is a process defined over an interval of an infinite number of infinitesimal time periods. The state space is the set of values in process {xj:

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214

The state space can be discrete or continuous. For example, if stock prices change in increments of eighths or sixteenths, the state space for stock prices is said to be discrete. The state space for prices is continuous if prices can assume any real value. Consider an example of a particular stochastic process, a discrete time random walk, also known as a discrete time Markov process. A random walk is a process whose future behavior, given by the sum of independent random variables, is independent of its past. Let z, be a random variable associated with time i and let St be a state variable at time t such that St = S0 + zx + z2 + ... + zt. Assume that random variables z{ are independent. The discrete time random walk is described as follows: (9.1)

E[Sl|S0,z1,z2,...,zt_1] = S,, + E[zJ

It is important to note that E[S J is a function only of SM and zt; the ordering of ^ where i < t (the price change history) is irrelevant to the determination of the expected value of St. A specific type of Markov process, the discrete martingale process with E[zJ = 0, is defined with respect to probability measure P and history or filtration Qf^ = {S0,z1,z2,...,zt.1} as follows: (9.2) which implies: (9.3) Note that E[zJ = E[zJ = 0. Thus, a martingale is a process whose future variations cannot be predicted with respect to direction given the process history. A martingale is said to have no memory and will not exhibit consistent trends. A submartingale is defined as:

A submartingale will trend upward over time such that E[zJ > 0, and a supermartingale will trend downward over time. 9.B: BINOMIAL PROCESSES (Background Reading: Section 7.C) Consider a Markov process which produces one of two potential outcomes St+At at each time t-f At. For example, suppose a stock price can increase (uptick) by a/AT (a>0) with probability p or decrease (downtick) by a/AT

Stochastic Processes

215

with probability (1-p). This process applies to each time period t. The expected value of St+At at time t+At is given by: (9.4) The variance of St+At at time t is given by:

(9.5) Note that if p=.5, the expected value and variance of St+At are St and a2At, respectively. The standard deviation of St+At equals a/AT. Also note that the coefficient a can be scaled such that av'At may assume any value. The binomial process described above may be applied to security prices, with prices increasing or decreasing by a specified monetary amount. However, over a specified period of time, one might expect that a security with a high price to be subject to greater monetary fluctuation than a security with a low price; a $1500 stock will probably experience greater price fluctuation than a $2 stock. It may be more reasonable to instead apply the binomial process to security returns, resulting in the following expected return: (9-6) This return-generating process also results in the following: (9.7) If we define u as (1 +av / AT) and d as (1-aV/JTt), we can extend the above one time period model to an n time period model as follows:

As we shall see in the following chapter, the binomial process is often used as an approximation for a Weiner process. This approximation procedure is particularly useful for the pricing of American options. When die Binomial process is used for this purpose, it is sometimes useful to revise the definition of d to 1/u, as we shall see in Section 10. A and Applications 10.1 and 10.2. 9.C: BROWNIAN MOTION, WEINER AND ITO PROCESSES (Background Reading: Section 6.C and Application 6.6) The models described in Section 9.B were expressed in terms of discrete time. However, since investors operate in an environment where time passes continuously, it is frequently useful to work with continuous time stochastic

Chapter 9

216

processes. Furthermore, continuous time modeling is frequently easier than discrete time modeling. This section extends the discussion of differential equations in Section 6.C (without random elements) to stochastic differential equations. One particular version of a continuous time/space random walk is a Wiener process. A Weiner process is a generalized form of a Brownian motion process. The Weiner process may allow for drift; the standard Brownian motion process does not. A process z is a standard Brownian motion process if: 1. changes in z over time are independent; COV(dzt,dzt.|) = 0 2. changes in z are normally distributed with E[dz] = 0 and E[(dz)2] = 1; dz ~ N(0, 1) 3. z is a continuous function of t 4. the process begins at zero, ZQ = 0 Brownian motion has a number of unique and very interesting traits. First, it is continuous everywhere and differentiable nowhere under Newtonian calculus (as discussed in Chapter 5); the Brownian motion process is not smooth and does not become smooth as time intervals decrease. We see in Figure 9.1 that Brownian motion is a fractal, meaning that regardless of the length of the observation time period, the process will still be Brownian motion. Consider the Brownian motion process represented by the top graph in Figure 9.1. If a short segment of is cut out and magnified as in the middle graph in Figure 9.1, the segment itself is a Brownian motion process; it does not smooth. Further magnification of cutouts as in the bottom graph continue to result in Brownian motion processes. Many other processes smooth as segments covering shorter intervals are magnified and examined such that they can be differentiated under Newtonian calculus. Once a Brownian motion hits a given value, it will return to that value infinitely often over any finite time period, no matter how short. Over a small finite interval, we can express the change in z (Az) over a finite period as follows:

A generalized Wiener process is defined as follows: (9.9) where a represents the drift in the value of St and dz is a standard Brownian motion process. Because prices of many securities such as stocks tend to have a predictable drift component in addition to randomness, generalized Wiener processes may be more applicable than standard Brownian motion, which only includes a random element. The generalized Weiner process expression can be applied to stock returns as follows:

217

Stochastic Processes Figure 9.1 Brownian Motion

(9.10) s

The drift term, \i, represents the instantaneous expected rate of return for the stock per unit of time and o is the instantaneous stock return standard deviation. We shall later derive the Black-Scholes Option Pricing Model from this Weiner process. Over a small finite interval, we can express AS as follows:

218

Chapter 9

This expression can be applied to stock returns as follows:

An ltd process is defined as follows:

where a and b represent drift and variability terms which may change over time. Note that both the drift and variance terms, a and b, are functions of both St and t, and may change over time. 9.D: ITO'S LEMMA (Background Reading: Section 5.F) In Section 5.F, we used the Taylor series expansion to estimate the change in a continuous differentiable function Ay = Af(x) as follows:

Now, suppose that y = f(x,t). The Taylor series expansion can be generalized to two independent variables as follows:

Now, consider an Ito process similar to that from Section 9.B:

In the finite period setting, this can be rewritten as:

Now, suppose that y is a function of this Ito process.

Stochastic Processes

219

When estimating Ay using the Taylor series expansion, our approximation procedure involves the dropping terms of a given or higher order. As Ax and At approach dx and dt, our approximation procedure provides estimates for Ay of increasing accuracy. In fact, under Newtonian calculus, solutions for dy based on dx and dt are exact; higher order terms such as (Axf and (At)2 are negligible because they tend towards zero faster than Ax and At as they approach dx and dt. This accuracy in differentiating does not hold for the Ito process because it does not become more smooth as Ax and At become smaller. Hence, the Taylor series expansion is necessary to compute dy. In the Ito process given above, all terms of higher order than two will be negligible since (Ax)' and (At)1 (with i > 2) will approach zero more quickly than Ax and At as they approach dx and dt. Now, write Ay as follows:

All first order terms in the first line will be non-negligible because they approach zero no faster than Ax and At as they approach dx and dt. Second order terms in the third line will be negligible because (At)2 and tv/AT approach a zero more quickly than do Ax and At as they approach dx and dt. In the second order term on the second line, terms become negligible for the same reason. Only b2dz2At remains non-negligible. Thus, we can write our expansion as follows: (9.11) As Ax and At approach zero, taking the limit of this result will provide us with Itd's Lemma, which provides us with a formula for stochastic calculus analogous to a Chain Rule: Define a composite function: yt =f[xt,t] and let dxt = a[xt,t]dt + b[xt,t]zt. By Itd's formula, we obtain:

Consider the following application of Ito's formula to the stochastic process followed by function ln(St) = F[S„t] such that dS, = /iS.dt + aS,dz:

220

Chapter 9

Following standard rules for differentiation from Chapter 5, the above equation is written:

(9.12) dln(S) Extending this differential equation to a finite period leads to the following:

Suppose, for example, that the following Ito process describes the price of a given stock:

This process describes the infinitesimal change in the price of the stock; the solution for this equation giving the actual price level at a point in time is given by (see also Section 6.C for details):

Suppose that one needed a single period return (or, log of price relative) and variance for the stock. The expected value and variance of the log of price relative are given by:

Stochastic Processes

221

APPLICATION 9.1: GEOMETRIC WEINER PROCESSES (Background Reading: Section 7.E) Suppose that the continuously compounded or logarithmic return a of a security follows a Weiner process: a = \xdt + odz such that over one period of time, the price of the security equals Stea. The mean instantaneous return equals \i and its variance equals o2. Using Ito's Lemma, we solve for dStea as follows:

Following standard rules for differentiation from Chapter 5, the above equation is written:

Thus, if the continuously compounded (logarithmic) return a follows a Weiner process with expected value fx, the price relative follows a geometric Weiner process with expected value [1+V202. Suppose the logarithmic return a on a stock follows a Weiner process with an expected value over one year equal to 10% and a variance equal to .16. That is, a - N0*,a2) with \L = .10 and a2 = .16. Based on Equations 7.20, the expected arithmetic return and variance for the stock will be computed as follows where T = l :

222

Chapter 9

APPLICATION 9.2: OPTION PRICES - ESTIMATING EXERCISE PROBABILITY (Background Reading: Applications 7.1 and 9.1) The value of a stock option is directly related to the probability that it will be exercised. That is, the option value is related to the probability that ST > X, the exercise price of the option. Assume that the price of a stock is based on the following geometric Weiner process over a T time period framework: (9.13) We wish to find the probability that ST > X:

(A) Next we will change the direction of the inequality (and several of the signs) in this equation, invoke the definition of the cumulative normal density function [N(*)], and make a slight algebraic change in presentation to obtain:

(B)

where r is defined as follows: (C) In an environment where investors are risk neutral, r can be interpreted as the logarithmic riskless rate of return. We will call the expression within the brackets in the last part of the equation above d r o\/T = d2: (9.14)

Stochastic Processes

223

APPLICATION 9.3: OPTION PRICES - ESTIMATING EXPECTED CONDITIONAL OPTION PRICES (Background Reading: Applications 7.1 and 9.2) First, we define a new term q which represents P multiplied by the expected stock price conditional upon its value exceeding the exercise price of the call option: (A) The stock price follows a geometric Weiner process. Making use of the algebraic manipulation "completing the squares," this equation is written as follows: (B) Note that when the stock price exceeds the exercise price of the call option, we can make the following statements about our randomly distributed variable z:

(C)

(D) In Application 9.2, we defined d^ which allows us to write Equation D as follows: (E) Using our definitions for q, dx and the cumulative normal density function, note the following:

224

Chapter 9

(F) which can be written: (9.15) This equation represents the expected stock price conditional upon its value exceeding the exercise price of the call option. In Application 9.2, we estimated the probability (P) that the call option will be exercised and that the investor will pay out exercise price X with present value XerT. Combine these two equations to obtain the Black-Scholes Options Pricing Model where d2 = dx - (p/T: (9.16)

APPLICATION 9.4: DERIVING THE BLACK-SCHOLES OPTION PRICING MODEL (Background Reading: Application 4.6 and Sections 6.C and 7.D) For this derivation, we shall assume that all standard Black-Scholes assumptions hold: 1. 2. 3. 4. 5. 6. 7.

There exist no restrictions on short sales of stock or writing of call options. There are no taxes or transactions costs. There exists continuous trading of stocks and options. There exists a constant riskless borrowing and lending rate. The range of potential stock prices is continuous. The underlying stock will pay no dividends during the life of the option. The option can be exercised only on its expiration date; that is, it is a European Option. 8. Shares of stock and option contracts are infinitely divisible. 9. Stock prices follow an Ito process; that is, they follow a continuous time random walk in two-dimensional continuous space.

We shall also assume that investors behave as though they are risk neutral. That is, investors price options as though they are risk neutral because they can always construct riskless hedges comprising options and their underlying securities. In Application 4.6, we found that the Law of One Price dictates that the current value of a call C0 on stock can be found from constructing a hedge portfolio: (9.17)

Stochastic Processes

225

S0 is the current value of its underlying stock and BQ is the current value of a riskless treasury instrument. Let #s be the number of shares of stock to purchase and #b be the number of treasury instruments to short in order to replicate the cash flow structure of the call. Similarly, we can replicate the cash flow structure of the bond as follows: (A) Let VH = B0 represent the value of a perfectly hedged portfolio. We can rewrite the above equation in terms of VH as follows: (B)

Since the hedge is riskless, its return should equal the riskless rate: (C) The hedge requires that we short sell #s shares of stock for each call that we purchase. The sensitivity of the call price to the stock price is dC/dS. Thus, the hedge will require that we short dC/dS shares for each purchased call. We write the value of the hedge portfolio and rewrite its differential equation as follows:

(D) (E) We rearrange the above differential equation by substituting for VH and solving for dC:

(F) which is rewritten as follows (G)

We shall assume that the instantaneous price change for the stock follows an ltd process: (H) which requires us to use Ito's Lemma to solve for dC. Substituting this Weiner process for dS into the preceding equation, we obtain the following: (I)

226

Chapter 9

Ito's formula is rewritten as follows for dC: (J) We shall use this particular formulation of Ito's Lemma to solve for dC as follows: (K)

(L) This equality can be rewritten as follows: (M)

(9.18) This is the Black-Scholes differential equation. Its particular solution, subject to the boundary condition CT = MAX[0, ST - X], is given by Equations 9.16, 9.16.aand9.16.b: (9.16)

(9.16a)

(9.16b) where N(d*) is the cumulative normal distribution function for (d*). This function is frequently referred to in a statistics setting as the "z" value for (d*). This solution to the Black-Scholes differential equation can be verified by finding the derivative of C0 in the Black-Scholes model with respect to t. From a computational perspective, one would first work through Equation 9.16.a and then Equation 9.16.b before valuing the call with Equation 9.16. The fifth order polynomial solving for N(Y) given in Section 7.D may be used for estimating density functions N(d^ and H(d^).

227

Stochastic Processes Example 1

Consider the following example of a Black-Scholes model application where an investor may purchase a six-month call option for $7.00 on a stock which is currently selling for $75. The exercise price of the call is $80 and the current riskless rate of return is 10% per annum. The variance of annual returns on the underlying stock is 16%. At its current price of $7.00, does this option represent a good investment? We will note the model inputs in symbolic form:

Our first step in solving for part a is to find d! and d2:

d,, = .09 - . 4 / 1 = .09-.2828 - -.1928 Next, by either using a Z-table or by using the polynomial estimating function above, we find normal density functions for dx and d2: N(dj) = N(.09) = .535864 N(dJ = N(-.1928) = .423549 Finally, we use N(d^ and N(d2) to value the call:

Since the 7.958 value of the call exceeds its 7.00 market price, the call represents a good purchase. Black-Scholes Model Sensitivities Option traders find it very useful to know how values of option positions change as factors used in the pricing model vary. Knowledge of sensitivities (sometimes called Greeks) are particularly useful to investors holding portfolios of options and underlying shares. For example, we mentioned above that the sensitivity of the call's value to the stock's price is given by delta: (9.19)

228

Chapter 9

Thus, a small increase in the value of the underlying stock would lead to approximately N(d^ times the amount of that increase in the price of the call option. A call investor may hedge his portfolio risk associated with infinitesimal share price changes by shorting N(d^ shares of underlying stock for each purchased call option. However, because this delta is based on a partial derivative with respect to the share price, it holds exactly only for an infinitesimal change in the share price; it holds only approximately for finite changes in the share price. This delta only approximates the change in the call value resulting from a change in the share price because any change in the price of the underlying shares would lead to a change in the delta itself:

(9.20)

This change in delta resulting from a change in the share price is known as gamma. Since gamma is positive, an increase in the share price will lead to an increase in delta. However, again, this change in delta resulting from a finite share price change is only approximate. Each time the share price changes, the investor must update his portfolio. Gamma indicates the number of additional shares which must be purchased or sold given a change in the stock's price. Since call options have a date of expiration, they are said to amortize over time. As the date of expiration draws nearer, the value of the European call option might be expected to decline as indicated by a positive theta: (9.21) Vega measures the sensitivity of the option price to the underlying stock's standard deviation of returns. One might expect the call option price to be directly related to the underlying stock's standard deviation: (9.22) In addition, one would expect that the value of the call would be directly related to the riskless return rate and inversely related to the call exercise price: (9.23)

Stochastic Processes

229

Sensitivities for the call option given in Example 1 are computed as follows:

SUGGESTED READINGS Elton and Gruber [1995] and Martin, Cox and MacMinn [1988] both provide a basic overview of the Binomial and Black-Scholes options pricing models. Martin, Cox and MacMinn also have prepared a mathematics appendix which includes a review of stochastic processes. Merton [1990] and Ingersoll [1987] cover stochastic processes along with a variety of applications to finance. Baxter and Rennie [1996], Wilmott, Dewynne and Howison [1993], Hull [1997], Pliska [1997] and Neftci [1996] all provide excellent informal introductions of stochastic processes and applications to the pricing of derivative securities. Jarrow and Rudd [1983] provide the source material for Application 9.3 as well as an excellent general introduction to option pricing. The text by Cox and Rubinstein [1985] to this day remains an excellent primer on options contracts, valuation and markets.

