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

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300 Actuarial Modelling of Claim Counts Table 8.2 (Continued). Level l r 0 l r 10 l r 20 l r 30 l r 40 l r 50 l r l 10 00% 831 % 1151 % 1266 % 1310 % 1327 % 1340% 9 00% 821 % 1107 % 1206 % 1238 % 1249 % 1255% 8 00% 826 % 1073 % 1150 % 1170 % 1174 % 1173% 7 00% 815% 947 % 1078 % 1109 % 1115 % 1114% 6 00% 791% 913 % 1038 % 1065 % 1070 % 1067% 5 00% 755% 876 % 1000 % 1026 % 1030 % 1028% 4 00% 774% 809% 833% 831% 829% 826% 3 00% 741% 782% 807% 806% 804% 802% 2 00% 704% 648% 767% 779% 779% 778% 1 00% 664% 618% 742% 754% 755% 755% 0 00% 515% 438% 435% 442% 446% 451% Table 8.3 Evolution of the transient relativities for the top initial distribution for the former compulsory Belgian bonus-malus scale. Level l r 0 l r 10 l r 20 l r 30 l r 40 l r 50 l r l 22 00 % 2227 % 2488 % 2587 % 2639 % 2668 % 2715% 21 00 % 1952 % 2217 % 2328 % 2387 % 2421 % 2474% 20 00 % 1749 % 2012 % 2131 % 2195 % 2233 % 2291% 19 00 % 1592 % 1850 % 1973 % 2041 % 2081 % 2141% 18 00 % 1470 % 1719 % 1843 % 1913 % 1954 % 2014% 17 00% 946 % 1330 % 1546 % 1678 % 1763 % 1903% 16 00 % 1093 % 1431 % 1594 % 1684 % 1735 % 1803% 15 00 % 1031 % 1354 % 1515 % 1603 % 1652 % 1714% 14 00% 980 % 1288 % 1446 % 1531 % 1577 % 1631% 13 00% 942 % 1234 % 1385 % 1464 % 1504 % 1549% 12 00% 460% 890 % 1116 % 1256 % 1344 % 1472% 11 00% 857 % 1026 % 1222 % 1317 % 1361 % 1402% 10 00% 831% 987 % 1177 % 1266 % 1305 % 1340% 9 00% 821% 954 % 1126 % 1199 % 1229 % 1255% 8 00% 826% 926 % 1074 % 1130 % 1152 % 1173% 7 00% 815% 589% 858 % 1001 % 1069 % 1114% 6 00% 791% 746% 944 % 1013 % 1039 % 1067% 5 00% 755% 722% 911% 975% 999 % 1028% 4 00% 774% 711% 743% 774% 794% 826% 3 00% 741% 685% 719% 751% 770% 802% 2 00% 704% 311% 632% 748% 776% 778% 1 00% 664% 618% 651% 691% 717% 755% 0 00% 515% 438% 361% 404% 425% 451%
Transient Maximum Accuracy Criterion 301 Table 8.4 Evolution of the transient relativities for the bottom initial distribution for the former compulsory Belgian bonus-malus scale. Level l r 0 l r 10 l r 20 l r 30 l r 40 l r 50 l r l 22 00 % 3049 % 2902 % 2818 % 2774 % 2750 % 2715% 21 00 % 2681 % 2629 % 2566 % 2529 % 2508 % 2474% 20 00 % 2429 % 2402 % 2361 % 2334 % 2318 % 2291% 19 00 % 2075 % 2240 % 2215 % 2189 % 2172 % 2141% 18 00 % 1901 % 2071 % 2066 % 2050 % 2038 % 2014% 17 00 % 1776 % 1929 % 1933 % 1926 % 1919 % 1903% 16 00 % 1813 % 1841 % 1827 % 1819 % 1814 % 1803% 15 00 % 1866 % 1788 % 1745 % 1729 % 1722 % 1714% 14 00 % 1408 % 1607 % 1633 % 1637 % 1636 % 1631% 13 00 % 1482 % 1566 % 1562 % 1557 % 1554 % 1549% 12 00 % 1341 % 1513 % 1499 % 1486 % 1480 % 1472% 11 00 % 1426 % 1493 % 1455 % 1430 % 1417 % 1402% 10 00 % 1486 % 1474 % 1419 % 1384 % 1364 % 1340% 9 00 % 1153 % 1293 % 1285 % 1272 % 1264 % 1255% 8 00 % 1223 % 1263 % 1231 % 1206 % 1192 % 1173% 7 00 % 1255 % 1203 % 1185 % 1158 % 1140 % 1114% 6 00 % 1268 % 1181 % 1152 % 1120 % 1099 % 1067% 5 00 % 1270 % 1159 % 1122 % 1087 % 1064 % 1028% 4 00% 994% 920% 884% 861% 847% 826% 3 00% 981% 901% 863% 839% 824% 802% 2 00% 966% 848% 838% 816% 801% 778% 1 00% 950% 827% 816% 794% 779% 755% 0 00% 566% 498% 475% 465% 460% 451% that even if the ultimate distribution of the policyholders in the scale is the same whatever the initial distributions, the transient relativities are affected by these distributions. Starting with a uniform distribution of the policyholders in the scale (Table 8.2), the r n l s increase to the r l s for levels 1 to 22, and they decrease to the limit for l = 0. The same phenomenon arises when starting with the top distribution (Table 8.3), but the difference between the r 10 l s and the asymptotic r l s is now larger. On the contrary, if the bottom distribution is used (Table 8.4) then the r n l s decrease to their limit r l . The relativities ¯r l computed using (8.6) are given in Table 8.5 according to the initial distribution of the policyholders in the scale and for a uniform distribution of age of policy over 50 years, that is, a n = 1/50 for n = 150. For the sake of completeness, the steady state relativities are displayed in the last column. When a uniform initial distribution of the policyholders is used, the relativities based on the transient maximum accuracy criterion are smaller than the steady state relativities, except for level 0. This is the case for all levels if the top distribution is assumed. On the contrary, if the bottom distribution is used then the ¯r l s are larger than the corresponding r l s. In order to figure out the impact of the maturity of the portfolio on the relativities, we considered two alternative age structures to the uniform age of policy distribution used so far. The three distributions (henceforth referred to as mature, young and old
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Actuarial Modelling of Claim Counts
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Actuarial Modelling of Claim Counts
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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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Part III Advances in Experience Rat
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Page 323 and 324: 8 Transient Maximum Accuracy Criter
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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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Bibliography 351 Norberg, R. (1976)
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Bibliography 353 Walhin, J.-F., & P
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