Markov chains and mixed poisson distributions in the no claims discount systems
Most NCD systems are unfair to either or both parties. Most systems are that of the simple random walk model, whereby in case of a claim, the policyholder moves down a discount level and vice versa. Then there are the extreme cases, whereby if a driver makes claim(s), he loses all the discounts accumulated over the years and goes back to the level of full premium payment. The other movements within the NCD systems are those of the in-between cases, these consider the frequency of claims, and can involve moving few steps back the discount level in case of claim(s). A fair NCD system, should take into consideration the frequency of claims and the non-homogeneity factor. In this paper, we have used the Markov chains to explain the movement between levels and used the Mixed Poisson distributions to calculate probabilities, with the mixing distributions being the exponential distribution, the one parameter gamma distribution, the two parameter gamma distribution and the Lindley distribution. The rules applied in different systems have been analysed and combined to take into consideration the claims frequency. The following distributions have been constructed, the Geometric, the Negative Binomial distribution with one and two parameters, and the Poisson Lindley, and their parameters estimated using the method of moments and the maximum likelihood method. The distributions have then been used to fit the claim frequency data and a comparison made with data from a Poisson – Inverse Gaussian distribution.