Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
CHAPTER FOUR
58
THE VALUE OF
COMMON STOCKS
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
WE SHOULD WARN you that being a financial expert has its occupational hazards. One is being cor-
nered at cocktail parties by people who are eager to explain their system for making creamy profits
by investing in common stocks. Fortunately, these bores go into temporary hibernation whenever the
market goes down.
We may exaggerate the perils of the trade. The point is that there is no easy way to ensure su-
perior investment performance. Later in the book we will show that changes in security prices are
fundamentally unpredictable and that this result is a natural consequence of well-functioning cap-
ital markets. Therefore, in this chapter, when we propose to use the concept of present value to
price common stocks, we are not promising you a key to investment success; we simply believe that
the idea can help you to understand why some investments are priced higher than others.
Why should you care? If you want to know the value of a firm’s stock, why can’t you look up the
stock price in the newspaper? Unfortunately, that is not always possible. For example, you may be
the founder of a successful business. You currently own all the shares but are thinking of going pub-

4.1 HOW COMMON STOCKS ARE TRADED
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
(NYSE).
1
This is the largest stock exchange in the world and trades, on an average
day, 1 billion shares in some 2,900 companies.
Suppose that you are the head trader for a pension fund that wishes to buy
100,000 GE shares. You contact your broker, who then relays the order to the floor
of the NYSE. Trading in each stock is the responsibility of a specialist, who keeps a
record of orders to buy and sell. When your order arrives, the specialist will check
this record to see if an investor is prepared to sell at your price. Alternatively, the
specialist may be able to get you a better deal from one of the brokers who is gath-
ered around or may sell you some of his or her own stock. If no one is prepared to
sell at your price, the specialist will make a note of your order and execute it as
soon as possible.
The NYSE is not the only stock market in the United States. For example, many
stocks are traded over the counter by a network of dealers, who display the prices at
which they are prepared to trade on a system of computer terminals known as
NASDAQ (National Association of Securities Dealers Automated Quotations Sys-
tem). If you like the price that you see on the NASDAQ screen, you simply call the
dealer and strike a bargain.
The prices at which stocks trade are summarized in the daily press. Here, for ex-
ample, is how The Wall Street Journal recorded the day’s trading in GE on July 2, 2001:
60 PART I

Stocks
© The McGraw−Hill
Companies, 2003
by the return that can be earned in the capital market on securities of comparable
risk. Shareholders receive cash from the company in the form of a stream of divi-
dends. So
PV(stock) ϭ PV(expected future dividends)
At first sight this statement may seem surprising. When investors buy stocks,
they usually expect to receive a dividend, but they also hope to make a capital gain.
Why does our formula for present value say nothing about capital gains? As we
now explain, there is no inconsistency.
Today’s Price
The cash payoff to owners of common stocks comes in two forms: (1) cash divi-
dends and (2) capital gains or losses. Suppose that the current price of a share is
P
0
, that the expected price at the end of a year is P
1
, and that the expected divi-
dend per share is DIV
1
. The rate of return that investors expect from this share
over the next year is defined as the expected dividend per share DIV
1
plus the ex-
pected price appreciation per share P
1
Ϫ P
0
, all divided by the price at the start

0
were less than $100,
the process would reverse. Fledgling’s stock would offer a higher rate of return
than comparable securities. In that case, investors would rush to buy, forcing the
price up to $100.
P
0
ϭ
5 ϩ 110
1.15
ϭ $100
Price ϭ P
0
ϭ
DIV
1
ϩ P
1
1 ϩ r
r ϭ
5 ϩ 110 Ϫ 100
100
ϭ .15, or 15%
Expected return ϭ r ϭ
DIV
1
ϩ P
1
Ϫ P
0

, and
we can express P
0
in terms of DIV
1
, DIV
2
, and P
2:
Take Fledgling Electronics. A plausible explanation why investors expect its
stock price to rise by the end of the first year is that they expect higher dividends
and still more capital gains in the second. For example, suppose that they are look-
ing today for dividends of $5.50 in year 2 and a subsequent price of $121. That
would imply a price at the end of year 1 of
Today’s price can then be computed either from our original formula
or from our expanded formula
We have succeeded in relating today’s price to the forecasted dividends for two
years (DIV
1
and DIV
2
) plus the forecasted price at the end of the second year (P
2
).
You will probably not be surprised to learn that we could go on to replace P
2
by
(DIV
3
ϩ P

