1 292d
Part 3 Financial Assets
weal th from the present stockholders to those who were allowed to purchase the
new shares. The preemptive right prevents this.
^sT
Identify some actions that companies have taken to make takeovers
more difficult.
What is the preemptive right, and what are the two primary reasons
for its existence?
9.2 TYPES OF COMMON STOCK
Classified Stock
Although most firms have only one type of common stock, in some instances
classified stock is used to meet special needs. Generally, when special classifica-
Common stock that is
given a special desig-
tions are used, one type is designated Class A, another Class B, and so on. Small,
new companies seeking funds from outside sources frequently use different
nation, such as Class A,
Class B, and so forth,
types of common stock. For example, when Genetic Concepts went public
recently, its Class A stock was sold to the public and paid a dividend, but this
to meet special needs
of the company.
Founders' Shares
Stock owned by the
firm's founders that has
sole voting rights but
restricted dividends for
a specified number of
years.
stock had no voting rights for five years. Its Class B stock, which was retained
by the organizers of the company, had full voting rights for five years, but the
legal terms stated that dividends could not be paid on the Class B stock until the
company had established its earning power by building up retained earnings to
a designated level. The use of classified stock thus enabled the public to take a
position in a conservatively financed growth company without sacrificing
income, while the founders retained absolute control during the crucial early
stages of the firm's development. At the same time, outside investors were protected against excessive withdrawals of funds by the original owners. As is often
the case in such situations, the Class B stock was also called founders' shares.
Note that "Class A," "Class B," and so on, have no standard meanings. Most
firms have no classified shares, but a firm that does could designate its Class B
shares as founders' shares and its Class A shares as those sold to the public,
while another could reverse these designations. Still other firms could use stock
classifications for entirely different purposes. For example, when General Motors
acquired Hughes Aircraft for $5 billion, it paid in part with a new Class H common, GMH, which had limited voting rights and whose dividends were tied to
Hughes's performance as a GM subsidiary. The reasons for the new stock were
that ( 1) GM wanted to limit voting privileges on the new classified stock because
of management's concern about a possible takeover and (2) Hughes employees
wanted to be rewarded more directly on Hughes's own performance than would
have been possible through regular GM stock. These Class H shares disappeared
in 2003 when GM decided to sell off the Hughes unit.
0
^^^
q,
What are some reasons why a company might use classified stock?
9.3 COMMON STOCK VALUATION
Common stock represents an ownership interest in a corporation, but to the typical investor, a share of common stock is simply a piece of paper characterized
by two features:
1. It entitles its owner to dividends, but only if the company has earnings out
of which dividends can be paid and management chooses to pay dividends
rather than retaining and reinvesting all the earnings. Whereas a bond con-
299
Chapter 9 Stocks and Their Valuation
293
tains a promise to pay interest, common stock provides no such promise-if
you own a stock, you may expect a dividend, but your expectations may not
in fact be met. To illustrate, Long Island Lighting Company (LILCO) had
paid dividends on its common stock for more than 50 years, and people
expected those dividends to continue. However, when the company encountered severe problems a few years ago, it stopped paying dividends. Note,
though, that LILCO continued to pay interest on its bonds, because if it had
not, then it would have been declared bankrupt and the bondholders could
have taken over the company.
2. Stock can be sold, hopefully at a price greater than the purchase price. If the
stock is actually sold at a price above its purchase price, the investor will
receive a capital gain. Generally, when people buy common stock they expect
to receive capital gains; otherwise, they would not buy the stock. However,
after the fact, they can end up with capital losses rather than capital gains.
LILCO's stock price dropped from $17.50 to $3.75 in one year, so the expected
capital gain on that stock turned out to be a huge actual capital loss.
Definitions of Terms Used in Stock Valuation Models
Common stocks provide an expected future cash flow stream, and a stock's
value is found as the present value of the expected future cash flows, which consist of two elements: (1) the dividends expected in each year and (2) the price
investors expect to receive when they sell the stock. The final price includes the
return of the original investment plus an expected capital gain.
We saw in Chapter 1 that managers should seek to maximize the value of
their firms' stock. Therefore, managers need to know how alternative actions are
likely to affect stock prices, and we develop some models to help show how the
value of a share of stock is determined. We begin by defining the following terms:
DT = dividend. the stockholder expects to receive at the end of
each. Year t. Do is the most recent dividend, which has
already been paid; Dz is the first dividend expected, and
it will be paid at the end of this year; DZ is the dividend
expected, at the end of two years; and so forth. :Dr represents the :first cash flow a new purchaser of the stock will
reeeiv.e. 'Note that Do, the dividend that has just been
paid, is lcnovkn with certainty. However, all future dividends are expected values, those expectations differ some.
what from investor to investor, and those differences lead
to differences in estimates of the stock s intrinsic value.
Pa = actual market price of the stock today
P, = expected price of the stock at the end of each Year t(pronou^ued 'T hat t"). P, is the intrinsic value of the stock
toda 'as seen by the particular investor doing the analysis; 1 is the price expected at the end of one year; and so
on. Note that P0 is the intrinsic value of the stock today
based on a particular investor's estimate of the stock's
expected dividend stream and the riskiness of that
stream. Hence, whereas the market price P0 is fixed and
2 Stocks generally pay dividends quarterly, so theoretically we should evaluate them on a quarterly
basis. However, in stock valuation, most analysts work on an annual basis because the data generaily are not precise enough to warrant refinement to a quarterly model. For additional information
on the quarterly model, see Charles M. Linke and J. Kenton Zumwalt, "Estimation Biases in Discounted Cash Flow Analysis of Equity Capital Cost in Rate Regulation," Financial Management,
Autuinn 1984, pp. 15-21.
300
^, ,'
Market Price, P0
The price at which a
stock sells in the
market.
Intrinsic Value, Po
that, in the mind of a
particular investor, is
justified by the facts;
r'^ may be different
from the asset's current
market price.
294
Part 3 Financial Assets
Growth Rate, g
The expected rate of
growth in dividends
per share.
Required Rate of
Return, r{
The minimum rate of
return on a common
stock that a stockholder considers
acceptable.
Expected Rate of
Return, i•,
The rate of return on a
common stock that a
stockholder expects to
receive in the future.
Actual Realized Rate
of Return, r,
The rate of return on a
common stock actually
received by stockholders in some past
period. i:, may be
greater or less than P,
and/or rs.
Dividend Yield
The expected dividend
divided by the current
price of a share of
stock.
Capital Gains Yield
The capital gain during
a given year divided by
the beginning price.
Expected Total Return
The sum of the
expected dividend
yield and the expected
capital gains yield.
g
is identical for all inr'estt?rs, P, ► could differ among
i.nvestors, depending on hc^%v optimistic they are regarding
the company. P,, the individual investor's estimate of the
intrinsic value today, could be above or below PO, the current stock price. An investor would buy the stock only if
their estimate of l;, were equal to or greater than P ►►.
As there are many investors in the market, there can be
many values for J. How-ever, we can think of an "average," or "marginal," investor whose actions actually determine the market price. For the marginal investor, PO must
equal Tj; otherwise, a disequilibrium would exist, and
bucing and selling in the market would change Pr0 until
PG = I;, for the marginal investor.
expected groKth rate in dividends as predicted by the
►narginal investor. If dividends are expected to grow at a
constant rate, g is also equal to the expected rate of
growth in earnings and in the stock's price. Different
investors use different g's to evaluate a firm's stock, but
the market price, P►a, is set on the basis of g as estimated
by the marginal investor.
r, = minimum acceptable, or required, rate of return on the
stock, considering both its riskiness and the returns
available on other investments. Again, this term generally relates to the marginal investor. The determinants of
r$ include the real rate of return, expected inflation, and
risk as discussed in Chapter S.
i•Y = expected rate of return that an investor who buys the
stock expects to receive in the future. r$ (pronounced "r
hat s") could be above or below r, but one would buy
the stock only if f5 were equal to or greater than rs.
i•s = actual, or realized, tz. fter-f7w ;fact rafie of retum, pronounced
"r bar s." You may expect to obtain a return of i•h = 10^'^ if
you buy a stock today, but if the market goes down, you
may end up next year with an actual realized return that
is much lower, perhaps even negative.
D11Pr0 = expected dividend yield on the stock during the coming
year. If the stock is expected to pay a dividend of Dl =
^1 during the next 12 months, and if its current price is
Po = $20, then the expected dividend yield is $1/$20 =
0.05 = 5%.
P' - Pn = expected
capital gains yield on the stock during the
coming
year.
If the stock sells for $20 today, and if it is
Pa
expected to rise to $21.00 at the end of one year, then the
expected capital gain is ].', - P ►0 = $21.00 - $20.00 = $1.00,
and the expected capital gains yield is $1.00J$20 = 0.05
ti9o.
Expected total return = iq = expected dividend yield (DI/Po) plus expected capital gains yield [(P1 - P,)/Pol. In our example, the expected
total return ='rK = 5% -t- 5^'^ = 10%.
Expected Dividends as the Basis for Stock Values
In our discussion of bonds, we found the value of a bond as the present value of
interest payments over the life of the bond plus the present value of the bond's
maturity (or par) value:
301
Chapter 9 Stocks and Their Valuation
V3
INT '+ INT 2+...T INT +M N
(1 + rd)
(1 ^ rd)tv
(1 = rd)
(1 + ra)
Stock prices are likewise determined as the present value of a stream of cash
flows, and the basic stock valuation equation is similar to the bond valuation
equation. What are the cash flows that corporations provide to their stockholders?
First, think of yourself as an investor who buys a stock with the intention of
holding it (in your family) forever. In this case, all that you (and your heirs)
would receive is a stream of dividends, and the value of the stock today is calculated as the present value of an infinite stream of dividends:
Value of stock = Pa = PV of expected future dividends
_
D,
+
(1 + rs)'
D"
^2.
+...+
(1 + rs)'
(1 T rs)2
Dt
t
t=1 (1 + rs)
(9-1)
What about the more typical case, where you expect to hold the stock for a finite
period and. then sell it-what will be the value of Pe in this case? Unless the company is likely to be liquidated or sold and thus to disappear, the value of the stock
i5 again defennitted by Equation 9-1. To see this, recognize that for any individual
investor, the expected cash flows consist of expected dividends plus the
expected sale price of the stock. However, the sale price the current investor
receives will depend on the dividends some future investor expects. Therefore,
for all present and future investors in total, expected cash flows must he based
on expected future dividends. Put another way, unless a firm. is liquidated or
sold to another concern, the cash flows it provides to its stockholders will consist
only of a stream of dividends; therefore, the value of a share of its stock must be
established as the present value of that expected dividend stream.
The general validity of Equation 9-1 can also be confirmed by asking the fol.lowing question: Suppose I buy a stock and expect to hold it for one year. I will
receive dividends during the year plus the value ?g, when I sell out at the end of the
year. But what will determine the value of f'1? The answer is that it will be determined as the present value of the dividends expected during Year 2 plus the stock
price at the end of that year, which, in turn, will be determined as the present value
of another set of future dividends and an even more distant stock price. This
process can be continued ad infinitum, and the ultimate result is Equation 9-1?
^y
Explain the following statement: 'Whereas a bond contains a promise to pay interest, a share of common stock typically provides an
expectation of, but no promise of, dividends plus capital gains."
What are the two parts of most stocks' expected total return?
If Dl = $2.00, g = 6%, and Pa = $40, what are the stock's expected
dividend yield, capital gains yield, and total expected return for the
coming year? (5%, 6%, 11`Yo)
'We should note that investors periodically lose sight of the long-run nature of stocks as
investments and forget that in order to seli n stock at a profit, one must find a buyer who will pay
the higher price- If you analyze a stock's value in accordance with Equation 9-1, conclude that the
stcuck's market price exceeds a reasonable value, and then buy the stock anywa, then you would be
following the "bigger fool" theory of investment-you. think you may be a fool to buy the stock at
its excessive price, but you also think that when you get ready to sell it, you can End someone who
is an even bigger fool. The bigger fool theory was widely followed in the summer of 1987, just
before the stock market lost more than one-third of its value in the October 1987 crash.
302
295
296
Part 3 Financial Assets
T-11 STOCKS
9.4 CONSTANT GROW T-11
Equation 9-1 is a generalized stock valuation model in the sense that the time
pattern of Dt can be anything: Dt can be rising, falling, fluctuating randomly, or
it can even be zero for several years and Equation 9-1 will still hold. With a computer spreadsheet we can easily use this equation to find a. stock's intrinsic value
for any pattern of dividends. In practice, the hard part is obtaining an accurate
forecast of the future dividends.
In many cases, the stream of dividends is expected to grow at a constant
rate. If this is the case, Equation 9-1 may be rewritten as follows:
_ D° (1 + g)'
P0
(1 + rs)'
= Do (1 + 9)
rs - g
Cons^tant Growth
(Gordon) Model
Used to find the value
of a constant growth
stock.
Do(1 + 9)x
D° (1 + g)2
+ (1 + rs)z
. .. + (1 + rs)s
D,
rs - g
(9-2)
The last term of Equation 9-2 is called the constant growth model, or the Gordon model, after Myron J. Gordon, who did much to develop and popularize it,
Illustration of a Constant Growth Stock
Assume that Allied Food Products just paid a dividend of $1.15 (that is, Do =
$1.15). Its stock has a required rate of return, r5, of 13.4 percent, and investors
expect the dividend to grow at a constant 8 percent rate in the future. The estimated dividend one year hence would be Dl = $1.15(1.08) _$1.24; Dz would be
$1.34; and the estimated dividend five years hence would be $1.69:
D5 = D0(1 + g)5 = $1.15(1.08)5 = $1.69
'We could use this procedure to estimate all future dividends, then use Equation
9-1 to determine the current stock value, 1'0. In other words, we could find each
expected future dividend, calculate its present value, and then sum all the present values to find the intrinsic value of the stock.
Such a process would be time consuming, but we can take a short cut-just
insert the illustrative data into Equation 9-2 to find the stock's intrinsic value, $23:
_ $1.15(1.08) _ $1.242
23.00
0.054 = $
P0
0.134 - 0.08
Note that a necessary condition for the derivation of Equation 9-2 is that the
required rate of return, r, be greater than the long-run growth rate, g. If the equation is used in situations where r;, is not greater than g, the results will be wrong, rneaningtess, and possibly misleading.
The concept underlying the valuation process for a constant growth stock is
graphed in Figure 9-1. Dividends are growing at the rate g = 8 percent, but
because r$ > g, the present value of each future dividend is declining. For example, the dividend in Year 1 is D, = D°(1 + g)l =$1.15(1.08) =$1.242. However,
the present value of this dividend, discounted at 13.4 percent, is PV(D3) =
$1.242/(1.130 =$1.095. The dividend expected in Year 2 grows to $1.242
(1.08) =$1.341, but the present value of this dividend falls to $1.043. Continuing,
` The last term in Equation 9-2 is derived in the Web/CD Extension of Chapter 5 of Eugene F.
Brigham and Phillip R Daves, lntrnr.ediate Financial Manageement, 8th ed. (Mason, OI-f: Thomson/
South-Western, 2004). In essence, Equation 9-2 is the sum of a geometric progression, and the final
result is the solution value of the progression.
303
k,_
Chapter 9 Stocks and Their Valuation
Present Values of Dividends of a Constant Growth
Stock where Do = $1.15, g = 8%, rs = 13.4%
FIGURE 9-1
Dividend
i$1
!
./{
,
4
-^
Dollar Amount of Each Dividend
= D,(1-- ^l
(-
..
1.15
PVD.
^
PV of Each Dividend =
Do (1 + g)'
t1 rbl^
--iP^ _jPV D, = Area under PV curve
•`'
=S23.D0
0
5
10
15
20
Years
.D; _$2.449 and PV(D) = $0.993, and so on. Thus, the expected dividends are
growing, but the present value of each successive dividend is declining, because
the dividend. growth rate (i;'percent) is less than the rate used for discounting
the dividends to the present (13.4 percent).
If we summed the present values of each future dividend, this summation
would be the value of the stock, PD. When g is a constant, this summation is
equal to Dl/(rs - g), as shown in Equation 9-2. Therefore, if we extended the
lower step-function curve in Figure 9-1 on out to infinity and added up the present values of each future dividend, the summation would be identical to the
value given by Equation 9-2, $23.00.
Note that if the growth rate exceeded the required return, the PV of each
Future dividend would exceed that of the prior year. If this situation were
graphed in Figure 9-1, both step-function curves would be increasing, suggesting an infinitely high stock price. Moreover, the stock price as calculated using
Equation 9-2 would be negatiae. Obviously, stock prices can be neither infinite
nor negative, and this ilJustrates why Equation 9-2 cannot be used unless rfi > g.
We will return to this point later in the chapter.
Dividend and Earnings Growth
Growth in dividends occurs primarily as a result of growth in earnings per share
(EPS). Earnings growth, in turn, results from a number of factors, including
(1) the amount of earnings the company retains and reinvests, (2) the rate of
304
297
T.1
i 2981
Part 3 F;nancir,l Assets
return the company earns on its equity (ROE), and (3) inflation. Regarding inflation, if output (in units) is stable but both sales prices and input costs rise at the
inflation rate, then EPS will also grow at the inflation rate. Even vvithout
inflation, EPS will also grow as a result of the reinvestment, or plowback, of
earnings. If the firm's earnings are not all paid out as dividends (that is, if some
fraction of earnings is retained), the dollars of inveshrient behind each share will
rise over time, which should lead to growth in earnings and dividends.
Even though a stock's value is derived from expected dividends, this does
not necessarily mean that corporations can increase their stock prices by simply
raising the current dividend. Shareholders care about all dividends, both current
and those expected in the future. Moreover, there is a trade-off between current
dividends, and future dividends. Companies that pay most of their current earnings out as dividends are obviously not retaining and reinvesting much in the
business, and that reduces future earnings and dividends. So, the issue is this:
Do shareholders prefer higher current dividends at the cost of lower future dividends, lower current dividends, and more growth, or are they indifferent
between growth and dividends? As we will see in the chapter on distributions to
shareholders, there is no simple answer to this question. Shareholders should
prefer to have the company retain earnings, hence pay less current dividends, if
it has highly profitable investment opportunities, but they should prefer to have
the company pay earnings out if investment opportunities are poor. Taxes also
play a role-since capital gains are tax deferred while dividends are taxed
immediately, this might lead to a preference for retention and growth over
current dividends. We will consider dividend policy in detail later in Part 5 of
this text.
When Can the Constant Growth Model Be Used?
