Collusion Between Retailers and Customers: The Case of Insurance Fraud in Taiwan

Abstract

This study analyzes how the insurance distribution channel can affect insurance fraud. It uses econometric models that confirm the existence of claim manipulation as a form of insurance fraud, whereby policyholders circumvent the bonus–malus system and reduce the actual burden of insurance deductibles. The econometric approach is based on joint regression models for the probability that a claim is manipulated on one hand, and the probability that the policyholder has strong incentives to do so, on the other hand. The estimation shows that there is a significantly positive residual correlation between these regressions, which establishes the likelihood of fraudulent claim manipulation. The econometric modelling of claim cost allows us to disentangle the manipulation of claims that correspond to true losses and small false claims filed at the end of the policy year, and also to highlight the role of the insurance distribution channel in these fraud mechanisms. Using data from two Taiwanese car insurers with very different distribution channels in 2010, we compare an insurer that relies heavily on dealer-owned agents (DOAs) with another insurer that does not rely on DOAs at all. We find strong evidence of severe claim manipulation when insurance is sold through DOAs. Moreover, as the first insurer significantly reduced its reliance on the DOA channel over time, we perform a before–after comparison using data from 2010 and 2018. The results show that the claim manipulation fraud previously observed in the DOA channel decreases as the market share of this distribution channel is reduced. All these results highlight the role of automobile insurance agencies in facilitating this fraud process. The theoretical underpinnings of our analysis are provided by a claim fraud model considering collusion and audit.

1. Introduction

Vertical relationships often involve outsourcing services from upstream companies to downstream retailers. Due to the way the latter operate, this can be the source of agency costs and may involve collusion between retailers and customers, who exploit loopholes in contracts between producers and customers. Discount fraud and warranty fraud are examples of such inappropriate customer behaviors that involve collusion with retailers or front-line employees. Discount fraud exploits special discounts that companies may offer in particular circumstances, for instance, when discounted products are used for a specific purpose, such as educational purposes for software. Warranty fraud occurs mainly when a service provider—such as an automobile repairer—replaces a defective part with a new spare part and triggers the manufacturer’s warranty, although the defective part was not original and therefore not protected by the warranty.

This study investigates another form of customer misbehavior facilitated by service providers acting on behalf of distributors: insurance fraud. We focus our attention on the Taiwanese car insurance market and on the role of car dealer-owned insurance agents (DOAs). In these cases, car dealers both sell automobile insurance to their customers and own an automobile repair shop. Understandably, this multi-faceted activity and the long-term connection with car owners favor collusion between policyholders and DOAs when the settlement of a claim gives rise to a negotiation with the insurer. With respect to insurance fraud itself, we will focus our attention on two cases of misconduct that have been observed in Taiwan, with both taking the form of a concentration of claims during the last months of the policy year. First of all, policyholders may file small false claims (i.e., claims that do not correspond to any loss) before the end of the policy year if they have not received compensation previously, a behavior highlighted by Li et al. (2013). Recouping part of the insurance premium paid to allegedly unfair insurers can be the psychological motivation behind this behavior. Secondly, per-claim deductibles and the bonus–malus mechanism can incentivize policyholders to manipulate claims that correspond to actual losses. This involves postponing these claims to the last month of the policy year and inflating them, either by bundling several losses in a single claim or by building up cost reports. Disentangling these two types of fraud (called premium recouping and claim manipulation, respectively) and assessing the role of DOAs in both of them are the main challenges addressed in the present study.

Our analysis is built on a database obtained from two major Taiwanese insurance companies. Company 1 provided information about the policyholders who filed an automobile claim in 2010 or 2018 and company 2 in 2010 only. Company 1 relied heavily on DOAs to sell policies, although the market share of this distribution channel decreased significantly from 2010 to 2018, while company 2 never used DOAs.

Starting with the year 2010, our results confirm that there was more manipulation of claims when insurance policies were sold through DOAs than other distribution channels and also that deductibles promoted fraud.1 As observed by Li et al. (2013), the logic of the claim manipulation mechanism can be found in the Taiwanese bonus–malus system, as well as in the incentives inherent in the design of deductible contracts. We will provide evidence on this mechanism indirectly, by showing that in 2010, the intertemporal pattern of claims was consistent with the manipulation of claims favored by DOAs, after controlling for other explanations, including moral hazard and premium recouping behavior.2 Things changed dramatically between 2010 and 2018: DOAs were less frequently used by insurers, with presumably lower bargaining power at the claims settlement stage. In other words, in 2018, it was more difficult for DOAs to collude with their clients at the expense of the insurer. As we will show, the role of DOAs as facilitators of insurance fraud through claim manipulation disappeared in 2018, in line with the decrease in their bargaining power.

The theoretical foundations of this analysis are provided by an insurance market model presented in the appendix that focuses on the strategic interaction between, on one hand, policyholders who file fraudulent claims by colluding with car repairers and/or insurance agents and, on the other hand, insurers who audit claims. The audit of claims is all the more costly as collusion is more difficult to detect, which is particularly the case when car repairers are sheltered by DOAs. In addition, if irregularities are detected by the insurer, the bargaining power of DOAs may allow them to dissuade insurers from enforcing penalties. The result is a higher fraud rate when insurance is distributed by DOAs than through other channels. This is reinforced in the case of deductible contracts, as deductibles increase the gain that policyholders obtain from fraud and weaken insurers’ incentives to monitor claims.

This study is organized as follows: In Section 2, we describe the background of our analysis, both from a factual and theoretical point of view. We introduce some factual observations about the timing of automobile insurance claims in Taiwan and describe regular fraud patterns by distinguishing premium recouping from claim manipulation. We also introduce a costly state verification model that provides the theoretical foundations of our approach to insurance fraud, featuring the interaction between a policyholder who may manipulate claims and an insurer who performs audits randomly. The model shows that this interaction leads to an equilibrium level of fraud determined by parameters related to the insurance contract and the insurance agent. We here only provide an intuitive presentation of the model, with more details being given in the appendix. In Section 3, we present the methodology of our empirical analysis, describing our objectives and conjectures, the data, and our econometric approach. In Section 4, we present and discuss our results. They contribute to the econometric analysis of insurance fraud in three ways: firstly, by disentangling premium recouping and claim manipulation in the Taiwanese car insurance market; secondly, by highlighting the role of DOAs in these mechanisms; and thirdly, by showing that this role deeply changed from 2010 to 2018. Conclusions are presented in Section 4. The insurance market model that supports our analysis is presented more completely in Appendix A.

2. Background

2.1. Factual Background

Our investigation will be based on information provided by two major Taiwanese insurers, called companies 1 and 2, about their automobile policyholders and claims in 2010 and 2018. In 2010, company 1 sold approximately 37% of its automobile insurance policies through DOAs, and this share dropped to about 20% in 2018. On the contrary, company 2 never sold insurance through the DOA channel.3

Insurance agents, whether DOAs or standard agents, are responsible for handling claims. These are often negotiations between, on the one hand, the insurer who aims to minimize the cost of claims and, on the other hand, agents, who can favor their clients, especially when they receive commissions based on sales. In this bargaining process, DOAs benefit from the size of their activity and the fact that they own the list of their clients. In particular, an insurer who discovers a claim manipulation by a DOA may be reluctant to take retaliatory actions because of this strategic advantage of the DOAs, who could switch to another insurer.4 In the case of company 1, it is likely that the bargaining power of the DOAs decreased between 2010 and 2018, as this insurer significantly reduced its dependence on DOAs. The specificity of DOAs also has an informational dimension because they work in partnership with car repairers, with both being sheltered by car dealers. This multi-faceted agency relationship creates an informational advantage: establishing that a claim has been falsified is particularly difficult and costly when filed through a DOA.

Some information about insurance contracts is useful for the present investigation. There are three different types of automobile physical damage insurance contracts in Taiwan: types A, B and C. Type A and B contracts cover all types of losses in collision and out-of-collision events, with more exclusions for B than for A,5 while type C contracts only cover the damages incurred in a collision involving two or more vehicles. These contracts also differ in terms of compensation: type A contracts offer low coverage with a deductible, type B contracts can be purchased with or without a deductible, and type C contracts provide full coverage without a deductible. Claims are per accident, with a specific 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 the premium proportional to the number of claims, without reference to their severity. Policyholders who switch to another insurance company bargain with the latter about the new starting point of their bonus–malus record.

Our study is related to specific forms of automobile insurance fraud in Taiwan. Li et al. (2013) observed that a large proportion of automobile insurance claims are filed during the last months of the policy year (as opposed to the calendar year). This is confirmed by our own database. Figure 1 presents the distribution of claims and their average cost in 2010 over the twelve months of the policy year, with a striking concentration of claims and a slight decrease in claim costs in the last months.6 Li et al. (2013) interpret this phenomenon as a “premium recouping” effect: some policyholders without an accident in the previous months tend to file small false claims during the last month of the policy year, to express the feeling that they were treated unfairly by the insurance company.7 They also observe that deductible contracts and the Taiwanese bonus–malus system can encourage policyholders to postpone true claims until the end of the policy year and to cumulate several losses in a single claim, a behavior that will be called “claim manipulation” in the following.8 Type A and B contracts may be subject to such claim fraud, regardless of their modalities, contrary to type C contracts, where claims are only filed in the case of a collision with another car, with a police report being required, which makes fraud very unlikely.

2.2. Theoretical Background

A costly state verification model provides the theoretical foundations of our approach to insurance fraud. It features the interaction between a policyholder who may file fraudulent claims and an insurer who randomly audits claims. The model shows that this interaction leads to an equilibrium level of fraud determined by parameters related to the insurance contract and the insurance agent. We here only provide an intuitive presentation of the model. More details can be found in the appendix of this article.

In this model, fraud corresponds to the manipulation of claims. It is committed by policyholders who may have one or two accidents that may be small or large during the same policy year. Policyholders may postpone small claims till their last policy month. They will file one single large claim for two minor accidents presented as a severe accident if another minor accident occurs later during the same policy year. Claim manipulation reduces the retained cost of the accidents since the deductible is paid only once. It also provides an additional gain by circumventing the bonus–malus system. The insurance agent (either a DOA or a standard agent) is in charge of claim handling.