230

Chapter 9 EXERCISES

9.1 A stock currently selling for $100 has a 75% probability of increasing by 20% in each time period and a 25% chance of decreasing by 20% in a given period. What is the expected value of the stock after four periods? 9.2 Suppose the logarithmic return a on a stock follows a Weiner process with an expected value over one year equal to 5% and a variance equal to .09; that is, a N(/i,a2) with /i = .05 and o2 = .09. Find the expected arithmetic return and variance for the stock. 9.3. Suppose that the following Ito process describes the price of a given stock:

a. What is the solution to this stochastic differential equation? b. Suppose that there are fifty-two periods in a year. What are the expected value and variance of the log of price relative for this stock over a fifty-two-week period? 9.4 Let all Black-Scholes assumptions hold and assume that a hedge portfolio can be constructed with a long position in one put and a long position in #s shares of the underlying stock. a. Define the hedge portfolio; that is, how many puts are required for each share of stock? Set up an appropriate equation for the value of the hedge portfolio. b. Based on this hedge portfolio, derive the Black-Scholes differential equation for puts. c. Find the particular solution to the Black-Scholes equation for puts subject to the boundary condition pT = MAX[0, X-ST]. 9.5 Evaluate calls and puts for each of the following European stock option series: Option 1 T = 1 S = 30 a = .3 r = .06 X = 25

Option 2 T = 1 S = 30 o = .3 r = .06 X = 35

Option 3 T = 1 S = 30 a = .5 r = .06 X = 35

Option 4 T = 2 S = 30 a = .3 r = .06 X = 35

9.6 Evaluate each of the European options in the series on ABC Company stock. Prices for each of the options are listed in the table. Determine whether each of the options in the series should be purchased or sold at the given market prices. The current market price of ABC stock is 120, the August options expire in nine days, September options in 44 days and October options in 71 days. The stock return standard deviations prior to expirations are projected to be .20 prior to August, .25 prior to September, and .20 before to October. The treasury bill rate is projected to be .06 for each of the three periods prior to expiration. Do not forget to convert the number of days given to fractions of 365-day years.

Stochastic Processes

X

no

115 120 125 130

CALLS AUG SEP 9.500 10.500 4.625 7.000 1.250 3.875 .250 2.125 .031 .750

231 OCT 11.625 8.125 5.250 3.125 1.625

a a a r S

= .20 FOR AUG = .25 FOR SEP = .20 FOR OCT =.06 = 120

PUTS X AUG SEP OCT 110 .031 .750 1.500 .375 1.750 115 2.750 1.625 6.750 120 4.500 5.625 6.750 125 7.875 10.625 10.750 11.625 130 Exercise prices for 15 calls and 15 puts are given in the ieft columns. Expiration dates are given in column headings and current market prices are given in the table interiors. 9.7 Using standard rules for calculus, derive sensitivities (Greeks) for a European put option. Then, using these formulas, determine option price sensitivities for a six-month European put on stock currently selling for $75 and with a return standard deviation equal to .4. Assume that the put has an exercise price equal to $80 and the riskless return equals 10%.

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10 Numerical Methods

10.A: INTRODUCTION Numerical analysis is a branch of mathematics concerned with finding numerical solutions to problems. Numerical methods are particularly concerned with problems for which analytical solutions are difficult, time-consuming or impossible to obtain. Many types of problems do not have closed form solutions (that is, one cannot isolate the unknown variable(s) by itself (themselves) to be solved for on one side of an equation^]); numerical methods are often necessary to obtain numerical solutions or estimates. The majority of analysts are familiar with a few elementary types of numerical methods such as interpolation. Mathematicians are very interested in the analysis and estimation of errors associated with various numerical techniques. They are also concerned with the speed of convergence to solutions. This chapter is intended to provide a small sampling of numerical techniques used in finance, with an emphasis on certain techniques used in options analysis. Several of the methods discussed here have been, to some extent, introduced earlier in the text. 10.B: THE BINOMIAL METHOD (Background Reading: Section 9.B and Application 9.1) Suppose that the appropriate stochastic process for a stock is the geometric Weiner process, but that it is either difficult or impossible to derive a formula for some function of that stochastic process. This is often the case for the pricing of certain derivative securities such as American options (options which can be exercised before expiration). A binomial approximation to the Weiner process can often provide a tractable pricing procedure. This is true, in part, because the binomial distribution is stable, and will tend toward the normal

234

Chapter 10

distribution as the number of segments within a given time period approaches infinity. There are several methods which are used to obtain parameters for the binomial method from the actual security returns generating process. For example, following Cox, Ross and Rubinstein [1979], we can begin the process of estimating the mean and variance to be used in the binomial distribution from Section 9.B and Application 9.1. Thus, we approximate the mean and variance for the binomial process from the historical Weiner process as follows: (10.1)

(10.2) Approximation 10.2 approaches equality as T approaches zero. Scaling S0 to one, the following can be used for returns variance of a binomial process: (10.3) Assume that the binomial process will lead to a one-time period expected return for a security equalling the riskless rate: (10.4) This enables us to write probabilities of an uptick p and downtick (1 - p) as:

(10.5) If we define d as 1/u such that ud = 1, the following represents the variance of returns: (10.6) Substituting o for 5 will provide a good approximation for variance (improving as the number of jumps in the binomial process, n approaches infinity): (10.7) Thus, we can use the following to estimate u and d in the Binomial approximation to the Weiner process: (10.8)

Numerical Methods

235

or, if n and T differ from 1: (10.9) Suppose, for example, that for a particular Weiner process, o = .30 and rf = .05. Using equations (10.5) and (10.8) above, we estimate p, u and d for a single time period binomial process as follows:

m

*"-.7408182 1.3498588-.7408182

=

5097409

We can verify our estimates with Equations (10.4) and (10.7) as follows:

= .5097409-1.3498588 + (1-.5097409)-.7408182 = 1.0512711

- .5097409
Chapter 10

236

a\ = .2253794 - .3/5 - .2253794-.212132 = -.4375114 C0 = 100N(-.2253794)

110 •#(-. 4375114) = 5.59

„.0S'.5

Next, we will compute the call's value using the binomial model. We will vary the number of jumps in the model as the example progresses. First, let n = 1 and use Equations 10.9 and 10.5 to compute p, u and d: u =e

.•h-fS

1.2363111

d = 1/1.2363111 = .8088578

Thus, there is a .5064 probability that the stock price will increase to 123.63 and a .4936 probability that the stock price will be 80.88678. There is a .5064 probability that the call will be worth 13.63; therefore, its current value is 6.73 = .5064 • 13.63 • e05"5. Call and put values are determined by the binomial model from Application 7.4 as follows where n = a = 1:

.5063881.49361191-1[1.2363111.80885781-1100-1101 (1+.05)5

.50638811-.4936119'[110-1.236311-1-.80885781100] (1+.05)5

= 6.73

= 14.02

Numerical Methods

237

Now, divide the single six-month interval into two three-month intervals; that is, n = 2. Next, use Equations 10.9 and 10.5 to compute p, u and d: u = *-V3* = 1.1618342 d = 1/1.1618342 = .8607079

Thus, there is a .50432 probability that the stock price will increase to 134.98, a .5 probability that the stock price will remain unchanged and a A9572 probability that the stock price will decline to 74.08. Thus, there is a .2543 probability that the call will be exercised, in which case, worth 34.98. Therefore, the call's current value is 6.20 = .2543 • .34.98 • e05"5. Call and put values are determined by the binomial model from Application 7.4 as follows where n = a = 2:

2 22 2 22 t5043 't4957 - »[1.16185 %8607 - a00-110] 5

(1+.05)-

.50432-2.49572{110-L16182-2'.86072400] (1+.05)5

= 6.20

13.48

As the six-month period is divided into more and finer subintervals, the values of the call and put will approach their Black-Scholes values. Table 10.1 extends this example to more than two subintervals. APPLICATION 10.2: AMERICAN PUT OPTION VALUATION Although the option pricing models that we have discussed up to this point apply to European contracts, most options that are traded in the United States (and in other financial markets) are of the American variety. This means that most actual options contracts can be exercised prior to expiration at the owner's discretion. Valuation of American options is made more difficult because of this option to exercise early.

Chapter 10

238

Table 10.1 Convergence of the Binomial Model to the Black-Scholes Model n

C0

Po

1

6.73

14.02

2

6.20

13.48

3

5.47

12.72

4

6.04

13.30

5

5.18

12.44

6

5.91

13.17

7

5.43

12.68

8

5.81

13.06

9

5.57

12.82

10

5.73

12.98

50

5.63

12.89

100

5.59

12.85

OO

5.59

12.85

Volatility Riskless rate Exercise price Initial stock price Term to expiration

a = rf = X = S0 = T =

.30 .05 110 100 .5

American calls on stock should never be exercised early if the underlying shares do not pay dividends. This is because interest to be earned on exercise money is lost once the option is exercised. Furthermore, retaining the exercise money provides some downside price protection on the stock; regardless of how much the share price might drop, the option holder can retain the exercise money associated with his call option. Thus, call options on nondividend paying shares are exercised only at expiration. However, American calls on dividend paying shares may be exercised on the ex-dividend date. This should occur only if the dividends to be paid by the firm exceeds the interest which may be earned on option exercise money. Exercise should ensure that the call owner receives the dividend with the shares that he purchases. On the other hand, early exercise of an American put results in its owner receiving exercise money and interest payments on that exercise money. An American put will be exercised before expiration whenever one or both of the following holds:

Numerical Methods

239

In the first case, the put should be exercised immediately when the stock price reaches zero because the intrinsic value of the put is maximized and interest can be received on exercise money. Second, the American put should always be exercised when the value of a European put with otherwise identical terms is exceeded by its current intrinsic value. Consider Figure 10.1 which lists in a two period binomial framework extending over six months for a $100 stock, a sixmonth European put option with an exercise price equal to $110, and a sixmonth American put option with otherwise identical terms. Note that if the stock declines during the first period, the value of the European option is exceeded by its intrinsic value; thus, it would be sensible to exercise the American put at the end of this period. Because there is a positive probability that the American put will be exercised prior to expiration, its value exceeds that of the European put at time zero. Figure 10.1 Stock Prices and European and American Put Values

128.40 113.315

4.42 4.42

100 1000 10 N/A

100 11.02

11.99

0 0

88.25 19.57 21.75

77.88 32.12

N/A

t=0

t=1

t=2

The top value at each vertex represents the value of the underlying stock. The second value at each vertex represents the value of an European put on that stock where X = 110. The bottom value at each vertex represents the value of an American put with an exercise price equal to 110. Note that a decrease in the stock price in the first period results in early exercise of the American put.

240

Chapter 10

10.C: THE METHOD OF BISECTION (Background Reading: Section 2.1) One type of problem frequently encountered in financial analysis is determining the root of an equation of the form f(x) = 0. Obvious examples of this include internal rate of return and yield to maturity. As we shall see, however, a large variety of other problems may be expressed in this manner. This type of problem is one which often does not have an easily obtainable closed form solution. For example, consider the following equation: (A)

g(x*) = 60.4153 = 30(1+JC*)5 + 10(1+x*)2

this equation is easily written as follows: (B)

f(x*) = 0 = 30(1 +**)5 + 10(1+JC*)2 - 60.4153

Solving for x in Equation A is identical to solving for the root of Equation B. One simple and reliable method for obtaining solutions for an equation such as this is the Method of Bisection, which is also known as the binary search method, the half-interval method or the Bolzano method. This method is based on the Intermediate Value Theorem from Calculus which states the following: If y* is a real number between f(a) and f(b), and y = f(x) is defined for all values between a and b, then there exists a value x* between a and b such that f(x*) = y*. If a and b have opposite signs, this value for y* might be zero. First, we need to determine whether there exist values for x, a! and bx such that f(ax) and f(bi) have opposite signs. This method searches for a solution by bisecting (dividing in half) the interval [a^fy] to determine in which half the true value of x, x* must be located. The process repeats until a solution of a given level of accuracy is found. For Equation B, we will select arbitrarily an initial trial solution of .05. We substitute .05 for x into Equation B to find that f(x) = -11.1018. Since this value is negative, we will use it as the left-most point of our segment, a,. Now, we will find a bx value which leads to f(b^ > 0. Since f(x) is increasing in x for this example, bx > &x. Again, we will arbitrarily, select .20 for bx. Substituting bx into Equation B, we determine that f(bj) = 28.6343. We now have the two endpoints of the segment that we will bisect: ax = .05, bx = .20. The midpoint of this segment equals xx = .5 (ax + bx) = .125. We now perform the same substitution with xx, finding that f(xx) = 6.301924. Because f(x}) > 0, X! must exceed x*. Therefore, xx can serve as the right endpoint of the segment (maximum) for the second iteration. This same process can be repeated with a2 = &x = .05 and b2 = xx = .125. For the second iteration, x2 = .0875 and f(x2) is determined to be -2.95693. Since this f(x2) < 0, for the third iteration, we set a3 = x2 = .0875 and b3 = b2 = .125, such that x3 =

Numerical Methods

241

.1065 and f(x3) = 1.52617. This process continues until either a desired level of accuracy is reached or until a maximum number of computations has been performed (i.e., it is not cost effective to continue computations). The root of Equation B equals .1. Table 10.2 demonstrates the series of calculations used to solve for this root. This table structure should work well in a spreadsheet format. APPLICATION 10.3: ESTIMATING BOND YIELDS (Background Reading: Section 4.D) Consider a ten-year bond making semiannual coupon payments at an annual rate of five percent. This bond has a face value of $1,000 and currently sells for $750. What is its yield to maturity? That is, we wish to solve for y in the following:

Table 10.2 The Method of Bisection Initial Equation: 0 = 30(1-I-x)5 + 10(1+x)2 - 60.4153 a, = 0.05 b, = 0.2 X, = 0.125 f(a^= -11.1019 f(bl)= 28.6343 b„ f(Xn) n K X. 6.301924 0.125 0.2 1 0.05 0.125 0.0875 -2.95694 2 0.05 0.125 0.10625 1.526171 3 0.0875 0.10625 -0.75106 0.096875 4 0.0875 0.10625 0.378524 0.101563 5 0.096875 0.101563 0.099219 -0.18851 6 0.096875 0.101563 0.100391 0.094443 7 0.099219 0.099805 -0.04717 0.100391 8 0.099219 0.100098 0.023599 0.100391 9 0.099805 0.100098 0.099951 -0.01180 10 0.099805 0.100024 0.100098 0.005899 11 0.099951 0.100024 0.099988 -0.00295 12 0.099951 0.100024 0.100006 0.001475 13 0.099988 0.099997 0.100006 -0.00074 14 0.099988 0.100006 0.100002 0.000369 15 0.099997 0.100002 0.099999 -0.00018 16 0.099997 0.100002 0.000092 17 0.099999 0.1 -0.00005 18 0.099999 0.1 0.1 0.000023 19 0.1 0.1 0.1 -0.00001 20 0.1 0.1 0.1

