JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2014.10.1109 MANAGEMENT OPTIMIZATION Volume 10, Number 4, October 2014 pp. 1109–1127 CVAR PROXIES FOR MINIMIZING SCENARIO-BASED VALUE-AT-RISK Helmut Mausser and Oleksandr Romanko Quantitative Research, Risk Analytics, Business Analytics, IBM 185 Spadina Avenue, Toronto, ON M5T2C6, Canada (Communicated by Hailiang Yang) Abstract. Minimizing VaR, as estimated from a set of scenarios, is a diffi- cult integer programming problem. Solving the problem to optimality may demand using only a small number of scenarios, which leads to poor out-of- sample performance. A simple alternative is to minimize CVaR for several different quantile levels and then to select the optimized portfolio with the best out-of-sample VaR. We show that this approach is both practical and effective, outperforming integer programming and an existing VaR minimiza- tion heuristic. The CVaR quantile level acts as a regularization parameter and, therefore, its ideal value depends on the number of scenarios and other problem characteristics. 1. Introduction. Minimizing a portfolio’s Value-at-Risk (VaR) is a challenging optimization problem. One reason for this is that, aside from certain special cases, such as when losses are elliptically distributed, VaR is not a simple function of the positions in the portfolio. As a result, it is common practice to approximate the portfolio’s loss distribution with a finite number of scenarios, and to optimize the VaR as estimated from this sample. Obtaining an accurate risk estimate, so that the optimized portfolio performs well on an out-of-sample basis, may demand an extremely large number of scenarios. This in itself is not necessarily problematic; optimizing other risk measures, such as the conditional VaR (CVaR), i.e., the av- erage loss exceeding the VaR, also requires scenario approximation. However, VaR presents an additional challenge in that minimizing its estimator, the sample quan- tile, entails integer programming. This makes it increasingly difficult to find an optimal solution as the number of scenarios increases, so that this approach offers limited practical benefit. In contrast, CVaR optimization is a linear program, which can be solved much more readily. This computational advantage motivates using CVaR as a substitute, or proxy, for VaR in optimization problems. 1 Further justification for optimizing CVaR in place of VaR stems from the fact that, for a given quantile level α, CVaR is an upper bound for VaR, i.e., CVaR α ≥ 2010 Mathematics Subject Classification. Primary: 91G10, 91G60, 91B30, 90C05; Secondary: 90C20, 90C11, 65K05. Key words and phrases. Value-at-risk, conditional value-at-risk, optimization, regularization. 1 Unlike CVaR, VaR generally is not subadditive, and thus only the former is a coherent risk measure [4]. While this makes CVaR preferable to VaR on theoretical grounds, herein we consider only their computational aspects, rather than their respective merits as risk measures per se. For discussion of the latter see, for example, Acerbi and Tasche [1] and Dan´ ıelsson et al. [6]. 1109