Testing Advantageous Selection by Hidden Action: Evidence from Automobile Liability Insurance. Rachel J. Huang. Larry Y. Tzeng. Kili C.

Transcription

1 Testing Advantageous Selection by Hidden Action: Evidence from Automobile Liability Insurance Rachel J. Huang Associate Professor, Department of Finance Yuan Ze University, Chung Li, Taiwan Larry Y. Tzeng Professor, Department of Finance, National Taiwan University, Taipei, Taiwan Kili C. Wang Associate Professor, Department of Insurance, Tamkang University, Taipei, Taiwan

2 Testing Advantageous Selection by Hidden Action: Evidence from Automobile Liability Insurance Abstract This paper examines advantageous selection in automobile liability insurance from the approach with hidden action, which argues that the individual s private information on his own characteristics will affect his decision on the investment on precautionary effort (hidden action) to reduce the loss probability and further results in a negative relationship between coverage and loss probability in equilibrium. We argue that individual s vehicle maintenance record could be a proper proxy for the precautionary effort. By combining insurance data from an insurance company and maintenance data from the largest car manufacturer in Taiwan, we demonstrate advantageous selection in automobile liability insurance in Taiwan. JEL classification: D80, G22, C30 Keywords: asymmetric information, advantageous selection, hidden action, automobile liability insurance. 1

3 1. Introduction Following Rothschild and Stiglitz (1976) and Shavell (1979), many theoretical papers have shown that adverse selection and moral hazard are essential in the insurance market. On the basis of Rothschild and Stiglitz (1976) and Shavell (1979), it is generally believed that the occurrence of the risk and the choice of the coverage are positively correlated. If the adverse selection problem exists in the insurance market and is settled on a separating equilibrium as suggested by Rothschild and Stiglitz (1976), then the high risks will choose high coverage while the low risks will choose low coverage. On the other hand, Shavell (1979) showed that the insured with higher coverage have less incentive to expend efforts on preventing the risk. To test whether adverse selection and/or moral hazard exist in the real world, many recent empirical studies using data from alternative insurance markets have examined the correlation between risk and coverage but they find mixed evidence for the theoretical prediction. Some of them 1 have found evidence that supports the existence of adverse selection and/or moral hazard. Other papers, by contrast, have found no evidence of adverse selection and/or moral hazard 2. Furthermore, some empirical papers, such as Cawley and Philipson (1999) and Finkelstein and McGarry (2006), have even found a significant negative correlation between risk and coverage. To explain why the risk and the coverage are negatively correlated, the theoretical side has developed models from two approaches: the first one adopts hidden action, and the second one assumes multi-dimensional hidden information. In the approach with hidden action, individual s risk type is determined by his/her investment on self-protection, and the investment is unobservable to insurers. This 1 For instance, Puelz and Snow (1994), Cardon and Hendel (2001), Finkelstein and Poterba (2004), and Cohen (2005), etc. 2 For example, Chiappori and Salanie (1997), Richaudeau (1999), Chiappori and Salanie (2000), Dionne, Gourieroux and Vanasse (2003), Dionne, Gourieroux and Vanasse (2001) and Saito (2006). 2

4 line of research assumes that individuals have the same ex ante risk probability and are heterogeneous in only one characteristic, which could be degree of risk aversion (de Meza and Webb, 2001), risk perception (Koufopoulos, 2002; Huang, Liu and Tzeng, 2008), patient (Sonnenholzner and Wamback, 2006), or regret (Huang, Muermann and Tzeng, 2007). This private information about the individual s characteristic will affect the individual s investment decision on self-protection, and further result in heterogeneous in the ex post risk probability. The above papers predict that the insurance market could settle on a separating equilibrium where the individuals with specific characteristic (i.e. more risk averse, less optimistic, more patient, less regret, respectively) will purchase more coverage and pay more attention to risk prevention. In other words, the high coverage is used to screen the low risk type. This phenomenon is named as advantageous selection 3 by de Meza and Webb (2001). In multi-dimensional private information approach, no hidden action is involved. Individuals are heterogeneous in ex ante risk probability as well as in other private characteristics. The risk type and the characteristics are independent. Individuals within one risk type could be further classified by degree of risk aversion (Smart, 2000; Villeneuve, 2003; Liu and Browne, 2007), or wealth (Wambach, 2000; Liu and Browne, 2007). Under certain condition, the equilibrium might also settle on advantageous selection. Fang, Keane and Silverman (2008) is the first one to directly test advantageous selection by adopting multi-dimensional private information approach. They argue that if the private characteristics is positively correlated with insurance coverage and negatively correlated with risk, then the characteristics could be a source of 3 Hemenway (1990, 1992) named this favorable selection for the insurance companies as propitious selection. 3

