COMSATS Virtual Campus
Islamabad
Handouts No 2
What Do Interest Rates Mean and What Is Their Role in Valuation?
Present value
Present value, also known as present discounted value, is a future amount of money that has
been discounted to reflect its current value, as if it existed today. The present value is always less
than or equal to the future value because money has interest-earning potential, a characteristic
referred to as the time value of money.Time value can be described with the simplified phrase,
“A dollar today is worth more than a dollar tomorrow”. Here, 'worth more' means that its value is
greater. A dollar today is worth more than a dollar tomorrow because the dollar can be invested
and earn a day's worth of interest, making the total accumulate to a value more than a dollar by
tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant, without
the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains
access to the money for a time before paying it back. By letting the borrower have access to the
money, the lender has sacrificed their authority over the money, and is compensated for it in the
form of interest. The initial amount of the borrowed funds (the present value) is less than the
total amount of money paid to the lender.
Present value calculations, and similarly future value calculations, are used to
value loans, mortgages, annuities, sinking funds, perpetuities, bonds, and more. These
calculations are used to make comparisons between cash flows that don’t occur at simultaneous
times. The idea is much like algebra, where variable units must be consistent in order to compare
or carry out addition and subtraction; time dates must be consistent in order to make comparisons
between values or carry out simple calculations. When deciding between projects in which to
invest, the choice can be made by comparing respective present values discounted at the same
interest rate, or rate of return. The project with the least present value, i.e. that costs the least
today, should be chosen.
Years' Purchase
The traditional method of valuing future income streams as a present capital sum is to multiply
the average expected annual cash-flow by a multiple, known as Years' purchase. Thus in selling
to a third party a property leased to a tenant under a 99 year lease at a rent of $10,000 per annum,
a deal might be struck at "20 years' purchase", which would value the lease at 20 * $10,000, i.e.
$200,000. This equates to a present value discounted in perpetuity at 5%. For a riskier
investment the purchaser would demand to pay a lower number of years' purchase. This was the
method used for example by the English crown in setting re-sale prices for manors seized at
the Dissolution of the Monasteries in the early 16th century. The standard usage was 20 years'
purchase.
Background
If offered a choice between 100 today or 100 in one year and there is a positive real interest rate
throughout the year ceteris paribus, a rational person will choose 100 today. This is described by
economists as time preference. Time preference can be measured by auctioning off a risk free
security—like a US Treasury bill. If a 100 note, payable in one year, sells for 80 now, then 80 is
the present value of the note that will be worth 100 a year from now. This is because money can
be put in a bank account or any other (safe) investment that will return interest in the future.
An investor who has some money has two options: to spend it right now or to save it. But the
financial compensation for saving it (and not spending it) is that the money value will accrue
through the compound interest that he will receive from a borrower (the bank account on which
he has the money deposited).
Therefore, to evaluate the real value of an amount of money today after a given period of time,
economic agents compound the amount of money at a given (interest) rate. Most actuarial
calculations use the risk-free interest rate which corresponds to the minimum guaranteed rate
provided by a bank's saving account for example. To compare the change in purchasing power,
the real interest rate (nominal interest rate minus inflation rate) should be used.
The operation of evaluating a present value into the future value is called a capitalization (how
much will 100 today be worth in 5 years?). The reverse operation—evaluating the present value
of a future amount of money—is called a discounting (how much will 100 received in 5 years—
at a lottery for example—be worth today?).
It follows that if one has to choose between receiving 100 today and 100 in one year, the rational
decision is to choose the 100 today. If the money is to be received in one year and assuming the
savings account interest rate is 5%, the person has to be offered at least 105 in one year so that
the two options are equivalent (either receiving 100 today or receiving 105 in one year). This is
because if 100 is deposited in a savings account, the value will be 105 after one year.
Interest Rates
Interest is the additional amount of money gained between the beginning and the end of a time
period. Interest represents the time value of money, and can be thought of as rent that is required
of a borrower in order to use money from a lender.[2][4] For example, when an individual takes
out a bank loan, they are charged interest. Alternatively, when an individual deposits money into
a bank, their money earns interest. In this case, the bank is the borrower of the funds and is
responsible for crediting interest to the account holder. Similarly, when an individual invests in a
company (through corporate bonds, or through stock), the company is borrowing funds, and must
pay interest to the individual (in the form of coupon payments, dividends, or stock price
appreciation).[1] The interest rate is the change, expressed as a percentage, in the amount of
money during one compounding period. A compounding period is the length of time that must
transpire before interest is credited, or added to the total.[2] For example, interest that is
compounded annually is credited once a year, and the compounding period is one year. Interest
that is compounded quarterly is credited 4 times a year, and the compounding period is three
months. A compounding period can be any length of time, but some common periods are
annually, semiannually, quarterly, monthly, daily, and even continuously.
