can be shown that the approximate relationshipin equation (5.11) becomes (see Problem 5.31)
, BD by
bB=l+y
More generally, if y is expressed with a compounding frequency of m times per year,
BDSssB= “TomyymA variable D* defined byDD* =——_l+y/m
is sometimes referred to as the bond’s modified duration. It allows the duration relationship to besimplified to8B = — BD" dy (5.13)
when y is expressed with a compounding frequency of m times per year. The following exampleinvestigates the accuracy of the modified duration relationship.
Example 5.7. The bond in Table 5.7 has a price of 94.213 and a duration of 2.653. The yield,expressed with semiannual compounding, is 12.3673%. The modified duration, D*, is
2.653~T+$0.123673/2
* = 2,499From equation (5.13),5B = —94.213 x 2.4985 dyor6B = —235.39 byInterest Rate Markets 115
When the yield (semiannually compounded) increases by 10 basis points (= 0.1%), dy = +0.001. Theduration relationship predicts that we expect 5B to be —235.39 x 0.001 = —0.235, so that the bondprice to goes down to 94.213 — 0.235 = 93.978. How accurate is this? When the bond yield(semiannually compounded) increases by 10 basis points to 12.4673% (or to 12.0941% withcontinuous compounding), an exact calculation similar to that in the previous example shows thatthe bond price becomes 93.978. This shows that the modified duration calculation gives good accuracy.
Bond Portfolios
The duration, D, of a bond portfolio can be defined as a weighted average of the durations of theindividual bonds in the portfolio, with the weights being proportional to the bond prices.Equations (5.11) to (5.13) then apply with B being defined as the value of the bond portfolio.They estimate the change in the value of the bond portfolio for a particular change dy in the yieldsof all the bonds.
It is important to realize that, when duration is used for bond portfolios, there is an implicitassumption {hat the yields of all bonds will change by the same amount. When the bonds havewidely differing maturities, this happens only when there is a parallel shift in the zero-coupon yieldcurve. We should therefore interpret equations (5.11) to (5.13) as providing estimates of the impacton the price of a bond portfolio of a parallel shift, dy, in the zero curve.
The duration relationship applies only to small changes in yields. This is illustrated in Figure 5.3,which shows the relationship between the percentage change in value and change in yield for twobond portfolios having the same duration. The gradients of the two curves are the same at the origin.
This meansthat both bond nortfolics changein value by the same sercentace for small vield changes2S Means tdat COTA CONG poruouos Change in yaiue oy tne Same percentage tor SMa yieid Changes
and is consistent with equation (5.12). For large yield changes, the portfolios behave differently.Portfolio X has more curvature in its relationship with yields than portfolio Y. A factor known asconvexity measures this curvature and can be used to improve the relationship in equation (5.12).
 
 
Figure 5.3 Two bond portfolios with the same duration116 CHAPTER 5
Hedging Portfolios of Assets and Liabilities
Financial institutions frequently attempt to hedge themselves against interest rate risk by ensuringthat the average duration of their assets equals the average duration of their liabilities. (Theliabilities can be regarded as short positions in bonds.) This strategy is known as duration matchingor portfolio immunization. When implemented, it ensures that a small parallel shift in interest rateswill have litile effect on the value of the portfolio of asseis and liabilities. The gain (loss) on theassets should offset the loss (gain) on the liabilities.
Duration matching does not immunize a portfolio against nonparallel shifts in the zero curve.This is a weakness of the approach. In practice, short-term rates are usually more volatile than, andare not perfectly correlated with, long-term rates. Sometimes it even happens that short- and long-term rates move in opposite directions to each other. Financial institutions often try to allow fornonparallel shifts by dividing the zero-coupon yield curve up into segments and ensuring that theyaré hedged agaist a movement in each segment. Suppose that ihe ith segment is the part of thezero-coupon yield curve between maturities t; and t;,,. A financial institution would examine theeffect of a small increase dy in all the zero rates with maturities between f; and t;,, while keepingthe rest of the zero-coupon yield curve unchanged. If the exposure were unacceptable, furthertrades would be undertaken in carefully selected instruments to reduce it. In the context of a bankmanaging a portfolio of assets and liabilities, this approach is sometimes referred to as GAPmanagement.
5.14 DURATION-BASED HEDGING STRATEGIES
 
