Table 46 BuhlmannStraubs Premiums ˆ Z j 09847 09276 08985 07279 09588 ˆ M a j

Table 46 buhlmannstraubs premiums ˆ z j 09847 09276

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Table 4.6: B¨uhlmann–Straub’s Premiums ˆ Z j 0.9847 0.9276 0.8985 0.7279 0.9588 ˆ M a j 2,055 1,524 1,793 1,443 1,603 The estimated parameters used in the above premiums are: (1) the collective premium for the whole portfolio, ¯ X ˆ ZW = $1 , 684, (2) the estimated within–class variance ˆ s 2 = 1 . 39 × 10 8 , (3) the estimated between–class variance (of conditional means) ˆ a = 89 , 639. The use of weights has a significant impact on these estimates. In Class 4, which has lowest credibility (see ˆ Z 4 ) the empirical credibility premium of $1,443 is substantially higher than the individual weighted average ¯ X 4 W of $1,353. Its low credibility is pushing this class premium towards the portfolio average. To a large extent this is due to the fact that, on average, only 2.4% of the portfolio drivers (i.e. claims) belong to Class 4. Finally, if we compare the premiums obtained previously with B¨uhlmann’s model to these B¨uhlmann–Straub’s premiums, we have: Note that under B: ¯ X = $1 , 671, ˆ s 2 = 46 , 000, and ˆ a = 72 , 300, while under the B–S model: ¯ X ˆ ZW = $1 , 684 (but ¯ X WW = 1 , 865), ˆ s 2 = 1 . 39 × 10 8 , and ˆ a = 89 , 639. We shall compare the estimates obtained with the above data to those produced with different models studied in the next few chapters.
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4.2. THE B ¨ UHLMANN–STRAUB MODEL 49 Table 4.7: B¨uhlmann’s vs B¨uhlmann–Straub’s Premiums B¨uhlmann’s model ˆ Z 0.9496 0.9496 0.9496 0.9496 0.9496 ˆ M a j ( B ) 2,044 1,519 1,814 1,376 1,602 B¨uhlmann–Straub’s model ˆ Z j 0.9847 0.9276 0.8985 0.7279 0.9588 ˆ M a j ( B - S ) 2,055 1,524 1,793 1,443 1,603
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  • Fall '09
  • Dr.D.Dryanov
  • Trigraph, Estimation theory, Mean squared error, Bias of an estimator, Credibility Models

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