7 + 08 Bept S69 + 004 437 9832 145} 93.53 + OF Dec 5 seve 9462 + 0.04 94.33 94.24 100B49 + 8 Est vol 29,985; vol Wed 25.957; open inf 239,338, +9,450.
 
 
(continued on next page)106 CHAPTER 5
 
Table 5.5 (continued)
 
10 Yr. Canadian Govt. Bonds {ME}-¢s 100,900 Sept 99.86 99.88 99.87 9987 - 001 9988 96.01 78,676Mar 10550 + 0.10 1m 101.60 5,505 Dec «99.85 9985 99.84 9984 - O01 9085 97.84 51494me 105.10 105.33 10494 10581 + 0.10 10540 108.30 6f.851 MrO2 98.88 39.65 99.84 9984 - 0.0} 9985 9617 37.581Est vol 2,838; vol Wed 3,951; open int 67,356, -723. dune 99.85 99.85 99.83 9983 - 0.02 9985 96.11 3390710 Yr. Euro Notional Bond (MATIF)-Euros 100,000 Sept 99.79 9879 9077 9976 - OOF 9979 96.35 14517Mar 90.32 90.40 9013 9030 + 005 9055 8533 175,083 Dec 99.70 99.70 9988 967 - GOt 970 %26 3264dune 90.42 90.70 90.26 90.45 + 0.06 9070 6919 26,706 MrQ3 so os wo» 9960 — OOF 9954 945 = 2102Est vol 224,549; vol Wed 218,960; open int 167.299, +6,100. dune 985 954 9640 21703 Month Eurlbor (MATIF)-Euros 1,000,000 Sept, OF ~. 9942 634 5010Mar we 95.24 0.02 9695 9463 6,063 Est vol 25,077; val Wed 34,493; fpen int 439,071, +82.
0.02 9665 9455 3.357 5 Yr, German Euro-Govt. Bond
03 9675 9447 = 2021 fre by UREX)-Euro 100,000; pts of 100%
03 9555 9437 2635 lune 40660 106.75 10653 10674 + O08 10675 10350 465,266
03 9648 9445 2617 Set . 10702 4 0.08 10888 teat 9.638
103 96.29 94.35 vol Thu 456.995: opan int 479,927, + iBEmr.
04 96.21 94.27 475 10 Yr, German Euro-Govt, Bon
48 95.97 94.10 330 (EURO. BUND) EUREXIE Euro 100, 000; pts of 100%
June 10946 1 096d + O08 109.70 705.00 687,796E)-as100,000 Sat 109.64 jer tthag 109.70 + OOF 10970 107.96 5747543 008 95.67 94.25 0 Thu 706,063; open ink 693.543, cats9550 95.8? 9548 9555 + 005 9662 9498 231,145 2. Yr, German Euro-Govt. Bon
 
iP
2Est vol 399,603; vol Wed 185,037; open int 231,145, — 295,318. ({EURO-SCHATZ) {EUAEX} Euro. 100,000; pts of 100%Euro-Yen (SGX)-Yen 100,000,000 pts Of 100% dune 103.03 403.08 102.97 (eer t ‘a93 “0822 mn seMar 8, 79 B77 BR - Got ge ge 96.662 Sept osdune 99.96 9966 90.67 99.67 we 17 148,646 vol Thu 378 open int sia Portes
 
