Rate Making Discussion and Computer Implementation in Auto Insurance
1Hao Ding
*1, First Author and Corresponding Author
The School of Economics, Xiamen University, Xiamen China
E-mail: [email protected]
Abstract
In the rate making of auto insurance, location factor is one of the most important risk categorical variables. How to bring the location factor into the rate making model of auto insurance and achieve it by computer is the focus point in the non- life contingencies filed. This article systematically introduces four important rate making methods. They are traditional method, GIS method, lattice method, and method of generalized linear model and the support of computer technology. Moreover, this article also does the compare evaluation about the advantages and disadvantages among the four methods. Finally, the article mainly introduces the generalized linear model and its realization in statistical software, namely, MATLAB and SAS.
Keywords:
region rate, grid point method, GIS method, generalized linear model, computer implementation1. Introduction
In the non-life insurance business, there usually classify the insurance that based on the independent risk characteristic. In addition, determine the rate of each risk classification on this basis [1]. In the rate making of auto insurance, location factor is the important risk categorical variable. Effectively consider about the location factor can increase the rationality of auto insurance price, and eliminate the adverse selection and moral risk.
In fact, in the rate making of auto insurance, there has been a long time to take applicant position as the risk categorical variable. In the turn of last century, the America insurance companies discover there have different accident rate in various places and divide the whole country into two ratemaking regions. One is the three big cities of New York, Boston, and Chicago. The other one is the else places. In 1917, the original two places were divided into 11 regions. In 1950, America has 14 ratemaking regions in every state [2]. However, the early region repartition is under the subjectivity judgment of actuary that cannot perfect affect the losing experience with the geographic variation. From the end of last century, with the algorithm development of non-life insurance and computer technology improvement, various correct ratemaking of region rate emerged as the times require. Serhat Guven(2004）first used multivariate analysis evaluation variables within the territory of the space, and put forward the application in different areas with the different measures [3].
America is the auto power country that has been formed one set of complicated and scientific rate calculation method during the years' development. This method represent to the highest level in the national auto insurance market. The auto insurance in America divides into personal vehicle insurance and commercial car insurance. When making the personal vehicle insurance there need to consider about the driver's age, sex, driving experiences, violet record, smoking history, marriage condition, and living position [4]. The unmarried male driver can get the highest rate if he covers the personal vehicle insurance. Because the insurance industry in America believes, the unmarried male driver is lack of responsibility that might cause the accident more easily.
The wide area of China makes the different geographic locations and social economy. Therefore, the auto insurance exist the location differences. However, the present auto insurance has not reflected this kind of difference. Not only in the commercial car insurance that formulated by China Insurance Association, but also in the vehicle compulsory insurance that started in 2006, there has not reflected the variation among different places’ cars [5]. This will lead to the unreasonable ratemaking.
However, in the plenty of existing researches about Chinese car insurance rates, the trendency that use different methods in different areas gradually emerges nowadays. Xiaohui WU, Zheng Zhang, Lei Liu, Lanlan Zhang (2011) analyzed the current of China motor vehicle insurance business development
[6]. In addtion, the information asymmetry also affects China's insurance system. Ying Zhang, Liangzi Zhang, Bingzhi Hu(2011) discussed the asymmetric information on the influence of the medical insurance system in China [7]; LIN Zhengkui, WANG Jian(2012) studied the experience of Chinain the "eleventh five-year plan" period of urban social liability insurance [8].
In a word, the way how to introduce the regional factors into car insurance rates, is significant to improving the accuracy of the determine results. And regional factor is the application of the geographical information system. GIS is a new system combined with the geographical science, computer technology, remote sensing technology and the information science. There are a lot of introductions and applications about GIS at home and abroad. Jinqu Zhang, YunQiang Zhu, Juanle Wang, Jiulin Sun, Yuyue Xu(2011) come up with the application in monitoring and evaluating Chinese regional development based on GIS [9]; Yingxia Wu，Jiejun Huang, Yanbin Yuan, Wei Cui, Yunjun Zhan(2012)simulation of land use/cover change for Wuhan city [10].
Discuss how to bring the region factor into the rate making of auto insurance has significant meaning for increasing the rate correctness. This article will systematically introduces four important rate making methods, compares the advantages and disadvantages based on the various method, and provides the suitable ratemaking method for the auto insurance industry.
2. Traditional method
The traditional ratemaking method is under the location. Each region has the independent ratemaking method. This is the simplest method that used in many insurance companies.
