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

296 Actuarial Modelling of Claim Counts
=
s∑
∣ ∑ k
l=0
∫ + (
w k
0
p n
l
)
∣
k − l k dF ∣ (8.1)
for n = 0 1 2 which is the sum on each level l of the absolute difference between the
probability for a policyholder to be in the level l after n periods and the probability for this
policyholder to be in level l when the stationary state is reached.
We can assume that the convergence to the steady state is acceptable when we reach n 0
such that
d TV p n 0 ≤ (8.2)
for some fixed >0. The convergence could then be checked by computing and analysing
the evolution of (8.1) with n.
Example 8.1 (Former compulsory Belgian bonus-malus scale) The convergence of the
former compulsory Belgian bonus-malus scale is assessed using the evolution of
C n = d TV p n
displayed in Figure 8.1. We notice a fast convergence during the first 20 years (C n decreasing
from 0.9 to 0.2) and then a very slow convergence to reach C n