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Springer Finance Editorial Board M. Avellaneda G. Barone-Adesi M. Broadie M.H.A. Davis E. Derman C. Klüppelberg E. Kopp W. Schachermayer

Springer Finance978-3-540-27904-4... · 2017-08-28 · Schmid B., Credit Risk Pricing Models (2004) Shreve S.E., Stochastic Calculus for Finance I (2004) Shreve S.E., Stochastic Calculus

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Page 1: Springer Finance978-3-540-27904-4... · 2017-08-28 · Schmid B., Credit Risk Pricing Models (2004) Shreve S.E., Stochastic Calculus for Finance I (2004) Shreve S.E., Stochastic Calculus

Springer Finance

Editorial BoardM. AvellanedaG. Barone-AdesiM. BroadieM.H.A. DavisE. DermanC. KlüppelbergE. KoppW. Schachermayer

Page 2: Springer Finance978-3-540-27904-4... · 2017-08-28 · Schmid B., Credit Risk Pricing Models (2004) Shreve S.E., Stochastic Calculus for Finance I (2004) Shreve S.E., Stochastic Calculus

Springer Finance

Springer Finance is a programme of books aimed at students, academics andpractitioners working on increasingly technical approaches to the analysis offinancial markets. It aims to cover a variety of topics, not only mathematical financebut foreign exchanges, term structure, risk management, portfolio theory, equityderivatives, and financial economics.

Ammann M., Credit Risk Valuation: Methods, Models, and Application (2001)Back K., A Course in Derivative Securities: Introduction to Theory and Computation (2005)Barucci E., Financial Markets Theory. Equilibrium, Efficiency and Information (2003)Bielecki T.R. and Rutkowski M., Credit Risk: Modeling, Valuation and Hedging (2002)Bingham N.H. and Kiesel R., Risk-Neutral Valuation: Pricing and Hedging of FinancialDerivatives (1998, 2nd ed. 2004)Brigo D. and Mercurio F., Interest Rate Models: Theory and Practice (2001, 2nd ed. 2006)Buff R., Uncertain Volatility Models-Theory and Application (2002)Carmona R.A. and Tehranchi M.R., Interest Rate Models: an Infinite Dimensional StochasticAnalysis Perspective (2006)Dana R.A. and Jeanblanc M., Financial Markets in Continuous Time (2003)Deboeck G. and Kohonen T. (Editors), Visual Explorations in Finance with Self-OrganizingMaps (1998)Delbaen F. and Schachermayer W., The Mathematics of Arbitrage (2005)Elliott R.J. and Kopp P.E., Mathematics of Financial Markets (1999, 2nd ed. 2005)Fengler M.R., Semiparametric Modeling of Implied Volatility (2005)Geman H., Madan D., Pliska S.R. and Vorst T. (Editors), Mathematical Finance–BachelierCongress 2000 (2001)

Gundlach M., Lehrbass F. (Editors), CreditRisk+ in the Banking Industry (2004)Jondeau E., Financial Modeling Under Non-Gaussian Distributions (2007)Kellerhals B.P., Asset Pricing (2004)Külpmann M., Irrational Exuberance Reconsidered (2004)Kwok Y.-K., Mathematical Models of Financial Derivatives (1998)Malliavin P. and Thalmaier A., Stochastic Calculus of Variations in Mathematical Finance(2005)Meucci A., Risk and Asset Allocation (2005)Pelsser A., Efficient Methods for Valuing Interest Rate Derivatives (2000)Prigent J.-L., Weak Convergence of Financial Markets (2003)Schmid B., Credit Risk Pricing Models (2004)Shreve S.E., Stochastic Calculus for Finance I (2004)Shreve S.E., Stochastic Calculus for Finance II (2004)Yor M., Exponential Functionals of Brownian Motion and Related Processes (2001)Zagst R., Interest-Rate Management (2002)Zhu Y.-L., Wu X., Chern I.-L., Derivative Securities and Difference Methods (2004)Ziegler A., Incomplete Information and Heterogeneous Beliefs in Continuous-time Finance(2003)Ziegler A., A Game Theory Analysis of Options (2004)

Page 3: Springer Finance978-3-540-27904-4... · 2017-08-28 · Schmid B., Credit Risk Pricing Models (2004) Shreve S.E., Stochastic Calculus for Finance I (2004) Shreve S.E., Stochastic Calculus

Attilio Meucci

Risk andAsset Allocation

With 138 Figures

123

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Attilio MeucciLehman Brothers, Inc.745 Seventh AvenueNew York, NY 10019USAE-mail: attilio [email protected]

Mathematics Subject Classification (2000): 15-xx, 46-xx, 62-xx, 65-xx, 90-xx

JEL Classification: C1, C3, C4, C5, C6, C8, G0, G1

Library of Congress Control Number: 2005922398

Corrected 3rd printing 2007

ISBN 978-3-540-22213-2 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction onmicrofilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof ispermitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version,and permission for use must always be obtained from Springer. Violations are liable to prosecution under theGerman Copyright Law.

