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

200 Actuarial Modelling of Claim Counts
4.7 Special Bonus Rule
4.7.1 The Former Belgian Compulsory System
In this section we will concentrate on the former compulsory Belgian bonus-malus system,
which all companies operating in Belgium have been obliged to use from 1992 to 2002. The
Belgian system consists of a scale of 23 levels (numbered from 0 to 22). A new driver starts
in class 11 if he uses his vehicle for pleasure and commuting and in class 14 if he uses his
vehicle for business. Each claim-free year is rewarded by a bonus point. The first claim is
penalized by four malus points and the subsequent ones by five malus points each.
According to the special bonus rule, a policyholder with four claim-free years cannot be
in a class above 14. This restriction is a concession to insureds with many claims in a few
years and who suddenly improve; very few policyholders are ever able to take advantage of
this rule.
Actually, the Belgian bonus-malus system is not Markovian due to the special bonus rule
(i.e. due to the fact that policyholders occupying high levels are sent to level 14 after four
claim-free years). Fortunately, it is possible to introduce fictitious classes in order to meet
the memoryless property by splitting the levels 16 to 21 into subclasses, depending on the
number of consecutive years without accident.
4.7.2 Fictitious Levels
Splitting the levels 16 to 21 into sub-levels, depending on the number of consecutive years
without accident, allows us to account for the special bonus rule. Let n j be the number of
sub-levels to be associated with bonus level j. A level ji is to be understood as level j
and i consecutive years without accidents. The transition rules are completely defined in
Table 4.17 and the different values for n j are given in Table 4.18. We take some liberty
with the notation by using the value 0 for the subscript i and by not using a subscript when
n j = 1.
4.7.3 Determination of the Relativities
The relativities r ji are obtained by minimizing the squared difference between the true relative
premium and the relative premium r L applicable to the policyholder when stationary state
has been reached.
The current situation is more complicated because some levels have to be constrained to
have the same relativity. Indeed the artificial levels ji have the property that
r j = r j1 =···=r jnj
j = 0s
We have to minimize E [ − r L 2] under these constraints.
The solution is given by
r j =
∑
k
∑
k
∫ ∑ nj
w k 0
w k
∫
0
i=1 ji k dF
∑ nj
i=1 ji k dF (4.23)