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Assets Liabilities and Shareholders™ Equity
Assets in place Market value of debt and other obligations
Investment opportunities Market value of shareholders™ equity
Valuing Stocks 287

Valuing Common Stocks
The cash payoff to owners of common stocks comes in two forms: (1) cash dividends
and (2) capital gains or losses. Usually investors expect to get some of each. Suppose
that the current price of a share is P0, that the expected price a year from now is P1, and
that the expected dividend per share is DIV1. The subscript on P0 denotes time zero,
which is today; the subscript on P1 denotes time 1, which is 1 year hence. We simplify
by assuming that dividends are paid only once a year and that the next dividend will
come in 1 year. The rate of return that investors expect from this share over the next year
is 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:
DIV1 + P1 “ P0
Expected return r
Let us now look at how our formula works. Suppose Blue Skies stock is selling for
$75 a share (P0 = $75). Investors expect a $3 cash dividend over the next year (DIV1 =
$3). They also expect the stock to sell for $81 a year hence (P1 = $81). Then the ex-
pected return to stockholders is 12 percent:
$3 + $81 “ $75
r= = .12, or 12%
Notice that this expected return comes in two parts, the dividend and capital gain:
Expected rate of return = expected dividend yield + expected capital gain
DIV1 P “ P0
P0 P0
$3 $81 “ $75
= +
$75 $75
= .04 + .08 = .12, or 12%
Of course, the actual return for Blue Skies may turn out to be more or less than
investors expect. For example, in 1998, one of the best performing stocks was
Amazon.com. Its price at the end of the year was $321.25, up from $30.125 at the be-
ginning of the year. Since the stock did not pay a dividend during the year, investors
earned an actual return of ($0 + $321.25 “ $30.125)/$321.25 = 9.66, or 966 percent.
This figure was almost certainly well in excess of investor expectations. At the other
extreme, the modem maker Hayes, which declared bankruptcy during the year, provided
an actual return of “98.5 percent, well below expectations. Never confuse the actual
outcome with the expected outcome.
We saw how to work out the expected return on Blue Skies stock given today™s stock
price and forecasts of next year™s stock price and dividends. You can also explain the
market value of the stock in terms of investors™ forecasts of dividends and price and the
expected return offered by other equally risky stocks. This is just the present value of
the cash flows the stock will provide to its owner:
DIV1 + P1
Price today = P0 =
For Blue Skies DIV1 = $3 and P1 = $81. If stocks of similar risk offer an expected re-
turn of r = 12 percent, then today™s price for Blue Skies should be $75:

$3 + $81
P0 = = $75
How do we know that $75 is the right price? Because no other price could survive in
competitive markets. What if P0 were above $75? Then the expected rate of return on
Blue Skies stock would be lower than on other securities of equivalent risk. (Check
this!) Investors would bail out of Blue Skies stock and substitute the other securities. In
the process they would force down the price of Blue Skies stock. If P0 were less than
$75, Blue Skies stock would offer a higher expected rate of return than equivalent-risk
securities. (Check this too.) Everyone would rush to buy, forcing the price up to $75.
When the stock is priced correctly (that is, price equals present value), the expected rate
of return on Blue Skies stock is also the rate of return that investors require to hold the

At each point in time all securities of the same risk are priced to offer the
same expected rate of return. This is a fundamental characteristic of prices in
well-functioning markets. It is also common sense.

Androscoggin Copper is increasing next year™s dividend to $5.00 per share. The fore-
Self-Test 3
cast stock price next year is $105. Equally risky stocks of other companies offer ex-
pected rates of return of 10 percent. What should Androscoggin common stock sell for?

We have managed to explain today™s stock price P0 in terms of the dividend DIV1 and
the expected stock price next year P1. But future stock prices are not easy to forecast
directly, though you may encounter individuals who claim to be able to do so. A for-
mula that requires tomorrow™s stock price to explain today™s stock price is not generally
As it turns out, we can express a stock™s value as the present value of all the forecast
future dividends paid by the company to its shareholders without referring to the future
stock price. This is the dividend discount model:
MODEL Discounted cash-
P0 = present value of (DIV1, DIV2, DIV3, . . ., DIVt, . . .)
flow model of today™s stock
price which states that share = + + +...+ +...
1 + r (1 + r)2 (1 + r)3 (1 + r)t
value equals the present
value of all expected future
How far out in the future could we look? In principle, 40, 60, or 100 years or more”
corporations are potentially immortal. However, far-distant dividends will not have sig-
nificant present values. For example, the present value of $1 received in 30 years using
a 10 percent discount rate is only $.057. Most of the value of established companies
comes from dividends to be paid within a person™s working lifetime.
How do we get from the one-period formula P0 = (DIV1 + P1)/(1 + r) to the dividend
discount model? We look at increasingly long investment horizons.
Let™s consider investors with different investment horizons. Each investor will value
the share of stock as the present value of the dividends that she expects to receive plus
the present value of the price at which the stock is eventually sold. Unlike bonds, how-
ever, the final horizon date for stocks is not specified”stocks do not “mature.” More-
Valuing Stocks 289