Let us denote by $\alpha$ and $\beta$ the fraud and audit strategy of the policyholder and the insurer, respectively. The policyholder’s strategy ($\alpha$) is the probability that he postpones a first small claim in order to file a large claim later, and the insurer’s strategy ($\beta$) is the probability of auditing large claims. We assume that an audit allows the insurer to detect with certainty whether the claim has been manipulated or not. If fraud is detected, then the insurer and the insurance agent (defending the interests of the policyholder) negotiate about the payment of the claim.

This defines a non-cooperative game where the policyholder chooses the fraud strategy to maximize his expected utility and the insurer chooses the audit strategy to minimize the cost of claims. At the Nash equilibrium of this game, both strategies are mutual best responses. The audit probability ($\beta$) makes the policyholder indifferent between manipulation and honesty, and the fraud probability ($\alpha$) makes the insurer indifferent between auditing and non-auditing. The equilibrium strategies of the policyholder and of the insurer may be written as ${\alpha }^{*}(d,\xi ,c)$ and ${\beta }^{*}(P,d,\xi$), respectively, where d is the deductible of the insurance contract, $\xi$ is the bargaining power of the insurance agent (DOA or standard agent) in case of negotiation with the insurer, c is the audit cost, and P is the insurance premium. Most importantly, the equilibrium fraud rate (${\alpha }^{*}$) is increasing with respect to $d, \xi$, and c, because a higher deductible, larger bargaining power of the insurance agent, or a larger audit cost makes auditing less attractive to the insurer. The larger $d, \xi$, or c, the larger the equilibrium fraud rate (${\alpha }^{*}$) that incentivizes the insurer to audit claims with positive probability.

This leads us to simple predictions about the effect of the type of contract and distribution channel on claim manipulation. Firstly, using ${\alpha }_d^{*\prime }>0$ shows that higher deductibles go along with more manipulation. In other words, for a given distribution channel, fraud is more prevalent when contracts include a deductible. Furthermore, using ${\alpha }_{\xi }^{*\prime }>0$ and ${\alpha }_c^{*\prime }>0$ shows that fraud occurs more frequently when insurance has been purchased through DOAs than standard insurance agents, either because it is more costly to audit a claim that goes through a DOA or because DOAs have larger bargaining power than standard insurance agents when they negotiate with the insurer.

3. Methodology

3.1. Objectives and Conjectures

Our overarching objective is to disentangle two distinct forms of automobile insurance fraud observed in Taiwan—premium recouping and claim manipulation—and to assess how these behaviors are shaped by contract design and insurance distribution channels, in particular, the role of dealer-owned agents (DOAs).

Building on the factual patterns documented by Li et al. (2013) and further descriptive evidence from our data, we focus on the concentration of insurance claims toward the end of the policy year. Premium recouping refers to the filing of small false claims near contract expiration by policyholders who have not previously claimed any indemnity, a behavior often interpreted as an expression of resentment or low moral cost of cheating. Claim manipulation, by contrast, involves strategic behavior surrounding real losses: policyholders postpone reporting accidents until the final policy month and inflate the reported cost of claims, either by bundling multiple losses into a single claim or by building up repair costs. Claim manipulation is particularly attractive under bonus–malus rules and per-claim deductibles.9

To operationalize these mechanisms empirically, we classify policyholders into groups based on their renewal behavior and contract characteristics. The recoup group ($RG$) includes policyholders who have not renewed their type A or B contract for more than one year after the current policy year and therefore face limited future consequences of filing a claim.10 Within $RG$, the suspicious group ($SG$) includes policyholders who have renewed their contract for exactly one additional year, a configuration under which the incentives to manipulate the bonus–malus system are the strongest.11 In 2010, we further distinguished two subgroups in $SG$: $SG1$ corresponds to policyholders with no-deductible contracts, and $SG2$ includes those with deductible contracts, reflecting differences in manipulation incentives. In 2018, all type A and B contracts include deductibles, and the distinction collapses accordingly.

In brief, $RG$ policyholders are those with presumably the lowest moral cost of fraud and potentially a propensity for trying to recoup part of their insurance premium by filing small false claims by the end of the policy year. Among them, $SG$ includes those who benefit the most from claim manipulation, by deferring actual claims to the last month of their policy year, and inflating them or bundling several losses within one single claim.

These considerations lead us to formulate testable hypotheses, where the suspicious period corresponds to the last month of the policy year, and the fraud rate is the number of claims per policyholder filed during the suspicious period.12

Hypothesis 1:

The fraud rate tends to be higher in the suspicious group than in the non-suspicious group, and this is particularly the case for individuals covered by deductible contracts.

Hypothesis 1 captures the timing dimension of fraud through claim manipulation or premium recouping. Distinguishing these two fraud patterns requires examining how the claim severity changes across the policy year. Indeed, if a substantial number of claims filed during the last policy month correspond in fact to first claims that have been postponed and inflated, possibly with the cumulated losses from several events, then these claims should be more expensive than the average. In other words, we should expect the ratio between “the average cost of first claims” and “the average cost of all claims” (hereinafter referred to as the first-claim cost ratio) to increase during this month, contrary to the premium recouping interpretation.13 Hence, conjecturing that there is a substantial amount of claim manipulation leads us to the following hypothesis.

Hypothesis 2:

In the suspicious group, the first-claim cost ratio is larger in the suspicious period than during the rest of the policy year.

Finally, our analysis emphasizes the role of DOAs in the fraud mechanism. DOAs combine insurance intermediation with close ties to repair shops and benefit from substantial bargaining power at the claims settlement stage. These features are expected to reduce the effectiveness of audits, facilitating claim manipulation, hence our third hypothesis.14

Hypothesis 3:

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

Figure 2. Distribution of claims (%) during the policy year 2010 for SG1 and SG2 according to the distribution channel, with type C contracts as the benchmark. The qualifiers “dealer” and “Ndealer” refer to the cases where the insurance policy has been sold through a DOA and through a standard agent, respectively.

Figure 2. Distribution of claims (%) during the policy year 2010 for SG1 and SG2 according to the distribution channel, with type C contracts as the benchmark. The qualifiers “dealer” and “Ndealer” refer to the cases where the insurance policy has been sold through a DOA and through a standard agent, respectively.

Figure 3. Changes in the first-claim cost ratio from months 1–11 to month 12 of the policy year 2010. SG1_first/SG1_all (dealer) is the first-claim cost ratio of contracts in the SG1 subset purchased through the DOA channel. The other variables have similar interpretations.

Figure 3. Changes in the first-claim cost ratio from months 1–11 to month 12 of the policy year 2010. SG1_first/SG1_all (dealer) is the first-claim cost ratio of contracts in the SG1 subset purchased through the DOA channel. The other variables have similar interpretations.

3.2. Data

The data of companies 1 and 2 include detailed information about the policyholders, their insurance contracts, and the claims that they filed. The available variables are listed in

Table 1. Definition of variables.

Variable	Definition
Explained variables
claim	Dummy variable equal to 1 when the insured has filed at least one claim during the policy year 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) and 0 otherwise.
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.
Explanatory variables
first group (underwriting and pricing variables)
female	Dummy variable equal to 1 if the insured is female and 0 otherwise.
age2025	Dummy variable equal to 1 if the insured is in the 20–25 age group and 0 otherwise.
age2530	Dummy variable equal to 1 if the insured is in the 25–30 age group and 0 otherwise.
age3060	Dummy variable equal to 1 if the insured is in the 30–60 age group and 0 otherwise.
ageabv60	Dummy variable equal to 1 if the insured is older than 60 and 0 otherwise.
carage0	Dummy variable equal to 1 when the car is less than one year old and 0 otherwise.
carage1	Dummy variable equal to 1 when the car is two years old and 0 otherwise.
carage2	Dummy variable equal to 1 when the car is three years old and 0 otherwise.
carage3	Dummy variable equal to 1 when the car is four years old and 0 otherwise.
carage4	Dummy variable equal to 1 when the car is five years old and 0 otherwise.
veh_m	Dummy variable equal to 1 when the capacity of the insured car is between 1800 and 2000 c.c. and 0 otherwise.
veh_l	Dummy variable equal to 1 when the capacity of the insured car is larger than 2000 and 0 otherwise.
sedan	Dummy variable equal to 1 when the car is a sedan and is for non-commercial or for long-term rental purposes and 0 otherwise. 4
bonus	Bonus–malus coefficient used to calculate the premium in the current contract year. It is a multiplier of the premium. Hence, it is a discount if it is smaller than 1, and it is a penalty if it is larger than 1.
tramak_j	Dummy variable equal to 1 when the brand of the insured car is j, with j = n, f, h, t, and c, and 0 otherwise. 5
second group (other control variables)
logprem	Logarithm 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 of company 1 and 0 otherwise.
company2	Dummy variable equal to 1 if the insurance contract is sold by company 2 and 0 otherwise. 6
AB	Dummy variable equal to 1 if the insured is covered by a type A or type B contract and 0 otherwise. 7
RG	Dummy variable equal to 1 when the insured belongs to the Recoup Group, 8 and 0 otherwise.

¹ The “suspicious group” (SG) includes individuals who renewed their contract with the same insurance company for only one year. The counter group for the SG includes the policyholders who did not renew their contract or renewed it for more than one year with the same insurance company. ² “Suspicious group 1” (SG1) includes SG policyholders with no-deductible contracts. The counter group for SG1 includes the policyholders with deductible contracts or who are not in SG. ³ “Suspicious group 2” (SG2) includes SG policyholders with deductible contracts. The counter group for SG2 includes the policyholders with no-deductible contracts or who are not in SG. ⁴ The counter group for sedan includes the insured whose cars are not small sedans, for example, small or large trucks, cargo, etc. ⁵ The counter group for tramak_j, j = n, f, h, t, and c, includes other brands (i.e., other than Nissan, Ford, Honda, Toyota, and China, respectively). ⁶ The counter groups for D and company 2 include the insurance contracts sold through other channels than DOAs and by company 1, respectively. ⁷ The counter group for type A and type B includes type C contracts. 8 The recoup group includes policyholders covered by type A or B contracts who did not renew their contract more than once after the current policy year.

