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Understanding Interest Rates
Interest rates are among the most closely watched variables in the economy. Their
movements are reported almost daily by the news media, because they directly affect
our everyday lives and have important consequences for the health of the economy.
They affect personal decisions such as whether to consume or save, whether to buy a
house, and whether to purchase bonds or put funds into a savings account. Interest
rates also affect the economic decisions of businesses and households, such as
whether to use their funds to invest in new equipment for factories or to save their
money in a bank.
Before we can go on with the study of money, banking, and financial markets, we
must understand exactly what the phrase interest rates means. In this chapter, we see
that a concept known as the yield to maturity is the most accurate measure of interest
rates; the yield to maturity is what economists mean when they use the term interest
rate. We discuss how the yield to maturity is measured and examine alternative (but
less accurate) ways in which interest rates are quoted. We’ll also see that a bond’s
interest rate does not necessarily indicate how good an investment the bond is
because what it earns (its rate of return) does not necessarily equal its interest rate.
Finally, we explore the distinction between real interest rates, which are adjusted for
inflation, and nominal interest rates, which are not.
Although learning definitions is not always the most exciting of pursuits, it is
important to read carefully and understand the concepts presented in this chapter.
Not only are they continually used throughout the remainder of this text, but a firm
grasp of these terms will give you a clearer understanding of the role that interest rates
play in your life as well as in the general economy.
Measuring Interest Rates
Different debt instruments have very different streams of payment with very different
timing. Thus we first need to understand how we can compare the value of one kind
of debt instrument with another before we see how interest rates are measured. To do
this, we make use of the concept of present value.
The concept of present value (or present discounted value) is based on the commonsense notion that a dollar paid to you one year from now is less valuable to you than
a dollar paid to you today: This notion is true because you can deposit a dollar in a
savings account that earns interest and have more than a dollar in one year.
Economists use a more formal definition, as explained in this section.
Let’s look at the simplest kind of debt instrument, which we will call a simple
loan. In this loan, the lender provides the borrower with an amount of funds (called
the principal) that must be repaid to the lender at the maturity date, along with an
additional payment for the interest. For example, if you made your friend, Jane, a simple loan of $100 for one year, you would require her to repay the principal of $100
in one year’s time along with an additional payment for interest; say, $10. In the case
of a simple loan like this one, the interest payment divided by the amount of the loan
is a natural and sensible way to measure the interest rate. This measure of the socalled simple interest rate, i, is:
If you make this $100 loan, at the end of the year you would have $110, which
can be rewritten as:
$100 (1 0.10) $110
If you then lent out the $110, at the end of the second year you would have:
$110 (1 0.10) $121
$100 (1 0.10) (1 0.10) $100 (1 0.10)2 $121
Continuing with the loan again, you would have at the end of the third year:
$121 (1 0.10) $100 (1 0.10)3 $133
Generalizing, we can see that at the end of n years, your $100 would turn into:
$100 (1 i)n
The amounts you would have at the end of each year by making the $100 loan today
can be seen in the following timeline:
$100 (1 0.10)n
This timeline immediately tells you that you are just as happy having $100 today
as having $110 a year from now (of course, as long as you are sure that Jane will pay
you back). Or that you are just as happy having $100 today as having $121 two years
from now, or $133 three years from now or $100 (1 0.10)n, n years from now.
The timeline tells us that we can also work backward from future amounts to the present: for example, $133 $100 (1 0.10)3 three years from now is worth $100
today, so that:
(1 0.10 )3
The process of calculating today’s value of dollars received in the future, as we have
done above, is called discounting the future. We can generalize this process by writing
Understanding Interest Rates
today’s (present) value of $100 as PV, the future value of $133 as FV, and replacing
0.10 (the 10% interest rate) by i. This leads to the following formula:
(1 i )n
Intuitively, what Equation 1 tells us is that if you are promised $1 for certain ten
years from now, this dollar would not be as valuable to you as $1 is today because if
you had the $1 today, you could invest it and end up with more than $1 in ten years.
The concept of present value is extremely useful, because it allows us to figure
out today’s value (price) of a credit market instrument at a given simple interest rate
i by just adding up the individual present values of all the future payments received.
This information allows us to compare the value of two instruments with very different timing of their payments.
