Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

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212 Actuarial Modelling of Claim Counts and from scale 2 to scale 1 d K d TV d E l 2 l 1 l 1 l 1 0 0 0 0 1 1 1 1 2 1 1 1 3 5 4 4 4 5 4 4 5 5 5 5 6 5 5 5 7 5 5 5 8 5 5 5 We here compare the distance between L 1 = l i = 01794 and L 2 = l j = 01794. We have f L=l= = l f ∫ 0 lf d The transition rules for a good policyholder (specifically, a male driver aged more than 30 years, with upfront premium, having private use of the car and living in a rural environment, whose a priori expected frequency is 0.0928) are given in the next tables: from scale 1 to scale 2 and from scale 2 to scale 1 d K d TV d E l 1 l 2 l 2 l 2 0 0 0 0 1 1 1 1 2 1 1 2 3 2 2 2 4 2 2 2 5 2 2 3 d K d TV d E l 2 l 1 l 1 l 1 0 0 0 0 1 2 2 1 2 2 3 2 3 5 5 5 4 5 5 5 5 5 5 5 6 5 5 5 7 5 5 5 8 5 5 5 We here compare the distance between L 1 = l i = 00928 and L 2 = l j = 00928.
Bonus-Malus Scales 213 We observe that the different metrics we have chosen do not provide very different results. Because the expected posterior random effect has a financial meaning (i.e. it is the multiplier of the average cost to get the a posteriori premium), we may be tempted to choose its corresponding metric to transfer a policyholder from one scale to the other. Note also that the a priori characteristics of the driver influence the way the policy is transferred from one scale to the other. This results in a number of rules according to the risk classification scheme applied by the insurance company. Remark 4.5 Transferring a policyholder from the bonus-malus scale of a given insurer to the bonus-malus scale of another insurer remains a more complicated task. Indeed, the actuary needs to know the a posteriori random effect in both situations. However the a priori random effect may be different due to another type of a priori tariff or due to adverse selection. The distance to minimize then extends to d 1 L 1 = l 1 2 L 2 = l 2 . 4.9 Dependence in Bonus-Malus Scales It is clear that the sequence L 1 L 2 is CIS, but we do not have in general that given L = l increases with l. The reason is as follows: Because of the finite number of levels, it does not automatically follow that a policyholder in a higher level filed more claims in the past. To show this, consider the scale −1/top. A policyholder in level 3 in year 3 may occupy that level having filed two claims in year 1 and no claim after. On the contrary, a policyholder in level 4 could have filed one claim in year 2. From (3.18) we conclude that should be larger (in the ≼ ST -sense) for policyholder 1 than for policyholder 2 despite the fact that policyholder 1 is in an inferior level. As a consequence, we cannot be sure that the Bayesian relativities obtained with a quadratic loss function are increasing with the level occupied in the scale. This is why linear relativities are so useful in practice. Remark 4.6 This counterintuitive fact becomes reasonable if we allow for random effects to vary in time. Provided the autocorrelogram is decreasing, old claims have less predictive power than recent ones. Then, we have to compare two old claims to one recent claim, and it becomes less obvious that policyholder 1 is more dangerous than policyholder 2. 4.10 Further Reading and Bibliographic Notes Chapter 7 in Rolski ET AL. (1999) offers an excellent introduction to Markov chains, with applications to bonus-malus systems. Several parts of this chapter are directly inspired from this source. For the most part, this chapter is based on Pitrebois, Denuit & Walhin (2003b), following on from Taylor (1997) for the extension of Norberg’s (1976) pioneering work on segmented tariffs; on Pitrebois, Denuit & Walhin (2004) for the linear relativities; on Pitrebois, Walhin & Denuit (2006c) for Section 4.8; and on Pitrebois, Denuit & Walhin (2003a) for the Belgian bonus-malus scale and its special bonus rule. Norberg (1976), Borgan, Hoem & Norberg (1981) and Gilde & Sundt (1989) assumed that the bonus-malus system forms a first order Markov chain. Centeno & Andrade e Silva (2002) considered bonus-malus systems that are not first order Markovian
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Contents Foreword Preface Notation
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Contents vii 2.4 Overdispersion 79
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Contents ix 4.2.3 Trajectory 170 4.
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Contents xi 7.4 Numerical Illustrat
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Part I Modelling Claim Counts Actua
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3 Credibility Models for Claim Coun
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Credibility Models for Claim Counts
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8 Transient Maximum Accuracy Criter
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Transient Maximum Accuracy Criterio
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Bibliography Albrecht, P. (1983a).
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Bibliography 347 Daengdej, J., Luko
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Bibliography 349 Greenwood, M., & Y
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Bibliography 351 Norberg, R. (1976)
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Bibliography 353 Walhin, J.-F., & P
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