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CHAPMAN & HALL/CRC FINANCIAL MATHEMATICS SERIES Portfolio Optimization and Performance Analysis CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the financial sector This series aims to capture new developments and summarize what is known over the whole spectrum of this field It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners The inclusion of numerical code and concrete real-world examples is highly encouraged Series Editors M.A.H Dempster Centre for Financial Research Judge Business School University of Cambridge Dilip B Madan Robert H Smith School of Business University of Maryland Rama Cont Center for Financial Engineering Columbia University New York Published Titles American-Style Derivatives; Valuation and Computation, Jerome Detemple Financial Modelling with Jump Processes, Rama Cont and Peter Tankov An Introduction to Credit Risk Modeling, Christian Bluhm, Ludger Overbeck, and Christoph Wagner Portfolio Optimization and Performance Analysis, Jean-Luc Prigent Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck Proposals for the series should be submitted to one of the series editors above or directly to: CRC Press, Taylor and Francis Group 24-25 Blades Court Deodar Road London SW15 2NU UK CHAPMAN & HALL/CRC FINANCIAL MATHEMATICS SERIES Portfolio Optimization and Performance Analysis Jean-Luc Prigent Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2007 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid‑free paper 10 International Standard Book Number‑10: 1‑58488‑578‑5 (Hardcover) International Standard Book Number‑13: 978‑1‑58488‑578‑8 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400 CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging‑in‑Publication Data Prigent, Jean‑Luc, 1958‑ Portfolio optimization and performance analysis / Jean‑Luc Prigent p cm ‑‑ (Chapman & Hall/CRC financial mathematics series ; 7) Includes bibliographical references and index ISBN‑13: 978‑1‑58488‑578‑8 (alk paper) ISBN‑10: 1‑58488‑578‑5 (alk paper) Portfolio management Investment analysis Hedge funds I Title II Series HG4529.5.P735 2007 332.6‑‑dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 2006100727 Preface Since the seminal mean-variance analysis was introduced by Markowitz (1952), the portfolio management theory has been expanded to take account of different features: • Dynamic portfolio optimization as per Merton (1962); • Choice of new decision criteria, based on risk aversion (utility functions) or risk measures (VaR, CVaR and beyond); • Market imperfections, e.g., transaction costs; and, • Specific portfolio strategies, such as portfolio insurance or alternative methods (hedge-funds) At the same time, many new financial products has been introduced, based in particular on financial derivatives Due to this intensive development and increasing complexity, this book has four purposes: • First, to recall standard results and to provide new insights about the axiomatics of the individual choice in an uncertain framework A concise introduction to portfolio choice under uncertainty based on investors’ preferences (usually represented by utility functions), and on several kinds of risk measures These theories are the fundamental basis of portfolio optimization - Chapter recalls the seminal approach of the utility maximization, introduced by Von Neumann and Morgenstern It also deals with further extensions of this theory, such as weighted expected utility theory, non-expected utility theory, etc - Chapter contains a survey about a new approach: the risk measure minimization Such risk measures have been recently introduced in particular to take better account of nonsymmetric asset return distributions • Second, to provide a precise overview on standard portfolio optimization Both passive and active portfolio management are considered Other results, such as risk measure minimization, are more recent V VI Portfolio Optimization and Performance Analysis - Chapter is devoted to the very well-known Markowitz analysis Some extensions are analyzed, in particular with risk minimization constraints such as safety criteria - Chapter deals with two important standard fund managements: managing indexed funds and benchmarked portfolio optimization In particular, statistical methods to replicate a financial index are detailed and discussed As regards benchmarking, the tracking error is computed and analyzed - Chapter recalls results about the main performance measures, such as the Sharpe and Treynor ratios and the Jensen alpha • Third, to make accessible the literature about stochastic optimization applied to mathematical finance (see for example Part III) to students, to researchers who are not specialists on this subject, and to financial engineers In particular, a review of the main standard results both for static and dynamic cases are provided For this purpose, precise mathematical statements are detailed without “too many” technicalities In particular: - Chapter provides an introduction to dynamic portfolio optimization The two main methods are the theory of stochastic control based on dynamic programming principle and, more recently, the martingale approach jointly used with convex duality - Chapter gives two important applications of previous results: the search for an optimal portfolio profile and the long-term management - Chapter is the more “technical” one It provides an overview on portfolio optimization with market frictions, such as incompleteness, transaction costs, labor income, random time horizon, etc • Finally, to show how theoretical results can be applied to practical and operational portfolio optimization (Part IV) This last part of the book deals with structured portfolio management which has grown significantly in the past few years Preface VII - Chapter is devoted to portfolio insurance and, in particular, to OBPI and CPPI strategies - Chapter 10 shows how common strategies, used by practitioners, may be justified by utility maximization under, for example, guarantee constraints It summarizes the main results concerning optimal portfolios when risk measures such as expected shortfall are introduced to limit downside risk - Chapter 11 recalls some problems when dealing with hedge funds, in particular the choice of appropriate performance measures As a by-product, special emphasis is put on: • Utility theory versus practice; • Active versus passive management; and, • Static versus dynamic portfolio management I hope this book will contribute to a better understanding of the modern portfolio theory, both for students and researchers in quantitative finance I am grateful to the CRC editorial staff for encouraging this project, in particular Sunil Nair, and for the help during the preparation of the final version: Michele Dimont and Shashi Kumar Jean-Luc PRIGENT, PARIS, February 2007 Contents List of Tables XIII List of Figures XV I Utility and risk analysis 1 Utility theory 1.1 Preferences under uncertainty 1.1.1 Lotteries 1.1.2 Axioms on preferences 1.2 Expected utility 1.3 Risk aversion 1.3.1 Arrow-Pratt measures of risk aversion 1.3.2 Standard utility functions 1.3.3 Applications to portfolio allocation 1.4 Stochastic dominance 1.5 Alternative expected utility theory 1.5.1 Weighted utility theory 1.5.2 Rank dependent expected utility theory 1.5.3 Non-additive expected utility 1.5.4 Regret theory 1.6 Further reading 7 11 13 15 17 19 24 25 27 32 33 35 Risk measures 2.1 Coherent and convex risk measures 2.1.1 Coherent risk measures 2.1.2 Convex risk measures 2.1.3 Representation of risk measures 2.1.4 Risk measures and utility 2.1.5 Dynamic risk measures 2.2 Standard risk measures 2.2.1 Value-at-Risk 2.2.2 CVaR 2.2.3 Spectral measures of risk 2.3 Further reading 37 37 38 39 40 41 43 48 48 54 59 62 IX ... IX X II Portfolio Optimization and Performance Analysis Standard portfolio optimization Static optimization 3.1 Mean-variance analysis 3.1.1 Diversification... standard portfolio optimization Both passive and active portfolio management are considered Other results, such as risk measure minimization, are more recent V VI Portfolio Optimization and Performance. .. MATHEMATICS SERIES Portfolio Optimization and Performance Analysis CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of