Chapter 71 CHAPTER 7 INTEREST RATE FUTURES In this chapter, we explore one of the most successful innovations in the history of futures markets; that is,

Chapter 7 1

CHAPTER 7INTEREST RATE FUTURES

In this chapter, we explore one of the most successful innovations in the history of futures markets; that is, interest rate futures contracts. This chapter is organized into the following sections:

1. Interest Rate Futures Contracts

2. Pricing Interest Rate Futures Contracts

3. Speculating With Interest Rate Futures Contracts

4. Hedging With Interest Rate Futures Contracts

Chapter 7 2

Interest Rate Futures Introduction

Interest rate futures contracts are one of the most successful innovations in futures trading.

Pioneered in the United States, they have expanded internationally with strong presence in Great Britain and Singapore.

The CBOT specializes in contracts with long-term maturity (e.g., 2-year, 5-year and 10-year T-notes, and 5-year LIBOR-based swaps).

The CME International Monetary Market (IMM) specializes in contracts with short-term maturity (e.g., 1-month, and 3-month Eurodollar deposits).

Chapter 7 3

Short-Term Interest Rates Contracts

In this section, four short-term interest rate futures contracts will be examined:

1. Eurodollar Futures

2. Euribor Futures

3. TIEE 28 Futures

4. Treasury Bill Futures

Chapter 7 4

Eurodollar Futures Product Profile

Product Profile: The CME=s Eurodollar Futures Contract Size: Eurodollar Time Deposit having a principal value of $1,000,000 with a three-month maturity. Deliverable Grades: Cash Settled to 3-month Dollar LIBOR Tick Size: 0.01=$25.00 Months 11 thru 40; 0.005=$12.50 Months 2 thru 10; 0.0025=$6.25 for nearest expiring month. Price Quote: Price is quoted in terms of the IMM 3-month Eurodollar index, 100 minus the yield on an annual basis for a 360-day year with each basis point worth $25. Contract Months: March, June, September, and December cycle for 10 years Expiration and final Settlement: Eurodollar futures cease trading at 5:00 a.m. Chicago Time (11:00 a.m. London Time) on the second London bank business day immediately preceding the third Wednesday of the contract month; final settlement price is based on the British Bankers= Association Interest Settlement Rate. Trading Hours: Floor: 7:20 a.m.-2:00 p.m; Globex: Mon/Thurs 5:00 p.m.-4:00 p.m.; Shutdown period from 4:00 p.m. to 5:00 p.m. nightly; Sunday & holidays 5:30 p.m.-4:00 p.m. Daily Price Limit: None

Chapter 7 5

Eurodollar Futures

1. Eurodollar futures currently dominate the U.S. market for short-term futures contracts.

2. Rates on Eurodollar deposits are usually based on LIBOR (London Interbank Offer Rate).

– LIBOR is the rate at which banks are willing to lend funds to other banks in the interbank market.

3. Eurodollars are U.S. dollar denominated deposits held in a commercial bank outside the U.S.

4. The Eurodollar contracts is for $1,000,000.

5. A Eurodollar futures contract is based on a time deposit held in a commercial bank (e.g., 3-month Eurodollar)

6. Eurodollar contracts are non-transferable.

Chapter 7 6

Eurodollar Futures

7. Eurodollar futures were the first contract to use cash settlement rather than delivery of an actual good for contract fulfillment.

8. To establish the settlement rate at the close of trading, the IMM determines the three-month LIBOR rate.

9. This settlement rate is then used to compute the amount of the cash payment that must be made.

10.The yield on the Eurodollar contract is quoted on an add-on basis as follows:

Chapter 7 7

Eurodollar Add-on Yield

In order to calculate the add-on yield, the price and discount must be computed as follows:

))(( 360$

DTMPrice

Discount Yield onAdd

DiscountValue FacePrice $

360

))(($

DTMValue FaceDYDiscount

360

))(( DTMValue FaceDYValue FacePrice

Or equivalently

Chapter 7 8


Suppose you have a 90-day Eurodollar deposit with a discount yield of 8.32%.

Step 1: Compute the discount and the price.

800,20$$Discount

800,20000,000,1 Price

200,979$Price

360

))(( DTMValue FaceDYValue FacePrice

360

)90)((0832.0 1,000,0001,000,000Price

Chapter 7 9


))(( 360$

DTMPrice

Discount Yield onAdd

))((90

360

200,979

800,20$ Yield onAdd

25$360900001.000,000,1$

Step 2: Compute the add-on yield using:

A one basis point change in the Add-on Yield, on a 3-month Eurodollar contract implies a $25 change in price. This amount can be compute using:

360 DTMYield onAddValue Face

Eurodollar futures contract prices are quoted using the IMM Index which is a function of the 3-month LIBOR rate:

IMM Index = 100.00 - 3-Month LIBOR

0.085 Yield onAdd

Chapter 7 10

Euribor Futures

Euribors are Eurodollar time deposits.

Swaps dealers use Euribor futures to hedge the risk resulting from their activities.

Euribor futures are traded at:

Euronex.liffe

– Contracts are based on a 3-month time deposit with a €1,000,000 notional value.

– Contracts are cash settled at expiration .

Eurex

– Contracts are based on a 3-month time deposit with a €3,000,000 notional value.

– Contracts are cash-settled at expiration.