The constant growth model is most appropriate for mature companies with a
stable history of growth and stable future expectations. Expected growth rates
vary somewhat among companies, but dividends for mature firms are often
expected to grow in the future at about the same rate as nominal gross domestic
product (real GDP plus inflation). On this basis, one might expect the dividends
of an average, or "normal," company to grow at a rate of 5 to 8 percent a year.
Zero Growth Stock
A common stock
whose future dividends
are not expected to
grow at all; that is,
g=0.
Note too that Equation 9-2 is sufficiently general to handle the case of a zero
growth stock, where the dividend is expected to remain constant over time. If
g = 0, Equation 9-2 reduces to Equation 9-3:
Po- D
rs
(9-3)
This is conceptually the same equation as the one we developed in Chapter 2
for a perpetuity, and it is simply the current dividend. divided by the discount
rate.
&Z '
7^
Write out and explain the valuation formula for a constant growth
stock.
Explain how the formula for a zero growth stock is related to that
for a constant growth stock.
A stock is expected to pay a dividend of $1 at the end of the year.
The required rate of return is rs = 11%. What would the stock's
price be if the growth rate were 5 percent? What would the price be
if g= 0%? ($16.67; $9.09)
305
T
Chapter 9 Stocks and Their Valuation
299
9.5 EXPECTED RATE OF RE`^UIRI-; ON ^.^
CONST-A.NT GROWTH STOCK
We can solve Equation 9-2 for r, again using the hat to indicate that we are dealing with an expected rate of return:_,;
Expected growth rate, or
Expected
Expected rate _
capital gains yield
dividend yield +
of return
(9-4)
rs = 1
0
Thus, if you buy a stock for a price P0 = $23, and if you expect the stock to
pay a dividend D, =$1.242 one year from now and to grow at a constant rate g
= 8% in the future, then your expected rate of return will be 13.4 percent:
rs
$1.242
8%=5.4%+8%=13.4%
$23 -r
In this form, we see that fs is the expected total returv and that it consists of an
expected dividend yield, D, /Pa = 5.4%, plus an expected growth rate or capital gains
yield, g = 8%.
Suppose this analysis had been conducted on January 1, 2006, so Po = $23 is
the January 1, 2006, stock price, and. D, =$1.242 is the dividend expected at the
end of 2006. What is the expected stock price at the end of 2006? We would
again apply Equation 9-2, but this time we would use the _year-end dividend,
D, = Dt(1 + g) = S1.242(1.08) = $1.3414:
P12,1a1rob =
D2007
=
$1.3414
_
rs - 9 0.134 - 0.08 -$24.84
Notice that $24.84 is 8 percent greater than P0, the 623 price on January 1, 2006:
$23(1.08) = $24.84
Thus, we would expect to make a capital gain of $24.84 - $23.00 =$1.84 during
2006, which would provide a capital gains yield of 8 percent:
Capital gain _ $1.84 _
Capital gains yield^ob = Beginning price
$23.00 - 0.08 = 8%
We could extend the analysis on out, and in each future year the expected
capital gains yield would always equal g, the expected dividend growth rate.
For example, the dividend yield in 2007 could be estimated as follows:
D2007 =$1.3414 ` 0.054 = 5.4%
$24.84
12131,i06
Dividend yield2007 = P
The dividend yield for 2008 could also be calculated, and again it would be 5.4
percent. Thus, for a constant growth stock, the following conditions must hold:
L The expected dividend yield is a constant.
2. The dividend is expected to grow forever at a constant rate, g.
' The T. value in Equation 9-2 is a required rate of return, but when we transform to obtain Equation
9-4, we are finding an expected rate of return Obviously, the transformation requires that rs=- P,
This equality holds if the stock market is in equilibrium, a condition that we discussed in Chapter 5.
306
••ild
The popular Motley
Foot Web site, http://
www. foof.com/school/
in troduction to valua ti on
.htm, provides a good
description of some of
the benefits and
drawbacks of a few of
the more commonly
used valuation
procedures.
300 1
Part 3 Financial Assets
3. The stock price is expected to grow at this same rate.
4.
The expected capital gains yield is also a consta: nt, and it is equal to g.
The term expected should be clarified-it.means expected in a probabilistic sen.se,
as the "statistically expected" outcome. Thus, when we say that the growth rate
is expected to remain constant at 8 percent, we mean that the best prediction for
the growth rate in any future year is 8 percent, not that we literally expect the
growth rate to be exactly 8 percent in each future year. In this sense, the constant
growth assumption is reasonable for many large, mature companies.
A. 5E
What conditions must hold if a stock is to be evaluated using the
constant growth model?
What does the term "expected" mean when we say expected growth
rate?
Suppose an analyst says that she values GE based on a forecasted
growth rate of 6 percent for earnings, dividends, and the stock price.
If the growth rate next year turns out to be 5 or 7 percent, would
this mean that the analyst's forecast was faulty? Explain.
9.6 VALUING STOCKS EXPECTED TO GROW
AT A NONCONSTANT RATE
Supernormal
(htonconstant) GrowtE,
The part of the firm's
life cycle in which it
grows much faster than
the economy as a
whole.
For many companies, it is not appropriate to assume that dividends will grow at
a constant rate because firms typically go through life cycles with different
growth rates at different parts of the cycle. During their early years, they
generally grow much faster than the economy as a whole; then they match the
economy's growth; and finally they grow at a slower rate than the economy.c'
Automobile manufacturers in the 1920s, computer software firms such as
Microsoft in the 1980s, and wireless firms in the early 2000s are examples of
firms in the early part of the cycle; these firms are called supernormal, or
nonconstant, growth firms. Figure 9-2 illustrates nonconstant growth and also
compares it with normal growth, zero growth, and negative grovvth.'
° The concept of life cycles could be broadened to product nyc{e, which would include both small
start-up companies and large companies like Microsoft and Procter & Gamble, which periodically
introduce new products that give sales and earnings a boost. We should also mention huainess c7icles,
which alternately depress and boost sales and profits. The growth rate just after a major new product has been introduced, or just after a firm emerges from the depths of a recession, is likely to be
much higher than the "expected long-run average growth rate," -which is the proper number for
DCF analysis.
' A negative growth rate indicates a declining company. A mining company whose profits are falling
because of a declining ore body is an example. Someone buying such a company would expect its
earnings, and consequently its dividends and stock price, to decline each year, and this would lead
to capital losses rather than capital gains. Obviously, a declining company's stock price will be relatively low, and its dividend yield must be high enough to offset the expected capital loss and still
produce a competitive total rehirn. Students sometimes argue that they would never be willing to
buy a stock whose price was expected to decline. However, if the present value of the expected dividends exceeds the stock price, the stock would still be a good investment that would provide a
good return.
307
^.
;^.
Chapter 9 Stocks and Their Valuation
1
301
Illustrative Dividend Growth Rates
FIGURE 9-2
Dividend
(S)
Normal Growth, 8%
End of Supernormal
Growth Period
Supernormal Growth, 30%
_,^& Normal Growth, 8%
^ 15 ^ f^ ' ,r
«.--m-^
•
- Zero Growth, 0%
Declining Growth,
1
2
3
A
5
Years
In the figure, the dividends of the supernormal growth firm are expected to
grow at a 30 percent rate for three years, after which the growth rate is expected
to fall to 8 percent, the assumed average for the economy. The value of this
firm's stock, like any other asset, is the present value of its expected future dividends as determined by Equation 9-1. When D, is growing at a constant rate, we
can simplify Equation 9-1 to Po = D, /(rs - g). In the supernormal case, however,
the expected growth rate is not a constant-it declines at the end of the period of
supernormal growth.
Because Equation 9-2 requires a constant growth rate, we obviously cannot
use it to value stocks that have nonconstant growth. However, assuming that a
company currently enjoying supernormal growth will eventually slow down
and become a constant growth stock, we can combine Equations 9-1 and 9-2 to
form a new formula, Equation 9-5, for valuing it.
First, we assume that the dividend will grow at a nonconstant rate (generally
y
relatively
Y high
g h rate) for N periods, after which it will grow at a constant
rate, g. N is often called the terminal date, or horizon date. Second, we can use
the constant growth formula, Equation 9-2, to determine what the stock's
horizon, or terminal, value will be N periods .from. today:
Terminal Date
(Horizon Date)
The date when the
growth rate becomes
constant. At this date
it is no longer necessary to forecast the
individual dividends.
Horizon (Terminal)
^NT'
Horizon value = ON =
Value
rs - g
The value at the
horizon date of all
The stock's intrinsic value today, Po, is the present value of the dividends during
the nonconstant growth period plus the present value of the horizon value:
308
dividends expected
thereafrer.
302
Part 3 Financial Assets
P0
T
N-1 1
+
©rs
t ...+
D2
D'
- +
(1
t
ry)N^^
(1 + rSjN
(1 + r,,)'
(1 + rs)Z
y
D=
r1 + r,)r
L
v
PV of dividends during the
Horizon value = PV of dividends
nonconstant growth
period, t=1,•••N
during the constant growth
period, t=N+1,•••
P0
(]Z
^
D,
(1 + rs)'
(1 + r5)2 ^ . .. +
PV of dividends during the
nonconstant growth period
t
1,
PN
DN
(^.^^
-1- rs)N + (1 + r5)"
PV of horizon
value, P,,:
i (Dnt+^)1^r5 - 9)]
N
(1 J.
)"
To implement Equation 9-5, we go through the following three steps:
1. Find the PV of each dividend during the period of nonconstant growth and
sum them.
2. Find the expected price of the stock at the end of the nonconstant growth
period, at which point it has become a constant growth stock so it can be
valued with the constant growth model, and discount this price back to the
present.
3. Add these two components to find the intrinsic value of the stock, l''°.
Figure 9-3 can be used to illustrate the process for valuing nonconstant
growth stocks. Here we assume the following five facts exist:
r, = stockholders' required rate of return = 13.4%. This rate is used to discount the cash flows.
N--= }'ears of nonconstant growth = 3.
g4 = rate of growth in both earnings and dividends during the nonconstant
arowth period = 30%. This rate is shown directly on the time line.
(Note: The growth rate during the nonconstant growth period could
vary from year to year. Also, there could be several different nonconstant growth periods, for example, 30 percent for three years, then
20 percent for three years, and then a constant 8 percent,)
?„ = rate of normal, constant growth after the nonconstant period = 89-e. This
rate is also shown on the time line, between Periods 3 and 4.
Do = last dividend the company paid =$1.15.
The valuation process as diagrammed in Fig-Lire 9-3 is explained in the steps set
forth below the time line. The value of the nonconstant growth stock is calculated to be $39.21.
Note that in this example we have assumed a relatively short three-year
horizon to keep things simple. When evaluating stocks, most analysts would use
a much longer horizon (for example, 10 years) to estimate intrinsic values. This
309
..: . .
-'^
Chapter 9 Stocks and Their Valuation
FIGURE 9-3
0
P
Process for Finding the Value of a Nonconstant
Growth Stock
gS = 30%
^
ot = 1.4950
13183
13.4%
1.5113
13.4%
13'4 %
36.3838
30%
2
30%
4
3
9,-B%
f
f----- - --i
DZ = 1.9435
C; = 2.5266
DQ = 2.7287
Pz = 50.5310
53.0576
39.2134 = S39.21 = Pa
Notes to Figure 9-3:
Step 1. Calculate the dividends expected at the end of each year during the noncor:stant
growth period. Calculate the first dividend, D; = D,(I = g,) = $1.15(1.30) = 51.4950.
Here g= is the growth rate during the three-year nonconstant g(oveth period, 30 percent.
Show the $1,4950 on the time line as the cash flow at Time 1. Then, calculate D: _
DI(1 y W = 51.4950;1.30) _$1.9435. and then D; = DZ(1 + g,) = 51,9435(1.30) _
$2.5266. Show these values on the time line as the cash flows at Time 2 and Time 3,
Note that Do is used only to calculate D,
Step 2.
The price of the stock is the PV of dividends from Time 1 to infinity, so in theory we
could project each future dividend, with the normal growth rate, g„ = 8%, used to calculate D, and subsequent dividends. However, we know that after D3 has been paid,
which is at Time 3, the stock becomes a constant growth stock. Therefore, we can use
the constant growth formula to find P;, which is the PV of the dividends from Time 4 to
infinity as evaluated at Time 3.
First, we determine DA =$2.5266(1.08) _$2.7287 for use in the formula, and then
we calculate P3 as follows:
P - Dt _ $2.7287 _ $50.5310
' rs - g, 0.134-COB
Step 3.
We show this $50.5310 on the time line as a second cash flow at Time 3. The $50.5310
is a Time 3 cash flow in the sense that the stockholder could sell it for 550,5310 at Time
3 and also in the sense that $50,5310 is the present value of the dividend cash flows
from Time 41o infinity. Note that the total cash flow at Time 3 consists of the sum of
D3 + P, _ $2.5266 + $50.5310 = 553.0576.
Now that the cash flows have been placed on the time line, we can discount each cash
flow at the required rate of return, rz = 13.4%. We could discount each cash flow by
dividing by (1.13+)`, where t= 1 for Time 1, t = 2 for Time 2, and t=- 3 for Time 3. This
produces the PVs shown to the left below the time line, and the sum of the PVs is the
value of the nonconstant growth stock, $39.21.
With a financial calculator, you can find the PV of the cash flows as shown on the
time line with the cash flow (CFLO) register of your calculator. Enter 0 for CFO because
you receive no cash flow at Time 0, CFj = 1 49.5, CF2 = 1.9435, and CF3 = 2.5266 +
50.5310 = 53.0576. Then enter I/YR = 13.4, and press the NPV key to find the value of
the stock, $39.21.
: L .
fi .
- ^r
-^_
requires a few more calculations, but analysts use spreadsheets so the arithmetic
is not a problem. In practice, the real limitation is obtaining reliable forecasts for
future growth.
=^'r
Explain how one would find the value of a nonconstant growth stock.
Explain what is rneant by "terminal (horizon) date" and "horizon
(terminal) value."
^..
310
I
303
Part 3 Financial Assets
304
^
^ =^
Evaluating Stocks That t7on`t
.
•
.
^
,
Pay Dividends
The dividend growth model assumes that the firm is
currently paying a dividend. However, many firms,
even highly profitable ones, including Cisco, Dell,
and Apple, have never paid a dividend. If a firm is
expected to begin paying dividends in the future, we
can modify the equations presented in the chapter
and use them to determine the value of the stock.
A new business often expects to have low sales
during its first few years of operation as it develops its
product. Then, if the product catches on, sales will
grow rapidly for several years. Sales growth brings
with it the need for additional assets-a firm cannot
increase sales without also increasing its assets, and
asset growth requires an increase in liability and/or
equity accounts. Small firms can generally obtain
some bank credit, but they must maintain a reasonable balance between debt and equity. Thus, additional bank borrowings require increases in equity, and
getting the equity capital needed to support growth
can be difficult for small firms. They have limited
access to the capital markets, and, even when they
can sell common stock, their owners are reluctant to
do so for fear of losing voting control. ThArefore, the
best source of equity for most small businesses is
retained earnings, and for this reason most small firms
pay no dividends during their rapid growth years.
Eventually, though, successful small firms do pay dividends, and those dividends generally grow rapidly at
first but slow down to a sustainable constant rate once
the firm reaches maturity.
If a firm currently pays no dividends but is
expected to pay dividends in the future, the value of
its stock can be found as follows:
1.
2.
3.
Estimate when dividends will be paid, the amount
of the first dividend, the growth rate during the
supernormal growth period, the length of the
supernormal period, the long-run (constant) growth
rate, and the rate of retum required by investors.
Use the constant growth model to determine the
price of the stock after the firm reaches a stable
growth situation.
Set out on a time line the cash flows (dividends
during the supernormal growth period and the
stock price once the constant growth state is
reached), and then find the present value of
these cash flows. That present value represents
the value of the stock today.
To illustrate this process, consider the situation
for MarvelLure inc., a company that was set up in
2004 to produce and market a new high-tech fishing
lure. MarvelLure's sales are currently growing at a
rate of 200 percent per year. The company expects
to experience a high but declining rate of growth in
sales and earnings during the next 10 years, after
which analysts estimate that it will grow at a steady
10 percent per year. The firm's management has
announced that it will pay no dividends for five years,
but if earnings materialize as forecasted, it will pay a
dividend of $0.20 per share at the end of Year 6,
$0.30 in Year 7, $0.40 in Year 8, $0.45 in Year 9, and
$0.50 in Year 10. After Year 10, current plans are to
increase dividends by 10 percent per year.
MarvelLure's investment bankers estimate that
investors require a 15 percent return on similar
stocks. Therefore, we find the value of a share of
MarvelLure's stock as follows:
pv=
$0
{1.15)^ -
+
$0
±__(1.15)`-
$0.30
$O.40
$0.45
(1,115)4
$0.50
;^.
;°#_,•
50.20
(1.15)`
U
^5^0.50(1.10) r
fi .0.15-0.10
5.115)'o
_ $3.30
The last term finds the expected price of the stock in
Year 10 and then finds the present value of that
price. Thus, we see that the dividend growth model
can be applied to firms that currently pay no dividends, provided we can estimate future dividends
with a fair degree of confidence. However, in many
cases we can have more confidence in the forecasts
of free cash flows, and in these situations it is better
to use the corporate valuation model as discussed in
the next section.
A_
3S ^,
mt
311
Chapter 9 Stocks and Their Valuation
305
9.7 VALUING THE EIN-TIRE ^^^^ ORA"I.^^N-3
Thus far we have discussed the discounted dividend approach to valuing a
firm's common stock. This procedure is widely used, but it is based on the
assumption that the analyst can forecast future dividends reasonably well. This
is often true for mature companies that have a history of steady dividend payments. The model can be applied to firms that are not paying dividends, but as
we show in the preceding box, this requires forecasting the time at which the
firm will commence paying dividends, the amount of the initial dividend, and
the growth rate of dividends once they commence. This suggests that a reliable
dividend forecast must be based on forecasts of the firm's future sales, costs, and
capital requirements.
An alternative approach, the total company, or corporate valuation, model,
can be used to value firms in situations where future dividends are not easily
predictable. Consider a start-up formed to develop and market a new product.
Such companies generally expect to have low sales during their first .few years
as they develop and begin to market their products. Then, if the products catch
on, sales will grow rapidly for several years. For example, eBav's sales were $48
million in 1998, the year it first went public, but in 1999 sales grew by nearly 400
percent and they hit $4.5 billion in 2005. Obviously, eBay has been more successful than most new businesses, but growth rates of 100, 500, or even 1,000 percent
are not uncommon during a firm's early years.
Growing sales require additional assets-and eBay could not have grown
without increasing its assets. Over the five-year period 1999-2004, its sales grew
by 658 percent, and that growth required a 583 percent increase in assets. The
increase in assets had to be financed, so eBay's liability and equity accounts also
grew by 583 percent as was required to keep the balance sheet in balance.
