A Reverse Expected Shortfall Optimization Formula

The celebrated Expected Shortfall (ES) optimization formula implies that ES at a fixed probability level is the minimum of linear real function plus scaled mean excess function. We establish reverse formula, which says any threshold maximum an curve minus Despite being simple result, this reveals elegant symmetries between and curve, as well their optimizers. closely related to Fenchel-Legendre transforms, our formulas are generalized from optimized certainty equivalents, popular class convex risk measures. analyze worst-case values under two settings model uncertainty illustrate usefulness further demonstrated with application using insurance datasets.