Robust Mean-Covariance Solutions for Stochastic Optimization

We provide a method for deriving robust solutions to certain stochastic optimization problems, based on mean-covariance information about the distributions underlying the uncertain vector of returns. We prove that for a general class of objective functions, the robust solutions amount to solving a c...

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Bibliographic Details
Published in:Operations research 2007-01, Vol.55 (1), p.98-112
Main Author: Popescu, Ioana
Format: Article
Language:English
Subjects:
Analysis
Applied sciences
Asset management
Covariance
Decision analysis
Decision theory. Utility theory
Decision-making
Determinism
Exact sciences and technology
Expected utility
finance
Financial management
Hierarchies
Mathematical functions
Mathematical moments
Mathematical vectors
Methods
Objective functions
Operational research and scientific management
Operational research. Management science
Optimization
portfolio
Portfolio management
Portfolio theory
programming
quadratic
Quadratic programming
risk
stochastic
Stochastic approximation
Stochastic models
Studies
Utility functions
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Summary:We provide a method for deriving robust solutions to certain stochastic optimization problems, based on mean-covariance information about the distributions underlying the uncertain vector of returns. We prove that for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program. We first prove a general projection property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean-variance robust objective. This allows us to use known univariate results in the multidimensional setting, and to add new results in this direction. In particular, we characterize a general class of objective functions (the so-called one- or two-point support functions), for which the robust objective is reduced to a deterministic optimization problem in one variable. Finally, we adapt a result from Geoffrion (1967a) to reduce the main problem to a parametric quadratic program. In particular, our results are true for increasing concave utilities with convex or concave-convex derivatives. Closed-form solutions are obtained for special discontinuous criteria, motivated by bonus- and commission-based incentive schemes for portfolio management. We also investigate a multiproduct pricing application, which motivates extensions of our results for the case of nonnegative and decision-dependent returns.
ISSN:0030-364X
1526-5463
DOI:10.1287/opre.1060.0353