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Robust PortfolioOptimization and

Management

FRANK J. FABOZZI

PETTER N. KOLM

DESSISLAVA A. PACHAMANOVA

SERGIO M. FOCARDI

John Wiley & Sons, Inc.

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Management

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Management

FRANK J. FABOZZI

PETTER N. KOLM

DESSISLAVA A. PACHAMANOVA

SERGIO M. FOCARDI

John Wiley & Sons, Inc.

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THE FRANK J. FABOZZI SERIES

Fixed Income Securities, Second Edition

by Frank J. Fabozzi

Focus on Value: A Corporate and Investor Guide to Wealth Creation

by James L. Grant andJames A. Abate

Handbook of Global Fixed Income Calculations

by Dragomir Krgin

Managing a Corporate Bond Portfolio

by Leland E. Crabbe and Frank J. Fabozzi

Real Options and Option-Embedded Securities

by William T. Moore

Capital Budgeting: Theory and Practice

by Pamela P. Peterson and Frank J. Fabozzi

The Exchange-Traded Funds Manual

by Gary L. Gastineau

Professional Perspectives on Fixed Income Portfolio Management, Volume 3

edited by Frank J. Fabozzi

Investing in Emerging Fixed Income Markets

edited by Frank J. Fabozzi and Efstathia Pilarinu

Handbook of Alternative Assets

by Mark J. P. Anson

The Global Money Markets

by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

The Handbook of Financial Instruments


Collateralized Debt Obligations: Structures and Analysis

by Laurie S. Goodman and Frank J. Fabozzi

Interest Rate, Term Structure, and Valuation Modeling


Investment Performance Measurement

by Bruce J. Feibel

The Handbook of Equity Style Management

edited by T. Daniel Coggin and Frank J. Fabozzi

The Theory and Practice of Investment Management

edited by Frank J. Fabozzi and Harry M. Markowitz

Foundations of Economic Value Added: Second Edition

by James L. Grant

Financial Management and Analysis: Second Edition

by Frank J. Fabozzi and Pamela P. Peterson

Measuring and Controlling Interest Rate and Credit Risk: Second Edition

by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

Professional Perspectives on Fixed Income Portfolio Management, Volume 4


The Handbook of European Fixed Income Securities

edited by Frank J. Fabozzi and Moorad Choudhry

The Handbook of European Structured Financial Products

edited by Frank J. Fabozzi and Moorad Choudhry

The Mathematics of Financial Modeling and Investment Management

by Sergio M. Focardi and Frank J. Fabozzi

Short Selling: Strategies, Risks, and Rewards


The Real Estate Investment Handbook

by G. Timothy Haight and Daniel Singer

Market Neutral Strategies

edited by Bruce I. Jacobs and Kenneth N. Levy

Securities Finance: Securities Lending and Repurchase Agreements

edited by Frank J. Fabozzi and Steven V. Mann

Fat-Tailed and Skewed Asset Return Distributions

by Svetlozar T. Rachev, Christian Menn, and Frank J. Fabozzi

Financial Modeling of the Equity Market: From CAPM to Cointegration

by Frank J. Fabozzi, Sergio M. Focardi, and Petter N. Kolm

Advanced Bond Portfolio Management: Best Practices in Modeling and Strategies

edited by Frank J. Fabozzi, Lionel Martellini, and Philippe Priaulet

Analysis of Financial Statements, Second Edition

by Pamela P. Peterson and Frank J. Fabozzi

Collateralized Debt Obligations: Structures and Analysis, Second Edition

by Douglas J. Lucas, Laurie S. Goodman, and Frank J. Fabozzi

Handbook of Alternative Assets, Second Edition

by Mark J. P. Anson

Introduction to Structured Finance

by Frank J. Fabozzi, Henry A. Davis, and Moorad Choudhry

Financial Econometrics

by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo Jasic

Developments in Collateralized Debt Obligations: New Products and Insights

by Douglas J. Lucas, Laurie S. Goodman, Frank J. Fabozzi, and Rebecca J. Manning

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Copyright © 2007 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

Wiley Bicentennial Logo: Richard J. Pacifico

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 oth-erwise, 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 Rose-wood 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 Per-missions 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 con-tained 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.

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ISBN: 978-0-471-92122-6

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

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www.wiley.com

FJF

To my wife Donna and my children,Francesco, Patricia, and Karly

PNK

To Åke and Gunilla, my parents, and to John and Carmen,my wife’s parents, for their unending love and support

DAP

To my husband, Christian Hicks,and in memory of my grandfather, Georgyi Milyankov

SMF

To the memory of Bertrand Russell to whom I owe the foundation of my intellectual development

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vii

Contents

Preface xiAbout the Authors xv

CHAPTER 1Introduction 1

Quantitative Techniques in the Investment Management Industry 1Central Themes of This Book 9Overview of This Book 12

PART ONE

Portfolio Allocation: Classical Theory and Extensions 15

CHAPTER 2Mean-Variance Analysis and Modern Portfolio Theory 17

The Benefits of Diversification 18Mean-Variance Analysis: Overview 21Classical Framework for Mean-Variance Optimization 24The Capital Market Line 35Selection of the Optimal Portfolio When There Is a Risk-Free Asset 41More on Utility Functions: A General Framework for Portfolio Choice 45Summary 50

CHAPTER 3Advances in the Theory of Portfolio Risk Measures 53

Dispersion and Downside Measures 54Portfolio Selection with Higher Moments through Expansions of Utility 70Polynomial Goal Programming for Portfolio

Optimization with Higher Moments 78Some Remarks on the Estimation of Higher Moments 80The Approach of Malevergne and Sornette 81Summary 86

