From: John Conover <john@email.johncon.com>
Subject: Re: Will the stock market
Date: 5 Feb 1999 21:53:41 -0000
Bill Terrell writes:
>
> Not quite sure if I understand you here; whether by saying "the next increment will be
> like the previous" you mean in the same direction. If so, no question there is such a
> thing as trend persistence, and without exhaustive study, 60% sounds reasonable. But if
> you generalize that to mean there is a dependence relationship (direction not necessarily
> specified) between the next increment and the previous, or better yet, the next increment
> and previous increments (plural), might not the correlation go higher?
>
Actually, (ignoring chaotic effects-ie., assuming a "traditional"
fractal model is adequate,) the solution space of fractals would be a
continuous spectrum from independent increments, (ie., a random walk
that is ergotic and things add root mean square as per a
Gaussian/normal distribution,) to dependent increments, (ie., those
fractals with a Cauchy distribution, and things add linearly.) The US
equity market indices are someplace in between, with about a 60%
dependency. The implication being that there is cross information
between stocks in a market, and short term (Markovian,)
persistence. The mathematics is slightly different, too. For example,
volatilities do not add root mean square, (ie., t^2 = a^2 + b^2 ...),
but t^1.67 = a^1.67 + n^1.67, instead.
I think most theoreticians, (and I don't speak for them,) consider the
equity markets to be a mechanistic, chaotic system of such endless
complexity that it can not be anticipated-so it exhibits random
properties, (random meaning unpredictable.) A common tossed die is
often cited as a similar example-it is a mechanistic system, (ie.,
"governed" by F=MA, etc.,) but exhibits random properties since the
outcome can not be anticipated. Two die tend to exhibit number
frequency distributions that is Gaussian in nature-in much the same
way the increments in a random walk fractal fall into a Gaussian
distribution.
>
> Okay, I won't take that to mean you are suggesting that a stock market bubble is in
> evidence. But I think further back the chain I challenged your argument as presenting
> what struck me as a rosier view than mine. So let me ask: Do you think we are not in a
> stock market bubble, or do you think we are in a stock market bubble, but not enough of
> one to justify putting most of one's long term assets in alternative vehicles at this
> time? Or ...?
>
Well, if we assume a fractal model is adequate, (and whether it is, or
not, depends on who is telling the story,) then we would be-by
definition. Fractals are made up of an infinite complexity of
"bubbles," at all time scales-and if it is a random walk fractal, then
the "bubbles" relate to each other, (ie., scale,) by the square root
function. Assuming a random walk fractal, if the current market
"bubble" started in 1995, then the chances of the "bubble" continuing
through at least 1999, ie., 4 years, would be about 1 / sqrt (4) =
0.5, or 50%.
How well does the random walk fractal fit the data? There is a graph
of the duration of "bubbles" in the DJIA, NYSE, S&P 500, a simulation,
and the random walk theoretical value at:
http://www.johncon.com/john/correspondence/981230002304.31518.html
It seems to fit reasonably well, (depending on one's POV, or course.)
John
BTW, the 1 / sqrt (t) is an approximation-easy enough to figure out in
one's head. The formula is erf (1 / sqrt (t)), which for t > 1, is
about 1 / sqrt (t).
--
John Conover, john@email.johncon.com, http://www.johncon.com/