Small firms can generally borrow some funds from their bank, but banks
insist that the debt/equity ratio be kept at a reasonable level, which means that
equity must also be raised. However, small firms have little or no access to the
stock market, so they generally obtain new equity by retaining earnings, which
means that they pay little or no dividends during their rapid growth years.
Eventually, though, most successful firms do pay dividends, and those dividends grow rapidly at first but then slow down as the firm approaches maturity.
°'-
pr '
It is difficult to forecast the future dividend stream of any firm that is expected
to go through such a transition, and even in the case of large firms such as Cisco,
Dell, and Apple that have never paid a dividend, it's hard to forecast when dividends will commence and how large they will be.
Another problem arises when it is necessary to find the value of a division
as opposed to an entire firm. For example, in 2005 Kerr-McGee, a large oil and
chemical company, decided to sell its chemical division. The parent company
had been paying dividends for many years, so the discounted dividend model
could be applied to it. However, the chemical division had no history of dividends, and it would likely be bought by another chemical company and folded
into the purchaser's other operations. How could Kerr-McGee's chemical division be valued? The answer is, "Use the corporate valuation model as discussed
in this section."
'The corporate valuation model presented in this section is widely used by analysts, and it is in
many respects superior to the discounted dividend model. However, it is rather involved as it
requires the estimation of sales, costs, and cash flows on out into the future before beginning the
discounting process. Therefore, some instructors may prefer to omit Section 9.7 and skip to Section
9.8 in the introductory course.
312
Total Company or
Corporate Valuation
Nrlode3
A valuation model
used as an alternative
to the dividend growth
model to determine
the value of a firm,
especially one with no
history of dividends or
a division of a larger
firm. This model first
calculates the firm's
free cash flows and
then finds their present
value to determine the
firm's value.
306
Part 3 Firar.cial Assets
The Corporate Valuation Model
In Chapter 3 we explained that a firm's value is determined by its ability to generate cash flow, both now and, in the future. Therefore, market value can be
expressed as follows:
Market = V
Com p an y
value
_
= PV of expected future free cash flows of company
FCFw
FCF2
FCF;
(1 + WACC)' + (1 + WACC) 2 T . . ^ (1 + WACC)F
(9-6)
Here FCFt is the free cash now in Year t and WACC is the weighted average cost
of the firm's capital.
Recall from Chapter 3 that free cash flow is the cash inflow during a given
year less the cash needed to finance required asset additions. Inflows are equal
to net after-tax operating income (also called NOPAT) plus noncash charges
(depreciation
and amortization),
which were deducted when calculating
NOPAT, while the required asset ad.ditions are the capital expenditures plus the
net addition to working capital. This was discussed in Chapter 3, where we
developed the following equation:
FCF = [EBIn1 - l) +
Depreciation '
and amortization
^-
Capital
expenditures
+
X
A, Net
operating
working
capital _j
4
Depreciation and amortization can be shifted from the first bracketed term to the
second term (and given a minus sign). Then the first term becomes EBITO - T),
also called NOPAT, and the second term becomes the net (rather than gross) new
investment in operating capital. The result is Equation 9-7, which shows that free
cash flow is equal to after-tax operating income (NOPAT) less the net new
investment in operating capital:
FCF = NOPAT - Net new investment in operating capital (9-7)
Turning to the discount rate, WACC, note first that free cash flow is the cash
generated before inaking any }'ayncerits to any investors-the common stockholders,
preferred stockholders, and bondholders-and that cash floul must provide a return to all
these investors. Each of these investor groups has a required rate of return that
depends on the risk of the particular secu.rity, and as we discuss in Chapter 10,
the average of those required returns is the WACC.
With this background, we can summarize the steps used to implement the
corporate valuation model. This type of analysis is performed both internally by
the firm's financial staff and also by external security analysts, who are generally
experts on the industry and quite familiar with the firm's history and future
plans. For illustrative purposes, we discuss an analysis conducted by Susan
Buskirk, senior food analyst for the investment banking firm Morton Staley and
Company. Her analysis is summarized in Table 9-1, which was reproduced from
the chapter Excel modeL
•
Based on Allied's history and her knowledge of the firm's business plan,
Susan estimated sales, costs, and cash flows on an annual basis for five
years. Growth will vary during those years, but she assumes that things will
stabilize and growth will be constant after the fifth year. She could have projected variability for more years if she thought it would take longer to reach
a steady-state, constant growth situation.
313
T
Chapter 9 Stocks and Their Valuation
TABLE -9 1:.:.;
Allied Food Products: Free Cash How Valuation
A
C
B
F
E
D
I4
,35
1 ae
137
138
G
H
Z009
9.0%
85.0%
8.0%
7.0%
2010
8.0%
85.0%
8.0%
7.0%
ForacastSdYsars
ts; Part 1. Key Inputs
2007
9.0%
87.0%
8.0%
8.0%
2006
10.0%
87.0%
8.0%
6.0%
Sales growth rate
Operating costs as a% of sales
Growth in operating capital
Depr'n as a% of operating capital
2008
9.0%
86.0%
8.0%
7.0%
40%
10%
139 Tax rate
140 WACC
6.0%
141 Long-run FCF growth, g,
142
143 Part 2. Forecast of Cash Flows During Period of Nonconstant Growth
144
Rftftal
145
2006
2005
Forecasted Yom
2008
2007
2009
2010
146
147 Sales
Operating costs
148
145
Depreciation
tso ESIT
7si NOPAT = EBff x (1-T)
152
153 Total operating capital
1c4 Net new operating cap
1;5 Free Cash Flow, FCF
56 PV of FCF9
53,000.0
2,616.2
100.0
$283.8
$3,300.0
2,871.0
116.6
$312.4
$3,597.0
3,129.4
168.0
$299.6
$3,920.7
3,371.8
158.7
$390.2
$4,273.6
3,632.6
171.4
5469.6
$4,615.5
3,923.2
185.1
$507.2
$170.3
$187.4
$179.8
$234.1
$281.8
$304.3
$1,800.0
280
-$109.7
$1,944.0
144.0
$43.4
$2,099.5
155.5
$24.3
$2,267.5
168.0
S66.1
$2,448.9
181.4
$100.4
$2,644.8
195.9
$108.4
$39.5
S20. 7
$49.7
£68.6
S67.3
N.A.
157
ise Part 3. Terminal Value and Intrinsic Value Estimation
FCF2010(1+gLR)
Estfmated Value at the Nodzon, 2010
159
15o Free Cash Flow (2011)
161 Terminal Value at 2010, TV
$114.9
$2,872.7 4-- -'
162 PV of the 2010TV
$1,783.7
^..--
Wane
FCF2011
WACC - g
TV/(1+WACC)N
iGS
j64
1es
16s
16^
168
169
170
Calculation of Firm's Intrinsic Value
Sum of PVs of FCFs, 2006-2010
PV of 2010 TV
Total corporate value
Less: rnarket value of debt and pfd
intrinsic value of common equity
Shares outstanding (millions)
$245.1
$1,783.7
2,028.8
$860.0
$1,168.8
50.0
171
172 Intrinsic Value Per Share
•
•
^:.
$23.38
Susan next calculated the expected free cash flows (FCFs) for each of the five
nonconstant growth years, and she found the PV of those cash flows, discounted at the WACC.
After Year 5 she assumed that FCF growth would be constant, hence the
constant growth model could be used to find Allied's total market value at
Year 5. This "horizon, or terminal, value" is the sum of the PVs of the FCFs
from Year 6 on out into the future, discounted back to Year 5 at the WACC.
•
Next, she discounted the Year 5 terminal value back to the present to find its
PV at Year 0.
•
She then summed all the PVs, the annual cash flows during the nonconstant
period plus the PV of the horizon value, to find the firm's estimated total
market value.
•
She then subtracted the value of the debt and preferred stock to find the
value of the common equity.
314
1
307
30^ Part 3 Financial Assets
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Other Approaches to Valuing
Common Stocks
While the dividend growth and the corporate value
models presented in this chapter are the most widely
used methods for valuing common stocks, they are
by no means the only approaches. Analysts often use
a number of different techniques to value stocks. Two
of these alternative approaches are described here.
The P/E Multiple Approach
Investors have long looked for simple rules of zhumb
to determine whether a stock is "Ciriv valued. One
such approach is to look at the stock's price-toearnings (P/E) ratio. Reca1l from Chapter 4 that a
company's P/E ratio shows how much investors are
willing to pay for each dollar of reported earnings. As
a staring point, you might conclude that stocks with
low PIE ratios are undervalued, since their price is
"[ow" given current earnings, while stocks with high
P/E ratios are overvalued.
Unfortunately, however, valuing stocks is not that
simple. We should not expect all companies to have
the same P/E ratio. P/E ratios are affected by riskinvestors discount the earnings of riskier stocks at a
higher rate. Thus, all else equal, riskier stocks should
have lower P/E ratios. In addition, when you buy a
stock, you not only have a claim on current earn-
ingsyou also have a claim on all future earnings. All
else equal, companies with stronger growth opportunities will generate larger future earnings and thus
should trade at higher PIE ratios. Therefore, eBay is
not necessarily overvalued just because its P/E ratio
is 52.8 at a time when the median firm has a PIE of
20.1. Investors believe that eBay's growth potential is
well above average. Whether the stock's future prospects Justifii its PiE ratio remains to be seen, but in
and of itself a hiah PIE ratio does not mean that a
stock is overvalued.
Nevertheless, PIE ratios can provide a useful starting point in stock valuation. If a stock's P/E razio is well
above its industry average, and if the stock's growth
potential and risk are similar to other firms in the industry, this may indicate that the stock's price is too high.
Likewise, if a company's P;E ratio falls well below its
historical average, this may signal that the stock is
undervalued-particularly if the company's growth
prospects and risk are unchanged, and if the overall
P/E for the market has remained constant or increased,
One obvious drawback of the PIE approach is
that it depends on reported accounting earnings. For
this reason, some analysts choose to rely on other
multiples to value stocks. For example, some analysts
Finally, she divided the equity value by the number of shares outstanding,
and the result was her estimate of Allied's intrinsic value per share. This value
was quite close to the stock's market price, so she concluded that Allied's
stock is priced at its equilibrium level. Consequently, she issued z"Hold" recommendation on the stock. If the estimated intrinsic value had been significantly below the market price, she would have issued a"Sell" recommendation, and had it been well above, she would have called the stock a"Buy."
Comparing the Total Company
and Dividend Growth Models
Analysts use both the discounted dividend model and the corporate model
when valuing mature, dividend-paying firms, and they generally use the corporate model when valuing firms that do not pay dividends and divisions. In principle, we should find the same intrinsic value using either model, but differences
are often observed. When a conflict exists, then the assumptions embedded in
the corporate model can be reexamined, and once the analyst is convinced they
are reasonable, then the results of that model are used. In our Allied example,
the estimates were extremely close-the dividend growth model predicted a
price of $23.00 per share versus $23.38 using the total company n•todel, and both
are essentially equal to Allied's actual $23 price.
315
fA
^^ -
Chapter 9 Stocks and Their Valuation
( 309
-4 1
look at a company's price-to-cash-flow ratio, while
others look at the price-to-sales ratio.
The EVA Approach
In recent years, analysts have looked for more rigorous alternatives to the dividend growth model. More
than a quarter of all stocks listed on the NYSE pay no
dividends. This proportion is even higher on Nasdaq.
While the dividend growth model can still be used
for these stocks (see box, "Evaluating Stocks That
Don't Pay Dividends"), this approach requires that
analysts forecast when the stock will begin paying
dividends, what the dividend will be once it is established, and the future dividend growth rate. In many
cases, these forecasts contain considerable errors.
An alternative approach is based on the concept
of Economic Value Added (E1,rA), which we discussed
back in Chapter 3. Also, recall from the box in Chapter
4 entitled, "EVA and ROE" that EVA can be written as
(Equity capital)(ROE - Cost of equity capital)
This equation suggests that companies can increase
their EVA by investing in projects that provide
shareholders with returns that are above their cost of
capital, which is the return they could expect to earn
},..
on alternative investments with the same level of risk.
When you buy stock in a company, you receive more
than just the book value of equity-you also receive
a claim on all future value that is created by the firm's
managers (the present value of all future EVAs). It follows that a company's market value of equity can be
wriiten as
Market value
of equity
Book
value
We car, find the "fundaments!" value of the
stock, Pt,, by simply dividing the above expression by
the number of shares outstanding
As is the case with the dividend growth model,
we can simplify the above expression by assuming
that at some point in time annual EVA becomes a
perpetuity, or grows at some constant rate over time.,,
'What we have presented here is a simplified version of
what is often referred to as the Edwards-Bell-Ghison (EBO)
model. For a more complete description of this t echni que
and an excellent summary of how it can be used in practice,
take a look at the article "Measuring Wealth,' by Charles
M. C. Lee, in CA Magazine, April 1996, pp. 32-37.
In practice, intrinsic value estimates based on the two models normally
deviate both from one another and from actual stock prices, leading different
analysts to reach different conclusions about the attractiveness of a given stock.
The better the analyst, the more often his or her valuations will turn out to be
correct, but no one can make perfect predictions because too many things can
change randomly and unpredictably in the future. Given all this, does it matter
whether you use the total company model or the dividend growth model to
value stocks? We would argue that it does. If we had to value, say, 100 mature
companies whose dividends were expected to grow steadily in the future, we
would probably use the dividend growth model. Here we would only need to
estimate the grox-vth rate in dividends, not the entire set of pro forma financial
statements, hence it would be more feasible to use the dividend model.
However, if we were studying just one or a few companies, especially companies still in the high-growth stage of their life cycles, we would want to project future financial statements before estimating future dividends. Then,
because we would already have projected future financial statements, we would
go ahead and apply the total company model. .Intel, which pays a quarterly dividend of 8 cents versus quarterly earnings of about $1.24, is an example of a company where either model could be used, but we think the corporate model
would be better.
Now suppose you were trying to estimate the value of a company that has
never paid a dividend, such as eBay, or a new firm that is about to go public, or
^
PV of all
future EVAs
316
310
Part 3 Financial Asse-'s
Kerr-McGee's chemical division that it plans to sell. [n all of these situations, vou
would be much better off using the corporate valuation, model. Actually, even if
a company is paying steady dividends, much can be learned from the corporate
valuation model, so analysts today use it for all types of valuations. The process
of projecting future financial statements can reveal a great deal about the company's operations and financing needs. Also, such an analysis can provide
insights into actions that might be taken to increase the company's value, and
for this reason it is integral. to the planning and forecasting process, as we discuss in a later chapter.
1Vrite out the equation for free cash flows, and explain it.
Why might
b ^ someone use the corporate valuation model even for
companies that have a history of paying dividends?
What steps are taken to find. a stock price as based on the firm's total
value?
Why might the calculated intrinsic stock value differ from the
stock's current market price? Which would be "correct," and what
does "correct" mean?
9.8 STOCK MARKET EQUILIBRIUM
Recall that rx, the required return on Stock X, can be found using the Security
Market Line (SML) equation from the Capital Asset Pricing Model (CAPM) as
)4
discussed back in Chapter 8:
rx - rRF + ( rM - rRF)bX - rRF + ( RPM)bX
If the risk-free rate is 6 percent, the market risk premium is 5 percent, and Stock
X has a beta of 2, then the marginal investor would require a return of 16 percent
on the stock:
rx = 6% + (5%)2.0
- 16%
Marginal Investor
A representative
investor whose actions
reflect Uhe beliefs of
those people who
are currently trading a
stock. It is the marginal
investor who
determines a stock's
price.
This 16 percent required return is shown as the point on the SML in Figure 9-4
associated with beta = 2.0.
a marginal investor will buy Stock X if its expected return is more than
16 percent, will sell it if the expected return is less than 16 percent, and will be
indifferent, hence will hold but not buy or sell, if the expected return is exactly
16 percent. Now suppose the investor's portfolio contains Stock X, and he or she
analyzes its prospects and concludes that its earnings, dividends, and price can
be expected to grow at a constant rate of 5 percent per year. The last dividend
was D. =$2.8571, so the next expected. dividend is
D, = $2.8571 (1.05) = $3
The investor observes that the present price of the stock, P0, is 530. Should he or
she buy more of Stock X, sell the stock, or maintain the present position?
The investor can calculate Stock X's expected rate of return as follows:
rX
P1
0 1 g $30 + 5% = 15%
317
,,
Chapter 9 Stocks and 'Their Valuation
311 ^
Expected and Required Returns on Stock X
` Fl G U R E 9-4
Rate of Return
t'•=.'
5tlL: r° ryF+ (r:,,- r^r1 b
---_y"
rt=16
r15 ------------ ^^^X
ru=11 ^--------.1
r
=6
0
2.0 Risk, bE
1.0
This value is plotted on Figure 9-4 as Point X, which is below the SML. Because
the expected rate of return is less than the required return, he or she, and many
other investors, would want to sell the stock. However, few people would want
to buy at the $30 price, so the present owners would be unable to find buyers
unless they cut the price of the stock. Thus, the price would decline, and the
decline would. continue until the price hit $27.27. At that point the stock would
be in equilibrium, defined as the price at which the expected rate of return,
16 percent, is equal to the required rate of return:
rx = _ $2727 +5%= 11%-r5%= 1 6 %
$b25, events would have
Had the stock initially sold for less than $27.27, say, $25,
been reversed. Investors would have wanted to purchase the stock because its
expected rate of return would have exceeded its required rate of return, buy
orders would have come in, and the stock's price would be driven up to $27.27.
To summarize, in equilibrium two related conditions must hold:
1. A stock's expected rate of return as seen by the marginal investor must equal
its required rate of retunL r"i = r;.
2. The actual market price of the stock must equal its intrinsic value as estimated by the marginal investor: P,
Of course, some individual investors may believe that fl > rl and Po > Pa , hence
they would invest most of their funds in the stock, while other investors might
have an opposite view and thus sell all of their shares. However, investors at the
margin establish the actual market price, and for these investors, we must have
Ti = rl and Po = Po. If these conditions do not hold, trading will occur until they do.
Changes in Equilibrium Stock Prices
Stock prices are not constant-they undergo violent changes at times. For example, on October 27,1997, the Dow Jones industrials fell. 554 points, a 7.18 percent
318
Equilibrium
The condition under
which the expected
return on a security is
just equal to its
required return, r = r.
Also, P= Pa, and the
price is stable.
