Two-element perturbation of Markov chain in analysis of bonus-malus system
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The purpose of the article is to apply the two-element perturbation of a Markov chain to the analysis of a bonus-malus system commonly employed in automobile insurance in order to classify policyholders. In the literature the bonus-malus system is modelled in the framework of the finite irreducible ergodic discrete Markov chain theory, which requires the assumption of a constant transition matrix and thus restricts the analysis of consequences of changes in the system's structure. In the article the application of the perturbed Markov chain is investigated. The two-element perturbation consists in an increase in one element of the transition matrix at the expense of an equal decrease in one other element in the same raw. In spite of its simplicity the perturbation allows for analyzing structure modification in the bonus-malus system due to specific transition matrix of its model. The perturbation proves to be an adequate tool in the study of consequences of changes in the transition rules i.e. rules determining the transfer of a policyholder from one class to another. It enables to examine the influence of their changes on stationary probabilities, mean first passage and return times and consequently on the system evaluation and performance. Hereby, it provides insurance companies with valuable information, indispensable for constructing a new system or modifying the old one.
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