Abstract
In a bonus-malus system in car insurance, the bonus class of a customer is updated from one 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.
Highlights
In the classical actuarial model for bonus-malus systems in automobile insurance (Denuit et al [8] or Lemaire [14], for example), there is a finite set of bonus classes= 1, . . . , K
The expected claims in a year of a typical customer is EΛ and his expected premiums are ErL∞ assuming that a typical customers bonus class is distributed as the stationary r.v
The explanation is natural: if the customer has a finite sojourn time, the system will have less time to learn about his risk characteristics in the form of λ than if he had been there for ever, as is the false assumption underlying the stationarity-based calculations
Summary
In the classical actuarial model for bonus-malus systems in automobile insurance (Denuit et al [8] or Lemaire [14], for example), there is a finite set of bonus classes. D log r(λ) d log λ λr ( ) r(λ) at λ (denoted elasticity outside the actuarial sciences); it measures to which extent r(λ) is linear at λ (as should ideally be the case), with e(λ) = 1 expressing ‘local linearity’ at λ Such stationary performance measures are only meaningful if the Markov chain L attains (approximate) stationarity within the typical time a customer spends in the portfolio. The expected claims in a year of a typical customer is EΛ and his expected premiums are ErL∞ assuming that a typical customers bonus class is distributed as the stationary r.v. L∞.
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