Interest Rate (i) Used in TVOM Calculations: the Issue
Of the five variables in a TVOM calculation,
the one that can be the most subjective, and
therefore the one with the greatest potential for
the introduction of error, is the
interest rate (i).
In some cases the interest rate is defined as part
of the problem; e.g., find the payment amount
on a 9%, 30-year $500,000 mortgage.
However, in other cases the interest rate is more
a matter of professional judgment; e.g., determine
the current price of a $10,000 payment due in 10 years.
Consider the following situation:
A fixed-income security with a face value of $1,000
pays $25 semi-annually and matures in 5 years.
What price would you pay for this security today?
This security pays a nominal rate of interest of 5%
calculated as (25 x 2)/1,000.
But if we want to know what it is worth today,
is 5% really an appropriate value for
the interest rate (i)?
What if the current yield on 30-day T-bills is 3%?
And the yield on 5-year Treasury notes is 6.5%?
And comparable corporate notes are paying 8%?
Which rate is most appropriate for valuing the security?
Interest rates (i) of 3%, 5%, 6.5% and 8% will produce
prices of $1,092, $1,000, $937, and
$878 respectively, a spread of $214 that equates to
more than 20% of the face value.
(If you are unsure how these prices (PVs)
were computed, see
example problem #35.)
But perhaps none of these rates is appropriate.
Perhaps some form of composite rate or combination of
rates should be used.
For example, what if we used the a 5-year note rate to price
the cash flows in years 4 and 5, a 2-year note rate to price the
the cash flows in years 2 and 3 and T-Bill yields to price the
cash flows in year 1?
Unfortunately there is no hard and fast rule that will
tell us what interest rate to use.
That choice is a matter of judgment.
The material presented here is intended to assist with
making an informed decision with regard to the choice of
Focus: Interest Rates and TVOM
It is important to keep in mind that the purpose of this
web site is to instruct students in, and provide a reference
on, time value of money (TVOM) calculations.
If we are to stay true to this purpose then it would
not be wise to stray too far into topics that are
related but fundamentally distinct from TVOM.
For example, we don't want to delve too far into economic
theory or go too deep into the practical aspects of the mortgage
or bond industry.
Instead we want to concentrate on the mechanics of the
TVOM calculations with our primary focus being on the mathematics.
Having said that, it is important to recognize that a
certain amount of background information is helpful
and probably even necessary to properly
apply TVOM theory.
And this is what I am attempting to do here with interest
rates; provide an overview of key issues that
are useful in understanding how TVOM calculations
To this end, the following topics related to interest
rates are presented:
- Interest Rate Definitions
- Benchmark Interest Rates
- Factors Affecting Interest Rates (Risks)
- Yield Curve
- Summary and Rule of Thumb
1. Interest Rate Definitions
Interest is the fee or cost paid or charged for borrowing
money (capital); it is the amount a borrower pays a lender for the
Interest is compensation to the lender and expense to
Principal is the amount of money (capital) which is owed by
the borrower to the lender and upon which interest is charged.
Rate of Interest.
The rate of interest or "interest rate" is the percentage
of principal charged or paid as fee over an interval of time,
typically a year.
For example, if $50 were charged to borrow
$1,000 principal for one year, the interest rate is 5%.
Interest rate is sometimes used interchangeably
with the term yield (see below).
Yield is a measure of return on a security.
Generally "return" refers to income return (interest and dividends)
as opposed to capital gain or loss.
There are two basic variations of yield:
Cost yield is equal to the annual payout (interest or dividends)
divided by the cost of the security.
Current yield is equal to the annual payout (interest or dividends)
divided by the current price of the security.
For example, if you pay $1,000 for a bond at issue that pays $50 annually
then the cost yield is 5% (50/1,000).
However, if interest rates rise to the point where the price of the
bond falls to $950, then the current yield is 5.26%
As mentioned previously, yield can be synonymous with interest rate
and there are many times when we will use a yield as our value for i in
a TVOM calculation.
Yield to Maturity(YTM).
The yield to maturity is the
return a bond holder earns under the assumptions that
1) the bond is held to maturity; 2) all coupon payments (interest)
and return of principal are timely; and 3) all
coupon payments are reinvested at the stated YTM.
In other words, "YTM" is the i in the equation below
that sets the sum of the present value of all interest payments plus
the present value of the return of principal at maturity equal to
the bond's current price...
Note that a significant assumption for the YTM is that
all interest payments are reinvested at the stated YTM.
Also note that the YTM is essentially the internal rate
of return (IRR) of the bond's cash flows.
