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

(^196) © The McGraw−Hill
Companies, 2002
VALUING LEVEL CASH FLOWS:
ANNUITIES AND PERPETUITIES
We will frequently encounter situations in which we have multiple cash flows that are
all the same amount. For example, a very common type of loan repayment plan calls for
the borrower to repay the loan by making a series of equal payments over some length
of time. Almost all consumer loans (such as car loans) and home mortgages feature
equal payments, usually made each month.
More generally, a series of constant or level cash flows that occur at the end of each
period for some fixed number of periods is called an ordinary annuity; or, more cor-
rectly, the cash flows are said to be in ordinary annuity form. Annuities appear very fre-
quently in financial arrangements, and there are some useful shortcuts for determining
their values. We consider these next.
Present Value for Annuity Cash Flows
Suppose we were examining an asset that promised to pay $500 at the end of each of the
next three years. The cash flows from this asset are in the form of a three-year, $500 an-
nuity. If we wanted to earn 10 percent on our money, how much would we offer for this
annuity?
From the previous section, we know that we can discount each of these $500 pay-
ments back to the present at 10 percent to determine the total present value:
Present value ($500/1.1^1 ) (500/1.1^2 ) (500/1.1^3 )
($500/1.1) (500/1.21) (500/1.331)
$454.55 413.22 375.66
$1,243.43
This approach works just fine. However, we will often encounter situations in which the
number of cash flows is quite large. For example, a typical home mortgage calls for
monthly payments over 30 years, for a total of 360 payments. If we were trying to de-
termine the present value of those payments, it would be useful to have a shortcut.
Because the cash flows of an annuity are all the same, we can come up with a very
useful variation on the basic present value equation. It turns out that the present value of
an annuity of Cdollars per period for tperiods when the rate of return or interest rate is
ris given by:
Annuity present value C()

C{}

[6.1]

The term in parentheses on the first line is sometimes called the present value interest

factor for annuities and abbreviated PVIFA(r, t).

The expression for the annuity present value may look a little complicated, but it isn’t

difficult to use. Notice that the term in square brackets on the second line, 1/(1 r)t, is

the same present value factor we’ve been calculating. In our example from the begin-

ning of this section, the interest rate is 10 percent and there are three years involved. The

usual present value factor is thus:

Present value factor 1/1.1^3 1/1.331 .75131

1 
[1/(1 r)t]

r

1 
Present value factor

r

166 PART THREE Valuation of Future Cash Flows

6.2

annuity
A level stream of cash
flows for a fixed period
of time.