In this paper, several car insurance claims problems are analyzed and solved via existing statistical models implementation for real-world datasets. The first problem which was studied is the problem of measuring the probability of a claim for a specific policy. This problem is solved by using a set of families of generalized linear models with an additional approach to analyze data by utilizing survival models. The best generalized linear model is then chosen according to statistical criteria. The second problem considers distinct classes of policies. A number of claims and prices are forecasted for the different groups. Same approach as for the first problem, generalized linear models are used and the best model is chosen according to statistical criterion. The third problem is the problem of scorecard generation. A brief interpretation and result of the built scorecard is also provided.
Usually, the insurance activity is aimed to protect the property interests of individuals and legal entities in the event at the expense of monetary funds, which are formed from the insurance premiums paid by policyholders. One of the main conditions for the effective functioning of the insurance market is the reliability of its participants ‒ insurance companies. Supporting the ability of insurance companies operating in the market to fulfill their obligations promptly and as a whole. That is their financial stability which is a special starting point for the actual manifestation and implementation of the insurance function. The current financial state of the insurance companies requires the search for new forms and methods of increasing their competitiveness and financial stability. They need to create special decision-support systems for more effective assessment of the policies, more precise forecasting of the probability of claims, evaluate the possible losses and develop more flexible conditions for insurance policy evaluation.
The variety of risk manifestation forms and the frequency and complexity of the consequences of
their implementation determine the need for an in-depth analysis of possible risks and
economicmathematical justification of the financial policy of insurance companies. For every car insurance
company importance of proper policy selection for a given client cannot be overestimated. The
insurance premium is formed according to the client's expectation to be prone to raise claims and the
size of those claims [
Information for the determination of terms and conditions of policies can be separated into two
parts: data concerning a driver and a car. Age, driving, and length of insurance policy are the values
that define a driver part of the information. However, some aspects like driver's habits are hard to
collect, describe and analyze. On the other hand, information about cars can be specified and collected
concerning some technical criteria [
A completely separate issue is creating models that can be used to predict some aspects of a claim based on selected data. One of the most important tasks is to predict the probability of a claim for a specific case. Insurance firms need to have a proper model for predicting and forecasting claims for different clients. Meanwhile, those methods should be easily interpreted and thus explained to clients or regulators about key factors that affect the terms and conditions of policies.
This work is concentrated on solving the main problems, which appear in the insurance field. The first and foremost task is that the claim expectation should be forecasted for a given client. It could be measured by the claim’s probability. Companies need a way to approximate the chances of claims to properly form policies' terms for a given client. This task requires taking into account the client's data and forming a decision based on it.
The second task is forecasting the number of claims for each group. The importance of this task is quite understandable while it is a part of company policy selection. By grouping clients by aggregating values, groups can be created. For these groups, the number of claims can be estimated and the models for forecasting can be built. The approach can follow two possible scenarios: modeling only the number of claims or total spending on a group.
Third task the model creation, which is usually paired with interpretation. This interpretation can provide valuable insight into what values increase the probability of the claim. This allows us to create scorecards that can be built to provide an easy tool to make decisions directly from data provided by a client. The main objective of this study is to define not only the probability and cost (value) of each claim but also the subset of the most damaged cases.
The appropriate approach usually depends on the task but the most important is that it is determined
by the flow of data extraction and preparation. The same method can be applied to the same data but
different approaches and pre-processing techniques may affect the results. For example, [
Generalized linear models (GLM) were the main tools used during our research. They provided a
unified framework for modeling and forecasting the target variables [
The generalized linear model is an extension of a simple linear regression model. A linear relationship between variables is the simplest case for researching links between factors. However, this is not true for most real-world processes where the relationship is more complicated than linear. In this case linking function is introduced. There are a number of different families that were used in the research.
The general way of writing down the generalized linear model is as follows: = ( ), where X denotes the independent variables and is a parameters vector is a link function to transform the scale of dependent variable to suit a linear relationship.
Generalized models can be used for discrete or continuous variables which provides it with a significant advantage.
Logistic regression (LR) is a statistical method that is used for classification values into different categories. In the scope of the research, the logistic regression was used for modeling claim probability for the one police. .
Normal or Gaussian generalized model uses an identity link function which is the same as simple linear regression.