Consider the situation where a position in an interest-rate-dependent asset such as a bond portfolioor a money market security is being hedged using an interest rate futures contract. Define:
Fo: Contract price for the interest rate futures contract
Dy: Duration of the asset underlying the futures contract at the maturity of the futurescontract
P: Forward value of the portfolio being hedged at the maturity of the hedge (in practice thisis usually assumed to be the same as the value of the portfolio today)
Dp: Duration of the portfolio at the maturity of the hedge
If we assume that the change in the yield, Sy, is the same for all maturities, which means that onlyparallel shifts in the yield curve can occur, it is approximately true that
8P = —PDp byTo a reasonable approximation, it is also true that8Fo = —FcoDr by
The number of contracts required to hedge against an uncertain Sy is therefore
 
Nt= PDp
= iDe (5.14)Interest Rate Markets 117
 
This is the duration-based hedge ratio. It is sometimes also called the price sensitivity hedge ratio.’Using it has the effect of making the duration of the entire position zero.
When the hedging instrument is a Treasury bond futures contract, the hedger must base Dr onan assumption that one particular bond will be delivered. This means that the hedger must estimatewhich of the available bonds is likely to be cheapest to deliver at the time the hedge is put in place.If, subsequently, the interest rate environment changes so that it looks as though a different bondwill be cheapest to deliver, the hedge has to be adjusted, and its performance may be worse thananticipated.
When hedges are constructed using interest rate futures, it is important to bear in mind thatinterest rates and futures prices move in opposite directions. When interest rates go up, an interestrate futures price goes down. When interest rates go down, the reverse happens, and the interestrate futures price goes up. Thus, a company in a position to lose money if interest rates dropshould hedge by taking a long futures position. Similarly, a company in a position to lose money ifinterest rates rise should hedge by taking a short futures position.
The hedger tries to choose the futures contract so that the duration of the underlying asset is asclose as possible to the duration of the asset being hedged. Eurodollar futures tend to be used forexposures to short-term interest rates, whereas Treasury bond and Treasury note futures contractsare used for exposures to longer-term rates.
Example5.8 It is August 2 and a fund manager with $10 million invested in government bonds isconcerned that interest rates are expected to be highly volatile over the next three months. The fundmanager decides to use the December T-bond futures contract to hedge the value of the portfolio.The current futures price is 93-02, or 93.0625. Because each contract is for the delivery of $100,000‘face value of bonds, the futures contract price is $93,062.50.
We suppose that the duration of the hand portfolio in three months will he 6.80 years. Thecheapest-to-deliver bond in the T-bond contract is expected to be a 20-year 12% per annum couponbond. The yield on this bond is currently 8.80% per annum, and the duration will be 9.20 years atmaturity of the futures contract.
The fund manager requires a short position in T-bond futures to hedge the bond portfolio. If interestrates go up, a gain will he made on the short futures positian and a loss will he made on the handportfolio. If interest rates decrease, a loss will be made on the short position, but there will be a gain onthe bond portfolio. The number of bond futures contracts that should be shorted can be calculatedfrom equation (5.14) as
10,000,000 6.80
93,062.50 * 9.20 ~ 194?
Rounding to the nearest whole number, the portfolio manager should short 79 contracts.
Convexity
For very small parallel shifts in the yield curve, the change in value of a portfolio depends solely onits duration. When moderate or large parallel shifts in interest rates are considered, a factor knownas convexity is important. Figure 5.3 shows the relationship between the percentage change in valueand change in yield, éy, for two portfolios having the same duration. The gradients of the twocurves are the same when dy = 0 (and equal to the duration of the portfolio). This means that bothportfolios change in value by the same percentage for small yield changes which is consistent with
'0 For a more detailed discussion of equation (5.14), see R. Rendleman, “Duration-Based Hedging with TreasuryBond Futures,” Journal of Fixed Income, 9, no. | (June 1999), 84-91.118 CHAPTER 5
equation (5.11). For large interest rate changes, the portfolios behave differently. Portfolio X hasmore convexity (or curvature) than portfolio Y. Its value increases by a greater percentage amountthan that of portfolio Y when yields decline, and its yalue decreases by less than that of portfolio Ywhen yields increase.
A measure of convexity is
cal eB Dice ™B ay" B
The convexity of a bond portfolio tends to be greatest when the portfolio provides payments evenlyover a long period of time. It is least when the payments are concentrated around one particularpoint in time.
When convexity is taken into account, it can be shown that equation (5.11) becomes
5By= TD by + Lo(yy
By matching convexity as well as duration, a company can make itself immune to relatively largeparallel shift in the zero curve. However, it is still exposed to nonparallel shifts.
SUMMARY
 
