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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management RSM332 Tutorial #4 Problem Set Oct.13/Oct.14 2011 1. (a) You are a bond trader for an investment bank. A client calls you up and asks for the quote of an annuity which pays off $1,000 each year in the next two years. Although no such annuity exists, you notice that a 10% coupon bond with face value of $1,000 and matures next year costs $1,016.76. You also find a 5% coupon bond with face value $1,000 and matures in two years costs $929.22. The coupons on both bonds are paid on an annual basis. Show how you can create the annuity for your client and at what price should you quote him for you to breakeven? (10 points) (b) Someone shows you a term structure of spot interest rates with the 1-year spot rate is 4%/year, the 2-year spot rate is 10%/year, and the 3-year spot rate is 6%/year. Compute the implied forward rates for the second year and third year. (4 points) (c) What is wrong with this term structure of interest rates in part (b)? If this is indeed the actual term structure of interest rates, show how you can make an arbitrage profit. Assume 1-year to 3-year pure discount bonds are available and they are priced based on the spot interest rates. (6 points) Solution: (a) We would like to use a portfolio of the two coupon bonds to replicate the annuity. Suppose we buy x units of the 1-year coupon bond and y units of the 2-year coupon bond. The cashflows at t = 1 and t = 2 are 1100 x + 50 y and 1050 y , respectively. We want to set both of them equal to 1000, so that they exactly replicate the cashflows of the annuity. Solving the following two equations 1100 x + 50 y = 1000 , 1050 y = 1000 , we obtain x = 0 . 8628 and y = 0 . 9524. Therefore, to create the annuity for your client, you need to buy 0.8628 unit of 1-year coupon bond and 0.9524 unit of 2-year coupon bond. You should at least charge your client . 8658 $1016 . 76 + 0 . 9524 $929 . 22 = $1765 . 3 for you to breakeven. (b) The forward rates for the second and third year are given by f 2 = (1 + r 2 ) 2 1 + r 1- 1 = (1 . 1) 2 1 . 04- 1 = 0 . 1635 , f 3 = (1 + r 3 ) 3 (1 + r 2 ) 2- 1 = (1 . 06) 3 (1 . 1) 2- 1 =- . 0157 ....
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This note was uploaded on 12/20/2011 for the course RSM 332 taught by Professor Raymondkan during the Fall '08 term at University of Toronto- Toronto.
- Fall '08