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ACTSC 231 Tutorial01_-_soln

# ACTSC 231 Tutorial01_-_soln - Solution to Problem Set 1 Q1...

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Solution to Problem Set 1 Q1. A K (0) = 16. So, K = 16 and a ( t ) = A K ( t ) /A K (0) = ( t 3 + 2 t 2 + 4 t ) / 16 + 1. Thus, i 3 = [ A K (3) - A K (2)] /A K (2) = [73 - 40] / 40 = 0 . 825, or i 3 = [ a (3) - a (2)] /a (2) = (4 . 5625 - 2 . 5) / 2 . 5 = 0 . 825. Q2. Let i be the simple interest rate, then we have 1 , 200(1 + iT ) = 1 , 320, so iT = 0 . 1. Thus, A 100 ( T/ 2) = \$100(1 + i · T/ 2) = \$100(1 . 05) = \$105 Q3. Let i be the compound interest rate per year, then 600(1 + i ) 2 - 600 = 264, so i = 0 . 2. Therefore, the amount of interest earned from \$2,000 for three years is \$2 , 000(1 . 2) 3 - 2 , 000 = \$1 , 456. Q4. At t = 10 the accumulated value of Tom’s savings is 900(1 + %2) 2 (1 + %3) 3 (1 + %4) 5 = 1 , 244 . 86. Q5. First note that A K ( n ) = A K ( n - 1)(1 + i n ). Thus, A K (5) = A K (4)(1 + i 5 ) = 1 , 000(1 + 0 . 01 × 5) = 1 , 050; A K (6) = A K (5)(1 + i 6 ) = 1 , 050(1 + 0 . 01 × 6) = 1 , 113; A K (7) = A K (6)(1 + i 7 ) = 1 , 050(1 + 0 . 01 × 6) = 190 . 91 . Or, A K (7) = A K (4)(1 + i 5 )(1 + i 6 )(1 + i 7 ) = 1 , 000(1 + 0 . 05)(1 + 0 . 06)(1 + 0 . 07) = 190 . 91. Q6. Grandparents A need to invest \$25 , 000( v 18 + v 19 ) while Grandparents B need to invest \$25 , 000( v 20 + v 21 ). Hence the difference is \$25 , 000 · v 18 [(1 + v ) - ( v 2 + v 3 )] = 1826 . 25, where where v = 1 / 1 . 04. Q7. The fact that \$500 will accumulate to \$566 in five years implies that 500(1+ i ) 5 = 566. Thus, v 5 = (1 + i ) - 5 = 500 / 566. The combined present value of three payments of 5,000 at time 10,15 and 20 is PV = 5 , 000 v 10 + 5 , 000 v 15 + 5 , 000 v 20 = 5 , 000 ( v 5 ) 2 + 5 , 000 ( v 5 ) 3 + 5 , 000 ( v 5 ) 4 1 0 , 393 . 804 Q8. Designate t = 0 for March 1, 1998.
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