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

Bonus-Malus Scales 215
No explicit expressions are available for the optimal and , which must be determined
numerically.
As pointed out by Subramanian (1998), the market shares of competitors in a given
market can be strongly affected when some insurers adopt an agressive competitive behaviour
by modifying the bonus-malus systems. In a deregulated market, insurers have an incentive
to innovate in their pricing decisions by partitioning their portfolios (a priori ratemaking)
and by designing new bonus-malus systems (a posteriori ratemaking). Viswanathan &
Lemaire (2005) examined the evolution of market shares and claim frequencies in a twocompany
market, when one insurer breaks off the existing stability by introducing a super
discount class in its bonus-malus system.
To end with, let us point out a final remark of primary importance. Merit-rating structures
in automobile insurance require the insured to decide whether to file a claim for an accident
when he is at fault. Since the penalties are independent of the claim amounts, one could
imagine that some policyholders prefer to carry the cost of the accident themselves in order
to avoid the premium increase in the future. Therefore, the data the actuary has at his disposal
are contingent on the actual bonus-malus system and are ‘censored’ in a very complicated
way. Hence, the policyholder might modify their behaviour when a new bonus-malus system
is introduced, resulting in an increasing (or decreasing) number of reported claims. This is
a particularly important side-effect when a bonus-malus system is modified. We will come
back to this issue in Chapter 5.
The numerical results presented in this chapter can be obtained using software such as
SAS R or R. There is also a specific software package, called BM-builder, which works in
the SAS R environment and allows for actuarial computations related to bonus-malus scales.
It has been developed by ReacFin SA, which is a spin-off of the Université Catholique de
Louvain, Louvain-la-Neuve, Belgium, created in January 2004 by the authors. Reacfin’s aim
is to provide actuarial solutions to its clients. Its strong link with the university guarantees
the use of up-to-date techniques. BM-builder aims to compute the relativities attached to
the different levels of a bonus-malus scale, taking into account the actual structure of the
insurance portfolio. In that respect, it properly integrates the interactions between a priori
and a posteriori ratemakings and allows for an efficient ratemaking. For more details, see
http://www.reacfin.com.