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CHAPTER 7
The Valuation and
Characteristics of Stock
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Common Stock
 Background
Stockholders own the corporation, but in
many instances the corporation is
widely held
• Stock ownership is spread among a
large number of people
Because of this, most stockholders are
only interested in how much money
they will receive as a stockholder
• Most equity investors aren’t
interested in a role as owners
The Return on an Investment in
Common Stock
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 The future cash flows associated with stock ownership consists
of
– Dividends and
– The eventual selling price of the shares
 If you buy a share of stock for price P0, hold it for one year during
which time you receive a dividend of D1, then sell it for a price P1,
your return, k, would be:
k=
D1+ P1 -P0 
P0
or
k=
D1
P0
dividend yield
+
P1-P0 
P0
capital gains yield
A capital gain (loss) occurs
if you sell the stock for a
price greater (lower) than
you paid for it.
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The Intrinsic (Calculated) Value and
Market Price
 A stock’s intrinsic value is based on
assumptions made by a potential investor
Must estimate future expected cash flows
• Need to perform a fundamental
analysis of the firm and the industry
 Different investors with different cash flow
estimates will have different intrinsic
values
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Price versus Earnings
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Developing Growth-Based Models
 Realistically most people tend to forecast growth rates rather
than cash flows
 A stock’s value today is the sum of the present values of the
dividends received while the investor holds it and the price for
which it is eventually sold
 An Infinite Stream of Dividends
Many investors buy a stock, hold for awhile, then sell, as
represented in the above equation
• However, this is not convenient for valuation purposes
D1
D2
P0 =


2
1  k  1  k 

Dn
1  k 
n

Pn
1  k 
n
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Developing Growth-Based Models
 A person who buys stock at time n will hold it
until period m and then sell it
Their valuation will look like this:
Pn =
Dn + 1
Dm
Pm
+…+
+
m-n
m-n
1 + k 
1
+
k
1
+
k




 Repeating this process until infinity results in:

P0  
i=1
Di
1 + k 
i
 Conceptually it’s possible to replace the final
selling price with an infinite series of dividends
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Stock Value = PV of Dividends:
D1
D2
D3
D
P̂0 


 ... 
1
2
3

1  k s  1  k s  1  k s 
1  k s 
What is a constant growth stock?
One whose dividends are expected to
grow forever at a constant rate, g.
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The Constant Growth Model
 If dividends are assumed to be growing at a constant rate forever
and we know the last dividend paid, D0, then the model simplifies
to:
 Which represents a series of fractions as follows
P0 =
D0 1  g
1  k 

D0 1  g
1  k 
2
2
D0 1  g
3

1  k 
3


 If k>g the fractions get smaller (approach zero) as the exponents get
larger
If k>g growth is normal
If k<g growth is supernormal
• Can occur but lasts for limited time period
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Constant Normal Growth
 Constant growth model can be simplified to
D1
P0 
k g
 1 
 D1 

 ks  g 
K must be
greater
than g.
 D1 = D0(1+g)
 The constant growth model is a simple
expression for forecasting the price of a
stock that’s expected to grow at a constant,
normal rate
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
The discount rate (ki) is the opportunity
cost of capital, i.e., the rate that could
be earned on alternative investments of
equal risk.
For Bonds:
ki = k* + IP + LP + MRP + DRP.
For Stocks:
Ks = KRF + (KM – KRF)s
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Constant Normal Growth—Example
Example
Q: Atlas Motors is expected to grow at a constant rate of 6% a year
into the indefinite future. It recently paid a dividends of $2.25 a
share. The rate of return on stocks similar to Atlas is about 11%.
What should a share of Atlas Motors sell for today?
A:
D1
P0 
k-g
$2.25 (1.06)

.11 - .06
 $47.70
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What happens if g > ks?
•If ks< g, get negative stock price, which is
nonsense.
D1
P̂0 
requires k s  g.
ks  g

We can’t use model unless (1) ks> g and
(2) g is expected to be constant forever.
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The Zero Growth Rate Case—A Constant
Dividend
 If a stock is expected to pay a constant,
non-growing dividend, each dollar
dividend is the same
 Gordon model simplifies to:
D
P0 
k
 A zero growth stock is a perpetuity to the
investor
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The Expected Return
 Can recast Constant Growth model to focus on
the return (k) implied by the constant growth
assumption
D1
k
g
P0
 g is the expected capital gains (%)
 The higher the expected growth in dividends the
faster the price is expected to grow
 Would this apply to farmland? How?
Basis for growth expectations
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 Just looking at past dividend growth is not very
informative because it can be distorted
 Growth in EPS is the fundamental driver of growth
in dividends (get data on EPS growth)
 EPS growth greater than sales growth is not
sustainable over a long period (so get data on
sales growth)
 Consider industry factors, including the general
economy, that affect growth and market share
 ROE times the retention rate is the fundamental
driver of EPS growth (get data on ROE and
retention rate)
 Also look for expert opinion
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What’s the stock’s market value?
D0 = 2.00, ks = 16%, g = 6%.
Constant growth model:
D1
$2.12
P0 

k s  g 0.16  0.06
$2.12

 $21.20.
0.10
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What is the stock’s market value one
^
year from now, P1?
 D1 will have been paid, so expected
dividends are D2, D3, D4 and so on. Then,
D2
$2.247
P1 

