On asymptotic properties of Bonus–Malus systems based on the number and on the size of the claims

Abstract

A general framework for a Bonus–Malus system (BMS) based on the number and the size of the claims is presented, the set of the bonus classes being an interval [a, b], say 0<a<1<b. The BMS is interpreted as a general Markov chain with state space [a, b]. It turns out that, under certain assumptions, the Markov chain possesses an invariant limit distribution to which it converges with a geometric rate. We show how the invariant distribution can be evaluated by means of simulation. We also deal with the best possible convergence rate and show how it can be presented by means of the spectral theory of Banach spaces.