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

Bonus-Malus Scales 193
Table 4.10 Linear relativities with a priori risk classification for the system −1/ + 2
and for Portfolio A.
Level l
Unconstrained relativities r l
with a priori ratemaking
Linear relativities rl
lin
priori ratemaking
with a
5 2714 % 2662%
4 2185 % 2269%
3 1925 % 1875%
2 1388 % 1482%
1 1286 % 1088%
0 685% 695%
Table 4.11 Linear relativities without a priori risk classification for the system
−1/ + 2 and for Portfolio A.
Level l Unconstrained relativities r l Linear relativities rl
lin
without a priori ratemaking without a priori ratemaking
5 3091 % 2983%
4 2414 % 2512%
3 2077 % 2041%
2 1429 % 1571%
1 1302 % 1100%
0 624% 629%
so that again the additional linear restriction does not really produce any deterioration in the
fit. Obviously, Q 1 and Q 2 are higher than before (when a priori risk classification was in
force). This was expected as is now more variable.
Example 4.19 (−1/+2 Scale, Portfolio B)
Portfolio B. The mean square errors are
Table 4.12 gives the linear relativities for
Q 1 = 04134 and Q 2 = 04182
We observe large discrepancies between r l and rl
lin , especially in the higher levels. The
constant penalty by step in the linear scale is 26.2 %.
4.5.5 Approximations
In Chapter 3, several discrete approximations to allowed us to derive simplified versions
of the credibility formulas (replacing integrals with sums). The same idea can be applied
here. Considering (4.13), the expression for r l when has support points 1 k with
respective probability masses p 1 p q as in (3.5) becomes
r l =
∑
k w k
∑ q
j=1 j l k j p j
∑k w k
∑ q
j=1 l k j p j