There are some problems that could result in duration-based hedging not working well in practice. We review some of these potential pitfalls here.
First, duration as a sensitivity measure has two shortcomings. It works well only for
small interest-rate changes and it presumes parallel shifts in the yield curve. Duration-based
hedging implicitly involves the same assumptions. To the extent that these assumptions are violated, duration-based hedging schemes will not perform well.
Careful choice of the futures contract can mitigate some of these problems. For instance, suppose the portfolio being hedged consists of bonds with roughly the same maturity. If we use a futures contract whose duration is “close” to the duration of these bonds, this will ensure that the portfolio value and the futures price depend on similar interest rates. If the portfolio consists of a large number of disparate bonds, we can separate it into blocks of roughly similar maturity and hedge each block separately with a futures contract matching it in duration.
Another problem in implementing a duration-based hedging scheme with a bond futures contract is that the duration DFof the futures contract may be hard to identify on account of
delivery options in the futures contract. For instance, in the Treasury bond futures contract on the CBoT, the short position may deliver any bond with at least 15 years to maturity (or first call) and any coupon. Using the duration of the standard bond in the contract is also problematic since the standard bond specifies only a coupon rate; its set of possible maturities remains large. One alternative in such a situation is to estimate the likely cheapest-to-deliver bond and use its duration.
6.9 Exercises
1. Explain the difference between the following terms: (a) Payoff to an FRA.(b) Price of an FRA. (c) Value of an FRA.
2. What characteristic of the eurodollar futures contract enabled it to overcome the settle- ment obstacles with its predecessors?
3. How are eurodollar futures quoted?
4. It is currently May. What is the relation between the observed eurodollar futures price of 96.32 for the November maturity and the rate of interest that is locked-in using the contract? Over what period does this rate apply?
5. What is the price tick in the eurodollar futures contract? To what price move does this correspond?
6. What are the gains or losses to a short position in a eurodollar futures contract from a 0.01 increase in the futures price?
7. You enter into a long eurodollar futures contract at a price of 94.59 and exit the contract a week later at a price of 94.23. What is your dollar gain or loss on this position? 8. What is the cheapest to deliver in a Treasury bond futures contract? Are there other
delivery options in this contract?
9. Describe the standard bond in each of the following contracts: (a) Treasury bond futures, (b) 10-year Treasury note futures, (c) 5-year Treasury note futures, and (d) Treasury bill futures.
10. Describe the conversion factor that applies if the delivered bond in a Treasury bond futures contract is different from the standard bond.
11. Explain the notion of duration of a bond. Under what conditions is this measure reason- ably accurate?
12. How does one measure the duration of a futures contract? That is, how is the duration of a futures contract related to the duration of the underlying bond?
13. Explain the principles involved in duration-based hedging. How does the computation of the hedge ratio here differ from that of the minimum-variance hedge computation? 14. On a $1,000,000 principal, 91-day investment, what is the interest payable if we use an
Actual/365 basis? What is the interest if the basis is Actual/360?
15. If the six-month interest rate is 6% and the one-year interest rate is 8%, what is the rate for an FRA over the period from six months to one year? Assume that the number of days up to six months is 182 and from six months to one year is 183.
16. If the three-month (91 days) Libor rate is 4% and the six-month (183 days) rate is 5%, what should be the 3× 6 FRA rate? If, at the end of the contract, the three-month Libor rate turns out to be 5%, what should the settlement amount be?
17. In Japan, if the three-month (91 days) interbank rate is 1% and the six-month (183 days) interbank rate is 0.25%, what is the 3× 6 FRA rate? Is this an acceptable rate? Why or why not?
18. If you expect interest rates to rise over the next three months and then fall over the three months succeeding that, what positions in FRAs would be appropriate to take? Would your answer change depending on the current shape of the forward curve?
19. A firm plans to borrow money over the next two half-year periods and is able to obtain a fixed-rate loan at 6% per annum. It can also borrow money at the floating rate of Libor + 0.5%. Libor is currently at 4%. If the 6× 12 FRA is at a rate of 6%, find the cheapest financing cost for the firm.
20. You enter into an FRA of notional 6 million to borrow on the three-month underlying Libor rate six months from now and lock in the rate of 6%. At the end of six months, if the underlying three-month rate is 6.6% over an actual period of 91 days, what is your payoff given that the payment is made right away? Recall that the ACT/360 convention applies.
21. You have entered into the 6× 9 FRA above at the rate of 6%. After three months, the FRA is now a 3× 6 FRA. If the three-month Libor rate is 5%, and the nine-month Libor rate is 7%, what is the current value of the FRA? Assume that the number of days from three to six months is 92.
