5 6 A workers compensation portfolio is made of 3 employers We have data for

# 5 6 a workers compensation portfolio is made of 3

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[5 ] 6. A worker’s compensation portfolio is made of 3 employers. We have data for the last 5 years, but not all employers were in business during the whole period; any year of inactivity is identified by a weight of 0 in the data table below: Ratios X jr /w jr (Weights w jr ) Year Employer 1 2 3 4 5 1 3 (1) 5 (1) 0 (0) 0 (0) 4 (1) 2 6 (2) 8 (2) 8 (2) 14 (2) 4 (2) 3 0 (0) 2 (3) 0 (3) 3 (3) 6 (3) Using the B¨uhlmann-Straub’s model and ˆ m = ¯ X WW , calculate the linear credibility premium for Employer 2. [5 ] 7. Hachemeister’s linear regression credibility model is given, for j = 1 , . . . , k , by: μ j ) = E [ X j | Θ j ] = Y β j ) = Y bracketleftbig β 0 j ) , . . . , β p - 1 j ) bracketrightbig prime , where μ j ) = bracketleftbig μ 1 j ) , . . . , μ n j ) bracketrightbig prime is a n × 1 vector, Y is a known n × p design matrix, and β j ) is an unknown regression vector of length p . Furthermore, Cov[ X j | Θ j ] = σ 2 j ) V j , where V j is a known positive semi–definite n × n (inverse) matrix of weights and σ 2 is a scalar function of Θ j . Also s 2 = E bracketleftbig σ 2 j ) bracketrightbig , A = Cov bracketleftbig β j ) bracketrightbig is a p × p matrix, and b = E bracketleftbig β j ) bracketrightbig is a p × 1 vector. Finally ˆ β j = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j X j = U j Y prime V - 1 j X j are the regression estimators, while the credibility estimators are B a j = Z j ˆ β j + ( I - Z j ) b , where Z j = A ( A + s 2 U j ) - 1 . Show that these formulas reduce to those of the B¨uhlmann-Straub’s credibility model when p = 1 and Y = (1 , . . . , 1) prime .

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ACTU 458 Final Examination December 2015 Page 3 of 3 [5 ] 8. Give three advantages that a Generalized Linear Model (GLM) credibility model may have over Hachemeister’s linear regression model. Can you also list two disadvantages of GLM’s.

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