TN1 Interest Rates and Financial Derivatives

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Document Date: November 2, 2006

An Introduction To Derivatives And Risk Management , 7th

Edition

Don Chance and Robert Brooks

Technical Note: Interest Rates and Financial Derivatives, Ch. 3, p. 56

This technical note supports the material in the Basic Notation and Terminology

section of Chapter 3 Principles of Option Pricing. We review here the appropriate

discount rate, calculating the accrual period, and interpreting interest rate quotations.

Appropriate Discount Rate

Financial derivatives valuation and management depend on the ability to estimate

the fair market price and current market value of derivatives. Most calculations involve

in some way the estimation of the present value or future value of cash flows. Hence,

there is a need to estimate the appropriate discount factor. The discount factor is aconstant when multiplied by a known future cash flow yields the present value (PV) of

that cash flow. The discount factor is a number less than one but greater than zero

(hence, the word discount). The discount factor (DF) can be used to compute the future

value (FV) of a current cash flow by dividing.

( )

DF

PVFV

DFFVPV

=

=

Traditionally, discount factors are expressed in terms of annualized interest rates, which

we describe in detail later. The important observation at this point is that the discount

factor is the number by which monetary value is transported through calendar time.

Historically in the United States, there have been two sources for interest rate data

used to compute discount factors, the London interbank offer rate (LIBOR) and the rate

on a benchmark United States treasury security (UST). Early in the history of financial

derivatives, the United States treasury rates were used widely. Increasingly, the financial

derivatives industry has migrated to using LIBOR for a wide variety of reasons. The

following is a partial summary of reasons LIBOR is preferred:

• USTs are presently exempt from state and local taxes. This tax exemption makes

USTs more attractive to investors in high tax brackets. This increased demand for

USTs drives up their prices and hence, lowers the yield to maturity.



• As we covered in chapter 1, there is an active repurchase agreement market for

USTs, facilitating the financing needs of derivative market participants and

providing securities for short sellers. The ability to use USTs in repurchase

agreements is an added bonus for investing in USTs, increasing the demand for

USTs and thus lowering the yield to maturity. At times, due to the high demand

by short sellers for a particular UST, the rate offered in the repo market is high,

providing an additional bonus.

• The cost of trading in USTs changes over time due to a variety of factors. This

change in liquidity makes USTs more or less attractive. Liquidity changes can be

driven by:

o Varying supply of USTs based on U. S. budget surpluses or deficits

o Varying demand for UST, during highly turbulent global events there is

flight to quality, meaning investors place a high demand on USTs, driving

their yield to maturity low.

o Perceived changes in default risk of USTs. High levels of U. S. debt

increase the likelihood of future financial distress.

• USTs offer some unique benefits to commercial banks, increasing the demand for

USTs. For example, when banks own USTs they do not have to set aside capital

as a provision for default, something banks have to do with commercial loans.• Although a rather obscure corner of financial markets, USTs offer several benefits

to municipal finance needs. Municipalities, such as state and local governments

can issue federally tax-exempt debt securities. These municipal securities have

very low interest rates, thus providing a cheap source of funding for

municipalities. When UST interest rates fall, municipalities are permitted to

engage in “advanced refunding” of existing debt by issuing new bonds, investing

the proceeds in USTs, and placing USTs in a bankruptcy-proof trust for old

bonds. Although a rather complex series of transactions, the net effect is to

increase the demand for USTs and hence lower the USTs yield to maturity.

IDRM7e, © Don M. Chance and Robert-Brooks Futures Risk Premiums2



Similar to the repo rate, LIBOR is a good proxy for the marginal dealer’s cost of

funds. By marginal we mean the derivative dealer that makes the next transaction or that

is the price setter. LIBOR is also the short-term opportunity cost of capital for financial

institutions. That is, when a financial institution needs to borrow additional money, they

can access the LIBOR market. The marginal dealer’s credit rating is usually assumed to

be AA (one notch below the high quality rating of AAA), so default risk is a possibility.