The traditional thinking is classifying the lost data that based on the planned area (general administrative region). One region calculates one rate. If the lost data can satisfy the standards of completely credibility, the rate in this region will be the historical lost rate divided by the total risk. The calculation formula is:
1 j N i i j j
L
R
N
jR
is the rate of region j, the region j hasN
j risk unit,L
istands for the lost of ith risk unit. When the lost data cannot satisfy the standards of completely credibility, we can use the more widely rate to modify the rate in region j. At this time, the rate in region j can be:1
(1
)
1 j N N i k i k j jL
L
R
Z
Z
N
N
Z
is the credibility factor andN
is the risk unit of the country.The advantage of the traditional method is the simple calculation with convenient practical operation that does not need the amount of algorithm. The actuary cost is very low.
3. GIS method
When do the region ratemaking, we usually pay attention to the geographic condition and social economy condition that whether there exist the obvious difference. In the traditional method of region ratemaking, there only use the lost data in the district line without consider about the neighbor lost date outside the district line. This pre-partition method will causes the right lost data cannot count into the rate making and leads the error in the ratemaking.
GIS system is the new type of system that with the development of geographic science, computer technology, remote sensing technology and information science. In the history of computer development, computer assisted drafting (CAD) makes people can use the computer to manage the graphic. One symbol of the graphic data is the clearly location of graphic factor and there exist various topologies among the different graphic. Simply speaking, the topology relation means the space and connection among graphic factors. The simple graphic factor has point, line, and polygon. The point has the coordinate (x, y). The line can be combined with amount of points. The line position can be the
set of coordinate (x1, y1),(x2, y2),……(xn, yn). On the plat, the polygon can be the range that formed by
the closed loop lines. There have various correlations among the graphic factors. For example, one point is on the line or in the polygon, one line can cross one polygon [11-12].
The following examples will introduce the basic thinking of this method. We set the post code area as the basic unit to divide the rate region. Of course, the post code region is only the same classification in the pre-partition and not the final risk region. In our region ratemaking, select the postcode is reasonable. First, the post code standard is the population density and the population density is the important standard for the geographic risk classification. Second, the postcode method is more clearly than the region division that can minimize the risk different in various locations. Third, for most insurance companies in china, the post code of insurant is the ready made data.
If we classify the lost data under the postcode classification and directly calculate the rate level in each postcode region, this has no difference that compared with the traditional method. Here we use other new thinking in GIS method.
For each post code region that named L, the neighbor place has the similar geographic character. Therefore, the losing experience near L and the losing experience of L are similar. During the ratemaking process, we usually have no enough lost data to estimate L rate. Therefore, we can calculate L’s rate by depending on the lost data that near L.
For the calculating convenient, each postcode region commands one center. Ordinary, it is the geographic center or the population calculation center. All the lost data that happened in the postcode region can be seen as happened in the geographic center. For the small range of postcode region, this assumption is reasonable. We have the geographic center that can calculate the distance between each region and do the weighting for the lost data. Weight function is confirmed by the calculator. It can be linear or non-linear. The basic principle is closer to the geographic center will get the bigger data weight. For example, if d is the distance from lost happen place to the postcode region of geographic center, we can determine the weight function of lost data is:
1, 0 5Kilometer (35 ) / 30, 5Kilometer 35Kilometer 0, 35Kilometer d W d d d
After determining the lost data, we can use the premium pure method to calculate the rate of each postcode region. If the weighting lost data cannot reach the standard of completely reliability, we can use the traditional method. We use the more range rate to modify the rate in this region. The formula is follow.
( , )x y local
(1
)
stateR
Z
R
Z
R
GIS method can calculate rate level for each postcode region. If the regions have the same rate, use the same shape to express them on the map. Then we can directly find out the different risk level of geographic areas on the map.
The advantage of GIS is not predefining the region limit. It uses the similar geographic to choose the lost data in the neighbor location. This method avoids the artificial factor influence and makes the rate making more scientific and reasonable.
4. Grid Point method
GIS method is not based on the existing geographic region for the risk level evaluation. However, in the primary period, there still need to use the visible geographic region (such as postcode region) to classify the data. The region center command has randomness. Grid point method does not need to divide the geographic region. Moreover, it can obtain the rate level of each point in finally. There will not exit the rate skip on the region boundary and ensure the rate continuity.
The grid point method needs the longitude and latitude of each point to calculate the rate of every geographic location. Therefore, the complicated data management has high requirement on the computer. However, the present geographic software can satisfy this question. In the practical application, there need to calculate the rate which on the limited and uniform distributed points. Moreover, through the insert method we can get the rate among points to points.
