Bonus-malus System Distinguishing Different Types Of Claims Based On Multivariate Distribution

With the rapid development of economy and technology,motor vehicles have become an indispensable part of the modern society,hence the demands for automobile insurance is increasing over time.Nowadays,the automobile insurance has become an important branch of non-life insurance and it even becomes the largest part of total premium income in many countries.In order to satisfy different drivers with different risks,the Bonus-Malus system has been adopted in automobile insurance to match the premium paid by a driver to the driver's^risk.In other words,the currently used premium calculation method largely depends on the number of historical claims made under each policy.In such an experience rating system,bonuses can be earned by not filing claims,which means that a policy holder can pay less premium in the upcoming term,also a malus is incurred when many claims have been filed,which means that the policy holder is going to pay more premium in the upcoming term.However,the majority of optimal Bonus-Malus Systems(BMS)presented in the actuarial literature so far are calculating the premium amount based on the number of claims made by each policyholder.This leads to the inequalities between different policyholders for the reason that a policyholder who has an accident with a small amount of loss will receive same penalty as a policyholder who has an accident that causes a large amount of loss.Therefore,in order to increase the fairness of the BMS,the impact of the claim amount should be taken into account so that when a claim occurs,an accident which causes larger claim amount will be punished more severely.Based on this idea,both the nrumber of claims and the size of loss for each incurred accident are taken into account in the designing process of the optimal BMS in this paper.In this paper,the Bayesian model is used to modify the bonus-malus system of tarification.The model allows us to differentiate claims based on the number of claims and the amount in each claim.To be more specific,in order to improve the fitting accuracy of claim number,this paper uses mixed distribution to fit claim number,which can adapt to dififerent samples and has higher fitting accuracy.Because the right-tail data of claim size in reality is less and easy to have a greater impact on parameters,it is not only difficult but also inefficient to directly fit claim size.Instead,we divide claims into different types according to the size of claims,and fit the number of claims for different types to consider the impact of the size of claims to obtain the posterior premium and the relative premium of BMS.In particular,we use the Dirichlet distribution as the prior distribution to avoid assuming the independence of parameters,in order to make the model more realistic and easier to extend to more multivariate scenarios.At the end,a practical example of the application of this developed model is presented,and the obtained result is compared with the amount derived from the classical model as well as other models.The comparison fully reflects the reasonability and advantages of the newly developed model.