Midterm-Test-1-S-331-2011-F

Midterm-Test-1-S-331-2011-F - . 83. (b) The policy value at...

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SOLUTIONS TO MIDTERM TEST # 1– ACTSC 331, FALL 2011 1. (a) Let the gross premium be P . Then, P ¨ a 45 = 2000 A 45 +(0 . 05 P +10)¨ a 45 +0 . 2 P +20. Hence, P = 42 . 49. (b) The gross premium policy value at the end of the first year is 1 V = 2000 A 46 + (0 . 05 P + 10)¨ a 46 - P ¨ a 46 = - 3 . 60 . (c) Let the net premium be π . Then, 2000 A 45 = π ¨ a 45 . Hence, π = 28 . 35. (d) The net premium benefit reserve at the end of the first year is 1 V = 2000 A 46 - π ¨ a 46 = 24 . 52. (e) i. 5 L = 2000 v K 45 +1 - 5 - π ¨ a K 45 +1 - 5 . ii. By (i), 5 L = (2000 + π d ) v K 45 +1 - 5 - π d . Thus, V ar ( 5 L | K 45 5) = (2000 + π d ) 2 V ar ( v K 45 +1 - 5 | K 45 5) = (2000 + π d ) 2 V ar ( v K 50 +1 ) = (2000 + π d ) 2 ( 2 A 50 - ( A 50 ) 2 ) = 203262 . 15 . 2. (a) We have π ¨ a 50: 4 = 7 vq 50 + 9 v 2 p 50 q 51 + 11 v 3 2 p 50 q 52 + 13 v 4 3 p 50 q 53 . Thus, π = 1
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Unformatted text preview: . 83. (b) The policy value at the end of the third year is 3 V = 13 vq 53- = 1 . 51. 3. (a) For h = 0 , 1 , 2 ,... , h V (4) = b h +1 vq x + h- h 4 + h 4 v 1 4 1 4 p x + h + h 4 v 2 4 2 4 p x + h + h 4 v 3 4 3 4 p x + h + h +1 V (4) vp x + h . (b) For h = 0 , 1 , 2 ,... and 1 4 < s < 2 4 , h + s V (4) = b h +1 v 1-s 1-s q x + h + s- h 4 v 2 4-s 2 4-s p x + h + s + h 4 v 3 4-s 3 4-s p x + h + s + h +1 V (4) v 1-s 1-s p x + h + s . 1...
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