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(The current fair market value is equal to the sum of the heights of all of the green bars,
which are the present values of the corresponding blue bars.)
(See more detail.)
To get the formula, we'll define some variables:
E = this year's Earnings per Share
G = growth rate of earnings (written as a decimal)
N = number of years earnings will grow
We're assuming that earnings will start to grow for N years, and then level off:
| Year | Earnings |
| 1 | E(1 + G) |
| 2 | E(1 + G)2 |
| N | E(1 + G)N |
| N + 1 | E(1 + G)N |
| N + 2 | E(1 + G)N |
Now we'll write R for our desired rate of return, and use it to find the present values of all of these earnings:
| Year | Present Value of Earnings |
| 1 | E(1 + G)/(1 + R) |
| 2 | E(1 + G)2/(1 + R)2 |
| N | E(1 + G)N/(1 + R)N |
| N + 1 | E(1 + G)N/(1 + R)N+1 |
| N + 2 | E(1 + G)N/(1 + R)N+2 |
What we've got here is two geometric series; one going from 1 to N, and the other going from N + 1 to infinity.
The result is basically too ugly to bother writing out; it's more sensible just to use the formula for the geometric series in a spreadsheet or computer program.
When people do write it out, they usually write it this way:
P = E1Q + E2Q2 + ... + ENQN + ENQN x Q/(1 - Q)
where E2 is the earnings in year 2 (or whatever) and Q is the so-called "discount factor" 1/(1 + R).
Zero-Growth Case
One special case is actually interesting to write out though.
If you assume that the stock is already in the "mature", zero-growth years -- ie, that N is zero -- the geometric series formula will simplify to:
P = E / R
or, equivalently,
P / E = 1 / R
So if you take a desired return of 11%, you find that the theoretical "fair" P/E ratio of the zero-growth stock is 1/.11 = 9.09, which sounds reasonable.
Constant-Growth Case
A second special case that people use is the "constant growth forever" case, meaning N is infinity.
The formula in this case simplifies to
P = E1 / (R - G)
where E1 is earnings over the next 12 months.
This approach can be dangerous.
Constant growth forever means the company is going to get infinitely big, which is a hard concept to fit into a common sense understanding of valuation.
The formula will give you a number as long as the growth rate G is less than the discount rate R;
but you can force it to give you a ridiculously huge number if you make G very close to R.
This graph won't let you try that - the blue bars could blow through the top of your screen and hurt somebody - but you can see it happen in the discounted cash flows calculator in the stock valuation article.
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