Hedging Interest Rate Risk
Treasury/Eurodollar Futures
Derivative Securities




Stocks and Bonds represent claims to specific
future cash flows
Derivative securities on the other hand represent
contracts that designate future transactions
Currently, there are approximately 300 million
derivative contracts outstanding with a market
value of around $50 Trillion
While equity trading is centered in New York
(NYSE, NASDAQ), derivative markets are
centered in Chicago (CME, CBOT, CBOE)
Futures Contracts
A futures contract describes a transaction (Commodity,
Price, and Quantity) that will be made in the future.
In “Trading Places” (1983), Eddie
Murphy and Dan Ackroyd were
trading Orange Juice Futures
Futures Contracts
Orange Juice futures (FCOJ) are
traded on the NYBOT (New York
Board of Trade)
Contract = 15,000 Lbs. ; Price = cents/lb
Exp
Open
High
Low
Settle
Change Interest
MAR
85.75
86.00
84.20
85.20
-1.25
18,849
MAY
88.20
88.40
86.60
87.70
-1.20
14,354
JUL
88.50
88.50
87.60
88.45
-1.15
1,889
NOV
91.50
91.50
90.00
89.95
-1.65
905
Every contract must have two participants (Long = Buy, Short = Sell)
Exp
Open
High
Low
Settle
Change Interest
MAR
85.75
86.00
84.20
85.20
-1.25
18,849
MAY
88.20
88.40
86.60
87.70
-1.20
14,354
JUL
88.50
88.50
87.60
88.45
-1.15
1,889
NOV
91.50
91.50
90.00
89.95
-1.65
905
A long position
in MAR FCOJ
would require
you to purchase
FCOJ in March
Now
Mar
A short position
in JUL FCOJ
would require
you to deliver
FCOJ in March
Apr
May
June
July
Aug
Dealers pass orders along to the
pit traders who create a
contract.
Long
Short
3 May Contracts (15k * 3 = 45k lbs.) @ 88 cents/lb.
The contracts are then passed along to the
exchange who will become the middleman
Note: the exchange is holding two contracts
with a zero net position
Short (3 contracts)
Long (3 Contracts)
Long (3 contracts)
Short (3 Contracts)
Contract Completion (FCOJ)
First Notice Date
Last Trading Day
Last Delivery Date
First Delivery Date
Last Notice Date
May 1
May 8
May 10
May 23
May 31
Contract Completion
Suppose that, on May 3, the short position decides that he
wants out of the contract. The current May futures price is .92
per Lb
3 Contracts
(Short) @ .88/LB
He could take a long position on 3 May
contracts at a price of .92/LB
This would effectively “cancel out” the
previous position at a loss of 3 cents/LB
.03*45,000 = $1,350 Loss
May 1
May 8
May 10
May 23
May 31
Contract Completion
Suppose that, on May 12, the short position opts for delivery
of the commodity. The current spot price is .84 per Lb
3 Contracts
(Short) @ .88/LB
The Exchange
Pairs up Longs
with Shorts
3 Contracts
(Long) @ .88/LB
Profit = (.88-.84)*45,000
Loss = (.88-.84)*45,000
= $1,800
= $1,800
May 1
May 8
May 10
May 23
May 31
Types of Futures
Currencies
Agriculture
Metals &
Energy
Financial
British Pound
Lumber
Copper
Treasuries
Euro
Milk
Gold
LIBOR
Japanese Yen
Cocoa
Silver
Municipal Index
Canadian Dollar
Coffee
Platinum
S&P 500
Mexican Peso
Sugar
Oil
DJIA
Cotton
Natural Gas
Nikkei
Wheat
Cattle
Soybeans
Eurodollar
Stock Index Futures

Stock Index Futures have no underlying
commodity
S&P 500
 NYSE Composite
 Value Line Index

These contracts are settled on a cash basis:
Short Position Profits = (F – S)*500
Long Position Profits = (S – F)*500
F = Futures Price, S = Current Spot Price