312 1
Part 3 Financial Assets
drop in value. Even worse, on October 19,1987, the Dow lost 508 points, causing
an average stock to lose 23 percent of its value on that one day, and some individual stocks lost more than 70 percent. To see what could cause such changes to
occur, assume that Stock X is in equilibrium, selling at a price of $27.27 per
share. If all expectations were exactly met, during the next year the price would
gradually rise to $28.63, or by 5 percent. However, suppose conditions changed
as indicated in the second column of the following table:
VARIABLE VALUE
New
Original
5%
4%
Risk free rate, raF
Market risk premium, rM - rPF
6%
5%
Stock X's beta coefficient, bX
2.0
1.25
5%
$2.B571
$27.27
6%
$2.8571
?
Stock X's expected growth rate, gx
p°
Price of Stock X
Now give yourself a test: How would the change in each variable, by itself,
affect the price, and what new price would result?
Every change, taken alone, would lead to an increase in the price. The first
three changes all lower rh, which declines from 16 to 10 percent:
Original rx = 6% + 5%(2.0) = 16%
New rx = 5% + 4%(1.25) = 10%
Using these values, together with the new g, we find that Ptt rises from $27.27 to
$75.71, Or by 178 percent:'
Original Po -
$2.8571(1.05) = $3 =
$27.27
0.11
0.16-0.05
$2.8571(1.06) - $3.0285
= $75.71
0.04
NewPo0.10-0.06
Note too that at the new price, the expected and required rates of return will be
equal:7Q
^X-$3.0285 ,
6%=10%=rX
$75.71
Evidence suggests that stocks, especially those of large companies, adjust
rapidly when their fundamental positions change. Such stocks are followed
closely by a number of security analysts, so as soon as things change, so does the
stock price. Consequently, equilibrium ordinarily exists for any given stock, and
required and expected returns are generally close to equal. Stock prices certainly
° A price change of this magnitude is by no means rare. The prices of many stocks double or halve
during a year. For example, during 2004, Starbucks Corporation, which operates a chain of retail
stores that sell whole bean coffees, increased in value by 88.1 percent. Noveilus Systems, a semiconductor equipment manufacturer, fell by 33.3 percent.
10 It should be obvious by now that actual realized rates of return are not necessarily equal to
expected and required retunu. Thus, an investor might have expected to receive a return of 15 percent if he or she had bought Novellus or Starbucks stock in 2004, but, after the fact, the realized
return on Starbuclcc was far above 15 percent, whereas that onNovellus was far below.
319
Chapter 9 Stocks and Their Valuation
change, sometimes violently and rapidly, but this simply reflects changing conditions and expectations. There are, of course, times when a stock will continue
to react for several months to unfolding favorable or unfavorable developments.
However, this does not signify a long adjustment period; rather, it simply i.ndicates that as more new information about the situation becomes available, the
market adjusts to it.
ts^
'
For a stock to be in equilibrium, what two conditions must hold?
If a stock is not in equilibrium, explain how financial, markets adjust
to bring it into equilibrium.
9.9 ^ N v'^ ^ TI1Q7G IN INTERNATIONAL
STOCKS
As noted in Chapter 8, the U.S. stock market amounts to only 40 percent of the
world stock market, and as a result many U.S. investors hold at least some foreign stock. Analysts have long touted the benefits of investing overseas, arguing
that foreign stocks both improve diversification and provide good growth
opportunities. For example, after the U.S. stock market rose an average of 17.5
percent a year during the 1980s, many analysts thought that the U.S. market in
the 1990s was due for a correction, and they suggested that investors should
increase their holdings of foreign stocks.
To the surprise of many, however, U.S. stocks outperformed foreign stocks in
the 1990s-they gained about 15 percent a year versus only 3 percent for foreign
stocks. However, the Dow Jones STOXX Index (which tracks 600 European companies) outperformed the S&P 500 from 2002 through 2004. Table 9-2 shows how
stocks in different countries performed in 2004. Column 2 indicates how stocks
in each country performed in terms of the U.S. dollar, while Column 3 shows
how the country's stocks performed in terms of its local currency. For example,
in 2004 Brazilian stocks rose by 25.12 percent, but the Brazilian real increased
over 11 percent versus the U.S. dollar. Therefore, if U.S. investors had bought
Brazilian stocks, they would have made 25.12 percent in Brazilian real terms, but
those Brazilian reals would have bought 11.1 percent more U.S. dollars, so the
effective return would have been 36.22 percent. Thus, the results of foreign
investments depend in part on what happens to the exchange rate. Indeed,
when you invest overseas, you are making two bets: (1) that foreign stocks will
increase in their local markets, and (2) that the currencies in which you will be
paid will rise relative to the dollar. For Brazil and most of the other countries
shown in Table 9-2, both of these situations occurred during 2004.
Although U.S. stocks have generally outperformed foreign stocks in recent
years, this by no means suggests that investors should avoid foreign stocks.
Holding some foreign investments still improves diversification, and it is inevitable that there will be years when foreign stocks outperform domestic stocks,
such as the period from 2002-2004. When this occurs, U.S. investors will be glad
they put some of their money into overseas markets.
What are the key benefits of adding foreign stocks to a portfolio?
When a U.S. investor purchases foreign stocks, what two things is
he or she hoping will happen?
320
313
^..
3 14
Part 3 Financial Assets
TABLE 9-2,:
Dow Jones Global Stock Indexes in 2004 (Ranked
by Performance in U.S.-Dollar Terms)
U.S. Dollars
Local Currency
Austria
+67.96%
-55.61%
South Africa
+52.17
+28.12
Country
Mexico
4-46.53
+45.45
Norway
+46.47
+32.96
Befoium
+43.07
+32.55
Greece
+40.02
+29.72
Ireland
+37.91
+27.77
Brazil
+36.22
+25.12
Sweden
+33.50
+23.15
Indonesia
+31.84
+45.30
Philippines
+30.09
+31.46
New Zealand
+30.03
+17.87
Denmark
+28.75
+19.01
Australia
+28.69
+23.38
Italy
+27.59
+18.21
Spain
+26.07
+16.80
Chile
+25.59
+17.68
South Korea
123.99
4-7.63
Canada
+21.98
+12.61
Portugal
+21.24
+12.32
Singapore
+ 19.09
+14.49
Hong Kong
+17.99
+18.13
France
+ 17.07
+8.47
United Kingdom
+16.93
+8.92
Japan
+16.62
+11.27
Germany
+14.29
+5.89
Switzerland
+ 14 18
+4.78
Netherlands
+ 11.84
43.62
Malaysia
+11.11
+11.12
United States
+10.16
+10.16
Taiwan
410.03
+2.68
Finland
+5,84
-1.94
Thailand
-8.77
-10.55
Venezuela
-18.83
+31.12
World
+14.47
-
World ex. U.S.
419.23
-
Source: Craig Karmin, "Currency Effect Enhances Overseas Returns," The Wall Street Journat,
January 3, 2005, p. R6.
321
Chapter 9 Stocks and Their Valuation
315
F_
9.1(} PREFERRED STOCK
Preferred stock is a hybrid-it is similar to bonds i n some respects and to common stock in others. This hybrid nature becomes apparent when we try to clasbonds, preferred
sify preferred in relation to bonds and common stock. Like
stock has a par value and a fixed dividend that must be paid before dividends
can be paid on the common stock. However, the directors can omit (or "pass")
the preferred dividend without throwing the company into bankruptcy. So,
although preferred stock calls for a fixed payment like bonds, not making the
payment will not lead to bankruptcy.
As noted earlier, a preferred. stock entitles its owners to regular, fixed dividend payments. If the payments last forever, the issue is a perpetuity whose
value, V. is found as follows:
(9-8)
uP = ^P
P
VP is the value of the preferred stock, D. is the preferred dividend, and rp is the
required rate of return on the preferred. Allied Food has no preferred outstandi.ng, but suppose it did, and this stock paid a dividend of $10 per year. If its
required return were 10.3 percent, then the preferred's value would be $97.09,
found as follows:
VP = $10.00 _ $97.09
0.103
In equilibrium, the expected return, r", must be equal to the required return, rP.
Thus, if we know the preferred's current price and dividend, we can solve for
the expected rate of return as follows:
rP =
P
VP
(9-8a)
Some preferreds have a stated maturity, often 50 years. Assume that our
illustrative preferred matured in 50 years, paid a $10 annual dividend, and had a
required return of 8 percent. We could then find its price as follows: Enter N=
50, I/YR = 8, PMT = 10, and FV = 100. Then press PV to find the price, VP =
$124.47. If rp= 10 percent, change I/YR to 10, in which case VF = PV = $100. If
you know the price of a share of preferred stock, you can solve for I/YR to find
the expected rate of return, Pt,.
^ Explain the following statement: "Preferred stock is a hybrid
security."
Is the equation used to value preferred stock more like the one used
to evaluate a bond or the one used to evaluate a "normal," constant
growth common stock? Explain.
322
Part 3 Financial Assets
3'f (s I
J^
Investing in Emerging Markets
Given the possibilities of better diversification. and
higher returns, U.S. investors have been putting more
and more money into foreign stocks. While most
investors limit their foreign holdings to developed
countries such as Japan, Germany, Canada, and the
United Kingdom, some have broadened their portfolios to include emerging markets such as South
Korea, Mexico, Singapore, Taiwan, and Russia.
Emerging markets provide opportunities for larger returns, but they also entail greater risks. For exampie, Russian stocks rose more than 150 percent in the
first halt of 1996, as it became apparent that Boris
Yeltsin would be reelected president. By contrast, if
you had invested in Taiwanese stocks, you would
have lost 30 percent in 1995-a year in which most
stock markets performed extremely well. Rapidly
declining currency values caused many Asian markets
to fall by more than 30 percent in 1997; however,
more recently most Asian markets have recovered,
and they ended 2004 on a positive note, Factors that
heiped these markets rise included peaceful elections, inflows of foreign capital, economic growth,
and the positive expectations for China's economy.
During 2004, only Thai and Chinese stocks in the
Asian region posted negative returns,
Stocks in emerging markets are intriguing for
two reasons. First, developing nations have the
ktr
^
-
t
greatest potential for growth. Second, while stock
returns in developed countries often move in sync
with one another, stocks in emerging markets generally march to their own drummers. Therefore, the correlations between U.S. stocks and those in emerging
markets are generally lower than between U.S. stocks
and those of other developed countries. Thus, correlations suggest that emerging markets impro-:e the
diversification of U.S. investors' portfolios. (Recall
from Chapter 8 that the lower the correlation, the
ethe benefit of diversification.}
greater
-
}e^,,,^.•°•
the other hand, stocks in emerging markets
are often extremely risky, iiliyuid, and involve higher
transactions costs, and most U.S. investors do not
have ready access to information on the companies
involved. To reduce these problems, mutual funds
focused on specific countries have, been createdthey are called "country funds." Country fvnds help
investors avoid the problem of picking individual
stocks, but they do little to protect you when entire
regions decline.
.4
Sources: "World Stock Markets Gamble--and Wm," The
Wall Street Journal, January 3, 2005, p. R6; and Mary Kissel,
"Asian Markets Post Gains on Solid Economic Growth:
China IPOs May Be '05 Hit,' The Wall Street Joumal,
January 3, 2005, p. R6.
-
%
.R ..
£
^T
^`
.
-
^
^-
Corporate decisions should be analyzed in terms of how alternative courses
of action are likely to affect a firm's value. However, it is necessary to know
how stock prices are established before attempting to measure how a given
decision will affect a specific firm's vaiue. This chapter discussed the rights
and privileges of common stockholders, showed how stock values are
determined, and explained how investors estimate stocks' intrinsic values
and expected rates of return.
Two types of stock valuation models were discussed: the discounted
dividend model and the corporate valuation model. The dividend model is
useful for mat-are, stable companies, and it is easier to use, but the corporate model is more flexible and better for use with companies that do not
pay dividends or whose dividends would be especially hard to predict.
323
I
Chapter 9 Stocks and Their Valuation
We a,,so discussed preferred stock, which is a hybrid security that has
some characteristics of a common stock and some, of a bond. Preferreds are
valued using models similar to those for perpetual and "regular" bonds.
't^i{e also discussed market equifi ^rltlrF?, noting that for a St J! ^ to be i l
equilibrium its price must he equal to its intrinsic vnEue as estimated by a
marginal investor, and its expected and required returns as seen by such
investors must also be equal. Finally, we noted that stocks are traded
vvor€dwide, that U.S. markets account for less than half of the value of all
stocks, that U.S. investors can benefit from global diversification, but also
that international investing can be risky and for most individuals should be
done through mutual funds whose managers have specialized knowledge of
foreign markets.
SELF-TEST QUESTIONS AND PROBLEMS
(Solutions Appear in Appendix A)
S')`1
Key terms Define each of the following terms:
a. Proxy; proxy fight; takeover
b. Preemptive right
c. Classified stock; founders' shares
d. Intrinsic value (Pd; market price (Pp)
e. Required. rate of return, rN expected rate of return, ry; actual, or realized, rate of
return, "rs
f. Capital gains yield; dividend yield; expected total return; growth rate, g
g. Zero growth stock
h. Normal, or constant, growth; supernormal (nonconstant) growth
i. Total company (corporate valuation) model
j. Terminal (horizon) date; horizon (terminal) value
k. Marginal investor
1. Equilibrium
m. Preferred stock
ST -2
Stock growth rates and valuation You are considering buying the stocks of two companies that operate in the same industry.ll-iey have very similar characteristics except for
their dividend payout policies. Both companies are expected to earn $3 per share this
year, but Company D(for "dividend") is expected to pay out all of its earnings as dividends, while Company G(for "growth") is expected to pay out only one-third of its
earnings, or $1 per share. D's stock price is $25. G and D are equally risky. Which of the
following statements is most likely to be true?
a. Company G will have a faster growth rate than Company D. Therefore, G's stock
price should be greater than $25.
b. Although G's growth rate should exceed D's, D's current dividend exceeds that of
G, and this should cause D's price to exceed G's.
c. A long-term investor in Stock D will get his or her money back faster because D
pays out more of its earnings as dividends. Thus, in a sense, D is like a short-term
bond, and G is like a long-term bond. Therefore, if economic shifts cause rd and. r5 to
increase, and if the expected streams of dividends from D and G remain constant,
both Stocks D and G will decline, but D's price should decline further.
d. D's expected and required rate of return is is = r5 = 12%. G's expected return will
be higher because of its higher expected growth rate.
e. If we observe that G's price is also $25, the best estimate of G's growth rate is
8 percent.
324
=^.
317
31$
.,^
y s
Part 3 Financial Ass=ts
S-1-3
.5-
Constant growth stock valuation Fletcher Company's current stock price is $36, its last
dividend was $2.40, and its required rate of return is 12 percent. If dividends are
expected to grow at a constant rate, g, in the future, and if r$ is expected to remain at
12 percent, what is Fletcher's expected stock price 5 years from now?
ST.-4
Nonconstant growth stock valuation Snyder Computers Inc. is experiencing rapid groi,^,th.
Earnings and dividends are expected to grow at a rate of 15 percent during the next 2
years,13 percent the following year, and at a constant rate of 6 percent during Year 4 and
thereafter. Its last dividend was $1.15, and its required rate of return is 12 percent.
a.
b.
c.
Calculate the value of the stock todav
Calculate A and Pz.
Calculate the dividend and capital gains yields for Years l., 2, and 3.
QUESTIONS
?-1
9-2
It is frequently stated that the one purpose of the preemptive right is to allow individuals to maintain their proportionate share of the ownership and control of a corporation.
a. How important do you suppose control is for the average stockholder of a firm
whose shares are traded on the New York Stock Exchange?
b. Is the control issue likely to be of more importance to stockholders of publicly
owned or closely held (private) firms? Explain.
a= '
Is the following the correct equation for finding the value of a constant growth stock?
Explain.
Do
Po= r5T9
9-3
If you bought a share of common stock, you would probably expect to receive dividends
plus an eventual capital gain. Would the distribution between the dividend yield and the
capital gain yield be influenced by the firm's decision to pay more dividends rather than
to retain and reinvest more of its earnings? Explain.
94
Two investors are evaluating GE's stock for possible purchase. They agree on the
expected value of D, and also on the expected future dividend growth rate. Further, they
agree on the riskiness of the stock. However, one investor normally holds stocks for 2
years, while the other holds stocks for 10 years. On the basis of the type of analysis done
in this chapter, should they both be willing to pay the same price for GE's stock?
Explain.
9-5
A bond that pays interest forever and has no maturity is a perpetual bond. In what
respect is a perpetual bond similar to a no-growth common stock? Are there preferred
stocks that are evaluated similarly to perpetual bonds and other preferred stocks that are
more like bonds with finite lives? Explain.
PROBLEMS
Easy
Problems 1-6
9-1
DPS calculation Warr Corporation just paid a dividend of $1.50 a share (that is, Dp =
$1.50). The dividend is expected to grow 7 percent a year for the next 3 years and then at
5 percent a year thereafter. What is the expected dividend per share for each of the next
5 years?
9-2
Constant growth valuation Thomas Brothers is expected to pay a $0.50 per share
dividend at the end of the year (that is, D, _$0.50). The dividend is expected to grow
at a constant rate of 7 percent a year. The required rate of return on the stock, r,,, is
15 percent. What is the stock's value per share?
9-3
Constant growth valuation Harrison. Clothiers' stock currently sells for $20 a share. It
just paid a dividend of $1.00 a share (that is, D,) =$1.00). The dividend is expected to
325
:^.
:^.
Chapter 9 Stocks and Their Valuation
9-4
grow at a constant rate of 6 percent a year. What stock price is expected 1 year from
now? What is the required rate of return?
Nonconstant growth valuation Hart Enterprises recently paid a dividend, Dp,, of $1:25. It
expects to have nonconstant growth of 20 percent for 2 years followed by a e.••onstant rate
of 5 percent thereafter. The firm's required return is 10 percent.
a.
b.
c.
ate
7-17
How far away is the terminal, or horizon, date?
What is the firm's horizon, or terminal, value?
What is the firm's intrinsic value today, P"?
9-5
Corporate value model Smith Technologies is expected to generate $150 million in free
cash flow next year, and FCF is expected to grow at a constant rate of 5 percent per year
indefinitely. Smith has no debt or preferred stock, and its VJACC is 10 percent If Smith
has 50 million shares of stock outstanding, what is the stock's value per share?
9-6
Preferred stock valuation Fee Founders has perpetual preferred stock outstanding that
sells for $60 a share and pays a dividend of $5 at the end of each year. What is the
required rate of return?
Preferred stock rate of return What will be the nominal rate of return on a perpetual
preferred stock with a 5100 par value, a stated dividend of 8 percent of par, and a current market price of (a) $60, ( b) S80, (c) $100, and (d) $140?
