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

Transient Maximum Accuracy Criterion 319
8.6 Super Bonus Level
8.6.1 Mechanism
As explained above, the companies operating in Belgium continue to use the former
compulsory bonus-malus scale, despite the deregulation of the a posteriori corrections. Slight
modifications have nevertheless been brought to the system. For marketing purposes, and
to keep the best drivers in the portfolio, many insurance companies operating in Belgium
have allowed the insured drivers reaching level 0 to ‘claim for free’: whatever the number of
accidents reported to the company, they stay in level 0. This makes the stationary distribution
degenerate: all its probability mass is concentrated at 0. Therefore, computations must be
based on the transient distribution.
The transition probabilities for the Belgian bonus-malus scale with a super bonus level 0
are the same as before, except for the line corresponding to level 0, which is replaced with
a line of 0s and a single 1 on the diagonal.
8.6.2 Initial Distributions
We use the following initial distributions: the uniform distribution (1/23 of the portfolio in
each level), the top distribution (78 % of the policyholders are concentrated in level 22 and
the remaining 22 % are spread over levels 0 to 21) and the steady-state distribution displayed
in Table 8.1. Starting with the stationary distribution gives an idea of the influence of the
introduction of a super bonus level in an existing bonus-malus scale.
8.6.3 Transient Relativities
The results for the uniform initial distribution are given in Tables 8.18 (transient distributions)
and 8.19 (corresponding relativities). We note that after 50 years, the majority of the
policyholders (76.0 %) are in the super bonus level. Most of the other policyholders are in
the upper levels. They correspond to the very bad drivers.
Concerning the transient relativities, we see that the super bonus level has a greater
relativity than level 1 during the first 20 years. This side-effect of the introduction of a super
bonus level is undesirable.
Table 8.20 presents the transient relativities for the top initial distribution. The relativities
also show particular patterns as policyholders principally move by clusters in the scale. For
example, the relativities of levels 0 to 2 are not ordered.
The transient relativities for the steady-state initial distribution are given in Table 8.21.
Once again, the transient relativities are not always in ascending order.
Table 8.22 shows the ¯r l s for the three initial distributions. The results for the steady-state
initial distribution are not in ascending order. Indeed, the relativity associated with the super
bonus level is larger than the relativities associated with levels 1 to 5.
Therefore, we see that the introduction of a super bonus level in an actual scale leads to
some undesirable side-effects: concentration of the majority of the policyholders in the super
bonus level after a few years and optimal relativities no longer ordered for the lowest levels
(these side-effects could be overcome with the introduction of a linear scale).