COMMON STOCK
The first thing that the organizers of a business do is to file the Articles of Incorporation
with the Secretary of State. Then, a board of directors is elected and from that point forward,
the company is run by the board of directors. The stockholders’ only say in managing the affairs
of the company is through the annual election of directors.
Stockholders do have certain rights that accrue to them as a group, known as collective
rights. Most of these require a two-thirds vote of shareholders and include:







Amendments to the charter.
Adopting and amending bylaws of the corporation (although sometimes this power is
delegated to the board of directors).
Selling the assets of the corporation that would materially change the life or nature of the
company.
Electing directors.
Authorizing new types of securities.
Changing the amount of authorized common stock.
Mergers. A “friendly” takeover is when shareholders approve. A “hostile” takeover is when
a company must buy enough of the outstanding stock in order to muster the vote necessary
to approve it.
In addition, each individual investor has certain specific rights, including

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A share of the profits or losses on a pro rata basis (in proportion to their ownership).
The right to sell their interest in the company. Some closely-held businesses will include a
right-of-first-refusal for the corporation to buy the stock of a selling shareholder in order to
keep the ownership within a certain group of investors.
The right to inspect the books of the company (within limitations). The annual report is the
shareholders’ look at the books. How do they know that the annual report is conveying the
truth? Because the financial statements will be audited.
The right to a residual share (if any) after liquidation.
The right to vote. Typically, each shareholder gets one vote for each share of stock that is
owned. A proxy allows the transfer of the right to vote. In addition, there is sometimes what
is called cumulative voting which provides for one voter per share per director being elected.
Thus, if five directors are being elected, each share gets five votes. What is the purpose of
cumulative voting when everybody gets additional votes? It allows minority shareholders to
concentrate their votes on a candidate in order to assure representation on the Board of
Directors.
Preemptive Rights
A preemptive right gives current stockholders the first option to purchase any new
shares that are issued on a pro rata basis. Then, if a shareholder owned 10% of a company
they would have the option to purchase 10% of any new shares issued in order to maintain their
10% overall ownership.
Common Stock Valuation
Valuing common stock is more difficult since the cash flows (dividends) are not constant.
Nonetheless, the principle remains the same: that the value of the stock is equal to the present
value of all of its future cash flows.
Suppose investors buy stocks utilizing a one-year planning horizon. The cash flows that
an investor would realize would be the dividend paid during the year and the price that they
realized upon selling the stock at the end of the year. The price they would be willing to pay at the
beginning of the year (time zero) would be equal to the present value of these cash flows:
P0 
D1
P1

(1  rS ) (1  rS )
But what is the price at the end of the year (P1)? The buyer would also have a one-period horizon
and would value the dividend and selling price that they would receive:
D2
P2

(1  rS ) (1  rS )
P1 
When P1 is substituted the current price (P0) becomes
P0 
D1
D2
P2


2
(1  rS ) (1  rS )
(1  rS ) 2
The question now becomes, what is P2?
P2 
D3
P3

(1  rS ) (1  rS )
which, when substituted, defines the current price as
P0 
P0 
D3
P3
D1
D2



(1  rS ) (1  rS ) 2 (1  rS ) 3 (1  rS ) 3
D3
D5
D1
D2
D4
D




  
2
3
4
5
(1  rS ) (1  rS )
(1  rS )
(1  rS )
(1  rS )
(1  rS ) 
As the process continues, it becomes clear that the value of the stock is simply the present value
of all future dividends forever.
The Gordon Growth (Dividend Valuation) Model
Assume that the growth in dividends occurs at a constant rate, g. Then, the dividends
from year 1 to infinity can be written as follows:
P0 
D0 (1  g )1 D0 (1  g ) 2
D0 (1  g )  1 D0 (1  g ) 








(1  rs )
(1  rs ) 2
(1  rs )  1
(1  rs ) 
Multiplying both sides by (1+r)/(1+g) yields
P0
(1  rS )
D (1  g )1
D (1  g ) 2 D0 (1  g )  1
 D0  0
   0

