Languages
Pages
Legal
Chapter 9
Stock Valuation
Learning Objectives
1. List and describe the four types of secondary markets.
2. Explain why many financial analysts treat preferred stock as a special type of bond
rather than as a true equity security.
3. Describe how the general dividend-valuation model values a share of stock.
4. Discuss the assumptions that are necessary to make the general dividend-valuation
model easier to use, and be able to use the model to compute the value of a firm’s
stock.
5. Explain why g must be less than R in the constant-growth dividend model.
6. Explain how valuing a preferred stock with a stated maturity differs from valuing a
preferred stock with no maturity date, and be able to calculate the price of a share
of preferred stock under both conditions.
I. Chapter Outline
9.1 The Market for Stocks
Equity securities are certificates of ownership of a corporation.
Households dominate the holdings of equity securities, owning more than 36 percent
of outstanding corporate equities.
A. Secondary Markets
1
In secondary markets, outstanding shares of stock are bought and sold among
investors.
An active secondary market enables firms to sell their new debt or equity issues at
a lower funding cost than firms selling similar securities that have no secondary
market.
B. Secondary Markets and Their Efficiency
In the United States, most secondary market transactions are conducted on one
of the many stock exchanges.
In terms of total volume of activity and total capitalization of the firms
listed, the NYSE is the largest in the world and NASDAQ is the second
largest.
In terms of the number of companies listed and shares traded on a daily
basis, NASDAQ is larger than the NYSE.
Firms listed on the NYSE tend to be, on average, larger in size and their
shares trade more frequently than firms whose securities trade on
NASDAQ.
There are four types of secondary markets, and each type differs according to the
amount of price information available to investors, which in turn, affects the
efficiency of the market.
1. Direct Search
The secondary markets farthest from our ideal of complete price information
are those in which buyer and seller must seek each other out directly.
2
It is too costly to perform a thorough search among all possible partners done
to locate the best price.
Securities that sell in direct search markets are usually bought and sold so
infrequently that no third party, such as a broker or dealer, finds it profitable
to serve the market.
The sales of common stock of small private companies and private placement
transactions are good examples of direct search markets.
2. Broker
Brokers bring buyers and sellers together to earn a fee, called a commission.
Brokers’ extensive contacts provide them with a pool of price information that
individual investors could not economically duplicate themselves.
By charging a commission fee less than the cost of direct search, brokers give
investors an incentive to make use of the information by hiring them as
brokers.
3. Dealer
Market efficiency is improved if there is someone in the marketplace to
provide continuous bidding (selling or buying) for the security.
Dealers provide this service by holding inventories of securities, which they
own, then buying and selling from the inventory to earn a profit.
Dealers earn their profits from the spread on the securities they trade—the
difference between their bid price (the price at which they buy) and their
offer price (the price at which they sell).
3
The advantage of a dealer over a brokered market is that brokers cannot
guarantee that an order will be executed promptly, while dealers can because
they have an inventory of securities.
A dealer market eliminates the need for a time-consuming search for a fair
deal by buying and selling immediately from the dealers’ inventory of
securities.
NASDAQ is the best-known example of a dealer market.
Electronic communications network (ECN) systems provide additional price
information to investors and increase marketability and competition, which
should improve NASDAQ efficiency.
4. Auction
In an auction market, buyers and sellers confront each other directly and
bargain over price.
The New York Stock Exchange is the best-known example of an auction
market.
In the NYSE the auction for a security takes place at a specific location on the
floor of the exchange, called a post.
The auctioneer in this case is the specialist who is designated by the exchange
to represent orders placed by public customers.
C. Reading the Stock Market Listings
The Wall Street Journal, the New York Times, and other newspapers in large
metropolitan areas provide stock listings for the major stock exchanges, such
as the NYSE and the relevant regional exchanges.
4
Exhibit 9.1 shows a section of the listing in the Wall Street Journal for the
NYSE.
D. Types of Equity Securities
The two types of equity securities are common stock and preferred stock.
Common stock represents the basic ownership claim in a corporation.
One of the rights of the owners is to vote on all important matters that
affect the life of the company, such as electing the board of directors or
voting on a proposed merger or acquisition.
Owners of common stock are not guaranteed any dividend payments and
have the lowest-priority claim on the firm’s assets in the event of
bankruptcy.
Legally, common stockholders enjoy limited liability.
Common stocks are perpetuities in the sense that they have no maturity.
Preferred stock also represents ownership interest in the corporation, but
preferred stock receives preferential treatment over common stock in certain
matters.
If a preferred dividend payment is not paid due to the firm’s financial
condition, the firm is not in default technically. However, the market
reacts as if the failure to make the dividend payment is a default and
punishes the stock accordingly.
Preferred stock owners are given priority treatment over common stock
with respect to dividends payments and the claims against the firm’s assets
in the event of bankruptcy or liquidation.
5
Dividends payments are paid with after-tax dollars subject to taxation.
Even though preferred stock is equity, the owners have no voting
privileges.
Preferred stocks are legally classified as perpetuities because they have no
maturity. However, most preferred stocks are not true perpetuities
because the shares contain a call provision and the share contract often
requires management to retire a certain percent of the stock annually
until the entire issue is retired.
E. Preferred Stock: Debt or Equity?
Legally, preferred stock is equity.
Like the dividends on common stock, preferred stock dividends are taxable.
A strong case can be made that preferred stock is really a special type of bond.
First, regular preferred stock confers no voting powers.
Second, preferred stockholders receive a fixed dividend, regardless of the
firm’s earnings, and if the firm is liquidated, they receive a stated value
(usually par) and not the residual value.
