Optimal Insurance Contracts in Two Dynamic Models

Abstract

As one type of principal-agent problem, the insurance contract models are closely related to the extent of information disclosure. We construct two new insurance contract models with full information and adverse selection respectively. The full information model is a continuous-time model in which an insurer proposes an insurance contract to a potential insured. Motivated by climate change and catastrophic events, we assume that the number of claims process is a shot-noise Cox process. The insurer selects the premium to be paid by the potential insured and the amount to be paid for each claim. In addition, the insurer can request some actions from the potential insured to reduce the number of claims. The insurer wants to maximize his expected total utility, while the potential insured signs the contract if his expected total utility for signing the contract is greater than or equal to his expected total utility when he does not sign the contract. We obtain an analytical solution for the optimal premium, the optimal amount to be paid for each claim, and the optimal actions of the insured. This leads to interesting managerial insights. The adverse selection model deals with multi-period insurance contracts between an insurer and the insureds of two risk types. The insurer offers a menu of contracts from which the insured can choose one that fits his type. We allow more than two outcome states. The loss amount is a positive random variable that can take two or more values. Accordingly, the traditional self-selection principal-agent model with pure adverse selection is not appropriate. To the traditional model, we add a new constraint that puts a boundary on the premium and the compensation. We obtain the optimal contracts that maximize the insurer’s utility and distinguish the types of insureds. We also explain why the traditional model is inappropriate and why the constraint of boundaries is necessary.