Let's look again at the basic DCF stock valuation
formulas 
General DCF formula
The value of shares of common stock, like any
other financial instrument, is often understood
as the present value of expected future returns.
Again we return to the discounted cash flow formula:
P
_{o} = D_{1}/(1+i_{1})
+ D_{2} /(1+i _{2} )2 +
D_{3}/(1+i_{3})3 + ... +
D _{inf} /(1+i _{inf})inf

P_{o}
= D_{n} / (1+i _{n})^{n} 
P _{o} 
= value of common stock 
D_{t} 
= pershare dividends expected at the end
of year t 
i_{t} 
= required return (discount rate) for each
year t 
inf 
= infinite time period 
This DCF formula leads to two particularized
formulas in situations of zero growth and constant
growth.
Zero growth
The simplest DCF model assumes constant dividends
 zero growth. In this artificial world (no inflation,
no variation, no change) the present value of
a constant dividend stream is the present value
of a perpetuity:
P
_{0} = D / i 
P _{0} 
= value of common stock 
D 
= pershare dividends expected at the end
of each year 
i 
= required return (discount rate) for each
year 
Constant
growth
In his 1937 Ph.D. thesis, John Burr Williams,
a Harvard graduate student, developed what he
called the Dividend Discount Model. Williams model
called for the investor to forecast the dividends
a company would pay and then discount them based
on the confidence in that forecast (from Capital
Ideas, Peter L. Bernstein). His model has
been modified many times. One such modification
is found in the GordonShapiro Dividend Discount
Model. We often expect that cash flows will grow,
with inflation and as the company progresses.
Thus, with the assumption that dividends will
also grow at a constant rate (g), Gordon and Shapiro
produced one of the most oftenused formulas in
stock valuation, known as the Gordon Shapiro Dividend
Discount Model, or Gordon Model for short.
P
_{o} =D_{o}(1+g)^{1}/(1+i)^{1}
+D _{o} (1+g) ^{2}/(1+i)^{2}
+ ... + D_{o}(1+g) ^{inf }/(1+i)^{
inf} 
P_{o}
= D_{1} / (i  g)  OR  P _{o}
= D_{o }(1 +g) / (i  g) 
P _{o} 
= present value of common stock
(with constant growth returns) 
D_{o} 
= most recent pershare dividend 
D_{1} 
= pershare dividend after one period of
growth [D1 = D0 (1+g)] 
i 
= required return (discount rate) for each
year t 
g 
= rate of growth 
inf 
= infinite time period 
This model makes heroic assumptions about the
flat continuity of growth, that extrapolation
from past dividends reflects likely future earnings,
and the stock's risk can be reflected in a single
discount rate. Such is valuation!

Example
Company is growing. It pays
most of its earnings as dividends, but retains
some earnings for future growth. The practice
has worked well, as its history of growth
suggests.
What is the value of Company stock, based
on dividend returns?

Year 
Dividend/share 
1 
1.00 
2 
1.06 
3 
1.15 
4 
1.25 
5 
1.36 
6 
1.44 
7 
1.59 

Answer: You can use the
Gordon model in three steps 
What is the
company's growth rate? This is not obvious
and requires a judgment call. Do we predict that
growth will continue at the average annual growth
rate for the last 6 periods? To compute this we
could find the average growth rate. Or do we predict
that growth will continue at the overall
rate of the last seven years? To compute this
we would find the growth rate over the
6 years of growth, in which the dividend
went from 1.00 to 1.59  what was the annual
growth rate that led to a compounded return of
59% over 6 years?
Year 
Dividend/share 
Annual growth 
Growth rate 
1 
1.00 


2 
1.06 
6.00% 

3 
1.15 
8.49% 

4 
1.25 
8.70% 

5 
1.36 
8.80% 

6 
1.44 
5.88% 

7 
1.59 
10.42% 


Average 
8.05% 
8.15% 
Often stock valuators use the growth rate (or
other measures such as internal rate of growth),
since it reflects the growth rate over a period
better than an average. In effect, it gives you
an idea of the "growth machine" propelling
the business.
Professor Palmiter,
I was looking for some information regarding dcf
valuation recently, and found your webpage
I'd like to bring to your attention what might
be an error (a small, but confusing one), if I'm
correct.
When you're discussing the growth rate, you make
a distinction between the average annual growth
rate and the growth rate over the period. First,
the growth rate over the period has a commonly
used name  CAGR, the Compound Annual Growth Rate.
Second, your CAGR is calculated using an ending
value of $1.60 per share, instead of the indicated
$1.59. The one penny difference causes a significant
difference in the CAGR calculation. At $1.60,
the CAGR = 8.15%, as you note, but at $1.59, it
equals 8.04%.
Perhaps I'm in error, and you're calculating the
growth rate in a way unfamiliar to me, but I thought
I'd point this out, just in case.
Regards,
J.D. Kern
What
is the required return (discount rate) for Company's
stock? This is even trickier. One method
is to use the CAPM  which predicts the company's
expected return (required return or discount rate)
based on the stock's expected volatility and the
market's expected return. For now, let's assume
the discount rate is 14%.
As we saw before in computing value under different
capitalization rate assumptions, the discount
rate will be one of the most important (and hotly
contested) issues in a valuation.
Apply the Gordon
model.
P_{0}
= D _{0} (1 + g)/(i  g) 
D _{0} 
= Most recent
pershare dividend 
$ 1.59 
i 
= Required return (discount
rate) 
14.0% 
g 
= Rate of growth 
8.15% 
P_{0} 
= Value of one
share of common stock 
$1.59(1 + .0815)
(.14  .0815)
= $29.39 
Notice that the formula requires that you compute
the return in the first period of growth [D_{0}(1
+ g) = $ 1.72] and then divide this by the difference
of the discount rate and the growth rate [.14
 .0815].
A Simpler Example
Company has preferred stock
that promises its holders a fixed annual
dividend of $8. Assuming a required return
of 12.8% for the preferred stock based
on the risk profile of Company, what is
the value of the preferred stock?
Answer.
Divide the annual dividend by the discount
rate ($ 8 / .128). The result is $62.50.

