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> -----Original Message-----
> From: Gene Pope [mailto:gene@xxxxxxxxxxxxx]
> Sent: Saturday, September 14, 2002 12:08 PM
> To: omega-list@xxxxxxxxxx
> Subject: Gauss vs. Cauchy
>
>
> Hey all,
>
> I'm trying to figure out how to write a variation on the StandardDev
> function in TS such that it reflects a non-Gaussian distribution, such as
> Cauchy or Paretian.
>
> I'm having difficulty translating the available formulas that describe the
> area under the distribution curves into the equivalent Deviation off of
> prices (or log of prices).
>
> Anyone out there who can throw me a bone (or point me to a better source for
> the math)? Much appreciated.
>
http://www.inv.com/79b4a.htm
LEPTOKURTOSIS
Compared to the Gaussian (bell-shaped) distribution, a leptokurtic distribution
of stock market returns has an excessive number of instances near the average
and at the extremes.
Leptokurtosis is a property of a class of statistical distributions called
stable Paretian. Whereas the Gaussian distribution is defined by two
parameters, the mean and standard deviation, Paretian distributions are defined
by four: a location parameter, a scale parameter, an index of skewness; and
alpha, a measure of the height of extreme tail areas of the distribution. The
Gaussian distribution is a special case, when alpha = 2, of a Paretian
distribution (Eugene Fama, The Journal of Business, 1965).
Fama measured the alpha of a portfolio of thirty large cap stocks. Using three
different estimating methods, he found an alpha range of the entire portfolio
between (1.73-1.94). The behavior of stocks is therefore more generally
Paretian rather than Gaussian, which is to say that the market is more volatile
and less at an equilibrium.
Statistical distributions, however, are descriptive and not causal. The Farmer
article suggests that trading between momentum and value investors causes the
leptokurtic behavior of the market. We further suggest that the direction of
the news and the deviation of the market from an equilibrium model affect the
short term direction of the stock market.
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