Ex Post Moral Hazard and Bayesian Learning in Insurance

Abstract

ABSTRACT We study a dynamic insurance market with asymmetric information and ex post moral hazard. In our model, the insurance buyer's risk type is unknown to the insurer; moreover, the buyer has the option of not reporting losses. The insurer sets premia according to the buyer's experience rating, computed via Bayesian estimation based on buyer's history of reported claims. Accordingly, the buyer has strategic incentive to withhold information about losses. We construct an insurance market information equilibrium model and show that a variety of reporting strategies are possible. The results are illustrated with explicit computations in a two-period risk-neutral case study. INTRODUCTION Buyers of casualty and property insurance possess varying levels of risk. Risk type is used by the insurer to set premia and is the main parameter in pricing the resulting insurance contract. Unfortunately, the intrinsic riskiness of the buyer is typically unknown from the point of view of the insurer. This leads ex ante to problems of moral hazard and adverse selection. Namely, higher-risk buyers will attempt to enter into contracts designed for low-risk buyers, and once they obtain insurance, all buyers have little incentive to act prudently. However, many insurance contracts (notably automobile insurance) have a recurring nature. Thus, the issue of information asymmetry is partially mitigated by implementing an experience-rating or bonus-malus system (see Lemaire, 1995, for a comprehensive review of such insurance schemes), through which the insurer gives incentives to the buyer to act in the best behavior. Through such incentives, a self-serving buyer may reveal his (1) risk type or exercise an optimal amount of preventive efforts. A second level of ex post information asymmetry arises in connection with reporting losses. After an insurable event occurs, the buyer has the option of not reporting the loss in the hopes of signaling that he is of a lower risk type and, hence, deserves a lower future premium. If the gain from lowering his perceived riskiness (and the corresponding future premia) outweighs the cost for settling the loss out-of-pocket, the buyer will not report the loss. This might happen, for instance, if the risk loading on the insurance is high enough and leads to ex post moral hazard. The presence of this nonreporting option has serious implications since it dramatically alters the information received by the insurer. Instead of acting to reveal his risk type, the buyer strategically manipulates reports. Although the insurer is not necessarily hurt by underreporting and may even encourage it to reduce processing costs, she needs a learning mechanism to infer the risk type of the buyer based on submitted claims. This is necessary in order to correctly determine the premium and the corresponding experience-rating update. In a competitive insurance market, failure to properly infer the buyer's risk profile would immediately result in losses. Thus, a rational insurer recognizes that nonreporting occurs and acts accordingly. The converse of nonreporting is insurance fraud whereby the buyer may manufacture false claims. The information asymmetry of insurance fraud is usually resolved by claim verification and monitoring; by contrast, verifying nonreported losses is usually impractical or against the law. There is anecdotal empirical evidence for such strategic behavior by both buyers and insurers, especially in the private passenger automobile and homeowner's insurance industries. For instance, after minor car accidents of the fender-bender type, it is commonplace that insurance agents advise their clients to pay for repairs themselves and not file a claim, so as to maintain their high rating. Conversely, it is often observed that the actual malus penalties for reporting auto claims are relatively severe and would be excessive in a world with perfect reporting (i. …