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

Bonus-Malus Scales 205
Table 4.21 Relativities r l = EL = l for the
Belgian bonus-malus system computed on the basis of
Portfolio A.
Level l
Relativities
Without special With special
bonus rule
bonus rule
22 2715 % 2843%
21 2475 % 2589%
20 2291 % 2389%
19 2141 % 2225%
18 2014 % 1994%
17 1903 % 1942%
16 1803 % 1869%
15 1714 % 1791%
14 1631 % 1924%
13 1550 % 1798%
12 1472 % 1684%
11 1402 % 1583%
10 1340 % 1495%
9 1255 % 1386%
8 1173 % 1283%
7 1114 % 1207%
6 1067 % 1147%
5 1028 % 1096%
4 826% 870%
3 802% 841%
2 778% 813%
1 755% 786%
0 451% 462%
explained by the fact that many drivers at level 14 have benefitted from the special bonus
rule. It is clear that such a situation is not acceptable from a commercial point of view. We
may constrain the scale to be linear in the spirit of Gilde & Sundt (1989). However we
propose a local adjustment to the scale in order to keep − r L 2 as small as possible.
Let us constrain the scale to be linear between levels 13 and 16. We are looking for
updated values for r ′ j , j = 1316. They are such that r ′ j = r ′ j−1
+a, j = 14 15 16 where
a = r ′ 16 − r ′ 13/3. We also want to keep the financial equilibrium of the system. Therefore
we constrain a local equilibrium :
∑16
j=13
r j j =
∑16
j=13
r ′ j j
Choosing r ′ 13 = 1798 %, we obtain r ′ 16 = 1937%.