Third, preferred stocks often have “credit” ratings that are similar to those
issued to bonds.
Fourth, preferred stock is sometimes convertible into common stock.
Finally, most preferred stock issues today are not true perpetuities.
Increasingly, preferred stock issues have the sinking fund feature, which
require mandatory annual retirement schedules.
6
9.2 Common Stock Valuation
Valuation of common and preferred stock is done by using the same basic
methodology that was discussed for bond valuations in Chapter 6.
Applying the valuation procedure to common stocks is more difficult than applying it
to bonds for various reasons.
First, in contrast to coupon payments on bonds, the size and timing of the
dividend cash flows are less certain.
Second, common stocks are true perpetuities in that they have no final
maturity date.
Finally, unlike the rate of return, or yield, on bonds, the rate of return on
common stock is not directly observable.
A. A One-Period Model
A one-period model provides an estimate of the market price.
The value of an asset is the present value of its future cash flows—the future
dividend and the end-of-period stock price.
B. A Two-Period Model
This model can be viewed as two one-period models strung together.
C. A Perpetuity Model
A series of one-period stock pricing models is strung together to arrive at a
stock perpetuity model.
Though theoretically sound, this model is not practical to apply because the
number of dividends could be infinite.
D. The General Dividend Valuation Model
7
Equation 9.1 is a general expression for the value of a share of stock. It says
that the price of a share of stock is the present value of all expected future
dividends.
The formula does not assume any specific pattern for future cash
dividends, such as a constant-growth rate.
It does not make any assumption about when the share of stock is going to
be sold in the future.
Finally, the model says that to compute a stock’s current value, we need to
forecast an infinite number of dividends.
Equation 9.1 implies that the underlying value of a share of stock is
determined by the market’s expectations of future cash flows that the firm can
generate.
In efficient markets, stock prices change constantly as new information
becomes available and is discounted into the firm’s market price.
For publicly traded companies, there is a constant stream of information about
the firm that reaches the market, with some having an impact on the stock
price while other information has no effect.
E. The Growth Stock Pricing Paradox
Growth stocks are typically defined as the stocks of companies whose earnings
are growing at above-average rates and are expected to continue to do so for
some time.
Fast growing companies typically pay no dividends on their stock during their
growth phase because management believes that the company has a number of
8
high-return investment opportunities and that both the company and its
investors will be better off if earnings are reinvested.
Equation 9.1 predicts and common sense says that if you own stock in a
company that will never pay you any cash, the market value of those shares of
stock are worth absolutely nothing.
In reality, these firms will eventually pay out dividends in the distant future.
If the internal investments succeed, the stock’s price should go up
significantly, and investors can sell their stock at a price much higher than
what they paid.
9.3 Stock Valuation: Some Simplifying Assumptions
To make Equation 9.1 more applicable, some simplifying assumptions about
the pattern of dividends are necessary.
Three different assumptions can cover most growth patterns.
Dividend payments remain constant over time; that is, they have a growth
rate of zero.
Dividends have a constant-growth rate.
Dividends have a mixed growth rate pattern; that is, dividends have one
payment pattern then switch to another.
A. Zero-Growth Dividend Model
The dividend payment pattern remains constant over time:
D1 = D2 = D3 = . . . = D∞
In this case the dividend-discount model (Equation 9.1) becomes:
9
This cash flow pattern essentially describes a perpetuity with a constant cash
flow. In Chapter 6 we developed the present value of a perpetuity with a
constant cash flow as CF/i, where CF is the constant cash flow and i is the
interest rate. Similarly, Equation 9.2 gives the valuation model for a zero-
growth stock.
B. Constant-Growth Dividend Model
Cash dividends do not remain constant but instead grow at some average rate g
from one period to the next forever.
Constant dividend growth is an appropriate assumption for mature companies
with a history of stable growth.
While an infinite horizon is still hard to comprehend, far-distant dividends
have a small present value and contribute very little to the price of the stock.
Deriving the constant-growth dividend model is fairly straightforward. First,
we need to build a model to compute the value of dividend payments for any
time period.
The constant-growth dividend model is easy to do because it is just an
application of Equation 6.6 from Chapter 6.
Recall that the equation for a growing perpetuity in Chapter 6 is given by
Equation 6.6:
PVA∞ = CF1/(i − g)
10
In other words, the constant-growth dividend model tells us that the current
price of a share of stock is the next period dividend divided by the difference
between the discount rate and the dividend growth rate.
Equation 9.4 shows how to value a constant-growth stock:
C. Computing Future Stock Prices
The constant-growth dividend model (Equation 9.4) can be modified to
determine the value, or price, of a share of stock at any point in time.
This results in Equation 9.5, which shows that the price of a share of stock at
time t is as follows:
D. The Relationship between R and g
The constant-growth dividend model yields solutions that are invalid anytime
the dividend growth rate equals or exceeds the discount rate (g ≥ R).
If g = R, the value of the denominator is zero and the value of the stock is
infinite, which makes no sense.
If g > R, the present value of the dividend gets bigger and bigger rather than
smaller and smaller, as it should. This implies that a firm that is growing at a
very fast rate does so forever.
E. Supernormal Growth Dividend Model
During the early part of their lives, very successful firms experience a
supernormal rate of growth in earnings.
11
To value a share of stock for a firm with supernormal dividend growth
patterns, we can apply Equation 9.1, our general dividend model, and Equation
9.5, which gives us the price of a share of stock with constant dividend growth
at any point in time.
Thus, our valuation model is (Equation 9.6):
P0 = PV (Mixed dividend growth) + PV (Constant dividend growth)
9.4 Valuing Preferred Stock
In computing the value of preferred stock, one needs to know whether the preferred
stock has an effective “maturity” because of a sinking fund option or call option.
