Automobile Insurance Contracts and Risk of Accident: An Empirical Test Using French Individual Data

Transcription

1 The Geneva Papers on Risk and Insurance Theory, 24: (1999) c 1999 The Geneva Association Automobile Insurance Contracts and Risk of Accident: An Empirical Test Using French Individual Data DIDIER RICHAUDEAU Laboratoire de Microéconomie Appliquée (LAMIA), Université Paris I Panthéon-Sorbonne, Département d Evaluation et de Recherche en Accidentologie, INRETS; Laboratoire de Finance-Assurance, Centre de Recherche en Economie et Statistique Timbre J boulevard Gabriel Péri Malakoff Cedex, France Abstract Insurance has for a long time been perceived as a way of transferring responsibility from insured agents to insurers and thus as potentially influencing insured agents behavior. Two particular opportunistic behaviors have been analyzed. First, the theory of adverse selection predicts that high-risk agents are likely to demand more insurance than are low-risk agents. Second, the theory of moral hazard predicts that the wider the insurance coverage, the less agents will try to prevent accidents. Both theories thus conclude that agents who are totally insured should have a higher probability of accident than those with only partial insurance, ceteris paribus. Nevertheless, one of the aims of insurance rating systems is to control for these opportunistic behaviors. In this article, we use individual data to see if the French automobile insurance rating system has achieved this aim. We do this using a two-step maximum-likelihood method. First, we compute a probit model to estimate the probability of taking out comprehensive versus third-party insurance. We then calculate the generalized residual, which is included as an independent variable in a negative binomial model estimating the probability of having an accident. The coefficient of this variable is argued to represent adverse selection and ex-ante moral-hazard behavior. Key words: adverse selection, moral hazard, automobile insurance contracts, risk of accident, econometrics 1. Introduction Insurance has for a long time been perceived as a way of transferring responsibility from insured agents to insurers and thus as potentially influencing insured agents behavior. Two particular opportunistic behaviors have been identified by insurers and modeled by economists. First, the theory of adverse selection (Rothschild and Stiglitz [1976]) shows that high-risk agents are more likely to demand insurance than are low-risk agents. Second, the theory of moral hazard (Arnott and Stiglitz [1988]) predicts that the wider the insurance coverage, the less agents will try to prevent accidents. These behaviors interfere with the development of the insurance market only when information is asymmetric: insured agents have better knowledge of both their probability of having an accident and of their efforts at prevention than do insurance companies. One of the main results of insurance theory in economics is that, in this context, the insurance market might be reduced. With perfect and symmetric information, constant risk aversion and no transaction costs, Mossin [1968] shows that optimal insurance contracts are complete, while, in the presence of asymmetric information, optimal insurance contracts

2 98 DIDIER RICHAUDEAU offer partial coverage to at least some of the insured agents. In the case of adverse selection, where insurers cannot observe the risk of their insured agents, Rothschild and Stiglitz [1976] demonstrate that there is an incentive for the low-risk agent to buy a partial insurance contract to be distinguished from the high-risk agent. In the case of moral hazard, where the individuals prevention efforts are not observable by the insurance company, Arnott and Stiglitz [1988] show that, in general, the optimal market response consists in offering partial insurance contracts to induce the insured to provide greater levels of self-protection. We conclude that neither insurance companies nor insured agents are satisfied by the presence of adverse selection or moral hazard on the market. A number of theorists have therefore developed models including new tools to overcome the problem of insured agents opportunism. Crocker and Snow [1986], for adverse selection, and Bond and Crocker [1991] or Winter [1992], for moral hazard, demonstrate that the equilibrium insurance contract is closer to full insurance, if the insurance company is able to discriminate between insured agents according to their observable characteristics, which are supposed to be highly correlated with their intrinsic risk or their level of preventive effort. Dionne and Lasserre [1985] and Cooper and Hayes [1987], for adverse selection, and Radner [1981] and Rubinstein and Yaari [1983], for moral hazard, find similar results if insurance companies use long-term contracts with an efficient bonus-malus system to punish agents who cheat. Although the efficiency of these insurance rating systems depends crucially on the assumptions chosen (an infinite horizon and a zero discount rate have to be imposed to come close to full insurance contracts), their use always improves market efficiency. As insurance companies have been using such tools for a long time (even before economists began to be interested in them), the market equilibrium with adverse selection or moral hazard should be, in theory, close to that with perfect information. This implies that unobservable determinants of the choice of insurance contracts should not be correlated with the risk of accident in the case of perfect information, while they would in the presence of adverse selection or moral hazard, as the intrinsic risk and self-protection efforts of drivers remain, in this case, unobservable. The aim of the current article is to test whether asymmetric information remains on the French automobile insurance market by estimating whether unobservables are correlated with risk. We use French individual data from 1995 and model both the choice of automobile insurance, using a probit model, and the number of accidents, using a negative binomial count data model. With cross-section data set, we are unable to capture moral hazard effects, only adverse selection. Nevertheless, if the level of preventive effort can be predicted before insurance contracts are signed, then this kind of ex-ante moral hazard can be modeled in our analysis as a kind of adverse selection (using the Rothschild and Stiglitz equilibrium). However, if this prediction is imperfect, and more particularly, if it is underestimated, then traditional moral hazard can remain in the market and we cannot account for it. This topic has recently been studied by Puelz and Snow [1994], Dionne, Gouriéroux, and Vanasse [1997], and Chiappori and Salanié [1997]. The two first contributions use individual data sets from Canadian insurance companies to test if high-risk drivers choose lower deductibles than do low-risk drivers. Since deductibles always remain limited (less than $500 Canadian), insured agents may not completely reveal their hidden risk by this choice;

