Interest rate futures
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DAY COUNT AND QUOTATION
CONVENTIONS
TREASURY BOND FUTURES
EURODOLLAR FUTURES
Duration-Based Hedging Strategies Using
Futures
HEDGING PORTFOLIOS OF ASSETS AND
LIABILITIES
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The interest earned between the two dates
Number of days between dates
 Interest earned in reference period
Number of days in reference period
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Day Count Conventions
in Actual/Actual
the U.S.
Treasury Bonds:
(in
period)
Corporate and municipal 30/360
Bonds:
Money Market Instruments: Actual/360
Bond Principal 100
Coupon Payment dates 3/1 , 9/1(reference
period)
Coupon Rate 8%
124
Calculate
the interest earned between 3/1 and 7/3
 4  2.6957
184
 Treasury bond
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122
 4  2.7111
180
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Actual/Actual (in period)
Corporate and municipal Bonds
30/360
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Day counts can be deceptive(business snapshot)
2/28 2005, 3/1 2005
Which would you prefer?
Treasury bond or Corporate and municipal Bonds
Answer: Corporate and municipal Bonds
30/360
5
360
P
 (100  Y )
n
P is the quoted price(discount rate)
Y is the cash price
n is the remaining life of the Treasury bill
measured in calendar days
360
P
 (100  Y )
n
Face Value = 100
Quoted Price = 8
Interest over the 91-day life=2.0222
91
( 100  0.08 
)
360
2.0222
Interest Rate for the 91 day
period=2.064%
(
)
100  2.0222
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Treasury Bond Price in the U.S are
quoted in dollars and thirty-second of a
dollar
The quoted price is for a bond with a
face value of $100
Cash price = Quoted price +Accrued
Interest
2010/1/10
2010/3/5
2010/7/10
Face Value = 100
Coupon Rate = 11%
Quoted Price = 95-16 or $95.50
54
 $5.5  $1.64
Accrued Interest=
181
Cash price= $95.5+$1.64=$97.14
The cash price of a $100000 bond is $97140
2018/7/10
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Treasury bond future price are quoted in the
same way as the Treasury bond prices
themselves.
One contract involves the delivery of $100000
face value of the bond
A $1 change in the quoted futures price would
lead to a $1000 change in the value of the
future contract
Delivery can take place at any time during the
Cash prices received by party with short
position=(Most Recent Settlement Price ×
Conversion factor) + Accrued interest
 Example
Settlement price of bond delivered = 90.00
Conversion factor = 1.3800
Accrued interest on bond =3.00
Price received for bond is
(1.3800×90.00)+3.00 = $127.20
per $100 of principal
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The party with the short position receives
= (Most recent settlement price ×
Conversion factor)+ Accrued interest
 The cost of purchasing a bond = Quoted
bond price + Accrued interest
 The cheapest-to-deliver is
Min [Quoted bond price – (Most recent
settlement price × Conversion factor)]
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The most recent settlement price =93-08, 93.25
Bond
1
2
3
Quoted bond
price($)
99.59
143.50
119.75
Conversion
factor
1.0382
1.5188
1.2615
The cost of delivering each of the bonds:
Bond1:99.59 – (93.25 ×1.0382)= $2.69
Bond2:143.50 – (93.25 ×1.5188)=$1.87
Bond3:119.75 – (93.25 ×1.2615)=$2.12
Quoted bond price – (Most recent settlement price ×
Conversion factor)]
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A number of factors determine the
cheapest-to-deliver bond [Quoted bond
price – (Most recent settlement price

× Conversion factor)]
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Bond Yields
6%  sloping
upward
downward  sloping
Yield Curve is
The Wild Card Play
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An exact theoretical future price for the
treasury bond contract is difficult to determine
Assume both the cheapest-to-delivery bond
and the delivery date are known
F0  ( S 0  I )e
rt
F: future price
S: spot price
I : present value of the coupons during the life of
future
Coupon
payment
Current
time
60 days
Coupon
payment
122days
148days
Cheapest-to-deliver coupon rate 12%
Conversion factor 1.4000
Current quoted bond price $120
Interest rate 10% annum
Delivery will take place in 270 days
F0  ( S 0  I )e
Maturity of
futures contract
rt
Coupon
payment
35days
Cash price
The present value of a coupon of$6
will be received after 122 days
(0.3342years)
Coupon
payment
Current
time
60 days
Coupon
payment
122days
Maturity of
futures contract
148days
The futures contract lasts for 270 days (0.7397years)
The cash price, If the contract were written on the 12%
F0  ( S 0  I )e
rt
Coupon
payment
35days
Coupon
payment
Current
time
60 days
Coupon
payment
122days
Maturity of
futures contract
148days
Coupon
payment
35days
There are 148 days of accrued interest. The quoted futures price, if the
contract were written on the 12% bond, is calculated by subtracting the
accrued interest
The quoted future price =
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A Eurodollar is a dollar deposited in a
