# Re: portfolio management

From: John Conover <john@email.johncon.com>
Subject: Re: portfolio management
Date: Sat, 19 Sep 1998 01:59:53 -0700

```John Conover writes:
>
> There is a formula, rms = 2P - 1, which if satisfied, will result in
> optimal and maximal growth. (All three of these formulas are derived
> in the tsinvest(1) manual page.) If rms > 2P - 1, the growth will be
> very large-for a while-and then "crash" down, (Netscape is an
> example.) If rms < 2P - 1, then the growth will be stilted, (Boeing is
> an example.)  Both solutions will yield less long term growth than the
> optimal, where rms = 2P - 1.
>

The tsinvest program is for simulating the optimal gains of multiple
equity investments.  The program decides which of all available
equities to invest in at any single time, by calculating the
instantaneous Shannon probability and statistics of all equities, and
then using statistical estimation techniques to estimate the accuracy
of the calculated statistics.

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/ntropix/archive/tsinvest.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/

DERIVATION
As a tutorial, the derivation will  start  with  a  simple
compound interest equation. This equation will be extended
to a first order  random  walk  model  of  equity  prices.
Finally,  optimizations  will  derived based on the random
walk model that are useful in optimizing equity  portfolio
performance.

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 con-
stant[1].)  In equities, a(t) is not constant, and fluctu-
ates, 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 invari-
ant, 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,  sin-
gle  coin game, F(t) is a fixed increment, (ie., either +1
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  price, over time.  That process is, for each
unit of time, subtract the value of the equity at the pre-
vious  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 value of  the  random  process.   The
average of these values are the average of the random pro-
cess, avgf.  The root mean square of these values  can  be
calculated  by  any  convenient  means, and will be repre-
sented 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 assuming that it does not vary
over time:

rms = f ........................................(1.7)

which, if there are sufficiently many samples, is a metric
of the equity's price "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 are the metrics of the equity's random process. 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 portfolio volatil-
ity is calculated as the root mean square sum of the indi-
vidual  volatilities  of  the  equities  in the portfolio.
Equation (1.9) states that the averages of the  normalized
increments  of  the equity prices add together linearly[6]
in the portfolio.  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" F(t) 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, irregard-
less  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 num-
ber 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
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 normal-
ized increments need to be measured, using the  "prescrip-
tion" 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.  A better investment
criteria is to choose the  equity  that  has  the  largest
growth,  while simultaneously having the smallest volatil-
ity.

Continuing with this line of  reasoning,  and  rearranging
Equation (1.12):

avg = rms * (2P - 1) ..........................(1.15)

which  is  an important equation since it states that avg,
(and the parameter that should be maximized,) is equal  to
rms,  which  is  the  measure  of  the  volatility  of the
equity's value, multiplied  by  the  quantity,  twice  the
likelihood  that  the  equity's value will increase in the
next time interval, minus unity.

As derived in the Section, OPTIMIZATION, below, the  opti-
mal  growth  occurs  when  f = rms = 2P - 1. Under optimal
conditions, Equation (1.14) becomes:

rms + 1
P = ------- ...................................(1.16)
2

or, sqrt (avg) = rms, (again, under  optimal  conditions,)
and substituting into Equation (1.14):

sqrt (avg) + 1
P = -------------- ............................(1.17)
2

giving three different computational methods for measuring
the statistics of an equity's value.

Note that, from Equations (1.14) and  (1.12),  that  since
avgf = avg / rms = (2P - 1), choosing the largest value of
the Shannon probability, P, will also choose  the  largest
value  of  the  ratio  of  avg / rms, rms, or avg, respec-
tively, in Equations (1.14), (1.16), or (1.17). This  sug-
gests  a method for determination of equity selection cri-
teria. (Note that  under  optimal  conditions,  all  three
equations  are  identical-only  the  metric methodology is
different. Under non-optimal conditions,  Equation  (1.14)
should  be  used. Unfortunately, any calculation involving
the average of the  normalized  increments  of  an  equity
value  time  series  will be very "sluggish," meaning that
practical issues may prevail, suggesting a preference  for
Equation  (1.17).)   However,  this  would  imply that the
equities are known to be optimal,  ie.,  rms  =  2P  +  1,
which,  although  it  is nearly true for most equities, is
not true for all equities. There is some possibility  that
optimality can be verified by metrics:

2
if avg < rms

then rms = f is too large in Equation (1.12)

2
else if avg > rms

then rms = f is too small in Equation (1.12)

2
else avg = rms

and the equities time series is optimal, ie., rms
= f = 2P - 1 from Equation (1.36), below

HEURISTIC APPROACHES
There  have  been  several heuristic approaches suggested,
for example, using the absolute value  of  the  normalized
increments  as  an  approximation to the root mean square,
rms, and calculating the Shannon probability, P  by  Equa-
tion (1.16), using the absolute value, abs, instead of the
rms. The statistical estimate in such a scheme should  use
the same methodology as in the root mean square.

