Simplifying the Dividend

Discount Model

THE DIVIDEND DISCOUNT MODEL

WITH NO GROWTH

Consider a company that pays out all its earnings to its common shareholders. Such a

company could not grow because it could not reinvest.6 Stockholders might enjoy a gen-

erous immediate dividend, but they could forecast no increase in future dividends. The

company™s stock would offer a perpetual stream of equal cash payments, DIV1 = DIV2

= . . . = DIVt = . . . .

6 We assume it does not raise money by issuing new shares.

292 SECTION THREE

The dividend discount model says that these no-growth shares should sell for the

present value of a constant, perpetual stream of dividends. We learned how to do that

calculation when we valued perpetuities earlier. Just divide the annual cash payment by

the discount rate. The discount rate is the rate of return demanded by investors in other

stocks of the same risk:

DIV1

P0 =

r

Since our company pays out all its earnings as dividends, dividends and earnings are

the same, and we could just as well calculate stock value by

EPS1

Value of a no-growth stock = P0 =

r

where EPS1 represents next year™s earnings per share of stock. Thus some people

loosely say, “Stock price is the present value of future earnings” and calculate value by

this formula. Be careful”this is a special case. We™ll return to the formula later in this

material.

Moonshine Industries has produced a barrel per week for the past 20 years but cannot

Self-Test 5

grow because of certain legal hazards. It earns $25 per share per year and pays it all out

to stockholders. The stockholders have alternative, equivalent-risk ventures yielding 20

percent per year on average. How much is one share of Moonshine worth? Assume the

company can keep going indefinitely.

THE CONSTANT-GROWTH

DIVIDEND DISCOUNT MODEL

The dividend discount model requires a forecast of dividends for every year into the fu-

ture, which poses a bit of a problem for stocks with potentially infinite lives. Unless we

want to spend a lifetime forecasting dividends, we must use simplifying assumptions to

reduce the number of estimates. The simplest simplification assumes a no-growth per-

petuity which works for no-growth common shares.

Here™s another simplification that finds a good deal of practical use. Suppose fore-

cast dividends grow at a constant rate into the indefinite future. If dividends grow at a

steady rate, then instead of forecasting an infinite number of dividends, we need to fore-

cast only the next dividend and the dividend growth rate.

Recall Blue Skies Inc. It will pay a $3 dividend in 1 year. If the dividend grows at a

constant rate of g = .08 (8 percent) thereafter, then dividends in future years will be

DIV1 = $3 = $3.00

DIV2 = $3 — (1 + g) = $3 — 1.08 = $3.24

DIV3 = $3 — (1 + g)2 = $3 — 1.082 = $3.50

Plug these forecasts of future dividends into the dividend discount model:

D1 D (1 + g) D1(1 + g)2 D1(1 + g)3 . . .

+1

P0 = + + +

1+r (1 + r)2 (1 + r)3 (1 + r)4

$3 $3.24 $3.50 $3.78

+...

= + + +

1.12 1.12 2 1.12 3 1.124

= $2.68 + $2.58 + $2.49 + $2.40 + . . .

Valuing Stocks 293

Although there is an infinite number of terms, each term is proportionately smaller

than the preceding one as long as the dividend growth rate g is less than the discount

rate r. Because the present value of far-distant dividends will be ever-closer to zero, the

sum of all of these terms is finite despite the fact that an infinite number of dividends

will be paid. The sum can be shown to equal

CONSTANT-GROWTH

DIVIDEND DISCOUNT

DIV1

P0 =

MODEL Version of the

r“g

dividend discount model in

This equation is called the constant-growth dividend discount model, or the Gor-

which dividends grow at a

don growth model after Myron Gordon, who did much to popularize it.7

constant rate.

Blue Skies Valued by the Constant-Growth Model

EXAMPLE 3

Let™s apply the constant-growth model to Blue Skies. Assume a dividend has just

been paid. The next dividend, to be paid in a year, is forecast at DIV1 = $3, the growth

rate of dividends is g = 8 percent, and the discount rate is r = 12 percent. Therefore, we

solve for the stock value as

DIV1 $3

P0 = = = $75

r“g .12 “ .08

The constant-growth formula is close to the formula for the present value of a per-

petuity. Suppose you forecast no growth in dividends (g = 0). Then the dividend stream

is a simple perpetuity, and the valuation formula is P0 = DIV1/r. This is precisely the

formula you used in Self-Test 5 to value Moonshine, a no-growth common stock.

The constant-growth model generalizes the perpetuity formula to allow for constant

growth in dividends. Notice that as g increases, the stock price also rises. However, the

constant-growth formula is valid only when g is less than r. If someone forecasts per-

petual dividend growth at a rate greater than investors™ required return r, then two things

happen:

1. The formula explodes. It gives nutty answers. (Try a numerical example.)

2. You know the forecast is wrong, because far-distant dividends would have incredi-

bly high present values. (Again, try a numerical example. Calculate the present value

of a dividend paid after 100 years, assuming DIV1 = $3, r = .12, but g = .20.)

ESTIMATING EXPECTED RATES OF RETURN

We argued earlier that in competitive markets, common stocks with the same risk are

priced to offer the same expected rate of return. But how do you figure out what that

expected rate of return is?

It™s not easy. Consensus estimates of future dividends, stock prices, or overall rates

of return are not published in The Wall Street Journal or reported by TV newscasters.

