Pierre Picard
email: pierre.picard@polytechnique.edu.

Kili C Wang

Insurance Fraud through Collusion between Policyholders and Car Dealers: Theory and Empirical Evidence

We analyze, from theoretical and empirical standpoints, how insurance distribution channels may a¤ect fraud when policyholders and service providers collude. The empirical analysis focuses on the Taiwan automobile insurance market. Striking forms of claims manipulation exist in this market: opportunistic policyholders tend to manipulate claim dates to reduce the burden of deductibles and to take advantage of the bonus-malus mechanism. We focus our attention on the role of dealer-owned agents (DOAs), since they have informational and bargaining advantages when faced with insurers and may be tempted to encourage collusion between their car repairers and policyholders. We develop an optimal contract model with claim auditing, where contracts are sold either through DOAs or through standard independent agents, and where policyholders and car repairers may collude to manipulate claims. We also use a database from the largest Taiwanese insurance company to test for the relevance of theoretical predictions. In particular, we verify that fraud occurs through the postponing of claims to the end of the policy year, possibly by …ling one single claim for several events, and we show that the fraud rate is larger among policyholders who purchase insurance through the DOA channel than among other policyholders.

Introduction

Claims fraud is widely considered to be an important source of ine¢ ciency in insurance markets, and deterring or detecting fraud is of crucial importance for the insurance industry.

In particular, the collusion between policyholders and service providers (e.g., car repairers or health care providers) facilitates the falsi…cation of claims, and it is one of the most important factors that encourage fraudulent practices. 1 This is similar to other fraud schemes in vertical relationships, such as discount fraud and warranty fraud, which are facilitated by retailers. 2,3 Such frauds are instances of customer misbehavior that involve collusion with retailers or frontline employees at the expense of producers and, ultimately, of honest customers. 4 In this paper, we further analyze how the collusion between policyholders and service providers facilitates fraud. Our empirical focus is on the Taiwan automobile insurance market and on the role of car dealer-owned insurance agents (DOAs) in this market. In Taiwan, a large percentage of automobile insurance contracts are sold through DOAs. Thus, DOAs sell not only cars, but also automobile insurance to their clients, and furthermore most of them own an auto repair shop. Understandably, the multi-faceted activity of DOAs and their long-term connection with car owners favor the creation of a mutually advantageous 1 The fact that the alliance between patients and physicians may lead to health care overspending is well documented in the health economics literature on ex-post moral hazard. Although medical ex-post moral hazard is not stricto sensu a fraudulent behavior, it highlights the importance of the policyholderservice provider coalition, and how this coalition may jeopardize the e¢ ciency of social or private risksharing mechanisms. See Alger and Ma (2003) on the deterrence of collusion between policyholders and service providers when some providers are collusive and others are honest. When there is a risk of collusion, the contractual relationship between insurers and service providers may improve the e¢ ciency of resource allocation, for instance through a¢ liated service provider networks, such as managed care organizations for health insurance (e.g., HMO and PPO in the USA) or Direct Repair Programs (DRP) for automobile insurance. See Gal-Or (1997) and Ma and McGuire (1997, 2002) on managed health care and Bourgeon et al. (2008) on a¢ liated service provider networks.

2 Discount fraud exploits the special discounts that companies may o¤er under particular circumstances, for instance when discounted products are used for a speci…c purpose, e.g., educational use for softwares. Although large scale discount fraud perpetrated by criminal enterprises does exist (in 2010, defrauders were sentenced to …ve years in prison and ordered to pay Microsoft $20 million in restitution for discount fraud), it probably remains exceptional. Small scale discount fraud involving collusion between retailers (who may guarantee the legitimacy of the discount) and opportunistic purchasers is probably much more widespread, although it is di¢ cult to evaluate its cost.

3 There are several patterns of warranty fraud (or service abuse). One of them exploits certain warranty policies that allow the end client to receive replacement parts for a product before returning the defective components, thus without veri…cation that they were original products. Warranty fraud also occurs when a service provider, e.g., a car repairer, replaces a defective part with a new spare part and triggers the producer's warranty, although the defective part was not original and thus was not protected by the warranty. 4 See Harris and Daunt (2013) on managerial strategies under the risk of customer misbehavior. Murthy and Djamaludin (2002) survey the literature on new product warranty. Insu¢ cient maintenance e¤ort by buyers and inadequate behavior of retailers are at the origin of a double moral hazard problem in warranty management. This is a particular case of transaction costs induced by the delegation of services to retailers; see Rey (2003) for an introductory survey.

policyholder-DOA alliance.

With the Taiwan case in mind, we will analyze the interaction between policyholders, car repairers and insurers when policyholders and repairers may collude to manipulate claims and insurers attempt to detect fraud through claims auditing. To do so, we will …rst elaborate a model that shows why deductible contracts and the bonus-malus system may provide incentives to falsify claims, and also why DOAs may lead to higher fraud rates. In short, claims auditing is all the more costly when collusion is more di¢ cult to detect, which is particularly the case for DOAs. In addition, should irregularities be detected by the insurer, DOAs have greater bargaining power than standard insurance agents because their market share (per agent) is larger, and they have a more captive clientele base, which they can credibly redirect toward another insurer. This may deter insurers from enforcing penalties against them, hence higher equilibrium fraud rates when claims are …led through DOAs than through standard insurance agents. This is reinforced in the case of deductible contracts, because deductibles weaken the insurers' incentives to monitor claims. Furthermore, the policyholders who intend to renew their contracts with the same insurer at the end of the policy year are even more incentivized to defraud because of speci…cities of the Taiwanese bonus-malus system. This will lead us to an empirical analysis based on a database of automobile physical damage insurance obtained from the largest insurance company in Taiwan. This data includes all of the policyholders who have …led a claim in 2010, amounting to nearly 11,000 …les. Our results sustain the prediction that fraud is greater when insurance policies have been sold through DOAs than through other distribution channels, and also that deductibles stimulate fraud.

This can be likened to the conclusions of Dionne and Gagné (2001). Using data from Québec, they showed that the amount of the deductible is a signi…cant determinant of the reported loss when no other vehicle is involved in the accident which led to the claim, and thus when the presence of witnesses is less likely. Miyazaki (2009) also focuses on the e¤ect of deductibles on insurance fraud. He shows through an experimental study that higher deductibles result in a weaker perception that claim padding is an unethical behavior, and thus in a larger propensity toward fraud. In this paper, we link deductibles to the falsi…cation of the number of claims and of the claim date (…ling one claim for two accidents and /or postponing claims to the last month of the policy year to take advantage of speci…c features of the Taiwanese bonus-malus system). Furthermore, we focus attention on the role of the policyholder-car repairer coalition and provide empirical evidence on the impact of such a coalition on the intensity of insurance fraud.

The remaining part of the paper is organized as follows. Section 2 provides further motivation for our analysis. We introduce some factual observations that should convince the reader that there is a signi…cant degree of claim manipulation in the Taiwanese car insurance market, and we describe regular fraud patterns. Section 3 develops our theoretical model to include claims auditing and the policyholder-car repairer coalition. Section 4 describes the data. Section 5 presents our empirical approach and discusses our results. Section 6 concludes. Complementary developments and mathematical proofs are in the Appendix.

Motivation

DOAs hold a substantial market share in the Taiwan automobile insurance market. For the insurance company that provides the base of our empirical analysis, 50.78% of vehicle damage insurance is sold through DOAs. 5 Furthermore, DOAs own the list of their customers, which increases their bargaining power when they negotiate contractual deals with insurance companies or when insurers monitor claims. An insurer who discovers a claim manipulation by a DOA may indeed hesitate to take retaliatory actions because of this strategic advantage of DOAs, who can choose to switch to another insurer. 6 In addition, DOAs also act as car repairers, and this position provides them with an informational advantage: establishing that a claim has been falsi…ed is particularly di¢ cult and costly when it has been …led through a DOA.

Our study is also related to the speci…c forms of automobile insurance fraud in Taiwan. 5 More precisely, 67.52% of type A contracts, 84,19% of type B contracts, and 43.71% of type C contracts are sold by DOAs. Read further for additional information on the three types of insurance contracts in Taiwan.

6 On average, DOAs sell more policies than other agents (three times more on average and considerably more for the largest DOAs), and their market power is particularly signi…cant for deductible contracts. They are independent agents, and, as emphasized by Mayers and Smith (1981), this status gives them more discretion in claim administration (e.g., they may intercede on behalf of their customers at the claim settlement stage) because they can credibly threaten to switch their business from one insurer to another. Actually, DOAs provide comparatively less stable customers to insurers than other insurance agents, with continuation rates (i.e., the fraction of customers who keep purchasing insurance from the same insurer one year after the other) which are about sixty percent for DOAs and seventy to eighty percent for other insurance agents. Li et al. (2013) have observed that a large proportion of automobile insurance claims are …led during the last months of the policy year. This is con…rmed by our own database.

Figure 1 presents the percentage distribution of claims and the average cost of claims (in hundred US dollars) over the twelve policy months. The heavy concentration of claims in the last month of the policy year is striking. Policy years and calendar years do not coincide and, as shown in Figures 2 and3, the concentration of claims during the last months of the policy year is compatible with seasonal ‡uctuations in the number of claims over the calendar year, with peaks during vacation months (January, March, July and December).

In addition, the average claim amount slightly decreases in the …nal policy months. Li et al. (2013) interpret this phenomena as a "premium recouping e¤ect": some policyholders without accident during the previous months tend to …le false smaller claims during the last month of the policy year in order to recoup a part of their premium. They might do so to express their feeling that they have been unfairly treated by the insurance company.

Whatever the interpretation we may have in mind, this distribution of claims over time strongly suggests that claim …ling is manipulated by a signi…cant number of policyholders.

(Insert Figures 1,2 and 3 here) Some factual information is needed to identify the factors that may lead policyholders to manipulate claims. There are three di¤erent types of automobile physical damage insurance contracts in Taiwan: types A, B and C. Type A and B contracts cover all kinds of collision and non-collision losses, with more exclusions for B than for A, 7 while type-C contracts only cover the damages incurred in a collision involving two or more vehicles. Contracts also di¤er in terms of indemnity: Type A contracts o¤er low coverage with a deductible, type B contracts may be purchased with or without deductible, and …nally type C contracts provide full coverage without deductible. Claims are per accident, with a speci…c deductible for each claim. The change in premium is ruled by a bonus-malus system when policyholders renew their contracts with the same insurance company, with a no-claim discount and an increase in premium proportional to the number of claims, without reference to their severity. The policyholders who switch to another insurance company bargain with this company about the new starting point of the bonus-malus record In this setting, opportunist policyholders may take advantage of manipulating claims for several reasons. According to the premium recouping interpretation of Li et al. (2013), defrauders are more likely to be among the policyholders who do not plan to keep a long term relationship with the same insurance company if, on average, such policyholders feel a lower moral cost of defrauding. 8 In our empirical analysis, this will lead us to de…ne a "recoup group" RG that includes the policyholders who have not renewed their contract more than one year after the policy year under consideration.9

The bonus-malus system and the per-claim deductibles also provide incentives to defraud.

Firstly, the claims …led during the last month of policy year t are not registered in the bonusmalus record of year t + 1 (they will be taken into account in the premium paid in year t + 2), and consequently, the policyholders who plan to renew their contract with the same insurer may be incited to postpone their claim to the last policy month, in order to delay the increase in premium. 10 Secondly, since the bonus-malus record depends on the number of claims and not on their severity, policyholders may be prompted to …le one unique claim for two accidents, should a second accident occur. This is even more pro…table in the case of deductible contracts, since deductibles are per-claim: the strategy that consists of postponing the …rst claim and merging any other accident with the …rst one within a unique claim yields full coverage for the part of the year that follows the …rst accident. Type A and B contracts are particularly subject to this kind of claims manipulation, because they include coverage for losses other than those associated with the collision between two cars. In our empirical analysis, the set of policyholders who renew their contract with the same insurer will be called the "suspicious group" SG because of this incentive to manipulate the bonus-malus system, with subgroups SG1 and SG2 for no-deductible and deductible contracts, respectively.

If we conjecture that some claims …led in the last policy month correspond in fact to postponed claims with the cumulated losses of two events, then we should expect that the ratio of "the average cost of …rst claims" over "the average cost of all claims" (hereafter called the …rst claim cost ratio) should increase during this month. Note however that this prediction could also be interpreted as the outcome of a moral hazard mechanism: this would be the case if a …rst accident made drivers more cautious, and thus they have less severe accidents should a second accident occur during the same policy year. To disentangle these two explanations, we may consider type C contracts as a benchmark to isolate the moral hazard e¤ect, since claims manipulation is unlikely for such a contract. 11 Figure 4 con…rms our intuition: the …rst claim cost ratios for SG1 and SG2 signi…cantly jump in the last month, and this is not the case for type C policies.