Springer is a part of Springer Science+Business Media

springer.com

© Springer-Verlag Berlin Heidelberg 2005

Scientific WorkPlace® is a trademark of MacKichan Software, Inc. and is used with permission. MATLAB® is atrademark of The Math Works, Inc. and is used with permission. The Math Works does not warrant the accuracyof the text or exercises in this book. This book’s use or discussion of MATLAB® software or related productsdoes not constitute endorsement or sponsorship by The Math Works of a particular pedagogical approach orparticular use of the MATLAB® software.

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective laws andregulations and therefore free for general use.

Cover design: WMX Design GmbH, HeidelbergCover illustration: courtesy of Linda GaylordTypesetting by the authorProduction: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig

Printed on acid-free paper 41/3180YL - 5 4 3 2 1 0

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to my true love,should she come

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XVAudience and style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XVIIStructure of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XVIIIA guided tour by means of a simplistic example . . . . . . . . . . . . .XIXAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XXVI

Part I The statistics of asset allocation

1 Univariate statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.3 Higher-order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.4 Graphical representations . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Taxonomy of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.3 Cauchy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.4 Student t distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.5 Lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.6 Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.3.7 Empirical distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

2 Multivariate statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.1 Building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2 Factorization of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.1 Marginal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .www1.E Exercises . . .. . . . . . . .

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VIII Contents

2.3 Dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4 Shape summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.1 Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.4.3 Location-dispersion ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . 542.4.4 Higher-order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.5 Dependence summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.5.1 Measures of dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.5.2 Measures of concordance . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.5.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.6 Taxonomy of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.6.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.6.2 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.6.3 Student t distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.6.4 Cauchy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.6.5 Log-distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.6.6 Wishart distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.6.7 Empirical distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.6.8 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.7 Special classes of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.7.1 Elliptical distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.7.2 Stable distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.7.3 In…nitely divisible distributions . . . . . . . . . . . . . . . . . . . . . 98

2.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

3 Modeling the market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.1 The quest for invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.1.1 Equities, commodities, exchange rates . . . . . . . . . . . . . . . . 1053.1.2 Fixed-income market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.1.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.2 Projection of the invariants to the investment horizon . . . . . . . . 1223.3 From invariants to market prices . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.3.1 Raw securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.3.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.4 Dimension reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.4.1 Explicit factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.4.2 Hidden factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.4.3 Explicit vs. hidden factors . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.4.4 Notable examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.4.5 A useful routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.5 Case study: modeling the swap market . . . . . . . . . . . . . . . . . . . . . 1503.5.1 The market invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.5.2 Dimension reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.5.3 The invariants at the investment horizon . . . . . . . . . . . . . 1603.5.4 From invariants to prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .wwwExercises . . .. . . . . . . ..E2

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Contents IX

3.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

Part II Classical asset allocation

4 Estimating the distribution of the market invariants . . . . . . . 1694.1 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

4.1.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724.1.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

4.2 Nonparametric estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1784.2.1 Location, dispersion and hidden factors . . . . . . . . . . . . . . 1814.2.2 Explicit factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.2.3 Kernel estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.3 Maximum likelihood estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.3.1 Location, dispersion and hidden factors . . . . . . . . . . . . . . 1904.3.2 Explicit factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.3.3 The normal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.4 Shrinkage estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.4.1 Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2014.4.2 Dispersion and hidden factors . . . . . . . . . . . . . . . . . . . . . . . 2044.4.3 Explicit factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

4.5 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.5.1 Measures of robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.5.2 Robustness of previously introduced estimators . . . . . . . . 2164.5.3 Robust estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

4.6 Practical tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2234.6.1 Detection of outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2234.6.2 Missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2294.6.3 Weighted estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2324.6.4 Overlapping data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.6.5 Zero-mean invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.6.6 Model-implied estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

5 Evaluating allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2375.1 Investor’s objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2395.2 Stochastic dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435.3 Satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495.4 Certainty-equivalent (expected utility) . . . . . . . . . . . . . . . . . . . . . 260

5.4.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2625.4.2 Building utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2705.4.3 Explicit dependence on allocation . . . . . . . . . . . . . . . . . . . 2745.4.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

5.5 Quantile (value at risk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775.5.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .wwwExercises . . .. . . . . . . ..E3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .www. . .. . . . . . . .Exercises4.E

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X Contents

5.5.2 Explicit dependence on allocation . . . . . . . . . . . . . . . . . . . 2825.5.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

5.6 Coherent indices (expected shortfall) . . . . . . . . . . . . . . . . . . . . . . . 2875.6.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2885.6.2 Building coherent indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 2925.6.3 Explicit dependence on allocation . . . . . . . . . . . . . . . . . . . 2965.6.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

