From: John Conover <john@email.johncon.com>
Subject: Re: Financial engineering article
Date: Mon, 23 Dec 1996 16:44:48 -0800
Same as before, typographical corrections, footnote [6] wordsmithing.
John
BTW, if you look at the "El Faro" bar problem, (new footnote [9],) it
is kind of interesting. It is what is known as a game-theoretic
prediction problem with a mixed strategy Nash equilibrium of 60%. It
turns out that one should predict more than 60% occupancy with a
probability of 0.4, and less than 60% occupancy with a probability of
0.6, and then just mix up your decisions with those probabilities in
an adequately random fashion, so that no one else can predict what you
are going to do. (If you think about it, this is an obvious solution.)
That is the best long term strategy for the "El Faro" bar problem,
(short of calling all hundred people and asking them what they are
going to do-which is a bit of an inconvenience just to go have a beer
every week.)
If we assume that everyone is real smart, (or by chance, over time,
"learns" how to play the "game,") then we would have a hundred random
processes, each with a probability of 0.4 of not going. And if we sum
all hundred of them together, we would have a fractal time series for
the occupancy of the bar over time. Obviously, a fractal (of the
random walk, or Brownian motion variety, in this case,) solution is
stable, over time, and is self reinforcing, ie., we would expect the
occupancy of El Faro's to be a fractal, forever.
Here is why. Now assume that some, maybe even many, of the people
wishing to go to the El Faro formulate new strategies in an attempt to
better their predictive position. Some strategies will work, some of
the time, but no strategy will work all of the time-ie., these
attempts will become the random process itself-and, over time, will
settle back to the Nash equilibrium. The very act of attempting to
better their predictive position reinforces the fractal solution, and
in fact becomes the "engine" of the fractal itself.
Obviously, the concept is applicable to many economic scenarios-like
the stock market, industrial markets, etc.
--
John Conover, john@email.johncon.com, http://www.johncon.com/
If we consider capital, V, invested in a savings account, and
calculate the growth of the capital over time:
V(t) = V(t - 1)(1 + a(t))........................(1.1)
where a(t) is the interest rate at time t, (usually a constant[1].)
In equities, a(t) is not constant, and varies, perhaps being negative
at certain times, (meaning that the value of the equity decreased.)
This fluctuation in an equity's value can be represented by modifying
a(t) in Equation 1.1:
a(t) = f(t) * F(T)..............................(1.2)
where the product f * F is the fluctuation in the equity's value at
time t. An equity's value, over time, is similar to a simple tossed
coin game [Sch91, pp. 128], where f(t) is the fraction of a gambler's
capital wagered on a toss of the coin, at time t, and F(t) is a random
variable[2], signifying whether the game was a win, or a loss, ie.,
whether the gambler's capital increased or decreased, and by how much.
The amount the gambler's capital increased or decreased is f(t) *
F(t).
In general, F(t) is a function of a random variable, with an average,
over time, of avgf, and a root mean square value, rmsf, of unity.
Note that for simple, time invariant, compound interest, F(t) has an
average and root mean square, both being unity, and f(t) is simply the
interest rate, which is assumed to be constant. For a simple, single
coin game, F(t) is a fixed increment, (ie., either +1 or -1,) random
V(t) = V(t - 1)(1 + f(t) * F(t))................(1.3)
and subtracting V(t - 1) from both sides:
V(t) - V(t - 1) = V(t - 1) (1 + f(t) * F(t)) -
V(t - 1)........................................(1.4)
and dividing both sides by V(t - 1):
V(t) - V(t - 1)
--------------- =
V(t - 1)
V(t - 1) (1 + f(t) * F(t)) - V(t - 1)
-------------------------------------...........(1.5)
V(t - 1)
and combining:
V(t) - V(t - 1)
--------------- =
V(t - 1)
(1 + f(t) * F(t) ) - 1 = f(t) * F(t)............(1.6)
We now have a "prescription," or process, for calculating the
characteristics of the random process that determines an equity's
value. That process is, for each unit of time, subtract the value of
the of the equity at the previous time from the value of the equity at
the current time, and divide this by the value of the equity at the
previous time. The root mean square[4] of these values are the root
mean square of the random process. The average of these values are
the average of the random process, avgf. The root mean square of
these values can be calculated by any convenient means, and will be
represented by rms. The average of these values can be found by any
convenient means, and will be represented by avg[5]. Therefore, if
f(t) = f, and does not vary over time:
rms = f.........................................(1.7)
which, if there are sufficiently many samples, is a metric of the
equity value's "volatility," and:
avg = f * F(t)..................................(1.8)
and if there are sufficiently many samples, the average of F(t) is
simply avgf, or:
avg = f * avgf..................................(1.9)
which is a metric on the equity value's rate of "growth." Note that
this is the "effective" compound interest rate from Equation 1.1.
