Pricing of Premium for Automobile Insurance using Bayesian Method

The aggregate claim model can be used to determine the amount of premium charged to the insured by the insurance company. This model consists of two mutually independent random variables, namely the number of claims that occur per period and the amount of claim for each event. In this study, the number of claims is Poisson distributed, and the amount of claim is distributed by generalized extreme value (GEV). The Bayes method is used to estimate the parameters of each distribution. Parameter estimation results are used to calculate the expectations and variances of the aggregate claim model which are then used to calculate insurance premiums. Based on the estimation results, the amount of premium charged to the insured ranges from IDR 3,831,480 to IDR 6,443,860.

AGGREGATE CLAIM ESTIMATION USING BIVARIATE HIDDEN MARKOV MODEL

In this paper, we propose an approach for modeling claim dependence, with the assumption that the claim numbers and the aggregate claim amounts are mutually and serially dependent through an underlying hidden state and can be characterized by a hidden finite state Markov chain using bivariate Hidden Markov Model (BHMM). We construct three different BHMMs, namely Poisson–Normal HMM, Poisson–Gamma HMM, and Negative Binomial–Gamma HMM, stemming from the most commonly used distributions in insurance studies. Expectation Maximization algorithm is implemented and for the maximization of the state-dependent part of log-likelihood of BHMMs, the estimates are derived analytically. To illustrate the proposed model, motor third-party liability claims in Istanbul, Turkey, are employed in the frame of Poisson–Normal HMM under a different number of states. In addition, we derive the forecast distribution, calculate state predictions, and determine the most likely sequence of states. The results indicate that the dependence under indirect factors can be captured in terms of different states, namely low, medium, and high states.

ON HETEROGENEITY IN THE INDIVIDUAL MODEL WITH BOTH DEPENDENT CLAIM OCCURRENCES AND SEVERITIES

It is a common belief for actuaries that the heterogeneity of claim severities in a given insurance portfolio tends to increase its dangerousness, which results in requiring more capital for covering claims. This paper aims to investigate the effects of orderings and heterogeneity among scale parameters on the aggregate claim amount when both claim occurrence probabilities and claim severities are dependent. Under the assumption that the claim occurrence probabilities are left tail weakly stochastic arrangement increasing, the actuaries' belief is examined from two directions, i.e., claim severities are comonotonic or right tail weakly stochastic arrangement increasing. Numerical examples are provided to validate these theoretical findings. An application in assets allocation is addressed as well.