(Insert Figure 4 here) At this stage, we may come back to the part played by DOAs. Figure 5 con…rms that DOAs may favor the manipulation of claims. While the claims …led by the policyholders of the two suspicious groups, SG1 and SG2, are signi…cantly concentrated in the last policy month, this pattern is even more obvious for the policyholders of each subgroup that have purchased insurance from DOAs. Figure 5 also shows that the last policy month pattern is much less obvious in the benchmark group, which includes those policyholders who are covered by no-deductible contracts and who have not renewed their contract with the same insurance company at the end of the policy year.

(Insert Figure 5 here) 3 The model

Setting

We consider an economy with a competitive insurance market, in which automobile insurance can be purchased either through car dealers who act as insurance agents (DOAs) or through independent insurance agents. Car dealers also own auto repair shops. Accidents may be minor or serious, with repair costs `and 2`whatever the car repairer, for minor and serious accidents respectively, and also an uninsurable loss " per accident. 12 Insurance policies consist of a premium P and possibly a deductible d for each accident. 13 Insurance pricing includes constant proportional loading , and insurers may o¤er di¤erent policies through car dealers and through other distribution channels.

Each individual may su¤er from 0, 1 or 2 accidents during the period covered by the insurance contract. 14 Let 1 and 2 be respectively the probability of 1 and 2 accidents, with 0 < 1 + 2 < 1. Each accident is minor with probability q m and serious with probability q s , with q m + q s = 1. There are two types of individuals with the same initial wealth w: type 1 has a smaller degree of absolute risk aversion than type 2. Let w f be the individual's …nal wealth. u h (w f ) denotes the type h von Neumann-Morgenstern utility function (with h = 1 or 2), and we assume u 0 h > 0 and u 00 h < 0, and

u 00 1 (w f ) u 0 1 (w f ) < u 00 2 (w f ) u 0 2 (w f )
; for all w f . Let h be the proportion of type h individuals, with 1 + 2 = 1. Car repairers are risk neutral.

We also assume that individuals have di¤erentiated preferences between purchasing insurance through a car dealer or through an independent agent. In particular, individuals who 1 2 Assuming that the insurable costs of serious accidents exactly double those of minor accidents simpli…es the notations of the model. We could more generally assume than serious accidents cost more than minor accidents. The repair shop market is competitive, so that policyholders can let their car be repaired at competitive price `or 2`whatever the insurance distribution channel. The uninsurable loss " corresponds to earnings losses, time value, daily life disruption or stress incurred in the case of an accident. This loss does not play a signi…cant role in our theoretical analysis, but it makes it possible for some individuals to choose a deductible contract while others prefer a full coverage contract (in what follows, the type 1 and 2 individuals respectively), which will …t our empirical analysis of the Taiwan automobile insurance market. The fact that the uninsurable loss does not depend on the size of the accident simply re ‡ects notational simplicity. More generally, we could assume that the individuals' wealth is a¤ected simultaneously by the insurable cost of accidents and by an uninsurable correlated background risk.

1 3 The fact that deductibles are per accident follows the usual practice of car insurance companies (of course not only in Taiwan), although it does not correspond to an optimal insurance contract design. This feature of automobile insurance probably re ‡ects the increase in transaction costs that would be induced by aggregate deductibles over the whole period covered by the contract. We exclude overcoverage, such as because of moral hazard, and thus we have d 0. For simplicity, we assume that the deductible is the same for the …rst and second claims. In Taiwan, second claims have larger deductibles than the …rst one that occurred during the same policy year. This increase in the level of deductible may be viewed as an incentive device in a moral hazard setting (see Li et al, 2007). The present model could be extended to a setting where deductibles would di¤er between …rst and second claims, without a¤ecting our qualitative conclusions.

1 4 Thus, we assume that policyholders cannot have more than two accidents during the policy year. A possible justi…cation for this assumption is that three accidents or more would reveal an abnormal behavior on the part of drivers that would lead insurers to deny the renewal of the contract at the end of the policy year, which would induce large transaction costs to the policyholder. If two accidents occur, then the policyholder would make unusual e¤ort (e.g., greatly reducing the use of his car) that would allow him to avoid any additional accidents during the same policy year.

have high search costs may prefer to purchase insurance through car dealers because often purchasing a new car goes together with taking out a new insurance policy. This is modelled as in a Hotelling game. Both types of individuals are uniformly located on the interval [0; 1].

A representative DOA and another representative independent insurance agent are located at the extremities of the [0; 1] segment: the DOA is at x = x D = 0 and owns a repair shop, while the the other distribution channel is at x = x A = 1.

Purchasing insurance entails a search disutility which is proportional at rate t to the distance covered to 0 and 1 according to the distribution channel. Thus, the expected utility of a type h customer located at x 2 [0; 1] with contract (P; d) is written as

u h (P; d) t jx x i j ;
where

u h (P; d) (1 1 2 )u h (w P ) + 1 u h (w P d ") + 2 u h (w P 2d 2"); (1)
for h = 1 or 2, with i = D if that customer purchases insurance through the representative DOA and i = A if he goes through the other distribution channel. 15 Type 2 individuals have a larger propensity to purchase insurance coverage than type 1 because they are more risk averse. Because of these di¤erentiated preferences, insurers o¤er menus of contracts. Let (P ih ; d ih ) be the insurance contract that is taken out by type h individuals, with i = A or D according to the distribution channel. 16

The fraud mechanism

Fraud is analyzed as the behavior of oppportunistic policyholders who delay their claims to the last month of the policy year, with the complicity of a car repairer. We consider a very 1 5 We assume that the degree of risk aversion and the preference between the two distribution channels are independently distributed among the population.

1 6 In practice, deductibles and premiums do not di¤er in Taiwan whether the contract is purchased through D or through A. Note however indemnities and premiums may include a less easily observed (or implicit) dimension associated either with delays or conditions in the payment of indemnities. For instance, given the value of time, a longer delay between the …ling of the claim and the payment of the indemnity is equivalent to a larger deductible. In this perspective, empirical facts suggest that this delay is shorter for D than for A. In other terms, policyholders may actually bene…t from a more generous coverage plan if their contract has been purchased through a dealer-owned agent than through another distribution channel, although the contractual indemnity is the same in both cases. simple form of the opportunistic policyholder-car repairer collusive game. The policyholder makes a take-it-or-leave-it o¤er to the car repairer in which he o¤ers to pay a …xed amount G to the repairer and he keeps the residual part of the collusive gain. Because of the bonus-malus system, type 1 and 2 policyholders may commit such a fraud in order to avoid paying a higher premium during the next policy year, and we denote as v the discounted value of the savings in future insurance premiums induced by such a bonus-malus fraud. 17

Only those individuals who plan to renew their contract with the same insurer may pro…t from such a bonus-malus fraud. We assume that they make up a proportion 2 (0; 1) of the policyholders (whatever their type). The last month of the policy year will be called the "suspicious period", because …ling a claim during this period may be a signal of fraud.

Accidents occur during the suspicious period with probability 2 (0; 1). 18 We also assume that postponing a minor claim to the suspicious period requires that another minor loss actually occurs during this period, so that the total losses may be presented as the outcome of a single major accident. Policyholders also get an additional advantage from fraud by reducing the retained cost from 2d ih to d ih . Thus, if fraud has been committed and is not detected, the collusive gain is d ih + v, and it is shared between repairer and policyholder as amounts G and d ih + v G respectively. Thus, if a minor accident occurs during the non-suspicious period, then the policyholder may decide not to immediately …le a claim for this accident. Two possible cases are then possible. 19 If another minor accident occurs in the suspicious period, then the policyholder may decide to …le a single large claim for the two accidents (called a "fraudulent claim" in what follows), which requires collusion with a car repairer. Auditing large claims …led during the suspicious period allows the insurer to detect such instances of fraud. We denote as c i the cost of an audit when insurance is purchased from i 2 fD; Ag. The fact that the car dealer owns the repair shop makes collusion all the easier. Thus, we assume that auditing 1 7 In Taiwan, bonus-malus fraud is pro…table for the policyholder for two reasons: …rstly, because claims …led during the suspicious period will a¤ect the premium with a one-year time lag, and secondly because the bonus-malus record only depends on the number of claims and not on their severity. For simplicity, we here aggregate these …nancial bene…ts derived from fraud and we assume that they have a common value for all policyholders.

1 8 Thus, = 1=12 if accidents are uniformly spread throughout the year. 1 9 Bear in mind that in what follows we neglect the possibility of more than two accidents for the same policyholder. We also assume that there are only two types of acccidents (minor or serious) with repair costs of `and 2`, respectively. Thus, we do not contemplate the possibility of presenting, say, a minor accident and a serious accident as an extreme accident with cost 3`. In other words, the falsi…cation of claims only consists of announcing one single serious accident instead of two minor accidents.

claims is more costly (or, put di¤erently, it is more di¢ cult to establish colluders' fraud) when insurance is purchased from D than from A. 20 We thus assume c D > c A . If no minor accident occurs in the suspicious period (i.e., if no other accident occurred or if it occurred during the non-suspicious period), then the insurer considers that any late claim is invalid and is dismissed. 21 If a policyholder is caught …ling a fraudulent claim through a collusive agreement with the repairer, then he has to pay a …ne B, and he does not receive a indemnity, and the repairer pays a …ne B 0 . 22 In practice, when fraud is discovered, the policyholder-repairer coalition has some bargaining power that may allow its members to escape the penalties. This is particularly the case when insurance has been purchased from a DOA, because the latter is in a position to threaten the insurer with redirecting its (presumably large) customer base toward another insurer. This is another reason why deterring fraud may be more di¢ cult when insurance has been taken out through a DOA than through a standard agent. The e¤ects of agents' bargaining power on the enforcement of fraud penalties is analyzed in Appendix 2, and for the sake of presentation simplicity is not taken into account here.

Fraud-audit interaction

Let ih 2 [0; 1] be the fraud rate of type h 2 f1; 2g individuals who purchase inurance from i 2 fA; Dg. This is the fraction of type h policyholders who decide not to immediately …le a claim when a minor accident occurs in the non-suspicious period, hoping for a future collusive agreement with a car repairer, should another minor accident occur in the suspicious period. 23 Let b = 2 =( 1 + 2 ) be the probability of having a second accident, conditionally on the occurence of a …rst accident in the non-suspicious period. 24 Such an accident will occur in the suspicious period with probability , and it will be minor with probability q m . Thus, if a 2 0 For example, in the DOA case, the hidden transfer G may take the form of a promise to purchase a new car in the near future.

2 1 This is a somewhat extreme assumption made for the sake of simplicity. A more realistic setting would consist of assuming that the policyholder can pretend to be in good faith. In that case, (depending on the circumstances of the accident) the law of insurance contracts would lead the insurer to consider that the claim is valid or invalid with some probabilities. Our modelling could be extended in this direction without qualitative change to the conclusions.

2 2 B and B 0 may also be interpreted as the litigation costs incurred by the policyholder and the car repairer when fraud is discovered.

2 3 We may check that policyholders would not take advantage of colluding with a repairer if a second accident occurs during the non-suspicious period.

2 4 For simplicity, we do not condition this probability on the exact date at which the …rst accident occurs. In other words, we consider the non-suspicious period as a whole.

…rst minor acccident occurs in the non-suspicious period, then a future collusive agreement with a car repairer will be possible with probability q m b . The audit of serious claims in the suspicious period may detect such fraud. These audits are triggered with probability ih 2 [0; 1]. 25 In short, ih and ih for i = A; D and h = 1; 2 are the policyholder's and insurer's strategies, respectively.

The expected utility of a type h policyholder who does not immediately …le a claim after a …rst (minor) accident in the non-suspicious period is written as 26

Eu F ih = q m b [(1 ih )u h (w P ih d ih 2" + v G) + ih u h (w P ih 2` 2" G B)] +(1 q m b )[b u h (w P ih ` d ih 2") + (1 b )u h (w P ih ` ")]:
If the policyholder does not immediately …le a claim after a minor accident in the nonsuspicious period, he will have the opportunity to defraud (i.e., to …le a single large claim in the suspicious period) with probability q m b . In that case, either the claim is audited or not, respectively with probabilities ih and 1 ih . If there is no audit, then the policyholder receives d ih + v G, which is his share of the collusive deal, in addition to his status quo net wealth w P ih 2d ih (i.e., the policyholder's wealth in the case of two accidents without fraud). If there is an audit, then no indemnity is paid by the insurer, and the policyholder pays the …ne B and does not recoup his side-payment G. If no fraudulent claim can be …led, then the late claim is dismissed: no insurance indemnity is paid for this claim, and any other accident (which occurs with probability b ) leads to another claim.

