**Robust Distortion Risk Measures**

In the presence of uncertainty, robustness of risk measures, which are prominent tools for the assessment of financial risks, is of crucial importance. Distributional uncertainty may be accounted for by providing bounds on the values of a risk measure, so-called worst- and best-case risk measures. Worst (best)-case risk measures are determined as the maximal (minimal) value a risk measure can attain when the underlying distribution is unknown – typically up to its first moments. However, these bounds as well as the (worst- and best-case) distributions that attain the worst- and best-case values are too large, respectively “unrealistic”, to be practically relevant.

We provide sharp bounds for the class of distortion risk measures with constraints on the first two moments combined with a constraint on the Wasserstein distance with respect to a reference distribution. Adding the Wasserstein distance constraint, leads to significantly improved bounds and more “realistic” worst-case distributions. Specifically, the worst-case distribution of the two most widely used risk measures, the Value-at-Risk and the Tail-Value-at-Risk, depend on the reference distribution and thus, are no longer two-point distributions.

This is a join publication by Carole Bernard, Silvana M. Pesenti, Steven Vanduffel