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

204 Actuarial Modelling of Claim Counts
Table 4.20 Distribution of L for the Belgian bonusmalus
system, computed on the basis of Portfolio A.
Level l
PrL = l
Without special
bonus rule
With special
bonus rule
22 54% 43%
21 38% 30%
20 29% 22%
19 23% 17%
18 19% 09%
17 16% 10%
16 15% 10%
15 13% 10%
14 13% 21%
13 12% 19%
12 12% 18%
11 12% 17%
10 12% 16%
9 14% 17%
8 16% 19%
7 17% 20%
6 18% 20%
5 18% 20%
4 43% 45%
3 38% 40%
2 34% 36%
1 31% 32%
0 503% 509%
We observe in Table 4.22 that the average a priori expected claim frequency EL = l
in level l is always higher with the special bonus rule than without that rule. The effect
is more pronounced in the highest levels of the scale and less pronounced in the lowest
levels of the scale. This fact is obvious from the definition of the special bonus rule. The
policyholders attaining the highest classes of the scale benefit from the special bonus rule.
Those staying in these highest classes show therefore a higher expected frequency. Even
below level 14 the effect remains true because the policyholders have benefitted from it
before attaining the lowest levels. Obviously the effect is less and less pronounced at the
bottom of the scale.
Some insurance companies use the bonus-malus scale as an underwriting tool. For instance,
they systematically refuse drivers with a bonus-malus level > 14. Our calculations show that
this is unreasonable because drivers at level 15 are on average less risky than drivers at level
14.
We see from Table 4.19 that without the special bonus rule, the relativities are always
increasing from level 0 to level 22. The same increasing pattern is observed for L = l.
When looking at the results for the bonus-malus system with special bonus rule (Table 4.21),
we observe that the relativities at levels 13–16 are not ordered any more. This can be