ACTSC232-Tut4-sol-post

ACTSC232-Tut4-sol-post - v k +1 k | q x ! = n-1 X k =0 kv k...

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ACTSC 232 Introduction to Actuarial Mathematics Tutorial 4 Solution, Winter 2012 1. (a) A (2) 50 = 0 . 3696 (b) A (2) 1 50: 20 = 0 . 2523 2. (a) A 1 20: 20 = 0 . 3 (b) ¯ A 1 20: 20 = 0 . 185 (c) 2 ¯ A 20: 20 = 0 . 293103 3. Using the SOA Illustrative Life Table, (a) E [ Z ] = 1 , 000 , 000 A 1 30: 20 = 29 , 324 . 053 (b) V ar ( Z ) = E [ Z 2 ] - ( E [ Z ]) 2 = 15 , 771 , 076 , 955 . 39 (c) E [ Z ] = 1 , 000 , 000 A (12) 1 30: 20 = 1 , 000 , 000 · i i (12) A 1 30: 20 = 30 , 121 . 98 4. Cov ( Z 1 ,Z 2 ) = E [ Z 1 Z 2 ] - E [ Z 1 ] · E [ Z 2 ] = 2 ¯ A 1 x : n - ¯ A 1 x : n · ¯ A x 5. (a) Proof RightSide = vq x + vp x n - 2 X k =0 ( k + 1) v k +1 k | q x +1 + n - 2 X k =0 v k +1 k | q x +1 ! = vq x + n - 2 X k =0 ( k + 1) v k +2 k +1 | q x + n - 2 X k =0 v k +2 k +1 | q x ! = n - 1 X k =1 kv k +1 k | q x + vq x + n - 1 X k =1
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Unformatted text preview: v k +1 k | q x ! = n-1 X k =0 kv k +1 k | q x + n-1 X k =0 v k +1 k | q x = n-1 X k =0 ( k + 1) v k +1 k | q x = LeftSide (b) Similar to (a), we have ( IA ) x = vq x + vp x (( IA ) x +1 + A x +1 ) Therefore, ( IA ) 51 = ( IA ) 50-vq x vp x-A 51 = 4 . 99675-. 00558 1 / 1 . 06-. 00558-. 24905 = 5 . 0731 1...
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