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D2 P2
1 k
This equation may be interpreted as the present value of dividends plus sales price for a
two-year holding period. Of course, now we need to come up with a forecast of P2. Continu-
ing in the same way, we can replace P2 by (D3 P3)/(1 k), which relates P0 to the value of
dividends plus the expected sales price for a three-year holding period.
Bodie’Kane’Marcus: IV. Security Analysis 12. Equity Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

12 Equity Valuation

More generally, for a holding period of H years, we can write the stock value as the pres-
ent value of dividends over the H years, plus the ultimate sales price, PH.
V0 ... (12.2)
k)2 k)H
1 k (1 (1
Note the similarity between this formula and the bond valuation formula developed in Chap-
ter 9. Each relates price to the present value of a stream of payments (coupons in the case of
bonds, dividends in the case of stocks) and a final payment (the face value of the bond or the
sales price of the stock). The key differences in the case of stocks are the uncertainty of divi-
dends, the lack of a fixed maturity date, and the unknown sales price at the horizon date. In-
deed, one can continue to substitute for price indefinitely to conclude
D1 D2 D3
V0 ... (12.3)
k)2 k)3
1 k (1 (1
Equation 12.3 states the stock price should equal the present value of all expected future
dividends into perpetuity. This formula is called the dividend discount model (DDM) of dividend discount
stock prices. model (DDM)
It is tempting, but incorrect, to conclude from Equation 12.3 that the DDM focuses exclu- A formula for the
sively on dividends and ignores capital gains as a motive for investing in stock. Indeed, we as- intrinsic value of a
sume explicitly in Equation 12.1 that capital gains (as reflected in the expected sales price, P1) firm equal to the
present value of all
are part of the stock™s value. At the same time, the price at which you can sell a stock in the fu-
expected future
ture depends on dividend forecasts at that time.
The reason only dividends appear in Equation 12.3 is not that investors ignore capital
gains. It is instead that those capital gains will be determined by dividend forecasts at the time
the stock is sold. That is why in Equation 12.2 we can write the stock price as the present value
of dividends plus sales price for any horizon date. PH is the present value at time H of all div-
idends expected to be paid after the horizon date. That value is then discounted back to today,
time 0. The DDM asserts that stock prices are determined ultimately by the cash flows accru-
ing to stockholders, and those are dividends.

The Constant Growth DDM
Equation 12.3 as it stands is still not very useful in valuing a stock because it requires dividend
forecasts for every year into the indefinite future. To make the DDM practical, we need to in-
troduce some simplifying assumptions. A useful and common first pass at the problem is to
assume that dividends are trending upward at a stable growth rate that we will call g. Then if
g 0.05, and the most recently paid dividend was D0 3.81, expected future dividends are
D1 D0(1 g) 3.81 1.05 4.00
g)2 (1.05)2
D2 D0(1 3.81 4.20
g)3 (1.05)3
D3 D0(1 3.81 4.41 etc.
Using these dividend forecasts in Equation 12.3, we solve for intrinsic value as
D0(1 g)2 D0(1 g)3
D0(1 g)
V0 ...
(1 k)2 (1 k)3
This equation can be simplified to
D0(1 g)
V0 (12.4)
k g
Bodie’Kane’Marcus: IV. Security Analysis 12. Equity Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

418 Part FOUR Security Analysis

Note in Equation 12.4 that we divide D1 (not D0) by k g to calculate intrinsic value. If the
market capitalization rate for Steady State is 12%, we can use Equation 12.4 to show that the
intrinsic value of a share of Steady State stock is
0.12 0.05
Equation 12.4 is called the constant growth DDM or the Gordon model, after Myron J.
constant growth
Gordon, who popularized the model. It should remind you of the formula for the present value
of a perpetuity. If dividends were expected not to grow, then the dividend stream would be a
A form of the dividend
simple perpetuity, and the valuation formula for such a nongrowth stock would be P0 D1/k.1
discount model that
Equation 12.4 is a generalization of the perpetuity formula to cover the case of a growing per-
assumes dividends
will grow at a petuity. As g increases, the stock price also rises.
constant rate.

Preferred stock that pays a fixed dividend can be valued using the constant growth dividend
discount model. The constant growth rate of dividends is simply zero. For example, to value
12.1 EXAMPLE a preferred stock paying a fixed dividend of $2 per share when the discount rate is 8%, we
Preferred Stock
and the DDM $2
V0 $25
0.08 0

High Flyer Industries has just paid its annual dividend of $3 per share. The dividend is expected
to grow at a constant rate of 8% indefinitely. The beta of High Flyer stock is 1.0, the risk-free rate
12.2 EXAMPLE is 6% and the market risk premium is 8%. What is the intrinsic value of the stock? What would
be your estimate of intrinsic value if you believed that the stock was riskier, with a beta of 1.25?
The Constant
Because a $3 dividend has just been paid and the growth rate of dividends is 8%, the
Growth DDM
forecast for the year-end dividend is $3 1.08 $3.24. The market capitalization rate is 6%
1.0 8% 14%. Therefore, the value of the stock is
D1 $3.24
V0 $54
k g 0.14 0.08
If the stock is perceived to be riskier, its value must be lower. At the higher beta, the mar-
ket capitalization rate is 6% 1.25 8% 16%, and the stock is worth only
0.16 0.08

