Introduction

• This chapter explains the popular Treasury bond and Eurodollar futures contracts that trade in the United States. Many of the other interest rate futures contracts throughout the world have been modeled on these contracts. The chapter also shows how interest rate futures contracts, when used in conjunction with “duration”, can be used to hedge a company's exposure to interest rate movements.

Day Counts

• The day count convention is usually expressed as X / Y. When we are calculating the interest earned between two dates, X defines the way in which the number of days between the two dates is calculated, and Y defines the way in which the total number of days in the reference period is measured. The interest earned between the two dates is

• Three-day count conventions that are commonly used in the United States are:
  • Actual/Actual (in period)
  • 30/360
  • Actual/360

Day Counts – Actual/Actual With Ba 2 Plus Calculator

• The actual / actual (in period) day count is used for Treasury bonds in the United States.

Example –

Semi-annual coupon paying bond with face value of $100,

Coupon payment dates are March 1 and September 1,

Coupon rate is 8% per annum. (This means that $4 of interest is paid on each of March 1 and September 1.)

Calculate the interest earned between March 1 and July 3 in 2019

The reference period is from March 1 to September 1.

There are ______ (actual) days in the reference period, and interest of $4 is earned during the period.

There are ______ (actual) days between March 1 and July 3.

The interest earned between March 1 and July 3 is

Day Counts – 30/360

• The 30 / 360 day count is used for corporate and municipal bonds in the United States. This means that we assume 30 days per month and 360 days per year when carrying out calculations.

Example –

In a corporate bond with the same terms as the Treasury bond just considered, with the 30 / 360 day count, the total number of days between March 1 and September 1 is _________

The total number of days between March 1 and July 3 is _________

The interest earned between March 1 and July 3 would therefore be

Day Counts – Actual/360 With Ba 2 Plus Calculator

• The actual/360 day count is used for money market instruments in the United States. This indicates that the 1-year reference period is 360 days. The interest earned during part of a year is calculated by dividing the actual number of elapsed days by 360 and multiplying by the rate.

Example –

In a money market instrument with the same terms as the Treasury bond just considered, with the actual/360 day count, the total number of days between March 1 and September 1 is _________

The total number of days between March 1 and July 3 is _________

The interest earned between March 1 and July 3 would therefore be

• It can be clearly observed from the actual/360 convention that the interest earned in 90 days is therefore exactly one-fourth of the quoted rate, and the interest earned in a whole year of 365 days is 365/360 times the quoted rate.

Day Counts Conventions In Different Countries

• Conventions vary from country to country and from instrument to instrument.
• Examples include
  • Money market instruments are quoted on an basis in Australia, Canada, and New Zealand.
  • LIBOR is quoted on an for all currencies except sterling, for which it is quoted on an basis.
  • Euro-denominated and sterling bonds are usually quoted on an basis

Price Quotations Of U.S. Treasury Bills

• The prices of money market instruments are sometimes quoted using a discount rate. This is the interest earned as a percentage of the final face value rather than as a percentage of the initial price paid for the instrument. An example is Treasury bills in the United States.
• If the price of a 91-day Treasury bill is quoted as _______, this means that the rate of interest earned is ____________ of the face value per 360 days. Suppose that the face value is $100. Over the 91-day life, interest earned is 

So this means that, for the 91-day period, the true rate of interest is

• In general, the relationship between the cash price per $100 of face value and the quoted price of a Treasury bill in the United States is

Where

• • P=quoted price,
  • Y=cash price
  • n=Remaining life of the Treasury bill measured in calendar days.

For example, when the cash price of a 90-day Treasury bill is ______, the quoted price is

• Treasury bond prices in the United States are quoted in dollars and thirty-seconds of a dollar. The quoted price is for a bond with a face value of $100. Thus, a quote of 120–05 or  indicates that the quoted price for a bond with a face value of $100,000 is
• The quoted price, which traders refer to as the clean price, is not the same as the cash price paid by the purchaser of the bond, which is referred to by traders as the dirty price or full price. In general,

Cash price = Quoted price + Accrued interest since last coupon date

• When the settlement date is a coupon payment date, then the accrued interest of a bond is zero, and hence the cash (dirty) price is equal to the quoted (clean) price.
• When accrued interest is not zero, the cash (dirty) price of the bond is equal the present value of its cash flows.