The most significant difference between preferred stock with a fixed maturity and a
bond is the risk of default.
Since preferred stock dividends are declared by the board of directors, failure to pay
dividends does not result in default.
The failure to pay a preferred stock dividend as promised is a serious financial breach
and signals to the market that the firm is in serious financial difficulty.
A. Preferred Stock with a Fixed Maturity
We can use the bond valuation model developed in Chapter 8 to determine its
price, or value.
Equation 8.2 can be restated as the price of a share of preferred stock (PS0):
Preferred stock price = PV (Dividend payments) + PV (Par value)
12
(9.7)
Since most preferred stock make quarterly dividend payments, m equals 4.
B. Perpetual Preferred Stock
Some preferred stock issues have no maturity.
Dividends are constant over time (g = 0).
Fixed dividend payments go on forever.
We can use Equation 9.2 to value such preferred stock issues.
13
II. Suggested and Alternative Approaches to the Material
This chapter focuses on equity securities and how they are valued. First, the fundamental factors
that determine a stock’s price or value are examined; then the authors construct several valuation
models that estimate this price. These models tell us what the stock’s price should be and can
then be used to compare our estimate against the actual market price.
This chapter begins with a discussion of the secondary markets for equity securities and
their market efficiency. Next, the authors explain how to read stock market price listings in the
newspaper, and introduce the types of equity securities that firms typically issue. Then they
develop a general valuation model and conclude that the value of a share of stock is the present
value of all expected future cash dividends. Some simplifying assumptions about dividend
payments are used to implement this valuation model. These assumptions correspond to actual
practice and allow for the development of several valuation models that are theoretically sound
and easy to apply.
Since market efficiency was discussed in Chapter 7, instructors can begin with a
overview of the market structure before proceeding to a discussion of stock valuation.
14
III. Summary of Learning Objectives
1. List and describe the four types of secondary markets.
The four types of secondary markets are (1) direct search, (2) broker, (3) dealer, and (4)
auction. In direct search markets, buyers and sellers seek each other out directly. In broker
markets, brokers bring buyers and sellers together for a fee. Trades in dealer markets go
through dealers who buy securities at one price and sell at a higher price. The dealers face the
risk that prices could decline while they own the securities. Auction markets have a fixed
location where buyers and sellers confront each other directly and bargain over the
transaction price.
2. Explain why many financial analysts treat preferred stock not as a true equity, but as a
special type of bond.
Preferred stock represents ownership in a corporation and entitles the owner to a guaranteed
dividend, which must be paid before dividends are paid to common stockholders. Similar to
bonds, preferred stock issues have credit ratings, are sometimes convertible to common stock,
and are often callable. Unlike owners of common stock, owners of nonconvertible preferred
stock owners do not have voting rights and do not participate in the firm’s profits beyond the
fixed dividends they receive. Because of their strong similarity to bonds, many financial
analysts treat preferred stock that are not true perpetuities as a form of debt rather than equity.
3. Describe how the general dividend-valuation model values a share of stock.
15
The general dividend-valuation model values of a share of stock as the present value of all
future cash dividend payments, where the dividend payments are discounted using the rate of
return required by investors for a particular risk class.
4. Discuss the assumptions that are necessary to make the general dividend-valuation
model easier to use, and be able to use the model to compute the value of a firm’s stock.
The problems with the general dividend-valuation model are that the exact discount rate that
should be used is unknown, dividends are often uncertain, and some companies do not pay
dividends at all. To make the model easier to apply, we make assumptions about the dividend
payment patterns of business firms. These simplifying assumptions allow the development of
more manageable models, and they also conform with the actual dividend policies of many
firms. Dividend patterns include the following: (1) dividends are constant (zero growth), as
computed in Learning by Doing Application 9.1; (2) dividends have a constant-growth
pattern (they grow forever at a constant rate g), as computed in Learning by Doing
Application 9.2; and (3) dividends grow first at a nonconstant rate then at a constant rate, as
computed in the Redteck example at the end of Section 9.3.
5. Explain why g must be less than R in the constant-growth dividend model.
The constant-growth dividend model assumes that dividends will grow at a constant rate
forever. The constant-growth model shows that if g = R, the value of the denominator is zero
and the value of the stock is infinite, which, of course, is nonsense. If g > R, the value of the
denominator is negative, as is the value of the stock, which also does not make economic
sense. Thus, g must always be less than R (g < R).
16
6. Explain how valuing a preferred stock with a stated maturity differs from valuing a
preferred stock with no maturity date, and be able to calculate the price of a share of
preferred stock under both conditions.
When preferred stock has a maturity date, financial analysts treat it as they treat any other
debt obligation—that is, like a bond. To value such preferred stock, we can use the bond
valuation model from Chapter 8. Before using the model, we need to recognize that we will be
using dividends in the place of coupon payments and that the par value of the stock will replace
the par value of the bond. In addition, while bond coupons are paid semiannually, preferred
dividends are paid quarterly. When a preferred stock has no stated maturity, it becomes a
perpetuity, with the dividend becoming the constant payment that goes on forever. We use the
perpetuity valuation model represented by Equation 9.2 to price such stocks. The calculations
appear in Learning by Doing Application 9.5 and the Northwest Airlines example at the end of
Section 9.4.
IV. Summary of Key Equations
Equation Description Formula
9.1The general dividend-
valuation model
55
44
33
221
0 )R1(
D
)R1(
D
)R1(
D
)R1(
D
)R1(
DP
9.2Zero-growth dividend
modelP0 = D/R
17
9.3
Value of a dividend at
time t in a constant-
growth scenario.