 
Source: Reprinted by permission of Dow Jones, Inc., via copyright Clearance Center, Inc.© 2001 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
that has more than 15 years to maturity on the first day of the delivery month and that is notcallable within 15 years from that day can be delivered. As will be explained later, the CBOT hasdeveloped a procedure for adjusting the price received by the party with the short positionaccording to the particular bond delivered.
The Treasury note and five-year Treasury ute futures contract in the United States are alsoactively traded. In the Treasury note futures contract, any government bond (or note) with amaturity between 64 and 10 years can be delivered. In the five-year Treasury note futures contract,any of the four most recently auctioned Treasury notes can be delivered.
The remaining discussion in this section focuses on Treasury bond futures. Many other bondfutures contracts traded in the United States and the rest of the world are designed in a similar wayto CBOT Treasury bond futures, so that many of the points we will make are applicable to thesecontracts as well.
Quotes
Treasury bond futures prices are quoted in the same way as the Treasury bond prices themselves(see Section 5.9). Table 5.5 shows that the settlement price on March 15, 2001 for the June 2001contract was 106-04, or 106 + One contract involves the delivery of $100,000 face value of thebond. Thus, a $1 change in the quoted futures price would lead to a $1,000 change in the value ofthe futures contract. Delivery can take place at any time during the delivery month.
Conversion Factors
As mentioned, the Treasury bond futures contract allows the party with the short position tochoose to deliver any bond that has a maturity of more than 15 years and that is not callablewithin 15 years. When a particular bond is delivered, a parameter known as its conversion factordefines the price received by the party with the short position. The quoted price applicable to thedelivery is the product of the conversion factor and the quoted futures price. Taking accruedinterest into account, as described in Section 5.9, the cash received for each $100 face value ofInterest Rate Markets 107
bond delivered is(Quoted futures price x Conversion factor) + Accrued interest
Each contract is for the delivery of $100,000 face value of bonds. Suppose the quoted futures priceis 90-00, the conversion factor for the bond delivered is 1.3800, and the accrucd intcrest on thisbond at the time of delivery is $3 per $100 face value. The cash received by the party with the shortposition (and paid by the party with the long position) is then
(1.3800 x 90.00) + 3.00 = $127.20
per $100 face value. A party with the short position in one contract would deliver bonds with facevalue of $100,000 and receive $127,200.
The conversion factor for a bond is equal to the value of the bond per dollar of principal on thefirst day of the delivery month on the assumption that the interest rate for all maturities equals6% per annum (with semiannual compounding).° The bond maturity and the times to the couponpayment dates are rounded down to the nearest three months for the purposes of the calculation.The practice enables the CBOT to produce comprehensive tables. If, after rounding, the bond lastsfor an exact number of six-month periods, the first coupon is assumed to be paid in six months. If,after rounding, the bond does not last for an exact number of six-month periods (i.e., there is anextra three months), the first coupon is assumed to be paid after three months and accrued interestis subtracted.
As a first example of these rules, consider a 10% coupon bond with 20 years and two months tomaturity. For the purposes of calculating the conversion factor, the bond is assumed to haveexactly 20 years to maturity. The first coupon payment is assumed to he made after six monthsCoupon payments are then assumed to be made at six-month intervals until the end of the 20 yearswhen the principal payment is made. Assume that the face value is $100. When the discount rate is6% per annum with semiannual compounding (or 3% per six months), the value of the bond is
 
 
40. « aA5 100- = $146.23» 1.03 + 1.03% $
Dividing by the face value gives a credit conversion factor of 1.4623.
As a second example of the rules, consider an 8% coupon bond with 18 years and four monthsto maturity. For the purposes of calculating the conversion factor, the bond is assumed to haveexactly 18 years and three months to maturity. Discounting all the payments back to a point intime three months from today at 6% per annum (compounded semiannually) gives a value of
4 100— +
1.037 7,933 = 9125-83
i Ms5
The interest rate for a three-month period is 1.03 — 1, or 1.4889%. Hence, discounting back tothe present gives the bond’s value as 125.83/1.014889 = $123.99. Subtracting the accrued interestof 2.0, this becomes $121.99. The conversion factor is therefore 1.2199.
® For contracts with maturities prior to March 2000, this interest rate was 8% in the CBOT conversion factorcalculation.108 CHAPTER 5
Cheapest-to-Deliver Bond
At any given time during the delivery month, there are many bonds that can be delivered in theCBOT Treasury bond futures contract. These vary widely as far as coupon and maturity areconcerned. The party with the short position can choose which of the available bonds is “‘cheapest”to deliver. Because the party with the short position receives
(Quoted futures price x Conversion factor) + Accrued interestand the cost of purchasing a bond isQuoted price + Accrued interestthe cheapest-to-deliver bond is the one for whichQuoted price — (Quoted futures price x Conversion factor)
is least. Once the party with the short position has decided to deliver, it can determine thecheapest-to-deliver bond by examining each of the bonds in turn.
Example 5.2 The party with the short position has decided to deliver and is trying to chooseamong the three bonds in Table 5.6. Assume the current quoted futures price is 93-08, or 93.25. Thecost of delivering each of the bonds is as follows:
Bond 1: 99,50 — (93.25 x 1.0382) = $2.69
Bond 2: 143.50 — (93.25 x 1.5188) = $1.87
Bond 3: 119.75 — (93.25 x 1.2615) = $2.12The cheapest-to-deliver bond is bond 2.
Table 5.6 Deliverable bonds in Example 5.2
 