. The data were collected over the periods 2010–2012 and 2018–2020, but our analysis of insurance claims will be limited to 2010 and 2018, in order to determine whether policyholders subsequently renewed their contracts for less or more than one year.15 We will start by considering the year 2010 in Section 4.1, Section 4.2, Section 4.3 and Section 4.4. As previously mentioned, company 1 has strongly reduced its dependence on DOAs from 2010 to 2018; therefore, in Section 4.5, comparing the results obtained for the years 2010 and 2018 will allow us to assess the consequence of this structural change.

We target owners of small sedans and small trucks for private use with type A, B, or C insurance contracts for physical car damage. In 2010, there were 141,739 policyholders in the whole sample, and among them, 9205 filed at least one claim, which defines what we call the “research sample”, that is to say, the sub-sample of policyholders who filed one or more claims during their policy year.16 The mean values of the variables in the two samples are displayed in the first two columns of

Table 2. Sample structure (2010).

	Whole Sample	Sub-Sample with Claim	DOAs in Company 1	No DOAs in Company 1	Company 2
claim	0.0649
SC	0.0294	0.4386	0.6628	0.2723	0.2954
RG	0.1903	0.2365	0.3165	0.2197	0.1739
AB	0.3396	0.7316	0.8979	0.6175	0.6228
C2	0.2770	0.4741	0.0000	0.0000	1.0000
SG	0.0890	0.7316	0.8979	0.6175	0.6228
SG1	0.0794	0.6670	0.8386	0.5589	0.5522
SG2	0.0095	0.0645	0.0593	0.0585	0.0706
D	0.3538	0.3978	1.0000	0.0000	0.0000
female	0.7128	0.7436	0.7758	0.7176	0.7236
age2025	0.0023	0.0022	0.0022	0.0025	0.0021
age2530	0.0303	0.0386	0.0317	0.0339	0.0456
age3060	0.8935	0.8943	0.8965	0.8872	0.8944
ageabv60	0.0087	0.0650	0.0696	0.0763	0.0580
carage0	0.1761	0.2983	0.4926	0.1383	0.1785
carage1	0.1537	0.2403	0.2387	0.2214	0.2468
carage2	0.0967	0.1010	0.0688	0.0882	0.1315
carage3	0.1205	0.1117	0.0699	0.1272	0.1425
carage4	0.1075	0.0749	0.0440	0.0941	0.0956
veh_m	0.2950	0.2589	0.2283	0.2807	0.2786
veh_l	0.2579	0.2678	0.2701	0.3070	0.2553
sedan	0.9090	0.9247	0.9612	0.8974	0.9015
lnprem	8.9909	9.5277	10.0894	9.5279	9.0563
bonus	0.8874	1.1140	0.8760	0.7154	1.4214
Observations	141,739	9205	3662	1179	4364

, with some significant differences. In particular, the percentages of type A or B contracts and particularly those in $SG1$ and $SG2$, are much larger in the research sample. It is not surprising to find that, on average, the bonus–malus coefficient and the insurance premium are higher in the research sample than in the whole sample, insofar as more frequent losses are reflected by bonus–malus records and insurance ratings. Also note that there is a higher proportion of cars under three years old in the research sample, which reflects the role of DOAs highlighted in this article. The three other columns of Table 2 separate the research sample into three subgroups, according to the insurance distribution channels (DOAs in company 1 and non-DOAs in companies 1 and 2), with a similar structure in terms of gender, age of the policyholder, size of the vehicle, and use. There is a much larger proportion of new vehicles for the DOA channel, reflecting the fact that, most of the time, a DOA sells an insurance contract when the corresponding dealer sells a new car.The percentage of claims filed in the last month of the policy year (measured by the average value of the $SC$ dummy), the percentage of type A or B contracts, and the share of $SG$ and $RG$ are higher in the DOA channel than in the two other channels.

3.3. Econometric Approach

Our econometric approach combines two complementary strands of the insurance economics literature. We rely on Dionne and Gagné (2002) for the identification of opportunistic fraud patterns, and we adopt the empirical testing framework developed by Chiappori and Salanie (2000) to implement our econometric analysis.

A central insight by Dionne and Gagné (2002) is that opportunistic insurance fraud is not randomly distributed over time but rather tends to concentrate near the end of the insurance contract, when experience-based pricing rules, deductibles, or contract renewal rules reduce the future cost of filing a claim. In such environments, the timing and cost of claims relative to the contract horizon provide informative signals of hidden opportunistic behavior, even when fraud is not directly observed. Following this logic, we leverage systematic patterns in claim timing and claim amounts near the end of the policy year to infer the presence of claim manipulation.

While this insight guides our identification strategy, our empirical implementation relies on the conditional correlation approach proposed by Chiappori and Salanie (2000). They show that under hidden information, unobserved incentives generate statistically testable conditional dependence between relevant outcomes once observable risk characteristics are controlled for. In this study, we adapt this framework to study claim manipulation under hidden information. Rather than testing for asymmetric information between insurers and policyholders in contract choice as performed by Chiappori and Salanie (2000), we examine whether unobserved incentives to manipulate claims are correlated with observable contract features and renewal behavior. Specifically, we leverage the timing of claims within the policy year and policyholders’ renewal decisions to construct groups for which the gains from manipulating the bonus–malus system and deductibles are plausibly higher. We estimate joint models for (i) the probability that a claim is filed during the suspicious period at the end of the policy year and (ii) the probability that a policyholder belongs to a group with strong incentives for intertemporal claim manipulation. A significantly positive residual correlation between these equations indicates the presence of claim manipulation behavior, consistent with the fraud patterns emphasized by Dionne and Gagné (2002). Our empirical identification of claim manipulation relies on several key assumptions that need to be highlighted. First, conditionally on observable underwriting and pricing characteristics (e.g., age, gender, vehicle age, etc.), it is assumed that the timing of actual accidents is not systematically concentrated in the final month of the policy year. Secondly, type C contracts can serve as a benchmark, because police reporting requirements and the lack of a deductible make it very unlikely that claims manipulation will occur in that case. Thirdly, our definition of the suspicious group is based on contract renewal behavior as a proxy for incentives embedded in the bonus–malus system. While renewal decisions may also reflect loyalty or switching costs, these factors alone should not produce a systematic end-of-year claim concentration if there was no intent to manipulate the insurance claims.

4. Results: Presentation and Discussion

4.1. Evidence on Claim Manipulation

Let us first test hypotheses 1 to 3 for the year 2010. Hypothesis 1 amounts to identifying whether there is conditional dependence between belonging to the suspicious group and filing a claim within the suspicious period, associated with the dummies $SG$ (or $SG1$ and $SG2$ for each subgroup) and $SC$, respectively. We do so with the following three bivariate probit models, where $\mathsf{\Phi }(.)$ is the cumulative normal distribution function and X is the vector of explanatory variables (with vectors of coefficients ${\beta }_{SC},{\beta }_{SG},\dots$), including the premium amount and all the variables used in pricing and underwriting decisions.17 In order to control for the recouping effect, the $RG$ dummy is also included in X.

Model 1

$$\mathrm{Prob}(SC=1)=\mathsf{\Phi }(X{\beta }_{SC}+\epsilon )$$

(1)

$$\mathrm{Prob}(SG=1)=\mathsf{\Phi }(X{\beta }_{SG}+\eta )$$

(2)

Model 2

$$\mathrm{Prob}(SC=1)=\mathsf{\Phi }(X{\beta }_{SC}+\epsilon )$$

(3)

$$\mathrm{Prob}(SG1=1)=\mathsf{\Phi }(X{\beta }_{SG1}+\eta )$$

(4)

Model 3

$$\mathrm{Prob}(SC=1)=\mathsf{\Phi }(X{\beta }_{SC}+\epsilon )$$

(5)

$$\mathrm{Prob}(SG2=1)=\mathsf{\Phi }(X{\beta }_{SG2}+\eta )$$

(6)

The results of these regressions are presented in

Table 3. Conditional dependence between SC and SG.

Variables	Model 1		Model 2		Model 3
	SC	SG	SC	SG1	SC	SG2
RG	0.1193 *** [0.0332]	1.5444 *** [0.0861]	0.1374 *** [0.0348]	1.3111 *** [0.0732]	0.3037 *** [0.1076]	2.1846 *** [0.1744]
female	−0.0138 [0.0317]	0.1493 *** [0.0386]	0.0034 [0.0330]	0.3138 *** [0.0393]	−0.0299 [0.0524]	0.0263 [0.0687]
age2025	0.0281 [0.2993]	−0.1512 [0.3506]	0.1413 [0.3045]	−0.1264 [0.3507]	0.2296 [0.4003]	−0.8458 [0.6399]
age2530	−0.2138 ** [0.0884]	−0.2505 ** [0.1071]	−0.4537 *** [0.0926]	−0.2802 *** [0.1083]	−0.2659 * [0.1464]	−0.7943 *** [0.2049]
age3060	0.0409 [0.0551]	0.0152 [0.0680]	−0.0292 [0.0568]	0.0367 [0.0685]	−0.0627 [0.0964]	−0.2297 * [0.1213]
tramak_n	0.1442 [0.1662]	0.3462 [0.2265]	0.1815 [0.1774]	0.4390 * [0.2277]	0.5884 * [0.3364]	0.3559 [0.3788]
tramak_f	−0.1785 *** [0.0623]	0.0285 [0.0749]	−0.1999 *** [0.0656]	0.0592 [0.0769]	−0.0827 [0.0979]	0.0165 [0.1249]
tramak_h	−0.1205 ** [0.0566]	−0.2087 *** [0.0652]	−0.0468 [0.0582]	−0.1026 [0.0664]	−0.1615 * [0.0897]	−0.3503 *** [0.1265]
tramak_t	0.0409 [0.0317]	0.2473 *** [0.0392]	0.0697 ** [0.0334]	0.3562 *** [0.0400]	−0.0694 [0.0563]	−0.2450 *** [0.0748]
tramak_c	−0.4594 *** [0.0765]	−0.1594 * [0.0818]	−0.3729 *** [0.0784]	−0.2453 *** [0.0829]	−0.2362 ** [0.1086]	−0.6231 *** [0.1738]
carage0	0.3822 *** [0.0506]	0.4696 *** [0.0600]	0.3307 *** [0.0536]	0.4260 *** [0.0611]	0.4480 *** [0.870]	0.5117 *** [0.1023]
carage1	0.1381 *** [0.0469]	0.0837 [0.0532]	0.1361 *** [0.0492]	0.1840 *** [0.0547]	0.0998 [0.0758]	0.3499 *** [0.0949]
carage2	0.0573 [0.0553]	−0.0801 [0.0626]	0.0060 [0.0576]	0.0526 [0.0643]	0.1059 [0.0865]	0.1412 [0.1164]
carage3	0.0956 * [0.0529]	0.0376 [0.0595]	−0.0547 [0.0551]	0.0272 [0.0609]	0.1182 [0.0793]	−0.1662 [0.1176]
carage4	−0.1447 ** [0.0607]	−0.1928 *** [0.0659]	−0.2558 *** [0.0637]	−0.1329 * [0.0681]	−0.2502 *** [0.0901]	0.0632 [0.1190]
veh_m	0.0783 ** [0.3456]	−0.2005 *** [0.0421]	0.1401 *** [0.0357]	−0.2903 *** [0.0430]	0.1156 * [0.0596]	0.1263 [0.0806]
veh_l	0.0636 [0.0403]	−0.0568 [0.0507]	0.0544 [0.0418]	−0.1804 *** [0.0519]	0.1670 ** [0.0771]	0.3927 *** [0.0922]
sedan	0.0685 [0.0585]	−0.3449 *** [0.0695]	0.0246 [0.0605]	−0.2958 *** [0.0700]	−0.0286 0.0951]	0.0040 [0.1277]
lnprem	0.1036 *** [0.0257]	0.4852 *** [0.0238]	0.1086 *** [0.0272]	0.4849 *** [0.0245]	0.0006 [0.0426]	0.3405 *** [0.0436]
bonus	−0.4794 *** [0.0345]	−0.1689 *** [0.0400]	−0.5341 *** [0.0371]	−0.1805 *** [0.0415]	−0.1382 ** [0.0559]	0.0550 [0.0682]
$\rho$	0.1395 *** [0.0319]		0.0873 *** [0.0337]		0.2608 *** [0.0514]