242

Chapter 10

Using the method of bisection, we arbitrarily select trial &x and bx values for y and proceed as demonstrated in Table 10.3. We find in Table 10.3 that the bond yield equals approximately .088126. APPLICATION 10.4: ESTIMATING IMPLIED VARIANCES (Background Reading: Application 9.4) Analysts often employ historical return variances to estimate the volatility of securities. However, one cannot always assume that variances will be constant over time or that historical data properly reflects current conditions. An alternative procedure to estimate security variances is based on the assumption that investors price options based on consideration of the underlying stock risk. If the price of the option is taken to be correct, and if the Black-Scholes Option Pricing Model is appropriate for valuing options, then one can infer the underlying stock standard deviation based on the known market price of the option and the option pricing model. Consider the following example pertaining to a six-month call currently trading for $8.20 and its underlying stock currently trading for $75: t = .5 X = 80

rf = .10 S 0 = 75

c0 = 8.20

Table 10.3 Computing Yield to Maturity 20 2,10 Initial Equation: 25
m

Numerical Methods

243

If investors have used the Black-Scholes Options Pricing Model to evaluate this call, the following must hold: 8.20 = 75-N(d!) - 80«" K5-N(d2) d, = {ln(75/80) + (.1 + .5a2)-.5} + ov^.5 d2 = d! - &/.5 Thus, we wish to solve the above system of equations for a. This is equivalent to solving for the root of: f(o*) = 0 = 75-N(d,) - 80*" U5-N(d2) - 8.20 based on equations above for dx and d2. There exists no closed form solution for o. Thus, we will use the Method of Bisection to search for a solution. We first arbitrarily select endpoints b, = .2 and a! = .5 such that f(b 1 )<0 and i(2iX)>0. Since these endpoints result in f(a) with opposite signs, our first iteration will use a1 = .5(.2-l-.5) = .35. We find that this estimate for sigma results in a value of -1.30009 for f(a). Since this f(a) is negative, we know that a* is in the segment b2 = .35 and a 2 =.5. We repeat the iteration process, finding after 19 iterations that a* = .411466. Table 10.4 details the process of iteration. A difficulty arising with estimating implied variances results from the fact that there will typically be more than one option trading on the same stock. Each option's market price will imply its own underlying stock variance, and these variances are likely to differ. How might we use this conflicting information to generate the most reliable variance estimate? Each of our implied variance estimates is likely to provide some information, yet has the potential for having measured with error. We may preserve much of the information from each of our estimates and eliminate some of our estimating error if we use for our own implied volatility a value based on an average of all of our estimates. However, because volatility might be expected to vary over time, one should average only those variances implied by options with comparable terms to expiration. The following suggests two means of averaging implied standard deviation estimates: 1. Simple average: Here, the final standard deviation estimate is simply the mean of the standard deviations implied by the market prices of the calls. 2. Average based on price sensitivities to a: Calls which are more sensitive to a as indicated by dcjda are more likely to imply a correct standard deviation estimate. Suppose we have n calls on a stock and each of which implies a stock standard deviation a}. Each call price will have a sensitivity to its implied underlying stock standard deviation dcQtJ/da}. The sensitivities can be summed, and a weighted average standard deviation estimate for the underlying stock can be computed from the following weighting scheme:

Table 10.4 Using the Bisection Method to Estimate Implied Volatility Initial Equation: SN^-Xe^NKdJ a, = 0.5 b,= 0.2 a,= 0.35 S0= 75 X = 80 rf = 0.1 T = 0.5 C0= 8.2 N(ld,l) d,(cO <M<0 0.1356555 -0.2178978 0.553953 0.5 f(a,)= 1.860465 f(b1)= -4.46788 0.487199 0.2 -0.0320922 -0.1735135 a. d,(cO d,(<0 N(|d,l) n a„ bn 0.06499919 -0.1824882 0.525913 0.35 1 0.5 0.2 0.10188237 -0.198638 0.425 2 0.5 0.35 0.540575 0.08394239 -0.1900615 0.533449 0.3875 3 0.425 0.35 0.40625 0.09302042 -0.1942417 0.537056 4 0.425 0.3875 5 0.425 0.40625 0.41562 0.09747658 -0.1964147 0.538826 6 0.41562 0.40625 0.41093 0.09525501 -0.1953217 0.537944 7 0.41562 0.41093 0.41328 0.09636739 -0.1958666 0.538386 8 0.41328 0.41093 0.41210 0.09581161 -0.1955937 0.538165 9 0.41210 0.41093 0.41152 0.09553341 -0.1954576 0.538054 10 0.41152 0.41093 0.41123 0.09539424 -0.1953896 0.537999 11 0.41152 0.41123 0.41137 0.09546383 -0.1954236 0.538027 12 0.41152 0.41137 0.41145 0.09549862 -0.1954406 0.538041 13 0.41152 0.41145 0.41148 0.09551602 -0.1954491 0.538048 14 0.41148 0.41145 0.41146 0.09550732 -0.1954449 0.538044 15 0.41146 0.41145 0.41145 0.09550297 -0.1954427 0.538042 16 0.41146 0.41145 0.41146 0.09550514 -0.1954438 0.538043 17 0.41146 0.41146 0.41146 0.09550623 -0.1954443 0.538044 18 0.41146 0.41146 0.41146 0.09550569 -0.1954441 0.538043 19 0.41146 0.41146 0.41146 0.09550596 -0.1954442 0.538044 20 0.41146 0.41146 0.41146 0.09550610 -0.1954443 0.538044 21 0.41146 0.41146 0.41146 0.09550616 -0.1954443 0.538044 22 0.41146 0.41146 0.41146 0.09550613 -0.1954443 0.538044 23 0.41146 0.41146 0.41146 0.09550615 -0.1954443 0.538044

N(ld,l) N(d,) 0.41375 0.553953 0.431122 0.487199 N(ld,l) N(d,) 0.427597 0.525913 0.42127 0.540575 0.424628 0.533449 0.42299 0.537056 0.42214 0.538826 0.42256 0.537944 0.42235 0.538386 0.42246 0.538165 0.42251 0.538054 0.42254 0.537999 0.42252 0.538027 0.42252 0.538041 0.538048 0.42251 0.538044 0.42251 0.538042 0.42252 0.538043 0.42252 0.538044 0.42252 0.538043 0.42252 0.538044 0.42252 0.538044 0.42252 0.538044 0.42252 0.538044 0.42252 0.538044 0.42252

N(d.) f((0 0.413754 1.860465 0.431124-4.46788 N(d,) f(a„) 0.4276 -1.29619 0.421273 0.284948 0.424630 -0.50501 0.422993 -0.10987 0.422143 0.087583 0.422571 -0.01113 0.422357 0.038229 0.422464 0.01355 0.422517 0.00121 0.422544 -0.00496 0.422531 -0.00188 0.422524 -0.00033 0.422521 0.000438 0.422522 0.000053 0.422523 -0.00014 0.422523 -0.00004 0.422523 0.000004 0.422523 -0.00002 0.00000 0.422523 0.00000 0.422523 0.00000 0.422523 0.00000 0.422523 0.00000 0.422523

Numerical Methods

245

(10.10)

where Wj represents the weight for the implied standard deviation estimate for call option i. Thus, the final standard deviation estimate for a given stock k based on all of the implied standard deviations from each of the call prices is: (10.11)

10.D: THE NEWTON-RALPHSON METHOD (Background Reading: Sections 5.F) Although the Method of Bisection will reliably generate solutions for a wide variety of types of problems, its chief drawback is that it frequently converges to a solution rather slowly. An alternative method for finding roots is the Newton-Ralphson Method (also known as Newton's Method of Tangents), which is based on Taylor Series Expansions. This method can be used in many cases where a derivative dy/dx exists for y = g(x). When the Newton-Ralphson Method can be used, it typically converges to a solution for x faster than does the Method of Bisection. Among the situations where this method will not reliably converge to a root is when the second derivative of the function changes sign near the root or when the initial trial solution for certain functions is too far away from the actual root. Suppose we wish to solve y* = g(x*) for x*; or equivalently, solve for the root x* of 0 = g(x*) - y* = f(x*). By the NewtonRalphson Method we first select an initial trial solution XQ. Based on a first order Taylor approximation, we would determine our next trial solution by solving for X! as follows: (10.12) Thus, we have the following for x,: (10.13) The process of iteration continues in much the same way: (10.14)

246

Chapter 10

Consider the function from Section 10.B:

We find f(x) as follows:

Suppose we arbitrarily select our first trial solution for x* to be XQ = .15 and f(Xo) = 13.150416. Our next trial solution xx is determined as follows:

Thus our next estimate for x equals .1039149. We continue the process as follows:

Table 10.5 demonstrates that only a small number of iterations is necessary for convergence. APPLICATION 10.4 (continued): ESTIMATING IMPLIED VARIANCES Consider the following example where we wish to estimate the volatility implied by a six month option with an exercise price of $80 currently selling for $8.20. Assume that the underlying stock price is currently $75 and that the riskless return rate is .10. We shall solve for the implied standard deviation using the Newton-Ralphson Method, with an arbitrarily selected initial trial solution of a0 = .6. We need the derivative of the Black-Scholes model with respect to the underlying stock return standard deviation, which is obtained in Application 9.4: Table 10.5 The Newton-Ralphson Method Initial Equation: 30(1+ x*)5 Xo = 0.15 f(Xo) = 13.15042 n x,, 0.15 1 2 0.103915 0.100026 3 0.100000 4

4- 10(l+x*) 2 -60.4153

f(x n ) 285.3509 244.8365 241.6361 241.6150

f(xn) 13.15042 0.952203 0.006234 0.000000

247

Numerical Methods

We see from Table 10.6 that this standard deviation results in a value of f(cr0) = 3.900906. Plugging .6 into Equation 10.15 for a0, we find that f(a 0 ) = 21.07361. Thus, our second trial value for a is determined by: ox = .6 (3.90090+21.0376) = .414891. This process continues until we converge to a solution of approximately .411466. Notice that the rate of convergence is much faster by using the Newton-Ralphson Method than by using the Method of Bisection. Table 10.6 The Newton-Ralphson Method and Implied Volatilities Initial Equation: SN(d,)-Xe-rtN(d2) rf = 0.1 S0 = 75 X = 80 c0 = 8.20 T = 0.5 cr0 = 0.6 n 1 2 3 4 5 6

o„

f(an)

d,(qn)

Uon)

N(ld,l)

.600000 20.82508.177864-.2463997 .57058527 .410319 21.06193 .094961 -.1951785 .53782718 .411466 21.06084 .095506-.1954443 .53804365 .411466 21.06084 .095506-.1954443 .53804365 .411466 21.06084 .095506-.1954443 .53804365 .411466 21.06084 .095506-.1954443 .53804365

N(ld,l) N(d,) .40268 .42262 .42252 .42252 .42252 .42252

.570585 .537827 .538044 .538044 .538044 .538044

N(d.)

f(o„)

0.402686 3.950117 0.422627-0.024150 0.422523 0.000000 0.422523 0.000000 0.422523 0.000000 0.422523 0.000000

SUGGESTED READINGS Hull [1997] provides descriptions of binomial trees and binomial approximations to Weiner Processes. This book also describes briefly the Newton-Ralphson procedure. The article by Cox, Ross and Rubinstein [1979] is the first to describe the binomial approach to option pricing. Jarrow and Rudd [1983] present a brief review of applications of numerical methods to option pricing.

248

Chapter 10 EXERCISES

10.1 A stock with a current market value equal to $50 has a standard deviation of returns equal to .60 and trades in a market with a riskless return rate of .08. Under the binomial framework, what would be the value of nine-month (.75 year) European calls and European puts with striking prices equal to $80 if the number of tree steps were: a. 2 b. 3 c. 8 10.2 A stock with a current market value equal to $50 has a multiplicative upward movement equal to 1.2776 and a multiplicative downward movement equal to .7828. Assuming the binomial framework, answer the following: a. What would be the value of a nine-month (.75 year) European call and a European put with striking prices equal to $60 if the number of tree steps were 2? b. What would be the value of a nine-month (.75 year) American call and an American put with striking prices equal to $60 if the number of tree steps were 2? c. Why is the American put worth more than the European put? d. Identify the vertex on the binomial lattice where the American put is exercised early. 10.3 Use the Method of Bisection to solve for the standard deviation implied by a oneyear call selling for $4,841 on a stock whose current market value equals $30 per share. Assume that standard Black-Scholes assumptions hold, the current riskless return rate equals .06 and that the exercise price of the option equals $35. Note: The solution given for this problem begins with at = 0.6 and bx = 0.2. 10.4 A stock with a current market value equal to $50 trades in a market with a riskless return rate of .06. Under the Black-Scholes framework, what would be the volatilities (standard deviations) implied by six-month (.5 year) European calls with current market values based on each of the following striking prices: a. X = 40; c0 = 11.50 b. X = 45; c0 = 8.25 c. X = 50; c0 = 4.75 d. X = 55; c0 = 2.50 e. X = 60; c0 = 1.25

APPENDIX A Solutions to End-of-Chapter Exercises

CHAPTER 2 2.1. TV8 = 10,500 • (1 + 8-.09) = 10,500 • 1.72 = 18,060 2.2. The following answer the five parts of 2.2: a.

TV% = 10,500 (1 + .09)8 = 10,500 x 1.99256 = 20,921.908

2.3. Many of the calculations for this problem will draw from the following expression:

a. Use the future annuity formula - assuming end of year payments. Set TVA equal to $1,000,000 then substitute for n to find that n = 41.25. Thus, the client must make payments for 42 years, or until he is 65 years old.

250

Appendix A

b. Use the same substitution process as in part a to find that n = 36.27. Thus, die client must make payments for 37 years. c. Now, using the TVA formula, and n equal to 17, substitute (or better still, solve algebraicly for CF) to find that the annual payment must be $24,664,134. d. Use the same process as in part c, except that n equals 27. The annual payment equals $8,257.6423. e. Use n equal to 27 and i equal to .12 to find that TVA equals $5,904.0937. f. Use the same process as in part e to find the following: a. the answer becomes: n = 77.63 or 78 years; c. the answer becomes: payment = $42,198,523; d. the answer becomes: payment = $21,238,541 g. 17: simply divide $1,000,000 by (1.03)17 to obtain $605,016.45; 27: simply divide $1,000,000 by (1.03)27 to obtain $450,189.06 h. 17: simply divide $1,000,000 by (1.09)17 to obtain $231,073.18 27: simply divide $1,000,000 by (1.09)27 to obtain $97,607,807 2.4. Solve a through c with the following annuity expression:

2.5. The investment should not be purchased due to the following:

2.6. Present Value is determined as follows:

2.7. The following is used to determine cash flows:

Solutions to End-of-Chapter Exercises c.

CF5 = 10,000(1+ .1)5"1 = 10,000 x 1.4641 = 14,641

d.

CFl0 = 10,000(1 + .1)1(M = 10,000 x 2.3579477 = 23,579.477

251

2.8. This problem can be solved with either of the following:

b. Either use the geometric expansion to obtain the following equation (multiplying both sides of the original function by (1 +g)/(l +k)):

or multiply the present value growing annuity function by (1 +k)n to obtain the function used in part a.

= (-100,000 + 50,000 - 50,000 + 75,000 + 75,000)/6(100,000) = .083 or 8.3%

b.

100,000 = 50,000/(1 +r)2 - 50,000/(1 +r)3 + 75,000/(1 +r)4 + 75,000/(1+ r)6

IRR = 9.32487405%, IRR = -227.776188859% c. There are actually two internal rates of return for this problem. However, 9.32487 % seems to be a reasonable rate. 2.11. a. Its annual interest payments: iy = Int/F0 ; Int = iy(F0) = (.12)(1000) = $120 b. Its current yield: cy = Int + P0 = 120 -*• 1,200 = .10 c. With Equation 4.8, yield to maturity is found to be .04697429 or 4.697429%

252

Appendix A

2.12.a. Its annual interest payments: $120, or $60 every six months. b. 120 -s- 1200 = Its current yield = .10 or 10%. c. Its yield to maturity: 1200 = 60/[l+(r/2)] 1 + 60/[l+(r/2)] 2 + ... + 60/[l+(r/2)] 5 + 1060/[l+(r/2)] 6 r = .0476634

CFX = $3,000, n = 20, g = .10 Solve for r above to obtain IRR = .11794166365 2.14.a. Each outcome has a one-third or .333 probability of being realized since the probabilities are equal and must sum to one. b. E[SALES] = (800,000 • .333)+ (500,000 • .333)+ (400,000 • .333) E[SALES] = 566,667 C.

var[sales] = [ (800,000 - 566,667)2 • .333 + (500,000 - 566,667)2 -.333 (400,000 - 566,667) • .333] = 28,888,000,000 = o)ak5

d. Expected return of Project A = (.3-333) + (.15-.333) + (.01-.333) = .15333 e. Variance of A's Returns = [(.3-.1533) 2 • .333 + (.15-.1533) 2 • .333 + (.01-.1533) 2 • .333] = .0140222 = a\ f. Expected Return of Project B = (.2-.333) + (.13-.333) + (.09-.333) = .14 Variance of B's Returns = [(.2-.14) 2 • .333 + (.13-.14) 2 • .333 + (.09-.14) 2 • .333] = .0020666 = a\ g. Standard deviations are square roots of variances. "SALES = !69,964

aA = .1184154 o B = .0454606

Solutions to End-of-Chapter Exercises

253

COV[SALESA] = (800,000 - 566,667) • (.3 - .1533) • .333 + (500,000 - 566,667) • (.15 - .1533) • .333 + (400,000 - 566,667) • (.01 - .1533) • .333 = 19,444 = oSALEStA

j . First, find the covariance between sales and returns on B. COV[SALEStB] = (800,000 - 566,667) • (.20 - .14) • .333 + (500,000 - 566,667) • (.13 - .14) • .333 + (400,000 - 566,667) • (.09 - .14) • .333 = 7666.67 = o

SALESfi

k. Coefficient of Determination is Correlation Coefficient squared: .9932 = .986 2.15. Project A has a higher expected return; however, it is riskier. Therefore, it does not clearly dominate Project B. Similarly, B does not dominate A. Therefore, we have insufficient evidence to determine which of the projects are better.

