T1-Answer-331F09

T1-Answer-331F09 - . 93. 3. By the aggregate mortality law,...

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Answers and Hints to Tutorial 1 – ACTSC 331, Fall 2009 Problem 1 Given T ( x ) > t , we have t L = v T ( x ) - t - ¯ P ( ¯ A x a T ( x ) - t . Thus, by the aggregate mortality law, we have V ar ( t L | T ( x ) > t ) = ± 1 + ¯ P ( ¯ A x ) δ ² 2 V ar ( v T ( x + t ) ) = " 1 + ¯ P ( ¯ A x ) δ # 2 [ 2 ¯ A x + t - ( ¯ A x + t ) 2 ] = 2 ¯ A x + t - ( ¯ A x + t ) 2 ( δ ¯ a x ) 2 , where the last equality follows from ¯ P ( ¯ A x ) = ¯ A x / ¯ a x and δ ¯ a x + ¯ A x = 1. Problem 2 Given constant forces of μ and δ , we have ¯ a x : n = Z n 0 v t t p x dt = Z n 0 e - δt e - μt dt = 1 - e - ( δ + μ ) n δ + μ . Thus, using ¯ A x : n = 1 - δ ¯ a x : n , we have ¯ P ( ¯ A 35: 25 ) = 0 . 04252 and 5 ¯ V ( ¯ A 35: 25 ) = 0 . 077 . Problem 3 (a) Under De Moivre’s law, we have ¯ A x : n = Z n 0 v t t p x μ x ( t ) dt + v n n p x = Z n 0 v t 1 100 - x dt + v n l n + x l x . Thus, using ¯ a x : n = 1 - ¯ A x : n δ , we have ¯ P ( ¯ A 35: 20 ) = 0 . 0384538 . (b) 10 ¯ V ( ¯ A 35: 20 ) = 0 . 32717 . (c) 20 ¯ V ( ¯ A 35: 20 ) = 1. Problem 4 1. By the EP, b ¯ A 50 = 200 ¯ a 50 = 200 1 - ¯ A 50 δ . Hence, b = 5225 . 32 . 2. The benefit reserve = b ¯ A 65 - 200 ¯ a 65 = b ¯ A 65 - 200 1 - ¯ A 65 δ = 1287
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Unformatted text preview: . 93. 3. By the aggregate mortality law, V ar ( 15 L | T (50) > 15) = b + 200 2 V ar ( v T (65) ) = b + 200 2 ( 2 A 65-( A 65 ) 2 ) . Thus, V ar ( 15 L | T (50) > 15) = 3 . 512 10 6 . Problem 5 1. By the aggregate mortality law, E ( 30 L | T (35) > 30) = 1000 E ( v T (65) ) = 1000 A 65 = 300 . 2. By the aggregate mortality law, V ( 30 L | T (35) > 30) = 1000 2 V ( v T (65) ) = 1000 2 ( 2 A 65-( A 65 ) 2 ) = 1000 2 + 2 - + ! 2 = 86470 . 60 . 1...
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