Standard errors in brackets; ***: p < 0.01; **: p < 0.05; *: p < 0.1.

, with a special interest in the residual correlation coefficient $\rho$. Hypothesis 1 should lead to a positive conditional correlation between filing a suspicious claim and belonging to a suspicious group. More formally, the estimated residual correlation coefficients of these models, i.e., ${\widehat{\rho }}_{SC,SG},{\widehat{\rho }}_{SC,SG1}$, and ${\widehat{\rho }}_{SC,SG2}$, should be positive and significantly different from 0, which leads us to test for null hypotheses $H_0:{\rho }_{SC,SG}\le 0,H_0:{\rho }_{SC,SG1}\le 0$, and $H_0:{\rho }_{SC,SG2}\le 0$ in models 1, 2 and 3, respectively.

The three estimated residual correlation coefficients are significantly positive, which allows us to reject the null hypothesis in each model and thus to state that there is a significantly positive conditional correlation between $SC$ and $SG, SG1$, or $SG2$ in each model. In other words, in accordance with Hypothesis 1, there is conditional dependence between belonging to the suspicious group and filing a claim within the suspicious period, regardless of whether the individual is covered by a deductible contract.18

We have characterized claim manipulation as the misbehavior that consists of postponing and inflating true claims, possibly by bundling several losses in a single claim, unlike the premium recouping misconduct, where false small claims are filed by the end of the policy year. If claim manipulation occurs more frequently than premium recouping, then the suspicious period should be characterized by high values of the first-claim cost ratio. This is expressed in Hypothesis 2, which we test through the regression

$$clmamt={\alpha }_{c}SC+{\alpha }_{f}first+{\alpha }_{fs}first\ast SC+{\alpha }_{X}X+e,$$

(7)

on the claims filed by members of $SG1$ and $SG2$. This corresponds to 6567 claims filed by 6140 policyholders from SG1 and 633 claims filed by 594 policyholders from SG2. In these regressions, $clmamt$ is the value of the claim (in US dollars), while $SC$ and $first$ are dummies indicating that the claim was suspicious (i.e., it was filed during the last month of the policy year) and that it was the first claim of the policyholder during this policy year, respectively. Regression (7) also includes the interaction variable $first\ast SC$. The results are reported in

Table 4. Testing hypothesis 2 (year 2010).

	SG1		SG2
	Est. Coeff.	p-Value	Est. Coeff.	p-Value
Intercept	−2869.3	<0.0001	−4642.1	<0.0001
SC	−198.9	0.0113	−742.1	0.0183
first	46.5	0.4172	−403.0	0.0630
first*SC	−113.3	0.1627	1465.7	<0.0001
female	17.1	0.4871	−145.3	0.0853
age2025	−237.2	0.3869	621.7	0.5280
age2530	−107.3	0.1077	−442.5	0.0865
age3060	−36.9	0.3693	223.1	0.1660
tramak_n	−201.7	0.0932	−620.8	0.1240
tramak_f	−184.5	0.0002	−134.0	0.3972
tramak_h	−117.8	0.0082	−138.6	0.4594
tramak_t	−193.9	<0.0001	−401.3	<0.0001
tramak_c	−219.1	0.0003	−836.5	0.0006
carage0	−149.2	0.0002	−108.0	0.3834
carage1	−103.0	0.0069	−192.3	0.1308
carage2	−25.8	0.5639	−159.7	0.2931
carage3	12.9	0.7677	−192.7	0.2024
carage4	103.0	0.0409	−30.5	0.8493
veh_m	−14.2	0.5944	−151.1	0.1689
veh_l	214.9	<0.0001	148.3	0.1818
sedan	269.6	<0.0001	305.2	0.0785
logprem	371.2	<0.0001	697.1	<0.0001
bonus	48.1	0.0681	−536.6	<0.0001
Adj R2	0.1138		0.4206
Observations	6567		633

.

The estimated coefficients of the interaction variable are ${\widehat{\alpha }}_{fs}=-113.3$ with a p-value $0.1627$ for $SG1$ and ${\widehat{\alpha }}_{fs}=1465.7$ with a p-value lower than $0.0001$ for $SG2$. This sustains Hypothesis 2 for $SG2$, but not for $SG1$, which confirms the fact that being covered by a deductible contract is a factor that encourages fraud through claim manipulation.

Hypothesis 3 focuses on the role of DOAs in this type of insurance fraud. We test this hypothesis with bivariate probit models 1, 2, and 3 in sub-samples that include policyholders who have purchased insurance through the DOA channel or through other distribution channels. This leads us to estimated residual correlation coefficients ${\widehat{\rho }}_{SC,SG}, {\widehat{\rho }}_{SC,SG1}$, and ${\widehat{\rho }}_{SC,SG2}$ in each sub-sample.

Detailed results are displayed in

Table 5. Conditional dependence between SC and SG in sub-samples—Model 1 (year 2010).

	Company 1 with DOAs		Company 1 Without DOAs		Company 2
	SC	SG	SC	SG	SC	SG
RG	0.2087 *** [0.0490]	1.4486 *** [0.1864]	−0.0127 [0.1068]	1.2273 *** [0.1624]	0.1663 *** [0.0560]	1.0515 *** [0.0861]
female	0.0602 [0.0535]	0.1776 ** [0.0790]	−0.1042 [0.0914]	0.1454 [0.1007]	−0.0456 [0.0463]	0.2719 *** [0.0507]
age2025	0.3982 [5405]	−0.3157 [0.6116]	−0.2333 [0.9014]	−0.4811 [0.8294]	0.0419 [0.4649]	−0.3576 [0.4989]
age2530	−0.4620 *** [0.1456]	−0.2670 [0.2155]	0.0615 [0.2624]	−0.6889 ** [0.3040]	−0.3168 ** [0.1310]	−0.5582 *** [0.1383]
age3060	−0.0284 [0.0856]	0.0917 [0.1254]	0.0773 [0.1563]	−0.2390 [0.1686]	−0.0226 [0.0869]	0.0069 [0.0947]
tramak_n	−0.0162 [0.3307]	0.0922 [0.4795]	0.5034 [0.4041]	0.4999 [0.5685]	0.3149 [0.2239]	0.2470 [0.2816]
tramak_f	−0.0451 [0.1189]	0.1857 [0.1755]	−0.0466 [0.1517]	−0.0111 [0.1682]	−0.0337 [0.0906]	−0.1600 [0.0993]
tramak_h	−0.0381 [0.1253]	0.0760 [0.1689]	−0.0707 [0.1447]	−0.0458 [0.1634]	−0.0103 [0.0750]	−0.3598 *** [0.0805]
tramak_t	−0.1260 ** [0.0538]	0.4904 *** [0.0760]	−0.1435 [0.0935]	0.2018 * [0.1037]	0.0418 [0.0491]	−0.0453 [0.0539]
tramak_c	−0.1936 [0.3462]	0.3031 [0.4113]	0.1873 [0.2121]	−0.3445 [0.2479]	−0.1218 [0.0872]	−0.3294 *** [0.0937]
carage0	−0.0363 [0.0975]	0.4796 *** [0.1268]	−0.1329 [0.1535]	0.6156 *** [0.1729]	0.0914 [0.0809]	0.4867 *** [0.0934]
carage1	0.0224 [0.0943]	0.1614 [0.1219]	−0.1268 [0.1240]	0.4121 *** [0.1320]	0.0151 [0.0695]	0.2196 *** [0.0703]
carage2	−0.1635 [0.1126]	−0.1748 [0.1472]	0.2890 * [0.1509]	0.4076 ** [0.1756]	−0.0292 [0.0754]	0.1062 [0.0790]
carage3	0.0092 [0.1111]	−0.0060 [0.1433]	−0.0056 [0.1332]	0.1812 [0.1457]	−0.1602 ** [0.0719]	0.1101 [0.0757]
carage4	−0.2389 * [0.1254]	−0.2915 * [0.1540]	−0.3620 ** [0.1597]	0.2705 * [0.1561]	−0.1641 ** [0.0810]	0.0986 [0.0848]
veh_m	−0.0862 [0.0585]	−0.2110 ** [0.0836]	−0.1666 * [0.1006]	−0.1394 [0.1110]	0.1415 ** [0.0564]	0.0495 [0.0618]
veh_l	−0.1789 *** [0.0689]	−0.2330 ** [0.0926]	−0.3471 *** [0.1242]	−0.0786 [0.1263]	0.0888 [0.0828]	0.2930 *** [0.0909]
sedan	−0.1914 [0.1258]	−0.2939 [0.1809]	0.0666 [0.1567]	−0.3395 * [0.1755]	0.0700 [0.0816]	−0.1867 ** [0.0884]
lnprem	0.2229 ** [0.0874]	0.6791 *** [0.0632]	−0.0554 [0.1058]	0.7074 *** [0.0752]	−0.0662 [0.0453]	0.5853 *** [0.5853]
bonus	−0.2188 [0.1636]	−1.2374 *** [0.1890]	0.4098 * [0.2408]	−1.0545 *** [0.2304]	−0.1504 * [0.080]	−0.4616 *** [0.0955]
Constant	−4.9188 * [0.7791]	−4.9188 *** [0.5876]	−0.2026 [0.9081]	−5.6378 *** [0.6592]	0.1742 [0.3822]	−4.6010 *** [0.3136]
$\rho$	0.5393 *** [0.0729]		0.1344 [0.1201]		0.0562 [0.0480]

Standard errors in brackets; ***: p < 0.01; **: p < 0.05; *: p < 0.1.