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viii

CONTENTS

CHAPTER 4Portfolio Selection in Practice 87

Portfolio Constraints Commonly Used in Practice 88Incorporating Transaction Costs in Asset-Allocation Models 101Multiaccount Optimization 106Summary 111

PART TWO

Robust Parameter Estimation 113

CHAPTER 5Classical Asset Pricing 115

Definitions 115Theoretical and Econometric Models 117Random Walk Models 118General Equilibrium Theories 131Capital Asset Pricing Model (CAPM) 132Arbitrage Pricing Theory (APT) 136Summary 137

CHAPTER 6Forecasting Expected Return and Risk 139

Dividend Discount and Residual Income Valuation Models 140The Sample Mean and Covariance Estimators 146Random Matrices 157Arbitrage Pricing Theory and Factor Models 160Factor Models in Practice 168Other Approaches to Volatility Estimation 172Application to Investment Strategies and Proprietary Trading 176Summary 177

CHAPTER 7Robust Estimation 179

The Intuition behind Robust Statistics 179Robust Statistics 181Robust Estimators of Regressions 192Confidence Intervals 200Summary 206

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ix

CHAPTER 8Robust Frameworks for Estimation: Shrinkage,

Bayesian Approaches, and the Black-Litterman Model 207

Practical Problems Encountered in Mean-Variance Optimization 208Shrinkage Estimation 215Bayesian Approaches 229Summary 253

PART THREE

Optimization Techniques 255

CHAPTER 9Mathematical and Numerical Optimization 257

Mathematical Programming 258Necessary Conditions for Optimality for Continuous

Optimization Problems 267Optimization Duality Theory 269How Do Optimization Algorithms Work? 272Summary 288

CHAPTER 10Optimization under Uncertainty 291

Stochastic Programming 293Dynamic Programming 308Robust Optimization 312Summary 332

CHAPTER 11Implementing and Solving Optimization Problems in Practice 333

Optimization Software 333Practical Considerations When Using Optimization Software 340Implementation Examples 346Specialized Software for Optimization Under Uncertainty 358Summary 360

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CONTENTS

PART FOUR

Robust Portfolio Optimization 361

CHAPTER 12Robust Modeling of Uncertain Parameters in Classical

Mean-Variance Portfolio Optimization 363

Portfolio Resampling Techniques 364Robust Portfolio Allocation 367Some Practical Remarks on Robust Portfolio Allocation Models 392Summary 393

CHAPTER 13The Practice of Robust Portfolio Management: Recent Trends and

New Directions 395

Some Issues in Robust Asset Allocation 396Portfolio Rebalancing 410Understanding and Modeling Transaction Costs 413Rebalancing Using an Optimizer 422Summary 435

CHAPTER 14Quantitative Investment Management Today and Tomorrow 439

Using Derivatives in Portfolio Management 440Currency Management 442Benchmarks 445Quantitative Return-Forecasting Techniques and Model-Based

Trading Strategies 447Trade Execution and Algorithmic Trading 456Summary 460

APPENDIX AData Description: The MSCI World Index 463

INDEX 473

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xi

Preface

n the past few years, there has been a notable increase in the use offinancial modeling and optimization tools in equity portfolio manage-

ment. In addition to the pressure on asset management firms to reducecosts and maintain a more stable and predictable performance in theaftermath of the downturn in the U.S. equity markets in 2002, three othergeneral trends have contributed to this increase. First, there has been arevived interest in predictive models for asset returns. Predictive modelsassume that it is possible to make conditional forecasts of futurereturns—an objective that was previously considered not achievable byclassical financial theory. Second, the wide availability of sophisticatedand specialized software packages has enabled generating and exploitingthese forecasts in portfolio management, often in combination with opti-mization and simulation techniques. Third, the continuous increase incomputer speed and the simultaneous decrease in hardware costs havemade the necessary computing power affordable even to small firms.

As the use of modeling techniques has become widespread amongportfolio managers, however, the issue of how much confidence practi-tioners can have in theoretical models and data has grown in impor-tance. Consequently, there is an increased level of interest in the subjectof robust estimation and optimization in modern portfolio manage-ment. For years, robustness has been a crucial ingredient in the engi-neering, statistics, and operations research fields. Today, these fieldsprovide a rich source of ideas to finance professionals. While robustportfolio management undoubtedly demands much more than therobust application of quantitative techniques, there is now a widespreadrecognition for the need of a disciplined approach to the analysis andmanagement of investments.

In this book we bring together concepts from finance, economic the-ory, robust statistics, econometrics, and robust optimization, and illustratethat they are part of the same theoretical and practical environment—in away that even a nonspecialized audience can understand and appreciate.At the same time, we emphasize a practical treatment of the subject, andtranslate complex concepts into real-world applications for robust return

I

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xii

PREFACE

forecasting and asset allocation optimization. Thereby, we address a num-ber of issues in portfolio allocation and rebalancing. In particular, we dis-cuss how to make portfolio management robust with respect to model risk,long-term views of the market, and market frictions such as trading costs.

The book is divided into four parts. Part I covers classical portfoliotheory and its modern extensions. We provide an up-to-date treatment ofmethods for advanced risk management, nonnormal distributions forasset returns, transaction costs, and multiaccount portfolio management.Part II introduces traditional and modern frameworks for robust estima-tion of returns. We address a number of topics that include dimensional-ity reduction, robust covariance matrix estimation, shrinkage estimators,and the Black-Litterman framework for incorporating investors’ views inan equilibrium framework. Part III provides readers with the necessarybackground for handling the optimization part of portfolio management.It covers major issues in numerical optimization, introduces widely usedoptimization software packages and modeling platforms, and discussesmethods for handling uncertainty in optimization models such as sto-chastic programming, dynamic programming, and robust optimization.Part IV focuses on applications of the robust estimation and optimizationmethods described in the previous parts, and outlines recent trends andnew directions in robust portfolio management and in the investmentmanagement industry in general. We cover a range of topics from portfo-lio resampling, robust formulations of the classical portfolio optimiza-tion framework under modeling uncertainty, robust use of factor models,and multiperiod portfolio allocation models—to the use of derivatives inportfolio management, currency management, benchmark selection,modern quantitative trading strategies, model risk mitigation, as well asoptimal execution and algorithmic trading.

