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Phd Dissertation ©Copyright 2010 Minfeng Zhu Portfolio Optimization with Tail Risk Measures and Non-Normal Returns Minfeng Zhu A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2010 Program Authorized to Offer Degree: Department of Statistics University of Washington Graduate School This is to certify that I have examined this copy of a doctoral dissertation by Minfeng Zhu and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Chair of the Supervisory Committee: R. Douglas Martin Reading Committee: R. Douglas Martin Andrew Clark Eric W Zivot Date: In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of the dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to ProQuest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106- 1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.” Signature ________________________ Date ____________________________ University of Washington Abstract Portfolio Optimization with Tail Risk Measures and Non-Normal Returns Minfeng Zhu Chair of the Supervisory Committee: Professor R. Douglas Martin Department of Statistics The traditional Markowitz mean-variance portfolio optimization theory uses volatility as the sole measure of risk. However, volatility is flawed both intuitively and theoretically: being symmetric it does not differentiate between gains and losses; it does not satisfy an expected utility maximization rationale except under unrealistic assumptions and is not a coherent risk measure. The past decade has seen considerable research on better risk measures, with the two tail risk measures Value-at-Risk (VaR) and Expected Tail Loss (ETL) being the main contenders, as well as research on modeling skewness and fat-tails that are prevalent in financial return distributions. There are two main approaches to the latter problem: (a) constructing modified VaR (MVaR) and modified ETL (METL) using Cornish-Fisher asymptotic expansions to provide non- parametric skewness and kurtosis corrections, and (b) fitting a skewed and fat-tailed multivariate parametric distribution to portfolio returns and optimizing the portfolio using ETL based on Monte Carlo simulations from the fitted distribution. It is an open question how MVaR and METL compare with one another and with empirical VaR and ETL, and also how much improvement can be obtained in fitting parametric distributions. In this dissertation, we first show that MVaR and METL are very sensitive to outliers, sometimes rendering complete failure of a portfolio. Then we propose new robust skewness and kurtosis estimates, study their statistical behavior and that of the resulting robust MVaR and METL through the use of influence functions, and show through extensive empirical studies that robust MVaR and METL can effectively curb the failure of the original estimates. We use the same experimental approach to show that the simple empirical ETL optimization yields portfolio performance essentially equivalent to that of the much more complex method of fitting multivariate fat-tailed skewed distributions. Finally we address the following important problem: VaR and ETL based portfolio optimization do not have expected utility maximization rationales. Thus we establish a method of designing coherent spectral risk measures based on non-satiated risk-averse utility functions. We show that the resulting risk measures satisfy second order stochastic dominance and their empirical portfolio performances are slightly improved over ETL. Table of Contents Page List of Figures ........................................................................................................................................... v List of Tables ........................................................................................................................................... ix 1. Chapter 1 Introduction .................................................................................................................... 1 1.1 Portfolio Optimization Theories .................................................................................................... 2 1.1.1 Mean‐Risk Optimization ........................................................................................................... 2 1.1.2 Expected Utility Maximization .................................................................................................. 5 1.1.3 Second‐Order Stochastic Dominance ........................................................................................ 7 1.2 Risk Measures ............................................................................................................................. 11 1.2.1 Value‐at‐Risk (VaR) and Modified VaR ................................................................................... 12 1.2.2 Expected‐Tail‐Loss (ETL) and Coherent Risk Measures ........................................................... 14 1.2.3 Lower Partial Moment (LPM) ................................................................................................. 16 1.2.4 Spectral Risk Measures (SRM) and SMCR ............................................................................... 17 1.3 Estimating Risk Measures with Higher Moments or Parametric Distributions ........................... 18 1.4 Remainder of Dissertation .......................................................................................................... 20 2. Chapter 2 Influence Functions and Robust Skewness and Kurtosis ................................................ 22 2.1 Skewness and Kurtosis and Robust Versions .............................................................................. 22 2.2 Influence Functions for Skewness and Kurtosis .......................................................................... 29 2.2.1 Classical Skewness and Kurtosis ............................................................................................. 30 2.2.2 Trimmed Skewness and Kurtosis ............................................................................................. 36 2.2.3 Quantile Based Skewness and Kurtosis ................................................................................... 45 2.2.4 Influence Functions of Skewness and Kurtosis under Skewed and Fat‐Tailed Distributions ... 48 2.2.5 Asymptotic Variance ............................................................................................................... 58 i 2.2.6 Finite Sample Variance ........................................................................................................... 60 2.2.7 Efficiency of Estimators ........................................................................................................... 64 2.2.8 Hedge Fund Examples ............................................................................................................. 68 3. Chapter 3 Portfolio Optimization with Modified VaR .................................................................... 72 3.1 Modified VaR Influence Functions .............................................................................................. 74 3.2 Finite Sample Variance ................................................................................................................ 80 3.3 Portfolio Optimization with Modified VaR .................................................................................. 85 3.3.1 The Basic Portfolio Optimization Problem .............................................................................. 85 3.3.2 Empirical Definitions of Risk Measures ................................................................................... 87 3.3.3 Optimization Methods ............................................................................................................ 89 3.4 Data and Experiment Settings ..................................................................................................... 92 3.4.1 Data ........................................................................................................................................ 92 3.4.2 Empirical Statistics on Hedge Fund Data ................................................................................ 95 3.5 Results ....................................................................................................................................... 101 3.5.1 Performance and Risk Measures .......................................................................................... 101 3.5.2 Experiments with Long‐Only Constraints .............................................................................. 103 3.5.3 Experiments with Upper Bounded Constraints ..................................................................... 114 3.5.4 Conclusions ........................................................................................................................... 119 4. Chapter 4 Portfolio Optimization with Modified ETL ..................................................................
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