,

Table 6. Conditional dependence between SC and SG1 in sub-samples—Model 2 (year 2010).

	Company 1 with DOAs		Company 1 Without DOAs		Company 2
	SC	SG1	SC	SG1	SC	SG1
RG	0.2010 *** [0.0501]	1.5341 *** [0.2253]	−0.0221 [0.1104]	1.1578 *** [0.1645]	0.0709 [0.0599]	1.2021 *** [0.0980]
female	0.1167 ** [0.0549]	0.2759 *** [0.0824]	−0.0620 [0.0935]	0.1638 [0.1028]	−0.1032 ** [0.0480]	0.2954 *** [0.0537]
age2025	0.4237 [0.5564]	−0.4132 [0.6221]	−0.1955 [0.8616]	−0.1888 [0.8311]	−0.3207 [0.4869]	−0.2477 [0.5016]
age2530	−0.4478 *** [0.1503]	−0.5637 ** [0.2213]	−0.0376 [0.2680]	−0.4613 [0.3076]	−0.2080 [0.1329]	−0.4562 *** [0.1452]
age3060	−0.0198 [0.0884]	0.0052 [0.1334]	0.0910 [0.1599]	−0.2645 [0.1707]	−0.0325 [0.0892]	−0.1003 [0.0987]
tramak_n	0.3639 [0.3716]	0.0218 [0.4651]	0.6697 * [0.4006]	0.3062 [0.5392]	0.3782 [0.2385]	0.1518 [0.2967]
tramak_f	−0.1312 [0.1246]	0.1808 [0.1894]	0.0723 [0.1563]	−0.1424 [0.1727]	−0.2251 ** [0.0958]	−0.2843 *** [0.1054]
tramak_h	−0.0022 [0.1272]	0.1606 [0.1764]	0.0368 [0.1485]	−0.2072 [0.1682]	−0.0244 [0.0769]	−0.4242 *** [0.0844]
tramak_t	−0.1027 * [0.0572]	0.5401 *** [0.0802]	−0.0549 [0.0972]	−0.0003 [0.1056]	0.0102 [0.0505]	−0.0394 [0.0565]
tramak_c	−0.3570 [0.3465]	0.1435 [0.4135]	0.0864 [0.2209]	−0.4660 * [0.2552]	−0.1067 [0.0892]	−0.4536 *** [0.0979]
carage0	−0.0659 [0.1021]	0.5069 *** [0.1329]	−0.0428 [0.1574]	0.5662 *** [0.1766]	0.0966 [0.0888]	0.6870 *** [0.0882]
carage1	0.0044 [0.0987]	0.2771 ** [0.1267]	−0.0676 [0.1287]	0.4466 *** [0.1354]	0.1138 [0.0749]	0.4128 *** [0.0739]
carage2	−0.0842 [0.1171]	0.0655 [0.1550]	0.2540 [0.1568]	0.4376 ** [0.1804]	0.0698 [0.0793]	0.2247 *** [0.0830]
carage3	−0.0432 [0.1142]	0.0676 [0.1472]	0.0953 [0.1352]	0.2778 * [0.1472]	0.0236 [0.0745]	0.2375 *** [0.0796]
carage4	−0.3611 *** [0.1289]	−0.1421 [0.1602]	−0.4584 *** [0.1689]	0.3412 ** [0.1630]	−0.0691 [0.0842]	0.1746 * [0.0896]
veh_m	0.0165 [0.0596]	−0.2147 ** [0.0864]	−0.1595 [0.1011]	−0.3025 *** [0.1143]	0.1506 *** [0.0584]	0.0094 [0.0652]
veh_l	−0.1736 ** [0.0699]	−0.2738 *** [0.0971]	−0.3776 [0.1246]	−0.2738 ** [0.1308]	0.1270 [0.0866]	0.2227 ** [0.0968]
sedan	−0.1373 [0.1279]	−0.3944 ** [0.1921]	0.0408 [0.1612]	−0.4320 ** [0.1846]	0.0674 [0.0838]	−0.3254 *** [0.0924]
lnprem	0.2181 ** [0.0890]	0.7433 *** [0.0660]	−0.0377 [0.1169]	0.7134 *** [0.0781]	−0.0475 [0.0484]	0.6283 *** [0.0385]
bonus	−0.1648 [0.1718]	−1.0223 *** [0.1953]	0.4459 [0.2571]	−1.0166 *** [0.2367]	−0.2510 *** [0.0932]	−0.4993 *** [0.1022]
Constant	−1.5136 * [0.7873]	−5.7331 *** [0.6162]	−0.4028 [1.0080]	−5.4420 *** [0.6751]	0.1944 [0.4086]	−4.8641 *** [0.3291]
$\rho$	0.5729 *** [0.0699]		0.0916 [0.1362]		−0.0610 [0.0534]

Standard errors in brackets; ***: p < 0.01; **: p < 0.05; *: p < 0.1.

and

Table 7. Conditional dependence between SC and SG2 in sub-samples—Model 3 (year 2010).

	Company 1 with DOAs		Company 1 Without DOAs		Company 2
	SC	SG2	SC	SG2	SC	SG2
RG	0.2622 [0.2004]	1.6147 *** [0.2621]	−0.4441 [0.4225]	2.3649 *** [0.4639]	0.2728 * [0.1594]	1.7827 *** [0.2152]
female	−0.1535 [0.1249]	−0.0666 [0.1494]	−0.0555 [0.1397]	0.3131 [0.2001]	−0.0175 [0.0669]	−0.0709 [0.0833]
age2025	0.2937 [0.7007]	−0.4864 [0.8008]	0.2417 [1.1300]	0.4348 [1.0916]	−0.0347 [0.6076]	−0.2253 [0.7808]
age2530	−0.6705 [0.3223]	−0.3919 [0.3941]	0.6021 [0.4272]	−0.7861 [0.7589]	−0.3573 * [0.1828]	−0.4311 * [0.2376]
age3060	0.0564 [0.2093]	−0.1246 [2562]	0.4362 * [0.2572]	−0.1269 [0.3146]	−0.1732 [0.1268]	−0.1471 [0.1547]
tramak_n	0.7126 [0.6845]	−0.1096 [0.7266]	−0.3747 [1.7463]	0.0300 [1.7441]	0.2852 [0.4113]	0.2575 [0.4415]
tramak_f	−0.1230 [0.2342]	0.3446 [0.2781]	−0.0555 [0.2220]	0.0392 [0.2885]	−0.0269 [0.1283]	−0.0740 [0.1624]
tramak_h	−0.0649 [0.2789]	−0.1577 [0.3410]	0.0791 [0.2231]	−0.5671 * [0.3424]	0.0696 [0.1063]	−0.2862 ** [0.1434]
tramak_t	−0.4971 *** [0.1352]	−0.8100 *** [0.1601]	−0.1005 [0.1473]	−0.1753 [0.1983]	0.0083 [0.0727]	−0.1858 ** [0.0946]
tramak_c	0.0140 [0.5654]	−0.0334 [0.8117]	−0.2963 [0.2935]	−0.2935 [0.4233]	−0.1097 [0.1245]	−0.8299 *** [0.1970]
carage0	0.1719 [0.1898]	0.4071 * [0.2236]	−0.0541 [0.2660]	0.2479 [0.3513]	0.3881 *** [0.1199]	0.3900 *** [0.1319]
carage1	−0.0991 [0.1911]	0.0391 [0.2321]	−0.0870 [0.1933]	0.4110 [0.2520]	0.0899 [0.0954]	−0.0666 [0.1193]
carage2	−0.0875 [0.2321]	−0.0744 [0.2908]	0.2821 [0.2407]	0.3666 [0.3243]	0.0783 [0.1058]	−0.0445 [0.1366]
carage3	0.3269 [0.2303]	−0.1232 [0.2926]	0.0080 [0.2028]	−0.2482 [0.3401]	0.2198 ** [0.0963]	−0.0513 [0.1284]
carage4	−0.2220 [0.2400]	−0.4267 [0.3248]	−0.3660 [0.2303]	0.4180 [0.2698]	0.0271 [0.1083]	−0.0476 [0.1430]
veh_m	0.0385 [0.1383]	0.0276 [0.1680]	0.1005 [0.1489]	−0.1990 [0.2141]	0.1787 ** [0.0815]	−0.0262 [0.1075]
veh_l	0.1327 [0.1801]	0.2949 [0.1809]	0.1798 [0.1944]	−0.0297 [0.2319]	0.2780 ** [0.1228]	0.3828 *** [0.1455]
sedan	−0.2460 [0.2635]	−0.2723 [0.3166]	0.0885 [0.2384]	−0.6997 ** [0.2843]	0.1115 [0.1199]	−0.3116 ** [0.1471]
lnprem	0.1987 [0.1692]	0.6067 *** [0.1097]	−0.1528 [0.2006]	0.4173 *** [0.1440]	−0.0067 [0.0713]	0.3557 *** [0.0688]
bonus	−0.0319 [0.3338]	−0.9141 ** [0.3744]	0.2821 [0.4282]	−0.9554 ** [0.4655]	−0.3427 ** [0.1385]	0.0301 [0.1585]
Constant	−1.4106 [1.4319]	−5.0827 *** [1.0026]	0.0842 [1.6199]	−3.8736 *** [1.2197]	−0.1057 [0.5649]	−3.6732 *** [0.5400]
$\rho$	0.7492 *** [0.1355]		−0.2020 [0.2206]		0.2076 *** [0.0702]

Standard errors in brackets; ***: p < 0.01; **: p < 0.05; *: p < 0.1.

