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

Risk Classification 55
Number of policyholders
7000
6722
.220 .213
6000
.210
.200
.190
.180
.170
5000
.160
.155
4686
.150
.140
.130
4000
.123
.120
.110
.108
3000
.100
.090
2295
.080
.070
2000
.060
1000
0
18–24 25–30 31–60
802
>60
.050
.040
.030
.020
.010
.000
18–24 25–30 31–60 >60
Age
Age
Figure 2.2 Composition of Portfolio A with respect to Age (left panel) and observed annual claim
frequencies according to Age (right panel).
Annual claim frequency
Assuming that the numbers of claims filed by the policyholders in the portfolio are
independent random variables, the likelihood then becomes
1 2 3 4 =
∝
n∏
i=1
) ki
( )
(d i Agei
exp − d i Agei
k i !
(
4∏
exp − j
j=1
∑
iAgei=j
)
∑
iAgei=j k
d i
i
j
where k i denotes the observed number of claims for policyholder i, and ‘∝’ reads ‘is
proportional to’. Differentiating L 1 2 3 4 = ln 1 2 3 4 with respect to j
and setting the derivative equal to 0 gives
−
∑
iAgei=j
d i + 1 j
∑
iAgei=j
k i = 0