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/

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