From: John Conover <john@email.johncon.com>
Subject: Re: portfolio management
Date: Sat, 19 Sep 1998 22:16:55 -0700
John Conover writes:
>
> Can the portfolio growth be made optimal? Yes, for example, by buying
> stocks on margin, (I'll show that margins are not a practical strategy
> when building a stock portfolio,) using the exact same methods we have
> been using all along-and it too can be optimized-ie., the portfolio
> growth can be maximized, while at the same time minimizing risk
> exposure to the margin calls. The tsstock program, (which is not part
> of the tsinvest suite-it is part of the fractal.tar.zip suite in
> http://www.johncon.com/ndustrix/archive/fractal.tar.gz,) can be used to do it.
> The trick is to buy enough, (but not more or less,) of the initial
> investment in a group of stocks that make up a portfolio on
> margin. How much on margin? It turns out to be:
>
The tsstock program is for simulating the gains of a stock investment
using Shannon probability.
Bottom line, it is for programmed trading (PT) of stocks. The C
sources are freely available as Open Source software on
http://www.johncon.com/ndustrix/archive/fractal.tar.gz.
A fragment, specific to this discussion, of the manual page is
attached ...
John
--
John Conover, john@email.johncon.com, http://www.johncon.com/
DESCRIPTION
This program is an investigation into whether a stock
price time series could be modeled as a fractal Brownian
motion time series, and, further, whether, a mechanical
wagering strategy could be devised to optimize portfolio
growth in the equity markets.
Specifically, the paradigm is to establish an isomorphism
between the fluctuations in a gambler's capital in the
speculative unfair tossed coin game, as suggested in
Schroeder, and speculative investment in the equity mar-
kets. The advantage in doing this is that there is a
large infrastructure in mathematics dedicated to the anal-
ysis and optimization of parlor games, specifically, the
unfair tossed coin game. See Schroeder reference.
Currently, there is a repository of historical price time
series for stocks available at,
http://www.ai.mit.edu/stocks.html
that contains the historical price time series of many
hundreds of stocks. The stock's prices are by close of
business day, and are updated daily.
The stock price history files in the repository are avail-
able via anonymous ftp, (ftp.ai.mit.edu,) and the programs
tsfraction(1), tsrms(1), tsavg(1), and tsnormal(1) can be
used to verify that, as a reasonable first approximation,
stock prices can be represented as a fractional Brownian
motion fractal, as suggested by Schroeder and Crownover.
(Note the assumption that, as a first approximation, a
stock's price time series can be generated by independent
increments.)
This would tend to imply that there is an isomorphism
between the underlying mechanism that produces the fluctu-
ations in speculative stock prices and the the mechanism
that produces the fluctuations in a gambler's capital that
is speculating on iterations of an unfair tossed coin.
If this is a reasonably accurate approximation, then the
underlying mechanism of a stock's price time series can be
analyzed, (by "disassembling" the time series,) and a
wagering strategy, similar to that of the optimal wagering
strategy in the iterated unfair coin tossing game, can be
formulated to optimize equity market portfolio growth.
As a note in passing, it is an important and subtile
point, that there are "operational" differences in wager-
ing on the iterated unfair coin game, and wagering on a
stock. Specifically, in the coin game, a fraction of the
gambler's capital is wagered on the speculative outcome of
the toss of the coin, and, depending on whether the toss
of the coin resulted in a win, (or a loss,) the wager is
added to the gambler's capital, (or subtracted from it,)
respectively. However, in the speculative stock game, the
gambler wagers on the anticipated FLUCTUATIONS of the
stock's price, by purchasing the stock. The important dif-
ference is that the stock gambler does not win or loose an
amount that was equal to the stock's price, (which was
equivalent to the wager in the iterated unfair coin game,)
but only the fluctuations of the stock's price, ie., it is
an important concept that a portfolio's value (which has
an investment in a stock,) and the stock's price do not,
necessarily, "track" each other.
In some sense, wagering on a stock is NOT like a gambler
wagering on the outcome of the toss of an unfair coin, but
like wagering on the capital of the gambler that wagered
on the outcome of the toss of an unfair coin. A very
subtile difference, indeed.
Note that the paradigm of the isomorphism between wagering
on a stock and wagering in an unfair tossed coin game is
that the graph, (ie., time series,) of the gambler's capi-
tal, who is wagering on the iterated outcomes of an unfair
tossed coin, and the graph of a stock's price over time
are statistically similar.