Equations 1.7 and 1.9 are important equations, since they can be used
in portfolio management. For example, Equation 1.7 states that the
volatility of the capital invested in many equities, simultaneously,
is calculated as the root mean square of the individual volatility of
the equities. Equation 1.9 states that the growths in the same equity
values add together linearly[6]. Dividing Equation 1.9 by Equation
1.7 results in the two f's canceling, or:
avg
--- = avgf.....................................(1.10)
rms
There may be analytical advantages to "model" avg as a simple tossed
coin game, (either played with a single coin, or multiple coins, ie.,
many coins played at one time, or a single coin played many times[7].)
The number of wins minus the number of losses, in many iterations of a
single coin tossing game would be:
P - (1 - P) = 2P - 1...........................(1.11)
where P is the probability of a win for the tossed coin. (This
probability is traditionally termed, the "Shannon probability" of a
win.) Note that from the definition of F(t) above, that P = avgf. For
a fair coin, (ie., one that comes up with a win 50% of the time,) P =
0.5, and there is no advantage, in the long run, to playing the game.
However, if P > 0.5, then the optimal fraction of capital wagered on
each iteration of the single coin tossing game, f, would be 2P - 1.
Note that if multiple coins were used for each iteration of the game,
we would expect that the volatility of the gambler's capital to
increase as the square root of the number of coins used, and the
growth to increase linearly with the number of coins used,
irregardless of whether many coins were tossed at once, or one coin
was tossed many times, (ie., our random generator, F(t) would assume a
binomial distribution and if the number of coins was very large, then
F(t) would assume, essentially, a Gaussian distribution.) Many
equities have a Gaussian distribution for the random process, F(t).
It may be advantageous to determine the Shannon probability to analyze
avg
--- = avgf = 2P - 1............................(1.12)
rms
or:
avg
--- + 1 = 2P...................................(1.13)
rms
and:
avg
--- + 1
rms
P = -------....................................(1.14)
2
where only the average and root mean square of the normalized
increments need to be measured, using the "prescription" or process
outlined above.
Interestingly, what Equation 1.12 states is that the "best" equity
investment is not, necessarily, the equity that has the largest
average growth, avgf. The best equity investment is the equity that
has the largest growth, while simultaneously having the smallest
volatility. In point of fact, the optimal decision criteria is to
choose the equity that has the largest ratio of growth to volatility,
where the volatility is measured by computing the root mean square of
the normalized increments, and the growth is computed by averaging the
normalized increments.
We now have a "first order prescription" that enables us to analyze
fluctuations in equity values, although we have not explained why
equity values fluctuate. For a formal presentation on the subject,
see the bibliography in [Art95] which, also, offers non-mathematical
insight into the explanation.
