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.
Highlights
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 $Q as the map that integrates the quantile function of a random variable in LQ with respect to Q
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
Summary
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 $Q as the map that integrates the quantile function of a random variable in LQ with respect to Q. In the case of a spectral risk measure, the domain of a quantile risk measure proposed in the present paper contains the class proposed by Acerbi (2002) and turns out to be a convex cone, which is of interest with regard to the subadditivity of the risk measure. We present a short proof of the subadditivity of expected shortfall and use this result to show that a quantile risk measure is subadditive if and only if it is spectral (Section 5). As a major issue of this paper, we proceed with a detailed comparison of the domain of a quantile risk measure with the classes of random variables proposed by Acerbi (2002) and Pichler (2013) in the spectral case (Section 6). As a complement, we briefly discuss related integrated quantile functions occurring in the measurement of economic inequality (Section 7)
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