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

192 Actuarial Modelling of Claim Counts
PrL = l = ∑ ∫ +
w k l k dF
k
0
s∑
EL = l PrL = l
l=0
s∑
VL = l − EL 2 PrL = l
l=0
Remark 4.3
It is interesting to note that the optimal and solving (4.16) also minimize
[ (EL ) ] [
E − r
lin 2 (EL ) ] 2
= E − − L
L
so that the linear relativities provide the best linear fit to the Bayesian ones. To check this
assertion, note that
[ ( ) ] 2
E − − L
[( ( ) ( ) ) 2]
= E − EL + EL − − L
[ ( ) ] 2
= E − EL
[ ( )( ) ]
+ 2E − EL EL − − L
[ (EL ) ] 2
+ E − − L
and the second term vanishes by definition of the conditional expectation EL, since
− EL is orthogonal to any function of L.
Example 4.18 (−1/+2 Scale, Portfolio A) Table 4.10 allows comparison of the relativities
of the two scales in the segmented case. As we can see, the values are close to each other.
The constant step between two levels in the linear scale is equal to 393%.
The values of the expected errors Q 1 = [ − r L 2] and Q 2 = [ − rL lin2]
are
respectively given by
Q 1 = 06219 and Q 2 = 06261
The small difference between Q 1 and Q 2 indicates that the additional linear restriction does
not really produce any deterioration in the fit. Note that the large differences in levels 1–5
are given low weights in the computation of Q 1 and Q 2 .
Table 4.11 displays the relativities without a priori segmentation. As can be observed, the
scale without a priori segmentation is more elastic, which is logical since it has to take into
account the full heterogeneity. The mean square errors are now given by
Q 1 = 06630 and Q 2 = 06688