optimum asset allocation with behavioral utilities - Stanford University

135 Pages·2004·0.58 MB·English

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
(cid:1)c Copyright 2000 by Aparna Gupta
All Rights Reserved
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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:
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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 theinvestment related decisions of anaverage
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.
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Acknowledgements
Iwish tosincerely thankmy advisors, Professor WalterMurray andProfessor 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
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Contents
Abstract iv
Acknowledgements v
1 Introduction 1
1.1 The Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Normative vs. Descriptive(Behavioral) Choice . . . . . . . . . 3
1.1.3 The Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Overview of Optimization Methods . . . . . . . . . . . . . . . 6
1.2 The Consumption Problem . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Basic Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Overview of Optimization Methods . . . . . . . . . . . . . . . 9
1.2.3 Contrast with the Investment Problem . . . . . . . . . . . . . 10
2 Modeling the Problem 12
2.1 Asset-price Dynamics - Binomial Tree Set-up . . . . . . . . . . . . . . 13
2.2 Optimal Asset Allocation Problem . . . . . . . . . . . . . . . . . . . 15
3 Solving the Problem 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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3.2 Computational Challenges . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 Univariate Optimization . . . . . . . . . . . . . . . . . . . . . 21
3.3.2 Multivariate Optimization . . . . . . . . . . . . . . . . . . . . 22
3.3.3 Quasi-Newton Algorithm . . . . . . . . . . . . . . . . . . . . . 23
3.3.4 Sequential Quadratic Programming Algorithm . . . . . . . . . 24
3.3.5 Application to the Investment Problem . . . . . . . . . . . . . 26
3.4 Modeling Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 Wealth as Variables . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.2 Non-Uniform Time Periods . . . . . . . . . . . . . . . . . . . 30
3.4.3 Intermediate Contributions . . . . . . . . . . . . . . . . . . . . 31
3.5 Comparison of Computational Effort . . . . . . . . . . . . . . . . . . 31
3.5.1 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . 31
3.5.2 Stochastic Optimal Control . . . . . . . . . . . . . . . . . . . 33
3.5.3 Stochastic Optimization . . . . . . . . . . . . . . . . . . . . . 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
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4.5 Comparisons and Conclusions . . . . . . . . . . . . . . . . . . . . . . 62
4.6 Contribution Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.1 Problem Formulation for Contributions . . . . . . . . . . . . . 65
4.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Consumption Problem 68
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Problem Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Computational Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.1 Negative Exponential utility . . . . . . . . . . . . . . . . . . . 77
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 An Alternate Approach 84
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 The Asset Allocation Problem using a Behavioral Utility . . . . . . . 85
6.2.1 Discovering the Structure in Solution . . . . . . . . . . . . . . 88
6.2.2 The Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.3 The Central Idea : Use the Structure . . . . . . . . . . . . . . 90
6.3 Applying Parametric Optimization . . . . . . . . . . . . . . . . . . . 91
6.4 Sensitivity to Asset-Dynamics Parameters . . . . . . . . . . . . . . . 94
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7 Future Work 99
7.1 Asset-return Dynamics Models . . . . . . . . . . . . . . . . . . . . . . 100
7.2 Solution Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 Consumption Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 103
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7.5 Improving Usability of Tool . . . . . . . . . . . . . . . . . . . . . . . 103
A Sequential Quadratic Programming Algorithms 105
A.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.3 Merit Function and Line Search . . . . . . . . . . . . . . . . . . . . . 107
A.4 Multiplier Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.5 Quasi-Newton Update of the Hessian . . . . . . . . . . . . . . . . . . 109
A.6 Active-set Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.7 A Simple SQP Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 110
B ADIFOR 111
C NPSOL Package 112
D SNOPT Package 114
Bibliography 116
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List of Tables
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