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

338 Actuarial Modelling of Claim Counts
a 12 = 20 %
a 20 = 10 %
a 25 = 10 %
a 30 = 10 %
a t = 0 for all other t
the minimization of E A gives
= 00694 and = 00132
The influence of the age structure on the optimal CRM coefficients is thus rather moderate.
9.3 Partial Liability
9.3.1 Reduced Penalty and Modelling Claim Frequencies
The French bonus-malus system possesses many particular rules. This section is devoted
to the study of one of them. Specifically, according to the terms of the French law, if the
policyholder is partially liable for the claim then the premium is multiplied by 1.125 instead
of 1.25. To take such a rule into account, we have to model the random couple N 1t N 2t
where N 1t counts the number of full liability claims filed during year t and N 2t counts the
number of partial liability claims filed during the same year. Clearly, N 1t + N 2t is the total
number of claims N t used in the preceding section.
Let q be the probability that the policyholder is only partially liable for the claim he
files. Further, let us assume a Bernoulli scheme for the claim types. This ensures that,
conditionally on , N 1t and N 2t are independent and both conform to the Poisson distribution
(see Property 6.1). Specifically, we have now
−1−q
1 − qk
PrN 1t = k = = e k= 0 1 2
k!
−q
qk
PrN 2t = k = = e k= 0 1 2
k!
The random variables N 1t and N 2t are obviously dependent if the risk proneness is
unknown. The joint probability mass for the random couple N 1t N 2t is given by
PrN 1t = k 1 N 2t = k 2 =
∫ +
This is a mixed bivariate Poisson model.
0
PrN 1t = k 1 = PrN 2t = k 2 = f d
9.3.2 Computations of the CRMs at Time t
Let us consider a policyholder covered for t years. In addition to the parameter t giving the
magnitude of the penalty in case of a full-liability claim, we introduce the new parameter t