Dt = D0 x (1 + g)t
9.4Constant-growth
dividend model
9.5
Value of a stock at time t
when dividends grow at a
constant rate.
9.6Supernormal growth
stock-valuation model
9.7Value of preferred stock
with a fixed maturity
18
V. Before You Go On Questions and Answers
Section 9.1
1. How do dealers differ from brokers?
A dealer differs from a broker in that a dealer takes ownership of assets and is exposed to
inventory risk, while a broker only facilitates a transaction on behalf of a client. Unlike
brokers, dealers are subject to capital risk, because they must finance their inventories of
securities.
2. What does the price-earnings ratio tell us?
The price-earnings ratio is the firm’s current price divided by the current earnings. When
valuing a company’s stock, it is useful to look at a company’s ratio and compare it to that of
similar firms. In general, a high P/E ratio means high projected earnings in the future. In
addition, it also tells us how much investors are willing to pay per dollar of earnings.
3. Why is preferred stock often viewed as a special type of a bond rather than a stock?
Just like debt, preferred stock is often callable and may be convertible into common stock.
Also, like debt, it has “credit” ratings that are similar to those issued to bonds. But most
important, preferred stock has no voting rights and just like bonds pays fixed dividends. For
these reasons, many analysts treat preferred stock as a special kind of debt rather than an
equity.
19
Section 9.2
1. What is the general formula used to calculate the price of a share of a stock? What does it
mean?
The general formula developed to value a share of stock is as follows:
5
54
43
32
210 )R1(
D
)R1(
D
)R1(
D
)R1(
D
)R1(
DP
It says that the price of a share of stock is the present value of all expected future dividends,
or: Stock price = PV (all future cash dividends).
2. What are growth stocks, and why do they typically pay little on dividends?
Growth stocks are defined as equity in any company whose earnings are growing faster than
the average firm and the higher growth rate is expected to continue for some time. Instead of
paying dividends, these firms reinvest the earnings back into the firm to pursue other high-
return investment opportunities.
Section 9.3
1. What three different models are used to value stocks based on different dividend patterns?
Based on dividend patterns, we can use the following three models to value stock: (1) zero-
growth dividend model, (2) constant-growth dividend model, or (3) supernormal dividend
growth model.
20
2. Explain why the growth rate g must always be less than the rate of return R.
For the valuation equation to have any meaning, the firm’s dividend growth rate g must be
less than the rate of return R. It also makes intuitive sense, since no firm can in the long run
grow faster than the rate of economy. If that was the case, this one firm would become 100
percent of the economy and that makes no sense.
Section 9.4
1. Why is it important for a firm not to skip payment of a preferred dividend?
A preferred dividend is treated like an interest payment on debt by financial markets and
investors. Although the firm will not be in default, any delay or failure to pay the dividends
by the firm will be treated seriously by the financial markets and make them think that the
firm is in financial difficulty.
2. How is a preferred stock with a fixed maturity valued?
Any preferred stock with a defined maturity date is similar to a bond with a fixed maturity
date. This similarity allows for the preferred stock to be valued in a similar fashion after
making adjustments for differences in the preferred stock relative to a bond. Using Equation
8.2, developed to value bonds, the coupons are replaced by the preferred dividends and the
number of payments is adjusted to four a year to accommodate the fact that firms pay the
21
dividends every quarter and not semiannually. This results in the model represented by
Equation 9.7.
22
VI. Self Study Problems
9.1 Ted McKay has just bought the common stock of Ryland Corp. The company expects to
grow at the following rates for the next three years: 30 percent, 25 percent, and 15
percent. Last year the company paid a dividend of $2.50. Assume a required rate of return
of 10 percent. Compute the expected dividends for the next three years and also the
present value of these dividends.
Solution:
Expected dividends for Ryland Corp and their present value:
0 10% 1 2 3
├─────────┼─────────┼─────────┼─────────┼──────────›
D0 = $2.50 D1 D2 D3
g1 = 30% g2 = 25% g3 = 15%
D1 = D0(1 + g1) = $2.50(1 + 0.30) = $3.25
D2 = D1(1 + g2) = $3.25(1 + 0.25) = $4.06
D3 = D2(1 + g3) = $4.06(1 + 0.15) = $4.67
Present value of the dividends = PV(D1) + PV(D2) + PV(D3)
= $2.96 + $3.36 + $3.51
= $9.83
9.2 Merriweather Manufacturing Company has been growing at a rate of 6 percent for the
past two years, and the CEO expects the company to continue to grow at this rate for the
23
next several years. The company paid a dividend of $1.20 last year. If your required rate
of return was 14 percent, what is the maximum price that you would be willing to pay for
this company’s stock?
Solution:
Present value of Merriweather stock:
0 14% 1 2 3
├─────────┼─────────┼─────────┼─────────┼──────────›
D0 = $1.20 D1 D2 D3
g = 6%
D1 = D0(1 + g) = $1.20(1 + 0.06) = $1.27
The maximum price you should be willing to pay for this stock is $15.88.
9.3 Clarion Corp. has been selling electrical supplies for the past 20 years. The company’s
product line has seen very little change in the past five years, and the company does not
expect to add any new items for the foreseeable future. Last year, the company paid a
dividend of $4.45 to its common stockholders. The company is not expected to grow its
revenues for the next several years. If your required rate of return for such firms is 13
percent, what is the current value of this company’s stock?