5 advantageous selection. Focusing on the Medigap insurance market in the United States, they combine Medicare Medicare Current Benecficiary Survey data and Health and Retirement Study and find that the sources of advantageous selection include income, education, longevity expectations, financial planning horizons, and cognitive ability. Differ from Fang, Keane and Silverman s (2008) testing on the multi-dimensional approach, this paper examines advantageous selection in automobile liability insurance from the approach with hidden action. Since most papers in the literature used data from insurance companies to examine asymmetric information in insurance markets, it is nearly impossible to have variables which the insurance companies can not observe. Our paper is valuable because it comes to investigating the evidence of advantageous selection by hidden actions which the previous literature has not yet explored. According to the literature, the hidden action has to satisfy two criteria: the first one is that the activity could reduce the likelihood that a loss will occur; the second criterion is that this activity to reduce risk is unobservable to insurers. In the automobile liability insurance we focus on, we propose that the individual s vehicle maintenance record could an adaptable proxy for precautionary action. If the vehicle is under proper maintenance, the car accident probability will decrease. Furthermore, insurance companies do not use the maintenance record under underwriting process. In other words, the record is a hidden action. Hence, the maintenance record could meet the requirement in the literature as an investment of self-protection. To collect information concerning the individual s car maintenance behavior, we have access to the data of a major car dealer. Since the car dealer has the largest market share in Taiwan, our data could still be representative even through we only focus on one particular brand of car. 4

6 The above data is combined with a data from a large insurance company. From the information of the insurance company, we classify the insured who only purchase compulsory third-party liability insurance as the insured with low coverage and treat individuals who extend to purchase voluntary liability insurance as the insured with high coverage. Our classification is similar as in Chiappori and Salanie (2000) who group liability insurance in France which covers damage inflicted to third parties as low coverage and the liability insurance which covers not only the third parties but also the insured as high coverage. Thus, we test whether the individuals who purchase both the compulsory and voluntary liability insurance are less likely to incur a liability claim than those who only purchase compulsory liability insurance. It is important to note that the investment on self-protection is an endogenous decision of individuals rather than an exogenous factor in this approach. Thus, we adopt the two-stage model as in Dionne et. al. (2001) to test the conditional correlation between the choice of coverage and claims, the choice of coverage and precautionary effort, as well as between precautionary effort and claims. We find that: (1) Individuals who purchase voluntary third party liability insurance tend to have a lower chance of filing a significant at-fault claim on the automobile liability insurance. (2) Individuals who purchase voluntary third party liability insurance would have greater tendency to maintain their cars properly. (3) Individuals who are more willing to invest in precautionary effort have a lower chance of filing a significant at fault claim on the liability insurance. Our findings suggest the existence of advantageous selection phenomenon under the approach with hidden action in the automobile liability insurance market in Taiwan. The first contribution of this paper is that we find that advantageous selection phenomenon is the driving force of the automobile liability insurance market of Taiwan. Our finding is inconsistent with Chiappori and Salanie (2000) who find 5

7 insignificant negative correlation between coverage and claim for the beginning drivers and insignificant positive correlation between them for the experienced drivers in France automobile liability insurance market. Second and moreover, we provide evidence for advantageous selection under the approach with hidden actions. We show that, in explaining advantageous selection in the insurance market, the theoretical model adopting hidden actions is at least as important as those with multiple-dimensional hidden information. Third, we find that individual s proper maintenance behavior on their insured car which can be treated as an index for the precautionary effort of the insured and could influence the loss probability of the driver. As far as we know, no other research has ever found this relationship yet. The remainder of this paper includes: the description of our data is in Section 2; the introduction of our empirical methodology are in section 3; the empirical outcomes are reported in section 4, and the conclusion of this paper is in section Data Our empirical data are combined from two sources. Both data sets cover the period from year 2002 to All the insurance data are obtained from one large insurance company in Taiwan. The insurance data variables include the content of the insurance contract and its claim record, as well as the characteristics of the insured individual and the insured car such as age, sex, marital status and sex of the insured, and the age, brand, registered area and usage of the insured car. The vehicle maintenance data are obtained from a car manufacturer which has the largest market share in Taiwan. To match these two sources, we restrict our sample on one particular brand. The market share of that brand of car in 2002 was 24%, and has gradually increased with time. In the year 2006, the share has increased to 29%. Thus, even through in the final sample, we focus on only one brand, the sample size is still large. We have 936,323 samples in total. 6