5.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

6 Optimizing allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.1 The general approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

6.1.1 Collecting information on the investor . . . . . . . . . . . . . . . . 3036.1.2 Collecting information on the market . . . . . . . . . . . . . . . . 3056.1.3 Computing the optimal allocation . . . . . . . . . . . . . . . . . . . 306

6.2 Constrained optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3116.2.1 Positive orthants: linear programming . . . . . . . . . . . . . . . . 3136.2.2 Ice-cream cones: second-order cone programming . . . . . . 3136.2.3 Semide…nite cones: semide…nite programming . . . . . . . . . 315

6.3 The mean-variance approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3156.3.1 The geometry of allocation optimization . . . . . . . . . . . . . . 3166.3.2 Dimension reduction: the mean-variance framework . . . . 3196.3.3 Setting up the mean-variance optimization. . . . . . . . . . . . 3206.3.4 Mean-variance in terms of returns . . . . . . . . . . . . . . . . . . . 323

6.4 Analytical solutions of the mean-variance problem . . . . . . . . . . . 3266.4.1 E¢cient frontier with a¢ne constraints . . . . . . . . . . . . . . . 3276.4.2 E¢cient frontier with linear constraints . . . . . . . . . . . . . . 3306.4.3 E¤ects of correlations and other parameters . . . . . . . . . . 3326.4.4 E¤ects of the market dimension . . . . . . . . . . . . . . . . . . . . . 335

6.5 Pitfalls of the mean-variance framework . . . . . . . . . . . . . . . . . . . . 3366.5.1 MV as an approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3366.5.2 MV as an index of satisfaction . . . . . . . . . . . . . . . . . . . . . . 3386.5.3 Quadratic programming and dual formulation . . . . . . . . . 3406.5.4 MV on returns: estimation versus optimization . . . . . . . . 3426.5.5 MV on returns: investment at di¤erent horizons . . . . . . . 343

6.6 Total-return versus benchmark allocation . . . . . . . . . . . . . . . . . . . 3476.7 Case study: allocation in stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

6.7.1 Collecting information on the investor . . . . . . . . . . . . . . . . 3556.7.2 Collecting information on the market . . . . . . . . . . . . . . . . 3556.7.3 Computing the optimal allocation . . . . . . . . . . . . . . . . . . . 357

6.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .wwwExercises . . .. . . . . . . ..E

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7 Estimating the distribution of the market invariants . . . . . . . 3637.1 Bayesian estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

7.1.1 Bayesian posterior distribution . . . . . . . . . . . . . . . . . . . . . . 3647.1.2 Summarizing the posterior distribution . . . . . . . . . . . . . . . 3667.1.3 Computing the posterior distribution . . . . . . . . . . . . . . . . 369

7.2 Location and dispersion parameters . . . . . . . . . . . . . . . . . . . . . . . . 3707.2.1 Computing the posterior distribution . . . . . . . . . . . . . . . . 3707.2.2 Summarizing the posterior distribution . . . . . . . . . . . . . . . 373

7.3 Explicit factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3777.3.1 Computing the posterior distribution . . . . . . . . . . . . . . . . 3777.3.2 Summarizing the posterior distribution . . . . . . . . . . . . . . . 380

7.4 Determining the prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3837.4.1 Allocation-implied parameters . . . . . . . . . . . . . . . . . . . . . . 3857.4.2 Likelihood maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

7.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

8 Evaluating allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3898.1 Allocations as decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

8.1.1 Opportunity cost of a sub-optimal allocation . . . . . . . . . . 3908.1.2 Opportunity cost as function of the market parameters . 3948.1.3 Opportunity cost as loss of an estimator . . . . . . . . . . . . . . 3978.1.4 Evaluation of a generic allocation decision . . . . . . . . . . . . 401

8.2 Prior allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4038.2.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4038.2.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4048.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

8.3 Sample-based allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4078.3.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4078.3.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4088.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

8.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

9 Optimizing allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4179.1 Bayesian allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

9.1.1 Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4199.1.2 Classical-equivalent maximization . . . . . . . . . . . . . . . . . . . 4219.1.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4229.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

9.2 Black-Litterman allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4269.2.1 General de…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4269.2.2 Practicable de…nition: linear expertise on normal

markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4299.2.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4339.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

Part III Accounting for estimation risk

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9.3.1 Practicable de…nition: the mean-variance setting . . . . . . 4389.3.2 General de…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4409.3.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4439.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

9.4 Robust allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4459.4.1 General de…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4459.4.2 Practicable de…nition: the mean-variance setting . . . . . . 4509.4.3

9.5 Robust Bayesian allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.5.1 General de…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4559.5.2 Practicable de…nition: the mean-variance setting . . . . . . 4579.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

9.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www

Part IV Appendices

A Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465A.1 Vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465A.2 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468A.3 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

