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

316 Actuarial Modelling of Claim Counts
Cn
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 1 2 3 4 5 6 7 8
Age of policy
Figure 8.3 Convergence of the transient policyholders’ distributions to the steady state distribution
for the system −1/ + 2.
The corresponding relativies ¯r l are presented in Table 8.15 for a uniform distribution of
age of policy. The last column gives the steady state relativities. We notice that the relativities
are more severe when the initial distribution is closer to the bottom initial distribution.
This seems reasonable since, with the bottom initial distribution, many policyholders are in
the bonus level from the beginning. Therefore, the bonus must be weaker and the maluses
stronger to ensure the financial balance. As for the −1/top bonus-malus scale, the relativities
obtained starting from the bottom initial distribution are the closest to the stationary
relativities.
Influence of the Maturity of the Portfolio
Let us consider the following three distributions of age of policy:
Age of Mature Young Old
policy portfolio portfolio portfolio
1 125 % 25 % 5 %
2 125 % 20 % 5 %
3 125% 15% 10%
4 125% 10% 10%
5 125% 10% 10%
6 125% 10% 15%
7 125% 5% 20%
8 125% 5% 25%