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.

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