Consider a very simple equity market, with only two people holding
equities. Equity value "arbitration" (ie., how equity values are
determined,) is handled by one person posting (to a bulletin board,) a
willingness to sell a given number of stocks at a given price, to the
other person. There is no other communication between the two
people. If the other person buys the stock, then that is the value of
the stock at that time. Obviously, the other person will not buy the
stock if the price posted is too high-even if ownership of the stock
is desired. For example, the other person could simply decide to wait
in hopes that a favorable price will be offered in the future. So the
stock seller must not post a price that the other person would
consider too high, and the other person would not buy at the price if
it is reasoned that the seller's pricing strategy will be to lower the
offering price in the future, which would be a reasonable deduction if
the posted price is considered too high. What this means is that the
seller must consider not only the behavior of the other person, but
what the other person thinks the seller's behavior will be, ie., the
seller must base the pricing strategy on the seller's pricing
strategy. Such convoluted logical processes are termed "self
referential," and the implication is that the market can never operate
in a consistent fashion that can be the subject of deductive analysis
[Pen89, pp. 101][8]. As pointed out by [Art95, Abstract], these types
of indeterminacies pervade economics[9]. What the two players do, in
absence of a deductively consistent and complete theory of the market,
is to rely on inductive reasoning. They form subjective expectations
or hypotheses about how the market operates. These expectations and
hypothesis are constantly formulated and changed, in a world that
forms from others' subjective expectations. What this means is that
equity values will fluctuate as the expectations and hypothesis
concerning the future of equity values change[10]. The fluctuations
created by these indeterminacies in the equity market are represented
by the term f(t) * F(t) in Equation 1.3, and since there are many such
indeterminacies, we would anticipate F(t) to have a Gaussian
distribution. This is a rather interesting conclusion, since
analyzing the actions of aggregately many "agents," each operating on
subjective hypothesis in a market that is deductively indeterminate,
can result in a system that can not only be analyzed, but optimized.
The only remaining derivation is to show that the optimal wagering
strategy is, as cited above:
f = rms = 2P - 1...............................(1.15)
where f is the fraction of a gambler's capital wagered on each toss of
a coin that has a Shannon probability, P, of winning. Following
[Rez94, pp. 450], consider that the gambler has a private wire into
the future who places wagers on the outcomes of a game of chance. We
assume that the side information which he receives has a probability,
P, of being true, and of 1 - P, of being false. Let the original
capital of gambler be V(0), and V(n) his capital after the n'th wager.
Since the gambler is not certain that the side information is entirely
reliable, he places only a fraction, f, of his capital on each wager.
Thus, subsequent to n many wagers, assuming the independence of
successive tips from the future, his capital is:
w l
V(n) = (1 + f) (1 - f) V (0).................(1.16)
where w is the number of times he won, and l = n - w, the number of
times he lost. These numbers are, in general, values taken by two
random variables, denoted by W and L. According to the law of large
numbers:
1
lim - W = P..........................(1.17)
n -> infinity n
1
lim - L = q = 1 - P..................(1.18)
n -> infinity n
The problem with which the gambler is faced is the determination of f
leading to the maximum of the average exponential rate of growth of
his capital. That is, he wishes to maximize the value of:
1 V(n)
G = lim - ln ----....................(1.19)
n -> infinity n V(0)
with respect to f, assuming a fixed original capital and specified P:
W L
G = lim - ln (1 + f) + - ln (1 - f)..(1.20)
n -> infinity n n
or:
G = P ln (1 + f) + q ln (1 - f)................(1.21)
which, by taking the derivative with respect to f, and equating to
zero, can be shown to have a maxima when:
dG P - 1 1 - P
-- = P(1 + f) (1 - f) -
df
1 - P - 1
(1 - P)(1 - f) (1 + f)P = 0...........(1.22)
combining terms:
P - 1 1 - P
0 = P(1 + f) (1 - f) -
P P
(1 - P)(1 - f) (1 + f ) ......................(1.23)
and splitting:
P - 1 1 - P
P(1 + f) (1 - f) =
P P
(1 - P)(1 - f) (1 + f) .......................(1.24)
then taking the logarithm of both sides:
ln (P) + (P - 1) ln (1 + f) + (1 - P) ln (1 - f) =
ln (1 - P) - P ln (1 - f) + P ln (1 + f).......(1.25)
and combining terms:
