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HW2 Solution

# HW2 Solution - PV = A r ± 1-1(1 r n ² = \$8882 74 08 ±...

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MIE375H1F - Financial Engineering - HW2 Solutions Nick Yeung September 29, 2010 Question 3.3 3.3a E [ age ] = 101 X x =90 p x · x = 0 . 07(90) + 0 . 08(91) + . . . + 0 . 04(101) = 95 . 13 3.3b Since the expected age is not an integer, we can take a weighted average of an annuity up to age 95 and one up to age 96: PV 95 = 10000 0 . 08 h 1 - 1 (1 + 0 . 08) 5 i = \$39 , 927 . 10 PV 96 = 10000 0 . 08 h 1 - 1 (1 + 0 . 08) 6 i = \$46 , 228 . 80 PV 95 . 13 = 0 . 87 PV 95 + 0 . 13 PV 96 = 0 . 87(\$39927 . 10) + 0 . 13(\$46228 . 80) = \$40746 . 32 3.3c Given age x , the annuity is: PV x = 10000 0 . 08 h 1 - 1 (1+0 . 08) ( x - 90) i , therefore, the expected annuity is: E [ annuity ] = 101 X x =90 p x · PV x = 101 X x =90 p x · 10000 0 . 08 h 1 - 1 (1 + 0 . 08) ( x - 90) i = \$38 , 387 1

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Question 3.5 Since the call is advantageous, it means that the face value +5% is worth less than the future cash flows given the new yield, y . Hence, assuming annual coupons: 105 100 (1 + y ) 15 + 10 y 1 - 1 (1 + y ) 15 The above polynomial can be solved using a computer to yield y 9 . 366% Question 3.8 In this mortgage, P = \$100 , 000, n = 30 years, r = 8%. 3.8a A = r (1 + r ) n P (1 + r ) n - 1 = (0 . 08)(1 . 08) 30 (\$100000) (1 + 0 . 08) 30 - 1 = \$8882
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Unformatted text preview: PV = A r ± 1-1 (1 + r ) n ² = \$8882 . 74 . 08 ± 1-1 (1 + 0 . 08) 25 ² = \$94821 . 26 3.8c Using the new Principal from 3.8b and the new r = 9% and new term of 25 years: A = r (1 + r ) n P (1 + r ) n-1 = (0 . 09)(1 + 0 . 09) 25 (\$94 , 821 . 26) (1 + 0 . 09) 25-1 = \$9653 . 40 2 3.8d Solving for n : 8882 . 74 = (0 . 09)(1 + 0 . 09) n (\$94 , 821 . 26) (1 + 0 . 09) n-1 n = 37 . 6 years It will take another 37.6 years before the mortgage is paid oﬀ for a total of 42.6 years on the mortgage. Question 3.9 Assuming semi-annual coupons, n = 36, r = 4 . 5%, Face Value of \$100, Coupon Payments = \$4 Price = C r ± 1-1 (1 + r ) n ² + F (1 + r ) n = 4 . 045 ± 1-1 (1 + 0 . 045) 36 ² + 100 (1 + 0 . 045) 36 = 91 . 17 3...
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HW2 Solution - PV = A r ± 1-1(1 r n ² = \$8882 74 08 ±...

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