, for models 1, 2, and 3, respectively, with conclusions on residual correlation summarized as follows:19

$\begin{array}{ccccc}& & \begin{array}{c}\mathbf{Company} \mathbf{1}\\ \mathbf{Dealer}\end{array}& \begin{array}{c}\mathbf{Company} \mathbf{1}\\ \mathbf{Non}\text{-}\mathbf{dealer}\end{array}& \mathbf{Company} \mathbf{2}\\ \mathbf{Model} \mathbf{1}& {\widehat{\rho }}_{SC,SG}& {0.5393 }^{***}& 0.1344& 0.0562\\ \mathbf{Model} \mathbf{2}& {\widehat{\rho }}_{SC,SG1}& {0.5729 }^{***}& 0.0916& -0.0610\\ \mathbf{Model} \mathbf{3}& {\widehat{\rho }}_{SC,SG2}& {0.7492 }^{***}& -0.2020& {0.2076 }^{***}\end{array}$

Hence, when the regressions are performed in the sub-sample of policyholders who purchased coverage through the DOAs of company 1, there is a significant positive residual correlation between $SC$ and $SG, SG1$, or $SG2$ at the 1% threshold. This correlation vanishes in the two other sub-samples, except between $SC$ and $SG2$ in company 2.

These econometric results highlight two intuitive and central conclusions about the process of claim manipulation. First of all, they reveal the role of facilitators played by DOAs in this process. Whether the manipulation is to defer claims or accumulate them over time, policyholders could hardly display such behavior on their own without the cooperation of car dealers and repair shops. Accordingly, evidence of this behavior is observed exclusively when insurance policies are sold through the DOA channel. Secondly, the results highlight the incentives for manipulation generated by contract design. Policies with deductibles, compared with those without, create stronger incentives for policyholders to avoid incurring claim costs. Therefore, policyholders facing deductibles are more inclined to engage in the manipulation of accumulated claims.

4.2. Categorizing Policyholders

It is worth examining whether categorizing policyholders allows us to better identify fraudulent behaviors. Let us first look at the behavioral differences between men and women. If fraudulent behaviors differ by gender, then the female indicator should be statistically significant with the same sign in both bivariate probit regressions. Among Table 3, Table 5, Table 6 and Table 7, this happens only in Table 6, which corresponds to the dealer channel of company 1; in this table, the female variable is statistically significant and positive simultaneously in the $SG1$ equation (the suspicious group with zero-deductible contracts) and in the $SC$ equation. In other words, women are more likely than men to belong to the zero-deductible suspicious group and to concentrate their claims in the final month of the policy year.

However, there remains an open question as to whether, in the dealer channel of company 1, women actually manipulate the timing of claims more intensely than men. In order to better understand this question, we performed model 2 regressions separately for male and female sub-samples, by limiting attention to the dealer channel of company 1. The following table shows the estimated residual correlation coefficients for these regressions.20 Wald tests show that the estimated difference between men and women is not statistically significant; thus, we may conclude that there is no evidence of gender differences in fraudulent behavior.

$\begin{array}{ccc}\mathrm{Residual} \mathrm{correlation} \mathrm{coefficient}& \mathrm{Men}& \mathrm{Women}\\ \rho & \begin{array}{c}{0.1491 }^{*}\\ \left[0.0789\right]\end{array}& \begin{array}{c}{0.2749 }^{***}\\ \left[0.0484\right]\end{array}\end{array}$

$\begin{array}{cc}\mathrm{Difference} \left(\mathrm{Men}\text{-}\mathrm{Women}\right)& \begin{array}{c}-0.1258\\ \left[0.0926\right]\end{array}\end{array}$

We can also examine whether fraudulent behavior differs according to age groups. To do so, we first examine the estimates of the bivariate probit regressions and observe that in most specifications, the age-group variables are not statistically significant in either equation. The only exception is the indicator for the ages of 25–30 (age2530), which shows some explanatory relevance. Since observations under the age of 25 represent only a small part of the sample, we combine individuals under 30 years old into one group. Individuals aged 30 years and over constitute the oldest sub-sample, while those under 30 years form the youngest sub-sample.

We then re-estimate the three bivariate probit models reported in Table 3 separately for these two sub-samples. The table below shows the estimated residual correlations for each sub-sample, and Wald tests were performed to assess whether the residual correlations differ significantly between the younger and older sub-samples. Some evidence of age-related differences in fraudulent behavior emerges only in model 1, with the younger cohort displaying significantly stronger signs of claim manipulation.

$\begin{array}{cccc}\begin{array}{c}\mathrm{Residual} \mathrm{correlation}\\ \mathrm{coefficient} \mathrm{for}:\end{array}& \mathrm{Model} 1& \mathrm{Model} 2& \mathrm{Model} 3\\ \mathrm{individuals} \mathrm{younger} \mathrm{than} 30& \begin{array}{c}{0.8334 }^{***}\\ \left[0.1537\right]\end{array}& \begin{array}{c}{0.2349 }^{***}\\ \left[0.0220\right]\end{array}& \begin{array}{c}0.0056\\ \left[0.0281\right]\end{array}\\ \mathrm{individuals} \left(\mathrm{weakly}\right) \mathrm{older} \mathrm{than} 30& \begin{array}{c}{0.3259 }^{***}\\ \left[0.0329\right]\end{array}& \begin{array}{c}0.1893\\ \left[0.1280\right]\end{array}& \begin{array}{c}0.1432\\ \left[0.1430\right]\end{array}\\ \mathrm{Difference} \left(\mathrm{Young}\text{-}\mathrm{Old}\right)& \begin{array}{c}{0.5075 }^{***}\\ \left[0.1572\right]\end{array}& \begin{array}{c}0.0456\\ \left[0.1299\right]\end{array}& \begin{array}{c}-0.1376\\ \left[0.1457\right]\end{array}\end{array}$

4.3. Sensitivity Analysis

Our definition of the suspicious period and therefore the identification of suspicious claims reflect the practice of insurers to update the insurance premium by excluding claims occurring during the last policy month. In order to strengthen the credibility of our conclusions, we conducted a sensitivity analysis by redefining the suspicious period as the second-to-last, third-to-last, and fourth-to-last months, successively. This was performed for Table 3, Table 5, Table 6 and Table 7, with results being presented in

Table 8. Residual correlation for alternative definitions of the suspicious period.

Suspicious Period	Residual Correlation Model 1 (SC, SG)	Residual Correlation Model 2 (SC, SG1)	Residual Correlation Model 3 (SC, SG2)
Second-to-last month claims	−0.0153 [0.0418]	0.0355 [0.0263]	−0.0269 [0.0328]
Third-to-last month claims	−0.3214 *** [0.0432]	−0.1393 *** [0.0321]	−0.0654 [0.0451]
Fourth-to-last month claims	−0.1824 *** [0.0466]	−0.1050 *** [0.0356]	−0.0518 [0.0505]

and

Table 9. Residual correlation according to the suspicious period and the distribution channel.

	SC
	Model 1 (SC, SG)	Model 2 (SC, SG1)	Model 3 (SC, SG2)
Panel 1: company 1, with DOAs
Second-to-last month claim	−0.3519 [0.9772]	−0.1004 * [0.0492]	−0.1264 ** [0.0562]
Third-to-last month claim	−0.9999 *** [0.0010]	−0.1379 ** [0.0694]	0.0590 [0.0855]
Fourth-to-last month claim	−0.2027 *** [0.0709]	−0.0976 ** [0.0428]	0.0120 [0.0502]
Panel 2: company 1, without DOAs
Second-to-last month claim	−0.9909 *** [0.0002]	−0.2006 *** [0.0678]	−0.0910 [0.0927]
Third-to-last month claim	0.1461 [5.1641]	−0.0571 [0.0853]	0.0999 [0.1051]
Fourth-to-last month claim	−0.1524 * [0.0800]	−0.0293 [0.0538]	−0.1103 [0.0666]
Panel 3: company 2
Second-to-last month claim	−0.9999 *** [0.0000]	−0.1218 [0.1002]	−0.3810 * [0.1637]
Third-to-last month claim	−0.9999 *** [0.0000]	−0.2691 *** [0.0869]	0.1029 [0.1209]
Fourth-to-last month claim	0.0880 [0.0820]	0.0660 [0.0572]	−0.0247 [0.0665]

. For the sake of brevity, we only present the residual correlation deduced from the bivariate probit regressions.

As shown in Table 8, regardless of whether $SC$ is defined using claims occurring in the second-to-last, third-to-last, or fourth-to-last month, there is no evidence of a statistically significant positive conditional correlation between belonging to the suspicious group and the occurrence of suspicious claims across models 1 to 3. In other words, we do not observe the same claim manipulation pattern when the suspicious period differs from the last policy month.

When the same sensitivity analysis is performed separately for each distribution channel—dealers of company 1, non-dealers of company 1, and company 2—with results reported in Table 9, we still fail to find a statistically significant positive conditional correlation between belonging to the suspicious group and the occurrence of suspicious claims. In other words, even after disaggregating our sample according to the distribution channel, there is no evidence that the members of the suspicious group tend to shift claims toward other policy months than the final month.

Taken together, the results from Table 8 and Table 9 not only support the robustness of our main findings, but they also reveal an additional pattern: as the definition of $SC$ varies across months, the estimated conditional correlations reported in these tables fluctuate substantially, in terms of both statistical significance and sign (positive versus negative). This suggests that, aside from the concentration of claims in the final month driven by manipulation within the suspicious group, claims occurring in the remaining eleven months are essentially random, providing additional evidence inconsistent with systematic manipulation.