3 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 99 the lower the deductibles, the higher the probability of rejecting adverse selection. Chiappori and Salanié [1997] use a more informative signal to test adverse selection with a French individual data set: the choice between the compulsory third-party insurance contract and comprehensive insurance, which includes both third-party and property-damage insurance. While Puelz and Snow do not reject adverse selection, these two others papers do. We use this latter approach. The originality of our contribution comes from the model itself and from the kind of data used. First, previous works have been based on probit models. The cost of this approach is that the occurrence of an accident had to be considered as dichotomous; the number of accidents was not taken into account. This article provides an original method for extending previous attempts to include the number of accidents. The idea is to include the generalized residuals of a probit model, which models the choice of an insurance contract, into a count data model that estimates the probability of having 0, 1, 2, and so on accidents. Second, these data do not come from an insurance company. Therefore, we can use additional information that could be crucial to the analysis of the choice of insurance contract and the probability of an accident, but which were not available in the previous studies. In particular, this data set records the annual number of kilometers driven, the income of the household and the total number of accidents, which includes accidents that have not been declared to insurance companies. The article proceeds as follows. Section 2 presents briefly the automobile insurance market in France and draws a parallel between existing contracts and theoretical predictions. Section 3 explains the model to be tested. Section 4 presents the description of the data set. Section 5 comments on the results, and Section 6 concludes. 2. The French automobile insurance market Automobile insurance is compulsory for all automobile owners in France, but the insured agent has the opportunity to choose between two general formula. Third-party insurance covers property damage and bodily injuries to a third party that are caused by an accident for which you are found responsible. This policy constitutes the basis of all insurance contracts. Comprehensive insurance includes third-party insurance, but it also covers damage to your own car when you are liable for the accident. 1 Among all vehicles, 70 percent are covered by comprehensive insurance. Note that a policy that covers your own bodily injuries, when you are considered responsible for the accident, is only proposed as an option. These two contracts represent the main choice that agents have to make. This article considers the choice of contract as a signal to discriminate between individuals in terms of unobservable risk. Two other signals could potentially be used to capture the risk of insured agents. The first signal concerns the choice of the level of deductible. This is an important choice variable, but it is beyond the scope of our model. First, different deductibles are proposed by different insurance companies. Deductibles can be fixed, proportional, or partly fixed and partly proportional. Second, deductibles are illegal for third-party insurance. Third, the level of the highest deductible (around 3000 French francs) is probably too low to reveal the agents intrinsic risk. The second signal regards the choice of personal injury insurance for automobile accidents. This policy could be a good way of revealing agents risk or risk aversion, but there remains the problem of information concerning this optional policy.

4 100 DIDIER RICHAUDEAU Insured agents are not perfectly informed about the existence of this policy and the risk that is running when it is not taken out. In particular, most people, when they have purchased a comprehensive insurance, are convinced that they are insured against bodily injuries that they might cause to themselves. This lack of information is the main reason why only 45 percent of insured agents take out this policy. As a result, the choice between third-party insurance and comprehensive insurance constitutes the most credible signal of the agents own risk. The deductible choice should have also been considered in this study for instance, by proposing a sequential decision process where the insured first chooses between third-party or comprehensive coverage and then, if comprehensive coverage is taken out, chooses among the menu of deductible schemes. Unfortunately, we do not have any information about deductibles in this data. This is one reason why it is interesting to work both with the total number of accidents, where deductibles do not matter, and with the number of declared accidents. The premiums on these contracts depend on numerous factors. First, insurance companies differentiate their prices according to car and driver characteristics. Prices are then adjusted by the bonus-malus coefficient to take into account the driver s experience. The French bonus-malus system is defined by law: the initial coefficient is one, and if you have an accident during a year, your coefficient increases by 25 percent. If you do not have any accident, your coefficient decreases by 5 percent. The coefficient is bounded between 0.5 and 3.5 (with some exceptions for the lower bound). The final premium is calculated by multiplying the premium determined by the differentiation and the bonus-malus coefficient. In theory, under certain hypotheses, these two techniques are supposed to be efficient in reducing the problem of asymmetric information. If residual asymmetric information remains, then the agent s choice between third-party and comprehensive insurance should not depend on unobservable variables correlated with risk. It should instead be completely determined by the tradeoff between the cost of insurance and the cost of expected damage (which depends on observable variables like risk exposure and car value) and other factors that could be unobservable (risk aversion for instance) but would not be correlated with risk. However, if some asymmetric information remains on the market, the choice of insurance policy still depends on this tradeoff, as well as on unobservable variables (intrinsic risk and the expected level of preventive effort) that are correlated with risk. This is what we will test empirically on a French cross-section data set. 3. The model Our main difficulty lies in revealing the unobservable variables that could influence the choice of insurance contract. If this choice was completely determined by the observable variables, these latter variables would indicate whether the agent would buy either comprehensive insurance or third-party insurance without error. However, this choice is not simple to analyze: there remains an error that can be interpreted as the impact that unobservable variables have on the choice of insurance policy. Our model is the following: C i = 1 ifci = X 1i β 1 + u i 0 C i = 0 ifci = X 1i β 1 + u i < 0,

