From: John Conover <john@email.johncon.com>
Date: 29 Dec 2000 22:58:10 -0000

```
Note that there is no single "best" way to invest. These are
engineered solutions, (they don't call it financial engineering for
nothing,) and the "best" solution depends on what one is trying to do.

The attached e-mail described an optimal framework for equity trading,
(i.e., exploiting market "bubbles.") But many are uncomfortable with
trading, and prefer long term investing-so we will engineer an optimal
solution for that, too.

As it turns out, (using the same numbers in the attached e-mail, rms =
0.02, avg = 0.0004-which are typical for the US equity markets,) the
number of long term equity investments in one's portfolio, (i.e., the
number of stocks held at any time,) times the number of days the
equities are held is a constant, 2500; for example, holding 10
nicely, (and it fits nicely into capital gains regulations, too.)

Virtually every fund manager in the US uses this framework for long
term investing-and it delivers about twice the performance of picking
and investing in stocks individually.

It works because equity prices are ergotic, (what that means is that
investing in 10 stocks at the same time for one day has the same
statistics as investing in one stock for 10 days; which kind of
implies that, in the long run, optimal trading and optimal long term
investing will achieve virtually identical results.)

Here's why.

For n many equities held in a portfolio, and an equal investment in
each, the averages of the marginal increments add linearly in the
portfolio:

avg    avg          avg
1      2            n
avgp = ---- + ---- + ... + ---- =
n      n            n

1
- (avg  + avg  + ... + avg ) ............ (1.0)
n     1      2            n

where avgp is the average of the marginal increments in portfolio
value. The root mean square of the marginal increments, rmsp, is:

2             2
rmsp = sqrt ((rms  / n)  + (rms  / n)  + ...
1             2

2    1          2      2
+ (rms  / n) ) = - sqrt (rms  + rms  + ...
n          n          1      2

2
rms  ) .................................. (1.1)
n

and the ratio:

avg  + avg  + ... + avg
avgp         1      2            n
---- = ------------------------------- ...... (1.2)
rmsp            2      2            2
sqrt (rms  + rms  + ... + rms )
1      2            n

is useful in the calculation of the Shannon entropy, (or probability,)
P, of the portfolio, P = (avgp / rmsp + 1) / 2, which is the
likelihood that the portfolio will increase in value on any given
day. Assuming all equities have identical fractal statistics, the
average of the marginal increments in the portfolio's value would be
avg, (ie., n many, divided by n,) and the root mean square of the
marginal increments, (ie., the volatility,) would be rms / sqrt (n),
(ie., sqrt (n) / n.)

For one equity, held N many days, the average of the marginal
increments at the end of the N'th day would be the sum of the daily
marginal increments:

avgp = avg  + avg  + ... + avg  ............. (1.3)

and the root mean square of the marginal increments at the end of the
N'th day would be:

rmsp = sqrt (rms  + rms  + ... + rms ) ...... (1.4)
1      2            N

and the ratio:

avg  + avg  + ... + avg
avgp         1      2            N
---- = ------------------------------- ...... (1.5)
rmsp   sqrt (rms  + rms  + ... + rms )
1      2            N

is useful in the calculation of the Shannon entropy, (or probability,)
P, of the portfolio, P = (avgp / rmsp + 1) / 2, which is the
likelihood that the portfolio will increase in value at the end of the
N'th day. If the statistics are stationary for N many days, then: the
average of the marginal increments in the portfolio's value would be N
* avg, (ie., N many,) and the root mean square of the marginal
increments would be sqrt (N) * rms, (ie., the square root of N many.)

Combining equations (1.2) and (1.5), the average of the marginal
increments, avgp, of the portfolio, for n many equities, held N many
days, would be:

avgp = N * avg .............................. (1.6)

and the root mean square, rmsp:

N
rmsp = sqrt (-) * rms ....................... (1.7)
n

Note that if rmsp = avgp, then the Shannon probability, (ie., the
likelihood of an up movement, from P = (avgp / rmsp + 1) / 2,) would
be unity, implying a no risk investment strategy:

N
avgp = rmsp = N * avg = sqrt (-) * rms ...... (1.8)
n

and solving:

rms
sqrt (n * N) = --- .......................... (1.10)
avg

which, for rms = 0.02 and avg = 0.0004, is:

0.02
sqrt (n * N) = ------ = 50 .................. (1.11)
0.0004

or:

n * N = 2500

justifying the statement "the number of equities in one's portfolio,
times the number of days the equities are held is a constant, 2500,"
for typical stocks in the US equity markets.

