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

Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720
Department of Industrial Engineering and Operations Research, University of California, Berkeley, California 94720
INRIA Rhone-Alpes ZIRST, 655 avenue de l'Europe 38330 Montbonnot Saint-Martin, France
elghaoui{at}eecs.berkeley.edu
max{at}ieor.berkeley.edu
francois.oustry{at}inria.fr

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.

Subject classifications: Finance, portfolio: Value-at-Risk, portfolio optimization; Programming, nonlinear: semidefinite programming, dualing, robust optimization.
History: Received September 2001; revision received July 2002; accepted July 2002.

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