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

Transient Maximum Accuracy Criterion 315
Table 8.13
Choice of the initial class for the −1/top scale.
Starting level Level of the resulting scale Efficiency ē
5 4 3 2 1 0
5 1812% 111% 946% 815% 709% 632% 11231
4 1812 % 1583 % 1002% 867% 757% 645% 1155
3 1812 % 1583 % 1399% 936% 818% 671% 11671
2 1812 % 1583 % 1399 % 1233% 897% 704% 11692
1 1812 % 1583 % 1399 % 1233 % 1051% 746% 11657
0 1812 % 1583 % 1399 % 1233 % 1051% 746% 11657
The results are given in Table 8.13. We conclude that level l = 2 is the ¯Q-optimal initial
level whereas level 5 is the usual starting level of this scale. The evolution of the expected
financial income I n when starting in level 2 is as follows:
I 0 = 1314%
I 1 = 1282%
I 2 = 877%
I 3 = 934%
I 4 = 972%
I 5 = 100 %
8.5.2 −1/+2 Scale
The transition rules of the −1/ + 2 bonus-malus scale are given in Table 4.2.
Initial Distribution
As for the −1/top scale, we have tested three different initial distributions of the policyholders
in the scale. In the first case (uniform distribution), the policyholders are uniformly spread
in the scale (16.67 % of the policyholders in each of the six levels). In the second case (top
distribution), 95 % of the policyholders are concentrated in the top of the scale (and the
remaining 5 % are spread over levels 0 to 4). In the last case (bottom distribution), 95 % of
the policyholders start from the bottom of the scale.
Convergence of the −1/+2 Scale
Figure 8.3 shows the convergence of the −1/ + 2 scale by plotting the evolution of C n . The
convergence is rather fast during the first five years and then slows down. The level = 005
is reached after about eight years (C 8