“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
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.