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Minyan Mailbag: Follow up - Of Fat Tails and Leptokurtosis



Note: Our goal in Minyanville is to remove intimidation from the financial markets and encourage an interactive dialogue among the Minyanship. We share this next discussion with that very intent.


I've got a tremendous amount of respect for the work you do, and my favorite columns on the site are your pontifications involving central banks, etc, but I've done research that contradicts a point you said the other day.

You said that a big down day increases the chances that the next day will be down, and vice versa, but I've found it to be the opposite. I don't have the data in front of me, but over the past 15 years the SPX's average 1-day return has been +0.04%. After a down day the average 1-day return increases, after another down day it increases again, and after a third down day it increases again.

The results are even more explicit when you lengthen the time frame of "holding the SPX" to a week. Consider the hypothetical meaning if your presumption were true. If an up day increased the odds of an up day, and a down day increased the odds of another down day, then the market will hypothetically want to go to infinity or zero in the long-run as it experiences successive up days or down days. If you've got data that disputes this then I'd love to see it.

Minyan Conor


Your research model is an autoregressive model, which is just a fancy way of saying it's measuring mean reversion - if stocks are down big, they will bounce back with a positive day the next day. A mean reversion model would produce a normal frequency distribution for returns (one day, 5 day or 20 day returns, doesn't matter). And the chart I showed clearly illustrates it does not: fatter tails and leptokurtosis are what the "real" SPX data illustrates.

I noted in my piece that the actual SPX return data...

"shows a much larger frequency of returns around the mean (where X = 0 in this chart) but a correspondingly smaller frequency of returns between 1 and 2 standard deviations from mean,..."

What this means is that yes, there are times when mean-reversion takes place - when your data (from only the last 15 yrs I might add) reverts to positive the day after a negative day. But there are times - times like 1987, like 2000-2002, like 1929, where those mean reverting tendencies do not operate. And in those instances, the probability of a down day is greater the day after a big down day than EMH would suggest.

Taken to the extreme what you illustrate would cause the stock market to go up to infinity or to zero respectively. But such an exercise rests on the assumption that markets are NEVER linear. I did not/do not say that at all: markets exhibit regimes of linearity and non-linearity which, taken together, make them 'complex". I am merely saying that the frequency distribution is not normal, which means the assumptions of efficiency are incorrect.

Hope that helps clear it up a touch.



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