4.4. Comments on the Role of Car Dealers in Claim Manipulation

A central conclusion of this study is that car dealers play an important role in facilitating the manipulation of claims by policyholders. To strengthen the credibility of this finding, we first check that for company 1, the evidence of manipulation in the policies sold through the car dealer channel—measured by the significantly positive residual correlation between suspicious groups and suspicious claims—is substantially stronger than that observed for company 1’s non-car dealer channel and company 2, which does not employ car dealers. To do so, we successively consider two null hypotheses $H_0:{\widehat{\rho }}_{SC,SG}^D\le {\widehat{\rho }}_{SC,SG}^{ND}$ and $H_0:{\widehat{\rho }}_{SC,SG}^D\le {\widehat{\rho }}_{SC,SG}^{C2}$ in model 1,21 and we proceed in the same way for models 2 and 3, hence with $SG1$ and $SG2$ instead of $SG$. The results reported in

Table 10. Difference in conditional dependence between SC and SG/SG1/SG2 in 2010.

	Model 1	Model 2	Model 3
	(SC, SG)	(SC, SG1)	(SC, SG2)
${\rho }^D-{\rho }^{ND}$	0.4049 *** [4.6650]	0.4814 *** [5.3121]	0.9513 *** [5.2735]
${\rho }^D-{\rho }^{C2}$	0.4831 *** [7.9626]	0.6339 *** [10.3059]	0.5416 *** [6.0383]
${\rho }^{C2}-{\rho }^{ND}$	−0.0782 [−1.1193]	−0.1526 [−1.9339]	0.4096 *** [3.4480]

show that both null hypotheses are rejected, regardless of the definition of the suspicious group. In other words, whatever the definition of suspicious group ($SG, SG1$, or $SG2$), the conditional correlation between filing a suspicious claim and belonging to the suspicious group is significantly greater when contracts are sold through car dealers associated with company 1 than through another distribution channel for company 1 or company 2.

Further evidence on the role of car dealers can be obtained by focusing on the first-claim cost ratio during the suspicious period (as in Hypothesis 2), considering sub-samples defined by the distribution channel and using type C contracts as a benchmark. A higher first-claim cost ratio during the suspicious period for $SG1$ or $SG2$ than for type C contracts would signal manipulation of claims with claim build-up or claim bundling by members of the suspicious groups. Symmetrically, a lower first-claim cost ratio would be compatible with the premium recouping mechanism highlighted by Li et al. (2013), with small claims being filed at the end of the policy year if no claim has been filed before. This leads us to consider regression (8) below, where the claim amount is the dependent variable, as in regression (7). In (8), $first, SC$, and X are identical to those in regression (7), and $S_1, S_2$, and $S_3$ are dummies indicating that the policy has been purchased from company 1 through the DOA channel, from company 1 through another distribution channel, and from company 2, respectively. Moreover, C is a dummy indicating that the insurance policy is a type C contract, used as a benchmark without claim manipulation.

$$\begin{array}{cc}\hfill clmamt& ={\alpha }_{c}SC+{\alpha }_{f}first+{\alpha }_{fs}first\ast SC\hfill \\ \hfill & +s_{SG11fs}SG1\ast S_1\ast first\ast SC\hfill \\ \hfill & +s_{SG21fs}SG2\ast S_2\ast first\ast SC\hfill \\ \hfill & +s_{SG23fs}SG2\ast S_3\ast first\ast SC\hfill \\ \hfill & +s_{Cfs}C\ast first\ast SC+{\alpha }_{X}X+e.\hfill \end{array}$$

(8)

The estimation of regression (8) shows that null hypothesis $H_0:$ $s_{SG21fs}\le s_{Cfs}$ is rejected at a significance level of 1%, in contrast to the results obtained when $s_{SG11fs}$ and $s_{SG23fs}$ are compared with $s_{Cfs}$.22 This means that the first-claim cost ratio is significantly higher during the last policy month when a deductible contract has been purchased from company 1 through the DOA channel. In total, in 2010, the deductible contracts sold through DOAs created the most favorable condition for insurance fraud through the postponing of claims, with claim build-up or bundling.

4.5. Smaller Bargaining Power for DOAs in 2018

From 2010 to 2018, company 1 reduced by almost half the share of its automobile insurance contracts sold through car dealers. They became less important partners of the insurer, with likely lower bargaining power in the claims settlement process.

To assess the consequences of this change, we gathered information on 269,475 type A, B, and C automobile insurance contracts sold by company 1 in 2018. The content of these contracts remained essentially the same as in 2010, with the only significant change being that, in 2018, following the liberalization of the Taiwanese insurance market, company 1 only sold type A and B contracts with a deductible. Therefore, the suspicious group ($SG$) should no longer be split into $SG1$ and $SG2$, and it coincides with what we called $SG2$ for the year 2010.

Table 11. Sample structure in 2018.

	Whole Sample	Sub-Sample with Claim	DOAs	No DOAs
claim	0.0371
SC	0.0090	0.2420	0.2695	0.2275
RG	0.3400	0.3614	0.4912	0.2931
AB	0.3500	0.3708	0.5039	0.3007
SG	0.2189	0.2697	0.3505	0.2272
D	0.2795	0.3452	1.0000	0.0000
female	0.5599	0.6515	0.6541	0.6502
age2025	0.0087	0.0057	0.0061	0.0055
age2530	0.0376	0.0401	0.0344	0.0430
age3060	0.7674	0.8107	0.7968	0.8180
carage0	0.0439	0.1970	0.2246	0.1825
carage1	0.0591	0.1702	0.1705	0.1701
carage2	0.0644	0.1477	0.1424	0.1504
carage3	0.0608	0.1132	0.1106	0.1146
carage4	0.0618	0.0993	0.1004	0.0987
veh_m	0.2800	0.3320	0.3372	0.3292
veh_l	0.1612	0.1881	0.1910	0.1866
sedan	0.9719	0.9951	0.9962	0.9945
lnprem	8.9993	9.0407	9.2992	8.9045
bonus	0.9370	0.7083	0.7316	0.6960
Observations	269,475	10,010	3455	6555

provides detailed information about the data. The comparison of Table 1 and Table 11 confirms the decrease in the proportion of contracts sold through DOAs, as well as other significant changes—notably the decrease in policyholders who filed a claim in 2010 and 2018 from 6.49% to 3.71%, respectively.23 Figure 4 also confirms that claim rates are still higher in the last month than in previous months of the policy year, with a significant decrease in the average claim cost during the last policy month, and Figure 5 shows a decrease in the first-claim cost ratio for all types of contracts, including policyholders in $SG$ going through DOAs, unlike what was observed for $SG2$ in 2010.

Does this mean that the claim manipulation favored by DOAs had disappeared by 2018? Formal tests were carried out to ascertain this. The results of bivariate probit regressions (similar to model 1 above) are presented in

Table 12. Conditional dependence between SC and SG (year 2018).

	DOAs		No DOAs
	SC	SG	SC	SG
RG	−0.1621 * [0.0833]	3.9976 *** [0.1734]	−0.0552 [0.0539]	4.1421 *** [0.1275]
female	0.1163 ** [0.0565]	0.1735 [0.1131]	0.1878 *** [0.0379]	0.1635 * [0.0839]
age2025	−0.4879 [0.4042]	1.7509 *** [0.6026]	−0.3606 [0.2516]	−0.2547 [0.7147]
age2530	−0.4798 *** [0.1719]	0.5993 ** [0.3043]	−0.6081 *** [0.1090]	0.1064 [0.2218]
age3060	−0.0701 [0.0716]	0.2061 [0.1511]	−0.0548 [0.0509]	−0.0222 [0.1054]
tramak_n	−0.5044 [0.5860]	1.2453 ** [0.5731]	0.4607 ** [0.2073]	−0.2498 [0.5385]
tramak_f	0.3979 *** [0.1484]	0.1127 [0.2855]	0.5452 *** [0.0839]	0.7828 *** [0.1915]
tramak_h	0.1188 [0.1440]	0.2980 [0.2660]	0.2119 *** [0.0795]	−0.3979 ** [0.1850]
tramak_t	0.5698 *** [0.0578]	0.1843 [0.1162]	0.3826 *** [0.0393]	0.3883 *** [0.0891]
tramak_c	0.4150 [0.2818]	−0.4958 [0.6022]	−0.0831 [0.1984]	0.3025 [0.4272]
carage0	0.2969 *** [0.0899]	−0.9596 *** [0.1991]	0.2489 *** [0.0581]	−0.6280 *** [0.1419]
carage1	−0.1035 [0.0878]	−0.9074 *** [0.1917]	−0.0309 [0.0576]	−0.3448 ** [0.1366]
carage2	−0.2374 *** [0.0891]	−0.1850 [0.1909]	−0.0843 [0.0579]	−0.3009 ** [0.1320]
carage3	−0.2848 *** [0.0980]	−0.2969 [0.1936]	−0.3812 *** [0.0658]	−0.5707 *** [0.1408]
carage4	−0.2814 *** [0.1003]	−0.3862 * [0.2060]	−0.2758 *** [0.0684]	−0.9843 *** [0.1550]
veh_m	−0.0768 [0.0593]	−0.0678 [0.1243]	0.1282 *** [0.0407]	0.2059 ** [0.0956]
veh_l	−0.3323 *** [0.0739]	−0.4531 *** [0.1601]	−0.1396 *** [0.0525]	0.3258 *** [0.1124]
sedan	−0.5778 [0.4758]	−1.3503 * [0.6892]	−0.4488 [0.2802]	−1.1315 ** [0.5508]
lnprem	0.1095 ** [0.0453]	−0.0162 [0.0916]	−0.0536 * [0.0291]	−0.0823 [0.0592]
bonus	0.9318 *** [0.1128]	0.3404 [0.2296]	0.7223 *** [0.0790]	−1.0135 *** [0.1906]
$\rho$	0.5229 ** [0.2575]		0.9277 *** [0.2172]

Standard errors in brackets; ***: p < 0.01; **: p < 0.05; *: p < 0.1. The difference in conditional dependence between SC and SG: ${\rho }^D-{\rho }^{ND}$ = −0.4048 (t = −1.7524; $H_0:{\rho }^D\le {\rho }^{ND}$ cannot be rejected).