H
P
0
ϭ
DIV
1
1 ϩ r
ϩ
DIV
2
11 ϩ r2
2
ϩ … ϩ
DIV
H
ϩ P
H
11 ϩ r2
H
P
0
ϭ
DIV
1
1 ϩ r
ϩ
DIV
2
ϩ P
2

ϭ
1
1 ϩ r
1DIV
1
ϩ P
1
2ϭ
1
1 ϩ r
aDIV
1
ϩ
DIV
2
ϩ P
2
1 ϩ r
bϭ
DIV
1
1 ϩ r
ϩ
DIV
2
ϩ P
2
11 ϩ r2
2
P

Period (H) Dividend (DIV
t
) Price (P
t
) Dividends Price Total
0 — 100 — — 100
1 5.00 110 4.35 95.65 100
2 5.50 121 8.51 91.49 100
3 6.05 133.10 12.48 87.52 100
4 6.66 146.41 16.29 83.71 100
10 11.79 259.37 35.89 64.11 100
20 30.58 672.75 58.89 41.11 100
50 533.59 11,739.09 89.17 10.83 100
100 62,639.15 1,378,061.23 98.83 1.17 100
TABLE 4.1
Applying the stock
valuation formula to
fledgling electronics.
Assumptions:
1. Dividends increase at
10 percent per year,
compounded.
2. Capitalization rate is
15 percent.
10050201043210
0
50
100
Present value, dollars
Horizon period

and working capital. Discounting earnings would recognize the rewards of that in-
vestment (a higher future dividend) but not the sacrifice (a lower dividend today).
The correct formulation states that share value is equal to the discounted stream of
dividends per share.
P
0
ϭ
a
∞
tϭ1
DIV
t
11 ϩ r2
t
64 PART I Value
4.3 A SIMPLE WAY TO ESTIMATE
THE CAPITALIZATION RATE
In Chapter 3 we encountered some simplified versions of the basic present value
formula. Let us see whether they offer any insights into stock values. Suppose,
for example, that we forecast a constant growth rate for a company’s dividends.
This does not preclude year-to-year deviations from the trend: It means only
that expected dividends grow at a constant rate. Such an investment would be
just another example of the growing perpetuity that we helped our fickle phi-
lanthropist to evaluate in the last chapter. To find its present value we must di-
vide the annual cash payment by the difference between the discount rate and
the growth rate:
Remember that we can use this formula only when g, the anticipated growth rate,
is less than r, the discount rate. As g approaches r, the stock price becomes infinite.
Obviously r must be greater than g if growth really is perpetual.
Our growing perpetuity formula explains P

These two formulas are much easier to work with than the general statement
that “price equals the present value of expected future dividends.”
2
Here is a prac-
tical example.
Using the DCF Model to Set Gas and Electricity Prices
The prices charged by local electric and gas utilities are regulated by state com-
missions. The regulators try to keep consumer prices down but are supposed to al-
low the utilities to earn a fair rate of return. But what is fair? It is usually interpreted
as r, the market capitalization rate for the firm’s common stock. That is, the fair rate
of return on equity for a public utility ought to be the rate offered by securities that
have the same risk as the utility’s common stock.
3
Small variations in estimates of this return can have a substantial effect on the
prices charged to the customers and on the firm’s profits. So both utilities and reg-
ulators devote considerable resources to estimating r. They call r the cost of equity
capital. Utilities are mature, stable companies which ought to offer tailor-made
cases for application of the constant-growth DCF formula.
4
Suppose you wished to estimate the cost of equity for Pinnacle West Corp. in
May 2001, when its stock was selling for about $49 per share. Dividend payments
for the next year were expected to be $1.60 a share. Thus it was a simple matter to
calculate the first half of the DCF formula:
The hard part was estimating g, the expected rate of dividend growth. One op-
tion was to consult the views of security analysts who study the prospects for each
company. Analysts are rarely prepared to stick their necks out by forecasting divi-
dends to kingdom come, but they often forecast growth rates over the next five
years, and these estimates may provide an indication of the expected long-run
growth path. In the case of Pinnacle West, analysts in 2001 were forecasting an
Dividend yield ϭ

