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

(^202) © The McGraw−Hill
Companies, 2002
value of $250,000, a $20,000 payment, and a 25-year period. If you grind through the cal-
culations (or get a little machine assistance), you should find that the unknown rate is
about 6.24 percent. You should take the annuity option if that rate is attractive relative to
other investments available to you. Notice that we have ignored taxes in this example, and
taxes can significantly affect our conclusion. Be sure to consult your tax adviser anytime
you win the lottery.
Future Value for Annuities
On occasion, it’s also handy to know a shortcut for calculating the future value of an an-
nuity. As you might guess, there are future value factors for annuities as well as present
value factors. In general, the future value factor for an annuity is given by:
Annuity FV factor (Future value factor
1)/r
[(1 r)t
1]/r

[6.2]

To see how we use annuity future value factors, suppose you plan to contribute $2,000

every year to a retirement account paying 8 percent. If you retire in 30 years, how much

will you have?

The number of years here, t,is 30, and the interest rate, r,is 8 percent, so we can cal-

culate the annuity future value factor as:

Annuity FV factor (Future value factor 
1)/r

(1.08^30 
1)/.08

(10.0627 
1)/.08

113.2832

The future value of this 30-year, $2,000 annuity is thus:

Annuity future value $2,000 113.28

$226,566.40

Sometimes we need to find the unknown rate, r,in the context of an annuity future

value. For example, if you had invested $100 per month in stocks over the 25-year pe-

riod ended December 1978, your investment would have grown to $76,374. This period

172 PART THREE Valuation of Future Cash Flows

CALCULATOR HINTS

Finding the Rate

Alternatively, you could use a financial calculator to do the following:

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

function RATE(nper,pmt,pv,fv); be sure to put in a zero for fv and to enter 1,000 as

the payment and 
3,000 as the pv.

N %i PMT PV FV

Enter 4 1,000 
3,000

Solve for 12.59