 $22.47.
k s  g 0.16  0.06
D3
$2.382
P2 

 $23.82.
k s  g 0.16  0.06
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Constant Growth Implications
Ks = 16%; g = 6%; 1/(.16-.06) = 10
Year
Dividend Multiple
Stock
Price
Growth
Rate
0
$2.12
10
$21.20
6%
1
$2.247
10
$22.47
6%
2
$2.382
10
$23.82
6%
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Find the expected dividend yield,
capital gains yield, and total return
during the first year.
D1 $2.12
Dividend yld 

 10%.
P0 $21.20
P1  P0
D1
Cap. gains yld 
 k s   6%.
P0
P0
Total return  10%  6%  16%.
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Why do the dividend and capital gains returns
add-up to the required rate of return?
Rearrange model to rate of return form:
D1
D1
P0 
 k s   g.
ks  g
P0
Then, ks = $2.12/$21.20+ 0.06
= 0.10 + 0.06 = 16%.
Dividend yield+capital gain
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What would P0 be if g = 0?
The dividend stream would be a perpetuity.
0
16%
1
2
3
2.00
2.00
2.00
PMT $2.00
P0 

 $12.50.
k
0.16
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Two Stage Growth
 At times a firm’s future growth may not be
expected to be constant
For example, a new product may lead to
temporary high growth
 The two-stage growth model allows us to value a
stock that is expected to grow at an unusual rate
for a limited time
Use the Gordon model to value the constant
portion
Find the present value of the non-constant
growth periods
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Problem 9 from lab
Frazier Inc. paid a dividend of $4 last year
(D0). The firm is expecting dividends to
grow at 21% in years 1–2 and 10% in Year
3. After that growth will be constant at
8% per year. Similar investments return
14%. Calculate the value of the stock
today.
a.$71.49b.$88.31c.$91.47d.$116.10
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ANS: C
D1 = 4(1.21) = 4.84
D2 = D1(1.21) = 5.8564
D3 = D2(1.10) = 6.442
D4 = D3(1.08) = 6.9574032
P3 = [D4]/(.14 – .08) = 115.95672
6.44 + 115.96 = 122.40
Calculator Steps:
CFo = 0, C01 = 4.84, C02 = 5.86, C03 = 122.40; I = 14
Solve for NPV = $91.37
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Example
Two Stage Growth—Example
Q: Zylon Corporation’s stock is selling for $48 a share according to
The Wall Street Journal. We’ve heard a rumor that the firm will
make an exciting new product announcement next week. By
studying the industry, we’ve concluded that this new product will
support an overall company growth rate of 20% for about two
years. After that, we feel growth will slow rapidly and level off at
about 6%. The firm currently pays an annual dividend of $2.00,
which can be expected to grow with the company. The rate of
return on stocks like Zylon is approximately 10%. Is Zylon a
good buy at $48?
A: We’ll estimate what we think Zylon should be worth given our
expectations about growth.
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Two Stage Growth—Example
Example
We’ll develop a schedule of expected dividend payments:
Year
Expected
Dividend
Growth
1
$2.40
20%
2
$2.88
20%
3
$3.05
6%
Next, we’ll use the constant growth model at the point in time
where the growth rate changes and constant growth begins.
That’s year 2, so:
D3
$3.05
P2 

 $76.25
k - g2
.10 - .06
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Do time line and PV on the board
 CF0 = 0
 CF1 = 2.40
 CF2 = 2.88 + 76.32 = 79.20
 Ks = 10%
 NPV =?= 67.64
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Another example: If we have
supernormal growth of 30% for 3 yrs,
^
then a long-run constant
g=6%, what is
P0? ks is still 16%.


