3: The Time Value of Money

Making this calculation for a longer annuity would become cumbersome. Fortunately, a shortcut

formula exists that simplifies the future-value calculation of an ordinary annuity. Using the symbol PMT

to represent the annuity’s annual payment, Equation 3.4 gives the future value of an annuity that lasts

for n years (FV), assuming an interest rate of r%:

Eq. 3.4 (^) FV PMT
r
r
n

[+ ]
×
 −





(1 )1
We can demonstrate that Equation 3.4 yields the
same answer obtained in the previous model by
plugging in the values PMT = $1,000, n = 5, and
r = 0.07:

FV $1,000

(1.07) 1

0.07

$1,000

1.4026 1

0.07

$1,000 5.75074 $5,750.74

5

=×

 −











=×

 −







=×=

Once again, we find the future value of this
ordinary annuity to be $5,750.74. You could also
obtain this value using Excel’s FV (future value)
function. Recall that the syntax of that function
is =fv(rate,nper,pmt,pv,type). Now that you are

dealing with problems involving cash flow streams

rather than lump sums, you need to input particular

values for the arguments ‘pmt’ and ‘type.’ The value

for ‘pmt’ is simply the periodic cash flow that the

annuity provides, and the value for ‘type’ equals 0

if the problem you are solving is an ordinary annuity

(‘type’ equals 1 for an annuity due). For an annuity

problem, the argument ‘pv’ requires some additional

explanation. Excel will interpret any value entered for

this argument as a lump sum payment that comes

before the annuity begins. In this example, there is no

initial lump sum (pv = 0), so you can obtain the future

value of the annuity by entering = fv(0.07,5,-1,000,0,0).

example

As in the previous example, you plan to save $1,000
at the end of each of the next five years to accumulate
money for a vacation, and you expect to earn 7% on
the money that you save. In addition, you just received
a bonus at work, which gives you another $5,000 to
invest immediately. How much can you accumulate in
five years if you invest your bonus in addition to the
$1,000 per year that you originally intended to save?
Previously, we found that the future value of
a $1,000 annuity invested over five years at 7%
was $5,750.74. To that total, we now want to add
the future value of a $5,000 lump sum invested
immediately. Using Equation 3.1, we have:

$5,000(1 + 0.07)^5 = $7,012.76

Adding that to the future value of the annuity

gives you a total of $12,763.50 ($7,012.76 +

$5,750.74) for your vacation. A quick way to

solve this is to enter into Excel = fv(0.07,5,–1,000,

–5,000,0). Notice here that you enter both the

$1,000 deposits and the initial $5,000 lump sum

as negative numbers. These inputs have the same

sign because they both represent money flowing in

the same direction (out of your wallet and into your

savings account). Entering the values this way causes

Excel to report the answer as a positive $12,763.50.

example

Next, consider a slight variation on the vacation saving problem that requires you to integrate what

you’ve learned about finding the future value of both a lump sum and an annuity.

Sometimes consumers know that they want to accumulate a certain amount of money by making

regular deposits into a savings account. In this situation, the uncertainty is not the future value of the

annuity, but rather the amount of time needed to accumulate that future value.