C Copyright 2012 Yun Zhang

c Copyright 2012 Yun Zhang ETG-ETL Portfolio Optimization Yun Zhang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2012 Reading Committee: James V. Burke, Chair R Douglas Martin Ka Kit Tung Hong Qian Program Authorized to Offer Degree: Applied Mathematics University of Washington Abstract ETG-ETL Portfolio Optimization Yun Zhang Chair of the Supervisory Committee: Professor James V. Burke Department of Mathematics Modern Portfolio Theory dates back to 1950s, when Markowitz proposed mean- variance portfolio optimization to construct portfolios. It provided a systematic approach to determine portfolio allocation when one is facing complicated risk structure that not only exists for individual assets but also across different assets. Since then there has been much research exploring better ways to quantify risk. In particular, asymmetric risk measures including the more recent downside risk measures. Here we use expected tail loss (ETL) as the risk measure which is a coherent risk measure, and define a reward measure, expected tail gain (ETG), to measure the upside return. We formulate the portfolio optimization problem using these two measures and developed an iterative algorithm to find its optimal solution. TABLE OF CONTENTS Page List of Tables . iii List of Figures . iv Chapter 1: Introduction . 1 Chapter 2: Risk, Reward and Performance Measures . 2 2.1 Risk Measures . 2 2.2 Reward Measures . 7 2.3 Performance Measures (Ratios) . 9 Chapter 3: Portfolio Optimization . 13 3.1 Optimization Review . 14 3.2 Mean-Variance Portfolio Optimization . 21 3.3 Mean-ETL Portfolio Optimization . 24 3.4 ETG-ETL Portfolio Optimization . 37 Chapter 4: An Algorithm for ETG-ETL Portfolio Optimization . 42 4.1 Support function expressions for ETGd and ETLd . 43 4.2 Subdifferential of ETGd β(w)........................ 46 4.3 Directions of ascent for ETGd β(w) when w is in the relative interior . 50 4.4 Steepest ascent direction of ETGd β(w) when w is on the relative boundary 54 4.5 Backtracking line search . 59 4.6 The algorithm scheme . 67 Chapter 5: Numerical Experiments . 71 5.1 An numerical example using three assets . 71 5.2 Convergence analysis . 77 i 5.3 A more realistic example . 80 5.4 Future Research . 83 Bibliography . 85 ii LIST OF TABLES Table Number Page 5.1 Optimality analysis for case 1. 75 5.2 Optimality analysis for case 2. 75 iii LIST OF FIGURES Figure Number Page 2.1 Illustration of VaR and ETL........................ 6 2.2 Illustration of ETG............................. 8 4.1 Illustration of I>(w; ~ k), I=(w; ~ k) and I≥(w; ~ k) . 48 4.2 Illustration of reachBD .......................... 60 4.3 Illustration of backTrack ......................... 61 4.4 Illustration of Bisection method for reachBD . 63 4.5 Illustration of Secant method for reachBD . 67 5.1 Returns time series for AMZN, ORCL and LLTC . 72 5.2 QQ-plots and kenel density estimations for AMZN, ORCL and LLTC 73 5.3 Searching paths for case 1. 74 5.4 Searching paths for case 2. 76 5.5 Weights for case 1 and case 2. 77 5.6 Boxplot of the max ascent values v.s. tolerance. 78 5.7 CPU time v.s. tolerance. 79 5.8 $1 portfolio cumulative return under scenario 1. 80 5.9 $1 portfolio cumulative return under scenario 2. 81 5.10 $1 portfolio cumulative return under scenario 3. 82 5.11 $1 portfolio cumulative return under scenario 4. 83 5.12 $1 portfolio cumulative return under scenario 5. 83 iv ACKNOWLEDGMENTS I would like to express my sincere appreciation to University of Washington, where I have had the opportunity to work with Professor James V. Burke from the Depart- ment of Mathematics, Professor Douglas Martin from the Department of Statistics, Professor Ka-Kit Tung and Professor Hong Qian from the Department of Applied Mathematics. Without their support, help and guidance, the accomplishment of the research represented in this thesis would not have been possible. 1 1 Chapter 1 INTRODUCTION A portfolio is a combination of individual investment assets. It has two primary characteristics: reward and risk. The higher the reward is, the higher the risk, and vice versa. Portfolio optimization solves the problem of allocating weight to each member asset such that the portfolio's reward is maximized while its risk is less than some given value, or the portfolio's risk is minimized while its reward is greater than or equal to some given value. This general approach is referred to as Modern Portfolio Theory (MPT), originated by Markowitz who won the Nobel Prize for his pioneering work in this area. MPT provides a systematic way to determine portfolio allocation when one is facing complicated risk structure that not only exists for individual asset but also across different assets. There has been much research exploring