8 2.1 Definition of variables The insurance we focus on is automobile liability insurance. There are two types of automobile liability insurance in Taiwan: compulsory and voluntary third party liability insurance. The purpose of compulsory liability is to protect the third parties life and health, which is damaged due to the usage of motor vehicle. The upper-limit indemnity for the medical expense of the injured third party is NT$200,000 (about $6,450), it for maim or life is NT$1,500,000 (about $48,380). The voluntary third party liability insurance compensates in the event of insufficient coverage under the compulsory insurance. The insured could purchase the insurance to reimburse the third parties bodily injury, or property damage. The written premium for the bodily injury is billion NT dollars in year 2007, and that for property damage is billion NT dollars. In order to keep the homogeneity of the insurable risk, we aim at the compulsory liability insurance and the voluntary third party bodily injury liability insurance. We use a dummy variable coverage to define the insurance coverage. It equals to one if the individual purchase voluntary third party bodily injury liability insurance, zero if the individual only purchase compulsory liability insurance. To define an accident, we focus on the claims with amount greater than one million NT dollars. Li, Liu and Yeh (2007) and Wang, Chung and Tzeng (2008) has suggested moral hazard exist in automobile insurance of Taiwan. Thus, we focus on the accidents with severe claim amount, since, in this kind of accident, moral hazard problem would be less severe. The dummy variable claim equals one if the individual has filed an at fault claim with claim amount greater than one million NT dollars, it equals zero if there is no claim. As mentioned in Introduction, the vehicle maintenance record from the car manufacturer could be a good proxy as the individual s investment in precautionary effort. In our data set, there are two variables could identify whether the vehicle is 7

9 properly maintained: one records whether the vehicle regularly comes back to the repair shop at the recommended mileage ( pm_mileage ), the other gets down whether the vehicle follows the maintenance instruction to replace or check the suggested items ( pm_item ). There is another variable in our date set which marks whether the vehicle comes back to the repair shop at the recommended maintenance time ( reminded ). We do not use this variable as a proxy of proper maintenance, because the repair shop will keep calling their customs to remind them at recommended maintenance time. If the insured come back at the suggested maintenance time, we will not be able to identify whether the insured maintain his/her car just from being reminded or the insured knows that he/she should maintain his/her car to reduce the probability of occurring an accident. However, no matter the insured voluntarily or passively come to the repair shop, the vehicle will be examined and the accident probability could be reduced. In other words, although the variable reminded is not a good proxy as hidden action, it should be considered as a control variable when we examine the relationship between claim, coverage and effort. The definitions of all other variables used in our paper are listed in Table Basic Statistics The basic statistics for the variables are presented in Table 2. In our sample, the percentage of the insured who extend their coverage to voluntary third party bodily injury liability is around 71% per year. The percentage is about 9% higher than that in a sample including all brands of vehicle. The percentage of at fault severe accident claims is about 0.01% to 0.03% per year, whereas the percentage of all at fault claim is about 0.69% to 0.98% per year. The percentage of properly maintained vehicle is increasing during our sample period no matter which variable is used to define proper maintenance. No matter which variable is used, we have at least 14% and at most 31% of the insured properly maintain their vehicle. 8

10 The sample is well-distributed. Over 90% of the cars are sedan. Around 50% of the car owners are female. Over 90% of the insured are married. 4 The insured age distribution is focused on those between the ages of 30 and 60. The high percentage of mid-aged, married female owner may be caused by the premium design. In Taiwan, the insurance premium depends on the characteristics of the car owner, and a mid-aged, married female owner could enjoy the most advantage in premium. 46% of the cars are registered in the north of Taiwan and about 50% of the cars are in the cities. The percentage of policies sold through the dealer-owned agent (the dummy variable of channel_d ) is the highest, around 33%. This may comes from the fact that our car maintenance data is from a large car manufactory in Taiwan, which is also a large dealer-owned agent in Taiwan. Hence, when we focus on the particular brand of cars, there should have higher percentage of insurance contracts are sold through this channel. 3. Methodology This paper adopts the methodology of two-stage method proposed by Dionne et al. (2001) to test the existence of asymmetric information in the insurance market. We first examine the pair wised relationship between coverage, claim and proper maintenance. Secondly, we re-examine the relationship by simultaneously treating coverage, claim and proper maintenance as endogenous variables. To test the pair wised relationship, we construct three models. In the first model, we examine the relationship between coverage and claim. The probability of filing an at fault claim with amount greater than one million NT dollars is estimated in the first stage by running the following Probit regression: 4 If we include all brands of the vehicle, the percentage of married insured is still over 88%. 9

