Risk and Asset Allocation

Risk and Asset AllocationSpringer financeSpringer Finance is a programme of books aimed at students, academicsand practitioners working on increasingly technical approaches to theanalysis of financial markets. It aims to cover a variety of topics, not onlymathematical finance but foreign exchanges, term structure, riskmanagement, portfolio theory, equity derivatives, and financial economicsK. Back, A Course in Derivative Securities: Introduction to Theory and 2oo1)M. Ammann, Credit Risk Valuation: Methods, Models, and ApplicationComputation(2005E. Barucci, Financial Markets Theory. Equilibrium, Efficiency and Information(2003)T.R. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging(2002)N H. Bingham and R Kiesel, Risk-Neutral Valuation: Pricing and Hedging of FinancialDerivatives(1998, 2nd ed 2004)D. Brigo and E. Mercurio, Interest Rate Models: Theory and Practice(2001)R. Buff, Uncertain Volatility Models-Theory and Application(2002)R.A. Dana and M. Jeanblanc, Financial Markets in Continuous Time(2002)G. Deboeck and T Kohonen(Editors), Visual Explorations in Finance withSelf-Organizing Maps(1998R. Elliott and P.E. Kopp, Mathematics of Financial Markets(1999, 2nd ed. 2005)H Geman, D. Madan, S.R. Pliska and T. Vorst(Editors), Mathematical FinanceBachelier Congress 2000(2001M Gundlach, F. Lehrbass(Editors), CreditRisk in the Banking Industry (2004B P Kellerhals, Asset Pricing(2004)Y.-K. Kwok, Mathematical Models of Financial Derivatives(1998)M. Kulpmann, Irrational Exuberance Reconsidered (200P Malliavin and a. Thalmaier, Stochastic Calculus of variations in MathematicalFinance(2005)A Meucci, Risk and Asset Allocation(2005)A Pelsser, Efficient Methods for Valuing Interest Rate Derivatives(2000)J.-L. Prigent, Weak Convergence of Financial Markets(2003)B. Schmid, Credit Risk Pricing Models(2004)S.E. Shreve, Stochastic Calculus for Finance I(200S.E. Shreve, Stochastic Calculus for Finance II (2004)M. Yor, Exponential Functionals of Brownian Motion and Related Processes(2001)RZagst, Interest-Rate Management(2002)Y.-L. Zhu, X Wu, I.-L. Chern, Derivative Securities and Difference Methods (2004)A. Ziegler, incomplete Information and Heterogeneous Beliefs in Continuous-timeFinance(2003)A Ziegler, A Game Theory Analysis of Options(2004)Attilio meucciRisk andAsset allocationWith 141 FiguresSpringerAttilio meucciLehman brothers inc745 Seventh AvenueNew York, NY 10019USAe-mail:attilio_meucci@symmys.comMathematics Subject Classification(2000): 15-xx, 46-XX,62-XX,65-XX,90-xxJEL Classification: Cl, C3, C4, C5, C6, C8, Go, GlLibrary of Congress Control Number: 2005922398ISBN-10 3-540-22213-8 Springer-Verlag Berlin Heidelberg New YorkISBN-13 978-3-540-22213-2 Springer-Verlag Berlin Heidelberg New YorkThis work is subject to copyright. All rights are reserved, whether the whole or part of the materiald, specifically the rightsof illustratiobroadcasting, reproduction on microfilm or in any other way, and storage in data banksDuplication of this publication or parts thereof is permitted only under the provisions of theGerman Copyright Law of September 9, 1965, in its current version, and permission for use mustalways be obtained from Springer-Verlag. Violations are liable to prosecution under the germanSpringer is a part of Springer Science+ Business Mediaspringeronline.comSpringer-Verlag Berlin Heidelberg 2005Printed in The netherlandsScientific WorkPlace@ is a trademark of MacKichan Software, Inc. and is used with permissionMATLABe is a trademark of The Math Works, Inc. and is used with permission. The Math Worksdoes not warrant the accuracy of the text or exercises in this book. This book's use or discussion ofMATLABe software or related products does not constitute endorsement or sponsorship by theMath Works of a particular pedagogical approach or particular use of the MATlAB@ softwareThe use of general descriptive names, registered names, trademarks, etc in this publication doesnot imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general useCover design: design e production, HeidelbergCover illustration: courtesy of linda gaylordTypesetting by the authorPrinted on acid-free paper 41/sz-543210to my true loveshould she comeContentsPrefaceAudnd styleXVIIStructure of the workXVIIIa guided tour by means of a simplistic exampleXIXAcknowledgmentsXXVIPart i The statistics of asset allocation1 Univariate statistics1.1 Building blocks1.2 Summary statistics3391.2.1 Location1.2.2 Dispersion111.2.3 Higher-order statistics141.2.4 Graphical representations1. 