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

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342 Actuarial Modelling of Claim Counts Table 9.5 Parameters and and financial equilibrium for different values of . = 15 = 20 = 25 Financial equilibrium Financial equilibrium Financial equilibrium 0.0507 0.0133 1.0419 0.0393 0.0133 1.0403 0.0321 0.0133 1.0393 9.4 Further Reading and Bibliographic Notes This chapter is based on Pitrebois, Denuit & Walhin (2006b). Despite its apparent difference with bonus-malus scales, the French bonus-malus system can be treated as a scale with many levels. Kelle (2000) followed this route, and used a Markov chain with 530 states to analyse the French system. The large amount of states needed is due to the malus reduction in the case of claims with shared responsibility, forcing the author to consider the pair (number of claims with whole responsibility, number of claims with partial liability) to make the computation. In this chapter, we did not consider all the characteristics of the bonus-malus system in force in France. We have disregarded the special bonus rule (which suppresses all the penalties after two claim-free years). The French law imposes other specific rules on insurance companies. For instance, the French bonus-malus system is such that drivers never pay more than 350 % of the base premium nor less than 50 % of the base premium. Therefore the minimization process has to be carried with an adapted CRM coefficient of the form r ∗ = max05 min35r Several simplifying assumptions can be considered to ease the numerical computations. For instance, we could work with binary annual claim numbers: either the policyholder does not report any claim or he reports a single claim to the company. Such an assumption replacing N t by minN t 1 leads to smaller discounts and higher penalties, which is a prudent strategy for the insurer. Even if the vast majority of bonus-malus systems appear as scales in which policyholders move according to their claims history, there are some exceptions (such as the system in force in France). We refer the reader to Neuhaus (1988) for another example, where the malus after a claim is expressed by a fixed monetary amount (instead of a relativity). This interesting mechanism restores some fairness in case of differentiated a priori price lists. The multivariate version of Panjer’s recursive formula has been derived by Sundt (1999) and Ambagaspitya (1999). Sundt (1999) provided a proof based on conditional expectations whereas Ambagaspitya (1999) used a proof based on generating functions. Sundt (1999) also showed that the following recursive formula can be used : f S s = 1 1 − af X 0 s∑ x≠0 ( a + b s ) 1 +···+s k f x 1 +···+x X xf S s − x k x > 0
Actuarial Analysis of the French Bonus-Malus System 343 The multivariate version of De Pril’s recursive formula for the convolution of independent and identically distributed random vectors has been derived by Sundt (1999) as a particular case of the multivariate Panjer algorithm, and by Walhin (2001) as a particular case of the multivariate version of Dhaene & Vandebroek’s (1995) recursive formula for the multivariate individual risk model. It can also be deduced from the multivariate extension of De Pril’s methodology as shown in Dickson & Waters (1999).
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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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Page 375 and 376: Bibliography Albrecht, P. (1983a).
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