As an example of how the present value concept can be used, let’s assume that
you just hit the $20 million jackpot in the New York State Lottery, which promises
you a payment of $1 million for the next twenty years. You are clearly excited, but
have you really won $20 million? No, not in the present value sense. In today’s dollars, that $20 million is worth a lot less. If we assume an interest rate of 10% as in the
earlier examples, the first payment of $1 million is clearly worth $1 million today, but
the next payment next year is only worth $1 million/(1 0.10) $909,090, a lot less
than $1 million. The following year the payment is worth $1 million/(1 0.10)2
$826,446 in today’s dollars, and so on. When you add all these up, they come to $9.4
million. You are still pretty excited (who wouldn’t be?), but because you understand
the concept of present value, you recognize that you are the victim of false advertising. You didn’t really win $20 million, but instead won less than half as much.
Four Types of
In terms of the timing of their payments, there are four basic types of credit market
1. A simple loan, which we have already discussed, in which the lender provides
the borrower with an amount of funds, which must be repaid to the lender at the
maturity date along with an additional payment for the interest. Many money market
instruments are of this type: for example, commercial loans to businesses.
2. A fixed-payment loan (which is also called a fully amortized loan) in which the
lender provides the borrower with an amount of funds, which must be repaid by making the same payment every period (such as a month), consisting of part of the principal and interest for a set number of years. For example, if you borrowed $1,000, a
fixed-payment loan might require you to pay $126 every year for 25 years. Installment
loans (such as auto loans) and mortgages are frequently of the fixed-payment type.
3. A coupon bond pays the owner of the bond a fixed interest payment (coupon
payment) every year until the maturity date, when a specified final amount (face
value or par value) is repaid. The coupon payment is so named because the bondholder used to obtain payment by clipping a coupon off the bond and sending it to
the bond issuer, who then sent the payment to the holder. Nowadays, it is no longer
necessary to send in coupons to receive these payments. A coupon bond with $1,000
face value, for example, might pay you a coupon payment of $100 per year for ten
years, and at the maturity date repay you the face value amount of $1,000. (The face
value of a bond is usually in $1,000 increments.)
A coupon bond is identified by three pieces of information. First is the corporation or government agency that issues the bond. Second is the maturity date of the
bond. Third is the bond’s coupon rate, the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond. In our example, the
coupon bond has a yearly coupon payment of $100 and a face value of $1,000. The
coupon rate is then $100/$1,000 0.10, or 10%. Capital market instruments such
as U.S. Treasury bonds and notes and corporate bonds are examples of coupon bonds.
4. A discount bond (also called a zero-coupon bond) is bought at a price below
its face value (at a discount), and the face value is repaid at the maturity date. Unlike
a coupon bond, a discount bond does not make any interest payments; it just pays off
the face value. For example, a discount bond with a face value of $1,000 might be
bought for $900; in a year’s time the owner would be repaid the face value of $1,000.
U.S. Treasury bills, U.S. savings bonds, and long-term zero-coupon bonds are examples of discount bonds.
These four types of instruments require payments at different times: Simple loans
and discount bonds make payment only at their maturity dates, whereas fixed-payment
loans and coupon bonds have payments periodically until maturity. How would you
decide which of these instruments provides you with more income? They all seem so
different because they make payments at different times. To solve this problem, we use
the concept of present value, explained earlier, to provide us with a procedure for
measuring interest rates on these different types of instruments.
Yield to Maturity
Of the several common ways of calculating interest rates, the most important is the
yield to maturity, the interest rate that equates the present value of payments
received from a debt instrument with its value today.1 Because the concept behind the
calculation of the yield to maturity makes good economic sense, economists consider
it the most accurate measure of interest rates.
To understand the yield to maturity better, we now look at how it is calculated
for the four types of credit market instruments.
Simple Loan. Using the concept of present value, the yield to maturity on a simple
loan is easy to calculate. For the one-year loan we discussed, today’s value is $100,
and the payments in one year’s time would be $110 (the repayment of $100 plus the
interest payment of $10). We can use this information to solve for the yield to maturity i by recognizing that the present value of the future payments must equal today’s
value of a loan. Making today’s value of the loan ($100) equal to the present value of
the $110 payment in a year (using Equation 1) gives us:
Solving for i,
This calculation of the yield to maturity should look familiar, because it equals
the interest payment of $10 divided by the loan amount of $100; that is, it equals the
simple interest rate on the loan. An important point to recognize is that for simple
loans, the simple interest rate equals the yield to maturity. Hence the same term i is used
to denote both the yield to maturity and the simple interest rate.
In other contexts, it is also called the internal rate of return.