9-7
'
9-8
Preferred stock valuation Ezzell Corporation issued perpetual preferred stock with a
10 percent annual dividend. The stock currently yields 8 percent, and its par value is $100.
a. What is the stock's value?
b. Suppose interest rates rise and pull the preferred stocks yield up to 12 percent.
What would be its new market value?
9-9
Preferred stock returns Bruner Aeronautics has perpetual preferred stock outstanding with
a par value of 5100. The stock pays a quarterly dividend of $2, and its current price is $80.
a. What is its nominal annual rate of return?
b. What is its effective annual rate of return?
$-`(E`i
Valuation of a declining growth stock Martell Mining Company's ore reserves are being
depleted, so its sales are falling. Also, its pit is getting deeper each year, so its costs are rising. As a result, the company's earnings and dividends are declining at the constant rate of
5 percent per year. If Do _$5 and r$ = 15%, what is the value of Martell Mining's stock?
9-11
Valuation of a constant growth stock A stock is expected to pay a dividend of 50.50 at
the end of the year ( that is, Dj = 0.50), and it should continue to grow at a constant rate
of 7 percent a year. If its required return. is 12 percent, what is the stock's expected price
4 years from today?
9-12
Valuation of a constant growth stock Investors require a 15 percent rate of return on
Levine Company's stock (that is, rs = 15%).
a. What is its value if the previous dividend was D. = $2 and investors expect dividends to grow at a constant annual rate of (1) -5 percent, ( 2) 0 percent, (3) 5 percent, or (4) lll percent?
b.
c.
Using data from part a, what would the Gordon (constant growth) model. value be
if the required rate of return were 15 percent and the expected growth rate were
( 1) 15 percent or (2) 20 percent? Are these reasonable results? Explain.
Is it reasonable to think that a constant growth stock could have g> rs?
9-13
Rates of return and equilibrium Stock C's beta coefficient is bc = 0.4, while Stock D's is
bD = -0.5. (Stock D's beta is negative, indicating that its return rises when returns on
most other stocks fall. There are very few negative beta stocks, although collection
agency stocks are sometimes cited as an example.)
a. If the risk-free rate is 7 percent and the expected rate of return on an average stock
is 11 percent, what are the required rates of return on Stocks C and D?
b. For Stock C, suppose the current price, P,„ is $25; the next expected dividend, D,, is
$1.50; and the stock's expected constant growth rate is 4 percent. Is the stock in equilibrium? Explain, and describe what would happen if the stock is not in equilibrium.
9-14
Constant growth You are considering an investment in Keller Corp's stock, which is
expected to pay a dividend of $2 a share at the end of the year (D, = $2.00) and has a
beta of 0.9. The risk-free rate is 5.6 percent, and the market risk premium is 6 percent.
Keller currently sells for $25 a share, and its dividend is expected to orow at some constant
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Part 3 Financial Assets
rate g. Assuming the market is in equilibrium, what does the market believe will be the
stock price at the end of 3 years? (That is, what is P3?)
9-15
Equilibrium stock price The risk-free rate of return, ry:, is 6 percent; the required rate of return on the market, r^,A, is 10 percent; and Upton Company's stock has a beta coefficient of 1.5.
a. If the dividend expected during the coming year, D1, is $2.25, and if g= a constant
5 percent, at what price should Upton's stock sell?
b. Now, suppose the Federal Reserve Board increases the money supply, causing the
risk-free rate to drop to 5 percent and r;a to fall to 9 percent. What would happen to
Upton's price?
c. In addition to the change in part b, suppose investors' risk aversion declines, and
this, combined with the decline in r", causes rM to fall to 8 percent. Now, what is
Upton's price?
d. Now suppose Upton has a change in management. The new group institutes policies that increase the expected constant growth rate from 5 to 6 percent. Also, the
new management smooths out fluctuations in sales and profits, causing beta to
decline from 1.5 to 1.:3. Assume that rRF and r,_, are equal to the values in part c.
^.:
After all these changes, what is its new equilibrium price? (Note: D, is now $2.27.)
Nonconstant growth M.icrotech Corporation is expanding rapidly and currently needs to
retain all of its earnings, hence it does not pay dividends- However, investors expect
Microtech to begin paying dividends, beginning with a dividend of $1.00 coming 3 years
from today. The dividend should grow rapidly-at a rate of 50 percent per year-during
Years 4 and 5, but after Year 5 growth should be a constant 8 percent per year. If the
required return on Microtech is 15 percent, what is the value of the stock today?
17•1 1 7
Corporate valuation Dozier Corporation is a fast-growing supplier of office products.
Analysts project the following free cash flows (FCFs) during the next 3 years, after which
FCF is expected to grow at a constant 7 percent rate. Dozier's WACC is 13 percent.
Year
FCF ($ millions)
a.
b.
c.
Challenging
Problems 1 B-24
0
1
2
3
NA
-$20
$30
$40
What is Dozier's terminal, or horizon, value? (Hint: Find the value of all free cash
flows beyond Year 3 discounted back to Year 3.)
What is the firm's value today?
Suppose Dozier has $100 million of debt and 10 million shares of stock outstanding.
What is your estimate of the price per share?
9-IS
Nonconstant growth Mitts Cosmetics Co.'s stock price is $58.88, and it recently paid a
$2 dividend. This dividend is expected to grow by 25 percent for the next 3 years, and
then grow forever at a constant rate, g, and r^, = 12%. At what constant rate is the stock
expected to grow after Year 3?
9-11?
Constant growth Your broker offers to sell you some shares of Bahnsen & Co. common
stock that paid a dividend of $2 yesterday. Bahnsen's dividend is expected to grow at
5 percent per year for the next 3 years, and, if you buy the stock, you plan to hold it for
3 years and then sell it. The appropriate discount rate is 12 percent.
a. Find the expected dividend for each of the next 3 years; that is, calculate D„ D„ and
D3. Note that Do = $2.00.
b. Given that the first dividend payment will occur 1 year from now, find the present
value of the dividend stream; that is, calculate the PV of D;, D,, and D;, and then
sum these PVs.
c. You expect the price of the stock 3 years from now to be $34.73; that is, you expect
I3 to equal $34.73. Discounted at a 12 percent rate, what is the present value of this
expected future stock price? In other words, calculate the PV of $34.73.
d_ If you plan to buy the stock, hold it for 3 years, and then sell it for $34.73, what is
the most you should pay for it today?
e. Use Equation 9-2 to calculate the present value of this stock. Assume that g = 5%,
and it is constant.
f. Is the value of this stock dependent upon how long you plan to hold it? In
other words, if your planned holding period were 2 years or 5 years rather than
3 years, would this affect the value of the stock today, P,)? Explain.
9-20
Nonconstant growth stock valuation Taussig Technologies Corporation (TTC) has been
growing at a rate of 20 percent per year in recent years. This same growth rate is
expected to last for another 2 years, then to decline to g„ = 6%.
. ,^
327
Chapter 9 Stocks and Their Valuation
If Da = $1.60 and rb = 10%, what is TTC's stock worth today? What are its expected
dividend and capital gains yields at this time, that is, during Year 1.?
b. Now assume that TT'C's period of supernormal growth is to last for. 5 years rather
than 2 years. How would this affect the price, dividend yield, and capital gains
yield? Answer in words only.
c. What will TTC's dividend and capital gains yields be once its period of supernormal
growth ends? (Hint: These values will be the same regardless of whether you examine the case of 2 or 5 years of supernormal growth; the calculations are very easy.)
d.- Of what interest to investors is the changi.ng relationship between dividend and
capital gains yields over time?
a.
9-21
Corporate value model Barrett Industries invests a lot of money in R&D, and as a result
it retains and reinvests all of its earnings. In other words, Barrett does not pay any dividends, and it has no plans to pay dividends in the near future. A major pension fund is
interested in purchasing Barrett's stock. The pension fund manager has estimated
Barrett's free cash flows for the next 4 years as follows: $3 million, $6 million, $10 million, and $15 million. After the 4th year, he cash flow is projected to grow at a constant
7 percent. Barrett's WACC is 12 percent, its debt and preferred stock total to $60 million,
and it has 10 million shares of common stock outstanding.
a. What is the present value of the free cash flows projected during the next 4 years?
b. Whatis the firm's terminal value?
c. What is the firm's total value today?
d. What is an estimate of Barrett's price per share?
9-22
Corporate value model Assume that today is December 31, 2005, and the following
infonnation applies to Vermeil Airline.,,:
• After-tax operating income tEBIT(1 - T), also called NOPAT) for 2006 is expected to
be $500 million.
•
•
•
The depreciation expense for 2006 is expected to be 5100 million.
The capital expenditures for 2006 are expected. to be $200 million.
No change is expected in net operating working capital.
• The free cash flow is expected to grow at a constant rate of 6 percent per year.
• The required return on equity is 14 percent.
• The WACC is 10 percenta The market value of the company's debt is $3 billion.
• 200 million shares of stock are outstanding.
Using the free cash flow approach, what should the company's stock price be today?
9-23
Beta coefficients Suppose Chance Chemical Company's management conducted a study
and concluded that if it expands its consumer products division (which is less risky than
its primary business, industrial chemicals), its beta would decline from 1.2 to 0.9. However, consumer products have a somewhat lower profit margin, and this would cause its
constant growth rate in earnings and dividends to fall from 6 percent to 4 percent. The
following also apply: rM = 9%; rRF = 617c; and Do = $2.00.
a. Should management expand the consumer products division?
b. Assume all the facts as given above except the change in the beta coefficient. How
low would the beta have to fall to cause the expansion to be a good one? (Hint: Set
1''0 under the new policy equal to lga under the old one, and find the new beta that
will produce this equality.)
9-24
Nonconstant growth Assume that it is now January 1, 2006. WraynelMartin Electric Inc.
(WME) has just developed a solar panel capable of generating 200 percent more electricity than any solar panel currently on the market. As a result, WME is expected to experience a 15 percent annual growth rate for the next 5 years. By the end of 5 years, other
firms will have developed comparable technology, and WME's growth rate will slow to
5 percent per year indefinitely. Stockholders require a return of 12 percent on WME's stock.
The most recent annual dividend (Da), which was paid yesterday, was $1.75 per share,
a. Calculate WTML"'s expected dividends for 2006, 2007, 2008, 2009, and 2010.
b. Calculate the value of the stock today, P. Proceed by finding the present value of
the dividends expected at the end of 2006, 2007, 2008, 2009, and 2010 plus the present value of the stock price that should exist at the end of 2010. The year-end 2010
stock price can be found by using the constant growth equation. Notice that to find
the December 31, 2010, price, you must use the dividend expected in 2011, which is
5 percent greater than the 2010 dividend.
c. Calculate the expected dividend yield, Dt/l a, capital gains yield, and. total return
(dividend yield plus capital gains yield) expected for 2006. (Assume that f'o= Po, and
328
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322, ^
Part 3 Financial Assets
e.
recognize that the capital gains yield is equal to the total return minus the dividend
yield.) Then calculate these same three yields for 2011.
How might an investor's tax situation affect his or her decision to purchase stocks of
companies in the early stages of their lives, when they are growing rapidly, versus
stocks of older, more mature firms? When does WME's stock become "mature" for
purposes of this question?
Suppose your boss tells you she believes that b'JME's annual growth rate will be
only 12 percent during the next 5 years and that the firm's long-run growth rate will
be only 4 percent. Without doing any calculations, what general effect would these
growth-rate changes have on the price of WME's stock?
Suppose your boss also tells you that she regards WME as being quite risky and that
she believes the required rate of return should be 14 percent, not 12 percent. Without
doing any calculations, how would the higher required rate of return affect the price
of the stock, the capital gains yield, and the dividend yield? Again, assume that the
long-run growth rate is 4 percent.
.`^
COMPR-E]EiENSTVE/SPItEADSPIEET
PROBLEM
9-25
{
Nonconstant growth and corporate valuation Rework Problem 9-20, parts a, b, and c,
using a spreadsheet model. For part b, calculate the price, dividend yield, and capital
gains yield as called for in the problem. After completing parts a through c, answer the
following additional question using the spreadsheet model.
d. 'ITC recently introduced a new line of products that has been wildly successful. On
the basis of this success and anticipated future success, the following free cash flows
were projected:
Year
1
FCF
$5.5
2
$121
3
4
5
S23.8
$44.1
$69.0
6
7
S8B.8 $107.5
8
9
10
$128.9
$147.1
$169.3
t` ^t
After the 10th year, TTC's financial planners anticipate that its free cash flow will grow
at a constant rate of 6 percent. Also, the firm concluded that the new product caused the
WACC to fall to 9 percent. The market value of TTC's debt is $1,200 million, it uses no
preferred stock, and there are 20 million shares of common stock outstanding. Use the
free cash flow method to value the stock.
Integrated Case
Mutual of Chicago Insurance Company
9-26
Stock valuation Robert Balik and Carol Kiefer are senior vice presidents of the Mutual of Chicago Insurance
Company. They are co-directors of the company's pension fund management division, with Balik having
responsibility for fixed income securities (primarily bonds) and Kiefer being responsible for equity investments. A major new client, the California League of Cities, has requested that Mutual of Chicago present an
investment seminar to the mayors of the represented cities, and Balik and Kiefer, who will make the actual
presentation, have asked you to help them.
To illustrate the common stock valuation process, Balik and Kiefer have asked you to analyze the Bon
Temps Company, an employment agency that supplies word-processor operators and computer programmers
to businesses with temporarily heavy workloads. You are to answer the following questions:
329
:^„
Chapter 9 Stocks and Their Valuation
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k'
k.
i,
I.
in,
1
323
Describe briefly the legal rights and privileges of common stockholders.
(1) Write out a formula that can be used to value any stock, regardless of its dividend pattern.
(2) What is a constant growth stock? How are constant growth stocks valued?
(3) What are the implications if a company forecasts a constant g that exceeds its rK? Mill many stocks
have expected g> r,s in the short run (that is, for the next few years)? In the long run (that is, forever)?
Assume that Bon Temps has a beta coefficient of 1.2, that the risk-free rate (the yield on T -bonds) is 7 percent, and that the required rate of return on the market is 12 percent: What is on Temps' required rate of
return?
Assume that Bon Temps is a constant growth company whose last dividend (Da, which was paid yesterday) was $2.00 and whose dividend is expected to grow indefinitely: at a 6 percent .rate.
(1) What is the firm's expected dividend stream over the next 3 years?
(?) What is its current stock price?
(3) 1•V hat is the stock's expected %.-alue 1 year from now?
(4) What are the expected dividend yield, capital gains yield, and total. return during the first year?
Now assume that the stock is currently selling at $30.29. What is its expected rate of return?
What would the stock price be if its dividends were expected to have zero growth?
Now assume that Bon Temps is expected to experience nornconstant gm-wth of 30 percent for the next 3
years, then to return to its long rur+ constant growth rate of 6 pezcent. Mat is the stock's value under
these conditions? What are its expected. dividend and capital gains yields in Year 1? Year 4?
Suppose Bon Temps is expected to experience zero growth during the,ftist 3)>ears and then to resume its
steady-state growth of 6 percent in the fourth year. What would its value be th.eii? What would its
expected dividend and capital gains yields be in Year 1? InYear:4Z
Finally, assume_ that Bon Temps' earnings and dividends are expected to decline:at a constant rate of 6
percent per }ea.r, that is, g= -b r°6. Why would anyone be willing to buy such a stock, and at what price
should it sel]? 'What would be its dividend and capital gains yields iii each year?
5uppn4 Ban Temps embarked on an aggressive expansion that requires additiiinaT capital. Management
decided to finance the expansion by borrowing $40 million and by halting dividbnd payments to increase
retained earnings. Its WACC is now 10 percent, and the projected free cash, flows, for the next 3 years are
-S.3 milliona $10 -million, and $20 million. After Year 3, free cash flow is projected to -groW at a constant 6
perrent: What is Bon Tem ps' total value? If it has 10 million shares of stock- and $40 'million of total. debt,
what is the price per share?
For Bon Temps' stock to be in equilibrium, what relationship must exist between its estimated intrinsic
value and.its-rurrent stock price and between its expected and required rates of,^return? Are the equi.librium intiincic value and expected rate of return the values that management estimates or values as estimnatedby sonie other entity? Explain.
if equlibriumdoes
not exi.st, how r, ill it be established?
i
Suppose Bon Tenips decided to issue preferred stock that would pay an'annualdividend of $5, and the
issue price was $50 per share. What would the expected return be on this stock? Would the expected rate
of return be the same if the preferred was a perpetual issue or if it had a 20-year maturity?
Please go to the ThamsanNOW Web site to access the
Cyberproblems.
cyberproblem
330
324
Part 3 Financial Assets
i.^
;,.
Access the Thomson ONE problems though the ThomsonNOW Web site.. Use the Thomson
ONE-Business School Edition online database to work this chapter's questions.
iY
Estimating ExxonMobll's Intrinsic Stock "Value
Ws^ _
In this chapter we described the various factors that influence stock prices and the
approaches that analysts use to estimate a stock's intrinsic value. By comparing these
intrinsic value estimates to the current price, an investor can assess whether it makes
sense to buy or sell a particular stock. Stocks trading at a price far below their estimated
intrinsic values may be good candidates for purchase, whereas stocks trading at prices far
in excess of their intrinsic value may be good stocks to avoid or sell.
While estimating a stock's intrinsic value is a complex exercise that requires reliable
data and good judgment, we can use the data available in Thomson One to arrive at a
quick "back of the envelope" calculation of intrinsic value.
Discussion Questions
1. For purposes of this exercise, let's take a closer look at the stock of ExxonMobil Corporation (XOM). Looking at the COMPANY OVERVIEW we can immediately see the
company's current stock price and its performance relative to the overall market in
recent months. What is ExxonMobil's current stock price? How has the stock performed relative to the market over the past few months?
2. Click on the "NEWS" tab to see the recent news stories for the company. Have there
been any recent events impacting the company's stock price, or have things been relatively quiet?
3. To provide a starting point for gauging a company's relative valuation, analysts often
look at a company's price-to-earnings (P/E) ratio. Returning to the COMPANY
OVERVIEW page, you can see XOM's current P/E ratio. To put this number in perspective, it is useful to compare this ratio with other companies in the same industry
and to take a look at how this ratio has changed over time. If you want to see how
XOM's P/E ratio stacks up to its peers, click on the tab labeled PEERS. Click on
FINANCIALS on the next row of tabs and then select KEY FINANCIAL RATIOS.
Toward the bottom of the table you should see information on the P/E ratio in the
section titled Market Value Ratios. Toward the top, you should see an item where it
says CLICK HERE TO SELECT NEW PEER SET-do this if you want to compare XOM
to a different set of firms. For the most part, is XOM's P/E ratio above or below that
of its peers? In Chapter 4, we discussed the various factors that may influence P/E
ratios. Off the top of your head, can these factors explain why XOM's P/E ratio differs
from its peers?