PG&E is no longer a suitable subject for the constant-growth DCF formula.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
annual growth of 6.6 percent.
5
This, together with the dividend yield, gave an esti-
mate of the cost of equity capital:
An alternative approach to estimating long-run growth starts with the payout
ratio, the ratio of dividends to earnings per share (EPS). For Pinnacle, this was fore-
casted at 43 percent. In other words, each year the company was plowing back into
the business about 57 percent of earnings per share:
Also, Pinnacle’s ratio of earnings per share to book equity per share was about
11 percent. This is its return on equity, or ROE:
If Pinnacle earns 11 percent of book equity and reinvests 57 percent of that, then
book equity will increase by .57 ϫ .11 ϭ .063, or 6.3 percent. Earnings and divi-
dends per share will also increase by 6.3 percent:
Dividend growth rate ϭ g ϭ plowback ratio ϫ ROE ϭ .57 ϫ .11 ϭ .063
That gives a second estimate of the market capitalization rate:
Although this estimate of the market capitalization rate for Pinnacle stock seems
reasonable enough, there are obvious dangers in analyzing any single firm’s stock
with the constant-growth DCF formula. First, the underlying assumption of regu-
lar future growth is at best an approximation. Second, even if it is an acceptable ap-
proximation, errors inevitably creep into the estimate of g. Thus our two methods
for calculating the cost of equity give similar answers. That was a lucky chance; dif-
ferent methods can sometimes give very different answers.

0
ϩ g ϭ .033 ϩ .066 ϭ .099, or 9.9%
66 PART I Value
5
In this calculation we’re assuming that earnings and dividends are forecasted to grow forever at the
same rate g. We’ll show how to relax this assumption later in this chapter. The growth rate was based
on the average earnings growth forecasted by Value Line and IBES. IBES compiles and averages fore-
casts made by security analysts. Value Line publishes its own analysts’ forecasts
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
Figure 4.2 shows DCF costs of equity estimated at six-month intervals for a sam-
ple of electric utilities over a seven-year period. The burgundy line indicates the
median cost-of-equity estimates, which seem to lie about 3 percentage points
above the 10-year Treasury bond yield. The dots show the scatter of individual es-
timates. Again, most of this scatter is probably noise.
Some Warnings about Constant-Growth Formulas
The simple constant-growth DCF formula is an extremely useful rule of thumb, but
no more than that. Naive trust in the formula has led many financial analysts to
silly conclusions.
We have stressed the difficulty of estimating r by analysis of one stock only. Try
to use a large sample of equivalent-risk securities. Even that may not work, but at
least it gives the analyst a fighting chance, because the inevitable errors in estimat-
ing r for a single security tend to balance out across a broad sample.
In addition, resist the temptation to apply the formula to firms having high cur-
rent rates of growth. Such growth can rarely be sustained indefinitely, but the

؉ g
American Corp. $41.71 $2.64 6.3% 3.8% 10.1%
CH Energy Corp. 43.85 2.20 5.0 2.0 7.0
CLECO Corp. 46.00 .92 2.0 8.8 10.8
DPL, Inc. 30.27 1.03 3.4 9.6 13.0
Hawaiian Electric 36.69 2.54 6.9 2.6 9.5
Idacorp 39.42 1.97 5.0 5.7 10.7
Pinnacle West 49.16 1.60 3.3 6.6 9.9
Potomac Electric 22.00 1.75 8.0 5.7 13.7
Puget Energy 23.49 1.93 8.2 4.8 13.0
TECO Energy 31.38 1.44 4.6 7.7 12.3
UIL Holdings 48.21 2.93 6.1 1.9 8.0
Average 10.7%
TABLE 4.2
DCF cost-of-equity estimates for electric utilities in 2001.
Source: The Brattle Group, Inc.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
But this is silly. No firm can continue growing at 20 percent per year forever, except
possibly under extreme inflationary conditions. Eventually, profitability will fall
and the firm will respond by investing less.
In real life the return on equity will decline gradually over time, but for sim-
plicity let’s assume it suddenly drops to 16 percent at year 3 and the firm responds
by plowing back only 50 percent of earnings. Then g drops to .50(.16) ϭ .08.
Table 4.3 shows what’s going on. Growth-Tech starts year 1 with assets of $10.00.

percent
20
15
10
5
0
Jan. 86
Jan. 87 Jan. 88 Jan. 89 Jan. 90 Jan. 91 Jan. 92
10-year Treasury
bond yield
Median
estimate
FIGURE 4.2
DCF cost-of-equity estimates for a sample of 17 utilities. The median estimates (burgundy line) track long-
term interest rates fairly well. (The blue line is the 10-year Treasury yield.) The dots show the scatter of
the cost-of-equity estimates for individual companies.
Source: S. C. Myers and L. S. Borucki, “Discounted Cash Flow Estimates of the Cost of Equity Capital—A Case Study,”
Financial Markets, Institutions and Instruments 3 (August 1994), pp. 9–45.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
We have to use trial and error to find the value of r that makes P
0
equal $50. It turns
out that the r implicit in these more realistic forecasts is approximately .099, quite
a difference from our “constant-growth” estimate of .21.