The first step of grid point method is establishing the grid point graphic. The grid point is the rate making point that need to predefine. Establish the grid point needs to consider about the distance from
point to point. This will get the reasonable rate after inserting. Moreover, this is depends on the data transform degree of geographic location. Various locations have the difference.
The second step is determine the used lost data when calculate each grid point rate. This method requires the lost data record has the longitude and latitude of lost happened place. Select the data standard is using the similar geographic principle. When calculate one grid point rate can use this point as the center, and then use the lost data in the region with 10 kilometers radius. Here we can use more complicated standard. We can select different radius for the different grid points. The data set shape cannot use circle. The data set selection can base on other standard such as the similar population density.
In the grid point method, we can use the GIS method to weighting the lost data. The lost data with close distance can obtain the bigger weight, and the far distance will obtain the smaller weight. We can use population to do the data weighting. The method to ensure the weight is following.
1 1 P W D
D
is the distance between lost happened place and grid point.P
is the distance sensitive for the weight. The laggerP
value will leads the smaller weight. In the last, standardW
as well as eachW
divide by the total
W
, we can get the weight of each lost data.In order to establish the rate model of each grid point, we use
( , )
x y
to be the longitude and latitude. Suppose there have N lost data can calculate the rate of this grid point,L
iis the ith warranty lose.( , )
x y
i is standard for the coordinate. Each warranty has one risk unit. The rate model of grid point( , )
x y
is in the following.( , ) ( , ) 1 i N x y x y i i
R
W
L
( , )x yiW
is the lost data weighting of warrantyi
, ( , )1
1
i N x y iW
Obtain each grid point rate and through the reasonable insert method, we can calculate the rate of random point. For the perceptual intuition, we can express the algorithm on the map and use the height of arbitrary point as the rate level.
Compare with the traditional method and GIS method, the grid point does not do the pre-grouping for the data. This method can correct each location and more scientific on the theory. However, the application needs the strong computer algorithm and geographic software support. This is difficult for the present insurance company.
5. Generalized linear model
5.1. Generalized linear model
In the region ratemaking, the region factor, driver sex, age, and other non-region factor has the inner relation. Therefore, before the ratemaking calculation, there need to do the regulation and eliminate other factors’ influence. However, during the practical operation, the complicated ratemaking needs to adapt some simplify method to adjust the data. For example, we can multiply by one adjustment factor. Under this method, the inner data will be influenced by the non-region factor and hardly deleted. From this part, the traditional method, GIS method and grid point has the disadvantage. In one model, we consider all the risks and the generalized linear model can solve this question.
In some countries there have been formed the industry standard of ratemaking classification. Especially in the personal insurance business, the generalized linear model obtains the widely acceptance by actuary. Compare with the traditional method, the advantage of the generalized linear model is not only finding the new rate factor and relative rate by ratemaking, but also examining the rate factor and relative rate.
The generalized linear model suppose the variable comes from exponential distribution, the variance is changing with the average change. In the calculation of non-life insurance, it is usually use to fit the
Poisson distribution of claim indemnity and fit the Gamma distribution of claim indemnity. They are all belonging to the exponential distribution. The explanatory variable through the linear adding can influence the transformation of dependent variable expectation.
The generalized linear model in combined with three parts:
(1) Random element means the probability distribution of dependent variable
Y
or error. Each observed value of dependentY
is independent and follow the distribution of exponential distribution.(2) System component means the linear combination of explanatory variable. It files as
1 1x
...
px
p.(3) Bind function means the function g is simple and derivable. It established the relation between random element and system component. It means
g
( )
or
1( )
E Y
g
. In the calculation of non-life insurance, the widest used bind function is the logarithm bind function. This will obtain one multiplication model. The dependent variable forecast value can express as the multiplication relation of explanatory variable.In the generalized linear model, each explanatory variable has the different value level. The many levels of region variable will lead the hard parameter estimate while establishing the generalized linear model. For example, if we use the postcode region as the basic unit, the entire postcode in China include too much parameters will lead the model validity questions.
The simplest way to solve the above questions is increasing the geographic range and reducing the region number. The question of this method is the geographic region division is influenced by the personal subjective judgment. The personal risk in the same region exist huge difference and will lead the error of ratemaking.
The other method to bring the region factor is regarding the region factor as the continuous variable. In fact, the close position has similar geographic condition and social economy condition. Therefore, from one place to the other place, the risk level is changing. Form the math view, this allows us to trade the region factor as the continuous variable. Bring the continuous variable into the generalized linear model do not need to bring many location parameters as the scatter parameter.
