Robust Distortion Risk Measures

TL;DR

This paper investigates the robustness of distortion risk measures under distributional uncertainty using Wasserstein balls, providing sharp bounds and applications to portfolio optimization and risk assessment.

Contribution

It introduces a method to quantify the robustness of distortion risk measures with respect to distributional changes, including explicit bounds for VaR and RVaR.

Findings

Derived sharp bounds for distortion risk measures under Wasserstein uncertainty.

Provided quasi-explicit bounds for Value-at-Risk and Range-Value-at-Risk.

Extended results to account for uncertainty in mean and variance.

Abstract

The robustness of risk measures to changes in underlying loss distributions (distributional uncertainty) is of crucial importance in making well-informed decisions. In this paper, we quantify, for the class of distortion risk measures with an absolutely continuous distortion function, its robustness to distributional uncertainty by deriving its largest (smallest) value when the underlying loss distribution has a known mean and variance and, furthermore, lies within a ball - specified through the Wasserstein distance - around a reference distribution. We employ the technique of isotonic projections to provide for these distortion risk measures a complete characterisation of sharp bounds on their value, and we obtain quasi-explicit bounds in the case of Value-at-Risk and Range-Value-at-Risk. We extend our results to account for uncertainty in the first two moments and provide applications to portfolio optimisation and to model risk assessment.