There is growing evidence that the entire debt market is moving to use some variation of

LIBOR as a benchmark.

Interest rate data is widely available. Presently, data sources include the Federal

Reserve’s site for U. S. Treasuries at

http://www.federalreserve.gov/RELEASES/H15/data.htm

and for LIBOR, the British Bankers Association site at

http://www.bba.org.uk/bba/jsp/polopoly.jsp?d=141&a=627.

We now turn to detailing interest rate quotation conventions and the calculation of

the accrual period. The accrual period is important because interest rates are often quoted

on an annualized basis. How one computes the fraction of the year under consideration

will impact the quoted interest rate.

Accrual Period

Although various accrual calculations are presented in the book, we summarize all

the major methodologies here. We also adopt a more detailed notation to be able to cover

a wide variety of accrual period calculations in one place.

The accrual period is the fraction of the year upon which an interest payment or

interest rate calculation is to be made. The accrual period can be expressed as:

NTD

NADAP =

where denotes the number of accrued days and denotes the number of totaldays in the year.

NAD NTD


There are numerous ways that debt securities require the calculation of the

number of days between two dates. This day count is used to compute the number of

days of accrued interest and the number of days in a coupon period. As we will see, the


http://www.bba.org.uk/bba/jsp/polopoly.jsp?d=141&a=627

http://www.bba.org.uk/bba/jsp/polopoly.jsp?d=141&a=627




method of counting days has an important influence on some derivative securities. We

review here several methods of computing the accrual period. This review is in no way

exhaustive, but represents many of the popular methods.

Day Type ACT/365

Intuitively, this method is fairly easy to grasp. You compute the literal number of

days between two dates. To arrive at the fraction of a year, you divide by 365 (that is,

ignore leap year). Counting the actual number of days is rather tedious, but an important

exercise when developing your understanding of financial derivatives. Most software

packages contain modules to do the various day-counting computations for you.1

Day Type 30/360

This day counting convention is much less intuitive. The general assumption is

that each month has 30 days and hence a year has 360. The rules for counting days are

fairly complex.

Day Type ACT/ACT

This day type requires the actual number of accrued days be computed for the

period as well as the accrual days. Hence, a leap year would make a difference.

Day Type ACT/360


accrual period but assumes a 360-day year. Eurodollar deposits and related futures

contracts use this method.

Day Type 30/ACT


entire year, but the accrual period is computed using the 30 days in a month.

1The only difficulty in counting days is knowing the number of days in a month. Children learn the old

saying “30 days hath September …” or the knuckle rule. However it is done, any educated person should

know how many days are in each month.




Day Type 30/365

This day type requires 30 days in a month for the number of accrued days but

assumes a 365 day year.

End-of-Month Rule

Many financial derivatives having cash flows at the end-of the month follow what

is called the end-of-month rule. For example, if a U.S. Treasury security matures on June

30th

and is semi-annual coupon paying, then the other coupon date is December 31 st .

Some securities issued by the Federal Home Loan Bank (FHLB) do not follow the end-

of-month rule. For some FHLB securities, if they are semi-annual coupon paying and it

matures on June 30th

, then the other coupon date is December 30th

. We now turn our

attention to interest rate quotations.

Interest Rate Quotations

In this section we will review several interest rate quotation methods. Remember

that compound frequency does not necessarily imply payment frequency. For example, a

continuously compounded interest rate can accompany a debt instrument that pays

interest quarterly. Our focus here is on the interest rate quotation convention, not the

interest payment convention. The notation is different here to provide easy comparison

across a wide variety of quotation methods.

Throughout this section, we will illustrate a calculation related to a 0.5-year, zero-

coupon, default-free bond that is trading at $975 and has a par value of $1,000. Assume

the actual number of days is 182. Thus,

000,1$FV

975$PV

=

=

Discount Interest

As we discussed in chapter 3, the discount interest method is used with U.S.