A.3.1 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470A.3.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

A.4 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472A.4.1 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472A.4.2 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474A.4.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

A.5 Spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475A.5.1 Analytical result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475A.5.2 Geometrical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 478

A.6 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480A.6.1 Useful identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480A.6.2 Tensors and Kronecker product . . . . . . . . . . . . . . . . . . . . . 482A.6.3 The "vec" and "vech" operators . . . . . . . . . . . . . . . . . . . . . 483A.6.4 Matrix calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

B Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487B.1 Vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487B.2 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490B.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

B.3.1 Kernel representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494B.3.2 Unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

B.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496B.5 Expectation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499B.6 Some special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

9.3 Resampled allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

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Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4534

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

List of …gures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

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Preface

In an asset allocation problem the investor, who can be the trader, or thefund manager, or the private investor, seeks the combination of securitiesthat best suit their needs in an uncertain environment. In order to determinethe optimum allocation, the investor needs to model, estimate, assess andmanage uncertainty.The most popular approach to asset allocation is the mean-variance frame-

work pioneered by Markowitz, where the investor aims at maximizing theportfolio’s expected return for a given level of variance and a given set of in-vestment constraints. Under a few assumptions it is possible to estimate themarket parameters that feed the model and then solve the ensuing optimiza-tion problem.More recently, measures of risk such as the value at risk or the expected

shortfall have found supporters in the financial community. These measuresemphasize the potential downside of an allocation more than its potential ben-efits. Therefore, they are better suited to handle asset allocation in modern,highly asymmetrical markets.All of the above approaches are highly intuitive. Paradoxically, this can be

a drawback, in that one is tempted to rush to conclusions or implementations,without pondering the underlying assumptions.For instance, the term "mean-variance" hints at the identification of the

expected value with its sample counterpart, the mean. Sample estimates makesense only if the quantities to estimate are market invariants, i.e. if they displaythe same statistical behavior independently across di erent periods. In equity-like securities the returns are approximately market invariants: this is why themean-variance approach is usually set in terms of returns. Consider insteadan investment in a zero-coupon bond that expires, say, in one month. Thetime series of the past monthly returns of this bond is not useful in estimatingthe expected value and the variance after one month, which are known withcertainty: the returns are not market invariants.

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XVI Preface

Similarly, when an allocation decision is based on the value at risk or onthe expected shortfall, the problem is typically set in terms of the portfolio’sprofit-and-loss, because the "P&L" is approximately an invariant.In general, the investor focuses on a function of his portfolio’s value at the

end of the investment horizon. For instance, the portfolio’s return or profit-and-loss are two such functions which, under very specific circumstances, alsohappen to be market invariants. In more general settings, the investor needsto separate the definition of his objectives, which depend on the portfoliovalue at a given future horizon, from the estimation of the distribution ofthese objectives, which relies on the identification and estimation of someunderlying market invariants.To summarize, in order to solve a generic asset allocation problem we need

to go through the following steps.

Detecting invarianceIn this phase we detect the market invariants, namely those quantities that

display the same behavior through time, allowing us to learn from the past.For equities the invariants are the returns; for bonds the invariants are thechanges in yield to maturity; for vanilla derivatives the invariants are changesin at-the-money-forward implied volatility; etc.

Estimating the marketIn this step we estimate the distribution of the market invariants from a

time series of observations by means of nonparametric estimators, parametricestimators, shrinkage estimators, robust estimators, etc.

Modeling the marketIn this phase we map the distribution of the invariants into the distribu-

tion of the market at a generic time in the future, i.e. into the distribution ofthe prices of the securities for the given investment horizon. This is achievedby suitable generalizations of the "square-root-rule" of volatility propagation.The distribution of the prices at the horizon in turn determines the distribu-tion of the investor’s objective, such as final wealth, or profit and loss, etc.

Defining optimalityIn this step we analyze the investor’s profile. We ascertain the features of a

potential allocation that are more valuable for a specific investor, such as thetrade-o between the expected value and the variance of his objective, or thevalue at risk of his objective, etc.; and we determine the investor’s constraints,such as budget constraints, reluctance to invest in certain assets, etc.

Only after performing separately the above steps can we proceed towardthe final goal:

Computing the optimal allocationAt this stage we determine exactly or in good approximation the allocation

that best suits the investor, namely the allocation that maximizes the valuablefeatures of the investor’s objective(s) given his constraints.

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Preface XVII

Nevertheless, the approach outlined above is sub-optimal: two additionalsteps are needed.

Accounting for estimation riskIt is not clear from the above that an allocation based on one month of data

is less reliable than an allocation based on two years of data. Nevertheless,the e ect of estimation errors on the allocation’s performance is dramatic.Therefore we need to account for estimation risk in the optimization process.Including experienceThe most valuable tool for a successful investor is experience, or a-priori

knowledge of the market. We need to include the investor’s experience in theoptimization process by means of a sound statistical framework.Purpose of this book is to provide a comprehensive treatment of all the

above steps. In order to discuss these steps in full generality and consistentlyfrom the first to the last one we focus on one-period asset allocation.

