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# Of Fat Tails and Leptokurtosis

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In my discussions with colleagues over the last year or so on the complexity theory of asset pricing, I have found that acceptance of the theory is not possible until one sees the severe limitations that the standard Efficient Market Hypothesis (EMH) operates under. More specifically, until folks see the rather wide gap between what EMH predicts and what the actual market does, investors have a difficult time accepting complexity as a viable model for why and how markets behave as they do.

We have spent lots of time writing about the fact that markets are not efficient, that investors are not entirely rational and, as a result of these two things, that stock prices are not some sort of random walk. By comparing how the market should 'act' if it were truly efficient and how it actually does 'act', we might be able to become comfortable with the idea that another model for asset price behavior may be operative.

Below is a chart of what a normal frequency distribution of returns would look like for an efficient stock market. While statisticians and mathematicians uniformly use the term "normal distribution", physicists sometimes call it a Gaussian distribution and, because of its curved flaring shape, social scientists refer to it as the "bell curve." Such a 'normal' distribution of returns for the S&P 500 would look like the below, where the X axis represents the standard deviation of the expected returns and the Y axis represents the probability of any given return's occurrence:

If we could illustrate that stock prices are not in fact normally distributed like the above theoretical chart, then we could then conclude both that investors are not rationale (something that Daniel Kahneman's and Amos Tversky's research has already concluded) and that markets are not efficient. By exhibiting insufficiency, theory would hold that there are, in the least, variables, characteristics, and potential models that exist that could help us better understand the market's dynamics and potentially predict future changes in asset prices more accurately.

So how does the actual distribution of S&P 500 returns look relative to the 'normal' distribution of returns predicted by the Efficient Market Hypothesis? The chart below overlays a representation of the actual 5 day returns data for the SPX from 1928 to 1989 (dashed line) on the 'normal' distribution chart from above.

Note the significant discrepancy between the two plots. The actual SPX return data is (1) skewed to the right, (2) 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, and (3) more frequent very large positive or negative returns than predicted. The term 'fat tails' refers to the higher-than-expected large positive or negative returns while the term leptokurtosis refers to the higher-than-expected peaks around the mean.

We wrote in an article back in December about this idea: the theoretical probability of seeing a 3 day return like that witnessed during the 1987 crash, the 1929 bear market and a few times during the 2000-2002 bear market is 1 occurrence in 7000 years. That it has happened more than 4 times speaks directly the discrepancy between the theoretical SPX returns and the actual returns. Again, the actual data doesn't fit the predicted data.

Importantly, the chart above is not novel at all; researchers (Turner & Weigel; "An analysis of stock market volatility", Russell Research Commentaries, Frank Russell Companies, 1990) have known about this 'problem' with EMH for decades. And whole careers have been made in an attempt to explain this phenomenon away. Too, further study of other asset markets - treasury bonds, currencies, other countries' stock markets, commodities - show the same characteristics of fat tails and leptokurtosis. And this similarity strongly suggests that the forces behind all negotiated financial instruments share a common framework. We happen to believe that common framework is complexity theory.

There are many important implications to what the two charts above suggest, both for existing linear models of stocks and economic statistics as well as for potential new non-linear models (on which we are working feverishly). We will go into more of these implications in the future but one we think will really cook your noodle and it is the idea of independence vs interdependence.

If markets are not efficient, if they are not a random walk, then they are interdependent (as opposed to independent). What does interdependent mean? It means the market has a memory - that one day's prices affect the next day's: that if the market sees a big down day, the probability is actually increased that the next day will be down (and vice versa). It means that one week's prices affect the next week's, that one year's prices affects the next year's and so on. The market remembers.

Think about that for a while. And then think about what it means for asset price prediction if you could develop a model for exploiting that memory.

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