5 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 101 where C is a dummy variable indicating the type of insurance contract (C = 1 for a comprehensive contract and C = 0 for third-party insurance). C is a latent variable representing the utility of taking out comprehensive insurance over third-party insurance. X 1 represents the matrix of observable independent variables, and u is the residual. We assume that u is independent of X 1. If we use the normal density function to estimate this equation, we obtain a traditional probit model: P(C = C i ) = (X 1i β 1 ) C i (1 (X 1i β 1 )) 1 C i, (1) with (.) being the normal cumulative density function. The generalized residual of this estimation, ε i, captures the impact of unobservable variables on the choice of insurance policy. ε i = E β1 (u i C i, X 1i ) = φ(x 1i ˆβ 1 ) (X 1i ˆβ 1 )(1 (X 1i ˆβ 1 )) (C i (X 1i ˆβ 1 )), where φ(.) is the density function of the normal distribution and (.) the cumulative distribution. If ε i is positive, then the agent bought comprehensive insurance (C i = 1). The more positive ε i, the lower the predicted probability of buying comprehensive insurance and hence the higher the agent s own expected risk (the predicted probability is low, but he bought comprehensive insurance anyway). If ε i is negative, then the agent bought thirdparty insurance (C i = 0). The more negative ε i, the higher the expected probability of buying a comprehensive insurance contract and the lower the agent s own expected risk (the predicted probability is high, but he bought third-party insurance anyway). Thus, in both cases, the greater ε i is, the greater the expected intrinsic risk of the agent. The second step of our model consists in estimating the probability of accident. We use a count data model and, more precisely, a negative binomial model of the number of accidents. In earlier work (Richaudeau [1998]), we showed that this model, a generalization of the Poisson model that allows the mean of the random variable to be lower than its variance, was the model that fit the data best out of all the count data models proposed by Gouriéroux, Monfort, and Trognon [1984a, 1984b]. In a negative binomial model, the probability of having y i accidents is equal to P(Y = y i ) = Ɣ ( ) y i + 1 σ [σ 2 exp(x 2 2i β 2 )] y i Ɣ ( ), (2) 1 Ɣ(yi + 1)[1 + σ σ 2 exp(x 2 2i β 2 )] y i + 1 σ 2 where Ɣ(.) is the Gamma function. ε i is included in the matrix X 2. If the estimated parameter of ε i is positive, we conclude that asymmetric information remains. This will be our test of the efficiency of the French automobile insurance market.

6 102 DIDIER RICHAUDEAU This kind of two-stage estimation can create biases in the estimated standard errors, and we may also be confronted with correlation between the residual of the second-step equation and ε i, which is included in X 2. We will correct this using the result of Murphy and Topel [1985], who found, in a context similar, the asymptotic variance-covariance matrix for a two-step maximum likelihood model, which is = R R 1 [ 2 R 3 R 1 1 R 3 R 4 R 1 1 R 3 R 3 ] R 1 1 R 4 R 1 2 with R 1 = E L 1 β 1 R 3 = E L 2 β 1 4. The data ( ) L1 R 2 = E L ( 2 L2 β 1 β 2 β 2 ) ( ) L2 R 4 = E L ( ) 1 L2. β 2 β 1 β 2 We use a cross-sectional data set from the Institut National de la Recherche sur les Transports et leur Sécurité, a French organization whose purpose is to analyze traffic safety. We used only 5703 observations (which correspond to vehicles) from the 8659 available in 1995 as many observations contained missing values for variable such as the bonus-malus coefficient or income. 2 This data set comes from a survey; therefore, a number of different insurance companies are represented. This allows for a more general test; working with a data set from a single company may create selection biases because each company has relatively homogeneous customers. Nevertheless, since we do not know which insurance company covers each vehicle, we have a problem of omitted fixed effects. As French insurance companies and in particular, mutual firms, are open only to certain professions, we suppose that dummy variables corresponding to occupational groups take into account these firm fixed effects. Premiums are also unrecorded on our data set. These premiums depends mainly on all observable variables that we include in our model, and thus they are implicitly, but imperfectly, controlled for. We have information on whether the accident had been declared to the insurer or not. We work on the total number of accidents (1226 accidents for 5703 vehicles), which includes both declared (848) and undeclared accidents (30 percent of all accidents in our data set) (see Table 1). These accidents that are not declared to insurers are likely to be less serious. There were to few undeclared accidents for a separate analysis to be undertaken on them. 4 We test our model both on declared accidents (which might involve insurance companies and are supposed to be more serious) and on total accidents (with which the government is more concerned). We also had another important variable that is not usually recorded in insurance files: the number of kilometers driven (see figure 1).