If the policyholder immediately …les a claim after his …rst minor accident, then his expected utility (after this …rst accident) is

Eu N ih = b u h (w P ih 2d ih 2") + (1 b )u h (w P ih d ih "):
The policyholder is willling to defraud by making a side-payment G to the car repairer

if Eu F ih Eu N ih , that is if ih h (P ih ; d ih ; G), where h (P ih ; d ih ; G) = q m b u h (w P ih d ih 2" + v G) q m b [u h (w P ih d ih 2" + v G) u h (w P ih 2` 2" G B)] + (1 q m b )[b u h (w P ih ` d ih 2") + (1 b )u h (w P ih ` ")] q m b [u h (w P ih d ih 2" + v G) u h (w P ih 2` 2" G B)] b u h (w P ih 2d ih 2") + (1 b )u h (w P ih d ih ") q m b [u h (w P ih d ih 2" + v G) u h (w P ih 2` 2" G B)] :
We observe that h (P ih ; d ih ; G) < 0 if d ih = v = 0, which re ‡ects the obvious fact that no audit is required to dissuade fraud if the defrauders have nothing to earn by postponing their claims. If d ih and/or v are large enough for auditing to be necessary, then we have h (P ih ; d ih ; G) 2 (0; 1) and @ h =@G < 0. We focus on this case in what follows. The repairer agrees to collude if his expected gain from collusion is positive, that is, if

G ih B 0 0:
The optimal side-payment o¤er from the policyholder to the car repairer is thus

G = ih B 0 .
The policyholder is indi¤erent between defrauding (through an optimal hidden agreement with the car repairer) and not defrauding if ih = h (P ih ; d ih ; ih B 0 ). This equation has a single solution ih = ih (P ih ; d ih ) 2 (0; 1) with ih > h (P ih ; d ih ; ih ) i¤ ih > ih (P ih ; d ih ).

We thus have ih = 1 -respect. ih 2 (0; 1)

; ih = 0 -if ih < ih -respect. ih = ih ; ih > ih .
Hence ih is the audit probability (for serious claims) above which type h individuals and repairers are deterred from colluding, when insurance has been purchased through distribution channel i.

Equilibrium fraud and audit

Let L 1 and L 2 be the expected repair costs, conditionally upon the occurrence of one or two accidents respectively, with 27

L 1 = (q m + 2q s )`; L 2 = 2(q 2 m + 2q 2 s + 3q m q s )`:
The expected cost of claims may be written as

C ih = L ( 1 + 2 2 )d ih + F C ih + AC ih ; (2)
where L = 1 L 1 + 2 L 2 is the expected repair cost, F C ih is the expected cost of fraudulent claims, and AC ih is the expected audit cost. Thus L ( 1 + 2 2 )d ih is the share of the expected repair cost borne by the insurer, and F C ih + AC ih is the total cost of fraud. Let us express F C ih and AC ih as functions of fraud and audit strategies. We have

F C ih = q m ih ( 1 + 2 )(1 ) fq m b [(1 ih )(d ih + v) 2 ih (` d ih )] (1 q m b )(` d ih )g: (3)
A policyholder who intends to renew his contract may try to defraud if his …rst accident is minor and if it is in the non-suspicious period, which occurs with probability

q m ( 1 + 2 )(1
). He then postpones his claim with probability ih , and he will actually have the opportunity to defraud with probability q m b . In that case, fraud will be detected with probability ih , and no insurance indemnity will be paid for the two minor claims. With probability 1 ih , fraud is not detected and the additional cost to the insurer is d ih + v. If the policyholder does not have the opportunity to defraud (which occurs with probability 1 q m b ), he just loses the indemnity for the …rst claim ` d ih .

Furthermore, we have AC ih = N ih c i , where N ih is the number of audits per type h policyholder for distribution channel i. Audits are concentrated on the …rst claims that 2 7 If one single accident occurs, it is minor with probability qm and serious with probability qs, with costs ànd 2`, respectively . In the case of two accidents, both of them are minor with probability q2 m and cost 2`, or both are severe with probability q 2 s and cost 4`, or one is minor and the other one is serious with probability 2qmqs and cost 3`. correspond to serious accidents. Policyholders have at least one accident, the …rst one being serious, with probability q s ( 1 + 2 ). In addition, opportunistic policyholders who intend to renew their contract …le a fraudulent claim with probability q 2 m ih 2 (1 ). 28 Serious accident claims are audited with probability ih . Thus, we have

AC ih = N ih c i = ih c i [q s ( 1 + 2 ) + q 2 m ih 2 (1 )]: (4)
The audit probabilitiy ih is chosen in [0; 1] by the insurer to minimize the expected cost of claims C ih . We thus have ih = 1 -respect. ih 2 (0; 1)

; ih = 0 -if ih < ih -respect. ih = ih ; ih > ih -where ih = (d ih ; c i ), with (d; c) q s c( 1 + 2 ) 2 q 2 m (1 )(2` d + v c) : (5)
ih is the threshold fraud rate such that the insurer is incentivized to audit claims if and only if ih ih . We have ih 2 (0; 1) if c i is not too large, and we focus attention on this case in what follows.

At equilibrium, the decisions of the policyholder-repairer coalition and of the insurer should be mutual best responses. The equilibrium is in mixed strategies: insurers audit claims with a probability that makes the potential defrauder (here the policyholder-repairer coalition) indi¤erent between defrauding and not defrauding, and symmetrically, the fraud rate makes insurers indi¤erent between auditing and not auditing. This is stated in Proposition 1.

Proposition 1 When insurers o¤ er contract (P i1 ; d i1 ); (P i2 ; d i2 ) through i 2 fD; Ag, the equilibrium fraud rates and the equilibrium audit strategies are ih = (d ih ; c i ) and ih = ih (P ih ; d ih ), respectively.

Corollary 1 For any distribution channel i 2 fA; Dg, we have i1 > i2 i¤ d i1 > d i2 , i.e., the larger the deductible, the larger the fraud rate.

Corollary 1 is a direct consequence of Proposition 1 because (d; c) is increasing in d.

The larger the deductible, the smaller the insurer's incentives to audit the claim, and thus the larger the minimal fraud rate that incentivizes the insurer to perform audits. In particular, everything else given (and in particular for a given distribution channel), the model predicts a larger fraud rate for deductible contracts than for full coverage contracts.

When ih = (d ih ; c i ), the expected cost of an insurance policy purchased by type h individuals through channel i is

C ih = L ( 1 + 2 2 )d ih + k 0 (d ih + k 1 ) (d ih ; c i );
where

k 0 = q m ( 1 + 2 )(1 ) 2 (0; 1) and k 1 = q m b v + `(1 q m b ).
Insurers price their contracts with the loading factor > 0. Thus, we have

P ih = (1 + )C ih = (1 + )[L ( 1 + 2 2 )d ih + k 0 (d ih + k 1 ) (d ih ; c i )];
which may be written more compactly as

P ih = (d ih ; (d i1h ; c i ));
where

(d; ) (1 + )[L ( 1 + 2 2 )d + k 0 (d + k 1 ) ]:
The equilibrium contract (P ih ; d ih ) maximizes u h (P; d) subject to P = (d; (d; c i )), and the equilibrium fraud rates are ih = (d ih ; c i ) for h 2 f1; 2g,i 2 fA; Dg.

Proposition 2 The optimal insurance contracts are such that

d i1 d i2 0, with d i1 > d i2 if d i2 > 0 for i = A or D.
The extent of coverage is the result of a trade-o¤ between the incentives to audit claims and the transaction costs, materialized by the fact that the fraud rate (d; c) is increasing in d and by the loading factor , respectively. In the absence of transaction costs, overcoverage would be optimal. 29 We have excluded overcoverage so that full coverage would be optimal if there were no transaction costs. However, transaction costs reduce the optimal insurance coverage. Type 1 individuals are less risk averse than type 2 individuals, and thus Proposition 2 states that their deductible is higher, as in the usual comparative statics of deductible contracts (see Schlesinger (2013)). The trade-o¤ between increasing audit incentives and reducing transaction costs may tip in favor of positive deductibles for type 1 and full coverage for type 2, and in that case deductible and no-deductible contracts are simultaneously o¤ered at equilibrium. 30 Proposition 3 At equilibrium, we have D1 > A1 ; D2 > A2 ; that is, for both types of individuals the fraud rate is larger among insurance policies purchased through D than through A.

The intuition of Proposition 3 is the following. Insurers need additional incentives to audit claims when insurance policies have been purchased through D than through A, because establishing the truth is more costly in the …rst case than in the second (i.e., c D > c A ).

These additional incentives emerge when the fraud rate is higher, which corresponds to the fact that (d; c) is increasing with c, hence at equilibrium there is a higher fraud rate for D than for A. The proof of Proposition 3 shows that this basic intuition remains valid if we take into account the fact that optimal deductibles may di¤er between both cases (i.e., we may have d Dh 6 = d Ah ), which also a¤ect incentives.

There is a threshold

x h 2 [0; 1] such that type h individuals located at x 2 [0; 1] choose i = D if x < x h ,

and they choose

i = A if x > x h . We have P ih = (d ih ; ih ) for i 2 fD; Ag. Hence u h ( (d Dh ; Dh ); d Dh ) tx h = u 1 ( (d Ah ; Ah ); d Ah ) t(1 x h );
and thus the market share of D and A are characterized by the threshold

x h = 1 2 + u h ( (d Dh ; Dh ); d Dh ) u h ( (d Ah ; Ah ); d Ah ) 2t ; for h = 1; 2. Let us consider the case d A1 > 0; d D1 > 0; d A2 = d D2 = 0, i.e.,
D = 2 x 2 1 x 1 + 2 x 2 = 2 2 1 x 1 + 2 ; A = 2 (1 x 2 ) 1 (1 x 1 ) + 2 (1 x 2 ) = 2 2 1 (1 x 1 ) + 2 ;
and x 1 < 1=2 gives D > A . We thus observe that fraud by type 1 policyholders causes a distortion between distribution channels.

Proposition 4 If d i1 > 0 and d i2 = 0 for i = A; D and if v is not large enough to induce fraud by type 2 individals, then the proportion of full coverage contracts is larger among insurance policies purchased through D than through A.

The data

Our data comes from the largest insurance company in Taiwan, with an automobile insurance market share of over 20%. Data is recorded at the individual level and provides detailed information about the policyholders, their insurance contracts and the claims they have …led. Available variables are listed in Table 1. Data has been collected over the 2010-2012 period, but our analysis will be restricted to 2010, so that we know whether policyholders subsequently renewed their contracts for less or more than one year. We target the owners of private usage small sedans and small trucks with type A, B or C insurance contracts for automobile physical damages. There are 109,461 policyholders, and 10.33% of them …led at least one claim in the year 2010, which corresponds to 11,248 observations. This subset de…nes our "research sample" (i.e., the sub-sample of policyholders with claims).

(Insert Tables 1, 2-1 and 2-2 here)

The mean values of the variables in the two samples are displayed in separates the research sample into two subgroups, according to the insurance distribution channels (DOA and non-DOA), with signi…cant di¤erences in terms of gender, usage, and vehicle size. There is also a higher proportion of new vehicles for the DOA channel, which re ‡ects the fact that DOAs sell both vehicles and insurance contracts. On average, the bonusmalus coe¢ cient is signi…cantly higher in the DOA group than in the non-DOA group, but insurance premiums do not signi…cantly di¤er between the two groups. 31 Furthermore, the percentage of insured parties who belong to the SG group is signi…cantly higher in the DOA channel than in the non-DOA channel, for SG1 (70.29% and 51.78%, respect.) as well as SG2 (6.35% and 3.01%, respect.) The percentage of claims …led in the suspicious period (de…ned as the last month of the policy year) is 54.66% in the non-DOA channel, and it rises to 67.19% in the DOA channel. We may also observe that the share of the RG group is signi…cantly lower in the DOA channel (34.28% vs 59.92%). Finally, the percentage of no-deductible contracts sold through the DOA channel is larger than that in the non-DOA channel (93.63% vs 91.34%).

3 1 It is indeed well known in Taiwan that individuals with less favorable claim records (and thus with a higher bonus-malus coe¢ cient) tend to purchase insurance through a DOA, and that some DOAs may unduly protect their customers from a strict enforcement of the bonus-malus rule.In Taiwan, a driver's bonus-malus coe¢ cient is public information and thus, in principle, an insurance company should be in the position to adjust the premium in each policy year even if policyholders switch from one insurance company to another. However, in practice, for the sake of keeping customers or winning new customers, insurance companies sometimes do not strictly enforce the bonus-malus rule. This is particularly the case when DOAs make use of their bargaining power to act on behalf of their customers so as to protect them from an increase in premium.

