Extreme-Case Distortion Risk Measures: A Unification and Generalization of Closed-Form Solutions

Extreme-case risk measures provide an approach for quantifying the upper and lower bounds of risk in situations where limited information is available regarding the underlying distributions. Previous research has demonstrated that for popular risk measures, such as value-at-risk and conditional value-at-risk, the worst-case counterparts can be evaluated in closed form when only the first two moments of the underlying distributions are known. In this study, we extend these findings by presenting closed-form solutions for a general class of distortion risk measures, which consists of various popular risk measures as special cases when the first and certain higher-order (i.e., second or more) absolute center moments, alongside the symmetry properties of the underlying distributions, are known. Moreover, we characterize the extreme-case distributions with convex or concave envelopes of the corresponding distributions. By providing closed-form solutions for extreme-case distortion risk measures and characterizations for the corresponding distributions, our research contributes to the understanding and application of risk quantification methodologies.