Introduction to Corporate Finance

PART 1: INTRODUCTION

3-5d FINDING THE FUTURE VALUE OF AN ANNUITY DUE

The calculations required to find the future value of an annuity due involve only a slight change to those

already demonstrated for an ordinary annuity. For the annuity due, the question is: how much money will

you have at the end of five years (to finance your exotic vacation) if you deposit $1,000 annually at the

beginning of each year into a savings account paying 7% annual interest?

Figure 3.9 graphically depicts this scenario on a time line. Note that the ends of years 0 to 4 are

respectively equivalent to the beginnings of years 1 to 5. As expected, the $6,153.29 future value of the

annuity due is greater than the $5,750.74 future value of the comparable ordinary annuity.^6 Because the

cash flows of the annuity due occur at the beginning of the year, the cash flow of $1,000 at the beginning

of year 1 earns 7% interest for five years, the cash flow of $1,000 at the beginning of year 2 earns 7%

interest for four years, and so on. Comparing this to the ordinary annuity, you can see that each $1,000

cash flow of the annuity due earns interest for one more year than the comparable ordinary annuity cash

flow. As a result, the future value of the annuity due is greater than the future value of the comparable

ordinary annuity.

We can convert the equation for the future value of an ordinary annuity, Equation 3.4, into an

expression for the future value of an annuity due, denoted FV (annuity due). To do so, we must take into

account that each cash flow of an annuity due earns an additional year of interest. Therefore, we simply

multiply Equation 3.4 by (1 + r), as shown in Equation 3.5:

Eq. 3.5 (^) FVannuityduePMT
r
r
r
n
()==×
 −





×
[(1+) 1]
(1+)
6 You can use the same Excel function to obtain the future value of an annuity due that you used to calculate the future value of an ordinary
annuity, except that the value of ‘type’ changes from 0 to 1. If you enter into Excel =fv(0.07,5,-1,000,0,1), then Excel produces the value
$6,153.29.
What is another application
of the annuity formula that
you might use in your personal
life? What are the important
data that you need to use the
formula in this application? Hint:
retirement is a potential long-
term investment.
thinking cap
question
Bruce and Mary Teiffel have just had their first child,
and they want to begin saving for private school
fees that that they anticipate starting when their
child turns 15. They estimate they will need a pool
of funds worth $200,000 to pay for three years of
private school fees. They plan to set aside $6,300 at
the end of each of the next 15 years, investing the
money to earn 10% interest. Given that plan, how
long will it take Bruce and Mary to accumulate the
money they need?
One way to solve this problem is to modify
Equation 3.4 to solve for the value n, the number
of periods in the annuity. A little algebra transforms
Equation 3.4 as follows:
In ÷In
FV r
PMT
rn

×

+









11 ()+=

Use $200,0000 for FV, 0.10 for r, and $6,300 for

PMT and you obtain:

In years

$200,000 0.10

$7,500

1÷In(1 0.10) 14.99

×

+









+=

Apparently, Bruce and Mary will have the money

they need just in time.

As we did with lump sums, we can employ

Excel’s ‘number of periods’ function to obtain a

quick solution to this problem. To apply this function,

enter =nper(0.10,–7,500,0,200,000,0), and Excel

will provide the answer, 14.99 years. In this syntax

of this function, we use 0 for pv, because Bruce and

Mary are just starting to save for the school fees and

have accumulated nothing so far. Notice also that

the payment (–$6,300) and the desired future value

($200,000) have opposite signs.

example