Testing hypotheses

Our …rst hypothesis is related to the fact that the perspective of contract renewal and the choice of a deductible contract are factors that stimulate fraud. Since we focus on the fraudulent behavior that consists of manipulating the claim date by moving it to the end of the policy year (possibly by …ling one claim for two events), we de…ne the fraud rate as the percentage of claims in the suspicious period. 32 Hypothesis 1: The fraud rate is higher in the suspicious group than in the nonsuspicious group, and this is particularly the case for individuals covered by deductible contracts.

According to Hypothesis 1, the policyholders from suspicious groups SG1 and SG2 have a larger propensity to …le a claim during the suspicious period than the other policyholders (the control group). Testing this hypothesis amounts to identifying whether there is a conditional dependence between belonging to the suspicious group SG (or to one of its subgroups SG1 and SG2), and …ling a claim within the suspicious period (evaluated by the dummy SC). To do so, we use a two-stage method, similar to the approach that has been followed by Dionne et al. (1997, 2001) in a setting with asymmetric information about risk types. 33 For notational simplicity, SG; SG1; SG2 also denote dummies for belonging to suspicious groups SG; SG1 and SG2; respectively. Testing the conditional dependence between SG1 and SC and between SG2 and SC leads us to instrument two decisions at the …rst stage.

One is the decision to renew the contract or not (SG = 0 or 1), 34 and the other refers to the contract choice (deduct = 0 or 1). These two decisions lead to the two endogenous variables:

SG1 = 1 if SG = 1 and deduct = 0; 0 otherwise. ; SG2 = 1 if SG = 1 and deduct = 1; 0 otherwise.
SG and deduct are estimated at the …rst stage by bivariate Probit regressions, with an instrumental variable approach. This requires …nding out some factors that are related to the decisions of renewing the contract and choosing full coverage, in addition to the underwriting and pricing variables, and that are unrelated to the decision of …ling a suspicious claim.

Firstly, the income of an individual may a¤ect his decisions about the contract and about his mobility between insurers, but there is no obvious reason for which income should be related to the decision of …ling a claim during the last policy month. The income level is thus a …rst candidate instrumental variable. Unfortunately, our database does not provide information about the income level of each policyholder. As an approximation, we will use a variable that corresponds to the average income level in the zip code area of the policyholder. We will separate the wealthiest areas from the poorest ones through a dummy variable income i , indicating that insured i lives in an area where the average income is in the top 25% tranche over the whole country.

Secondly, the insurance distribution channel may a¤ect the contract choice as well as the renewal decision. In particular, DOAs own large market shares in Taiwan, they have tight relationships with their customers, and they are in a favorable position to push customers to renew their contracts and to choose more expensive no-deductible contracts. This potential in ‡uence of DOAs will be taken into account through the dummy variable dealer i , indicating that individual i lives in a zip code area where the density of DOAs is in the top 25% tranche.

There is no obvious reason for which the propensity to …le a claim during the suspicious period would be a¤ected by the local density of DOAs, which leads us to treat dealer i as another candidate instrumental variable. 35 Accordingly, at stage 1 Bivariate Probit regressions are written as

Pr(SG i = 1jincome i ; dealer i ; X i ) = ( inc income i + d dealer i + X i + " SGi ); (6)
Pr(deduct i = 1jincome i ; dealer i ; X i ) = ( inc income i + d dealer i + X i + " dedti ); (7) cov(" SGi ; " dedti ) = ; (8)
where X i is the column vector of underwriting and pricing variables for policyholder i, including: gender and age of the policyholder, usage, brand, size and age of the insured vehicle. This is the …rst group of explanatory variables in Table 1.

At stage 2, we estimate the probability that policyholders …le their …rst claim during the suspicious period. We explore the conditional dependence between SC and SG1 and between SC and SG2, by considering Pr(SG1 i ) Pr(SG i = 1; deduct i = 0) and Pr(SG2 i )

Pr(SG i = 1; deduct i = 1) as explanatory variables in a second stage Probit regression, which is written as

Pr(SC i = 1j Pr(SG1 i ); Pr(SG2 i ); RG i ; X i ) = ( instr1 Pr(SG1 i ) + instr2 Pr(SG2 i ) + r RG i + X i ); (9)
where SC i = 1 when policyholder i …les his …rst claim during the suspicious period and SC i = 0 otherwise. To control for the possibility that last policy-month claims may result from a premium recouping behavior, we also use the control variable RG i (with RG i = 1 when the contract is of type A or B and has not been subsequently renewed for more than one year, and RG i = 0 otherwise). Here, also, X i is the column vector that contains …rst group explanatory variables of Table 1. Dionne et al. (2001) argue that the explanatory variables of stage 2 regression should also include dummies for the variables instrumented at stage 1. We refer to this method as the DGV approach. It leads to the following stage 2 regression:

Pr(SC i = 1j Pr(SG1 i ); Pr(SG2 i ); SG1 i ; SG2 i ; RG i ; X i ) = ( instr1 Pr(SG1 i ) + instr2 Pr(SG2 i ) + s1 SG1 i + s2 SG2 i + r RG i + X i ): (10)
In the 2SLS approach (regression (9)), the conditional dependence between SG1 and SC as well as between SG2 and SC is evaluated through the estimated coe¢ cients of Pr(SG1 i )

and Pr(SG2 i ), i.e., by instr1 and instr2 , respectively. In the DGV approach (regression (10)), the conditional dependence is evaluated by the overall sum of the estimated coe¢ cients of Pr(SG1 i ) and SG1 i and the sum of the estimated coe¢ cients of Pr(SG2 i ) and SG2 i , i.e., by instr1 + s1 and instr2 + s2 , respectively. 36 (Insert Table 3 here)

The …rst stage bivariate Probit estimations are listed in the two …rst columns of Table 3, with intuitive results. The inhabitants of higher income areas have a lower probability of continuing the same contract, and a higher probability of purchasing a deductible contract. This is consistent with a decreasing absolute risk aversion assumption: in a setting where individuals may have partial information on the quality of insurance contracts, less risk averse individuals are less reluctant to move from an insurer to another one, and they also tend to choose lower coverage. Furthermore, inhabitants of high DOA density areas are more likely to continue the same contract and their probability to choose a deductible contract is lower, which is consistent with the presumption that DOAs tend to steer their customers in a direction that is in their own interest. Policyholders from the premium recouping group are less likely to continue the same contract for at least one year (which simply re ‡ects the de…nition of RG), and they tend to opt for deductible contracts. We also see that the owners of larger vehicles are comparatively more likely to renew their insurance contract and to opt for a contract with a deductible.

The results of the second-stage estimation by the 2SLS approach with the Probit model are reported in the third column of Table 3. They show the conditional dependence between SC and either SG1 or SG2, with coe¢ cients 1:5688 and 2:0479 that are signi…cant at the 1% level. The fourth column corresponds to the second stage of the DGV approach. The estimated coe¢ cients of Pr(SG1) and Pr(SG2) are signi…cantly di¤erent from 0 at the 1% level, 3 6 As a preliminary step, the 2SLS approach requires testing (1) whether there is a weak instrument problem by the Anderson-Rubin test, (2) whether the instrument is over-identi…ed by Sargan's J test, and …nally (3) whether the instrumental variable method is relevant by the Durbin-Wu-Hausman test. Dionne et al. (2014) state that estimating the conditional probability of the instrumented variable through LPM or through the Probit model is qualitatively consistent with the 2SLS approach. Estimating Pr(SGi = 1jincomei; dealeri; Xi) and Pr(deducti = 1jincomei; dealeri; Xi) by two LPMs and performing these three tests validates our instrumental variable approach. The results of these tests are in Table 8 in the Appendix. which con…rms the existence of an endogeneity problem. The dummy variables SG1 and SG2 are also signi…cantly di¤erent from 0 at the 5% and 1% threshold, respectively, which con…rms the conditional dependency between SC and SG1 or SG2, with total coe¢ cients 1:3591 = 1:0427 + 0:3164 and 2:1910 = 1:7074 + 0:4836, respectively. Thus, the 2SLS and DGV approaches lead to similar conclusions, and they con…rm our presumption of a positive conditional dependence between belonging to SG1 or SG2 and …ling a …rst claim during the suspicious period, which supports Hypothesis 1.

Remark 1: Table 3 also o¤ ers some interesting byproducts that are worth mentioning.

Firstly, the owners of vehicles that are new or less than three years old tend to …le their …rst claim during the suspicious period, which re ‡ects the "car wash" phenomenon in Taiwan's insurance market. Secondly, the policyholders from the premium recouping group also tend to …le their …rst claims in the suspicious period, which echoes the conclusions of Li et al. (2013). Thirdly, females also …le their …rst claim during the suspicious period more frequently than males, but that does not necessarily re ‡ect a gender e¤ ect in fraudulent behavior. It is usual in Taiwan to register cars under the name of females (e.g., a wife or mother), even when the primary driver is a male, in order to bene…t from cheaper insurance premiums.

Hence, instead of a gender e¤ ect, the above mentioned correlation may just re ‡ect the fact that the policyholders who carefully manage their insurance budget may also try to obtain undue advantage from their insurance company.

For the sake of robustness veri…cation, we have also followed the approach of Chiappori and Salanié (2000). They use a pair of Probit regressions to explain the probability of …ling a claim and the probability of choosing partial coverage, and they appraise the conditional dependence between these two variables by submitting the residuals of the two regressions to a W test. Similarly, two sets of pairwise Probit regressions have been run. The …rst ones are performed among the policyholders who are either in SG1 or in neither SG1 nor SG2, and they aim at estimating the conditional correlation between SG1 and SC. The second set of pairwise regressions are performed among the policyholders who are either in SG2 or in neither SG1 nor SG2, and they allow us to estimate the conditional correlation between SG2 and SC. Two W statistics, calculated with the residuals of the two regressions in each set, are signi…cantly di¤erent from 0 at the 1% threshold. We have also calculated the correlation coe¢ cient of these residuals for each set, and …nd that both are positive and signi…cantly di¤erent from 0 at the 1% level. 37 Hence, the empirical results from this robustness test also support to our Hypothesis 1.

If defrauders postpone their claims to the suspicious period and if they may cumulate losses in a unique claim, then the suspicious period should be characterized by high values of …rst-claim cost ratios. This is expressed in Hypothesis 2.

Hypothesis 2: The …rst-claim cost ratio is larger in the suspicious period than during the rest of the policy year, particularly for the suspicious group.

Hypothesis 2 is tested through the following regression:

clmamt i = c SC i + f f irst i + f s f irst SC i + X i ; (11)
which is performed among the claims …led by members of the SG group, where clmamt i is the value (in US dollars) of the claims …led by policyholder i. In regression (11), we use two additional variables (f irst i and f irst i SC i ) besides SC i and vector X i . f irst i = 1 when this is the …rst claim …led by policyholder i during the policy year, otherwise f irst i = 0 and f irst SC i is an interaction variable. We perform the above test separately for SG1 and SG2. In our sample, this corresponds to 9,741 claims …led by 7,489 policyholders from the SG1 group, and 763 claims …led by 639 policyholders from the SG2 group. The estimated coe¢ cient of the interaction term b f s is the key to test. We obtain b f s = 149:37 with p value 0.1014 for SG1, and b f s = 249:33 with p value 0.0658 for SG2. To some extent, these results con…rm the validity of Hypothesis 2, with a lower signi…cance level for SG1 than for SG2. 38 To complete this veri…cation, we run the regression that explains the value of the claims over the whole sample (not only the SG group) by including dummies

SG1 i ; SG2 i ; SC i ; f irst i ;
and their double and triple interaction terms in the explanatory variables. Furthermore, in order to be able to identify fraud (as de…ned above, that is claims manipulation) and the premium recouping behavior, we also include RG i , and the double and 3 7 Computing the W statistic with the residuals of the regressions for SG1 and SC yields W = 201:76, which is signi…cantly di¤erent from 0 at the 1% level. The correlation coe¢ cient between the residuals of these regressions is = 0:003611, and it is also signi…cantly di¤erent from 0 at the 1% level. Likewise, using the residuals from the regressions for SG2 and SC gives W = 257:99 and = 0:03221, and these statistics are signi…cantly di¤erent from 0 at the 1% level. The full regression results are not reported here because of space constraints, but they are available from the authors upon request.