7 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 103 Table 1. Distribution of number of accidents. All accidents Declared accidents Number of accidents Number Percent Number Percent Total Figure 1. Mean annual number of kilometers driven by age. We estimated our model both with and without this variable for two purposes. First, without this variable, we test for the presence of asymmetric information. Second, with this variable, we test if agents who are fully insured have a higher probability of accident than those who are only partially insured, ceteris paribus. This is one way of seeing if insurance could be a factor of risk. To include the maximum number of observable variables, 3 we use 17 types of information that allow us to create around 90 variables. The list of variables and their definitions are reported in the appendix. Detailed descriptive statistics are available from the author.

8 104 DIDIER RICHAUDEAU 5. The results 5.1. The choice of insurance The results (Appendix B, column 1) are interesting for two main reasons. First, there are few empirical references on this subject. Second, there is always a theoretical ambiguity between a risk-effect (for example when you have just obtained your driving license you are considered, and you generally consider yourself, to be risky, and therefore you should take out comprehensive insurance) and a price-effect (as you are a young driver, and therefore supposed to be riskier than older drivers, your premium is higher). Whereas we have no prior beliefs about which effect dominates, an empirical investigation can provide useful information. For instance, the price-effect dominates (i.e., the parameter is negative) for those who have a high bonus-malus coefficient 4 (i.e., those who had an accident for which they were considered as responsible, or those who are young drivers) or for cars that are frequently driven by a young secondary driver, while the risk-effect dominates (i.e., the parameter is positive) for those who are new drivers. The insurance rating process seems to be dissuasive enough to limit opportunistic behavior from insured agents. Other results seem to be consistent with what one would expect. We note that females have a greater probability of buying comprehensive insurance than males, indicating perhaps that their risk-aversion is greater. New cars those that are often used and accumulate many kilometers during a year and those at the top end of the group-classification used by insurance companies are more likely to be insured by comprehensive insurance than are old cars, rarely driven cars, and lower-end cars. Income has a positive effect, as does population density (the frequency of accidents is higher, even if they are less serious). Inhabitants of southern France seem to be dissuaded from buying comprehensive contracts, most likely because premiums are much higher there. One interesting point to note is that occupational groups also have a role in the choice of insurance. This can be explained by the characteristics of the French automobile insurance market and, in particular, by the presence of mutual insurance companies specialized in different occupational groups. To benefit from the mutual insurance system, these groups must have a lower frequency of accidents that allows insurers to offer them attractive premiums The probability of having an accident We choose to present first the comparison between results concerning the total number of accidents and those relative to the number of declared accidents. In fact, it is not easy to compare the two: a one-car accident, for instance, will never be declared when the contract is a third part only, whereas it will be declared under comprehensive coverage if the cost is large enough. Nevertheless, this is one of the main interests of our article, as all other papers have worked only on declared accidents, and we show that, overall, there is no great difference between the two regressions.

9 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 105 There are three specific differences between the two regressions (Appendix B, columns 2 and 3). First, the home population density has a very significant effect 5 only when all accidents are modeled. This suggests that accidents are generally less serious in urban areas. Second, the same relation appears for the parameter associated with the driver does not own the car he drives variable. This might reveal that agents with this characteristic change their behavior when the seriousness of the accident they could have increases. Third, the age of car, in terms of years or of total kilometers driven, has a greater negative impact on declared accidents than on accidents in general. When a car ages, its monetary value decreases, and it becomes less worthwhile to declare accidents. For all of the other risk factors, the results are quite similar regardless of the type of accident analyzed. We note that risk decreases with age until the end of the working life and increases thereafter, to reach the probability level of a male driver between ages 31 and 40. More driving experience does not have a significant impact until between 5 and 10 years after a driver s license is obtained, when each additional year decreases the probability of having accidents. As with age, the number of kilometers driven per year seems to have a more significant impact when it is coded with dummy variables (DKMINF5, etc.) than with continuous variables (KMINF5, etc.). The less a person drives, the less risky he is, but within each class of mobility, one additional kilometer has no additional effect on the probability of having an accident, except in the class 5000 to kilometers per year. Finally, small cars, in the sense of the group classification used by French insurance companies, have fewer accidents than others, whereas income has a positive effect, which could reflect decreasing risk-aversion, a lower relative cost of an accident that could induce the driver to take less care, or a higher opportunity cost of time that could induce insured agents to drive faster. It is interesting to note that some of the variables that are used by insurance companies to determine premiums are not significant in our model. This is the case for occupational groups, home area characteristics (which can double or triple premiums), or the existence of a secondary driver, even a young one (which can double premiums) The test of persistence of asymmetric information The result of the test for the persistence of adverse-selection and moral-hazard effects in the French automobile insurance market is to an extent consistent with the results of Chiappori and Salanié [1997] and Dionne, Gouriéroux, and Vanasse [1997]: the parameter associated with the generalized residual is positive but not statistically significant. Agents who buy the widest insurance coverage are not riskier than others, ceteris paribus (see Table 2). The use Table 2. The test with all observable variables. Estimated coefficient t-statistic All accidents (0.0538) Declared accidents (0.0671)