Regardless, futures positions are making
“bets” on the price of the underlying
commodity.
Long Position
Profits from
price increases
Short Position
Profits from price
decreases
Treasury Futures
Treasury futures first began trading on the CME in 1976.
The underlying commodity is a Treasury Bill, Note, or Bond.
Remember, when interest rates rise, Treasury prices fall!
Long Position
Profits from
price
increases
Profits from
decreasing
interest rates
Short Position
Profits from
price
decreases
Profits from
increasing
interest rates
T-Bill Futures
With T-Bill Futures, the commodity is a $1M
Treasury Bill with 3 months left until maturity
Contracts exist for February, March, April, June,
September, and December delivery
First
Trading Day
Nov 16, 2004
Last Trading
Day (T-Bill
Auction)
Delivery
Day
Feb 14
Feb 18
T-Bill Yields
We have already calculated the Yield to Maturity for
90 Day Treasury Bills
YTM =
Face Value - Price
Price
365
t
*100
Days left until
maturity
Annualized
Often, the yield referred to for Treasury Bills is the
discount yield
Annualized with a 360 day
year
Face Value - Price
360
DY =
*100
Face
t
Value
Interest As a percentage of
Face Value rather than Price
Pricing T-Bill Futures
T-Bill futures are listed using the IMM (International
Monetary Market) Index
IMM = 100 – Discount Yield
For example, if the Price of a $100, 90 Day Treasury were $98.
DY =
$100 - $98
$100
360
90
*100 = 8%
IMM = 100 – 8 = 92
Every .005 increase in the IMM raises the value of a long T-Bill
position by $12.50 ($25 per basis point).
Eurodollar

The term Eurodollar refers to deposits
denominated in a currency other than the
bank’s home currency
European banks offer Eurodollar time
deposits (terms can range from overnight to
several years)
 European banks will lend dollar reserves to
each other at the LIBOR rate (London Interbank Offering Rate)

Eurodollar Futures (1981)


