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How is the interest you pay or
receive calculated?
How do these calculations affect your interest?
What's the difference between simple interest and compound interest?
Does repaying a loan early save you money?
Although Shakespeare
cautioned "neither a borrower nor a lender be," using and providing
credit has become a way of life for many individuals in today's economy.
Examples of borrowing by individuals are numerous—home mortgages, car loans,
credit cards, etc. While perhaps more commonly thought of as investing, many
examples of lending by individuals can be identified. By opening a savings
account, an individual makes a loan to the bank; by purchasing a savings
bond, an individual makes a loan to the government.
As
with goods and services that an individual might buy or sell, the use or
extension of credit has a price attached to it, namely the interest paid or
earned. And, just as consumers shop for the best price on a particular item
of merchandise, so too should consumers "comparison shop" for
credit—whether borrowing or lending. But comparing prices for credit can, at
times, be confusing. Although the price of credit is generally stated as a
rate of interest, the amount of interest paid or earned depends on a number
of other factors, including the method used to calculate interest.
Two
federal laws have been passed to minimize some of the confusion consumers
face when they borrow or lend money. The
Truth in Lending Act, passed in 1968, has made it easier for consumers to
comparison shop when they borrow money. Similarly, the purpose of the Truth
in Savings Act, passed in 1991, is to assist consumers in comparing deposit
accounts offered by depository institutions.
Provisions
of the Truth in Lending Act have been implemented through the Federal
Reserve's Regulation Z, which defines creditor responsibilities. Most
importantly, creditors are required to disclose both the Annual Percentage
Rate (APR) and the total dollar Finance Charge to the borrowing consumer.
Simply put, the APR is the relative cost of credit expressed in percentage
terms on the basis of one year. Just as "unit pricing" gives the
consumer a basis for comparing prices of different-sized packages of the same
product, the APR enables the consumer to compare the prices of different
loans regardless of the amount, maturity, or other terms.
Similarly,
provisions of the Truth in Savings Act have been implemented through the
Federal Reserve's Regulation
DD. These provisions include a requirement that depository institutions
disclose an annual percentage yield (APY) for interest-bearing deposit
accounts. Like the APR, an APY will provide a uniform basis for comparison by
indicating, in percentage terms on the basis of one year, how much interest a
consumer receives on a deposit account.
While
federal laws make it easier to comparison shop for credit and deposit
accounts, a variety of methods continue to be used to calculate the amount of
interest paid or earned by a consumer. To make an informed decision, it is
useful to understand the relationships between these different methods.
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Interest
Calculations
Interest
represents the price borrowers pay to lenders for credit over specified
periods of time. The amount of interest paid depends on a number of factors:
the dollar amount lent or borrowed, the length of time involved in the
transaction, the stated (or nominal) annual rate of interest, the repayment
schedule, and the method used to calculate interest.
If,
for example, an individual deposits $1,000 for one year in a bank paying 5
percent interest on savings, then at the end of the year the depositor may
receive interest of $50, or some other amount, depending on the way interest
is calculated. Alternatively, an individual who borrows $1,000 for one year
at 5 percent and repays the loan in one payment at the end of a year may pay
$50 in interest, or some other amount, again depending on the calculation
method used.
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Simple
Interest
The various
methods used to calculate interest are basically variations of the simple
interest calculation method. The basic concept underlying simple interest is
that interest is paid only on the original amount borrowed for the length of
time the borrower has use of the credit. The amount borrowed is referred to
as the principal. In the simple interest calculation, interest is computed
only on that portion of the original principal still owed.
Example 1
Suppose $1,000
is borrowed at 5 percent and repaid in one payment at the end of one year.
Using the simple interest calculation, the interest amount would be 5 percent
of $1,000 for one year or $50 since the borrower had use of $1,000 for the
entire year.
When more than one payment is made on a simple
interest loan, the method of computing interest is referred to as
"interest on the declining balance." Since the borrower only pays
interest on that amount of original principal that has not yet been repaid,
interest paid will be smaller the more frequent the payments. At the same
time, of course, the amount of credit at the borrower's disposal is also
smaller.
Example 2
Using simple
interest on the declining balance to compute interest charges, a 5 percent,
$1,000 loan repaid in two payments--one at the end of the first half-year and
another at the end of the second half-year would accumulate total interest
charges of $ 37.50. The first payment would be $500 plus $25 (5 percent of
$1,000 for one-half year), or $525; the second payment would be $500 plus
$12.50 (5 percent of $500 for one-half year), or $512.50. The total amount
paid would be $525 plus $512.50, or $1,037.50. Interest equals the difference
between the amount repaid and the amount borrowed, or $37.50. If four
quarterly payments of $250 plus interest were made, the interest amount would
be $31.25; if 12 monthly payments of $83.33 plus interest were made, the
interest amount would be $27.08.
