Interest rate futures. DAY COUNT AND QUOTATION CONVENTIONS TREASURY BOND FUTURES EURODOLLAR FUTURES Duration-Based Hedging Strategies Using Futures HEDGING.

Interest rate futures

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%

Maturity of Futures

Convexity Adjustment

(bps)

2 3.2

4 12.2

6 27.0

8 47.5

10 73.8

212

2

1tt

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

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.