  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.
