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CHAPTER 4 Time Value of Money 155
TABLE 4.7 The Future Value at the End of
Years 1 and 2 from Investing $100
at 8% Interest, Given Various
Compounding Periods
will have at the end of each period. At the end of 12 months (1 year), with 8%
quarterly compounding, Fred will have $108.24; at the end of 24 months (2
years), he will have $117.16.
Table 4.7 compares values for Fred Moreno’s $100 at the end of years 1
and 2 given annual, semiannual, and quarterly compounding periods at the 8
percent rate. As shown, the more frequently interest is compounded, the greater
the amount of money accumulated. This is true for any interest rate for any
period of time.
A General Equation for
Compounding More Frequently Than Annually
The formula for annual compounding (Equation 4.4) can be rewritten for use
when compounding takes place more frequently. If m equals the number of times
per year interest is compounded, the formula for annual compounding can be
rewritten as
i mn
FVnPV 1 (4.18)
m
If m1, Equation 4.18 reduces to Equation 4.4. Thus, if interest is compounded
annually (once a year), Equation 4.18 will provide the same result as
Equation 4.4. The general use of Equation 4.18 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, m would equal 2 in
Equation 4.18; for quarterly compounding, m would equal 4. Substituting the
appropriate values for semiannual and quarterly compounding into Equation
4.18, we find that
1. For semiannual compounding:
0.08
2
FV 2$100 1 22
Compounding period
End of year Annual Semiannual Quarterly
1 $108.00 $108.16 $108.24
2 116.64 116.99 117.16
$100(1 0.04) 4 $116.99