Price Quotations Of U.S. Treasury Bonds – Example

• Suppose that it is March 5, 2018, and the bond under consideration is an 11% coupon bond maturing on July 10, 2038, with a quoted price of 155–16 or

  $155.50. Because coupons are paid semiannually on government bonds (and the final coupon is at maturity), the most recent coupon date is January 10, 2018, and the next coupon date is July 10, 2018.

• The (actual) number of days between January 10, 2018, and March 5, 2018, is _____
• The (actual) number of days between January 10, 2018, and July 10, 2018, is _______.
• On a bond with $100 face value, the coupon payment is ______ on January 10 and July 10.
• The accrued interest on March 5, 2018, is the share of the July 10 coupon accruing to the bondholder on March 5, 2018. Since actual/actual in period is used for Treasury bonds in the United States, the accrued interest is
• The cash price per $100 face value for the bond is therefore __________
• Thus, the cash price of a $100,000 bond is ____________

Treasury Bond Futures

• One of the most popular long-term interest rate futures contracts is the Treasury bond futures contract traded by the CME Group. In this contract, any government bond that has between 15 and 25 years to maturity on the first day of the delivery month can be delivered by the short position. The exchange has developed a procedure for adjusting the price received by the party with the short position according to the particular bond or note it chooses to deliver.
• When a particular bond is delivered, a parameter known as its conversion factor defines the price received for the bond by the party with the short position. The applicable quoted price for the bond delivered is the product of the conversion factor and the most recent settlement price for the futures contract. Taking accrued interest into account the cash received for each $100 face value of the bond delivered is

Conversion Factors – Example

• Each contract is for the delivery of $100,000 face value of bonds. Suppose that the most recent settlement price is 120–00, the conversion factor for the bond delivered is 1.3800, and the accrued interest on this bond at the time of delivery is $3 per $100 face value. The cash received by the party with the short position (and paid by the party with the long position) is then per $100 face value. A party with the short position in one contract would deliver bonds with a face value of _____________ and receive ___________

Conversion Factors

• The conversion factor for a bond is set equal to the quoted price the bond would have per dollar of principal on the first day of the delivery month on the assumption that the interest rate for all maturities equals 6% per annum (with semiannual compounding). The bond maturity and the times to the coupon payment dates are rounded down to the nearest 3 months for the purposes of the calculation.
  • If, after rounding, the bond lasts for an exact number of 6-month periods, the first coupon is assumed to be paid in 6 months.
  • If, after rounding, the bond does not last for an exact number of 6-month periods (i.e., there are an extra 3 months), the first coupon is assumed to be paid after 3 months and accrued interest is subtracted.

• As a first example of these rules, consider a 10% coupon bond with 20 years and 2 months to maturity. For the purposes of calculating the conversion factor, the bond is assumed to have exactly 20 years to maturity. The first coupon payment is assumed to be made after 6 months. Coupon payments are then assumed to be made at 6-month intervals until the end of the 20 years when the principal payment is made. Assume that the face value is $100. When the discount rate is 6% per annum with semiannual compounding (or 3% per 6 months), the value of the bond is

Dividing by the face value gives a conversion factor of ___________

• As a second example of the rules, consider an 8% coupon bond with 18 years and 4 months to maturity. For the purposes of calculating the conversion factor, the bond is assumed to have exactly 18 years and 3 months to maturity. Discounting all the payments back to a point in time 3 months from today at 6% per annum (compounded semiannually) gives a value of

The interest rate for a 3-month period is

Hence, discounting back to the present gives the bond's value as

Subtracting the accrued interest of _________ this becomes __________

The conversion factor is therefore ______________

Cheapest-To-Deliver Bond

• At any given time during the delivery month, there are many bonds that can be delivered in the Treasury bond futures contract. These vary widely as far as coupon and maturity are concerned. The party with the short position can choose which of the available bonds is “cheapest” to deliver.