Many different interest rates are quoted in financial markets and calculated by analysts. The n-yearzero rate or n-year spot rate is the rate applicable to an investment lasting for n years when all ofthe return is realized at the end. Forward rates are rates applicable to futures periods of timeimplied by today’s zero rates. The par yield on a bond of a certain maturity is the coupon rate thatcauses the bond to sell for its par value.
Treasury rates should be distinguished from LIBOR rates. Treasury rates are the rates at whichthe government of the country borrows. LIBOR rates are the rates at which one large internationalbank is prepared to lend to another large international bank. LIBOR rates are often used as therisk-free rates when derivatives are valued.
Two very popular interest rate contracts are the Treasury bond and Eurodollar futures contractsthat trade in the United States. In the Treasury bond futures contracts, the party with the shortposition has a number of interesting delivery options:
1. Delivery can be made on any day during the delivery month.2. There are a number of alternative bonds that can be delivered.
3. On any day during the delivery month, the notice of intention to deliver at the 2:00 p.m.settlement price can be made any time up to 8:00 p.m.
These options all tend to reduce the futures price.
The Eurodollar futures contract is a contract on the three-month rate starting on the thirdWednesday of the delivery month. Eurodollar futures are frequently used to estimate LIBORforward rates for the purpose of constructing a LIBOR zero curve. When long-dated contracts areused in this way, it is important to make what is termed a convexity adjustment to allow for themarking to market in the futures contract.
The concept of duration is important in hedging interest rate risk. Duration measures how longon average an investor has to wait before receiving payments. It is a weighted average of the timesInterest Rate Markets 119
until payments are received, with the weight for a particular payment time being proportional tothe present value of the payment.A key result underlying the duration-based hedging scheme described in this chapter is
8B = —BD by
where B is a bond price, D is its duration, dy is a small change in its yield (continuouslycompounded), and 5B is the resultant small change in B. The equation enables a hedger to assessthe sensitivity of a bond price to small changes in its yield. It also enables the hedger to assess thesensitivity of an interest rate futures price to small changes in the yield of the underlying bond. Ifthe hedger is prepared to assume that dy is the same for all bonds, the result enables the hedger tocalculate the number of futures contracts necessary to protect a bond or bond portfolio againstsmall changes in interest rates.
The key assumption underlying the duration-based hedging scheme is that all interest rateschange by the same amount. This means that only parallel shifts in the term structure are allowedfor. In practice, short-term interest rates are generally more volatile than are long-term interestrates, and hedge performance is liable to be poor if the duration of the bond underlying the futures
enat anata At aim the duratinn af the aece theta hedge
contract differs marxeary from the duration of the asset being hedged.
SUGGESTIONS FOR FURTHER READING
 
