, 27 (October 1960), 139-51.
McCabe, G.M., and C. T. Franckle, ‘The Effectiveness of Rolling the Hedge Forward in the TreasuryBill Futures Market,” Financial Management, \2 (Summer 1983), 21-29.
Miller, M., and C, Culp, “Risk Management Lessons from Metallgesellschaft,” Journal of AppliedCorporate Finance, 7, no. 4 (Winter 1995), 62-76.
Nikkhah, S., “How End Users Can Hedge Fuel Costs in Energy Markets,” Futures, October 1987, 66--67.
Stulz, R. M., “Optimal Hedging Policies,” Journal of Financial and Quantitative Analysis, 19 (June 1984),127-40.
mrecat ions Amp cnan
GUESTIONS AND PROBLEMS(ANSWERS IN N SOLUTIONS MANUAL)
 
 
4.1. Under what circumstances are (a) a short hedge and (b) a long hedge appropriate?4.2, Explain what is meant by basis risk when futures contracts are used for hedging.
4.3. Explain what is meant by a perfect hedge. Does a perfect hedge always lead to a better outcomethan an imperfect hedge? Explain your answer
imnper
4.4. Under what circumstances does a minimum variance hedge portfolio lead to no hedging at all?
4.5. Give three reasons that the treasurer of a company might not hedge the company’s exposure to aparticular risk.
4.6. Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65,the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and thecoefficient of correlation between the two changes is 0.8. What is the optimal hedge ratio for athree-month contract? What does it mean?
4.7. A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts onthe S&P 500 to hedge its risk. The index is currently standing at 1080, and each contract is forHedging Strategies Using Futures 89
48.
49,
4.10.
41
4.12.
4.14.
4.15.
4.16.
4.17.
4.20.
delivery of $250 times the index. What is the hedge that minimizes risk? What should the companydo if it wants to reduce the beta of the portfolio to 0.6?
In the Chicago Board of Trade’s corn futures contract, the following delivery months areavailable: March, May, July, September, and December. State the contract that should be usedfor hedging when the expiration of the hedge is in (a) June, (b) July, and (c) January.
Does a perfect hedge always succeed in locking in the current spot price of an asset for a futuretransaction? Explain your answer.
Explain why a short hedger’s position improves when the basis strengthens unexpectedly andworsens when the basis weakens unexpectedly.
. Imagine you are the treasurer of a Japanese company exporting electronic equipment to the
United States. Discuss how you would design a foreign exchange hedging strategy and thearguments you would use to sell the strategy to your fellow executives.
Suppose that in Example 4.2 of Section 4.3 the company decides to use a hedge ratio of 0.8. Howdoes the decision affect the way in which the hedge is implemented and the result?
. “If the minimum variance hedge ratio is calculated as 1.0, the hedge must be perfect.” Is this
statement true? Explain your answer.
“Tf there is no basis risk, the minimum variance hedge ratio is always 1.0.” Is this statement true?Explain your answer.
“When the convenience yield is high, long hedges are likely to be particularly attractive.’ Explainthis statement. Illustrate it with an example.
The standard deviation of monthly changes in the spot price of live cattle is 1.2 (in cents perpound). The standard deviation of monthly changes in the futures price of live cattle for theclosest contract is 1.4. The correlation between the fulures price chauges and the spot pricechanges is 0.7. It is now October 15. A beef producer is committed to purchasing 200,000 poundsof live cattle on November 15. The producer wants to use the December live-cattle futurescontracts to hedge its risk. Each contract is for the delivery of 40,000 pounds of cattle. What
strategy should the beef producer follow?
A corn farmer argues: ‘‘I do not use futures contracts for hedging. My real risk is not the price ofcorn. It is that my whole crop gets wiped out by the weather.” Discuss this viewpoint. Should thefarmer estimate his or her expected production of corn and hedge to try to lock in a price forexpected production? .
. On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per share.
The investor is interested in hedging against movements in the market over the next month anddecides to use the September Mini S&P 500 futures contract. The index is currently 1,500 and onecontract is for delivery of $50 times the index. The beta of the stock is 1.3. What strategy shouldthe investor follow?
. Suppose that in Example 4.4 the company decides to use a hedge ratio of 1.5. How does the
decision affect the way the hedge is implemented and the result?
A U.S. company is interested in using the futures contracts traded on the CME to hedge its
Australian dollar exposure. Define r as the interest rate (all maturities) on the U.S. dollar and r,¢
as the interest rate (all maturities) on the Australian dollar. Assume that r and r; are constant
and that the company uses a contract expiring at time T to hedge an exposure at time t (T > t).a. Using the results in Chapter 3, show that the optimal hedge ratio is
alts)90
CHAPTER 4
b. Show that, when ¢ is one day, the optimal hedge ratio is almost exactly So/Fy, where Sp isthe current spot price of the currency and Fy is the current futures price of the currency forthe contract maturing at time T.
c. Show that the company can take account of the daily settlement of futures contracts for ahedge that lasts longer than one day by adjusting the hedge ratio so that it always equalsthe spot price of the currency divided by the futures price of the currency.
. An airline executive has argued: “There is no point in our using oil futures. There is just as much
chance that the price of oil in the future will be less than the futures price as there is that it will begreater than this price.” Discuss this viewpoint.
ASSIGNMENT QUESTIONS
 