Annual Percentage Rate (APR).
The Annual Percentage Rate
is the annual interest rate after inclusion of fees and other costs.
The concept of an APR is an attempt by regulators to create
a standard means of expressing the true cost of borrowing.
However, because lenders have some discretion as to what
costs are included in the calculation (e.g., points are required
but application fees are not), APRs reported by different lenders
may not be directly comparable.
The Federal Truth in Lending law requires mortgage lenders
to disclose the APR in their advertisements.
Nominal rate of interest.
The nominal interest rate is the rate applied to
principal each compounding period.
Regardless of the compounding frequency, the nominal rate is
almost always given as an annual rate.
For example, when we refer to a nominal rate of "8% quarterly"
we actually mean 2% applied to principal each quarter.
The nominal or stated rate of 8% is applied
proportionately across all four quarters.
The nominal rate does not directly take into account compounding;
it is simply the rate applied to principal each compounding
period. (See effective interest rate below.)
The nominal rate is also referred to as the stated or applied
In TVOM problems the value for i is typically a nominal rate.
Effective rate of interest (ieff).
The effective interest rate is the nominal interest rate
(i) taking into account the effect of compounding
as defined here:
So if we have a nominal rate of "12% monthly,"
we calculate an annual effective rate of interest of 12.68%
as shown below:
Just to reiterate, in the equation above 12% is the nominal rate
of interest and 12.68% is the effective rate of interest.
The effective rate is higher than the nominal rate because of compounding.
In the example above we are applying 12% on a monthly basis but the
additional interest on interest results in an annual effective return that
is larger than 12%.
The effective rate is almost always given as an annual rate
and is sometimes referred to as the effective annual
The effective rate provides a means of comparing
interest rates of different compounding frequencies.
It also serves as a means of equivalence in that it is possible
through the annual effective rate to find a monthly rate of
interest that is equivalent to a known quarterly rate
The concepts of nominal and effective interest rates - and the
distinction between the two - are critical to understanding TVOM theory.
The definitions presented above may not be sufficient for readers
to fully grasp all of the subtleties associated with nominal
and effective interest rates.
If you really want to gauge your comprehension of this
critical issue, you can review the following example problems:
You have an option to purchase a $5,000 note that is due in 3 years
for $4,100. Alternatively you can invest the $4,100 in a CD that pays 3% every 6
months over the same period. Which offers the higher return? Which is the
View solution here.
Issue: Calculate the nominal interest rate (i) and effective interest rate
(ieff) and compare alternative investments.
Find the present value of $1,000 due at the end of 10 years if interest is
calculated at a) a nominal annual rate of 6% compounded monthly and b) an
effective quarterly rate of 1.5%.
View solution here.
Issue: Compare the present value (PV) of a single sum under different
compounding frequencies; distinguish between nominal and effective interest
What is the effective annual rate of interest being earned by an
investor who receives $5 interest each month on a $500 note? How much would the
monthly payment be if the investor were earning 10% compounded quarterly?
View solution here.
Issue: Calculate the effective rate of interest (ieff) given a fixed
monthly interest amount and principal balance; calculate the amount of the
interest payment under quarterly compounding.
An investor loaned $5,000 10 years ago. The investment earned an
effective annual rate of interest of 9% for the past 6 years and is now worth
$9,950. a) Assuming monthly compounding, what is the annual nominal interest
rate earned for the first 4 years; b) what is the annual effective rate of
interest for the first 4 years?
View solution here.
Issue: Calculate the PV of an investment and the nominal (i) and
effective (ieff) rates of interest; distinguish between
nominal and effective rates of interest.
One institution offers an annual rate of 5% compounded monthly
while another offers a rate of 5.1% compounded quarterly. Which offers the
View solution here.
Issue: Calculate the effective interest rate (ieff) under different
compounding frequencies and nominal interest rates.
How much interest will be earned on $1,000 at 5.5% compounded daily
over 180 days? What is the effective annual rate of interest earned? (Assume a
360 day year).
View solution here.
Issue: Calculate the dollar amount of interest earned and the effective annual rate of
interest (ieff) .
2. Benchmark Interest Rates
Listed below are various rates that can be considered when
choosing an appropriate value for i in a TVOM
Historical data on a variety of interest rates is available through
the Federal Reserve's
Data Download Program.
Inflation is the rate at which the general level of
prices are rising in the economy.
Inflation is commonly expressed as the change in the level of
prices as measured against an aggregate of base prices.
There are various means of making this measurement but the
most common are the Consumer Price Index (CPI)
and the broader GDP Deflator.
