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

TLDR

Classical portfolio optimization methods, such as mean‑variance or VaR, are highly sensitive to errors in estimated means and covariances. The paper proposes a tractable approach to mitigate this sensitivity by defining worst‑case VaR under partial distributional knowledge. The authors formulate worst‑case VaR with bounded mean and covariance as a semidefinite program and extend the framework to factor‑model uncertainty, support constraints, and relative entropy. They demonstrate that worst‑case VaR computation and optimization can be solved efficiently as semidefinite programs.

Abstract

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.