# Quantitative Analysis of Non-Linear High Entropy Economic Systems I

From: John Conover <john@email.johncon.com>
Subject: Quantitative Analysis of Non-Linear High Entropy Economic Systems I
Date: 14 Feb 2002 07:38:46 -0000

### Introduction

Much of applied economics has to address non-linear high entropy systems-those systems characterized by random fluctuations over time-such as equity prices, gross domestic product, industrial markets, etc.

An optimal wagering strategy for high entropy systems was first suggested by Claude Shannon in the mid 1940's in his pioneering work on information theory, The Mathematical Theory of Communication, (pp. 39-42.) The concept was further refined by J. L. Kelly Jr. in the Bell System Tech. J. vol. 35, 1956, A New Interpretation of Information Rate, (pp. 917-926), which established the isomorphism between the information-theoretic concept of information rate in a binary symmetric channel and speculation under uncertainty. More recently, Fazlollah M. Reza in An Introduction To Information Theory, (pp. 450-452,) has offered an alternative derivation that requires only the concepts of the law of large numbers, and the maxima from calculus-both taught in freshman mathematics-to derive the same optimal wagering strategy.

### Optimization

Following Reza, suppose a gambler wagers on an iterated game of chance. Further, suppose there is a chance, `P`, of the gambler winning any iteration of the game, and of `1 - P`, of losing. Let the capital the gambler starts with be `V(0)`, and `V(t)` the capital after the `t`'th iteration of the game. Since the gambler is not certain of the outcome of any iteration of the game, only a fraction, `f`, of the capital is wagered on each iteration of the game. Thus, after `t` many iterations the capital is:

``````

w          l
V(t) = (1 + f)  * (1 - f)  * V(0) ..................(1.1)

``````

where `w` is the number of times the gambler won, and `l = t - w`, the number of times the gambler lost. These numbers are functions of two random variables, denoted by `W` and `L`. Then, by the law of large numbers:

``````

1
lim            - W = P .............................(1.2)
t->infinity t

``````

and:

``````

1
lim            - L = q = 1 - P .....................(1.3)
t->infinity t

``````

The gambler must determine the fraction, `f`, of the capital to wager on each iteration of the game, which will maximize the average exponential growth rate, `G`, of the capital, i.e., maximize the value of:

``````

1     V(t)
G = lim            - ln (----) .....................(1.4)
t->infinity t     V(0)

``````

with respect to `f`:

``````

W              L
G = lim            - ln (1 + f) + - ln (1 - f) .....(1.5)
t->infinity t              t

``````

or:

``````

G = P ln (1 + f) + q ln (1 - f) ....................(1.6)

``````

which, by taking the derivative with respect to `f`, and equating to zero, is maximal when:

``````

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

1 - P - 1        P
- (1 - P)          (1 + f)  = 0 ................(1.7)

``````

and, combining terms:

``````

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

P        P
- (1 - P) (1 - f)  (1 + f)  = 0 ................(1.8)

``````

and, splitting:

``````

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

P        P
= (1 - P) (1 - f)  (1 + f)  ....................(1.9)

``````

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) .....(1.10)

``````

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) ...........(1.11)

``````

or:

``````

ln (1 - f) - ln (1 + f) = ln (1 - P) - ln (P) ......(1.12)

``````

and performing the logarithmic operations:

``````

1 - f        1 - P
ln (-----) = ln (-----) ............................(1.13)
1 + f          P

``````

and exponentiating:

``````

1 - f   1 - P
----- = ----- ......................................(1.14)
1 + f     P

``````

reduces to:

``````

P (1 - f) = (1 - P)(1 + f) .........................(1.15)

``````

and expanding:

``````

P - P f = 1 - P f - P + f ..........................(1.16)

``````

or:

``````

P = 1 - P + f ......................................(1.17)

``````

and, finally:

``````

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

``````

Substituting, `q`, from Equation (1.3) into Equation (1.6):

``````

G = P ln (1 + f) + (1 - P) ln (1 - f) ..............(1.19)

``````

where `G` is the average exponential growth rate of the gambler's capital, per iteration of the game. As a useful form, exponentiating both sides:

``````

P        (1 - P)    G
g = (1 + f)  (1 - f)        = e  ...................(1.20)

``````

where the gain, `g`, is the average multiplicative gain per iteration of the game, i.e., after `t` many iterations, the gambler's capital would have increased by a factor of `g^t`.

