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

Credibility Models for Claim Counts 145
Table 3.10 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 (expected annual claim frequency of
9.28 %) and for a mixture of mixed uniform approximations of i (improved 4-convex extrema).
T i
Number of claims k •
0 1 2 3 4 5
1 91.98 % 178.36 % 264.74 % 351.04 % 437.76 % 524.92 %
2 85.16 % 165.12 % 245.13 % 325.04 % 404.57 % 486.88 %
3 79.28 % 153.70 % 228.23 % 302.76 % 376.26 % 451.61 %
4 74.16 % 143.77 % 213.51 % 283.40 % 352.14 % 420.74 %
5 69.66 % 135.06 % 200.54 % 266.38 % 331.27 % 394.52 %
6 65.67 % 127.37 % 189.00 % 251.28 % 312.83 % 372.29 %
7 62.18 % 120.55 % 178.65 % 237.81 % 296.27 % 353.06 %
8 58.93 % 114.47 % 169.29 % 225.73 % 281.26 % 335.95 %
9 56.05 % 109.04 % 160.80 % 214.82 % 267.62 % 320.35 %
10 53.43 % 104.17 % 153.05 % 204.90 % 255.23 % 305.90 %
Table 3.11 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 (expected annual claim frequency
of 14.09 %) and for a mixture of mixed uniform approximations of i (improved 4-convex
extrema).
T i
Number of claims k •
0 1 2 3 4 5
1 88.32 % 171.25 % 254.21 % 337.07 % 419.97 % 505.05 %
2 79.09 % 153.34 % 227.69 % 302.05 % 375.37 % 450.47 %
3 71.60 % 138.83 % 206.16 % 273.75 % 340.28 % 405.73 %
4 65.41 % 126.87 % 188.25 % 250.30 % 311.63 % 370.88 %
5 60.21 % 116.90 % 173.05 % 230.58 % 287.30 % 342.82 %
6 55.76 % 108.51 % 159.96 % 213.75 % 266.28 % 318.81 %
7 51.93 % 101.38 % 148.58 % 199.16 % 248.12 % 297.44 %
8 48.58 % 95.27 % 138.64 % 186.29 % 232.40 % 278.28 %
9 45.63 % 89.96 % 129.95 % 174.75 % 218.71 % 261.30 %
10 43.01 % 85.30 % 122.35 % 164.32 % 206.58 % 246.42 %
is constrained ex ante to be linear in past observations, i.e. the predictor ˆN iTi +1 of N iTi +1 is
of the form
T
∑ i
ˆN iTi +1 = c i0 + c it N it
The coefficients c i0 and the c it s involved in ˆN iTi +1 are obtained from the minimization of an
expected squared difference.
t=1