The underlying commodity is a $1M 3 month Eurodollar
time deposit. However, these deposits are not
marketable. Therefore, Eurodollar futures are settled on
a cash basis
Eurodollar futures can be treated like a T-Bill Future
IMM = 100 – LIBOR
Every .005 increase in the IMM raises the value of the long
position by $12.50. ($25 per basis point)
Eurodollar Futures vs. T-Bill
Futures
T-Bill Futures
Volume (2001)
123
($123M)
Eurodollar
Futures
730,000
($730B)
•As the Eurodollar market grew, it became more liquid
relative to the T-Bill market
•LIBOR is a “risky” rate. Therefore, it correlates better
with other risks
Pricing T-Bill/Eurodollar Futures
Suppose that a march Eurodollar future (expires in 47
days) was currently selling for 94.555
We also have the current money
rates (LIBOR)
Term
1 Month
3 Months
6 Months
1 Year
Yield
5.18%
5.3125
5.6438
5.8163
IMM = 100 - LIBOR
This contract is paying
an annualized (yield) of
100 – 94.555 = 5.445%
Term
1 Month
3 Months
6 Months
1 Year
Purchase/Sale
of Eurodollar
Future
Now
Yield
5.18%
5.3125
5.6438
5.8163
The Eurodollar Future
currently has an annual
yield of 5.445%
5.445
= 1.3613%
4
$1M (1.013613) = $1,013,613
Delivery of a
$1M 90Day
Eurodollar
account
Receipt of
$1,013,613
Day 47
Day 137
90 Days
Use a linear interpolation
to get the 47 day spot rate
5.2175%
Yield
47
360
Term
1 Month
3 Months
6 Months
1 Year
= .6811%
47 Day Return
5.3125%
5.2175%
5.18%
Term
1 Month
3 Months
47 Days
Yield
5.18%
5.3125
5.6438
5.8163
Use a linear interpolation to
get the 137 day spot rate
5.4855%
Yield
137
360
Term
1 Month
3 Months
6 Months
1 Year
= 2.0875%
137 Day Return
5.6438%
5.4855%
5.3125%
Term
3 Months
6 Months
137 Days
Yield
5.18%
5.3125
5.6438
5.8163
Term
S(47)
S(137)
Yield
.6811%
2.0875%
The Eurodollar Future
currently has an annual
yield of 5.445%
5.445
= 1.3613%
4
S(47) = .6811%
S(137) = 2.0875%
Now
Day 47
1.020875
1.006811 =1.01397 = 1.3970% = F(47,90)
Day 137
The Eurodollar Future
currently has an annual
yield of 5.445% (1.3613%)
The implied no-arbitrage
interest rate between 47 and
137 days is 5.588% (1.3970%)
IMM = 100 – 5.445 = 94.555
IMM = 100 – 5.588 = 94.412
The interest rate on the futures contract is to low!!
or, alternatively
The price of the futures contract is too high!!!
Borrow at Futures Rate (Sell
a Futures contract)
Now
Day 47
Profit = 1.013970 – 1.013613 $1M = $357
Day 137
Lend at the implied forward
rate
How do you lend at the implied
forward rate?
By lending for the entire 137 day period and
borrowing for the first 47 days, your net position
is as a lender for the last 90 day period!
Borrow
Lend
Now
Day 47
Day 137
Go Short on a the futures contract
at a price of 94.555
Lend $992,885 for 137 days at the
spot rate of 5.4855% (You will be
paid $1,013,613 in 137 days)
Borrow $992,885 for 47 days at the
spot rate of 5.2175%
Receive $1,013,613 from the
original 137 day loan
Pay $1,013,613 on the 90 day loan
Borrow $1,000,000 at the rate
established by the futures contract
(5.445%)
Pay back the $992,885 Loan +
interest ($999,643)
Now
Day 47
Day 137
On the 47th day, you get a
net cash flow of $352. This
is the present value of $357
dollars to be received in 90
Days (you get the profits on
day 47 rather than day 137)
Date
Cash In
Cash Out
Now
$992,855
$992,855
47 Days
$1,000,000
$999,648
137 Days
$1,013,613
$1,013,613
The no arbitrage price of a price of a futures
contract will reflect the forward rate implied by
the yield curve. But remember, the forward rate
is the expected future spot rate
Futures Rate = Expected Future Spot Rate
Treasury Note/Bond Futures
Contract
Underlying Asset
20 Year T-Bond
(FV = $100,000)
10 Year T-Note
(FV = $100,000)
5 Year T-Note
(FV = $100,000)
2 Year T-Note
(FV = $200,000)
15-20 Year T-Bond with
a 6% coupon
6.5 – 10 Year T-Note
with a 6% coupon
4.25 – 5 Year T-Note
with 6% coupon
1.75 – 2 Year T-Note
with 6% coupon
The commodity for T-Note/Bond futures is a
Treasury with a 6% annual coupon. What if
there are no 6% bonds available?
Treasury Note/Bond futures are based on
cheapest to deliver (CTD) basis.
Requirements for Delivery
1. The Face value of the
delivered notes must sum to
$100,000 (per contract)
2. All the notes must have the
same characteristics (term,
coupon)
It’s the short position’s
option to deliver whatever
has the lowest cost
