Bonus-Malus Scales in Segmented Tariffs: Gilde & Sundt’s Work Revisited

In a recent paper (Pitrebois, Denuit & Walhin (2003)), the authors have shown how to compute analytically the Bayesian relativities for a Bonus-Malus scale superimposed on a segmented tariff. However, the percentages associated with the levels of such a scale may exhibit an abrupt rise or drop from one level to another. For commercial reasons, it is sometimes preferable that each level inflicts the same relative penalty. In this paper, the authors will show how to obtain such linear relativities for a given Bonus-Malus System. The model allows for a priori ratemaking. It is applied to a real-life portfolio. Keywords: Bonus-Malus System, Markov Chain, stationary distribution, a priori ratemaking, linear scale