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```A most excellent question, I was wondering the same thing myself!
The PEG idea, on which the Fool Ratio is based, came from the
Wise.  Naturally us Fools should be concerned.  Here's what I
came up with.

For the following:
P = Price per share

E = Earnings per share in the last twelve months (Trailing Earnings)

G = Average Annual Growth in earnings

F = Earnings per share next year (Forward Earnings, predicted by analysts)

Also, recall that:
G = 100 * ((F/E) - 1)

First, consider what the following expression translates to
mathematically: "In a fully and fairly valued situation, a growth
stock's price-to-earnings ratio should equal the percentage of
the growth rate of its company's earnings per share."  That is:

P/E = G

And since:
G = 100 * ((F/E) - 1)

From both of the above, one gets:

F = E + (P/100)

(Trust me on this, that is what the algebra reveals.)

So the above indicates the assumption that, in a perfect world,
earnings shall increase by 1% of the present share price.  So why 1%?

Since PEG and Fool Ratio are intended for growth stocks, let's
begin with some basics.  A company on the grow, young and not yet
ready to pay dividends, shall plow back its earnings into its
assets.  Hence its stock price should naturally grow with the
earnings, because the earnings shall increase its book value.  In
a perfect world, share price and book value (what a company would
be worth if everything it had was sold off or cashed in, all
bills collected, and all debts paid off) should be in lock step.

Ok, then, the above translates to this math:

P[1] = share price in year 1
E[1] = earnings per share in year 1

"P" increases exactly with the earnings made - hence next year's
share price reflects last year's share price plus last year's
earnings:
P[2] = share price in year 2 = P[1] + E[1]

We rewrite the math expression "F = E + (P/100)" to:
E[2] = earnings per share in year 2 = E[1] + ((1/100) * P[1])

You'll note then that in year 3:
E[3] = E[2] + ((1/100) * P[2])
P[3] = P[2] + E[2]

And you could go on for year 4, 5, etc.  For any year i + 1:
E[i + 1] = E[i] + ((1/100) * P[i])
P[i + 1] = P[i] + E[i]

Now, what's so special about that "1/100" next to the "P[i]"?
Nothing that I can see - maybe it's just something arbitrarily
chosen, and that we could pick some other value.  Let's replace
it with an "a" for "arbitrary":
E[i + 1] = E[i] + (a * P[i])
P[i + 1] = P[i] + E[i]

And there you have it, a perfect growth company expressed
mathematically!  At this point, it may be good to set up a
spreadsheet to iterate the years of a perfectly valued
enterprise.  For instance, starting with P[1] = \$1.00 and E[1] = \$0.10:

Let a =  0.01
Year  Price  EPS    P/E    G      PEG  Gain
1     \$1.00  \$0.10  10.00
2     \$1.10  \$0.11  10.00  10.00  1    10.00%
3     \$1.21  \$0.12  10.00  10.00  1    10.00%
4     \$1.33  \$0.13  10.00  10.00  1    10.00%
5     \$1.46  \$0.15  10.00  10.00  1    10.00%
6     \$1.61  \$0.16  10.00  10.00  1    10.00%
7     \$1.77  \$0.18  10.00  10.00  1    10.00%
8     \$1.95  \$0.19  10.00  10.00  1    10.00%
9     \$2.14  \$0.21  10.00  10.00  1    10.00%
10    \$2.36  \$0.24  10.00  10.00  1    10.00%

("Gain" is the annual percentage increase of share price.)

What if "a" were something different, such as a = 0.02?

Let a =  0.02
Year  Price   EPS    P/E   G      PEG  Gain
1     \$1.00   \$0.10  10.00
2     \$1.10   \$0.12  9.17  20.00  0.5  10.00%
3     \$1.22   \$0.14  8.59  18.33  0.5  10.91%
4     \$1.36   \$0.17  8.19  17.18  0.5  11.64%
5     \$1.53   \$0.19  7.89  16.37  0.5  12.22%
6     \$1.72   \$0.22  7.68  15.79  0.5  12.67%
7     \$1.95   \$0.26  7.52  15.36  0.5  13.02%
8     \$2.20   \$0.30  7.41  15.05  0.5  13.29%
9     \$2.50   \$0.34  7.32  14.82  0.5  13.50%
10    \$2.84   \$0.39  7.26  14.65  0.5  13.65%

It would seem one's choice of "a" changes things dramatically -
for one thing the "In a fully and fairly valued situation,..."
clause from above no longer applies.  So what is this "a" really about?

I included a "Gain" column (not to be confused with growth!) to
see what is happening to the valuation of the stock.  What I
found out, emperically with my spreadsheet, was that the "Gain"
eventually converges to a constant value.  For example, the
"Gain" settles to 14.14% after 25 years in the above example.

Furthermore, I found that this convergence value of the "Gain" is
equal to the square root of "a".  In other words, if you had a
particular "Gain" in mind, you would set the "a" to the square
root of that "Gain".  It wouldn't matter what P[1] or E[1] is,
"Gain" always goes to the same value, that of "a" times "a".

Since the historical gain of the stock market is in the
neighborhood of 10%, then "a" = Square_Root(10%) =
Square_Root(0.10) = 0.01.  With this "a", we get back where we started.

My opinion:  The "In a fully and fairly valued situation,..."
saying is a mnemonic of truth rather than the Truth itself.
While misleading, it is nonetheless a useful way in remembering
the PEG, and of course the Fool Ratio too.  They should work so
long as the market gains always average out to about 10% a year.
And if it doesn't, we Fools can fix our beloved Ratio before the
Wise can figure out what's going on with their PEG!

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