The discount rate is the rate charged
to banks on short-term borrowings directly
from the US Federal Reserve Bank (the Fed).
The central banks of other nations also charge interest
on borrowings by their member banks.
The discount rate is set directly by the Federal Reserve Bank
and generally applies to overnight borrowings.
Note because of imprecise terminology the term "discount rate" may
also refer to the value used for i in a TVOM calculation.
This is particularly true in the case of present value (PV)
calculations where some future value is being "discounted" back
to its current value.
Federal Funds Rate.
The federal funds rate is the rate charged among banks
for overnight loans. Thus, it is the rate charged when banks borrow
from each other, not from the Fed.
Unlike the discount rate, the fed funds rate is not set
directly by the Fed but is a target rate
(as illustrated by the variations in the chart below)
that the Fed may influence through
its open market operations.
LIBOR is the London Interbank Offered Rate
at which banks lend money to other banks in the London
The rate is published daily and is an average of inter-bank
rates paid on Eurodollar CDs (US dollar denominated CDs issued
by major international banks).
The LIBOR rate is generally regarded as the proxy for
the global rate at which large international banks can borrow
from each other.
It is increasingly popular as the reference rate for
international financial transactions.
In addition, derivative securities such as
futures, currencies, swaps, and forwards are increasingly
priced by reference to a LIBOR rate.
T-Bills are US government securities issued
by the Department of the Treasury with maturities
of 30, 91, and 182 days.
They are issued at a discount meaning they do not
pay interest during their term but are redeemed at
maturity for their face value.
The amount of interest earned on a T-Bill is equal to the
face value less the discounted purchase price.
The interest rate or yield is calculated as:
T-Bill rates are set through weekly auctions and such
rates are viewed internationally as proxies for
the riskless rate for their given maturities.
The Repo rate is the rate implied or explicit in a
repurchase agreement (repo).
In a repo one party sells securities to another party
for cash with the obligation
to repurchase the securities for a greater price in the
The excess of the repurchase price over the original price
is deemed to be interest, the amount of which is determined
by the repo rate.
Call Money Rate.
The Call Money Rate is the rate banks charge
to brokers on loans secured by stock and securities collateral.
Brokers typically add 100 basis points to this rate
as the margin rate they charge on loans to their clients.
The call money rate may also be referred to as the "broker rate".
Traditionally the US prime rate is the rate
charged by banks to their most creditworthy customers.
It is normally 300 basis points above the federal funds rate
and typically changes only when the Fed changes the fed funds rate
or when the Fed injects or withdraws reserves
from the banking system through its "open market" operations.
The prime rate is typically used as a benchmark for setting
rates on other borrowings such as credit cards and
adjustable rate mortgages (ARMs).
As illustrated below, prime rates will vary (sometimes considerably)
3. Factors Affecting Interest Rates (Risks)
In identifying the impact that prevailing interest rates have
on the value of i used in our TVOM calculations,
we are generally more concerned with the differences
among interest rates than with the
general level of interest rates.
There are three primary factors that account for differences
among interest rates:
While there are other factors that
contribute to differences among interest rates, for purposes
of this discussion we will focus on the three mentioned above.
Each of these factors is considered a
risk for which a premium must be paid
to compensate the investor for assuming that risk.
These premiums and the effect on interest rates
are illustrated in the chart below:
Starting at the bottom of the chart, the
real rate of interest
is determined by economic activity.
It represents the true cost of borrowed
funds and is based on the rate of return
a business earns on its capital
as well as consumers' preference
to consume rather than save.
Generally speaking, real rates are low in
recession and high during periods of
1. Inflation is the rate of change
Investors demand an inflation premium
to compensate them for the loss of purchasing power
they incur when they loan funds.
The real rate of interest plus the inflation
premium is referred to as the
nominal rate of interest.
Again we encounter some imprecise terminology.
The "nominal" rate mentioned here refers to
an economic concept and should not
be confused with the nominal interest rate
(i.e., the stated or applied interest rate) used
in TVOM calculations that we defined earlier.
The 90-day T-Bill rate is often used
as a proxy for the nominal rate of interest.
If the difference between the nominal and real
rates of interest is equal to the rate of inflation,
then the chart above reflects an expected
rate of inflation of 3.2%.
Note the real rate of
interest cannot be observed directly.
Typically it is "backed into" by subtracting
the expected rate of inflation from the T-Bill
2. The term premium reflects the
risk introduced by the investment's time
horizon, its time to maturity.
In general, the greater the time to maturity, the
higher the premium.
