Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionIII. Valuation of Future
Cash Flows- Discounted Cash Flow
Valuation
(^204) © The McGraw−Hill
Companies, 2002
012345
$400 $400 $400 $400 $400
Notice how the cash flows here are the same as those for a four-year ordinary annuity,
except that there is an extra $400 at Time 0. For practice, check to see that the value of
a four-year ordinary annuity at 10 percent is $1,267.95. If we add on the extra $400, we
get $1,667.95, which is the present value of this annuity due.
There is an even easier way to calculate the present or future value of an annuity due.
If we assume cash flows occur at the end of each period when they really occur at the
beginning, then we discount each one by one period too many. We could fix this by sim-
ply multiplying our answer by (1 r), where ris the discount rate. In fact, the relation-
ship between the value of an annuity due and an ordinary annuity is just:
Annuity due value Ordinary annuity value (1 r) [6.3]
This works for both present and future values, so calculating the value of an annuity due
involves two steps: (1) calculate the present or future value as though it were an ordi-
nary annuity, and (2) multiply your answer by (1 r).Perpetuities
We’ve seen that a series of level cash flows can be valued by treating those cash flows
as an annuity. An important special case of an annuity arises when the level stream of
cash flows continues forever. Such an asset is called a perpetuitybecause the cash
flows are perpetual. Perpetuities are also called consols, particularly in Canada and the
United Kingdom. See Example 6.7 for an important example of a perpetuity.
Because a perpetuity has an infinite number of cash flows, we obviously can’t com-
pute its value by discounting each one. Fortunately, valuing a perpetuity turns out to be
the easiest possible case. The present value of a perpetuity is simply:
PV for a perpetuity C/r [6.4]
For example, an investment offers a perpetual cash flow of $500 every year. The return
you require on such an investment is 8 percent. What is the value of this investment?
The value of this perpetuity is:
Perpetuity PVC/r$500/.08 $6,250
This concludes our discussion of valuing investments with multiple cash flows. For
future reference, Table 6.2 contains a summary of the annuity and perpetuity basic cal-
culations we described. By now, you probably think that you’ll just use online calcula-
tors to handle annuity problems. Before you do, see our nearby Work the Webbox!174 PART THREE Valuation of Future Cash Flows
Time value applications
abound on the Web.
See, for example,
http://www.collegeboard.com,
http://www.1stmortgagedirectory.
com, and
personal.fidelity.com.
perpetuity
An annuity in which the
cash flows continue
forever.
consol
A type of perpetuity.
Preferred Stock
Preferred stock(or preference stock) is an important example of a perpetuity. When a corpo-
ration sells preferred stock, the buyer is promised a fixed cash dividend every period (usually
every quarter) forever. This dividend must be paid before any dividend can be paid to regular
stockholders, hence the term preferred.
Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar is-
sue of preferred stock already outstanding has a price of $40 per share and offers a dividendEXAMPLE 6.7