There are several types and terms associated with interest rates.
Compound Interest
Compound interest is multiplicative. Interest is earned on the interest that has already accrued
(credited) in addition to the principal (initial amount).
The term
interest.
is the factor often used to make various calculations concerning compound
Simple Interest
Simple interest is additive. Simple interest is earned only on the principal amount, and there is no
interest earned on interest already accrued.
The term
is the factor used to make various calculations when simple interest is
applied. Simple interest is often used when calculating interest earned during a fraction of the
year. Simple interest earns a higher return during the first year, and compound interest earns a
higher return after the first year. At the end of the first year, simple interest and compound
interest earn the same return.
Effective Annual Rate of Interest
Effective annual rate of interest is the percentage increase in an amount of money after a year of
accumulating interest. This does not depend on what happens during the year. It can be
calculated by
Where
represents the amount of money at time .
Nominal Annual Rate of Interest
Nominal annual rate of interest is NOT the same as effective annual rate of interest. Instead,
nominal annual interest is compounded more than once a year, say,
times. For instance, if the
nominal annual interest rate is 8% and interest is compounded quarterly (4 times a year, or every
three months), then the interest rate for ¼ of the year is (8/4)%=2%. Because it is compounded
more often, the effective annual interest rate is actually more than 8%
The formula to convert between effective annual interest rate and nominal annual interest rate is
Where is the effective annual interest rate and
is the nominal annual interest rate, and
the number of times interest is compounded in a year
is
Rate of Discount
The rate of discount,
transferred.
, refers to interest that is payable in advance, before funds have been
There also exists a nominal annual discount rate,
, which is analogous to nominal interest rate
The formula to convert between effective annual interest rate,
rate is
Where is the effective annual interest rate, and
and effective annual discount
is the effective annual discount rate
Continuous Compounding
Continuous compounding is nominal interest that is compounded infinitely throughout the year.
It is the instantaneous growth rate in the principal amount, i.e. the interest rate can change as a
function of time. A force of interest, which describes the accumulated amount of money as a
function of time, characterizes continuous compounding:
Where is the force of interest, and
its derivative.
is the amount of money as a function of time, , and
The formula to convert between effective annual interest rate, , and the constant force of
interest, , is
Real Rate of Interest
Real rate of interest refers to the interest rate that has been adjusted to account for inflation; it is
the percent change in buying power due to inflation. The real return refers to the surplus
associated with an interest rate compared to an investment rate that matches the inflation rate.
Where
is the inflation rate.
Calculation
The operation of evaluating a present sum of money some time in the future called a
capitalization (how much will 100 today be worth in 5 years?). The reverse operation—
evaluating the present value of a future amount of money—is called discounting (how much will
100 received in 5 years be worth today?).
Spreadsheets commonly offer functions to compute present value. In Microsoft Excel, there are
present value functions for single payments (=NPV) and series of equal, periodic payments
(=PV). Programs will calculate present value flexibly for any cash flow and interest rate, or for a
schedule of different interest rates at different times.
Present Value of a Lump Sum
The most commonly applied model of present valuation uses compound interest. The standard
formula is:
Where is the future amount of money that must be discounted, is the number of
compounding periods between the present date and the date where the sum is worth , is the
interest rate for one compounding period (the end of a compounding period is when interest is
applied, for example, annually, semiannually, quarterly, monthly, daily). The interest rate, , is
given as a percentage, but expressed as a decimal in this formula.
Often,
is referred to as the Present Value Factor
This is also found from the formula for the future value with negative time.
For example if you are to receive $1000 in 5 years, and the effective annual interest rate during
this period is 10% (or 0.10), then the present value of this amount is
The interpretation is that for an effective annual interest rate of 10%, an individual would be
indifferent to receiving $1000 in 5 years, or $620.92 today.
The purchasing power in today's money of an amount
be computed with the same formula, where in this case
of money, years into the future, can
is an assumed future inflation rate.