Investors demand a term premium
because the longer a security is
held, the greater the price change for
a given change in interest rates and
therefore the greater the market risk.
Thus, investors usually demand higher yields at
This is why a "normal" yield curve is
(See Yield Curve below.)
3. The risk premium reflects
the likelihood of default.
Less creditworthy borrowers have a higher
risk so a higher yield is required
to compensate investors.
U.S. Treasuries have virtually no credit risk
so their yields will always be lower than
comparable private sector securities.
Spreads between Treasuries and corporate securities tend
to widen when interest rates are high as
investors feel other issues are vulnerable and seek
a safe place for their money.
When choosing an appropriate interest rate
(i), we should be careful
to consider the context in which the TVOM
calculation is being performed and the possible
influence that each of the three risk premiums
(credit, term, and inflation) may introduce.
Example. We need a interest rate to
value a corporate payout that will be made in
In this case we might use the current
yield on similar corporate obligations because
this rate should reflect all three risk premiums,
just like the payout we are trying to value.
But what if we can't find a comparable corporate
The only quotes we have are for AAA
rated corporations while the payout we are
valuing will be made by a BBB company.
In order to take this difference into account,
we might observe the risk premium differential
between AAA and BBB corporate obligations
and use it to adjust the AAA rate upward to
reflect the additional risk assumed.
Note that not all three risks are present in every
There is no credit premium, at least not in theory,
associated with Treasuries and there is no term
premium inherent in T-Bill rates.
4. Yield Curve
The effect of the "term premium" discussed previously
can be observed in the yield curve.
The yield curve is a graphical
representation of the relationship between interest rates
In other words, it shows the particular yield associated
with a given maturity, say 5% for a 30 year bond.
While yield curves are available for all
types of issues, it is the yield curve for
US Treasuries, published daily in the Wall Street
Journal, that most people associate
with the term "yield curve."
The graphic below shows the US Treasury yield curve before and
after the Federal Reserve cut both the discount rate
and the fed funds rate in September of 2007:
Note that between July 2006 and July 2007
the yield curve was relatively flat implying
that there was in effect no term premium
associated with Treasury issues during this period.
In other words, 3-month T-Bills were paying the
same 5% as 30-year bonds.
A flat yield curve is traditionally considered an anomaly
and typically does not persist for long as can be seen
in the October curve.
The October curve also illustrates that the Fed's influence
is primarily on short term rates and may not necessarily
extend to long term rates which are set by market forces.
Since the yield curve provides a means to measure the
term premium it is not uncommon for bonds and other
fixed income contracts to be "priced off the yield curve"
by setting their yield equal to the yield at the same maturity
point on the yield curve and then adjusting for credit
quality and other factors.
5. Summary and Rule of Thumb
In concluding I will try, as best I can,
to put this complicated subject
into perspective and offer a rule of thumb that,
hopefully, can serve as an aid in selecting the
appropriate interest rate (i)
for use in your TVOM calculation.
The first question to ask in choosing an interest rate
is: What is the objective?
In other words, what are you trying to accomplish?
The objective will determine the need for
precision and, thus, the degree of effort
expended in selecting a value for i.
If the objective is a quick and
dirty comparison of two alternative investments
on a relative basis, then
it can be argued that the value chosen for i
is less critical than if the objective is to
obtain a precise valuation.
Any inaccuracies resulting
from using a reasonable but less than
perfect i will tend to cancel
out as the same error will be present
in both valuations.
On the other hand, if the objective
is to value a security
or liability as precisely as possible,
then the need for accuracy
in choosing a value for i takes on
much greater significance.
Still, the amount of effort that goes into this
process can vary considerably.
As a minimum, the credit, term, and
inflation factors and their associated premiums
discussed previously should be considered.
The degree of effort into how these premiums
are determined and applied can vary considerably.
In some cases, it may be sufficient just to add a
credit premium to the yield on a Treasury security
of appropriate maturity.
In other cases, complex economic models may be required to
forecast future interest rates and economic activity.
If all this seems confusing, a
simple but less than perfect rule of thumb
is to set the interest rate equal to
the "required yield."
The required yield is the rate
the investor requires, or can reasonably expect,
from the investment.
This rate can be determined by observing the
yields on comparable investments in the market
("comparable" loosely meaning
those of the same maturity and credit quality)
and, with consideration for current and
anticipated economic activity, choosing the one
that is most reasonable.
Granted this rule will rarely result in a
precise valuation but, when faced with a lack of
detailed information, it does at least provide
something to work with and can be refined as
the need dictates.