 
 
 
 
4.24,
. The following table gives data on monthly changes in the spot price and the futures price for a
certain commodity. Use the data to calculate a minimum variance hedge ratio.x
Spot price change +0.50 +0.61 —0.22 —0.35 +0.79Futures price change +0.56 +0.63 -0.12 —0.44 +0.60
Spot price change +0.04 +0.15 +0.70 —0.51 —0.41Futures price change —0.06 +0.01 +0.80 —0.56 —0.46
 
 
 
 
. It is July 16. A company has a portfolio of stocks worth $100 million..The beta of the portfolio
is 1.2. The company would like to use the CME December futures contract on the S&P 500 tochange the beta of the portfolio to 0.5 during the period July 16 to November 16. The index iscurrently 1,000, and each contract is on $250 times the index.a. What position shouid the company take?b. Suppose that the company changes its mind and decides to increase the beta of the portfoliofrom 1.2 to 1.5, What position in futures contracts should it take?
It is now October 2002. A company anticipates that it will purchase | million pounds of copper ineach of February 2003, August 2003, February 2004, and August 2004. The company has decidedto use the futures contracts traded in the COMEX division of the New York Mercantile Exchangeto hedge its risk. One contract is for the delivery of 25,000 pounds of copper. The initial margin is$2,000 per contract and the maintenance margin is $1,500 per contract. The company’s policy isto hedge 80% of its exposure. Contracts with maturities up to 13 months into the future areconsidered to have sufficient liquidity to meet the company’s needs. Devise a hedging strategy forthe company. Assume the market prices (in cents mee pound) today and at future dates are as
 
