Optimal Insurance Menu Design under the Expected-Value Premium Principle

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.

• The paper introduces a Stackelberg model where an insurer designs a menu of contracts to screen for both risk aversion and risk type under asymmetric information.
• It demonstrates that under risk-attitude uncertainty, the optimal contract yields a linear pricing structure, while risk-type uncertainty leads to a nonlinear loading schedule.
• Numerical analysis confirms that higher-risk types receive lower loadings and supports the model’s implications for market design and regulatory policy.

Optimal Insurance Menu Design under the Expected-Value Premium Principle

Problem Formulation and Theoretical Framework

The paper addresses optimal insurance contract design in the presence of asymmetric information, within a Stackelberg game structure where the insurer acts as the leader. The key innovation is allowing both the insured's risk attitude (parameterized by risk aversion $\gamma$) and risk type (encoded in the loss distribution parameter $\theta$) to be private information. Importantly, the risk aversion may be a function of the risk type, reflecting possible dependencies between risk exposure and preferences. The insurer, assumed to be a monopoly, offers a menu of contracts from which each customer self-selects, revealing private information via their choice.

The premium principle adopted is the expected-value premium, which computes premiums as a linear loading over expected indemnities. Both the insurer and the insured have mean-variance (MV) utilities, a choice that facilitates tractable closed-form results in many classical insurance models, but here leads to new analytical challenges under information asymmetry.

The Stackelberg equilibrium is formulated rigorously with the insurer announcing the menu (pricing schedule) and the agent responding optimally by picking contract parameters (in particular, coverage). Truth-telling/self-selection constraints are enforced to guarantee incentive compatibility, preventing agents from mimicking other types.

Risk-Attitude Uncertainty

When only risk attitudes are privately known to the insured (the loss distribution is common knowledge), the optimal contract structure is characterized explicitly. The main result is that the optimal coverage takes an excess-of-loss (stop-loss) form with a deductible inversely related to risk aversion. The risk loading, however, is constant across agent types—a linear pricing structure. Analytical derivations show that the truth-telling/incentive compatibility constraint mathematically enforces flat pricing: there is no informational rent or cross-subsidization across types.

Formally, the optimal indemnity for a risk aversion $\gamma$ is:

$$\hat{l}(\gamma; y) = \left(y - \frac{\xi^*}{\gamma}\right)_+,$$

where $\xi^*$ is the risk loading optimized from the insurer's MV utility aggregated over the risk aversion distribution. The flat risk loading arises as a unique solution from the first-order condition of the agent's maximization.

Strong claim: The risk loading does not depend on individual risk aversion; all types face the same marginal price, and screening is effected solely via coverage choices.

Monotonicity results for deductible and premium are also established: more risk-averse agents choose lower deductibles (higher coverage) and pay higher total premiums, but the per-unit premium is invariant.

Risk-Type Uncertainty

When risk type (parameter $\theta$ in the loss distribution) is privately known and possibly correlated with risk aversion, the optimal menu’s structure is fundamentally altered. The expected-value premium principle, under the MV utility, produces contracts of excess-of-loss form, but now with a nonlinear risk loading schedule. The contract for type $\theta$ is:

$$\hat{l}(\theta; y) = \left(y - \frac{\xi(\theta)}{\gamma(\theta)}\right)_+,$$

with the risk loading function $\xi(\theta)$ determined by an ordinary differential equation (ODE) derived from the truth-telling constraint:

$$\xi'(\theta)\Psi(\theta, \xi(\theta)) + (1+\xi(\theta))\Psi_\theta(\theta, \xi(\theta)) = 0,$$

where $\theta$0 and its derivative capture how increments in type affect the expected coverage.

Through fixed-point arguments, existence and uniqueness of the solution for $\theta$1 are established under broad regularity conditions. Notably, high-risk types receive lower risk loadings, inducing them to truthfully select the intended contract. This result—the risk loading is strictly decreasing in risk type—exemplifies nonlinear pricing utilized for screening where risk type impacts claim costs directly.

Contradictory claim to classical intuition: In competitive markets, higher risk is typically penalized; here, to counter adverse selection, higher risk is ‘rewarded’ with premium discounts.

The paper extends this analysis to allow for risk aversion that is a function of risk type ($\theta$2). While this alters the absolute risk loading levels, the qualitative finding that loadings decrease in risk type persists. The premium-volume effect (total premium paid) remains monotonic with risk level when $\theta$3 is increasing.

Numerical Analysis

Comprehensive numerics support and illustrate the analytic characterizations. Key findings include:

• Under risk-attitude uncertainty, deducible and premium functions behave monotonically. The spread of the risk aversion distribution increases the equilibrium risk loading.
• Under risk-type uncertainty, the equilibrium risk loading schedule exhibits strict monotonicity, decreasing with respect to risk type regardless of the loss distribution's parametric family.
• The level and shape of loadings are sensitive to both distributional properties of the loss process and heterogeneity in risk aversion, with richer dynamics when loss distributions are heavy-tailed.
• The effect of increased uncertainty (e.g., wider support for $\theta$4 or $\theta$5) can nontrivially raise or lower equilibrium loadings, depending on parameter configurations and tail properties.

Comparison with full-information benchmarks highlights how asymmetric information situates the equilibrium between competitive (zero profit) and monopoly (maximal loadings) solutions.

Implications and Future Directions

This work rigorously delineates how contract menus can be designed to elicit private information under the expected-value premium principle, distinguishing sharply between linear (risk-attitude uncertainty) and nonlinear (risk-type uncertainty) optimal pricing. The demonstration that high-risk types optimally face lower loadings as a truth-telling incentive is particularly pertinent for empirical studies of insurance pricing and adverse selection.

Practically, the results suggest that insurance pricing structures observed in the market—such as nonlinear pricing schedules and cross-subsidization—are reconcilable with optimal screening in the face of multi-dimensional heterogeneity. The findings also have regulatory implications in contexts where linearity in premiums is legislated or debated.

Theoretically, the comprehensive characterization of Stackelberg equilibria under expected-value versus variance premium principles serves as a reference for future research, including the exploration of multi-dimensional types, time-dynamic contracts, or more general utility frameworks (such as CARA or rank-dependent utility). The tractable ODE characterization for the loading function under type uncertainty provides a template for both extensions and for empirical calibration to real-world data.

An immediate extension is to consider joint uncertainty in both risk type and attitude, possibly with endogenously correlated distributions, and to analyze the welfare and efficiency losses due to discrete menu constraints found in practice.

Conclusion

The paper rigorously advances the theory of insurance menu design under asymmetric information, establishing clear distinctions in optimal contract structure depending on whether uncertainty lies in risk preferences or in risk types. Analytical and numerical methods confirm that while risk-aversion heterogeneity leaves the risk loading flat (linear pricing), heterogeneity in risk type compels a strictly decreasing loading schedule (nonlinear pricing), aligning with the need to elicit costly hidden information. The results enhance both the theory and practice of insurance economics, with broad implications for market design, regulatory policy, and the empirical characterization of adverse selection effects.

Reference: "Optimal Insurance Menu Design under the Expected-Value Premium Principle" (2604.15881)