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

206 Actuarial Modelling of Claim Counts
Table 4.22 Average a priori claim frequency
EL = l in level l for the Belgian bonus-malus
system, computed on the basis of Portfolio A.
Level l
EL = l
Without special With special
bonus rule
bonus rule
22 184% 187%
21 177% 180%
20 172% 174%
19 168% 170%
18 165% 164%
17 162% 163%
16 160% 161%
15 158% 160%
14 156% 163%
13 155% 160%
12 153% 158%
11 152% 156%
10 151% 154%
9 149% 152%
8 148% 150%
7 147% 149%
6 146% 148%
5 146% 147%
4 143% 144%
3 143% 143%
2 142% 143%
1 142% 142%
0 138% 138%
Now let us compare the value of the expected error Q = [ − r L 2] with the original
model, Q 1 , and with the constrained model, Q 2 :weget
Q 1 = 041121 and Q 2 = 041150
This shows that the error induced by the commercial constraint is really small. So we may
adapt the scale locally without resorting to a full linear scale constraint.
We can perform the local minimization numerically without imposing a linear scale
between levels 13 and 16. We use the following constraints :
r ′ 13 ≤ r ′ 14
r ′ 14 ≤ r ′ 15
r ′ 15 ≤ r ′ 16