Stock Valuation
(Chapter 9)
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
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
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
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
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.
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 ) (1 + g 2 )
N
.
.
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 ) N
…
Div N (1 + g 2 )
= Div 0 (1 + g1 ) N (1 + g 2 )
…
N
N+1
Differential Growth
We can value this as the sum of:
 a T-year annuity growing at rate g1
C  (1 + g1 )T 
PA =
1 −
T 
R − g1  (1 + R ) 
 plus the discounted value of a perpetuity
growing at rate g2 that starts in year T+1
 Div T +1 


R − g2 

PB =
T
(1 + R )
Differential Growth

Consolidating gives:
 Div T +1 




T
C  (1 + g1 )   R − g 2 
+
P=
1 −
T 
T
R − g1  (1 + R )  (1 + R )
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%.
Apply the formula
 $2(1.08) 3 (1.04) 


3
.12 − .04
$2 × (1.08)  (1.08)  

1
−
P=
+


.12 − .08  (1.12) 3 
(1.12) 3
(
$32.75)
P = $54 × [1 − .8966] +
3
(1.12)
P = $5.58 + $23.31
P = $28.89
Using Cash flow to calculate price
$2(1.08)
0
1
$2.16
0
1
$2(1.08)
2
$2(1.08) $2(1.08) (1.04)
…
2
$2.33
2
3
3
3
4
The constant
$2.62
$2.52 +
growth phase
.12 − .04beginning in year 4
3
can be valued as a
growing perpetuity
at time 3.
$2.16 $2.33 $2.52 + $32.75
P0 =
+
+
= $28.89
2
3
1.12 (1.12)
(1.12)
$2.62
P3 =
= $32.75
.08
Estimate 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
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.
Using DGM to find R

Start with the DGM:
D 0 (1 + g)
D1
=
P0 =
R -g
R -g
D 0 (1 + g)
D1
+g=
+g
R=
P0
P0
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% of its earnings as dividends plus
the net present value of the growth
opportunities.
EPS
P=
+ NPVGO
R
Example of NPVGO
Consider a firm that has forecasted EPS of $5, a discount
rate of 16%, and is currently priced at $75 per share.
 We can calculate the value of the firm as a cash cow.
EPS $5
P0 =
=
= $31.25
R
.16
So, NPVGO must be: $75 - $31.25 = $43.75
How Retention Rate affect Firm
value

An increase in the retention rate will:
◦ Reduce the dividend paid to shareholders
◦ Increase the firm’s growth rate


These have offsetting influences on stock
price
Which one dominates?
◦ If ROE>R, then increased retention increases
firm value since reinvested capital earns more
than the cost of capital.
Suggested Exercise

Chapter 9: 4,7,10