Chapter 7 11

Euribor Futures Product Profile

Product Profile: Euronext-Liffe Euribor Futures Contract Size: 1,000,000 with a three-month maturity. Deliverable Grades: Cash Settled to Euopean Bankers Federation=s Euribor Offered Rate (EBF Euribor) for three-month euro time deposits. Tick Size: .005 percent representing 12.5. Price Quote: 100 minus the Euribor rate of interest carried out to three decimal places. Contract Months: March, June, September, and December and four serial months so that 24 delivery months are available for trading, with the nearest six expirations being consecutive calendar months. Expiration and final Settlement: The last trading day is two business days prior to the third Wednesday of the contract month. Final settlement is based on Euopean Bankers Federation=s Euribor Offered Rate (EBF Euribor) for three-month euro time deposits at 10:00 a.m. London time on the last trading day. Trading Hours: 7:00 a.m. to 6:00 p.m. Daily Price Limit:

Chapter 7 12

TIEE 28 Futures

The TIEE 28 futures contract is based on the short-term (28-day) Mexican interest rate.

The contract is traded on the Mexican Derivatives Exchange (Mercado Mexicano de Derivados, or MexDer)

A 28-day TIIE futures contract has a face value of 100,000 Mexican pesos.

The contract is cash settled based on the 28-day Interbank Equilibrium Interest Rate (TIIE), calculated by Banco de México.

Chapter 7 13

TIEE 28 Futures TIEE 28 Futures

Product Profile: The MexDer=s TIEE Futures Contract Size: Each 28-Day TIIE Futures Contract covers a face value of One Hundred Thousand Mexican Pesos. Deliverable Grade Cash settled based of the 28-Day Interbank Equilibrium Interest Rate (TIIE), calculated by Banco de México based on quotations submitted by full-service banks using a mechanism designed to reflect conditions in the Mexican Peso Money Market. Tick Size: One basis point of the annualized percentile rate of yield Price Quote: Trading of 28-Day TIIE futures contracts use the annualized percentile rate of yield expressed in percentile terms, with two decimal places. Contract Months: MexDer lists different Series of the 28-Day TIIE Futures Contracts on a monthly basis for up to sixty months (five years). Expiration and final Settlement: The last trading day is the bank business day after Banco de México holds the primary auction of government securities in the week corresponding to the third Wednesday of the Maturity Month. Trading Hours: Bank businessdays from 7:30 a.m. to 3:00 p.m., Mexico City time. Daily Price Limit: None

Chapter 7 14

Treasury Bill Futures

1. A T-bill is the U.S. government borrowing money for a short period of time.

– Treasury bills have original maturities of 13 weeks and 26 weeks.

2. The Treasury bill futures contract calls for the delivery of T-bills having a face value of $1,000,000 and a time to maturity of 90 days at the expiration of the futures contract.

– 91-day and 92 day T-bills may also be delivered with a price adjustment.

– The contracts have delivery dates in March, June, September, and December.

– The delivery dates are chosen to make newly issued 13 week T-bills immediately deliverable against the futures contract.

Chapter 7 15


Price quotations for T-bill futures use the International Monetary Market Index (IMM).

IMM Index = 100 - DY

Where:

DY = Discount Yield

Example

A discount Yield of 7.1% implies an IMM Index of:

IMM Index = 100 - 7.1

IMM Index = 92.9

Chapter 7 16


Recall that a bill with 90 days to maturity and a 8.32% discount yield, has a price of $979,200 and a $discount of $20,800. For a futures contract with a discount yield of 8.32%, the price to be paid for the T-bill at delivery would be $979,200.

A one basis point shift implies a $25 change on a $1,000,000, 3-month futures contract.

If the futures yield rose to 8.35%, the delivery price would be $979,125.

Chapter 7 17

Other Short-Term Interest Rate Futures

Insert Figure 7.1 here

Chapter 7 18

Longer-Maturity Interest Rate Futures

Longer-maturity interest rate futures are based on coupon-bearing debt instruments as the underlying good.

These instruments require the delivery of an actual bond.

In this section, long-term interest rate futures contracts will be examined, including:

1. Treasury Bond Futures

2. Treasury Note Futures

3. Non-US Longer Maturity Interest Rate Futures

Chapter 7 19

Treasury Bond Futures

Traded at the CBOT, the Treasury bond futures contract is one of the most successful futures contracts.

Requires the delivery of T-bonds with a $100,000 face value and with at least 15 years remaining until maturity or until their first permissible call date.

T-bond contracts trade for delivery in March, June, September, and December.

Delivery against the T-bond contract is a several day process that the short trader can trigger to cause delivery on any business day of the delivery month.

– First Position Day

First permissible day for the short to declare his/her intentions to make delivery, with delivery taking place 2 business days later.

– Position Day

Short declares his/her intentions to make delivery. This may occur on the first position day or some other later day.

Delivery Day

Clearinghouse matches the short and long traders and requires them to fulfill their responsibilities.

