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

238 Actuarial Modelling of Claim Counts
As explained in Chapter 2, overdispersion can be detected by using the statistics T 1 ,
T 2 or T 3 presented in Section 2.4.6. The values obtained with Portfolio C are T 1 = 2004,
T 2 = 1456 and T 3 = 1289. All the associated p-values are less than 10 −4 leading to the
rejection of the null hypothesis in favour of a mixed Poisson model.
In order to take the residual heterogeneity into account, we fitted a Negative Binomial
regression model to the Portfolio C. The results are given in Table 5.7. All the variables and
levels are still relevant. The log-likelihood is now equal to −61 393.3 and is better than with
Poisson regression.
Resulting Price List
The resulting price list is obtained thanks to the Negative Binomial model for the frequency
(presented in Table 5.7) and to the LogNormal model for the average cost of the standard
claims (presented in Table 5.5).
Neglecting the large claims, the pure premium of the reference class is obtained by
(
)
exp freq
0 + cost
0
+ 2
= E31676 (5.6)
2
This amount corresponds to the pure premium of a male policyholder living in an urban
area, aged between 31 and 60, driving a car older than 10 years, using gasoil and with a
power greater than 110 kW, paying his premium in several installments and having opted for
a more extensive coverage than only the compulsory motor third party liability insurance.
To obtain the pure premium (neglecting the large claims) for a policyholder belonging to
another risk class, the percentages of Table 5.8 must be applied to the base pure premium
Table 5.7 Results of the Negative Binomial regression of the claim counts recorded in Portfolio C.
Variable Level Coeff Std error Wald 95 % conf limit Chi-sq Pr>Chi-sq
Intercept −14626 00720 −16037 −13216 41306