Solution:
24
Present value of Clarion Corp. stock:
0 13% 1 2 3
├─────────┼─────────┼─────────┼─────────┼──────────›
D0 = $4.45 D1 D2 D3
g = 0%
Since the company’s dividends are not expected to grow,
D0 = D1 =D2=……..D∞ = $4.45 = D
Present value of the stock = D/R
= $4.45/0.13
= $34.23
9.4 Barrymore Infotech is a fast growing communications company. The company did not
pay a dividend last year and is not expected to do so for the next two years. Last year the
company’s growth accelerated, and it expects to grow at a rate of 35 percent for the next
five years before slowing down to a more stable growth rate of 7 percent for the next
several years. In the third year, the company has forecasted a dividend payment of $1.10.
Calculate the price of the company’s stock at the end of its rapid growth period (i.e., at
the end of five years). Your required rate of return for such stocks is 17 percent. What is
the current price of this stock?
Solution:
Present value of Barrymore Infotech stock:
25
0 17% 1 2 3 4 5 6 7 8
├────┼────┼────┼────┼────┼────┼────┼────┤
D0 D1 D2 D3 D4 D5 D6
g1 – g5 = 35% g6 = 7%
D0 = D1 =D2 = 0
D3 = $1.10
D4 = D3(1 + g4) = $1.10(1 + 0.35) = $1.485
D5 = D4(1 + g5) = $1.485(1 + 0.35) = $2.005
D6 = D5(1 + g6) = $2.005(1 + 0.07) = $2.145
Price of stock at t = 5 P5:
Present value of the dividends = PV(D1) + PV(D2) + PV(D3) + PV(D4) + PV(D5)
Present value of stock = PV(Dividends) + PV(P5)
= $2.39 + [$21.45/(1.17)5]
= $2.39 + $9.78
= $ 12.17
26
9.5 You are interested in buying the preferred stock of a bank that pays a dividend of $1.80
every quarter. If you discount such cash flows at 8 percent, what is the price of this
stock?
Solution:
Present value of preferred bank stock:
Quarterly dividend on preferred stock = D = $1.80
Required rate of return = 8%
Current price of stock = (D/R )
= $1.80/0.02
= $90.00
27
VII. Critical Thinking Questions
9.1 Why does the market price of a security vary from the true equilibrium price?
Let us start by first defining the market equilibrium price of a security as the price that
equates the demand for a security with the supply of the security. The role of the security
markets is to bring buyers and sellers together in the most efficient way such that
securities are bought and sold at the true equilibrium price. In reality, however, barriers
of various kinds including the geographic separation of the two parties make the market
price of a security slightly different than the true equilibrium price. The more efficient the
market place, the smaller the deviation between the two.
9.2 Why are investors and managers concerned about market efficiency?
The role of secondary markets is to bring buyers and sellers together. Ideally, we would
like security markets to be as efficient as possible. Markets are efficient when current
market prices of securities traded reflect all available information relevant to the security.
If this is the case, security prices will be near or at their equilibrium price. The more
efficient the market, the more likely this is to happen. This makes it easier for managers
to price the securities close to the equilibrium price.
What investors are most concerned about is having complete information
regarding a security’s current price and where that price information can be obtained.
Efficient markets allow them to trade at prices that are closer to the true equilibrium price
than otherwise possible.
28
Thus, both investors who provide funds and managers (firms) who raise money
are concerned when high transaction costs lead to inefficient markets.
9.3 Why are common stockholders considered to be more at risk than the holders of other
types of securities?
In the hierarchy of lenders of funds to a firm, common stockholders have the most to
lose. In the event of a firm becoming bankrupt, the law requires that creditors of different
types, including bondholders, be paid off first. Next, preferred stockholders are paid off.
Finally, common stockholders receive their investment if any funds are still available.
Thus, common stockholders receive their money back last and are placed at most risk.
This feature of common equity is referred to as residual claim.
9.4 How can individual stockholders avoid double taxation?
Double taxation refers to the fact that in the United States a firm’s income is taxed first
and then any dividends paid to investors get taxed at the personal tax rate. Thus, investors
pay taxes twice. Some investors who desire to get around this problem try to invest in
growth firms that do not pay out dividends but instead reinvest in the firm. This allows
the firms to grow with internal capital and leads to the firm’s value growing faster.
Stockholders benefit from rising stock prices and can sell some or all of their holdings
and generate capital gains, which are taxed at a lower rate than income.
29
9.5 What does it mean when a company has a very high P/E ratio? Give examples of
industries in which you believe high P/E ratios are justified.
A high P/E ratio implies that investors believe that the firm has good prospects for
earnings growth in the future. In fact, they believe that the firm will have higher growth
potential than firms with lower P/E ratios. Companies in industries that are fast growing
like biotech or any hi-tech industry have high P/E ratios. In the past, firms like Cisco and
Intel had very high P/E ratios. As these firms matured and settled to annual growth rates
of 15 percent or less, their P/E ratios have declined.
9.6 Preferred stock is considered to be nonparticipating because
a. investors do not participate in the election of the firm’s directors.
b. investors do not participate in the determination of the dividend payout policy.
c. investors do not participate in the firm’s earnings growth.
d. none of the above.
c. Nonparticipating implies that the preferred dividend remains constant regardless
of any increase in the firm’s earnings. Thus, investors in a firm’s preferred stock will not
see higher dividends when the firm’s earnings increase. Nor will they see a decrease if
the firm’s earnings decrease.
30
9.7 Explain why preferred stock is considered to be a hybrid of equity and debt securities.
The law considers preferred stock as equity. Thus, holders are treated as the firm’s
owners. Also, like common stockholders, preferred stockholders have to pay taxes on
their dividend income. However, preferred stockholders do not have any voting rights. In
addition, they receive only a fixed dividend just like bondholders. If a firm is liquidated,
then they receive a stated value (par value) similar to bondholders. Preferred stock is
rated by credit rating agencies just like bonds. Some preferred issues are convertible to
the firm’s common stock just as convertible bonds. Some preferred issues are not
perpetual and have a fixed maturity just like bonds.
