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158 PART 2 Important Financial Concepts
0.16 2nd Input Function
100
1.1735
Solution
117.35
e x
nominal (stated) annual rate
Contractual annual rate of
interest charged by a lender or
promised by a borrower.
effective (true) annual rate (EAR)
The annual rate of interest
actually paid or earned.
Next multiply this value by $100 to get the future value of $117.35 as shown at the
left. (Note: On some calculators, you may not have to press 2nd before pressing e x .)
Spreadsheet Use The future value of the single amount with continuous compounding
also can be calculated as shown on the following Excel spreadsheet.
The future value with continuous compounding therefore equals $117.35. As
expected, the continuously compounded value is larger than the future value of
interest compounded semiannually ($116.99) or quarterly ($117.16). Continuous
compounding offers the largest amount that would result from compounding
interest more frequently than annually.
Nominal and Effective Annual Rates of Interest
Both businesses and investors need to make objective comparisons of loan costs
or investment returns over different compounding periods. In order to put interest
rates on a common basis, to allow comparison, we distinguish between nominal
and effective annual rates. The nominal, or stated, annual rate is the contractual
annual rate of interest charged by a lender or promised by a borrower. The
effective, or true, annual rate (EAR) is the annual rate of interest actually paid or
earned. The effective annual rate reflects the impact of compounding frequency,
whereas the nominal annual rate does not.
Using the notation introduced earlier, we can calculate the effective annual
rate, EAR, by substituting values for the nominal annual rate, i, and the compounding
frequency, m, into Equation 4.21:
EAR 1 m i
1 (4.21)
We can apply this equation using data from preceding examples.
EXAMPLE Fred Moreno wishes to find the effective annual rate associated with an 8% nominal
annual rate (i0.08) when interest is compounded (1) annually (m1);
(2) semiannually (m2); and (3) quarterly (m4). Substituting these values into
Equation 4.21, we get
1. For annual compounding:
0.08
1
EAR 1 1
m
1(1 0.08) 1 110.08 10.08 8%