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

80 Actuarial Modelling of Claim Counts
In the class C 1 ∪ C 2 , the expected claim number equals
m = p 1 m 1 + p 2 m 2
where p 1 and p 2 denote the respective weights of C 1 and C 2 (the ratio of the class exposure
to the sum of the exposures of both classes, say). Considering the variance of the number of
claims in C 1 ∪ C 2 , it can be decomposed as the average of the conditional variances 2 1 and
2 2 plus the variance of the conditional means m 1 and m 2 (that is, the weighted sum of their
squared difference with respect to the grand mean m). The variance thus becomes
p 1 2 1 + p 2 2 2
+p
} {{ } 1 m 1 − m 2 +p
} {{ } 2 m 2 − m 2 >m
} {{ }
=m
>0
>0
which exceeds the mean. Hence, omitting relevant ratemaking variables induces
overdispersion.
2.4.3 Consequences of Overdispersion
As mentioned in Section 2.3.14, misspecification of the variance function does not affect the
consistency of ̂, but leads to misspecification of the asymptotic variance-covariance matrix
of ̂. As a result, we have a loss of efficiency.
Overdispersion leads to underestimates of standard errors and overestimates of Chi-square
statistics (as demonstrated in Table 2.3), which in turn may imply artificial statistical
significance for the parameters. Consequently, some explanatory variables may become not
significant after overdispersion has been accounted for. In practice, failing to account for
overdispersion might produce too many risk classes in the portfolio.
2.4.4 Modelling Overdispersion
Many explanatory variables are unknown to the insurance company or cannot be incorporated
in the price list (for legal, moral or economic reasons). There are thus unobservable
characteristics Z i that may influence the number of claims filed by policyholder i as explained
in Section 2.1.2. Of course, some Z ij s may be correlated with the observable characteristics
X i . To remove these correlations, we could think of first regressing the Z i sontheX i s, with
a linear regression model
p∑
Z ij = 0 + k X ik + ij
Then, the score becomes
0 +
p∑
j=1
dimZ i
∑
j X ij + j Z ij = 0 +
j=1
for appropriate ˜ 0 , ˜ j s and ˜ i .
= ˜ 0 +
k=1
p∑
j=1
dimZ i
∑
j X ij + j
( 0 +
p∑
˜j X ij +˜ i
j=1
j=1
p∑ )
k X ik + ij
k=1