Allen, S. L., and A. D. Kleinstein, Valuing Fixed-Income Investments and Derivative Securities, NewYork Institute of Finance, New York, 1991.
Fabozzi, F. J., Fixed-Income Mathematics: Analytical and Statistical Techniques, McGraw-Hill, NewYork, 1996.
Fabozzi, F. J., Duration, Convexity, and Other Bond Risk Measures, Frank J. Fabozzi Assoc., 1999,
Figlewski, S., Hedging with Financial Futures for Institutional Investors, Ballinger, Cambridge, MA,1986.
Gay, G. D., R. W. Kolb, and R. Chiang, “Interest Rate Hedging: An Empirical Test of AlternativeStrategies,” Journal of Financial Research, 6 (Fall 1983), 187-97.
Klemkosky, R. C., and D. J. Lasser, “An Efficiency Analysis of the T-bond Futures Market,” Journalof Futures Markets, 5 (1985), 607-20.
Kolb, R. W., and R. Chiang, “Improving Hedging Performance Using Interest Rate Futures,”Financial Management, 10 (Fall 1981), 72-79.
Resnick, B. G. “The Relationship between Futures Prices for U.S. Treasury Bonds,” Review ofResearch in Futures Markets, 3 (1984), 88-104.
Resnick, B. G., and E. Hennigar, “The Relationship between Futures and Cash Prices for U.S.Treasury Bonds,” Review of Research in Futures Markets, 2 (1983), 282-99.
Senchak, A. J., and J. C. Easterwood, ‘Cross Hedging CDs with Treasury Bill Futures,” Journal ofFutures Markets, 3 (1983), 429-38.
Veit, W. T., and W. W. Reiff, “Commercial Banks and Interest Rate Futures: A Hedging Survey,”Journal of Futures Markets, 3 (1983), 283-93.120 CHAPTER 5
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
 
5.1. Suppose that zero interest rates with continuous compounding are as follows:
 
Maturity (years) Rate (% per annum)1 8.0153 7.24 7.05 69
Calculate forward interest rates for the second, third, fourth, and fifth years.
§.2.. The term structure is upward sloping. Put the following in order of magnitude:a. The five-year zero rateb. The yield on a five-year coupon-bearing bondc. The forward rate corresponding to the period between 5 and 5.25 years in the futureWhat is the answer io this question when the ierm structure is downward sioping?
5.3. The six-month and one-year zero rates are both 10% per annum. For a bond that lasts 18 monthsand pays a coupon of 8% per annum (with a coupon payment having just been madc), the yield is10.4% per annum. What is the bond’s price? What is the 18-month zero rate? All rates are quotedwith semiannual compounding.
5.4. It is January 9, 2003. The price of a Treasury bond with a 12% coupon that matures on October 12,2009, is quoted as 102-07. What is the cash price?
5.5. The price of a 90-day Treasury bill is quoted as 10.00. What continuously compounded return (onan actual/365 basis) does an investor earn on the Treasury bill for the 90-day period?
5.6. What assumptions does a duration-based hedging scheme make about the way in which the termstructure of interest rates moves?
5.7. It is January 30. You are managing a bond portfolio worth $6 million. The duration of theportfolio in six months will be 8.2 years. The September Treasury bond futures price is currently108-15, and the cheapest-to-deliver bond will have a duration of 7.6 years in September. Howshould you hedge against changes in interest rates over the next six months?
5.8. Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 4%, 4.2%,4.4%, 4.6%, and 4.8% per annum with continuous compounding respectively. Estimate the cashprice of a bond with a face value of 100 that will mature in 30 months pays a coupon of 4% perannum semiannually.
5.9. A three-year bond provides a coupon of 8% semiannually and has a cash price of 104. What is thebond yield?
5.10. Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6.5%, and7%, respectively. What is the two-year par yield?
5.11. The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0. A 1.5-year bond thatwill pay coupons of $4 every six months currently sells for $94.84. A two-year bond that will paycoupons of $5 every six months currently sells for $97.12. Calculate the 6-month, I-year, 1.5-year,and 2-year zero rates.Interest Rate Markets 121
5.12. Suppose that zero interest rates with continuous compounding are as follows:
Maturity (years) Rate (% per annum)1 12.02 13.03 13.74 14.25 14.5
Calculate forward interest rates for the second, third, fourth, and fifth years.
5.13. Suppose that zero interest rates with continuous compounding are as follows:
 