PV of a Bond
A corporation issues a bond, an interest earning debt security, to an investor to raise funds. The
bond has a face value, , coupon rate, , and maturity date which in turn yields the number of
periods until the debt matures and must be repaid. A bondholder will receive coupon payments
semiannually (unless otherwise specified) in the amount of
, until the bond matures, at which
point the bondholder will receive the final coupon payment and the face value of a
bond,
. The present value of a bond is the purchase price.The purchase price is equal
to the bond's face value if the coupon rate is equal to the current interest rate of the market, and
in this case, the bond is said to be sold 'at par'. If the coupon rate is less than the market interest
rate, the purchase price will be less than the bond's face value, and the bond is said to have been
sold 'at a discount', or below par. Finally, if the coupon rate is greater than the market interest
rate, the purchase price will be greater than the bond's face value, and the bond is said to have
been sold 'at a premium', or above par. The purchase price can be computed as:
Technical details
Present value is additive. The present value of a bundle of cash flows is the sum of each one's
present value.
In fact, the present value of a cashflow at a constant interest rate is mathematically one point in
the Laplace transform of that cashflow, evaluated with the transform variable (usually denoted
"s") equal to the interest rate. The full Laplace transform is the curve of all present values,
plotted as a function of interest rate. For discrete time, where payments are separated by large
time periods, the transform reduces to a sum, but when payments are ongoing on an almost
continual basis, the mathematics of continuous functions can be used as an approximation.
These calculations must be applied carefully, as there are underlying assumptions:

That it is not necessary to account for price inflation, or alternatively, that the cost of
inflation is incorporated into the interest rate.
 That the likelihood of receiving the payments is high — or, alternatively, that the default
risk is incorporated into the interest rate.
See time value of money for further discussion.
Variants/Approaches
There are mainly two flavors of PV. Whenever there will be uncertainties in both timing and
amount of the cash flows, the expected present value approach will often be the appropriate
technique.
Traditional Present Value Approach – in this approach a single set of estimated cash
flows and a single interest rate (commensurate with the risk, typically a weighted
average of cost components) will be used to estimate the fair value.
 Expected Present Value Approach – in this approach multiple cash flows scenarios
with different/expected probabilities and a credit-adjusted risk free rate are used to
estimate the fair value.

Choice of interest rate
The interest rate used is the risk-free interest rate if there are no risks involved in the project. The
rate of return from the project must equal or exceed this rate of return or it would be better to
invest the capital in these risk free assets. If there are risks involved in an investment this can be
reflected through the use of a risk premium. The risk premium required can be found by
comparing the project with the rate of return required from other projects with similar risks. Thus
it is possible for investors to take account of any uncertainty involved in various investments.
Present Value Method of Valuation
An investor, the lender of money, must decide the financial project in which to invest their
money, and present value offers one method of deciding. A financial project requires an initial
outlay of money, such as the price of stock or the price of a corporate bond. The project claims to
return the initial outlay, as well as some surplus (for example, interest, or future cash flows). An
investor can decide which project to invest in by calculating each projects’ present value (using
the same interest rate for each calculation) and then comparing them. The project with the
smallest present value – the least initial outlay – will be chosen because it offers the same return
as the other projects for the least amount of money.
Introduction to the Present Value of a Single Amount (PV)
If you received $100 today and deposited it into a savings account, it would grow over time to be
worth more than $100. This fact of financial life is a result of the time value of money, a
concept which says it's more valuable to receive $100 now rather than a year from now. To put it
another way, the present value of receiving $100 one year from now is less than $100.
Accountants use Present Value (PV) calculations to account for the time value of money in a
number of different applications. For example, assume your company provides a service in
December 2011 and agrees to be paid $100 in December 2012. The time value of money tells us
that the part of the $100 is interest you will earn for waiting one year for the $100. Perhaps only
$91 of the $100 is service revenue earned in 2011 and $9 is interest that will be earned in 2012.
The calculation of present value will remove the interest, so that the amount of the service
revenue can be determined. Another example might involve the purchase of land: the owners
will either sell it to you for $160,000 today, or for $200,000 if you pay at the end of two years.
To help analyze the alternatives, you would use a PV calculation to tell you the interest
rate implicit in the second option.
PV calculations can also tell you such things as how much money to invest right now in return
for specific cash amounts to be received in the future, or how to estimate the rate of return on
your investments. Our focus will be on single amounts that are received or paid in the future.
We'll discuss PV calculations that solve for the present value, the implicit interest rate, and/or the
length of time between the present and future amounts.
Calculations for the Present Value of a Single Amount
At the outset, it's important for you to understand that PV calculations involve cash amounts—
not accrualamounts.
In present value calculations, future cash amounts are discounted back to the present time.
(Discounting means removing the interest that is imbedded in the future cash amounts.) As a
result, present value calculations are often referred to as a discounted cash flow technique.
