From: John Conover <john@email.johncon.com>
Subject: Re: Cycles or random shocks?
Date: 1 Nov 1999 00:36:46 -0000
j. tyler writes:
>
> Currently there is another thread on the likelihood of another 1929. Of
> course, that is the $64,000 question - no one has any real informatiion.
> The standard theory on the subject now is that "cycles" are due to
> random shocks. That means now one really knows when another crisis may
> strike. And it is doubtful that there is an economist in the country
> who has a clear explanation why we are enjoying unusually good fortune
> right now.
>
Ok, I'll give it a shot.
Between 7 September, 1929, and 6 June, 1932, the US equity markets
lost about 90% of their value, (DJIA values of 375.44 to 42.68.) Most
of that was the in the 400 trading days following October, 28'th and
29'th, 1929, (where the market dropped about 12% in each of the two
days, bounced back to being only about 10% down, and then just
constantly deteriorated for two years.)
The day-to-day fluctuations in the DJIA are about +/- 1%, rms,
(meaning that for 68% of the time, the fluctuations are less than +/-
1%,) so assuming a first order approximation of equity prices as a
Brownian motion fractal, (ie., no leptokurtosis,) at the end of a 400
trading day interval, we would expect the DJIA to be about 0.01 * sqrt
(400) , or about 0.2, (meaning that if we look at all possible 400 day
time intervals of the DJIA, we would expect the increase, or decrease,
to be less than 20%, 68% of the time.
So, 90% would be 0.9 / 0.2 = 4.5 standard deviations, or a probability
of 0.00000339767, or once in 294,000 trading days, or about once every
1,200 years.
John
BTW, the "unusually good fortune right now", assuming the last bad
year, (where the DJIA turned in negative numbers on the year,) was
1993, has a chance of continuing into its seventh year of about of 1 /
sqrt (7), or about 37.8%-again assuming a first order approximation of
a Brownian fractal as a mathematical expediency-ie., it was virtual
certainty that at least one bull 8 year run would happen in the
century, (likewise, for a bear market, too, as per 1930-1956, when the
DJIA had recovered to its 7 September, 1929 value, in non-real index
value, was a virtual certainty.)
However, the *_depth_* of the 1930 recession was quite rare, (although
its length would be considered rather common.)
Assuming a first order approximation of a Browning fractal, the depth
of things move on a 0.01 * sqrt (t), scenario, and the chances of
something continuing move on erf (1 / sqrt (t)) which is about 1 /
sqrt (t), for t >> 1.
Which means that the average bear or bull market would last, on
average, about 4.3 years, (but there would be a lot of variability
since the 1/ sqrt (t) function has tails that initially fall off
rapidly, then fall off very sluggishly.)
There are empirical graphs of all possible bubble run lengths and
magnitudes of the DJIA, S&P, and NYSE for the century, compared with
the theoretical values of a Brownian fractal at:
http://www.johncon.com/john/correspondence/981230002304.31518.html
http://www.johncon.com/john/correspondence/990212211510.16959.html
and it does seem to be a useful and reasonable first order
approximation to the dynamics of the markets, (depending on who is
telling the story, of course.)
--
John Conover, john@email.johncon.com, http://www.johncon.com/