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Example 3
When interest
on the declining balance method is applied to a 5 percent, $1,000 loan that
is to be repaid in two equal payments, payments of $518.83 would be
made at the end of the first half-year and at the end of the second
half-year. Interest due at the end of the first half-year remains $25;
therefore, with the first payment the balance is reduced by $493.83 ($518.83
less $25), leaving the borrower $506.17 to use during the second half-year.
The interest for the second half-year is 5 percent of $506.17 for one-half
year, or $12.66. The final $518.83 payment, then, covers interest of $12.66
plus the outstanding balance of $506.17. Total interest paid is $25 plus
$12.66, or $37.66, slightly more than in Example
2.
This equal payment variation is commonly used
with mortgage payment schedules. Each payment over the duration of the loan
is split into two parts. Part one is the interest due at the time the payment
is made, and part two–the remainder–is applied to the balance or amount still
owed. In addition to mortgage lenders, credit unions typically use the simple
interest/declining balance calculation method for computing interest on
loans. A number of banks also offer personal loans using this method.
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Other Calculation Methods
Add-on
interest, bank discount, and compound interest calculation methods differ
from the simple interest method as to when, how, and on what balance interest
is paid. The "effective annual rate" for these methods is that
annual rate of interest which, when used in the simple interest rate formula,
equals the amount of interest payable in these other calculation methods. For
the declining balance method, the effective annual rate of interest is the
stated or nominal annual rate of interest. For the methods described below,
the effective annual rate of interest differs from the nominal rate.
Add-on
interest
When the add-on
interest method is used, interest is calculated on the full amount of the
original principal. The interest amount is immediately added to the original
principal, and payments are determined by dividing principal plus interest by
the number of payments to be made. When only one payment is involved, this
method produces the same effective interest rate as the simple interest
method. When two or more payments are to be made, however, use of the add-on
interest method results in an effective rate of interest that is greater than
the nominal rate. True, the interest amount is calculated by applying the
nominal rate to the total amount borrowed, but the borrower does not have use
of the total amount for the entire time period if two or more payments are
made.
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Example 4
Consider,
again, the two-payment loan in Example
3. Using the add-on interest method, interest of $50 (5 percent of $1,000
for one year) is added to the $1,000 borrowed, giving $1,050 to be repaid;
half (or $525) at the end of the first half-year and the other half at the
end of the second half-year.
Recall
that in Example 3, where the declining balance method was used, an effective
rate of 5 percent meant two equal payments of $518.83 were to be made. Now
with the add-on interest method each payment is $525. The effective rate of
this 5 percent add-on rate loan, then, is greater than 5 percent. In fact,
the corresponding effective rate is 6.631 percent. This rate takes into
account the fact that the borrower does not have use of $1,000 for the entire
year, but rather use of $1,000 for the first half-year and use of about $500
for the second half-year.
To
see that a one-year, two equal-payment, 5 percent add-on rate loan is
equivalent to a one-year, two equal-payment, 6.631 percent declining balance
loan, consider the following. When the first $525 payment is made, $33.15 in
interest is due (6.631 percent of $1,000 for one-half year). Deducting the
$33.15 from $525 leaves $491.85 to be applied to the outstanding balance of
$1,000, leaving the borrower with $508.15 to use during the second half-year.
The second $525 payment covers $16.85 in interest (6.631 percent of $508.15
for one-half year) and the $508.15 balance due.
In this
particular example, using the add-on interest method means that no matter how
many payments are to be made, the interest will always be $50. As the number
of payments increases, the borrower has use of less and less credit over the
year. For example, if four quarterly payments of $262.50 are made, the
borrower has the use of $1,000 during the first quarter, around $750 during
the second quarter, around $500 during the third quarter, and around $250
during the fourth and final quarter. Therefore, as the number of payments
increases, the effective rate of interest also increases. For instance, in
the current example, if four quarterly payments are made, the effective rate
of interest would be 7.922 percent; if 12 monthly payments are made, the
effective interest rate would be 9.105 percent. The add-on interest method is
sometimes used by finance companies and some banks in determining interest on
consumer loans.
Bank discount
When the bank
discount calculation method is used, interest is calculated on the amount to
be paid back, and the borrower receives the difference between the amount to
be paid back and the interest amount. The bank discount method is also
referred to as the discount basis.
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Example 5
Consider the
loan in Example
1 where a 5 percent, $1,000 loan is to be repaid at the end of one year.
If the bank discount method is used, the interest amount of $50 would be
deducted from the $1,000, leaving the borrower with $950 to use over the
year. At the end of the year, the borrower pays $1,000. The interest amount
of $50 is the same as in Example 1. The borrower in Example 1, however, had
the use of $1,000 over the year. Thus, the effective rate of interest here
would be 5.263 percent—$50 divided by $950—compared to an effective rate of 5
percent in Example 1.
