From: John Conover <john@email.johncon.com>

Subject: Quantitative Analysis of Non-Linear High Entropy Economic Systems V

Date: 14 Feb 2002 07:38:46 -0000

As mentioned in Section I, Section II, Section III and Section IV much of applied economics has to address non-linear high entropy systems-those systems characterized by random fluctuations over time-such as net wealth, equity prices, gross domestic product, industrial markets, etc.

The dynamics of non-linear high entropy systems are probabilistic
in nature, and the understanding of the mathematics involved permits
*engineered solutions* in the field of finance, such as
development of strategies for portfolio growth optimization.

Note: the C source code to all programs used is available from the NtropiX Utilities page, or, the NdustriX Utilities page, and is distributed under License.

As a demonstration of *financial engineering*, a simple
portfolio management strategy will be developed. From Section
I, Important
Formulas, Equation
(1.24):

**
avg
--- + 1
rms
P = ------- ........................................(1.24)
2
**

where

and
**avg**

are the average and deviation of the
marginal increments, respectively, and **avg**

is the likelihood of an up movement in the price of an equity. From Equation
(1.20):**P**

**
P (1 - P)
g = (1 + rms) (1 - rms) ....................(1.20)
**

is the average increase in price of
an equity per unit time, for example, after
**g**

many days, an equity's value would
have increased in value by a factor of
**n**

.**g^n**

Equation
(1.24) and Equation
(1.20) work on a portfolio's value, too. Let

and
**avgp**

be the average and the deviation of
the marginal increments of the portfolio's value, and
**rmsp**

be the the likelihood of an up
movement in the portfolio's value. Then:**Pp**

**
avgp
---- + 1
rmsp
Pp = ------- ........................................(5.1)
2
**

and:

**
Pp (1 - Pp)
gp = (1 + rmsp) (1 - rmsp) ................(5.2)
**

where

is the average increase in
value of the portfolio per unit time.**gp**

To increase

,
**gp**

must increase, and, to increase
**Pp**

, **Pp**

must increase, and/or, **avgp**

decrease.**rmsp**

Knowing that for

,
the averages of the marginal increments of the equities in a portfolio
add linearly:**avg << rms**

**
1 1 1
avgp = - avg + - avg + ... + - avg ............(5.3)
K 1 K 2 K K
1 2 2 n
**

where the subscripts denote the

of the **avg**

individual equities in the
portfolio, and **n**

is the fraction of the
portfolio allocated to an individual equity.**K**

Knowing that for

,
deviations of the fluctuations in equity prices add root-mean-square
in the portfolio:**avg << rms**

**
1 2 1 2
rmsp = sqrt ((- rms ) + (- rms ) + ...
K 1 K 2
1 2
1 2
... + (- rms )) ............................(5.4)
K n
n
**

The median values of the daily average and deviation of the
marginal increments of equities on the US exchanges,

and **avg
= 0.0004**

,
respectively, can be used as an approximation for all equities in the
portfolio to simplify the equations:**rms = 0.02**

**
1 1 1
avgp = - avg + - avg + ... + - avg ...............(5.5)
n n n
= avg
**

since there are

many of terms, and:**n**

**
1 2 1 2
rmsp = sqrt ((- rms ) + (- rms ) + ...
n n
1 2
... + (- rms )) .............................(5.6)
n
1
= -------- rms
sqrt (n)
**

since there are

many of terms.**n**

Figure I is a plot of the average daily gain,

of a portfolio consisting of
**g**

many identical equities, each having
an average and deviation of the marginal increments of its price,
**n**

, and **avg = 0.0004**

, respectively, with equal asset allocation.**rms =
0.02**

Of interest is that the average daily gain,

, of the portfolio can be almost
doubled, (which is what we set out to do,) by maintaining an equal
asset allocation in the equities in the portfolio-but the marginal
utility of more than ten assets diminishes rapidly.**g**

The simple portfolio management strategy:

Always maintain about ten equities in the portfolio.

Always maintain about equal asset allocation between the ten equities.