10 106 DIDIER RICHAUDEAU Table 3. The test without the number of kilometers driven. Estimated coefficient t-statistic All accidents (0.0537) Declared accidents (0.0670) of a risk-classification and a bonus-malus system seems to be an efficient way of controlling insured agents behavior. However, if we estimate our model without controlling for the number of kilometers driven, our conclusions are less clear (see Table 3). This framework is closer to the one faced by insurance companies, as French insurers do not usually observe this information. It is also the framework most closely related to previous contributions, since they used data sets provided by insurance companies. In this context, although the other parameters remain relatively stable, 6 both the parameter associated with the generalized residual and its significance level increase to reach a probability of rejection of the null hypothesis of no effect close to 95 percent. Here, agents who take out the widest insurance coverage do seem to have a higher probability of accident, not because the insurance induces them to pay less attention (moral hazard) but because they tend to drive more than agents who have only third-party insurance. Thus, we seem to be observing adverse selection (agents who buy comprehensive insurance are more mobile than the others), which may be different from Rothschild and Stiglitz s concept (adverse selection according to intrinsic risk), and that is not eliminated by the French automobile insurance rating system. We have attempted to introduce more cross-effects between variables to check if this conclusion is affected by nonlinearities in the variables, as Dionne, Gouriéroux, and Vanasse [1997] have suggested. We do not find any significant change in our results. Nevertheless, we can not introduce as many cross-effects as we want: for instance, our sample contains only six sport cars driven by young drivers. 6. Conclusion The results of our study have shown that the pricing methods that are used by French insurance companies to protect themselves from insured agents opportunistic behavior seem to be efficient. Agents who take out the widest insurance coverage do not have a higher probability of accident than other agents, ceteris paribus. Nevertheless, since French automobile insurance companies do not use annual kilometers driven in the calculation of their premiums, an adverse-selection problem may remain on the market: agents who take out the broadest policies are those who drive the most. Because these agents are more exposed to traffic hazards, they are riskier than others, but this is not a result of their insurance coverage.

11 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 107 These results have to be qualified. First, we take neither the seriousness of damage nor risk-aversion into account. Furthermore, the insurance-rating system uses much more sophisticated processes than those that we have considered, and, in particular, a lot of cross-effects are considered in calculating premiums. For instance, young male drivers with brand-new sports cars pay higher premiums than do others. We cannot include those effects in our model because we only have 5703 observations, and thus few young drivers, and almost no drivers with this kind of car. It could be of interest to improve our estimations with a larger data set. Nevertheless, our results can provide some interesting insights for theorists, insurance companies, and also the government. For instance, we have found that high-risk agents who are correctly identified by insurance companies (agents with a high bonus-malus coefficient or vehicles with a young secondary driver) are more likely to buy third-party insurance than are other drivers. Therefore, it seems that the insurance-rating system is particularly dissuasive for these agents and may even be too dissuasive: if premiums were actuarially fair, agents should be indifferent between the two contracts. This kind of overrating phenomenon may also reduce the size of the French insurance market. Appendix A. List of variables used Age and sex of the main driver DM1821 = 1 if the driver is a male between 18 and 21 years old; DF1821 = 1 if the driver is a female between 18 and 21 years old; etc. NOVICE1 = 1 if the driver obtained his driver s license in 1995; NOVICE2 = 1 if the driver obtained his driver s license in 1994; etc. LICE9495 is the year the driver s license was obtained if this year is 1994 or 1995 and zero otherwise; LICE46L is the year the driver s license was obtained if this year is anterior to 1946 and zero otherwise; etc. Driver s experience BONMALUS = bonus-malus coefficient Occupational groups (in order) Farmer, craftsman, private-sector executive, public-sector executive, professor, public-sector intermediate profession, private-sector intermediate profession, teacher, public-sector employee, private-sector employee, worker, other professions (reference), retired Home area DENSSUP = 1 if the population density is greater than 100,000 inhab/km 2. Paris (reference), the northern part of France, eastern, western, southwestern, southeastern, Mediterranean France Car GROU7L = 1 if the vehicle is classified in a group lower than or equal to 7 according to the group-classification used by French insurance companies (smallest cars); GROU14H = 1 if the vehicle is classified in a group higher or equal to 14 (biggest cars). NONOWN = 1 if the driver is not the owner of the car. DIESEL = 1 if the type of fuel used is diesel. NKMNAGE = 1 if total kms are lower than 50,000 and the car was manufactured after 1994; MOKMNAGE = 1 if total kms are greater than 50,000 and the car was manufactured after 1994; NKMMAGE = 1 if total kms are lower than 50,000 and the car was manufactured between 1991 and 1993; (Continued on next page.)