Have to project out the
assumptions on a time line.
Apply the constant growth
model after 3 years.
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High growth followed by constant lower growth:
0
ks=16% 1
g = 30%
D0 = 2.00
2
g = 30%
2.60
4
3
g = 30%
3.380
g = 6%
4.394
4.658
46.58
CF0=0
CF1=2.6
CF2=3.38
4.658
P3 
 $46.58
0.16  0.06
CF3=50.97
I=16
NPV = 37.410
= P0
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What is the expected dividend yield
and capital gains yield at t = 0?
At t = 4?
$2.60
Div. yield0 
 6.95%.
$37.41
Therefore the capital gain must be :
Cap. gain0  16%  6.95%  9.055%.
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

During nonconstant growth,
dividend and cap. gains
yields are not constant, and
capital gains yield is less
than g.
After t = 3, g = constant = 6%
= capital gains yield; k =
16%; so D/P = 16 - 6 = 10%.
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If g = -6%, would anyone buy the
stock? If so, at what price?
Firm still has earnings and still pays
dividends, so P0 > 0:
D01 g
D1
P0 

.
ks  g
ks  g
^
$2.00 0.94
$1.88


 $8.55.
0.16   0.06 0.22
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What is the annual D/P and capital
gains yield?
Capital gains yield = g = - 6.0%,
Dividend yield = 16.0% - (-6.0%) =
22%.
D/P and cap. gains yield are
constant, with high dividend yield
(22%) offsetting negative cap.
gains yield.
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super normal growth example
The XYZ corporation has had annual
earnings and dividends increase at the rate
of 75% recently and recently paid
dividends of $4/share. The outlook is for
continued high growth at 50% per year for
the next three years, then a more modest
growth rate of 5% per year for all future
years.
The required return for a company of this
risk is 15%.
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SUPER NORMAL GROWTH ANSWER
 D1=6, D2=9, D3=13.50, D4=13.5(1.05)=14.175
 P3=price of the stock at time 3, when the
constant growth begins
P3=14.175[1/.15-.05]=14.175(10)
=$141.75
 Draw a time line.
 Using the CFj part of your calculator:
CF0=0, CF1=6, CF2=9,
CF3=13.5+141.75=155.25, I=15
NPV=P0=$114.10
Super Normal Growth
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Base
Dividend
Growth
rate
Num- Constant ks
ber of growth
years rate
$1
20%
5
3%
15%
$30
40%
3
4%
16%
$2
30%
2
5%
14%
2
30%
3
5%
14%
7- 38
What is market equilibrium?
In equilibrium, stock prices are stable.
There is no general tendency for
people to buy versus sell.
In equilibrium, expected returns must
equal required returns:
k  D1 P0  g  k  k RF  k M  k RF  b.
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How is equilibrium established?
If ^
k = (D1/ P0) + g > k, then
P0 is “too low,” a bargain.
Buy orders > sell orders;
P0 bid up; D1/P0 falls until
D1/P0 + g = ^k = k.
Why do stock prices change?
7- 40
D1
P0 
ki  g
1. ki could change:
ki = kRF + (kM - kRF )bi
kRF = k* + IP
2. g could change due to economic or firm
situation
3. D could change – usually due to earnings
4. Super-normal growth expectations could
be formed or change.
7- 41
Feb. 4, 1994:
Fed announced
increase in interest rates at 11 a.m.
Result:
Dow Jones fell 95 points.
ki = kRF + (kM - kRF )bi
kRF = k* + IP
D1
P0 
ki  g
7- 42
Securities Analysis
 Securities analysis is the art and science of
selecting investments
 Fundamental analysis looks at a company and its
business to forecast value
 Technical analysis bases value on the pattern of
past prices and volumes
 The Efficient Market Hypothesis says information
moves so rapidly in financial markets that price
changes occur immediately, so it is impossible to
consistently beat the market to bargains
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What’s the Efficient
Market Hypothesis?
EMH: Securities are normally
in equilibrium and are “fairly
priced,” given what is currently
known. One cannot “beat
the market” except through
good luck or inside info.
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1. Weak-form EMH:
Can’t profit by looking at
past trends. A recent decline
is no reason to think stocks
will go up (or down) in the
future. Seems empirically
true, but “technical analysis”
is still used.
7- 45
2. Semi-strong form EMH:
All publicly available info. is
reflected in stock prices, so
doesn’t pay to pore over
annual reports looking for
undervalued stocks. Largely
true, but superior analysts
can still profit by finding and
using new information.
It is VERY hard to tell good luck from
superior stock picking/timing ability.
7- 46
3. Strong-form EMH:
All information, even inside
info, is embedded in stock
prices. Not true--insiders
can gain by trading on the
basis of insider information,
but that’s illegal.
7- 47
Markets are efficient because:
1.
2.
3.
4.
15,000 or so trained analysts; MBAs,
CFAs, Technical PhDs.
Work for firms like Merrill, Morgan,
Prudential, which have much money.
Have similar access to data.
Thus, news is reflected in P0 almost
instantaneously.