Moral hazard in insurance markets with distortion risk measures

This paper studies optimal insurance design in a competitive market with one policyholder and multiple insurers. The policyholder’s risk preferences are modeled by a distortion risk measure, and the loss distribution depends on prevention effort, which reduces the loss amount; insurers price contracts using a common distortion premium principle. Insurers price contracts based on an effort benchmark. Because effort is unobservable, premiums cannot be conditioned on realized effort, creating moral hazard. The policyholder selects the optimal effort and indemnity to minimize the risk measure, taking into account the loss distribution, insurance cost, and effort cost; insurers design contracts to induce the policyholder to match the effort level assumed in the premium. Under this alignment, all insurers earn zero risk-adjusted profits in equilibrium, making them indifferent between offering coverage and not offering it, which is consistent with full competition among insurers. We focus on cases in which the policyholder’s distortion risk measure is value-at-risk or tail value-at-risk. The paper also examines relationships among parameters under exponentially distributed losses and extends the analysis to a general strictly concave distortion function for the policyholder. Under symmetric information, where insurers directly observe actual effort, we provide sufficient conditions under which the policyholder’s objective value is strictly lower than that under asymmetric information, thereby demonstrating moral hazard.