Fixed Income Mathematics

EXAMPLE 22.14. Using this equation, calculate the conversion factor of

the 5 1/4s of 02/15/29 U.S. T-bond deliverable into the Chicago Board of Trade’s

U.S. T-bond futures contract expiring in March of 2000.

C = 0.0525

N = 28 (March 1, 2000, through March 1, 2028; remember, full years)

X = 9 months (March 1 through February 15, rounded down from 11 months

to the number of months in the nearest full quarter, which in this case is

9 months.

䊏

UNDERSTANDING THE EQUATIONS FOR

COMPUTING CONVERSION FACTORS

Like many ﬁxed-income mathematics equations, these conversion factor equa-

tions appear more fearsome than they actually are. We’ll look at the one for

the 10-year Treasury notes and explain the terms. The other three equations

are comparable.

Here is the equation for the 10-year T-note:

where

CF = Conversion factor

C = Annual coupon on $100 face amount (in decimal form)

N = Complete years to call or maturity

X = Number of months in excess of the whole N (rounded down to com-

plete quarters).

The annual interest rate (yield) is 6%, or 0.06 in decimal form, compounded

semiannually. The true yield is therefore 3%, or 0.03, semiannually.

X is rounded down to complete quarters, so X can equal only 3, 6, or 9. If

X = 9, then set X = 3 and increase the 2N factor to 2N + 1. Similarly, if X = 6,

set X = 0 and increase 2N to 2N + 1. You can see that if X is 6 or 9, we increase

the number of full periods by 1 (one-half year) and reduce X by 6 (months).

CC C X

XNN

=+¥-

-¥

()

103

2006

103

=+¥-

-¥

()

=+¥

[]

103

0 0525

006

103

0 0525

0 98533 0 0263 0 875 0 814528 0 18547193 0 013125

0 8977

57 57

.... . .

302 Futures Contracts

All calculations for this conversion factor are to a quarter. Note that all calcu-

lations for 2-year and 5-year notes are rounded down to the next whole month,

so all calculations for these conversion factors are to a whole month. These all

differ from the standard bond price calculation, which is to an exact day, as

shown in Chapter 6, Equation 6.1.

The conversion factor applies to a unit amount (1). The conversion factor

is therefore multiplied by the actual par amount of the trade and by the futures

contract price, to determine the actual cash for the principal amount of the

bonds delivered. Accrued interest is added as well, as usual for bond trades.

The last term inside the outer square brackets is the present

value of the ﬁnal maturity amount. It shows the future value (1) discounted

at 3% for 2N semiannual periods. If the original calculation for X gave X = 6

or X = 9, then X was reduced by 6 and 2N increased to 2N + 1.

The middle term inside the outer square brackets is the present value of the

annuity of coupon payments, as displayed in Equation 5.6 in Chapter 5 and

as shown in the following development:

Once more, if the original value was X = 6 or X = 9, X was reduced by 6

and 2N increased to 2N + 1.

We have now the two parts of the standard bond price calculation.

The seller of the bond is entitled to the accrued interest on the bond. The

first term inside the outer square bracket, , is the accrued interest due for

one-half year. If the note is priced on an even semiannual period, this is the

accrued interest.

The term outside the outer square brackets, , equals 1 if the con-

version factor is priced on a semiannual period, because X = 0. If the conver-

sion factor is priced to a quarterly period, the factors inside the outer square

brackets are discounted back to the previous semiannual period at a 3% semi-

annual rate.

103

006

103

2003

103

003

¥-

()

¥-

103

Understanding the Equations for Computing Conversion Factors 303

Get Fixed Income Mathematics now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Start your free trial

Fixed Income Mathematics by Robert Zipf

Don’t leave empty-handed

It’s yours, free.

Check it out now on O’Reilly