Forms
of borrowing that use the bank discount method often have no intermediate
payments. For example, the bank discount method is used for Treasury bills
sold by the U.S. government and commercial paper issued by
businesses. In addition, U. S. savings
bonds are sold on a discount basis, i.e., at a price below their face value.
How many days in a year?
In the above
examples, a year was assumed to be 365 days long. Historically, in order to
simplify interest calculations, lenders and borrowers often assumed that each
year had twelve 30-day months, resulting in a 360-day year. For any given
nominal rate of interest, the effective rate of interest will be greater when
a 360-day year is used in the interest calculation than when a 365-day year
is used.
Example
6
Suppose that a
$1,000 loan is discounted at 5 percent and payable in 365 days. This is the
situation in Example
5 where, based on a 365-day year, the effective rate of interest was
5.263 percent. If the bank discount calculation assumes a 360-day year, then
the length of time is computed to be 365/360 or 1-1/72 years instead of
exactly one year; the interest deducted (the discount) equals $50.69 instead
of $50; and the effective annual rate of interest is 5.34 percent. Some of
the examples cited earlier that use the bank discount method, namely Treasury
bills sold by the U.S.
government and commercial paper issued by businesses, assume a 360-day year
in calculating interest.
Compound interest
When the
compound interest calculation is used, interest is calculated on the original
principal plus all interest accrued to that point in time. Since interest is
paid on interest as well as on the amount borrowed, the effective interest
rate is greater than the nominal interest rate. The compound interest rate
method is often used by banks and savings institutions in determining
interest they pay on savings deposits "loaned" to the institutions
by the depositors.
Example
7
Suppose $1,000
is deposited in a bank that pays a 5 percent nominal annual rate of interest,
compounded semiannually (twice a year). At the end of the first half-year,
$25 in interest (5 percent of $1,000 for one-half year) is payable. At the
end of the year, the interest amount is calculated on the $1,000 plus the $25
in interest already paid, so that the second interest payment is $25.63 (5
percent of $1,025 for one-half year). The interest amount payable for the
year, then, is $25 plus $25.63, or $50.63. The effective rate of interest is
5.063 percent, which is greater than the nominal 5 percent rate.
The
more often interest is compounded within a particular time period, the
greater will be the effective rate of interest. In a year, a 5 percent nominal
annual rate of interest compounded four times (quarterly) results in an
effective annual rate of 5.0945 percent; compounded 12 times (monthly),
5.1162 percent; and compounded 365 times (daily), 5.1267 percent. When the
interval of time between compoundings approaches zero (even shorter than a
second), then the method is known as continuous compounding. Five percent
continuously compounded for one year will result in an effective annual rate
of 5.1271 percent.
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When Repayment Is Early
In the above
examples, it was assumed that periodic loan payments were always made exactly
when due. Often, however, a loan may be completely repaid before it is due.
When the declining balance method for calculating interest is used, the
borrower is not penalized for prepayment since interest is paid only on the
balance outstanding for the length of time that amount is owed. When the
add-on interest calculation is used, however, prepayment implies that the
lender obtains some interest that is unearned. The borrower then is actually
paying an even higher effective rate since the funds are not available for
the length of time of the original loan contract.
Some
loan contracts make provisions for an interest rebate if the loan is prepaid.
One method used in determining the amount of the interest rebate is referred
to as the "Rule of 78". Application of the Rule of 78 yields the
percentage of the total interest amount that is to be returned to the
borrower in the event of prepayment. The percentage figure is arrived at by
dividing the sum of the integer numbers (digits) from one to the number of
payments remaining by the sum of the digits from one to the total number of
payments specified in the original loan contract. For example, if a
five-month loan is paid off by the end of the second month (i.e., there are
three payments remaining), the percentage of the interest that the lender
would rebate is (1+2+3)÷(1+2+3+4+5) = (6÷15), or 40 percent. The name derives
from the fact that 78 is the sum of the digits from one to 12 and, therefore,
is the denominator in calculating interest rebate percentages for all
12-period loans.
Application
of the Rule of 78 results in the borrowers paying somewhat more interest than
would have been paid with a comparable declining balance loan. How much more
depends on the total number of payments specified in the original loan
contract and the effective rate of interest charged. The greater the
specified total number of payments and the higher the effective rate of
interest charged, the more the amount of interest figured under the Rule of
78 exceeds that under the declining balance method.
The difference between the Rule of 78 interest
and the declining balance interest also varies depending upon when the
prepayment occurs. This difference over the term of the loan tends to
increase up to about the one-third point of the term and then decrease after
this point. For example, with a 12-month term, the difference with prepayment
occurring in the second month would be greater than the difference that would
occur with prepayment in the first month; the third-month difference would be
greater than the second-month difference; the fourth month (being the
one-third point) would be greater than both the third month-difference and
the fifth-month difference. After the fifth month, each succeeding month's
difference would be less than the previous month's difference.