If this is the case, at least in principle, it should be
possible to "dissect" the time series of both "games," and
determine the underlying statistical mechanism of both.
Further, it should be, at least in principle, possible to
optimize portfolio growth of speculative investments in
the equity markets using information-theoretic entropic
techniques. See Kelly, Pierce, Reza, and Schroeder.
Under these assumptions, the amount of capital won or lost
in each iteration of the unfair tossed coin game would be:
V(t) - V(t - 1)
for all data points in the gambler's capital time series.
This would correspond to the amount of money won or lost
on each share of stock at each interval in the stock price
time series.
Likewise, the normalized increments of the gambler's capi-
tal time series can be obtained by subtracting the value
of the gambler's capital in the last interval from the
value of the gambler's capital in the current interval,
and dividing by the value of the gambler's capital in the
last interval:
V(t) - V(t - 1)
---------------
V(t - 1)
for all data points in the gambler's capital time series.
This would correspond to the fraction of the gambler's
capital that was won or lost on each iteration of the
game, or, alternatively, the fraction that the stock price
increased or decreased in each interval.
The normalized increments are a very useful "tool" in ana-
lyzing time series data. In the case of the unfair coin
tossing game, the normalized increments are a "graph," (or
time series,) of the fraction of the capital that was won
or lost, every iteration of the game. Obviously, in the
unfair coin game, to win or lose, a wager had to be made,
and the graph of the absolute value, or more appropri-
ately, the root mean square, (the absolute value of the
normalized increments, when averaged, is related to the
root mean square of the increments by a constant. If the
normalized increments are a fixed increment, the constant
is unity. If the normalized increments have a Gaussian
distribution, the constant is ~0.8 depending on the accu-
racy of of "fit" to a Gaussian distribution,) of the nor-
malized increments is the fraction of the capital that was
wagered on each iteration of the game. As suggested in
Schroeder, if an unfair coin has a chance, P, of coming up
heads, (winning) and a chance 1 - P, of coming up tails,
(loosing,) then the optimal wagering strategy would be to
wager a fraction, f, of the gambler's capital, on every
iteration of the game, that is:
f = 2P - 1
This would optimize the exponential growth of the gam-
bler's capital. Wagering more than this value would result
in less capital growth, and wagering less than this value
would result in less capital growth, over time. The vari-
able f is also equal to the root mean square of the nor-
malized increments, rms, and the average, avg, of the nor-
malized increments is the constant of the average exponen-
tial growth of the gambler's capital:
t
C(t) = (1 + avg)
where C(t) is the gambler's capital. It can be shown that
the formula for the probability, P, as a function of avg
and rms is:
avg
--- + 1
rms
P = -------
2
where the empirical measurement of avg and rms are:
n
-----
1 \ V(t) - V(t - 1)
avg = - > ---------------
n / V(t - 1)
-----
i = 0
and,
n
----- 2
2 1 \ [ V(t) - V(t - 1) ]
rms = - > [ --------------- ]
n / [ V(t - 1) ]
-----
i = 0
respectively, (additionally, note that these formulas can
be used to produce the running average and running root
mean square, ie., they will work "on the fly.")
The formula for the probability, P, will be true whether
the game is played optimally, or not, ie., the game we are
"dissecting," may not be played with f = 2P - 1. However,
the formula for the probability, P:
rms + 1
P' = -------
2
will be the same as P, only if the game is played opti-
mally, (which, also, is applicable in "on the fly" method-
ologies.)
Interestingly, the measurement, perhaps dynamically, (ie.,
"on the fly,") of the average and root mean square of the
normalized increments is all that is necessary to optimize
the "play of the game." Note that if P' is smaller than P,
then we need to increase rms, by increasing f, and, like-
wise, if P' is larger than P, we need to decrease f. Thus,
without knowing any of the underlying mechanism of the
game, we can formulate a methodology for an optimal wager-
ing strategy. (The only assumption being that the capital
can be represented as an independent increment fractal-
and, this too can, and should, be verified with meticulous
application of fractal analysis using the programs tsfrac-
tion(1), tsrms(1), tsavg(1), and tsnormal(1).)
At this point, it would seem that the optimal wagering
strategy and analytical methodology used to optimize the
growth of the gambler's capital in the the unfair tossed
coin gain is well in hand. Unfortunately, when applying
the methodology to the equity markets, one finds that, for
almost all stocks, P is greater than P', perhaps tending
to imply that in the equity markets, stocks are over
priced.