Chapter 7 20

Treasury Bond FuturesPrice Quotation for Major Interest Rate Futures

Contracts

Insert Figure 7.1 Here

Chapter 7 21

Treasury Bond Futures Delivery Process


Chapter 7 22

Treasury Bond Futures Product Profile

Product Profile: The CBOT=s 30 Year Treasury Bond Futures Contract Size: One U.S. Treasury bond with face value at maturity of $100,000 Deliverable Grades: U.S. Treasury bonds that, if callable, are not callable for at least 15 years from the first day of the delivery month or, if not callable, have a maturity of at least 15 years from the first day of the delivery month. The invoice price equals the futures settlement price times a conversion factor plus accrued interest. The conversion factor is the price of the delivered note ($1 par value) to yield 6 percent. Tick Size: 1/32 of a point ($31.25/contract); par is on the basis of 100 points. Price Quote: Points ($1,000) and thirty seconds a point; i.e., 84-16 equals 84 16/32. Contract Months: March, June, September, and December Expiration and final Settlement: The last trading day is the seventh business day preceding the last business day of the delivery month. The contract is settled with physical delivery. The last delivery day is the last business day of the delivery month. Trading Hours: Open Auction: 7:20 am - 2:00 pm, Central Time, Monday - FridayElectronic: 7:00 pm - 4:00 pm, Central Time, Sunday - FridayTrading in expiring contracts closes at noon, Chicago time, on the last trading day. Daily Price Limit: None.

Chapter 7 23

Treasury Bond Futures Conversion Factor

The T-bond contract does not specify exactly which bond must be delivered to fulfill the futures contract. Rather, a number of different bonds can be delivered to fulfill the futures contract.

Because the short trader chooses whether to make delivery, and which bond to deliver, the short trader will want to deliver the bond that is least expensive for him/her to obtain. This bond is called the cheapest-to-deliver bond.

To address this issue, a conversion factor is computed to equate the bonds.

Chapter 7 24


Where:

DSP = Decimal Settlement Price

(The decimal equivalent of the quoted price)

CF = Conversion Factor

(the conversion factor as provided by the CBOT)

AI = Accrued Interest

(Interest that has accrued since the last coupon payment onthe bond)

This system is effective as long as the term structure of interest rates is flat and the bond yield is 6%. However, if the term structure of interest rates is not flat, or if bond yields are not 6%, some bonds will still be less expensive to deliver against the futures contract than others.

AICFDSP ))(000,100($Amount Invoice

Chapter 7 25

T-Bond and T-Notes Delivery Sequence

Table 7.1 shows key dates in the delivery process for T-bond and T-note futures contracts in 1997.

Table 7.1 The Delivery Sequence for T-Bond & T-Note Futures Expiring in 1997

Contract Expiration

First Position

First Notice

First Delivery

Last Trading

Last Delivery

MAR 97 FEB 27 FEB 28 MAR 3 MAR 21 MAR 31 JUN MAY 29 MAY 30 JUN 2 JUN 20 JUN 30 SEPT AUG 28 AUG 29 SEPT 2 SEP 19 SEP 30 DEC NOV 26 NOV 28 DEC 1 DEC 19 DEC 31

Chapter 7 26


Table 7.1

Conversion Factors for Treasury-Bond Futures for September and December 2004

Coupon

Maturity Date

Sep-04

Dec-04

5 1/4

11/15/28

0.9052

0.9056

5 1/4 02/15/29 0.9047 0.9052 5 1/2 08/15/28 0.9370 0.9374 6 02/15/26 0.9999 1.0000 6 1/8 11/15/27 1.0155 1.0153 6 1/8 08/15/29 1.0159 1.0159 6 1/4 08/15/23 1.0278 1.0277 6 1/4 05/15/30 1.0324 1.0322 6 3/8 08/15/27 1.0461 1.0460 6 1/2 11/15/26 1.0606 1.0602 6 5/8 02/15/27 1.0761 1.0758 6 3/4 08/15/26 1.0903 1.0899 6 7/8 08/15/25 1.1029 1.1024 7 1/8 02/15/23 1.1236 1.1228 7 1/4 08/15/22 1.1352 1.1343 7 1/2 11/15/24 1.1734 1.1721 7 5/8 11/15/22 1.1774 1.1759 7 5/8 02/15/25 1.1889 1.1878 7 7/8 02/15/21 1.1928 1.1911 8 11/15/21 1.2113 1.2094 8 1/8 05/15/21 1.2206 1.2185 8 1/8 08/15/21 1.2224 1.2206 8 1/2 02/15/20 1.2474 1.2450 8 3/4 05/15/20 1.2750 1.2721 8 3/4 08/15/20 1.2775 1.2750

Source: Chicago Board of Trade web site: www.cbot.com.

Chapter 7 27

Treasury Note Futures

Treasury note futures are a shorter maturity version of a Treasury bond.

• T-note Futures are very similar to Treasury bond futures.

• T-note futures contracts are available for 2-year, 5-year, and 10-year maturities.

Contract Size

2-year contract $200,000

5-year & 10 year contract $100,000

Deliverable Maturities

2-year contract 21 -24 month

5-year contract 4 yrs 3 mos. to 5 yrs 3 mos.

10-year contract 6 yrs 6 mos. to 10 years

Chapter 7 28

CBOT’s 10-Year Treasury Note FuturesProduct Profile

Product Profile: The CBOT=s 10 Year Treasury Note Futures

Contract Size: One U.S. Treasury Note with face value at maturity of $100,000 Deliverable Grades: U.S. Treasury notes maturing at least 6.5 years, but not more than 10 years, from the first day of the delivery month. The invoice price equals the futures settlement price times a conversion factor plus accrued interest. The conversion factor is the price of the delivered note ($1 par value) to yield 6 percent. Tick Size: One half of 1/32 of a point ($15.625/contract) rounded up to the nearest cent; par is on the basis of 100 points. Price Quote: Points ($1,000) and one half of 1/32 of a point; i.e., 84-16 equals 84 16/32, 84-165 equals 84 16.5/32 Contract Months: March, June, September, and December Expiration and final Settlement: The last trading day is the seventh business day preceding the last business day of the delivery month. The contract is settled with physical delivery. The last delivery day is the last business day of the delivery month. Trading Hours: Open Auction: 7:20 am - 2:00 pm, Central Time, Monday - FridayElectronic: 7:00 pm - 4:00 pm, Central Time, Sunday - FridayTrading in expiring contracts closes at noon, Chicago time, on the last trading day. Daily Price Limit: None.