3 8 This is consistent with the fact that the policyholders with deductible contracts (i.e., the SG2 subgroup) have a greater incentive to …le a unique claim for two events than the policyholders with no-deductible contracts (the SG1 subgroup).

triple interaction variables RG i SC i , and RG i SC i f irst i among explanatory variables. Remark 2: It is worth observing that these conclusions derived from regression ( 12)

clmamt i = s1 SG1 i + s2 SG2 i + RG RG i + c SC i + f f irst i + cf SC i f irst i + sf 1 SG1 i f irst i + sf 2 SG2 i f irst i + Rf RG i f irst i + sc1 SG1 i SC i + sc2 SG2 i SC i + Rc RG i SC i + scf 1 SG1 i SC i f irst i + scf 2 SG2 i SC i f irst i + Rcf RG i SC i f irst i + X i : (12
should not be attributed to (ex ante) moral hazard or adverse selection. Ex ante moral hazard explains why a more comprehensive insurance coverage may make a driver less cautious. This incentive e¤ ect is even stronger for policyholders who had no accidents before the suspicious period, because the bonus-malus system forgives the …rst accident. Hence, under the moral hazard hypothesis, the policyholders from the SG1 group (i.e., those with a no-deductible contract) should be less cautious than those from SG2 (the policyholders with a deductible contract), and according to the moral hazard interpretation, they should have more severe …rst accidents in the last policy month. Regression (12) predicts exactly the contrary.

Let us investigate now how adverse selection may a¤ ect our results. Firstly, in a setting with hidden information about risk types, past and future claim experiences may be linked, but man-made claim manipulation should reduce the predictive power of this link. To check if this is actually the case, we use the 2010 data to run two Probit regressions that estimate the probability of …ling a claim either in any month of 2011 or in the suspicious period of 2011, respectively. The regressions were run separately for the suspicious and non-suspicious groups. 40 Observing the policyholders'2011 claim records allows us to calculate the prediction error for the claims …led in all of 2011 and for the claims …led in the suspicious period of 2011. In a second stage, we use a t-test to evaluate whether this prediction error is smaller for the claims …led over the whole year than for those …led in the suspicious period. 41 Panel A of Table 4 con…rms that this is the case, at the same time for both the suspicious and non-suspicious groups. Furthermore, the di¤ erence of the prediction error is signi…cantly di¤ erent and larger in absolute value in the suspicious groups, especially in SG2, than in the non-suspicious group. This con…rms the manipulation of claims, beyond any possible hidden information about policyholders' risk types.

Secondly, we know that adverse selection may lead to a positive correlation between the contract coverage and the probability of …ling claims, but it does not induce any particular timing for claims such as the one on which we are focusing. Panel B of Table 4 provides the hazard rate in the suspicious groups SG1 and SG2, and in the non-suspicious group.

In SG1 and SG2, the hazard rates are signi…cantly higher in the last policy month than in the other months, and these last month hazard rates are signi…cantly higher than in the non-suspicious group, which con…rms that claim manipulation does occur. The fact that the last month hazard rate is even larger for SG2 than for SG1 con…rms that our observations cannot be attributed to adverse selection.

(Insert Table 4 here)

Remark 3: We may also be worried by the fact that SG2 includes two types of deductible contracts, with more extensive exclusions for type B than for type A. To control for any disturbances that may be linked to this di¤ erence in the scope of coverage, we perform a robustness test, in which we limit our sample to type-B contracts. 42 The empirical results are listed in Table 5. Basically, the results are consistent with those of Table 3, which con…rms the robustness of our results.

(Insert Table 5 here)

Beyond the mere fact that fraud does exist, estimating its cost is also important. To included in the vector of observable variables Xi.

do this, we refer to the empirical results from the DGV model. The estimated coe¢ cients of Pr(SG1) and SG1 are 1:0427 and 0:3164 (see the fourth column in Table 3), and their marginal e¤ects are 0:3806 and 0:1155, respectively. This implies that, overall, the probability of …ling a claim in the suspicious period increases by 49:61% when comparing a policyholder from the SG1 group to those in the non-suspicious group. The average cost of non-detected fraudulent claims is NT$5; 027 if we presume that fraud is committed by …ling a unique claim for two events, postponed to the last month of the policy year to avoid the penalty from the bonus-malus rule. 43 This implies that the di¤erence in annual fraud cost between members of the SG1 group and policyholders from the non-suspicious group is about NT$2; 494.

Likewise, the estimated coe¢ cients of Pr(SG2) and SG2 are 1:7074 and 0:4836, with marginal e¤ects 0:6232 and 0:1765, respectively, which implies that the probability of …ling a claim in the suspicious period increases by 79:97% when we compare members of the SG2 group to policyholders from the non-suspicious group. The average cost of a fraudulent claim is NT$10; 027, once again with the assumption that defrauders …le a unique claim for two events and postpone their claim to the last month of their policy year. 44 This implies that the policyholders from the SG2 group entail an expected cost of fraud that is about NT$8; 019 higher than for the insured from the non-suspicious group. Since there are 7; 489 and 639 policyholders in SG1 and SG2, respectively, we may deduce that the expected cost of fraud is about NT$23; 801; 707, which represents 8:74% of the total premiums paid by the policyholders from our sample (NT$2:724 billion). These are of course very crude estimates, but they give an idea of the cost of fraud through claims manipulation in Taiwan.

Our third hypothesis is inspired by Proposition 3; it relates the fraud rate to the insurance distribution channel.

Hypothesis 3: The fraud rate in the suspicious group is comparatively even larger when insurance has been purchased through the DOA channel than through other distribution channels.

Testing the validity of Hypothesis 3 will follow the same approach as for Hypothesis 1. Dummy D i indicates that policyholder i has purchased insurance through the DOA channel, and now three endogenous variables, SG i ; deduct i and D i ; must be instrumented in the 2SLS approach. As previously, SG i and deduct i are instrumented by income i and dealer i through bivariate Probit regressions, leading to SG1 i = Pr(SG i = 1; deduct i = 1)

and SG2 i = Pr(SG i = 1; deduct i = 0): Furthermore, dealer i and income i are also candidate instruments for D i , because a large DOA density may be an encouragement to purchase insurance from a DOA, and because people with high income may have larger search costs, which may lead them to purchase insurance from a DOA. This is particularly the case when individuals purchase a new car, hence a third instrumental variable new i , which indicates that the insured vehicle is less than three years old. Thus, D i is instrumented by:

Pr(D i = 1jincome i ; dealer i ; new i ; X 2i ) = ( icm income i + d dealer i + new new i + X 2i ): (13)
Stage 2 of the 2SLS approach is now written as:

Pr(SC i = 1j Pr(SG1 i ); Pr(SG2 i ); Pr(D i ); RG i ; Pr(D i ) Pr(SG1 i ); Pr(D i ) Pr(SG2 i ); Pr(D i ) RG i ; X 2i ) = ( instr1 Pr(SG1 i ) + instr2 Pr(SG2 i ) + D Pr(D i ) + r RG i + Dinstr1 Pr(D i ) Pr(SG1 i ) + Dinstr2 Pr(D i ) Pr(SG2 i ) + Dr Pr(D i ) RG i + X 2i ); (14)
with Pr(SG1 i ); Pr(SG2 i ), and Pr(D i ) Pr(D i = 1) being estimated at stage 1. In particular, we include interaction terms Pr(D i ) Pr(SG1 i ) and Pr(D i ) Pr(SG2 i ) in order to evaluate whether the conditional dependence between SG1 and SC and between SG2 and SC are comparatively higher in the DOA channel. The premium recouping e¤ect and its interaction with the DOA channel are also taken into account through RG i and Pr(D i ) RG i , respectively.

At Stage 2 of the DGV approach, the explanatory variables include the dummy variables SG1 i ; SG2 i and the two estimated variables Pr(SG1 i ); Pr(SG2 i ), with interaction terms to assess whether the conditional dependence between SG1 and SC and between SG2 and SC is a¤ected by the DOA channel. This is written as:

Pr(SC i = 1j Pr(SG1 i ); Pr(SG2 i ); SG1 i ; SG2 i ; Pr(D i ); RG i ; Pr(D i ) Pr(SG1 i ); Pr(D i ) Pr(SG2 i ); Pr(D i ) SG1 i ; Pr(D i ) SG2 i ; Pr(D i ) RG i ; X 2i ) = ( instr1 Pr(SG1 i ) + instr2 Pr(SG2 i ) + S1 SG1 i + S2 SG2 i + D Pr(D i ) + r RG i + Dinstr1 Pr(D i ) Pr(SG1 i ) + Dinstr2 Pr(D i ) Pr(SG2 i ) + DS1 Pr(D i ) SG1 i + DS2 Pr(D i ) SG2 i + Dr Pr(D i ) RG i + X 2i ):
The results are in Table 6. 45 The …rst column lists the estimated coe¢ cients of the …rst stage regression for Pr(D): they con…rm that individuals living in areas with high average income and high DOA density tend to purchase insurance through the DOA channel. This is also the case for the owners of vehicles that are less than three years old.

The 2SLS and DGV Probit regressions for SC are in the second and third columns. In the 2SLS Probit model, the estimated coe¢ cients of Pr(SG1) and Pr(SG2) are 0:6843 and 1:0644, and they are signi…cantly di¤erent from 0 at the 10% and 5% levels, respectively.

The estimated coe¢ cients of SG1 Pr(D )and SG2 Pr(D) are 1:0477 and 1:1005, and they are signi…cantly di¤erent from 0 at the 1 % level. All in all, this con…rms that there is a signi…cant conditional dependence between belonging to the suspicious group and …ling claims during the suspicious period. This conditional dependence is even stronger among the insured who have purchased insurance through the DOA channel, and these e¤ects are stronger for SG2 than for SG1. In other words, we may conclude that the fraud phenomenon associated with the claim date manipulation does exist, and that it is more severe among those individuals with deductible contracts and who have purchased insurance through the DOA channel, which con…rms the prediction from Hypothesis 3.

(Insert Table 6 here)

The third column of Table 6 corresponds to the DGV model. The most relevant results are the following. The estimated coe¢ cients of Pr(SG2) and SG2 are equal to 0.8930 and 0.1930, with signi…cance level 1% and 10%, respectively, and an overall e¤ect of 1.0860. The estimated coe¢ cients of Pr(SG1) and SG1 are equal to 0.4762 and 0.0845, respectively, but the second one is not signi…cant. The coe¢ cients of interaction terms Pr(D) Pr(SG1) and Pr(D) Pr(SG2) are equal to 0.7134 and 0.9773 and they are signi…cant at the 1% level.

Similar conclusions are obtained for Pr(D) SG1 and Pr(D) SG2, with signi…cance levels of 10% and 5%, respectively. It is particularly interesting to observe that each coe¢ cient in the DOA channel is larger than its equivalent in the non-DOA channel, which con…rms the role of DOAs in the fraud process. Furthermore, whatever the distribution channel, the SG2 coe¢ cients are larger than their SG1 equivalents, which con…rms that deductible contracts exacerbate fraudulent behaviors.

Calculation shows that the marginal e¤ect of the estimated coe¢ cients of Pr(D i ) SG1 i and Pr(D i ) Pr(SG1 i ) are equal to 0:1016 and 0:2517, which implies that, in the SG1 group, the probability of …ling a claim during the suspicious period is 35.33% larger when policyholders have purchased insurance through the DOA channel than through another channel. Thus, if the expected cost of a fraudulent claim by an SG1 policyholder is NT$5; 027, as we have already estimated, then the expected fraudulent claim cost of such policyholders is NT$1; 776 larger when insurance has been purchased through the DOA channel than through another channel. The marginal e¤ect of the estimated coe¢ cients of Pr(D i ) SG2 i and Pr(D i ) Pr(SG2 i ) are equal to 0:1214 and 0:3448, thus with a 46.62% larger probability of …ling a claim in the suspicious period for a member of the SG2 group who has purchased insurance through the DOA channel rather than through another channel. For an expected cost of fraudulent claims in the SG2 group equal to NT$10; 027, this amounts to an increase of NT$4; 675 in the expected cost of fraud when an SG2 policyholder takes out insurance from a DOA rather than through another distribution channel.

At the end, we may calculate the increase in fraud cost for each suspicious subgroup by comparison with the non-suspicious group. For example, 6; 323 policyholders in SG1

have taken out insurance through the DOA channel, with an expected increase in fraudulent claiming of NT$2; 770, and hence a total additional cost of Hence the suspicious policyholders in SG1 and SG2 who have purchased insurance through the DOA channel are at the origin of an increase in the cost of fraud that can be estimated at NT $22; 377; 917, which corresponds to about 8.22% of the premium written by this company for this line of business. The estimated increase in fraud cost is only $1; 420; 260, that is 0.52% of the premium written.

A legitimate question that may arise is whether the higher expected cost of claims in the DOA channel comes from fraudulent behaviors, as we have argued so far, or whether it rather re ‡ects the fact that, on average, the individuals who take out insurance from DOAs have higher risks. This issue may be clari…ed by estimating the claim amount using the following OLS regression:

claimamt i = 0 + D D i + A A i + B B i + DA D i A i + DB D i B i + X i X + " i : (15)
A i and B i are dummies for type A or B contracts, respectively, with type C contract as counterpart, and X i includes the underwriting and pricing variables as in the previous regressions. The estimated results are in Table 7. Type A and B contracts are associated with claim costs that are signi…cantly larger than for type C contracts. The estimated coe¢ cient of D i is negative, but it is not signi…cantly di¤erent from 0. Likewise, the estimated coe¢cients of interaction terms D i A i , and D i B i are not signi…cantly di¤erent from 0. In other words, the policyholders of the DOA channel do not have higher claim costs than others, whatever their contract. Hence, the increase in claim costs is not an intrinsic characteristic of the distribution channel: it re ‡ects the fraudulent behaviors of some policyholders (the suspicious groups) who may take advantage of the manipulation of claims, and this behavior is facilitated by DOAs.

