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to a thumping $1.6 billion.
then a further 15 days to consider their final numbers. If
Fifteen days later the two sides met again to ex-
the two companies™ valuations were less than 10 per-
change their final valuations. There was an air of shock
cent apart, AT&T would be obliged to buy at the aver-
in the room; despite hearing the other side™s arguments,
age of the two prices. If they were more than 10 percent
the difference in their valuations had barely narrowed. It
apart, an independent arbitrator would be appointed. If
seemed that an independent arbitrator was required
the arbitrator decided that the true value was about
and so another investment bank, Wasserstein Perella,
midway between the two companies™ valuations, then
was called in to provide an independent valuation.
the arbitrator™s valuation would be used. If it was close
Some weeks later a herd of about 50 investment
to AT&T™s valuation, then the arbitrator™s price and
bankers and lawyers crowded into the offices of
AT&T™s price would be averaged and LIN™s valuation
Wasserstein Perella to defend their estimates of the
would be ignored. Conversely, if it was close to LIN™s
value of LIN. Comparisons were made with the value of
figure, then the arbitrator™s price would be averaged in
other cellular communications companies. Each side
with LIN™s valuation and AT&T™s figure would be ig-
presented projections of LIN™s future profits and divi-
dends. There were also arguments about the rate at
Each company appointed an investment bank to
which these future dividends should be discounted. For
prepare and argue its case. AT&T™s case was presented
example, each side argued that the other had failed to
by Morgan Stanley while LIN™s case was prepared by
measure properly the risk of the stock.
Bear Stearns and Lehman Brothers. Each side faced a
The final upshot: After hearing the arguments from
quandary. AT&T™s advisers were tempted to go for a low
both sides, Wasserstein Perella placed a value of
figure, while LIN™s advisers were tempted to come up
$127.50 on each share of LIN. This meant that the total
with a high figure. But if the dispute went to arbitration,
cost of the shares to AT&T was about $3.3 billion.
then an extreme valuation was more likely to be out of
line with the arbitrator™s figure and therefore was more Source: The story of the valuation of LIN Broadcasting is set out in
likely to be ignored. It seemed to make sense to take an S. Neish, “Wrong Number,” Global M&A (Summer 1995).

The dramatic appreciation in stock prices in the late 1990s was attributed by many
investors to a “new paradigm,” where the revolution in information technology would
boost company profitability. But the skeptics argued that the run-up in stock prices may
be due to accounting problems. The nearby box discusses the possibility that part of the
run-up of stock prices relative to earnings in the 1990s, which has worried many stock
market observers, may be due to other accounting problems. The article focuses on the
distortions created in income statements when investments in research, development,

Valuing Stocks 301

software, and training are treated as expenses which reduce reported earnings, rather
than as investments in intangible assets, which would then be gradually depreciated
over time.

Investors routinely buy and sell shares of common stock. Companies frequently buy
and sell entire businesses. So it is natural to ask whether the formulas that we have pre-
sented in this material can also be used to value these businesses.
Sure! Take the case of Blue Skies. Suppose that it has 2 million shares outstanding.
It plans to pay a dividend of DIV1 = $3 a share. So the total dividend payment is 2 mil-
lion — $3 = $6 million. Investors expect a steady dividend growth of 8 percent a year
and require a return of 12 percent. So the total value of Blue Skies is
$6 million
PV = = $150 million
.12 “ .08
Alternatively, we could say that the total value of the company is the number of shares
times the value per share:
PV = 2 million — $75 = $150 million
Of course things are always harder in practice than in principle. Forecasting cash
flows and settling on an appropriate discount rate require skill and judgment. As the
nearby box shows, there can be plenty of room for disagreement.

What information about company stocks is regularly reported in the financial
pages of the newspaper?
Firms that wish to raise new capital may either borrow money or bring new “partners” into
the business by selling shares of common stock. Large companies usually arrange for their
stocks to be traded on a stock exchange. The stock listings report the stock™s dividend yield,
price, and trading volume.