 As a side bar, if `P = 1`, i.e., an absolute certainty, then Equation (1.20) degenerates into the formula for compound interest, ```g = (1 + f)```, and the value of the capital at the end of `n` many iterations would be increased by a factor of ```g^n = (1 + f)^n```, where `f` is the interest rate. For the initiated, the equations can be derived using Ito Calculus, which is the preferred methodology in Geometric Brownian Motion analysis. From Ito's Lemma, then the average, `avg`, is the mean, which is the sum of `f(t)`, over `n` many iterations, divided by `n`, and the root-mean-square, `rms`, is the square root of the sum of `f(t)` squared, over `n` many iterations, divided by `n`: `````` srms = sqrt (rms^2 - avg^2) G = avg - (srms^2 / 2) g = e^(avg - (srms^2 / 2)) `````` Which has the interesting implication, for the sum of many Geometric Brownian Motion trajectories, (for example, a portfolio, or GDP,) that the sum is not ergodic, in the sense of the time-average of the sum not being equal to the ensemble average, (the ensemble having a Log Normal distribution.)

Equation (1.18) and Equation (1.20) are important. As a simple example, suppose each outcome of an iterated game depends on a throw of a fair six sided die-if the die comes up four, or less, the gambler wins what was wagered; else loses it. The gambler's optimal strategy is to put ```1 / 3``` of the capital at risk, (by wagering it,) on each iteration of the game, since `P = 4 / 6`, and from Equation (1.18), `f = 2 P - 1 = 1 / 3`. The gambler's capital would increase, from Equation (1.20), on average, by a factor of ```g = 1.05826736798``` per iteration of the game, or about `6%` per game.

But how does the gambler play the game if the value for P is unknown?

Equation (1.20) offers a metric methodology. For the `t`'th iteration of the game, the fractional gain, `f(t)`, in the gambler's capital for the iteration is:

``````

V(t) - V(t - 1)
f(t) = --------------- .............................(1.21)
V(t - 1)

``````

The absolute value of `f(t)` is the fraction of the capital the gambler wagered on the `t`'th iteration of the game-it is the fraction of capital that the gambler put at risk, `f`. The capital increased if the gambler won on the game in the `t`'th iteration, and decreased if the gambler lost-by the fraction of the capital the gambler put at risk, `f`.

 As a side bar, this is one of the fundamental concepts in the quantitative analysis of high entropy economic systems-risk is the root-mean-square, `rms`, of the values of `f(t)`, (for example, the absolute value of the fluctuations in the gambler's capital, `|f|`,) and the average, `avg`, of `f(t)`, is the average gain in value of the high entropy economic system, (for example, the average increase in value of the gambler's capital.) If the frequency distribution of `f(t)` has a normal/Gaussian frequency distribution, (as many high entropy economic systems do,) then the average, `avg`, is the mean, which is the sum of `f(t)`, over `n` many iterations, divided by `n`, and the root-mean-square, `rms`, is the square root of the sum of `f(t)` squared, over `n` many iterations, divided by `n`. This presents a definition issue for the term root-mean-square: Physics: the square root of the average (mean) value of the square of a quantity. Statistics: the square root of the arithmetic mean of the squares of the deviation of observed values from their arithmetic mean. The term root-mean-square is used here in the engineering or physics sense of noise power from electronic communication theory; instantaneous power depends on the absolute value of a quantity, (like `f`,) and is the effective constant value that would produce identical characteristics when calculated using the law of large numbers. There is a potential conflict of definition between this derivation and the concept of (implicit,) risk = statistical root-mean-square, `srms`, (e.g., the deviation, which is the square root of the variance from the mean,) as defined in classic Black-Scholes-Merton methodology. However, the two can be reconciled. If the definition of risk = `srms` is used, square `srms`, (to obtain the variance,) add the square of the mean, `avg`, and take the square root of the sum to obtain `rms`, (e.g., biased sample variance.) The issue concerns computational methodologies for measuring the "effective average" of the absolute value of `f` using the law of large numbers, and their interpretations. (Note, also, that the ratio of `avg` and `srms` is the Sharpe Ratio.) The relationship between the absolute deviation, (half-normal distribution,) and `rms` is a factor of ```sqrt (2 / pi) = 0.79788456```, (IFF the probability distribution of the marginal increments of the gambler's capital, `f(t)`, has a normal/Gaussian frequency distribution,) reconciling this derivation with the concepts of Black-Scholes-Merton and the Sharpe Ratio, (and interpretations from both physics and formal mathematics.) For the initiated, the running average of the marginal increments, `avg`, (and their deviation, `rms`,) from Equation (1.21) is a linear Kalman filter, (i.e., a one-pass algorithm.) See the documentation to the `tskalman` program for particulars.