And, it would be expected that one's portfolio value would grow about
twice as fast as the value of any stock in the portfolio, too. For a
single typical stock, P = (avg / rms + 1) / 2 = (0.0004 / 0.02 + 1) /
2 = 0.51, the average gain in value per day, G, would be:

G = (1 + rms)^P * (1 - rms)^(1 - P)

= (1.02)^0.51 * (0.98)^0.49 = 1.0002 .... (1.12)

year.

However, for a portfolio of 10 such stocks, the rmsp would be 0.02 /
sqrt (10) = 0.00632, giving a Pp = (avg / rmsp + 1) / 2 = 0.532, and
the average gain in value per day, Gp, of the portfolio would be:

G = (1 + rmsp)^Pp * (1 - rmsp)^(1 - Pp)

= (1.00632)^0.532 * (0.99375)^0.468

= 1.00038 ............................... (1.13)

which is about twice as much.

John

John Conover writes:
>
> Well, the NASDAQ had to finish up 83 today to avoid year 2000 having
> the dubious distinction of being the worst year of its 30 year
> history. It didn't make it, and is down about 50% from its highs of
> the year, and about 35% on the calendar year.
>
> The NASDAQ lost about 50% of its value since mid March, (about a 180
> trading days,) and the chances of that happening has a standard
> deviation of about 0.02 * sqrt (180) = 0.268, or about 25%; 50% would
> once in 30 years, (the NASDAQ is 29 years old-started in 1971.)
>
> All the indices turned in negative numbers for the calendar year,
> (which from the attached e-mail should be about a once a decade chance
> if there is correlation in the indices-it works out; the last time
> this happened was 1990, and before that, 1981.)
>
> So, what is the prognostication for the future?
>
> The current bear market "bubble" started in about the first quarter of
> 1999, (the S&P and DJIA haven't really moved since then,) and half of
> the bear markets last less than 4.3 years, (and half more,) so there
> is about a 50/50 chance that the current bear market will bottom in
> about early 2001, with the indices being back on track by about late
> 2002. During that two year interval, there is about a 50/50 chance
> that the indices will turn in about twice their average growth, (which
> is a little under 10% per year,) or there is about a 50/50 chance that
> the increase in the value of the indices will be about 20% for the
> years 2001 and 2002, or so.
>
> But such things are not all that rosy-these types of probability
> functions have very sluggish tails; for example, there is, also, a 1 /
> sqrt (25) = 20% chance that the current bear market will last at least
> as long, (but not as deep,) as the bear market of the Great
> Depression-25 years.
>
> So, how does one use such information?
>
> The chances of 2001 being a bull year is about 1 / sqrt (3) = 0.577,
> or about 58%; and one should place about 2P - 1 = 2 * 0.58 - 1 = 0.16,
> or about 15% of one's wealth at risk on the chance that it will be a
> bull year.
>
> For 2002, its about a 1 / sqrt (2) = 0.707, or about a 70% chance for
> a bull year; so, one should place about 2 * 0.7 - 1 = 0.4, or about
> 40% of one's wealth should be placed at risk on the chance that it
> will be a bull year.
>
>         John
>
> BTW, note what is happening; since the markets have been negative,
> (i.e., less than their average 10% growth,) for about two years,
> (e.g. one should not have been invested-another of mathematics most
> profound insights,) money is being moved back into market,
> gradually. One doesn't want to "wager" everything, since there is a
> significant probability that the bear market will last much longer
> than four years, and all would be lost. However, investing nothing
> would preclude taking advantage of the average anticipated gain of 20%
> in the indices per year for the next two years, too.  Obviously, the
> optimal fraction of wealth to put at risk lies between these two
> limits, and the "magic" number is 2P - 1, if one wants to attempt to
> exploit "bubbles."
>
> John Conover writes:
> >
> > What's the chances of any arbitrage system, (like an equity market,)
> > plummeting 22% in a month of 20 trading days?
> >
> > You measure the risk, (e.g., the deviation, which is the square root
> > of the variance of the fluctuations,) and multiply that by the
> > square root of the number of days. The deviation for the NASDAQ is
> > about 2%, (meaning that for 1 sigma = 68% of the time, the
> > fluctuations are less than 2%,) so, the standard deviation of the
> > fluctuation measured on a time scale of 20 days would be 0.02 *
> > sqrt (20) = 9%, or so, (meaning that for 68% of the time, the
> > fluctuations on a time scale of 20 days would be less than 9%).
> >
> > Its that fractal stuff.
> >
> > So, 22% / 9% = 2.44 sigma, which has a probability, (using the
> > normal probability tables, or your handy dandy calculator,) of
> > between such things, on average.
> >
> > And just when we had decided that it was a new economy.
> >
> >       John
> >
> > BTW, it works out about right. The last such rogue month was
> > in 1987, and there were 9 of them in the 20'th century.
> >

--

John Conover, john@email.johncon.com, http://www.johncon.com/

```