Conversion Factors
Suppose that you have a short position on a a Treasury bond future
that expires this month (any bond with an expiration date between 2020
and 2030 would be acceptable for delivery:
Maturity
May 2020
August 2023
August 2025
Coupon
8.75%
7.25%
6.875%
Bid Price
149:16
134:21
132:21
The cheapest to deliver bond will always be the lowest coupon,
longest maturity bond
Conversion Factors
Maturity
Coupon
May 2020
August 2023
August 2025
8.75%
7.25%
6.875%
Conversion
Factor
1.2695
1.1331
1.1017
The conversion factors are meant to make all deliverable bonds
“equally attractive”
Invoice
Amount
=
Contract
Size
Futures
Price
Conversion
Factor
Accrued
Interest
Requirements for Delivery
1. The Face value of the
delivered notes must sum to
$100,000 (per contract)
It’s the short position’s
option to deliver whatever
has the lowest cost
2. All the notes must have the
same characteristics (term,
coupon)
To Find the cheapest to deliver bond/note
Maximize
Conversion
Factor
Current
Futures
Price
-
Spot
Price
Note: This will always be negative
Pricing T-Note/Bond Futures
20 Year
Treasury
Delivered
Now
20 Year
Treasury
Expires
March
March
2025
The Logic behind pricing treasury note/bond futures is the
same as with T-Bill futures. The price should reflect
expectation of future spot rates. However, note that
expectations of future spot rates are already incorporated in
bond prices!
Futures Price =
Expected
Future
+ (Carry Costs – Carry Return)
Treasury
Price
Arbitrage Costs
Hedging
Lets return to the 5 year Treasury Note example. Interest rates are
currently 5% and are expected to stay at 5% (the yield curve is flat). A
5 year treasury note with $500,000 of face value and a 5% annual
coupon.
$25,000
$25,000
$25,000 $25,000 $525,000
+
+
+
+
P =
= $500,000
2
3
4
(1.05)
(1.05)
(1.05)
(1.05)
(1.05) 5
We already calculated the Modified Duration for this
bond
MD = 4.3
That is, a 100 basis point increase in the interest rate
lowers this bond’s price by (.043)($500,000) = $21,500
Hedging with T-Bill Futures
Short Position
(Futures)
Profits from
price
decreases
Profits from
increasing
interest rates
If you are long in bonds, you are worried about rising interest rates
(rising interest rates lower the value of your bond). Therefore, you
could hedge this risk by holding short positions in T-Bill futures
(Short positions make money when interest rates drop)
Hedging with T-Bill Futures
Short Position
(Futures)
Profits from
price
decreases
Profits from
increasing
interest rates
A perfect hedge eliminates all your interest rate risk
Change in value
of Value of
=
Futures position
Change in
# of Futures value of
Contracts
each
contract
$2,500
=
Change in
value of
bond
position
$21,500
$21,500/$2,500 = 8.6 Contracts
Hedging with T-Bill Futures
Change in value
of Value of
=
Futures position
Change in
# of Futures value of
Contracts
each
contract
=
Change in
value of
bond
position
$2,500
$21,500
$21,500/$2,500 = 8.6 Contracts
Dollar Duration of
Bonds
Hedge Ratio =
Dollar Duration of
Futures
=
MD(B)
MD(F)
FV(B)
FV(F)
=
4.3
.25
$500K
$1M
One Problem…..
156.71
160
140
123.41
120
100
86.38
80
60
45.35
39.18
40
20
0
1Yr
2Yr
3Yr
4Yr
5Yr
Here we have the 5 year Treasury key durations.
Note that this bond’s price is most sensitive to the 5
Year spot rate. The future’s value is based on the 90
day treasury rate
X 100
Change in value
of Value of
=
Futures position
Change in
# of Futures value of
Contracts
each
contract
$2,500
=
Change in
value of
bond
position
$21,500
$21,500/$2,500 = 8.6 Contracts
We assumed that the 90 Day T-Bill rate and the 5 Year Rate were
perfectly correlated. Suppose, instead, that we have
Change in
Change in
= (.5) 90 Day
5 Year Rate
Treasury
Rate
The hedge ratio drops to 4.3!
Hedging with T-Note/Bond Futures

The strategy would be the same. If you are
worried about increasing interest rates, take a
short position in futures contracts. The hedge
ratio for T-Note/Bond futures depends on




Size of bond position relative to the size of a futures
contract
Duration of your bond position relative to the duration
of the underlying asset in the futures contract
Correlation between the interest rate affecting your
bond portfolio and the interest rate influencing the
futures price
Impact of interest rate on CTD bond