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The expectile can be considered as a generalization of quantile. Whileexpected shortfall is a quantile based risk measure, we study its counterpart-- the expectile based expected shortfall -- where expectile takes the place ofquantile. We provide its dual representation in terms of Bochner integral.Among other properties, we show that it is bounded from below in terms ofconvex combinations of expected shortfalls, and also from above by the smallestlaw invariant, coherent and comonotonic risk measure, for which we give theexplicit formulation of the corresponding distortion function. As a benchmarkto the industry standard expected shortfall we further provide its comparativeasymptotic behavior in terms of extreme value distributions. Based on theseresults, we finally compute explicitly the expectile based expected shortfallfor some selected class of distributions.

Dual Representation of Expectile based Expected Shortfall and Its Properties