follows: \Ho asvwow  35353)9))8)8)9)9)9)9.0.0.0QDate Oct 2002 Feb 2003. Aug 2003 Feb 2004 Aug 2004Spot price 72.00 69.00 “68,00 77.00 88.00Mar 2003 futures price 72.30 69.10 a
Sept 2003 futures price 72.80 70.20 64.80 ae
Mar 2004 futures price 70.70 64.30 76.70 —
Sept 2004 futures price 64.20 76.50 88.20
What is the impact of the strategy you propose on the price the company pays for copper? What isthe initial margin requirement in October 2002? Is the company subject to any margin calls?Hedging Strategies Using Futures 91
4.25. A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is concernedabout the performance of the market over the next two months and plans to use three-monthfutures contracts on the S&P 500 to hedge the risk. The current level of the index is 1250, onecontract is on 250 times the index, the risk-free rate is 6% per annum, and the dividend yield onthe index is 3% per annuln.
a. What is the theoretical futures price for the three-month futures contract?
b. What position should the fund manager take to eliminate all exposure to the market over thenext two months?
c. Calculate the efféct of your strategy on the fund manager’s returns if the level of the marketin two months is 1,000, 1,100, 1,200, 1,300, and 1,400.92 CHAPTER 4
APPENDIX 4A
Proof of the Minimum Variance Hedge Ratio Formula
As in Section 4.4 we define:
5S: change in spot price, S, during a period of time equal to the life of the hedgedF: change in futures price, F, during a period of time equal to the life of the hedgeos: standard deviation of 6Sop: standard deviation of dF
p: coefficient of correlation between 6S and 6F
A: hedge ratio
When the hedger is long the asset and short futures, the change in the value of the hedger’s
position during the life of the hedge is6S ~hdF
for each unit of the asset held. For a long hedge the change ishdF -5SIn either case the variance, v, of the change in value of the hedged position is given by
v= 0% + h’ap — 2hposopso thaton = 2ho}y — 2pos0¢
Setting this equal to zero, and noting that 3°v/dh” is positive, we see that the value of h that
minimizes the variance is1 osA= p—Orgn LIA rT cP
CHAPTER 5
INTEREST RATE MARKETS
 
In this chapter we cover many different aspects of interest rate markets. We explain zero rates, paryields, forward rates, and the relationships between them. We cover day count conventions and theway in which the prices of Treasury bonds, corporate bonds, and Treasury bills are quoted in theUnited States. We show how forward rate agreements can be valued. We discuss the durationmeasure and explain how it can be used to quantify a company’s exposure to interest rates. We alsoconsider interest rate futures markets. We describe in some detail the popular Treasury bondfutures and Eurodollar futures contracts that trade in the United States, and we examine how theycan be used for duration-based hedging.
5.1 TYPES OF RATES
 
 
For any given currency many different types of interest rates are regularly quoted. These includemortgage rates, deposit rates, prime borrowing rates, and so on. The interest rate applicable in asituation depends on the credit risk. The higher the credit risk, the higher the interest rate. In thissection, we introduce three interest rates that are particularly important in options and futuresmarkets.
Treasury Rates
Treasury rates are the interest rates applicable to borrowing by a government in its own currency. Forexample, U.S. Treasury rates are the rates at which the U.S. government can borrow in U.S. dollars;Japanese Treasury rates are the rates at which the Japanese government can borrow in yen; and soon. It is usually assumed that there is no chance that a government will default on an obligationdenominated in its own currency. For this reason, Treasury rates are often termed risk-free rates.
LIBOR Rates
Large international banks actively trade with each other 1-month, 3-month, 6-month, and 12-monthdeposits denominated in all of the world’s major currencies. At a particular time Citibank mightquote a bid rate of 6.250% and an offer rate of 6.375% to other banks for six-month deposits inAustralian dollars. This means that it is prepared to pay 6.250% per annum on six-month depositsfrom another bank or advance deposits to another bank at the rate of 6.375% per annum. The bidrate is known as the London Interbank Bid Rate, or LIBID. The offer rate is known as the London
' The reason for this is that the government can always meet its obligation by printing more money.
9394 CHAPTER 5
Interbank Offer Rate, or LIBOR. The rates are determined in trading between banks and change aseconomic conditions change. If more banks want to borrow funds than lend funds, LIBID andLIBOR increase. If the reverse is true, they decrease.
LIBOR is a widely used reference rate. LIBOR rates are generally higher than the correspondingTreasury rates because they are not risk-free rates. There is always some chance (albeit small) thatthe bank borrowing the money will default. As mentioned in Chapter 3, banks and other largefinancial institutions tend to use the LIBOR rate rather than the Treasury rate as the “risk-freerate” when they evaluate derivatives transactions. The reason is that financial institutions investsurplus funds in the LIBOR market and borrow to meet their short-term funding requirements inthis market. They regard LIBOR as their opportunity cost of capital.
Repo Rate
Sometimes an investment dealer funds its trading activities with a repo or repurchase agreement.This is a contract where an investment dealer who owns securities agrees to sell them to anothercompany now and buy them back later at a slightly higher price. The company is providing a loanto the investment dealer. The difference between the price at which the securities are sold and theprice at which they are repurchased is the interest it earns. The interest rate is referred to as the reporate. If structured carefully, the loan involves very little credit risk. If the original owner of thesecurities does not honor the agreement, the lending company simply keeps the securities. If thelending company does not keep to its side of the agreement, the original owner of the securitieskeeps the cash.
The most common type of repo is an overnight repo, in which the agreement is renegotiated eachday. However, longer-term arrangements, known as term repos, are sometimes used.
5.2 ZERO RATES
 