Remark 4: Apart from these main results, Table 6 also provides two interesting byproducts that are common to the 2SLS and DGV Probit models. Firstly, the estimated coe¢ cient of RG i is positive and signi…cantly di¤ erent from 0, at least at the 10% level, which con…rms the existence of the premium recouping behavior. However, the estimated coe¢ cients of the interaction term RG i Pr(D i ) have negative signs that are not signi…cant.

Thus, compared to other distribution channels, DOAs do not particularly help opportunistic policyholders to recoup premiums at the end of the policy year. Their behavior, as an act of collusion, rather focuses on the manipulation of the claim date. Secondly, the estimated co-e¢ cient of f emale i is positive and signi…cant. This con…rms that fraudulent behaviors may be widespread among those individuals who carefully manage their budget, since the declared gender of the owner of the car may be manipulated to take advantage of a lower premium. This is consistent with our previous observation made regarding Table 3.

Hypothesis 4: The proportion of no-deductible contracts is larger when contracts are sold through the DOA channel.

Finally, Hypothesis 4 re ‡ects Proposition 4. We test this hypothesis by using the whole sample (not only the subsample of policyholders with claims). In a …rst step, we estimate the probability of choosing the DOA channel (i.e., Pr(D i )) by regression ( 13), and in a second step we test the following regression:

Pr(nodedt i = 1j Pr(D i ); B i ; X 2i ) = ( D Pr(D i ) + B B i + X 2i ); (16)
where nodedt i = 1 when policyholder i chooses a no-deductible contract, and nodedt i = 0 otherwise. Contracts may also di¤er through di¤erent exclusions, and the additional dummy B i is used to control for this heterogeneity. Estimation gives b D = 1:2499, with a p value smaller than 0.0001, which con…rms the validity of Hypothesis 4.

We may also control for the unobservable heterogeneity between individuals, particularly in terms of risk type through a two-stage method. In the …rst stage, we test

Pr(clm i = 1jincome i ; dealer i ; new i ; X 2i ) = ( icm income i + d dealer i + new new i + X 2i ); (17)
where clm i = 1 if individual i has …led a claim and clm i = 0 otherwise. In the second stage,

we use the DGV model to control for unobservable heterogeneity. This is written as :

Pr(nodedt i = 1j Pr i (clm i ); clm i ; Pr(D i ); B i ; X i ) = ( ec Pr(clm i ) + c clm i + D Pr(D i ) + B B i + X i ); (18)
where Pr(clm i ) Pr(clm i = 1) and Pr(D i ) is the estimated probability of choosing the DOA channel. We obtain b D = 1:0111 with a p value smaller than 0.0060, which once again con…rms the previous result, but at a lower signi…cance level.

Conclusion

This paper has focused attention on the policyholder-service provider coalition in insurance mechanisms: how it can a¤ect the credibility of claim auditing, how several patterns of fraud may emerge in the car insurance market, and how service providers and policyholders may draw bene…t from such a coalition. The important role of car dealers in Taiwan provides an exceptional opportunity to analyze this interaction between insurer, policyholder and provider.

Indeed, the economic analysis of insurance fraud is usually based on a very abstract picture of claims fraud (…ling a fraudulent claim although no accident has occurred, or exagerating a claim), but in practice understanding insurance fraud often requires a much more speci…c analysis of the claims fraud process. The Taiwan case o¤ers such a possibility, with fraud frequently taking place through the manipulation of the claim's date in order to avoid a penalty from the bonus-malus system and to reduce the burden of a second deductible, should another accident occur.

On the theoretical front, we have analyzed how claim monitoring is a¤ected by the collusion between policyholders and car repairers, how deductible contracts and the speci…c rules of the Taiwanese bonus-malus regime may be incitements to fraud, and how fraud may induce distortions in the proportions of deductible and no-deductible contracts depending on the distribution channel. On the empirical front, we have established that the intertemporal manipulation of claims was actually a signi…cant determinant of insurance fraud in Taiwan.

In particular, we have shown that policyholders with deductible contracts who intend to renew their policies (the suspicious group) have a larger propensity to defraud than other policyholders, by postponing their claims until the last month of the policy year, and possibly by merging two events into a single claim. Consequently, there is an increase in the average cost of …rst claims …led by the suspicious group in the last month of the policy year. We have also shown that the collusion between policyholders and DOAs is a crucial mechanism that contributes to the development of fraud in the Taiwanese car insurance market. Finally, we have also shown that such mechanisms induce a distortion toward a lower proportion of deductible contracts among the DOAs'customers than among individuals who purchase insurance through other distribution channels.

Our analysis has made progress toward a more complete analysis of claims fraud behavior, but it also opens new perspectives for further research. The theoretical modelling could be re…ned, for instance, by considering a more realistic set of possible losses (say a continuum instead of two levels) or by analyzing the intertemporal distribution of claims in a more satisfactory way than considering each subperiod (suspicious and non-suspicious) as a whole. It would also be of great interest to analyze the insurer-car repairer vertical relationship (including vertical integration) which would be the most e¢ cient when there is a risk of fraud and collusion. Finally, insurance fraud is just one case where collusion induces additional transaction costs in the vertical relationships between producers and retailers.

Other examples include discount fraud and warranty fraud. More theoretical and empirical research is required to evaluate the costs associated with such fraudulent behaviors, and to better understand how they can be detected and prevented most e¢ ciently. ih = ih . Symmetrically, ih = ih 2 (0; 1) is an optimal choice of the policyholder if ih = ih . Thus ih = ih ; ih = ih is the unique equilibrium.

Proof of Proposition 2

Let the expected utility of type h = 1; 2 policyholders who purchase insurance through

(d; ) = (1 + )[L ( 1 + 2 2 )d + k 0 (d + k 1 ) ]; (d; c) = Kc(2` d + v c) 1 ; with K q s ( 1 + 2 )= 2 q 2 m (1 ). Let e i (d) (d; (d; c i )):
We have

e 0 i (d) = (1 + )[ ( 1 + 2 2 ) + k 0 (d; c i ) +k 0 (d + k 1 )Kc(2` d + v c) 2 ]; e 00 i (d) = 2Kck 0 (1 + )(2` d + v c) 3 (2`+ v c + k 1 ) > 0:
Thus, we have

e u ih (d) = (1 1 2 )u h (w e i (d)) + 1 u h (w e i (d) d ") + 2 u h (w e i (d) 2d 2"); e u 0 ih (d) = (1 1 2 )u 0 h (w e i (d)) e 0 i (d) 1 u 0 h (w e i (d) d ")(1 + e 0 i (d)) 2 u 0 h (w e i (d) 2d 2")(2 + e 0 i (d)):
Using e 00 i (d) > 0 and u 00 h < 0 shows that e u ih (d) is a concave function. Let d ih be the optimal deductible for type h individuals, i.e., d ih maximizes e u ih (d) with respect to d 0. Assume …rst that d i1 > 0, which implies e u 0 i1 (d i1 ) = 0 and e 0 i (d i1 ) < 0 . We have

e u 0 i2 (d i1 ) = (1 1 2 )u 0 2 (w e i (d i1 )) e 0 i (d i1 ) 1 u 0 2 (w e i (d i1 ) d i1 ")(1 + e 0 i (d i1 )) 2 u 0 2 (w e i (d i1 ) 2d i1 2")(2 + e 0 i (d i1 )):
Since type 2 individuals are more risk averse than type 1 individuals, we know from Pratt (1964) that there exists a function g : R ! R such that u 2 (y) g(u 1 (y)), with g 0 > 0 and g 00 < 0. This allows us to write

e u 0 i2 (d i1 ) = (1 1 2 )g 0 (u 1 (y 0 ))u 0 1 (y 0 ) e 0 i (d i1 ) 1 g 0 (u 1 (y 1 ))u 0 1 (y 1 )(1 + e 0 i (d i1 )) 2 g 0 (u 2 (y 2 ))u 0 1 (y 2 )(2 + e 0 i (d i1 ));
where y 0 = w e i (d i1 ); y 1 = w e i (d i1 ) d i1 " and y 2 = w e i (d i1 ) 2d i1 2", with y 2 < y 1 < y 0 . Let us …rst consider the case where 1 + e 0 i (d i1 ) > 0. Let y 2 (y 1 ; y 0 ). Using g 00 < 0 and u 0 1 > 0 yields g 0 (u 1 (y 0 )) < g 0 (u 1 (y )) < g 0 (u 1 (y 1 )) < g 0 (u 1 (y 2 )):

Using e 0 i (d i1 ) < 0 < 1 + e 0 i (d i1 ) then gives e u 0 i2 (d i1 ) < g 0 (u 1 (y ))e u 0 i1 (d i1 ) = 0; which implies d i2 < d i1 because of the concavity of e u i2 (d). Similarly, when 1 + e 0 i (d i1 ) < 0 < 2 + e 0 i (d i1 )
, we let y 2 (y 2 ; y 1 ) and a similar argument also yields d i2 < d i1 . Similarly, if d i1 = 0, we have e u 0 i1 (0) 0, and the same argument gives e u 0 i2 (0) < 0 and thus d i2 = 0.

Example with mean-variance preferences

The case d i1 > 0; d i2 = 0 can be conveniently illustrated by a mean-variance example.

Assume that u 1 (w f ) and u 2 (w f ) are quadratic, so that we may write

u h (w f ) = E(w f ) h V ar(w f );
with 2 > 1 . When insurance is purchased through distribution channel i, we have

E(w f ) = w e 0 i (d) ( 1 + 2 2 )(d + "); V ar(w f ) = I(d + ") 2 ; with I = 1 (1 1 ) + 4 2 (1 1 2 ) > 0. We have d ih > 0 i¤ e 0 i (0) ( 1 + 2 2 ) 2 h I" 2 > 0;
and thus we have

d i1 > 0; d i2 = 0 if 1 < < 2 ; where = 1 2I" 2 [ ( 1 + 2 2 ) (1 + )k 0 Kc(2`+ v c) 2 (2`+ v c + k 1 )]: Proof of Proposition 3 Let u h ( (d; ); d) h (d; ):
Let us write the equilibrium fraud rate as

(d; c) = Kc(2` d + v c) 1 ; with K q s ( 1 + 2 )= 2 q 2 m (1 ). Assume u h ( (d; (d; c)); d) = h (d; (d; c)) is maximized w.r.t. d at d = b d h (c) with fraud rate b h (c) ( b d h (c); c). Thus, we have ih = b h (c i ) ( b d h (c i ); c i ) for i 2 fA; Dg.
The …rst-order and second-order optimality conditions for this maximization are respectively written as

F h 0 hd + 0 h @ @d = 0; (19)
and

S h 00 hd 2 + 00 hd @ @d + 00 h 2 @ @d 2 + 0 h @ 2 @d 2 < 0; (20)
where 0 hd ; 0 h ; 00 hd 2 ; 00 hd ; 00 h 2 denote …rst and second derivatives of h and all functions are evaluted at

d = b d h (c). Di¤erentiating (19) gives b d 0 h (c) = F 0 hc S h ;
where

F 0 hc = @F h @c = 00 hd @ @c + 00 h 2 @ @c @ @d + 0 h @ 2 @d@c : After simplication we get b 0 h (c) = @ @d b d 0 h (c) + @ @c = (1=S h ) 0 h @ 2 @d 2 @ @c @ 2 @d@c @ @d + 00 hd 2 @ @c : (21)
We have

@ @d = Kc(2` d c) 2 ; @ @c = K(2` d c) 1 + Kc(2` d c) 2 ; @ 2 @d 2 = 2Kc(2` d c) 3 ; @ 2 @d@c = K(2` d c) 2 + 2Kc(2` d c) 3 : Hence @ 2 @d 2 @ @c @ 2 @d@c @ @d = K 2 c(2` d c) 4 > 0: Furthermore, h (d; ) = (1 1 2 )u h (w (d; )) + 1 u h (w (d; ) d) + 2 u h (w (d; ) 2d):
(d; ) is linear in d, and thus h (d; ) is concave in d, which implies 00 hd 2 < 0. We also have 0 h = @u h @P @ @ < 0:

Using (21) and 0 h < 0; 00 hd 2 < 0; @ =@c > 0 then yields b 0 h (c) > 0. Thus, we have

D1 = b (c D ) > b (c A ) = A1 . 47 such that Eu F ih = Eu N ih with G = ih (1 i )B 0 :
Since defrauders who are caught are not punished with probability i , the expected actuarial cost of a deductible insurance policy is now written as

F C ih = q m ih ( 1 + 2 )(1 ) fq m b [(1 ih )(d ih + v) 2 ih (1 i )(` d ih )] (1 q m b )(` d ih )g: (24)
The equilibrium audit and fraud strategies are ih = (d ih ; c i ; i ) and ih = ih (P ih ; d ih ; i ),

with ih (d ih ; c i ; i ) q s c i ( 1 + 2 ) 2 q 2 m (1 )[(1 )(2` d ih ) + i d + v c i ) ; (25)
which can be interpreted in the same way as (5) in Section 3.4 and extends Proposition 1 in a straightforward way.

