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Elements of Financial Risk Management ELEMENTS OF FINANCIAL RISK MANAGEMENT ELEMENTS OF FINANCIAL RISK MANAGEMENT Peter F. Christoffersen Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo This book is printed on acid-free paper. Copyright © 2003, Elsevier Science (USA). All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier Science homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Academic Press An imprint of Elsevier Science 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press 84 Theobald’s Road, London WC1X 8RR, UK http://www.academicpress.com Library of Congress Catalog Card Number: 2003107899 International Standard Book Number: 0-12-174232-6 PRINTED IN THE UNITED STATES OF AMERICA 0304050607 7654321 To Susan CONTENTS PREFACE XI ACKNOWLEDGMENTS XIII 1 Risk Management and Financial Returns 1.1. Chapter Outline 1 1.2. Learning Objectives 1 1.3. Risk Management and the Firm 2 1.4. A Brief Taxonomy of Risks 4 1.5. Stylized Facts of Asset Returns 6 1.6. Overview of the Book 9 1.7. Further Resources 9 1.8. Empirical Exercises on CD-ROM 10 References 18 vii viii CONTENTS 2 Volatility Modeling 2.1. Chapter Overview 19 2.2. Simple Variance Forecasting 20 2.3. The GARCH Variance Model 23 2.4. Extensions to the GARCH Model 26 2.5. Maximum Likelihood Estimation 28 2.6. Variance Model Evaluation 30 2.7. Using Intraday Information 32 2.8. Summary 38 2.9. Further Resources 38 2.10. Empirical Exercises on CD-ROM 39 References 46 3 Correlation Modeling 3.1. Chapter Overview 47 3.2. Value at Risk for Simple Portfolios 48 3.3. Portfolio Variance 51 3.4. Modeling Conditional Covariances 52 3.5. Modeling Conditional Correlations 54 3.6. Quasi-Maximum Likelihood Estimation 58 3.7. Realized and Range-Based Covariance 59 3.8. Summary 61 3.9. Further Resources 61 3.10. Appendix: VaR from Logarithmic versus Arithmetic Returns 62 3.11. Empirical Exercises on CD-ROM 63 References 70 4 Modeling the Conditional Distribution 4.1. Chapter Overview 71 4.2. Visualizing Non-Normality 73 4.3. The Standardized t(d) Distribution 74 4.4. The Cornish-Fisher Approximation to VaR 79 4.5. Extreme Value Theory (EVT) 80 4.6. The Expected Shortfall Risk Measure 85 4.7. Summary 87 4.8. Further Resources 88 4.9. Empirical Exercises on CD-ROM 89 References 97 CONTENTS ix 5 Simulation-Based Methods 5.1. Chapter Overview 99 5.2. Historical Simulation (HS) 100 5.3. Weighted Historical Simulation (WHS) 103 5.4. Multi-Period Risk Calculations 105 5.5. Monte Carlo Simulation (MCS) 108 5.6. Filtered Historical Simulation (FHS) 110 5.7. Summary 112 5.8. Further Resources 113 5.9. Empirical Exercises on CD-ROM 113 References 119 6 Option Pricing 6.1. Chapter Overview 121 6.2. Basic Definitions 122 6.3. Option Pricing Under the Normal Distribution 123 6.4. Allowing for Skewness and Kurtosis 129 6.5. Garch Option Pricing Models 133 6.6. Implied Volatility Function (IVF) Models 138 6.7. Summary 139 6.8. Further Resources 140 6.9. Appendix: The CFG Option Pricing Formula 141 6.10. Empirical Exercises on CD-ROM 142 References 151 7 Modeling Option Risk 7.1. Chapter Overview 153 7.2. The Option Delta 154 7.3. Portfolio Risk Using Delta 159 7.4. The Option Gamma 161 7.5. Portfolio Risk Using Gamma 163 7.6. Portfolio Risk Using Full Valuation 166 7.7. A Simple Example 168 7.8. Pitfall in the Delta and Gamma Approaches 171 7.9. Summary 173 7.10. Further Resources 174 7.11. Empirical Exercises on CD-ROM 175 References 180 x CONTENTS 8 Backtesting and Stress Testing 8.1. Chapter Overview 181 8.2. Backtesting VaRs 184 8.3. Increasing the Information Set 189 8.4. Backtesting Expected Shortfall 190 8.5. Backtesting the Entire Distribution 191 8.6. Stress Testing 194 8.7. Summary 197 8.8. Further Resources 198 8.9. Empirical Exercises on CD-ROM 198 References 208 Index 209 PREFACE INTENDED READERS This book is intended for three types of readers with an interest in financial risk management: first, master’s and Ph.D. students specializing in finance and economics; second, market practitioners with a quantitative undergraduate or graduate degree; third, advanced undergraduates majoring in economics, engi- neering, finance, or another quantitative field. I have taught the less technical parts of the book in a fourth-year undergraduate finance elective course and an MBA elective on financial risk management. The more technical material I have covered in a Ph.D. course on options and risk management and in technical training courses on market risk designed for market practitioners. In terms of