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

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Actuarial Modelling of Claim Counts
There are various bonus-malus systems used around the world. A typical form of no-claim
bonus in the United Kingdom is as follows:
one claim-free year 25 % discount
two claim-free years 40 % discount
three claim-free years 50 % discount
four claim-free years 60 % discount.
Drivers earn an extra year of bonus for each year they remain without claims at fault up to a
maximum of four years, but lose two years bonus each time they report a claim at fault. In
such a system, maximum bonus is achieved in only a few years and the majority of mature
drivers have maximum bonus.
Bonus-malus systems used in Continental Europe are often more elaborate. Bonus-malus
scales consist of a finite number of levels, each with its own relativity (or relative premium).
The amount of premium paid by a policyholder is then the product of a base premium with
the relativity corresponding to the level occupied in the scale. New policyholders have access
to a specified level. After each year, the policy moves up or down according to transition
rules of the bonus-malus system. If a bonus-malus system is in force, all policies in the same
tariff class are partitioned according to the level they occupy in the bonus-malus scale. In
this respect, the bonus-malus mechanism can be considered as a refinement of a priori risk
evaluation splitting each risk class into a number of subcategories according to individual
past claims histories.
As explained in Chapter 4, bonus-malus systems can be modelled using (conditional)
Markov chains provided they possess a certain memoryless property that can be summarized
as follows: the knowledge of the present level and of the number of claims of the present
year suffices to determine the level to which the policy is transferred. In other words, the
bonus-malus system satisfies the famous Markov property: the future (the level for year t +1)
depends on the present (the level for year t and the number of accidents reported during year
t) and not on the past (the claim history and the levels occupied during years 1 2t−1).
This allows us to determine the optimal relativities in Chapter 4 using an asymptotic criterion
based on the stationary distribution, and in Chapter 8 using transient distributions. Several
performance measures for bonus-malus systems are reviewed in Chapters 5 and 8.
During the 20th century, most European countries imposed a uniform bonus-malus system
on all the companies operating in their territory. In 1994, the European Union decreed that
all its member countries must drop their mandatory bonus-malus systems, claiming that such
systems reduced competition between insurers and were in contradiction to the total rating
freedom implemented by the Third Directive. Since that date, Belgium, for instance, dropped
its mandatory system, but all companies operating in Belgium still apply the former uniform
system (with minor modifications for the policyholders occupying the lowest levels in the
scale). In other European countries, however, insurers compete on the basis of bonus-malus
systems. This is the case for instance in Spain and Portugal.
However, the mandatory French system is still in force. Quite surprisingly, the European
Court of Justice decided in 2004 that both the French and Grand Duchy of Luxembourg
mandatory bonus-malus systems were not contrary to the rating freedom imposed by the
European legislation. The French law thus still imposes on the insurers operating in France
a unique bonus-malus system. That bonus-malus system is not based on a scale. Instead the