CHAPTER 7
INTEREST 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
1
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 3month Eurodollar deposits).
Chapter 7
2
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
3
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 threemonth 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
4
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
5
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
6
Eurodollar Add-on Yield
Add  on Yield 
(
$ Discount
Price
)( )
360
DTM
In order to calculate the add-on yield, the price and
discount must be computed as follows:
$ Discount 
DY ( Face Value)( DTM )
360
Price  Face Value  $Discount
Or equivalently
Price  Face Value 
DY ( Face Value)( DTM )
360
Chapter 7
7
Eurodollar Add-on Yield
Suppose you have a 90-day Eurodollar deposit with a
discount yield of 8.32%.
Step 1: Compute the discount and the price.
Price  Face Value 
Price  1,000,000 
DY ( Face Value)( DTM )
360
0.0832(1,000,000)(90)
360
Price  1,000,000  20,800
Price  $979,200
$Discount  $20,800
Chapter 7
8
Eurodollar Add-on Yield
Step 2: Compute the add-on yield using:
Add  on Yield 
360
($Discount
)(
)
Price
DTM
Add  on Yield 
$20,800 360
(979
)( 90 )
,200
Add  on Yield  0.085
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:
Face Value   Add  on Yield  DTM  360
$1,000,000  .0001 90  360  $25
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
Chapter 7
9
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
10
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
11
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
12
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
13
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
14
Treasury Bill Futures
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
15
Treasury Bill Futures
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
16
Other Short-Term Interest Rate Futures
Insert Figure 7.1 here
Chapter 7
17
Longer-Maturity Interest Rate Futures
Longer-maturity interest rate futures are based on couponbearing 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
18
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
19
Treasury Bond Futures
Price Quotation for Major Interest Rate Futures
Contracts
Insert Figure 7.1 Here
Chapter 7
20
Treasury Bond Futures Delivery Process
Insert Figure 7.2 here
Chapter 7
21
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
22
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
23
Treasury Bond Futures Conversion
Factor
Invoice Amount  DSP ($100,000)(CF )  AI
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 on
the 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.
Chapter 7
24
T-Bond and T-Notes Delivery Sequence
Table 7.1 shows key dates in the delivery process for Tbond and T-note futures contracts in 1997.
Table 7.1
Contract
Expiration
MAR 97
JUN
SEPT
DEC
The Delivery Sequence for T-Bond & T-Note Futures Expiring in
1997
First
First
First
Last
Last
Position
Notice
Delivery
Trading
Delivery
FEB 27
FEB 28
MAR 3
MAR 21
MAR 31
MAY 29
MAY 30
JUN 2
JUN 20
JUN 30
AUG 28
AUG 29
SEPT 2
SEP 19
SEP 30
NOV 26
NOV 28
DEC 1
DEC 19
DEC 31
Chapter 7
25
Treasury Bond Futures Conversion
Factor
Table 7.1
Conversion Factors for Treasury-Bond Futures
for September and December 2004
Coupon
Maturity Date
Sep-04
Dec-04
5 1/4
5 1/4
11/15/28
02/15/29
0.9052
0.9047
0.9056
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
26
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
27
CBOT’s 10-Year Treasury Note Futures
Product 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, 84165 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
28
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
29
Pricing Interest Rate Futures Contracts
Because, interest rate futures trade in a full carry market,
the foundation for pricing interest rate futures is the Costof-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
30
Cost-of-Carry Rule 3
Recall: the cost-of-carry rule #3 says:
F 0, t  S 0(1  C 0, t )
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
Chapter 7
31
Cost-of-Carry Rule 6
Recall: the cost-of-carry rule #6 says:
F 0, d  F 0, n(1  Cn , d )
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=n
to t=d
Chapter 7
32
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
33
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:
F 0, t
 1  C 0, t
S0
or
F 0, t
 1 C 0, t
S0
Interest rate futures conform almost perfectly to the Costof-Carry Model. However, we must take into account
some of the peculiar aspects of debt instruments.
Chapter 7
34
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
35
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
36
Cash-and-Carry Arbitrage for Interest
Rate Futures
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.