ϩ
.65
11.12
3
ϩ
1
11.12
3

.67
1.10 Ϫ .042
ϭ $9.13
ϭ
.50
1 ϩ r
ϩ
.60
11 ϩ r2
2
ϩ
1.15
11 ϩ r2
3
ϩ
1
11 ϩ r2
3

1.24
r Ϫ .08

r Ϫ .08
CHAPTER 4
The Value of Common Stocks 69
Year
1234
Book equity 10.00 12.00 14.40 15.55
Earnings per share, EPS 2.50 3.00 2.30 2.49
Return on equity, ROE .25 .25 .16 .16
Payout ratio .20 .20 .50 .50
Dividends per share, DIV .50 .60 1.15 1.24
Growth rate of dividends (%) — 20 92 8
TABLE 4.3
Forecasted earnings and dividends for
Growth-Tech. Note the changes in year
3: ROE and earnings drop, but payout
ratio increases, causing a big jump in
dividends. However, subsequent
growth in earnings and dividends falls
to 8 percent per year. Note that the
increase in equity equals the earnings
not paid out as dividends.
ͭ
ͭ
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003

distinctions make sense.
Imagine first the case of a company that does not grow at all. It does not plow
back any earnings and simply produces a constant stream of dividends. Its stock
would resemble the perpetual bond described in the last chapter. Remember that
the return on a perpetuity is equal to the yearly cash flow divided by the present
value. The expected return on our share would thus be equal to the yearly dividend
divided by the share price (i.e., the dividend yield). Since all the earnings are paid
out as dividends, the expected return is also equal to the earnings per share di-
vided by the share price (i.e., the earnings–price ratio). For example, if the dividend
is $10 a share and the stock price is $100, we have
Expected return ϭ dividend yield ϭ earnings–price ratio
The price equals
P
0
ϭ
DIV
1
r
ϭ
EPS
1
r
ϭ
10.00
.10
ϭ 100
ϭ .10ϭ
10.00
100
ϭ

Therefore, once again the market capitalization rate equals the earnings–price ratio:
Table 4.5 repeats our example for different assumptions about the cash flow gen-
erated by the new project. Note that the earnings–price ratio, measured in terms of
EPS
1
, next year’s expected earnings, equals the market capitalization rate (r) only
when the new project’s NPV ϭ 0. This is an extremely important point—managers
frequently make poor financial decisions because they confuse earnings–price ra-
tios with the market capitalization rate.
In general, we can think of stock price as the capitalized value of average earnings
under a no-growth policy, plus PVGO, the present value of growth opportunities:
P
0
ϭ
EPS
1
r
ϩ PVGO
r ϭ
EPS
1
P
0
ϭ
10
100
ϭ .10
Net present value per share at year 1 ϭϪ10 ϩ
1
.10

I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
The earnings–price ratio, therefore, equals
It will underestimate r if PVGO is positive and overestimate it if PVGO is negative.
The latter case is less likely, since firms are rarely forced to take projects with nega-
tive net present values.
Calculating the Present Value of Growth Opportunities
for Fledgling Electronics
In our last example both dividends and earnings were expected to grow, but
this growth made no net contribution to the stock price. The stock was in this
sense an “income stock.” Be careful not to equate firm performance with the
growth in earnings per share. A company that reinvests earnings at below
the market capitalization rate may increase earnings but will certainly reduce
the share value.
Now let us turn to that well-known growth stock, Fledgling Electronics. You may
remember that Fledgling’s market capitalization rate, r, is 15 percent. The company
is expected to pay a dividend of $5 in the first year, and thereafter the dividend is
predicted to increase indefinitely by 10 percent a year. We can, therefore, use the
simplified constant-growth formula to work out Fledgling’s price:
Suppose that Fledgling has earnings per share of $8.33. Its payout ratio is then
In other words, the company is plowing back 1 Ϫ .6, or 40 percent of earnings. Sup-
pose also that Fledgling’s ratio of earnings to book equity is ROE ϭ .25. This ex-
plains the growth rate of 10 percent:
Growth rate ϭ g ϭ plowback ratio ϫ ROE ϭ .4 ϫ .25 ϭ .10
The capitalized value of Fledgling’s earnings per share if it had a no-growth pol-
icy would be
But we know that the value of Fledgling stock is $100. The difference of $44.44 must
be the amount that investors are paying for growth opportunities. Let’s see if we