= q 2 m ih [d ih + 2(1 i )(` d ih )]d ih .
We consider the case where the decrease in cost is larger when the deductible is lower, which requires i < 1=2.

consistent with the results obtained in the 2SLS approach.

The second stage regressions show that …ling a suspicious claim and continuing the contract (or choosing a deductible contract) are conditionally dependent decisions, with a positive signi…cant dependence. The estimated coe¢ cient of Pr(SG) is 1.5457, and it is significantly di¤erent from 0 at the 1% level. The estimated coe¢ cient of Pr(deduct) is 1.2647, and it is also signi…cantly di¤erent from 0 at the 1% level, which indicates that policyholders who renew their contracts and have chosen a deductible contract have a larger propensity to …le claims during the suspicious period. bonus Bonus-malus coefficient used to calculate the premium in the current contract year. It is a multiplier on the premium. Hence, it is a discount if it is smaller than 1 and it is a penalty if it is larger than 1.

Explanatory variables (Second group): income

Dummy variable equal to 1 if the insured lives in an area with average income in the top 25% tranche, and 0 otherwise.

B

Dummy variable equal to 1 when the insured is covered by a type B contract, and 0 otherwise. 6

RG

Dummy variable equal to 1 when the insured belongs to the "recoup group",7 and 0 otherwise.

Notes

(1) In the above models Pr(SG1 i ) and Pr(SG2 i ) are the estimated probabilities of belonging to the suspicious groups SG1 and SG2, respectively, calculated at the first stage, that is Pr(SG1 i )= Pr(SG i =1, deduct i =0), and Pr(SG2 i ) = Pr(SG i =1, deduct i =1). In the DGV-Probit model, SG1 and SG2 are dummy variables equal to 1 if the policyholder belongs to the suspicious groups SG1 and SG2, respectively, and 0 otherwise.

(2) In all the above regressions, we have also controlled for the brand of the insured car. This is not reported for reasons of confidentiality.

(3) ***,** and * indicate that the estimated coefficient is significantly different from 0 at the 1%, 5% and 10% level, respectively.

(4) We have also performed two sets of the 2SLS-LPM to confirm the validity of our IV model. In both sets, the null hypothesis of irrelevant model is rejected by the Durbin-Wu-Hausman test, the null hypothesis of exogenous instrumental variable cannot be rejected by the Anderson-Rubin test, the null hypothesis of no over identification cannot be rejected by the J test.

  type 1 individuals chooose deductible contracts and type 2 individuals choose full coverage contracts, because the former type is less risk averse than the latter. We have u 1 ( (d D1 ; D1 ); d D1 ) < u 1 ( (d A1 ; A1 ); d A1 ) because c D > c A , which gives x 1 < 1=2. Suppose that v is not large enough to induce fraud by type 2 individuals. In this case, we would have D2 = A2 = 0, and thus x 2 = 1=2. The proportion of full coverage contracts among the contracts sold through D and A are respectively

)

  Performing this regression among the 14,797 claims …led by the members of the research sample gives b scf 1 = 369:97 with p value <0.0154, b scf 2 = 532:57 with p value <0.0001, and b Rcf = 10:11 with p value <0.0581. 39 The inequalities b scf 2 > b scf 1 > 0 once again validate Hypothesis 2. Symmetrically, b Rcf < 0 con…rms that members of the RG group tend to …le small claims at the end of the policy year, when they have not …led any claim during the previous months.

i

  = A; D be written as e u ih (d) u h ( (d; (d; c i )); d); where (:) and (:) are de…ned by

Figure 1 :

 1 Figure 1: Distribution of claims and average claim cost (first claims) in the policy year

Figure

  Figure 5: Distribution of claims in the policy year

dealer

  Dummy variable equal to 1 if the insured lives in an area with DOA density in the top 25% tranche, and 0 otherwise.

new

  Dummy variable equal to 1 when the insured car is less or equal to three year old, and 0 otherwise.logpremLogarithm of the premium of the contract in the current contract year.

D

  Dummy variable equal to 1 if the insurance contract is sold through the DOA channel, and 0 otherwise.

A

  Dummy variable equal to 1if the insured is covered by a type A contract, and 0 otherwise.

  Table2-1, with some signi…cant di¤erences. In particular, the research sample includes a larger proportion of female owners, medium-sized and new vehicles, with a larger share for one brand. More importantly, the research sample includes a larger fraction of policyholders who belong to the SG2 group and who have purchased insurance through the DOA channel than the whole sample ( 5.68% vs 2.85%, and 79.98% vs 50.78%, respectively). The share of the RG group also increases from 19.8% in the whole sample to 39.42% in the research sample. Table2-2

  The equilibrium fraud rates are ih = e h (c i ; i ) ( e d h (c i ; i ); c i ; i ) for i 2 fA; Dg. We have (similarly to the proof of Proposition 3, with an unchanged de…nition for S h ): 4 8 For a given fraud rate ih , the decrease in actuarial cost dC ih < 0 induced by a small increase in the audit probability d ih > 0 is dC ih

The equilibrium contract (P ih ; d ih ) maximizes u h (P; d) subject to P = (d; (d; c i ; i )), and the equilibrium fraud rates are ih = (d ih ; c i ; i ) for i = A or D. In the same way as in the proof of Proposition 3, we can then show that Dh > Ah if c D = c A and D > A . To establish this result, we make the additional assumption D < 1=2. 48 The de…nition of (d; ) is unchanged, and we still denote h (d; ) u h ( (d; ); d). h (d; (d; c; )) is maximized w.r.t. d at d = e d h (c; ) with fraud rate e h (c; ) ( e d h (c; ); c; ).

Table 2 -1: Structure of the full sample and of the sub-sample with claims

 2

		Full sample	Sub-sample with	Difference
		(1)	claims (2)	(2)-(1)
	claim	0.1033
	SC		0.6468
	RG	0.1980	0.3942	0.1962***
	nodedt	0.9514	0.9317	-0.0167***
	A	0.0103	0.0140	0.0037***
	B	0.3882	0.8876	0.4994***
	SG	0.7179	0.7226	-0.1262***
	SG1	0.6894	0.6658	-0.0236***
	SG2	0.0285	0.0568	0.0283***
	D	0.5078	0.7998	0.2920***
	female	0.7118	0.7600	0.0482***
	age2025	0.0032	0.0028	-0.0004
	age2530	0.0342	0.0464	0.0122
	age3060	0.8947	0.8929	-0.0018
	ageabv60	0.0679	0.0579	-0.0100
	carage0	0.2192	0.5116	0.2924***
	carage1	0.1381	0.2023	0.0705***
	carage2	0.0915	0.0642	-0.0273***
	carage3	0.1109	0.0665	-0.0444***
	carage4	0.0986	0.0438	-0.0548***
	veh_m	0.2875	0.2256	-0.0619***
	veh_l	0.2692	0.2696	0.0004
	tramak_n	0.0069	0.0057	-0.0012
	tramak_f	0.0609	0.0586	-0.0552
	tramak_h	0.0805	0.0531	-0.0274
	tramak_t	0.4692	0.6262	0.1570***
	tramak_c	0.0415	0.0120	-0.0295
	sedan	0.9166	0.9398	0.0232
	logprem	9.2346	10.0632	0.8286***
	bonus	0.7180	0.8773	0.1593***
	No of obs. No of obs. No of obs. No of obs.	109,461	11,248

Table 2 -2: Structure of the DOA and non-DOA subsamples DOA (1)

 2

	Non-DOA	Difference
	(2)	(1)-(2)

Table 3 : Empirical evidence of fraud First stage (bivariate Probit) Second stage

 3

		SG	deduct	2SLS-Probit	DGV-Probit
	constant	1.4264***	-1.7669***	0.2160	0.0735
	Pr(SG1)			1.5688***	1.0427***
	Pr(SG2)			2.0479***	1.7074***
	SG1				0.3164**
	SG2				0.4836***
	income	-0.0752**	0.3093***
	dealer	0.0161***	-0.1339***
	RG	-2.0748***	0.3028***	1.0057***	0.1517***
	female	0.0941***	-0.1455***	0.0847***	0.0821***
	age2530	0.2405	0.3783	-0.2256	-0.2375
	age3060	0.2930	0.1659	-0.1790	-0.1920
	ageabv60	0.0973	0.1112	-0.3126	-0.3187
	carage0	0.4225***	0.0570**	0.2332***	0.2513***
	carage1	0.1009*	0.0440*	0.1966***	0.1944***
	carage2	0.1192	-0.0603	0.2427***	0.2404***
	carage3	0.1052	-0.0768	0.2076***	0.2067***
	carage4	0.0425	-0.0313	0.0552	0.0555
	veh_m	0.0127	0.0147	-0.0097	-0.0100
	veh_l	0.1046***	0.2310***	-0.0557*	-0.0605*
	sedan	-0.0350	0.2751***	-0.0201	-0.0200
	Pseudo		0.0542	0.0451	0.0465

Table 4 : Additional evidence of fraud

 4

		SG1	SG2	non-SG
		Panel A: Predicted errors
	filing a claim	0.1201	0.1213	0.1330
	filing SC	0.3072	0.3969	0.2639
	t test	-130 (<0.0001)	-170 (<0.0001)	-31.3673 (<0.0001)
		Panel B: Baseline hazard in each policy month
	1 st month		-	-
	2 nd month		-	0.00003
	3 rd month		-	0.00003
	4 th month		-	0.00003
	5 th month		-	0.00003
	6 th month	0.000002		0.00003
	7 th month	0.0001	0.0003	0.0015
	8 th month	0.0021	0.0019	0.0024
	9 th month	0.0033	0.0042	0.0041
	10 th month	0.0055	0.0104	0.0069
	11 th month	0.0191	0.0326	0.0345
	12 th month	0.0538	0.0874	0.0457

Table 5 : Empirical evidence of fraud -Focus on type B contracts First stage (bivariate probit) Second stage

 5

		SG	deduct	2SLS-Probit	DGV-Probit
	constant	1.2881***	-1.8345***	0.7050**	0.6609**
	Pr(SG1)			1.3076***	1.0288***
	Pr(SG2)			1.9137***	1.6741***
	SG1				0.2407**
	SG2				0.3573***
	income	-0.0610***	0.3272***
	dealer	0.0172*	-0.0966**
	RG	-2.9190***	0.1962***	1.0022***	0.1487***
	female	0.1065***	-0.1667***	0.0828***	0.0822***
	age2530	0.3279	0.7018	-0.3714	-0.3701
	age3060	0.3536	0.4452	-0.3829	-0.3814
	ageabv60	0.0529	0.3848	-0.5057	-0.5007
	carage0	0.8014***	-0.0019	0.2920***	0.0150
	carage1	0.1876**	-0.0781	0.1607***	0.1603***
	carage2	-0.1434	-0.1629	0.1543***	0.1557***
	carage3	-0.0972	-0.2115	0.0981	0.0985
	carage4	-0.2555	-0.0338	-0.0446	-0.0424
	veh_m	0.0339	0.0078	-0.0338	-0.0326
	veh_l	0.0642*	0.0351	-0.0951***	-0.0945***
	sedan	0.0448	0.1995**	-0.0281	-0.0278
	Pseudo		0.0521	0.0352	0.0373
	Same notes as in Table 3

Table 6 : Empirical evidence of fraud through DOAs First stage (Instrument on D)

 6

		Second stage
	2SLS-Probit	DGV-Probit

Table 7 : Comparing the policyholders' risk between the DOA and non-DOA channels

 7 In the above regression, we have also controlled the brand of the insured car. The results are not reported for reasons of confidentiality.

