On the spectral properties and convergence of the bonus-malus Markov chain model

In this paper, we study the bonus-malus model denoted by BMk(n). It is an irreducible and aperiodic finite Markov chain but it is not reversible in general. We show that if an irreducible, aperiodic finite Markov chain has a transition matrix whose secondary part is represented by a nonnegative, irreducible and periodic matrix, then we can estimate an explicit upper bound of the coefficient of the leading-order term of the convergence bound. We then show that the BMk(n) model has the above-mentioned periodicity property. We also determine the characteristic polynomial of its transition matrix. By combining these results with a previously studied one, we obtain essentially complete knowledge on the convergence of the BMk(n) model in terms of its stationary distribution, the order of convergence, and an upper bound of the coefficient of the convergence bound.