# The Automobile Insurance Pricing Model, Combining Static Premium with Dynamic Premium. Based on the Generalized Linear Models

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1 The Automobile Insurance Pricing Model, Combining Static Premium with Dynamic Premium Based on the Generalized Linear Models Chenghui Han Dan Yao Sujin Zheng School of Insurance, Central University of Finance and Economics Beijing, China Abstract Since 2015, the reformation of automobile insurance has been restarting in China, which makes the rate of the automobile insurance more and more marketable. In this background, with the development of the Telematics technology, the dynamic factors will be added into the automobile insurance pricing model so that the automobile insurance pricing will become more variable. We combine the static premium with the dynamic premium and built a pricing model based on the Generalized Linear Models (GLM), then compare it with other automobile insurance pricing models: the model considering the static factors such as car models, driving zone and so on; the static premium model considering the factor of driving kilometers; and the dynamic premium model. We discuss the difference between these models in the imitative effect, the influence on cash flows of insurance companies, and so on. Finally, we try to give some advice to the future development of automobile insurance pricing in China. Key words: market-based pricing, dynamic premium, GLM

6 the following formula: E(Y i ) = μ i = g 1 (η i ) = g 1 (X i β) In this formula, Y i means the dependent variable, which can be frequency and severity of claims; g( ) is called linked function; X i means the independent variable, which is a vector of risk ranking variables; β is a coefficient vector of the risk ranking vector. The keys in application of GLM are the choices of distribution function of dependent variable Y and the linked function g( ). Based on the previous researches of many scholars, the linked function which is often used to estimate frequency and severity of claims is the logarithmic linked function. So we choose logarithmic function as our linked function directly. Because of the non-negativity, the distribution of frequency and severity both can t be the normal distribution. We plan to compare some common distribution of frequency and severity, showing in the Table 2, and choose the better one to build the GLM. Table 2 Assumption of Distribution and Linked Function to Compare Frequency Severity Distribution Linked Function Distribution Linked Function Poisson Logarithmic Gamma Logarithmic Negative Binomial Logarithmic Inverse Gaussian Logarithmic Model Test To choose a distribution When we want to know the goodness-of-fit of the GLM, we often compare the used model with Saturated Model. The two models have the same assumptions of distribution and linked function, and their only difference is the number of independent variables: the saturated model has the maximum quantity of independent variables. If there is no or little difference between the goodness-of-fit of these two models, it means that the used model can explain the data well. Using mathematical symbol language, we can assume the likelihood function of used model by l(b; y), the likelihood function of saturated model by l(b max ; y). Then, Deviance (D) is defined as following: D = 2 [l(b max ; y) l(b; y)] When the used model fits well, the expectation of deviance should be close to its freedom degree. We build all the models by the R statistical software. The statistical results from R include the deviance of a null model (no independent variables) and a certain model, and we can denoted them by D n and D c. The difference between them is denoted as D = D n D c, which obeys Chi-Square distribution with a freedom degree as (n c) (n and c are the freedom degrees of D n and D c ). So we can choose a better model by comparing the difference between the models and a null model, deviances of models, and the value of AIC. To test a nested model

7 The nested model is a model which has the same distribution but more independent variables than another model. When judging the significance of variables in nested model, we always use the deviances of two models. We will assume that: β 1 H 0 :β = β 0 = ( ) β q β 1 H 1 :β = β 1 = ( ),q < p < n β p where n is the number of sample; q and p is the number of independent variables of primary model and nested model. We can use the difference between their deviances: D = D 0 D 1 to test the 2 hypothesis. If the value of D is bigger than the 95% quantile of the distribution χ p q, then we suggest that the nested model fits better. We use this method to compare the model of static premium with or without the kilometer factor. 3.2 Data Because the current Telematics technology in China is developing, and there is not enough public data, so we choose the foreign data, which give details of third party motor insurance claims in Sweden for the year The data were compiled by a Swedish Committee on the Analysis of Risk Premium in Motor Insurance. This data classified customers based on four factors for pricing, Kilometers (5 ranks), Zone (7 ranks), Bonus (7 ranks), and Type (9 ranks). There were Insured (about 2.38 million), Claim number and Payment statistics in the data, so that we could calculate frequency and severity. This data was the summarized data with 2,182 pieces in total To Add Person s Factor There is no person s factor in this data, such as gender and age. For enriching the factors, we add the gender factor into this data. In the data, the Insured, Claim number and Payment should be separated based on gender of drivers. We assume that the number of men and women is equal. There have been some research results about the frequency and severity of Chinese men and women (Zhang Y, 2007). The information is shown in the Table 3. Table 3 Information of Auto Insurance Claim in China (From Zhang Y, 2007) Frequency Severity Average Loss Male 17.30% 2, Female 19.80% 1, Based on Table 3, we can see that the frequency of female is higher, the severity of male is higher, and in total the average loss of male is higher. This conclusion is the same as common sense that the female drivers may have more small claims but few of large accidents. So we can use the information to calculate the separated Claim number and Payment

