Common Stock Valuation
Fundamental analysis is a term for studying a
company’s accounting statements and other financial
and economic information to estimate the economic
value of a company’s stock.

The basic idea is to identify “undervalued” stocks to
buy and “overvalued” stocks to sell.

In practice however, such stocks may in fact be
correctly priced for reasons not immediately apparent
to the analyst.

The Dividend Discount Model (DDM) is a method to estimate the value of
a share of stock by discounting all expected future dividend payments. The
basic DDM equation is:

D(1)
D(2)
D(3)
D(T)
V(0) 



2
3
1  k  1  k  1  k 
1  k T

In the DDM equation:
 V(0) = the present value of all future dividends
 D(t) = the dividend to be paid t years from now

k = the appropriate risk-adjusted discount rate
Suppose that a stock will pay three annual dividends
of $200 per year, and the appropriate risk-adjusted
discount rate, k, is 8%.


In this case, what is the value of the stock today?
V(0) 
D(1)
D(2)
D(3)


1  k  1  k 2 1  k 3
V(0) 
$200
$200
$200


 $515.42
2
3
1  0.08  1  0.08  1  0.08 
Assume that the dividends will grow at a constant growth rate g. The
dividend next period (t + 1) is:

Dt  1  Dt   1  g
So, D(2)  D(1)  (1  g)  D(0)  (1  g)  (1  g)

For constant dividend growth, the DDM formula becomes:
T
D(0)(1  g)   1  g  
V(0) 
 
1  
k  g   1  k  
if k  g
V(0)  T  D(0)
if k  g
Suppose the current dividend is $10, the dividend growth rate is 10%,
there will be 20 yearly dividends, and the appropriate discount rate is 8%.


What is the value of the stock, based on the constant growth rate model?
T
D(0)(1  g)   1  g  
V(0) 
 
1  
k  g   1  k  
if k  g
20
$10  1.10    1.10  
V 0  
   $243.86
1  
.08  .10   1.08  
Assuming that the dividends will grow forever at a
constant growth rate g.

For constant perpetual dividend growth, the DDM
formula becomes:

D0  1  g D1
V 0 

kg
kg
(Important : g  k)



Think about the electric utility industry.
In 2009, the dividend paid by the utility company, DTE Energy Co.
(DTE), was $2.12.
Using D0 =$2.12, k = 5.75%, and g = 2%, calculate an estimated value for
DTE.
P0 
$2.12  1.02
 $57.66
.0575  .02
Note: the actual mid-2009 stock price of DTE was $40.29.
What are the possible explanations for the difference?
The growth rate in dividends (g) can be estimated in
a number of ways:

 Using the company’s historical average growth rate.
 Using an industry median or average growth rate.
 Using the sustainable growth rate.
Sustainabl e Growth Rate  ROE  Retention Ratio
 ROE  (1 - Payout Ratio)

Return on Equity (ROE) = Net Income / Equity

Payout Ratio = Proportion of earnings paid out as dividends
Retention Ratio = Proportion of earnings retained for
investment


In 2009, American Electric Power (AEP) had an ROE of 10%,
projected earnings per share of $2.90, and a per-share
dividend of $1.64. What was AEP’s:
 Retention rate?
 Sustainable growth rate?

Payout ratio = $1.64 / $2.90 = .566 or 56.6%

So, retention ratio = 1 – .566 = .434 or 43.4%

Therefore, AEP’s sustainable growth rate = .10  .434 =
.0434, or 4.34%

What is the value of AEP stock using the perpetual growth
model and a discount rate of 5.75%?
P0 



$1.64  1.0434 
 $121.36
.0575  .0434
The actual late-2009 stock price of AEP was $31.83.
In this case, using the sustainable growth rate to value the
stock gives a reasonably poor estimate.
What can we say about g and k in this example?
The two-stage dividend growth model assumes that
a firm will initially grow at a rate g1 for T years, and
thereafter grow at a rate g2 < k during a perpetual
second stage of growth.


The Two-Stage Dividend Growth Model formula is:
T
T
D(0)(1  g1 )   1  g1    1  g1  D(0)(1  g2 )
V(0) 
 

1  
k  g1   1  k    1  k 
k  g2
Although the formula looks complicated, think of it as
two parts:

 Part 1 is the present value of the first T dividends (it is the same formula
we used for the constant growth model).
 Part 2 is the present value of all subsequent dividends.
So, suppose MissMolly.com has a current dividend of
D(0) = $5, which is expected to “shrink” at the rate g1 - -10%
for 5 years, but grow at the rate g2 = 4% forever.

