Interest Rate Futures

Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012

So far we have covered futures contracts on

commodities and stock indices.

We have seen how they work, how they are used for

hedging, and how futures prices are determined.

We now move on to consider interest rate futures.

We will focus on the government bond futures

contracts.

Many of the other interest rate futures contracts

throughout the world have been modeled on this sort of

contracts.

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Introduction

The day count defines the way in which interest

accrues over time.

In general, we know the interest earned over some

reference period…

…and we are interested in calculating the interest

earned over some other period.

The day count convention is usually expressed as X/Y.

When we are calculating the interest earned between

two dates, X defines the way in which the number of

days between the two dates is calculated, and Y

defines the way in which the total number of days in

the reference period is measured.

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Day Count

The interest earned between the two dates is:

(X/Y) x Interest earned in the reference period

Three types of day count conventions that are

commonly used:

1. Actual number of days/actual number of days

2. 30 days/360 days

3. Actual number of days/360 days

Conventions vary from instrument to instrument and

from country to country.

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Day Count

The actual/actual day count is mainly used for

government bonds.

A coupon payment on a bond is a periodic interest

payment that the bondholder receives during the time

between when the bond is issued and when it matures.

Assume that a government bond is issued on

September 1, 2013.

The bond principal is $100.

The coupon payment dates are March 1 and

September 1 (reference period).

The coupon rate is 8% per annum (i.e., 4% per 6

months).

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Day Count: Actual/Actual

We wish to calculate the interest earned between

March 1 and July 3.

There are 184 (actual) days in the reference period.

There are 124 (actual) days between March 1 & July 3.

An interest of $4 is earned during the reference period.

Therefore:

(124/184) x 4 = 2.6957 or 0.026957%

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Day Count: Actual/Actual

The 30/360 day count is mainly used for corporate

bonds.

We assume 30 days per month and 360 days per year

when carrying out calculations.

Hence, the total number of days between March 1 and

September 1 (reference period) is 180.

The total number of days between March 1 and July 3

is (4 x 30) + 2 = 122.

In a corporate bond with the same terms as the

government bond considered, the interest earned

between March 1 and July 3 would be:

(122/180) x 4 = 2.7111 or 0.027111%

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Day Count: 30/360

Between February 28 and March 1, 2013, you have a choice

between owning a government bond and a corporate bond.

Both pay the same coupon and have the same quoted price.

Assuming no risk of default, which would you prefer?

You should be indifferent, but in fact you should have a

marked preference for the corporate bond:

Under the 30/360 day count convention used for corporate

bonds, there are 3 days between February 28, 2013, and

March 1, 2013.

Under the actual/actual (in period) day count convention used

for government bonds, there is only 1 day.

You would earn approximately three times as much interest

by holding the corporate bond!

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Day Count: Actual/Actual vs 30/360

The actual/360 day count is used for money market

instruments (debts that mature in less than a year).

The reference period is 360 days.

The interest earned during part of a year is calculated by

dividing the actual number of elapsed days by 360 and

multiplying by the relevant interest rate.

E.g., the interest earned in 90 days is exactly one-fourth

of the quoted rate.

Moreover, the interest earned in a whole year of 365

days is 365/360 times the quoted rate.

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Day Count: Actual/360

One of the most popular long-term interest rate futures

contracts is the government (Treasury) bond futures

contract.

In this contract, any government bond that has more

than 15 years to maturity and is not callable within 15

years can be delivered.

A callable bond (redeemable bond) allows the issuer of

the bond to retain the privilege of redeeming the bond at

some point before the bond reaches its date of maturity.

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Government Bond Futures Contracts

A procedure has been developed for adjusting the price

received by the party with the short position according to

the particular bond it chooses to deliver.

In other words, the Treasury bond futures contract

allows the party with the short position to choose to

deliver the underlying bond.

When a particular bond is delivered, a parameter known

as its conversion factor defines the price received for the

bond by the party with the short position.

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Government Bond Futures Contracts

The cash received for each $100 face value of the bond

delivered is:

(Most recent settlement price x Conversion factor) +

Accrued interest

The conversion factor for a bond is set equal to the

quoted price the bond would have per dollar of principal

on the first day of the delivery month on the assumption

that the interest rate for all maturities equals 6% per

annum (with semiannual compounding).

The accrued interest on a bond is the proportion of the

bond’s coupon which has been accrued (earned but not

yet paid), since the last coupon payment.

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Government Bond Futures Contracts

Each contract is for the delivery of $100,000 face value

of bonds.

Suppose that the most recent settlement price is 90.00.

The conversion factor for the bond delivered is 1.3800.

The accrued interest on this bond at the time of delivery

is $3 per $100 face value.

The cash received by the party with the short position

(and paid by the party with the long position) is:

(1.3800 x 90.00) + 3.00 = $127.20 per $100 face value.