3. Deliver T-bill (that has
now 90 days to maturity)
against futures contract.
167
4. T-bill
matures
Table 7.2 and 7.3 further develop this example.
Chapter 7
37
Cash-and-Carry Arbitrage for Interest
Rate Futures
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.
Table 7.2
Interest Rate Futures and Arbitrage
Today's Date: January 5
Discount
Yield
Price
($1,000,000
Face Value)
Futures
MAR Contract (Matures in 77 days on March 22)
Cash Bills:
167Bday TBbill (Deliverable on MAR futures)
77Bday TBbill
12.50%
$968,750
10.00
953,611
6.00
987,167
How was the bill price of $987,167 from Table 7.2
calculated?
Chapter 7
38
Cash-and-Carry Arbitrage for Interest
Rate Futures
The bill prices were calculated as follows:
Bill Price  Face Value 
DY ( Face Value)( DTM )
360
For the March Futures Contract
Bill Price  1,000,000 
0.125(1,000,000)(90)
360
Bill Price  968,750
For the March 167-day T-bill
Bill Price  1,000,000 
0.10(1,000,000)(167)
360
Bill Price  953,611
For the 77-day T-bill with $1,000,000 face value
Bill Price  1,000,000 
0.06(1,000,000)(77)
360
Bill Price  987,166
Chapter 7
39
Cash-and-Carry Arbitrage for Interest
Rate Futures
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
40
Cash-and-Carry Arbitrage for Interest
Rate Futures
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:
Bill Price  Face Value 
DY ( Face Value)( DTM )
360
Rearranging the equation results:
Face Value 
360 Bill Price
360  DY ( DTM )
Face Value 
360($953,611)
360  0.06(77)
Face Value 
343,299,960
355.38
FaceValue  966,008.10
Chapter 7
41
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.
Time
Mar 22
Mar 22
Profit/contract
Transaction
Cash Flow
Deliver 167-day T-bill
$968,750
(that now has 90 days to
maturity) against the
futures contract
Repay debt on 77-day
$966,008
T-Bill that matures today
$2,742
Combined, these transactions appear as follows on a
timeline:
0
1
1. Borrow money
2. Buy 167-day T-bill
3. Sell a futures contract
4. Deliver the T-bill
against the futures
contract
5. Pay off the loan
Chapter 7
42
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
43
Reverse Cash-and-Carry Arbitrage to
Interest Rate Futures
Combined, these transactions appear as follows on a
timeline:
Jan 5
1. Borrow money
2. Buy 77-day T-bill
3. Buy a futures
contract
Mar 22
4. Collect from maturing T-bill
5. Accept delivery on 90-day
contract
Chapter 7
Jun 20
6. Collect 1 M
from mature
T-bill
7. Pay off loan
44
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
45
Interest Rate Futures Rate Relationships
Method 1
Jan 5
Mar 22
1. Buy 167-day T-bill
Jun 20
2. Collect from maturing T-bill
Method 2
Jan 5
1. Buy 77-day T-bill
2. Buy a future
contract for 90-day
T-bill w/ 77 days to
maturity
Mar 22
3. Collect from maturing
T-bill
4. Buy a 90-day T-bill using
the futures contract
Jun 20
5. Collect from
maturing
T-bill
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.
Chapter 7
46
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
MAR Contract
(Matures on March 22 or 77 days)
Discount Yield
12.5%
Price
$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:
NA Yield 
Price of Futures Contract  Long Term T  Bill Price
DTMFC
Price of Futures Contract X
360
Where:
NA Yield = the no arbitrage Yield
DTMFC = days to maturity of the futures contract
Chapter 7
47
Financing Cost and Implied Repo Rate
NA Yield 
NA Yield 
$968,750  953,611
77
$968,750 X
360
$15,139
$207 ,204 .86
NA Yield  0.07306
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
48
Financing Cost and Implied Repo Rate
We can also calculate the implied repo rate as follows:
F 0, t
 1 C 0, t
S0
In our case the spot price is the price of the 167-day to
maturity T-bill, so:
$968,750
 1  C 0, t
$953,611
1  C 0, t  1.015875
The implied repo rate (C) is 1.5875%
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.
