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COPYRIGHT 2001 MIT Press Journals
I. Introduction
SINCE the significant contribution of Townsend (1979), an insurance contract with a deductible is described as an optimal contract in the presence of costly state verification problems. To minimize auditing costs and guarantee insurance protection against large losses to risk-averse policyholders, this optimal contract reimburses the total reported loss less the deductible when the reported loss is above the deductible, and pays nothing otherwise. The contract specifies that the insurer commits itself to audit all claims with probability 1. This deductible contract is optimal only for the class of deterministic mechanisms. Consequently, we should not observe any fraud, notably in the form of build-up, in markets with deductible contracts, because the benefits of such activity are nil. However, fraud is now a significant problem in automobile insurance markets for property damages while deductible contracts are often observed.
Different extensions have been proposed in the recent literature on security design to take into account different issues regarding the optimal insurance contracts (Townsend, 1988; Lacker & Weinberg, 1989; Mookherjee & Png, 1989; Crocker & Morgan, 1998; Bond & Crocker, 1997; Crocker & Tennyson, 1996; Picard, 1996, 2000; Boyer, 1997, 1998). Three main issues related to our empirical model are discussed in this literature. First, the deductible model implies that the principal fully commits to the contract in the sense that he will always audit all claims even if the perceived probability of lying is nil. It is clear that this contract is not renegotiation proof: at least for small losses above the deductible, the insurer has an incentive not to audit the claim and save the auditing cost. However, if the client anticipates such a behavior from the insurer, he or she will not necessarily tell the truth when filing the claim.
One extension to the basic model was to suggest that random audits are more appropriate to reduce auditing costs (Mookherjee & Png, 1989; Townsend, 1988). However, the optimal insurance contract is no longer a deductible contract, and the above commitment issue remains relevant. Another extension is to suggest that costly state falsification is more pertinent than costly state verification for insurance contracting with ex post moral hazard. The optimal contract under costly state falsification leads to insurance overpayments for small losses and undercompensation for severe accidents (Lacker & Weinberg, 1989; Crocker & Tennyson, 1996; Crocker & Morgan, 1998). We do not yet observe such contracts for property damages in automobile insurance markets, although they seem to be present for bodily injuries in some states or provinces (Crocker & Tennyson, 1996).
The object of this study is to verify how the presence of a deductible may affect the optimal falsification behavior of an insured. This is an important test, because it is now documented that approximately 10% of the claims in the studied market contain some fraud (Caron & Dionne, 1997). From the above literature, we already know that a straight deductible is not optimal under costly state falsification.(1) We will show that this type of insurance contract can indeed introduce perverse effects when falsification behavior is potentially present: a higher deductible may create incentives to fraud or cheat, particularly when the insured anticipates that the claim has a small probability of being audited.
This paper mainly concerns build-up, which is an attempt by the insured to inflate the damages resulting from a true automobile accident. At least two examples can be used to illustrate how build-up can take place. The first one is when the intermediary (for example, the body shop) has a passive role. In that case, the claimant can exaggerate the losses associated with an accident by convincing the body shop to include, in the claim, damages from a previous accident (not claimed because the repair cost was below the deductible) or other expenses such as painting the whole car. The second example is when the intermediary offers to the insured more than the needed repair activities or services. The benefits of this type of build-up may be split between the insured and the intermediary. (The insured may recover his deductible.) This type of behavior is observed even when the insured must ask quotations from several workshops because he may communicate the information between them, particularly when there is no commitment about auditing. According to a recent survey, 39% of fraud activities in the market studied are in the form of build-up (Dionne & Belhadji, 1996).
Our empirical hypothesis is as follows: when the success probability of defraud is sufficiently high, the observed loss following an accident is higher when the deductible of the insurance contract is higher. Because we have access only to reported losses, a higher deductible also implies a lower probability of reporting small losses to the insurer. To isolate the fraud effect related to the presence of a deductible in the contract, we must therefore introduce some corrections in the data to eliminate potential bias explained by incomplete information. We will use the method of censored dependent variable developed by Nelson (1977) and extended to the truncation case by Maddala (1983).
The paper is organized as follows. In section II, we show how the presence of a deductible in an insurance contract may affect the optimal falsification behavior. Our objective is not to derive the optimal insurance contract under costly state falsification, but to show how the parameters of an observed given contract with a deductible may affect the incentives for falsification, at equilibrium. In section III, we present the econometric model developed to take into account potential bias in the data explained by the fact that we do not have access to all accidents of the insured, but only to their claims made to insurance firms. Section IV describes the data and variables used in the various specifications considered. Results in section V indicate, among other things, that with an appropriate correction for selectivity the amount of the deductible remains a significant determinant of the reported loss, at least when no other vehicles are involved in the accident or when the success probability of defraud is sufficiently high. Section VI concludes the paper by discussing how the commitment issue can be addressed by the insurance market.
II. Theoretical Model
Consider a risk-averse individual who is making the marginal decision of falsifying his true accident cost (A [is greater than] 0). As already discussed in the introduction, our objective is to analyze the effect of a deductible on this decision. Consequently, we suppose that the agent has already signed a deductible contract for property damages. Without falsification, the individual's wealth determined by nature (or after his accident) is W - D [equivalent] [W.sub.0] - A - P + (A - D), where (A - D) is the insurance coverage of the accident cost A, and D is the amount of deductible.(2) W is the level of wealth not contingent: this is the initial wealth, [W.sub.0], minus P, the insurance premium. Under falsification, the total claim is C = L + A instead of A, where L is the level of falsification or claim inflation, and (C - D) is the insurance coverage. Ex post, the decision is then to choose the level of falsification in order to maximize
(1) pU(W - D + L) + (1 - p)U(W - A),
where A + L - D is the payment from the insurer in case of success; U(*) is the standard von Neuman-Morgenstern utility function with U'([multiplied by]) [is greater than] 0, U"([multiplied by]) [is less than] 0; and p is the success probability of falsification.
Falsification is costly. There is a penalty cost (A - D) when the activity is discovered with probability (1 - p) by the insurer. In other words, we assume that there is no insurance coverage when falsification is discovered by the insurer.(3) However, the important behavioral assumption in equation (1) is that falsification is not found with probability 1, as implicitly suggested in standard contracts with a deductible (Townsend, 1979). The probability 1 - p is lower than 1 for at least two reasons: either the insurer does not audit the file (absence of full commitment or random auditing), or it audits but does not find any evidence of fraud even when there is fraud. (See Dionne and Belhadji (1996) and Caron and Dionne (1997) for detailed analyses of claim auditing in the Quebec automobile insurance market. Caron and Dionne show that only 33% of the fraudulent claims are detected when audited.)
We now extend the model of Picard (1996) to our application.(4) Because the empirical analysis is oriented toward the effects of different parameters of the insurance contract, we will limit the theoretical analysis to the equilibrium of the audit game. For matter of space, we consider only the noncommitment case because this case corresponds to the market studied.
As suggested by Picard (1996), the audit game can be described as a three-stage game. The proportion of opportunists in the market is assumed to be [Sigma]. Because at equilibrium all insurers offer the same insurance contracts, it is also clear that this proportion is the same in the portfolio of each insurer. The nature determines whether the insured is honest or opportunist with probability 1 - [Sigma] and [Sigma], respectively. Honest policyholders always tell the truth whereas opportunists defraud with probability [Alpha]. Finally, the insurer may decide to audit with probability [Delta]. His success probability of detecting a fraudulent claim when auditing is equal to m. Therefore, p (the success probability of fraud) is equal to 1 - [Delta]m.
Before considering...
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