On a family of risk measures based on largest claims

Abstract

Given a set of n >= 2 independent and identically distributed claims, the expected average of the n−i largest claims, with 0 <= i<= n−1, is shown to be a distortion risk measure with concave distortion function that can be represented in terms of mixtures of tail value-at-risks with beta mixing distributions. This result allows to interpret the tail value-at-risk in terms of the largest claims of a portfolio of independent claims. As an application, we provide sufficient conditions for stochastic comparisons of premiums in the context of large claims reinsurance.