We believe that practitioners and analysts who have to develop anduse portfolio management applications will find these themes—alongwith the numerous examples of applications and sample computercode—useful. At the same time, we address the topics in this book in atheoretically rigorous way, and provide references to the original works,so the book should be of interest to academics, students, and research-ers who need an updated and integrated view of the theory and practiceof portfolio management.

TEACHING USING THIS BOOK

This book can be used in teaching courses in advanced econometrics,financial engineering, quantitative investments and portfolio manage-

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Preface

xiii

ment, as the main course book, as supplemental reading on advanced top-ics, and/or for student projects. The material in Chapters 2 through 11 ofthe book is appropriate for undergraduate advanced electives on invest-ment management, and all topics in the book are accessible to graduatestudents in finance, economics or in the mathematical and physical sci-ences. The material is also appropriate for use in advanced graduate elec-tives in the decision sciences and operations research that focus onapplications of quantitative techniques in finance.

For a typical course, it is natural to start with Chapters 2, 5, and 6where modern portfolio and asset pricing theory and standard estima-tion techniques are covered. Basic practical considerations are presentedin Chapters 4 and 11. Chapters 3, 7, 8, 10, 12, and 13 are moreadvanced and do not have to be covered in full. A possibility is to focuson the most common techniques used in portfolio management today,such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) (inChapter 3), shrinkage estimators and the Black-Litterman model (inChapter 8), robust optimization (in Chapters 10 and 12), and transac-tion costs and portfolio rebalancing (in Chapter 13). Student projectscan be based on specialized topics such as multiaccount optimization (inChapter 4), numerical optimization techniques (in Chapter 9), moderntrading strategies, optimal execution, and algorithmic trading (in Chap-ter 14).

ACKNOWLEDGMENTS

In writing a book that covers a wide range of topics in portfolio manage-ment theory and practice, applied mathematics, statistics, and operationsresearch, we were fortunate to have received valuable comments and sug-gestions from the following individuals (listed below in alphabetical order):

■

Sebastian Ceria and Robert Stubbs of Axioma, Inc. reviewed Chapter12.

■

Eranda Dragoti-Cela of Siemens—Fin4Cast reviewed Chapter 12.

■

Dashan Huang of Kyoto University reviewed Chapters 10, 12, and 13.

■

Ivana Ljubic of the University of Vienna reviewed Chapter 12.

■

John M. Manoyan of CYMALEX Advisors reviewed Chapter 14.

■

Jeff Miller of Millennium Partners reviewed Chapters 13 and 14.

■

Bernd Scherer of Morgan Stanley reviewed Chapter 4.

■

Melvyn Sim of the National University of Singapore Business Schoolreviewed Chapter 12.

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PREFACE

■

Reha Tütüncü of Goldman Sachs Asset Management reviewed Chap-ters 10 and 12.

We thank Morgan Stanley Capital International, Inc., http://www.msci.com, for providing us with the MSCI World Index data setused in some of the computational examples throughout the book. Inparticular, we are indebted to Nicholas G. Keyes for answering all ofour questions in regards to the data set.

Megan Orem typeset the book and provided editorial assistance. Weappreciate her patience and understanding in working through numer-ous revisions of the chapters and several reorganizations of the table ofcontents.

Frank J. FabozziPetter N. KolmDessislava A. PachamanovaSergio M. Focardi

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xv

About the Authors

Frank J. Fabozzi

is Professor in the Practice of Finance in the School ofManagement at Yale University. Prior to joining the Yale faculty, he was aVisiting Professor of Finance in the Sloan School at MIT. Frank is a Fel-low of the International Center for Finance at Yale University and on theAdvisory Council for the Department of Operations Research and Finan-cial Engineering at Princeton University. He is the editor of the

Journal ofPortfolio Management

and an associate editor of the

Journal of FixedIncome

. He earned a doctorate in economics from the City University ofNew York in 1972. In 2002 was inducted into the Fixed Income AnalystsSociety’s Hall of Fame and is the 2007 recipient of the C. Stewart Shep-pard Award given by the CFA Institute. He earned the designation ofChartered Financial Analyst and Certified Public Accountant. He hasauthored and edited numerous books in finance.

Petter N. Kolm

is a doctoral student in Finance at the School of Manage-ment, Yale University, a financial consultant in New York City, and amember of the editorial board of the

Journal of Portfolio Management

.Previously, he worked in the Quantitative Strategies Group at GoldmanSachs Asset Management where his responsibilities included researchingand developing new quantitative investment strategies for the group’shedge fund. Petter coauthored the books

Financial Modeling of theEquity Market: From CAPM to Cointegration

and

Trends in QuantitativeFinance

. His research interests include various topics in finance, such asequity and fixed income modeling, delegated portfolio management,financial econometrics, risk management, and optimal portfolio strate-gies. Petter received a doctorate in mathematics from Yale University in2000. He also holds an M.Phil. in applied mathematics from the RoyalInstitute of Technology in Stockholm and an M.S. in mathematics fromETH Zürich.

