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ACTSC232-A2-sol

# ACTSC232-A2-sol - ACTSC 232 Introduction to Actuarial...

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ACTSC 232 Introduction to Actuarial Mathematics Assignment 2 Solution, Winter 2012 1. Proof. Recall that e x = X k =1 k p x , (1) and E [ K 2 x ] = X k =1 (2 k - 1) k p x (2) Substituting (1) and (2) into the right side, we have RightSide = p x + 2 X k =1 p x · k p x +1 + X k =1 (2 k - 1) p x · k p x +1 = p x + 2 X k =1 k +1 p x + X k =1 (2 k - 1) k +1 p x = p x + X k =1 (2 k + 1) k +1 p x = (2 · 1 - 1) p x + X k =2 (2 k - 1) k p x = X k =1 (2 k - 1) k p x = LeftSide 2. see p.4 3. This special life table has a 3-yr selected period, that is l * [ x ]+3 = l * x +3 . (a) The required table is as follows: x l * [ x ] l * [ x ]+1 l * [ x ]+2 l * x +3 x + 3 20 99180 98942 98700 98529 23 21 99135 98866 98615 98444 24 22 99079 98785 98529 98363 25 (i) We first derive the column for l * x +3 . Since p * x = p x +1 , we have l * x = l * x +1 · l x +1 l x +2 Let x = 24 and x = 23, we can obtain l * 23 , l * 24 from l * 25 which is known to be 98363. For instance, let x = 24, l * 24 = l * 25 · l 25 l 26 = 98363 · 98444 98363 = 98444 1

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(ii) For the column of l * [ x ]+2 , we know that p * [ x ]+2 = 2 p x +2 . That is l * [ x ]+2 = l * x +3 · l x +2 l x +4 Let x = 22, x = 21 and x = 20, l * [ x ]+2 can be obtained from l * x +3 . For example, let x = 21, we have l * [21]+2 = l * 24 · l 23 l 25 = 98444 · 98615 98444 = 98615
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ACTSC232-A2-sol - ACTSC 232 Introduction to Actuarial...

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