Maturity (months) Rate (% per annum)3 8.06 8.29 8.412 8.515 8.618 8.7
Calculate forward interest rates for the second, third, fourth, fifth, and sixth quarters.
5.14. A 10-year, 8% coupon bond currently sells for $90. A 10-year, 4% coupon bond currently sellsfor $80, What is the 10-year zero rate? (Hint: Consider taking a long position in two of the 4%coupon bonds and a short position in one of the 8% coupon bonds.)
5.15. Explain carefully why liquidity preference theory is consistent with the observation that the termstructure tends to be upward sloping more often than it is downward sloping.
5.16. It is May 5, 2003. The quoted price of a government bond with a 12% coupon that matures onJuly 27, 2011, is 110-17. What is the cash price?
5.17. Suppose that the Treasury bond futures price is 101-12. Which of the following four bonds ischeapest to deliver?
 
Bond Price Conversion factor1 125-05 1.21312 142-15 1.37923 115-31 1.11494 144-02 1.4026
5.18. It is July 30, 2002. The cheapest-to-deliver bond in a September 2002 Treasury bond futurescontract is a 13% coupon bond, and delivery is expected to be made on September 30, 2002.Coupon payments on the bond are made on February 4 and August 4 each year. The termstructure is flat, and the rate of interest with semiannual compounding is 12% per annum. Theconversion factor for the bond is 1.5. The current quoted bond price is $110. Calculate the quotedfutures price for the contract.
5.19. An investor is looking for arbitrage opportunities in the Treasury bond futures market. Whatcomplications are created by the fact that the party with a short position can choose to deliver anybond with a maturity of over 15 years?122
5.20.
5.26.5.27.
5.29.
5.31.5.32.
CHAPTER 5
Assuming that zero rates are as in Problem 5.12, what is the value of an FRA that enables theholder to earn 9.5% for a three-month period starting in one year on a principal of $1,000,000?The interest rate is expressed with quarterly compounding.
. Suppose that the nine-month LIBOR interest rate is 8% per annum and the six-month LIBOR
interest rate is 7.5% per annum (both with continuous compounding). Estimate the three-monthEurodollar futures price quote for a contract maturing in six months.
. A five-year bond with a yield of 11% (continuously compounded) pays an 8% coupon at the end
of each year.a. What is the bond’s price?b. What is the bond’s duration?c. Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in its yield.d. Recalculate the bond’s price on the basis of a 10.8% per annum yield and verify that theresult is in agreement with your answer to (c).
. Suppose that a bond portfolio with a duration of 12 years is hedged using a futures contract in
which the underlying asset has a duration of four years. What is likely to be the impact on thehedge of the fact that the 12-year rate is less volatile than the four-year rate?
. Suppose that it is February 20 and a treasurer realizes that on July 17 the company will have to
issue $5 million of commercial paper with a maturity of 180 days. If the paper were issuedtoday, the company would realize $4,820,000. (In other words, the company would receive$4,820,000 for its paper and have to redeem it at $5,000,000 in 180 days’ time.) The SeptemberEurodollar futures price is quoted as 92.00. How should the treasurer hedge the company’sexposure?
. On August I a portfolio manager has a bond portfolio worth $10 million. The duration of the
portfolio in October will be 7.1 years. The December Treasury bond futures price is currently91-12 and the cheapest-to-deliver bond will have a duration of 8.8 years at maturity. Howshould the portfolio manager immunize the portfolio against changes in interest rates over thenext two months?
How can the portfolio manager change the duration of the portfolio to 3.0 years in Problem 5.25?Between February 28, 2002, and March 1, 2002, you have a choice between owning a governmentbond paying a 10% coupon and a corporate bond paying a 10% coupon. Consider carefully theday count conventions discussed in this chapter and decide which of the two bonds you wouldprefer to own. Ignore the risk of default.
. Suppose that a Eurodollar futures quote is 88 for a contract maturing in 60 days. What is the
LIBOR forward rate for the 60- to 150-day period? Ignore the difference between futures andforwards for the purposes of this question.
“When the zero curve is upward sloping, the zero rate for a particular maturity is greater than thepar yield for that maturity. When the zero curve is downward sloping the reverse is true.” Explainwhy this is so.
. The three-month Eurodollar futures price for a contract maturing in six years is quoted as 95.20.
The standard deviation of the change in the short-term interest rate in one year is 1.1%.Estimate the forward LIBOR interest rate for the period between 6.00 and 6.25 years in thefuture.
Prove equation (5.13).
Explain why the forward interest rate is less than the corresponding futures interest rate calculatedfrom a Eurodollar futures contract.Interest Rate Markets 123
ASSIGNMENT QUESTIONS
 
