Statistical Analysis Of The Steady State And Severity Of The Automobile Insurance Reward And Punishment Syste

The Bonus-Malus System is widely used in automobile insurance industry in various countries and is an important part of automobile insurance.This paper studies BMSs from three different perspectives.In this paper,we first consider the influence of claim amount on BMS and build a BMS based on the number of claims and accumulated claim amount.An empirical study with actual data shows that the hierarchical generalized linear model is better than generalized linear model in calculating the BMS’s coefficients.To illustrate the advantages of a BMS including the number of claims and cumulative claim amount,a steady-state analysis and a severity analysis are conducted.It’s found that a BMS including the number of claims and cumulative claim amount is more severe than a BMS only including number of claims,contains more information about the policyholder and gives new policyholders lower premium concessions.Secondly,this paper combines risk expectation and risk measure index to establish two new premium calculation models: Conditional Tail Expectation premium calculation model and Conditional Value at Risk premium calculation model.The empirical results show that the two premium calculation models established in this paper are meaningful.The BMS built through the Conditional Value at Risk premium calculation model is more severe,its stability of premium is relatively good,and the average premium decreases faster with time before reaching stability.Finally,under the assumption that the number of claims obeys the negative binomial distribution,and the parameters of negative binomial distribution are quantified into triangular fuzzy numbers to construct a fuzzy BMS.The average premium and severity indexs are calculated by the non-fuzzy number corresponding to the probability distribution of fuzzy BMS,which are used to analyze the steady-state and severity of fuzzy BMS.Through empirical analysis,it is found that ordinary BMS is a special case of fuzzy BMS and is not related to the distribution of claim number.The uncertainty is introduced into BMS,which has little influence on the average premium and severity of BMS.