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# Q3-Answer-2010W - 2 = K x 1 v K x 1(b We have ¯ I ¯ A x =...

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ANSWERS TO QUIZ 3 – ACTSC 232, WINTER 2010 1. (a) i. The present value is Z = ( 5000 v T 90 , 0 < T 90 3 , 5000 v 3 , T 90 > 3 . ii. The single benefit premium is E ( Z ) = 5000 ¯ A 90: 3 = 5000 i δ A 1 90: 3 + v 3 3 p 90 = 4549 . 48 , where A 1 90: 3 = 2 k =0 v k +1 k p 90 q 90+ k = 0 . 803585 . (b) i. The present value is Z = 3000( K 90 + 1) v T 90 . ii. The single benefit premium is E ( Z ) = 3000( I ¯ A ) 90 = 3000 i δ ( IA ) 90 = 6799 . 38, where ( IA ) 90 = X k =0 ( k + 1) v k +1 k p 90 q 90+ k = 4 X k =0 ( k + 1) 1 1 + i k +1 l 90+ k l 90 1 - l 90+ k +1 l 90+ k = 2 . 21162 . 2. (a) The present value is Z = ( 0 , 0 < T 35 5 , 3000 v T 35 , T 35 > 5 . (b) The APV is E ( Z ) = 3000 R 55 5 e - 0 . 03 t 1 55 dt = 1215 . 74 . (c) The probability is Pr { Z > 1000 } = Pr { 3000 v T 35 > 1000 , T 35 > 5 } = Pr { 5 < T 35 < 36 . 62 } = 0 . 575 . 3. (a) Z 1 = T x v T x
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Unformatted text preview: 2 = ( K x + 1) v K x +1 . (b) We have ( ¯ I ¯ A ) x = E h T x v T x i = E h ( K x + 1) v K x +1 v-1+ R x i-E h (1-R x ) v K x +1 v-1+ R x i = ( IA ) x E h v-1+ R x i-A x E h (1-R x ) v-1+ R x i . Hence, the formula follows from E h v-1+ R x i = Z 1 e-δ ( t-1) dt = i δ and E h (1-R x ) v-1+ R x i = Z 1 (1-t ) e-δ ( t-1) dt = Z 1 se δs ds = 1-e δ + δe δ δ 2 = i δ ± 1 d-1 δ ² . 1...
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