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 L Q of random variables, we define the quantile risk measure ϱ Q as the map that integrates the quantile function of a random variable in L Q with respect to Q. The definition of L Q ensures that ϱ Q cannot attain the value + ∞ and cannot be extended beyond L Q 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.

Keywords

• integrated quantile functions
• quantile risk measures
• spectral risk measures
• subadditivity
• value at risk
• expected shortfall