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

134 Actuarial Modelling of Claim Counts
Table 3.3 Values of E i N i• = k • for different combinations of observed periods T i
and number of past claims k • for an average driver from Portfolio A (average annual
claim frequency of 14.09 %).
T i
Number of claims k •
0 1 2 3 4 5
1 88.3 % 171.2 % 254.2 % 337.1 % 420.0 % 503.0 %
2 79.1 % 153.3 % 227.6 % 301.8 % 376.1 % 450.4 %
3 71.6 % 138.8 % 206.0 % 273.3 % 340.5 % 407.7 %
4 65.4 % 126.8 % 188.2 % 249.6 % 311.0 % 372.5 %
5 60.2 % 116.7 % 173.2 % 229.8 % 286.3 % 342.8 %
6 55.8 % 108.1 % 160.5 % 212.8 % 265.2 % 317.5 %
7 51.9 % 100.7 % 149.4 % 198.2 % 247.0 % 295.7 %
8 48.6 % 94.2 % 139.8 % 185.5 % 231.1 % 276.7 %
9 45.7 % 88.5 % 131.4 % 174.3 % 217.1 % 260.0 %
10 43.1 % 83.5 % 123.9 % 164.3 % 204.8 % 245.2 %
Table 3.4 Values of E i N i• = k • for different combinations of observed periods T i
and number of past claims k • for a bad driver from Portfolio A (average annual claim
frequency of 28.40 %).
T i
Number of claims k •
0 1 2 3 4 5
1 78.9 % 1531 % 227.2 % 301.3 % 375.5 % 449.6 %
2 65.2 % 1264 % 187.7 % 248.9 % 310.2 % 371.4 %
3 55.5 % 1077 % 159.9 % 212.0 % 264.2 % 316.4 %
4 48.4 % 938 % 139.2 % 184.7 % 230.1 % 275.5 %
5 42.9 % 831 % 123.3 % 163.6 % 203.8 % 244.0 %
6 38.5 % 746 % 110.7 % 146.8 % 182.9 % 219.0 %
7 34.9 % 676 % 100.4 % 133.1 % 165.9 % 198.6 %
8 31.9 % 619 % 91.8 % 121.8 % 151.8 % 181.7 %
9 29.4 % 570 % 84.6 % 112.3 % 139.9 % 167.5 %
10 27.3 % 529 % 78.5 % 104.1 % 129.7 % 155.3 %
Table 3.4 shows higher discounts for the bad driver, with percentages of 78.9 % after one
claim-free year, 65.2 % after two claim-free years and 27.3 % after ten claim-free years.
The discounts awarded to policyholders who do not report any accident to the insurance
company are thus increasing with the a priori annual expected claim frequency. The more
claims are expected by the insurance company on the basis of observable characteristics, the
higher the discount in case no claims are reported. Note that the a posteriori expected claim
frequencies remain large for a priori bad drivers since reporting no claims happens with a
smaller probability.