(1  g )
(1  rs )
(1  rs )  2
(1  rs )  1
Subtracting the first equation from the second equation, and recognizing that the first term of the
first equation equals the second term of the second equation, while the second-to-last term of the
first equation equals the last term of the second equation (i.e., they cancel), we get
P0
(1  rS )
(1  g ) 
 P0  D0 
(1  g )
(1  rS ) 
If k>g, then the denominator of the last term becomes infinitely greater than the numerator and the
whole term becomes zero, leaving
P0
(1  rS )
 P0  D0
(1  g )
Or
P0 (1  rS )  P0 (1  g )  D0 (1  g )
P0 (1  rS  1  g )  D1
P0 
D1
rS  g
This model is known as the Gordon Growth Model, or the Dividend Valuation Model. As
we have seen in its derivation, k must be greater than g for the model to be valid, and we have
assumed that the dividends grow at a constant rate of growth forever.
Another Approach
Suppose a company pays out all of its earnings as dividends. Then the yield to the stockholder is
rS 
E
P0
If a company pays all of its earnings out as dividends, it cannot grow since there is no
reinvestment to support increases in assets. Allowing for reinvestment of earnings, we can
define
b = Retained Earnings/Earnings
(1-b) = Dividends/Earnings (Dividend Payout ratio)
Then,
rS 

E (1  b) E bE


P0
P0
P0
Div RE

P0
P0
 Dividend Yield  ??????
Suppose we put $100 in a bank today earning 10%.
Today
Bank Acct $100
Income
$ 10
Withdraw
4 (40% payout ratio)
“Retained” 6
End of Year $106
What does the $6/$100 = 6% represent?
The growth in the value of the Bank account.
Thus,
rS 

Div RE

P0
P0
Div
g
P0
 Dividend Yield  Capital Gain Yield
The total yield is composed of two portions: the current income yield and the increase in price,
just as a bond’s total return is calculated.
Since we’re interested in the Price of the stock, let’s rearrange and solve for Po
P0 
D1
rS  g
This is the same result we obtained previously.
Suppose a stock just paid a dividend of $1.00 per share. Our required rate of return is
10% and the dividends are not growing. What is the value of the stock?
D0 =
r=
g=
$1.00
10%
0%
D1 = D0*(1+g)
= 1.00*(1.00)
= 1.00
P0 
$1.00
 $10.00
.1  0
This is the same as the valuation of Preferred stock or a perpetuity.
Suppose the dividend is growing at a 5% rate. What is the value of the stock?
D0 =
r=
g=
$1.00
10%
5%
D1 = D0*(1+g)
= 1.00*(1.05)
= 1.05
P0 
$1.05
 $21.00
.1  .05
Why is this stock worth more than twice as much?
Suppose the stock is growing at 10%. What is the value of the stock?
D0 =
r=
g=
$1.00
10%
10%
D1 = D0*(1+g)
= 1.00*(1.10)
= 1.10
P0 
$1.10
 Undefined
.1  .1
So the equation does not work if k=g.
What if the growth rate is 15%?
D0 =
r=
g=
$1.00
10%
15%
D1 = D0*(1+g)
= 1.00*(1.15)
= 1.15
P0 
$1.15
 $23.00 
.1  .15
What does a negative price imply?
Can companies grow at 15% per year?
Suppose the company is expected to grow at 15% for two years, followed by 10% growth for two
more years, and finally slowing to 5% growth thereafter. How do we value this stock?
Since the dividends are not growing at a constant rate in the first four years, wee must
calculate the present values of the first four years’ worth of dividends one by one. As seen in the
table, the present value of the first four years’ dividends amounts to $4.32 in today’s dollars.
D1=
D2 =
D3=
D4 =
D5 =
1.00
1.15
1.32
1.45
1.60
*
*
*
*
*
1.15 =
1.15 =
1.10 =
1.10 =
1.05 =
1.15
1.32
1.45
1.60
1.68
*
*
*
*
0.9091
0.8264
0.7513
0.6830
=
=
=
=
1.05
1.09
1.09
1.09
-------4.32
To Capture the value of the remaining dividends beyond year 4, we can utilize the Gordon Growth
Model as a short-cut since the requirements for its implementation will be satisfied by the fifth
year, i.e., a constant growth rate that is less than the required rate of return.
P4 
D5
$1.68