(P - 1) ln (1 + f) - P ln (1 + f) +
(1 - P) ln (1 - f) + P ln (1 - f) =
ln (1 - P) - ln (P)............................(1.26)
or:
ln (1 - f) - ln (1 + f) =
ln (1 - P) - ln (P)...........................(1.27)
and performing the logarithmic operations:
1 - f 1 - P
ln ----- = ln -----............................(1.28)
1 + f P
and exponentiating:
1 - f 1 - P
----- = -----..................................(1.29)
1 + f P
which reduces to:
P(1 - f) = (1 - P)(1 + f)......................(1.30)
and expanding:
P - Pf = 1 - Pf - P + f........................(1.31)
or:
P = 1 - P + f..................................(1.32)
and, finally:
f = 2P - 1.....................................(1.33)
Footnotes:
[1] For example, if a = 0.06, or 6%, then at the end of the first time
interval the capital would have increased to 1.06 times its initial
value. At the end of the second time interval it would be (1.06),
and so on. What Equation 1.1 states is that the way to get the value,
V in the next time interval is to multiply the current value by
1.06. Equation 1.1 is nothing more than a "prescription," or a process
to make an exponential, or "compound interest" mechanism. In general,
exponentials can always be constructed by multiplying the current
value of the exponential by a constant, to get the next value, which
in turn, would be multiplied by the same constant to get the next
value, and so on. Equation 1.1 is nothing more than a construction of
V (t) = exp(kt) where k = ln(1 + a). The advantage of representing
exponentials by the "prescription" defined in Equation 1.1 is
analytical expediency. For example, if you have data that is an
exponential, the parameters, or constants, in Equation 1.1 can be
determined by simply reversing the "prescription," ie., subtracting
the previous value, (at time t - 1,) from the current value, and
dividing by the previous value would give the exponentiating constant,
(1 + at). This process of reversing the "prescription" is termed
calculating the "normalized increments." (Increments are simply the
difference between two values in the exponential, and normalized
increments are this difference divided by the value of the
exponential.) Naturally, since one usually has many data points over a
time interval, the values can be averaged for better precision-there
is a large mathematical infrastructure dedicated to precision
enhancement, for example, least squares approximation to the
normalized increments, and statistical estimation.
[2] "Random variable" means that the process, F(t), is random in
nature, ie., there is no possibility of determining what the next
value will be. However, F can be analyzed using statistical methods
[Fed88, pp. 163], [Sch91, pp. 128]. For example, F typically has a
Gaussian distribution for equity values [Cro95, pp. 249], in which
case the it is termed a "fractional Brownian motion," or simply a
"fractal" process. In the case of a single tossed coin, it is termed
"fixed increment fractal," "Brownian," or "random walk" process. In
any case, determination of the statistical characteristics of Ft are
the essence of analysis. Fortunately, there is a large mathematical
infrastructure dedicated to the subject. For example, F could be
verified as having a Gaussian distribution using Chi-Square
techniques. Frequently, it is convenient, from an analytical
standpoint, to "model" F using a mathematically simpler process
[Sch91, pp. 128]. For example, multiple iterations of tossing a coin
can be used to approximate a Gaussian distribution, since the
distribution of many tosses of a coin is binomial-which if the number
of tosses is sufficient will represent a Gaussian distribution to
within any required precision [Sch91, pp. 144], [Fed88, pp. 154].
[3] Equation 1.3 is interesting in many other respects. For example,
adding a single term, m * V(t - 1), to the equation results in V(t) =
v(t - 1) (1 + f(t) * F(t) + m * V(t - 1)) which is the "logistic," or
'S' curve equation,(formally termed the "discreet time quadratic
equation,") and has been used successfully in many unrelated fields
such as manufacturing operations, market and economic forecasting, and
analyzing disease epidemics [Mod92, pp. 131]. There is continuing
research into the application of an additional "non-linear" term in
Equation 1.3 to model equity value non-linearities. Although there
have been modest successes, to date, the successes have not proved to
be exploitable in a systematic fashion [Pet91, pp. 133]. The reason
for the interest is that the logistic equation can exhibit a wide
variety of behaviors, among them, "chaotic." Interestingly, chaotic
behavior is mechanistic, but not "long term" predictable into the
future. A good example of such a system is the weather. It is an
important concept that compound interest, the logistic function, and
fractals are all closely related.
