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.
T. Boonen, Li-Wei Zheng· ASTIN Bulletin: The Journal...· 1 citation
We study peer-to-peer (P2P) insurance contracting between a risk-averse P2P reinsurer and multiple risk-averse peers in an asymmetric Nash-bargaining framework, where all agents seek to improve expected utility relative to their disagreement points. Consistent with the expected value premium principle, we impose a price-fairness condition requiring each peer's expected contribution to be based on a common loading applied to the peer's expected loss. To justify the bargaining formulation relative to a standard fixed-weight weighted-sum optimization problem, we provide an axiomatic characterization showing that the Nash bargaining solution satisfies properties well suited to voluntary P2P insurance contracting in small pools. We establish the existence and uniqueness of the optimal contract and derive first-order characterizations for the full-, partial-, and zero-reinsurance regimes. To address subgroup formation, we develop computationally tractable sufficient conditions that rule out viable coalitional deviations, both with and without price fairness. Our numerical study investigates the impact of price fairness and pool size on the optimal contract and agents'welfare. Price fairness reduces dispersion in risk allocations and certainty-equivalent loadings among peers. Regarding pool size, welfare need not increase monotonically, highlighting that risk-pool expansion depends not only on diversification but also on the evolution of bargaining power.
T. Boonen, W. Chong, Kenneth Tsz Hin Ng et al.· 0 citations
