Robustifying Convex Risk Measures: A Non-Parametric Approach

This paper introduces a framework for robustifying convex, law invariant risk measures, to deal with ambiguity of the distribution of random asset losses in portfolio selection problems. The robustified risk measures are defined as the worst-case portfolio risk over the ambiguity set of loss distributions, where an ambiguity set is defined as a neighborhood around a reference probability measure representing the investors beliefs about the distribution of asset losses. Under mild conditions, the infinite dimensional optimization problem of finding the worst case risk can be solved analytically and closed-form expressions for the robust risk measures are obtained. Using these results robustified versions of several risk measures, including the standard deviation, the Conditional Value-at-Risk, and the general class of distortion functionals. The resulting robust policies are of similar computational complexity as their non-robust counterparts. Finally, a numerical study shows that in most instances the robustified risk measures perform significantly better out-of-sample than their non-robust variants in terms of risk, expected losses, and turnover.