interest rate futures. day count and quotation conventions treasury bond futures eurodollar 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
The interest earned between the two dates
Day Count Conventions in the U.S.
period referencein earnedInterest period referencein days ofNumber
datesbetween days ofNumber
Treasury Bonds: Actual/Actual (in period)
Corporate and municipal Bonds:
30/360
Money Market Instruments: Actual/360
Bond Principal 100Coupon Payment dates 3/1 , 9/1(reference period)Coupon Rate 8%Calculate the interest earned between 3/1 and 7/3 Treasury bond Actual/Actual (in period)
Corporate and municipal Bonds 30/360
6957.24184
124
7111.24180
122
5
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
P is the quoted price(discount rate)Y is the cash pricen is the remaining life of the Treasury bill
measured in calendar days
)100(360
Yn
P
Face Value = 100
Quoted Price = 8
Interest over the 91-day life=2.0222
Interest Rate for the 91 day period=2.064%
)360
9108.0100(
)0222.2100
0222.2(
)100(360
Yn
P
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
2010/3/52010/1/10 2010/7/10
Face Value = 100Coupon Rate = 11%Quoted Price = 95-16 or $95.50
2018/7/10
Accrued Interest=
Cash price= $95.5+$1.64=$97.14
The cash price of a $100000 bond is $97140
64.1$5.5$181
54
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 delivery month
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 =
$127.20per $100 of principal
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)]
The most recent settlement price =93-08, 93.25
The cost of delivering each of the bonds:Bond1:99.59 – (93.25 ×1.0382)= $2.69Bond2:143.50 – (93.25 ×1.5188)=$1.87Bond3:119.75 – (93.25 ×1.2615)=$2.12 Quoted bond price – (Most recent settlement price × Conversion
factor)]
Bond Quoted bond price($)
Conversionfactor
1 99.59 1.0382
2 143.50 1.5188
3 119.75 1.2615
A number of factors determine the cheapest-to-deliver bond [Quoted bond price – (Most recent settlement price × Conversion factor)]
Bond Yields 6%
Yield Curve is
The Wild Card Play
slopingdownward
slopingupward
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
F: future priceS: spot price I : present value of the coupons during the life of future contract
rteISF )( 00
Cheapest-to-deliver coupon rate 12%Conversion factor 1.4000Current quoted bond price $120Interest rate 10% annumDelivery will take place in 270 days
Coupon payment
Coupon payment
Coupon payment
Current time
Maturity of futures contract
60 days 122days 148days 35days
Cash price
The present value of a coupon of$6 will be received after 122 days (0.3342years)
rteISF )( 00
Coupon payment
Coupon payment
Coupon payment
Current time
Maturity of futures contract
60 days 122days 148days 35days
The futures contract lasts for 270 days (0.7397years)The cash price, If the contract were written on the 12%
rteISF )( 00
Coupon payment
Coupon payment
Coupon payment
Current time
Maturity of futures contract
60 days 122days 148days 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 =
A Eurodollar is a dollar deposited in a bank outside the United States
Eurodollar futures are futures on the 3-month Eurodollar deposit rate (same as 3-month LIBOR rate)
One contract is on the rate earned on $1 million
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
If Q is the quoted price of a Eurodollar futures contract, the value of one contract is 10,000[100-0.25(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
1000000×0.0001(0.01%) ×0.25(3個月期 )=25 or $25
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 settled at the end of the underlying three- month period
)012.0about is (typicallyyear per
changes rateshort theofdeviation standard
theis and ) later than days (90
contract futures theunderlying rate the
ofmaturity theis contract, futures
theofmaturity to time theis where2
1rate Futures=rate Forward
is madeoften "adjustmentconvexity "A
1
2
1
212
t
t
t
tt
Consider the situation where σ=0.012 and we wish to calculate the forward rate when the 8-year Eurodollar futures price quote is 94.
1. In this case T1=8, T2=8.25, and convexity adjustment is
or 0.475%(47.5 basis points)
00475.025.88012.02
1 2
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%
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 the forward rates can then be used to extend the zero
curve It is usually assumed that the forward interest rate calculated from the future contract applies to the
1iitoTT
Suppose that Fi is the forward rate calculate from the ith Eurodollar futures contract and is the zero rate for a maturity Ti
equation(4.5)
So that
ii
iiiii TT
TRTRF
1
11
1
11
)(
i
iiiiii T
TRTTFR
iR
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%, the 491 days rate is 4.893%
04893.0491
400048.091053.0
1
11
)(
i
iiiiii T
TRTTFR
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
FC
P
DF
PDN
Assumes that only parallel shift in yield curve take place
Assumes that yield curve changes are small It is approximately true that
(4.15)
It is also approximately true
The number of contracts required to hedge against an uncertain is
yDFF FCC yPDP p
y
When the hedge instrument is a Treasury bond futures contract , the hedger must 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
FD
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
The number of contracts that should be shorted is10,000,000 6.8079.42
93,062.50 9.20
FC
P
DF
PDN
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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