Bond Quoted price (8) Conversion factor1 99.50 1.03822 143.50 1.51883 119.75 1.2615
A number of factors determine the cheapest-to-deliver bond. When bond yields are in excess of6%, the conversion factor system tends to favor the delivery of low-coupon, long-maturity bonds.When yields arc less than 6%, the system tends to favor the delivery of high-coupon, short-maturity bonds. Also, when the yield curve is upward sloping, there is a tendency for bonds withalong time to maturity to be favored, whereas when it is downward sloping, there is a tendency forbonds with a short time to maturity to be delivered.
The Wild Card Play
Trading in the CBOT Treasury bond futures contract ceases at 2:00 p.m. Chicago time. However,Treasury bonds themselves continue trading in the spot market until 4:00 p.m. Furthermore, atrader with a short futures position has until 8:00 p.m. to issue to the clearinghouse a notice ofintention to deliver. If the notice is issued, the invoice price is calculated on the basis of theInterest Rate Markets 109
settlement price that day. This is the price at which trading was conducted just before the closingbell at 2:00 p.m.
This practice gives rise to an option known as the wild card play. If bond prices decline after2:00 p.m., the party with the short position can issue a notice of intention to deliver and proceed tobuy cheapest-to-deliver bonds in preparation for delivery at the 2:00 p.m. futures price. If the bondprice does not decline, the party with the short position keeps the position open and waits until thenext day when the same strategy can be used.
As with the other options open to the party with the short position, the wild card play is notfree. Its value is reflected in the futures price, which is lower than it would be without theoption.
Determining the Quoted Futures Price
An exact theoretical futures price for the Treasury bond contract is difficult to determine becausethe short party’s options concerned with the timing of delivery and choice of the bond that isdelivered cannot easily be valued. However, if we assume that both the cheapest-to-deliver bondand the delivery date are known, the Treasury bond futures contract is a futures contract on asecurity providing the holder with known income.’ Equation (3.6) then shows that the futuresprice, Fo, is related to the spot price, Sp, by
Fy = (Sp — De™ (5.5)
where / is the present value of the coupons during the life of the futures contract, T is the time untilthe futures contract matures, and r is the risk-free interest rate applicable to a time period oflength T.
Example 5.3 Suppose that, in a Treasury bond futures contract, it is known that the cheapest-to-deliver bond will be a 12% coupon bond with a conversion factor of 1.4000. Suppose also that it isknown that delivery will take place in 270 days. Coupons are payable semiannually on the bond. Asillustrated in Figure 5.2, the last coupon date was 60 days ago, the next coupon date is in 122 days,and the coupon date thereafter is in 305 days. The term structure is flat, and the rate of interest (withcontinuous compounding) is 10% per annum. Assume that the current quoted bond price is $120.‘The cash price of the bond is obtained by adding to this quoted price the proportion of the nextcoupon payment that accrues to the holder. The cash price is therefore
60120 + 4 122 x 6 = 121.978
A coupon of $6 will be received after 122 days (= 0.3342 year). The present value of this is
The futures contract lasts for 270 days (0.7397 year). The cash futures price if the contract werewritten on the 12% bond would therefore be
(121.978 — 5.803)e"1*°.7397 — 125.094
7 In practice, for the purposes of determining the cheapest-to-deliver in this calculation, analysts usually assume thatzero rates at the maturity of the futures contract will equal today’s forward rates.110 CHAPTER 5
 
MaturityofCoupon Current Coupon futures Couponpayment time payment contract paymentf + + t+60 122 148 35days days days days
Figure 5.2. Time chart for Example 5.3
At delivery there are 148 days of accrued interest. The quoted futures price if the contract werewritten on the 12% bond is calculated by subtracting the accrued interest:
148094 — = = 120.2125.09 6x 5-4 0.242From the definition of the conversion factor, 1.4000 standard bonds are considered equivalent toeach 12% bond. The quoted futures price should therefore be
120.242
74000 = 85.887
 