Understanding Interest Rates
The key to understanding the calculation of the yield to maturity is equating today’s
value of the debt instrument with the present value of all of its future payments. The
best way to learn this principle is to apply it to other specific examples of the four types
of credit market instruments in addition to those we discuss here. See if you can develop
the equations that would allow you to solve for the yield to maturity in each case.
Fixed-Payment Loan. Recall that this type of loan has the same payment every period
throughout the life of the loan. On a fixed-rate mortgage, for example, the borrower
makes the same payment to the bank every month until the maturity date, when the
loan will be completely paid off. To calculate the yield to maturity for a fixed-payment
loan, we follow the same strategy we used for the simple loan—we equate today’s
value of the loan with its present value. Because the fixed-payment loan involves more
than one payment, the present value of the fixed-payment loan is calculated as the
sum of the present values of all payments (using Equation 1).
In the case of our earlier example, the loan is $1,000 and the yearly payment is
$126 for the next 25 years. The present value is calculated as follows: At the end of
one year, there is a $126 payment with a PV of $126/(1 i); at the end of two years,
there is another $126 payment with a PV of $126/(1 i)2; and so on until at the end
of the twenty-fifth year, the last payment of $126 with a PV of $126/(1 i)25 is made.
Making today’s value of the loan ($1,000) equal to the sum of the present values of all
the yearly payments gives us:
. . . $126
(1 i )
(1 i )
(1 i )25
More generally, for any fixed-payment loan,
. . . FP
(1 i )
(1 i )
(1 i )n
LV loan value
FP fixed yearly payment
n number of years until maturity
For a fixed-payment loan amount, the fixed yearly payment and the number of
years until maturity are known quantities, and only the yield to maturity is not. So we
can solve this equation for the yield to maturity i. Because this calculation is not easy,
many pocket calculators have programs that allow you to find i given the loan’s numbers for LV, FP, and n. For example, in the case of the 25-year loan with yearly payments
of $126, the yield to maturity that solves Equation 2 is 12%. Real estate brokers always
have a pocket calculator that can solve such equations so that they can immediately tell
the prospective house buyer exactly what the yearly (or monthly) payments will be if
the house purchase is financed by taking out a mortgage.2
Coupon Bond. To calculate the yield to maturity for a coupon bond, follow the same
strategy used for the fixed-payment loan: Equate today’s value of the bond with its
present value. Because coupon bonds also have more than one payment, the present
The calculation with a pocket calculator programmed for this purpose requires simply that you enter
the value of the loan LV, the number of years to maturity n, and the interest rate i and then run the program.
value of the bond is calculated as the sum of the present values of all the coupon payments plus the present value of the final payment of the face value of the bond.
The present value of a $1,000-face-value bond with ten years to maturity and
yearly coupon payments of $100 (a 10% coupon rate) can be calculated as follows:
At the end of one year, there is a $100 coupon payment with a PV of $100/(1 i );
at the end of the second year, there is another $100 coupon payment with a PV of
$100/(1 i )2; and so on until at maturity, there is a $100 coupon payment with a
PV of $100/(1 i )10 plus the repayment of the $1,000 face value with a PV of
$1,000/(1 i )10. Setting today’s value of the bond (its current price, denoted by P)
equal to the sum of the present values of all the payments for this bond gives:
. . . $100 $1,000
(1 i )
(1 i )
(1 i )10
(1 i )10
More generally, for any coupon bond,3
(1 i )
(1 i )
(1 i )
(1 i )n
P price of coupon bond
C yearly coupon payment
F face value of the bond
n years to maturity date
In Equation 3, the coupon payment, the face value, the years to maturity, and the
price of the bond are known quantities, and only the yield to maturity is not. Hence
we can solve this equation for the yield to maturity i. Just as in the case of the fixedpayment loan, this calculation is not easy, so business-oriented pocket calculators
have built-in programs that solve this equation for you.4
Let’s look at some examples of the solution for the yield to maturity on our 10%coupon-rate bond that matures in ten years. If the purchase price of the bond is
$1,000, then either using a pocket calculator with the built-in program or looking at
a bond table, we will find that the yield to maturity is 10 percent. If the price is $900,
we find that the yield to maturity is 11.75%. Table 1 shows the yields to maturity calculated for several bond prices.
