OPTIMUM ASSET ALLOCATION WITH BEHAVIORAL UTILITIES : A PLAN FOR ACQUIRING AND CONSUMING RETIREMENT FUNDS a dissertation submitted to the program in scientific computing & computational mathematics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy Aparna Gupta May 2000 c Copyright 2000 by Aparna Gupta All Rights Reserved ii I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Walter Murray (Principal Co-Advisor) I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. William F. Sharpe (Principal Co-Advisor) I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gene H. Golub Approved for the University Committee on Graduate Studies: iii Abstract The question of optimal strategic asset allocation for investors with behavioral util- ities saving for retirement is addressed. To date this problem has been addressed assuming that an investor is rational in the sense when making investment decisions the preference relation of the investor satisfies all the axioms of choice. Research in behavioral science indicates that not all the investment related decisions of an average person satisfy the axioms of choice. In this thesis a broader class of smooth utilities is considered that includes those satisfying the axioms of choice, but allows some to be violated. We also discuss the optimal consumption problem of how “best” a retiree should invest and consume her retirement funds. We show that mathematically this problem and that of how best to save for retirement can be formulated as large-scale nonlinear constrained optimization problems. Although both can be solved using the same algorithm, there are important differences in how the two problems are modeled. iv Acknowledgements I wish to sincerely thank my advisors, Professor Walter Murray and Professor William F. Sharpe, for their constant inspiration and guidance. I owe most to Professor Murray for his insights into the problem, and his immense patience. Despite being a very busy person, he most willingly made room for discussions with me. Most pleasurable aspect of which was that my discussions with him did not stay restricted to the research problem, I got much valuable professional advice from him, discussed scores of topics such as, Indian food, the amazing variety of English accents, etc., and for the last one year, his most adorable daughter Sofia. Professor Sharpe gave me not only his valuable time, but will be a constant source of inspiration throughout my life. I am truly grateful to him for making time for dis- cussions with me in his extremely busy schedule, helping me structure the problem addressed in this thesis and giving key insights into the financial concepts of the prob- lem. An additional attraction of visiting him was to admire the beautiful paintings by his wife in his office. I am also very thankful to him for his guidance in chalking out my future career plans. Many thanks to Professor Gene Golub for not only taking the time to serve on my reading committee, but also to provide me the opportunity of benefiting from his ocean of experience. As for many of his students, he will always be a guiding force for me. I also wish to thank Professor Bernard Roth, Professor Steve Grenadier and Professor Manju Puri for serving on my oral committee, and especially Professor Puri v Contents Abstract iv Acknowledgements v 1 Introduction 1 1.1TheInvestmentProblem......................... 2 1.1.1 Utility Theory ........................... 2 1.1.2 Normativevs.Descriptive(Behavioral)Choice......... 3 1.1.3 ThePortfolioProblem...................... 5 1.1.4 OverviewofOptimizationMethods............... 6 1.2TheConsumptionProblem........................ 8 1.2.1 BasicIssues............................ 8 1.2.2 OverviewofOptimizationMethods............... 9 1.2.3 ContrastwiththeInvestmentProblem............. 10 2 Modeling the Problem 12 2.1Asset-priceDynamics-BinomialTreeSet-up.............. 13 2.2OptimalAssetAllocationProblem................... 15 3 Solving the Problem 17 3.1Introduction................................ 17 vii 3.2ComputationalChallenges........................ 18 3.3OptimizationTechniques......................... 21 3.3.1 UnivariateOptimization..................... 21 3.3.2 MultivariateOptimization.................... 22 3.3.3 Quasi-NewtonAlgorithm..................... 23 3.3.4 SequentialQuadraticProgrammingAlgorithm......... 24 3.3.5 ApplicationtotheInvestmentProblem............. 26 3.4ModelingVariations........................... 27 3.4.1 WealthasVariables........................ 28 3.4.2 Non-UniformTimePeriods................... 