Using poisson and exponential mixtures in estimating automobile insurance premiums.

Abstract

The main purpose of this project was to design an optimal bonus malus system that incorporates both the number of claims and the claim size. Majority of insurance companies charge premiums based on the number of accidents. This way a policyholder who had an accident with a small size of loss is penalized in the same way with a policyholder who had an accident with a big size of loss, thus the need to develop a model that incorporates both the frequency and the severity components. The frequency component was modelled using Poisson mixtures where the number of claims is Poisson distributed and the underlying risk for each policyholder or group of policyholders is the mixing distribution. We considered the mixing distribution to be gamma, exponential, Erlang and Lindely distribution. For the severity component we used exponential gamma mixture (Pareto distribution) where the claim amount is exponential distributed and the mean claim amount is inverse Gamma. Using the Bayes theory we obtain the posterior structure function for the frequency and the severity component. The premium was estimated as the mean of the posterior structure function for the frequency component if we compute premiums based on the number of claims only. The premium based on both frequency and severity components was estimated as the product of the mean of the posterior structure function of the frequency component and the mean of the posterior structure function of the severity component. We applied the data presented by Walhin and Paris (2000) with some adjustment of the claim amount data to fit the Pareto distribution. The study established that if we consider only the frequency component, the system was unfair to policyholders with small claim amounts. However optimal BMS based on frequency and severity component was found to be fair to all policyholder since policyholders with large claim amounts were charged higher malus due to the risk they pose to portfolio. Therefore we recommend a system that considers both frequency and severity components. Keywords. BMS, Poisson mixtures, exponential mixtures, frequency component, severity component.