Risk measures induced by efficient insurance contracts

TL;DR

This paper characterizes risk measures derived from efficient insurance contracts, showing that a mixture of mean and Expected Shortfall (ES) naturally arises in Pareto optimal contracts with deductibles, linking risk measures to contract efficiency.

Contribution

It provides a novel theoretical justification for the role of ES and other risk measures in insurance design by characterizing them through efficient contracts.

Findings

Mixture of mean and ES characterizes risk in efficient contracts.

Characterization of risk measures like mean and distortion risk measures.

Links between contract efficiency and specific risk measures.

Abstract

The Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify the special role of ES in insurance and actuarial science. To fill this gap, we study characterization of risk measures induced by efficient insurance contracts, i.e., those that are Pareto optimal for the insured and the insurer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of the insured and the insurer, when contracts with deductibles are efficient. Characterization results of other risk measures, including the mean and distortion risk measures, are also presented by linking them to different sets of contracts.