As a side bar, the simple portfolio management strategy is
not new. From " Investors should ... take time to research five to 10 companies ... buy good firms and hold the stock for the long run ... As a generalization, the strategy is used as a
However, most funds use For example, adjusting allocations for equal risk, (the
deviation of the marginal increments of an equity's price,
equal for all For less volatile times, Equation (1.18), from Section I, Important Formulas, can be used:
A reasonable approximation to Equation
(1.18) is when the asset allocation for all
If it is assumed, as an approximation, that There is an interesting corollary to all this-it is possible to make money on equities that are decreasing in price without going short, or doing puts. Setting the average multiplicative gain per iteration of the
game,
for an equity to have a positive gain in value. The
implication is that an equity's gain in value It may seem counter intuitive, but just because the average
daily gain in value of an equity is greater than zero, and, the
equity's value moves up more than it does down, is Is it a mathematical curiosity, or do we really see equities with these kinds of price characteristics? In fact, the answer is that we do. During the To explore further, it is recommended that something like the
for 100,000 days, and use the -D 0 -M 0 -m 0 options for The simulation is quite surprising; even though it is a
manufactured data set, the portfolio value increases quite
nicely, even though the equities at the end of the simulation
have virtually zero value-the portfolio made money on equities
that declined in value enough to be virtually worthless. (If one
is skeptical-and one should be-and wants to try it on real data,
there is a file supplied with the Its a very compelling example on the utility of balancing one's portfolio. As a concluding remark to this side bar, there is a remaining
question; what makes a company's equity pro forma
suboptimal-where The answer is "We have met the enemy and he is us." The investors are the ones that paid more for companies than
they were worth. No one else, (and the phenomena is not new or
unique, for example, see Extraordinary
Popular Delusions And The Madness Of Crowds, 1841.) The
market, (that's us,) set the value of the equities, not the
market makers and CEOs. We should have known better, (or at
least we should have known how to make money-and keep
it-exploiting the |

As an approximation, from above, we also know that our investment horizon should be about a calendar year in the future. But what about the past? How long should we wait to invest in an equity?

From Appendix
I, Quantitative
Analysis of Non-Linear High Entropy Economic Systems Addendum for
a *typical*, (i.e., median value,) equity on the US markets
which has a median and deviation of the marginal increments of,

and **avg = 0.0004**

, respectively, has metrics that will converge on
the average, **rms =
0.02**

, as a function of
**avg**

, which is
approximately **0.02 * erf (1 / sqrt (t))**

for
**0.02 * (1 / sqrt (t))**

for
**n >> 1**

many days in the time series, and the
deviation of the error in the measurement of
**t**

will be **avg**

.**(rms / avg) /
sqrt (t)**

As a side bar, what is being considered here is very subtle. The market determines the value of an equity. Its like using equity prices as a measuring device, (or metric,) for the value of a company. But the measuring device has stochastic errors as the market responds to information, sometimes under correcting, sometimes over correcting. The market's response determining the price of the equity, is
serial on sequence of the information, moving the price as
If the information occurs randomly, (which apparently it
does-there seems to be a limit on the accuracy of prediction of
such things as future dividends for P/E ratios,) then it would
be expected that duration of fluctuations in equity prices
follows an The market determination of the value of a company, and thus its fair market equity price, is very similar to convergence in Bernoulli P Trials, which has similar characteristics-at least as far Brownian motion/random walk equivalent of an equity's price goes. Its as if the aggregate, (whatever that is,) of the market is used as test equipment which has measuring errors. |

Suppose that we want the *total* probability that an equity
price will move up on any given day. For the *typical*
characteristics, i.e., a median and deviation of the marginal
increments of,

and
**avg = 0.0004**

, respectively, then from Section
I, Important
Formulas, Equation
(1.24):**rms = 0.02**

**
avg 0.0004
--- + 1 ------ + 1
rms 0.02
P = ------- = ---------- = 0.51.....................(1.24)
2 2
**

And, letting

be the corresponding
probability of the number of standard deviations in the error of the
measurement of **Pc**

, (one standard
deviation is approximately **avg**

.) Then the total probability of an up movement,
compensated for data set size would be **0.02 * (1 / sqrt
(t))**

, which must be greater than 0.5 for a worthwhile
investment. So, all we have to do is find out how many standard
deviations, **P *
Pc**

