Optimal Insurance Menu Design under the Expected-Value Premium Principle

Paper: arXiv 2604.15881

Authors: Xia Han, Bin Li

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using ODEs, continuous-state Stackelberg games, and mean-variance utility optimization under adverse selection. However, it lacks empirical backtesting or real-world dataset validation, relying instead on synthetic numerical illustrations.

Research Flowchart

  flowchart TD
    A[Goal: Design Optimal Insurance Menu under Asymmetric Information] --> B[Model Setup: Stackelberg Game & Mean-Variance Utility]
    B --> C[Inputs: Loss Distributions, Risk Aversion Parameters & Premium Principle]
    C --> D[Case 1: Private Risk Attitude<br>Screening Risk Preferences]
    C --> E[Case 2: Private Risk Type<br>Solve ODE for Optimal Risk Loading]
    D --> F[Findings: Excess-of-Loss Coverage with Linear Pricing Schedule]
    E --> G[Findings: Nonlinear Pricing with Decreasing Risk Loadings for High-Risk Types]

Cited References (62)

1. Monopoly, non-linear pricing and imperfect information: The insurance market (1977)
2. On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game between an insurer and a reinsurer (2018)
3. Stochastic Stackelberg differential reinsurance games under time-inconsistent mean-variance framework (2019)
4. Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information (1976)
5. Impact of Information Asymmetry and Limited Production Capacity on Business Interruption Insurance (2022)
6. Optimal insurance with adverse selection (2012)
7. Insurance as a lemons market: Coverage denials and pooling (2020)
8. Optimal Insurance: Dual Utility, Random Losses, and Adverse Selection (2023)

1 of 62 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.