Table 1 Yields to Maturity on a 10%-Coupon-Rate Bond Maturing in Ten
Years (Face Value = $1,000)
Price of Bond ($)
Yield to Maturity (%)
Most coupon bonds actually make coupon payments on a semiannual basis rather than once a year as assumed
here. The effect on the calculations is only very slight and will be ignored here.
The calculation of a bond’s yield to maturity with the programmed pocket calculator requires simply that you
enter the amount of the yearly coupon payment C, the face value F, the number of years to maturity n, and the
price of the bond P and then run the program.
Understanding Interest Rates
Three interesting facts are illustrated by Table 1:
1. When the coupon bond is priced at its face value, the yield to maturity equals the
2. The price of a coupon bond and the yield to maturity are negatively related; that
is, as the yield to maturity rises, the price of the bond falls. As the yield to maturity falls, the price of the bond rises.
3. The yield to maturity is greater than the coupon rate when the bond price is
below its face value.
These three facts are true for any coupon bond and are really not surprising if you
think about the reasoning behind the calculation of the yield to maturity. When you
put $1,000 in a bank account with an interest rate of 10%, you can take out $100 every
year and you will be left with the $1,000 at the end of ten years. This is similar to buying the $1,000 bond with a 10% coupon rate analyzed in Table 1, which pays a $100
coupon payment every year and then repays $1,000 at the end of ten years. If the bond
is purchased at the par value of $1,000, its yield to maturity must equal 10%, which
is also equal to the coupon rate of 10%. The same reasoning applied to any coupon
bond demonstrates that if the coupon bond is purchased at its par value, the yield to
maturity and the coupon rate must be equal.
It is straightforward to show that the bond price and the yield to maturity are negatively related. As i, the yield to maturity, rises, all denominators in the bond price formula must necessarily rise. Hence a rise in the interest rate as measured by the yield
to maturity means that the price of the bond must fall. Another way to explain why
the bond price falls when the interest rises is that a higher interest rate implies that
the future coupon payments and final payment are worth less when discounted back
to the present; hence the price of the bond must be lower.
There is one special case of a coupon bond that is worth discussing because its
yield to maturity is particularly easy to calculate. This bond is called a consol or a perpetuity; it is a perpetual bond with no maturity date and no repayment of principal
that makes fixed coupon payments of $C forever. Consols were first sold by the
British Treasury during the Napoleonic Wars and are still traded today; they are quite
rare, however, in American capital markets. The formula in Equation 3 for the price
of the consol P simplifies to the following:5
The bond price formula for a consol is:
(1 i )2
(1 i )3
which can be written as:
P C (x x 2 x 3 . . . )
in which x 1/(1 i). The formula for an infinite sum is:
1 x x2 x3 . . .
1 x 1 C c1 1(1 i ) 1 d
which by suitable algebraic manipulation becomes:
P = price of the consol
C = yearly payment
One nice feature of consols is that you can immediately see that as i goes up, the
price of the bond falls. For example, if a consol pays $100 per year forever and the
interest rate is 10%, its price will be $1,000 $100/0.10. If the interest rate rises to
20%, its price will fall to $500 $100/0.20. We can also rewrite this formula as
We see then that it is also easy to calculate the yield to maturity for the consol
(despite the fact that it never matures). For example, with a consol that pays $100
yearly and has a price of $2,000, the yield to maturity is easily calculated to be 5%
Discount Bond. The yield-to-maturity calculation for a discount bond is similar to
that for the simple loan. Let us consider a discount bond such as a one-year U.S.
Treasury bill, which pays off a face value of $1,000 in one year’s time. If the current
purchase price of this bill is $900, then equating this price to the present value of the
$1,000 received in one year, using Equation 1, gives:
and solving for i,
(1 i ) $900 $1,000
$900 $900i $1,000
$900i $1,000 $900
More generally, for any one-year discount bond, the yield to maturity can be written as:
F face value of the discount bond
P current price of the discount bond
In other words, the yield to maturity equals the increase in price over the year
F – P divided by the initial price P. In normal circumstances, investors earn positive
returns from holding these securities and so they sell at a discount, meaning that the
current price of the bond is below the face value. Therefore, F – P should be positive,
and the yield to maturity should be positive as well. However, this is not always the
case, as recent extraordinary events in Japan indicate (see Box 1).
An important feature of this equation is that it indicates that for a discount bond,
the yield to maturity is negatively related to the current bond price. This is the same
conclusion that we reached for a coupon bond. For example, Equation 6 shows that
a rise in the bond price from $900 to $950 means that the bond will have a smaller