Bayesian Approach for Bonus-Malus Systems with Gamma Distributed Claim Severities in Vehicles Insurance

Abstract

Aims: This paper aims at developing a model to calculate the net premiums under Bonus-Malus System. This model depends on the following components; frequency, severity and policy year(s) of an individual policyholder in vehicle insurance. Study Design: This study is an empirical research and was based on a secondary data regarding the number of claims and severities of vehicle insurance. Place and Duration of Study: The study was conducted within the period of the year 2013 to 2015. During the stated period, a random sample consisting of 3,000 vehicle insurance policies was selected from a Saudi Insurance company. The detailed data collected included; number of claims and their severities. Methodology: This study is based on the use of the Bayesian approach to formulate a compound model, which includes three variables; the number of claims, amount of claims and time. This model is used to calculate the net vehicle insurance premiums for individual policyholders, according to the policyholder’s total number of claims and total amount of claims during a specific period of time. The data were analyzed using IBM SPSS Statistics 22 and MathCad 2001 professional software. Results: This paper presents the design of optimal Bonus-Malus Systems using finite mixture models, probability distributions and considering the number of claims following Poisson distribution compound with time. Severities presented are in accordance with gamma distribution. The proposed model shows a directly proportional relationship between accident occurrence, increase in size of the loss incurred and increase in premium. This also reflects proportionately in situations, where premium decreases in the absence of accidents over a period of time. The proposed model achieves a stability and fairness in the premium of vehicle insurance for all policyholders under different levels of total losses and different levels of total number of claims realized during a given period of time. Currently this fairness is lacking in the Saudi vehicle insurance market.