Introduction to Finance

Rule of 72 
a shortcut method used 
to approximate the time required for 
an investment to double in value 
annuity 
a series of equal payments 
(or receipts) that occur over a 
number of time periods
ordinary annuity 
equal payments 
(or receipts) occur at the end of 
each time period
annuity due 
exists when equal 
periodic payments (or receipts) 
occur at the beginning of each 
time period
2
Annuity due problems are discussed in the Learning Extension at the end of this chapter.

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9.5 Future Value of an Annuity
233
While the computational process was relatively easy for the three-year ordinary annuity 
example, the required calculations become much more cumbersome as the time period is 
lengthened. As a result, the following equation was derived for fi nding the future value of an 
ordinary annuity (FVA):
FVA
n
= PMT{[(1 + 
r
)
n
– 1] ÷ 
r
} (9.5)
PMT is the periodic equal payment, 
r
is the compound interest rate, and 
n
is the total num-
ber of periods. Inserting the data from the preceding three-year annuity example results in 
the following answer:
FVA
3
= $1,000{[(1 + 0.08)
3
– 1] ÷ 0.08}
= $1,000[(1.2597 – 1) ÷ 0.08]
= 
$1,000(3.246)
= 
$3,246
Most fi nancial calculators are programmed to readily fi nd future values of annuities. 
In addition to the previously identifi ed keys, fi nancial calculators also will have a payments 
(PMT) key for purposes of working problems involving ordinary annuities. The result for the 
three-year ordinary annuity can be verifi ed using a fi nancial calculator.
First, clear the calculator. Next, enter –1000 (for both TI and HP calculators) and press 
the PMT key. Then, enter 8 and press the %i key, and enter 3 and press the N key. Finally, 
press the CPT key and then the FV key to calculate the FVA of 3246.40, which rounds to 
$3,246. Note that because this problem involves a periodic outfl ow of $1,000, most fi n-
ancial calculators require that the payment be entered as a negative number to solve for a 
positive FVA.