bank outside the United States
Eurodollar futures are futures on the 3month Eurodollar deposit rate (same as
3-month LIBOR rate)
One contract is on the rate earned on $1
million
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Eurodollar futures contracts last as long
as 10 years
When it expires (on the third
Wednesday of the delivery month) the
final settlement price is 100 minus the
actual three month deposit rate
100-R
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If Q is the quoted price of a Eurodollar futures
contract, the value of one contract is 10,000[1000.25(100-Q)]
A change of one basis point or 0.01 in a Eurodollar
futures quote corresponds to a contract price change
of $25
The $25 per basis point rule is consistent that an
interest rate per year changes by 1 basis point, the
interest earned on 1 million dollar for 3 months
change by
For Eurodollar futures lasting beyond two
years we cannot assume that the forward rate
equals the futures rate
 There are Two Reasons reduce the forward
rate
1.Futures is settled daily where forward is settled
once
2.Futures is settled at the beginning of the
underlying three-month period; forward is
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A " convexity adjustment " often made is
1 2
 t1t 2
2
where t1 is the time to maturity of the
Forward rate = Futures rate 
futures contract, t 2 is the maturity of
the rate underlying the futures contract
(90 days later than t1 ) and  is the
standard deviation of the short rate changes
per year (typically  is about 0.012)
Consider the situation where σ=0.012 and we
wish to calculate the forward rate when the 8year Eurodollar futures price quote is 94.
1. In this case T1=8, T2=8.25, and convexity
1 is 2
adjustment
 0.012  8  8.25  0.00475
2
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or 0.475%(47.5 basis points)
2.The future rate is 6% per annum on an actual/360
basis, annual rate of
6%(365/360) = 6.083%
3.The estimate of the forward rate is
6.083-0.475=5.608%
1
 2 t1t 2
2
Maturity of
Futures
2
4
6
8
10
Convexity
Adjustment
(bps)
3.2
12.2
27.0
47.5
73.8
LIBOR deposit rates define the LIBOR zero
curve out to one year
 Once the convexity adjustment just described
has been
made, Eurodollar futures are often used to
extend the zero curve
 Eurodollar futures can be used to determine
forward rates
and
forward rates can then be used to extend
Ti toTthe
i 1
the zero curve
 It is usually assumed that the forward interest
rate
calculated from the future contract applies to the
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Suppose that Fi is the forward rate calculate
from the ith Eurodollar futuresR icontract and
is the zero rate for a maturity Ti
Ri 1Ti 1  RiTi
Fi 
Ti 1  Ti
equation(4.5)R  Fi (Ti 1  Ti )  RiTi
i 1
T
i 1
So that
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If the 400 day LIBOR rate has been
calculated as 4.80% with continuous
compounding and the forward rate for the
period between 400 and 491 days is 5.30%,
0.053
 91
 0is
.048
 400
the 491
days
rate
4.893%
491
Fi (Ti 1  Ti )  RiTi
Ri 1 
Ti 1
 0.04893
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Define:
FC Contract price for interest rate futures
DF Duration of asset underlying futures
at maturity
P Value of portfolio being hedged
DP Duration of portfolio at hedge
maturity
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Assumes that only parallel shift in yield curve
take place
Assumes that yield curve changes are small
P   PD p y
It is approximately true thatF   F D y
C
C F
(4.15)
It is also approximately
true
y
PDP
N  required to hedge
The number of contracts
FC DF
against an uncertain
is
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When the hedge instrument is a Treasury bond
F
futures contract , the hedgerDmust
base
on an
assumption that one particular bond will be
delivered, this mind the hedger must estimate
the cheapest-to-deliver bond
the interest rates and future prices move in
opposite direction
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It is August 2. A fund manager has $10 million
invested in a portfolio of government bonds with a
duration of 6.80 years and wants to hedge against
interest rate moves between August and December
The manager decides to use December T-bond
futures. The futures price is 93-02 or 93.0625 and the
duration of the cheapest to deliver bond is 9.2 years
000, 000 that
6.80 should be shorted
The number of10,
contracts
PDis
P

 79.42
93, 062.50
9.20
N
FC DF
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This involves hedging against interest rate
risk by matching the durations of assets and
liabilities
It provides protection against small parallel
shifts in the zero curve
Duration matching does not immunize a
portfolio against nonparallel shifts in the zero
curve
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This is a more sophisticated approach used by
banks to hedge interest rate. It involves
Bucketing the zero curve
Hedging exposure to situation where rates
corresponding to one bucket change and all
other rates stay the same.