Introduction to Corporate Finance

Ross et al.: Fundamentals

of Corporate Finance, Sixth

Edition, Alternate Edition

III. Valuation of Future

Cash Flows

6. Discounted Cash Flow
   Valuation

© The McGraw−Hill^203

Companies, 2002

had the worststretch of stock returns of any 25-year period between 1925 and 2001.
How bad was it?

Here we have the cash flows ($100 per month), the futurevalue ($76,374), and the
time period (25 years, or 300 months). We need to find the implicit rate, r:

$76,374 $100 [(Future value factor 
1)/r]

763.74 [(1 r)^300 
1]/r

Because this is the worst period, let’s try 1 percent:

Annuity future value factor (1.01^300 
1)/.01 1,878.85

We see that 1 percent is too high. From here, it’s trial and error. See if you agree that r
is about .55 percent per month. As you will see later in the chapter, this works out to be
about 6.8 percent per year.

A Note on Annuities Due

So far, we have only discussed ordinary annuities. These are the most important, but
there is a variation that is fairly common. Remember that with an ordinary annuity, the
cash flows occur at the end of each period. When you take out a loan with monthly pay-
ments, for example, the first loan payment normally occurs one month after you get the
loan. However, when you lease an apartment, the first lease payment is usually due im-
mediately. The second payment is due at the beginning of the second month, and so on.
A lease is an example of an annuity due. An annuity due is an annuity for which the
cash flows occur at the beginning of each period. Almost any type of arrangement in
which we have to prepay the same amount each period is an annuity due.
There are several different ways to calculate the value of an annuity due. With a fi-
nancial calculator, you simply switch it into “due” or “beginning” mode. It is very im-
portant to remember to switch it back when you are done! Another way to calculate the
present value of an annuity due can be illustrated with a time line. Suppose an annuity
due has five payments of $400 each, and the relevant discount rate is 10 percent. The
time line looks like this:

CHAPTER 6 Discounted Cash Flow Valuation 173

annuity due

An annuity for which the

cash flows occur at the

beginning of the period.

CALCULATOR HINTS

Future Values of Annuities

Of course, you could solve this problem using a financial calculator by doing the

following:

Notice that we put a negative sign on the payment (why?). With a spreadsheet, use the

function  FV(rate,nper,pmt,pv); be sure to put in a zero for pv and to enter 
2,000 as the

payment.

N %i PMT PV FV

Enter 30 8 
2,000

Solve for 226,566.42