# AssignmentTwoAnswers - ActSc 445/845 Assignment Two Due...

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ActSc 445/845 Assignment Two Due Date: Thursday, October 28, 2010. 1. You want to construct a portfolio formed of 2-year, 5-year and 8-year zero-coupon bonds. All bonds have a face value of \$100. The initial amount of money to be invested is \$1 , 000. Assume that the continu- ously compounded spot rates for 2, 5, and 8 years are 0.02, 0.04, and 0.06 respectively. (a) Determine how many units of each zero-coupon bond you need to buy if you want the Fisher-Weil duration and convexity to be 5 and 30, respectively. (b) Assume all spot rates instantaneously increase by 10bp. Use the Fisher-Weil duration and convexity set in part (a) to approximate the change of value (in dollars) of the portfolio determined in part (a). Compare your approximation to the true change in value. (c) View the portfolio determined in part (a) as a single fixed income security with cash flows at 2, 5, and 8. Determine its annual effective yield and compute its modified duration and convexity. (d) Using the modified duration and convexity found in (c), what is the approximate change in value of the security mentioned in (c) if the effective annual yield increases by 0.005. Answer: (a) Let x 2 , x 5 and x 8 be the numbers of units of the 2, 5 and 8 year zero coupon bonds respectively, that need to be purchased. Matching price, and Fisher-Weil duration and convexity leads to the following linear system: 100 e - 0 . 02 · 2 x 2 + 100 e - 0 . 04 · 5 x 5 + 100 e - 0 . 08 · 8 x 8 = 1000 2 · 100 e - 0 . 02 · 2 x 2 + 5 · 100 e - 0 . 04 · 5 x 5 + 8 · 100 e - 0 . 08 · 8 x 8 = 5000 4 · 100 e - 0 . 02 · 2 x 2 + 25 · 100 e - 0 . 04 · 5 x 5 + 64 · 100 e - 0 . 08 · 8 x 8 = 30 , 000 The solution to this linear system is x 2 = 2 . 8911, x 5 = 5 . 4285, x 8 = 5 . 2680. 1

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(b) The approximate new price is: P new P - P · D FW · ε + P · 1 2 C FW + ε 2 = 1000 - 1000 · 5 · (0 . 001) + 1000 2 · 30 · (0 . 001) 2 = 995 . 015 (1) So that the approximate change is -4.985. The true new price is: P new = x 2 e - 0 . 021 · 2 + x 5 e - 0 . 041 · 5 + x 8 e - 0 . 081 · 8 = 995 . 01497 . (2) (c) The effective annual yield y solves: 1000 = x 2 100 (1 + y ) 2 + x 5 100 (1 + y ) 5 + x 8 100 (1 + y ) 8 A nonlinear solver yields y = 0 . 05867. The modified duration and convexity are: D m = 1 1000 2 · x 2 · 100 (1 + y ) 3 + 5 · x 5 · 100 (1 + y ) 6 + 8 · x 8 · 100 (1 + y ) 9 = 4 . 937956 C m = 1 1000 2 · 3 · x 2 · 100 (1 + y ) 4 + 5 · 6 · x 5 · 100 (1 + y ) 7 + 8 ·
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