On Partial Stochastic Comparisons Based on Tail Values at Risk

Abstract

The tail value at risk at levelp, withp is an element of(0,1),is a risk measure that captures the tail risk of losses and asset return distributions beyond thepquantile. Given two distributions, it can be used to decide which is riskier. When the tail values at risk of both distributions agree, whenever the probability levelp is an element of(0,1),about which of them is riskier, then the distributions are ordered in terms of the increasing convex order. The price to pay for such a unanimous agreement is that it is possible that two distributions cannot be compared despite our intuition that one is less risky than the other. In this paper, we introduce a family of stochastic orders, indexed by confidence levelsp0 is an element of(0,1),that require agreement of tail values at risk only for levelsp>p0. We study its main properties and compare it with other families of stochastic orders that have been proposed in the literature to compare tail risks. We illustrate the results with a real data example.