To illustrate a simple stock wagering strategy, suppose
that analytical measurements are made on a stock's price
time series, and it is found, conveniently, that P = P',
implying that f = rms, (after computing the normalized
increments of the stock's price time series and calculat-
ing avg, rms, P, and P'.) Note that in the optimized
unfair coin tossing game, that wagering a fraction, f =
rms, of the gambler's capital would optimize the exponen-
tial growth of the gambler's capital, and that the fluctu-
ations, over time, of the gambler's capital would simply
be the normalized increments of the gambler's capital. The
root mean square of the fluctuations, over time, are the
fraction of that the gambler's capital wagered, over time.
To achieve an optimal strategy when wagering on a stock,
the objective would be that the normalized increments in
the value of the portfolio, and the root mean square value
of the normalized increments of the portfolio, also, sat-
isfy the criteria, f = rms. Note that the fraction of the
portfolio that is invested in the stock will have normal-
ized increments that have a root mean square value that
are the same as the root mean square value of the normal-
ized increments of the stock.
The issue is to determine the fraction of the stock port-
folio that should be invested in the stock such that that
fraction of the portfolio would be equivalent to the gam-
bler wagering a fraction of the capital on a coin toss. It
is important to note that the optimized wagering strategy
used by the gambler, when wagering on the outcome of a
coin toss, is to never wager the entire capital, but to
hold some capital in reserve, and wager only a fraction of
the capital-and in the optimum case this wager fraction is
f = rms. In a stock portfolio, even though the investment
is totally in stocks, it could be considered that some of
this value is wagered, and the rest held in reserve. The
amount wagered would be the root mean square of the nor-
malized increments of the stocks price, and the amount
held in reserve would be the remainder of the portfolio's
value. (Note the paradigm-there is an isomorphism between
the fluctuating gambler's capital in the unfair coin toss-
ing game, and the fluctuating value of a stock portfolio.)
In the simple case where P = P', the fraction of the port-
folio value that should be invested in the stock is f =
root mean square of the stock's normalized increments,
which would be the same as f = 2P - 1, where P =
((avg/rms) + 1) / 2 or P = (rms + 1) / 2. Note that the
fluctuations in the value of the portfolio do to the fluc-
tuations in the stocks price would be statistically simi-
lar to the fluctuations in the gambler's capital when
playing the unfair coin tossing game.
This also leads to a generality, where P and P' are not
equal. If the root mean square of the normalized incre-
ments of the stock price time series are too small, say by
a factor of 2, then the fraction of the portfolio invested
in the stock should be increased, by a factor of 2 (in
this example.) This would make the root mean square of the
fluctuations in the value of the portfolio the same as the
the root mean square of the fluctuations in the gambler's
capital under similar statistical circumstances, (albeit
with twice as much of the portfolio's equivalent "cash
reserves" tied up in the investment in the stock.
To calculate the ratio by which the fraction of the port-
folio invested in a stock must be increased:
avg
--- + 1
rms
P = -------
2
and,
f = 2P - 1 = rms
and letting the measured rms by rms ,
m
avg
m
---- + 1
rms avg
m m
f = 2P - 1 = 2 -------- - 1 = ---- = rms
2 rms
m
(Note that both of the values, avg and rms, are
functions of the probability, P, but their ratio is
not.)
and letting F be the ratio by which the fraction of the
portfolio invested in a stock must be increased to accom-
modate P not being equal to P':
avg
rms m
F = ---- = ----
rms 2
m rms
m
and multiplying both sides of the equation by f, to get
the fraction of the portfolio that should be invested in
the stock while accommodating P not being equal to P':
2
avg avg avg
m m m
F * f = ---- * ---- = ----
2 rms 3
rms m rms
m m
which can be computed, dynamically, or "on the fly," and
where avg and rms are the average and root mean square of
the normalized increments of the stock's price time
series, and assuming that the stock's price time series is
composed of independent increments, and can be represented
as a fractional Brownian motion fractal.