, are
equal to **0.02 * (1 / sqrt (t))**

, look up what probability
that is, and use that for **avg**

.**Pc**

Working *backwards* through the numbers for a single equity,

, or
**P * Pc = 0.5 = 0.51 * Pc**

, and the probability of
an up movement in the equity, compensated for data set size, would be
0.5, and the compensated **Pc = 0.980392157**

,
meaning that playing this strategy would yield zero gain. But
**avg = 0**

is 2.06 standard deviations,
and the deviation of the error in the measurement of
**0.980392157**

has to be at least that good.**avg**

So, for the compensated

to be
zero, 2.06 standard deviations of **avg**

per standard deviation has to equal
**0.02 * (1 / sqrt
(t))**

, or where
**avg = 0.0004**

, which
means **0.0004 / 2.06 = 0.02 / sqrt (t)**

trading days, which is
about 42 years at 253 trading days per calendar year, which is
obviously not a workable solution in light of the durability
of US companies from Section
IV.**t = 10609**

However, suppose we construct a portfolio of ten equities. Then how long should we wait to invest in an equity?

Doing the same thing:

**
avg
sqrt (10) --- + 1
rms
P = ----------------- = 0.5316227766 ...............(1.24)
2
**

And, again, working *backwards* through the numbers for the
portfolio of ten statistically similar equities, (assuming the equity
prices are statistically independent,)

, or **P * Pc = 0.5 =
0.5316227766 * Pc**

, and the probability of an up movement in
the equity, compensated for data set size, would be 0.5, and the
compensated **Pc =
0.940516513**

, meaning that
playing this strategy would yield zero gain. But
**avg = 0**

is 1.555 standard
deviations, and the deviation of the error in the measurement of
**0.940516513**

has to be at least that good.**avg**

So, for the compensated

to be
zero, 1.555 standard deviations, of **avg**

per standard deviation, has to equal
**(0.02 / sqrt (10)) *
(1 / sqrt (t))**

, or where
**avg = 0.0004**

, which means **0.0004 / 1.555 = (0.02 / sqrt (10)) * (1 / sqrt
(t))**

trading days, which is about 2.4 years at 253
trading days per calendar year-a much more workable scenario.**t =
605**

Note that 2.4 years is the minimum; buying equities with less than
a 2.4 year history, and one would, eventually, lose all one's money,
(as many did in the *dot-com* mania.) Requiring a 4.4 year
history of an equity to be listed before investing would be even
better-since 4.4 years is the median value of *bubbles* in
equity prices.

Which completes the simple portfolio management strategy:

Always maintain about ten, or more, equities in the portfolio.

Always maintain about equal asset allocation between the ten equities.

Always consider the investment horizon to be about a calendar year.

Always be skeptical of investing in companies with less than a two and a half year history of being a publicly traded-preferably, a minimum of four and a half years.

Note that the simple portfolio management strategy-a financially
engineered solution-is nothing more than four simple policies; an
investment framework, (that's what financial economics is all about.)
The policies are listed in order of importance, and the second policy
is the one that makes the money-its the *engine* of the
strategy.

As a side bar, the strategy is an equity investment
architecture-it does not pick equities, nor does it attempt to
Why don't more people know about it? Few have the intellectual tenacity to struggle through the abstract mathematical models, which are axiomatic, and sometimes counter intuitive to many. The strategy can be beat, in the short run, (50% of the time,
for less than 4.4 years,) on shear luck alone-but in the long
run, it is very difficult to do better. (For example, it was
beat during the |

How well does the simple portfolio management strategy work?

To find out, the price history of the first ten equities, (in alphabetical order,) in the DJIA were downloaded from Yahoo!'s Historical Prices database, (the prices are adjusted for splits,) and the investment strategy simulated. The simulation is to simply to maintain equal asset allocation, in each of the ten equities, every day, and the annual returns measured. Since, by luck alone, any strategy-no matter how irrelevant-could show good returns for 4.4 years, 50% of the time, (that's the implication of Equation 3.2,) the simulation will consist of a quarter of a century of daily returns.

The construction of the simulation with the historical data is outlined in Appendix I, for those so inclined.