30 3.4.3 IntermediateContributions.................... 31 3.5ComparisonofComputationalEffort.................. 31 3.5.1 DynamicProgramming...................... 31 3.5.2 StochasticOptimalControl................... 33 3.5.3 StochasticOptimization..................... 34 4 Application to Investment Problem 36 4.1 Introduction - Utility Theory ...................... 36 4.1.1 Descriptive, Normative, and Prescriptive Decision Making . 39 4.2 Behavioral Utilities - Challenge ..................... 42 4.3 Optimal Results for Behavioral Utilities ................. 43 4.3.1 Loss Aversion Utility ....................... 44 4.3.2 Piece-wise Linear Utility ..................... 46 4.3.3 α - t Semivariance Utility . ................. 51 4.3.4 Probability of Reaching a Goal ................. 53 4.3.5 Variant of Probability of Reaching a Goal ........... 57 4.4 Verification of Results from Standard Utilities ............. 58 viii 4.5ComparisonsandConclusions...................... 62 4.6ContributionSchemes.......................... 64 4.6.1 ProblemFormulationforContributions............. 65 4.6.2 Results............................... 66 5 Consumption Problem 68 5.1Introduction................................ 68 5.2ProblemSet-up.............................. 71 5.3ComputationalIssues........................... 74 5.4Results .................................. 75 5.4.1 Negative Exponential utility . ................. 77 5.5Conclusions................................ 83 6 An Alternate Approach 84 6.1Introduction................................ 84 6.2 The Asset Allocation Problem using a Behavioral Utility . ...... 85 6.2.1 DiscoveringtheStructureinSolution.............. 88 6.2.2 TheStructure........................... 89 6.2.3 TheCentralIdea:UsetheStructure.............. 90 6.3ApplyingParametricOptimization................... 91 6.4SensitivitytoAsset-DynamicsParameters............... 94 6.5Conclusion................................. 95 7 Future Work 99 7.1Asset-returnDynamicsModels...................... 100 7.2SolutionTechnique............................ 101 7.3InvestmentProblem........................... 102 7.4ConsumptionProblem.......................... 103 ix 7.5 Improving Usability of Tool ....................... 103 A Sequential Quadratic Programming Algorithms 105 A.1Problem.................................. 105 A.2Introduction................................ 106 A.3MeritFunctionandLineSearch..................... 107 A.4MultiplierEstimates........................... 108 A.5Quasi-NewtonUpdateoftheHessian.................. 109 A.6Active-setApproach........................... 109 A.7ASimpleSQPAlgorithm......................... 110 B ADIFOR 111 C NPSOL Package 112 D SNOPT Package 114 Bibliography 116 x List of Tables xi List of Figures 2.1Oneperiodtransitionruleforbinomial-treemodel........... 14 2.2Orderingofnodesinthebinomialtree................. 16 4.1 Plot of loss-aversion utility against wealth, W ............. 45 4.2 Optimal Stocks investment weights for a 9 periods problem using Loss aversion utility . ............................ 47 4.3 Optimal Stocks investment weights for a 9 periods problem using Loss aversion utility, plotted by wealth levels at nodes for each time period 47 4.4 Optimal Stocks investment weights for a 9 periods problem using Loss aversion utility, with different reference points (continuous for 8% yearly return,dashedlinefor5%yearlyreturn)................ 48 4.5 Optimal Stocks investment weights for a 9 periods problem using Loss aversion utility, with different reference points (continuous for 8% yearly return,dashedlineforstartingwealth)................. 48 4.6 Optimal Stocks investment weights for a 9 periods problem using Loss aversion utility, with non-uniform time periods (4 quarters, 2 half year, 1oneyear,1twoyearand1fouryearperiods)............. 49 4.7 Plot of piece-wise linear utility against wealth ............. 49 4.8 Optimal Stocks investment weights for a 9 periods problem using Piece- wise Linear utility, plotted by wealth levels at nodes for each time period 51 xii 4.9 Plot of α-t semivariance utility against wealth ............. 52 4.10 Optimal Stocks weights for a 9 periods problem using α-t semivariance utility, plotted by wealth levels at each time period .......... 52 4.11 Optimal Stocks investment weights for a 9 periods problem using α-t semivariance utility, where slope is zero above the reference ...... 53 4.12 Optimal Stocks weights for a 7 periods problem using Probability of Goal utility, plotted by wealth levels for each time period . ...... 55 4.13 Histogram for wealth at