How can one calculate the present value of a stock given forecasts of future divi-
dends and future stock price?
Stockholders generally expect to receive (1) cash dividends and (2) capital gains or losses.
The rate of return that they expect over the next year is defined as the expected dividend per
share DIV1 plus the expected increase in price P1 “ P0, all divided by the price at the start of
the year P0.
Unlike the fixed interest payments that the firm promises to bondholders, the dividends
that are paid to stockholders depend on the fortunes of the firm. That™s why a company™s
common stock is riskier than its debt. The return that investors expect on any one stock is
also the return that they demand on all stocks subject to the same degree of risk. The present
value of a stock equals the present value of the forecast future dividends and future stock
price, using that expected return as the discount rate.

How can stock valuation formulas be used to infer the expected rate of return on
a common stock?

The present value of a share is equal to the stream of expected dividends per share up to
some horizon date plus the expected price at this date, all discounted at the return that
investors require. If the horizon date is far away, we simply say that stock price equals the
present value of all future dividends per share. This is the dividend discount model.
If dividends are expected to grow forever at a constant rate g, then the expected return on
the stock is equal to the dividend yield (DIV1/P0) plus the expected rate of dividend growth.
The value of the stock according to this constant-growth dividend discount model is P0 =
DIV1/(r “ g).

How should investors interpret price-earnings ratios?
You can think of a share™s value as the sum of two parts”the value of the assets in place
and the present value of growth opportunities, that is, of future opportunities for the firm
to invest in high-return projects. The price-earnings (P/E) ratio reflects the market™s
assessment of the firm™s growth opportunities.

www.ganesha.org/invest/index.html Links to information useful for valuing securities
www.nasdaq.com/ Information about Nasdaq and Amex-traded stocks
Related Web www.nyse.com/ Information about stocks and trading on the New York Stock Exchange
www.fool.com/School/HowtoValueStocks.htm How investors value firms
Links www.zacks.com Information and analyses from Zacks Investment Research
www.Investools.com Investing tools, links to research reports on public companies and invest-
ment newsletters
www.morningstar.net Morningstar is a premier source of information on mutual funds
www.brill.com More information on mutual funds, as well as articles and other educational re-

liquidation value plowback ratio
common stock
Key Terms
market-value balance sheet present value of growth
initial public offering (IPO)
secondary market dividend discount model opportunities (PVGO)
sustainable growth rate
dividend constant-growth dividend
price-earnings (P/E) multiple discount model
book value payout ratio

Quiz 1. Dividend Discount Model. Amazon.com has never paid a dividend, but its share price is
$66 and the market value of its stock is $22 billion. Does this invalidate the dividend dis-
count model?
2. Dividend Yield. Favored stock will pay a dividend this year of $2.40 per share. Its dividend
yield is 8 percent. At what price is the stock selling?
3. Preferred Stock. Preferred Products has issued preferred stock with a $7 annual dividend
that will be paid in perpetuity.

a. If the discount rate is 12 percent, at what price should the preferred sell?
b. At what price should the stock sell 1 year from now?
c. What is the dividend yield, the capital gains yield, and the expected rate of return of the
Valuing Stocks 303

4. Constant-Growth Model. Waterworks has a dividend yield of 8 percent. If its dividend is
expected to grow at a constant rate of 5 percent, what must be the expected rate of return on
the company™s stock?
5. Dividend Discount Model. How can we say that price equals the present value of all future
dividends when many actual investors may be seeking capital gains and planning to hold
their shares for only a year or two? Explain.
6. Rate of Return. Steady As She Goes, Inc., will pay a year-end dividend of $2.50 per share.
Investors expect the dividend to grow at a rate of 4 percent indefinitely.

a. If the stock currently sells for $25 per share, what is the expected rate of return on the
b. If the expected rate of return on the stock is 16.5 percent, what is the stock price?
7. Dividend Yield. BMM Industries pays a dividend of $2 per quarter. The dividend yield on
its stock is reported at 4.8 percent. What price is the stock selling at?