Thus preferred stock is a hybrid security—like equity in some ways and like debt
security in others.
9.8 Why is stock valuation more difficult than bond valuation?
Despite the availability of mathematical models to value stocks, it is more difficult to
apply valuation techniques to stocks than to bonds. First, unlike bonds, firms are not in
default if dividends are not declared. This makes it difficult to determine the size and
timing of the cash flows. Second, common stock, unlike bonds, does not have a fixed
maturity, and hence, it is difficult to determine a terminal value unlike bonds, which have
a maturity value. Next, it is easier to calculate the present value of a bond because the
31
required rate of return is observable. In the case of stocks, it is rather difficult to estimate
a required rate of return for many stocks and classify them into different risk groups.
9.9 You are currently thinking about investing in a stock valued at $25.00 per share. The
stock recently paid a dividend of $2.25 and is expected to grow at a rate of 5 percent for
the foreseeable future. You normally require a return of 14 percent on stocks of similar
risk. Is the stock overpriced, underpriced, or correctly priced?
This stock is underpriced at $25. Using the constant-growth model, we can arrive at a
price of $26.25 for this stock. This makes the stock underpriced, and it should be
considered a good buy.
9.10 Stock A and Stock B are both priced at $50 per share. Stock A has a P/E ratio of 17,
while Stock B has a P/E ratio of 24. Which is the more attractive investment, considering
everything else to be the same, and why?
Stock A is the more attractive investment because it has a lower P/E ratio. The lower the
P/E ratio, the larger the amount of earnings supporting the stock price. This makes Stock
A a more attractive investment than Stock B.
32
VIII. Questions and Problems
BASIC
9.1 Present value of dividends: Fresno Corp is a fast growing company. The company
expects to grow at a rate of 30 percent over the next two years and then slow down to a
growth rate of 18 percent for the following three years. If the last dividend paid by the
company was $2.15, estimate the dividends for the next five years. Compute the present
value of these dividends if the required rate of return was 14 percent.
Solution:
0 1 2 3 4 5
├───────┼────────┼───────┼────────┼───────┤
D0 = $2.15 g1-2 = 30%; g3-5 = 18%; kCS = 14%
D1 = D0(1 + g1) = $2.15(1.30) = $2.795
D2 = D1(1 + g2) = $2.795(1.30) = $3.634
D3 = D2(1 + g3) = $3.634(1.18) = $4.288
D4 = D3(1 + g4) = $4.288(1.18) = $5.06
D5 = D4(1 + g4) = $5.06(1.18) = $5.97
33
9.2 Zero growth: Nynet, Inc., paid a dividend of $4.18 last year. The company does not
expect to increase its dividend for the next several years. If the required rate of return is
18.5 percent, what is the current price of the stock?
Solution:
D0 = $4.18; g = 0; R = 18.5%
9.3 Zero growth: Knight Supply Corp. has seen no growth for the last several years and
expects the trend to continue. The firm last paid a dividend of $3.56. If you require a rate
of return of 13 percent, what is the current stock price?
Solution:
D0 = $3.56; g = 0; R = 13%
9.4 Zero growth: Ron Santana is interested in buying the stock of First National Bank.
While the bank expects no growth in the near future, Ron is attracted by the dividend
income. Last year, the bank paid a dividend of $5.65. If Ron Santana requires a return of
14 percent on such stocks, what is the maximum price he should be willing to pay?
Solution:
34
D0 = $5.65; g = 0; R = 14%
9.5 Zero growth: The current stock price of Largent, Inc., is $44.72. If the required rate of
return is 19 percent, what is the dividend paid by this firm, which is not expected to grow
in the near future?
Solution:
P0 = $44.72; R = 19%; D = ?;
9.6 Constant growth: Moriband Corp. just declared a dividend of $2.15 yesterday. The
company is expected to grow at a steady rate of 5 percent for the next several years. If
stocks such as these require a rate of return of 15 percent, what should be the market
value of this stock?
Solution:
D0 = $2.15; g = 5%; R = 15%
35
9.7 Constant growth: Nyeil, Inc., is a consumer products firm growing at a constant rate of
6.5 percent. The firm’s last dividend was $3.36. If the required rate of return was 18
percent, what is the market value of this stock?
Solution:
D0 = $3.36; g = 6.5%; R = 18%
9.8 Constant growth: Reco Corp. is expected to pay a dividend of $2.25 next year. The
forecast for the stock price a year from now is $37.50. If the required rate of return is 14
percent, what is the current stock price? Assume constant growth.
Solution:
D1 = $2.25; P1 = $37.50; R = 14%
36
9.9 Constant growth: Proxicam, Inc., is expected to grow at a constant rate of 7 percent. If
the company’s next dividend is $1.15 and its current price is $22.35, what is the required
rate of return on this stock?
Solution:
D1 = $1.15; P0 = $23.00; g = 7%
9.10 Preferred stock valuation: X-Centric Energy Company has issued perpetual preferred
stock with a par of $100 and a dividend of 4.5 percent. If the required rate of return is
8.25 percent, what is the stock’s current market price?
Solution:
D = 4.5% ($100) = $4.50; R = 8.25%
9.11 Preferred stock valuation: The First Bank of Ellicott City has issued perpetual preferred
stock with a $100 par value. The bank pays a quarterly dividend of $1.65 on this stock.
37
What is the current price of this preferred stock given a required rate of return of 11.6
percent?