over, both dividends and final sales price can only be estimated. But the general valua-
tion approach is the same. For a one-period investor, the valuation formula looks like
DIV1 + P1
P0 =
A 2-year investor would value the stock as
DIV1 DIV2 + P2
P0 = +
1+r (1 + r)2
and a 3-year investor would use the formula
P0 = + +
1+r (1 + r)2 (1 + r)3
In fact we can look as far out into the future as we like. Suppose we call our horizon
date H. Then the stock valuation formula would be
P0 = + +...+
(1 + r)2 (1 + r)H

In words, the value of a stock is the present value of the dividends it will pay
over the investor™s horizon plus the present value of the expected stock price
at the end of that horizon.

Does this mean that investors of different horizons will all come to different conclu-
sions about the value of the stock? No! Regardless of the investment horizon, the stock
value will be the same. This is because the stock price at the horizon date is determined
by expectations of dividends from that date forward. Therefore, as long as the investors
are consistent in their assessment of the prospects of the firm, they will arrive at the
same present value. Let™s confirm this with an example.

Valuing Blue Skies Stock
Take Blue Skies. The firm is growing steadily and investors expect both the stock price
and the dividend to increase at 8 percent per year. Now consider three investors, Erste,
Zweiter, and Dritter. Erste plans to hold Blue Skies for 1 year, Zweiter for 2, and Drit-
ter for 3. Compare their payoffs:
Year 1 Year 2 Year 3
Erste DIV1 = 3
P1 = 81
Zweiter DIV1 = 3 DIV2 = 3.24
P2 = 87.48
Dritter DIV1 = 3 DIV2 = 3.24 DIV3 = 3.50
P3 = 94.48

Remember, we assumed that dividends and stock prices for Blue Skies are expected to
grow at a steady 8 percent. Thus DIV2 = $3 — 1.08 = $3.24, DIV3 = $3.24 — 1.08 =
$3.50, and so on.

Erste, Zweiter, and Dritter all require the same 12 percent expected return. So we can
calculate present value over Erste™s 1-year horizon:
DIV1 + P1 $3 + $81
PV = = = $75
1+r 1.12
or Zweiter™s 2-year horizon:
DIV1 DIV2 + P2
PV = +
1+r (1 + r)2
$3.00 $3.24 + $87.48
= +
1.12 (1.12)2
= $2.68 + $72.32 = $75
or Dritter™s 3-year horizon:
PV = + +
1 + r (1 + r) 2 (1 + r)3
$3 $3.24 $3.50 + $94.48
= + +
1.12 (1.12)2 (1.12)3
= $2.68 + $2.58 + $69.74 = $75
All agree the stock is worth $75 per share. This illustrates our basic principle: the value
of a common stock equals the present value of dividends received out to the investment
horizon plus the present value of the forecast stock price at the horizon. Moreover, when
you move the horizon date, the stock™s present value should not change. The principle
holds for horizons of 1, 3, 10, 20, and 50 years or more.

Refer to Self-Test 3. Assume that Androscoggin Copper™s dividend and share price are
Self-Test 4
expected to grow at a constant 5 percent per year. Calculate the current value of An-
droscoggin stock with the dividend discount model using a 3-year horizon. You should
get the same answer as in Self-Test 3.

Look at Table 3.5, which continues the Blue Skies example for various time hori-
zons, still assuming that the dividends are expected to increase at a steady 8 percent

Horizon, Years PV (Dividends) + PV (Terminal Price) = Value per Share
Value of Blue Skies
1 $ 2.68 $72.32 $75.00
2 5.26 69.74 75.00
3 7.75 67.25 75.00
10 22.87 52.13 75.00
20 38.76 36.24 75.00
30 49.81 25.19 75.00
50 62.83 12.17 75.00
100 73.02 1.98 75.00
Valuing Stocks 291

Value of Blue Skies for
different horizons.
70 PV (terminal price)


Value per share, dollars
PV (dividends)






1 2 3 10 20 30 50 100
Investment horizon, years

compound rate. The expected price increases at the same 8 percent rate. Each row in the
table represents a present value calculation for a different horizon year. Note that pres-
ent value does not depend on the investment horizon. Figure 3.12 presents the same data
in a graph. Each column shows the present value of the dividends up to the horizon and
the present value of the price at the horizon. As the horizon recedes, the dividend stream
accounts for an increasing proportion of present value but the total present value of div-
idends plus terminal price always equals $75.
If the horizon is infinitely far away, then we can forget about the final horizon
price”it has almost no present value”and simply say
Stock price = PV (all future dividends per share)


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