In order to bring the region continuous variable into the generalized linear model, first we should identify the geographic unit for the evaluation and give each unit a geographic coordinate (two-dimensional number). Use two-(two-dimensional number to express the region variable can make it as the continuous variable and bring the multinomial function into the generalized linear model.
The most insurance companies grasp the postcode information of assured, to use postcode region is feasible. However, there exist two questions. First, the postcode is established for the postal service system’s location and deliver but not for the division of assured risk level. Second, the postcode is under the variation with the postal service requirement. This will damage the serviceability of historical data. Third, the coordinate appoint has the subjectivity. It can select the geographic center as the region coordinate and the population-weighting center can be the region coordinate too.
In the grid point, the region is divided into the geographic unit by grid point. It avoids all the questions by selecting the postcode region. However, how to establish the grid point and put the lost data “into” the grid point is depends on the personal subjective judgment. The best method to select the geographic unit is determining the two-dimension coordinate for each lost data. However, this cost is very high and hard to apply. Consider about the development level of auto insurance, the present way to use postcode region is feasible.
For the convenient discussion, here we use one simple sample to introduce the basic process of generalized linear model in ratemaking. Suppose we have the pricing formula:
Insurance premium Basic rate Region factor Auto model factor Age factor
The age variable has two levels (young and old) and “old” is the basic level. Auto model variable has three levels (large, middle, small) and “large” is the basic level. Region variable has 15 levels (1-15) and each level stands for one geographic unit, we select “4” as the basic level.
The next we will separately trade the region variable as the discrete variable and continuous variable and bring them into the generalized linear model.
The generalized linear model can be expressed as the following matrix form:
1
( )
g
,
XB
The generalized linear model can be expressed in the following formula:
0 1 1 1 2 2 3 1 4 2 5 3 6 5 7 15 17 ,
i A i S i S i T i T i T i T i T i i
y
x
x
x
x
x
x
x
x
i1, 2,, 900
is the constant,
A1 is the parameter of age variable,
S1,
S2 are the parameter of auto model variable.
T1,
T2,
T3,
T5,
T15 are the parameters of region variables.This example will increase 17 (1+2+14) explanatory variables. Use
x
1 to explain the age, use2
x
andx
3 to explain the auto model, and usex x
4, , ,
5
x
17 to explain the explanatory variables. The value of these 17 explanatory variables is 0 or 1. Each variable value is in table 1-3. This will cause 90(2 3 15)
equations to represent the various combinations among age, auto model and region.Table 1. Age value
Age x1
Yong 1 Old 0
Table 2. Auto model value
Auto model x2 x3
Small 1 0
Middle 0 1
Large 0 0
Table 3. Region value
Region x4 x5 x6 x7 …… x16 x17 1 1 0 0 0 0 0 2 0 1 0 0 0 0 3 0 0 1 0 0 0 …… 14 0 0 0 0 1 0 15 0 0 0 0 0 1
In the following we will bring the continuous variable as the region variable into the generalized linear model. For this, each geographic unit should set one coordinate at first. Suppose there have two variable, the value of x is 1,2,3,4,5, and the value of y is 1,2,3. Therefore, each region coordinate is:
Region 1: (1, 1) Region 2: (1, 2) Region 3: (1, 3)
Region 4: (2, 1) basic fundamental ……
Region 15: (5, 3)
In this example, we use the simplest function of simple polynomial to bring the region into the generalized linear model. The matrix expression is in the following.
0 1 1 2 1 1 1 0 1 1 1 1 1 0 1 2 1 1 1 0 1 3 1 1 1 0 2 1 . . . . 1 0 0 0 5 1 1 0 0 0 5 2 1 0 0 0 5 3 A S S X Y XB
The above generalized linear model can be expressed by:
0 1 1 1 2 2 3 4 5
,
i A i S i S i X i Y i i
y
x
x
x
x
x
i1, 2, ,90The meaning of
0, A1, S1,
S2 is similar with the variable model in the discrete region.
X, Y are the parameters of region coordinate variables.From the above we can find out, if we regard region variable as the discrete variable, the generalized linear model will create 18(1+1+2+14)parameters. If we regard it as the continuous variable, there are only 6(1+1+2+2) parameters. When the region variable level is much more, the latter will obviously reduce the parameter number. In fact, in the ratemaking of auto insurance, there exist thousands of region units. If we regard the region variable as the continuous variable, we can greatly reduce the model parameter number, simplify the model form and increase the goodness of fittest and predictive ability. Certainly, during the process of bringing region factor into the generalized linear model, we can select second-order polynomials or other polynomial functions that based on the goodness of fittest.