Another  alternative  is to model equity value time series
as a fixed increment fractal,  ie.,  by  counting  the  up
movements  in  an equity's value. The Shannon probability,
P, is then calculated by the quotient of the up movements,
divided  by  the  total  movements. There is an issue with
this model, however. Although not  common,  there  can  be
adjacent  time  intervals where an equity's value does not
change, and it is not clear how the  accounting  procedure
should  work. There are several alternatives. For example,
no changes can be counted as  up  movements,  or  as  down
movements,  or  disregarded  entirely, or counted as both.
The statistical estimate should be performed as  in  Equa-
tion  (1.14), with an rms of unity, and an avg that is the
Shannon probability itself-that is  the  definition  of  a
fixed increment fractal.

MARKET
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 the way they do.
For a formal presentation on the subject, see the bibliog-
raphy  in  [Art95]  which,  also,  offers non-mathematical
insight into the subject.

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 equities at a given price,  to  the  other
person.   There  is no other communication between the two
people. If the other person buys the equity, then that  is
the  value  of  the  equity  at that time.  Obviously, the
other person will not buy the equity if the  price  posted
is  too  high-even  if ownership of the equity is desired.
For example, the other person could simply decide to  wait
in  hopes  that  a  favorable price will be offered in the
future.  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  inde-
terminacies  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 expecta-
tions  and hypothesis concerning the future of equity val-
ues change[10]. The fluctuations created by these indeter-
minacies  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  con-
clusion,  since  analyzing  the  aggregate actions of 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.

OPTIMIZATION
The only remaining derivation is to show that the  optimal
wagering strategy is, as cited above:

f = rms = 2P - 1 ..............................(1.18)

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,  (ie.,  an
inductive  hypothesis,)  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 frac-
tion, 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) ................(1.19)

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 .........................(1.20)
n -> infinity n

1
lim           - L = q = 1 - P .................(1.21)
n -> infinity n

The  problem with which the gambler is faced is the deter-
mination of f leading to the maximum of the average  expo-
nential  rate of growth of his capital. That is, he wishes
to maximize the value of:

1    V(n)
G = lim           - ln ---- ...................(1.22)
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) .(1.23)
n -> infinity n              n

or:

G = P ln (1 + f) + q ln (1 - f) ...............(1.24)

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 ..........(1.25)

combining terms:

P - 1        1 - P
0 = P(1 + f)      (1 - f)      -

P         P
(1 - P)(1 - f)  (1 + f )  .....................(1.26)

and splitting:

P - 1        1 - P
P(1 + f)      (1 - f)      =

P        P
(1 - P)(1 - f)  (1 + f)  ......................(1.27)

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) ......(1.28)

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) ...........................(1.29)

or:

ln (1 - f) - ln (1 + f) =

ln (1 - P)  - ln (P)...........................(1.30)

and performing the logarithmic operations:

1 - f      1 - P
ln ----- = ln ----- ...........................(1.31)
1 + f        P

and exponentiating:

1 - f   1 - P
----- = ----- .................................(1.32)
1 + f     P

which reduces to:

P(1 - f) = (1 - P)(1 + f) .....................(1.33)

and expanding:

P - Pf = 1 - Pf - P + f .......................(1.34)

or:

P = 1 - P + f .................................(1.35)

and, finally:

f = 2P - 1 ....................................(1.36)

Note that Equation (1.24), which, since rms =  f,  can  be
rewritten:

G = P ln (1 + rms) + (1 - P) ln (1 - rms) .....(1.37)

where  G  is  the average exponential rate of growth in an
equity's value, from one time interval to the next,  (ie.,
the  exponentiation  of  this value minus unity[11] is the
"effective  interest  rate",  as  expressed  in   Equation
(1.1),) and, likewise, Equation (1.36) can be rewritten:

rms = 2P - 1 ..................................(1.38)

and substituting:

G = P ln (1 + 2P - 1) +

(1 - P) ln (1 - (2P - 1)) .................(1.39)

or:

G = P ln (2P) +

(1 - P) ln (2 (1 - P)) ....................(1.40)

using a binary base for the logarithm:

G = P ln (2P) +
2

(1 - P) ln (2 (1 - P)) ....................(1.41)
2

and carrying out the operations:

G = P ln (2) + P ln (P) +
2          2

(1 - P) ln (2) + (1 - P) ln (1 - P)) ......(1.42)
2                2

which is:

G = P ln (2) + P ln (P) +
2          2

ln (2) - P ln (2) + (1 - P) ln (1 - P) ....(1.43)
2          2                2

and canceling:

G = 1 + P ln (P) + (1 - P) ln (1 - P) .........(1.44)
2                2

if  the  gambler's  wagering strategy is optimal, ie., f =
rms = 2P - 1,  which  is  identical  to  the  equation  in
[Schroder, pp. 151].

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.

[Pen89]  Roger  Penrose.  "The Emperor's New Mind." Oxford
University Press, New York, New York, 1989.

[Rez94]  Fazlollah  M. Reza.  "An Introduction to Informa-
tion 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.

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