5.11 EURODOLLAR FUTURES
 
Another very popular United States interest rate futures contract is the three-month Eurodollarfutures contract traded on the Chicago Mercantile Exchange (CME). A Eurodollar is a dollardeposited in a U.S. or foreign bank outside the United States. The Eurodollar interest rate is therate of interest earned on Eurodollars deposited by one bank with another bank. It is essentiallythe same as the London Interbank Offer Rate (LIBOR) mentioned in earlier chapters.
Three-month Eurodollar futures contracts are futures contracts on the three-month Eurodollarinterest rate. The contracts have maturities in March, June, September, and December for up to10 years into the future. In addition, as can be seen from Table 5.5, the CME trades short-maturitycontracts with maturities in other months.
If Q is the quoted price for a Eurodollar futures contract, the exchange defines the value of one
contract as10,000[100 — 0.25(100 — Q)] (5.6)
Thus, the settlement price of 95.53 for the June 200! contract in Table 5.5 corresponds to a
contract price of10,000[100 — 0.25(100 — 95.53)] = $988,825
It can be seen from equation (5.6) that a change of one basis point or 0.01 in a Eurodollar futuresquote corresponds to a contract price change of $25.
When the third Wednesday of the delivery month is reached the contract is settled in cash. The °final marking to market sets Q equal to 100 — R, where R is the actual three-month Furodollarinterest rate on that day, expressed with quarterly compounding and an actual/360 day countconvention. Thus, if the three-month Eurodollar interest rate on the third Wednesday of thedelivery month is 8%, the final marking to market is 92, and the final contract price, fromequation (5.6), is
10,000[160 — 0.25(100 — 92)] = $980,000Interest Rate Markets 111
 
If Q is a Eurodollar futures quote, (100 — Q)% is the Eurodollar futures interest rate for a three-month period beginning on the third Wednesday of the delivery month. Thus Table 5.5 indicatesthat on March 15, 2001, the futures interest rate for the three-month period beginning WednesdayJune 20, 2001, was 100 — 95.53 = 4.47%. This is expressed with quarterly compounding and anactual/360 day count convention.
Other contracts similar to the CME Eurodollar futures contracts trade on interest rates in othercountries. As shown in Table 5.5, CME and SGX trade Euroyen contracts, LIFFE and MATIF tradeEuribor contracts (i.c., contracts on the three-month LIBOR rate for the euro), and LIFFE tradesthree-month Euroswiss futures (see Table 2.2 for the meaning of these exchange abbreviations),
Forward vs. Futures Interest Rates
For short maturities (up to one year) the Eurodollar futures interest rate can be assumed to be thesame as the corresponding forward interest rate. For longer-dated contracts the differences betweenfutures and forward interest rates mentioned in Section 3.9 become important.
Analysts make what is known as a convexity adjustment to convert Eurodollar futures rates toforward interest rates. One way of doing this is by using the formula
Forward rate = Futures rate — dort, ty
where ¢, is the time to maturity of the futures contract, ft) is the time to the maturity of the rateunderlying the futures contract, and o is the standard deviation of the change in the short-terminterest rate in one year. Both rates are expressed with continuous compounding.® A typical valuefor o is 1.2% or 0.012.
Example 5.4 Consider the situation where o = 0.612, and we wish io calculate the forward ratewhen the eight-year Eurodollar futures price quote is 94. In this case t; = 8, t) = 8.25, and theconvexity adjustment is
$x 0.012? x 8 x 8.25 = 0.00475
or 0.475%, The futures rate is 6% per annum on an actual/360 basis with quarterly compounding.This is 6 x 365/360 = 6.083% per annum on an actual/365 basis with quarterly compounding or6.038% with continuous compounding. The forward rate is, therefore, 6.038 — 0.475 = 5.563% perannum with continuous compounding.
The forward rate is less than the futures ratc.” The size of the adjustment is roughly proportional tothe square of the time to maturity of the futures contract. Thus the convexity adjustment for theeight-year contract is approximately 64 times that for a one-year contract.
5.12 THE LIBOR ZERO CURVE
 
 
The LIBOR zero curve (which is also sometimes referred to as the swap zero curye) is frequentlyused as the risk-free zero curve when derivatives are valued. Spot LIBOR rates (see Section 5.1) are
5 This formula is based on an interest rate model known as the Ho—Lee model.
° The reason for this can be seen from the arguments in Section 3.9. The variable underlying the Eurodollar futurescontracts is an interest rate and therefore ts highly positively correlated to other interest rates. See Problem 5.32.112 CHAPTER 5
used to determine very short-term LIBOR zero rates. After that Euro futures (i-e., Eurodollarfutures, Euroyen futures, Euribor futures, etc.) are frequently used. Once a convexity adjustmentsuch as that just described is made, the Euro futures contracts define forward rates for future three-month time periods. .
In the United States, March, June, September, and December Eurodollar futures are often usedto determine the LIBOR zero curve out to five years. Suppose that the ith Eurodollar futurescontract matures at time 7; (i= 1,2,...). We usually assume that the forward interest ratecalculated from this futures contract applies to the period 7; to 7,,. (There is at most a smallapproximation here.) This enables a bootstrap procedure to be used to determine zero rates.Suppose that F; is the forward rate calculated from the ith Eurodollar futures contract and R;, is thezero rate for a maturity 7;. From equation (5.1), we have
r- Ris Tia — RiT; ‘: Ti — 7;so thatF(T,4; — 7; RT;Rig _ i itt iD b RT; (5.7)
Thay
Example 5.5 The 400-day LIBOR zero rate has been calculated as 4.80% with continuouscompounding and, from a Eurodollar futures quote, it has been calculated that the forward ratefor a 91-day period beginning in 400 days is 5.30% with continuous compounding. We can useequation (5.7) to obtain the 491-day rate as
0.053 x 91 + 0.048 x 400
1 = 0.04893
or 4.893%.
5.13 DURATION
 
