Chapter 6
Stock Valuation
McGraw-Hill/Irwin
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Key Concepts and Skills
Understand how stock prices depend on future
dividends and dividend growth
 Be able to compute stock prices using the
dividend growth model
 Understand how growth opportunities affect
stock values
 Understand the PE ratio
 Understand how stock markets work

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Chapter Outline
6.1
6.2
6.3
6.4
6.5
6.6
6.7
McGraw-Hill/Irwin
The Present Value of Common Stocks
Estimates of Parameters in the Dividend-Discount
Model
Growth Opportunities
The Dividend Growth Model and the NPVGO
Model
Price-Earnings Ratio
Some Features of Common and Preferred Stock
The Stock Markets
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
6.1 The PV of Common Stocks


The value of any asset is the present value of its
expected future cash flows.
Stock ownership produces cash flows from:



Dividends
Capital Gains
Valuation of Different Types of Stocks



Zero Growth
Constant Growth
Differential Growth
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Case 1: Zero Growth

Assume that dividends will remain at the same level
forever
Div 1  Div 2  Div 3  
 Since future cash flows are constant, the value of a zero
growth stock is the present value of a perpetuity:
Div 3
Div 1
Div 2
P0 



1
2
3
(1  R) (1  R) (1  R)
Div
P0 
R
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Case 2: Constant Growth
Assume that dividends will grow at a constant rate, g,
forever, i.e.,
Div 1  Div 0 (1  g )
Div 2  Div 1 (1  g )  Div 0 (1  g ) 2
Div 3  Div 2 (1  g )  Div 0 (1  g )3
Since future cash flows grow at a constant rate forever,
the value of a constant growth stock is the present value
of a growing perpetuity:
Div 1
P0 
Rg
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Constant Growth Example
Suppose Big D, Inc., just paid a dividend of
$.50. It is expected to increase its dividend by
2% per year. If the market requires a return of
15% on assets of this risk level, how much
should the stock be selling for?
 P0 = .50(1+.02) / (.15 - .02) = $3.92

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Case 3: Differential Growth
Assume that dividends will grow at different rates
in the foreseeable future and then will grow at a
constant rate thereafter.
 To value a Differential Growth Stock, we need to:

Estimate future dividends in the foreseeable future.
 Estimate the future stock price when the stock
becomes a Constant Growth Stock (case 2).
 Compute the total present value of the estimated
future dividends and future stock price at the
appropriate discount rate.

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Case 3: Differential Growth
 Assume that dividends will grow at rate g1 for N
years and grow at rate g2 thereafter.
Div 1  Div 0 (1  g1 )
Div 2  Div 1 (1  g1 )  Div 0 (1  g1 ) 2
..
.
Div N  Div N 1 (1  g1 )  Div 0 (1  g1 ) N
Div N 1  Div N (1  g 2 )  Div 0 (1  g1 ) N (1  g 2 )
..
.
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Case 3: Differential Growth
Dividends will grow at rate g1 for N years and grow
at rate g2 thereafter
Div 0 (1  g1 ) Div 0 (1  g1 ) 2
…
0
1
2
Div 0 (1  g1 )
…
 Div 0 (1  g1 ) N (1  g 2 )
…
N
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N
Div N (1  g 2 )
N+1
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Case 3: Differential Growth
We can value this as the sum of:
 an N-year annuity growing at rate g1
T

C
(1  g1 ) 
PA 
1 
T 
R  g1  (1  R) 
 plus the discounted value of a perpetuity growing at
rate g2 that starts in year N+1
 Div N 1 


R  g2 

PB 
N
(1  R)
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Case 3: Differential Growth
Consolidating gives:
 Div N 1 


C  (1  g1 )T   R  g 2 
P

1 
T 
N
R  g1  (1  R)  (1  R)
Or, we can “cash flow” it out.
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A Differential Growth Example
A common stock just paid a dividend of $2. The
dividend is expected to grow at 8% for 3 years,
then it will grow at 4% in perpetuity.
What is the stock worth? The discount rate is 12%.
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With the Formula
 $2(1.08)3 (1.04) 



.12  .04
$2  (1.08)  (1.08)3  

P

1 
3
3
.12  .08  (1.12) 
(1.12)

$32.75
P  $54  1  .8966 
3
(1.12)
P  $5.58  $23.31
McGraw-Hill/Irwin
P  $28.89
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With Cash Flows
$2(1.08)
0
1
$2.16
0
$2(1.08)
1
2
$2(1.08)3 $2(1.08)3 (1.04)
…
2
3
$2.33
$2.62
$2.52 
.08
2
3
4
The constant
growth phase
beginning in year 4
can be valued as a
growing perpetuity
at time 3.
$2.16 $2.33 $2.52  $32.75
P0 


 $28.89
2
3
$2.62
1.12 (1.12)
(1.12)
P 
 $32.75
3
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.08
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6.2 Estimates of Parameters

The value of a firm depends upon its growth
rate, g, and its discount rate, R.

