Introduction to Finance

DISCUSSION QUESTION 2
Let’s assume that you want to become a millionaire before you retire. How do you plan 
build a net worth of one million dollars?
9.5
Future Value of an Annuity
The previous discussion focused on cash payments, or receipts, that occurred only as lump 
sum present and future values. However, many fi nance problems involve equal payments or 
receipts over time, referred to as 
annuities
. More specifi cally, an 
annuity
is a series of equal 
payments, or receipts, that occur over a number of time periods.
An 
ordinary annuity
exists when the equal payments (or receipts) occur at the end of 
each time period. An 
annuity due
exists when equal periodic payments (or receipts) start at 
the end of time period zero, or in other words, at the beginning of each time period. We focus 
on problems involving ordinary annuities in this chapter.
2
For example, suppose you want to 
invest $1,000 per year for three years at an 8 percent interest rate. However, since you will not 
make your fi rst payment until the end of the fi rst year, this will be an ordinary annuity.
This problem also can be illustrated using a time line as follows: 
$1,000 × 1.08 × 1.08 =
$1,000 × 1.08 =
$1,000 =
Begin:
Year
Payments 1
2
3
0
1
2
3
$1,000
1,080
1,166
FV annuity = $3,246
End:
Notice that to calculate the future value of this ordinary annuity we must add the future val-
ues of the fi rst payment ($1,166), the second payment ($1,080), and the third payment ($1,000). 
This results in a future value of $3,246. To summarize, since the fi rst payment is made at the end 
of the fi rst year, it is compounded for two years. The second payment is compounded for one year 
and the third payment earns zero interest since the payment is made at the end of the third year.
The future value of this annuity can also be determined by making the following computations:
FV ordinary annuity = $1,000(1.08)
2
+ $1,000(1.08)
1
+ $1,000(1.08)
0
= $1,000(1.166) + $1,000(1.080) + $1,000(1.000)
= 
$1,000(3.246)
= 
$3,246