RL = .062

2.16.a.

RY = .106 RM = .098 b.

o\ = .000696 (Remember to convert returns to percentages) o\ = .008824 o2M = .001576

254

Appendix A

c. COV[L,Y] = [(.04-.062X.19-.106) + (.07-.062)<.04-.106) + (.11-.062X-.04-. 106) + (.04-.062X.21-. 106) + (.05-.062) <.13-.106)]/5 = -.002392

d. COV[L,M] = [(.04-.062X.15-.098) + (.07-.062)-(.10-.098) + (.11-.062X.03-.098) + (.04-.062X.12-.098) + (.05-.062)-(.09-.098) ] + 5 = -.000956

e. COV[M,Y] = [(.15-.098X.19-.106) + (.10-.098X.04-. 106) +(.03-.098X-.04-.106)+(.12-.098X.21-.106)+(.09-.098X.13-.106)]+5 = .003252

2.17. Assuming variance and correlation stability, the forecasted values would be the same as the historical values in Problem (2.16). 2.18. Expected Return, Variance and Standard Deviation for Stock A I E^ £ Eifi Rj - EfRJ (R, - EfRJ)2 (R, - EfR,])^ .00125 1 .10 .50 .05 • .05 .0025 .00125 2 .20 .50 .10 .05 .0025 E[RJ = .15 o2a = .0025; aa = .05 Calculations for Stock B 1 Ei £ RP, R5 - EHRJ 1 .15 .50 .075 .025 2 .10 .50 .05 -.025 E[RJ = .125 Covariance between Returns on Stocks A and B 1 Eai Ebi £ ILrEITU RbcEIRJ {Rai^IEalXEbcEIEbl)Pi 1 .10 .15 .50 -.05 .025 -.000625 2 .20 .10 .50 .05 -.025 -.000625 COV(A,B) = -.00125 2.19.

Coefficient of determination is never negative because it is a squared value of the correlation coefficient.

Solutions to End-of-Chapter Exercises 2.20). a. Date 1/09 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/19 1/20 b., c.

255

Compan>'X Company Y Company Z Return Return Return 0 0 .00207 .00249 .00625 -.00413 0 .00621 -.00207 .00248 .00617 -.00207 -.00248 0 .00208 .03980 .04907 .04158 .00239 -.00584 -.02994 -.00238 .00588 0 .00239 .00584 .00205 .00238 -.00581 0 -.00238 .00584 0 Average Standard Return Stock Deviationi .004064 X .011479 Y .006693 .014150 .000869 Z .015537

CHAPTER 3

3.1.a b.

= .002025 + .0225 + .0054 = .029925 c

op = ^029925 = .1729884 since standard deviation is the square root of variance.

d

3.3. As proportions of funds invested in the Tilden Company increase, both expected portfolio return and portfolio variance (risk) levels will increase. Portfolio expected return increases because Tilden Company stock has a higher expected return. Portfolio variance increases because the correlation coefficient of .4 is not low enough to offset

256

Appendix A

the high variance of returns on the Tilden Company stock. The slope of the curve should be positive, although, it should be more steep at the bottom. 3.4.a.

F = .075,

a

b.

F = .075,

a

p

'

= .16

p

= .14

c.

F = .075,

a

d.

F = .075,

o„ = .0871770

p

e.

'

F = .075,

p

a

= .116619

= .04

3.5. Correlation coefficients have no effect on the expected return of the portfolio. However, a decrease in the correlation coefficients between security returns will decrease the variance or risk of that portfolio. 3.6.a. b.

Rp! = .25, Rp2 = .11, Rp3 = -.045; Since portfolio weights are equal, each weight is .5. Yp = (.20 • .25) + (.50 • .11) + (.30 • -.045) = .0915

c. Calculate standard deviation as follows: o 2 = (.25-.0915)2 • .20 + (.ll-.0915)2 • .50+ (-.045-.0915)2 • .30 o 2 = .0050244+ .0001711+ .0055896 = .0107851 ; ap = .1038517

d.

RA = .093 ; RB = .09

e. Calculate standard deviations as follows: Q\ = (.30-.093)2 • .20 + (.12-.093)2 • .50 + (-.09-.093) 2 • .30 o2A = .018981 ; aA = .1377715 a\ = (.20-.09)2 • .20 + (.10-.09)2 • .50 + (0-.09) 2 • .30 o 2 = .0049 ; o D = .07 f

g.

oAB = (.30-.093)-(.20-.09)-.20 + (.12-.093)-(.l-.09)-.50 + (-.09-.093)-(0-.09)-.30 = .00998548 Wp = (.5 • .093) + (.5 • .09) = .0915 ; It is the same, though found using portfolio weights and expected security returns rather than portfolio return outcomes and associated probabilities.

Solutions to End-of-Chapter Exercises

h.

257

op = .52 • .13777152 + .52 • .072 + 2 • .5 • .5 • .1377715 • .07 • .9985 a2 = .0107851 ; op = .1038517 ; the same as part c.

3.7. Security weights are: w x = .167, wY = .333, w z = .5 Tp = (.167 • .10) + (.333 • .15) + (.5 • .20) = .167 2

a

+ + + +

= (.167-.167-.12-.12-1) + (.167-.333-.12-.18-.8) (.167-.5-.12-.24-.7) + (.333-.167-.18-.12-.8) (.333-.333-.18-.18-1) + (.333-.5-.18-.24-.6) (.5\167\24\12\7) + (.5\333\24\18\6) (.5-.5-.24-.24-1) = .0323144; ap = .179762

3.8. Here, we want to find that wA value that will set portfolio variance equal to zero. Remember that portfolio weights must sum to one. Thus, wB is simply 1 - wA. o2p = w\ • .102 + w\ • .182 + 2 • wA • wB • .10 • .18 • -1 = 0 0 = .01 w2 + .0324 • (l-wA)2 2

- .036 • wA • (l-wA)

2

0 = .Olw + .0324 + .0324w - .0648vv< - .036w^ + .036wj 0 = .0784w2 - .1008^ + .0324; Solve for wA using the quadratic formula:

Plugging in for a,b and c, we find that the portfolio is riskless when wA = .64286. Thus, wfl = .35714. Riskless portfolios can be constructed from risky securities only when their returns are perfectly inversely correlated. Even in this case, only one combination of weights results in a riskless portfolio. 3.9. This would be a perfectly diversified portfolio; its standard deviation will be zero. Portfolio variance is determined as follows:

3.11.a. Technically the only correct method, but where do we get the probabilities?

258

Appendix A

b. OK if betas are stable, but requires a lot of inputs c. OK if only one index explains returns, fewer computations may be required for a large sample of securities d. OK if more than 1 index explains returns, but is more likely to generate errors in measurement or be less significant e. Updates for new information better than b through e, but is time-consuming and assumes that all securities react identically to changes in fundamental factors 3.12. Increasing # improves when: more sources of covariance are picked up in the model; mere is relatively little covariance between old and new indices. Decreasing # improves when: there is less measurement error.

CHAPTER 4

.02 .0004 .0032 .0020 .16 [.02 .16 .10] = .0032 .0256 .0160 .10 .0020 .0160 .0100 4.2.a.

The weights matrix is given as follows:

W b.

.25 .35 .40 ,60 .20 .20

The returns vector is given as follows: .10 r = .18 .26

c. The funds' returns are computed as follows:

Solutions to End-of-Chapter Exercises

259

.10 .25 .35 .40 .192 .18 _ .60 .20 .20] 26 .148 • . d-

.192 .148 Form a returns vector as follows: [.5 .5]

4.3.a.

.17

.05 r = .08 .18 b. The covariance matrix is as follows: .01 .02 .04

v= .02 .16 .10 .04 .10 .25 c. The weights vector is given: .20 .30 .50 d. Expected portfolio return is given as follows:

w=

.05 £[iy = [.20 .30 .50] .08 = .124 .18 r £[jy = w' e. Portfolio variance is found as follows: .01 .02 .04 .20 a2 = [.20 .30 .50] 02 .16 .10 .30 = .1177 .04 .10 .25. .50 4.4.a.

•i-

W

w

V

First, augment the matrix with the Identity Matrix: row 1 row 2

1 2 3 4

1 0 0 1

Now use the Gauss-Jordan Method to transform the original matrix to an identity matrix; the resulting right-hand side will be the inverse of the original matrix:

260

Appendix A

la

1

2

|

0 1 row l x l

1

2a

0 -.6 | -1

lb

1 0 | - 2

2b

0 1 | 1.5 -.5 (2a) x 1/-.6

1 row 2 x 31 1

-2 1 1.5 -.5 .04 .04 | 1 0 .04 .16 | 0 1 1

1 1 25

0

0 -3 | 25 -25 1 0 | 33.3 -8.3 0 1 | -8.3

The inverse matrix is

c. The inverse matrix is

d. The inverse matrix is

e.

8.3

33.3 -8.3 -8.3

(la)

(la) - 2 x (2b)

Thus, the inverse matrix is:

b.

3

8.3.

1 2 3 4

.04 .04 .04 .16 2 0 0 | 1 0 0 2 4 0 | 0 1 0 4 8 20 | 0 0 ll 1 0 0 | .5 0 0 0 4 0 | - 1 1 0 0 8 20 | -2 0 1.

Solutions to End-of-Chapter Exercises

261

1 0 0 |

.5

0

0'

0 1 0 | -.25 .25 0 0 0 2|

0

1 0 0 |

.5

-21 0

0 1 0 | -.25 .25 0 0 1 | .5 The inverse matrix is :

0

0

-.1 .05 33.3 -8.3

c-1

-8.3

33.3 -8.3 -8.3

0

-.1 .05

0

-.25 .25 0

4.5.

0

0

8.3

.01 • .11

8.3

-.583

x

i

= x2

=

.83

See (4.b) above for the inverse matrix. 4.6. Our original system of equations is represented: .08 .08 .1 1

x

i

0

.08 .32 .2 1

x2

0

.1

x3

.15

.2

0 0

1 1 0

0

\

1

The elements of C and s are known; our problem is to find the weights in vector x. Thus we will rearrange the system in part 3 from Cx = s to C"*s = x, where C"1 is the inverse of matrix C. So, the time-consuming part of our problem is to find C l . We will begin by augmenting Matrix C with the Identity Matrix I: Row 1

.08 .08 .1 1 i 1 0 0 0

Row 2

.08 .32 .2 1 i 0 1 0 0

Row 3

.1

Row 4

1

. 2 0 0 : 0 0 1 0 1

0 0 i 0 0 0 1

Original System

262

Appendix A 1 1

1.25

12.5

2a

0 3

1.25

0

3a

0 1 -1.25 -12.5 | -12.5

0

10 ol (row3) • 10-(la)

4a

0 0 -1.25 -12.5 | -12.5

0

0

|

12.5

0

| -12.5 12.5 0

lb

1 0

.83

12.5

2b

0 1

.416

0

3b

0 0 -1.6

4b

0

Ol (rowl) • 12.5

la

|

16.6

0 (row2) • 12.5 - (la) l | (row4) • 1 - (la)

-4.16 0 0 (la) - (2b)

| -4.16

4.16

0 (2a) • 1/3

0

-4.16 10 0 (3a) - (2b) 0 0 1 (4a) 0 0 -1.25 -12.5 | -12.5 6.25

-12.5 |

-8.3

|

-6.25

12.5

Ic

[10 0

2c

0 1 0 -3.125 | -6.25 3.125 7.5

5

0' (lb) - (3c) • .83

2.5

0 (2b) - (3c) • .416

-6

0 (3b) • -1/1.6

3c

0 0 1

4c

[0 0 0 -3.125 | -6.25 3.125 -7.5 1 (4b) - (3c) • -1.25

|

5

2.5

Id

1 0 0 0 |

0

0-10

2

(Ic) - (4d) • 6.2

2d

0 1 0 0 |

0

0

10

-1

(2c) - (4d) • -3.125

3d

0 0 1 0 | -10 10 -24

2.4

4d

0 0 0 1 |

2

(3c) - (4d) • 7.5 -.32 (4c) • -1/(3.125)

-1 2.4 C'1

/ 0

0

--10

2

0

*i

0

0

10

-1

0

x2

-10 10 --24

2.4

.15

x3

-1 '2.4

-.32

1

2

c

l

.

s

f ,5 ' .5

-1.2 [.04

x

*

—

X

=

X

Now it is clear that: xx = (00) + (OO) + (-10-. 15) 4- (2-1) = .5 x2 == (00) 4- (00) 4- (10-.15) + (-1-1) = .5 x3 = (-100) + (100) 4- (-24-.15) + (2.4 • 1) = -1.2 x4 = (20) + (-10) 4- (2.4-.15) 4- (-.32-1) = .04 4.7.a. First, solve the following system for the discount functions d:

Solutions to End-of-Chapter Exercises

50

50

80

80

263

1050

I>i

1080

*>2

110 1110

878.9172 955.4787

=

.D>.

0 •

CF

1055.4190 =

d

We find that D{ = ,943396, D 2 = .857338 and D3 obtained as follows

.751314. The spot rates are

.06

b. The weights are found by solving for w as follows: 50

80

110

50

80

1110

W

0

w

1050 1080

W

A

B

=

30 30 1030

c .

CF -.03996

.00396

.03885

-.00385 -.001667

-.00100

.00100 CF1

.002666

30 30

0

W

A

=

1030 -

PQ

W

B

. wc . =

w

We find that wA = 1.666666, wR = -.666666 and that w r 0. This means that bond D is replicated by a portfolio consisting of 1.666666 of bond A and -.666666 of bond B. 4.8. The following system may be solved for b to determine exactly how many of each of the bonds are required to satisfy the fund's cash flow requirements:

264

Appendix A

1100 100 0

110

120

1100 110

120

0

0

0

0

1110 120 0

b

i

30,000,000

K

15,000,000

'

1120

\b3

25,000,000

• vi

35,000,000

CF First, we invert Matrix CF to obtain CF'1: .000909 -.000083 -.00008 -.000079 0

.000909

0

0

0

0

-.00009 -.000087 .00090

-.000096

0

.000892

1

CF

Thus by inverting matrix CF to obtain CF"1, and premultiply ing vector P0 by CF"1 to obtain solutions vector b, we find that the purchase of 21,193.5 Bonds 1; 8,312.858 Bonds 2; 19,144.14 Bonds 3; and 31,250 Bonds 4 satisfy the insurance company's exact matching requirements. 4.9. First, find the values of Pure Securities 1, 2 and 3 as follows: 5 7 9

Pi

2 4 8

Pi

9 .0227272

13

5 3

=

5

.p\

-.0681818 .11363636

5

.375

-.375

-.125

3

-.1931818

.3295454

.03409090

5

.4772727 >•' p 2 .125 = =

A

.1931818

Thus, we find that Px = .477272727, P2 = .125 and P3 = .193181818. The value of Security D equals 1-.477272727 4- 1-.125 4- 1- .193181818 = .7954545454. 4.10. The following are the two equations to be solved simultaneously: 1. EFN = $400,000 - $60,000 - RE 2. RE = $100,000 - .04EFN We will first rewrite the two equations as follows: 340,000 = 1-EFN 4- IRE 100,000 = .04-EFN + 1-RE " 340,000 100,000

=

1 1 .04 1

•

EFN RE

Solutions to End-of-Chapter Exercises

265

We will now invert C and postmultiply it by s to obtain x: 250,000

1.04166667

-1.04166667

340,000

90,000

-.04166667

1.04166667

100,000

Thus, EFN = 250,000 and RE

90,000.

4.11. Since the riskless return rate is .125, the current value of a security guaranteed to pay $1 in one year would be $1/1.125 = .8888889. The security payoff vectors are as follows:

10

1 '

b =

16

c =

1

2 * 8 .

Portfolio holdings are determined as follows: 10 1

•

16 1

#r # b

=

'2 ' 8 .

