Midterm-S-2010W

# Midterm-S-2010W - Solutions to Midterm Test ACTSC 232...

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Solutions to Midterm Test – ACTSC 232, WINTER 2010 1. (a) The present value of the beneﬁts is Z = ( 1000 v T 60 , 0 < T 60 8 , 700 v 8 , T 60 > 8 . (b) The APV is E [ Z ] = 1000 R 8 0 e - 0 . 05 t × 1 50 dt + 700 e - 0 . 05 × 8 × 42 50 = 526 . 02 . (c) We have E [ Z 2 ] = 1000 2 R 8 0 e - 2 × 0 . 05 t × 1 50 dt +700 2 e - 2 × 0 . 05 × 8 × 42 50 = 295078 . 009 . Thus, V ar [ Z ] = 18380 . 97. (d) We know that Z ∈ { 700 v 8 } ∪ [1000 v 8 , 1000) = { 469 . 22 } ∪ [670 . 32 , 1000) . Thus, the probability is Pr { 500 < Z < 900 } = Pr { 500 < 1000 v T 60 < 900 ,T 60 8 } = Pr { 670 . 32 < 1000 v T 60 < 900 } = 0 . 117856 . 2. (a) The present value is 600 ¨ a 10 + 1000 ± ¨ a K x +1 - ¨ a ( K x +1) 10 ² . (b) The APV of the annuity is 600 ¨ a 10 + 1000 10 | ¨ a x . We have ¨ a 10 = 1 - v 10 d = 8 . 43533 , 10 | ¨ a x = X k =10 v k k p x = X k =10 e - δk e - μk = e - 10( μ + δ ) 1 - e - ( μ + δ ) = 11 . 0176 . Hence, the APV is 16078.80. 3. It follows from A 50 = A 1 50: 15 + 15 E 50 A 65 and A 50: 15 = A 1 50: 15 + 15 E 50 that 15 E 50 = 0 . 491722 and A 1 50: 15 = 0 . 052778. (a) The APV is 2000 15 E 50 = 983.44. (b) The APV is 2000 A 1 50: 15 = 105 . 556. (c) The APV is 2000 ¯ a 50: 15 = 2000(1 - ¯ A 50: 15 ) . Under the UDD assumption, ¯ A 50: 15 = i δ A 1 50: 15 + 15 E 50 . Thus, the APV is 18166.30. 4. (a) Y = 10 ¨ a K 50 +1 + 20
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