[4] In this section, "root mean square" is used to mean the variance
of the normalized increments. In Brownian motion fractals, this is
computed by sigmatotal^2 = sigma1^2 + sigma2^2 ... However, in many
fractals, the variances are not calculated by adding the squares,
(ie., a power of 2,) of the values-the power may be "fractional," ie.,
3 / 2 instead of 2, for example [Sch91, pp. 130], [Fed88, pp.
178]. However, as a first order approximation, the variances of the
normalized increments of equity values can successfully be added root
mean square [Cro95, kpp. 250]. The so called "Hurst" coefficient,
which can be measured, determines the process to be used. The Hurst
coefficient is range of the equity values over a time interval,
divided by the standard deviation of the values over the interval, and
its determination is commonly called "R / S" analysis. As pointed out
in [Sch91, pp. 157] the errors committed in such simplified
assumptions can be significant-however, for analysis of equities,
squaring the variances seems to be a reasonable simplification.
[5] For example, many calculators have averaging and root mean square
functionality, as do many spreadsheet programs-additionally, there are
computer source codes available for both. See the programs tsrms and
tsavg. The method used is not consequential.
[6] There are significant implications do to the fact that equity
volatilities are calculated root mean square. For example, if capital
is invested in N many equities, concurrently, then the volatility of
the capital will be rms / sqrt (N) of an individual equity's
volatility, rms, provided all the equites have similar statistical
characteristics. But the growth in the capital will be unaffected,
ie., it would be statistically similar to investing all the capital in
only one equity. What this means is that capital, or portfolio,
volatility can be minimized without effecting portfolio growth-ie.,
volatility risk can addressed. Further, it does not make any
difference, as far as portfolio value growth is concerned, whether the
individual equities are invested in concurrently, or serially, ie., if
one invested in 10 different equities for 100 days, concurrently, or
one could invest in only one equity, for 10 days, and then the next
equity for the next 10 days, and so on. The capital growth would have
the same characteristics for both agendas. (Note that the concurrent
agenda is superior since the volatility of the capital will be the
root mean square of the individual equity volatilities divided by the
square root of the number of equities. In the serial agenda, the
volatility of the capital will be simply the root mean square of the
individual equity volatilities.) Almost all equity wagering strategies
will consist of optimizing variations on combinations of serial and
concurrent agendas. There are further applications. For example,
Equation 1.6 could be modified by dividing both the normalized
increments, and the square of the normalized increments by the daily
trading volume. The quotient of the normalized increments divided by
the trading volume is the instantaneous growth, avg, of the equity, on
a per-share basis. Likewise, the square root of the square of the
normalized increments divided by the daily trading volume is the
instantaneous root mean square, rmsf, of the equity on a per-share
basis, ie., its instantaneous volatility of the equity. (Note that
these instantaneous values are the statistical characteristics of the
equity on a per-share bases, similar to a coin toss, and not on time.)
Additionally, it can be shown that the range-the maximum minus the
minimum-of an equity's value over a time interval will increase with
the square root of of the size of the interval of time [Fed88,
pp. 178]. Also, it can be shown that the number of expected stock
value "high and low" transitions scales with the square root of time,
meaning that the probability of an equity value "high or low"
exceeding a given time interval is proportional to the square root of
the time interval [Schroder, pp. 153].
[7] Here the "model" is to consider two black boxes, one with a stock
"ticker" in it, and the other with a casino game of a tossed coin in
it. One could then either invest in the equity, or, alternatively,
invest in the tossed coin game by buying many casino chips, which
constitutes the starting capital for the tossed coin game. Later,
either the equity is sold, or the chips "cashed in." If the statistics
of the equity value over time is similar to the statistics of the coin
game's capital, over time, then there is no way to determine which box
has the equity, or the tossed coin game. The advantage of this model
is that gambling games, such as the tossed coin, have a large
analytical infrastructure, which, if the two black boxes are
statistically the same, can be used in the analysis of equities. The
concept is that if the value of the equity, over time, is
statistically similar to the coin game's capital, over time, then the
analysis of the coin game can be used on equity values. Note that in
the case of the equity, the terms in f(t) * F(t) can not be
separated. In this case, f = rms is the fraction of the equity's
value, at any time, that is "at risk," of being lost, ie., this is the
portion of a equity's value that is to be "risk managed." This is
usually addressed through probabilistic methods, as outlined below in
the discussion of Shannon probabilities, where an optimal wagering
strategy is determined. In the case of the tossed coin game, the
optimal wagering strategy is to bet a fraction of the capital that is
equal to f = rms = 2P - 1 [Sch91, pp. 128, 151], where P is the
Shannon probability. In the case of the equity, since f = rms is not
subject to manipulation, the strategy is to select equities that
closely approximate this optimization, and the equity's value, over
time, on the average, would increase in a similar fashion to the coin
game. The growth of either investment would be equal to avg = rms^2,
on average, for each iteration of the coin game, or time unit of
equity investment. This is an interesting concept from risk management
since it maximizes the gain in the capital, while, simultaneously,
minimizing risk exposure to the capital.