Amount received by the party with the short position is

and the cost of purchasing a bond to the short position is

Quoted bond price + Accrued interest

The cheapest-to-deliver bond is the one for which

is least.

Once the party with the short position has decided to deliver, it can determine the cheapest-to-deliver bond by examining each of the deliverable bonds in turn.

Cheapest-To-Deliver Bond – Example

• The party with the short position has decided to deliver and is trying to choose between the three bonds in the table below. Assume the most recent settlement price is 93–08, or 93.25.

Bonds	Quoted Bond Price ($)	Conversion Factor
1	99.50	1.0382
2	143.50	1.5188
3	119. 75	1.2615

The cost of delivering each of the bonds is as follows:

Bond 1:

Bond 2:

Bond 3:

Hence, the cheapest-to-deliver bond is Bond ________

Cheapest-To-Deliver Bond (Self Study)

• A number of factors determine the cheapest-to-deliver bond.

1. Yield Levels – When bond yields are in excess of 6%, the conversion factor system tends to favor the delivery of low-coupon long-maturity bonds. When yields are less than 6%, the system tends to favor the delivery of high-coupon short-maturity bonds.
2. Yield Curve Slope – When the yield curve is upward-sloping, there is a tendency for bonds with a long time to maturity to be favored, whereas when it is downward-sloping, there is a tendency for bonds with a short time to maturity to be delivered.

Determining The Futures Price

• An exact theoretical futures price for the Treasury bond contract is difficult to determine because the short party's options concerned with the timing of delivery and choice of the bond that is delivered cannot easily be valued. However, if we assume that both the cheapest-to-deliver bond and the delivery date are known, the Treasury bond futures contract is a futures contract on a traded security (the bond) that provides the holder with known income and hence, the futures price, , is related to the spot price, , by

where

I is the present value of the coupons during the life of the futures contract,

T is the time until the futures contract matures,

r is the risk-free interest rate applicable to a time period of length T.

Determining The Futures Price – Example

• 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 1.6000.

Suppose also that it is known that delivery will take place in 270 days. Coupons are payable semiannually on the bond. As illustrated in this figure, 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 (with continuous compounding) is 10% per annum. Assume that the current quoted bond price is $115. The cash price of the bond is obtained by adding to this quoted price the proportion of the next coupon payment that accrues to the holder. The cash price is therefore

A coupon of $6 will be received after 122 days (= ________ years). The present value of this is

The futures contract lasts for 270 days (=_________ years). The cash futures price, if the contract were written on the 12% bond, would therefore be

At delivery, there are 148 days of accrued interest. The quoted futures price, if the contract were written on the 12% bond, is calculated by subtracting the accrued interest

From the definition of the conversion factor, 1.6000 standard bonds are considered equivalent to each 12% bond. The quoted futures price should therefore be

Eurodollar Futures

• The most popular interest rate futures contract in the United States is the three-month Eurodollar futures contract traded by the CME Group. A Eurodollar is a dollar deposited in a U.S. or foreign bank outside the United States. The Eurodollar interest rate is the rate of interest earned on Eurodollars deposited by one bank with another bank. It can be regarded as the same as the London Interbank Offered Rate (LIBOR).
• A three-month Eurodollar futures contract is a futures contract on the interest that will be paid (by someone who borrows at the LIBOR interest rate) on $1 million for a future three-month period. It allows a trader to speculate on a future three-month interest rate or to hedge an exposure to a future three-month interest rate. Eurodollar futures contracts have maturities in March, June, September, and December for up to 10 years into the future. This means that in 2017 a trader can use Eurodollar futures to take a position on what interest rates will be as far into the future as 2027. Short-maturity contracts trade for months other than March, June, September, and December.

