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Modeling and Performance of Bonus-Malus Systems: Stationarity versus Age-Correction

By Søren Asmussen
Thiele Research Reports
No. 05, October 2013

In a bonus-malus system in car insurance, the bonus class of a customer is updated from a year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes of a customer in consecutive years forms a Markov chain, and most of the literature measures performance of the system in terms of the stationary characteristics of this Markov chain. However, the rate of convergence to stationarity may be slow in comparison to the typical sojourn time of a customer in the portfolio. We suggest an age-correction to the stationary distribution and present an extensive numerical study of its effects. An important feature of the modeling is a Bayesian view, where the Poisson rate according to which claims are generated for a customer is the outcome of a random variable specific to the customer.

Keywords: actuarial mathematics; Bayes premium; equilibrium distribution; experience rating; insurance portfolio; Markov chain; motor insurance; Poisson claims; stationary distribution

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