Bayesian bonus-malus premium with Poisson-Lindley distributed claim frequency and Lognormal-Gamma distributed claim severity in automobile insurance

Abstract

The traditional automobile insurance bonus-malus system (BMS) merit-rating depends on the number of claims. An insured individual who makes a small severity claim is penalized unfairly compared to an insured person who makes a large severity claim. A model for assigning the bonus-malus premium was proposed. Consideration was based on both the number and size of the claims that were assumed to follow a Poisson-Lindley distribution and a Lognormal-Gamma distribution, respectively. The Bayesian method was applied to compute the bonus-malus premiums, integrated by both frequency and severity components based on the posterior criteria. Practical examples using a real data set are provided. This approach offers a fairer method of penalizing all policyholders in the portfolio.