Dessislava A. Pachamanova

is an Assistant Professor of OperationsResearch at Babson College where she holds the Zwerling Term Chair.Her research interests lie in the areas of robust optimization, portfolio

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ABOUT THE AUTHORS

risk management, simulation, and financial engineering. Dessislava’s aca-demic research is supplemented by consulting and previous work in thefinancial industry, including projects with quantitative strategy groups atWestLB and Goldman Sachs. She holds an A.B. in Mathematics fromPrinceton University and a Ph.D. in Operations Research from the SloanSchool of Management at MIT.

Sergio Focardi

is a founding partner of the Paris-based consulting firmThe Intertek Group and consults and trains on quantitative methods inequity portfolio management. Sergio is a member of the Editorial Boardof the

Journal of Portfolio Management

, co-author of the CFA Institute’smonograph

Trends in Quantitative Finance

(Fabozzi, Focardi and Kolm,2006) of the books

Financial Econometrics

(Rachev, Mittnik, Fabozzi,Focardi, Jasic, Wiley, 2007),

Financial Modeling of the Equity Market

(Fabozzi, Focardi and Kolm, Wiley, 2006),

The Mathematics of FinancialModeling and Investment Management

(Focardi and Fabozzi, Wiley,2004),

Risk Management: Framework, Methods and Practice

(Focardiand Jonas, Wiley, 1998), and

Modeling the Markets: New Theories andTechniques

(Focardi and Jonas, Wiley, 1997). Sergio has implementedlong-short equity portfolio selection applications based on dynamic factoranalysis. His research interests include the econometrics of large equityportfolios and the modeling of regime changes. Sergio holds a degree inElectronic Engineering from the University of Genoa and a postgraduatedegree in Communications from the Galileo Ferraris ElectrotechnicalInstitute (Turin).

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CHAPTER

1

1

Introduction

s the use of quantitative techniques has become more widespread inthe financial industry, the issues of how to apply financial models

most effectively and how to mitigate model and estimation errors havegrown in importance. This book discusses some of the major trends andinnovations in the management of financial portfolios today, focusingon state-of-the-art robust methodologies for portfolio risk and returnestimation, optimization, trading, and general management.

In this chapter, we give an overview of the main topics in the book.We begin by providing a historical outlook of the adoption of quantita-tive techniques in the financial industry and the factors that have con-tributed to its growth. We then discuss the central themes of the book inmore detail, and give a description of the structure and content of itsremaining chapters.

QUANTITATIVE TECHNIQUES IN THE INVESTMENT

MANAGEMENT INDUSTRY

Over the last 20 years there has been a tremendous increase in the use ofquantitative techniques in the investment management industry. Thefirst applications were in risk management, with models measuring therisk exposure to different sources of risk. Nowadays, quantitative mod-els are considered to be invaluable in all the major areas of investmentmanagement, and the list of applications continues to grow: option pric-ing models for the valuation of complicated derivatives and structuredproducts, econometric techniques for forecasting market returns, auto-mated execution algorithms for efficient trading and transaction costmanagement, portfolio optimization for asset allocation and financial

A

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2

ROBUST PORTFOLIO OPTIMIZATION AND MANAGEMENT

planning, and statistical techniques for performance measurement andattribution, to name a few.

Today, quantitative finance has evolved into its own discipline—anexample thereof is the many university programs and courses beingoffered in the area in parallel to the “more traditional” finance andMBA programs. Naturally, many different factors have contributed tothe tremendous development of the quantitative areas of finance, and itis impossible to list them all. However, the following influences and con-tributions are especially noteworthy:

■

The development of modern financial economics, and the advances inthe mathematical and physical sciences.

■

The remarkable expansion in computer technology and the inventionof the Internet.

■

The maturing and growth of the capital markets.

Below, we highlight a few topics from each one of these areas and dis-cuss their impact upon quantitative finance and investment managementin general.

Modern Financial Economics and the Mathematical and Physical Sciences

The concepts of portfolio optimization and diversification have beeninstrumental in the development and understanding of financial marketsand financial decision making. The major breakthrough came in 1952with the publication of Harry Markowitz’s theory of portfolio selec-tion.

1

The theory, popularly referred to as

modern portfolio theory,

pro-vided an answer to the fundamental question: How should an investorallocate funds among the possible investment choices? Markowitz sug-gested that investors should consider

risk and return

together and deter-mine the allocation of funds among investment alternatives on the basisof the trade-off between them. Before Markowitz’s seminal article, thefinance literature had treated the interplay between risk and return in acasual manner.

The idea that sound financial decision making is a quantitativetrade-off between risk and return was revolutionary for two reasons.First, it posited that one could make a quantitative evaluation of risk

1

Harry M. Markowitz, “Portfolio Selection,”

Journal of Finance

7, no. 1 (March1952), pp. 77–91. The principles in Markowitz’s article were later expanded uponin his book

Portfolio Selection

, Cowles Foundation Monograph 16 (New York: JohnWiley & Sons, 1959). Markowitz was awarded the Nobel Prize in Economic Sciencesin 1990 for his work.


Introduction

3

and return

jointly

by considering portfolio returns and their comove-ments. An important principle at work here is that of portfolio diversifi-cation. It is based on the idea that a portfolio’s riskiness depends on thecovariances of its constituents, not only on the average riskiness of itsseparate holdings. This concept was foreign to classical financial analy-sis, which revolved around the notion of the value of single investments,that is, the belief that investors should invest in those assets that offerthe highest future value given their current price. Second, it formulatedthe financial decision-making process as an optimization problem. In par-ticular, the so-called mean-variance principle formulated by Markowitzsuggests that among the infinite number of portfolios that achieve a par-ticular return objective, the investor should choose the portfolio thathas the smallest variance. All other portfolios are “inefficient” becausethey have a higher variance and, therefore, higher risk.

