From: John Conover <john@email.johncon.com>
Subject: Re: A year ago today, the NASDAQ was at an all time high
Date: 12 Mar 2001 05:59:03 -0000
BTW, the probabilities are more than just intellectual curiosities.
Here's why.
It looks like a major disaster, (where a stock portfolio can lose
virtually all of its value,) in an equity market happens about every
300 to 400 years-call it 350 years.
That means that the probability of a major disaster happening during a
lifetime of investing, (where one's retirement just disappears, for
example,) is about 60 / 350 = 17%, for 60 years of investing. Or,
there is an 83% chance that such a disaster won't happen.
What fraction of one's investments should be in an equity portfolio
that both maximizes the gain of the portfolio, while, at the same time
minimizes risk exposure to catastrophic economic disasters?
That fraction is 2 * 0.83 - 1 = 66%.
Which is why all the fund managers run around 65%, for most of the
time.
John
BTW, most modern fund managers have folks on staff that do such
quantitative analysis. Its an entrenched discipline in economics and
finance these days. But about 2/3'rds of one's total investments
allocated to high risk speculative markets being optimal was known far
before the information theorists formally proved it in 1956. It was a
closely guarded secret of the Dutch banking system in the 16'th
century. That it could be applied to commodity futures-like tulip
bulbs-was a bit of a revelation, without which, the downfall of the
entire European banking system was inevitable-the Dutch saved the day
by forcing 33% collateral in physical assets to back the speculation
in tulip bulbs near the end of the tulip "bubble." (During the tulip
"bubble", people were trading whole farms for 3 tulip bulbs, if you
can believe that-dot tulip mania.)
John Conover writes:
>
> Ooops, a typo. The S&P 500's drop in a year ended on September 6,
> 1930, when its value was 21.65, for a decrease in value of 32%-not
> September 25, 1930, as previously mentioned.
>
> Using Black-Scholes-Merton methodology to calculate the probabilities,
> (assuming the standard deviation of the daily marginal returns has
> statistically independent increments with a normal distribution, and
> was static over the twentieth century,) the probabilities for the
> first year of the Great Depression:
>
> The DJIA's standard deviation of the marginal returns is 0.010947,
> (about a percent a day, measured from 2 January 1900 to 1 March
> 2001, or 27700 trading days,) so the standard deviation at the end
> of the first calendar year, (of 291 trading days,) would be about
> 0.01 * sqrt (291) = 19%, or 38% would be about 2 standard
> deviations, or a probability of about 2.0%. Call it once every 50
> years, or so.
>
> For the S&P 500, the standard deviation of the marginal returns of
> 0.011262, (measured from 3 January 1928 to 1 March 2001, or 19455
> trading days,) so the standard deviation at the end of the first
> calendar year, (of 291 trading days,) would be about the same as the
> DJIA's at 19%, or 32% would be about 1.7 standard deviations, or a
> probability of about 4.7%. Call it once every 20 years, or so.
>
> And the NASDAQ has a standard deviation of 0.013083, (measured from
> 11 October 1984 to 1 March 2001, or 4141 trading days,) so the
> standard deviation of the NASDAQ's value between a year ago and now,
> (of 251 trading days-we don't trade on Saturday's any more,) would
> be about 0.013 * sqrt (251) = 21%, or 59% would be about 2.9
> standard deviations, or a probability of about 0.21%. About once
> every 476 years; call it 500 years, or so.
>
> And, to where the low occurred in the Great Depression:
>
> The DJIA dropped to its low, from its high, after 842 trading days,
> or the standard deviation would be 0.01 * sqrt (842) = 32%, or 89%
> would be 2.8 standard deviations, or a probability of about 0.25%.
> Call it once every 400 years, or so.
>
> The S&P 500 dropped to its low, from its high, after 811 trading
> days, or the standard deviation would be 0.01 * sqrt (811) = 32%, or
> 86% would be about 2.7 standard deviations, or a probability of
> about 0.36%. About once every 278 years; call it 300 years, or so.
>
> For the NASDAQ to follow a similar scenario to the DJIA and the S&P
> 500 in the Great Depression, and drop 87.5% from its high in 826.5
> trading days, the standard deviation would be 0.013 * sqrt (826.5) =
> 38%, or 87.5% would be 2.3 standard deviations, or a probability of
> about 1%. Call it once every 100 years, or so.
>
> (Note: Adjusting the 826.5 trading days to 688.75 since the NASDAQ
> has never traded on Saturdays-which was the case for the DJIA and
> S&P 500 in the Great Depression-the standard deviation would be
> 0.013 * sqrt (688.75) = 34%, or 87.5% would be 2.5 standard
> deviations, or a probability of 0.54%. About once every 185 years;
> call it 200 years, or so. Its an attempt to reconcile comparing
> apples and oranges.)
>
> John
>
> BTW, estimating the accuracy on the standard deviations of the
> daily marginal returns:
>
> For the DJIA, 0.010947 at 27700 trading days could be as low as
> 0.009778, and as high as 0.012116, or about a +/- 12% accuracy.
>
> For the S&P 500, 0.011262 at 19455 trading days could be as low as
> 0.010092 and as high as 0.012432, or about a +/- 12% accuracy.
>
> For the NASDAQ, 0.013083, at 4141 trading days could be as low as
> 0.01191 and as high as 0.014256, or about a +/- 9.8% accuracy.
>
> And, estimating the chances that the standard deviations of the daily
> marginal returns were measured, by serendipity, in a "bubble" of
> duration at least as long as the time they were measured:
>
> For the DJIA, erf (1 / sqrt (27700)) = 0.72%.
>
> For the S&P 500, erf (1 / sqrt (19455)) = 0.88%.
>
> For the NASDAQ, erf (1 / sqrt (4141)) = 1.8%.
>
> John Conover writes:
> > March 10, 2000 the NASDAQ closed at an all time high of 5048.62.
> >
> > It closed today at 2052.78 -115.95 (-5.35%), or its value is about 41%
> > of what it was a year ago today, for a loss of 59%.
> >
> > John
> >
> > BTW, when was the last time a major American market index fell at
> > least 59% in a year?
> >
> > The answer? Not in the last century, and not during the Great
> > Depression:
> >
> > The DJIA, on September 3, 1929 closed at a high of 381.17, which it
> > did not reach again again until November 23, 1954, (382.74). On
> > September 3, 1930, it closed at 237.54, for a drop of 38% in a year.
> > It continued to deteriorate until July 8, 1932 when it closed at low
> > of 41.22, for a loss in value of 89%.
> >
> > The S&P 500 closed at a high of 31.92 September 7, 1929, which it
> > did not reach again until October 25, 1954, (31.96). It was 21.65 on
> > September 6, 1930, for a drop of 32% in a year. It continued to
> > deteriorate until June 1, 1932 when it closed at a low of 4.40, for
> > a loss in value of 86%.
> >
--
John Conover, john@email.johncon.com, http://www.johncon.com/