12 108 DIDIER RICHAUDEAU (Continued). MKMMAGE = 1 if total kms are between 50,000 and 100,000 and the car was manufactured between 1991 and 1993; OKMMAGE = 1 if total kms are greater than 100,000 and the car was manufactured between 1991 and 1993; NKMOAGE = 1 if total kms are lower than 50,000 and the car was manufactured before 1991; MKMOAGE = 1 if total kms are between 50,000 and 100,000 and the car was manufactured before 1991; OKMOAGE = 1 if total kms are greater than 100,000 and the car was manufactured before RANK = rank of the car in the household s fleet. Mobility DKMINF5 = 1 if the number of kilometers driven is less than 5,000 kms; DKM510 = 1 if the number of kilometers driven is between 5,000 and 10,000 kms; etc. KMINF5 = number of kilometers driven if this number is less than 5,000, and 0 otherwise; etc. ROAD1 = 1 if one type of road (urban, local roads and highways) is used for more than 10 percent of the kilometers; ROAD2 = 1 if two types of roads are used for more than 10 percent of the kilometers; etc. URBSUP59 = 1 if urban roads are used for more than 59 percent of the kilometers; LOCSUP59 = 1 if local roads are used for more than 59 percent of the kilometers; HIGSUP29 = 1 if highways are used for more than 29 percent of the kilometers. EVERYD = 1 if the car is used daily; ADAILY almost daily; casually (reference); WEEKEND week-end; ANEVER almost never. Income and wealth INCOMEUC = annual income per unit of consumption. HOUSOWN = 1 if living in owner-occupied housing. NBCAR = number of cars used by the household. Casual driver NOCDRIV = no casual driver; FLCDRIV = 1 if female casual driver driving fewer than 10 percent of the kms; FHCDRIV = 1 if female casual driver driving more than 10 percent of the kms; MLCDRIV = 1 if male casual driver driving fewer than 10 percent of the kms; MHCDRIV = 1 if male casual driver driving more than 10 percent of the kms; YLCDRIV = 1 if young casual driver driving fewer than 10 percent of the kms; YHCDRIV = 1 if young casual driver driving more than 10 percent of the kms. Appendix B. Results (parameter, standard-error, and significance level) Equation (1) Equation (2) Variable (1) Variable All accidents (2) Declared accidents (3) Age and sex of the main driver Age and sex of the main driver (reference DM27HIGH) (reference DM3140) DM1821 0, DM1821 0, , ,259 0,3426 0,3857 (Continued on next page.)

13 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 109 (Continued). Equation (1) Equation (2) Variable (1) Variable All accidents (2) Declared accidents (3) DF1821 0, DF1821 0, , ,3081 0,4391 0,5150 DM2226 0, DM2226 0, , ,1276 0,2019 0,2557 DF2226 0, DF2226 0, , ,1249 0,2148 0,2696 DF27HIGH 0, DM2730 0, , ,0580 0,1650 0,2164 NOVICE1 0, DF2730 0, , ,3219 0,1826 0,2449 NOVICE2 0, DF3140 0,0475 0, ,3151 0,1521 0,1801 NOVICE3 0, DM4150 0, , ,2513 0,1583 0,1886 Driver s experience DF4150 0, , BONMALUS 1, ,1755 0,2030 0,1495 DM5160 0, , Occupational groups 0,1904 0,2296 (reference OTHERPROF) DF5160 0, ,42947 FARMER 0, ,1794 0,2166 0,1719 DM6165 0, , CRAFTSMAN 0,1438 0,2563 0,3134 0,1476 DF6165 0, , PRIVEXEC 0, ,2864 0,3350 0,1350 DM6670 0, , PUBEXEC 0, ,2395 0,2866 0,2843 DF6670 0, ,48284 PROF 0, ,3175 0,4746 0,1833 DM7175 0, , PUBINT 0, ,2610 0,3147 0,1705 DF7175 0, , PRIINT 0, ,3848 0,4941 0,0942 DMF76HIG 0, , TEACHER 0, ,3240 0,3782 0,1381 LICE9495 1, , PUBEMP 0, ,1362 5,2279 0,1080 LICE9193 0, , PRIEMP 0, ,7854 2,0728 0,0911 LICE8690 1, ,28474 WORKER 0, ,7260 1,0812 0,0963 LICE6685 0, ,1196 RETIRED 0, ,1805 0,2180 0,0924 LICE4665 0, , (Continued on next page.)

