PART 1: INTRODUCTION
3 -7b STATED VERSUS EFFECTIVE ANNUAL INTEREST RATES
Both consumers and businesses need to make objective comparisons of loan costs or investment returns
over different compounding periods. To put interest rates on a common basis for comparison, we must
distinguish between stated and effective annual interest rates. The stated annual rate is the contractual
annual rate of interest charged by a lender or promised by a borrower. The effective annual rate (EAR), or true
annual return, is the annual rate of interest actually paid or earned. Why the difference? The effective
annual rate reflects the effect of compounding frequency; the stated annual rate does not.
Using the notation introduced earlier, we can calculate the effective annual rate by substituting values
for the stated annual rate (r) and the compounding frequency (m) into Equation 3.14:Eq. 3.14 (^) EAR
r
m
m
= 1+ 1
−
We can apply this equation using data from preceding examples.
LO3.6
To find the value at the end of two years of your
$100 deposit in an account paying 8% annual
interest compounded continuously, substitute PV =
$100, r = 0.08 and n = 2 into Equation 3.13:
FV (continuous compounding) = $100 × (e0.08×2)
= $100 × 2.71830.16
= $100 × 1.1735
= $117.35
The future value with continuous compounding
therefore equals $117.35, which, as expected,
is larger than the future value of interest
compounded semiannually ($116.99) or quarterly
($117.17).^12
example
Find the effective annual rate associated with an
8% stated annual rate (r = 0.08) when interest is
compounded annually (m = 1), semiannually (m = 2)
and quarterly (m = 4). Substituting these values into
Equation 3.14 obtains the following results:
For annual compounding:
EAR=+
−=+−=−
=
1
008
1
1100811081
00
1. (.). 1
.888= .% 0
For semiannual compounding:EAR=+
−=+−=−
=
1
008
2
110041 1 0816 1
0
2. (.). 2
... 0816 = 816 %
For quarterly compounding:EAR=+
−=+−=−
=
1
008
4
110021 1 0824 1
0
4. (.). 4
... 0824 = 824 %
The results mean that 8% compounded
semiannually is equivalent to 8.16% compounded
annually, and 8% compounded quarterly is
equivalent to 8.24% compounded annually. These
values demonstrate two important points:
(1) the stated and effective rates are equivalent
for annual compounding; and (2) the
effective annual rate increases with increasing
compounding frequency.examplestated annual rate
The contractual annual rate of
interest charged by a lender or
promised by a borrower
effective annual rate
(EAR)
The annual rate of interest
actually paid or earned,
reflecting the impact of
compounding frequency. Also
called the true annual return