With a discount rate of k = 10%, what is the present value
of the stock?

T
T
D(0)(1  g1 )   1  g1    1  g1  D(0)(1  g 2 )
V(0) 
 

1  
k  g1
1

k
1

k
k  g2
  

 
5
5
$5.00(0.90 )   0.90    0.90  $5.00(1  0.04)
V(0) 
 

1  
0.10  ( 0.10)   1  0.10    1  0.10 
0.10  0.04
 $14.25  $31.78
 $46.03.
The total value of $46.03 is the sum of a $14.25 present value of the first
five dividends, plus a $31.78 present value of all subsequent dividends.

Chain Reaction, Inc., has been growing at a phenomenal rate of 30%
per year.


You believe that this rate will last for only three more years.

Then, you think the rate will drop to 10% per year.

Total dividends just paid were $5 million.

The required rate of return is 20%.

What is the total value of Chain Reaction, Inc.?
First, calculate the total dividends over the “supernormal” growth
period:

Year
Total Dividend: (in $millions)
1
$5.00 x 1.30 = $6.50
2
$6.50 x 1.30 = $8.45
3
$8.45 x 1.30 = $10.985
Using the long run growth rate, g, the value of all the shares at
Time 3 can be calculated as:

V(3) = [D(3) x (1 + g)] / (k – g)
V(3) = [$10.985 x 1.10] / (0.20 – 0.10) = $120.835
Therefore, to determine the present value of the firm today, we need the
present value of $120.835 and the present value of the dividends paid in the
first 3 years:
D(1)
D(2)
D(3)
V(3)
V(0) 



1  k  1  k 2 1  k 3 1  k 3

V(0) 
$6.50
$8.45
$10.985
$120.835



1  0.20  1  0.20 2 1  0.20 3 1  0.20 3
 $5.42  $5.87  $6.36  $69.93
 $87.58 million.
The discount rate for a stock can be estimated using the capital asset
pricing model (CAPM ).
 We will discuss the CAPM in a later chapter.
 However, we can estimate the discount rate for a stock using this
formula:

Discount rate = time value of money + risk premium
= U.S. T-bill rate + (stock beta x stock market risk premium)
T-bill rate: return on 90-day U.S. T-bills
Stock Beta: risk relative to an average stock
Stock Market Risk Premium: risk premium for an average stock
Constant Perpetual Growth Model:






Simple to compute
Not usable for firms that do not pay dividends
Not usable when g > k
Is sensitive to the choice of g and k
k and g may be difficult to estimate accurately.
Constant perpetual growth is often an unrealistic assumption.
Two-Stage Dividend Growth Model:





More realistic in that it accounts for two stages of growth
Usable when g > k in the first stage
Not usable for firms that do not pay dividends
Is sensitive to the choice of g and k
k and g may be difficult to estimate accurately.

We have valued only companies that pay dividends.
 But, there are many companies that do not pay dividends.
 What about them?
 It turns out that there is an elegant way to value these
companies, too.

The model is called the Residual Income Model (RIM).
 Major Assumption (known as the Clean Surplus Relationship, or CSR): The
change in book value per share is equal to earnings per share minus
dividends.

Inputs needed:





Earnings per share at time 0, EPS0
Book value per share at time 0, B0
Earnings growth rate, g
Discount rate, k
There are two equivalent formulas for the Residual Income Model:
P0  B 0 
EPS0 (1  g)  B 0  k
kg
or
P0 
EPS1  B 0  g
kg
BTW, it turns out that the
RIM is mathematically the
same as the constant
perpetual growth model.



National Beverage Corporation (FIZ)
It is July 1, 2005—shares are selling in the market for $7.98.
Using the RIM:





EPS0=$0.47
DIV = 0
B0=$4.271
g = 0.09
K = .103
What can we say
about the market
price of FIZ?
P0  B0 
EPS0  (1  g)  B0  k
k g

P0  $4.271 
$.47  (1  .09)  $4.271 .103
.103  .09
P0  $4.271  $5.57  $9.84.
Using the information from the previous slide, what growth rate results
in a FIZ price of $7.98?

P0  B0 
EPS0  (1  g)  B0  k
k g
$7.98  $4.271 
$.47  (1  g)  $4.271 .103
.103  g
$3.709  (.103  g)  .47  .47g  .4399
$.3820  3.709g  .47g  .0301
.3519  4.179g
g  .0842 or 8.42%.

