Generalized Expected-Shortfalls Based on Distortion Risk Measures

In this talk, we present explicit representations of generalized Expected-Shortfall (ES) based on a distortion risk measure with a general distortion function, not necessarily differentiable, as well as the reverse generalized ES optimization formula. Applying this reverse optimization formula, we obtain the expressions of the worst-case value and the worst-case distribution function of a distortion risk measure of a stop-loss random variable over a distributional uncertainty set induced via the Wasserstein distance of order 2 and the first two moments constraints. Properties of the buffered probability of exceedance (bPOE) based on the generalized ES are also investigated, and it is shown that the corresponding bPOE can be computed by solving a smooth one-dimensional convex optimization problem.

This is a joint work with Zhenfeng Zou, Shuyu Gong and Meng Guan.