= .
Poisson regression is a statistical model that is used when the dependent variable is a count of occurrence. Its link function is following:
To predict claims or similar events like death or accidents, survival models can be used. They can
be utilized when the outcome can be traced along some period of time [
The simplest form of survival model is a table with all events noted with timestamp of occurrences. It may give a significant insight into the time periods when most events occur.
Scorecards are special tables constructed in a way to provide scores for every feature, summing up the scores for a record, the total points can be estimated. It is possible to move records to one of the preselected categories by assigning levels to the score.
Scorecards are powerful practical tools that can be used to fast identify policies with high risks
[
3.4. Other methods and models
The prediction of the insurance field is huge and rich with many approaches and methods that are effective for forecasting the probability of claims [5‒9]. Some methods cover not only the same objective as the current study but are also applied to handling more financially oriented data, missing data, and combining results of the several models [5‒9]. Let's make a brief overview of these methods and present results in a general table Table 1. (Abdelhadi et al.
2020) [21]
Classification to predict claims occurrence
A decision tree (DT) and its variation is a family of classification methods that are built on a tree structure for handling the decision-making process based on binary decisions on each step. This allows to apply of the method to data with non-linear relations between features and target variables. There are several extensions of the basic model: random forest, CART models as part of multivariable trees. An example of research is in the work [22].
The random tree is used for classification tasks so a direct comparison of this method with the regression family of methods doesn't seem to be direct. There are tasks like determination of whether the claim will happen at all which can be approached by both methods but with prediction of continuous variable only one method could be used.
The simplest model is straightforward: each node checks features and directs the pipeline to one of two possible branches till the final is reached. However, this model is not suitable for complex data since it tends to overfit and variable selection can be biased.
One of the very popular extensions that was also covered by work [22] is CART or Classification and Regression Trees. It overcomes the limitation of the original model by allowing to model and predict regression variables without restricting original capabilities for categorical methods.
Another method is Random Forest. It combines several decision trees which in turn can be regressive together and via weighting of their output comes up with a single decision. It can be seen as a statistical-machine learning algorithm.
A further development that might not be so widespread in the Insurance topic but noteworthy is multivariable trees which use multivariable values for response variables.
Support Vector Machines (SVM) is a method dedicated to providing solutions for both classification and regression problems. It is a supervised learning algorithm in which the idea is based on a hyperplane. This hyperplane of space of fewer dimensions is target one and is used for decisionmaking and boundary creation for target point separation.
One of the SVM key features is that in cases when it is not possible to find a plane in the current domain it can transfer inputs to higher dimensions in order to find a hyperplane in a new, higher dimension. This allows us to overcome obstacles that the target dimension possesses. SVM can be modified to solve regression tasks in [23].
Necessary data for model creation were obtained from a Car Insurance dataset provided on a Kaggle web site [24]. The mentioned dataset is oriented on technical aspects of the car with most variables featuring physical parts of a machine for which police is formed.
Dataset consists of two parts which were used for training and testing. It contained 58592 and 39063 records for each part respectively. Each record is a unique policy with information about the owner of the policy and the car. The dataset has information about whether there was a claim during the upcoming 6 months for the insurance. This was a target value during the first stage of modeling. Additionally, the dataset had information about a range of different features with the total number of variables being equal to 44.
For further research, the grouping by several variables has been made with the aim of forming groups of special clients and policies for which modeling was made.
It's important to note that the dataset doesn't contain information about financial data. There is no information about the price of cars and insurance premiums for a policy.
Modeling and forecasting have been done using generalized linear models for binomial, gaussian, and Poisson types. Scorecard was also generated to assist in decision-making and interpretation of the results.
Modeling for the probability of claim was done by building two models – binomial and Gaussian. The comparison presented in Table 2 has shown that Gaussian performs significantly better. age_of_car – how old is the car; policy_tenure – the length of the policy up to date; area_cluster – the area where most driving by the policy holder is done; make – the car’s manufacturer; ncap_rating – rating the car’s safety given by the agency.
atr – synthesized variable based on the car's features: extra safety bags, lamps, etc. The target relationship is then represented by the following formula: _ = ( + × × confusion matrix for the normal distribution which is presented in Table 4.
