From: John Conover <john@email.johncon.com>
Subject: Re: forwarded message from Kimberly Bodelson
Date: Thu, 23 Jan 1997 02:48:56 -0800
I made a typo!
"sigma, we have more drops, (October of 1927 and 1989 for example,)"
should obviously been 1987, (October 19'th to be exact, and the 1927
date was October 29'th.)
My apologies-I was typing out of memory. So did I make any other
mistakes? Well, I wondered about that, and had johncon re-run the
data to make sure.
Some interesting facts:
Since 2 January, 1900, to date, there have been 53 declines of more
than 10% in the market indices. Of those 53, 15 were more than 25%.
To put that into perspective, every 2 years we would expect a decline
of 10%, and every 6 years, a decline of 25%, or so[1].
And how do the declines work? That is interesting. In the biz days
following the 1929 crash, the market dropped about 10%, (doesn't sound
like much does it?) However, the market then went out of steep decline
and into a slow grinding decline until 1932, at which time it had
dropped about 90%!!!! (The DJIA was 381 the date of the 1929 crash,
dropping 30, points, or so in the next few days, ending at 41 in 1932,
ie., about a 10 to 1 drop.) And how long did it take to get back to
381? Not until 1954. (To put that into perspective, radio became
popular, the TV was invented, we went through WWII, the baby boomer
generation was conceived, the Korean War started and almost finished,
and then the market regained its losses.) Is this an anomaly? No. If
you invested in 1968, you would have to wait until 1982 to make any
money, (not to mention how much you lost in the high inflation of the
era while you were waiting those 14 years-don't forget indices are not
in real dollars, whatever those are.) And then there was 73-74, which
was only a 2 year bear market, and, of course, 19 October, 1987, which
lasted about 2-3 years, or so[2].
Interestingly, there is more known about the American equity markets
than any other actuary[3]. The only things we have more knowledge
about is the speed of light, (now to 11 decimal places, and possibly
12,) and the element silicon, (for example, the band gap of silicon is
known to 9, and possibly 10 decimal places.)
John
[1] The root mean square, ie. one sigma, of the market indices runs
about 3% on a day-to-day time scale, or so. So we would expect a 9%,
or 3 sigma, drop in the indices every every 746 business days, or
about 2.87 years-if the distribution of the increments was represented
accurately as a Gaussian distribution. (2 years, or 520 biz days is
2.9 sigma for the same numbers, so we are close.) However, we would
expect a 25% drop, ie., about 8 sigma every 3 x 10^15 days, or about
about a thousand trillion years!!!!! (Which was my statement earlier,
that to 3 sigma we have a good model, beyond that ...) Note that these
sorts of things drive the numerical people up the wall. They would
attack the 3% number with a vengeance, and not the 3 sigma error of
3%, (which is considered "reasonable" accuracy in statistics.) They
would point out that these things are very sensitive, and the Gaussian
distribution highly non-linear at these levels, so itty bitty errors
make big differences. For example, a 3% error in the 3% rms value
would account for the discrepancy. And how many days of data do we
have to have for our measurements to have less than a 3% error in the
3% value? 188 days worth, to a 95% confidence level, which is the
confidence level the political polesters use. (And most statistics
folks consider that schlocky methodology.) So, you think the market
has been rising since 1982, right? Is it a short term "bubble" or is
it a genuine long term phenomena. How many days of data do we have to
have for a 95% confidence level that the market is really rising, and
not just a bubble? The market would have to have this trend over
824862 days, or about 3 millenia!!!!!! (We aren't sure what the
Babylonian markets were doing, but 3003 BC was a bumper crop for
barley, of which 85% was used to make beer. They kept records in
cuneiform clay tablets, which were organized into a relational
database, like a general ledger, much like we use today, only on clay
tablets instead of magnetic media.) Babylonian affairs aside, the
point is that one must be very careful about what one says about
markets, and their mastery thereof. As a general number, trends in
markets assert their self over a period that is measured in decades,
not years[2]. As you can see, multi-year run ups (and run downs,) in
the indices are common, and statistically understandable-at least to 3
sigma limits, or so.
[2] The duration of bear/bull markets is interesting. The nature of
things fractal is that they have these long time intervals above, or
below where they should be, (not periodic, but cyclic.) If we assume a
Gaussian distribution of the increments, these "intervals" when
averaged over a "sufficiently long time," will have probability of
duration that decrease with the reciprocal of the square root of time,
which is a very sluggish function-thats why they last so long.
Further, the number of bear or bull markets in a given time interval
is expected to be proportional to the square root of the time
interval. Note that although we can't predict when they will happen,
(there stochastic, and that means we can't,) we can tell a lot about
their statistical properties, which is what programmed trading is all
about. And what does this mean about the current market? The DJIA is
about twice what it should be to fit into the data of the rest of the
century. And are such things to be expected? Yes, they are, because we
would also would expect that the range of the indices in a bull minus
bear markets to increase with the square root of time.
[3] Of these things, the equity markets are the most difficult to
measure because they are stochastic. For example, the rms is about 3%,
or so, on a day to day basis. Squaring this number is approximately
the growth of the market in a day, or about 0.0009, or about a tenth
of a percent, in very rough numbers. Rms is the scale with which one
measures randomness. So, the problem is how do you measure a tenth of
a percent in something that is rattling around 3%? The answer is it
takes a lot of data. For example, suppose you wanted to be 99%
confident that you measured 0.0009 to and error level that is within
1%, (not an unreasonable expectation,) it would take 82,486,132 days,
(or about a third of a million years,) of data!!!! (We have several
centuries of data, BTW. The Spanish were good chroniclers, as were the
Dutch in their bull/bear market of tulip bulbs, which eventually went
bust-which is the 3'rd most highly studied fractal, next to the 1927
and 1987 market busts.) Just to give you a feel for such things, if we
drop our confidence level requirements to 50%, and and our error
requirements to 0.0009, it would require only 494 days, or about 2
years of data. So, in a 2 year run up, we are only 50% confident that
it is really even a run up, ie., a crap shoot, if we bet on that.
BTW, the way these things are handled by the programmed traders is to
calculate P for the equation f = 2P - 1. Then a statistical estimate
is made on the accuracy of P, and P is reduced by this amount, which
essentially takes the uncertainty of measurement into the stochastic
process of the stock's day-to-day rattling around. (For example, ASND
has been running very hot, with a measured P of 0.6. Suppose we did a
statistical estimate, and found that over the time interval, we were
90% confident that this value was not below 0.57, we would play P as
0.57 * 0.9 = 0.513. So we skirted the problem of not having enough
data with risk management. However, this will only work in bubbles,
and if the long term market P is above about 0.52, which it isn't, so
there is still some risk involved, but it is minimal, and deceasing as
the PT's gain more data with time. As you can see, they don't
gamble-they reduced their long term uncertainty risk to over 3 sigma.)
--
John Conover, john@email.johncon.com, http://www.johncon.com/