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

294 Actuarial Modelling of Claim Counts
where the transient regime must be considered. This concerns new entrants in a bonusmalus
system (especially the young drivers). Often, age is included in the a priori
ratemaking, raising the premium for young drivers (especially young males). The large
premium surcharges imposed on young drivers pose social problems in many countries.
As shown by Boucher & Denuit (2006), the heterogeneity is huge inside classes of
young drivers. Once individual factors have been accounted for (on the basis of a fixed
effect model for panel data), young drivers even became less risky on average than
mature and old ones in the empirical study conducted by Boucher & Denuit (2006). In
fact, the vast majoriy of claims reported by young drivers is concentrated on just a few
policies.
In addition to the severe explicit penalties contained in the a priori tariff, young drivers
enter the bonus-malus scale far above the average level they should occupy given their
annual expected claim frequency. There is thus an implicit penalty for new drivers (added to
the explicit penalty found in most commercial price lists), since the relativity corresponding
to the access level of all bonus-malus systems is in every case substantially higher than the
average stationary relativity. The implicit surcharge paid by newcomers can be evaluated by
comparing the access level to the stationary level for the sub-population of the policyholders
insured for a period of 20 years, say.
Young inexperienced drivers generally cause many more accidents than the other
categories of the policyholders. At the same time, classes composed of young
drivers are more heterogeneous than the other ones: the numerous claims are filed
by a minority of insured drivers (causing several claims per year). There are
basically two ways to take this phenomenon into account when designing bonus-malus
systems:
• Either the more important residual heterogeneity is recognized (by a larger variance of the
random effect) and particular transition rules (i.e. heavier penalties when a claim is filed)
are imposed on young drivers during the first few years.
• Or young drivers are first placed in a special −1/top scale, and once the bottom level is
attained they are sent to the regular bonus-malus scale (entering the scale at their average
stationary level).
Let us follow the second approach. The bonus-malus scale is as follows: young drivers
are first placed in the highest level of a −1/top scale (with six levels, say). Careful young
drivers then reach level 0 in five years, and enter the regular scale at that time (the regular
bonus-malus scale can be of the −1/+2 type, for instance).
In such a case, the levels of the initial −1/top scale form a transient class in the
Markov chain describing the trajectory of the policyholders accross the bonus-malus
scales. The policyholders will all leave the initial scale sooner or later and never
come back to it, so that the associated stationary probabilities are all equal to 0.
Applying the asymptotic criterion to compute the relativities of such a hybrid bonusmalus
system does not account for the initial scale. The transient distribution will be
influenced by the −1/top scale, and should therefore be used in this case to determine
the relativities associated with the −1/top initial scale and with the regular −1/+2
scale.