Insurance Contract Design When the Insurer Has Private Information on Loss Size

ABSTRACT This article examines the optimal indemnity contract in an insurance market, when the insurer has private information about the size of an insurable loss. Both parties know whether or not a loss occurred, but only the insurer knows the true value of the loss and/or to what extent the losses are covered under the policy. The insured may verify the insurer's loss estimate for a fixed auditing cost. The optimal contract reimburses the auditing costs in addition to full insurance for losses less than some endogenous limit. For losses exceeding this limit, the contract pays a fixed indemnity and requires no monitoring. The optimal contract is compared with the contracts obtained in cases where it is only the insured who can observe the loss size. INTRODUCTION Following a loss, the insurer and the insured generally have opposing interests with regards to the amount of an insurance claim. For example, Doherty and Muermann (2005) note, "Insurers are now more likely to dispute large claims, to offer less than 100 cents on the dollar, or to try to get away without paying." Indeed, larger claims tend to be underpaid, as documented by Crocker and Tennyson (2002). On the other side, the insured wants to obtain as high an indemnity as possible. To date, the theoretical literature on insurance fraud only has considered private loss information held by the insured, who can then inflate loss claims. The insurer can verify the claim, but only at a cost. (1) However, insurers often have as much, if not more, information about the true loss size as the insured. Moreover, any indemnity payment for a loss depends not only upon the economic value of damages but also upon contract terms and stipulations about what losses are covered and to what extent. This article analyzes a model of costly state verification in an insurance market where the insurer knows the magnitude of the covered damages but the insured does not. This assumption regarding information asymmetry is the reverse of that usually assumed in the fraud literature. We derive the optimal indemnity schedule and compare it with those in the extant fraud literature. Assuming that there are no nonmonitoring transactions costs, this schedule is almost a "mirror image" of that obtained by Bond and Crocker (1997) when the insured has private information about the size of the loss, although the derivations are somewhat different. Aspects of our optimal schedule can be seen in real-world contracting. For example, automobile collision damages are typically deemed a "total loss" once repair costs reach a high enough percent of the car's actual cash value. (2) The same rationale applies to marine insurance, where a "constructive total loss" applies to situations in which the projected losses are high enough not to assess the damages further and simply "write them off" as a total loss. Of course, an even more realistic model would incorporate private information by both the insurer and the insured, but this goes beyond our scope in this article. As an example of the type of asymmetry that we have in mind, consider an automobile collision accident that had minimal visible damage to the car but may have weakened the frame of the car. The structural damage might not be noticeable immediately, but it can subject the car to further breakdowns and costly repairs in the future. Oftentimes the insurer will have its own mechanics (or subcontracted mechanics) assess the damage. Should the car's owner accept the insurer's assessment of the damage and the corresponding indemnity payment? Oftentimes, such acceptance also requires the insured to sign a waiver to claim additional damages. As a second example, consider an accident that involves a physical injury. The insured may feel much pain, but she might not know whether the pain is just a minor sprain or something more serious. Some minor treatment might ease the pain and correct the condition temporarily. …