M07_MCDO8122_01_ISM_C07 - Chapter 7 Interest Rate Forwards...

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Chapter 7 Interest Rate Forwards and Futures ± Question 7.1 We can use (7.1) and solve for the effective annual yield as follows: 1 1/ 1 (0, ) [1 (0, )] (0, )] (0, ) (0, ) (0, ) 1 n n n Pn rn rn Pn rnPn = + += =− We can determine the continuous rate for maturity T , (0, ), cc rT from the zero-coupon bond prices; specifically, (0, ) (0, ) 1 1 (0, ) ln [1/ (0, )] 1 ln [ (0, )] cc T cc eP T P T T PT T = = For the forward rates and par coupon rates, we can use Equations (7.3) and (7.6). We obtain the following yields and prices: Maturity Zero-Coupon Bond Yield Zero Coupon Bond Price One-Year Implied Forward Rate Par Coupon Cont. Comp. Zero Yield 1 0.04000 0.96154 0.04000 0.04000 0.03922 2 0.04500 0.91573 0.05003 0.04489 0.04402 3 0.04500 0.87630 0.04500 0.04492 0.04402 4 0.05000 0.82270 0.06515 0.04958 0.04879 5 0.05200 0.77611 0.06003 0.05144 0.05069 Note that we have chosen to put the one year implied forward rate from year n to n + 1 in maturity n + 1 row. We demonstrate the calculations for a maturity of four years. The zero-coupon bond yield is derived from Equation (7.1) and the above result: 1/3 ( 0 , 4 ) ( 0 , 4 ) 10 . 8 2 2 7 . 0 5 rP −− =
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80 McDonald • Fundamentals of Derivatives Markets The one year implied forward rate from Year 3 to Year 4 is determined by Equation (7.3): 0 (0, 3) 0.8763 (3, 4) 1 1 0.06515 (0, 4) 0.8227 P r P −= The par coupon rate for a four year maturity uses Equation (7.6) with 0, t = 4, T = and four yearly coupons: 1( 0 , 4 ) (0,1) (0, 2) (0, 3) (0, 4) 1 0.8227 0.96154 0.91573 0.8763 0.8227 0.04958 P c PP = +++ = ++ + = The four year continuous yield is: 4 ln(0.8227) 0.04879 4 r == ± Question 7.2 The coupon bond pays a coupon of $60 each year plus the principal of $1,000 after five years. We have cash flows of 60, 60, 60, 60,1060. To obtain the price of the coupon bond, we multiply each cash flow by the zero-coupon bond price of that year. Specifically, Bond Price 60 ( (0,1) (0, 2) (0, 3) (0, 4) (0, 5)) 1000 (0, 5) 60 (0.96154 0.91573 0.87630 0.8227 0.77611) 1000 0.77611 60 4.3524 776.11 261.14 776.11 1037.25 P P + + + + + × + + + + + × + = + = This yields a bond price of $1,037.25. ± Question 7.3 This is a straightforward application of Equations (7.1), (7.3), and (7.6). We also need the continuous rate calculation (derived in Question 7.1): ln(1/ (0, )) ln( (0, )) (0, ) cc P TP T rT TT Maturity Zero-Coupon Bond Yield Zero Coupon Bond Price One-Year Implied Forward Rate Par Coupon Cont. Comp. Zero Yield 1 0.03000 0.97087 0.03000 0.03000 0.02956 2 0.03500 0.93351 0.04002 0.03491 0.03440 3 0.04000 0.88900 0.05007 0.03974 0.03922 4 0.04500 0.83856 0.06014 0.04445 0.04402 5 0.05000 0.78353 0.07024 0.04903 0.04879
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Chapter 7 Interest Rate Forwards and Futures 81 ± Question 7.4 To solve this problem, we first generate the zero-coupon bond prices from the implied one-year forward rates that are given. All other rates follow as in Question 7.1. In order to derive bond prices from one-year forward rates, we have to work recursively. The initial one- year implied forward rate is equivalent to the one year zero-coupon bond yield (i.e., it is the rate that you can lock in from Year 0 to Year 1). Hence the zero-coupon bond price is (0,1) 1/1.05 0.95238. P == We then use Equation (7.3) to find the two zero-coupon bond price as follows: [] 0 0 (0,1) (0,1) 1( 1 , 2 ) ( 0 , 2 ) (0, 2) 1 (1, 2) PP rP Pr += = + One can then use this (0, 2) P and the three-year implied forward rate given to find (0, 3). P The four and five year zero-coupon prices are derived similarly.
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M07_MCDO8122_01_ISM_C07 - Chapter 7 Interest Rate Forwards...

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