It has been said  not by me  that the best predictor of future stock returns is the arithmetic
mean of historical returns. Or, to put if differently, the Expected Return is best estimated
by using the Mean of historical returns.
Intrigued by this statement (based upon probabilistic/theoretical considerations) I decided to
test the thesis as follows:
 Starting in January, 1928, calculate the AVERAGE return over the next 5 years (ending Dec, 1932).
 Starting in January, 1928, calculate the ANNUALIZED return over the next 5 years (ending Dec, 1932).
 Get the ACTUAL annual return for the next year, starting in January, 1933.
 Repeat steps 1, 2 and 3 for January, 1929 then January, 1930 etc. etc. ending January 1995 (or thereabouts).
 Determine the correlation between the AVERAGE and ACTUAL returns and between the
ANNUALIZED and ACTUAL returns, for the collection of returns generated by the above steps.
 Repeat the above steps for 10 years and/or 15 years (or whatever).
>What's that probabilistic/theoretical stuff?
If the historical returns are distributed according to some probability distribution,
with an AVERAGE return of, say, 7.89%, then the AVERAGE of all possible returns for next year,
selected from the same distribution of returns, will also be 7.89%, and that means that ...
>That means the EXPECTED return, for next year, is also 7.89%, eh?
Exactly.
>So, how did the correlation go?
As in Figure 1.
>Conclusion?
I have no conclusion, except that both predictors seem okay for short time periods.
Who knows whether the annual historical returns are distributed in
some ... uh, convenient way ... and whether that distribution remains constant as the decades go by ... and whether ...
>Yeah, but what it they were distributed, say, Normally?
Okay, I'll try that. I'll generate a random set of about a thousand annual returns and do the same thing
as we did above ... except, of course, the time periods 1928, 1929, etc. don't mean
anything. We just look at a 2year moving window, a 5year window, etc.
I got Figure 2, below.
>Similar, eh?
Yes.
>Did you use Normal or Lognormal distribution?
You decide
 Figure 1

Figure 2
>What about the S&P 500. Normal or Lognormal?
Let's not start that debate. Pick whatever you want. The point is, should one use the AVERAGE
of historical returns, or the ANNUALIZED return ... and over what previous period? Forget about
how the returns are distributed.
>What would you do?
Neither. I'd look at the newspaper.
>The newspaper? To get an analyst's opinion?
No. To get news on the economy and what war is happening where and between whom and what CEO
is being accused of fraud and ...
>Yeah, yeah. So why are you doing this predictor stuff?
For its entertainment value.
 Figure 3

>What about Nday moving averages? Maybe years is too long.
Moving averages of previous daily returns? Okay, let's try that.
We'll consider the last two years or so and look at the correlation between the Nday moving
average of daily returns, and tomorrow's return. We get Figure 4.
>That looks like a negative correlation!
Yes.
>How about the volumeweighted return? Have you tried that?
It's the same. Small negative correlation.
 Figure 4

>Bollinger bands?
Okay, we can check to see if tomorrow's S&P goes UP if today's S&P falls below the
lower Bolliband ... and whether it goes DOWN if it crosses above the upper Bolliband.
>So?
If we use a Bollinger Band with 2 1/2 Standard Deviations above and below the 20day
Moving Average, we get Figure 5 where the wee dots indicate crossings.
>So?
Tomorrow's S&P goes UP/DOWN (according to Bolliband crossings) about 50% of the time.
>Hmm ... might as well toss a coin, eh?
 Figure 5