 
5,33. Assume that a hank can borrow or lend money at the same interest rate in the LIBOR market.The 91-day rate is 10% per annum, and the 182-day rate is 10.2% per annum, both expressed withcontinuous compounding. The Eurodollar futures price for a contract maturing in 91 days isquoted as 89.5. What arbitrage opportunities are open to the bank?
5.34. A Canadian company wishes to create a Canadian LIBOR futures contract from a US.Eurodollar futures contract and forward contracts on foreign exchange. Using an example,explain how the company should proceed. For the purposes of this problem, assume that a futurescontract is the same as a forward contract.
5.35. Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-yearzero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year zero-couponbond with a face value of $5,000. The current yield on all bonds is 10% per annum.
a.b.
a
5.36, The
pp
5.37. It isa,
b.
Show that both portfolios have the same duration.Show that the percentage changes in the values of the two portfolios for a 0.1% per annumincrease in yields are the same.
. What are the percentage changes in the values of the two portfolios for a 5% per annum
increase in yields?
following table gives the prices of bonds:Bond principal Time to maturity Annual coupon* Bond price(3) (years) (8) (3)100 0.50 0.0 98100 1.00 0.0 95100 1.50 6.2 101100 2.00 8.0 104
* Half the stated coupon is assumed to be paid every six months.
. Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months.. What are the forward rates for the periods: 6 months to 12 months, 12 months to 18 months,
18 months to 24 months?
. What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds that provide
semiannual coupon payments?
. Estimate the price and yield of a two-year bond providing a semiannual coupon of 7% per
annum.
June 25, 2002. The futures price for the June 2002 CBOT bond futures contract is 118-23.Calculate the conversion factor for a bond maturing on January |, 2018, paying a couponof 10%.
Calculate the conversion factor for a bond maturing on October 1, 2023, paying a couponof 7%.
. Suppose that the quoted prices of the bonds in (a) and (b) are 169.00 and 136.00,
respectively. Which bond is cheaper to deliver?
. Assuming that the cheapest-to-deliver bond is actually delivered, what is the cash price
received for the bond?
5.38. A portfolio manager plans to use a Treasury bond futures contract to hedge a bond portfolio overthe next three months. The portfolio is worth $100 million and will have a duration of 4.0 years in124 CHAPTER 5
three months. The futures price is 122, and each futures contract is on $100,000 of bonds. Thebond that is expected to be cheapest to deliver will have a duration of 9.0 years at the maturity ofthe futures contract. What position in futures contracts is required?a. What adjustments to the hedge are necessary if afler one month the bond that is expected tobe cheapest to deliver changes to one with a duration of seven years?b. Suppose that all rates increase over the three months, but long-term rates increase less thanshort-term and medium-term rates. What is the effect of this on the performance of thehedge? 
CHAPTER 6SWAPS
A swap is an agreement between two companies to exchange cash flows in the future. The agreementdefines the dates when the cash flows are to be paid and the way in which they are to be calculated.Usually the calculation of the cash flows involves the future values of one or more market variables.
A forward contract can be viewed as a simple example of a swap. Suppose it is March I, 2002,and a company enters into a forward contract to buy 100 ounces of gold for $300 per ounce in oneyear. The company can sell the gold in one year as soon as it is received. The forward contract istherefore equivalent to a swap where the company agrees that