The n-year zero rate (short for zero-coupon rate) is the rate of interest earned on an investment that
starts today and lasts for m vears. All the interest and srincipal is realized at the end of m yearsstarts today and tasts for 7 years. Au tne interest and principai is reauzed at the end Of # years.
There are no intermediate payments. The n-year zero rate is sometimes also referred to as then-year spot rate. Suppose the five-year Treasury zero rate with continuous compounding is quoted as5% per annum. This means that $100, if invested at the risk-free rate for five years, would grow to
100 x ¢9*5 — 128.40
Many of the interest rates we observe directly in the market are not pure zero rates. Consider a five-year government bond that provides a 6% coupon. The price of this bond does not exactlydetermine the five-year Treasury zero rate because some of the return on the bond is realized in theform of coupons prior to the end of year five. Later in this chapter we will discuss how we candetermine Treasury zero rates from the prices of traded instruments.
5.3 BOND PRICING
 
 
Most bonds provide coupons periodically. The owner also receives the principal or face value ofthe bond at maturity. The theoretical price of a bond can be calculated as the present value of allInterest Rate Markets 95
Table 5.1 Treasury zero rates
 
Maturity Zero rate (%)(years) (continuously compounded)0.5 5.01.0 5.8“1S 6.42.0 6.8
the cash flows that will be received by the owner of the bond using the appropriate zero rates asdiscount rates. Consider the situation where Treasury zero rates, measured with continuouscompounding, are as in Table 5.1 (we explain later how these can be calculated). Suppose thata two-year Treasury bond with a principal of $100 provides coupons at the rate of 6% per annumsemiannually. To calculate the present value of the first coupon of $3, we discount it at 5.0% forsix months; to calculate the present value of the second coupon of $3, we discount it at 5.8% forone year; and so on. The theoretical price of the bond is therefore
3e70:05%0.5 4 39-0.058%1.0 4 3,-0.064x1.5 4 193 971.068 «2.0 _ 9g 39or $98.39.
Bond Yield
The yield on a coupon-bearing bond is the discount rate that equates the cash flows on the bond toits market value. Suppose that the theoretical price of the bond we have been considering, $98.39,is also its market value (i.e., the market’s price of the bond is in exact agreement with the data inTable 5.1). If y is the yield on the bond, expressed with continuous compounding, we must have
Ber BET 4 BETS 4 103777? = 98.39
This equation can be solved using an iterative (“trial-and-error”) procedure to give y = 6.76%.
Par Yield
The par yield for a certain maturity is the coupon rate that causes the bond price to equal its facevalue. Usually the bond is assumed to provide semiannual coupons. Suppose that the coupon onatwo-year bond in our example is c per annum (or te per six months). Using the zero rates inTable 5.1, the value of the bond is. equal to its face value of 100 when
sens +5e note +e tmnt + (100 + 5)enos2e = 100
This equation can be solved in a straightforward way to give c = 6.87. The two-year par yieldis therefore 6.87% per annum with semiannual compounding (or 6.75% with continuouscompounding).
? One way of solving nonlinear equations of the form f(y) = 0, such as this one, is to use the Newton-Raphsonmethod. We start with an estimate yo of the solution and produce successively better estimates 9), y2, y3,... using theformula yj, = 3) — f(y) /f'(y), where f’(y) denotes the partial derivative of f with respect to y.96 CHAPTER 5
More generally, if d is the present value of $1 received at the maturity of the bond, A is the valueof an annuity that pays one dollar on each coupon payment date, and m is the number of couponpayments per year, the par yield c must satisfy
100 = A~ + 100diia
so that_ (100 — 100dync= A
“In our example, m = 2, d= 7 0068x2 _ 9.87284, andAm cH 00SH05 4 p-0.0581.0 | 4-0.064%1.5 4 4-0.068%2.0 _ 3 99997
The formula contirms that the par yield is 6.87% per annum with semiannual compounding.
5.4 DETERMINING TREASURY ZERO RATES
 