The following includes the inverse matrix: -.166667

0.166667 -8

2.666667 -1.666667

We find that #r = 1 and #b = -8. This implies that the payoff structure of a single call can be replicated with a portfolio comprising 1 share of stock for a total of $12 and short-selling 8 T-Bills for a total of $7.11111111. This portfolio requires a net investment of $4.8888889. Since the call has the same payoff structure as this portfolio, its current value must be $4.8888889. 4.12. Since the riskless return rate is .125, the current value of a security guaranteed to pay $1 in one year would be $1/1.125 = .8888889. The security payoff vectors are as follows:

h =

10 16 25

b =

1 1 1

0 cl5

1

1

c9

10

7 16

Portfolio holdings are determined as follows: 10 1 0

#/i

16 1 1

# b

=

# cl5 .

25 1 10

1 7 16

The following includes the inverse of the securities payoff matrix: .2

.22222

-.02222

3.0 -2.22222

.22222

.2

.13333

-.33333

1 7 16

1.0

= -9.0 0

266

Appendix A We find that a portfolio replicating a call with an exercise price of 9 can be constructed with the following numbers of shares, T-Bills and calls with an exercise price of 15: #h = 1 and #b = -9 and #cl5 = 0. Thus, the payoff structure of a single call with an exercise price of 9 can be replicated with a portfolio comprising 1 share of stock for a total $12, short-selling 9 T-Bills for a total of $8 and selling 0 calls with an exercise price of 15 for a total of 0. Thus, this portfolio requires a net investment of $4. Since the call has the same payoff structure as this portfolio, its current value must be $4.

4.13.

1 [1 9 -6] 1 = 4 1

1'

i

1

1

[-1 -4 5] 1 = 0

[-6 10 -4] 1 = 0

= 4

1

1 ii = 0

2

3'

i

= 0

Vectors 2 and 3 are orthogonal to the unit vector. 4.14. We may write the expected returns vector E[r], the beta matrix #, the unit vector i and the factor risk premia vector h as follows: .08 E[r] =

.18

?

.08

1.5

2

1

3

2.5

1

1

i =

1

?1 6 =

1.

?2

The expected returns vector might be expressed as follows: .08

m

.18

1 = *o

.08

+

1 1

1.5

2

3

2.5

1

1

*i

6.L

We may solve the above for 50, dx and 62 for factor risk premia. This system may be expressed as a system of equations as follows: .08 16 t + 1.56j + 2.062 .18 = 160 + 3.061 + 2.56 2 .08 = 160 + 1.061 + 1.062 In matrix format, our system of equations is as follows: 1 1.5 2.0 \

.08

1 3.0 2.5

»2

1 1.0 1.0

6

3.

= .18 .08

We invert the coefficients (betas) matrix and solve for 6 as follows:

Solutions to End-of-Chapter Exercises

267 8o

-0.4 -0.4

1.81.08

-1.2

0.8

0.4 .18 = «, =

1.6

-0.4 -1.21.08 =

.04 .08 -.04

.**. The riskless return rate and two factor risk premia are determined to be as follows: 8Q = .04 = rf, 61 = .08, 62 = -0.04. One might suppose that the riskless return rate is quite high in this economy. CHAPTER 5 5.1. a. b. c. d.

The derivatives are found as follows: dy/dx = 3-1-x11 = 3 dy/dx = 70-x0"1 = 0 dy/dx = 5-2-x21 4- 3-1-x11 = lOx 4- 3 dy/dx = 4-2-x21 4- 13-1-x51 4- 9-1-x11 + 70-x0"1 4- 11-1-x31 - 8x 4- 65x4 + 9 4- 33x2 e. dy/dx = 2 • (1/7) • x 1 ' 7 " l = (2/7)x"6/7 = (2/7)/(Vx6) f. dy/dx = 2-2-x2"1 4- 7-1/2-x1'2'1 + 3-1-x'1"1 + 5-3-x 3 1 4- 3-1-x-1"1 = 4x 4- (7/2)x1/2 - 3x 2 - 15x'4 - 3x"2 = 4x + (l/2)h/x - 3/x2 - 15/x4 - 3/x2

5.2. a. b. c. d. e. f.

Second derivatives are found as follows: dy/dx = 0 dy/dx = 0 dy/dx = 10 dy/dx = 8 + 260x3 + 66x dy/dx = (-12/49)x"13/7 dy/dx = 4 - (7/4)A/x3 + 6/x3 4- 60/x3 4- 6/x3

5.3. Find first derivatives, set them equal to zero and solve for x. Then check second derivatives to ensure they are positive: a. dy/dx = 6x ; d2y/dx2 = 6 ; xmin = 0 b. dy/dx = 7 ; d2y/dx2 = 0 ; There is no finite minimum c. dy/dx = lOx + 3 ; d2y/dx2 = 10 ; xmin = -.3 d. dy/dx = 3x2 - 6x 4- 2 ; d2y/dx2 = 6x - 6 ; using the quadratic formula, we find that x = .4226497 and 1.5773503. The second derivative is negative when x < 1, therefore xmax = .4226497 and xmin = 1.5773503. e. dy/dx = 6x2 ; d2y/dx2 = 12x ; There is no finite minimum; the first derivative is zero when x = 0, but when x = 0, y is not positive. f. dy/dx = -2x 4- 7 = 0 ; d2y/dx2 = -2 ; There is no finite minimum 5.4. Only functions d and f have finite maximums: d. dy/dx = 3x2 - 6x + 2 = 0 ; d2y/dx2 = 6x - 6 ; xmax = -.4226497 f. dy/dx = -2x + 7 = 0 ; d2y/dx2 = -2 ; xmax = 3.5 5.5.a. i. First, find the yield to maturity (ytm) of the bond:

Appendix A

268

ytm = .111 ii. Use ytm from Part i in the duration formula:

Note: Negative signs are omitted.

ytm = .118

3

c.

ytm = .126 ;

^

=

.

2,000 (1.126)3

= 3

1,400 d. There are several ways to work this problem. First, consider the cash flows of the portfolio: P0 = 900 4- 800 4- 1400 = 3100 CF! = 1000 ; CF2 = 1000 ; CF3 = 2000 I0 °° + 10Q0 + 20QQ 3 - 3 1 0 0 (1+yfm)1 (1+yfm)2 (1+yfm)

Q-NPV--

A Dur

_

M

,

i . 1000

CFt (UytmY P0

=

1-122

Dur = 2.161 years

; ytm - . 1 2 2

2 • 1000 (1.122) 3100

2

3 • 2000 (1.122)3

Solutions to End-of-Chapter Exercises

269

Second, notice that the portfolio duration is a weighted average of the bond durations: (900/3100) • 1 4- (800/3100) • 2 4- (1400/3100) • 3 = 2.161 5.6. The duration of a pure discount bond equals its maturity. 5.7.a. i. First find the bond's ytm:

ii.

Now, use ytm to find Duration: 1 - 100 U 4 3

Dur -

b.

0

120

= NPV -

1

(1+yfm)

+

2 - 100 +

<

U43

> 900

120 2

(1+yfm)

2

3 - 1100 +

3

112Q

•

(1+yfm)3

= 2.722

- 900

ytm = .165 1 • 120 U 6 5

Dur =

c.

+ +

2 • 120 (U65)2 900

+ +

3 • 1120 (L165>3

. 2.672

n MM, 100 100 100 1100 0 = NPV = (l+ytm)1 + (l+ytm)2 + (1+yfm)3 + (1+yfm)4

nnn

900

ytm = .134

Dur -

d.

0

= NPV -

1-100

2-100

3-100

1134

2

3 900

10 ° > 10° + (1 +ytm)l (1 +ytm)2

+

4 • 1100 t1-134)4

= 3.456

112 ° - 800 (1 +ytm)3

ytm = .194 1 - 100 Dur -

Ll94

2 • 100 +

(U94)2

800

3 - 1100 +

(U94)3

- 2.703

5.8.a. %AP0 = Dur • %A(l4-r); %AP0 = 2.722 • .10 = .2722 %AP0 = .2722; AP0 = .2722 • 900 = 244.98; The new price is 900 + 244.98 = 1144.98 b. %AP0 = 2.672 • .10 = .2672; AP0 = 235.98; Price = 1135.98 c. %AP0 = 3.456 • .10 = .3456; AP0 = 311.04; Price = 1211.04 d. %AP0 = .2703; AP0 = 216.24; new price = 1016.24

Appendix A

270

5.9.

Dur = 2 . 7 2 2 - — + 2 . 6 7 2 - — + 3 . 4 5 6 - - ^ 3500 3500 3500 5.10. Partial derivatives are found as follows: i. a. dy/dx = 2 d. dy/dx = 15x2 + 7z b. dy/dx = 6x e. dy/dx = 36x2z5 4- 6z 6 c. ay/3x = 14x f. dy/dx = Ef^nix^z2 ii. a. dy/dz = 0 d. dy/dz = 6z 4- 7x b. dy/dz = 4 e. dy/dz = 60x3z4 4- 6x c. dy/dz = 0 f. dy/dz = 2zE?=1nxi

+

2 . 7 0 3 - — = 2.893 3500

5.11. Derivatives are found as follows: a. dy/dx = 27(9x + 7)2 b. dy/dx = 4x//(4x 2 4- 9) c. dy/dx = 9x(21x24-16x)4-9(7x34-8x2+6) = 252x3 4- 216x2 4- 54 d. dy/dx=6(3x-8)(5x-7)3 + 15(3x-8)2(5x-7)2 e. dy/dx = -x'2 = -1/x2 f. dy/dx = 19/(5x - 7)2 5.12. Solve the following for z(l) and z(2): .16z(l) + .05z(2) = (.25-.05) .05z(l) + .04z(2) = (.15-.05) z(l) = .7692308 ; z(2) = 1.5384616 Thus, w(l) = .333333333 and w(2) = .666666667 E[R(m)] = .1833 ; a 2 = .0577 ; om = .2403 ; G = .5547 5.13.a.

c. If the derived CML is for a single investor, an increase in risk aversity will lead to an increased required risk-premium on the market (Rm), increasing the slope of the CML. If the derived CML is for a market of many investors, no single investor will be able to affect its slope. In this case, the slope will remain unchanged; an investor will simply vary his holdings of the riskless asset. 5.14. Increase the cost of borrowing: if the borrower pays this cost, borrowing rates increase. Decrease the return from lending: lending rates decrease if the lender pays this cost. In summary, transactions costs increase interest rates; get two separate lines, one for borrowing and one for lending.

Solutions to End-of-Chapter Exercises

271

5.15. Use borrowing rate of .06 since he is likely to be a borrower. His 18% required return exceeds the return of either security. .02 = .09z! 4- 0z2 ; z{= .2222 .06 = 0 z , 4- .36z2 ; z2 = .1667 ; w2 = .571 ; w2 = .429 E[RJ = .097 ; w f = ( l - w j ; E[Rp] = .18 = (l-wj.06 + wm.097 .18 = .06 4- wm-.037 ; wm = 3.243 ; Borrow $67,297 Invest $97,297 in the market: $55,556 in security 1 and $41,741 in Security 2. 5.16. Answers are as follows: a. U10)00o = 10,000,000 - .01 • 100,000,000 = 9,000,000 b. I V ™ = 12,000,000 - .01 • 144,000,000 = 10,560,000 c tVooo + 2i000 = 9,000,000 4- (1000 - .02-10,000) • 2000 4- (1/2) • (-.02) • 20002 = 10,560,000 5.17. Durations and convexities are as follows: a. Dur = 4.203743015 ; Con = 20.31015 b. Dur = 4.0373493 ; Con = 17.86034 P1B = 1072.095524 5.18.a. P1A = 995.7906904 P1B = 1075.667592 b. P1A = 1000.1770910 P1R = 1075.815735 c. P1A = 1000.0000000 The new bond values given in 18.c are precise. Note how much better the bond convexity model in 18.b estimates revised bond prices than the duration model in 18.a. 5.19. First the Lagrange function: L = 15x2 - 3x + X(10 - .5x) Next, find the first order conditions: ay/ax = 30x - .5X = 3

ay/ax = -.5x

= -10

We find that x = 20 and X = 1194 20. (1) First set up LaGrange Function: L = (50 + 3x 4- 10X2) 4- X(100 - 5x) (2) Now, find first order conditions (partial derivatives with respect to each of the unknowns, setting them equal to zero): dL dx dL dz

3 + 20* - 5A. = 0 ; 100 - 5x = 0

20x - 5A. = - 3

; -5JC - 0k = -100

(3) Next, solve the above system of equations for x and X: 20 -5

-3

-5

-100

0 C x = 20; X

S

272

Appendix A

5.21. a. i. In this case, our problem is defined: min : o2 = .04A2 4- ,16wB2 4- .08wAwB (objective function) s.t.: .15 = .lw A 4- .2wB (constraint 1) 1 = lwA + lwB (constraint 2) The process of solving this problem by the method of Lagrange can be summarized: 1. Set up the Lagrange Function. 2. For minimization, set series of partial derivatives equal to zero. 3. Set up matrices from the equations to solve them. 4. Solve system of equations for portfolio weights. Thus, we solve the problem as follows: I. First, set up the Lagrange Function: L = (.04wA2 + .16wB2 + .08wAwB) - X^.15 - .lw A -.2wB) - X2(l - lw A - lwB) Notice that we have simply included the Objective Function along with the two constraints in L. X! and X2 are Lagrange multipliers; the values in the last two sets of parentheses must equal zero. II. Find partial derivatives with respect to each of the unknowns and set equal to zero: 6L/6wA = .08wA + .08wB 4- .lXj 4- 1\ 2 = 0 6L/6wB = .32wB 4- .08wA + 2\x 4- 1X2 = 0 6Lt6\x = .15 - . l w A - ,2wB = 0 <5L/6X2 = 1 - lwA - lwB = 0 Now, we have four equations with four unknowns. For simplicity, lets rearrange these equations: .08wA + .08wB -f .IX! 4- 1X2 = 0 .08wA + .32wB + .2Xj + 1X2 = 0 .lw A 4- .2wB 4- 0\x + 0X2 = .15 lwA + lwB 4- OX! 4- 0X2 = 1 III. One way to solve the system in part II is to represent it with a system of matrices. Three are needed-a coefficient matrix (C), an unknowns vector (x) and a solutions vector (s). .08 .08 .1 I

W

.08 .32 .2 1

W

1

.2

I

1 0 C

0 0 .15

A B

0 0

*.

0.

V -

x

1 =

s

IV. The elements of C and s are known; our problem is to find the weights in vector x. Thus, we will rearrange the system in part III from Cx = s to C^s = x, where C 1 is the inverse of matrix C. So, the time-consuming part of our problem is to find C 1 . We will begin by augmenting Matrix C with the Identity Matrix I:

Solutions to End-of-Chapter Exercises

273

Row 1

.08 .08 .1 1 | 1 0 0 0

Row 2

.08 .32 .2 1 | 0 1 0 0

Row 3

.1

Row 4

1

Original System

. 2 0 0 | 0 0 1 0 1

0 0 | 0 0 0 1

la

1 1

1.25

12.5

2a

0 3

1.25

0

3a

0 1 -1.25 -12.5 | -12.5

0

10 0 (row3) • 10-(la)

4a

0 0 -1.25 -12.5 | -12.5

0

0 1 (row4) • 1 - (la)

lb

1 0

2b

0 1 .416

3b

0 0

4b

|

12.5

0

0 0 (rowl) • 12.5

| -12.5 12.5 0 0 (row2) • 12.5 - (la)

.83

12.5

|

0

16.6

| -4.16

-4.16 0

0 (1<0 - (2b)

4.16

0 (2a) • 1/3

0

-4.16 10 0 (3a) - (2*) 0 0 -1.25 -12.5 | -12.5 0 0 1 (4a)

lc

10 0

2c

-1.6

-12.5 | -8.3

6.25

|

5

0 (lb) - (3c) • .83

0 1 0 -3.125 | -6.25 3.125

2.5

0 (2b) - (3c) • .416

3c

0 0 1

-6

0 (3b) • [-1/(1.6)]

4c

0 0 0 -3.125 | -6.25 3.125 -7.5 1 (4b) - (3c) • -1.25

7.5

12.5

|

-6.25

5

2.5

Id

10 0 0|

0

0-10

2

(lc) - (4d) • 6.2

2d

0 1 0 0 |

0

0

10

-1

(2c) - (4d) • -3.125

3d

0 0 1 0 | -10 10 -24

2.4

4d

0 0 0 1|

2

-1 2.4

/

C

1

0

0

-10

2

0

0

10

-1

-10 10 -24

2.4

2

-1 2.4

(3c) - (4d) • 7.5 -.32 (4c) • [-1/(3.125)]

-.32

0 0

.15 1

W

A

W

B

*1

.V

Now it is clear that: wA = (00) 4- (00) 4- (-10-.15) 4- (2-1) = .5 wB = (00) 4- (OO) 4- (10-.15) 4- (-1-1) = .5 X! = (-10O) 4- (10O) 4- (-24-.15) 4- (2.4 • 1) = -1.2 X2 = (20) 4- (-10) 4- (2.4-.15) 4- (-.32-1) = .04 Thus, our portfolio weights are:

Appendix A

274

wA = .5 and wB = .5 Note, that since only two securities will be included in the portfolio, we can see from the expected return constraint that wA must equal wB, which must equal .5. The Lagrange method is usually quite useful since it can be used with any number of securities; generally, the solution is not trivial as was die case here, a. ii. min: o2 = .04wA2 4- .16wB2 4- .08wAwB s.t.: .12 = .lwA 4- .2wB 1 = lwA 4- lwB L = .04wA2 4- 16wB2 4- .08wAwB - XJC12-.1WA-.2WB) - X2(l-lwA-lwB) 5L/6wA = .08wA + .08wB 4- .IX, 4- 1X2 = 0 6L/5wB = .08wA 4- .32wB 4- 2\x 4- 1X2 = 0 6L/5\X = .lwA 4- .2wB 4- 0\x 4- 0X2 = .12 6L/5X2 = lwA 4- lwB 4- OX, 4- 0X2 = 1 Thus, our system is: .08 .08 .1 1" .08 .32 .2 1 .1 .2 0 0 1 1 0 0. C

0 0 .12 1

W

A

W

B

K

h

- x

= S

Notice that the coefficients matrix (C) is identical to that in 21 .a.i above. Thus, its inverse is identical to C 1 in part 21.a.i. Note also that only element 3 in the solutions vector has changed. Thus, we determine our weights as follows: 0 0 -10 2

0-10 2 ' 0 10 -1 10 -24 2.4 -1 2.4 -.32

W

A 0 .8 .2 0 »B — — .12 -.48 K -.032 1. A.„

a. iii. Note that only the third element in x has changed. Thus: 2 .8 1.92 .112 . wA = .2 wn = .8 b. Our asset returns and standard deviations are:

275

Solutions to End-of-Chapter Exercises Asset A B

i.

