The Stochastic Programming Approach to Asset, Liability and Wealth Management William T. Ziemba Alumni Professor of Financial Modeling and Stochastic Optimization Faculty of Commerce University of British Columbia Vancouver, BC, V6T 1Z2 Canada and Nomura Visiting Senior Research Fellow in Financial Mathematics Mathematical Institute Oxford University Oxford OX1 3LB UK AIMR Publisher May 30, 2003 —- Contents Preface ....................................... v 1 The Issues 1 2 The fundamentals of asset allocation 3 Mean-variance analysis . 3 Estimation of utility functions . 9 The importance of means . 9 CAPM and its extensions . 12 Sharpe ratios . 15 Mean variance in practice . 17 3 The stochastic programming approach 20 Discrete scenarios/fat tails . 20 Extreme scenario examples . 25 A crash danger model . 27 The 2000-2003 crash in the S&P500 . 29 Procedures for scenario generation . 40 Fixed mix and strategic asset allocation . 43 SP versus fixed mix . 46 More evidence on SP vs fixed mix models . 51 How to make good multiperiod SP models . 51 4 Insurance Company Applications 55 Elements of the insurance business . 55 The Russell-Yasuda Kasai Model . 56 Insurance products with guarantees . 66 5 Pension Fund Applications 72 Ageing of the world’s populations . 72 European pension funds and bonds . 74 InnoALM . 78 Risk in 401(k) and other company pension plans . 89 A continuous time approach to surplus management . 96 iii 6 Individual asset-liability planning models 100 Models for wealthy individuals and families . 100 Retirement . 103 Endowments . 104 7 Hedge Fund Applications 109 Keynes as a hedge fund manager . 112 Gamblers as hedge fund managers . 115 Typical hedge fund trades . 117 How to lose money in derivatives . 121 The Failure of Long Term Capital Management . 124 The Kelly and fractional Kelly wagering strategies . 126 Commodity trading . 131 Calculating the optimal Kelly fraction . 133 8 The top ten points to remember 135 Bibliography 137 Material from Chapter 2 1 Certainty equivalent method . 2 Material from Chapter 3 4 Some approaches for scenario generation . 5 Material from Chapter 4 19 Russell-Yasuda Kasai model . 19 The Prometeia insurance guarantees model . 21 Material from Chapter 5 26 ALM as a multistage SP model . 26 Own company stock . 29 The Rudolf-Ziemba (2003) model . 29 Dybvig’s (1988, 1995, 1999) protective spending model . 33 Material from Chapter 7 36 A convergence trade . 36 NSA puts and calls . 37 Kelly criterion . 40 Kelly and fractional Kelly strategies for the game of blackjack . 42 Lotteries . 43 Optimal Kelly . 49 Properties of capital growth criterion . 52 iv PREFACE v Preface Dedicated to the memory of my two most admired purists whose work stood the test of time against the critics: Theodore Samuel Williams, baseball player and fisherman and Merton H. Miller, finance theorist, financial mar- ket colleague and professional keynote speaker. Also to Chris Hensel my late Frank Russell colleague and Teppo Martikainen my late University of Helsinki colleague who were most supportive of our joint work on anoma- lies and portfolio management as well as the stochastic programming work discussed in this monograph. This monograph presents an easily readable, up-to-date treatment of asset and wealth management in the presence of liabilities and other portfolio complexities such as transactions costs, liquidity, taxes, investor preferences including downside risk control, policy and other constraints, uncertain returns and the timing of returns and commit- ments. The presentation discusses the issues involved in the management of investment portfolios for large financial institutions such as pension funds, insurance companies and hedge funds and individuals concerned with life cycle planning. For most practical purposes, discrete time, multiperiod stochastic programming models provide a superior alternative to other approaches such as mean-variance, simulation, control theory and continuous time finance. This approach leads to models that take into account investor preferences in a simple understandable way. One typically maximizes the concave risk averse utility function composed of the discounted expected wealth in the final period less a risk measure composed of the Arrow-Pratt risk aversion index times the sum of convex penalties for target violations relating to investor goals of various types in various periods. The convexity means that the larger the target violation, the larger the penalty cost. Hence risk is measured as the non-attainment of investor goals and this risk is traded off against expected returns. This is similar to a mean-variance pref- erence structure except it is on final wealth and risks are downside and across several periods and several investor goal targets. Discrete