4. Now to see how XOM's P/E ratio has varied over time-return back to the COMPANY
OVERVIEW page. Next click FINANCIALS-GROWTH RATIOS and then select
WORLDSCOPE-INCOME STATEMENT RATIOS. Is XOM's current P/E ratio well
above or well below its historical average? If so, do you have any explanation for why
the current P/E deviates from its historical trend? On the basis of this information,
does XOM's current P/E suggest that the stock is undervalued or overvalued? Explain.
5. In the text, we discussed using the dividend growth model to estimate a stock's intrinsic value. To keep things as simple as possible, let's assume at first that XOM's dividend is expected to grow at some constant rate over time. If so, the intrinsic value
equals Dl/(rs - g), where D, is the expected annual dividend 1 year from now, rs is the
stock's required rate of return, and g is the dividend's constant growth rate. To estimate the dividend growth rate, it's first helpful to look at XOM's dividend history.
331
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Chapter 9 Stocks and Their Valuation
Staying on the current Web page (WORLDSCOPE-INCOME STATEMENT RATIOS)
you should immediately find the company's annual dividend over the past several
years. On the basis of this information, what has been the average annual dividend
growth rate? Another way to get estimates of dividend growth rates is to look at analysts' forecasts for future dividends, which can be found on the ESTIMATES tab.
Scrolling down the page you should see an area marked "Consensus Estimates" and
a tab under "Available Measures." Here you click on the down arrow key and select
Dividends Per Share (DPS). What is the median year-end dividend forecast? You can
use this as an estimate of D; in your measure of intrinsic value. You can also use this
forecast along with the historical data to arrive at a measure of the forecasted dividend growth rate, g.
6. The required return on equity, rs, is the final input needed to estimate intrinsic value.
For our purposes you can either assume a number (say, 8 or 9 percent), or you can
use the CAPM to calculate an estimate of the cost of equity using the data available
in Thomson One. (For more details take a look at the Thomson One exercise for
Chapter 8). Having decided on your best estimates for D,, rs, and g, you can calculate
XOM's intrinsic value. How does this estimate compare with the current stock price?
Does your preliminary analysis suggest that XOM is undervalued or overvalued?
Explain.
7. It is often useful to perform a sensitivity analysis, where you show how your estimate
of intrinsic value varies according to different estimates of DI, r, and g. To do so,
recalculate your intrinsic value estimate for a range of different estimates for each of
these key inputs. One convenient way to do this is to set up a simple data table in
Excel. Refer to the Excel tutorial accessed through the ThomsonNOW Web site for
instructions on data tables. On the basis of this analysis, what inputs justify the current
stock price?
8. On the basis of the dividend history you uncovered in question 5 and your assessment of XOM's future dividend payout policies, do you think it is reasonable to
assume that the constant growth model is a good proxy for intrinsic value? If not, how
would you use the available data in Thomson One to estimate intrinsic value using
the nonconstant growth model?
9. Finally, you can also use the information in Thomson One to value the entire corporation. This approach requires that you estimate XOM's annual free cash flows. Once
you estimate the value of the entire corporation, you subtract the value of debt and
preferred stock to arrive at an estimate of the company's equity value. Divide this
number by the number of shares of common stock outstanding, and you calculate an
alternative estimate of the stock's intrinsic value. While this approach may take some
more time and involves more judgment concerning forecasts of future free cash flows,
you can use the financial statements and growth forecasts in Thomson One as useful
starting points. Go to Worldscope's Cash Flow Ratios Report (which you find by clicking on FINANCIALS, FUNDAMENTAL RATIOS, and WORLDSCOPE RATIOS) and you
will find an estimate of "free cash flow per share." While this number is useful, Worldscope's definition of free cash flow subtracts out dividends per share; therefore, to
make it comparable to the measure in this text, you must add back dividends. To see
Worldscope's definition of free cash flow (or any term), click on SEARCH FOR COMPANIES from the left toolbar, and select the ADVANCED SEARCH tab. In the middle
of your screen, on the right-hand side, you will see a dialog box with terms. Use the
down arrow to scroll through the terms, highlighting the term for which you would
like to see a definition, Then, click on the DEFINITION button immediately below the
dialog box.
A
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f :.
-.-
-
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333
The Market Risk Premium:
Expectational Estimates Using
Analysts' Forecasts
Robert S. Harris and Felicia C. Marston
Using expectationat data from financial analysis, we estimate a market risk premium for US stocks.
Using the ScPcP 500 as a proxy for the market portfolio, the average market risk premium is found 10 be
7.14"/o above yields on long-term US go vernment bonds over the period 1982-1998. This risk premium
varies over time; much of this variation can be explained by either the level of interest rates or readily
available forward-looking proxies for risk The market risk premium appears to move inversely with
government interest rates suggesting that required returns on stocks are more stable than interest
rates themselves. [JEL: G31, G12]
RI'he notion of a market risk premium (the spread
between investor required returns on safe and average
risk assets) has long played a central role in finance. It
is a key factor in asset allocation decisions to determine
the portfolio mix of debt and equity instruments.
Moreover, the market risk premium plays a critical role
in the Capital Asset Pricing Model (CAPM), the most
widely used means of estimating equity hurdle rates by
practitioners. In recent years, the practical significance
of estimating such a market premium has increased as
firms, financial analysts, and investors employ financial
frameworks to analyze corporate and investment
performance. For instance, the increased use of
Economic Value Added (EVA®) to assess corporate
performance has provided a new impetus for estimating
capital costs.
•
The most prevalent approach to estimating the market
risk premium relies on some average of the historical
spread between returns on stocks and bonds.' This
'Roben S. Harris is the C. Stewart Sheppard Professor of Business
Administration and Felicia C. Marston is an Associate Professor
at the University or Virginia, Charlottesville, VA 22906.
The authors thank Erik Benrud, an anonymous reviewer, and
seminar participants at the University of Virginia, the
University of Connecticut and at the SEC for comments.
Thanks to Darden Sponsors, TVA, the Walker Family Fund,
and McIntire Associates for support of this research and to
IBES, Inc. for supplying data.
choice has some appealing characteristics but is
subject to many arbitrary assumptions such as the
relevant period for taking an average. Compounding
the difficulty of using historical returns is the well
noted fact that standard models of consumer choice
would predict much lower spreads between equity and
debt returns than have occurred in US markets-the
so called equity risk premium puzzle (see Welch, 2000
and Siegel and Thaler, 1997). In addition, theory calls
for a forward-looking risk premium that could well
change over time.
This paper takes an alternate approach by using
expectational data to estimate the market risk premium.
The approach has two major advantages for
practitioners. First, it provides an independent
estimate that can be compared to historical averages.
At a minimum, this can help in understanding likely
ranges for risk premia. Second, expectational data allow
investigation of changes in risk prernia over time. Such
time variations in risk premia serve as important signals
from investors that should affect a host of financial
decisions. This paper provides new tests of whether
changes in risk premia over time are linked to forwardlooking measures of risk. Specifically, we look at the
'Bruner, Eades, Harris, and Higgins (1998) provide survey
evidence on both textbook advice and practitioner methods
for estimating capital costs. As testament to the market for
cost of capital estimates, Ibbotson Associates (1998) publishes
a "Cost of Capital Quarterly."
334
HARRIS & MARSTOAI-THE MARKET RISK PREMIUM
relationship between the risk premium and four exante measures of risk: the spread between yields on
corpora•te and government bonds, consumer sentiment
about future economic conditions, the average level
of dispersion across analysts as they forecast
corporate earnings, and the implied volatility on the
S&P500 Index derived from options data.
Section I provides background on the estimation of
equity required returns and a brief discussion of
current practice in estimating the market risk premium.
In Section II, models and data are discussed. Following
a comparison of the results to historical returns in
Section III, we examine the time-series characteristics
of the estimated market premium in Section IV. Finally,
conclusions are offered in Section V.
et al. (1998) point out, few respondents cited use of
expectational data to supplement or replace historical
returns in estimating the market premium.
Survey evidence also shows substantial variation
in empirical estimates. When respondents gave a
precise estimate of-the market premium, they cited
figures from 4% to over 7% (Bruner et al., 1998). A
quote from a survey respondent highlights the range
in practice. "In 1993, we polled various investment
banks and academic studies on the issue as to the
appropriate rate and got anywhere between 2 and 8%,
but most were between 6% and 7.4%." (Bruner et al.,
1998). An informal sampling of current practice also
reveals large differences in assumptions about an
appropriate market premium. For instance, in a 1999
application of EVA analysis, Goldman Sachs
1. Background
Investment Research specifies a market risk premium
of"3% from 1994-1997 and 3.5% from 1998-1999E for
The notion of a "market" required rate of return is a the S&P Industrials" (Goldman Sachs, 1999). At the
convenient and widely used construct. Such a rate (k)
same time, an April 1999 phone call to Stem Stewart
is the minimum level of expected return necessary to
revealed that their own application of EVA typically
compensate investors for bearing the average risk of
employed a market risk premium of 6°/a. In its application
equity investments and receiving dollars in the future
of the CAPM, lbbotson Associates ( 1998) uses a market
rather than in the present. In general, k will depend on
risk premium of7.8%. Not surprisingly, academics do not
returns available on alternative investments (e.g.,
agree on the risk premium either. Welch (2000) surveyed
bonds). To isolate the effects of risk, it is useful to
leading financial economists at major universities. For a
work in terms of a market risk premium (rp), defined as
30-year horizon, he found a mean risk premium of 7.1%
but a range from 1.5% to 15% with an interquartile range
(1)
rp = k-i,
of 2.4% (based on 226 responses).
To provide additional insight on estimates of the
where i= required return fora zero risk investment.
market premium, we use publicly available
Lacking a superior alternative, investigators often
expectational data. This expectational approach
use averages of historical realizations to estimate a
employs the dividend growth model (hereafter referred
market risk premium. Bruner, Eades, Harris, and Higgins
to as the discounted cash flow (DCF) model) in which
(1998) provide recent-survey results on best practices
a consensus measure of financial analysts' forecasts
by corporations and financial advisors. While almost
(FAF) of earnings is used as a proxy for investor
all respondents used some average of past data in
expectations. Earlier work has used FAF in DCF models=
estimating a market risk premium, a wide range of but generally has covered a span of only a few years
approaches emerged. "While most of our 27 sample due to data availability.
companies appear to use a 60+ year historical period
to estimate returns, one cited a window of less than
ten years, two cited windows of about ten years, one
began averaging with 1960, and another with 1952 data"
(p. 22). Some used arithmetic averages, and some used
geotri'etric. This historical approach requires the
assumptions that past realizations are a good surrogate
for future expectations and, as typically applied, that
the risk premium is constant over time. Carleton and
Lakonishok (I985) demonstrate empirically some of the
problems with such historical premia when they are
disaggregated for different time periods or groups of
firms. Siegel (1999) cites additional problems of using
historical returns and argues that equity premium
estimates from past data are likely too high. As Bruner .
H. Models and Data
The simplest and most commonly used version of
the DCF model is employed to estimate shareholders'
required rate of return, k, as shown in Equation (2):
'See Malkiel (1982), Brigham, Vinson, and Shome (1985),
Harris (1986), and Harris and Marston (1992). The DCF
approach with analysts' forecasts has been used frequently in
regulatory settings. Ibbotson Associates (1998) use a variant
or the DCF model with forward-looking growth rates; however,
they do this as a separate technique and not as part of the
CAPM. For their CAPM estimates, they use historical averages
for the market risk premium.
335
.1;
^y .
,.^
t, .
Y.
JOURNAL OF APPLIED FINANCE- 2001
k = p` + g,
k 0
(2)
where Dt= dividend per share expected to be received
at time one, Pa = current price per share (time 0), and g
= expected growth rate in dividends per share? A
primary difficulty in using the DCF model is obtaining
an estimate of g, since it should reflect market
expectations of future performance. This paper uses
published FAF of long-run growth in earnings as a
proxy for g. Equation (2) can be applied for an
individual stock or any portfolio of companies. We
focus primarily on its application to estimate a market
premium as proxied by the S&P500.
FAF comes from IBES Inc. The mean value of
individual analysts' forecasts of five-year growth rate
in EPS is used as the estimate of gin the DCF model.
The five-year horizon is the longest horizon over which
such forecasts are available from IBES and often is the
longest horizon used by analysts. IBES requests
"normalized" five-year growth rates from analysts in
order to remove short-term distortions that might stem
from using an unusually high or low earnings year as
a base. Growth rates are available on a monthly basis.
Dividend and other firm-specific information come
from COMPUSTAT. D, is estimated as the current
indicated annual dividend times (1+g). Interest rates
(both government and corporate) are from Federal
Reserve Bulletins and Moody's Bond Record. Exhibit I
describes key variables used in the study. Data are
used for all stocks in the Standard and Poor 's 500
stock (S&P500) index followed by IBES. Since fiveyear growth rates are first available from IBES beginning
in 1982, the analysis covers the period from January
1982-December 1998.
The approach used is generally the same approach
as used in Harris and Marston (1992). For each month,
'Our methods follow Harris (1986) and Harris and Marston
(1992) who discuss earlier research and the approach employed
here, including comparisons of single versus multistage growth
models. Since analysis' forecast growth in earnings per share,
t heir projections should incorporate the anticipated effects of
here repurchase programs. Dividends per share would grow at
the same rate as EPS as long as companies manage a constant
ratio of dividends to earnings on a per sharp basis. Based an
S&P500 figures (see the Standard and Poor's website for their
procedures), the ratio or DPS to EPS was .5I during the period
1982-89 and .52 for the period 1990-98. Lamdin (2001)
discusses some issues if share repurchases destroy the
equivalence of EPS and DPS growth rates. Theoretically, i is a
risk-free rate, though its empirical proxy is only a"least risk"
alternative that is it'self subject to risk. For instance, Asness
(2000) shows that over the 1946-1998 period, bond volatility
(in monthly realized returns) has increased relative to stock
volatility, which would be consistent with a drop in the equity
market premium.
a market required rate of return is calculated using
each dividend-paying stock in the S&P500 index for
which data are available. As additional screens for
reliability of data, in a given month we eliminate a firm
if there are fewer than three analysts' forecasts or if
the standard deviation around the mean forecast
exceeds 20%. Combined, these two screens eliminate
fewer than 20 stocks a month. Later we report on the
sensitivity of the results to various screens. The DCF
model in Equation (2) is applied to each stock and the
results weighted by market value of equity to produce
the market-required return. The risk premium is
constructed by subtracting the interest rate on
government bonds.
We weighted 1998 results by year-end 1997 market
values since the monthly data on market value did not
extend through this period. Since data on firm-specific
dividend yields were not available for the last four
months of 1998 at the time of this study, the market
dividend yield for these months was estimated using
the dividend yield reported in the Wall Street Journal
scaled by the average ratio of this figure to the
dividend yield for our sample as calculated in the first
eight months of 1998. Adjustments were then made
using growth rates from 1BES to calculate the market
required return. We also estimated results using an
average dividend yield for the month that employed
the average of the price at the end of the current and
prior months. These average dividend yield measures
led to similar regression coefficients as those reported
later in the paper.
For short-term horizons (quarterly and annual), past
research (Brown, 1993) finds that on average analysts'
forecasts are overly optimistic compared to
realizations. However, recent research on quarterly
horizons (Brown, 1997) suggests that analysts'
forecasts for S&P500 firms do not have an optimistic
bias for the period 1993-1996. There is very little
research on the properties of five-year growth
forecasts, as opposed to shorter horizon predictions.
Boebel (1991) and Boebel, Harris, and Gultekin (1993)
examine possible bias in analysts' five-year growth
rates. These studies find evidence of optimism in IBES
growth forecasts. In the most thorough study to date,
Boebel (1991) reports that this bias seems to be getting
smaller over time. His forecast data do not extend into
the 1990s.
Analysts' optimism, if any, is not necessarily a
problem for the analysis in this paper. If investors share
analysts' views, our procedures will still yield
unbiased estimates of required returns and risk premia.
In light of the possible bias, however, we interpret the
estimates as "upper bounds" for the market premium.
This study also uses four very different sources to
create ex ante measures of equity risk at the market
336
HARRIS & MARSTON-THE MARKET RISK PREMIUM
Exhibit 1. Variable Definitions
k
=
Equity required rate return.
Po
=
Price per share.
DI
=
Expected dividend per share measured as current indicated annual
dividend from COMPUSTAT multiplied by (I + g).
Average financial analysts' forecast of five-year growth rate in earnings
per share (from IBES).
Yield to maturity on long-term US government obligations (source:
Federal Reserve, 30-year constant maturity series).
rp
=
Equity risk premium calculated as rp = k - I.
BSPREAD
=
spread between yields on corporate and government bonds, BSPREAD =
yield to maturity on long-term corporate bonds (Moody's average across bond rating categories)
minus i.
CON
=
Monthly consumer confidence index reported by the Conference Board
(divided by 100).
DJSP
=
Dispersion of analysts' forecasts at the market level.
VOL
=
Volatility for the S+P500 index as implied by options data.
level. The first proxy comes from the bond market and
is calculated as the spread between corporate and
government bond yields (BSPREAD). The rationale is
that increases in this spread signal investors'
perceptions of increased riskiness of corporate activity
that would be translated to both debt and equity
owners. The second measure, CON, is the consumer
confidence index reported by the Conference Board at
the end of the month. While the reported index tends
to be around 100, we rescale CON as the actual index
divided by 100. We also examined use of CON as of
the end of the prior month; however, in regression
analysis, this lagged measure generally was not
statistically. significant in explaining the level of the
market risk premium.' The third measure, DISP,
measures the dispersion of analysts' forecasts. Such
analyst disagreement should be positively related to
perceived risk since higher levels of uncertainty would
likely #enerate a wider distribution of earnings
forecasts for a given firm. DISP is calculated as the
average of firm-specific standard deviations for each
stock in the S&P500 covered by IBES. The firm-specific
standard deviation is calculated based on the
dispersion of individual analysts' growth forecasts
'We examined two other proxies for Consumer Confidence.
The Conference Board's Consumer Expectations Index yielded
essentially the same results as those reported. The University
or Michigan's Consumer Sentiment Indices tended to be less
significantly linked to the market risk premium, though
coefficients were still negative.
around the mean of individual forecasts for that
company in that month. DISP also was estimated using
a value-weighted measure of analyst dispersion for
the firms in our sample. The results reported use the
equally weighted version but similar patterns were
obtained with both constructions.' Our final measure,
VOL, is the implied volatility on the S&P500 index. As
of the beginning of the month, a dividend-adjusted
Black Scholes formula is used to estimate the implied
volatility in the S&P500 index option contract, which
expires on the third Friday of the month. The call
premium, exercise price, and the level of the S&P500
index are taken from the Wall Street Journal, and
treasury yields come frotn'the Federal Reserve.
