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

Credibility Models for Claim Counts 133
Recall that E i = 1 and EN iTi +1 = iTi +1. Hence, the Bayesian credibility premium is
obtained by multiplying the a priori expected number of claims EN iTi +1 by an appropriate
correction factor. This factor appears as a weighted average of the prior expectation of
i , receiving weight a/a + i• , and the average claim frequency for that policy k • / i•
receiving a weight i• /a + i• .
3.3.6 Numerical Illustration
Let us now compute the coefficients to apply to the pure premium according to the number
of claims reported in the past. To this end, let us consider the Negative Binomial fit
of Portfolio A given in Table 2.7. Table 3.2 displays the values of E i N i• = k • for
different combinations of observed periods T i and number of past claims k • for a good
driver (with observable characteristics: man, age 35, rural area, upfront premium, private
use, i = 00928). Tables 3.3–3.4 are the analogues for an average driver (with observable
characteristics: woman, age 25, urban area, upfront premium, private use, i = 01408) and
a bad driver (with observable characteristics: man, age 22, rural area, split premium, private
use, i = 02840), respectively.
If the good driver does not report any accident during the first year, we see from Table 3.2
that he will pay 92 % of the base premium to be covered during the second year. If, in
addition, he does not file any claim during the second year, the premium decreases to 85.2 %
of the base premium. After ten claim-free years, he will have to pay 53.4 % of the base
premium to be covered by the insurer.
Considering the average driver, we see from Table 3.3 that he will be awarded more
discount than the good driver if he does not file any claim. Indeed he will have to pay
88.3 % (instead of 92 %) of the base premium after one claim-free year, 79.1 % (instead of
85.1 %) of the base premium after two claim-free years, and 43.1 % (instead of 53.4 %) after
ten claim-free years.
Table 3.2 Values of E i N i• = k • for different combinations of observed periods T i
and number of past claims k • for a good driver from Portfolio A (average annual claim
frequency of 9.28 %).
T i
Number of claims k •
0 1 2 3 4 5
1 92.0 % 178.4 % 264.7 % 351.1 % 437.5 % 523.8 %
2 85.2 % 165.1 % 245.1 % 325.0 % 405.0 % 485.0 %
3 79.3 % 153.7 % 228.2 % 302.6 % 377.0 % 451.5 %
4 74.2 % 143.8 % 213.4 % 283.0 % 352.7 % 422.3 %
5 69.7 % 135.1 % 200.5 % 265.9 % 331.3 % 396.7 %
6 65.7 % 127.3 % 189.0 % 250.6 % 312.3 % 374.0 %
7 62.1 % 120.4 % 178.8 % 237.1 % 295.4 % 353.7 %
8 58.9 % 114.3 % 169.6 % 224.9 % 280.2 % 335.6 %
9 56.0 % 108.7 % 161.3 % 213.9 % 266.6 % 319.2 %
10 53.4 % 103.6 % 153.8 % 204.0 % 254.1 % 304.3 %