11 Prob( claimi = 1 X1 i) = Φ( X1 iδclm) (1) where X 1 i is the vector of variables for the insured s information, including the characteristics of the insured and the characteristics of the insured s car which are used for calculating the premium. These independent variables are listed in the first part of Table 1. δ clm is the vector of regression coefficients, and Φ is the density and cumulative distribution function of N (0,1). After regression (1) is run, the estimated probability of a claim can be calculated. Accordingly, we can run the second-stage regression to estimate the probability of choosing a high-coverage insurance contract. The explanatory variables in the second-stage regression include the estimated probability of a claim, the dummy variable of a claim, and the relevant characteristic variables of the insured X 2 i, which include the first part and second part variables in Table 1. Hence, the second-stage Probit regression is as follows: Prob( coverage = 1 claim ˆ, claim, X ) =Φ ( β claim + β claim ˆ + X β ), i i i 2i 1, cov i 2, cov i 2i 3, cov (2) where claim ˆ i is the estimated probability of filing a claim which is estimated using the first-stage Probit regression. When we test the significance of the conditional correlation as the criterion for asymmetric information, it is the coefficient of β 1,cov that we seek to test. If β 1,cov is significantly positive, then the driving force behind the asymmetric information in this insurance market is either adverse selection or moral hazard, or both of them. However, if β 1,cov is significantly negative, then the driving force behind the asymmetric information in this insurance market is advantageous selection. In the second model examining the pair wised relationship, we estimate the 10

12 probability of proper maintenance at the first stage: Prob( pm _ ki = 1 X1 i) = Φ( X1 iδ pm), (3) where k = mileage, item. At the second stage, we examine the relationship between claim and proper maintenance: Prob( claim = 1 pm_k, pm_k ˆ, X ) =Φ ( β pm_k + β pm_k ˆ + X β ), i i i 3i 1, clm i 2, clm i 3i 3, clm (4) where X 3 i includes the first part and third part variables in Table 1 and pm_k ˆ i is the estimated probability of proper maintenance from Equation (3). In the first stage of the third model, we estimate the probability of choosing voluntary third party bodily injury liability insurance: Prob( coveragei = 1 X1 i) = Φ( X1 iδcov). (5) We further test the correlation between coverage and proper maintenance at the second stage: Prob( pm_k = 1 coverage, coverage ˆ, X ) =Φ ( β coverage + β coverage ˆ + X β ), i i i 1i 1, pm i 2, pm i 1i 3, pm (6) where covˆ eragei is the estimated probability of the insured choosing voluntary liability insurance from Equation (5). Further, we test the conditional correlation in regard to the relationship among risk, coverage and precautionary effort by regress each endogenous variable by the other two endogenous variables as well as other independent variables. In the methodology of two stage method, we first regress the three endogenous variables, precautious effort of proper maintain, filing claim and coverage choice, separately: Prob( coveragei = 1 X1 i) = Φ( X1 iδcov) (7) Prob( claimi = 1 X1 i) = Φ( X1 iδclm) (8) 11

