Modern Portfolio Theory - Jack Clark Francis 4g624

Modern Portfolio Theory

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Modern Portfolio Theory Foundations, Analysis, and New Developments + Website

JACK CLARK FRANCIS DONGCHEOL KIM

John Wiley & Sons, Inc.

Cover Design: Leiva-Sposato. c Ekely / iStockphoto. Cover Image: c 2013 by Jack Clark Francis and Dongcheol Kim. All rights reserved. Copyright Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical , please our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993, or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our website at www.wiley.com. Library of Congress Catag-in-Publication Data: Francis, Jack Clark. Modern portfolio theory : foundations, analysis, and new developments + website / Jack Clark Francis, Dongcheol Kim. p. cm. – (Wiley finance series) Includes index. ISBN 978-1-118-37052-0 (cloth); ISBN 978-1-118-41763-8 (ebk); ISBN 978-1-118-42186-4 (ebk); ISBN 978-1-118-43439-0 (ebk) 1. Portfolio management. 2. Risk management. 3. Investment analysis. I. Kim, Dongcheol, 1955– II. Title. HG4529.5.F727 2013 332.601–dc23 2012032323

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To Harry Markowitz

Contents Preface

xvii

CHAPTER 1 Introduction 1.1 1.2 1.3

The Portfolio Management Process The Security Analyst’s Job Portfolio Analysis 1.3.1 Basic Assumptions 1.3.2 Reconsidering the Assumptions 1.4 Portfolio Selection 1.5 The Mathematics is Segregated 1.6 Topics to be Discussed Appendix: Various Rates of Return A1.1 Calculating the Holding Period Return A1.2 After-Tax Returns A1.3 Discrete and Continuously Compounded Returns

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PART ONE

Probability Foundations CHAPTER 2 Assessing Risk

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2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

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Mathematical Expectation What Is Risk? Expected Return Risk of a Security Covariance of Returns Correlation of Returns Using Historical Returns Data Input Requirements Portfolio Weights A Portfolio’s Expected Return Portfolio Risk Summary of Notations and Formulas

CHAPTER 3 Risk and Diversification 3.1

Reconsidering Risk 3.1.1 Symmetric Probability Distributions 3.1.2 Fundamental Security Analysis

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viii 3.2

3.3 3.4

3.5

Utility Theory 3.2.1 Numerical Example 3.2.2 Indifference Curves Risk-Return Space Diversification 3.4.1 Diversification Illustrated 3.4.2 Risky A + Risky B = Riskless Portfolio 3.4.3 Graphical Analysis Conclusions

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PART TWO

Utility Foundations CHAPTER 4 Single-Period Utility Analysis 4.1 4.2 4.3 4.4 4.5

Basic Utility Axioms The Utility of Wealth Function Utility of Wealth and Returns Expected Utility of Returns Risk Attitudes 4.5.1 Risk Aversion 4.5.2 Risk-Loving Behavior 4.5.3 Risk-Neutral Behavior 4.6 Absolute Risk Aversion 4.7 Relative Risk Aversion 4.8 Measuring Risk Aversion 4.8.1 Assumptions 4.8.2 Power, Logarithmic, and Quadratic Utility 4.8.3 Isoelastic Utility Functions 4.8.4 Myopic, but Optimal 4.9 Portfolio Analysis 4.9.1 Quadratic Utility Functions 4.9.2 Using Quadratic Approximations to Delineate Max[E(Utility)] Portfolios 4.9.3 Normally Distributed Returns 4.10 Indifference Curves 4.10.1 Selecting Investments 4.10.2 Risk-Aversion Measures 4.11 Summary and Conclusions Appendix: Risk Aversion and Indifference Curves A4.1 Absolute Risk Aversion (ARA) A4.2 Relative Risk Aversion (RRA) A4.3 Expected Utility of Wealth A4.4 Slopes of Indifference Curves A4.5 Indifference Curves for Quadratic Utility

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PART THREE

Mean-Variance Portfolio Analysis CHAPTER 5 Graphical Portfolio Analysis 5.1 5.2 5.3 5.4 5.5