DIV
1
r Ϫ g
ϭ
5
.15 Ϫ .10
ϭ $100
EPS
P
0
ϭ r a1 Ϫ
PVGO
P
0
b
72 PART I Value
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
Everything is the same in year 2 except that Fledgling will invest $3.67, 10 percent
more than in year 1 (remember g ϭ .10). Therefore at t ϭ 2 an investment is made
with a net present value of
Thus the payoff to the owners of Fledgling Electronics stock can be represented
as the sum of (1) a level stream of earnings, which could be paid out as cash divi-
dends if the firm did not grow, and (2) a set of tickets, one for each future year, rep-
resenting the opportunity to make investments having positive NPVs. We know

NPV
1
r Ϫ g
ϭ
2.22
.15 Ϫ .10
ϭ $44.44
Present value of level stream of earnings ϭ
EPS
1
r
ϭ
8.33
.15
ϭ $55.56
NPV
2
ϭϪ3.33 ϫ 1.10 ϩ
.83 ϫ 1.10
.15
ϭ $2.44
CHAPTER 4 The Value of Common Stocks 73
6
Michael Eisner, the chairman of Walt Disney Productions, made the point this way: “In school you had
to take the test and then be graded. Now we’re getting graded, and we haven’t taken the test.” This was
in late 1985, when Disney stock was selling at nearly 20 times earnings. See Kathleen K. Wiegner, “The
Tinker Bell Principle,” Forbes (December 2, 1985), p. 102.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition

Cost of PVGO PVGO, percent
Stock (October 2001) EPS* Equity, r
†
؍ P
0
؊ EPS/r of Stock Price
Income stocks:
Chubb $77.35 $4.90 .088 $21.67 28
Exxon Mobil 42.29 2.13 .072 12.71 30
Kellogg 29.00 1.42 .056 3.64 13
Weyerhaeuser 50.45 3.21 .128 25.37 50
Growth stocks:
Amazon.com 8.88 Ϫ.30 .24 10.13 114
Dell Computer 23.66 .76 .22 20.20 85
Microsoft 56.38 1.88 .184 46.16 82
Wal-Mart 52.90 1.70 .112 37.72 71
TABLE 4.6
Estimated PVGOs.
*EPS is defined as the average earnings under a no-growth policy. As an estimate of EPS, we used the forecasted earnings per share for
2002. Source: MSN Money (moneycentral.msn.com).
†
The market capitalization rate was estimated using the capital asset pricing model. We describe this model and how to use it in Sections
8.2 and 9.2. For this example, we used a market risk premium of 8 percent and a risk-free interest rate of 4 percent.
7
However, Amazon’s reported earnings probably understate its earnings potential. Amazon is growing
very rapidly, and some of the investments necessary to finance that growth are written off as expenses,
thus reducing current income. Absent these “investment expenses,” Amazon’s current income would
probably be positive. We discuss the problems encountered in measuring earnings and profitability in
Chapter 12.
Brealey−Meyers:

measures r only if PVGO ϭ 0 and only if reported EPS is
the average future earnings the firm could generate under a no-growth policy. An-
other reason P/Es are hard to interpret is that the figure for earnings depends on
the accounting procedures for calculating revenues and costs. We will discuss the
potential biases in accounting earnings in Chapter 12.
CHAPTER 4
The Value of Common Stocks 75
4.5 VALUING A BUSINESS BY DISCOUNTED CASH FLOW
Investors routinely buy and sell shares of common stock. Companies frequently
buy and sell entire businesses. In 2001, for example, when Diageo sold its Pillsbury
operation to General Mills for $10.4 billion, you can be sure that both companies
burned a lot of midnight oil to make sure that the deal was fairly priced.
Do the discounted-cash-flow formulas we presented in this chapter work for
entire businesses as well as for shares of common stock? Sure: It doesn’t matter
whether you forecast dividends per share or the total free cash flow of a business.
Value today always equals future cash flow discounted at the opportunity cost of
capital.
Valuing the Concatenator Business
Rumor has it that Establishment Industries is interested in buying your company’s
concatenator manufacturing operation. Your company is willing to sell if it can get
the full value of this rapidly growing business. The problem is to figure out what
its true present value is.
Table 4.7 gives a forecast of free cash flow (FCF) for the concatenator business. Free
cash flow is the amount of cash that a firm can pay out to investors after paying for
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill

Earnings 1.20 1.44 1.73 2.07 2.49 2.81 3.18 3.36 3.57 3.78
Investment 2.00 2.40 2.88 3.46 2.69 3.04 1.59 1.68 1.78 1.89
Free cash flow Ϫ.80 Ϫ.96 Ϫ1.15 Ϫ1.39 Ϫ.20 Ϫ.23 1.59 1.68 1.79 1.89
Earnings growth
from previous
period (%) 20 20 20 20 20 13 13 6 6 6
TABLE 4.7
Forecasts of free cash flow, in $ millions, for the Concatenator Manufacturing Division. Rapid expansion in years 1–6
means that free cash flow is negative, because required additional investment outstrips earnings. Free cash flow turns
positive when growth slows down after year 6.
Notes:
1. Starting asset value is $10 million. Assets required for the business grow at 20 percent per year to year 4, at 13 percent in years 5 and
6, and at 6 percent afterward.
2. Profitability (earnings/asset values) is constant at 12 percent.
3. Free cash flow equals earnings minus net investment. Net investment equals total capital expenditures less depreciation. Note that
earnings are also calculated net of depreciation.
8
Table 4.7 shows net investment, which is total investment less depreciation. We are assuming that in-
vestment for replacement of existing assets is covered by depreciation and that net investment is de-
voted to growth.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
PV(free cash flow) PV(horizon value)
Of course, the concatenator business will continue after the horizon, but it’s not
practical to forecast free cash flow year by year to infinity. PV

ϭϪ3.6
PV1cash flows2ϭϪ
.80
1.1
Ϫ
.96
11.12
2
Ϫ
1.15
11.12
3
Ϫ
1.39
11.12
4
Ϫ
.20
11.12
5
Ϫ
.23
11.12
6
PV1horizon value2ϭ
1
11.12
6
a
1.59

11.12
6
a
1.06
.10 Ϫ .08
bϭ $29.9
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
roughly match those projected for the concatenator business in year 6. Suppose fur-
ther that these companies tend to sell at price–earnings ratios of about 11. Then you
could reasonably guess that the price–earnings ratio of a mature concatenator op-
eration will likewise be 11. That implies:
PV(business) ϭϪ3.6 ϩ 19.7 ϭ $16.1 million
Horizon Value Based on Market–Book Ratios Suppose also that the market–book
ratios of the sample of mature manufacturing companies tend to cluster around
1.4. (The market–book ratio is just the ratio of stock price to book value per share.)
If the concatenator business market–book ratio is 1.4 in year 6,
PV(business) ϭϪ3.6 ϩ 18.5 ϭ $14.9 million
It’s easy to poke holes in these last two calculations. Book value, for example, of-
ten is a poor measure of the true value of a company’s assets. It can fall far behind
actual asset values when there is rapid inflation, and it often entirely misses im-
portant intangible assets, such as your patents for concatenator design. Earnings
may also be biased by inflation and a long list of arbitrary accounting choices. Fi-
nally, you never know when you have found a sample of truly similar companies.
But remember, the purpose of discounted cash flow is to estimate market value—

11.12
6
111 ϫ 3.182ϭ 19.7
78 PART I Value
10
We cover this point in more detail in Chapter 11.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
We know that present value in any period equals the capitalized value of next
period’s earnings, plus PVGO:
But what if PVGO ϭ 0? At the horizon period H, then,
In other words, when the competition catches up, the price–earnings ratio equals
l/r, because PVGO disappears.
Suppose competition is expected to catch up by period 8. We can recalculate the
value of the concatenator business as follows:
11
ϭ $16.7 million
PV(business) ϭϪ2.0 ϩ 16.7 ϭ $14.7 million
We now have four estimates of what Establishment Industries ought to pay for
the concatenator business. The estimates reflect four different methods of estimat-
ing horizon value. There is no best method, although in many cases we put most
weight on the last method, which sets the horizon date at the point when manage-
ment expects PVGO to disappear. The last method forces managers to remember
that sooner or later competition catches up.
Our calculated values for the concatenator business range from $14.7 to $18.8