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Example 8
Suppose that
there are two $1,000 loans that are to be repaid over 12 months. Interest on
the first loan is calculated using a 5 percent add-on method, which results
in equal payments of $87.50 due at the end of each month ($1,000 plus $50
interest divided by 12 months). The effective annual rate of interest for
this loan is 9.105 percent. Any interest rebate due because of prepayment is
to be determined by the Rule of 78.
Interest
on the second loan is calculated using a declining balance method where the
annual rate of interest is the effective annual rate of interest from the
first loan, or 9.105 percent. Equal payments of $87.50 are also due at the
end of each month for the second loan.
Suppose
that repayment on both loans occurs after one-sixth of the term of the loan
has passed, i.e., at the end of the second month, with the regular first
month's payment being made for both loans. The interest paid on the first loan
will be $14.74, while the interest paid on the second loan will be $14.57, a
difference of 17 cents. If the prepayment occurs at the one-third point,
i.e., at the end of the fourth month (regular payments having been made at
the end of the first, second, and third months), interest of $26.92 is paid
on the first loan and interest of $26.69 on the second loan, a difference of
23 cents. If the prepayment occurs later, say at the three-fourths point,
i.e., at the end of the ninth month (regular payments having been made at the
end of the first through eighth months), $46.16 in interest is paid on the
first loan and $46.07 in interest is paid on the second loan, a difference of
but 9 cents.
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Charges Other than Interest
In addition to
the interest that must be paid, loan agreements often will include other
provisions which must be satisfied. Two examples of these provisions are
mortgage points and required (compensating) deposit balances.
Mortgage
points
Mortgage
lenders will sometimes require the borrower to pay a charge in addition to
the interest. This extra charge is calculated as a percentage of the mortgage
amount and is referred to as mortgage points. For example, if 2 points are
charged on a $100,000 mortgage, then 2 percent of $100,000, or $2,000, must
be paid in addition to the stated interest. The borrower, therefore, is
paying a higher price than if points were not charged–i.e., the effective
rate of interest is increased. In order to determine what the effective rate
of interest is when points are charged, it is necessary to deduct the dollar
amount resulting from the point calculation from the mortgage amount and add
it to the interest amount to be paid. The borrower is viewed as having use of
the mortgage amount less the point charge amount rather than the entire
mortgage amount.
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Example
9
Suppose that 2
points are charged on a 20-year, $100,000 mortgage where the rate of interest
(declining balance calculation) is 7 percent. The payments are to be $775.30
per month. Once the borrower pays the $2,000 point charge, there is $98,000
to use. With payments of $775.30 a month over 20 years, the result of the
2-point charge is an effective rate of 7.262 percent.
The longer the
time period of the mortgage, the lower will be the effective rate of interest
when points are charged because the point charge is spread out over more
payments. In the above example, if the mortgage had been for 30 years instead
of 20 years, the effective rate of interest would have been 7.201 percent.
Required (compensating) deposit balances
A bank may
require that a borrower maintain a certain percentage of the loan amount on
deposit as a condition for obtaining the loan. The borrower, then, does not
have the use of the entire loan amount but rather the use of the loan amount
less the amount that must be kept on deposit. The effective rate of interest
is greater than it would be if no compensating deposit balance were required.
Example 10
Suppose that
$1,000 is borrowed at 5 percent from a bank to be paid back at the end of one
year. Suppose, further, that the lending bank requires that 10 percent of the
loan amount be kept on deposit. The borrower, therefore, has the use of only
$900 ($1,000 less 10 percent) on which an interest amount of $50 (5 percent
of $1,000 for one year) is charged. The effective rate of interest is,
therefore, 5.556 percent as opposed to 5 percent when no compensating balance
is required.
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Summary
Although not an
exhaustive list, the methods of calculating interest described here are some
of the more common methods in use. They indicate that the method of interest
calculation can substantially affect the amount of interest paid, and that
savers and borrowers should be aware not only of nominal interest rates but
also of how nominal rates are used in calculating total interest charges.
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Additional Reading
Further
information on the factors determining interest rates is contained in the
publication, Points
of Interest (PDF
7.74MB).
Ordering
Information
This
publication, as well as other Federal Reserve materials on money and banking,
the financial system, the economy, consumer credit, and related topics, can
be ordered on-line using our on-line
order form, or by contacting our Publications
Department.
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Through time, the level of interest rates may fluctuate, but the
methods of calculation remain constant. Thus, the concepts of figuring
interest explained in this publication apply regardless of whether the
specific numerical examples used are representative of today's market rates.
This booklet is
a revised version of ABCs of Interest originally published by the
Federal Reserve Bank of Chicago in the
September 1973 issue of its monthly review, Business Conditions.
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