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

Credibility Models for Claim Counts 153
Table 3.13 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 good driver
(expected annual claim frequency equal to 9.28 %) from Portfolio A, with c = 1.
T i
Number of claims k •
0 1 2 3 4 5
1 92.3 % 175.4 % 258.5 % 341.5 % 424.6 % 507.7 %
2 85.7 % 162.8 % 240.0 % 317.1 % 394.2 % 471.4 %
3 80.0 % 151.9 % 223.9 % 295.9 % 367.9 % 439.9 %
4 75.0 % 142.4 % 209.9 % 277.4 % 344.8 % 412.3 %
5 70.5 % 134.0 % 197.5 % 261.0 % 324.5 % 388.0 %
6 66.6 % 126.6 % 186.5 % 246.5 % 306.5 % 366.4 %
7 63.1 % 119.9 % 176.7 % 233.5 % 290.3 % 347.1 %
8 59.9 % 113.9 % 167.9 % 221.8 % 275.8 % 329.7 %
9 57.1 % 108.5 % 159.8 % 211.2 % 262.6 % 314.0 %
10 54.5 % 103.5 % 152.6 % 201.6 % 250.7 % 299.7 %
Table 3.14 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 = 1.
T i
Number of claims k •
0 1 2 3 4 5
1 89.0 % 1674 % 245.8 % 324.3 % 402.7 % 481.1 %
2 80.1 % 1507 % 221.4 % 292.0 % 362.6 % 433.2 %
3 72.9 % 1371 % 201.3 % 265.5 % 329.8 % 394.0 %
4 66.8 % 1257 % 184.6 % 243.5 % 302.4 % 361.3 %
5 61.7 % 1161 % 170.5 % 224.9 % 279.2 % 333.6 %
6 57.3 % 1078 % 158.3 % 208.9 % 259.4 % 309.9 %
7 53.5 % 1007 % 147.8 % 195.0 % 242.1 % 289.3 %
8 50.2 % 944 % 138.6 % 182.8 % 227.0 % 271.3 %
9 47.2 % 889 % 130.5 % 172.1 % 213.7 % 255.4 %
10 44.6 % 839 % 123.3 % 162.6 % 201.9 % 241.2 %
(the values listed in the columns entitled k • = 1 to 5 are smaller in Table 3.13 compared
to Table 3.2). Since the financial balance is fulfilled by the credibility premiums obtained
with an exponential loss function, the discounts for claim-free policyholders are also
reduced (the values in the column entitled k • = 0 are higher in Table 3.13 compared to
Table 3.2).
Note however that the a posteriori corrections obtained with an exponential loss function
with c = 1 are very similar to those coming from a quadratic loss function. To see this, let
us now increase the value of c to 5 in Table 3.16. Increasing c results in reduced discounts
and also in reduced penalties.