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

Credibility Models for Claim Counts 155
Table 3.17 Values of the a posteriori corrections obtained from (3.17) for different
combinations of observed periods T i and number of past claims k • for an average driver
(expected annual claim frequency equal to 14.09 %) from Portfolio A, with c = 5.
T i
Number of claims k •
0 1 2 3 4 5
1 90.8 % 1561 % 221.4 % 286.7 % 352.0 % 417.4 %
2 83.2 % 1429 % 202.6 % 262.4 % 322.1 % 381.8 %
3 76.7 % 1318 % 186.8 % 241.9 % 296.9 % 351.9 %
4 71.2 % 1223 % 173.3 % 224.3 % 275.4 % 326.4 %
5 66.5 % 1141 % 161.7 % 209.2 % 256.8 % 304.4 %
6 62.3 % 1069 % 151.5 % 196.0 % 240.6 % 285.2 %
7 58.7 % 1006 % 142.5 % 184.4 % 226.3 % 268.2 %
8 55.4 % 950 % 134.5 % 174.1 % 213.7 % 253.2 %
9 52.5 % 900 % 127.4 % 164.9 % 202.4 % 239.8 %
10 49.9 % 855 % 121.0 % 156.6 % 192.2 % 227.8 %
Table 3.18 Values of the a posteriori corrections obtained from (3.17) for different
combinations of observed periods T i and number of past claims k • for a bad driver
(expected annual claim frequency equal to 28.40 %) from Portfolio A, with c = 5.
T i
Number of claims k •
0 1 2 3 4 5
1 85.6 % 1363 % 186.9 % 237.5 % 288.2 % 338.8 %
2 75.0 % 1190 % 163.1 % 207.2 % 251.2 % 295.3 %
3 66.7 % 1058 % 144.8 % 183.8 % 222.9 % 261.9 %
4 60.2 % 952 % 130.3 % 165.3 % 200.4 % 235.5 %
5 54.8 % 866 % 118.5 % 150.3 % 182.1 % 213.9 %
6 50.3 % 795 % 108.6 % 137.8 % 166.9 % 196.1 %
7 46.5 % 734 % 100.3 % 127.2 % 154.1 % 181.0 %
8 43.3 % 682 % 93.2 % 118.2 % 143.1 % 168.1 %
9 40.4 % 637 % 87.0 % 110.3 % 133.6 % 156.9 %
10 38.0 % 598 % 81.6 % 103.5 % 125.3 % 147.2 %
3.5 Dependence in the Mixed Poisson Credibility Model
3.5.1 Intuitive Ideas
The main focus of this section is to formalize intuitive ideas with the help of stochastic
orderings. Every actuary intuitively feels that the a posteriori claim frequency distribution
must become more dangerous as more claims are reported. Here we precisely define ‘more
dangerous’ and explain that the a posteriori premium must increase with the total claim
number in the mixed Poisson model.