Annual Percentage Rate Versus Eff ective Annual Rate
Banks, fi nance
companies, and other lenders are required by the Truth in Lending law to disclose their lend-
ing interest rates on credit extended to consumers. Such a rate is called a contract or stated rate
or, more frequently, an
annual percentage rate (APR)
. The method of calculating the APR
on a loan is set by law. The APR is the interest rate,
r
, charged per period multiplied by the
number of periods in a year,
m
:
APR
=
r
×
m
(9.10)
Thus, a car loan that charges interest of 1 percent per month has an APR of 12 percent
(i.e., 1 percent times 12 months). An unpaid credit card balance that incurs interest charges of
1.5 percent per month has an APR of 18 percent (1.5 times 12 months).
However, the APR misstates the true interest rate. The
eff ective annual rate (EAR)
,
sometimes called the
annual eff ective yield
, is the true opportunity cost measure of the interest
rate, as it considers the eff ects of periodic compounding. For example, say an unpaid January
balance of $100 on a credit card accumulates interest at the rate of 1.5 percent per month.
The interest charge is added to the unpaid balance; if left unpaid, February’s balance will
be $101.50. If the bill remains unpaid through February, the 1.5 percent monthly charge is
levied based on the total unpaid balance of $101.50. In other words, interest is assessed on
previous months’ unpaid interest charges. Thus, since interest compounds, the APR formula
will
understate
the true or eff ective interest cost. This will always be true, except in the special
case where the number of periods is one per year—that is, in annual compounding situations.
If the periodic interest charge,
r
, is known, the EAR is found by using equation 9.11:
EAR = (1 +
r
)
m
– 1
(9.11)
where
m
is the number of periods per year. If the APR is known instead, divide the APR by
m
and use the resulting number for
r
in equation 9.11.
3
As an example of the eff ective annual rate concept, let’s fi nd the true annual interest cost
of a credit card that advertises an 18 percent APR. Since credit card charges are, typically,
assessed monthly,
m
(the number of periods per year) is 12. Thus, the monthly interest rate is,
r
= APR ÷
m
= 18% ÷ 12 = 1.5%
From equation 9.11, the EAR is,
(1 + 0.015)
12
– 1 = 1.1956 – 1 = 0.1956, or 19.56%
The true interest charge on a credit card with an 18 percent APR is really 19.56 percent!
When the annual stated rate stays the same, more frequent interest compounding helps
savers earn more interest over the course of a year. For example, is it better to put your money
in an account off ering (option 1) 8 percent interest per year, compounded quarterly, or (option 2)
8 percent interest per year, compounded monthly?
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