Chapter 7 29

Non-US Long Maturity Interest Rate Futures

Product Profile: Eurex=s Euro Bund Futures

Contract Size: One German bund with a par value of 100,000 euros. Deliverable Grades: A long-term debt instrument issued by the German Federal Government with a term of 82 to 102 years and an interest rate of 6 percent. The invoice price equals the futures settlement price times a conversion factor plus accrued interest. Tick Size: 0.01 percent, representing 10 euros. Price Quote: In a percentage of par value, carried out two decimal places. . Contract Months: The three successive months within the March, June, September, and December delivery cycle. Expiration and final Settlement: The last trading day is two trading days prior to the delivery day of the contract month. The delivery day is the 10th calendar day of the contract month, if this day is an exchange trading day; otherwise, the immediately following exchange trading day. Trading Hours: Eurex operates in three trading phases. In the pre-trading period users may make inquiries or enter, change or delete orders and quotes in preparation for trading. This period is between 7:30 and 8:00 a.m. The main trading period is between 8:00 a.m. and 7:00 p.m. Trading ends with the post-trading period between 7:00 p.m. and 8:00 p.m. Daily Price Limit: None

Chapter 7 30

Pricing Interest Rate Futures Contracts

Because, interest rate futures trade in a full carry market, the foundation for pricing interest rate futures is the Cost-of-Carry-Model that we discussed in Chapter 3.

This section introduces a review of the Cost-of-Carry Model as discussed in Chapter 3, including:

1. Cost-of-Carry Rule 3

2. Cost-of-Carry Rule 6

3. Features that Promote Full Carry

4. Repo Rates

5. Cost-of-Carry Model in Perfect Market

6. Cash-and-Carry Arbitrage for Interest Rate Futures

Chapter 7 31

Cost-of-Carry Rule 3

Recall: the cost-of-carry rule #3 says:

Where:

S0 = The current spot price

F0,t = The current futures price for delivery of the product at time t

C0,t= The percentage cost required to store (or carry) the commodity from today until time t

)1( ,00,0 tt CSF

Chapter 7 32

Cost-of-Carry Rule 6

Recall: the cost-of-carry rule #6 says:

F0,d = the futures price at t=0 for the the distant delivery contract maturing at t=d

Fo,n= the futures price at t=0 for the nearby delivery contract maturing at t=n

Cn,d= the percentage cost of carrying the good from t=nto t=d

)1( ,,0,0 dnnd CFF

Chapter 7 33

Full Carry Features

Recall from Chapter 3 that there are five features that promote full carry:

1. Ease of Short Selling

2. Large Supply

3. Non-Seasonal Production

4. Non-Seasonal Consumption

5. High Storability

Interest rates futures have each of these features and thus conform well to the Cost-of-Carry Model.

Chapter 7 34

Repo Rate

Recall from Chapter 3 that if we assume that the only carrying cost is the financing cost, we can compute the implied repo rate as:

tt

CS

F,0

0

,01

tt

CS

F,0

0

,01

or

Interest rate futures conform almost perfectly to the Cost-of-Carry Model. However, we must take into account some of the peculiar aspects of debt instruments.

Chapter 7 35

Cost-of-Carry Model in Perfect Market

Assumptions

1. Markets are perfect.

2. The financing cost is the only cost of carrying charge.

3. Ignore the options that the seller may possess such as the option to deliver differing securities.

4. Ignore the differences between forward and futures prices.

Chapter 7 36

Cash-and-Carry Arbitrage for Interest Rate Futures

Recall from Chapter 3 that in order to earn an arbitrage profit, a trader might want to try a cash-and-carry arbitrage.

Recall further that a cash-and-carry arbitrage involves selling a futures contract, buying the commodity and storing it until the futures delivery date. Then you would deliver the commodity against the futures contract.

Applying the cash-and-carry arbitrage to interest rate futures requires careful selection of the commodity’s interest rate (T-bill, T-bond etc) that will be purchased.

Each of the interest rate futures contracts specifies the maturity of the interest rate instrument to be delivered. The interest rate instrument must have this maturity on the delivery date.

Chapter 7 37


Example, a T-bill futures contract requires the delivery of a T-bill with 90 days to maturity on the delivery date.

So, if you sell a T-bill futures contract that calls for delivery in 77 days, we must purchase a T-bill that will have 90 days to maturity, 77 days from today, in order to meet your obligations. That is, you must purchase a T-bill that has 167 days to maturity today.

0 77

1. Sell futures Contract.2. Buy T-bill Futures contract w/ 167 days to maturity.

4. T-bill matures

167

3. Deliver T-bill (that has now 90 days to maturity) against futures contract.

Table 7.2 and 7.3 further develop this example.