better ways to quantify risk since the establishment of MPT. From the earliest symmetric risk measures to later asymmetric risk that focus on the down-side risk. Here we use Expected Tail Loss (ETL), or Conditional Value-at-Risk (CVaR), as the risk measure, and defined a reward measure, Expected Tail Gain (ETG), in a manner similar to ETL. The resulting portfolio optimization problem turns out to be a concave programming problem. This problem can be converted into a mixed integer optimization problem, but in this form the problem requires an exhaustive search over all possible combinatorial options and is thus prohibitively time consuming even for small portfolios. We suggest a heuristic algorithm that is time efficient and provides excellent results based on empirical data for portfolios containing hundreds of stocks. 2 Chapter 2 RISK, REWARD AND PERFORMANCE MEASURES 2.1 Risk Measures In finance, risk is the uncertainty that an investment's actual return will be different from expected. There are different types of risk associated with different financial assets, such as interest rate risk, credit risk, liquidity risk, currency risk etc. This thesis specifically focuses on equity risk, i.e., the uncertainty of future prices for equity securities. For a single asset, let Pt denote its price at t, and its simple return is defined as Pt − Pt−1 Rt = ; (2.1.1) Pt−1 which is also called holding period return. Its log return is defined as Pt rt = log : (2.1.2) Pt−1 These formulas assume the price has been dividend and split adjusted. Notice that simple return has a minimum bound of −1. This is hard to handle when we use parameterized distributions, such as normal distribution or skewed t distribution, to model the return. The log return doesn't have this issue as its value can spread over the entire real axis. Therefore, the log return is preferred. Since rt is a random variable given the price at Pt−1, risk can be defined as rt's uncertainty. Throughout this thesis, we use r as the random return, f(r) as its probability density function and F (r) as its cumulative density function in the formulas for various risk measures. This section lists a few concepts that are essential in understanding risk measures, including symmetric and asymmetric/downside risk measures. In particular, we focus on coherent risk measures of risk. 3 2.1.1 Classical Risk Measures Volatility Volatility is defined as the standard deviation of an asset's return. It is the most widely used measure of uncertainty and is usually denoted by σ, σ = pE(r − Er)2; where E stands for expectation. Volatility has the same units as the return and is a symmetric risk measure in the sense that it treats both the upside and downside deviation from the mean identically. Mean Absolute Deviation (MAD) Mean-absolute deviation is another symmetric measure, defined as the expected absolute deviation from the mean, MAD = Ejr − Erj: It serves as a substitute for volitility in measuring the statistical dispersion when volitility is infinite. However, it is not everywhere differential and so is not preferred as an objective function in portfolio optimization. Volatility and MAD are both symmetric risk measures, i.e., they treat positive and negative deviations from the mean the same. However, historical returns indicate that return is not normally distributed, but rather fat-tailed and skewed. In addition, investors favor a scenario where realized return is higher than expected, and consider as risk only a downside return that is less than a certain expected return. In this regard, numerous downside risk measures have been proposed. 4 Semi-standard Deviation (SSD) Semi-standard deviation is similar to standard deviation except that it only takes into account the negative deviations from the mean, sZ µ SSD = (r − µ)2f(r)dr: −∞ Here µ is the expectation of r. Semi-standard Deviation was first proposed by Markowitz [25]. Quirk and Saposnik[33] provided a theoretical background for SSD being superior than variance/volitility. Mao [23] advocates using SSD in adapting Markowitz's model to capital budgeting. Lower Partial Moments (LPM) A key development in the research of downside risk measures was Lower Partial Moments (LPM) by Bawa[4] and Fishburn[12]. Z c LPM(α; c) = (c − r)αf(r)dr α > 0 −∞ where c is the minimum acceptable return, a predetermined threshold, and α is the degree of the lower partial moment. We can set c to be a constant, say 0, or the risk-free rate, and α can take non-integer values. The LPM with α < 1, α = 1 and α > 1 captures risk seeking behavior, risk neutral behavior and risk averse behavior, respectively. When c = µ, the expectation of return r, and α = 2, then LPM(2; µ) is just SSD.

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