 $33.60
rS  g .1  .05
The present value of all the dividends from year five through infinity has a value of $33.60 in year
four dollar terms. This needs to be discounted back to year zero:
$33.60*.6830 = $22.95 Present Value of Dividends Year 5 through Infinity
4.32 Present Value of Dividends Year 1 through Year 4
$27.27 Present value of Dividends Year 1 through Infinity
Graphically, we have calculated the present value of all the future dividends as follows:
STOCK VALUATION
D4
D5
D6
***
6
***
D3
D2
P4=D5/(k-g)
D1
P0
0
1
2
3
4
5
Year
If one were to pay $27.27 for a share of this stock today, and received a dividend of $1.15
in the first year, $1.32 in the second year, $1.45 as a dividend in the third year, etc., a rate of
return of exactly 10% would be earned. Of course this assumes that the stock is being held
forever.
Suppose you only intended to hold the stock for two years. How would this change what
you would be willing to pay for the stock? To determine what we would be willing to pay for the
stock, we must again calculate the present value of the cash flows that we would receive. In this
case, we would receive the first two years’ dividends and then the price of the stock at the end of
two years.
D1 =
$
1.15
D2 =
P2 =
$
1.32
???
The question that remains is what can we expect the stock to sell for in two years? Given our
expectations with regard to the growth in future dividends and what we consider to be a fair rate of
return for the stock, let’s put ourselves in the buyer’s position two years from now. The buyer
would estimate the future dividends that they would receive and discount them to a present value.
D1' = D3 =
D2' = D4 =
D3' = D5 =
$
$
$
1.45 *.9091 =
1.60 *.8264 =
1.68
P2' = P4 = $1.68/(.1-.05) =
$
$
$
1.32
1.32
2.64
$ 33.60
The present value of Dividends from the Buyer’s year 3 through infinity is captured using the
Gordon Model, but this must also be discounted back to the Buyer’s year zero.
P0’ = P4 = $ 33.60 * .8264 = $ 27.77 Present Value to Buyer of Dividends from Year 2 to infinity
2.64 Present Value to Buyer of Dividends in Year 1 and Year 2
$ 30.41 Present Value to Buyer of Dividends in Year 1 to inifinity
Now that we have estimated the Price we can sell the stock for in two years, we can calculate the
present value of the cash flows that we will receive:
D1 = $ 1.15 * .9091 = $ 1.05 Present value of first year dividend
D2 = $ 1.32 * .8264 = 1.09 Present value of second year dividend
P2 = $ 30.41* .8264 = 25.13 Present value of second year stock price
Total Present Value = $27.27 Total present value of cash flows we receive
Notice that the price of the stock is the same. Why? What are we selling in two years when we
sell the stock? We’re selling the rights to the dividends beyond that point in time. Then when we
discount that price to today, we have just calculated the present value of the dividends from year 3
on to today and added them to the first two years’ worth of dividends. Graphically,
STOCK VALUATION -- TWO-YEAR HOLDING PERIOD
D5
D4
D6
***
D3
D2
P2'=P4=(D3'=D5)/(k-g)
D1
P0'=P2
P0
0
1
2 (0')
1'
2'
3'
4'
***
Year
The value of the stock is the present value of all future dividends—it does not matter who
receives those dividends.
Do people really sit down and try to estimate future dividends? Yes. Stock analysts pour
through annual reports and other information to estimate what sales will be, how much money
needs to be reinvested to have the capacity to meet demand, and how much money is left over
for paying dividends. The resulting value is very dependent upon the assumptions that must
necessarily be made. The assumptions employed must be realistic.