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

308 Actuarial Modelling of Claim Counts
8.4 Exponential Loss Function
Under an exponential loss function, the aim is to minimize the objective function
[
n = E exp ( − c − r n
L n
)]
under the constraint Er n
L n
= E. This yields
r n
l
= E + 1 c
( [
]
)
E ln Eexp−cL n − ln Eexp−cL n = l
for l = 0 1s.
Of course, there is no reason to focus on the particular nth period. Then, nonnegative
weights a 1 a 2 a 3 summing to 1 representing the age distribution of the policies in the
portfolio are introduced and the aim of the actuary is to minimize
=
+∑
n=1
a n n
Minimizing means minimizing the expected squared rating error for a randomly chosen
policy.
8.5 Numerical Illustrations
8.5.1 Scale −1/Top
The transition rules of the −1/top scale are given in Table 4.1.
Initial Distribution
We have tested three different initial distributions PrL 0 = l. In the first case (uniform
distribution), the policyholders are uniformly spread in the scale (16.67 % of the policyholders
in each of the six levels). In the second case (top distribution), 95 % of the policyholders
are concentrated in the top of the scale (and the remaining 5 % are evenly spread over
levels 0 to 4). In the last case (bottom distribution), 95 % of the policyholders start
from the bottom of the scale (and the remaining 5 % are evenly spread over levels
1to5).
Convergence of the −1/Top Scale
Figure 8.2 represents the evolution of C n with n for a uniform initial distribution. It gives an
idea of the speed of convergence of the −1/top scale. We clearly see that C n = 0 for n ≥ 5,
i.e. that the stationary state is reached after 5 years. This was known from Chapter 4.
Transient Relativities
Table 8.9 gives the evolution with n of the corresponding transient relativities for the three
starting distributions. We see that the transient relativities do not depend on the initial