Solution:
Quarterly dividend = $1.65
Required rate of return = R = 11.6%
9.12 Preferred stock: The preferred stock of Axim Corp. is selling currently at $47.13. If
your required rate of return is 12.2 percent, what is the dividend paid by this stock?
Solution:
P0 = $47.13; R = 12.2%
9.13 Preferred stock: Each quarter, Sirkota, Inc., pays a dividend on its perpetual preferred
stock. Today, the stock is selling at $63.37. If the required rate of return for such stocks is
15.5 percent, what is the quarterly dividend paid by this firm?
Solution:
P0 = $63.37; R = 15.5%
38
Annual dividend = $9.82
Quarterly dividend = $9.82 /4 = $2.46
INTERMEDIATE
9.14 Constant growth: Kay Williams is interested in purchasing the common stock of
Reckers, Inc., which is currently priced at $37.45. The company expects to pay a
dividend of $2.58 next year and expects to grow at a constant rate of 7 percent.
a. What should the market value of the stock be if the required rate of return is 14
percent?
b. Is this a good buy?
Solution:
a.
b. The stock is overpriced and not a good buy.
9.15 Constant growth: Your required rate of return is 23 percent. Ninex Corp. has just paid a
dividend of $3.12 and expects to grow at a constant rate of 5 percent. What is the
expected price of the stock three years from now?
39
Solution:
R = 23%; D0 = $3.12; g = 5%
9.16 Constant growth: Jenny Banks is interested in buying the stock of Fervan, Inc., which is
growing at a constant rate of 6 percent. Last year, the firm paid a dividend of $2.65. Her
required rate of return is 16 percent. What is the current price for this stock? What would
be the price of the stock in year 5?
Solution:
g = 6%, D0 = $2.65, R = 16%
9.17 Nonconstant growth: Tre-Bien, Inc., is a fast growing technology company. The firm
projects a rapid growth of 30 percent for the next two years, then a growth rate of 17
percent for the following two years. After that, the firm expects a constant growth rate of
8 percent. The firm expects to pay its first dividend of $2.45 a year from now. If your
required rate of return on such stocks is 22 percent, what is the current price of the stock?
40
Solution:
g1 = g2 = 30%, g3 = g4 = 17%, g = 8%, D1 = $2.45, R = 22%
D1 = $2.45, D2 = $2.45(1.30) = $3.19, D3 = $3.19(1.17) = $3.73
D4 = $3.73(1.17) = $4.36, D5 = 4.36(1.08) = $4.71
9.18 Nonconstant growth: ProCor, a biotech firm, forecasted the following growth rates for
the next three years: 35 percent, 28 percent, and 22 percent. The company then expects to
grow at a constant rate of 9 percent for the next several years. The company paid a
dividend of $1.75 last week. If the required rate of return is 20 percent, what is the market
value of this stock?
Solution:
g1 = 35%; g2 = 28%; g3 = 22%; g4 = 9%; D0 = $1.75; R = 20%
41
9.19 Nonconstant growth: Revarop, Inc., is on a fast growth stock and expects to grow at a
rate of 23 percent for the next four years. It then will settle to a constant-growth rate of 6
percent. The first dividend will be paid in year 3 and be equal to $4.25. If the required
rate of return is 17 percent, what is the current price of the stock?
Solution:
g1-4 = 23%; g = 6%; D3 = $4.25; R = 17%
D4 = D3 (1.23) = $4.25(1.23) = $5.23
42
9.20 Nonconstant growth: Quansi, Inc., expects to pay no dividends for the next six years. It
has projected a growth rate of 25 percent for the next seven years. After seven years, the
firm will grow at a constant rate of 5 percent. Its first dividend to be paid in year 7 will be
worth $3.25. If your required rate of return is 24 percent, what is the stock worth today?
Solution:
gconstant = 5%; R = 24%; D7 = $3.25; D1 – D6 = 0
9.21 Nonconstant growth: Staggert Corp. will pay dividends of $5.00, $6.25, $4.75, and
$3.00 for the next four years. Thereafter, the company expects its growth rate to be at a
constant rate of 6 percent. If the required rate of return is 18.5 percent, what is the current
market price of the stock?
Solution:
D1 = $5; D2 = $6.25; D3 = 4.75; D4 = $3; g = 6%; R = 18.5%;
43
9.22 Nonconstant growth: Diaz Corp. is growing rapidly at a rate of 35 percent for the next
seven years. The first dividend to be paid three years from now will be worth $5. After
seven years the company will settle to a constant growth rate of 8.5 percent. What is the
market value for this stock given a required rate of return of 14 percent?
Solution:
g1-7 = 35%; D3 = $5.00; g = 8.5%; R = 14%
D1 = D2 = 0 ; D3 = $5
D4 = 5(1.35) = $6.75
D5 = $6.75(1.35) = $9.11
D6 = 9.11 (1.35) = $12.30
D7 = $12.30(1.35) = $16.61
D8 = 16.61(1.085) = $18.02
44
9.23 Nonconstant growth: Tin-Tin Waste Management, Inc., is growing rapidly. Dividends
are expected to grow at rates of 30 percent, 35 percent, 25 percent, and 18 percent over
the next four years. Thereafter the company expects to grow at a constant rate of 7
percent. The stock is currently selling at $47.85 and the required rate of return is 16
percent. Compute the dividend for the current year (D0).
Solution:
g1 = 30%; g2 = 35%; g3 = 25%; g4 = 18%; g = 7%; R = 16%; P0 = $47.85
45
ADVANCED
9.24 Riker Departmental Stores has forecasted a high growth rate of 40 percent for the next
two years, followed by growth rates of 25 percent and 20 percent for the following two
years. It then expects to stabilize its growth to a constant rate of 7.5 percent for the next
several years. The firm paid a dividend of $3.50 recently. If the required rate of return is
18 percent, what is the current market price of the stock?