14 110 DIDIER RICHAUDEAU (Continued). Equation (1) Equation (2) Variable (1) Variable All accidents (2) Declared accidents (3) Home area 0,2488 0,2939 reference (DENSINF, PARIS) LICE46M 0, , DENSSUP 0, ,9158 1,0926 0,0494 Driver s experience NORTH 0, BONMALUS 0, , ,0837 0,2201 0,2656 EAST 0, Occupational groups 0,0790 (reference OTHERPROF) WEST 0, FARMER 0, , ,0711 0,3316 0,4033 SOUTHWEST 0, CRAFTSMAN 0, , ,0734 0,2725 0,3166 SOUTHEAST 0, PRIVEXEC 0, , ,0686 0,2098 0,2473 MEDITERA 0,42738 PUBEXEC 0, , ,0720 0,3743 0,4326 Vehicle (references GROU1011, MKMMAGE) PROF 0, , GROU7L ,2309 0,2848 0,0892 PUBINT 0, , GROU89 0, ,3205 0,3508 0,0597 PRIINT 0, , GROU1213 0, ,1512 0,1800 0,0568 TEACHER 0, , GROU14H 0, ,1920 0,2315 0,0867 PUBEMP 0, , NONOWN 0, ,1732 0,2024 0,1768 PRIEMP 0, , DIESEL 0, ,1618 0,1923 0,0571 WORKER 0, , NKMNAGE 0, ,1628 0,2016 0,1195 RETIRED 0, , MOKMNAGE 0, ,1963 0,2338 0,2994 Home area NKMMAGE 0, (references DENSINF, PARIS) 0,1141 DENSSUP 0, , OKMMAGE 0, ,0766 0,0948 0,1768 NORTH 0, ,06404 NKMOAGE 0, ,1278 0,1550 0,1195 EAST 0, , MKMOAGE 0, ,1281 0,1560 0,0937 WEST 0, , (Continued on next page.)

15 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 111 (Continued). Equation (1) Equation (2) Variable (1) Variable All accidents (2) Declared accidents (3) OKMOAGE 1, ,1138 0,1292 0,0881 SOUTHWST 0, , RANK 0, ,1150 0,1436 0,0595 SOUTHEAST 0, , ,1046 0,1239 Mobility MEDITERA 0, , (references DKM1015, ROAD1, 0,1235 0,1481 CASUAL) DKMINF5 0, Vehicle 0,0809 (references GROU1011, MKMMAGE) DKM510 0, GROU7L 0, , ,0601 0,1724 0,2317 DKM1520 0, GROU89 0, , ,0686 0,1010 0,1215 DKM2030 0, GROU1213 0, , ,0759 0,0853 0,1015 DKMSUP30 0, GROU14H 0, , ,1110 0,1180 0,1421 ROAD2 0, NONOWN 0, , ,0554 0,1974 0,2625 ROAD3 0, DIESEL 0, , ,0765 0,0843 0,0998 URBSUP59 0, NKMNAGE 0, , ,0730 0,1219 0,1389 LOCSUP59 0,0316 MOKMNAGE 0,3736 0, ,0684 0,2622 0,3421 HIGSUP29 0, NKMMAGE 0, , ,0658 0,1290 0,1553 EVERYD 0, OKMMAGE 0, , ,0640 0,2011 0,2280 ADAILY 0, NKMOAGE 0, , ,0634 0,2347 0,2645 WEEKEND 0, MKMOAGE 0, , ,1284 0,1265 0,1477 ANEVER 0, OKMOAGE 0, , ,1549 0,1093 0,1264 RANK 0, , ,0845 0,1016 Income and wealth Mobility (references DKM1015, ROAD1, CASUAL) INCOMEUC 0, KMINF5 3, , ,0614 2,4854 3,0558 HOUSOWN 0, KM510 2, , ,0529 1,1839 1,4549 (Continued on next page.)

16 112 DIDIER RICHAUDEAU (Continued.) Equation (1) Equation (2) Variable (1) Variable All accidents (2) Declared accidents (3) NBCAR 0, KM1015 0, , ,0422 0,9438 1,1311 Occasional drivers KM1520 0, , (reference FLCDRIV) 1,0003 1,1795 NOCDRIV 0, KM2030 0, , ,0551 0,5589 0,6500 FHCDRIV 0, KMSUP30 0, , ,0793 0,4078 0,4556 MLCDRIV 0, DKMINF5 1, , ,0862 0,4470 0,5521 MHCDRIV 0, DKM510 0, , ,2101 0,1871 0,2407 YLCDRIV 0, DKM1520 0, , ,2402 0,1335 0,1634 YHCDRIV 0, DKM2030 0, , ,2635 0,1484 0,1842 DKMSUP30 0, ,55653 Constant 0,1943 0,2264 CONSTANT 1, ROAD2 0, , ,2092 0,0949 0,1159 ROAD3 0, , ,1177 0,1449 URBSUP59 0, , ,1122 0,1331 LOCSUP59 0, , ,1066 0,1246 HIGSUP29 0, ,1025 0,0931 0,1110 EVERYD 0, ,055 Likelihood at zero for (1): 0,1106 0, ,01837 ADAILY 0, , Likelihood at the optimum for (1): 0,1073 0, , WEEKEND 0, , ,2637 0,3520 Likelihood at zero for (2): ANEVER 0, , , ,3777 0,4258 Likelihood at the optimum for (2): Income 3042,6937 INCOMEUC 0, , Likelihood at zero for (3): 0,0944 0, ,8072 Casual drivers Likelihood at the optimum for (3): (reference FLCDRIV) 2370,80078 NOCDRIV 0, , ,0849 0,1029 (Continued on next page.)

