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