Price-earnings ratio (P/E ratio)
 Current stock price divided by annual earnings per share (EPS)

Earnings yield
 Inverse of the P/E ratio: earnings divided by price (E/P)

Price-cash flow ratio (P/CF ratio)
 Current stock price divided by current cash flow per share
 In this context, cash flow is usually taken to be net income plus
depreciation.
Most analysts agree that in examining a company’s
financial performance, cash flow can be more
informative than net income.

Earnings and cash flows that are far from each other
may be a signal of poor quality earnings.


Price-sales ratio (P/S ratio)
 Current stock price divided by annual sales per share
 A high P/S ratio suggests high sales growth, while a low P/S ratio
suggests sluggish sales growth.

Price-book ratio (P/B ratio)
 Market value of a company’s common stock divided by its book
(accounting) value of equity
 A ratio bigger than 1.0 indicates that the firm is creating value
for its stockholders.
Intel Corp (INTC) - Earnings (P/E) Analysis
5-year average P/E ratio
Current EPS
EPS growth rate
37.30
$1.16
17.5%
Expected stock price = historical P/E ratio  projected EPS
$50.84 = 37.30  ($1.16  1.175)
Mid-2005 stock price = $26.50
Intel Corp (INTC) - Cash Flow (P/CF) Analysis
5-year average P/CF ratio
Current CFPS
CFPS growth rate
19.75
$1.94
13.50%
Expected stock price = historical P/CF ratio  projected CFPS
$43.49 = 19.75  ($1.94  1.135)
Mid-2005 stock price = $26.50
Intel Corp (INTC) - Sales (P/S) Analysis
5-year average P/S ratio
Current SPS
SPS growth rate
6.77
$5.47
10.50%
Expected stock price = historical P/S ratio  projected SPS
$40.92 = 6.77  ($5.47  1.105)
Mid-2005 stock price = $26.50
The next few slides contain a financial
analysis of the McGraw-Hill Company, using
data from the Value Line Investment Survey.

Based on the CAPM, k = 3.1% + (.80  9%) = 10.3%

Retention ratio = 1 – $.66/$2.65 = .751

Sustainable g = .751  23% = 17.27%
Because g > k, the constant growth rate model
cannot be used. (We would get a value of -$11.10 per
share)


Let’s assume that “today” is January 1, 2005, g = 7%, and k = 10.3%.
Using the Value Line Investment Survey (VL), we can fill in column two (VL) of
the table below.

We use column one and our growth assumption for column three (CSR) of the
table below.

2005 (time 0)
2006 (VL)
2006 (CSR)
Beginning BV per share
NA
9.45
9.45
EPS
2.25
2.65
2.4075
DIV
0.66
0.66
1.7460
Ending BV per share
9.45
11.50
10.1115
2.251.07
9.451.07
" Plug"  2.4075 - (10.1115 - 9.45)

Using the CSR assumption:
P0  B0 
EPS0  (1  g)  B0  k
k g
P0  $9.45 
Stock price at the time = $47.04.
What can we say?
Using Value Line numbers for
EPS1=$2.65, B1=$11.50
B0=$9.45; and using the actual change in
book value instead of an estimate of the
new book value, (i.e., B1-B0 is = B0 x k)

$2.25  (1  .07)  $9.45  .103
.103  .07
P0  $52.91.
P0  B0 
EPS0  (1  g)  B0  k
k g
P0  $9.45 
P0  $27.63
$2.65  ($11.50 - 9.45)
.103  .07
Quick calculations used: P/CF = P/E  EPS/CFPS
P/S = P/E  EPS/SPS




www.nyssa.org (the New York Society of Security Analysts)
www.aaii.com (the American Association of Individual Investors)
www.eva.com (Economic Value Added)
www.valueline.com (the home of the Value Line Investment Survey)

Websites for the companies analyzed in this chapter:
• www.aep.com
• www.americanexpress.com
• www.pepsico.com
• www.starbucks.com
• www.sears.com
• www.intel.com
• www.disney.go.com
• www.mcgraw-hill.com

Security Analysis: Be Careful Out There

The Dividend Discount Model
 Constant Dividend Growth Rate Model
 Constant Perpetual Growth
 Applications of the Constant Perpetual Growth Model
 The Sustainable Growth Rate

The Two-Stage Dividend Growth Model
 Discount Rates for Dividend Discount Models
 Observations on Dividend Discount Models

Residual Income Model (RIM)

Price Ratio Analysis
 Price-Earnings Ratios
 Price-Cash Flow Ratios
 Price-Sales Ratios
 Price-Book Ratios
 Applications of Price Ratio Analysis

An Analysis of the McGraw-Hill Company