scenarios that represent the possible occurrences of the return and other random parameter outcomes in various periods are generated from econometric and other models such as those related to market dangers with increasing risk and from expert modeling. Mean return estimation and inclusion of extreme events are very important for model success. The scenario approach has a number of advantages such as : • Normality or lognormality used in other approaches which is not an accurate representation of actual asset prices especially for losses need not be assumed. • Tail events can be included easily; studies show that downside probabilities esti- mated from actual option prices are 10 to 100 to 1000 times fatter than lognormal. • Scenario dependent correlation matrices across assets can be modeled and used in the decision making process so that normal and crisis economic times (with higher and differing signed correlations) can be considered separately. vi Preface • The exact scenario that will occur and the probabilities and values of all the scenarios do not need to be accurately determined to provide superior model performance compared to other models, and strategies such as mean variance, fixed mix, portfolio insurance, etc. • It is an optimization approach over time to determine the best decisions, taking into account relevant constraints, uncertainties and preferences of the decision maker. • Most natural practical aspects of asset liability applications such as those men- tioned can be modeled well in the multiperiod stochastic programming approach. Solution methods to solve such models are now at a high level of development and can be accomplished on high performance PCs. Model output is easy to un- derstand and interpret using good graphical interfaces that are user friendly and understandable by non-optimization experts such as pension fund trustees. The models can be tested via simulation and statistical methods and considerable in- dependent evidence demonstrates their superiority to other standard approaches and strategies. • The approach tends to protect investors from large market losses by considering the effects of extreme scenarios while doing this in a model that accounts for the other key aspects of the problem. The world are more dangerous now with many more extreme scenarios. For example, derivatives worldwide is now a $142 trillion industry. While much of this trade is for hedging and reduces risk, the shear size adds new risks. • Determining the right sized and truly diversified positions across time is crucial in the protection against extreme scenarios so the results will be good in normal times and avoid disasters. Special thanks go to Mark Kritzman, Jaynee Dudley, and the Trustees of AIMR for inviting me to write this monograph which puts together many of my institu- tional investor lectures presented over the past ten plus years. Thanks also to the organizers and participants at the Frank Russell Consulting Client Conference, DAIS Group, Berkeley Program in Finance, PIMCO, Yamaichi Securities, San Francisco Insti- tute of Chartered Financial Analysts Quantative Investment Program, Swiss Institute of Banking in St Gallen, the Mexican Bolza, Chicago Board of Trade, University of Bergamo, Financier Association (Turin), Isaac Newton Institute (Cambridge Univer- sity), Den Norske Dataforening (Oslo), UBC Faculty Association, Charles University, UBC Global Investment Conference Lake Louise, UNICOM (London), Bendheim Cen- ter Princeton University, Portfolioakatemia Investment Management Seminar Helsinki, Euro Plus Dublin, Uni Credit Milan, Centre for Financial Engineering National Uni- versity of Singapore, Helsinki School of Economics, Hermes Centre of Excellence in Computatiional Finance University of Cyprus, IIASA, the Nomura Centre for Quanti- tative Finance at the Mathematical Institute, Oxford University, Arrowstreet Capital Client Conference and at many academic conferences and universities around the world for help in encouraging and developing my ideas. PREFACE vii The ideas in this monograph are the intersection of my theoretical and applied work on stochastic programming and portfolio theory and consulting and money management activities and my interest in various types of investments and gambling. My stochas- tic programming colleagues Roger Wets, Chanaka Edirisinghe, Markus Rudolf, Stavros Zenios, John Mulvey, Stein Wallace, Michael Dempster, Karl Frauendorfer, Alan King, Jitka Dupaˇcov`a,Marida Bertocchi, Rita D’Eclessia, Alexei Gavaronski, Julie Higle, Suvrajeet Sen, Georg Pflug, Hercules Vladimirou, Leonard MacLean, Gautam Mitra, Yonggan Zhao and Horand Gassmann all have contributions to ALM and have helped encourage my work discussed here. My nine years as a consultant to the Frank Russell