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

248 Actuarial Modelling of Claim Counts
transferred after the first claim has been reported. Here, we only use the observations related
to the policyholders having filed a single standard claim during the observation period (so
that we have the exact cost of this claim at our disposal). Also, the estimation of f could
be performed in a nonparametric way, allowing for an effect fl i of being in level l i (and
imposing some smoothness in the fls if needed) before selecting an appropriate parametric
specification.
Let us now apply this methodology to Portfolio C. The policyholders have been subject
to the 23-level former compulsory Belgian bonus-malus scale. The estimation of the
regression parameters in the model containing all the explanatory variables are displayed in
Table 5.12. Here, we take fl = l − 13 2 . The log-likelihood is −130 1542565. We see
from this table that several covariates are not significant. Therefore, we adopt a backward
selection procedure, and exclude the irrelevant covariates. This yields the results displayed in
Table 5.13. The log-likelihood is now −130 1554197. The parameters and are estimated
at ̂ = 16821 and ̂ = 10286.
Compared to the LogNormal fit to the claim costs displayed in Table 5.5, we see that
the intercept is now smaller, as expected. The age classes have been modified, and the
young drivers seem to cause more expensive accidents. The effect of the covariate City
remains approximately the same. The categories for the age of the vehicle have also been
modified.
Table 5.12 Fit of the model for the accident costs subject to bonus hunger in Portfolio C, containing
all the explanatory variables.
Variable Level Coeff Std error Wald 95 % conf limits Chi-sq Pr>Chi-sq
Intercept 58028 00453 57122 58934 1640911 < 00001
Ageph 18−24 02624 00640 01344 03903 1682 < 00001
Ageph > 60 00717 00500 −00282 01716 206 01512
Ageph 25−60 0 0 0 0 . .
City Rural 00459 00275 −00091 01009 279 00948
City Urban 0 0 0 0 . .
Agev 0−2 00327 00652 −00977 01631 025 06158
Agev 3−5 −01721 00465 −02652 −00791 1369 00002
Agev 6−10 −01445 00357 −02159 −00732 1640 00001
Agev > 10 0 0 0 0 . .
Variable Level Coeff Std error Wald 95 % conf limits Chi-sq Pr>Chi-sq
Intercept 35045 00904 33238 36852 150445 < 00001
Ageph 18−24 −03244 01637 −06519 00031 393 00476
Ageph > 60 05419 01105 03209 07630 2405 < 00001
Ageph 25−60 0 0 0 0 . .
City Rural 00796 00416 −00036 01628 366 00558
City Urban 0 0 0 0 . .
Agev 0−2 −07825 02621 −13067 −02582 891 00028
Agev 3−5 −03406 01107 −05620 −01193 947 00021
Agev 6−10 00190 00449 −00708 01087 018 06725
Agev > 10 0 0 0 0 . .
BM level −00011 00006 −00022 00000 367 00555