We now discuss how Treasury zero rates can be caiculated from the prices of instruments thattrade. One approach is known as the bootstrap method. To illustrate the nature of the method,consider the data in Table 5.2 on the prices of five bonds. Because the first three bonds pay nocoupons, the zero rates corresponding to the maturities of these bonds can be easily calculated.The three-month bond provides a return of 2.5 in three months on an initial investment of 97.5.With quarterly compounding, the three-month zero rate is (4 x 2.5)/97.5 = 10.256% per annum.Equation (3.3) shows that, when the rate is expressed with continuous compounding, it becomes
 
4in(1 +o
) = 0.10127
or 10.127% per annum. The six-month bond provides a return of 5.1 in six months on an initial
investment of 94.9, With semiannual compounding, the six-month rate is (2 x 5.1)/94.9 = 10.748%
per annum. Equation (3.3) shows that, when the rate is expressed with continuous compounding, it
becomes
0.107482
 
2in( 1+ ) = 0.10469
or 10.469% per annum. Similarly, the one-year rate with continuous compounding is
10a(t +5) = 0.10536or 10.536% per annum.
The fourth bond lasts 1.5 years. The payments are as follows:6months: $4
1 year: $4
1.5 years: $104
.
From our earlier calculations, we know that the discount rate for the payment at the end of sixInterest Rate Markets 97
Table 5.2. Data for bootstrap method
Bond principal Time to maturity Annual coupon* Bond price
 
(8) (years) (3) (3)
100 0.25 0 97.5100 0.50 0 94.9100 1.00 0 90.0100 1.50 8 96.0100 2.00 12 101.6
* Half the stated coupon is assumed to be paid every six months.
months is 10.469%, and the discount rate for the payment at the end of one year is 10.536%. Wealso know that the bond’s price, $96, must equal the present value of all the payments received bythe bondholder. Suppose the 1.5-year zero rate is denoted by R. It follows that
e70-104609%0.5 4 44-0.10536K1.0 4 1Q4gy-KXLS _ 96This reduces to
e 1k — 9.85196or_|n 0.85196
1.5
= 0.10681
The 1.5-year zero rate is therefore 10.681%. This is the only zero rate that is consistent with theé-month rate, l-ycar rate, and the data in Table 5.2.
The 2-year zero rate can be calculated similarly from the 6-month, 1-year, and |.5-year zero ratesand the information on the last bond in Table 5.2. If R is the two-year zero rate, then
670: 10469x0.5 + 6e70-10536x 1.0 + 6e7 0 10681 x15 +4 106e7**29 = 101.6
This gives R = 0.10808, or 10.808%.
The rates we have calculated are summarized in Table 5.3. A chart showing the zero rate as afunction of maturity is known as the zero curve. A common assumption is that the zero curve islinear between the points determined using the bootstrap method. (This means that the 1.25-yearzero rate is 0.5 x 10.536 + 0.5 x 10.681 = 10.6085% in our example.) It is also usually assumed
Table 5.3 Continuously compounded zerorates determined from data in Table 5.2
 