	Variables	Est. Ceoff.	P value
	Intercept	23.2634	0.4353
	D	-1.8105	0.3308
	A	17.6766	0.0001
	B	4.1090	0.0050
	D*A	1.7407	0.7093
	D*B	1.6078	0.6758
	Female	-2.0447	0.0028
	age2025	4.0914	0.8924
	age2530	1.3474	0.9639
	age3060	-5.8571	0.0097
	ageabv60	1.4598	0.9609
	carage0	0.2970	0.7804
	carage1	1.6699	0.1395
	carage2	1.4368	0.3232
	carage3	3.4444	0.0147
	carage4	3.6011	0.0264
	veh_m	2.8894	0.0001
	veh_l	12.6377	<.0001
	sedan	10.6254	<.0001
	Adjusted	0.0605

Note:

Table 8 : Empirical results from 2SLS-LM 2SLS-LPM

 8

		SG	SC	deduct	SC
	constant	0.9114***	9.7965***	0.0654	0.0956
	Pr(SG)		1.5457***
	Pr(deduct)				1.2647***
	income	-0.0326*		0.3541***
	dealer	0.0622**		-0.1790***
	RG	-0.6093***	0.1829***	0.0362***	0.1977***
	female	0.0213***	0.3130 ***	-0.0183***	0.1126***
	age2530	0.0487	0.2796	0.0597	-0.3104
	age3060	0.0603	0.4492	0.0308	-0.2286
	ageabv60	0.0112	-0.2022	0.0246	-0.3570
	carage0	0.1186***	1.0114***	0.0060*	0.2330***
	carage1	0.0166	0.3753**	0.0044	0.1922***
	carage2	0.0200	0.4569**	0.0070	0.2509***
	carage3	0.0144	0.3641**	-0.0096	0.2208***
	carage4	-0.0011	0.0441	0.0001	0.0523
	veh_m	0.0036	-0.0271	0.0026	-0.0096
	veh_l	0.0229***	-0.1785***	0.0244***	-0.0978***
	sedan	0.0043	0.0716	0.0371***	0.0798
	J test	0.3066		0.1591
	Anderson-Rubin test	0.4732		0.3820
	Durbin-Wu-Hausman test	0.0220		0.0420
	Pseudo/Adjusted	0.4570	0.0510	0.0713	0.0420

Type B contracts cover all the areas of type-A contracts, except the non-collision losses caused by intentional damage, vandalism, and any unidenti…ed reasons.

It is well known that insurance fraud is often associated with the feeling that the insurance company is unfair; seeFukukawa et al. (2007),[START_REF] Miyazaki | Perceived ethicality of insurance claim fraud : do higher deductibles lead to lower ethical standards[END_REF] and[START_REF] Tennyson | Economic institutions and individual ethics : a study of consumer attitudes toward insurance fraud[END_REF][START_REF] Tennyson | Insurance experience and consumers'attitudes toward insurance fraud[END_REF]. The premium recouping phenomenon could re ‡ect a kind of resentment against insurers, particularly from policyholders who have not …led any claim during the policy year.

Because of the bonus-malus system (see below), the policyholders who renew their contract only one year have the same incentive to defraud as the policyholders who switch insurers at the end of the policy year.1 0 In addition, the bonus-malus system forgives the …rst accident for drivers who have had no other accidents for three years, which provides an even larger manipulation gain.

1 Type C contracts only cover the risk of collision. Thus, their claims involve a third party, which makes manipulation di¢ cult.

5 We will assume that all serious claims (for i and h given) are audited with the same probability ih . In other words, the audit frequency is not conditional on whether the claim is …led during the suspicious or non-suspicious period. This seems to be a realistic assumption insofar as the beginning of the policy year varies across individuals, and conditioning auditing on the date of the claim in the policy year of each individual would probably entail substantial transaction costs. Be that as it may, concentrating audits on the suspicious period individual by individual would increase the e¢ ciency of the fraud deterrence mechanism, but this would not qualitatively a¤ect our conclusions.2 6 The formula would be almost unchanged if the …rst accident also occurs in the suspicious period. In such a case, the gain from collusion would be lower (v should be replaced by a lower collusive gain v 0 ) because the advantage from bonus-malus fraud would be lower. Consequently, defrauding by …ling a single claim for two minor accidents in the suspicious period does not occur for the equilibrium audit strategy.

8 Indeed, the policyholder has two minor accidents, the …rst one in the non-suspicious period and the second one in the suspicious period, with probability q 2 m 2 (1 ). He does not …le a claim immediately after the …rst accident with probability ih .

9 See[START_REF] Boyer | Overcompensation as a partial solution to commitment and renegotiation problerms: the case of ex post moral hazard[END_REF].

0 See the illustrative example with mean-variance preferences after the proof of Proposition 2 in Appendix 1.

2 Of course, this does not mean that all the claims …led in the suspicious period are fraudulent. 3 3 They aim at appraising whether the correlation between claims and coverage re ‡ects individuals'unobservable characteristics, which are not used by insurers in underwriting and pricing decisions. In order to avoid a spurius correlation caused by misspeci…cation, they add the conditional expectation of one decision variable (such as …ling a claim) when regressing on the other one (such as choosing the insurance coverage). To avoid endogeneity problems,[START_REF] Dionne | Asymmetric information and adverse selection in Mauritian slave auctions[END_REF] and[START_REF] Dionne | An eye for an eye : investigating the impact of consumer perception of corporate unfairness on aberrant consumer behavior[END_REF] estimate this conditional expectation through an instrumental variable approach.

3 4 We de…ne the suspicious group (SG ) as the set of individuals who renew their contract at the end of the policy year.

5 Other thresholds (10%, 20%, 30%

40%, and 50%) have been tested, and it turned out that the 25% criterion yields the best instrumental variable.

9 The full estimated results of regressions (11) and (12) are available from the authors upon request.

0 In other words, these Probit regressions regress clmi and SCi, respectively, on the explanatory variables

1 The prediction error is the absolute value of the di¤erence between the estimated probability of …ling a claim and the dummy equal to 1 if the individual has …led a claim in 2011 and 0 otherwise. We calculate the di¤erence between the prediction errors over the whole 2011 year and over the suspicious period, and we test whether this di¤erence is negative.4 2 In other words, in this test, the suspicious group includes the policyholders (from the SG1 group) with a no-deductible type B contract that has been renewed at the end of the policy year, and the policyholders (from the SG2 group) with a deductible type B contract that has been renewed at the end of the policy year, while the control group contains the other policyholders with a type B contract that has not been renewed.

3 The average insurance premium in our research sample is NT$ 25,136. We may roughly estimate that the defrauders who …le a unique claim for two events and postpone their claim to the last month of their policy year avoid about 20% of this amount: 10% because of the decrease in the number of claims, and 10% because the increase in premium will be postponed for one year.4 4 This cost includes the avoided deductible and the avoided bonus-malus penalty. The deductibles of …rst and second claims are NT$3; 000 and NT$5; 000 respectively, hence there is a NT$5; 000 fraud cost when policyholders …le a unique claim to cover two accidents. Adding the NT$5; 027 avoided penalty due to the increase in premium to the deductible of the second claim yields a total fraud cost of NT$10; 027.

5 Here we have also checked the robustness of our IV method by using two sets of 2SLS-LPM and by checking that the null hypothesis of irrelevant model is rejected by the Durbin-Wu-Hausman test, the null hypothesis of exogenous instrumental variable cannot be rejected by the Anderson-Rubin test, and the null hypothesis of no-over identi…cation cannot be rejected by the J test.

6 In our research sample,

individuals from the SG2 group have taken out insurance from DOAs. There are 1,166 and 68 members in the Non-DOA group, for SG1 and SG2 respectively.

7 Note that if transaction costs prevent insurers from o¤ering contracts with deductibles that depend on the distribution channel, i.e., if we assume dA1 = dD1 d1, then the fraud rates are D1 = (d1; cD) > (d1; cA) = A1, and the proposition is obviously still valid.

The "suspicious group" (SG) includes the individuals who renew their contract with the same insurance company. The counter group for SG includes the policyholders who do not renew their contract with the same insurance company.

The "suspicious group 1" (SG1) includes the policyholders with no-deductible contract who renew their contract with the same insurance company. The counter group for SG1 includes the policyholders with deductible contract or who do not renew their contract with the same insurance company.

The "suspicious group 2" (SG2) includes the policyholders with deductible contract who renew their contract with the same insurance company. The counter group for SG2 includes the policyholders with no-deductible contract or who do not renew their contract with the same insurance company.

The insured cars in counter group for tramak_j, j= n, f, h, t, c , are other brands (other thanNissan, Ford, Honda, Toyota, and China.)

The counter group includes the insured cars are not small sedan, for example small or large truck, cargo…etc.

The contracts in the counter groups for A and B are type C contracts.

The "recoup group" includes the policyholders covered by type A or B contracts who do not renew their contract or renew it for only one year.

Appendix 1

Proof of Proposition 1

If ih > ih , then the optimal choice of the policyholder is ih = 0 < ih , which gives ih = 0 for the optimal choice of the insurer, hence a contradiction. Symmetrically, if ih < ih , then the optimal choice of the policyholder is ih = 1 > ih , which gives ih = 1 for the optimal choice of the insurer, hence once again a contradiction. Thus we necessarily have ih = ih 2 (0; 1) at equilibrium. ih = ih is an optimal choice of the insurer if Appendix 2

In this appendix, we show how the bargaining power of the policyholder-repairer coalition a¤ects the scale of fraud. The bargaining power of the colluders is taken into account in a very crude way by assuming that the defrauders will not be punished with probability i 2 (0; 1), with i = D or A. Intuitively, the insurance agent is incentivized to stand up for its customer (and possibly also for the repairer in the case of a DOA that owns the repair shop), and it may threaten the insurer to redirect its customers toward another insurer. This may deter the insurer from enforcing the penalty. A larger bargaining power for D than for A corresponds to D > A . Thus, if the colluders are spotted (which occurs if the claim is audited), then with probability 1 i the penalties are enforced (no indemnity is paid by the insurer and the colluders pay the …nes B and B 0 , respectively, and with probability i the insurer interprets the fraud as an involuntary error, i.e., the policyholder receives the total cumulated contractual indemnity 2(` d ih ) and no …nes are paid. Under these assumptions, a type h policyholder with two minor accidents and a repairer are willing to defraud (with the policyholder making a side-payment G to the repairer on a take-it-or-leave-it basis) if

and

respectively, with

Fraud is deterred (i.e., ih = 0) if ih > ih where ih (P i1 ; d ih ; i ) is the value of ih @e (c; )

We have

, and thus

Using S h < 0; 0 h < 0; 00 hd 2 < 0; @ =@ > 0; (26) and ( 27) yields @e (c; )=@ > 0. Thus, Appendix 3

(Insert Table 8 here)

As shown in Table A1, the p-values of the Anderson-Rubin test in the two …rst-stage LPMs are 0.4732 and 0.3820, respectively, and the null hypothesis of the exogenous instrumental variable cannot be rejected. Secondly, the p-values of the J test in these two LPMs are 0.3066 and 0.1591, and we cannot reject the null hypothesis that the instruments are not over identi…ed either. Thirdly, the p-values of the Durbin-Wu-Hausman test in these two LPMs are 0.0220 and 0.0420, respectively, and we can reject the null hypothesis that the instrumental variable method is irrelevant at the 5% level for each instrumented variable.

All in all, this con…rms the properness of our IV approach. Table A1 also yields some byproducts: policyholders living in high income areas and in areas with a high density of DOAs signi…cantly tend to renew their contracts and to choose policies with deductibles, which is

Explained variable: SG

Dummy variable equal to 1 when the insured belongs to the "suspicious group", 1 and 0 otherwise.

SG1

Dummy variable equal to 1 when the insured belongs to "suspicious group 1", 2 and 0 otherwise.

SG2

Dummy variable equal to 1 when the insured belongs to "suspicious group 2", 3 and 0 otherwise.

nodedt

Dummy variable equal to 1 when the insured has taken out a no-deductible contract, and 0 otherwise.

SC

Dummy variable equal to 1 when the insured has filed his or her first claim during the suspicious period (in the last policy month), 0 otherwise.

Explanatory variables: First group (underwriting and pricing factors) female

Dummy variable equal to 1 if the insured is a female, 0 otherwise.

age2025

Dummy variable equal to 1 if the insured is in the 20-25 age group, 0 otherwise.

age2530

Dummy variable equal to 1 if the insured is in the 25-30 age group, 0 otherwise .

age3060

Dummy variable equal to 1 if the insured is in the 30-60 age group, 0 otherwise.

ageabv60

Dummy variable equal to 1 if the insured is older than 60, 0 otherwise.

carage0

Dummy variable equal to 1 when the car is less than one year old, 0 otherwise.

carage1

Dummy variable equal to 1 when the car is two years old, 0 otherwise.

carage2

Dummy variable equal to 1 when the car is three years old, 0 otherwise.

carage3

Dummy variable equal to 1 when the car is four years old, 0 otherwise.

carage4

Dummy variable equal to 1 when the car is more than four years old, 0 otherwise.