8 by the following equations: m + n = c x + y = p m n a b = 17.3% 19.8% x y { a b = where a and b are the numbers of male and female insured; m and n are the claim numbers of men and women; x and y are the payments of men and women; the total claim numbers is c; and the total payment is p. After adding the gender factor into the data, there are 4,364 pieces in total, and next, we will use this adjusted data To Simulate Kilometers per Insured In consideration of the model with dynamic premium, we need the actual kilometers of every risk groups. Because of the lack of actual kilometers data, we will make some assumption by stochastic simulation. Table 4 Information of Kilometers in Data Rank Kilometers Interval < 1,000 1,000~15,000 15,000~20,000 20,000~25,000 > 25,000 Insured 806, , , , ,673 Table 4 shows the classification of Kilometers factor. We can see that the range of kilometers is large and it is difficult to simulate a distribution of kilometers based on the insured. So we assume that there is the uniform distribution in each kilometers interval, and then the simulated actual kilometers can be calculated based on the insured in each interval. 3.3 Descriptive Statistics of Single Factor Except the gender factor which is assumed (G0 Female, G1 Male), we analyze the frequency and severity of all the independent variables one by one by descriptive statistics of single factor. The results are shown in Picture 1-4.

9 9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% Kilometers--Frequency 8.3% 6.5% 7.2% 7.0% 5.6% K1 K2 K3 K4 K5 SEK 5,000 4,500 4,000 3,500 3,000 2,500 2,000 1,500 1, Kilometers--Severity 4,586 4,668 4,263 4,080 3,828 K1 K2 K3 K4 K5 Picture 1 Loss Distribution (Classified by Kilometers) In Picture 1, rank K1 K5 mean fewer than 1000, , , and more than kilometers. Picture 1 shows that the more kilometers there are, the higher frequency is, and there is not a clear trend in the severity. 12.0% 10.0% 8.0% 10.4% Zone--Frequency 7.9% 7.2% SEK Zone--Severity 6,000 5,257 5,000 4,645 4,806 4,779 4,279 4,241 4, % 4.0% 5.8% 6.3% 5.7% 5.0% 3,000 2,000 1, % 1, % Z1 Z2 Z3 Z4 Z5 Z6 Z7 - Z1 Z2 Z3 Z4 Z5 Z6 Z7 Picture 2 Loss Distribution (Classified by Zone) In Picture 2, rank Z1 Z7 mean the 7 zones of Sweden: Z1 Stockholm, Göteborg, Malmöwith surroundings; Z2 Other large cities with surroundings; Z3 Smaller cities with surroundings in southern Sweden; Z4 Rural areas in southern Sweden; Z5 Smaller cities with surroundings in northern Sweden; Z6 Rural areas in northern Sweden and Z7 Gotland. Picture 2 shows that the frequency of cities is higher than the one of rural areas, and frequency in southern Sweden is higher than northern Sweden. There are also differences between the severities, but the trend is not the same as frequency, which may be related to the labor cost and price in different zones.

10 14.0% 12.0% 10.0% 8.0% 6.0% 4.0% 2.0% 0.0% Bonus--Frequency 12.9% 7.9% 6.8% 6.6% 5.5% 5.2% 3.6% B1 B2 B3 B4 B5 B6 B7 SEK 6,000 5,000 4,000 3,000 2,000 1,000 - Bonus--Severity 4,033 4,298 3,888 5,112 4,070 4,094 4,501 B1 B2 B3 B4 B5 B6 B7 Picture 3 Loss Distribution (Classified by Bonus) In Picture 3, rank B1 B7 mean the no claims bonus, equal to the number of years since last claim, and plus 1. Picture 3 shows that the bigger the number of years since last claim are, the higher the frequency is, and the severities of the middle ranks are lower. We suggest the reasons may be that when these insured have an accident, the accident may be a large one, because they may want to hold the smaller claims in order to get a no-claims discount. 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% Type--Frequency 9.2% 7.6% 8.0% 8.4% 7.1% 7.3% 5.8% 5.4% 3.3% T1 T2 T3 T4 T5 T6 T7 T8 T9 SEK 6,000 5,000 4,000 3,000 2,000 1,000 - Type--Severity 4,710 4,376 4,062 3,176 4,512 4,209 3,874 4,820 4,829 T1 T2 T3 T4 T5 T6 T7 T8 T9 Picture 4 Loss Distribution (Classified by Type) In Picture 4, rank T1 T9 mean the types of vehicles, where T1 T8 represent 8 different common vehicle types and T9 represent all other types. Picture 4 shows that there are different frequencies and severities between different vehicle types. It is important to pricing the auto insurance with consideration of vehicle types.

