Until now we have been assuming that future
values arose from only one initial investment.
What happens when there is a stream of multiple
investments? An annuity describes a stream
of equal annual cash flows. (Annuities come in
two flavors: a stream of outflows invested
to produce future returns or a stream of inflows
of investment returns.)
Consider an example. What would be the future
value of $1,000 invested at the end of each year
in a bank account that paid 8% interest, compounded
annually? (Notice the importance of whether the
amounts are invested at the beginning or end of
each year.)

Investment 
Future value
(in five years) 
Year 1 
$1,000 x (1 + .08)^{4}
= $1,360.49 
Year 2 
$1,000 x (1 + .08)^{3}
= $1,259.71 
Year 3 
$1,000 x (1 + .08)^{2}
= $1,166.40 
Year 4 
$1,000 x (1 + .08)^{1}
= $1,080.00 
Year 5 
$1,000 x (1 + .08)^{0}
= $1,000.00 
Total 
$5,886.60 
Now consider what equal payment could be made
at the end of each year if you started with $5,866.60,
assuming the principal earned interest at an annual
rate of 8%. You guessed it! An annuity of $1,000.

Simplifying calculation of annuity's
future value
Annuity calculations are simplified by using
tables, as well as calculators and spreadsheets.
Look at the attached tables, created using a spreadsheet,
which show:
From the attached future
value table, you will notice that if you invest
$1 each year at 12% interest for 25 years (assuming
annual compounding) you would have $133.33. Now
imagine that you started a savings plan in which
you put $2,000 each year into a mutual fund that
on average has 12% annual returns, after fees
and taxes. In 25 years, you would have $266,667.74.
(Notice the power of steady savings and compounding!)
Or, if you are adventuresome, you can calculate
the present value of an annuity using a formula
(imbedded in the attached present
value table):
FV
= Pymt_{n} [ (1 + i)^{n}
 1 ] / i 
FV 
future value at the end of
period 
Pymt_{n} 
payment made at the end of each of n
periods 
i 
interest rate for each period (assumed
equal throughout) 
n 
number of periods 

Example
King Tobacco Company is negotiating to compensate
the state of New Columbia for the state's healthcost
payments related to tobacco use. The company says
it will pay the State of New Columbia $200,000,000
for the payments that the state incurred over
the last twenty year  which have averaged $10
million per year.
New Columbia's attorney general understands that
this offer does not take into account the time
value of money and seeks a settlement that will
reflect prejudgment interest. What should the
attorney general seek, assuming the following
interest rates (averaged over the last 20 years):
 the riskfree rate of US Treasury notes/bills/bonds
 5.1%
 the state's cost of borrowing  6.5%
 the tobacco companies' cost of debt  8.9%
 the tobacco companies' cost of equity  13.4%
Answer:
There are at least two steps in calculating prejudgment
interest. First, what interest rate is appropriate?
Second, using this interest rate, what is the
current value of the state's past losses (a future
value computation)? See "Should
Tobacco Companies Pay the Present Value of Damages?"
An interest rate that focuses on the state's
loss may underdeter the culpable party, here
King Tobacco. On the other had, an interest rate
that looks at the loss from the tobacco company's
perspective (particularly that of equity shareholders)
may overstate the company's responsibility.
Furthermore, these computations assume a flat
interest rate during the entire prejudgment period.
That is, payments made 20 years ago are treated
the same as those last year, even though their
future value when made may have been different
given different interest expectations. 