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

Bonus-Malus Scales 181
= 1 1 1 1 1 1
⎛
2305672 −0678486 −0565648 −0440943 −0303122 −0150806
1305672 0321514 −0565648 −0440943 −0303122 −0150806
×
0400834 0226351 0434352 −0440943 −0303122 −0150806
⎜ −0417897 0140245 0339189 0559057 −0303122 −0150806
⎝ −1158715 0062332 0253083 0463895 0696878 −0150806
−1829035 −0008166 0175170 0377788 0601716 0849194
= 0606531 0063789 0070498 0077913 0086107 0095163.
Coming back to Example 4.7, we see that the matrices P k 01, k ≥ 5, have all their rows
equal to T 01.
⎞
⎟
⎠
Example 4.10 (−1/+2 scale)
Formula (4.9) gives here
T = 1 1 1 1 1 1
⎛
( )
2 − exp− 1 1− exp− 1 1− 2
exp−exp− 1 + + 2
2 2
1 − exp− 2 1 1− exp− 1 exp−1 +
×
1 1− exp− 2 1 1− exp− exp−1 +
⎜ 1 1 1− exp− 2 1 exp−
⎝ 1 1 1 1− exp− 2 exp−
1 1 1 1 1− exp− 1 + exp−
⎞
−1
⎟
⎠
Applying this formula in the particular case = 01, we get
T 01 = 1 1 1 1 1 1
⎛
⎞
1095163 1 0909516 1 0995476 0999845
0095163 2 1 0909516 1 0995321
×
1 0095163 2 1 0909516 0995321
⎜ 1 1 0095163 2 1 0904837
⎟
⎝ 1 1 1 0095163 2 0904837 ⎠
1 1 1 1 0095163 1904837
= 1 1 1 1 1 1
−1
⎛
2759072 −0630802 −0512948 −0658608 −0480599 −0309449
1659534 0358730 −0524518 −0561440 −0472165 −0293474
×
0578107 0244995 0454957 −0475982 −0366345 −0269065
⎜−0478699 0133850 0332122 0599117 −0272234 −0147489
⎝−1434937 0033282 0220977 0571906 0812921 −0037482
−2300176 −0057717 0120409 0547285 0794810 1062056
⎞
⎟
⎠
= 0782901 0082338 0090998 0022278 0016387 0005097