“Lastly, the cotton story shows the strange liaison among different branches of the economy, and between economics and nature. That cotton prices should vary the way income does; that income variations should look like Swedish fire-insurance claims; that these, in turn, are in the same mathematical family as formulae describing the way we speak, or how earthquakes happen–this is, truly, the greatest mystery of all.” –*Benoit Mandelbrot 2004*

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If there are any economists, investors, or scientists who still believe the movements of the financial markets follow the standard (Gaussian) distribution, 2008 has likely shaken their foundations and fractured their paradigms.

A quick calculation of the Standard Deviation of one day’s close to the following day’s close (in percentage terms), from October 1, 1928 to October 24, 2008, is below. The calculations show that the average daily change on the Dow Jones has been .024%, with one standard deviation being 1.15%. Already, we see the daily change can vary greatly from the average change.

Above is the confidence intervals (orange background) for various standard deviations beyond the mean. If a data distribution is approximately normal then about 68% of the values are within 1 standard deviation of the mean, about 95% of the values are within two standard deviations and about 99.7% lie within 3 standard deviations.

In October alone, there have been 3 days with close-to-close changes greater than 6 standard deviations beyond the mean. The change on October 13th was 11.08%, better than 9 standard deviations from the mean. The crash in October 1987 would be greater than 18 standard deviations.

*These large variances in the Dow Jones data should not be present, unless the day-to-day changes are not normally distributed.*

For the rest of this article, I will then assume, as I believe it to be true, that the financial markets cannot be described within a Gaussian distribution.

Of course, to know that financial markets do not conform to a standard distribution is to understand that the Capital Asset Pricing Model, along with Value At Risk and Beta; Black-Scholes Formula and the Modern Portfolio Theory, are hopelessly and inherently flawed.

Indeed, the recent bear market has proven the financial markets to be much more volatile and risky, and anyone using the above models and theories to manage risk are likely finding themselves having to re-work all their models and risk-management formulae. Should they continue to rely on a standard distribution, they will forever be re-working their models, assuming they are not first ruined by them.

Since I have thrown out the idea that the market’s movements will be contained within X standard deviations from the mean, it is then crucial to digest and internalize this:

*There is nothing to stop the markets from experiencing large and devastating losses, of a magnitude never before witnessed.*

*_________________________________________________________________________________________________*

Part 2 of this series will describe fractals and how understanding them may be the key to understanding the recent market dislocations.

Part 3 will synthesize parts 1 and 2 in a discussion of why and how markets crash.

Benoit Mandelbrot is the expert on this topic. His latest book, *The (Mis) Behavior of Markets* is a must read for anyone interested in fractals and the financial markets.

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my head… it hurts!!

yoo speld dufos rong

Very provocative article. Hopefully, we get more from you soon, Wood — on this subject. Fuck, I love this website.

-Phil

Good stuff and thanks for the book reference.

Sorry but I have an off topic question. Have you done any work with Fibonacci numbers and what we are seeing in the market? I am a bit skeptical myself but then again I am also ignorant.

Phil, thanks!

F-ing Loser, I have done very little with Fibo numbers. Flys Nemesis, Mr. Tim Knight uses them often.

This is why I like Elliot Wave Theory. Love it or hate it, it seems to work best (at least to me) when market participants are trading on the emotions of the times. Being able to count these fractals as they appear in real time, and being able to assign a probability to the next form to appear, is the best edge I have in a market like this.

Also, love him or hate him, Prechter has taken Mandelbrot to task a few times, and IMHO, comes out ahead. I’ve read both of them and although I have enjoyed Mandelbrot’s writing and give him credit for bringing the subject to a wider audience, Prechter gives you something to work with by providing more answers than questions.

10* Wood.

Excellent post. Having not read “(mis)beaviour of the markets” – I must now add that to my list.

The more I look at the market, the more Taleb’s ‘barbell theory’ of investing makes sense.

Markets like this were made for his theories.

Wood – I’m curious what software you run that gives you the close-to-close SD going back to the 30’s – I love this type of stuff.

Ozark, I’ll have to check out Prechter. Thanks for the heads-up about him and Mandelbrot. Sounds very interesting.

Of course Mandelbrot picks at EW theory a little bit…

TSS, I downloaded the data from Yahoo finance into an Excel spreadsheet. From there, just calculate the close to close changes, and run the standard deviation function over the column of changes. Takes about 10 minutes if you have any experience with Excel.

Wood,

Have you done a runs test lately and determined Z-scores? Curious to see if there is positive or negative dependency with the markets now.

wood, obvious now that you told me – many thanks.

we are done…

Wood,

NOVA (PBS) is broadcasting a series on fractal geometry right now … “Hunting the Hidden Dimension” with interviews of Mandelbrot and his time at IBM, another engineer that invented fractal mountain making with the computer and ended up at Lucas Films as well as other mathmaticians like Julia.

Maybe you’ve already seen it or can find it On Demand if you have comcast or even youtube.

Thought you and others might be interested … Boo!

On again here (MA) at 7pm tonight and 4pm tomorrow …. particularly enjoyed the part about “fractal antennas” invented by a Boston man after attending a Mandelbrot symposium on fractals and the universe.

Awesome 8! Thanks for letting us know.