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Markov Set-Chain - Application to Analysis of Bonus-Malus System

100%
Niemiec M.
Przegląd Statystyczny
|
2005
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vol. 52
|
issue 1
87-102
EN
After describing theoretical basis and properties of the Markov set-chains, their application to the analysis of an automobile insurance system is presented. The bonus-malus system is a system of assigning premiums on the basis of the premium paid in the preceding period and the number of claims reported by a policyholder at that time. In the literature this system is commonly modelled with the use of homogeneous Markov chains, which requires often unrealistic assumption of constant transition matrix and consequently unchanged loss number distribution. The basic parameter of the loss number distribution is its mean called an average claim frequency. Its value may fluctuate from time to time due to insurance companies' actions, changes in the behaviour of a policyholder as well as external factors such as weather conditions or state of roads. A model of a bonus-malus system is constructed in the framework of the Markov set-chain theory. It enables to examine consequences of average claim frequency changes. It is shown how the fluctuation of the average claim frequency may influence both a stationary probability that a policyholder belongs to the class of a distinct premium and expected time that is needed by an insured from a particular class to reach another or once again the same class. The results of the study are crucial to insurance companies having interest not only in system evaluation but also in predicting changes in its performance.
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Two-element perturbation of Markov chain in analysis of bonus-malus system

100%
Niemiec M.
Roczniki Kolegium Analiz Ekonomicznych
|
2005
|
vol. 14
59-74
EN
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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