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m48-ch07 - Chapter 7 Interest Rate Forwards and Futures...

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Chapter 7 Interest Rate Forwards and Futures Question 7.1. Using the bond valuation formulas (7.1), (7.3), (7.6) we obtain the following yields and prices: Maturity Zero-Coupon Bond Zero Coupon One-Year Par Cont. Comp. Zero Yield Bond Price Implied Forward Coupon Yield Rate 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 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. This yields a bond price of $1,037.25280. Question 7.3. Maturity Zero-Coupon Zero Coupon One-Year Par Coupon Cont. Comp. Zero Bond Yield Bond Price Implied Forward Yield Rate 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 89
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Part 2 Forwards, Futures, and Swaps Question 7.4. Maturity Zero-Coupon Zero Coupon One-Year Par Coupon Cont. Comp. Zero Bond Yield Bond Price Implied Forward Yield Rate 1 0.05000 0.95238 0.05000 0.05000 0.04879 2 0.04200 0.92101 0.03406 0.04216 0.04114 3 0.04000 0.88900 0.03601 0.04018 0.03922 4 0.03600 0.86808 0.02409 0.03634 0.03537 5 0.02900 0.86681 0.00147 0.02962 0.02859 Question 7.5. Maturity Zero-Coupon Zero Coupon One-Year Par Coupon Cont. Comp. Zero Bond Yield Bond Price Implied Forward Yield Rate 1 0.07000 0.93458 0.07000 0.07000 0.06766 2 0.06000 0.88999 0.05009 0.06029 0.05827 3 0.05000 0.86384 0.03028 0.05065 0.04879 4 0.04500 0.83855 0.03016 0.04578 0.04402 5 0.04000 0.82193 0.02022 0.04095 0.03922 Question 7.6. In order to be able to solve this problem, it is best to take equation (7.6) of the main text and solve progressively for all zero-coupon bond prices, starting with year one. This yields the series of zero-coupon bond prices from which we can proceed as usual to determine the yields. Maturity Zero-Coupon Zero Coupon One-Year Par Coupon Cont. Comp. Zero Bond Yield Bond Price Implied Forward Yield Rate 1 0.03000 0.97087 0.03000 0.03000 0.02956 2 0.03500 0.93352 0.04002 0.03491 0.03440 3 0.04000 0.88899 0.05009 0.03974 0.03922 4 0.04700 0.83217 0.06828 0.04629 0.04593 5 0.05300 0.77245 0.07732 0.05174 0.05164 90
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Chapter 7 Interest Rate Forwards and Futures Question 7.7. a) We are looking for r 0 ( 1 , 3 ) . We will use equation (7.3) of the main text, and the known one-year and three-year zero-coupon bond prices. We have to solve the following equation: 1 + r 0 ( 1 , 3 ) 3 1 = P ( 0 , 1 ) P ( 0 , 3 ) r 0 ( 1 , 3 ) = P ( 0 , 1 ) P ( 0 , 3 ) 1 = 0 . 943396 0 . 816298 1 = 0 . 07504 b) Let’s calculate the zero-coupon bond price from year 1 to 2 and from year 1 to 3, they are: P 0 ( 1 , 2 ) = P ( 0 , 2 ) P ( 0 , 1 ) = 0 . 881659 0 . 943396 = 0 . 93456 P 0 ( 1 , 3 ) = P ( 0 , 3 ) P ( 0 , 1 ) = 0 . 816298 0 . 943396 = 0 . 86528 Now, we have the relevant implied forward zero-coupon prices and can find the coupon of the par 2-year coupon bond issued at time 1 according to formula (7.6). c = 1 P 0 ( 1 , 3 ) P 0 ( 1 , 2 ) + P 0 ( 1 , 3 ) = 0 . 13472 0 . 93456 + 0 . 86528 = 0 . 074851 Question 7.8. a) We have to take into account the interest we (or our counterparty) can earn on the FRA settlement if we settle the loan on initiation day, and not on the actual repayment day. Therefore, we tail the FRA settlement by the prevailing market interest rate of 5%. The dollar settlement is: ( r annually r FRA ) 1 + r annually × notional principal = ( 0 . 05 0 . 06 ) 1 + 0 . 05 × $500 , 000 . 00 = − $4 , 761 . 905 b) If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement amount is determined by: ( r annually r FRA ) × notional principal = ( 0 . 05 0 . 06 ) × $500 , 000 . 00 = − $5 , 000 We have to pay at the settlement, because the interest rate we could borrow at is 5%, but we have agreed via the FRA to a borrowing rate of 6%. Interest rates moved in an unfavorable direction.
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