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

Bonus-Malus Scales 191
4.5.4 Linear Relativities
As demonstrated in the numerical examples with the −1/top, −1/+2 and −1/+3 scales, the
relativities obtained above may exhibit a rather irregular pattern, and this may be undesirable
for commercial purposes. It may therefore be interesting to smooth this scale in order to
obtain relativities which are regularly increasing according to the level. As suggested by
Gilde & Sundt (1989), a linear scale of the form r lin
l
= + l l = 0 1s could
then be desirable. Then, Norberg’s maximum accuracy criterion becomes a constrained
minimization:
[ ] ]
min E − r lin
L 2 = min E
[ − − L 2 (4.16)
Setting the derivative of the objective function with respect to equal to 0 yields
= E − EL
Doing the same with the derivative of the objective function with respect to gives
[
]
0 = E L − − L
= EL − EL − EL 2
Replacing the value of with the expression found above, we get
0 = EL − ELE + EL 2 − EL 2
= CL − VL
The solution of the optimization problem (4.16) is thus given by:
=
CL
VL
and = −
CL
EL (4.17)
VL
The linear relative premium scale is thus of the form
where
r lin
l
= 1 +
CL
l − EL
VL
CL = EL − EL
= ∑ s∑
w k lE l k − EL
k l=0
= ∑ s∑ ∫ +
w k l l k dF − EL
k l=0
0