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

202 Actuarial Modelling of Claim Counts
Table 4.18 Sub-levels of the
Belgian bonus-malus system.
j
n j
22 1
21 2
20 3
19 4
18 4
17 3
16 2
15 1
14 1
13 1
12 1
11 1
10 1
9 1
8 1
7 1
6 1
5 1
4 1
3 1
2 1
1 1
0 1
We also immediately verify that
s∑
r j PrL = j = 1
j=0
which ensures that the bonus-malus scale is financially balanced at stationary state.
4.7.4 Numerical Illustration
The numerical results for the Belgian bonus-malus system and for Portfolio A are displayed in
Table 4.19. In this table, the special bonus rule has not been taken into account. Specifically,
the values in the third column are computed with the help of (4.14) with â = 0889 and
̂ = 01474. Those values were obtained by fitting a Negative Binomial distribution to the
portfolio’s observed claims frequencies. The fourth column is based on (4.13) with â = 1065
and the ̂ k s obtained from the a priori risk classification described in Table 2.7. The last
column is computed with the help of (4.15).
Once the stationary state has been reached, more or less half of the policies occupy level
0 and enjoy the maximum discount. This is due to the fact that the transition rules of the