Maturity Zero rate (%)(years) (continuously compounded)0.25 10.1270.50 10.4691.00 10.5361.50 10.681
2.00 10.80898 CHAPTER 5
 
Interest rate (% per annum)
Maturity (years)
1 1 L 1 L L L L 1 pi»1 2
Figure 5.1 Zero rates given by the bootstrap method
that the zero curve is horizontal prior to the first point and horizontal beyond the last point.Figure 5.1 shows the zero curve for our data. By using longer maturity bonds, the zero curve wouldbe more accurately determined beyond two years.
In practice, we do not usually have bonds with maturities equal to exactly 1.5 years, 2 years,2.5 years, and so on. The approach often used by analysts is to interpolate between the bond pricedata before it is used to calculate the zero curve. For example, if they know that a 2.3-year bondwith a coupon of 6% sells for 98 and a 2.7-year bond with a coupon of 6.5% sells for 99, they
misht assume that 9 9 S_vear bond with a counen of 625% world cell far O28 5Mignt assume that 4 2.5-year bona With 4 Coupon OF 0.25 7o Would Séu Tor 70.5.
5.5 FORWARD RATES
 
Forward interest rates are the rates of interest implied by current zero rates for periods of time inthe future. To illustrate how they are calculated, we suppose that the zero rates are as shown in thesecond column of Table 5.4. The rates are assumed to be continuously compounded. Thus, the10% per annum rate for one year means that, in return for an investment of $100 today, theinvestor receives 100e"' = $110.52 in one year; the 10.5% per annum rate for two years meansthat, in return for an investment of $100 today, the investor receives 100e°!°*? = $123.37 in twoyears; and so on.
Table 5.4 Calculation of forward rates
 
Zero rate Forward rate
Year for an n-year investment for n-th year(n) (% per annum) (% per annum)
I 10.0
2 10.5 11.0
3 10.8 11.4
4 11.0 11.6
5 Wl 11.5Interest Rate Markets 99
 
The forward interest rate in Table 5.4 for year 2 is 11% per annum. This is the rate of interestthat is implied by the zero rates for the period of time between the end of the first year and the endof the second year. It can be calculated from the one-year zero interest rate of 10% per annum andthe two-year zero interest rate of 10.5% per annum. It is the rate of interest for year 2 that, whencombined with 10% per annum for year 1, gives 10.5% overall for the two years. To show that the
orrarm oun o% ner suppose a is ins oC rate o 9, ‘or o first voarcorrect answer is 11% per annum, suppose that $100 is invested. A rate of 10% for the first year
and 11% for the second year yields100e"!¢%!! = $123.37
at the end of the second year. A rate of 10.5% per annum for two years yields
100e" 105x2
which is aiso $123.37. This exampie illustrates the generai result that, when inierest raies arecontinuously compounded and rates in successive time periods are combined, the overall equival-ent rate is simply the average rate during the whole period. (In our example, 10% for the first yearand 11% for the second year average to 10.5% for the two years.) The result is only approximatelytrue when the rates are not continuously compounded.
The forward rate for the year 3 is the rate of interest that is implied by a 10.5% per annum two-year zero rate and a 10.8% per annum three-year zero rate. It is 11.4% per annum. The reason isthat an investment for two years at 10.5% per annum combined with an investment for one year at11.4% per annum gives an overall average return for the three years of 10.8% per annum. Theother forward rates can be calculated similarly and are shown in the third column of the table. Ingeneral, if R, and R, are the zero rates for maturities T, and 7, respectively, and Rr is the forwardinterest rate for the period of time between 7, and 7, then
_— BT, - RT;
RF Th
(5.1)To illustrate this formula, consider the calculation of the year 4 forward rate from the data inTable 5.4: 7; = 3, T, = 4, Ry = 0.108, and R, = 0.11, and the formula gives R, = 0.116.
Assuming that the zero rates for borrowing and investing are the same (which is not toounreasonable for a large financial institution), an investor can lock in the forward rate for a futuretime period. Suppose, for example, that the zero rates are as in Table