Extremal cases of distortion risk measures with partial information

TL;DR

This paper develops a unified framework to determine the extremal bounds of distortion risk measures under partial distributional information, such as moments and shape constraints, using probability inequalities and the Schwarz inequality.

Contribution

It introduces a comprehensive method to find worst- and best-case bounds for distortion risk measures with limited distributional data, extending previous results to a broader class of measures.

Findings

Derived theoretical bounds for Value-at-Risk under partial info.

Extended bounds to a wide class of distortion risk measures.

Characterized distributions achieving extremal risk scenarios.

Abstract

This paper investigates the impact of distributional uncertainty on key risk measures under the partial knowledge of underlying distributions characterized by their first two moments and shape information (specifically symmetry and/or unimodality). We first employ probability inequalities to establish the theoretical best- and worst-case bounds on Value-at-Risk, reflecting the most extreme tail risk achievable within the moment and shape constraints, and then we extend this worst-case/best-case analysis to a broad class of distortion risk measures by the modified Schwarz inequality, deriving their corresponding robust bounds under the same partial information setting concerning moments and distribution shapes of the underlying distributions. In addition, we give a clear characterization of the distributions that attain the best- and worst-case scenarios. The proposed approach provides a unified framework for extremal problems of distortion risk measures.