In the next stage the modelling was made based on survival theory. It is possible to construct a survival model where each claim is treated as the death of a member of the population. We will count the length of the policy as a measure of time. Thus, the claims population "survives" during the policy length interval. It was decided that high-quality prognoses cannot be derived from existing data when claims prediction is done in the scope of the simple policy.
Let's build a Cox proportional hazards model:
ℎ( = (_, _) ~ ( + _, = __) ), + where n = 58592, number of events = 3748.
It can be seen however that length of policy indeed has an effect on the claims number amounts but this observation is rather trivial and cannot be used to make a decision since only short-range policies should be preferred (Figure 1). Therefore, the relationship between the length of the policy and the frequency of lawsuits was revealed. At the moment of time 1, 1.6 and 1.7 year duration there is a sharp increase in claims. It is possible to perform separation and in the future to focus on the threshold values found. Also from the survival model is easier to determine the duration of the most risky policies and to define the possible new policies politics.
The calculation of individual cases (a claim for each policy separately) showed the absence of parameters and characteristics that would accurately indicate the onset of a claim. All probabilities for each policy lie between 0.001 and 0.12. In this case, a decision was made to proceed to the consideration of individual segments.
Grouping of data by segment, manufacturer, and machine brand was performed. Thus, we have moved from looking at an individual car to the segment as a whole, where individual characteristics are of little importance.
Two values can be calculated for segments: 1. The number of claims in the segment. 2. Amount of payment by segments.
It was decided to implement a further approach to working with groups. The Poisson generalized linear model was chosen as the model to forecast the number of cases. It showed a high level of accuracy. The equation for modelling relationships was presented in the such way: _ +
As can be seen in Figures 3 and 4 the claims' number prediction across groups has a higher quality degree. This also shows that despite the low ability to predict each unique case, prediction of the group is a much easier task.
Another approach was chosen for dealing with the group. It was about forecasting the price of all cars for which claims were issued. The gaussian model was used as the most appropriate. This also showed significant accuracy (Table 6).
Additionally, in the dataset, the car's price was missing data. For this model, the following approach was used: 1. To find the average price for every class. 2. Adjust it according to the attribute feature. 3. To group price per category to create a new feature – total (price).
The equation for modeling is as follows: _ = ( + + _ _ ).
_ + + _ +
We need to understand which variables and intervals for these variables are the most significant in the aim of our insurance task. Information value (IV) is one of the most useful techniques for selecting important variables in a predictive model. This helps to rank the variables based on their importance. On Figure 5 it is presented how many claims cases were and how they correlated in accordance to different values of the car's age.
Finally, the scorecard was built (Table 7). It provided information about values that are associated with high risk of a claim for this dataset. Non-significant values have been filtered out. The remaining variables describe continuous data – age of policyholder and policy tenure for which binning is made. Categorical variables were also presented in the work – area of clusters which were named in the initial dataset and ranges from C1 to C22 and variables that related to technical aspects: rear mirror availability and functionality, brakes type, and transmission type.
Today car insurance companies require a lot of information to decide policies and conditions [5]. Even though a vast amount of information can be collected it doesn't guarantee the ability to create a model that can predict a claim for a specific policy with a significant level of accuracy due to the randomness the of claim's nature. Some special cases can be chosen, less or more prone to claims cases can be selected but it doesn't allow to make a robust prediction according to the results. From a built model for probability prediction, the gaussian generalized model has been chosen. It shows that claims' nature cannot be determined based on some specific features or its combinations since for same key variables. There are examples of policies with and without claims. Obtained values show a high level of centering which doesn't allow to select intervals for confident claim selection and hence undermines the usefulness of such an approach.
The problem of single-claim prediction is the hardest one. For the claims risk management, we need to forecast the probability of each claim, of each type of claim, and to develop a special scoring card in an understandable and easily interpretable manner with the key features automatically.
More promising are results for a group of claims where policies are selected and combined under the same group with similar features. Such groups have a higher degree of an accuracy and can be modeled and forecasted with respect to number of claims or total cars' price for which claims have been made. Overall, the results show a low ability to predict specific cases but relatively high confidence in forecasting in big groups.
It is worth noting that different methods like Random Forest could perform better with the task of predicting claims per observation which can be examined in consequent researchers.
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