17 AUTOMOBILE INSURANCE CONTRACTS AND RISK OF ACCIDENT 113 (Continued). Equation (1) Equation (2) Variable (1) Variable All accidents (2) Declared accidents (3) FHCDRIV 0, , ,1149 0,1390 MLCDRIV 0, , ,1356 0,1626 MHCDRIV 0, , ,3213 0,3562 YLCDRIV 0, , ,4101 0,4499 YHCDRIV 0, , ,3895 0,7253 Constant CONSTANT 1, , ,3249 0,3957 Generalized residual RESIDU 0, , ,0538 0,0671 Sigma2 SIGMA2 0, , ,1168 0, %; 5%; 1% Acknowledgments This article has benefited from useful remarks of LAMIA seminar participants (especially Marc Gurgand, Louis Lévy-Garboua, and David Margolis). The comments of Bernard Salanié and of two anonymous referees also led to substantial improvements. Notes 1. Third-party and property damage insurance cannot be taken out from different insurance companies. 2. We checked that the observations used in the final sample had the same characteristics as those deleted. No significant difference was found using chi-square tests and a direct comparison of descriptive statistics. 3. Driver s age and sex, number of years since obtaining a driving licence, bonus-malus coefficient, occupational groups, population density of home area and home region, rating of car in terms of power, speed, and weight according to the classification used by French insurance companies, car ownership, type of petrol used, age of the vehicle in terms of years and kilometers driven, car rank among the household s fleet, number of annual kilometers driven, type of road used, use frequency, income, and characteristics of occasional drivers. 4. The introduction of the bonus-malus coefficient in the matrix of independent variables can generate a problem of endogeneity. Nevertheless, we note that the accidents that modify the bonus-malus coefficient are very specific and often differ from the kind of accidents we have used as our dependent variable. First, these accidents must be declared. Second, you have to be recognized as responsible for it, at least partially. Without this variable, results remain stable, and therefore we expect the bias to be small. 5. Murphy and Topel s correction does not change the estimation of standard errors in any major way.

18 114 DIDIER RICHAUDEAU 6. Detailed results of this estimation are not reported here. All parameters have the same sign as in the previous estimation. Only marginal effects and significance levels of parameters associated with variables that are correlated with the number of kilometers driven (use frequency, car age, rank of the car among the household s fleet, type of fuel used) increase. This group of variables related to kilometers driven is significant according to a likelihood-ratio test between the two specifications. Results are available from the author. References ARNOTT, R.J. and STIGLITZ, J.E. [1988]: The Basic Analytics of Moral Hazard, Scandinavian Journal of Economics, 90, BOND, E.W. and CROCKER, K.J. [1991]: Smoking, Skydiving and Knitting: The Endogoneous Categorization of Risks in Insurance Markets with Asymmetric Information, Journal of Political Economy, 99, CHIAPPORI, P.A. and SALANIÉ, B. [1997]: Empirical Contract Theory: The Case of Insurance Data, European Economic Review, 41, COOPER, R. and HAYES, B. [1987]: Multi-Period Insurance Contracts, International Journal of Industrial Organization, 5, CROCKER, K.J. and SNOW, A. [1986]: The Efficiency Effects of Categorical Discrimination in the Insurance Industry, Journal of Political Economy, 94, DIONNE, G., GOURIÉROUX, C., and VANASSE, C. [1997]: The Informational Content of Household Decisions with Application to Insurance Under Adverse Selection, CREST Working Paper DIONNE, G. and LASSERRE, P. [1985]: Adverse Selection, Repeated Insurance Contracts and Announcement Strategy, Review of Economic Studies, 52, GOURIÉROUX, C., MONFORT, A., and TROGNON, A. [1984a]: Pseudo-Maximum Likelihood Methods: Theory, Econometrica, 52, GOURIÉROUX, C., MONFORT, A., and TROGNON, A. [1984b]: Pseudo-Maximum Likelihood Methods: Application to Poisson Models, Econometrica, 52, MOSSIN, J. [1968]: Aspects of Rational Insurance Purchasing, Journal of Political Economy, 76, MURPHY, K.M. and TOPEL, R.H. [1985]: Estimation and Inference in Two-Step Econometric Models, Journal of Business and Economic Statistics, 3, PUELZ, R. and SNOW, A. [1994]: Evidence on Adverse Selection: Equilibrium Signaling and Cross- Subsidization in the Insurance Market, Journal of Political Economy, 102, RADNER, R. [1981]: Monitoring Cooperative Agreements in a Repeated Principal-Agent Relationship, Econometrica, 49, RICHAUDEAU, D. [1998]: Modélisation du risque routier, Recherche Transports et Securité, 60, ROTHSCHILD, M. and STIGLITZ, J. [1976]: Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information, Quarterly Journal of Economics, 4, RUBINSTEIN, A. and YAARI, M.E. [1983]: Repeated Insurance Contracts and Moral Hazard, Journal of Economic Theory, 30, WINTER, R.A. [1992]: Moral Hazard and Insurance Contracts, in Contributions to Insurance Economics, G. Dionne (Ed.), Boston: Kluwer,