If m1, Equation 4.20 reduces to Equation 4.4. Thus, if interest is com-
pounded annually (once a year), Equation 4.20 will provide the same result as
Equation 4.4. The general use of Equation 4.20 can be illustrated with a simple
example.EXAMPLE The preceding examples calculated the amount that Fred Moreno would have at
the end of 2 years if he deposited $100 at 8% interest compounded semiannually
and compounded quarterly. For semiannual compounding, mwould equal 2 in
Equation 4.20; for quarterly compounding, mwould equal 4. Substituting the
appropriate values for semiannual and quarterly compounding into Equation
4.20, we find that- For semiannual compounding:
FV 2 $100 1
2 2
$100(10.04)^4 $116.99- For quarterly compounding:
FV 2 $100 1
4 2
$100(10.02)^8 $117.16These results agree with the values for FV 2 in Tables 4.5 and 4.6.If the interest were compounded monthly, weekly, or daily, mwould equal 12,
52, or 365, respectively.Using Computational Tools for Compounding
More Frequently Than Annually
We can use the future value interest factors for one dollar, given in Table A–1,
when interest is compoundedmtimes each year. Instead of indexing the table
foripercent andnyears, as we do when interest is compounded annually, we
index it for (im) percent and (mn) periods. However, the table is less
useful, because it includes only selected rates for a limited number of periods.
Instead, a financial calculator or a computer and spreadsheet is typically
required.EXAMPLE Fred Moreno wished to find the future value of $100 invested at 8% interest
compounded both semiannually and quarterly for 2 years. The number of com-
pounding periods, m,the interest rate, and the number of periods used in each
case, along with the future value interest factor, are as follows:Compounding Interest rate Periods Future value interest factor
period m (i ÷ m)(mn) from Table A–1Semiannual 2 8% 2 4% 2 2 4 1.170
Quarterly 4 8% 4 2% 4 2 8 1.1720.08
40.08
2178 PART 2 Important Financial Concepts