Chapter 7 38


Table 7.2

Interest Rate Futures and Arbitrage

Today's Date: January 5

Futures

Discount Yield

Price ($1,000,000 Face Value)

MAR Contract (Matures in 77 days on March 22)

12.50%

$968,750

Cash Bills: 167Bday TBbill (Deliverable on MAR fu-tures)

10.00

953,611

77Bday TBbill

6.00

987,167

Assume that markets are perfect including the assumption of borrowing and lending at a risk-less rate represented by the T-bill yields. Suppose that you have gathered the information in Table 7.2 and wish to determine if an arbitrage opportunity is present.

How was the bill price of $987,167 from Table 7.2 calculated?

Chapter 7 39


The bill prices were calculated as follows:

360

))((Price Bill

DTMValue FaceDYValue Face

For the March Futures Contract

360

)90)(000,000,1(125.0000,000,1Price Bill

750,968Price Bill

For the March 167-day T-bill

360

)167)(000,000,1(10.0000,000,1Price Bill

611,953Price Bill

For the 77-day T-bill with $1,000,000 face value

360

)77)(000,000,1(06.0000,000,1Price Bill

166,987Price Bill

Chapter 7 40


The transactions necessary to earn an arbitrage profit are given in Table 7.3.

Table 7.3

CashBandBCarry Arbitrage Transactions

January 5 Borrow $953,611 for 77 days by issuing a 77Bday TBbill at 6%. Buy 167Bday TBbill yielding 10% for $953,611. Sell MAR TBbill futures contract with a yield of 12.50% for $968,750.

March 22 Deliver the originally purchased TBbill against the MAR futures contract and collect $968,750. Repay debt on 77Bday TBbill that matures today for $966,008.

Profit: $968,750 B 966,008 $ 2,742

How was the $966,008 from Table 7.3 calculated?

Chapter 7 41


The $966,008 is the face value of a 77-day T-bill with a current price of $953,611. To calculate this value, rearrange the bill price formula:

)(360 DTMDY

Price Bill 360Value Face

)77(06.0360

)611,953($360

Value Face

38.355

960,299,343Value Face

10.008,966Value Face

Rearranging the equation results:

360

))((Price Bill


Chapter 7 42

Cash-and-Carry Arbitrage to Interest Rate Futures

When delivery is due on the futures contract on March 22, you deliver the T-bill (which now has 90 days to maturity) against the futures contract.

0

1. Borrow money2. Buy 167-day T-bill3. Sell a futures contract

4. Deliver the T-bill against the futures contract5. Pay off the loan

1

Combined, these transactions appear as follows on a timeline:

Time Transaction Cash Flow Mar 22 Deliver 167-day T-bill

(that now has 90 days to maturity) against the futures contract

$968,750

Mar 22 Repay debt on 77-day T-Bill that matures today

$966,008

Profit/contract $2,742

Chapter 7 43

Reverse Cash-and-Carry Arbitrage to Interest Rate Futures

Using the same values as shown in Table 7.2, now assume that the rate on the 77-day T-bill is 8%.

Given this new information and Table 7.2 prices, a reverse cash-and-carry arbitrage opportunity is present. Table 7.4 shows the result.

To calculate the values in Table 7.4 follow the steps shown for the previous cash-and-carry example.

Table 7.4

Reverse CashBandBCarry Arbitrage Transactions

January 5 Borrow $952,174 by issuing a 167Bday TBbill at 10%. Buy a 77Bday TBbill yielding 8% for $952,174 that will pay $968,750 on March 22. Buy one MAR futures contract with a yield of 12.50% for $968,750.

March 22 Collect $968,750 from the maturing 77Bday TBbill. Pay $968,750 and take delivery of a 90Bday TBbill from the MAR futures contract.

June Collect $1,000,000 from the maturing 90Bday TBbill that was delivered on the futures contract. Pay $998,493 debt on the maturing 167Bday TBbill.

Profit: $1,000,000 B 998,493 $ 1,507

Chapter 7 44

Reverse Cash-and-Carry Arbitrage to Interest Rate Futures

Combined, these transactions appear as follows on a timeline:

Jan 5 Mar 22

1. Borrow money2. Buy 77-day T-bill3. Buy a futures contract

6. Collect 1 M from mature

T-bill7. Pay off loan

Jun 20

4. Collect from maturing T-bill5. Accept delivery on 90-day contract

Chapter 7 45

Interest Rate Futures Rate Relationships

Rate relationship that must exist between interest rates to avoid arbitrage:

Consider two methods of holding a T-bill for 167 days.

Method 1:

Buy a 167 day T-bill

Method 2:

Buy a 77 day T-bill.

Buy a futures contract for delivery of a 90 day T-bill in 77 days.

Use the futures contract to buy a 90-day T-bill.

These investment appear as follows on a timeline.

Chapter 7 46

Interest Rate Futures Rate Relationships

Method 2

Jan 5 Mar 22

1. Buy 77-day T-bill2. Buy a future contract for 90-day T-bill w/ 77 days to maturity

5. Collect from maturing T-bill

Jun 20

3. Collect from maturing T-bill4. Buy a 90-day T-bill using the futures contract

Either of these two methods of investing in T-bills has exactly the same investment and exactly the same risk.

Since both investment have exactly the same risk and exactly the same investment, they must have exactly the same yield to avoid arbitrage.