22. Given a 3× 6 FRA with a rate of 10% and a time interval between three and six months of 92 days, plot the settlement amount if the three-month rate after three months ends up anywhere from 1% to 20%. Is your plot linear, convex, or concave? Why? If you are using FRAs to hedge your borrowing risk, does the shape of the payoff function cause you concern and why?
23. You anticipate a need to borrow USD 10 million in six-months’ time for a period of three months. You decide to hedge the risk of interest-rate changes using eurodollar futures contracts (=90 days). Describe the hedging strategy you would follow. What if you decided to use an FRA instead?
24. In the question above, suppose that the underlying Libor rate for three months after six months (as implied by the eurodollar futures contract price) is currently at 4%. Say
the underlying period is 91 days. Using the same numbers from the previous question and adjusting for tailing the hedge, how many futures contracts are needed? Assume fractional contracts are permitted.
25. Using the same numbers as in the previous two questions, compute the payoff after six months (i.e., at maturity) under (a) an FRA and (b) a eurodollar futures contract if the Libor rate at maturity is 5%. Also compute the payoffs if the Libor rate ends up at 3%. Comment on the difference in payoffs of the FRA versus the eurodollar futures. 26. The “standard bond” in the Treasury bond futures contract has a coupon of 6%. If,
instead, delivery is made of a 5% bond of maturity 18 years, what is the conversion factor for settlement of the contract?
27. Suppose we have a flat yield curve of 3%. What is the price of a Treasury bond of remain- ing maturity seven years that pays a coupon of 4%? (Coupons are paid semiannually.) What is the price of a six-month Treasury bond futures contract?
28. What is the price of a Treasury bill with a discount rate of 6% and maturity of 182 days? What is the price of a 91-day futures contract on the 91-day Treasury bill if the 91-day Treasury bill is trading at 95?
29. In the previous question, write down an expression for the payoff of the futures contract if after 91 days the discount rate of the remaining 91-day Treasury bill varies from 1% to 8%. Is the payoff function linear, convex, or concave? Why?
30. Suppose you own a zero-coupon bond with face value $3 million that matures in one year. The bond is priced off the continuously compounded zero-coupon rate that is currently at
r= 7%. Suppose you want to hedge the price of the bond six months from now using the
three-month eurodollar futures contract that expires in six-months’ time, assuming that the rate at that time remains unchanged for the shorter maturity. How many contracts will you need to trade to construct this hedge? Can you explain intuitively why this number is in the ballpark expected?
31. If we wish to hedge a bond that pays a cash flow of 2 million after six months and another cash flow of 102 million after twelve months, suggest a hedging scheme using eurodollar futures contracts. Assume that the bond is priced on a semiannual compounding basis and has a current yield to maturity of 4% per annum.
32. Qualitatively discuss how you would hedge a portfolio of bonds using eurodollar futures contracts.
33. (Difficult) Assume that the yield curve is flat at 6%. All bonds pay semiannually. Bond A has a coupon of 5.5% and a maturity of seven years. Bond B has a coupon of 6.2% and a maturity of five years. We wish to short bond B to offset the risk (duration-based hedging) of a long position in bond A. How many units of bond B do we need to short for every unit of bond A to achieve this?
34. Refer to the previous question. A futures contract on bond B trades as well. What is the price of the one-year bond futures contract on bond B? How many units of this contract do we need to short to offset a one-unit long position in bond A over the next year?
35. We are given a portfolio of bonds with value P = 100 and duration DP = 1. The
six-month Treasury bill future trades at price F1= 95 and duration DF1 = 0.4. Also,
the twelve-month Treasury bill future trades at price F2= 92 and duration DF2= 0.9.
Suggest a duration-based hedging strategy for portfolio P. State clearly the assumptions for your choice.
36. The following market-based FRA rates are provided.
Period (months) Forward Rates (%)
0–6 3.00
6–12 4.00
12–18 5.00
18–24 6.00
Answer the following questions:
(a) Find the price of a two-year maturity security with a coupon of 4.5%. (b) Find the price of a six-month bond future on this bond.
(c) What is the price of a twelve-month bond future on this bond? (d) Find the durations of all the three instruments above.
(e) If we invest $100 in the two-year bond, then how many units of the two futures contracts should we buy such that we have equal numbers of units in each contract, and we optimize our duration-based hedge?
(f) After setting up the hedge, the next instant, the entire forward curve shifts up by 1% at all maturities. What is the change in the value of the hedged portfolio? Is it zero? If not, explain the sign of the change.