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

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