Building on Markowitz’s work, William Sharpe,

2

John Lintner,

3

andJan Mossin

4

introduced the first asset pricing theory, the capital assetpricing model—CAPM in short—between 1962 and 1964. The CAPMbecame the foundation and the standard on which risk-adjusted perfor-mance of professional portfolio managers is measured.

Modern portfolio theory and diversification provide a theoreticaljustification for mutual funds and index funds, that have experienced atremendous growth since the 1980s. A simple classification of fundmanagement is into active and passive management, based upon the

effi-cient market hypotheses

introduced by Eugene Fama

5

and Paul Samuel-son

6

in 1965. The efficient market hypothesis implies that it is notpossible to outperform the market consistently on a risk-adjusted basisafter accounting for transaction costs by using available information. Inactive management, it is assumed that markets are not fully efficient andthat a fund manager can outperform a market index by using specificinformation, knowledge, and experience. Passive management, in con-

2

William F. Sharpe, “Capital Asset Prices,”

Journal of Finance

19, no. 3 (September1964), pp. 425–442. Sharpe received the Nobel Prize in Economic Sciences in 1990for his work.

3

John Lintner, “The Valuation of Risk Assets and the Selection of Risky Investmentsin Stock Portfolio and Capital Budgets,”

Review of Economics and Statistics

47(February 1965), pp. 13–37.

4

Jan Mossin, “Equilibrium in a Capital Asset Market,”

Econometrica

34, no. 4 (Oc-tober 1966), pp. 768–783.

5

Eugene F. Fama, “The Behavior of Stock Market Prices,”

Journal of Business

38(January 1965), pp. 34–105.

6

Paul A. Samuelson, “Proof that Properly Anticipated Prices Fluctuate Randomly,”

Industrial Management Review

6, no. 2 (Spring 1965), pp. 41–49. Samuelson washonored with the Nobel Prize in Economic Sciences in 1970.


4


trast, relies on the assumption that financial markets are efficient andthat return and risk are fully reflected in asset prices. In this case, aninvestor should invest in a portfolio that mimics the market. John Bogleused this basic idea when he proposed to the board of directors of thenewly formed Vanguard Group to create the first index fund in 1975.The goal was not to outperform the S&P 500 index, but instead to trackthe index as closely as possible by buying each of the stocks in the S&P500 in amounts equal to the weights in the index itself.

Despite the great influence and theoretical impact of modern portfo-lio theory, today—more than 50 years after Markowitz’s seminalwork—full risk-return optimization at the asset level is primarily doneonly at the more quantitatively oriented firms. In the investment man-agement business at large, portfolio management is frequently a purelyjudgmental process based on qualitative, not quantitative, assessments.The availability of quantitative tools is not the issue—today’s optimiza-tion technology is mature and much more user-friendly than it was atthe time Markowitz first proposed the theory of portfolio selection—yetmany asset managers avoid using the quantitative portfolio allocationframework altogether.

A major reason for the reluctance of investment managers to applyquantitative risk-return optimization is that they have observed that itmay be unreliable in practice. Specifically, risk-return optimization isvery sensitive to changes in the inputs (in the case of mean-variance opti-mization, such inputs include the expected return of each asset and theasset covariances). While it can be difficult to make accurate estimates ofthese inputs, estimation errors in the forecasts significantly impact theresulting portfolio weights. It is well-known, for instance, that in practi-cal applications equally weighted portfolios often outperform mean-vari-ance portfolios, mean-variance portfolios are not necessarily well-diversified, and mean-variance optimization can produce extreme ornon-intuitive weights for some of the assets in the portfolio. Such exam-ples, however, are not necessarily a sign that the

theory

of risk-returnoptimization is flawed; rather, that when used

in practice

, the classicalframework has to be modified in order to achieve reliability, stability,and robustness with respect to model and estimation errors.

It goes without saying that advances in the mathematical and physi-cal sciences have had a major impact upon finance. In particular, mathe-matical areas such as probability theory, statistics, econometrics,operations research, and mathematical analysis have provided the neces-sary tools and discipline for the development of modern financial eco-nomics. Substantial advances in the areas of robust estimation androbust optimization were made during the 1990s, and have proven to be


Introduction

5

of great importance for the practical applicability and reliability of port-folio management and optimization.

Any statistical estimate is subject to error—estimation error. A robustestimator is a statistical estimation technique that is less sensitive to outliersin the data. For example, in practice, it is undesirable that one or a fewextreme returns have a large impact on the estimation of the average returnof a stock. Nowadays, Bayesian techniques and robust statistics are com-monplace in financial applications. Taking it one step further, practitionersare starting to incorporate the uncertainty introduced by estimation errorsdirectly into the optimization process. This is very different from the classi-cal approach, where one solves the portfolio optimization problem as aproblem with deterministic inputs, without taking the estimation errorsinto account. In particular, the statistical precision of individual estimates isexplicitly incorporated in the portfolio allocation process. Providing thisbenefit is the underlying goal of

robust portfolio optimization

. First introduced by El Ghaoui and Lebret

7

and by Ben-Tal andNemirovski,

8

modern robust optimization techniques allow a portfoliomanager to solve the robust version of the portfolio optimization prob-lem in about the same time as needed for the classical portfolio optimi-zation problem. The robust approach explicitly uses the distributionfrom the estimation process to find a robust portfolio in

one single

opti-mization, thereby directly incorporating uncertainty about inputs in theoptimization process. As a result, robust portfolios are less sensitive toestimation errors than other portfolios, and often perform better thanclassical mean–variance portfolios. Moreover, the robust optimizationframework offers great flexibility and many new interesting applica-tions. For instance, robust portfolio optimization can exploit the notionof statistically equivalent portfolios. This concept is important in large-scale portfolio management involving many complex constraints such astransaction costs, turnover, or market impact. Specifically, with robustoptimization, a manager can find the best portfolio that (1) minimizestrading costs with respect to the current holdings and (2) has anexpected portfolio return and variance that are statistically equivalentto those of the classical mean-variance portfolio.

