Dual Representation of Expectile based Expected Shortfall and Its Properties

TL;DR

This paper introduces the expectile-based expected shortfall, a risk measure generalizing expected shortfall using expectiles, and explores its properties, dual representation, and asymptotic behavior compared to traditional measures.

Contribution

It provides the first dual representation of expectile-based expected shortfall and analyzes its bounds, properties, and asymptotic behavior relative to expected shortfall.

Findings

Bounded by convex combinations of expected shortfalls

Upper bounded by a law invariant, coherent risk measure

Explicit calculations for certain distribution classes

Abstract

The expectile can be considered as a generalization of quantile. While expected shortfall is a quantile based risk measure, we study its counterpart -- the expectile based expected shortfall -- where expectile takes the place of quantile. We provide its dual representation in terms of Bochner integral. Among other properties, we show that it is bounded from below in terms of convex combinations of expected shortfalls, and also from above by the smallest law invariant, coherent and comonotonic risk measure, for which we give the explicit formulation of the corresponding distortion function. As a benchmark to the industry standard expected shortfall we further provide its comparative asymptotic behavior in terms of extreme value distributions. Based on these results, we finally compute explicitly the expectile based expected shortfall for some selected class of distributions.