Solution:
g1-2 = 40%; g3 = 25%; g4 = 20%; g = 7.5%; D0 = $3.50; R = 18%
9.25 Courtesy Bancorp issued perpetual preferred stock a few years ago. The bank pays an
annual dividend of $4.27, and your required rate of return is 12.2 percent.
46
a. What is the value of the stock given your required rate of return?
b. Should you buy this stock if its current market price is $34.41? Explain.
Solution:
a. D = $4.27; R = 12.2%
b. Since the stock is worth $35.00 but can be bought for $34.41, you should buy this
stock.
9.26 Rhea Kirby owns shares in Ryoko Corp. Currently, the market price of the stock is
$36.34. The company expects to grow at a constant rate of 6 percent for the foreseeable
future. Its last dividend was worth $3.25. Roger Kirby’s required rate of return for such
stocks is 16 percent. She wants to find out whether she should sell his shares or add to her
holdings.
a. What is the value of this stock?
b. Based on your answer above, should she buy additional shares in Ryoko Corp?
Why or why not?
Solution:
a.
b. No, she should not buy more shares. This stock is overpriced with the stock
selling at a higher price than what it is worth. She should sell her shares.
47
9.27 Perry, Inc., declared a dividend of $2.50 yesterday. You are interested in investing in this
company, which has forecasted a constant-growth rate of 7 percent for the next several
years. Your required rate of return is 18 percent.
a. Compute the expected dividends D1, D2, D3, and D4.
b. Find the present value of these four dividends.
c. What is the price of the stock four years from now (i.e., P4)?
d. Calculate the present value of P4. Add the answer you got in part (b). What is the
price of the stock today?
e. Use the equation for constant growth (Equation 9.4) and compute the price of the
stock today.
Solution:
a. D0 = $2.50 g = 7% R = 18%
b.
c.
48
d.
e. For a constant-growth stock:
9.28 Zweite Pharma is a fast growing drug company. The company forecasts that in the next
three years, its growth rates will be 30 percent, 28 percent, and 24 percent, respectively.
Last week it declared a dividend of $1.67. After three years the company expects a more
stable growth rate of 8 percent for the next several years. Your required rate of return is
14 percent.
a. Compute the dividends for the next three years and find their present value.
b. Calculate the price of the stock at the end of year 3 when the firm settles to a
constant-growth rate.
c. What is the current price of the stock?
Solution:
g1 = 30%; g2 = 28%; g3 = 24%; g = 8%; D0 = $1.67; R = 14%
49
a.
b.
c.
9.29 Triton, Inc., expects to grow at a rate of 22 percent for the next five years and then settle
to a constant-growth rate of 6 percent. The company’s most recent dividend was $2.35.
The required rate of return is 15 percent.
a. Find the present value of the dividends during the rapid growth period.
b. What is the price of the stock at the end of year 5?
c. What is the price of the stock today?
Solution:
g1-5 = 22%; g = 6%; D0 = $2.35; R = 15%
50
a.
b.
c.
9.30 Ceebros Builders are expanding very fast and expect to grow at a rate of 25 percent for
the next four years. They recently declared a dividend of $3.60 but do not expect to pay
any dividends for the next three years. In year 4, they intend to pay a $5 dividend and
thereafter grow it at a constant-growth rate of 6 percent. The required rate of return on
such stocks is 20 percent.
a. Calculate the present value of the dividends during the fast growth period.
b. What is the price of the stock at the end of the fast growth period (P4)?
51
c. What is the stock price today?
d. Would today’s stock price be driven by the length of time you intend to hold the
stock?
Solution:
a. g1-4 = 25% g5 = 6% D0 = $3.60 D4 = $5.00 kcs = 20%
b.
c.
d. No, the length of the holding period has no bearing on today’s stock price.
52
Sample Test Problems
9.1 Mason Corp. is a manufacturer of consumer staples and has experienced no growth for
the last five years while paying out a dividend of $3.50 every year. The CFO of the firm
expects the firm to have no growth for the foreseeable future. If your required rate of
return is 10 percent, what is the price of this stock today?
Solution:
D0 = $3.50; g = 0; R = 10%
9.2 Bucknell, Inc., recently paid a dividend of $2.10. The firm forecasts a growth of 6 percent
for the next several years. What is the price of the stock today given your discount rate of
13 percent?
Solution:
D0 = $2.10; g = 6%; R = 13%
53
9.3 Bradley Corp. is growing at a constant rate of 7.2 percent every year. Last week the
company paid a dividend of $1.85. If your required rate of return is 15 percent, what will
be the stock’s price four years from now?
Solution:
D0 = $1.85; g = 7.2%; R = 15%
9.4 Wichita Technologies is expected to grow at a rate of 35 percent for the next three years
and then settle to a constant-growth rate of 7 percent. The company will pay no dividend
for the first two years and then pay a dividend of $1.25 in year three. What is the
company’s price when the company’s supernormal growth ends? What is the price of the
stock today? The firm’s required rate of return is 12 percent.
Solution:
g1-3 = 35%; g = 7%; D1 = D2 = $0; D3 = $1.25; R = 12%
54
9.5 UNC Bancorp has issued preferred stock with no maturity date. It has a par value of $100
and pays a quarterly dividend of $2.25. If your required rate of return is 8 percent, what is
the price of the stock today?
Solution:
Quarterly dividend = $2.25
Required rate of return = R = 8%
55