An important area of quantitative finance is that of modeling assetprice behavior, and pricing options and other derivatives. This field can

7

Laurent El Ghaoui, and Herve Lebret, “Robust Solutions to Least-Squares Prob-lems with Uncertain Data,”

SIAM Journal on Matrix Analysis and Applications

18(October 1997), pp. 1035–1064.

8

Aharon Ben-Tal, and Arkadi S. Nemirovski, “Robust Convex Optimization,”

Mathematics of Operations Research

23, no. 4 (1998), pp. 769–805; and AharonBen-Tal, and Arkadi S. Nemirovski, “Robust Solutions to Uncertain Linear Pro-grams,”

Operations Research Letters

25, no. 1 (1999), pp. 1–13.


6


be traced back to the early works of Thorvald Thiele

9

in 1880, LouisBachelier

10

in 1900, and Albert Einstein

11

in 1905, who knew nothingabout each other’s research and independently developed the mathemat-ics of Brownian motion. Interestingly, while the models by Thiele andBachelier had little influence for a long time, Einstein’s contribution hadan immediate impact on the physical sciences. Historically, Bachelier’sdoctoral thesis is the first published work that uses advanced mathemat-ics in the study of finance. Therefore, he is by many considered to be thepioneer of financial mathematics—the first “quant.”

12

The first listed options began trading in April 1973 on the ChicagoBoard Options Exchange (CBOE), only one and four months, respec-tively, before the papers by Black and Scholes

13

and by Merton

14

onoption pricing were published. Although often criticized in the generalpress, and misunderstood by the public at large, options opened thedoor to a new era in investment and risk management, and influencedthe introduction and popularization of a range of other financial prod-ucts including interest rate swaptions, mortgage-backed securities, call-able bonds, structured products, and credit derivatives. New derivativeproducts were made possible as a solid pricing theory was available.Without the models developed by Black, Scholes, and Merton and manyothers following in their footsteps, it is likely that the rapid expansion

9

Thorvald N. Theile, “Sur la Compensation de Quelques Erreurs Quasi-Systématiquespar la Méthodes de Moindre Carrés [On the Compensation of Some Quasi-SystematicErrors by the Least Square Method],”

Vidensk. Selsk. Skr.

5 (1880), pp. 381–408.

10

Louis Bachelier, “Théorie de la Speculation [Theory of Speculation],”

Annales Sci-entifiques de l’École Normale Supérieure Sér.

, 3, 17 (1900), pp. 21–86

11

Albert Einstein, “On the Movement of Small Particles Suspended in StationaryLiquid Demanded by the Molecular-Kinetic Theory of Heat,” in R. Fürth (ed.),

In-vestigations of the Theory of Brownian Movement

(New York: Dover Publications,1956).

12

The term “quant” which is short for

quantitative analyst

(someone who works inthe financial markets developing mathematical models) was popularized, amongother things, by Emanuel Derman in his book

My Life as a Quant

(Hoboken, NJ:John Wiley & Sons, 2004). On a lighter note, a T-shirt with the words “Quants DoIt with Models” circulated among some quantitative analysts on Wall Street a fewyears ago.

13

Fischer S. Black and Myron S. Scholes, “The Pricing of Options and Corporate Li-abilities,”

Journal of Political Economy

81, no. 3 (1973), pp. 637–659. Scholes re-ceived the Nobel Prize of Economic Science in 1997 for his work on option pricingtheory. At that time, sadly, Fischer Black had passed away, but he received an hon-orable mention in the award.

14

Robert C. Merton, “Theory of Rational Option Pricing,”

Bell Journal of Econom-ics and Management Science

4, no. 1 (Spring 1973), pp. 141–183. Merton receivedthe Nobel Prize of Economic Science in 1997 for his work on option pricing theory.


Introduction

7

of derivative products would never have happened. These moderninstruments and the concepts of portfolio theory, CAPM, arbitrage andequilibrium pricing, and market predictability form the foundation notonly for modern financial economics but for the general understandingand development of today’s financial markets. As Peter Bernstein so ade-quately puts it in his book

Capital Ideas

: “Every time an institution usesthese instruments, a corporation issues them, or a homeowner takes outa mortgage, they are paying their respects, not just to Black, Scholes,and Merton, but to Bachelier, Samuelson, Fama, Markowitz, Tobin,Treynor, and Sharpe as well.”

15

Computer Technology and the Internet

The appearance of the first personal computers in the late 1970s andearly 1980s forever changed the world of computing. It put computa-tional resources within the reach of most people. In a few years everytrading desk on Wall Street was equipped with a PC. From that point on,computing costs have declined at the significant pace of about a factor of2 every year. For example, the cost per gigaflops

16

is about $1 today, tobe compared to about $50,000 about 10 years ago.

17

At the same time,computer speed increased in a similar fashion: today’s fastest computersare able to perform an amazing 300 trillion calculations per second.

18

This remarkable development of computing technology has allowedfinance professionals to deploy more sophisticated algorithms used, forinstance, for derivative and asset pricing, market forecasting, portfolioallocation, and computerized execution and trading. With state-of-the-art optimization software, a portfolio manager is able to calculate theoptimal allocation for a portfolio of thousands of assets in no more thana few seconds—on the manager’s desktop computer!

15

Peter L. Bernstein,

Capital Ideas

(New York: Free Press,1993).

16

Flops is an abbreviation for

floating point operations per second

and is used as ameasure of a computer’s performance. 1 gigaflops = 10

9

flops.

