From: John Conover <john@email.johncon.com>
Date: 29 Dec 2000 23:42:08 -0000

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As a side bar, note the requirement that equal investments in each
stock in the portfolio had to be maintained at all times for long term
investing.

What this says is that moving money around in the portfolio is more
important than picking the right stocks-which is counter intuitive,
(although one would do better picking the right stocks, too.) Fund
managers typically do this on a day-do-day operational basis.

Note that equal investments in each stock is a requirement imposed by
all stocks being "typical," and having the same statistics, (which is
not a bad approximation, BTW.) Fund managers alter the amounts
invested in each stock by altering equations 1.0 through 1.5, (they
put a Kn in front of each of the avg and rms values, where 0 < Kn < 1,
and solve for the optimal Kn's.) An optimal solution is, in fact,
quite difficult to compute-it requires mathematical programming
techniques, (one of the things the quantitative analysts for a fund do
for a living.) For this reason, fund managers usually have more, or
less than equal, invested in the various assets in the portfolio, (but
not far from equal.)

Moving money around in the portfolio is called "balancing asset
allocations."

What happens if one does not balance a portfolio?

The marginal increments of the portfolio's value will evolve into a
leptokurtotic distribution as the value of the stocks in the portfolio
evolve into a log-normal distribution. In the extreme long range, it
reduces the portfolio's rate of gain by about half, (the log-normal
distribution of stock values means that one, and only one, stock would
be dominating the portfolio value at any given time-which is almost
like investing in only one stock-so we lose the 2X advantage we gained
through portfolio management as described in equations 1.12 and 1.13,
below.)

Note that the indices are not balanced by their very nature, so are
leptokurtotic-fluctuations in GE or MSFT have a far more significant
impact on the indices than other stocks.

So, as a lemma, it is always possible-in the long run-for a long term
investor's portfolio performance to "beat the market."

Which is a nice engineered solution for a long term investing
framework.

John

BTW, there is a second long term investing lemma, too. And that is
mitigating risk, (which is equivalent to rms in the attached e-mail,)
is far more important than picking winning stocks. By only mitigating
risk alone-through balancing and combining stocks in the portfolio-the
portfolio's growth rate doubled.

John Conover writes:
> 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
> equities for 250 trading days, (about a calendar year,) would work
> 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/

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