[8] Penrose, referencing Russell's paradox, presents a very good
example of logical contradiction in a self-referential system.
Consider a library of books. The librarian notes that some books in
the library contain their titles, and some do not, and wants to add
two index books to the library, labeled "A" and "B," respectively; the
"A" book will contain the list of all of the titles of books in the
library that contain their titles; and the "B" book will contain the
list of all of the titles of the books in the library that do not
contain their titles. Now, clearly, all book titles will go into
either the "A" book, or the "B" book, respectively, depending on
whether it contains its title, or not. Now, consider in which book,
the "A" book or the "B" book, the title of the "B" book is going to be
placed-no matter which book the title is placed, it will be
contradictory with the rules. And, if you leave it out, the two books
will be incomplete.)
[9] [Art95] cites the "El Farol Bar" problem as an example. Assume one
hundred people must decide independently each week whether go to the
bar. The rule is that if a person predicts that more than, say, 60
will attend, it will be too crowded, and the person will stay home; if
less than 60 is predicted, the person will go to the bar. As trivial
as this seems, it destroys the possibility of long-run shared,
rational expectations. If all believe few will go, then all will go,
thus invalidating the expectations. And, if all believe many will go,
then none will go, likewise invalidating those expectations.
Predictions of how many will attend depend on others' predictions, and
others' predictions of others' predictions. Once again, there is no
rational means to arrive at deduced a-priori predictions. The
important concept is that expectation formation is a self-referential
process in systems involving many agents with incomplete information
about the future behavior of the other agents. The problem of
logically forming expectations then becomes ill-defined, and rational
deduction, can not be consistent or complete. This indeterminacy of
expectation-formation is by no means an anomaly within the real
economy. On the contrary, it pervades all of economics and game theory
[Art95].
[10] Interestingly, the system described is a stable system, ie., if
the players have a hypothesis that changing equity positions may be of
benefit, then the equity values will fluctuate-a self fulfilling
prophecy. Not all such systems are stable, however. Suppose that one
or both players suddenly discover that equity values can be "timed,"
ie., there are certain times when equities can be purchased, and
chances are that the equity values will increase in the very near
future. This means that at certain times, the equites would have more
value, which would soon be arbitrated away. Such a scenario would not
be stable.
Bibliography:
[Art95] W. Brian Arthur. "Complexity in Economic and Financial
Markets." Complexity, 1, pp. 20-25, 1995. Also available from
http://www.santafe.edu/arthur, February 1995.
[Cro95] Richard M. "Crownover. Introduction to Fractals and Chaos."
Jones and Bartlett Publishers International, London, England, 1995.
[Fed88] Jens Feder. "Fractals." Plenum Press, New York, New York,
1988.
[Mod92] Theodore Modis. "Predictions." Simon & Schuster, New York, New
York, 1992.
[Pen89] Roger Penrose. "The Emperor's New Mind." Oxford University
Press, New York, New York, 1989.
[Pet91] Edgar E. Peters. "Chaos and Order in the Capital Markets."
John Wiley & Sons, New York, New York, 1991.
[Rez94] Fazlollah M. Reza. "An Introduction to Information Theory."
Dover Publications, New York, New York, 1994.
[Sch91] Manfred Schroeder. "Fractals, Chaos, Power Laws."
W. H. Freeman and Company, New York, New York, 1991.