• The contract is designed so that a one-basis-point (= 0.01) move in the futures quote corresponds to a gain or loss of $25 per contract. When a Eurodollar futures quote increases by one basis point, a trader who is long one contract gains $25 and a trader who is short one contract loses $25. Similarly, when the quote decreases by one basis point a trader who is long one contract loses $25 and a trader who is short one contract gains $25. Suppose, for example, a settlement price changes from 99.325 to 99.285. Traders with long positions ______ _____________ per contract; traders with short positions gain __________ per contract. A one-basis-point change in the futures quote corresponds to a 0.01% change in the underlying interest rate. This in turn leads to a change in the interest that will be earned on $1 million in three months. The $25 per basis point rule is therefore consistent with the point made earlier that the contract locks in an interest rate on $1 million for three months.

• The futures quote is 100 minus the futures interest rate. A trader who is long gains when interest rates fall and one who is short gains when interest rates rise.

• This table shows a possible set of outcomes for the June 2016 contract for a trader who takes a long position on May 3, 2016, at the last trade price of 99.330.

The contract price is defined as

where Q is the quote.

Thus, the initial price of 99.330 for the June 2013 contract corresponds to a contract price of

Date	Trade Price	Settlement Future Price	Change	Gain Per Long Contract
May 3, 2016	99.330
May 3, 2016		99.325	-0.005	-12.50
May 4, 2016		99.275	-0.050	-125.00
:		:	:	:
June 13, 2016		99.220	+0.010	+25.00
Total			-0.110	-275.00

In this table, the final contract price corresponds to a settle price of 99.220 and is equal to

and the difference between the initial and final contract price is ______ This is consistent with the loss calculated in this table using the “$25 per one-basis-point move” rule.

Eurodollar Futures – Example (Optional – Self Study)

• An investor wants to lock in the interest rate for a three-month period beginning two days before the third Wednesday of September, on a principal of $100 million.

Suppose that the September Eurodollar futures quote is 96.50, indicating that the investor can lock in an interest rate of 100-96.5 or 3.5% per annum. The investor hedges by buying 100 contracts. Suppose that, two days before the third Wednesday of September, three-month LIBOR turns out to be 2.6%. The final settlement in the contract is then at a price of 97.40. The investor gains

or $225,000 on the Eurodollar futures contracts.

Convexity Adjustment

• The daily marking to market of the futures contract may lead to differences between actual forward rates and the rates implied by futures contracts. Generally, long-dated Eurodollar futures contracts lead to implied forward rates larger than actual forward rates. Analysts make what is known as a convexity adjustment to account for the total difference between the two rates. One popular adjustment is

where

is the time to maturity of the futures contract

is the time to the maturity of the rate underlying the futures contract.

is the standard deviation of the change in the short-term interest rate in 1 year.

Both rates are expressed with continuous compounding

Convexity Adjustment – Example

• Consider the situation where =0.012 and the forward rate is to be calculated when the 8-year Eurodollar futures price quote is 94.

In this case T1=______, T2=_________________, and the convexity adjustment is

or _________( ________ basis points).

The table below shows how the size of the adjustment increases with the time to maturity

Maturity Of Futures (Years)	Convexity Adjustment (Basis Points)
2	3.2
4	12.2
6	27.0
8	47.5
10	73.8

Using Eurodollar Futures To Extend The Libor Zero Curve

• The LIBOR zero curve out to 1 year is determined by the 1-month, 3-month, 6-month, and 12-month LIBOR rates. Once the convexity adjustment just described has been made, Eurodollar futures are often used to extend the zero curve. Suppose that the Eurodollar futures contract matures at time ( = 1, 2, …). It is usually assumed that the forward interest rate calculated from the futures contract applies to the period to . This enables a bootstrap procedure to be used to determine zero rates. Suppose that is the forward rate calculated from the Eurodollar futures contract and is the zero rate for a maturity . It is already known that

which gives

Other Euro rates such as Euroswiss, Euroyen, and Euribor are used in a similar way.

