A review and some complements on quantile risk measures and their domain

Abstract

In the present paper, we study quantile risk measures and their domain. Our starting point is that, for a probability measure Q on the open unit interval and a wide class LQ of random variables, we define the quantile risk measure ϱQ as the map that integrates the quantile function of a random variable in LQ with respect to Q. The definition of LQ ensures that ϱQ cannot attain the value +∞ and cannot be extended beyond LQ without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. We also present a particularly short proof of the subadditivity of expected shortfall.