E(R) .10 .20 .09 ff Our Lagrange Function is now:

a

.20 .40 0

PA,B

= -5

PA,rf = 0

PB,rf = 0

L = (.04wA24-.16wB24-.08wAwB)-Xl(.15-.lwA-.2wB-.09wRF)-X2(l -1WA-1WB-1WRF)

Because arf = p Brf = p Arf = 0, wrf terms are dropped from the first set of parentheses. The partial derivatives are: 5L/5wA = .08wA 4- .08wB 4- A\x 4- 1X2 = 0 6L/6wA = .32wB 4- .08wA 4- .2X! 4- 1X2 = 0 6L/6WRF = -.09Xt - 1X2 = 0 8L/8\X = .15 - .lw A - .2wB - .09wRF = 0 5L/6X2 = 1 - lw A - lw B - IWRF = 0

Now we have a (5 x 5) coefficients matrix: .08 .08 0

.1 1

W

A

0

.08 .32 0

.2 1

W

B

0

.09 1

0

0

.1

.2 .09

0

0

0

xl

1

1

0

0

V

1

W

RF

0

=

.15 1.

Simply solve this system to obtain: 14.68447 -1.33495 -13.3495 -6.79612

.61165

0

W

-1.33495

-.87379

0

W

-13.3495 1.213592 12.13592 -2.91262 1.262136

0

.121359

1.213592 9.708738 -23.301

2.097087

0.15

1.262136 2.097087

-.18874

1 .

-6.79612 9.708738 -2.91262 .61165

-.87379

A

B

=

W

rf

*1

V

Thus, we have: wA = -.40777 wB = .582524 Wi

RF

.825243

ii. If the return constraint were to decrease to .12, only the fourth element in the solutions vector(s) would change:

276

Appendix A

0 wA = -.20388 0 which results in the wft = .291262 0 following weights : .12 w^ = .912621 1 iii. Increasing the return constraint to .18 results in the following weights: wA =-.61165 wB = .873786 w RF = .737864 5.22. 6L dwx 6L 6w2 6L 6w3 6L 5Xl SL

L = .16w22 4- .64w32 4- X^.20 - .05wj - .15w2 - .25w3) 4- X2(l - w t - w2 - w3) = =

-.05X! - 1X2 -.15X! - 1X2

=0

1.28W3-.25X! - 1X2

=0

.32w2

=

= -.05wj - .15w2 - .25w3

ax 2 = - W , x 2 = -.0500X!

w2

=0

•

w3

= -.20 = -1

w2 = .3125Xt w3 = .1563\ Wi = -.4688Xt 4- 1 .20 = .05(-.4688Xi 4- 1) + .15C3125X0 4- .25(.1563X!) Xi = 2.3962 w2 = .7488 w3 = .3745 Wi = -.1233 CHAPTER 6 6.1.a. F(x) = k b. F(x) = 7x + k c. F(x) = 7x3 4- k d. F(x) = 7x3 4- 5x + k e. F(x) = ex 4- k f. F(x) = e5x 4- k g. F(x) = 5X 4- k h. F(x) = ln(x) 4- k

Solutions to End-of-Chapter Exercises

277

= lOOO^-lOOO*0 = 1000(e-l) = 1,718.28 6.3.a. P(x) = J p(x)dx = x3 b. The distribution function for x will be J f(x)p(x), which, since f(x) = x, equals { x-3x2 = f 3x3 = .75x4. The probability that x will be in the range .5 to 1 equals P[.5 < x < 1] = x 3 | ! 5 = [l 3 - .53] = .875. The expected value of x given that it falls within this range is a conditional distribution determined by: E[x|.5 < x < 1] = { j J5xp(x)}/{P[.5 < x < 1]} = .75-14 - .75-.54 = .703125/.875 = .8035714 c. The probability that x will be in the range 0 to .5 equals X3|Q5 = [.53 - 03] = .125. The expected value of x given that it falls within this range is a conditional distribution determined by: E[x|0 < .5 < x] = { J o5*POO}/{P[0 < x < .5]} = .75-.54 - .75-04 = .046875/. 125 = .375 d. E[x] = J Jxp(x) = .75-14 - .750 4 = .75. Notice that this value equals the sum of { {65xp(x)} and { J l5xp(x)}. 2 e. a = J 1 [f(x)] 2 p(x)dx-(J 1 f(x)p(x)dx) 2 = JJx2p(x)dx - (J Jxp(x)dx)2 = f ix2-3x2dx - (J Jx^xMx)2 And since the term to the right of the minus sign is expected value: a2 = J J3x4dx - .75 2 = .6x5|J - .75 2 = .6 - 0 - .5625 = .0375 6.4.a. P(x) = J p(x)dx = .Olx b. E[x | 50 < x ^ 100] = { J $J°xp(x)dx}/{P[50 < x < 100]} { f }g°.01xdx}/{ J $°p(x)dx} = {.005 • 1002 - .005 • 502}/.5 = 75 c. E[x | 0 <; x <; 50] = { J J°xp(x)dx}/{P[0 < x <£ 50]} { {J°.01xdx}/{ {J°p(x)dx} = {.005 • 502 - .005 • 02}/.5 = 25 d. E[x] = { 100f(x)p(x)dx = Ji°°.01xdx = .005x2|100 = 50 e. a2 = J J00[f(x)]2p(x)dx - ({ 100f(x)p(x)dx)2 = { i°Vp(x)dx - (f J°°xp(x)dx)2 li^.OlxMx-d^.Olxdx)2 Since the term to the right of the minus sign is expected value: a2 = {J°°.01x2dx . 502 = (l/300)x3|i°° - .5 2 = (1/300) • 1,000,000 - 0 - 2500 833.33 f. E[x | 50 < x < 100] = 75 g. E[CFJ = {.5 -0} 4- {.5 -E[x-50 | 50 <£ x <> 100]} = 12.5 6.5. First, we integrate the density functions (ignoring constants of integration): a.

Pf (x) = J3x2 = JC3

for 0 z x * 1

Pg (x) = j(2x3 + x) = —JC4 + - x 2 b.

- V + ±x2 zx* 2

2

for 0 s x s 1

for 0 z x * 1

J

= = = =

Appendix A

278

Thus, security f exhibits first order stochastic dominance over security g.

c.

Thus, security f exhibits second order stochastic dominance over security g. Note that first order stochastic dominance always implies second order stochastic dominance. 6.6. The amount of dividend payment to be received at any infinitesimal time interval dt equals f(t)dt = 2,000,000dt. The present value of this sum equals f(t)e"tadt = 2,000,000e' u . To find the present value of a sum received over a finite interval beginning with t = 0, one may apply the definite integral as follows:

Pn0,20] = f2°2,0O0,0O0e Jo

AOt

dt = 2,000,000-

-.10*

.10 2,000,000 -(I-*" 2 ) = $17,293,294 .10

6.7. First, we set up the discount function to be integrated over time:

Integrating and canceling constants of integration, we obtain the following annuity function:

6.8.a. This differential equation can be constructed from two parts. First, the account should generate interest each period as follows:

However, deductions will be made from the account each period. These deductions will reduce the amount of interest that the account will draw in the future. Assume (T-t) relevant years in the future. The payments deducted from the account each period plus interest that would otherwise have accumulated is found as follows:

Solutions to End-of-Chapter Exercises

279

The difference between these represents the differential equation representing the "path" of the account:

b. To find the state of the system at any time T, integrate the final equation from part 7.a as follows:

From Equations (6.27) to (6.32), we obtain the following:

where FV0 = PV. This integral is solved as follows:

c. Substitute numbers in the solution to part 7.b as follows: FV7n = $l,000,000e 057 +

$100>000

.05

[ l - e 0 5 7J]

= $1,419,067.5 - $838,135.10 = $580,932.4 d. Solve the following for T to find that the retiree runs out of money in approximately 13.86 years. FVTT = 0 = $l,000,000* 05r +

$100>OOQ

.05

[1 - eM*\J L

= -$1,000,000* MT + $2,000,000 ln(2,000,000) = ln(l,000,000)+.05J ; T = 13.862944

CHAPTER 7 7.1. a. b. c.

Use Equation 7.4 to obtain the following: .01024 .07680 .23040

280

Appendix A

d. .34560 e. .25920 f. .07776

7.2. Use Equation 7.5 to obtain the following: a. b. c. d. e. f.

.07776 .92224 .66304 .31744 .08704 .01024

7.3.a.

At least 5 increases are necessary for the price at the end of the day to exceed $100. The probability of this occurring is computed as follows:

= .2456 + .213 + .1267 + .0494 + .0114 + .0012 = .5979 b. At least 9 increases (.50-7-.0625 = 8, plus one more gain to offset one loss out of 10) are necessary for the price at the end of the day to exceed $100.50. The probability of this occurring is computed as follows:

c. zero: the stock would require 160 up-jumps in price to increase by 10 points, yet it only has 100 opportunities to up-jump. 7.4.a. 0; The highest potential stock price equals $85.2. b. 2 up-jumps are required; .412 = .1681 c. 1.42-$60 = $85.2; .7-$60 = $42. d. 1.422-$60 = $120.98; 1.42-.7-$60 = $60; .72-$60 = $29.4. e. MAX[$120.98-$50,0] = $70.98; MAX[$60-$50,0] = $10; MAX[$29.4-$50,0] =0 f. Potential stock prices are $120.98 and $60 Potential call values are $70.98 and $10 The discounted call value would be [.41-$70.98 4- .59-$10]e 05 ' 5 = $36.28 g. Potential stock prices are and $60 and $29.58 Potential call values are $10 and 0 The discounted call value would be [.41 -$10 4- .590]e' 05,5 = $4 h. The current discounted call value is [.41-$36.28 4- .59-$4]e'05'5 = $17,235 7.5.a. cT = MAX[0,ST-X]; cT = $0 or $15 b. $100/$90- 1 = .1111 c. One can construct a one period hedge for a call option by shorting a shares of stock per option contract such that cu-auS0 = cd-adS0. We solve for the hedge ratio a as follows:

Solutions to End-of-Chapter Exercises

281

d. Since shorting o; shares of stock ensures that the portfolio is perfectly hedged, the hedged portfolio must earn the riskless rate of return: cu-auS0 = cd-o:dSo = (c0aS^e*^. Thus, the current value of the call can be solved for as follows:

This solution can be verifed with the solution methodology given in Exercise 7.4. 7.6. Compute the expected return and variance as follows:

7.7. The following table answers 5.a and 5.b: Day t

Price

Price Relative St/SM

0

100

NA

1

104

1.0400000

.0392207

2

102

0.9807692

-.019418

3

104

1.0196078

.019418

4

102

0.9807692

-.019418

5

106

1.0392157

.0384662

c. /i = .0582689 4- 5 = .0116537 d. a2 = E(at - /i)2 -s- 4 = .0008675 CHAPTER 8 8.1. First, find mean Y and X values:

Log of Price Relative ln(St/St.,)

282

Appendix A

Next, find standard deviations of Y and X:

Find the covariance between the dependent and independent variables. If desired, find the correlation coefficient:

Finally, determine regression coefficients and predicted Y values:

8.2. Solve in order for X, Y, ax, ay, axy, fi, a and Y1986: X = 635.71 Y = 97 ax = 197.69 ay = 74 axy = 14,602 0 = .373 OL = -140.52 Yl986 = 307.08 8.3.a. First, calculate return standard deviations for each of the stocks and die market portfolio:

=(.12 + .18 + .07 + .03 + .10)/5 = .10

Solutions to End-of-Chapter Exercises

283

= (.10 + .14 + .06 + .02 + .08)+5 = .08

b. Calculate correlation coefficients between returns on each of the stocks and returns on the market portfolio.

=[(.12-.10)(.10-.08)4-(.18-.10)(.14-.08)+(.07-.10)(.06-.08)+ (.03-.10)(.02-.08)+(.10-.10)(.08-.08)]/5 _(.02)(.02) + (.08)(.06) + (-.03)(-.02)^-.07)(-.06) + 0 _ 5

{m

=[(.12-.10)(.04-.064)+(.18-.10)(.20-.064)+(.07-.10)(.02-.064)-H (.03-.10)(.03-.064)+(.10-.10)(.09-.064)]/5 _(.02)(-.024) + (.08)(.136)^(-.03)(-.044)^(-.07)(-.094)^0 _ 5

m 6 6

=[(.04-.064)(.10-.08)+(.14-.08)(.20-.064)+(.06-.08)(.02-.064)+ (.02-.08)(-.03-.064)+(.08-.08)(.09-.064)]/5 = .00284 PKm =C0Vh,J"h"m = "hj"h"m =.002/(.0502)(.04) = .996

284

Appendix A

=.00284/(.078)(.04) = .909 c. The slopes of the lines are the stock Betas. d. Calculate Betas for each of the stocks. How do your betas compare to the slopes of the stock characteristic lines?

=(.0502)(.04)(.996)/(.04)2 =.05/.04 =1.25

=(.078)(.04)(.909)/(.04)2 = 1.775

The betas are the slopes of the characteristic lines. 8.4. The following table summarizes preliminary computations: Year Ct Yt C t -T Y t -T (Ct-77)(Yt-T)

350 1 325 -105 2 335 364 -95 3 355 385 -75 -55 4 375 405 5 401 -29 438 6 433 473 3 7 466 512 36 8 492 547 62 9 537 590 107 10 576 630 146 Preliminary Computations:

-119 -105

-84 -64 -31 4 43 78 121 161

12,495 9,975 6,300 3,520

14,691 11,025 7,056 4,096

899 12

961 16

1,548 4,836 12,947 23,506

1,849 6,084 14,641 25,921

EC| =4,295

c =430

E

y =469

=4,694

S 2 =2,289,172 St

(Y t -T) 2

285

Solutions to End-of-Chapter Exercises

Regression Coefficients: b{ =76,038 + 85,810 =0.89 b0 = C - ^ F =430-(0.89)(469) =13 Testing for Significance (Regression Diagnostics): Year Ct E[CJ=b 0 4-b 1 Y t e t =C t -E[C] 1 325 2 335 3 355 4 375 5 401 6 433 7 466 8 492 9 537 10 576 Computations:

325 337 356 373 403 434 469 500 538 574

0 -2 -1 2 -2 -1 -3 -8 -1 2

e2 0 4 1 4 4 1 9 64 1 4

286

Appendix A

df = (n-2) = 8 Formal test of hypotheses: H 0 : b 0 = 0 ; HA: b 0 * 0 H0: bx = 0 ; HA: bx * 0 Test for significance at the 95 % confidence level (two tailed test) t(b0) = 2.32 > 2.306 ; therefore, b0 is significantly different from zero; reject the first null hypothesis t(bO = 89 > 2.306 ; therefore, bx is significantly different from zero; reject the second null hypothesis E[cJ = 13 4- 0.89*E[yJ p 2 y = .99 (2.32) (89) df = 8 Thus, we may conclude that the Canseco Stock price is directly related to the number of employees that the firm employs. 8.5. To estimate the 4.5 year yield, we shall perform an OLS regression of Dt on t and t2. The first step in our computations is to calculate each value for D, from yt. We find that D, = .943396, D2 = .854172, D3 = .751315, D4 = .649321 and D5 = .554929. We regress Dt against t and t2 to obtain the following regression equation: Dt = 1.040426 - .09412* - .00068*2 (217.0) (12.0) (.53) Inserting t=4.5 into this equation, we find that D 45 = .60319. This leads to a yield y 45 solution of .118892. Also, note that our estimates for a and bx are statistically significant at the .01 level; b2 is not. Although our computations for y4>5 are correct based on the relationship defined by the fixed income analyst, one should question use of the t2 independent variable since it was not statistically significant. One might obtain more accurate forecasts for y by omitting this t2 term from his regressions, assuming that his data and measurements are reliable. CHAPTER 9 9.1. The expected value of the stock is computed with Equation 9.8 as follows: .316406-$207.36 + .421875-$138.24 4- .2109375-$92.16 4- .046875-$61.44 4- .0039063-$40.96 = $146.4141 9.2. The mean and variance of arithmetic returns are computed as follows:

Solutions to End-of-Chapter Exercises

9.3.a.