PV calculations involve the compounding of interest. This means that any interest earned is
reinvested anditself will earn interest at the same rate as the principal. In other words, you
"earn interest on interest." The compounding of interest can be very significant when the interest
rate and/or the number of years is sizeable.
We will use present value (PV) to mean a single future amount such as one receipt or one
payment. Here are the components of a present value (PV) calculation:
1. Present value amount (PV)
2. Future value amount (FV)
3. Length of time before the future value amount occurs (n)
4. Interest rate used for discounting the future value amount (i)
If you know any three of these four components, you will be able to calculate the unknown
component. Accountants are often called upon to calculate this unknown component.
Visualizing the Present Value (PV) Amount
Let's assume that Customer X provides your company with a promissory note for $1,000 in
exchange for service your company provided. The note is due at the end of two years and it does
not specify any interest. The fair market value of the note and the fair market value of the service
are not known. Because of the time value of money, you know that some interest is involved in a
two-year note, even though it is not stated explicitly. You estimate the interest rate by
considering both the length of the loan and the credit worthiness of Customer X. If Customer X
is a reputable company like Google, you know there is minimal risk and a low interest rate would
be used. If, however, Customer X has a bad credit history, then a high interest rate would be
used. Let's assume that you have determined 10% to be the appropriate rate for Customer X. We
now know three of the four components we need: (1) the future value amount ($1,000), (2) the
length of time (2 years), and (3) the interest rate (10%). With these three components, we know
enough to calculate the fourth component, present value.
A timeline can help us visualize what is known and what needs to be computed. The present time
is noted with a "0," the end of the first period is noted with a "1," and the end of the second
period is noted with a "2."
The following timeline depicts the information we know, along with the unknown component,
(PV):
The letter "n" refers to the length of time (in this case, two years). The letter "i" refers to the
percentage interest rate used to discount the future amount (in this case, 10%). Both (n) and (i)
are stated within the context of time (e.g., two years at a 10% annual interest rate).
(Later on we will give examples where (n) and (i) pertain to a half-year, a quarter of a year, or a
month.)
Visualizing The Length of Time (n)
Sometimes the present value, the future value, and the interest rate for discounting are known,
but the length of time before the future value occurs is unknown. To illustrate, let's assume that
$1,000 will be invested today at an annual interest rate of 8% compounded annually. The
investment will be sold when its future value reaches $5,000. Because we know three
components, we can solve for the unknown fourth component—the number of years it will take
for $1,000 of present value to reach the future value of $5,000.
The following timeline depicts the known components and the unknown component (n):
Visualizing The Interest Rate (i)
What will our timeline look like when our unknown component is the interest rate? For this
example, let's assume that we know the following: the present value is $900, the future value
amount is $1,000, and the length of time before the future value occurs is two years. Since we
know three of the components, the fourth one—the interest rate that will discount the future
value amount to the present value—can be calculated.
The following timeline depicts the known components and the unknown component (i):
Visualizing The Future Value Amount (FV)
Let's assume that the interest rate, the length of time, and the present value are known,
and the future value is the component we don't know. If a present value of $1,000 is
invested at 6% per year and compounded annually for four years, what is the future
value amount?
The following timeline depicts the known components and the unknown component
(FV):
Basic Types of Debt Instruments
Mishkin defines debt instruments (equivalently, credit market instruments) to be particular
types of contractual agreements that require the borrower to pay the lender certain fixed dollar
amounts at regular intervals until a specified time is reached. Mishkin provides a more detailed
discussion of five basic types of debt instruments that are distinguished from one another by their
payment provisions: simple loan contracts; fixed-payment loans; coupon bonds; discount bonds
(or "zero-coupon bonds"); and consols (or "perpetuities") that pay a fixed amount in each
payment period forever ("in perpetuity").
The notes below concentrate on the first four types of debt instruments.
Important Remark: It is assumed below that all bond sales and purchases are for newly
issued bonds, so that the sellers and buyers are enabling new borrowing. Bonds can also be resold in secondary markets. In secondary market exchanges, the sellers are NOT borrowers; i.e.,
they are NOT acquiring command over additional purchasing power. Rather, they are simply
engaging in portfolio rebalancing, meaning they are adjusting the composition of their asset
portfolios (e.g., more money and less bonds). Moreover, buyers of bonds in secondary markets
are not enabling any new borrowing (i.e., they are not original lenders); they are simply
acquiring entitlements to payment streams that had previously been owned by others.