13 Prob( pm _ ki = 1 X1 i) = Φ( X1 iδ pm) (9) From regression (7), (8) and (9), we can calculate the estimated probability of proper maintenance, a claim and choosing a high-coverage insurance contract; i.e. coveˆ rage, claˆ im, p mˆ _ k. And then, we can run the second-stage regression to estimate the probability of a claim, choosing a high-coverage insurance contract and precautionary effort of proper maintain by regress on the other two endogenous variables, the estimated probability of the other two endogenous variables and their vectors of independent variables in regression (7), (8) and (9). The three regressions in the second stage are as follows: Prob( coverage 1 _,, ˆ _, ˆ i = pm ki claimi pm ki claimi, X 2i), =Φ ( α pm _ k + α claim + α pmˆ _ k + α claim ˆ + X α ) 1, cov i 2,cov i 3, cov i 4, cov i 2i 5, cov Prob( claim ˆ ˆ i = 1 coveragei, pm _ ki, coveragei, pm _ ki, X 3i), =Φ ( α coverage + α pm _ k + α coverage ˆ + α pmˆ _ k + X α ) 1, clm i 2, clm i 3, clm i 4, clm i 3i 5, clm (10) (11) Prob( pm _ k 1,, ˆ, ˆ i = coveragei claimi coveragei claimi, X1 i). =Φ ( α coverage + α claim + α coverage ˆ + α claim ˆ + X α ) 1, pm i 2, pm i 3, pm i 4, pm i 1i 5, pm (12) The key variables that we should observe in these three regressions are α 1, j and α, j = cov, clm, pm. According to the theory of advantageous selection, the 2, j hypotheses first infer a negative conditional correlation between risk and coverage, i.e., α 1, clm and α 2,cov are significantly negative. Secondly, the conditional correlation between precautionary effort and risk is negative, i.e., α 2, clm and α 2. pm are significantly negative. Third, the conditional correlation between precautionary effort and coverage is positive, i.e., α 1,cov and α 1, pm are significantly positive. 12

14 4. Empirical Results The empirical results are listed in Tables 3 to 6. The results from the coefficient β of pair wised two stage Probit method, j = cov, clm, pm, are listed in Tables 3 1,j and 4. The results from the coefficients α 1,j and α 2,j of Equations (10) to (12) are listed in Tables 5 and 6. The results from the pair wised two stage method show that the conditional correlation between risk and coverage is significantly negative in year 2002, year 2003 and year 2005, but insignificant negative in year 2004 and year Worth to be mentioned, although the coefficients of year 2004 and year 2006 are statistically insignificant, but the p-value of them are still around 10%. Secondly, the theory of advantageous selection also predict that precautionary effort and claim is negatively correlated. This theoretical prediction is supported in our test. Tables 3 and 4 shows that the coefficients β 1,clm are significantly negative in all years. Thirdly, the other concerned prediction in the theory of advantageous selection is that precautionary effort and coverage choice should be positive correlated. This prediction is also supported in our empirical test. When we adopt the pair wised two stage method, no matter which proxy variable of precautionary effort we use, all the coefficient of β 1, pm are significantly positive in Tables 3 and 4. In addition, after controlling the hidden action, Tables 5 and 6 demonstrate that the negative correlation between coverage and claim becomes insignificant. Hence, the triangular relationships predicted by the theory of advantageous selection basically are sustained in our empirical test. 5. Conclusions 13

15 Through the data from one of the large insurance company in Taiwan, we collect the data of automobile compulsory liability insurance as well as voluntary third party liability insurance. This data let us own the information concerning the contract content, the claim record, as well as the characters of the insured. Through one of repair workshops owned by a particular brand of car manufacturer which is with fairly large market share in Taiwan, we collect unique information concerning the individual s car maintenance behavior which include whether the car owners properly maintain their cars according to the recommended mileage and the recommended maintain items. Together with these two data resources, we could have three important endogenous variables, including: risk, coverage and precautionary effort, which let us have the opportunity to test directly the prediction of advantageous selection. By using two-stage method of Dionne et al. (2001) to check the pair wised relationship between coverage, claim and precautionary effort, we find that: (1) Individuals who are willing to extend to purchase voluntary third party liability insurance have a lower chance of filing an significant at-fault claim on the automobile compulsory liability insurance. (2) Individuals who are willing to extend to purchase voluntary third party liability insurance would also have greater tendency to pay the precautionary effort. (3) Individuals who are more willing to pay attention to precautionary effort are also have lower chance of filing an significant at fault claim on the compulsory liability insurance. After consider precautionary effort, the correlation between coverage and claim becomes insignificant. Hence, the advantageous selection is supported from the approach of hidden action. 14

