Robust distortion risk metrics and portfolio optimization

We establish sharp upper and lower bounds for distortion risk metrics under distributional uncertainty. The uncertainty sets are characterized by four key features of the underlying distribution: mean, variance, unimodality, and Wasserstein distance to a reference distribution. We first examine very general distortion risk metrics, assuming only finite variation for the underlying distortion function and without requiring continuity or monotonicity. This broad framework includes notable distortion risk metrics such as range value-at-risk, glue value-at-risk, Gini deviation, mean-median deviation and inter-quantile difference. In this setting, when the uncertainty set is characterized by a fixed mean, variance and a Wasserstein distance, we determine both the worst- and best-case values of a given distortion risk metric and identify the corresponding extremal distribution. When the uncertainty set is further constrained by unimodality with a fixed inflection point, we establish for the case of absolutely continuous distortion functions the extremal values along with their respective extremal distributions. We apply our results to robust portfolio optimization and model risk assessment offering improved decision-making under model uncertainty.