Therefore, a party with the short position in one contract

would deliver bonds with a face value of $100,000 and

receive $127,200.

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Government Bond Futures Contracts: Example

At any given time during the delivery month, there are

many bonds that can be delivered in the government

bond futures contract.

These vary widely as far as coupon and maturity are

concerned.

The party with the short position can choose which of

the available bonds is ‘‘cheapest’’ to deliver.

To decide about the cheapest contract, a cost/revenue

analysis needs to be conducted.

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Cheapest-to-Deliver Bond

The quoted price, which traders refer to as the clean

price, is not the same as the cash price paid by the

purchaser of the bond, which is referred to by traders as

the dirty price:

Cash price = Quoted price +

Accrued interest since last coupon date

Note that the quoted price is for a bond with a face value

of $100.

The dirty price is the all-in price actually paid for the

bond.

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Cheapest-to-Deliver Bond

Because the party with the short position receives:

(Most recent settlement price x Conversion factor) + Accrued

interest

and the cost of purchasing a bond is:

Quoted bond price + Accrued interest

the cheapest-to-deliver bond is the one for which:

min [Quoted bond price - (Most recent settlement price x

Conversion factor)]

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Cheapest-to-Deliver Bond

The party with the short position has decided to

deliver and is trying to choose between the three

bonds in the table below.

Assume the most recent settlement price is $93.25.

Once the party with the short position has decided

to deliver, it can determine the cheapest-to-deliver

bond by examining each of the deliverable bonds in

turn.

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Cheapest-to-Deliver Bond: Example

The cost of delivering each of the bonds is as

follows:

Bond 1: 99.50 – (93.25 x 1.0382) = $2.69

Bond 2: 143.50 – (93.25 x 1.5188) = $1.87

Bond 3: 119.75 – (93.25 x 1.2615) = $2.12

Therefore, the cheapest-to-deliver bond is Bond 2.

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Cheapest-to-Deliver Bond: Example

A number of factors determine the cheapest-to-

deliver bond.

When bond yields are in excess of 6%, the

conversion factor system tends to favour the

delivery of low-coupon long-maturity bonds.

When yields are less than 6%, the system tends to

favour the delivery of high-coupon short-maturity

bonds.

When the yield curve is upward-sloping, there is a

tendency for bonds with a long time to maturity to be

favoured, whereas when it is downward-sloping,

there is a tendency for bonds with a short time to

maturity to be delivered.

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Cheapest-to-Deliver Bond: Determinants

A curve that plots the interest rates, at a set point in

time, of bonds having equal credit quality, but differing

maturity dates.

The most frequently reported yield curve compares the

three-month, two-year, five-year and 30-year U.S.

Treasury debt.

There are three main types of yield curve shapes:

1) A normal yield curve is one in which longer maturity

bonds have a higher yield compared to shorter-term

bonds due to the risks associated with time.

2) An inverted yield curve is one in which the shorter-term

yields are higher than the longer-term yields, which

can be a sign of upcoming recession.

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The Yield Curve

3) A flat yield curve is one in which the shorter- and

longer-term yields are very close to each other,

which is also a predictor of an economic transition.

The slope of the yield curve is also seen as

important:

the greater the slope, the greater the gap between

short- and long-term rates.

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The Yield Curve

An exact theoretical futures price for a government

bond contract is difficult to determine because the

short party’s options concerned with the timing of

delivery and choice of the bond that is delivered

cannot easily be valued.

However, if we assume that both the cheapest-to-

deliver bond and the delivery date are known, the

government bond futures contract is a futures

contract on a traded security (the bond) that

provides the holder with known income given by:

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Determining the Futures Price

where:

I is the present value of the coupons during the life

of the futures contract

T is the time until the futures contract matures

r is the risk-free interest rate applicable to a time

period of length T.

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Determining the Futures Price

A investor is looking for arbitrage opportunities in the

Treasury bond futures market.

What complications are created by the fact that the party

with a short position can choose to deliver any bond with

a maturity of over 15 years?

Answer

If the bond to be delivered and the time of delivery were

known, arbitrage would be straightforward.

When the futures price is too high, the arbitrageur buys

bonds and shorts an equivalent number of bond futures

contracts.

When the futures price is too low, the arbitrageur shorts

bonds and goes long an equivalent number of bond

futures contracts.

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Determining the Futures Price: Arbitrage

Uncertainty as to which bond will be delivered introduces

a series of complications.

Maybe the most important one is that the bond that

appears cheapest-to-deliver now may not in fact be

cheapest-to-deliver at maturity.

In the case where the futures price is too high, this is not

a major problem since the party with the short position

(i.e., the arbitrageur) determines which bond is to be

delivered.

However, in the case where the futures price is too low,

the arbitrageur’s position is far more difficult since he or

she does not know which bond to short; it is unlikely that

a profit can be locked in for all possible outcomes.

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Determining the Futures Price: Arbitrage