D1

V0

k)2

(1

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.

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Essentials of Investments, Companies, 2003

Fifth Edition

417

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.

D1 D2 DH 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.

dividends.

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

1k

This equation can be simplified to

D1

D0(1 g)

V0 (12.4)

k g

kg

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

$4.00

$57.14

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

DDM

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

compute

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

$3.24

$40.50

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:

1

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

419

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

D1

P0

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

kg

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

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

EXAMPLE 12.3

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

(continued)

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

D1

(concluded) $4.00

$66.67

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-

Concept

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