3 Taxonomy of distributions161.3.1 Uniform distribution1.3.2 Normal distribution181.3.3 Cauchy distribution201.3.4 Student t distribution221.3.5 Lognormal distribution241.3.6 Gamma distribution261.3.7 Empirical distribution1.T Technical appendixWWW1.E ExercisesWWW2 Multivariate statistics2.1 Building blocks342.2 Factorization of a distribution382.2.1 Marginal distribution382.2.2 Copulas40VIII Contents2.3 Dependence452.4 Shape summary statistics.482. 4.1 Location482.4.2 Dispersion502.4.3 Location-dispersion ellipsoid542.4.4 Higher-order statistics..572.5 Dependence summary statistics592.5.1 Measures of dependence...592.5.2 Measures of concordance642.5.3 Correlation.672.6 Taxonomy of distributions.702.6.1 Uniform distribution702.6.2 Normal distribution722.6.3 Student t distribution772.6.4 Cauchy distribution2.6.5 Log.2.6.6 Wishart distribution842.6.7 Empirical distribution872.6.8 Order statistics.892.7 Special classes of distributions2.7.1 Elliptical distributions2.7.2 Stable distributions962.7.3 Infinitely divisible distributions982.T Technical appendix2.E ExercisesWWWModeling the market1013.1 The quest for invariance1033.1.1 Equities, commodities, exchange rates...1053.1.2 Fixed-income market3.1.3 Derivatives.1143.2 Projection of the invariants to the investment horizon1223.3 From invariants to market prices..1263.3.1 Raw securities1263.3.2 Derivatives1293.4 Dimension reduction..1313.4.1 Explicit factors1333.4.2 Hidden factors1383.4.3 Explicit vs. hidden factors.1433.4.4 Notable examples.1453.4.5 A useful routine...1473.5 Case study: modeling the swap market1503.5.1 The market invariants1503.5.2 Dimension reduction1513.5.3 The invariants at the investment horizon3.5.4 From invariants to prices162Contents3.T Technical appendixWwW3.E ExercisesWWWPart ii Classical asset allocation4 Estimating the distribution of the market invariants1694.1 Estimators1714.1.1 Definition1724.1.2 Evaluation.......,,,.1734.2 Nonparametric estimators1784.2.1 Location, dispersion and hidden factors1814.2.2 Explicit fact1844.2.3 Kernel estimators1854.3 Maximum likelihood estimators1864.3. 1 Location, dispersion and hidden factors1904.3.2 Explicit factors1924.3.3 The normal case1934.4 Shrinkage estimators2004.4.1 Location..2014.4.2 Dispersion and hidden factors2044.4.3 Explicit facto2094.5 Robustness2094.5.1 Measures of robustness2114.5. 2 Robustness of previously introduced estimators2164.5.3 Robust estimators2214.6 Practical til2234.6. 1 Detection of outliers2234.6.2 Missing data224.6.3 Weighted estimates2324.6.4 Overlapping2344.6.5 Zero-mean invariants2344.6.6 Model-implied estimation2354.T Technical appendixWWW4.EEⅹ excisesWWW5 Evaluating allocations..2375.1 Investor's objectives2395.2 Stochastic dominance2435.3 Satisfaction5.4 Certainty-equivalent(expected utility)2605.4.1 Properties2625.4.2 Building utility functions2705.4.3 Explicit dependence on allocation2745.4.4 Sensitivity analysis2765.5 Quantile(value at risk2775.5.1 Properties278Contents5.5.2 Explicit dependence on allocation2825.5.3 Sensitivity analysis....2855.6 Coherent indices(expected shortfall2875.6.1 Properties5.6.2 Building coherent indices2925.6.3 Explicit dependence on allocation2965.6.4 Sensitivity analysis5.T Technical appendixwwW5.E ExercisesWww6 Optimizing allocations..3016.1 The general approach...3026.1.1 Collecting information on the investor3036.1.2 Collecting information on the market3056.1.3 Computing the optimal allocation6.2 Constrained optimization3116.2.1 Positive orthants: linear programming3136.2.2 Ice-cream cones: second-order cone programming3136.2.3 Semidefinite cones: semidefinite programming3156.3 The mean-variance approach6.3.1 The geometry of allocation optimization6.3.2 Dimension reduction the mean-variance framework3163196.3.3 Setting up the mean-variance optimization3206.3.4 Mean-variance in terms of returns326.4 Analvtical solutions of the mean-variance problem3266.4.1 Efficient frontier with affine constraints3276.4.2 Efficient frontier with linear constraints3306.4.3 Effects of correlations and other parameters3326.4.4 Effects of the market dimension3356.5 Pitfalls of the mean-variance framework3366.5.1 MV as an approximation..3366.5.2 MV as an index of satisfaction3386.5. 3 Quadratic programming and dual formulation3406.5.4 MV on returns: estimation versus optimization3426.5.5 MV on returns: investment at different horizons3436.6 Total-return versus benchmark allocation3476.7 Case study: allocation in stocks3546.7.1 Collecting information on the investor3556.7.2 Collecting information on the market3556.7.3 Computing the optimal allocation3576.T Technical appendi6.E ExercisesWWW