Figure II is a plot of the pro forma of the individual equities picked from the DJIA, by ticker symbol. Each equity started with an initial investment of $1000. The plot represents the growth in value of a portfolio consisting of exactly one equity, with an initial investment of $1000, held for a quarter of a century. It will be used for comparative purposes.

Figure III is a plot of the index value of the same ten equities picked from the DJIA, and the result of balancing the asset allocation between the ten equities, equally, every day, for a quarter of a century, starting with an initial investment of $1000.

The index graph represents the pro forma of a portfolio with an
initial investment of $1000, a quarter of a century ago, in the ten
equities, and then just letting the investment *ride*.

The balanced graph represents the pro forma of a portfolio with an initial investment of $1000, a quarter of a century ago, in the same ten equities, but the asset allocation between the equities was made equal, each, and every day.

And, how did it work out?

The simple portfolio management strategy:

Increased the portfolio value faster than the value of any equity in the portfolio.

Beat the index of all equities in the portfolio.

The particulars:

The equity value with the fastest growth, over the quarter of a century, was GE, which increased in value by a factor of 41.56566, (about 16.1% per year.) The simple portfolio management strategy increased the portfolio value by a factor of 52.68776, (about 17.2% per year,) resulting in a 26.757905444% improvement in the quarter century investment over

*any*single equity, (i.e., the simple portfolio management strategy grew the value of the portfolio faster than the value any equity in the portfolio grew, which is in line with theoretical expectations.)The index value of the ten equities, (i.e., the summed value of the equities,) increased by a factor of 12.655270655 for the quarter century-about 10.7% per year vs. the 17.2% for the simple portfolio management strategy, (which is reasonably near the theoretical expectations of about 2X.) The simple portfolio management strategy beat the index by about 4.1633056642-about 4X = 400%-over the quarter century.

As a side bar, the simple portfolio management strategy
performed well because it mitigated risk-which is often
overlooked by investors. The time period of the simulated
investment included two bear market downturns/adjustments,
(October of 1987 and 1999-2001,) and performed well. Risk
management is an important part of Although the simulation balanced the portfolio every day, this is really not necessary for the casual long term investor. Many balance periodically-once a week, or once a month. Others balance asset allocations when one asset's value exceeds the others by 5-10%. What is necessary is to not have too much of the portfolio's value dependent on a single asset, making the portfolio's value vulnerable to fluctuations in the value of a single investment. Although, through most of the quarter century simulation, at any one time, there was at least one equity that out performed the simple portfolio management strategy, in the long run, the strategy out performed all equities in the portfolio. The metric of risk is the deviation of the marginal
increments in an investment's value-i.e., how large the
fluctuations in value are. One can look at the graphs, above,
and see how the simple portfolio management strategy worked-it
was a Almost all hedging strategies work that way. |

What are the chances of a company funded by a venture capitalist being a high value asset after five years?

Although the log-normal characteristics of a venture capital fund can be computed analytically[1] using the techniques outlined in Section II, simulation has the advantage of developing intuition.

The marginal increments of a typical equity's growth in value has
an average and deviation,

,
and, **avg = 0.0004**

, respectively, which are
the the median values for equities listed on the US markets for the
Twentieth Century. Choosing 1,000 companies to get an adequate
statistical distribution, and identical statistics for all companies,
a **rms = 0.02**

program data file, *tsinvestsim*

, would
look like:*sim*

**
0, p = 0.51, i = 10000000
1, p = 0.51, i = 10000000
.
.
.
998, p = 0.51, i = 10000000
999, p = 0.51, i = 10000000
**

where, from Equation (1.24):

**
avg 0.02
--- + 1 ------ + 1
rms 0.0004
P = ------- = ---------- = 0.51 ....................(1.24)
2 2
**

and the companies are numbered,

, with an equal initial investment of ten million
dollars in each company-which was a typical value in the 1990's:**0, 1, ... 998,
999**

**
***tsinvestsim* *sim* 1265 | tail -1000 | cut -f3 | sort -n > *value*

where the simulation ran for five calendar years, of 253 working
days per calendar year,

. The value of the 1,000 companies, at the end of
five years is numerically sorted, smallest value to largest value.**5 * 253 =
1265**