Dividend yield comes from DRI. The option contract
that is closest to being at the money is used.
Ill. Estimates of the Market Premium
Exhibit 2 reports both required returns and risk
premia by year (averages of monthly data). The
estimated risk premia are positive, consistent with
equity owners demanding additional rewards over and
above returns on debt securities. The average
expectational risk premium (1982 to 1998) over
)For the regressions reported in Exhibit 6, the valueweighted dispersion measure actually exhibited more
explanatory power. For regressions using the Prais-Winsten
method (see footnote 7), the coefficient on DISP was not
significant in 2 of the 4 cases.
337
;.
JOURNAL OF APPLIED FINANCE - 2001
10
Exhibit 2. Bond Market Yields, Equity Required Return, and Equity Risk Premium, 1982-1998
Values are averages of monthly figures in percent. i is the yield to maturity on long-term government bonds, k is the required return
on the S&P500 estimated as a value weighted average using a discounted cash flow model with analysts' growth forecasts. The risk
Div yield is expected dividend per share divided
premium rp = k- i. The average of analysts' growth forecasts is g.
by price per share.
Year
Div. Yield
k
g
F
rp = k- i
1982
6.89
12.73
19.62
12.76
6.86
1983
5.24
12.60
17.86
11.18
6.67
1984
5.55
12.02
17.57
12.39
5.18
1985
4.97
11.45
16.42
10.79
5.63
1986
4.08
11.05
15.13
7.80
7.34
1967
3.64
11.01
14.65
8.58
6.07
1988
4.27
11.00
15.27
8.96
6.31
1989
3.95
11.09
15.03
8.45
6.58
1990
4.03
11.69
15.72
8.61
7.11
1991
3.64
11.99
15.63
8.14
7.50
1992
3.35
12.13
15.47
7.67
7.81
1993
3.15
11.63
14.7B
6.60
8.18
1994
3.19
11.47
14.66
7.37
7.29
1995
3.04
11.51
14.55
6.88
7.67
1996
2.60
11.89
14.49
6.70
7.79
1997
2.18
12.60
14.78
6.60
8.17
1998
1.LO
12.95
14,75
5.58
.17
Average
3.86
11.81
15.67
8.53
7.14
-government bonds is 7.14%, slightly higher than the
6.47% average for 1982 to 1991 reported by Harris and
Marston (1992). For comparison purposes, Exhibit 3
/ contains historical returns and risk premia. The average
expectational risk premium reported in Exhibit 2 is
approximately equal to the arithmetic (7.5%) long-term
differential between returns on stocks and- long-teFm
government bonds.'
binterestingly, for the 1982-I996 period the arithmetic spread
between large company stocks and long-term government
bonds was only 3.3% per year. The downward trend in interest
rates resulted in average annual returns of 14.1% on longterm government bonds over this horizon. Some (e.g.,
Ibbotson, 1997) argue that only the income ( not total) return
on bonds should be subtracted in calculating risk premia.
Exhibit 2 shows the estimated risk premium changes
over time, suggesting changes in the market's
perception of the incremental risk of investing in equity
rather than debt securities. Scanning the last column
of Exhibit 2, the risk premium is higher in the 1990s
than earlier and especially so in late 1997 and 1998.
Our DCF results provide no evidence to support the
notion of a declining risk premium in the 1990s as a
driver of the strong run up in equity prices.
A striking feature in Exhibit 2 is the relative stability
of the estimates of k. After dropping (along with
interest rates) in the early and mid-1980s, the average
annual value ofk has remained within a 75 basis point
range around 15% for over a decade. Moreover, this
stability arises despite some variability in the
338
11
HARRIS & MARSTON-THE MARKET RISK PREMIUM
Exhibit 3. Average Historical Returns on Bonds, Stocks, Bills, and Inflation in the US, 1926-1998
Historical Return Realizations
Geometric Mean
Arithmetic Mean
Common Stock (Large Cort>patty)
11.2%
13.2%
Long-tetm Govemrrent Bonds
5.3
5.7
Treasury Bills
3.8
3.8
Inflation Rate
3.1
3.2
Source: Ibbotson Associates, Inc., 1999 Stocks, Bonds, Bills and Inflorion, 1999 Yearbook.
underlying dividend yield and growth components of
k as Exhibit 2 illustrates. The results suggest that k is
more stable than government interest rates. Such
relative stability of k translates into parallel changes
in the market risk premium. In a subsequent section,
we examine whether changes in our market risk premium
estimates appear linked to interest rate conditions and
a number of proxies for risk.
We explored the sensitivity of the results to our
screening procedures in selecting companies. The
reported results screen out all non-dividend paying
stocks on the premise that use of the DCF model is
inappropriate in such cases. The dividend screen
eliminates an average of 55 companies per month. In' a
given month, we also screen out firms with fewer than
three analysts' forecasts, or if the standard deviation
around the mean forecast exceeds 20%. When the
analysis is repeated without any of the three screens,
the average risk premium over the sample period
increased by only 40 basis points, from 7.14% to 7.54%.
The beta of the sample firms also was estimated and
the sample average was one, suggesting that the
screens do not systematically remove low or high-risk
firms. (Specifically, using firms in the screened sample •
as of December 1997 (the last date for which we had
CRSP return data), we used ordinary least squares
regressions to estimate beta for each stock using the
prior 60 months ofdata and the CRSP return (SPRTRN)
as the msrket index. The value-weighted average of the
individual betas was 1.00.)
The results reported here use firms in the S&P500 as
reportetl by COMPUSTAT in September 1998. This
could create a survivorship bias, especially in the earlier
months of the sample. We compared our current results
to those obtained in Harris and Marston (1992) for
which there was data to update the S&P500
composition each month. For the overlapping period,
January 1982-May 1991, the two procedures yield the
same average market risk premium, 6.47%. This
suggests that the firms departing from or entering the
S&P500 index do so for a number of reasons with no
discernable effect on the overall estimated S&P500
market risk premium.
IV. Changes in the Market Risk
Premium Over Time
With changes in the economy and financial markets,
equity investments may be perceived to change in risk.
For instance, investor sentiment about future business
conditions likely affects attitudes about the riskiness
of'equity investments compared to investments in the
bond markets. Moreover, since bonds are risky
investments themselves, equity risk premia (relative
to bonds) could change due to changes in perceived
riskiness of bonds, even if equities displayed no shifts
in risk.
In earlier work covering the 1982-1991 period, Harris
and Marston ( 1992) reported regression results
indicating that the market premium decreased with the
level of government interest rates and increased with
the spread between corporate and government bond
yields (BSPREAD). This bond yield spread was
interpreted as a time series proxy for equity risk. In
this paper, we introduce three additional ex ante
measures of risk shown in Exhibit 1: CON, DISP, and
VOL. The three measures come from three independent
sets of data and are supplied by different agents in the
economy (consumers, equity analysts, and investors
(via option and share price data)). Exhibit 4 provides
summary data on all four of these risk measures.
Exhibit 5 replicates and updates earlier analysis by
Harris and Marston (I992).1 The results confirm the
earlier patterns. For the entire sample period, Panel A
shows that risk premia are negatively related to interest
rates. This negative relationship is also true for both
'OLS regressions with levels of variables generally showed
severe autocorrelation. As a result, we used the Prais-Winsten
method (on levels of variables) and also OLS regressions on
first differences of variables. Since both methods yielded similar
results and the latter had more stable coefficients across
specifications, we report only the results using first differences.
Tests using Durbin-Watson statistics from regressions in
Exhibits 5 and 6 do not accept the hypothesis of autocorrelated
errors ( tests at . 01 significance level, see Johnston, 1984).
We also estimated the first difference model without an intercept
and obtained estimates almost identical to those reported.
339
JOURNAL OF APPLIED FINANCE - 2001
12
Exhibit 4. Descriptive Statistics on Ex Ante Risk Measures
Entries are based on monthly data. BSPREAD is the spread between yields on long-term corporate and government bonds. CON
is the consumer confidence index. DISP measures the dispersion of analysts' forecasts of earnings growth. VOL is the volatility on
the S&P500 index implied by options data. Variables are expressed in decimal form, (e.g.,12% _.12).
Panel A. Variables are Monthly Levels
Mean
Standard Deviation
Minimum
Maximum
BSPREAD
.0123
.0040
.0070
.0254
CON
.9504
.2242
.473
1.382
DISP
.0349
.0070
.0285
.0687
VOL
.1599
.0697
.0765
.6085
Mean
Pane! B. Variables are Monthly Changes
Minimum
Standard Deviation
Maximum
-.00001
.0011
-.0034
.0036
CON
.0030
.0549
-.2300
.2170
D1SP
-.00002
.0024
-.0160
.0154
VOL
-.0008
.0592
-.2156
.4081
Panel C. Correlation Coefficients for Monthly Changes
DISP
CON
VOL
BSPREAD
BSPREAD
BSPREAD
1.00
-.16**
.054
CON
-.16**
1.00
.065
-.09
.065
1.00
.027
-.09
.027
1.00
DISP
VOL
.054
.22*
.22*
,*Significantly different from zero at the .05 level.
*Significantly different from zero at the . 01 level.
the 1980s and 1990s as displayed in Panels B and C.
For the entire 1982 to 1998 period, the addition of the
yield spread *risk proxy to the regressions lowers the
magnitude of the coefficient on government bond
yields, as can be seen by comparing Equations (1) and
(2) of Panel A. Furthermore, the coefficient of the yield
spread (0.488) is itself significantly positive. This
pattern suggests that a reduction in the risk differential
between investment in government bonds and in
corporate bonds is translated into a lower equity
market risk premium.
In major respects, the results in Exhibit 5 parallel
earlier findings. The market risk premium changes over
time and appears inversely related to government
interest rates but is positively related to the bond yield
spread, which proxies for the incremental risk of
investing in equities as opposed to government bonds.
One striking feature is the large negative coefficients
on government bond yields. The coefficients indicate
the equity risk premium declines by over 70 basis
points for a 100 basis point increase in government
interest rates.' This inverse relationship suggests
'The Exhibit 5 coefficients on i are significantly different
from -1. 0 suggesting that equity required returns do respond
to interest rate changes. However, the large negative
coefficients imply only minor adjustments of required returns
to interest rate changes since the risk premium declines. in
earlier work (Harris and Marston, 1992) the coefficient was
significantly negative but not as large in absolute value. In that
earlier work, we reported results using the Prais-Winsten
estimators. When we use that estimation technique and recreate
the second regression in Exhibit 5, the coefficient for i is -.584 (f
-- 12.23) for the entire sample period 1982-1998.
340
13
HARRIS & MARSTON THE MARKET RISK PREMIUM
Exhibit S. Changes in the Market Equity Risk Premium Over Time
The exhibit reports regression coefficients (1-values). Regression estimates use all variables expressed as monthly changes to
correct for autocon$lation. The dependent variable is the market equity risk premium for the S&P500 index. BSPREAD is the
spread between yields on long-term corporate and government bonds. The yield to maturity on long-term government bonds is
denoted as i. For purposes of the regression, variables are expressed in decimal form, (e:g, 12% =.12).
A. 1982-1998
B. 1980s
C. 19903
BSPREAD
t
Intercept
Time Period
-.0002
(-1.49)
-.869
(-16.54)
-.0002
(-1.11)
-.749
(-11.37)
-.0005
(-1.62)
-.887
(-10.97)
-.0004
(-1.24)
-.759
(-7.42)
-.0000
(-0.09)
-.840
(-13.78)
-.0000
(0.01)
-.757
(-9.85)
R?
.57
.59
.488
(2.94)
.56
.57
.508
(1.99)
.64
.347
(1.76)
.65
Exhibit 6. Changes in the Market Equity Risk Premium Over Time and Selected Measures of Risk
The exhibit reports regression coefficients (t-values). Regression estimates use all variables expressed as monthly changes
to correct for autocorrelation. The dependent variable is the market equity risk premium for the S&P500 index. BSPREAD
is the spread between yields on long-term corporate and government bonds. The yield to maturity on long-term government
bonds is denoted as f. CON is the consumer confidence index. DISP measures the dispersion of analysts' forecasts of
earnings growth. VOL is the volatility on the S&P500 index implied by options data. For purposes of the regression,
12).
variables are expressed in decimal form, (e.g., 12%
Time Period
intercept
i
BSPREAD
CON
DISP
VOL
Adj. R2
A. 1982-1998
/
B. May 1986-1998
(1)
0.0002
(.97)
(2)
-0.0001
(-.96)
(3)
0.0002
(.79)
(4)
-0.0001
(-.93)
-0.733
(-11.49)
0.433
(2.69)
(5)
0.0000
(.06)
-0.818
(-11.21)
0.420
(2S2)
(6)
0.0001
(S3)
(7)
0.0000
(.02)
-0.737
(-11.31)
-0.831
(-11.52)
0.453
(2.76)
0.326
(1.95)
-0.014
(-3.50)
0.05
-0.007
(-2.48)
0.60
0.224
(2.38)
0.02
-0.007
(-2.77)
0.185
(3.13)
0.62
-0.005
(-2.23)
0.378
(3.77)
0.68
-0.005
(-2.12)
341
0.372
(3.77)
0.011
(2.89)
0.05
0.006
(2.66)
0.69
14
.j
JOURNAL OF APPLIED FINANCE - 2001
much greater stability in equity required returns than
is often assumed. For instance, standard application
of the CAPM suggests a one-to-one change in equity
returns and government bond yields.
Exhibit 6 introduces three additional proxies for risk
and explores whether these variables, either
individually or collectively, are correlated with the
market premium. Since the estimates of implied volatility
start in May 1986, the exhibit shows results for both
the entire sample period and for the period during which
we can introduce all variables. Entered individually
each of the three variables is significantly linked to
the risk premium with the coefficient having the
expected sign. For instance, in regression (1) the
coefficient on CON is -.014, which is significantly
different from zero (t = -3.50). The negative coefficient
signals that higher consumer confidence is linked to a
lower market premium. The positive coefficients on
VOL and DISP indicate the equity risk premium
increases with both market volatility and disagreement
among analysts. The effects of the three variables appear
largely unaffected by adding other variables. For
instance, in regression (4) the coefficients on CON and
DISP both remain significant and are similar in magnitude
to the coefficients in single variable regressions.'
Even in the presence of the new risk variables,
Exhibit 6 shows that the market risk premium is affected
by interest rate conditions. The large negative
coefficient on government bond rates implies large
reductions in the equity premium as interest rates rise.
One feature of our data may contribute to the observed
negative relationship between the market risk premium
and the level of interest rates. Specifically, if analysts
are slow to report updates in their growth forecasts,
changes in the estimated k would not adjust fully with
changes in the interest rate even if the true risk premium
were constant. To address the impact of "stickiness"
in the measurement of k, we formed "quarterly"
measures of the risk premium that treat k as an average
over the quarter. Specifically, we take the value of k at
the end of a quarter and subtract from it the average
value of i for the months ending when k is measured.
For instance, to form the risk premium for March 1998,
'Realized equity returns are difficult to predict out of sample
(see Goyal and Welch, 1999). Our approach is different in
that we look at expectational risk premia which are much
more stable. For instance, when we estimate regression
coefficients (using the specification shown in regression 7 of
Exhibit 6) and apply them out of sample we obtain
"predictions" of expectational risk premia that are
significantly more accurate (better than the .01 level) than a
no change forecast. We use a"rolting regression" approach
using data through December 1991 to get coefficients to predict
the risk premium in January 1992, We repent the procedure
moving forward a month and dropping the oldest month of
data from the regression. Details are available from the authors.
342
the average value of i for January, February, and March
is subtracted from the March value of k. This approach
assumes that, in March, k still reflects values ofg that
have not been updated from the prior two months.
The quarterly measure of risk premium then is paired
with the average values of the other variables for the
quarter. For instance, the March 1998 "quarterly" risk
premium would be paired with averaged values of
BSPREAD over the January through March period. To
avoid overlapping observations for the independent
variables, we use only every third month (March, June,
September, December) in the sample.
As reported in Exhibit 7, sensitivity analysis using
"quarterly" observations suggests that delays in
updating may be responsible for a portion, but not all,
of the observed negative relationship between the
market premium and interest rates. For example, when
quarterly observations are used, the coefficient on r in
regression (2) of Exhibit 7 is -.527, well below the earlier
estimates but still significantly negative.'a
As an additional test, movements in the bond risk
premium (BSPREAD) are examined. Since BSPREAD is
constructed directly from bond yield data, it does not
have the potential for reporting lags that may affect
analysts' growth forecasts. Regression 3 in Exhibit 7
shows BSPREAD is negatively linked to government
rates and significantly so." While the equity premium
need not move in the same pattern as the corporate
bond premium, the negative coefficient on BSPREAD
suggests that our earlier results are not due solely to
"stickiness" in measurements of market required returns.
The results in Exhibit 7 suggest that the inverse
relationship between interest rates and the market risk
premium may not be as pronounced as suggested in
earlier exhibits. Still, there appears to be a significant
negative link between the equity risk premium and
government interest rates. The quarterly results in
Exhibit 7 would suggest about a 50 basis point change
in risk premium for each 100 basis point movement in
interest rates.
Overall, the ex ante estimates of the market risk
premium are significantly linked to ex ante proxies for
risk, Such a link suggests that investors modify their
required returns in response to perceived changes in
the environment. The findings provide some comfort
that our risk premium estimates are capturing, at least
"Sensitivity analysis for the 1982-1989 and 1990-1998
subperiods yields results similar to those reported.
"We thank Bob Conroy for suggesting use of BSPREAD.
Regression 3 in Exhibit 7 appears to have autocorrelated
errors: The Durbin-Watson (DW) statistic rejects the hypothesis
of no autocorrelation. However, in subperiod analysis, the
DW statistic for the 1990-98 period is consistent with no
autocorrelation and the coefficient on i is essentially the same
(-.24, / A-8.05) as reported in Exhibit 7.
HARRIS & MARSTON-THE MARKET RISK PREMIUM
15
Exhibit 7. Regressions Using Alternate Measures of Risk Premia to Analyze Potential Effects of
Reporting Lags in Analysts' Forecasts
The exhibit reports regression coefficients (t-values). Regression estimates use all variables expressed as changes (monthly
or quarterly) to correct for autocorrelation. BSPREAD is the spread between yields on long-term corporate and government
bonds. rp is the risk premium on the S&P500 index. The yield to maturity on long-term government bonds is denoted as
1. For purposes of the regression, variables are expressed in decimal form, (e.g., 12%
12).
