Introduction to Corporate Finance

3: The Time Value of Money

These techniques allow you to calculate the future value of any cash flow stream. However, one category

of cash flow, known as annuities, is very common in finance, and there are some helpful shortcuts that you

can use to calculate the future value of an annuity.

3-5b TYPES OF ANNUITIES

Before looking at future-value computations for annuities, we distinguish between the two basic types of annuities:

the ordinary annuity and the annuity due. An ordinary annuity is an annuity for which the payments occur at the

end of each period, whereas an annuity due is one for which the payments occur at the beginning of each period.

To demonstrate these differences, assume that you wish to choose the better of two annuities as a personal

investment opportunity. Both are five-year, $1,000 annuities. Annuity A is an ordinary annuity, and annuity

B is an annuity due. Although the amount of each annuity totals $5,000, the timing of the cash flows differs;

each cash flow arrives one year sooner with the annuity due than with the ordinary annuity. As you might

expect (given the core principle of the time value of money), for any positive interest rate, the future value

of an annuity due is always greater than the future value of an otherwise identical ordinary annuity.^45 Why?

Because you receive the first cash flow today in the annuity due, giving you a longer time to earn interest.

3-5c FINDING THE FUTURE VALUE OF AN ORDINARY ANNUITY

The future value of an ordinary annuity can be calculated using the same method demonstrated earlier

for a mixed stream.

5 Because ordinary annuities arise frequently in finance, we use the term ‘annuity’ throughout this book to refer to ordinary annuities, unless
otherwise specified.

The time line below shows a mixed stream of cash flows that has a future value in four years of $846.95 if the
interest rate is 7%. The mixed stream starts with an immediate cash flow of $100, but the question mark in year
2 indicates that you do not know the value of the cash flow that arrives in that period. How can you find that
value of the missing piece of the mixed stream?

0 1 2 3 4

Year

Cash flow $100 $150? $200 $175

FV of mixed stream in year 4 = $846.95 (r = 7%)

First, calculate the future value of the four cash flows that you know using Equation 3.3:

FV = $100(1 + 0.07)^4 + $150(1 + 0.07)^3 + $200(1 + 0.07)^1 + $175 = $703.84

If the future value of the entire stream is $846.95, and the future value of the four cash flows shown on
the time line is $703.84, then the difference must be the future value of the missing cash flow in year 2. That
difference is $143.11. In other words, the cash flow in year 2 must grow to $143.11 after earning interest for
two years. We could also say that the missing cash flow on the time line equals the present value in year 2 of
$143.11 to be received in year 4. Therefore, we can use Equation 3.2 as follows:

PV=

+

=

$.

(.)

$

14311

1007

2 125

example

ordinary annuity

An annuity for which the

payments occur at the end of

each period

annuity due

An annuity for which the

payments occur at the

beginning of each period