Optimal Insurance Menu Design under the Expected-Value Premium Principle

2026-04-17

AI Summary

This study addresses the design of optimal insurance contracts by a monopolist facing dual private information: policyholders’ risk types and their risk aversion levels, under asymmetric information. Within a Stackelberg framework and adopting the expected-value premium principle, the authors formulate a mean-variance utility maximization problem subject to incentive compatibility and individual rationality constraints. Leveraging mechanism design theory, calculus of variations, and ordinary differential equations (ODEs), they derive the optimal contract structure. The key contributions include the novel finding that when risk aversion depends on risk type, the optimal contract takes the form of excess-of-loss coverage; the derivation of an ODE characterizing the risk-loading component, along with a proof of its existence and uniqueness; and the discovery of a nonlinear pricing pattern wherein higher-risk individuals face lower risk-loading premiums. Numerical simulations further illustrate how heterogeneity in risk distributions shapes the optimal contract design.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Cryptocurrency and smart contracts
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This paper studies optimal insurance design under asymmetric information in a Stackelberg framework, where a monopolistic insurer faces uncertainty about both the insured's risk attitude, captured by a risk-aversion parameter, and the insured's risk type, characterized by the loss distribution. In particular, when the risk type is unobservable, we allow the risk-aversion parameter to depend on the risk type. We construct a menu of contracts that maximizes the mean-variance utilities of both parties under the expected-value premium principle, subject to a truth-telling constraint that ensures the truthful revelation of private information. We show that when risk attitude is private information, the optimal coverage takes the form of excess-of-loss insurance with linear pricing in terms of the risk loading (defined as the premium minus the expected loss), designed to screen risk preferences. In contrast, when risk type is unobserved, we restrict the coverage function to an excess-of-loss form and derive an ordinary differential equation that characterizes the optimal risk loading. Under mild conditions, we establish the existence and uniqueness of the solution. The results show that equilibrium contracts exhibit nonlinear pricing with decreasing risk loadings, implying that higher-risk individuals face lower risk loadings in order to induce self-selection. Finally, numerical illustrations demonstrate how parameter values and the distributions of unobserved heterogeneity affect the structure of optimal contracts and the resulting pricing schedule.

Problem

• optimal insurance design
• asymmetric information
• risk aversion
• risk type
• truth-telling constraint

Innovation

• asymmetric information
• optimal insurance design
• excess-of-loss insurance
• risk loading
• incentive compatibility