16 References: Cardon, J.H., & Hendel, I Asymmetric information in health insurance: Evidence from the national medical expenditure survey. Rand Journal of Economics. 32(3): Cawley, J., & Philipson, T.J An empirical examination of information barriers to trade in insurance. American Economic Review. 89(4): Chiappori, P.A., & Salanie, B Empirical contract theory: The case of insurance data. European Economic Review. 41(3): Chiappori, P.A., & Salanie, B Testing for asymmetric information in insurance markets. Journal of Political Economy. 108(1): Cohen, A Asymmetric information and learning: Evidence from the automobile insurance market. The Review of Economics and Statistics. 87(2): De Meza, D., & Webb, D. C Advantageous selection in insurance markets. Rand Journal of Economics. 32(2): Dionne, G., Gourieroux, C., & Vanasse, C Testing for evidence of adverse selection in the automobile insurance market: A comment. Journal of Political Economy. 109(2): Dionne, G., Gourieroux, C. & Vanasse, C The informational content of household decisions with applications to insurance under adverse selection, Working Paper. Venice International University, San Servolo. Fang, H., Keane, M. & Silverman, D Sources of advantageous selection: Evidence from the Medigap insurance market. Journal of Political Economy, 116(2): Finkelstein, A., & McGarry, K Multiple dimensions of private information: Evidence from the long-term care insurance market. American Economic Review. 96(4): Finkelstein A., & Poterba, J Adverse selection in insurance markets: Policyholder evidence from the U.K. annuity market. Journal of Political Economy. 112(1): Huang, R.J., Liu, Y, & Tzeng, L.Y Hidden overconfidence and advantageous selection. Working paper. Huang, R.J., Muermann, A., & Tzeng, L.Y Hidden regret in insurance markets: adverse and advantageous selection. Working paper. Koufopoulos, K Asymmetric information, heterogeneity in risk perceptions and insurance: An explanation to a puzzle. Working paper. Puelz, R., & Snow, A Evidence on adverse selection: Equilibrium signaling and cross-subsidization in the insurance market. Journal of Political Economy. 102(2):

17 Richaudeau, D Automobile insurance contracts and risk of accident: An empirical test using French individual data. The GENEVA Papers on Risk and Insurance Theory. 24(1): Rothschild, M., & Stiglitz, J.E Equilibrium in competitive insurance markets: An essay on the economics of imperfect information. Quarterly Journal of Economics. 90(4): Saito, K Testing for asymmetric information in the automobile insurance market under rate regulation. Journal of Risk and Insurance. 73(2): Shavell, S On moral hazard and insurance. Quarterly Journal of Economics. 93(4): Sonnenholzner, M. & Wambach, A On the role of patience in an insurance market with asymmetric information. Working paper Wang, K.C., Huang, R.J. & Tzeng, L.Y Which firm gets burned? Conference of American Risk and Insurance Association. Washington D.C. USA. 16

18 Table 1 Definitions of Variables Variable Definition Dependent variables: coverage a dummy variable that equals 1 when an individual purchases voluntary third party bodily injury liability insurance, otherwise it equals 0 claim pm_mileage pm_item a dummy variable that equals 1 when the insured has filed an at fault claim with amount greater than one million NT dollars, otherwise it equals 0 a dummy variable that equals 1 when the insured s car is maintained according to the recommended mileage in each time maintain each time during the year, otherwise it equals 0. a dummy variable that equals 1 when the insured s car is maintained not below the recommended maintain items in each time maintain during the year, otherwise it equals 0. Independent variables: First part: carage0 a dummy variable that equals 1 when the car is less than one year old, otherwise it equals 0 carage1 a dummy variable that equals 1 when the car is one year old, otherwise it equals 0 carage2 a dummy variable that equals 1 when the car is two years old, otherwise it equals 0 carage3 a dummy variable that equals 1 when the car is three years old, otherwise it equals 0 carage4 a dummy variable that equals 1 when the car is four years old, otherwise it equals 0 carage5 a dummy variable that equals 1 when the car is five years old, otherwise it equals 0 carage6 a dummy variable that equals 1 when the car is six years old, otherwise it equals 0 carage7 a dummy variable that equals 1 when the car is seven years old, otherwise it equals 0 carage8 a dummy variable that equals 1 when the car is eight years old, otherwise it equals 0 catpcd_1 a dummy variable that equals 1 when the car is a sedan and is for non-commercial or for long-term rental purposes, otherwise it equals 0 catpcd_2 a dummy variable that equals 1 when the car is a small freight-truck and is for non-commercial purposes or for business use, otherwise it equals 0 sexf a dummy variable that equals 1 when the owner of the car is female, otherwise it equals 0 married a dummy variable that equals 1 when the owner of the car is married, otherwise it equals 0 age25 a dummy variable that equals 1 when the insured is between the ages of 30 and 25, otherwise it equals 0 age30 a dummy variable that equals 1when the insured is between the ages of 60 and 30, otherwise it equals 0 age60 a dummy variable that equals 1 when the insured is over the age of 60, otherwise it equals 0 city a dummy variable that equals 1 when the owner of the car lives in a city, otherwise it equals 0 north a dummy variable that equals 1 when the car is registered in the north of Taiwan, otherwise it equals 0 south a dummy variable that equals 1 when the car is registered in the south of Taiwan, otherwise it 17

