Extremal cases of distortion risk measures with partial information

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.