earnings in period 9
r
b
PV
H
ϭ
earnings
Hϩ1
r
PV
t
ϭ
earnings
tϩ1
r
ϩ PVGO
CHAPTER 4 The Value of Common Stocks 79
11
The PV of free cash flow before the horizon improves to Ϫ$2.0 million because inflows in years 7 and
8 are now included.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
There are two approaches to valuing a company’s existing shares when new shares
will be issued. The first approach discounts the net cash flow to existing shareholders
if they buy all the new shares issued. In this case the existing shareholders would pay

On reflection, you will see that our two valuation methods must give the same an-
swer. The first assumes that the existing shareholders provide all the cash whenever
the firm needs cash. If so, they will also receive every dollar the firm pays out. The sec-
ond method assumes that new investors put up the cash, relieving existing sharehold-
ers of this burden. But the new investors then receive a share of future payouts. If in-
vestment by new investors is a zero-NPV transaction, then it doesn’t make existing
stockholders any better or worse off than if they had invested themselves. The key as-
sumption, of course, is that new shares are issued on fair terms, that is, at zero NPV.
12
Existing shares
Existing ϩ new shares
ϭ
1,000,000
1,191,500
ϭ .839, or 83.9%
80 PART I
Value
12
The same two methods work when the company will use free cash flow to repurchase and retire out-
standing shares. We discuss share repurchases in Chapter 16.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
I. Value 4. The Value of Common
Stocks
© The McGraw−Hill
Companies, 2003
In this chapter we have used our newfound knowledge of present values to exam-
ine the market price of common stocks. The value of a stock is equal to the stream
of cash payments discounted at the rate of return that investors expect to receive

case, you may wish to use a two-stage DCF formula, where near-term dividends are
forecasted and valued, and the constant-growth DCF formula is used to forecast
the value of the shares at the start of the long run. The near-term dividends and the
future share value are then discounted to present value.
The general DCF formula can be transformed into a statement about earnings
and growth opportunities:
The ratio EPS
1
/r is the capitalized value of the earnings per share that the firm would
generate under a no-growth policy. PVGO is the net present value of the investments
P
0
ϭ
EPS
1
r
ϩ PVGO
r ϭ
DIV
1
P
0
ϩ g
P
0
ϭ
DIV
1
r Ϫ g
P

J. B. Williams: The Theory of Investment Value, Harvard University Press, Cambridge,
Mass., 1938.
The following articles provide important developments of Williams’s early work. We suggest, how-
ever, that you leave the third article until you have read Chapter 16:
D. Durand: “Growth Stocks and the Petersburg Paradox,” Journal of Finance, 12:348–363
(September 1957).
M. J. Gordon and E. Shapiro: “Capital Equipment Analysis: The Required Rate of Profit,”
Management Science, 3:102–110 (October 1956).
M. H. Miller and F. Modigliani: “Dividend Policy, Growth and the Valuation of Shares,”
Journal of Business, 34:411–433 (October 1961).
Leibowitz and Kogelman call PVGO the “franchise factor.” They analyze it in detail in:
M. L. Leibowitz and S. Kogelman: “Inside the P/E Ratio: The Franchise Factor,” Financial
Analysts Journal, 46:17–35 (November–December 1990).
Myers and Borucki cover the practical problems encountered in estimating DCF costs of equity for
regulated companies; Harris and Marston report DCF estimates of rates of return for the stock
market as a whole:
S. C. Myers and L. S. Borucki: “Discounted Cash Flow Estimates of the Cost of
Equity Capital—A Case Study,” Financial Markets, Institutions and Instruments, 3:9–45
(August 1994).
82 PART I Value
that the firm will make in order to grow. A growth stock is one for which PVGO is
large relative to the capitalized value of EPS. Most growth stocks are stocks of rap-
idly expanding firms, but expansion alone does not create a high PVGO. What mat-
ters is the profitability of the new investments.
The same formulas that are used to value a single share can also be applied to
value the total package of shares that a company has issued. In other words, we can
use them to value an entire business. In this case we discount the free cash flow
thrown off by the business. Here again a two-stage DCF model is deployed. Free
cash flows are forecasted and discounted year by year out to a horizon, at which
point a horizon value is estimated and discounted.