Jan 5 Mar 22

1. Buy 167-day T-bill

2. Collect from maturing T-bill

Jun 20

Method 1

Chapter 7 47

Financing Cost and Implied Repo Rate

Calculate the rate that must exist on the 77-day T-bill to avoid the arbitrage as follows:

Event Discount Yield Price MAR Contract (Matures on March 22 or 77 days)

12.5% $968,750

Cash T-bill 167-day T-bill (deliverable on MAR futures)

10% $953,611

77-Day T-bill ??% $???

Use the no arbitrage equation to determine the appropriate yield on the 77-day T-bill by, using the following equation:

360XContract Futures of Price

Price BillT Term LongContract Futures of PriceYieldNA

DTMFC

Where:NA Yield = the no arbitrage YieldDTMFC = days to maturity of the futures contract

Chapter 7 48


36077

750,968$

611,953750,968$

XYield NA

86.204,207$

139,15$Yield NA

07306.0Yield NA

So in order for there to be no arbitrage opportunities available, the yield on the 77 day T-bill must be 7.3063%.

If the yield on the 77 day T-bill is greater than 7.3063%, then engage in a reverse cash-and-carry arbitrage. If the yield on the 77 day T-bill is less than 7.3063%, engage in a cash-and-carry arbitrage.

Chapter 7 49


We can also calculate the implied repo rate as follows:

tt

CS

F,0

0

,01

In our case the spot price is the price of the 167-day to maturity T-bill, so:

tC ,01611,953$

750,968$

015875.11 ,0 tC

The implied repo rate is the cost of holding the commodity for 77 days, between today and the time that the futures contract matures, assuming this is the only financing cost, it is also the cost of carry.

The implied repo rate (C) is 1.5875%

Chapter 7 50


1. If the implied repo rate exceeds the financing cost, then exploit a cash-and-carry arbitrage opportunity

2. If the implied repo rate is less than the financing cost, then exploit a reverse cash-and-carry arbitrage.

Borrow funds Buy cash bond Sell futures

Hold bondDeliver against futures

Realize profit

Buy futures Sell bond short Invest proceeds until futures exp.

Take deliveryRepay short sale obligation

Realize profit

Chapter 7 51

Cost-of-Carry Model for T-Bond Futures

The cost of carry concepts for T-bill futures that we have just examined also apply to T-bond futures. However, the computation must be adjusted to reflect the coupon payment and accrued interests.

Chapter 7 52

Cost-of-Carry Model in Imperfect Markets

In this section, the borrowing and lending assumptions are relaxed, and the Cost-of-Carry Model is explored under the following assumption:

1. The borrowing rate exceeds the lending rate.

2. The financing cost is the only carrying charge.

3. Ignore the options that the seller may possess.

4. Ignore the differences between forward and futures prices.

Recall that allowing the borrowing and lending rates to differ leads to an arbitrage band around the futures price. Now assume that the borrowing rate is 25 basis points, or one-fourth of a percentage point, higher than the lending rate. Continuing to use our T-bill example.

Chapter 7 53

Cash-and-Carry Strategy

Instrument

Lending Rate

Borrowing Rate

77-day bill

7.3063

7.5563 167-day bill

10.0000

10.2500

Table 7.6 CashBandBCarry Transactions

with Unequal Borrowing and Lending Rates January 5 Borrow $953,611 for 77 days at the 77Bday borrowing rate of 7.5563. Buy 167Bday TBbill yielding 10% for $953,611. Sell one TBbill futures contract with a yield of 12.29% for $969,275. March 22 Deliver the originally purchased TBbill against the MAR futures contract and collect $969,275. Repay debt on 77Bday TBbill that matures today for $969,277. Profit: -$2 0

Notice that the entire arbitrage profit disappears when these differential borrowing and lending rates are considered.

Chapter 7 54

Reserve Cash-and-Carry Transaction

Table 7.7

Reverse CashBandBCarry Transactions with Unequal Borrowing and Lending Rates

January 5 Borrow $952,454 at the 167-day borrowing rate of 10.25%. Buy a 77-day T-bill yielding 7.3063% for $952,454. Buy 1 MAR futures contract with a futures yield of 12.97% for $967,575. March 22 Collect $967,575 from the maturing 77-day T-bill. Pay $967,575 and take delivery of a 90-day T-bill on the futures contract. June 20 Collect $1,000,000 from the maturing 90-day T-bill that was delivered on the futures contract. Pay $1,000,003 debt on the maturing 167-day T-bill. Profit: -$3 0

Again notice that the entire arbitrage profit disappears when these different borrowing and lending rates are considered.

Chapter 7 55

A Practical Survey of Interest Rate Futures Pricing

Recall from Chapter 3 that transaction costs lead to a no-arbitrage band of possible futures prices. In essence, transaction costs increase the no-arbitrage band just as unequal borrowing and lending rates do.

Impediments to short selling as a market imperfection would frustrate the reverse cash-and-carry arbitrage strategy.

From a practical perspective, restrictions on short selling are unimportant in interest rate futures pricing because:

– Supplies of deliverable Treasury securities are plentiful and government securities have little (or zero) convenience yield.

– Treasury securities are so widely held, many traders can simulate short selling by selling T-bills, T-notes, or T-bonds from inventory. Therefore, restrictions on short selling are unlikely to have any pricing effect.

