Optimal Insurance Menu Design under the Expected-Value Premium Principle

TL;DR

This paper develops an optimal insurance menu design under asymmetric information, using a Stackelberg model to derive contracts that screen risk preferences and types, with nonlinear pricing and numerical analysis.

Contribution

It introduces a new model for optimal insurance menus considering private risk attitudes and types, deriving conditions for optimal contracts with nonlinear pricing.

Findings

Optimal contracts are excess-of-loss with linear pricing in risk loadings.

Risk loadings decrease with higher risk types, enabling self-selection.

Numerical examples illustrate the impact of heterogeneity on contract structure.

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.