Simple Loan Contracts:
Under the terms of a simple loan contract, the borrower (contract issuer) receives from the lender
(contract buyer) a specified amount of funds (the loan value LV or principal) for a specified
period of time (the maturity). The borrower agrees that, at the end of this period of time -referred to as the maturity date -- the borrower will repay the loan value to the lender together
with an additional payment referred to as the interest payment.
Borrower
Receives: Loan Value LV
|
START |___________________________ MATURITY DATE
|
|
Lender
Loan Value LV
Receives:
+ Interest Payment I
The annual borrowing fee for a simple loan with a loan value LV, an interest payment I, and a
maturity of N years is measured by the simple interest rate given by I divided by [LV times N].
Important Remark: Mishkin always implicitly assumes that the maturity N on simple loans is
one year (N=1). As will be clarified further below, his assertion "for simple loans, the simple
interest rate equals the yield to maturity" is only true if N=1 is assumed for the simple loans and
the yield to maturity is calculated as an annual rate.
Example of a Simple Loan Contract:
A borrower receives a loan on January 1, 1999, in amount $500.00, and agrees to pay the lender
$550.00 on January 1, 2001. Thus, the loan value is $500.00, the maturity is two years, the
maturity date is January 1, 2001, and the interest payment is $50.00. The simple (annual) interest
rate for this loan is then $50/[$500*2] = .05, or 5 percent.
Real-World Examples of Simple Loan Contracts: Standard bank deposit accounts take this
form. Also, various money market instruments (e.g., commercial loans to businesses) can take
this form.
Fixed-Payment Loan Contracts:
Under the terms of a fixed-payment loan contract, the borrower (contract issuer) receives from
the lender (contract buyer) a specified amount of funds -- the loan value -- and, in return, makes
periodic fixed payments to the lender until a specified maturity date. These periodic fixed
payments include both principal (loan value) and interest, so at maturity there is no lump sum
repayment of principal.
Borrower
Receives: Loan Value LV
|
MATURITY
START |__________________________________ DATE
|
|
Lender
Receives:
|
|
|
|
Fixed
Fixed
Fixed
Payment FP Payment FP Payment FP
Example of a Fixed-Payment Loan Contract:
Joe arranges a 15-year installment loan with a finance company to help pay for a new car. Under
the terms of this loan, Joe receives $20,000 now to finance the purchase of a new car but must
make payments of $2000 every year for the next 15 years to the finance company.
Real-World Examples of Fixed Payment Loan Contracts: Installment loans (e.g., auto loans)
and home mortgages typically take this form.
Coupon Bond:
Under the terms of a coupon bond, the borrower (bond issuer) agrees to pay the lender (bond
buyer) a fixed amount of funds (the coupon payment) on a periodic basis until a specified
maturity date, at which time the borrower must also pay the lender the face value (or par value)
of the bond. The coupon rate of a coupon bond is, by definition, the amount of the coupon
payment divided by the face value of the bond.
As will be clarified in the next section, below, the purchase price of a coupon bond depends on
the "present value" of the stream of anticipated coupon payments plus the final face value
payment promised by the bond. Coupon bonds that sell above their face value are said to sell at a
premium, and those that sell below their face value are said to sell at a discount.
Borrower Purchase
Receives: Price Pb
|
MATURITY
START |_______________________ /\/\/\ _____ DATE
|
|
|
|
|
|
Lender
Coupon
Coupon ... Coupon
Receives:
Payment C Payment C
Payment C
+ Face Value F
Example of a Coupon Bond:
Suppose a coupon bond has a face value of $1000, a maturity of five years, and an annual
coupon payment of $60. Then, at the end of each year for the next five years, the borrower (bond
issuer) must pay the lender (bond buyer) a coupon payment of $60. In addition, at the end of five
years (the maturity date), the borrower must pay the lender the face value of the bond, $1000.
The coupon rate for this coupon bond is $60/$1000 = .06, or 6 percent.
Real-World Examples of Coupon Bonds: Capital market instruments such as U.S. Treasury
notes and bonds take this form. Corporate bonds also typically take this form.
Discount Bond (or Zero-Coupon Bond):
Under the terms of a discount bond, the borrower (bond issuer) immediately receives from the
lender (bond buyer) the purchase price Pd of the bond, which is typically less than the face value
F of the bond. In return, the borrower promises that, at the bond's maturity date, he will pay the
lender the face value F of the bond.
