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

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290 Actuarial Modelling of Claim Counts Table 7.7 Results for a deductible per claim varying according to the level occupied in the malus zone, combined with reduced relativities r l , for the scale −1/top. Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or 5 197.3 % 157.8 % 4610 4604 4 170.9 % 136.7 % 4610 4604 3 150.7 % 120.6 % 4610 4604 2 134.8 % 107.8 % 4610 4604 1 122.0 % 97.6 % 4610 4604 0 54.7 % 54.7 % 0 0 Table 7.8 Results for a deductible per claim varying according to the level occupied in the malus zone, combined with reduced relativities r l , for the scale −1/ + 2. Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or 5 309.1 % 247.3 % 4610 4604 4 241.4 % 193.1 % 4610 4604 3 207.7 % 166.2 % 4610 4604 2 142.9 % 114.3 % 4610 4604 1 130.2 % 104.2 % 4610 4604 0 62.4 % 62.4 % 0 0 As already mentioned, since this equation does not depend on the level l, the amount of deductible will be the same for each level of the scale. Tables 7.7 and 7.8 display the numerical results. The third column gathers the relativities: those in the malus zone have been reduced by 20 % compared to column 2. The last two columns display the amounts of deductible. In this case, the amounts of deductible are reasonable and can be implemented in practice (about 150 % of the annual pure premium). Quite surprisingly, the LogNormal distribution now produces smaller deductibles than its Negative Exponential counterpart. 7.5 Further Reading and Bibliographic Notes This chapter is based on Pitrebois, Denuit & Walhin (2005) for the most part. The Panjer family of counting distributions is known as the Katz family in applied probability. This family attracted a lot of attention in the literature, due to the fact that it contains underdispersed (Binomial), equidispersed (Poisson), and overdispersed (Negative Binomial) distributions. As a result of Panjer’s (1981) publication, a lot of other articles have appeared in the actuarial literature covering similar recursion relations. Multivariate versions of the Panjer algorithm will be used in Chapter 9. Claim amounts have also been taken into account by Bonsdorff (2005), who studied bonus-malus systems where the transitions between bonus levels in the entire interval a b
Bonus-Malus Systems with Varying Deductibles 291 are determined by the number of claims of the previous year and the total amount of claims of the previous year. Vandebroek (1993) analysed the efficiency of bonus-malus systems and partial coverages in preventing moral hazard problems, by means of stochastic dynamic programming. See also Holtan (1994) and Lemaire & Zi (1994a) for the trade-off between bonus-malus systems and deductibles.
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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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Part III Advances in Experience Rat
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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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