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

312 Actuarial Modelling of Claim Counts
The corresponding relativities are obtained by
⎛
⎜
⎝
r 2
0
r 2
1
r 2
2
r 2
3
r 2
4
r 2
5
⎛ ∫ +
⎞
∑k w k p 0
0 0
+ p 0
1
+ p 0
2 exp−2 ⎞
kdF
∫ +
∑k w k p 0
0 0
+ p 0
1
+ p 0
2 exp−2 kdF
∫ +
∑k w k p 0
0 3
exp−2 k dF
∫ +
∑k w k p 0
0 3
exp−2 k dF
∫ +
∑k w k p 0
0 4
exp−2 k dF
∫ +
∑k w =
k p 0
0 4
exp−2 k dF
∫ +
∑k w k p 0
0 5
exp−2 k dF
∫ +
∑k w k p 0
0 5
exp−2 k dF
∫ +
∑k w k exp−
0 k 1 − exp− k dF
∫ +
⎟
∑k w
⎠
k exp−
0 k 1 − exp− k dF
⎜
∫ +
⎟
⎝
∑k w k 1 − exp−
0 k dF ⎠
∫ + ∑k w k 1 − exp−
0 k dF
⎛
=
∫ +
∑k w k ⎜
⎝
∑k w k
∫ +
0
exp−2 k dF
∑k w k
∫ +
0
exp−2 k dF
∑k w k
∫ +
0
exp−2 k dF
∑k w k
∫ +
0
exp−2 k dF
∑k w k
∫ +
0
exp−2 k dF
∑k w k
∫ +
0
exp−2 k dF
∑k w k
∫ +
0
exp−2 k dF
∑k w k
∫ +
0
exp−2 k dF
0
exp− k 1 − exp− k dF
∑k w k
∫ +
0
exp− k 1 − exp− k dF
∑k w k
∫ +
0
1 − exp− k dF
∑k w k
∫ +
0
1 − exp− k dF
⎞
⎟
⎠
Once again, we can see that the relativities at time 2 do not depend on the initial distribution.
Now, r 2
4 and r 2
5
are equal to the stationary relativities r 4 and r 5 , respectively, and the
values r 2
0 to r 2
3 are equal. Similar expressions can be computed for time 3, 4 and 5 to
show that the transient relativities do not depend on the initial distribution.
The relativities ¯r l are displayed in Table 8.10 for the three initial distributions assuming
a uniform distribution of age of policy a n = 1/5, for n = 1 to 5 . The relativies computed
from the bottom initial distribution are close to the steady state relativities given in the last
column (except for level 0) whereas the relativities computed from a uniform or a top initial
distribution are weaker than the stationary relativities for levels 1 to 5. So we see that the
initial distribution can have a great influence on the resulting ¯r l s.