Delineating Efficient Portfolios Portfolio Analysis Inputs Two-Asset Isomean Lines Two-Asset Isovariance Ellipses Three-Asset Portfolio Analysis 5.5.1 Solving for One Variable Implicitly 5.5.2 Isomean Lines 5.5.3 Isovariance Ellipses 5.5.4 The Critical Line 5.5.5 Inefficient Portfolios 5.6 Legitimate Portfolios 5.7 ‘‘Unusual’’ Graphical Solutions Don’t Exist 5.8 Representing Constraints Graphically 5.9 The Interior Decorator Fallacy 5.10 Summary Appendix: Quadratic Equations A5.1 Quadratic Equations A5.2 Analysis of Quadratics in Two Unknowns A5.3 Analysis of Quadratics in One Unknown A5.4 Solving an Ellipse A5.5 Solving for Lines Tangent to a Set of Ellipses

CHAPTER 6 Efficient Portfolios 6.1 6.2

Risk and Return for Two-Asset Portfolios The Opportunity Set 6.2.1 The Two-Security Case 6.2.2 Minimizing Risk in the Two-Security Case 6.2.3 The Three-Security Case 6.2.4 The n-Security Case 6.3 Markowitz Diversification 6.4 Efficient Frontier without the Risk-Free Asset 6.5 Introducing a Risk-Free Asset 6.6 Summary and Conclusions Appendix: Equations for a Relationship between E(rp ) and σp

CHAPTER 7 Advanced Mathematical Portfolio Analysis 7.1

Efficient Portfolios without a Risk-Free Asset 7.1.1 A General Formulation 7.1.2 Formulating with Concise Matrix Notation

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x 7.1.3 The Two-Fund Separation Theorem 7.1.4 Caveat about Negative Weights 7.2 Efficient Portfolios with a Risk-Free Asset 7.3 Identifying the Tangency Portfolio 7.4 Summary and Conclusions Appendix: Mathematical Derivation of the Efficient Frontier A7.1 No Risk-Free Asset A7.2 With a Risk-Free Asset

CHAPTER 8 Index Models and Return-Generating Process 8.1

Single-Index Models 8.1.1 Return-Generating Functions 8.1.2 Estimating the Parameters 8.1.3 The Single-Index Model Using Excess Returns 8.1.4 The Riskless Rate Can Fluctuate 8.1.5 Diversification 8.1.6 About the Single-Index Model 8.2 Efficient Frontier and the Single-Index Model 8.3 Two-Index Models 8.3.1 Generating Inputs 8.3.2 Diversification 8.4 Multi-Index Models 8.5 Conclusions Appendix: Index Models A8.1 Solving for Efficient Portfolios with the Single-Index Model A8.2 Variance Decomposition A8.3 Orthogonalizing Multiple Indexes

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PART FOUR

Non-Mean-Variance Portfolios CHAPTER 9 Non-Normal Distributions of Returns 9.1 9.2 9.3

9.4 9.5

9.6

Stable Paretian Distributions The Student’s t-Distribution Mixtures of Normal Distributions 9.3.1 Discrete Mixtures of Normal Distributions 9.3.2 Sequential Mixtures of Normal Distributions Poisson Jump-Diffusion Process Lognormal Distributions 9.5.1 Specifications of Lognormal Distributions 9.5.2 Portfolio Analysis under Lognormality Conclusions

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CHAPTER 10 Non-Mean-Variance Investment Decisions 10.1

Geometric Mean Return Criterion 10.1.1 Maximizing the Terminal Wealth 10.1.2 Log Utility and the GMR Criterion 10.1.3 Diversification and the GMR 10.2 The Safety-First Criterion 10.2.1 Roy’s Safety-First Criterion 10.2.2 Kataoka’s Safety-First Criterion 10.2.3 Telser’s Safety-First Criterion 10.3 Semivariance Analysis 10.3.1 Definition of Semivariance 10.3.2 Utility Theory 10.3.3 Portfolio Analysis with the Semivariance 10.3.4 Capital Market Theory with the Semivariance 10.3.5 Summary about Semivariance 10.4 Stochastic Dominance Criterion 10.4.1 First-Order Stochastic Dominance 10.4.2 Second-Order Stochastic Dominance 10.4.3 Third-Order Stochastic Dominance 10.4.4 Summary of Stochastic Dominance Criterion 10.5 Mean-Variance-Skewness Analysis 10.5.1 Only Two Moments Can Be Inadequate 10.5.2 Portfolio Analysis in Three Moments 10.5.3 Efficient Frontier in Three-Dimensional Space 10.5.4 Undiversifiable Risk and Undiversifiable Skewness 10.6 Summary and Conclusions Appendix A: Stochastic Dominance A10.1 Proof for First-Order Stochastic Dominance A10.2 Proof That FA (r) ≤ FB (r) Is Equivalent to EA (r) ≥ EB (r) for Positive r A10.3 Proof for Second-Order Stochastic Dominance A10.4 Proof for Third-Order Stochastic Dominance Appendix B: Expected Utility as a Function of Three Moments