Figure IV is a plot of the simulated values of 1,000 companies,
each with identical statistics, and each with an initial investment of
$10,000,000, after five years. The companies are sorted by value,
least, to most, and numbered,

. This graph is compared with the initial
investment in each company of $10,000,000, and the VC rule-of-thumb
that a company should be able to show at least a factor of three
increase in value in five years to be a desirable investment.**0, 1, ... 998,
999**

In the simulation, about 40% of the companies lost money, about
half met the factor of three requirement, and about 10% were
*star* performers. Note that a fund would be *carried*
by less than 10% of the investment assets-about 1 in 12 is the
empirical value for the VC industry, as a whole; that is why it is
considered *high-risk* investing.

As a side bar, the graph in Figure IV was relatively stable
through the Twentieth Century-from the At the end of the market So,
initial investing in |

[1]Computing the equity value of the 90'th percentile companies, for example, working with the Brownian motion/random walk equivalent of the system as described in Section II, and from Equation (2.3), the natural logarithm of the median value of all companies at the end of five years will have increased by:

**
0.0002 * 1265 = 0.253 ...............................(5.9)
**

where, the the gain in value, per business day,

from Equation
(1.20):
**g**

**
0.51 (1 - 0.51)
g = (1 + 0.02) (1 - 0.02) = 1.0002 ....(5.10)
**

has a natural logarithm of

, and the natural logarithm of the deviation of
the values of all companies in the fifth year, from Equation
(2.4):**ln (1.0002) =
0.0002**

**
0.02 * sqrt (1265) = 0.71133677 .....................(5.11)
**

The 90'th percentile is about ** 1.28**
deviations, (from the

*normal probability
function*

table of values,) so:**
ln (M) = 0.253 + (1.28 * 0.71133677)
= 1.16351106528 ..............................(5.12)
**

Or an increase in value by at least a factor of

is the 90'th percentile in Figure
IV, above, (i.e., 900 companies have less, 100 have more-or one in
nine, on average, will have a market capitalization worth more than
$32,011,530 at the end of five years, based on an initial investment
of $10,000,000.)**M =
3.2011530252**

The price history for each of the ten equities were downloaded from
Yahoo!'s Historical Prices database, (the
prices are adjusted for splits,) and the

program used to convert the price histories from csv spread sheet
format to *csv2tsinvest*

time series format, for example, for GE's price history:*tsinvest*

**
***csv2tsinvest* GE *ge.csv* > *ge*

There is an issue with the csv spreadsheet data; it is not Y2K compliant, (the years are listed as 77, 78, ... 01, 02,) which was fixed with sed(1), or a text editor could be used to insert "19" before any year not beginning with 0, else, insert 20.

The

time series for all equities were combined using the sort(1) command,
to make a combined time series of all equities,
*tsinvest*

, (which is why the date/time stamp
is the first column of a **djia**

database.)*tsinvest*

The

program was used for the simulations:*tsinvest*

**
***tsinvest*

-D 0 -m 10 -M 10 -i *djia* | cut -f1 > *balanced*
*tsinvest*

-D 0 -m 10 -M 10 -i -j *djia* | cut -f1 > *index*

The Unix cut(1) command was used to cut all except the first column
of the

output file, which is the index, (that's all we are interested in for
this simulation.) The *tsinvest*

,
**-D 0**

, and, **-m 0**

options shut down any optimization that the **-M
0**

program would normally make, (they tell the program to unconditionally
invest in an equity, whether it normally would, or not, and, in no
less than 10 equities at any one time, and no more than 10-i.e., no
matter what, invest in the 10 equities.) The
*tsinvest*

option tells the program to produce
an index-which by default is always balanced for comparison with the
other strategies that the program is capable of-and the
**-i**

tells the program to not to balance
the index values.) The files **-i**

and **balanced**

were then plotted.**index**

The

program was used to multiply a *tsmath*

time series to adjust for an initial investment of $1000 for plotting
comparisons of the individual equities, which were then plotted.*tsinvest*

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

Copyright © 2002-2004 John Conover, john@email.johncon.com. All Rights Reserved. Last modified: Mon Jan 8 18:17:29 PST 2001 $Id: 020326213456.9658.html,v 1.0 2004/10/13 07:04:52 conover Exp $