Borrower
Receives:
Purchase
Price Pd
|
START |_________________________ MATURITY DATE
|
|
Lender
Face Value F
Receives:
Important Cautionary Remarks:
The above definition of a discount bond follows the definition used in Mishkin. Some other
authors refer to zero-coupon bonds as PURE discount bonds, labelling as a "discount bond" any
bond that sells at a discount in the sense that its market price is less than its face value.
Also, Mishkin asserts that discount bonds make no interest payments. While this is literally true,
in the sense that only a face value payment is made, it is NOT true that the interest RATE on
discount bonds is zero. Indeed, as will be seen below, the most basic measure of interest rates in
use today is the annual "yield to maturity" i. For a one-year discount bond, the formula for
calculating i reduces to i = [F-Pd]/Pd, hence i is only zero in the highly unlikely event that F=Pd.
Discount Bond Example:
On January 1, 1999, a borrower gives a lender a discount bond with a face value of $200 and a
maturity of 2 years, and the lender gives $150 to the borrower. The borrower must then pay the
lender $200 on January 1, 2001.
Real-World Examples of Discount Bonds: U.S. Treasury bills and U.S. savings bonds take this
form.
Measuring Interest Rates by Yield to Maturity
By definition, the current yield to maturity for a marketed debt instrument is the
particular fixed annual interest rate i which, when used to calculate the present value
of the debt instrument's future stream of payments to the instrument's holder, yields a
present value equal to the current market value of the instrument.
Mishkin discusses and illustrates the calculation of the yield to maturity for the four
basic types of debt instruments introduced in the first section of these notes, above.
Below we review this calculation for two of these debt instrument types: fixedpayment loan contracts and coupon bonds.
Yield to Maturity for Fixed-Payment Loan Contracts:
Recall from previous discussion the general form of a fixed-payment loan contract:
Borrower
Receives:
Loan Value LV
|
MATURITY
START |__________________________________ DATE
|
|
|
|
|
|
Lender
Fixed
Fixed
Fixed
Receives:
Payment FP
Payment FP
Payment FP
Consider a particular fixed-payment loan contract with a loan value LV = $5000,
annual fixed payments FP = $660.72, and a maturity of N = 20 years. What is the
yield to maturity for this loan contract?
The first question that must be answered is what is the current value of this loan
contract at the date of its issuance?
The borrower (contract issuer) receives from the lender (contract buyer) the $5000
loan value at the date the loan contract is issued. This $5000 loan value, then,
constitutes the current value of the loan contract. It is, in effect, the price paid by the
lender to purchase the loan contract from the borrower.
By definition, then, the yield to maturity of this fixed-payment loan contract is the
particular fixed annual interest rate i which, when used to calculate the present value
of the loan contract, results in a present value that is exactly equal to $5000, the
current value of the loan contract.
More precisely, for any fixed annual interest rate i, let PV(i) denote the present value
of the lender's payment stream under this fixed payment loan contract when calculated
using this interest rate i. Then the way you determine the yield to maturity on this
fixed payment payment loan contract is you calculate the particular interest rate i that
satisfies the formula
(9)
$5000 = PV(i) .
This formula will now be developed step by step.
Using the discussion in the previous section, given any fixed annual interest rate i, the
present value of the fixed-payment loan contract at hand -- that is, the present value of
the payment stream to the lender generated by this loan contract -- is found as follows.
The payment stream to the lender generated by this loan contract consists of twenty
successive yearly fixed payments, each having the nominal value FP=$660.72. Using
formula (3), given any year n, n = 1,...,20, and any fixed annual interest rate i, the
present value of the particular fixed payment FP = $660.72 received at the end of year
n is
FP/(1+i)n .
Consequently, given any fixed annual interest rate i, the present value PV(i) for the
fixed payment loan contract as a whole is given by the sum of all of these separate
present value calculations for the fixed payments FP received by the lender (debt
instrument holder) at the end of years 1 through 20, i.e.,
(10)
PV(i) = FP/(1+i) + FP/(1+i)2 + ... + FP/(1+i)20.
Since the current value of the loan contract is $5000, the desired yield to maturity is
then found by solving equation (9) for i with PV(i) given explicitly by equation (10).
Because the present value PV(i) depends in a rather complicated way on i, the
determination of i from formula (9) is not straightforward. To make life easier, tables
have been published that can be used to determine yields to maturity for various types
of fixed-payment loan contracts once the current value and fixed payments of the loan
are known. For example, using such tables, it can be shown that the solution for i in
equation (9) above is approximately i = .12. That is, the yield to maturity i for a fixed-
payment loan contract with a current value of $5000, with annual fixed payments of
$660.72, and with a maturity of twenty years, is approximately 12 percent.