. The estimated residual correlation between $SG$ and $SC$ is still significantly positive regardless of the distribution channel, but null hypothesis $H_0:{\rho }^D\ge {\rho }^{ND}$ is rejected at the 1% significance threshold. In other words, the positive residual correlation between belonging to the suspicious group and filing a claim in the suspicious period is still valid, which confirms claim manipulation, but the role of DOAs in this fraud process has disappeared. For completeness, we verified that the difference ${\rho }^D-{\rho }^{ND}$ significantly decreased between 2010 and 2018, meaning that the higher conditional correlation between $SC$ and $SG$ for the $DOA$ channel, compared with other distribution channels, significantly decreased from 2010 to 2018.24

We performed a robustness check with a two-stage method in order to confirm this change from 2010 to 2018. To do so, we created a new dataset that included $SG2$ and type C contracts sold by company 1 in 2010 or 2018, with the $y_{2018}$ dummy being used to indicate that the contract was sold in 2018.25 The first stage consists of estimating the probit regression

$$Pr[SG=1]=\mathsf{\Phi }(X{\beta }_{SG}+\eta ),$$

and the estimated probability of belonging to the suspicious group ($\widehat{SG}$) and the D dummy for the DOA channel are used as explanatory variables in the second-stage regression:

$$\begin{array}{cc}\hfill Pr[SC& =1]=\mathsf{\Phi }({\beta }_{estSG}\widehat{SG}+{\beta }_{SG}SG+{\beta }_{D}D+{\beta }_{2018}{y}_{2018}\hfill \\ \hfill & +{\beta }_{SGD}SG\ast D+{\beta }_{SG2018}SG\ast y_{2018}+{\beta }_{D2018}D\ast y_{2018}\hfill \\ \hfill & +{\beta }_{SGD2018}SG\ast D\ast y_{2018}+X{\beta }_{SC}+\epsilon )\hfill \end{array}$$

The results are presented in

Table 13. Comparative manipulation ability of DOAs (years 2010 and 2018).

	First Stage		Second Stage
	Est. Coeff.	p-Value	Est. Coeff.	p-Value
Intercept	−37.0773	<0.0001	−0.7793	0.0027
SG			−0.4212	0.0899
$\widehat{SG}$			−0.1565	0.5312
dealer			0.00618	0.9478
y2018			−0.2399	0.0005
SG*dealer			1.8147	<0.0001
SG*y2018			0.7457	0.0029
dealer*y2018			0.0443	0.6633
SG*dealer*y2018			−1.7265	<0.0001
recoup	16.7495	0.8673	−0.1304	0.5139
female	0.3715	0.0198	0.0640	0.0322
age2025	−4.3641	0.9975	−0.4786	0.0182
age2530	0.0644	0.9055	−0.5400	<0.0001
age3060	0.6748	0.0241	−0.0957	0.0182
tramak_n	−4.9743	0.9972	0.1025	0.575
tramak_f	0.6856	0.0085	0.0525	0.4566
tramak_h	−0.3143	0.3305	0.0618	0.3446
tramak_t	−0.3328	0.0447	0.2423	<0.0001
tramak_c	0.3515	0.4887	−0.2892	0.0562
carage0	0.2408	0.2913	0.4741	<0.0001
carage1	0.1337	0.5468	0.1807	<0.0001
carage2	−0.2893	0.2825	0.1914	<0.0001
carage3	−0.3436	0.2325	0.0823	0.1051
carage4	0.3167	0.1901	0.0314	0.5563
veh_m	−0.1722	0.3223	0.0300	0.3498
veh_l	−0.1807	0.3411	0.0333	0.3941
sedan	−1.4823	<0.0001	−0.1291	0.3234
logprem	3.8113	<0.0001	−0.0223	0.3687
bonus	−0.8332	0.0087	0.5239	<0.0001
−2logL	12,270.286		11,655.898

. The estimated coefficient of the triple interaction term $SG\ast D\ast y_{2018}$ is ${\widehat{\beta }}_{SGD2018}=-1.7265$, and it is significantly different from 0 at the 1% significance threshold. In other words, DOAs’ incentivizing of policyholders from the suspicious group ($SG$) to manipulate claims significantly decreased from 2010 to 2018.

Considering that DOAs played a crucial role in the manipulation of claims in 2010, one may wonder whether the decrease in their bargaining power fully canceled the fraud process in 2018, be it under the form of claim manipulation or of the premium recouping behavior. To verify this, we estimated regression (7) for $SG$ and for type C contracts, with data from the year 2018. The results are presented in

Table 14. Testing Hypothesis 2 for the year 2018.

	SG		Type C
	Est. Coeff.	p-Value	Est. Coeff.	p-Value
Intercept	20.8341	0.0259	30.3130	0.0009
SC	−4.8308	0.0581	−9.0929	0.0006
first	−0.8095	0.5265	−3.7417	0.0036
first*SC	−0.7615	0.7789	1.8227	0.5221
female	−3.7137	<0.0001	−0.7140	0.4280
age2025	−10.3178	0.0970	11.9403	0.0314
age2530	−3.5532	0.1216	3.4655	0.1534
age3060	−2.5520	0.0254	−0.9612	0.4385
tramak_n	−1.7228	0.7422	−7.9312	0.1331
tramak_f	−4.9029	0.0172	−9.3758	<0.0001
tramak_h	−7.9695	<0.0001	−11.4974	<0.0001
tramak_t	−7.0401	<0.0001	−11.1717	<0.0001
tramak_c	−8.6608	0.0585	−10.9192	0.0065
carage0	2.7724	0.0447	3.4785	0.0142
carage1	5.1712	<0.0001	4.1478	0.0029
carage2	3.7767	0.0039	5.8271	<0.0001
carage3	2.5255	0.0692	5.5491	0.0003
carage4	2.3239	0.1084	4.1172	0.0100
veh_m	5.0363	<0.0001	4.5882	<0.0001
veh_l	9.7725	<0.0001	15.1139	<0.0001
sedan	6.2842	0.3710	7.3519	0.2293
logprem	−0.6929	0.2462	−1.5460	0.0395
bonus	2.1328	0.2304	−0.0887	0.9615
Adj. ${\mathrm{R}}^2$	0.0773		0.0549
Observations	3149		7530

. The estimated coefficient ${\widehat{\alpha }}_{fs}$ is not significantly different from 0 in both subsets of contracts. In other words, in 2018, unlike in 2010, there was no significant change in the cost of first claims filed during the last month of the policy year compared with previous months.

4.6. Summary and Implications of the Results

The results presented in Section 4.1, Section 4.2, Section 4.3, Section 4.4 and Section 4.5 convey a coherent picture of how contractual incentives and distribution structures jointly shape manipulation behavior in automobile insurance. First, the evidence consistently points to intertemporal manipulation of real losses, rather than random concentration of claims or purely psychological premium recouping, as the dominant form of fraud in this market. Second, contract design determines the payoff from manipulation, but it does not by itself make manipulation feasible. Indeed, deductibles and bonus–malus rules create incentives to postpone and bundle claims, yet these incentives translate into economically meaningful fraud only when policyholders can coordinate claim reporting and repair activities. Third, the analysis underscores the central role of distribution channels as enabling institutions. Dealer-owned agents combine insurance intermediation, claim handling, and access to repair services, thereby lowering coordination costs and weakening insurers’ enforcement capacity. Fraud, in this sense, is an outcome mediated by organizational arrangements and bargaining power at the claims settlement stage. Fourth, the comparison between 2010 and 2018 illustrates that fraud is responsive to organizational change. The attenuation of claim manipulation following the reduction in insurers’ reliance on dealer-owned agents shows that fraud can be substantially deterred without altering core contract parameters.

Taken together, the results yield a coherent interpretation of insurance fraud in this market. Fraud primarily takes the form of opportunistic manipulation of real losses rather than small false claims. Contractual incentives such as deductibles and bonus–malus rules create the motive for manipulation, but distribution channels determine whether these incentives can be exploited. When intermediaries have great bargaining power while defending the interests of their customers and control complementary services such as repair shops, as in the DOA channel, then manipulation flourishes. Conversely, organizational changes that reduce intermediary power can substantially deter fraud without altering core contract design.

5. Conclusions

This study has shown that insurance fraud in automobile insurance is shaped not only by the individual incentives embedded in deductibles and bonus–malus rules but also—crucially –by the structure of insurance distribution channels. Using Taiwanese car insurance data, we documented two distinct fraud mechanisms: premium recouping through small false claims and the manipulation of claims through strategic deferral and inflation of real losses. Our econometric analysis has combined the approach of Dionne and Gagné (2002) to the identification of opportunistic fraud patterns on one hand, and the empirical testing framework of Chiappori and Salanie (2000) on the other hand. Our results provide strong evidence that, in 2010, claim manipulation was significantly more prevalent when insurance policies were sold through dealer-owned agents (DOAs), while this channel-specific effect largely disappeared in 2018 after insurers significantly reduced their reliance on DOAs.

The results highlight the importance of vertical relationships in the economics of insurance fraud. DOAs combine insurance intermediation with close links to repair services, which increases the difficulty of detecting manipulation and weakens the ability of insurers to enforce penalties. As a result, distribution channels with greater intermediary bargaining power can significantly amplify incentives for fraud, particularly under contracts with deductibles and experience-based pricing rules. Conversely, the decrease in the DOA effect over time suggests that organizational changes in distribution can significantly reduce fraud, even in the absence of major reforms in contract design or audit technology.

From a managerial and regulatory perspective, the results suggest that anti-fraud strategies should not focus exclusively on policyholder behavior or contractual parameters. Insurers can also reduce exposure to fraud by limiting dependence on powerful intermediaries, redefining agency relationships, or separating sales from claims-related services. More broadly, the distribution structure must be considered an essential element of fraud risk management by insurers.

This study also has limitations that should be acknowledged. Our analysis is based on indirect evidence of fraud inferred from claim timing and cost patterns, rather than identified fraudulent claims. Although this approach is well established in the literature and appropriate given data constraints, it necessarily captures behavioral patterns consistent with fraud rather than verified misconduct.

Future research could extend this framework by incorporating richer information on repair networks, audit outcomes, or intermediary contracts, as well as examining similar mechanisms in other insurance markets or jurisdictions. More generally, the integration of organizational and contractual characteristics in empirical analyses of insurance fraud seems essential to understanding how fraud arises—and how it can be effectively deterred.