Audience and style

A few years ago I started teaching computer-based graduate courses on assetallocation and risk management with emphasis on estimation and modelingbecause I realized the utmost importance of these aspects in my experienceas a practitioner in the financial industry. While teaching, I felt the needto provide the students with an accessible, yet detailed and self-contained,reference for the theory behind the above applications. Since I could not findsuch a reference in the literature, I set out to write lecture notes, which overthe years and after many adjustments have turned into this book.In an e ort to make the reader capable of innovating rather than just

following, I sought to analyze the first principles that lead to any given recipe,in addition to explaining how to implement that recipe in practice. Once thosefirst principles have been isolated, the discussion is kept as general as possible:the many applications detailed throughout the text arise as specific instancesof the general discussion.I have tried wherever possible to support intuition with geometrical argu-

ments and practical examples. Heuristic arguments are favored over math-ematical rigor. The mathematical formalism is used only up to (and notbeyond) the point where it eases the comprehension of the subject. TheMATLAB

R°applications downloadable from symmys.com allow the reader to

further visualize the theory and to understand the practical issues behind theapplications.A reader with basic notions of probability and univariate statistics could

learn faster from the book, although this is not a prerequisite. Simple con-cepts of functional analysis are used heuristically throughout the text, but thereader is introduced to them from scratch and absolutely no previous knowl-edge of the subject is assumed. Nevertheless the reader must be familiar withmultivariate calculus and linear algebra.

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XVIII Preface

For the above reasons, this book targets graduate and advanced under-graduate students in economics and finance as well as the new breed of prac-titioners with a background in physics, mathematics, engineering, finance oreconomics who ever increasingly populate the financial districts worldwide.For the students this is a textbook that introduces the problems of the finan-cial industry in a format that is more familiar to them. For the practitioners,this is a comprehensive reference for the theory and the principles underlyingthe recipes they implement on a daily basis.Any feedback on the book is greatly appreciated. Please refer to the website

symmys.com to contact me.

Structure of the work

This work consists of the printed text and of complementary online resources.

• Printed text

The printed text is divided in four parts.Part IIn the first part we present the statistics of asset allocation, namely the

tools necessary to model the market prices at the investment horizon. Chap-ters 1 and 2 introduce the reader to the formalism of financial risk, namelyunivariate and multivariate statistics respectively. In Chapter 3 we discusshow to detect the market invariants and how to map their distribution intothe distribution of the market prices at the investment horizon.Part IIIn the second part we discuss the classical approach to asset allocation.

In Chapter 4 we show how to estimate the distribution of the market invari-ants. In Chapter 5 we define optimality criteria to assess the advantages anddisadvantages of a given allocation, once the distribution of the market isknown. In Chapter 6 we set and solve allocation problems, by maximizing theadvantages of an allocation given the investment constraints.Part IIIIn the third part we present the modern approach to asset allocation,

which accounts for estimation risk and includes the investor’s experience inthe decision process. In Chapter 7 we introduce the Bayesian approach toparameter estimation. In Chapter 8 we update the optimality criteria to assessthe advantages and disadvantages of an allocation when the distribution ofthe market is only known with some approximation. In Chapter 9 we pursueoptimal allocations in the presence of estimation risk, by maximizing theiradvantages according to the newly defined optimality criteria.Part IVThe fourth part consists of two mathematical appendices. In Appendix A

we review some results from linear algebra, geometry and matrix calculus.

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Preface XIX

In Appendix B we hinge on the analogies with linear algebra to introduceheuristically the simple tools of functional analysis that recur throughout themain text.

• Online resources

The online resources consist of software applications and ready-to-printmaterial. They can be downloaded freely from the website symmys.com.Software applicationsThe software applications are in the form of MATLAB programs. These

programs were used to generate the case studies, simulations and figures inthe printed text.Exercise bookThe exercise book documents the above MATLAB programs and discusses

new applications.Technical appendicesIn order to make the book self-contained, the proofs to almost all the

technical results that appear in the printed text are collected in the form ofend-of-chapter appendices. These appendices are not essential to follow thediscussion. However, they are fundamental to a true understanding of thesubjects to which they refer. Nevertheless, if included in the printed text,these appendices would have made the size of the book unmanageable.The notation in the printed text, say, "Appendix www.2.4" refers to the

technical appendix to Chapter 2, Section 4, which is located on the internet.On the other hand the notation, say, "Appendix B.3" refers to the mathemat-ical Appendix B, Section 3, at the end of the book.