17

See Michael S. Warren, John K. Salmon, Donald J. Becker, M. Patrick Goda, Th-omas Sterling, and Grégoire S. Winckelmans, “Pentium Pro Inside: I. A Treecode at430 Gigaflops on ASCI Red. II. Price/Performance of $50/Mflop on Loki andHyglac,”

Supercomputing ’97,

Los Alamitos, 1997, IEEE Computer Society; andWikipedia contributors, “FLOPS,”

Wikipedia, The Free Encyclopedia

, http://en.wikipedia.org/w/index.php?title=FLOPS&oldid=90585825 (accessed December1, 2006).

18

As of November 2006, the IBM BlueGene/L system with 131072 processor unitsheld the so-called Linpack record with a remarkable performance of 280.6 teraflops(that is, 280.6 trillions of floating-point operations per second). See TOP500,www.top500.org.


8


But computational power alone is not sufficient for financial applica-tions. It is crucial to obtain market data and other financial informationefficiently and expediently, often in real time. The Internet and the WorldWide Web have proven invaluable for this purpose. The World WideWeb, or simply the “Web,” first created by Tim Berners-Lee working atCERN in Geneva, Switzerland around 1990, is an arrangement of inter-linked, hypertext documents available over the Internet. With a simplebrowser, anybody can view webpages that may contain anything fromtext and pictures, to other multimedia based information, and jump frompage to page by a simple mouse click.

19

Berners-Lee’s major contributionwas to combine the concept of hypertext with the Internet, born out ofthe NSFNet developed by the National Science Foundation in the early1980s. The Web as we know it today allows for expedient exchange offinancial information. Many market participants—from individuals toinvestment houses and hedge funds—use the Internet to follow financialmarkets as they move tick by tick and to trade many different kinds ofassets such as stocks, bonds, futures, and other derivatives simultaneouslyacross the globe. In today’s world, gathering, processing, and analyzingthe vast amount of information is only possible through the use of com-puter algorithms and sophisticated quantitative techniques.

Capital Markets

The development of the capital markets has of course had a significantimpact on quantitative finance and the investment management industryas a whole. Investors today have a vast number of assets available in thecapital markets, from more traditional assets such as stocks, bonds, com-modities (precious metals, etc.) and real estate to derivative instrumentssuch as options, futures, swaps, credit linked securities, mortgage-backedsecurities and other structured products, and specialized financial insur-ance products. These securities and products allow market participants toget exposure to, or to hedge risks—sometimes very specific risks. Forexample, a corporate bond portfolio manager may decide to hedge spe-cific credit risks in his portfolio using a credit default swap, or a propri-etary trader can short equity volatility by selling a volatility swap.

However, the number of assets available alone is not enough to guar-antee success, if the assets are only traded infrequently and in small vol-umes. Successful capital markets have to be liquid, allowing marketparticipants to trade their positions quickly and at low cost. An asset is

19

A recent study concluded that as of January 2005 there are over 11.5 billion publicwebpages available on the Internet, see Antonio Gulli and Alessio Signorini, “The In-dexable Web is More than 11.5 billion pages,” 2005, Dipartimento di Informaticaat Universita’ di Pisa and Department of Computer Science at University of Iowa.


Introduction

9

said to be liquid if it can be converted to cash quickly at a price close tofair market value. The U.S. capital markets are the most liquid in theworld with Japan and the United Kingdom following. Cash, being thebasic liquid asset, does not fluctuate in value—it itself defines price. Allother assets can change in value and have an uncertain future price, mak-ing them risky assets. Naturally, informed investors will only hold less liq-uid and risky assets if they can expect to earn a premium, a risk premium.

With the tremendous increase in the number of assets—and with it,the amount of investment opportunities—it is hard, even for largerinvestment houses, to track and evaluate the different markets. Quanti-tative techniques lend themselves for automatic monitoring and analysisof the full multitude of securities. These tools give quantitative analysts,portfolio managers, and other decision makers the opportunity to sum-marize the vast amount of information available, and to present it in acohesive manner. Modern financial and the econometric models rely onthe access to accurate data, often with as long history as possible. It istypically much easier to obtain clean and trustworthy financial datafrom mature and liquid markets. In fact, the lack of reliable data is oneof the inherent problems in applying sophisticated quantitative modelsto more illiquid markets. In these cases, practitioners are forced to relyon simulated data, make stronger assumptions in their models, or useless data-intensive models.

CENTRAL THEMES OF THIS BOOK

The purpose of this book is to provide a comprehensive introductionand overview of the state-of-the-art of portfolio management and opti-mization for practitioners, academics, and students alike. We attempt tobridge the gap from classical portfolio theory, as developed in the early1950s, to modern portfolio optimization applications used in practicetoday. In particular, we provide an up-to-date review of robust estima-tion and optimization methods deployed in modern portfolio manage-ment, and discuss different techniques used in order to overcome thecommon pitfalls associated with classical mean-variance optimization.We discuss recent developments in quantitative trading strategies, tradeexecution, and operations research. While we focus on real world prac-tical usability, and emphasize intuition and fundamental understanding,we try not to sacrifice mathematical rigor whenever possible.

We note that the concept of robustness in investment science extendsbeyond statistical and modeling methods. It suggests a new approach tofinancial forecasting, asset allocation, portfolio management, and trad-


10


ing. As a matter of fact, the concept of a

robust quantitative investmentframework

seems to be gaining ground in the quantitative investmentcommunity, and is loosely defined by the following four stages:

1. Estimate reliable asset forecasts along with a measure of their confi-dence.

2. Deploy a robust model for portfolio allocation and risk management.3. Manage portfolio rebalancing and trading costs efficiently as market

conditions change. 4. Monitor and review the entire investment process on a regular basis.