Chapter 7 56

Speculating with Interest Rate Futures

There are several ways that you can speculate with interest rate futures:

1. Outright Position.

2. Intra-Commodity T-Bill Spread

3. A T-bill/Eurodollar (TED) Spread

4. Notes over Bonds (NOB)

Chapter 7 57

Speculating with Outright Position

Two ways to speculate with outright positions are:

1. Purchase an interest rate futures contract: a bet that interest rates will go down.

2. Sell an interest rate futures contract: a bet that interest rates will go up.

Suppose you think that interest rates will go up.

The transactions necessary to bet on your hunch are outlined in Table 7.80.

Table 7.8

Speculating with Eurodollar Futures Date

Futures Market

September 20 Sell 1 DEC 90 Eurodollar futures at 90.30.

September 25 Buy 1 DEC 90 Eurodollar futures at 90.12.

Profit: 90.30 B 90.12 = .18 Total Gain: 18 basis points $25 = $450

Chapter 7 58

Speculating with Outright Position

Interest rates have gone up as you predicted. Your profit (based on $25 per basis point contract) is:

Profit = (Sell Rate – Buy Rate)($25)

Profit = (90.30 – 90.12) = 0.18

0.18 is 18 basis points, each of which implies a $25 change in contract value so:

Profit = (Basis Points)(Value per Basis Point)

Profit = (18)($25) = $450

Chapter 7 59

Intra-Commodity T-Bill Spread

If you don’t know if rates will rise or fall, but do think that the shape of the yield curve will change, (that is the relationship between short term interest rates and long term interest rates will change) you might engage in an Intra-commodity T-bill spread.

If you think that the spread will narrow (the yield curve will become flatter) you would buy the longer term contract and sell the shorter term contract.

If you think that the spread will widen (the yield curve will become steeper), you would buy the shorter term contract and sell the longer term contract.

Chapter 7 60


Suppose you have the following information (Table 7.9) regarding T-bills and T-bill futures contracts for March 20. The left 2 columns are T-bills, and the right 3 columns are futures contracts. You think that the yield curve will flatten and wish to trade to make a profit.

Table 7.9

Spot and Futures Eurodollar Rates for March 20 Time to Maturity or Fu-tures Expiration

Add-on Yield

Futures Contract

Futures

Yield

IMM In-

dex 3 months

10.00%

JUN

12.00%

88.00

6

10.85

SEP

12.50

87.50 9

11.17

DEC

13.50

86.50

12

11.47

Chapter 7 61


Notice that the T-bills exhibit an upward sloping yield curve.

Notice that the futures contract yields also exhibit and upward sloping yield curve.

If the yield curve flattens, the yield spread between subsequent maturing futures contracts must narrow. That is, the difference between the yield on the December contract and on the September contract must narrow.

Since you think that the spread will narrow (the yield curve will become flatter) you would buy the longer term contract and sell the shorter term contract, as it is demonstrated in Table 7.10.

Chapter 7 62


Table 7.10

Speculation on Eurodollar Futures Date

Futures Market

March 20

Buy the DEC Eurodollar futures at 86.50. Sell the SEP Eurodollar futures at 87.50.

April 30

Sell the DEC Eurodollar futures at 88.14. Buy the SEP Eurodollar futures at 89.02.

Profits: DEC

88.14

B86.50 1.64

SEP

87.50 B89.02 B 1.52

Total Gain: 12 basis points $25 = $300

Gain in Basis Points

Change in December Contract 1.64Change in September Contract -1.52

Net Change in Positions 12 basis points

Each Basis Point is worth $25

Profit

Net Change in Positions 12Basis Point Value $25Profit $300

Chapter 7 63

T-Bill/Eurodollar (TED) Spread

The TED spread is the spread between Treasury bill contracts and Eurodollar contracts.

In theory, Treasury bills should always have a lower yield than Eurodollar deposits.

T-bills are backed by the full taxing authority of the U.S. government.

Eurodollar deposits are generally not backed by the respective governments.

Thus, T-bills are a safer investment and as such, should pay a lower interest rate. Eurodollars are riskier and should pay a higher rate of interest.

How much lower/higher?

The amount of the difference depends upon world events. To the extent that the world situation is considered safe, the difference should be low. To the extent that the world situation is unsafe, the difference should be high.

Table 7.11 shows the transactions necessary to engage in a TED spread when you wish to bet that the spread will widen.

Chapter 7 64

T-Bill/Eurodollar (TED) Spread

Table 7.11

InterBCommodity Spread in ShortBTerm Rates Date

Futures Market

February 17

Sell one DEC Eurodollar futures contract with an IMM Index value of 90.29. Buy one DEC TBbill futures contract yielding 8.82% with an IMM Index value of 91.18.

October 14 Buy one DEC Eurodollar futures contract with an IMM Index value of 89.91. Sell one DEC TBbill futures contract yielding 8.93% with an IMM Index value of 91.07.

Profits:

Eurodollar 90.29 B89.91 .38

TBbill

91.07

B91.18 B .11

Total Profit: 27 basis points $25 = $675

Notice that the spread widened as the trader expected, allowing him/her to earn a $675 profit.

Chapter 7 65

Notes over Bonds (NOB)

The NOB is a speculative strategy for trading T-note futures against T-bond futures.

NOB spreads exploit the fact that T-bonds underlying the T-bond futures contract have a longer duration than the T-notes underlying the T-note futures contract. A given change in yields will cause a greater price reaction for the T-bond futures contract.