5.2. The computer realization of the generalized linear model
Generalized linear model based on Matlab: from the late 20th century on, the application of generalized linear model to classification ratemaking has developed rapidly, such as medical statistics, biostatistics, social statistics, etc. The application of generalized linear model has complicated calculation, which depends on computer and mathematical software. Matlab is one of the greatest mathematical softwares in the world, and generalized linear model is the application of the GLM of Matlab statistics toolbox，mainly glmfit function and glmfval function [13-14]. Grammar is as follows：
b=glmfit(X, Y, ′distr′)
b=glmfit(X, Y, ′distr′, ′link′, ′estdisp′, offset, pwts, ′const′) [b, dev, stats]=glmfit (…)
The function sees [3]. Parameters ′offset′ and ′pwts′ have been further explained：vector ′offset′ and ′pwts′ are vectors that have the same length as Y，and they can omit or be endowed with empty vector [ ] or ′′.
Vector offset is a special variable，and its is coefficient 1.0. To build the model of the ratio of a certain subclass possessing total amount N，parameter offset should be used. For instance，to build the model of various curved surfaces’ defects，to build a model of the ratio of the expected value of defect number on the curved surface and the curved surface area, the number of the defects can be used as response variable，which follows Poisson distribution and has logarithmic link function and regards the logarithm of the curved surface area as offset.
Parameter pwts is weight vector. For example, if the weight of Y(i) is f(i)，f(i) can be used as weight pwts.
Glmva()is a function that is relative to glmft() Matlab function，which calculates the predictive values of the generalized linear model.
Grammar is as follows： yfit = glmval (b, X, ′iink′)
[yfit, dlow, dhi]=glmval (b, X, ′link′, stats, clev)
[yfit, dlow, dhi]=glmval (b, X, ′link′, stats, clev, N, offset, ′const′) Function is as follows：
yfit=glmval(b, X, ′link′）is to calculate the corresponding predictive variable X, coefficient b and the predictive value of Y in link function ′link′ ; b is the coefficient vector estimated by glmfi （）; ′link′ must be the same as that used in glmfi(); yfit is the predictive value.
[yfit, dlow, dhi]=glmval(b, X, ′link′, stats, clev) is the confidence limits that return to the predictive value; dlow is the lower bound; dhi is the upper bound; stats is the statistical structure returned by glmfit; clev is confidence levels (default value is 0.95). Interval [yfitdlow, yfit+dhi] is the confidence interval of yfit.
Apart from MATLAB, the basic statistical software SAS can also slove thus problems. GENMOD process is to make a maximum likelihood estimation of parameter vector to fit generalized linear model
[15]. Generally speaking, and the parameter’s maximum likelihood estimation has no closed solution, GENMOD process is to estimate the value of model parameters by an iterative fitting process. Discret parameter
also can be estimated by maximum likelihood estimation, residual migration and Pearson
2 statistics resolved by freedom. Parameter estimation is based on the basis of maximum likelihood estimation’s asymptotic normality, calculating the estimated parameters’ covariance, standard deviation and relevant P value at the same time.GENMOD process can produce Type1, that is to say, in its MODEL sentences, the item with more regulations can fit a series models; moreover, one table summarizes double likelihood difference of the logarithm between each pair models to test the model’s statistical significance. Type1 analysis is similar to type (ordinal) square in GLM process, as a result, the result of the process depends on model item’s fitting order.
6. Summary
The traditional method, the GIS method, the grid point method and the generalized linear method show the gradual perfection process of ratemaking in automobile insurance. Relatively speaking, the generalized linear model is the perfect method theoretically, which solves all the common questions among the three preceding methods: under the condition of multiplied by adjusting factor, the influence caused by non-regional factors in data can not be eliminated completely [16-17]. Moreover, the constant upgrade of MATLAB makes the generalized linear method that is complicated develop rapidly in the field of classification ratemaking on non-life insurance with the help of computer and software.
In the auto insurance, we consider the postcode is easily obtained and its division is very meticulousness. The geographic condition and social economy condition has not much differences in the same postcode region. Therefore, we can select the postcode region as the basic geographic unit and classify the relative lost data to establish the generalized linear model. Certainly, the generalized linear model has the limitation. It needs the ceaseless improvement and perfection during the application.
Otherwise, the various region models have important inner relation. Therefore, use different model to do the calculation for the same practical question, then to select the most practical result.
7. References
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