A guided tour by means of a simplistic example

To better clarify the content of each chapter in the main text we present amore detailed overview, supported by an oversimplified example which, westress, does not represent a real model.Part IA portfolio at a given future horizon is modeled as a random variable

and is represented by a univariate distribution: in Chapter 1 we review uni-variate statistics. We introduce the representations of the distribution of ageneric random variable , i.e. the probability density function, the cumula-tive distribution function, the characteristic function and the quantile, and wediscuss expected value, variance and other parameters of shape. We presenta graphical interpretation of the location and dispersion properties of a uni-variate distribution and we discuss a few parametric distributions useful inapplications.

For example, we learn what it means that a variable is normally dis-tributed:

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XX Preface

2¢, (0.1)

where is the expected value and 2 is the variance.

The market consists of securities, whose prices at a given future horizoncan be modeled as a multivariate random variable: in Chapter 2 we discussmultivariate statistics. We introduce the representations of the distributionof a multivariate random variable X, namely the joint probability densityfunction, the cumulative distribution function and the characteristic function.We analyze the relationships between the di erent entries of X: the marginal-copula factorization, as well as the concepts of dependence and of conditionaldistribution.We discuss expected value, mode and other multivariate parameters of

location; and covariance, modal dispersion and other multivariate parametersof dispersion. We present a graphical interpretation of location and dispersionin terms of ellipsoids and the link between this interpretation and principalcomponent analysis.We discuss parameters that summarize the co-movements of one entry of

X with another: we introduce the concept of correlation, as well as alternativemeasures of concordance. We analyze the multivariate generalization of thedistributions presented in Chapter 1, including the Wishart and the matrix-variate Student distributions, useful in Bayesian analysis, as well as verygeneral log-distributions, useful to model prices. Finally we discuss specialclasses of distributions that play special roles in applications.

For example, we learn what it means that two variables X ( 1 2)0 are

normally distributed:X N( ) , (0.2)

where ( 1 2)0 is the vector of the expected values and where the covari-

ance matrix is the identity matrix, i.e. I. We represent this variable asa unit circle centered in : the radius represents the two eigenvalues and thereference axes represent the two eigenvectors. As it turns out, the normal dis-tribution (0 2) belongs to the special elliptical, stable and infinitely divisibleclasses.

In Chapter 3 we model the market. The market is represented by a setof securities that at time trade at the price P . The investment decisionis made at the time and the investor is interested in the distribution ofthe prices P + at a determined future investment horizon . Modeling themarket consists of three steps. First we need to identify the invariants hiddenbehind the market data, i.e. those random variables X that are distributedidentically and independently across time.

For example suppose that we detect as invariants the changes in price:

X P P , (0.3)

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Preface XXI

where the estimation horizon e is one week.Secondly, we have to associate a meaningful parametric distribution to

these invariants

For example suppose that the normal distribution (0 2) with the identityas covariance is a suitable parametric model for the weekly changes in prices:

X N( I) . (0.4)

In this case the market parameters, still to be determined, are the entries of.

Finally, we have to work out the distribution of the market, i.e. the pricesP + at the generic horizon , given the distribution of the invariants Xat the specific horizon e. This step is fundamental when we first estimateparameters at a given horizon and then solve allocation problems at a di erenthorizon.

For example, suppose that the current market prices of all the securitiesare normalized to one unit of currency, i.e. P 1, and that the invest-ment horizon is one month, i.e. four weeks. Then, from (0 3) and (0 4) thedistribution of the market is normal with the following parameters:

P + N(m 4I) , (0.5)

wherem 1+ 4 . (0.6)

In a market of many securities the actual dimension of risk in the mar-ket is often much lower than the number of securities: therefore we discussdimension-reduction techniques such as regression analysis and principal com-ponent analysis and their geometrical interpretation in terms of the location-dispersion ellipsoid. We conclude with a detailed case study, which covers allthe steps involved in modeling the swap market: the detection of the invari-ants; the "level-slope-hump" PCA approach to dimension reduction of theswap curve invariants, along with its continuum-limit interpretation in termsof frequencies; and the roll-down, duration and convexity approximation ofthe swap market.Part IIIn the first part of the book we set the statistical background necessary

to formalize allocation problems. In the second part we discuss the classicalapproach to solve these problems, which consists of three steps: estimatingthe market distribution, evaluating potential portfolios of securities and op-timizing those portfolios according to the previously introduced evaluationcriteria.

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XXII Preface

In Chapter 4 we estimate from empirical observations the distribution ofthe market invariants. An estimator is a function that associates a number,the estimate, with the information that is available when the investmentdecision in made. This information is typically represented by the time seriesof the past observations of the market invariants.

For example, we can estimate the value of the market parameter in (0 4)by means of the sample mean:

{x1 x } 7 b 1 X=1

x , (0.7)

where we dropped the estimation interval from the notation.