The duration of a bond, as its name implies, is a measure of how long on average the holder of thebond has to wait before receiving cash payments. A zero-coupon bond that matures in 7 years hasa duration of » years. However, a coupon-bearing bond maturing in 7 years has a duration of lessthan n years, because the holder receives some of the cash payments prior to year 7.
Suppose that a bond provides the holder with cash flows ¢; at time ¢; (1 <i < 7). The price, B,and yield, y (continuously compounded), are related by
B= Vice (5.8)i=lThe duration, D, of the bond is defined asnm yhD= Dani ese (5.9)
 
". fee MiD= Dal F ]
i=linterest Rate Markets 113
The term in square brackets is the ratio of the present value of the cash flow at time ¢; to the bondprice. The bond price is the present value of all payments. The duration is therefore a weightedaverage of the times when payments are made, with the weight applied to time 1; being equal to theproportion of the bond’s total present value provided by the cash flow at time t;. The sum of theweights is 1.0. We now show why duration is a popular and widely used measure.
From equation (5.8),n
oB=-)5 pp Teay i=1 ene
and from equation (5.9) this can be written=-=-BD (5.10)
If we make a small parallel shift to the yield curve, increasing all interest rates by a small amount,by, the yields on all bonds also increase by dy. Equation (5.10) shows that the bond’s price increasesby 5B, where
8B _ _pp (S11)byor6BF = TP oy (5.12)
Equation (5.12) is the key duration relationship. It shows that the percentage change in a bondprice is (approximately) equal to its duration multiplied by the size of the parallel shift in the yieldcurve.
Consider 4 three-year 10% coupon bond with a face value of $100. Suppose that the yield on thebond is 12% per annum with continuous compounding. This means that y = 0.12. Couponpayments of $5 are made every six months. Table 5.7 shows the calculations necessary to determinethe bond’s duration. The present values of the bond’s cash flows, using the yield as the discountrate, are shown in column 3 (e.g., the present value of the first cash flow is Se~°!?*° = 4.709). Thesum of the numbers in column 3 gives the bond’s price as 94.213. The weights are calculated bydividing the numbers in column 3 by 94.213. The sum of the numbers in column 5 gives theduration as 2.653 years.
Table 5.7 Calculation of duration
Time (years) Cash flow ($) Present value Weight Time x weight
 
0.5 5 4.709 0.050 0.0251.0 5 4.435 0.047 0.0471.5 5 4.176 0.044 0.0662.0 5 3.933 0.042 0.0832.5 5 3.704 0.039 0.0983.0 105 73.256 0.778 2.333
 
Total 130 94.213 1.000 2.653114 CHAPTER 5
Small changes in interest rates are often measured in basis points. A basis point is 0.01%per annum. The following example investigates the accuracy of the duration relationship inequation (5.11).
Example 5.6 For the bond in Table 5.7, the bond price, B, is 94.213 and the duration, D, is 2.653,
so that equation (5.11) gives$B = —94.213 x 2.653 dy
or5B = —249.95 dy
When the yield on the bond increases by 10 basis points (= 0.1%), 6y = +0.001. The durationrelationship predicts that 6B = —249.95 x 0.001 = —0.250, so that the bond price goes down to94.213 — 0.250 = 93.963. How accurate is this? When the bond yield increases by 10 basis points to12.1%, the bond price is
§e 0.121 0.5 +4 5e70-121 4 5e70 RI xLs +4 570-121 x2.0 + Se7 0-121 «2.5 +4 105¢e70 21x15 = 93.963
which is (to three decimal places) the same as that predicted by the duration relationship.
Modified Duration
The preceding analysis is based on the assumption that y is expressed with continuous compound-ing. If y is expressed with annual compounding, it