The last stage includes the ability to evaluate past performance, aswell as to measure and analyze portfolio risk. The role of quantitativemodels for econometric forecasting and optimization at each of thesestages is very important, especially in large-scale investment manage-ment applications that require allocating, rebalancing, and monitoringof thousands of assets and portfolios.

From a broad perspective, the topics in this book can be categorizedin the following four main areas: robust estimation, robust portfolioallocation, portfolio rebalancing, and management of model risk.

Robust Estimation

Models to predict expected returns of assets are routinely used by majorasset management firms. In most cases, these models are straightforwardand based on factors or other forecasting variables. Since parameterestimation in these financial models is data-driven, they are inevitablysubject to estimation error. What makes matters worse, however, is thatdifferent estimation errors are accumulated across the different stages inthe portfolio management process. As a result, the compounding ofsmall errors from the different stages may result in large aggregateerrors at the final stage. It is therefore important that parameters esti-mated at the different stages are reliable and robust so that the aggre-gate impact of estimation errors is minimized.

Given the existing plethora of financial forecasting models, the entiretopic of robust statistical estimation is too extensive to cover in thisbook.

20

We will, however, touch upon several major topics. In particular,we review some fundamental statistical techniques for forecastingreturns, show how robust statistical estimators for important inputs inthe portfolio optimization process can be obtained, and how a robust

20

For an overview of equity forecasting models, see Frank J. Fabozzi, Sergio M. Fo-cardi, and Petter N. Kolm,

Financial Modeling of the Equity Market: From CAPMto Cointegration

(Hoboken, NJ: John Wiley & Sons, 2006).


Introduction

11

portfolio allocation framework minimizes the impact of estimation andmodel errors. We describe robust frameworks for incorporating theinvestor’s views such as shrinkage techniques and the Black-Littermanmodel to produce informed forecasts about the behavior of asset returns.

Robust Portfolio Allocation

Robust asset allocation is one of the most important parts of the invest-ment management process, and the decision making is frequently basedon the recommendations of risk-return optimization routines. Severalmajor themes deserve attention. First, it is important to carefully con-sider how portfolio risk and return are defined, and whether these defi-nitions are appropriate given observed or forecasted asset returndistributions and underlying investor preferences. These concerns giverise to alternative theories of risk measures and asset allocation frame-works beyond classical mean-variance optimization. Second, the issue ofhow the optimization problem is formulated and solved in practice iscrucial, especially for larger portfolios. A working knowledge of thestate-of-the-art capabilities of quantitative software for portfolio man-agement is critical. Third, it is imperative to evaluate the sensitivity ofportfolio optimization models to inaccuracies in input estimates. Wecover the major approaches for optimization under uncertainty in inputparameters, including a recently developed area in optimization—

robustoptimization

—that has shown a great potential and usability for portfo-lio management and optimization applications.

Portfolio Rebalancing

While asset allocation is one of the major strategic decisions, the decisionof how to achieve this allocation in a cost-effective manner is no lessimportant in obtaining good and consistent performance. Furthermore,given existing holdings, portfolio managers need to decide how to rebal-ance their portfolios efficiently to incorporate new views on expectedreturns and risk as the economic environment and the asset mix change.There are two basic aspects of the problem of optimal portfolio rebalanc-ing. The first one is the robust management of the trading and transactioncosts in the rebalancing process. The second is successfully combiningboth long-term and short-term views on the future direction and changesin the markets. The latter aspect is particularly important when taxes orliabilities have to be taken into account. The two aspects are not distinct,and in practice have to be considered simultaneously. By incorporatinglong-term views on asset behavior, portfolio managers may be able toreduce their overall transaction costs, as their portfolios do not have to berebalanced as often. Although the interplay between the different aspects


12


is complex to evaluate and model, disciplined portfolio rebalancing usingan optimizer provides portfolio managers with new opportunities.

Managing Model Risk

Quantitative approaches to portfolio management introduce a newsource of risk—model risk—and an inescapable dependence on histori-cal data as their raw material. Financial models are typically predic-tive—they are used to forecast unknown or future values on the basis ofcurrent or known values using specified equations or sets of rules. Theirpredictive or forecasting power, however, is limited by the appropriate-ness of the inputs and basic model assumptions. Incorrect assumptions,model identification and specification errors, or inappropriate estima-tion procedures inevitably lead to model risk, as does using modelswithout sufficient out-of-sample testing. It is important to be cautious inhow we use models, and to make sure that we fully understand theirweaknesses and limitations. In order to identify the various sources ofmodel risk, we need to take a critical look at our models, review themon a regular basis, and avoid their use beyond the purpose or applica-tion for which they were originally designed.

OVERVIEW OF THIS BOOK

We have organized the book as follows. Part I (Chapters 2, 3, and 4)introduces the underpinnings of modern portfolio theory. Part II (Chap-ters 5, 6, 7, and 8) summarizes important developments in the estima-tion of parameters such as expected asset returns and their covariancesthat serve as inputs to the classical portfolio optimization framework.Part III (Chapters 9, 10, and 11) describes the tools necessary to handlethe optimization step of the process. Part IV (Chapters 12, 13, and 14)focuses on applications of the methods described in the previous parts,and outlines new directions in robust portfolio optimization and invest-ment management as a whole.

We start out by describing the classical portfolio theory and theconcepts of diversification in Chapter 2. We introduce the concepts ofefficient sets and efficient frontiers, and discuss the effect of long-onlyconstraints. We also present an alternative framework for optimal deci-sion making in investment—expected utility optimization—and explainits relationship to classical mean-variance optimization.

Chapter 3 extends classical portfolio theory to a more generalmean-risk setting. We cover the most common alternative measures ofrisk that, in some cases, are better suited than variance in describing


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