prerequisites, the reader should have taken as a minimum a course on investments, a course on options, and one or two courses on econometrics or mathematical statistics. In addition, certain chapters require some previous exposure to matrix algebra. xi xii PREFACE SOFTWARE A number of empirical exercises are listed at the end of each chapter. Excel spread- sheets with the data underlying the exercises can be found on the CD-ROM that is shipped with the book. The CD-ROM also contains Excel files with answers to all the exercises. This way, virtually every technique discussed in the main text of the book is implemented on the CD using actual asset return data. The material on the CD is an essential part of the book. Previews of the spreadsheets are therefore shown following the exercise questions at the end of each chapter. Updates to the material in the book including the data and the answers to the exercises as well as links to further risk management resources can be found at www.christoffersen.ca. Any suggestions regarding improvements to the book are most welcome. Please e-mail these suggestions to [email protected]. ACKNOWLEDGMENTS Many people have played an important part—knowingly or unknowingly—in the writing of this book. Without implication, I would like to acknowledge the fol- lowing people for stimulating discussions on topics covered in this book: Torben Andersen, Gurdip Bakshi, Jeremy Berkowitz, Tim Bollerslev, Bryan Campbell, Francesca Carrieri, Susan Christoffersen, Frank Diebold, Jin Duan, Rob Engle, Vihang Errunza, John Galbraith, Rene Garcia, Eric Ghysels, Silvia Goncalves, Jinyong Hahn, Steve Heston, Derek Hulley, Atsushi Inoue, Kris Jacobs, Eric Jacquier, Chris Jones, Michael Jouravlev, Philippe Jorion, Ohad Kondor, Jose Lopez, Simone Manganelli, Tom McCurdy, James MacKinnon, Nour Meddahi, Saikat Nandi, Andrew Patton, Andrey Pavlov, Matthew Pritsker, Eric Renault, Garry Schinasi, Jean-Guy Simonato, Norm Swanson, Til Schuermann, Torsten Sloek, and Jonathan Wright. I have had a team of outstanding students working with me on the manuscript and on the CD-ROM in particular. They are Roustam Botachev, Thierry Koupaki, Stefano Mazzotta, Daniel Neata, and Denis Pelletier. For financial support of my research in general and of this book in particular, I would like to thank CIRANO, CIREQ, FCAR, IFM2, and SSHRC. I would also like to thank my editor at Academic Press, Scott Bentley, for his encouragement during the process of writing this book. Finally, I would like to thank Susan for constant moral support, and Nicholas and Phillip for helping me keep perspective. xiii 1 RISK MANAGEMENT AND FINANCIAL RETURNS 1.1. CHAPTER OUTLINE 1.2. LEARNING OBJECTIVES 1.3. RISK MANAGEMENT AND THE FIRM 1.3.1. Why Should Firms Manage Risk? 1.3.2. Evidence on Risk Management Practices 1.3.3. Does Risk Management Improve Firm Performance? 1.4. A BRIEF TAXONOMY OF RISKS 1.5. STYLIZED FACTS OF ASSET RETURNS 1.6. OVERVIEW OF THE BOOK 1.7. FURTHER RESOURCES 1.8. EMPIRICAL EXERCISES ON CD-ROM 1.1. CHAPTER OUTLINE This chapter begins by listing the learning objectives of the book. We then ask why firms should be occupied with risk management in the first place. In answering this question, we discuss the apparent contradiction between standard investment theory and the emergence of risk management as a field, and we list theoretical reasons for why managers should give attention to risk management. We also discuss the empirical evidence of the effectiveness and impact of current risk management practices in the corporate as well as financial sectors. Next, we list a taxonomy of the potential risks faced by a corporation, and we briefly discuss the desirability of exposure to each type of risk. After the risk taxonomy discussion, we list the stylized facts of asset returns, which are illustrated by the S&P 500 equity index. Finally, we present an overview of the remainder of the book. 1.2. LEARNING OBJECTIVES The book is intended as a practical handbook for risk managers as well as a textbook for students. It suggests a relatively sophisticated approach to risk mea- surement and risk modeling. The idea behind the book is to document key features 1 2 RISK MANAGEMENT AND FINANCIAL RETURNS of risky asset returns and then construct tractable statistical models that capture these features. More specifically, the book is structured to help the reader to do the following: • Become familiar with the range of risks facing corporations and learn how to measure and manage these risks.
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