Yield to Maturity for Coupon Bonds:
Recall from previous discussion the basic contractual terms of a coupon bond:
Borrower Purchase
Receives: Price Pb
|
MATURITY
START |_______________________ /\/\/\ _____ DATE
|
|
|
|
|
|
Coupon
Coupon ... Coupon
Lender
Payment C Payment C
Payment C
Receives:
+ Face Value F
Consider a coupon bond whose purchase price is Pb=$94, whose face value is F = $100, whose
coupon payment is C = $10, and whose maturity is 10 years. By definition, the coupon rate for
this bond is equal to C/F = $10/$100 = .10 (i.e., 10 percent).
The payment stream to the lender generated by this coupon bond is given by
(11)
( $10, $10, $10, $10, $10, $10, $10, $10, $10, [$10 + $100] ).
For any given fixed annual interest rate i, the present value PV(i) of the payment stream (11) is
given by the sum of the separate present value calculations for each of the payments in this
payment stream as determined by formula (5). That is,
(12)
PV(i) = $10/(1+i) + $10/(1+i)2 + ... + $10/(1+i)9 + [$10 + $100]/(1+i)10 .
The current value of the coupon bond is its current purchase price Pb = $94. It then follows by
definition that the yield to maturity for this coupon bond is found by solving the following
equation for i:
(13)
Pb = PV(i) ,
where PV(i) is as given in (12). The calculation of the yield to maturity i from formula (13) can
be difficult, but tables have been published that permit one to read off the yield to maturity i for a
coupon bond once the purchase price, the face value, the coupon rate, and the maturity are
known.
For example, using such tables, it can be shown that the yield to maturity i for the coupon bond
currently under consideration, which has a purchase price of $94 per $100 of face value, a
coupon rate of 10 percent, and a maturity of 10 years, is approximately equal to 11 percent.
More generally, given any coupon bond with purchase price Pb, face value F, coupon payment
C, and maturity N, the yield to maturity i is found by means of the following formula:
(14a)
Pb = PV(i) ,
where the present value PV(i) of the coupon bond is given by
(14b)
PV(i) = C/(1+i) + C/(1+i)2 + ... + C/(1+i)N-1 + [C+F]/(1+i)N .
Some Final Important Observations on Yield to Maturity:
For any coupon bond with a given coupon payment C, face value F, and maturity N, the purchase
price Pb of the bond is equal to the face value F if and only if the yield to maturity i for the bond
is equal to the coupon rate C/F.
This observation follows directly from the structure of a coupon bond. When the purchase price
equals the face value, the coupon bond essentially functions as a bank deposit account into which
a principal amount (the face value) is deposited by a lender, earns a fixed annual interest rate (the
coupon rate) for N years, and is then recovered by the lender.
Illustration for a One-Period Coupon Bond:
For any coupon bond with a given coupon payment C, a given face value F, a given
maturity N=1, and a given purchase price Pb, the formula Pb = PV(i) for determining the
yield to maturity i can be written as
(15)
F + C
Pb = ---------- .
(1+i)
Dividing each side of formula (15) by the face value F, one obtains
(16)
1 + C/F
Pb/F = ---------- .
(1+i)
Given C, F, and N=1, formula (16) implies that Pb equals F (i.e., the left-hand side equals
1) IF AND ONLY IF i equals C/F (i.e., the right-hand side equals 1).
More generally, for any coupon bond with a given coupon payment C, given face value F, and
given maturity N, the purchase price Pb of the bond is lower (higher) than F if and only if the
yield to maturity i is higher (lower) than the coupon rate C/F. This follows directly from formula
(14) for determination of the yield to maturity, using the previously noted fact that the purchase
price Pb is equal to F if and only if the yield to maturity i is equal to the coupon rate C/F.
For example, suppose formula (14) holds with Pb = F and i = C/F. Taking C, F, and N=1 as
given, consider what happens if the yield to maturity i now increases, so that i exceeds the
coupon rate C/F. Since C and F are given, PV(i) decreases, which implies that Pb must also
decrease. Since F is given, and Pb was originally equal to F, this implies that Pb must now
be lower than F.
Moreover, for any coupon bond with a given coupon payment C, face value F, and maturity N,
the yield to maturity i of the bond is inversely related to the purchase price Pb of the bond. That
is, the higher the yield to maturity i, the lower the purchase price Pb, and conversely. This
inverse relationship also follows directly from formula (14).
