From: John Conover <john@email.johncon.com>
Subject: Re: U.S. recovery: Recession is certain, economists say
Date: Fri, 21 Sep 2001 14:21:46 -0700
The attached expresses a lot of concern about the number of
consecutive "down days" in the US equity market indices.
Bear in mind that the current tragic circumstances are a four sigma
catastrophic event-it would be expected that the marginal increments
of the market indices, assuming statistical independence, (a
reasonable assumption,) have enough nearly consecutive negative
movements to equal the probability of a four sigma event-about one
in 50,000, or so, or about 16, since 2^16 = 65,536, (working with
integer values, and round numbers.)
So, we would give the markets a 50% chance a positive movement before
mid-to-end of next week, and a 50% chance of no positive movements
until after.
So, the market's reaction to the recent tragic events is not
to be unexpected.
John
BTW, all I'm doing here is working with the tails of the distributions.
I counted how many catastrophic events, (of at least a certain
significance,) there were in a large time interval. The probability
of such a catastrophic event would be the number of events divided
by the length of the time interval.
I assumed that the equity markets react to events, which occur
randomly, (as opposed to generating randomness their self,) meaning
that the tails of the distributions of the increments of the market
indices will have the same probability distribution, as the events
that created them. I assumed statistical independence, meaning a
normal distribution-since, historically, market indices can be
expediently modeled in the short term, using such a distribution.
The tails and standard deviation of the normal curve have to be
related by sigma values, (otherwise, its not a normal curve's
frequency distribution-by definition-and, empirically, we know
that it is a reasonable assumption based on substantial
theoretical foundations.)
Knowing the historical value of the standard deviation of the
indices, (via metrics,) and that the statistically independent
increments are summed root mean square, the probabilities of
expected values can be calculated over a short time interval.
Note: Working with the tails of distributions is a very important
concept. Extrapolating measured values of the standard deviation
of the increments of a time series into the tails of the
distribution is often a leap of faith producing misleading and
erroneous probability values for catastrophic events. A good
verification of standard deviation metrics is to count the number
of 2 sigma, 3 sigma, 4 sigma, etc., events in the time series, and
see if this frequency count can be justified with the empirical value
of the standard deviation.
Many applied mathematics folks prefer working with tail counts as
opposed to standard deviation-they derive the standard deviation from
the tail counts; not the other way around.
> Certain is a big word. The US equity markets, with all the negative
> sentiment in the media, are not doing that bad-at least considering
> the difficult circumstances of the immediate past.
>
> More than 6,000 soles perished in the WTC attack. That would make
> September 11, 2001, the bloodiest day in American history, (replacing
> the battle of Antietam, September 15, 1862, in the US Civil War-in
> which it is estimated that about 6,000 were killed, also.)
>
> Considering the US to be about 4 centuries old, or about 146,000
> days, with two catastrophic instances of at least 6,000 soles perishing
> in a single day, which would represent a probability of 2 / 146,000 =
> 1.37E-5; i.e., the WTC attack was about a 4.2 sigma catastrophe.
>
> Since the standard deviation of the increments in an equity index is
> about 2% per day, and if the increments are statistically independent,
> (a reasonable assumption,) then in four days the standard deviation
> of the value of the index would be 0.02 * sqrt (4) = 4%, (meaning that
> in any 4 day interval, an indice would be within +/- 4% of its
> starting value, 68% of the time.)
>
> The market's reaction, at the end of 4 days to a 4 sigma event, would
> be about 4% * 4 = 16%, meaning that for 84% of the 4 sigma catastrophic
> events, the market's reaction would be to drop less than a 16%, and for
> 16%, they would drop more.
>
> Since the markets opened on Monday, the index values have dropped about
> 12% in the last four days.
>
> So, the markets, although significantly down, are reacting a little
> better than would be expected to the catastrophe of September 11,
> 2001.
>
> John
>
> BTW, this does not mean that a recession is not imminent-it might be,
> and might not. Such things are unknowable.
>
> http://www.infoworld.com/articles/hn/xml/01/09/20/010920hnrecession.xml
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--
John Conover, john@email.johncon.com, http://www.johncon.com/