11 3.4 Models and Analysis Based on the adjusted data, we will build the models and analyze the results To Choose Distributions of Frequency and Severity First, we compare the distributions of frequency and severity, and choose the better one. For the frequency, we build GLMs with static premium (including kilometers) based on Poisson and Negative Binomial distribution separately, and the other assumptions of them are completely the same. The results of statistic testing is shown in Table 5. Table 5 Statistic test of Frequency (Different Distributions) Negative Binomial Poisson DF Value DF Value Null Deviance Residual Deviance D AIC In Table 5, Null Deviance represents the deviance of a null model which includes intercept and deviant only; Residual Deviance represents the deviance of the model with static premium (not including kilometers); D represents the difference between the two deviances; DF means the degree of freedom. The values of D of Negative Binomial model and Poisson model are very close, and their degrees of freedom are the same. It is obvious that the value of D, 31623, is bigger than the 95% quantile of χ So both Negative Binomial model and Poisson model are significant. In consideration of the residual deviances of two models, the value of deviance of the Poisson model is more close to its degree of freedom. The value of deviance of the Negative Binomial is so much bigger than its degree of freedom, which is a result from the problems of this model. As a consequence, the Poisson distribution is better for our GLM of frequency. For the severity, it is similar to frequency, but the distributions are Gamma and Inverse Gaussian. The results of statistic testing is shown in Table 6. Table 6 Statistic test of Severity (Different Distributions) Gamma Inverse Gaussian DF Value DF Value Null Deviance Residual Deviance D AIC Similar to Table 5, from Table 6 we can see that the value of D of the Gamma model, , is bigger than the 95% quantile of χ 2 25, but the value of D of the Inverse Gaussian model, , is smaller than the 95% quantile of χ So the Inverse Gaussian model can be regarded as a null model, and the Gamma model is significant. However, the value of residual deviance of the Gamma model is more close to its degree

12 of freedom, and the AIC of Gamma is smaller, so that the Gamma distribution is better for our GLM of severity Comparison of Models with Static Premium (Including Kilometers or Not) Using the Poisson and Gamma as the distribution of frequency and severity, we build the GLMs with static premium (not including kilometers and including kilometers) by R statistical software. The results of model is shown in Table 7 and Table 8. Table 7 Models of Frequency and Severity (Not Including Kilometers) Frequency Severity Value Pr(> z ) Value Pr(> t ) (Intercept) <2e-16*** <2e-16*** G <2e-16*** <2e-16*** Z <2e-16*** Z <2e-16*** *** Z <2e-16*** <2e-16*** Z <2e-16*** * Z <2e-16*** <2e-16*** Z <2e-16*** B <2e-16*** ** B <2e-16*** e-05*** B <2e-16*** *** B <2e-16*** * B <2e-16*** e-08*** B <2e-16*** <2e-16*** T e-08*** T e-14*** * T <2e-16*** e-10*** T e-12*** *** T <2e-16*** * T ** e-05*** T e-07*** T <2e-16*** e-06***

13 Table 8 Models of Frequency and Severity (Including Kilometers) Frequency Severity Value Pr(> z ) Value Pr(> z ) (Intercept) <2e-16*** <2e-16*** G <2e-16*** <2e-16*** K <2e-16*** ** K <2e-16*** * K <2e-16*** ** K <2e-16*** * Z <2e-16*** Z <2e-16*** *** Z <2e-16*** <2e-16*** Z <2e-16*** ** Z <2e-16*** <2e-16*** Z <2e-16*** B <2e-16*** ** B <2e-16*** e-05*** B <2e-16*** ** B <2e-16*** B <2e-16*** e-07*** B <2e-16*** <2e-16*** T *** T <2e-16*** * T <2e-16*** e-09*** T E-14*** *** T <2e-16*** T * e-05*** T e-07*** T E-12*** e-06*** The models including kilometers are the nested models of the ones not including kilometers. So we can use the ANOVA test by R statistical software to compare nested models and original models. The results of test is shown in Table 9 and Table 10. Table 9 ANOVA of Frequency Original Model (1): Frequency ~ Gender + Zone + Bonus + Type Nested Model (2): Frequency ~ Gender + Kilometers + Zone + Bonus + Type DF of Deviance Deviance DF D Pr(>Chi) Significance:0 *** ** 0.01 * < 2.2e-16***

15 Table 11 Static Premium Rate (Including Kilometers) Rate Table (Including Kilometers) Frequency (1) Severity (2) Pure Premium (1) (2) Basic Rate Adjustment Coefficient Gender Gender Kilometres Kilometres Kilometres Kilometres Kilometres Zone Zone Zone Zone Zone Zone Zone Bonus Bonus Bonus Bonus Bonus Bonus Bonus Type Type Type Type Type Type Type Type Type Table 12 Mean Kilometers Kilometers Ranks Kilometers Interval Mean Kilometers K1 < 1, K2 1,000 15,000 8,000 K3 15,000 20,000 17,500 K4 20,000 25,000 22,500 K5 > 25,000 30,000

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