Treasury bills and some other money market securities. The accrual period is actual over

360 ( 360ACTAP = ). The present value or price of the security is related to its par value

or future value and the discount interest rate (DI) based on the following equation:




( )[ ]APr 1FVPV DI−=

or

AP

FV

PV1

r DI

−

=

Using the data above,

049451.0360182

000,1

9751

AP

FV

PV1

r DI =

−

=

−

=

or 4.9451%.

Add-on or Money Market Interest

The add-on interest method (also known as the money market method) is used

with Eurodollar CDs and some other money market securities. The accrual period is

actual over 360 ( 360ACTAP = ). The present value or current price of the security is

related to its par value or future value and the add-on interest rate (AO) based on the

following equation:

( )[ ]APr 1

FVPV

AO+=

or

AP

1PV

FV

r AO

−

= .


050719.0360182

1975

000,1

AP

1PV

FV

r AO =

−

=

−

=

or 5.0719%. Notice the significant difference in quoted interest rate between discount

and add-on interest rates.




( ) ( )( )

051282.01975

000,121

PV

FV2r

36018021AP21

SA =

−

=

−

=

or 5.1282%. Again, this is yet a different rate quote.

Monthly Interest:

The monthly interest method is used with mortgages and mortgage-related

securities. The accrual period is 30 over 360 ( 36030AP = ). The present value or current

price of the security is related to its par value or future value and the monthly

compounded interest rate (MO) based on the following equation:

AP12

MO

12

r 1

FVPV

+

=

or

( )

−

= 1

PV

FV12r

AP121

MO .


( ) ( )( )

050743.01975

000,1121

PV

FV12r

360180121AP121

MO =

−

=

−

=


Daily Interest

The daily interest method is used by bank securities and some mortgage-backed

securities. The accrual period is often actual over 365 ( 365ACTAP = ), but also takes

many other forms. The present value or current price of the security is related to its par

value or future value and the daily compounded interest rate (DA) based on the following

equation:

AP365

DA

365

r 1

FVPV

+

=

or




( )

−

= 1

PV

FV365r

AP3651

MO .


( ) ( )( )

050778.01975000,13651

PVFV365r

3651823651AP3651

MO =

−

=

−

=


Continuous Interest

The continuously compounded interest method is used by some bank securities

and most derivative pricing models. The accrual period is actual over 365

( 365ACTAP = ). The present value or current price of the security is related to its par

value or futures value and the continuously compounded interest rate (CO) based on the

following equation:

APr COFVePV −=

or

AP

PV

FVln

r CO

= .


050775.0365182

975

000,1ln

r CO =

=


Concluding Comments

While many people, especially students, find it confusing if not frustrating that the

financial markets use so many different interest rate quotation conventions. Rest assured

that these diverse methods are not used just to create stress for market participants and

students. The diversity of methods is, as often the case, a product of the lore and history

of the markets. The use of 30-day months and 360-day years reflect the days when

calculators did not exist and interest could be calculated more easily that way. For





example, a 90-day loan at 6% interest would result in the payment of .06 * 90/360 =

0.015 or 1.5% interest times the principal. Naturally old conventions die slowly and

often not at all.

Perhaps the most important thing to remember, however, is that an interest rate is

just a means of expressing the relationship between present value and future value. In the

example illustrated in this note, a six-month zero coupon bond with a face value of

$1,000 has a price of $975. The interest rate on this bond is the figure that relates the

present value ($975) to the future value ($1,000). We saw that this rate can be 4.9451%,

5.0719%, 5.2086%, 5.1282%, 5.0743%, 5.07785%, or 5.0775%. All of these rates are

nothing more than seven different but related mathematical functions that connect $975

now to $1,000 six months later. What matters most is that $975 is the present value of

$1,000.

References

Mayle, Jan. Standard Securities Calculation Methods Fixed Income Securities

Formulas for Price, Yield, and Accrued Interest Volume I, Third Edition, 1993.

TN1 Interest Rates and Financial Derivatives

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