To see this, consider what happens when i increases in formula (14), keeping C, F, and N fixed.
When i increases, the denominator (1+i) of the discounted coupon payment C/(1+i) appearing in
PV(i) in formula (14) increases, implying that the ratio C/(1+i) is smaller than before, and
similarly for each of the other discounted coupon payments that are summed to obtain PV(i) in
(14). Consequently, PV(i) decreases. It then follows from formula (14) that Pb also decreases.
This inverse relationship between the yield to maturity of a debt instrument and its purchase
price actually holds in general. For any debt instrument with any given payment stream, when
the yield to maturity for the debt instrument rises, the purchase price of the debt instrument must
fall, and vice versa. This follows directly from the general definition for the yield to maturity,
applicable to all debt instruments.
Nominal interest rate
In finance and economics, nominal interest rate or nominal rate of interest refers to two
distinct things: the rate of interest before adjustment for inflation (in contrast with thereal interest
rate); or, for interest rates "as stated" without adjustment for the full effect of compounding (also
referred to as the nominal annual rate). An interest rate is callednominal if the frequency of
compounding (e.g. a month) is not identical to the basic time unit (normally a year).
Nominal versus real interest rate
The real interest rate is the nominal rate of interest minus inflation. In the case of a loan, it is this
real interest that the lender receives as income. If the lender is receiving 8 percent from a loan
and inflation is 8 percent, then the real rate of interest is zero because nominal interest and
inflation are equal. A lender would have no net benefit from such a loan because inflation fully
diminishes the value of the loan's profit.
The relationship between real and nominal interest rates can be described in the equation:
where r is the real interest rate, i is the inflation rate, and R is the nominal interest rate.[1]
A common approximation for the real interest rate is:
real interest rate = nominal interest rate - expected inflation
In this analysis, the nominal rate is the stated rate, and the real interest rate is the interest after the
expected losses due to inflation. Since the future inflation rate can only be estimated, the ex
ante and ex post (before and after the fact) real interest rates may be different; the premium paid
to actual inflation may be higher or lower. In contrast, the nominal interest rate is known in
advance.
Nominal versus effective interest rate
The nominal interest rate (also known as an Annualised Percentage Rate or APR) is the periodic
interest rate multiplied by the number of periods per year. For example, a nominal annual interest
rate of 12% based on monthly compounding means a 1% interest rate per month
(compounded).] A nominal interest rate for compounding periods less than a year is always lower
than the equivalent rate with annual compounding (this immediately follows from elementary
algebraic manipulations of the formula for compound interest). Note that a nominal rate without
the compounding frequency is not fully defined: for any interest rate, the effective interest
rate cannot be specified without knowing the compounding frequency and the rate. Although
some conventions are used where the compounding frequency is understood, consumers in
particular may fail to understand the importance of knowing the effective rate.
Nominal interest rates are not comparable unless their compounding periods are the
same; effective interest rates correct for this by "converting" nominal rates into annual compound
interest. In many cases, depending on local regulations, interest rates as quoted by lenders and in
advertisements are based on nominal, not effective interest rates, and hence may understate the
interest rate compared to the equivalent effective annual rate.
Confusingly, in the context of inflation, 'nominal' has a different meaning. A nominal rate can
mean a rate before adjusting for inflation, and a real rate is a constant-prices rate. The FIsher
equation is used to convert between real and nominal rates. To avoid confusion about the term
nominal which has these different meanings, some finance textbooks use the term 'Annualised
Percentage Rate' or APR rather than 'nominal rate' when they are discussing the difference
between effective rates and APR's.
The term should not be confused with simple interest (as opposed to compound interest) which is
not compounded.
The effective interest rate is always calculated as if compounded annually. The effective rate is
calculated in the following way, where r is the effective rate, i the nominal rate (as a decimal,
e.g. 12% = 0.12), and n the number of compounding periods per year (for example, 12 for
monthly compounding):
Monthly compounding
Example 1: A nominal interest rate of 6%/a compounded monthly is equivalent to an effective
interest rate of 6.17%.
Example 2: 6% annually is credited as 6%/12 = 0.5% every month. After one year, the initial
capital is increased by the factor (1+0.005)12 ≈ 1.0617.
Daily compounding
A loan with daily comp have a substantially higher rate in effective annual terms. For a loan with
a 10% nominal annual rate and daily compounding, the effective annual rate is 10.516%. For a
loan of $10,000 (paid at the end of the year in a single lump sum), the borrower would pay
$51.56 more than one who was charged 10% interest, compounded annually.