CHAPTER 11 Risk Management: Value at Risk 11.1 11.2 11.3

11.4

VaR of a Single Asset Portfolio VaR Decomposition of a Portfolio’s VaR 11.3.1 Marginal VaR 11.3.2 Incremental VaR 11.3.3 Component VaR Other VaRs 11.4.1 Modified VaR (MVaR) 11.4.2 Conditional VaR (CVaR)

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Methods of Measuring VaR 11.5.1 Variance-Covariance (Delta-Normal) Method 11.5.2 Historical Simulation Method 11.5.3 Monte Carlo Simulation Method 11.6 Estimation of Volatilities 11.6.1 Unconditional Variance 11.6.2 Simple Moving Average 11.6.3 Exponentially Weighted Moving Average 11.6.4 GARCH-Based Volatility 11.6.5 Volatility Measures Using Price Range 11.6.6 Implied Volatility 11.7 The Accuracy of VaR Models 11.7.1 Back-Testing 11.7.2 Stress Testing 11.8 Summary and Conclusions Appendix: The Delta-Gamma Method

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PART FIVE

Asset Pricing Models CHAPTER 12 The Capital Asset Pricing Model 12.1 12.2

Underlying Assumptions The Capital Market Line 12.2.1 The Market Portfolio 12.2.2 The Separation Theorem 12.2.3 Efficient Frontier Equation 12.2.4 Portfolio Selection 12.3 The Capital Asset Pricing Model 12.3.1 Background 12.3.2 Derivation of the CAPM 12.4 Over- and Under-priced Securities 12.5 The Market Model and the CAPM 12.6 Summary and Conclusions Appendix: Derivations of the CAPM A12.1 Other Approaches A12.2 Tangency Portfolio Research

CHAPTER 13 Extensions of the Standard CAPM 13.1

Risk-Free Borrowing or Lending 13.1.1 The Zero-Beta Portfolio 13.1.2 No Risk-Free Borrowing 13.1.3 Lending and Borrowing Rates Can Differ

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13.2

Homogeneous Expectations 13.2.1 Investment Horizons 13.2.2 Multivariate Distribution of Returns 13.3 Perfect Markets 13.3.1 Taxes 13.3.2 Transaction Costs 13.3.3 Indivisibilities 13.3.4 Price Competition 13.4 Unmarketable Assets 13.5 Summary and Conclusions Appendix: Derivations of a Non-Standard CAPM A13.1 The Characteristics of the Zero-Beta Portfolio A13.2 Derivation of Brennan’s After-Tax CAPM A13.3 Derivation of Mayers’s CAPM for Nonmarketable Assets

CHAPTER 14 Empirical Tests of the CAPM 14.1 14.2

14.3

14.4

14.5 14.6

Time-Series Tests of the CAPM Cross-Sectional Tests of the CAPM 14.2.1 Black, Jensen, and Scholes’s (1972) Tests 14.2.2 Fama and MacBeth’s (1973) Tests 14.2.3 Fama and French’s (1992) Tests Empirical Misspecifications in Cross-Sectional Regression Tests 14.3.1 The Errors-in-Variables Problem 14.3.2 Sensitivity of Beta to the Return Measurement Intervals Multivariate Tests 14.4.1 Gibbons’s (1982) Test 14.4.2 Stambaugh’s (1982) Test 14.4.3 Jobson and Korkie’s (1982) Test 14.4.4 Shanken’s (1985) Test 14.4.5 Generalized Method of Moment (GMM) Tests Is the CAPM Testable? Summary and Conclusions

CHAPTER 15 Continuous-Time Asset Pricing Models 15.1 15.2

Intertemporal CAPM (ICAPM) The Consumption-Based CAPM (CCAPM) 15.2.1 Derivation 15.2.2 The Consumption-Based CAPM with a Power Utility Function 15.3 Conclusions Appendix: Lognormality and the Consumption-Based CAPM A15.1 Lognormality A15.2 The Consumption-Based CAPM with Lognormality

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