Chapter 7
49
Financing Cost and Implied Repo Rate
1. If the implied repo rate exceeds the financing cost, then
exploit a cash-and-carry arbitrage opportunity
Borrow funds
Buy cash bond
Sell futures
Realize profit
Deliver against
futures
Hold bond
2. If the implied repo rate is less than the financing
cost, then exploit a reverse cash-and-carry
arbitrage.
Buy futures
Sell bond short
Invest proceeds
until futures exp.
Realize profit
Repay short sale
obligation
Take delivery
Chapter 7
50
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
51
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
52
Cash-and-Carry Strategy
Instrument
77-day bill
167-day bill
Lending Rate
7.3063
10.0000
Borrowing Rate
7.5563
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
53
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
54
A Practical Survey of Interest Rate
Futures Pricing
Recall from Chapter 3 that transaction costs lead to a noarbitrage 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 Tbonds from inventory. Therefore, restrictions on short
selling are unlikely to have any pricing effect.
Chapter 7
55
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
56
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
September 20
September 25
Futures Market
Sell 1 DEC 90 Eurodollar futures at
90.30.
Buy 1 DEC 90 Eurodollar futures at
90.12.
Profit: 90.30 B90.12 = .18
Total Gain: 18 basis points  $25 = $450
Chapter 7
57
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
58
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
59
Intra-Commodity T-Bill Spread
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 Futures Expiration
Add-on
Yield
Futures
Contract
3 months
6
9
12
10.00%
10.85
11.17
11.47
JUN
SEP
DEC
Chapter 7
Futures IMM InYield
dex
12.00%
12.50
13.50
88.00
87.50
86.50
60
Intra-Commodity T-Bill Spread
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
61
Intra-Commodity T-Bill Spread
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.
Sell the DEC Eurodollar futures at 88.14.
Buy the SEP Eurodollar futures at 89.02.
April 30
Profits:
DEC
SEP
88.14
B86.50
1.64
Total Gain: 12 basis points  $25 = $300
87.50
B89.02
B 1.52
Gain in Basis Points
Change in December Contract
Change in September Contract
1.64
-1.52
Net Change in Positions
12 basis points
Each Basis Point is worth $25
Profit
Net Change in Positions
Basis Point Value
Profit
12
$25
$300
Chapter 7
62
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
63
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.
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.
October 14
Profits:
Eurodollar
TBbill
90.29
B89.91
.38
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
64
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 Tnotes 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
65
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
66
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
67
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
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.
June 15
Manager receives $970,000 to
invest.
Market yield: 10%
$1,000,000 face value of TBbills
now costs $975,000.
Loss = -$5,000
Net wealth change = 0
Chapter 7
Futures Market
The manager buys one TBbill
futures contract to mature in six
months.
Futures price: $970,000
The manager sells one TBbill
futures contract maturing
immediately.
Futures yield: 10%
Futures price: $975,000
Profit = $5,000
68
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:
Bill Price  Face Value 
DY ( Face Value)( DTM )
360
Bill Price  $1,000,000 
0.12($1,000,000)(90)
360
Bill Price  $970,000
On June 15, the 90-day T-bill yield has fallen to 10%.
Thus, the price of a 90 day T-bill is:
Bill Price  Face Value 
DY ( Face Value)( DTM )
360
Bill Price  $1,000,000 
0.10($1,000,000)(90)
360
Bill Price  $975,000
Thus, if the manager were to purchase the T-bill in
the market, he would be $5,000 short.
Chapter 7
69
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.
Insert Figure 7.7 here
The idealized yield Curve Shit for the long Hedge.
Chapter 7
70
Short Hedge
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.
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
Because the bank hedged, its profits were not affected
by a change in interest rates.
Chapter 7
71
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
72
Cross-Hedge
Table 7.14
A CrossBHedge Between
T-bill Futures and Commercial Paper
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 contract 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
Date
Net wealth change = ?
Chapter 7
73