This document proves the Gordon Growth Model formula for valuing stocks. It shows that the present value of a stock is equal to the next period's expected dividend divided by the difference between the required rate of return and the growth rate of dividends. The proof begins by setting up a summation of future dividends discounted back to the present. Algebraic manipulations are used to simplify the summation and take the limit as the number of periods approaches infinity. This results in the Gordon Growth Model formula where present value equals dividends divided by the difference between the required return and growth rates. The derivation requires that the growth rate be less than the required return for the summation to converge. It also notes that if the growth

1. Prove for the Gordon Growth Model for Stocks
PV Stock =
Where:
Now, since is a constant we can take it out of the summation and calling = and
= we substitute those new variables getting:
Or,
Calling = we now have and noting that we do not have to be concerned with for now,
,
Let’s call that summation,
Now, if we multiply by , we will get

2. If we subtract
less
We can observed that only the first term of the first summation and the last
term of the second summation will not cancel each other; that is, we will
have:
Then, factoring S out in the term on the left

3. Solving for ,
Now since is really an infinite summation over the n terms, we take the limit for as -> ∞
Or
Which will converge as long as < …in such case, -> 0; then,
But = therefore,
With a little of algebra,
Substituting back in the original equation gives our demonstration. Note that
this demonstration requires < ; meaning

4. = or equivalently
Which makes mathematical sense since the sum cannot converge if and will
increase without boundaries when ( implies that
We should further note that the sum could have =0 meaning it will pay the exact
same dividend every time…like the case for preferred stocks…in such a case
the final formula becomes
With =0
Which is the formula for capitalized cost.
Ismae
l
Digitally signed by
Ismael
DN: cn=Ismael, o, ou,
email=ismaeltorres20
02@yahoo.com, c=US
Date: 2013.06.08
12:25:00 -04'00'