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

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312 Actuarial Modelling of Claim Counts The corresponding relativities are obtained by ⎛ ⎜ ⎝ r 2 0 r 2 1 r 2 2 r 2 3 r 2 4 r 2 5 ⎛ ∫ + ⎞ ∑k w k p 0 0 0 + p 0 1 + p 0 2 exp−2 ⎞ kdF ∫ + ∑k w k p 0 0 0 + p 0 1 + p 0 2 exp−2 kdF ∫ + ∑k w k p 0 0 3 exp−2 k dF ∫ + ∑k w k p 0 0 3 exp−2 k dF ∫ + ∑k w k p 0 0 4 exp−2 k dF ∫ + ∑k w = k p 0 0 4 exp−2 k dF ∫ + ∑k w k p 0 0 5 exp−2 k dF ∫ + ∑k w k p 0 0 5 exp−2 k dF ∫ + ∑k w k exp− 0 k 1 − exp− k dF ∫ + ⎟ ∑k w ⎠ k exp− 0 k 1 − exp− k dF ⎜ ∫ + ⎟ ⎝ ∑k w k 1 − exp− 0 k dF ⎠ ∫ + ∑k w k 1 − exp− 0 k dF ⎛ = ∫ + ∑k w k ⎜ ⎝ ∑k w k ∫ + 0 exp−2 k dF ∑k w k ∫ + 0 exp−2 k dF ∑k w k ∫ + 0 exp−2 k dF ∑k w k ∫ + 0 exp−2 k dF ∑k w k ∫ + 0 exp−2 k dF ∑k w k ∫ + 0 exp−2 k dF ∑k w k ∫ + 0 exp−2 k dF ∑k w k ∫ + 0 exp−2 k dF 0 exp− k 1 − exp− k dF ∑k w k ∫ + 0 exp− k 1 − exp− k dF ∑k w k ∫ + 0 1 − exp− k dF ∑k w k ∫ + 0 1 − exp− k dF ⎞ ⎟ ⎠ Once again, we can see that the relativities at time 2 do not depend on the initial distribution. Now, r 2 4 and r 2 5 are equal to the stationary relativities r 4 and r 5 , respectively, and the values r 2 0 to r 2 3 are equal. Similar expressions can be computed for time 3, 4 and 5 to show that the transient relativities do not depend on the initial distribution. The relativities ¯r l are displayed in Table 8.10 for the three initial distributions assuming a uniform distribution of age of policy a n = 1/5, for n = 1 to 5 . The relativies computed from the bottom initial distribution are close to the steady state relativities given in the last column (except for level 0) whereas the relativities computed from a uniform or a top initial distribution are weaker than the stationary relativities for levels 1 to 5. So we see that the initial distribution can have a great influence on the resulting ¯r l s.
Transient Maximum Accuracy Criterion 313 Table 8.10 Relativies computed on the basis of the transient maximum accuracy criterion for the different initial distributions (−1/top scale). Level l Uniform Top Bottom Steady state distribution ¯r l distribution ¯r l distribution ¯r l distribution r l 5 1812 % 1812 % 1812 % 1812% 4 1403 % 1111 % 1583 % 1599% 3 1114% 946 % 1399 % 1439% 2 928% 815 % 1233 % 1312% 1 815% 709 % 1051 % 1209% 0 711% 632% 746% 612% Influence of the Maturity of the Portfolio Let us now examine the influence of the maturity of the portfolio. To this end, let us consider the three different distributions of the age of the policies PrA = n = a n that are displayed in the following table: Age of policy Mature Young Old portfolio portfolio portfolio 1 20% 30% 10% 2 20% 25% 15% 3 20% 20% 20% 4 20% 15% 25% 5 20% 10% 30% The first one is a uniform distribution which represents a mature portfolio. The second distribution represents a relatively young portfolio or a portfolio which is growing, i.e. PrA = n decreases with n. Finally, the third distribution represents an old portfolio or a portfolio which is declining, i.e. PrA = n increases with n. Asssuming a uniform initial distribution of the policyholders in the scale, we get the ¯r l s displayed in Table 8.11. Table 8.11 Relativies computed on the basis of the transient maximum accuracy criterion for different maturities of the portfolio (-1/top scale). Level l Mature Young Old Steady state portfolio ¯r l portfolio ¯r l portfolio ¯r l distribution r l 5 1812 % 1812 % 1812 % 1812% 4 1403 % 1318 % 1496 % 1599% 3 1114 % 1030 % 1214 % 1439% 2 928% 883% 984 % 1312% 1 815% 811% 819 % 1209% 0 711% 744% 683% 612%
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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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3 Credibility Models for Claim Coun
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Credibility Models for Claim Counts
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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)
Page 383: Bibliography 353 Walhin, J.-F., & P
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