Dependent Variable
Intercept
I
BSPREAD
Adj. RZ
(1)
Equity Risk Premium (rp)
Monthly Observations
(same as Table V)
-.0002
-.749
(-11.37)
.488
(2.94)
.59
(2)
Equity Risk Freudian (rp)
"Quarterly" nonoverlapping
observations to account for
lags in analyst reporting
-.0002
(-.49)
-.527
(-6.18)
.550
(2.20)
.60
(3)
Corporate Bond Spread (BSPREAD)
Monthly Observations
-.0001
(-1.90)
-.247
(41.29)
in part, underlying changes in the economic
environment. Moreover, each of the risk measures
appears to contain relevant information for investors.
The market risk premium is negatively related to the
level of consumer confidence and positively linked to
interest rate spreads between corporate and
government debt, disagreement among analysts in their
forecasts of earnings growth, and the implied volatility
of equity returns as revealed in options data.
V. Conclusions
flShareholder required rates of return and risk premia
should be based on theories about investors'
expectations for the future. In practice, however, risk
premia are typically estimated using averages of
historical returns. This paper applies an alternate
approach to estimating risk premia that employs
publicly *available expectational data. The resultant
average market equity risk premium over government
bonds is comparable in magnitude to long-term
differences (1926 to 1998) in historical returns between
stocks and bonds. As a result, our evidence does not
resole the equity premium puzzle; rather, the results
suggest investors still expect to receive large spreads
to invest in equity versus debt instruments.
There is strong evidence, however, that the market
risk premium changes over time. Moreover, these
changes appear linked to the level of interest rates as
well as ex ante proxies for risk drawn from interest rate
spreads in the bond market, consumer confidence in
future economic conditions, disagreement among
financial analysts in their forecasts and the volatility
.38
of equity returns implied by options data. The significant
economic links between the market premium and a wide
array of risk variables suggests that the. notion of a
constant risk premium over time is not an adequate
explanation of pricing in equity versus debt markets.
These results have implications for practice. First,
at least on average, the estimates suggest a market
premium roughly comparable to long-term historical
spreads in returns between stocks and bonds. Our
conjecture is that, if anything, the estimates are on the
high side and thus establish an upper bound on the
market premium. Second, the results suggest that use
of a constant risk premium will not fully capture
changes in investor return requirements. As a specific
example, our findings indicate that common application
of models such as the CAPM will overstate changes
in shareholder return requirements when government
interest rates change. Rather than a one-for-one
change with interest rates implied by use of constant
risk premium, the restllts indicate that equity required
returns for average risk stocks likely change by half
(or less) of the change in interest rates. However, the
picture is considerably more complicated as shown by
the linkages between the risk premium and other
attributes of risk.
Ultimately, our research does not resolve the answer
to the question "What is the right market risk
premium?" Perhaps more importantly, our work
suggests that the answer is conditional on a number
of features in the economy-not an absolute. We hope
that future research will harness ex ante data to provide
additional guidance to best practice in using a market
premium to improve financial decisions.®
343
^• :
JOURNAL OF APPLIED FINANCE - 2001
16
References
Asness, C.S., 2000, "Stocks versus Bonds: Explaining the
Equity Risk Premium," Financial Analysts Journal 36 (No.
2, March/April), 96-113.
Harris, R. and F. Marston, 1992, "Estimating Shareholder Risk
Premia Using Analysts' Growth Forecasts," Financial
Management 21 (No. 2, Summer), 63-70.
Boebel, R.B., 1991, "The Information Content of Financial
Analysts' Forecasts: Short-term vs. Long-term," University
of North Carolina at Chapel Hill, Dissertation. University
.
Microfilms International order number 9207936.
Ibbotson Associates, Inc., 1998 Cost of Capital Quarterly,
1998 Yearbook.
Boebel, R.B., M.N. Gultekin and R.S. Harris, 1993, "Financial
Analysts' Forecast of Corporate Earnings Growth: Top
Down Versus Bottom Up;' in Handbook of Security Analyst
Forecasting and Asset Allocation, John B. Guerard, Jr. and
Mustafa N. Gultekin (eds.), London, JAI Press, 185-192.
Brigham, D., D. Shome, and S. Vinson, 1985, "The Risk
Premium Approach to Measuring A Utility's Cost of
Equity," Financial Management 14 (No. I. Spring), 33-45.
Brown, L.D., 1993, " Earnings Forecasting Research: Its
Implications for Capital Market Research," International
Journal of Forecasting 9 (No. 3, November), 295-320.
Brown, L.D., 1997, "Analyst Forecasting Errors: Additional
Evidence," Financial Analysts Journal 53 (No. 6,
November/December), 81-90.
Bruner, R.F., K. Eades, R. Harris and R. Higgins, 1998, "Best
Practices in Estimating the Cost of Capital: Survey and
Synthesis." Financial Practice and Education 8 (No. 1.
Spring/Summer), 13-28.
Ibbotsoti Associates, Inc., 1997 Stocks, Bonds, Bills, and
Inflation, 1997 Yearbook.
Johnston, J., 1984, Econometric Methods, New York, McGrawHill, 3fd edition.
Kennedy, P., 1985, A Guide to Econometrics, Cambridge, MIT
Press, 2^d edition.
Lamdin, D., 2001, "Estimating the Cost of Equity for
that
Repurchase:
and
Theory
Corporations
Application,"Engineering Economist, 46 (No. 1), 53-63.
Madalla, G.S., 1977, Econometrics, New York, McGraw-Hill.
Malkiel, B., 1979, "The Capital Formation Problem in the
United States," Journal of Finance 34 (No. 2, May), 291306.
Malkiel, B., 1982, "Risk and Return: A New Look," in The
Changing Role of Debt and Equity in Financing US Capitol
Formation, B.B. Friedman ( ed.), National Bureau of
Economic Research, Chicago, University of Chicago Press.
Carleton, W.T. and J. Lakonishok, 1985, "Risk and Return on
Equity: The Use and Misuse of Historical Estimates,"
Financial Analysts Journal 41 (No. 1, January/February),
38-47.
Marston, F., R. Harris, and P. Crawford, 1992, "Risk and Return
in Equity Markets: Evidence Using Financial Analysts'
Forecasts," in Handbook of Security Analysts' Forecasting
and Asset Allocation, J. Guerard and M. Gultekin (eds.),
Greenwich, CT, JAI Press, 101-122.
Chan, L., Y. Hamao, and J. Lakonishok. 1991, "Fundamental
and Stock Returns in Japan," Journal of Finance, 46 (No.
5, December) 1739-64.
Marston, F. and R. Harris, 1993, "Risk, Return, and Equilibrium:
A Revisit Using Expected Returns," Financial Review, 28
(No. 1, February).
Cragg, J. and B.G. Malkiel, 1992, Expectations and the
Structure of Shore Prices, National Bureau of Economic
Research, Chicago, University of Chicago Press.
Siegel, J., 1999, "The Shrinking Equity Premium," The Journal
of Portfolio Management (Fall), 10-17.
Goldman Sachs, 1999, EVA Company Analysis, (January).
Siegel, J., and R. Thaler, 1997, "Anomalies: The Equity
Premium Puzzle," Journal of Economic Perspectives, I 1
(No. 1, Winter), 191-200.
Goyal, A. and 1. Welch, 1999, "Predicting the Equity
Premium," Anderson Graduate School of Management at
UCLA (August 14).
VanderWeide, J. and W.T. Carleton, 1998, "Investor Growth
Expectations: Analysts vs. History," Journal of Portfolio
Management (Spring), 78-82.
Harris, R., 1986, "Using Analysts' Growth Forecasts to
I Estimate Shareholder Required Rates of Return," Financial
Management 15 (No. l, Spring), 58-67.
Welch, 1., 2000, "Views of Financial Economists on the Equity
Premium and on Professional Controversies," Journal of
Business 73 (No. 4, October), 501-537.
344
WORKP APER 7
345
Cost of Capital Estimation The Risk Premium Approach to Measuring a Utility's...
Eugene F Brigham; Dilip K Shome; Steve R Vinson
Financial Management (pre-1986); Spring 1985; 14, 1; ABU[NFORM Global
pg. 33
Cost of Capital Estimation
The Risk Premium Approach to Measuring
a Utility's Cost of Equity
Eugene F. Brigham, Dilip K. Shome, and Steve R. Vinson
Eugene F. Brigham and Dilip K. Shame are faculty members of the
Unh'ersin of Florida and the Virginia Polytechnic Institute and State
Uni►'etsin, respectnr►v: Steve R. Vinson is affiliated ndtli AT&T
Comtmmications.
n In the mid-1960s, Myron Gordon and others began
applying the theory of finance to help estimate utilities'
costs of capital. Previously, the standard approach in
cost of equity studies was the "comparable earnings
method," which involved selecting a sample of unregulated companies whose investment risk was judged to
be comparable to that of the utility in question, calculating the average return on book equity (ROE) of
these sample companies, and setting the utility's service rates at a level that would permit the utility to
achieve the same ROE as comparable companies. This
procedure has now been thoroughly discredited (see
Robichek 115]), and it has been replaced by three market-oriented (as opposed to accounting-oriented) approaches: (i) the DCF method, (ii) the bond-yield-plusrisk-premium method, and (iii) the CAPM, which is a
specific version of the generalized bond-yield-plusrisk-premium approach.
mated risk premiums since 1965. Third, we examine
the relationship between equity risk premiums and the
level of interest rates, because it is important, for purposes of estimating the cost of capital, to know just
how stable the relationship between risk premiums and
interest rates is over time. If stability exists, then one
can estimate the cost of equity at any point in time as a
function of interest rates as reported in The Wall Street
Journal, the Federal Reserve Bulletin, or some similar
source.' Fourth, while we do not discuss the CAPM
directly, our analysis does have some important implications for selecting a market risk premium for use in
that model. Our focus is on utilities, but the methodology is applicable to the estimation of the cost of
'For example. the Federal Energy Regulatory Cmnmission's Staff tecmtly proposed that a risk premium be estimated every two years and
that. between estimation dates, the lastddemdned risk preadurn be
added to the current yield on ten-year Treasury bands to obtain an
estimate of the cast of equity to an average utility (Docket Rbt 90-36).
Suhseqnently, the FCC made a similar proposal ("Notice of proposed
Rukmakiny," August 13. 1984. Docket No. 84-BOQ). Obviousty, the
validity of such procedWes depends on (i) the accuracy of the risk
paemium estimate and 01) do stability of the selattionship between HA
premiums and interest rates. Both proposals am sGN under review.
Our purpose in this paper is to discuss the riskpremium approach, including the market risk premium
that is used in the CAPM. First, we critique the various
procedures that have been used in the past to estimate
risk premiums. Second, we present some data on esti33
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346
34
PtttANaAL MANA3EMItVrrsPntiVG 198s
equity for any publicly traded firm, and also for nontraded firms for which an appropriate risk class can be
assessed, including divisions of publicly traded corporations?
Alternative Procedures for Estimating
Risk Premiums
In a review of both rate cases and the academic
literature, we have identified three basic methods for
estimating equity risk premiums: (i) the ex post, or
historic. yield spread method-, (ii) the survey method;
and (iii) an er ante yield spread method based on DCP
analysis? In this section, we briefly review these three
methods.
Historic Risk Premiums
A number of researchers, most notably Ibbotson and
Sinquefietd 1121, have calculated historic holding period returns on different securities and then estimated
risk premiums as follows:
Historic
=
Risk
Premium
Average of the
annual returns on
a stock index for
a particular
past period
-
Average of the
annual returns on
a bond index for .(1)
the same
past period
mate of its cost of equity, and (ii) indirectly, where
I&S data are used to estimate the market risk premium
in CAPM studies.
There are both conceptual and measurement problems with using I&S data for purposes of estimating
the cost of capital. Conceptually, there is no com?elling reason to think that investors expect the same
relative returns that were earned in the past. Indeed,
evidence presented in the following sections indicates
that relative expected returns should, and do, vary
significantly over time. Empirically, the measured historic premium is sensitive both to the choice of estimation horizon and to the end points. These choices are
essentially arbitrary, yet they can result in significant
differences in the final outcome. These measurement
problems are common to most forecasts based on time
series data.
The Survey Approach
One obvious way t7 estimate equity risk premiums
is to poll investors. Charles Benore I I], the senior
utility analyst for Paine Webber Mitchell Hutchins, a
leading institutional brokerage house, conducts such a
survey of major institutional investors annually. His
1983 results are reported in Exhibit 1.
Exhibit 1. Results of Risk Premium Survey. 1983'
Ibbotson and Sinquefield (I&S) calculated both arithmetic and geometric average returns, but most of their
risk-premium discussion was in terms of the geometric
averages. Also, they used both corporate and Treasury
bond indices, as well as a T-bill index, and they analyzed all possible holding periods since 1926. The I&S
study has been employed in numerous rate cases in two
ways: (i) directly, where the I&S historic risk premium
is added to a company's bond yield to obtain an esti-
`r6e FCC is particularly interested in risk-premium methodologies.
because (i) only eighteen of the 1.400 telephone companies it regulates
have publicly-traded stock, and hence offer the possibility of DCF
anatysis. MW(ii) mast of the publiety-trrded telephone companies have
bath regulated and unregulated assets, to a corporate qCF cost might
not be applicable to the regulated units of the companies.
'In rate cases, some witnesses also have calculated the difforential
between the yield to maturity (YTM) of it company's bonds and its
concurrent ROE, and Wen called this differential a risk ptamium. In
general. this procedure is unsound, because the YTM on a bond is a
future expected return on the bond's market nulae, while the ROE is the
past reuff:ed return on the stock's beak twine. Thus, comparing YTMs
and ROFs is like comparing apples and oranEes.
Assuming a double A. long-term utility bond currently yields 121/•Ar.
the common stock for the same company v6vuld be fairly priced relative
to the bond if its expected return was ag follows:
Total Return
Indicated Risk premium
Percent Of
( basis points)
Respondents
over 20'/•.'3
20'/_^
19'/•%
18116%
171/2%
16'/•%
15%s%
14'/ %
t3'1:%
under 13Y2%
over 800
800
70D
600
500
400
300
200
[00
under 10D
ttl%
8%
29%
35%
16%
0%
1%
Weighted
average
358
IO(ixr
'Betmte's questionnaire included the first two cvtumns, while his third
column provided a space for the respondents to Indicate which risk
premium they thought applied. We summarized Beaom's responses in
the frequency distribution given in Column 3. Also, in his questionnaire
each year. Benore adjusts the double A bond yield and the total returns
(Column I) to reflect current market conditions. Both the questiou
above and the responses to it were taken from t he survey conducted in
April 1983.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
347
alIONAaA. SHOM6, VMtSOP11CO5T OF I:QUITY MtiSUtt[sMtdNT
Benore's results, as measured by the average risk
premiums, have varied over the years as follows:
Average RP
Year
1978
(basis points)
491
1979
1980
1981
1982
475
423
349
275
1983
358
The survey approach is conceptually sound in that it
attempts to measure investors' expectations regarding
risk premiums, and the Benore data also seem to be
carefully collected and processed. Therefore, the Benore studies do provide one useful basis for estimating
risk premiums. However, as with most survey results,
the possibility of biased responses and/or biased sampling always exists. For example, if the responding
institutions are owners of utility stocks (and many of
them are), and if the respondents think that the survey
results might be used in a rate case, then they might
bias upward their responses to help utilities obtain
higher authorized returns. Also. Benore surveys large
institutional investors, whereas a high percentage of
utility stocks are owned by individuals rather than institutions, so there is a question as to whether his
reported risk premiums are really based on the expectations of the "representative" investor. Finally, from a
pragmatic standpoint, there is a question as to how to
use the Benore data for utilities that are not rated AA.
The Benore premiums can be applied as an add-on to
the own-company bond yields of any given utility only
if it can be assumed that the premiums are constant
across bond rating classes. A priori, there is no reason
to believe that the premiums will be constant.
DCF-Based Ex Ante Risk Premiums
In a number of studies, the DCF model has been
used to estimate the ex ante market risk premium,
RP,,. Here, one estimates the average expected future
return on equity for a group of stocks, k,,,, and then
subtracts the concurrent risk-free rate. RF, as proxied
by the yield to maturity on either corporate or Treasury
securities:'
RPy=km -RF.
(2)
Conceptually, this procedure is exactly like the I&S
approach except that one makes direct estimates of
future expected returns on stocks and bonds rather than
35
assuming that investors expect future returns to mirror
past returns.
The most difficult task, of course, is to obtain a valid
estimate of kM, the expected rate of return on the market. Several studies have attempted to estimate DCF
risk premiums for the utility industry and for other
stock market indices. Two of these are summarized
next.
Vandell and Koster. In a recently published
monograph, Vandell and Kesler 118) estimated er ante
risk premiums for the period from 1944 to 1978. RF
was measured both by the yield on 90-day T-bills and
by the yield on-the Standard and Poor's AA Utility
Bond Index. They measured kM as the average expected return on the S&P's 500 Index, with the expected
return on individual securities estimated as follows:
k °^21- ^, + gt•
Po
(3)
where,
D, = dividend per share expected over the next
twelve months,
P. = current stock price,
g = estimated long-term constant growth rate,
and
i = the i'" stock.
To estimate g;, Vandell and Kesler developed fifteen
forecasting models based on both exponential smoothing and trend-line forecasts of earnings and dividends,
and they used historic data over several estimating
horizons. Vandell and Kesler themselves acknowledge
that, like the Ibbotson-Sinquefield premiums, their
analysis is subject to potential errors associated with
trying to estimate expected future growth purely from
past data. We shall have more to say about this point
later.
In this analysis. most people hove used yields on long-term bonds
rather than short-terra money market instruments. It is recognized that
limg-term bonds. even Treasury bonds. me not risk fiee, so an RPa
based an these debt instruments is smaller than it would be if there were
some better proxy to the long-lerm riskless rale. People have attempted
to use the T-bill rate for RF, but the 7-bill rate embodies a different
average Inflation premium than stock•s, and it is subject to random
fluctuations caused by monetary policy. international currency flows.
and other factors. Thus, many penpk believe that for cost of capital
purposes. Rp should be based an Ing-term securities.
We ddlest tosee how debt mamtities would affect ourcalculatedfisk
ptnnimns. If a short-term rate such as the 30-dsy T -bill Me is used.
measured risk premiums jump around widely and. no far as we coald
tell. randomly. The chnice or & maturity in the to. to 30-year range line
little effect, as the yield cmve is generally faidy Rat in Mai range.
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348