Eurodollar Futures To Extend The Libor Zero Curve – Example

• Assume that the 400-day LIBOR zero rate has been calculated as 4.80% with continuous compounding and, from Eurodollar futures quotes, it has been calculated that
  • the forward rate for a 90-day period beginning in 400 days is 5.30%
  • the forward rate for a 90-day period beginning in 491 days is 5.50%
  • the forward rate for a 90-day period beginning in 589 days is 5.60%

all with continuous compounding.

• The 491-day rate can be calculated as
• The 589-day rate can be calculated as

The next forward rate of ________ would be used to determine the zero curve out to the maturity of the next Eurodollar futures contract. (Note that, even though the rate underlying the Eurodollar futures contract is a 90-day rate, it is assumed to apply to the 91 or 98 days elapsing between Eurodollar contract maturities.)

Duration-Based Hedging Strategies Using Futures

• Interest rate futures can be used to hedge the yield on a bond portfolio at a future time. The number of contracts required to hedge against an uncertain is given by

where

is the contract price for one interest rate futures contract

is the duration of the asset underlying the futures contract at the maturity of the futures contract

P is the forward value of the portfolio being hedged at the maturity of the hedge (in practice, this is usually assumed to be the same as the value of the portfolio today)

is the duration of the portfolio at the maturity of the hedge.

• This is the duration-based hedge ratio. It is sometimes also called the price sensitivity hedge ratio.

Duration-Based Hedging Strategies Using Futures (Optional – Self Study)

• When the hedging instrument is a Treasury bond futures contract, the hedger must base DF on an assumption that one particular bond will be delivered. This means that the hedger must estimate which 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 bond will be cheapest to deliver, then the hedge has to be adjusted and as a result its performance may be worse than anticipated.
• When hedges are constructed using interest rate futures, it is important to bear in mind that interest rates and futures prices move in opposite directions. When interest rates go up, an interest rate futures price goes down. When interest rates go down, the reverse happens, and the interest rate futures price goes up. Thus, a company in a position to lose money if interest rates drop should hedge by taking a long futures position. Similarly, a company in a position to lose money if interest 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 as close as possible to the duration of the asset being hedged. Eurodollar futures tend to be used for exposures to short-term interest rates, whereas ultra T-bond, Treasury bond, and Treasury note futures contracts are used for exposures to longer-term rates.

Duration-Based Hedging Strategies Using Futures – Example

• It is August 2 and a fund manager with $10 million invested in government bonds is concerned that interest rates are expected to be highly volatile over the next 3 months. The fund manager 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. Suppose that the duration of the bond portfolio in 3 months will be 6.80 years. The cheapest-to-deliver bond in the T-bond contract is expected to be a 20-year 12% per annum coupon bond. The yield on this bond is currently 8.80% per annum, and the duration will be 9.20 years at maturity of the futures contract.

The fund manager requires a short position in T-bond futures to hedge the bond portfolio. If interest rates go up, a gain will be made on the short futures position, but a loss will be made on the bond portfolio. If interest rates decrease, a loss will be made on the short position, but there will be a gain on the bond portfolio. The number of bond futures contracts that should be shorted can be as

Hedging Portfolios Of Assets And Liabilities

• Financial institutions sometimes attempt to hedge themselves against interest rate risk by ensuring that the average duration of their assets equals the average duration of their liabilities. (The liabilities can be regarded as short positions in bonds.) This strategy is known as duration matching or portfolio immunization. When implemented, it ensures that a small parallel shift in interest rates will have little effect on the value of the portfolio of assets and liabilities. The gain (loss) on the assets 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, and are not perfectly correlated with, long-term rates.
  • Sometimes it even happens that short- and long-term rates move in opposite directions to each other.
• Duration matching is therefore only a first step and financial institutions have developed other tools to help them manage their interest rate exposure. Other instruments like swaps, FRAs, bond futures, Eurodollar futures, etc. might be required to fully manage interest rate exposure.