Correlation     and related pseudo-periodicity

In a paper by Grinblatt & Moskowitz, it is noted that future returns seem to be negatively correlated in the short term (1 month past returns) and long term (3 - 5 year past returns), but positively correlated in the medium term (3 - 12 months past returns). This seems an intriguing observation, so I did the following, using monthly S&P 500 returns, starting in Jan, 1960 (for example):

  1. I calculate the recent monthly returns by averaging the monthly returns over the previous 12 months.
  2. I compare this recent return to the monthly return 1 month in the future.
  3. I move ahead one month to Feb, 1960 and repeat steps 1 & 2 (calculating the average of 12 previous months and the return 1 month in the future).
  4. After umpteen years of "moving ahead one month", I have two sequences of returns: the 12-month recent averages and the returns 1 month in the future.
  5. I calculate the Pearson Correlation coefficient for these two sets of returns.
  6. Then I repeat steps 1-5, comparing recent returns with the return 2 months in the future.
  7. Then 3 months in the future.
  8. etc. etc.
I get this interesting result:

>Why?
Yes. That's the question ...

>And what's the answer?
I have no idea. Yet.
However, it has nothing to do with "the market" or investor psychology or Greenspan or ...

>So what does it have to do with?
I don't know ... yet. But if we replace the monthly S&P 500 returns with 'invented' returns, generated by a random number generator, we get Figure 2

Now, I'll have to repeat all this with, say, 6-month past returns, or 24 months or ...

>And what does all this mean?
I suspect there's a mathematical explanation so I'll ...

>Mathematical? Uh ... can we change the subject?
Pay attention.


Fig. 1

Fig. 2

Fig. 3

>Okay, but the paper by Grinblatt & Moskowitz talks about future returns compared the average return over the past 1 month, 12 months, 36 months ...

Right, so I did that, too.

Figure 3 shows the correlation between future returns (which I take as the average monthly return over the NEXT 3 months) and past returns (which I take as the average monthly return over the past M months).
>It still looks sort of periodic.
Let's call it pseudo-periodic, okay?


>What about your invented, random monthly returns? Does it ...?
In Figure 4, instead of S&P 500 returns, I go through the same ritual, but I use a set of randomly generated monthly returns.

>So, what're you going to do with all this?
Let me think on it.


Fig. 4

See also Correlation: does it matter?

for Part II