Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach

• Published in: Operations research Vol. 51; no. 4; pp. 543 - 556
• Main Authors: Ghaoui, Laurent El, Oks, Maksim, Oustry, Francois
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.07.2003; Institute for Operations Research and the Management Sciences
• Subjects: Analysis; Approximation; Componentwise operations; Covariance; Covariance matrices; Ellipsoids; Finance, portfolio: Value-at-Risk, portfolio optimization; Financial planning; Financial portfolios; Management science; Mathematical models; Mathematical moments; Mathematical vectors; Matrices; Nonlinear programming; Operations research; Optimization; Polytopes; Portfolio management; Probability; Probability distribution; Probability distributions; Programming, nonlinear: semidefinite programming, dualing, robust optimization; Semidefinite programming; Studies; United States
• ISSN: 0030-364X; 1526-5463
• DOI: 10.1287/opre.51.4.543.16101

Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that only bounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problem of computing and optimizing the worst-case VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information.