Robust distortion risk metrics with Wasserstein and Unimodal Constraints

Peng Liu, Steven Vanduffel (), Yi Xia and Jing Yao

Papers from arXiv.org

Abstract: 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 a broad class of distortion functions, assuming only finite variation and imposing neither continuity nor monotonicity. This class includes important examples such as the Gini deviation, the mean-median deviation, and inter-quantile differences. When the uncertainty set is defined by a fixed mean, variance, and Wasserstein distance, we derive the worst- and best-case values of the distortion risk metric and identify the corresponding extremal distributions. We then impose an additional unimodality constraint. In this case, for absolutely continuous distortion functions, we again characterize the worst- and best-case values and explicitly determine the optimal distributions attaining these bounds. Moreover, an algorithm is developed to compute sharp upper bounds and the worst-case quantiles. We apply our results to a robust portfolio optimization problem of interest.

Date: 2025-11, Revised 2026-09
New Economics Papers: this item is included in nep-rmg
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