The constant growth DDM is valid only when g is less than k. If dividends were expected
to grow forever at a rate faster than k, the value of the stock would be infinite. If an analyst de-
rives an estimate of g that is greater than k, that growth rate must be unsustainable in the long
run. The appropriate valuation model to use in this case is a multistage DDM such as that dis-
cussed below.
The constant growth DDM is so widely used by stock market analysts that it is worth ex-
ploring some of its implications and limitations. The constant growth rate DDM implies that
a stock™s value will be greater:

Recall from introductory finance that the present value of a $1 per year perpetuity is 1/k. For example, if k 10%,
the value of the perpetuity is $1/0.10 $10. Notice that if g 0 in Equation 12.4, the constant growth DDM formula
is the same as the perpetuity formula.
Bodie’Kane’Marcus: IV. Security Analysis 12. Equity Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

12 Equity Valuation

1. The larger its expected dividend per share.
2. The lower the market capitalization rate, k.
3. The higher the expected growth rate of dividends.
Another implication of the constant growth model is that the stock price is expected to
grow at the same rate as dividends. To see this, suppose Steady State stock is selling at its in-
trinsic value of $57.14, so that V0 P0. Then
k g
Note that price is proportional to dividends. Therefore, next year, when the dividends paid
to Steady State stockholders are expected to be higher by g 5%, price also should increase
by 5%. To confirm this, note
D2 $4(1.05) $4.20
P1 D2/(k g) $4.20/(0.12 0.05) $60.00
which is 5% higher than the current price of $57.14. To generalize
D2 D1
D1(1 g)
P1 (1 g)
k g k g
P0(1 g)
Therefore, the DDM implies that, in the case of constant expected growth of dividends, the
expected rate of price appreciation in any year will equal that constant growth rate, g. Note
that for a stock whose market price equals its intrinsic value (V0 P0) the expected holding-
period return will be
E(r) Dividend yield Capital gains yield
D1 D1
P1 P0
g (12.5)
P0 P0
This formula offers a means to infer the market capitalization rate of a stock, for if the stock
is selling at its intrinsic value, then E(r) k, implying that k D1/P0 g. By observing the
dividend yield, D1/P0, and estimating the growth rate of dividends, we can compute k. This
equation is known also as the discounted cash flow (DCF) formula.
This is an approach often used in rate hearings for regulated public utilities. The regulatory
agency responsible for approving utility pricing decisions is mandated to allow the firms to
charge just enough to cover costs plus a “fair” profit, that is, one that allows a competitive re-
turn on the investment the firm has made in its productive capacity. In turn, that return is taken
to be the expected return investors require on the stock of the firm. The D1/P0 g formula
provides a means to infer that required return.

Suppose that Steady State Electronics wins a major contract for its revolutionary computer
chip. The very profitable contract will enable it to increase the growth rate of dividends from
5% to 6% without reducing the current dividend from the projected value of $4.00 per share.
What will happen to the stock price? What will happen to future expected rates of return on The Constant
the stock? Growth Model
Bodie’Kane’Marcus: IV. Security Analysis 12. Equity Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

420 Part FOUR Security Analysis

The stock price ought to increase in response to the good news about the contract, and
indeed it does. The stock price rises from its original value of $57.14 to a postannouncement
12.3 EXAMPLE price of
(concluded) $4.00
k g 0.12 0.06
Investors who are holding the stock when the good news about the contract is announced
will receive a substantial windfall.
On the other hand, at the new price the expected rate of return on the stock is 12%, just
as it was before the new contract was announced.
D1 $4.00
E(r) g 0.06 0.12, or 12%
P0 $66.67
This result makes sense, of course. Once the news about the contract is reflected in the stock
price, the expected rate of return will be consistent with the risk of the stock. Since the risk of
the stock has not changed, neither should the expected rate of return.

2. a. IBX™s stock dividend at the end of this year is expected to be $2.15, and it is ex-
pected to grow at 11.2% per year forever. If the required rate of return on IBX
CHECK stock is 15.2% per year, what is its intrinsic value?
b. If IBX™s current market price is equal to this intrinsic value, what is next year™s
expected price?
c. If an investor were to buy IBX stock now and sell it after receiving the $2.15 div-
idend a year from now, what is the expected capital gain (i.e., price apprecia-
tion) in percentage terms? What is the dividend yield, and what would be the
holding-period return?

Stock Prices and Investment Opportunities
Consider two companies, Cash Cow, Inc., and Growth Prospects, each with expected earnings
in the coming year of $5 per share. Both companies could in principle pay out all of these
earnings as dividends, maintaining a perpetual dividend flow of $5 per share. If the market
capitalization rate were k 12.5%, both companies would then be valued at D1/k $5/0.125
$40 per share. Neither firm would grow in value, because with all earnings paid out as div-
idends, and no earnings reinvested in the firm, both companies™ capital stock and earnings ca-
pacity would remain unchanged over time; earnings2 and dividends would not grow.
Now suppose one of the firms, Growth Prospects, engages in projects that generate a return
on investment of 15%, which is greater than the required rate of return, k 12.5%. It would
be foolish for such a company to pay out all of its earnings as dividends. If Growth Prospects
retains or plows back some of its earnings into its highly profitable projects, it can earn a 15%
rate of return for its shareholders, whereas if it pays out all earnings as dividends, it forgoes


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