Representing such an algorithm in pseudo code:
1) for each data point in the stock's price time
series, find the, possibly running, normalized
increment from the following equation:
V(t) - V(t - 1)
---------------
V(t - 1)
2) calculate the, possibly running, average of all
normalized increments in the stock's price time series
by the following equation:
n
-----
1 \ V(t) - V(t - 1)
avg = - > ---------------
n / V(t - 1)
-----
i = 0
3) calculate the, possibly running, root mean square
of all normalized increments in the stock's price time
series by the following equation:
n
----- 2
2 1 \ [ V(t) - V(t - 1) ]
rms = - > [ --------------- ]
n / [ V(t - 1) ]
-----
i = 0
4) calculate the, possibly running, fraction of the
portfolio to be invested in the stock, F * f:
2
avg
m
F * f = ----
3
rms
m
To reiterate what we have so far, consider a gambler,
iterating a tossed unfair coin. The gambler's capital,
over time, could be a represented as a Brownian fractal,
on which measurements could be performed to optimize the
gambler's wagering strategy. There is supporting evidence
that stock prices can be "modeled" as a Brownian fractal,
and it would seem reasonable that the optimization
techniques that the gambler uses could be applied to stock
portfolios. As an example, suppose that it is desired to
invest in a stock. We would measure the average and root
mean square of the normalized increments of the stock's
price time series to determine a wagering strategy for
investing in the stock. Suppose that the measurement
yielded that the the the fraction of the capital to be
invested, f, was 0.2, (ie., a Shannon probability of 0.6,)
then we might invest the entire portfolio in the stock,
and our portfolio would be modeled as 20% of the portfolio
would be wagered at any time, and 80% would be considered
as "cash reserves," even though the 80% is actually
invested in the stock. Additionally, we have a metric
methodology, requiring only the measurement of the average
and root mean square of the increments of the stock price
time series, to formulate optimal wagering strategies for
investment in the stocks. The assumption is that the
stock's price time series is composed of independent
increments, and can be represented as a fractional Brown-
ian motion fractal, both of which can be verified through
a metric methodology.
Note the isomorphism. Consider a gambler that goes to a
casino, buys some chips, then plays many iterations of an
unfair coin tossing game, and then cashes in the chips.
Then consider investing in a stock, and some time later,
selling the stock. If the Shannon probability of the time
series of the unfair coin tossing game is the same as the
time series of the stock's value, then both "games" would
be statistically similar. In point of fact, if the toss of
the unfair coin was replaced with whether the stock price
movement was up or down, then the two time series would be
identical. The implication is that stock values can be
modeled by an unfair tossed coin. In point of fact, stock
values are, generally, fractional Brownian motion in
nature, implying that the day to day fluctuations in price
can be modeled with a time sampled unfair tossed coin
game.
There is an implication with the model. It would appear
that the "best" portfolio strategy would be to continually
search the stock market exchanges for the stock that has
the largest value of the quotient of the average and root
mean square of the normalized increments of the stock's
price time series, (ie., avg / rms,) and invest 100% of
the portfolio in that single stock. This is in contention
with the concept that a stock portfolio should be "diver-
sified," although it is not clear that the prevailing con-
cept of diversification has any scientific merit.
To address the issue of diversification of stocks in a
stock portfolio, consider the example where a gambler,
tossing an unfair coin, makes a wager. If the coin has a
60% chance of coming up heads, then the gambler should
wager 20% of the capital on hand on the next toss of the
coin. The remaining 80% is kept as "cash reserves." It can
be argued that the cash reserves are not being used to
enhance the capital, so the gambler should play multiple
games at once, investing all of the capital, investing 20%
of the capital, in each of 5 games at once, (assuming that
the coins used in each game have a probability of coming
up heads 60% of the time-note that the fraction of capital
invested in each game would be different for each game if
the probabilities of the coins were different, but could
be measured by calculating the avg /rms of each game.)
Likewise, with the same reasoning, we would expect that
stock portfolio management would entail measuring the quo-
tient of the average and root mean square of the normal-
ized increments of every stock's price time series, (ie.,
avg / rms,) choosing those stocks with the largest quo-
tient, and investing a fraction of the portfolio that is
equal to the this quotient. Note that with an avg / rms =
0.1, (corresponding to a Shannon probability of 0.55-which
is "typical" for the better performing stocks on the New
York Stock Exchange,) we would expect the portfolio to be
diversified into 10 stocks, which seems consistent with
the recommendations of those purporting diversification of
portfolios. In reality, since most stocks in the United
States exchanges, (at least,) seem to be "over priced,"
(ie., P larger than P',) it will take more capital than is
available in the value of the portfolio to invest, opti-
mally, in all of the stocks in the portfolio, (ie., the
fraction of the portfolio that has to be invested in each
stock, for optimal portfolio performance, will sum to
greater than 100%.) The interpretation, I suppose, in the
model, is that at least a portion of the investment in
each stock would be on "margin," which is a relatively low
risk investment, and, possibly, could be extended into a
formal optimization of "buying stocks on the margin."