9.4.a. The hedge portfolio can be defined as follows:

b. The put differential equations are derived as follows:

Assume that the stock price follows a geometric Weiner process:

which requires us to use Ito's Lemma to solve for dp:

Invoking Ito's Lemma, we have:

287

Appendix A

288

This is the Black-Scholes differential equation which is to be solved subject to the boundary condition:

c. We can guess for the particular solution to the Black-Scholes differential equation:

where N(d*) is the cumulative normal distribution function for (d*). We can use standard rules for calculus to verify this solution. Note that the first of the three equations is consistent with the put-call parity relation: 9.5. First, value the calls using the Black-Scholes Model, then use put-call parity to value the puts: Thus, we will first compute dj, d2, N(dx), N(d2) for each of the calls; then we will compute each call's value. Finally, we will use put-call parity to value each of the puts. d(l) d(2) N(d,) N(d2) Call Put

Option 1 .957739 .657739 .830903 .744647 7.395 0.939

Option 2 -.163836 -.463836 .434930 .321383 2.455 5.416

Option 3 .061699 -.438301 .524599 .330584 4.841 7.803

Option 4 .131632 -.292626 .552365 .384904 4.623 5.665

9.6. First, value the calls using the Black-Scholes Model, then use put-call parity to value the puts. Thus, we will first compute dt, d2, N ^ ) , N(d2) for each of the calls; then we

Solutions to End-of-Chapter Exercises

289

will compute each call's value. Finally, we will use put-call parity to value each of the puts. First find for each of the 15 calls values for dp _X Aug Sep Oct 1.129163 1.162841 110 2.833394 115 1.417978 .617046 .658904 120 .126728 .062811 .176418 125 -1.237028 -.343571 -.286369 130 -2.485879 -.795423 -.731003 Next, find for each of the 15 calls values for d2: x Oct Aug Sep 110 2.801988 1.042362 1.074632 115 1.386572 .530245 .570695 120 .031405 .039928 .088208 125 -1.268433 -.430371 -.374578 130 -2.517284 -.882222 -.819212 Now, find N(d^ for x Aug 110 .997697 115 .921901 120 .525041 125 .108038 130 .006462

each of the Sep .870585 .731398 .550422 .365584 .213184

15 calls: Oct .877553 .745021 .570017 .387298 .232388

Next, determine N(d2) for each of the 15 calls: x Sep Oct Aug 110 .851378 .858730 .997461 115 .702029 .715897 .917214 120 .515925 .535145 .512527 125 .333463 .353987 .102322 130 .188828 .206333 .005913 Finally, use N(dj) and N(d2) to value the calls and puts: PUTS CALLS Oct Aug Aug Sep X 0.003 110 10.165 11.494 11.942 0.134 115 5.305 7.616 8.030 1.415 120 1.593 4.586 4.930 5.009 .193 2.488 2.741 125 1.375 9.816 130 1.211 .008

use put-call parity to value the

Sep .701 1.787 3.721 6.587 10.274

Oct 0.666 1.685 3.537 6.290 9.866

The options whose values are underlined are overvalued by the market; they should be sold. Other options are undervalued by the market; they should be purchased. 9.7. Sensitivities for the put option are as follows:

290

Appendix A

CHAPTER 10 10.1. The following are call and put values: n Cn Pn 2 4.62 29.96 3 3.91 29.25 8 3.87 29.21 10.2.a. c0 = $5.10; p0 = $11.61 b. c0 = $5.10; p0 = $12.47 c. The American put can be exercised early. d. The American put will be exercised at the end of the first period if the stock price declines. The stock's value, given a decline will be $39.14, making the intrinsic value of the put equal to $20.86. However, at this vertex, the European put would only be worth $19.09. Thus, $20.86 profit from early put exercise exceeds the value of the European put at that vertex. 10.3. The solution follows the algorithm below: Initial Equation: SN(d1)-XertN(d2) a! = 0.6 bx = 0.2 ox = 0.4 rf = .06 S0 = 30 X = 35 C0 = 4.841 T = 1

Solutions to End-of-Chapter Exercises d,(an) dn(aa) N(dt) N(d2) f(an) f(at) = 1.190459 0.6 0.1430822 -0.4569178 0.556887 0.323865 1.190459 f(b,) =-3.54257 0.2 -0.3707534 -0.5707534 0.3554110.284083-3.54257 d,(a„) (UcO N(d,) n a„ b„ N(d,) f(o„) crn .2 -.0353767 -.4353767 .48589 .331645 -1.19590 .4 1 .6 .06169864 -.4383014 .524599 .330584 0.000332 .5 .4 2 .6 .01577627 -.4342237 .506294 .332063 -0.59757 .45 .4 3 .5 .475 .03928804 -.435712 .51567 .331523 -0.29849 .45 4 .5 .4875 .0506204 -.4368796 .520186 .331099 -0.14904 5 .5 .475 .4875 .49375 .05619008 -.4375599 .522405 .330853 -0.07434 6 .5 .49375 .49687 .05895186 -.4379231 .523505 .330721 -0.03700 7 .5 .496875 .49843 .0603271 -.4381104 .524053 .330653 -0.01833 8 .5 .498438 .49922 .06101333 -.4382054 .524326 .330619 -0.009 9 .5 .499219 .49960 .0613561 -.4382533 .524462 .330601 -0.00433 10.5 .499609 .49980 .0615274 -.4382773 .524530 .330593 -0.002 11 .5 .499805 .49990 .06161303 -.4382893 .524565 .330588 -0.00083 12.5 .499902 .49995 .06165584 -.4382953 .524582 .330586 -0.00025 13 .5 .499951 .49998 .06167724 -.4382983 .52459 .330585 0.00004 14 .5 15 .49998 .49995 .49996 .06166654 -.4382968 .524586 .330586 -0.00011 16 .49998 .49996 .49997 .06167189 -.4382976 .524588 .330585 -0.00003 , 0.000004 17 .49998 .49997 .49997 .06167456 -.438298 .524589 .330585 18 .49997 .49997 .49997 .06167323 -.4382978 .524589 .330585 -0.00001 10.4. Implied volatilities are given as follows: a. x = 40; a = .2579 b. X = 45; a = .3312 c. X = 50; o = .2851 d. X = 55; a = .2715 e. X = 60; a = .2704

291

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APPENDIX B

Statistics Tables Table B.l Standard Normal Distribution z

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

0.00 .0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4986

0.01 0.02 0.03 .0040 .0080 .0120 .0438 .0478 .0517 .0832 .0871 .0909 .1217 .1255 .1293 .1591 .1628 .1664 .1950 .1985 .2019 .2291 .2324 .2356 .2611 .2642 .2673 .2910 .2939 .2967 .3186 .3212 .3238 .3437 .3461 .3485 .3665 .3686 .3708 .3869 .3888 .3906 .4049 .4066 .4082 .4207 .4222 .4236 .4345 .4357 .4370 .4463 .4474 .4484 .4564 .4573 .4582 .4649 .4656 .4664 .4719 .4726 .4732 .4778 .4783 .4788 .4826 .4830 .4834 .4864 .4868 .4871 .4896 .4898 .4901 .4920 .4922 .4925 .4940 .4941 .4943 .4955 .4956 .4957 .4966 .4967 .4968 .4975 .4976 .4977 .4982 .4982 .4983 .4987 .4987 .4988

0.04 .0159 .0557 .0948 .1331 .1700 .2054 .2389 .2703 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988

0.05 .0199 .0596 .0987 .1368 .1736 .2088 .2421 .2734 .3023 .3289 .3531 .3749 .3943 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989

0.06 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989

0.07 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2793 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989

0.08 .0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990

0.09 .0358 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990

Appendix B

294 Table B.2

t-Distribution Right-tail

area, a

df

0 100

0 050

0.025

0010

0005

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.282

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.645

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 1.960

31.821 6.695 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.326

63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.576

00

Examples: The t value for oo degrees of freedom that bounds a right-tail area of 0.025 is 1.960. The t value for oo degrees of freedom that bound left and right-tail areas (two tails) summing to 0.05 is 1.960.

APPENDIX C Notation Definitions

This glossary defines notation used in this book. Since the book covers a wide range of topics, certain symbols will have more than one definition. Definitions in such cases are noted in the text. In addition, certain notation used only in a short portion of the text may be defined in the text only and not here. GREEK LETTERS (1) Continuously compounded or logarithmic return; (2) vertical intercept in Single Index Model; (3) minimum number of up-jumps required to exercise call in binomial model. 6: (1) Beta in Capital Asset Pricing Model; (2) beta in Single Index Model. 7: Regression coefficient in Fundamental Beta estimation Procedure. A: Change. 5: (1) Partial derivative notation; (2) factor risk premia in Chapter 4. e: (1) Error term; (2) is an element of. 0: Slope of the Capital Market Line. X: (1) Market price of risk as measured by variance; (R p -r f )/a 2 ; (2) LaGrange multiplier. /x: (1) Continuously compounded security return; (2) in Chapter 2, population mean. II: Denotes product operation. 7r: (1) The number pi, with approximate value 3.141; (2) insurance or risk premium. p: Correlation coefficient. E: Denotes summation operation. a.: Standard deviation. ox-{. Covariance. o]\ Variance. : The set of all events 0 . <j>: An event taken from 4>. a:

296

Appendix C

LATIN LETTERS a: B: b: C: c:

CFt: COV: cy: d: Dt: df. d2: DD: Dur: E[*]: e: F: f(*): FVn: g: I: i: INT: IRR: K: k: L: lim: in: m: n: N[*]: P: p: P^ P0: Pt: PV: q: R: r: rf:

(1) Vertical intercept of a regression line; (2) drift in a Weiner Process. Brokerage cost in Application 5.4. (1) Beta in Capital Asset Pricing Model; (2) beta in Single Index Model; (3) slope of a regression line; (4) variability in a Weiner Process. (1) Call value; (2) consumption in Applications 2.3 and 5.16. (1) Usually denotes constant or coefficient; (2) in Application 2.3, denotes marginal propensity to consume; (3) call value; (4) coupon rate; (5) cash balance in Application 5.4. Cash flow at time t. Covariance. Current yield. Multiplicative downward movement. Discount fmction for time t. Parameter in Black Scholes formula. Parameter in Black Scholes formula. Demand deposits. Duration. Expected value of [*]. The number e with an approximate value equal to 2.7182818. Face value of debt instrument. Function of (*). Future value of cash flow received in n periods, Growth rate. Index value. (1) Interest rate; (2) counter in a summation or product. Interest payment. Internal rate of return. Currency in an economy, (1) Discount rate; (2) constant of integration. LaGrange function, Limit, Natural log. Number of compounding intervals per period, (1) Ending or stopping value or time; (2) sample size; (3) number of securities in portfolio. Cumulative normal density function for [*]. (1) Price; (2) probability, (1) Put value; (2) probability. Probability associated with outcome i. Purchase price of asset. Price of asset at time t. Present value. Probability of option exercise implied by Black-Scholes Option Pricing model. Return, (1) Usually denotes rate of return or interest rate; (2) in Application 2.3, represents reserve requirement. Risk-free rate of return.

Notation Definitions

297

ROI: Return on investment. S: In Chapter 2, denotes the value of a series. St: (1) Usually denotes stock value at time t; (2) in Chapters 7 and 9, may denote random variable value at time t. s: Sample standard deviation. T: Usually denotes maturity or expiration date of an instrument, t: Usually denotes time, u: Multiplicative upward movement. U: Utility. V: Value. Wj: Portfolio weight for security i. W 0 : Initial wealth. X: (1) Usually denotes striking price of an option; (2) in Chapter 2, denotes a cash flow; (3) in Chapter 9, a random variable threshold; (4) in Application 5.4, cash demand. X0: Cash flow, time zero, x: Usually denotes random variable. Y: Income or output. y: (1) Yield to maturity; (2) may denote random variable. y 01 : Spot rate over t periods. yitl: Forward rate on note originated at time i and maturing at time t. z:

X • Wj.

OTHER MATHEMATICS SYMBOLS x:iMt: A function x mapping the set 12 of outcomes onto the real number space R. The real number space. R: w,*,P: Probability space. V: O:

1*.

1; -*]

For all. And. Given. Factoral. (1) Approaches; (2) mapping on to.

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GLOSSARY

Acceptance region. The range marked by the critical value or values which contains the null hypothesis value of the population parameter. Alternative hypothesis. The claim that the population parameter differs from that specified by the null hypothesis. American option. Option which can be exercised prior to expiration. Annuity. A series of equal payments made at regular intervals. Antidifferentiation. The inverse process of differentiation. Arbitrage. The simultaneous purchase and sale of assets with identical cash flow structures. Arbitrage Pricing Theory. A theory of market equilibrium where expected security returns are linearly related to a series of factors. Bankrupt. The situation arising when a firm is unable to fulfill its obligations and its assets are surrendered to a court for management and distribution. Basis. The set of n or more vectors which can, through linear combinations, express any other vector in that n dimensional space. Beta. Coefficient which measures the risk of a security relative to the risk of some factor (usually the market). Binomial process. A process which results in one of two potential outcomes at each stage or vertex. Black-Scholes Option Pricing Model. A continuous time-space option pricing formula. Bond. Financial security which makes fixed payment(s) at specified interval(s). Brownian motion. A Newtonian non-differentiable stochastic continuous time-space process whose increments are independent over time. Call. A security or contract granting its owner the right to purchase a given asset at a specified price on or before the expiration date of the contract. Capital Asset Pricing Model. A theory of market equilibrium where security expected returns are related to their covariances (or betas) with the market portfolio. Capital market. The market for financial resources. Capital Market Line. The line plotting risk and return combinations of the most efficient portfolios of assets, including the riskless asset.

300

Glossary

Central Limit Theorem. States that the distribution of the mean of n independent and identically distributed random variables tends to a normal distribution as n approaches infinity. Coefficient of correlation. A measure of the strength and direction of the relationship between two sets of variables. Ranges between zero and one and may be regarded as a "standardized" covariance (dividing covariance by the product of the standard deviations of the two variable sets). Coefficient of determination. Correlation coefficient squared (often called "r-squared" or "p-squared"). May be interpreted as the proportion of variability in one data set which may be "explained" be a second data set. Complete market. A market in which all claims in a time-state space are hedgable and priced. Convexity. (1) The slope of the slope of a function; (2) The sensitivity of the duration of a bond to changes in the market rate of interest. Correlation. The strength and the direction of the relationship between two variables. See coefficient of correlation. Coupon. The interest rate on debt as a percentage of its face value. Covariance. A statistical measure of the comovement between two sets of variables. Critical value. A value which marks the boundary between acceptance and rejection regions for the null hypothesis. Degree (of a differential equation). The power to which its highest order derivative is raised. Density. The probability that a continuous random variable assumes a value between y* and y*+dy where dy -* 0. Derivative. The instantaneous slope of a function. Derivative security. An instrument whose payoff or value is a function of that of another security, index or value. Diagonal matrix. A symmetric matrix whose elements off the principal diagonal are zero. Differential equation. A function which represents the derivative of another function. Discount rate. A rate used to discount (usually reduce) future cash flows to express their values relative to current cash flows. Discrete. A variable which can be assigned only a countable number of values. Distribution function. The probability that a continuous random variable assumes a value no greater than y*. Diversify. To accumulate a variety of different types of assets. Drift. The predictable change component of a stochastic process. Duration. Measures the proportional sensitivity of a bond to changes in the market rate of interest. Efficient. (1) Produces maximum profit relative to investment amount; (2) has highest return given risk; (3) has least risk given return. Efficient Frontier. The curve plotting risk and return combinations of the most efficient portfolios of risky assets. Efficient market. Security prices instantly adjust to fully reflect all available information. Efficient portfolio. (1) Portfolio with the highest return at its risk level, or (2) portfolio with the lowest risk level at its return level. European option. An option which can be exercised only at expiration.

