Principles of Corporate Finance_ 12th Edition

88 Part One Value

bre44380_ch04_076-104.indd 88 09/30/15 12:46 PM

Dangers Lurk in Constant-Growth Formulas

The simple constant-growth DCF formula is an extremely useful rule of thumb, but no more

than that. Naive trust in the formula has led many financial analysts to silly conclusions.

We have stressed the difficulty of estimating r by analysis of one stock only. Try to use a

large sample of equivalent-risk securities. Even that may not work, but at least it gives the

analyst a fighting chance, because the inevitable errors in estimating r for a single security

tend to balance out across a broad sample.

Also, resist the temptation to apply the formula to firms having high current rates of

growth. Such growth can rarely be sustained indefinitely, but the constant-growth DCF for-

mula assumes it can. This erroneous assumption leads to an overestimate of r.

Example The U.S. Surface Transportation Board (STB) tracks the “revenue adequacy” of

U.S. railroads by comparing the railroads’ returns on book equity with estimates of the cost

of the equity. To estimate the cost of equity, the STB traditionally used the constant-growth

formula. It measured g by stock analysts’ forecasts of long-term earnings growth. The for-

mula assumes that earnings and dividends grow at a constant rate forever, but the analysts’

“long-term” forecasts looked out five years at most. As the railroads’ profitability improved,

the analysts became more and more optimistic. By 2009, their forecasts for growth averaged

12.5% per year. The average dividend yield was 2.6%, so the constant-growth model esti-

mated the industry-average cost of capital at 2.6 + 12.5 = 15.1%.

So the STB said, in effect, “Wait a minute: railroad earnings and dividends can’t grow at

12.5% forever. The constant-growth formula no longer works for railroads. We’ve got to find a

more accurate method.” The STB now uses a two-stage growth model, which we now discuss.

DCF Models with Two Stages of Growth Consider Growth-Tech, Inc., a firm with

DIV 1  = $.50 and P 0  = $50. The firm has plowed back 80% of earnings and has had a return

on equity (ROE) of 25%. This means that in the past

Dividend growth rate = plowback ratio × ROE = .80 × .25 = .20

The temptation is to assume that the future long-term growth rate g also equals .20. This

would imply

r = _____.50

50.00

+ .20 = .21

But this is silly. No firm can continue growing at 20% per year forever, except possibly

under extreme inflationary conditions. Eventually, profitability will fall and the firm will

respond by investing less.

In real life the return on equity will decline gradually over time, but for simplicity let’s

assume it suddenly drops to 16% at year 3 and the firm responds by plowing back only 50% of

earnings. Then g drops to .50 × .16 = .08.

Table 4.4 shows what’s going on. Growth-Tech starts year 1 with book equity of $10.00 per

share. It earns $2.50, pays out 50 cents as dividends, and plows back $2. Thus it starts year 2

with book equity of $10 + 2 = $12. After another year at the same ROE and payout, it starts

year 3 with equity of $14.40. However, ROE drops to .16, and the firm earns only $2.30. Divi-

dends go up to $1.15, because the payout ratio increases, but the firm has only $1.15 to plow

back. Therefore subsequent growth in earnings and dividends drops to 8%.

Now we can use our general DCF formula:

P 0 =

DIV 1

_____

1 + r

+

DIV 2

_______

(1 + r)^2

+

DIV 3 + P 3

_________

(1 + r)^3