19 equals 0 east a dummy variable that equals 1 when the car is registered in the east of Taiwan, otherwise it equals 0 channel_i a dummy variable that equals 1 when the policy is sold through the channel i, i = D,T,L,F,A, otherwise it equals 0. The definition of channel_d (channel_t, channel_l, channel_f, channel_a) is the insurance policy distribution in which the insurance policies are sold by the dealer-owned agency (telephone service center, auto leasing companies, financial institutions, agency, respectively). Second part: lnprem logarithm of the total premium for the year of each policy. Third part: reminded a dummy variable that equals 1 when the insured s car is maintained according to the recommended time in each time maintain during the year, otherwise it equals 0. 18

20 Table 2 Basic Statistics Year Variable Mean Std. Dev Mean Std. Dev Mean Std. Dev Mean Std. Dev Mean Std. Dev coverage claim pm_mileage pm_item reminded carage carage carage carage carage carage carage carage carage catpcd_ catpcd_ sexf married age age age city north south east channel_d channel_t channel_l channel_f channel_a lnprem observations Note: All the variables in Table 2 are dummy variables except for lnprem. The maximum value of lnprem from year 2002 to year 2006 are: , , , , The minimum value of lnprem 19

21 from year 2002 to year 2006 are: , , , ,

22 Table 3 The pair-wised relationship between risk, coverage and effort from the coefficients in second stage regression of two stage method ---use pm_mileage as the proxy variable of effort Dependent Variable coverage claim pm_mileage Year Independent variable β 1,cov β 1,clm β 1, pm claim (0.0999) pm_mileage (0.0609) coverage claim (0.0376) pm_mileage (0.0785) coverage claim (0.1233) pm_mileage (0.0170) coverage claim (0.0032) pm_mileage (0.0941) coverage claim (0.1272) pm_mileage (0.0787) coverage

23 Note: P-values are in parentheses. 22

24 Table 4 The pair-wised relationship between risk, coverage and effort from the coefficients in second stage regression of two stage method ---use pm_item as the proxy variable of effort Year Independent variable Dependent Variable coverage β 1,cov claim β 1,clm pm_item β 1, pm claim (0.0999) pm_item (0.0954) coverage claim (0.0376) pm_item (0.0515) coverage claim (0.1233) pm_item (0.0992) coverage claim (0.0032) pm_item (0.0285) coverage claim (0.1272) pm_item (0.0126) coverage

25 Note: P-values are in parentheses 24

26 Table 5 The triangular relationship between risk, coverage and effort from the coefficients in second stage regression of two stage method ---use pm_mileage as the proxy variable of effort Year Independent variable Dependent Variable coverage claim pm_mileage claim (0.3420) pm_mileage (0.3094) coverage (0.2112) claim (0.4899) pm_mileage (0.4425) coverage (0.4471) claim (0.7964) pm_mileage (0.2027) coverage (0.6715) claim (0.0240) pm_mileage (0.9651) coverage (0.1352) claim (0.1489) pm_mileage (0.0915) coverage (0.0860) (0.3569) (0.4324) (0.2158) (0.8798) (0.0637) Note: P-values are in parentheses 25

27 Table 6 The triangular relationship between risk, coverage and effort from the coefficients in second stage regression of two stage method ---use pm_item as the proxy variable of effort Year Independent variable Dependent Variable coverage claim pm_item claim (0.3501) (0.8389) pm_item (0.8633) coverage (0.3245) claim (0.4873) (0.3897) pm_item (0.4921) coverage (0.4430) claim (0.7956) (0.6430) pm_item (0.5433) coverage (0.6810) claim (0.0231) (0.5189) pm_item (0.4022) coverage (0.1511) claim (0.1599) (0.0063) pm_item (0.0153) coverage (0.0985) Note: P-values are in parentheses 26