Assets in place Market value of debt and other obligations

Investment opportunities Market value of shareholders™ equity

Valuing Stocks 287

Valuing Common Stocks

TODAY™S PRICE AND TOMORROW™S PRICE

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

P0

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%

$75

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

+1

=

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 =

1+r

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:

288 SECTION THREE

$3 + $81

P0 = = $75

1.12

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

stock.

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?

THE DIVIDEND DISCOUNT MODEL

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

helpful.

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:

DIVIDEND DISCOUNT

MODEL Discounted cash-

P0 = present value of (DIV1, DIV2, DIV3, . . ., DIVt, . . .)

flow model of today™s stock

DIV1 DIV2 DIV3 DIVt

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”

dividends.

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

this:

DIV1 + P1

P0 =

1+r

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

DIV1 DIV2 DIV3 + P3

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

DIV1 DIV2 DIVH + PH

P0 = + +...+

(1 + r)2 (1 + r)H

1+r

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

EXAMPLE 2

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.

290 SECTION THREE

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:

DIV1 DIV2 DIV3 + P3

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

TABLE 3.5

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

FIGURE 3.12

Value of Blue Skies for

80

different horizons.

70 PV (terminal price)

60

Value per share, dollars

PV (dividends)

50

40

30

20

10

0

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)