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

168 Actuarial Modelling of Claim Counts
insurer, he must first obtain a certificate from the former insurer stating his attained bonusmalus
level and whether pending claims could affect this level. The new insurer must then
award the same discount or apply the same surcharges. The competition between insurance
companies is limited to the services offered and the a priori premiums.
Things become more complicated in a deregulated market, where each insurer is free to
design its own bonus-malus system. Then, insurers compete also on the basis of a posteriori
corrections. It rapidly becomes extremely difficult for policyholders to determine the optimal
insurance provider, since companies apply different penalties when claims are reported. In
Section 4.8, we consider a policyholder switching from insurer A to insurer B. He occupies
the level l 1 in the bonus-malus scale used by company A, and the question is where to place
him in the bonus-malus scale used by company B.
Section 4.9 examines the dependence properties existing between the successive levels
occupied by the policyholders and the random risk parameter. It is argued that contrarily to
the results obtained with credibility models, the risk parameters do not necessarily increase
with the level occupied in the scale.
The final Section 4.10 gives references and addresses further issues.
4.2 Modelling Bonus-Malus Systems
4.2.1 Typical Bonus-Malus Scales
Before embarking on an abstract definition of bonus-malus systems, let us discuss a couple
of examples that will be used throughout this chapter.
Example 4.1 (−1/Top Scale) This bonus-malus scale has 6 levels (numbered 0 to 5).
Policyholders are classified according to the number of claim-free years since their last claim
(0, 1, 2, 3, 4 or at least 5). After a claim all premium reductions are lost. The transition rules
are described in Table 4.1. Specifically, the starting class is the highest level 5. Each claimfree
year is rewarded by one bonus class. When an accident is reported, all the discounts are
lost and the policyholder is transferred to level 5.
Note that the philosophy behind such a bonus-malus system is different from credibility
theory. Indeed, this bonus-malus scale only aims to counteract moral hazard: it is in fact
more or less equivalent to a deductible which is not paid at once but smoothed over the time
Table 4.1 Transition rules for
the scale −1/top.
Starting Level occupied if
level 0 ≥ 1
claim is reported
0 0 5
1 0 5
2 1 5
3 2 5
4 3 5
5 4 5