Where does g come from?
g = Retention ratio × Return on retained earnings
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Where does R come from?

The discount rate can be broken into two parts.
The dividend yield
 The growth rate (in dividends)


In practice, there is a great deal of estimation
error involved in estimating R.
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Using the DGM to Find R

Start with the DGM:
D 0 (1  g)
D1
P0 

R -g
R -g
Rearrange and solve for R:
D 0 (1  g)
D1
R
g
g
P0
P0
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6.3 Growth Opportunities
Growth opportunities are opportunities to
invest in positive NPV projects.
 The value of a firm can be conceptualized as
the sum of the value of a firm that pays out
100-percent of its earnings as dividends and
the net present value of the growth
opportunities.
EPS
P
 NPVGO
r

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6.4 The NPVGO Model

We have two ways to value a stock:
The dividend discount model
 The sum of its price as a “cash cow” plus the per
share value of its growth opportunities

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The NPVGO Model: Example
Consider a firm that has EPS of $5 at the end of the
first year, a dividend-payout ratio of 30%, a discount
rate of 16%, and a return on retained earnings of
20%.
The dividend at year one will be $5 × .30 = $1.50 per share.
 The retention ratio is .70 ( = 1 -.30), implying a growth rate in
dividends of 14% = .70 × 20%.
From the dividend growth model, the price of a share is:

Div 1
$1.50
P0 

 $75
R  g .16  .14
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The NPVGO Model: Example
First, we must calculate the value of the firm as a
cash cow.
EPS $5
P0 
R

.16
 $31.25
Second, we must calculate the value of the growth
opportunities.
3.50  .20 

 3.50  .16 
$.875
P0 

 $43.75
Rg
.16  .14
Finally, P0  31.25  43.75  $75
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6.5 Price-Earnings Ratio


Many analysts frequently relate earnings per share to
price.
The price-earnings ratio is calculated as the current
stock price divided by annual EPS.

The Wall Street Journal uses last 4 quarter’s earnings
Price per share
P/E ratio 
EPS
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6.6 Features of Common Stock
Voting rights (Cumulative vs. Straight)
 Proxy voting
 Classes of stock
 Other rights

Share proportionally in declared dividends
 Share proportionally in remaining assets during
liquidation
 Preemptive right – first shot at new stock issue to
maintain proportional ownership if desired

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Features of Preferred Stock

Dividends
Stated dividend must be paid before dividends can
be paid to common stockholders.
 Dividends are not a liability of the firm, and
preferred dividends can be deferred indefinitely.
 Most preferred dividends are cumulative – any
missed preferred dividends have to be paid before
common dividends can be paid.


Preferred stock generally does not carry voting
rights.
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6.7 The Stock Markets
Dealers vs. Brokers
 New York Stock Exchange (NYSE)

Largest stock market in the world
 Members

 Own
seats on the exchange
 Commission brokers
 Specialists
 Floor brokers
 Floor traders
Operations
 Floor activity

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NASDAQ
Not a physical exchange – computer-based
quotation system
 Multiple market makers
 Electronic Communications Networks
 Three levels of information

Level 1 – median quotes, registered representatives
 Level 2 – view quotes, brokers & dealers
 Level 3 – view and update quotes, dealers only


Large portion of technology stocks
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Stock Market Reporting
52 WEEKS
YLD
VOL
NET
HI
LO STOCK SYM DIV % PE 100s CLOSE CHG
25.72 18.12 Gap Inc GPS 0.18 0.8 18 39961 21.35 …
Gap has
been as high
as $25.72 in
the last year.
Gap pays a
dividend of 18
cents/share.
Given the current
price, the dividend
yield is .8%.
Gap has been as
low as $18.12 in
the last year.
McGraw-Hill/Irwin
Gap ended trading at
$21.35, which is
unchanged from yesterday.
Given the current
price, the PE ratio is
18 times earnings.
3,996,100 shares traded
hands in the last day’s
trading.
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Quick Quiz
What determines the price of a share of stock?
 What determines g and R in the DGM?
 Decompose a stock’s price into constant
growth and NPVGO values.
 Discuss the importance of the PE ratio.
 What are some of the major characteristics of
common and preferred stock?

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