8. Stock Values. Integrated Potato Chips paid a $1 per share dividend yesterday. You expect
Practice the dividend to grow steadily at a rate of 4 percent per year.
Problems a. What is the expected dividend in each of the next 3 years?
b. If the discount rate for the stock is 12 percent, at what price will the stock sell?
c. What is the expected stock price 3 years from now?
d. If you buy the stock and plan to hold it for 3 years, what payments will you receive? What
is the present value of those payments? Compare your answer to (b).

9. Constant-Growth Model. A stock sells for $40. The next dividend will be $4 per share. If
the rate of return earned on reinvested funds is 15 percent and the company reinvests 40 per-
cent of earnings in the firm, what must be the discount rate?
10. Constant-Growth Model. Gentleman Gym just paid its annual dividend of $2 per share,
and it is widely expected that the dividend will increase by 5 percent per year indefinitely.
a. What price should the stock sell at? The discount rate is 15 percent.
b. How would your answer change if the discount rate were only 12 percent? Why does the
answer change?

11. Constant-Growth Model. Arts and Crafts, Inc., will pay a dividend of $5 per share in 1
year. It sells at $50 a share, and firms in the same industry provide an expected rate of re-
turn of 14 percent. What must be the expected growth rate of the company™s dividends?
12. Constant-Growth Model. Eastern Electric currently pays a dividend of about $1.64 per
share and sells for $27 a share.

a. If investors believe the growth rate of dividends is 3 percent per year, what rate of return
do they expect to earn on the stock?
b. If investors™ required rate of return is 10 percent, what must be the growth rate they ex-
pect of the firm?
c. If the sustainable growth rate is 5 percent, and the plowback ratio is .4, what must be the
rate of return earned by the firm on its new investments?

13. Constant-Growth Model. You believe that the Non-stick Gum Factory will pay a dividend
of $2 on its common stock next year. Thereafter, you expect dividends to grow at a rate of 6
percent a year in perpetuity. If you require a return of 12 percent on your investment, how
much should you be prepared to pay for the stock?

14. Negative Growth. Horse and Buggy Inc. is in a declining industry. Sales, earnings, and div-
idends are all shrinking at a rate of 10 percent per year.

a. If r = 15 percent and DIV1 = $3, what is the value of a share?
b. What price do you forecast for the stock next year?
c. What is the expected rate of return on the stock?
d. Can you distinguish between “bad stocks” and “bad companies”? Does the fact that the
industry is declining mean that the stock is a bad buy?

15. Constant-Growth Model. Metatrend™s stock will generate earnings of $5 per share this
year. The discount rate for the stock is 15 percent and the rate of return on reinvested earn-
ings also is 15 percent.

a. Find both the growth rate of dividends and the price of the stock if the company reinvests
the following fraction of its earnings in the firm: (i) 0 percent; (ii) 40 percent; (iii) 60 per-
b. Redo part (a) now assuming that the rate of return on reinvested earnings is 20 percent.
What is the present value of growth opportunities for each reinvestment rate?
c. Considering your answers to parts (a) and (b), can you briefly state the difference between
companies experiencing growth versus companies with growth opportunities?

16. Nonconstant Growth. You expect a share of stock to pay dividends of $1.00, $1.25, and
$1.50 in each of the next 3 years. You believe the stock will sell for $20 at the end of the
third year.

a. What is the stock price if the discount rate for the stock is 10 percent?
b. What is the dividend yield?

17. Constant-Growth Model. Here are data on two stocks, both of which have discount rates
of 15 percent:

Stock A Stock B
Return on equity 15% 10%
Earnings per share $2.00 $1.50
Dividends per share $1.00 $1.00

a. What are the dividend payout ratios for each firm?
b. What are the expected dividend growth rates for each firm?
c. What is the proper stock price for each firm?


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