Robust distortion risk metrics and portfolio optimization

TL;DR

This paper develops bounds for distortion risk metrics under distributional uncertainty considering key distribution features, and applies these results to robust portfolio optimization and model risk assessment.

Contribution

It introduces a comprehensive framework for bounding distortion risk metrics under various distributional constraints, including mean, variance, unimodality, and Wasserstein distance, with applications to finance.

Findings

Derived sharp bounds for distortion risk metrics under multiple distributional constraints.

Identified extremal distributions for worst- and best-case risk assessments.

Enhanced decision-making in portfolio optimization under model uncertainty.

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 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.