Glossary

301

Event study. Testing security price reactions to events affecting corporations. Expected return. Weighted average return, where weights are determined by probabilities associated with potential return outcomes. Expected value. Weighted averages, where weights are probabilities associated with outcomes. Face value. The principal or par value of debt. Feasible region. Investment opportunity set, in terms of return and risk. Filtration. History or recording of the path of a process. Forward contract. A contract for the future purchase, sale and/or exchange of an asset at a price which is set when the contract is agreed to. Fractal. A geometric shape which maintains its essential features when decomposed and magnified. Function. A rule which assigns a unique second number to each number in a set. Futures contract. A publicly (exchange) traded forward contract providing for the exchange or transfer of an asset or assets at a price which is set when positions are taken in the contract. Future value. The value of a sum of money after it has been invested for a period of time. Gamble. To take a risk. Geometric mean return. The "n111" root minus one of the product of the sum of one plus periodic returns, where n is the number of returns to be averaged. It is an average return which has been adjusted for the impact of compounding. Hedge. To take a position to reduce risk. Heteroscedasticity. This exists when error terms are correlated with the independent variable. Identity matrix. A diagonal matrix consisting of ones along the principal diagonal. Immunization. A fixed income strategy concerned with matching the present values of asset portfolios with the present values of cash flows associated with future liabilities. Index. An indicator (e.g., The Dow Jones Industrials Average may be regarded as a market index). Infinitesimal. Value approaching zero. Interest. A charge imposed on borrowers by lenders. Internal rate of return. The discount rate which sets the Net Present Value (NPV) of an investment equal to zero. It is a measure of the profitability of an investment. Inverse matrix. A"1 exists for the square matrix A if the product A'A or AA"1 equals the identity matrix I. Kurtosis. Concentration of distribution function about the mean and about the tails relative to between the mean and tails. Fourth moment about the mean. Law of One Price. Assets or portfolios with identical cash flow structures must have the same market price. Linear combinations. Combinations of vector addition and scalar multiplication. Linearly dependence. Exists when a vector in a given n dimensional space can be expressed as a linear combination of a set of other vectors in the same space. Liquid. Easily converted into cash or sold. Market. The arena for buying and selling. Market portfolio. The combination of all assets held by investors and institutions. Markov process. See Random walk.

302

Glossary

Martingale process. A random walk with zero drift or whose expected change equals zero. Matrix. An ordered rectangular array of numbers. Maturity. Payments cease on a debt security. The maturity date is the date on which payments cease. Mean. Average; sum of data points divided by the number of data. Median. Value in the middle of a ranked data set. Mode. Value which appears most frequently in a data set. Multi-factor model. A model of market returns which are driven by more than one Brownian motion. Mutual fund. An institution which pools investors' funds into a single portfolio. Newtonian calculus. Classical differential and integral calculus pertaining to smooth functions. Null hypothesis. The claim that die population parameter equals the maintained value or values. Objective function. The function whose value is to be minimized or maximized in an optimization problem. Option. A security which grants its owner the right to buy or sell an asset at a specific price on or before the expiration date of the security. See Call; Put. Order (of a differential equation). The order of its highest derivative. Portfolio. A collection of investment holdings. Power set. The power set of a given set X is comprised of all sets included in X. If X has n elements, its power set will have 2n elements including the empty set. Present value. The value of a future cash flow or series of cash flows expressed in terms of money received now. Principle diagonal. Contains the series of elements where Row i = Column j . Probability. Likelihood; likely to be expressed in percentage or decimal terms. Pure discount note. A debt security paying no interest; it only pays its face value or principal. Pure security. Also known as an elementary, primitive or Arrow-Debreu security. An investment which pays 1 if and only if a particular outcome or state of nature is realized and nothing otherwise. Put. A security or contract granting its owner the right to sell a given asset at a specified price on or before the expiration date of the contract. Random walk. A process whose future behavior, given by the sum of independent random variables, is independent of its past. Range. The difference between high and low values in a data set. Return. Profit relative to initial investment amount. Risk. Uncertainty. Risk premium. Return offered or demanded as compensation for accepting uncertainty. Scalar. A matrix with exactly one element. Security. A marketable certificate denoting a financial claim; that is, a paper or contract with underlying value which can be bought and sold. Semi-strong form efficiency. Exists when prices reflect all publicly available information. Set. A collection of any type of objects. Short sell. To borrow and sell. Presumably, the short-sold security will be repurchased and returned to its original owner.

Glossary

303

Significance level. The probability of rejecting the null hypothesis when it is true. Skewness. Asymmetry of a distribution. Third moment about the mean of a distribution. Solution (to a differential equation). A function which, when substituted for die dependent variable, satisfies the equality. Space. A system of entities such as outcomes or points in time. Spot rate. The yield at present prevailing for zero coupon bonds of a given maturity. Square matrix. Matrix with the same number of rows and columns. Standard deviation. A measure of dispersion, risk and uncertainty. It is the square root of variance. State space. The set of values generated by a process. Statistics. A branch of mathematics concerning the collection, organization, interpretation and presentation of numerical facts and data. Stochastic. Random. Stochastic processes. Processes generating outcomes which are influenced by random effects over time. Strong form efficiency. Exists where prices reflect all information, public or private. Submartingale. Random walk with positive drift or positive expected change. Symmetric matrix. A square matrix where Cy equals cjfi for all i and j . Systematic risk. Risk that is common to the market or a large number of securities. Taylor series. The expression of the value of a function f(x) near x in terms of f(x) and its first and higher order derivatives. Technical stock analysis. Concerns the examination of historical price sequences. Terminal value. The value of a sum of money after it has been invested for a period of time. Term Structure of Interest Rates. The relationship between yields to maturity of debt securities and the length of time before the securities mature. Transpose. To interchange the rows and columns of a matrix. Treasury bill. Short-term pure discount note issued by the Treasury of the United States federal government. Considered to be relatively free of risk. Unit matrix. See Identity matrix. Unsystematic risk. Risk that is unique or specific to one firm. Variance. A measure of dispersion, risk and uncertainty. It is the expected value of the squared deviation of a data point from the expected value of the data set. It is the square of standard deviation. Vector. A matrix with either only one row or one column. Vector space. The set of all vectors with n real valued elements or coordinates. Warrant. An option on the treasury stock of a firm. Weiner Process. A Newtonian non-differentiable stochastic continuous time-space process whose increments are independent over time. Weak form efficiency. Exists when security prices reflect all data regarding historical prices. Yield to maturity. The internal rate of return for a bond. Zero coupon bond. A bond which makes no interest payments. See Pure discount note.

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REFERENCES

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INDEX

American options, (See Options) Annuity, 18-20 Antidifferentiation, 137-138 APT, (See Arbitrage Pricing Theory) Arbitrage, 7, 40, 56, 59, 60, 64, 65, 68, 70, 74-77 Arbitrage Pricing Theory (APT), 79, 40, 74-77 Area, 137-142 Baumol Cash Management Model, 91-94, 126 Beta, 8, 11, 3 0 , 4 2 , 4 3 , 185-187, 192, 210 Binary variables, 208 Binomial distribution, 164-166, 170171, 214-215,233-237 Binomial method, 233-237 Binomial option pricing, (See Binomial Option Pricing Model) Binomial Option Pricing Model, 10, 69, 77, 164-166, 170-172, 233235 Binomial process, (See Binomial distribution) Bisection, method of, 240-245, 247, 248 Black-Scholes Option Pricing Model, 5, 7, 10, 26, 217, 224-229, 235238, 242, 243, 246

Bond yields, 9, 22, 57-59, 94-97, 114, 198-199, 240-242 Brownian motion, 5-6, 215-218 Call, (See Options) Capital Asset Pricing Model (CAPM): applying, 40, 184-189; deriving, 122-124; history, 79; research papers on, 7-11, 77, 126; unrealistic assumptions, 145; Capital Market Line, 6, 104-112, 119-122 CAPM, (See Capital Asset Pricing Model) Cash management, 12, 91-94, 126 Chain rule, 102-104, 111 Coefficient of determination, 27, 182, 183, 188, 194-196 Comovement, 26-28,31,44 Complete capital market, 67 Constrained Utility Maximization, 6, 124-126 Contingent claims analysis, 65 Continuous dividend payments, 148 Convexity, 113-115, 129 Correlation coefficient, 27, 28, 31, 39, 98, 107, 196 Coupon bond, 57, 59, 114 Covariance, 26-29, 38-44, 50, 76, 98, 130, 133, 186

314 Cumulative density, 143, 147, 166 Dedication, 60, 64, 97 Default, 12, 60, 94, 95 Definite integral, 140, 143, 149, 153, 166 Density function, 5, 142-144, 147, 161, 166,222,223 Derivative, 84-91,94,96,98,99, 102-104, 110, 111, 113, 114, 123, 131, 132, 137, 138, 151, 171,226,228,229,233,245, 246 Derivative securities, 9, 69, 171, 229, 233 Diagonal matrix, 49, 50 Differential calculus, 13, 83 Differential equation, 151, 152,220, 225, 226, 230 Differential equations, 13, 151, 152, 216 Distribution function, 142-144, 147, 161, 163, 164,213,226 Diversification, 6, 28, 39, 97, 104 Duration, 9,94-97, 113-115, 126, 129

Index Geometric Weiner process, 221, 222, 223, 233 Gordon Stock Pricing Model, 6, 20 Hindu-Arabic numerical system, 1 Hypothesis testing, 175, 179 Identity matrix, 50, 54-56 Immunization, 9, 97, 113, 115, 126 Implied variance, 243 Income multiplier, 21 Indefinite integral, 137, 143, 144 Index models, (See Factor models) Integral calculus, 13,137,141,154 Interest, 1, 6, 9, 44, 110, 120, 124, 153, 208, 238, 239; and the Baumol model, 91-94; sensitivity of bond prices to, 9497,113-114; and time value, 15, 16, 18, 22, 84; yield curve, 57-59 Inverse, 16, 27, 28, 54, 55, 58, 77, 83, 117, 133, 135, 137 It6 process, 218-220, 225, 230 Ito's Lemma, 219,221,225,226 Kurtosis, 23, 161, 167

Efficiency, 6, 7, 10, 11, 21, 189, 199, 201 Event study, 11, 199-202 Exact matching, 60, 61, 77 Expected return, 2 4 , 2 5 , 3 7 , 4 0 , 4 1 , 5 2 , 7 6 , 7 7 , 106, 107, 110, 118, 119, 120-122, 129, 130, 133, 144,201,215,234 Expected value, 5, 7, 23, 27, 29, 74, 76, 115, 144, 150, 154, 161, 163, 164-167, 169, 170, 174, 182,214,215,220,221,230 Factor models, 8 Financial models, 3, 16 Fixed income, 2, 9, 60, 97, 198 Function, 83-84 Future value, 15, 16, 18, 147, 150, 169-171 Geometric expansion, 17, 18, 20

LaGrange function, 116,118,119, 120, 122, 125, 129, 133 LaGrange multiplier, 116-119,125 LaGrange optimization, 120, 122, 126 Law of One Price, 6, 7, 56, 57, 64, 71,224 Limit, 42, 83, 84, 104, 138, 143, 161, 166, 168, 219 Linear combinations, 61,65 Linear dependence, 61,62 Marginal utility, 5, 7, 88, 89, 91 Markov process, 214 Martingale, 7, 214 Matrix, 4, 13, 44, 49-56, 58, 59, 61, 70, 7 1 , 7 4 , 7 6 , 7 7 , 102, 112, 117, 119, 126, 133, 134, 135 Maxima, 84 Mean, 8, 9, 22-25, 73, 77, 98, 120,

Index

315

122, 161, 167, 174, 177-179, 201,203,208,221,234,243 Median, 23 Merger returns, 201 Minima, 84 Mode, 23 Moment, 161 Multi-Index Model, 43, 44 Multiple regression, 29, 101, 193, 195, 196, 198, 212 Multiplier, 21, 116-119, 125

Portfolio selection, 118,145 Present value, 12, 16, 18-20, 58, 71, 95, 96, 114, 115, 149, 153, 171, 224 Probability spaces, 159 Product rule, 102, 103 Pure security, 66-70 Put, 10, 39, 71-73,77,230,231, 235-239, 248

Natural log, 16, 84, 168 Newton-Ralphson Method, 245-247 Normal distribution, 100,142,161, 166-168, 226, 234 Normal equations, 100, 101 Numerical methods, 3, 13, 233, 247

R-square, (See Coefficient of determination) Random variables, 159, 161, 213, 214 Random walk, 214, 216, 224 Regression, 8, 29, 30, 40, 41, 43, 99-102, 181-185, 189, 190, 192, 193-198, 208, 209, 210-212 Regression coefficients, 29, 100, 101, 181, 193, 194, 196-198, 210 Return measurement, 21 Risk aversion coefficients, 115 Risk measurement, 7, 24, 30 Risk minimization on a spreadsheet, 133 Rules: for finding derivatives, 132; for finding integrals, 157

Option pricing, (See Options) Option values, (See Options, pricing) Options: American, 237-239; call, 10, 26, 54, 69-73, 77, 150, 164166, 169-171, 223-229, 235-239, 242, 243, 245; pricing, 8, 10, 26, 69,71,77, 150, 164, 165, 169, 170-172, 217, 224, 229, 235, 237, 242, 247 Orthogonal vectors, 73-74 Parkinson, 8, 25, 31 Perpetuity, 19, 20 Polynomial: estimation with, 157, 166, 177, 226-227; polynomial rule, 89-91, 94, 96, 102, 131 Portfolio analysis, 7, 8, 37, 40, 77, 145 Portfolio performance evaluation, 191 Portfolio return, 30, 37-39, 44 Portfolio risk: bonds, 97-98; constrained minimization, 118122; index models, and, 42; measurement of, 38-40, 50; options, with, 228; performance, and, 191-193; risk aversion, and, 115-116; spreadsheets, and, 133-135

Quotient rule, 102, 103, 110

Scalar, 49,61, 77 Scientific method, 1, 2 Simple regression, 29, 101, 181, 196 Single index model, 40-42, 44 Single population, 23, 179 Skewness, 23, 161, 167 Slope, 84-89,99, 108, 110, 121, 137, 181, 183-186, 193, 194 Solving systems of equations, 55 Spanning, 61, 64, 67 Spot rate, 57, 58, 198 Square matrix, 49, 54 Standard deviation, 23-27, 37, 104, 106-108, 110, 118, 119-122, 129, 144, 163, 166, 177, 178, 205-207, 215, 217, 228, 235, 242, 243, 245, 246-248

316 State Preference Model, 65, 66, 68 Stochastic dominance, 9, 145, 146, 147, 148, 154 Stochastic processes, 3, 13, 159, 213, 215, 229 Submartingale, 7 Symmetric matrix, 49 Taylor series, 112-115,129,218, 219, 245 Time value, 13, 15-17, 31, 106, 210 Transpose, 51 t-Table, 181,294 Two populations, 179 Unit matrix, 50 Utility, 5-9, 65, 87-89, 91, 115, 116, 124-126, 129, 145, 146

Index Variance, 8,23-27,29, 31, 37, 38-44, 52, 53, 77, 97, 98, 106, 119, 120, 122, 133, 134, 144, 161, 166-168, 174, 181, 186, 193, 197,215,218,220,221, 227, 230, 234, 243 Vector, 49,51, 53, 56,58, 59, 61-68,73-76, 119, 133, 135 Vector space, 61, 63-67 Weak form market efficiency, 6, 189 Weiner process, 5, 215-217, 221-223, 225, 230, 233, 234, 235 z-table, 167, 177, 178, 227, 293

About the Author JOHN L. TEALL is Associate Professor of Finance at Pace University and has served on the faculties of New York University, Fordham University, Dublin City University and others. He is a former member of the American Stock Exchange and has consulted with numerous financial institutions including Goldman Sachs, National Westminster Bank, and Citicorp.