7 Notice that the first dividend is assumed to come at the end of the first period and is discounted for a full

period. If the stock has just paid its dividend, then next year™s dividend will be (1 + g) times the dividend just

paid. So another way to write the valuation formula is

DIV0 — (1 + g)

DIV1

P0 = =

r“g r“g

294 SECTION THREE

Economists argue about which statistical models give the best estimates. There are nev-

ertheless some useful rules of thumb that can give sensible numbers.

One rule of thumb is based on the constant-growth dividend discount model. Re-

member that it forecasts a constant growth rate g in both future dividends and stock

prices. That means forecast capital gains equal g per year.

We can calculate the expected rate of return by rearranging the constant-growth for-

mula as

DIV1

r= +g

P0

= dividend yield + growth rate

For Blue Skies, the expected first-year dividend is $3 and the growth rate 8 percent.

With an initial stock price of $75, the expected rate of return is

DIV1

r= +g

P0

$3

= + .08 = .04 + .08 = .12, or 12%

$75

Suppose we found another stock with the same risk as Blue Skies. It ought to offer

the same expected rate of return even if its immediate dividend or expected growth rate

is very different. The required rate of return is not the unique property of Blue Skies or

any other company; it is set in the worldwide market for common stocks. Blue Skies

cannot change its value of r by paying higher or lower dividends or by growing faster

or slower, unless these changes also affect the risk of the stock. When we use the rule

of thumb formula, r = DIV1/P0 + g, we are not saying that r, the expected rate of return,

is determined by DIV1 or g. It is determined by the rate of return offered by other

equally risky stocks. That return determines how much investors are willing to pay for

Blue Skies™s forecast future dividends:

DIV1 expected rate of return offered

+g=r=

P0 by other, equally risky stocks

Given DIV1 and so that Blue Skies offers an

g, investors set adequate expected rate of

the stock price return r

Blue Skies Gets a Windfall

EXAMPLE 4

Blue Skies has won a lawsuit against its archrival, Nasty Manufacturing, which forces

Nasty Manufacturing to withdraw as a competitor in a key market. As a result Blue

Skies is able to generate 9 percent per year future growth without sacrificing immedi-

ate dividends. Will that increase r, the expected rate of return?

This is very good news for Blue Skies stockholders. The stock price will jump to

DIV1 $3

P0 = = = $100

r “ g .12 “ .09

But at the new price Blue Skies will offer the same 12 percent expected return:

Valuing Stocks 295

DIV1

r= +g

P0

$3

= + .09 = .12, or 12%

$100

Blue Skies™s good news is reflected in a higher stock price today, not in a higher ex-

pected rate of return in the future. The unchanged expected rate of return corresponds

to Blue Skies™s unchanged risk.

Androscoggin Copper can grow at 5 percent per year for the indefinite future. It™s sell-

Self-Test 6

ing at $100 and next year™s dividend is $5.00. What is the expected rate of return from

investing in Carrabasset Mining common stock? Carrabasset and Androscoggin shares

are equally risky.

Few real companies are expected to grow in such a regular and convenient way as

Blue Skies or Androscoggin Copper. Nevertheless, in some mature industries, growth

is reasonably stable and the constant-growth model approximately valid. In such cases

the model can be turned around to infer the rate of return expected by investors.

NONCONSTANT GROWTH

Many companies grow at rapid or irregular rates for several years before finally settling

down. Obviously we can™t use the constant-growth dividend discount model in such

cases. However, we have already looked at an alternative approach. Set the investment

horizon (Year H) at the future year by which you expect the company™s growth to settle

down. Calculate the present value of dividends from now to the horizon year. Forecast

the stock price in that year and discount it also to present value. Then add up to get the

total present value of dividends plus the ending stock price. The formula is

DIV1 DIV2 DIVH PH

+...+

P0 = + +

2 H (1 + r)H

1+r (1 + r) (1 + r)

PV of dividends from PV of stock price

Year 1 to horizon at horizon

The stock price in the horizon year is often called terminal value.

Estimating the Value of United Bird Seed™s Stock

EXAMPLE 5

Ms. Dawn Chorus, founder and president of United Bird Seed, is wondering whether

the company should make its first public sale of common stock and if so at what price.

The company™s financial plan envisages rapid growth over the next 3 years but only

moderate growth afterwards. Forecast earnings and dividends are as follows:

Year: 1 2 3 4 5 6 7 8

Earnings

per share $2.45 3.11 3.78 5% growth thereafter

Dividends

per share $1.00 1.20 1.44 5% growth thereafter

296 SECTION THREE

Thus you have a forecast of the dividend stream for the next 3 years. The tricky part

is to estimate the price in the horizon Year 3. Ms. Chorus could look at stock prices for

mature pet food companies whose scale, risk, and growth prospects today roughly

match those projected for United Bird Seed in Year 3. Suppose further that these com-

panies tend to sell at price-earnings ratios of about 8. Then you could reasonably guess

that the P/E ratio of United will likewise be 8. That implies

P3 = 8 — $3.78 = $30.24

You are now in a position to determine the value of shares in United. If investors de-

mand a return of r = 10 percent, then price today should be

P0 = PV (dividends from Years 1 to 3) + PV (forecast stock price in Year 3)

$1.00 $1.20 $1.44

PV (dividends) = + + = $2.98

1.10 1.102 1.103

$30.24

PV(PH) = = $22.72