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

Bonus-Malus Scales 203
Table 4.19 Numerical characteristics for the Belgian bonus-malus system without the special bonus
rule, computed on the basis of Portfolio A.
Level l PrL = l r l = EL = l r l = EL = l EL = l
without a priori with a priori
ratemaking
ratemaking
22 54 % 3060 % 2715% 184%
21 38 % 2739 % 2475% 177%
20 29 % 2487 % 2291% 172%
19 23 % 2283 % 2141% 168%
18 19 % 2112 % 2014% 165%
17 16 % 1965 % 1903% 162%
16 15 % 1838 % 1803% 160%
15 13 % 1726 % 1714% 158%
14 13 % 1625 % 1631% 156%
13 12 % 1528 % 1550% 155%
12 12 % 1439 % 1472% 153%
11 12 % 1359 % 1402% 152%
10 12 % 1289 % 1340% 151%
9 14 % 1198 % 1255% 149%
8 16 % 1111 % 1173% 148%
7 17 % 1048 % 1114% 147%
6 18% 998 % 1067% 146%
5 18% 956 % 1028% 146%
4 43% 752% 826% 143%
3 38% 727% 802% 143%
2 34% 703% 778% 142%
1 31% 681% 755% 142%
0 503% 376% 451% 138%
Belgian system are not severe enough in comparison with the average claims frequency. And
the situation is even more serious when looking at the actual figures on the market. Indeed
the market average claims frequency is even smaller than the one of the analysed portfolio.
If we compute the relativities without taking into account the a priori ratemaking system,
the relativities vary between 376 % for level 0 and 3060 % for level 22. On the other hand,
if we adapt the relativities to the a priori risk classification, these relativities vary between
451 % and 2715 %; the severity of the a posteriori corrections is thus weaker in this case.
Tables 4.20, 4.21 and 4.22 display the results for the Belgian bonus-malus system when the
special bonus rule is taken into account. Table 4.20 compares the probability mass functions
of L when the special bonus rule is taken into account, with respect to the case without
special bonus rule. This rule decreases the probabilities associated with upper levels. The
decrease for level 18 is even more apparent, since policyholders in that level benefit from
the special bonus rule.
Table 4.21 gives the relativities computed with and without the special bonus rule. We
see that the relativities are larger with the rule, since it boils down to softening the penalty
in the case where claims are filed to the insurance company.