The astute reader would note that the fractions of the
portfolio invested in each stock was added linearly, when
these values are really the root mean square of the nor-
malized increments, implying that they should be added
root mean square. The rationale in linear addition is that
the Hurst Coefficient in the near term is near unity, and
for the far term 0.5. (By definition, this is the charac-
teristic of a Brownian motion fractal process.) Letting
the Hurst Coefficient be H, then the method of summing
multiple processes would be:
H H H
V = V + V + ...
tot 1 2
so in the far term, the values would be added root mean
square, and in the near term, linearly. Note that this is
also a quantitative definition of the terms "near term"
and "far term." Since the Hurst Coefficient plot is on a
log-log scale, the demarcation between the two terms is
where 1 - ln (t) = 0.5 * ln (t), or when ln (t) = 2, or t
= 7.389... The important point is that the "root mean
square formula" used varies with time. For the near term,
H = 1, and linear addition is used. For the far term, a
root mean square summation process is used. (Note, also,
that a far term H of 0.5 is unique to Brownian motion
fractals. In general, it can be different than 0.5. If it
is larger than 0.5, then it is termed fractional Brownian
motion, depending on who is doing the defining.)
There are some interesting implications to this near
term/far term interpretation. First, the "forecastability"
is better in the near term than far term-which could be
interpreted as meaning that short term strategies would
yield better portfolio performance than long term strate-
gies-see the Peters reference, pp. 83-84. Secondly, it can
be used to optimize portfolio long term strategy. For
example, suppose that a stock's Shannon probability is
0.52, and all stocks in the portfolio have the same Shan-
non probability. This means that the portfolio should con-
sist of 25 stocks. However, in the long run, the portfolio
would have a root mean square value of the square root of
25 times 0.04, or 0.2. This would tend to imply that, on
the average, over the long run, the stock portfolio would
be one fifth of the total investments. Naturally, this
ratio could be adjusted, over time, depending on the
instantaneous value of the Shannon probabilities of all
different investments, like bonds, metals, etc.
This would imply that "timing of the market" would have to
be initiated to adjust the ratio of investment in stocks.
One of the implications of entropic theory is that this is
impossible. However, as the Shannon probability of the
various investments change, statistical estimation can be
used to asses the statistical accuracy of these movements,
and the ratios adjusted accordingly. This would tend to
suggest that adaptive computational control system method-
ology would be an applicable alternative.
As a note in passing, the average and root mean square of
the normalized increments of a stock's price time series,
avg and rms, respectively, represent a qualitative metric
of the stock. The average, avg, is an expression of the
stock's growth in price, and the root mean square, rms, is
a expression of the stock's price volatility. It would
seem, incorrectly, at first glance that stocks should be
selected that have high price growth, and low price
volatility-however, it is a more complicated issue since
avg and rms are interrelated, and not independent of each
other. See the references for theoretical concepts.
In the diversified portfolio, the "volatilities" of the
individual stocks add root mean square to the volatility
of the portfolio value, so, everything else being equal,
we would expect that the volatility of the portfolio value
to be about 1 /3 the volatility of the stocks that make up
the portfolio. (The ratio 1 / came from square root of 1 /
10, which is about 1 / 3.) (There is a qualification here,
it is assumed that all stock price time series are made up
of independent increments, and can be represented as a
fractional Brownian motion fractal-note that this state-
ment is not true if the time series is characterized as
simple Brownian motion, like the gambler's capital in the
unfair coin toss game-see Schroeder, pp. 157 for details.)
So, it can be supposed, if one desires maximum performance
in a stock portfolio, then one should search the stock
market exchanges for the stock that has the highest quo-
tient of the average and root mean square of the normal-
ized increments of stock price time series, and invest
100% of the portfolio in that stock. As an alternative
strategy, one could diversify the portfolio, investing in
multiple stocks, and lower the portfolio volatility at the
expense of lower portfolio performance. Arguments can
probably be made for both strategies.