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

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310 Actuarial Modelling of Claim Counts Table 8.9 Transient relativities for the different initial distributions (−1/top scale). Level l r 0 l r 1 l Uniform initial distribution r 2 l r 3 l r 4 l r 5 l r l 5 00 % 1812 % 1812 % 1812 % 1812 % 1812 % 1812% 4 00% 882 % 1599 % 1599 % 1599 % 1599 % 1599% 3 00% 882% 792 % 1439 % 1439 % 1439 % 1439% 2 00% 882% 792% 720 % 1312 % 1312 % 1312% 1 00% 882% 792% 720% 661 % 1209 % 1209% 0 00% 882% 792% 720% 661% 612% 612% Level l r 0 l r 1 l r 2 l Top initial distribution r 3 l r 4 l r 5 l r l 5 00 % 1812 % 1812 % 1812 % 1812 % 1812 % 1812% 4 00% 882 % 1599 % 1599 % 1599 % 1599 % 1599% 3 00% 882% 792 % 1439 % 1439 % 1439 % 1439% 2 00% 882% 792% 720 % 1312 % 1312 % 1312% 1 00% 882% 792% 720% 661 % 1209 % 1209% 0 00% 882% 792% 720% 661% 612% 612% Level l r 0 l r 1 l Bottom initial distribution r 2 l r 3 l r 4 l r 5 l r l 5 00 % 1812 % 1812 % 1812 % 1812 % 1812 % 1812% 4 00% 882 % 1599 % 1599 % 1599 % 1599 % 1599% 3 00% 882% 792 % 1439 % 1439 % 1439 % 1439% 2 00% 882% 792% 720 % 1312 % 1312 % 1312% 1 00% 882% 792% 720% 661 % 1209 % 1209% 0 00% 882% 792% 720% 661% 612% 612% The corresponding relativities are obtained by (8.4). They are given by ⎛ ⎜ ⎝ r 1 0 r 1 1 r 1 2 r 1 3 r 1 4 r 1 5 ⎛ ∫ ⎞ + ∑k w k p 0 0 0 + p 0 1 exp− ⎞ kdF ∫ + ∑k w k p 0 0 0 + p 0 1 exp− kdF ∫ + ∑k w k p 0 0 2 exp− k dF ∫ + ∑k w k p 0 0 2 exp− k dF ∫ + ∑k w k p 0 0 3 exp− k dF ∫ + ∑k w k p 0 0 3 exp− k dF = ∫ + ∑k w k p 0 0 4 exp− k dF ∫ + ∑k w k p 0 0 4 exp− k dF ∫ + ∑k w k p 0 0 5 exp− k dF ∫ + ∑k w k p 0 0 5 exp− k dF ⎟ ⎠ ⎜ ∫ + ⎟ ⎝ ∑k w k 1 − exp− 0 k dF ⎠ ∫ + ∑k w k 1 − exp− 0 k dF
Transient Maximum Accuracy Criterion 311 ⎛ ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF = ⎜ ∫ + ⎝ ∑k w k ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF ∑k w k ∫ + 0 exp− k dF 0 1 − exp− k dF ∑k w k ∫ + 0 1 − exp− k dF ⎞ ⎟ ⎠ We see that the relativities at time 1 do not depend on the initial distribution p 0 l l = 0 15. Moreover, we observe that the values r 1 0 to r 1 4 are all equal. Only the value of r 1 5 is different from the others and is equal to the steady-state value r 5 . The distribution of the policyholders in the −1/top scale after two years is then given by ⎛ ⎜ ⎝ p 2 0 ⎞ p 2 1 p 2 2 p 2 3 p 2 4 ⎟ ⎠ p 2 5 ⎛ = ⎜ ⎝ p 0 0 p 0 1 p 0 2 p 0 3 p 0 4 p 0 5 ⎞T ⎛ ⎞ exp−2 0 0 0 exp−1 − exp− 1 − exp− exp−2 0 0 0 exp−1 − exp− 1 − exp− exp−2 0 0 0 exp−1 − exp− 1 − exp− 0 exp−2 0 0 exp−1 − exp− 1 − exp− ⎟ ⎜ ⎠ ⎝ 0 0 exp−2 0 exp−1 − exp− 1 − exp− ⎟ ⎠ 0 0 0 exp−2 exp−1 − exp− 1 − exp− ⎛ p 0 0 + p 0 1 + p 0 2 exp−2 ⎞ p 0 3 exp−2 p 0 4 exp−2 = p 0 5 exp−2 ⎜ ⎝ exp−1 − exp− ⎟ ⎠ 1 − exp−
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Contents Foreword Preface Notation
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Contents vii 2.4 Overdispersion 79
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Contents ix 4.2.3 Trajectory 170 4.
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Contents xi 7.4 Numerical Illustrat
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Part I Modelling Claim Counts Actua
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Page 375 and 376: Bibliography Albrecht, P. (1983a).
Page 377 and 378: Bibliography 347 Daengdej, J., Luko
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Page 381 and 382: Bibliography 351 Norberg, R. (1976)
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