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

314 Actuarial Modelling of Claim Counts
Financial Balance
Finally, we compare the evolution of the expected financial income in two different cases.
First, Table 8.12 presents the evolution of I n (computed using (8.7)) when three different
initial distributions are used. We see that, in each situation, the financial balance is reached
after 5 years (the time needed to reach the steady state). We notice that the uniform and the
top initial distribution ensure profit in the first years whereas the bottom initial distribution
causes losses in the first years. Too many policyholders (with respect to the steady state
situation) are in the malus levels of the scale in the first two cases whereas too many
policyholders are in the bonus level in the last case.
Table 8.12 also gives the evolution of the expected financial income Ī n computed using
(8.8). The varying parameter is the distribution of the age of the policies. We see that this
parameter has a little influence on the results. The most interesting point to notice is that the
expected financial income does not converge to 100 % but goes down under 100 %. This is
the result of the use of the ¯r l s.
Choice of the Initial Level
On the basis of the concepts presented in Section 8.3.4, we now try to find the most
efficient initial level for the −1/top scale. The procedure can be summarized as follows:
for each initial distribution p 0 = e k (where e k is the vector with 095 in the kth entry
and 001 elsewhere), we compute the relativities ¯r l with the help of (8.6), as well as the
¯Q-efficiency of each solution. The optimal initial level is then the one which maximises the
¯Q-efficiency.
Table 8.12 Evolution of the expected financial income I n
based on the r l s and Ī n based on the ¯r l s.
n Uniform Top Bottom
distribution I n distribution I n distribution I n
0 1330 % 1783% 655%
1 1217 % 1601% 792%
2 1135 % 1480% 877%
3 1075 % 1393% 934%
4 1032 % 1329% 972%
5 1000 % 1000 % 1000%
n Mature Young Old
portfolio Ī n portfolio Ī n portfolio Ī n
0 1131 % 1100 % 1168%
1 1057 % 1035 % 1086%
2 1012 % 1000 % 1031%
3 987% 982% 998%
4 975% 974% 981%
5 969% 970% 973%