Thus, the NOB spread is an attempt to take advantage of either changing levels of yields or a changing yield curve by using an inter-market spread.

Chapter 7 66

Hedging with Interest Rate Futures

There are several ways that you can hedge with interest rate futures, including:

1. Long Hedges

2. Short Hedges

3. Cross-Hedges

Chapter 7 67

Hedging with Interest Rate Futures

Recall that the goal of a hedger is to reduce risk, not to generate profits.

Using interest rate futures to hedge involves taking a futures position that will generate a gain to offset a potential loss in the cash market.

This also implies that a hedger takes a futures position that will generate a loss to offset a potential gain in the cash market.

Chapter 7 68

Long Hedges

On December 15, a portfolio manager learns that he will have $970,000 to invest in 90-day T-bills six months from now, on June 15. Current yields on T-bills stand at 12% and the yield curve is flat, so forward rates are all 12% as well. The manager finds the 12% rate attractive and decides to lock it in by going long in a T-bill futures contract maturing on June 15, exactly when the funds come available for investment as Table 7.12 shows:

Table 7.12

A Long Hedge with TBBill Futures Date

Cash Market

Futures Market

December 15

A portfolio manager learns he will receive $970,000 in six months to invest in TBbills. Market Yield: 12% Expected face value of bills to purchase $1,000,000.

The manager buys one TBbill futures contract to mature in six months. Futures price: $970,000

June 15

Manager receives $970,000 to invest. Market yield: 10% $1,000,000 face value of TBbills now costs $975,000.

The manager sells one TBbill futures contract maturing immediately. Futures yield: 10% Futures price: $975,000

Loss = -$5,000

Profit = $5,000

Net wealth change = 0

Chapter 7 69

Long Hedges

With current and forward yields on T-bills at 12 percent, the portfolio manager expects to be able to buy $1,000,000 face -value of T-bills for $970,000 because:

360

))((Price Bill


360

)90)(000,000,1($12.0000,000,1$Price Bill

On June 15, the 90-day T-bill yield has fallen to 10%. Thus, the price of a 90 day T-bill is:

360

)90)(000,000,1($10.0000,000,1$Price Bill

000,975$Price Bill

000,970$Price Bill

Thus, if the manager were to purchase the T-bill in the market, he would be $5,000 short.

360

))((Price Bill


Chapter 7 70

Long Hedges

The futures profit exactly offsets the cash market loss for a zero change in wealth. With the receipt of the $970,000 that was to be invested, plus the $5,000 futures profit, the original plan may be executed, and the portfolio manager purchases $1,000,000 face value in 90-day T-bills.


The idealized yield Curve Shit for the long Hedge.

Chapter 7 71

Short Hedge

Table 7.13

Hedging a Bank=s Cost of Funds Using Interest Rate Futures

Date

Cash Market

Futures Market

March

Bank makes nine-month fixed rate loan financed by a six-month CD at 3.0 percent and rolled over for three months at an expected rate of 3.5 percent.

Establish a short position in SEP Eurodollar futures at 96.5 reflecting a 3.5 percent futures yield.

September

Three-month LIBOR is now at 4.5 percent. The bank=s cost of funds are one percent above its expected cost of funds of 3.5 percent. The additional cost equals $2,500, i.e., 90/360 x .01 x $1 million..

Offset one SEP Eurodollar futures contract at 95.5 reflecting a 4.5 percent futures yield. This produces a profit of $2,500 = 100 basis points x $25 per basis point x 1 contract.

Total Additional Cost of Funds: $2,500

Futures Profit: $2,500

Net Interest Expense After Hedge: 0

Banks may wish to hedge their interest rate positions to lock in profits. Table 7.13 demonstrates how a bank that makes a one million dollar fixed rate loan for 9 months, and can only finance the loan with 6-month CDs, can hedged its position.

Because the bank hedged, its profits were not affected by a change in interest rates.

Chapter 7 72

Cross-Hedge

Recall that a cross-hedge occurs when the hedged and hedging instruments differ with respect to:

1. Risk level

2. Coupon

3. Maturity

4. Or the time span covered by the instrument being hedged and the instrument deliverable against the futures contract.

To illustrate how a cross-hedge is conducted, assume that a large furniture manufacturer has decided to issue one billion 90-day commercial paper in 3 months. Table 7.14 illustrate the cross-hedge.

Chapter 7 73

Cross-Hedge

Table 7.14

A CrossBHedge Between T-bill Futures and Commercial Paper

Date

Cash Market

Futures Market

Time = 0

The Financial V.P. plans to sell 90Bday commercial paper in 3 months in the amount of $1 billion, at an expected yield of 17%, which should net the firm $957,500,000.

The V.P. sells 1,000 TBbill futures contracts to mature in 3 months with a futures yield of 16%, a futures price per con-tract of $960,000, and a total futures price of $960,000,000.

Time = 3 mos. The spot commercial paper rate is now 18%, the usual 2% above the spot TBbill rate. Consequently, the sale of the $1 billion of commercial paper nets $955,000,000, not the expected $957,500,000.

The TBbill futures contract is about to mature, so the TBbill futures rate = spot rate = 16%. The futures price is still $960,000 per contract, so there is no gain or loss.

Opportunity loss = ?

Gain/loss = 0

Net wealth change = ?

Chapter 71 CHAPTER 7 INTEREST RATE FUTURES In this chapter, we explore one of the most successful innovations in the history of futures markets; that is,

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