We discuss general rules to evaluate the quality of an estimator. The mostimportant feature of an estimator is its replicability, which guarantees that asuccessful estimation does not occur by chance. An estimator’s replicability ismeasured by the distribution of its loss and is summarized by error, bias andine ciency. Then we introduce di erent estimators for di erent situations:nonparametric estimators, suitable in the case of a very large number of ob-servations; maximum likelihood estimators under quite general non-normalassumptions, suitable when the parametric shape of the invariants’ distribu-tion is known; shrinkage estimators, which perform better when the amountof data available is limited; robust estimators, which the statistician shoulduse when he is not comfortable with a given parametric specification of themarket invariants. Throughout the analysis we provide the geometrical inter-pretation of the above estimators. We conclude with practical tips to deal,among other problems, with outliers detection and missing values in the timeseries.In Chapter 5 we show how to evaluate an allocation. The investor can al-

locate his money in the market to form a portfolio of securities. Therefore, theallocation decision is defined by a vector whose entries determine the num-ber of units (e.g. shares) of the respective security that are being purchasedat the investment time . The investor focuses on his primary objective, arandom variable whose distribution depends on the allocation and the marketparameters: di erent objectives corresponds to di erent investment priori-ties, such as benchmark allocation, daily trading (profits and losses), financialplanning, etc.

For example, assume that the investor’s objective is final wealth. If themarket is distributed as in (0 5) the objective is normally distributed:

0P + N¡ 0m 2

¢, (0.8)

where m is given in (0 6) and 2 is a simple function of the allocation.

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Preface XXIII

Evaluating an allocation corresponds to assessing the advantages and dis-advantages of the distribution of the respective objective. We start consideringstochastic dominance, a criterion to compare distributions globally: neverthe-less stochastic dominance does not necessarily give rise to a ranking of thepotential allocations. Therefore we define indices of satisfaction, i.e. functionsof the allocation and the market parameters that measure the extent to whichan investor appreciates the objective ensuing from a given allocation.

For example, satisfaction can be measured by the expected value of finalwealth: a portfolio with high expected value elicits a high level of satisfaction.In this case from (0 6) and (0 8) the index of satisfaction is the followingfunction of the allocation and of the market parameters:

( ) 7 E { } = 0 (1+ 4 ) . (0.9)

We discuss the general properties that indices of satisfaction can or shoulddisplay. Then we focus on three broad classes of such indices: the certainty-equivalent, related to expected utility and prospect theory; the quantile ofthe objective, closely related to the concept of value at risk; and coherentand spectral measures of satisfaction, closely related to the concept of ex-pected shortfall. We discuss how to build these indices and we analyze theirdependence on the underlying allocation. We tackle a few computational is-sues, such as the Arrow-Pratt approximation, the gamma approximation, theCornish-Fisher approximation, and the extreme value theory approximation.In Chapter 6 we pursue the optimal allocation for a generic investor. For-

mally, this corresponds to maximizing the investor’s satisfaction while keepinginto account his constraints. We discuss the allocation problems that can besolved e ciently at least numerically, namely convex programming and inparticular semidefinite and second-order cone programing problems.

For example, suppose that transaction costs are zero and that the investorhas a budget constraint of one unit of currency and can purchase only positiveamounts of any security. Assume that the market consists of only two securi-ties. Given the current market prices, from (0 9) the investor’s optimizationproblem reads:

argmax1+ 2=1

0

0 (1+ 4b) , (0.10)

where b are the estimated market parameters (0 7). This is a linear program-ming problem, a special case of cone programing. The solution is a 100%investment in the security with the largest estimated expected value. Assum-ing for instance that this is the first security, we obtain:

b1 b2 1 1 2 0. (0.11)

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XXIV Preface

In real problems it not possible to compute the exact solution to an allo-cation optimization. Nevertheless it is possible to obtain a good approximatesolution by means of a two-step approach. The core of this approach is themean-variance optimization, which we present in a general context in termsmarket prices, instead of the more common, yet more restrictive, representa-tion in terms of returns. Under fairly standard hypotheses, the computationof the mean-variance frontier is a quadratic programming problem. In specialcases we can even compute analytical solutions, which provide insight intothe e ect of the market on the allocation in more general contexts: for ex-ample, we prove wrong the common belief that uncorrelated markets providebetter investment opportunities than highly correlated markets. We analyzethoroughly the problem of managing assets against a benchmark, which is theexplicit task of a fund manager and, as it turns out, the implicit objective ofall investors. We discuss the pitfalls of a superficial approach to the mean-variance problem, such as the confusion between compounded returns andlinear returns which gives rise to distortions in the final allocation. Finally,we present a case study that reviews all the steps that lead to the optimalallocation.Part IIIIn the classical approach to asset allocation discussed in the second part

we implicitly assumed that the distribution of the market, once estimated,is known. Nevertheless, such distribution is estimated with some error. As aresult, any allocation implemented cannot be truly optimal and the truly opti-mal allocation cannot be implemented. More importantly, since the optimiza-tion process is extremely sensitive to the input parameters, the sub-optimalitydue to estimation risk can be dramatic.

