Closed-Form Solutions for Extreme-Case Distortion Risk Measures and Applications to Robust Portfolio Management

Abstract

Extreme-case (worst-case and best-case) risk measures refer to the extreme (maximal and minimal) values for risk measures when only partial information of the underlying risk is available. We obtain closed-form solutions for extreme-case distortion risk measures with the first two moments of the underlying risks available, which completely generalizes the worst-case Value-at-Risk and Expected Shortfall in some classic literature. We characterize the extreme-case distributions that offers great intuition related to the choice of distortion functions.

Keywords: model uncertainty, extreme case, distortion risk measure, robust portfolio selection