The parameter b in the optimization (0 10) is only an estimate of the trueparameter that defines the distribution of the market (0 4). The true expectedvalue of the second security could be larger than the first one, as opposed towhat stated in (0 11). In this case the truly optimal allocation would read:

1 2 1 0 2 1. (0.12)

This allocation is dramatically di erent from the allocation (0 11), which wasimplemented.

As a consequence, portfolio managers, traders and professional investors ina broader sense mistrust the "optimal" allocations ensuing from the classicalapproach and prefer to resort to their experience. In the third part of the bookwe present a systematic approach to tackle estimation risk, which also includeswithin a sound statistical framework the investor’s experience or models.Following the guidelines of the classical approach, in order to determine

the optimal allocation in the presence of estimation risk we need to introducea new approach to estimate the market distribution, update the evaluation

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Preface XXV

criteria for potential portfolios of securities and optimize those portfolios ac-cording to the newly introduced evaluation criteria.In Chapter 7 we introduce the Bayesian approach to estimation. In this

context, estimators are not numbers: instead, they are random variables mod-eled by their posterior distribution, which includes the investor’s experienceor prior beliefs. A Bayesian estimator defines naturally a classical-equivalentestimator and an uncertainty region.

For example, the Bayesian posterior, counterpart of the classical estimator(0 7), could be a normal random variable:

b N

µ1

2b + 1

2 0 I

¶, (0.13)

where 0 is the price change that the investor expects to take place. Thenthe classical-equivalent estimator is an average of the prior and the sampleestimator: bce 1

2b + 1

2 0; (0.14)

and the uncertainty region is a unit circle centered in bce:E

©such that ( bce)0 ( bce) 1

ª. (0.15)

Since it is di cult for the investor to input prior beliefs directly in themodel, we discuss how to input them implicitly in terms of ideal allocations.In Chapter 8 we introduce criteria to evaluate the sub-optimality of a

generic allocation. This process parallels the evaluation of an estimator. Theestimator’s loss becomes in this context the given allocation’s opportunitycost, i.e. a positive random variable which represents the di erence betweenthe satisfaction provided by the true, yet unattainable, optimal allocation andthe satisfaction provided by the given allocation.

In our example, from (0 9) the opportunity cost of the sub-optimal allo-cation (0 10) reads:

OC( ) = (1+ 4 )0 (1+ 4b)0 , (0.16)

where is the truly optimal allocation (0 12).

We analyze the opportunity cost of two extreme approaches to allocation:at one extreme the prior allocation, which completely disregards any informa-tion from the market, relying only on prior beliefs; at the other extreme thesample-based allocation, where the unknown market parameters are replacedby naive estimates.In Chapter 9 we pursue optimal allocations in the presence of estimation

risk, namely allocations whose opportunity cost is minimal. We present allo-

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XXVI Preface

cations based on Bayes’ rule, such as the classical-equivalent allocation andthe Black-Litterman approach. Next we present the resampling technique byMichaud. Then we discuss robust allocations, which aim at minimizing themaximum possible opportunity cost over a given set of potential market pa-rameters. Finally, we present robust Bayesian allocations, where the set ofpotential market parameters is defined naturally in terms of the uncertaintyset of the posterior distribution.

In our example, the sub-optimal allocation (0 10) is replaced by the fol-lowing robust Bayesian allocation:

argmin1+ 2=1

0

½max

EOC( )

¾, (0.17)

where the opportunity cost is defined in (0 16) and the uncertainty set isdefined in (0 15). The solution to this problem is a balanced allocation where,unlike in (0 11), both securities are present in positive amounts.

In general it is not possible to compute exactly the optimal allocations.Therefore, as in the classical approach to asset allocation, we resort to thetwo-step mean-variance setting to solve real problems.

Acknowledgments

I wish to thank Carlo Favero, Carlo Giannini, John Hulpke, Alfredo Pastorand Eduardo Rossi, who invited me to teach finance at their institutions,thereby motivating me to write lecture notes for my courses.A few people provided precious feedback on di erent parts of the draft

at di erent stages in its development, in particular Davide DiGennaro, LucaDona’, Alberto Elices, Silverio Foresi, Davide Guzzetti, Philip Stark, DirkTasche, Kostas Tryantafyllapoulos and an anonymous referee. FrancescoCorielli and Gianluca Fusai furnished insightful comments and suggested newmaterial for the book during many pleasant conversations throughout the lastfew years.I am indebted to Catriona Byrne, Susanne Denskus and Stefanie Zoeller

at Springer for their active support; to EDV-Beratung Frank Herweg for care-fully correcting the proofs; and to George Pearson and John MacKendrick atMacKichan Software, Inc. for helping me discover the capabilities of ScientificWorkPlace

R°, which I used to write this book.

A special thank is due to Jenifer Shiu, for her support during the last yearof writing.

Greenwich, January 2005,Attilio Meucci