If `n` has a sufficiently large data set size, then by the law of large numbers, in `n` many iterations of the game, the total number of wins for the gambler would be expected to be ```n P``` and the total number of losses would be expected to be `n (1 - P)`, for an average number of wins of ```(n P - n (1 - P)) / n = 2 P - 1```. The average gain, `avg` would then be:

``````

avg = (2 P - 1) f ..................................(1.22)

``````

where `f` is the return on the fraction of capital wagered on each iteration of the game, (`srms`,) and is equal to `rms`, the root-mean-square of the marginal increments of the capital, and `avg` is the mean of the marginal increments of the capital. Solving for `P`:

``````

avg
--- = 2 P - 1 ......................................(1.23)
rms

``````

or:

``````

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

``````

The interpretation of `P` is that it is the likelihood of up movement in the characteristics of a high entropy economic system.

Substituting Equation (1.18) for `P` in Equation (1.24), and letting `f = rms`:

``````

avg
---- + 1
rms + 1    rms
------- =  -------- ................................(1.25)
2          2

``````

or, maximal growth occurs when:

``````

2
avg = rms  .........................................(1.26)

``````
 As a side bar, note that the entropic model used is non-linear; it is not a random walk model. A random walk is a cumulative sum of a random variable-the noise is additive. In the entropic model used, the noise is multiplicative. The long term characteristics of the model exhibit a log-normal distribution, as mentioned in Section II. As an equivalent model, the iterative equation: `````` V(t) = V(t - 1) * (1 + a(t)) .....................(1.27) `````` can be used, where `a(t)` is a function of a random variable, with a normal/Gaussian frequency distribution that has a mean equal to `avg` and deviation equal to `rms`. The non-zero mean creates the exponential contour of the characteristics, and the random variable the fluctuations. Note the similarity to the simplist non-linear dynamical system function, the iterated parabolic/logistic function: `````` V(t) = V(t - 1) * (a + b * V(t - 1)) = 2 a V(t - 1) + b V (t - 1) .................(1.28) `````` The paradigm of the model is that the higher order terms, (of which their may be many,) of the economic non-linear dynamical system can be modeled with a high entropy stochastic methodology-if the complexity of the economic system is sufficiently high. A concept not unlike pertubation theory.

The concept is similar to automating risk arbitrage, based on historical performance, where an expected-value table of the difference between an investment's potential upside, (multiplied by the probability of achieving the potential upside,) and the the potential downside, (multiplied by the probability of the downside occurring,) is the expected value of the return of the investment-the difference between the potential upside and potential downside corresponding to the root-mean-square, `rms`, of the value of the investment, and the average, `avg`, the expected value, where `P` is the likelihood of the potential upside occurring.

### Important Formulas

Reiterating the important equations for high entropy economic systems:

``````

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

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

``````

and for maximal growth:

``````

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

``````

which is when:

``````

2
avg = rms  .........................................(1.26)

``````

where `avg` is the mean, and `rms` the root-mean-squre, of the system's marginal increments.

Of interest is that only two variables,`avg` and `rms`, are involved in optimizing the performance of a high entropy economic system. Typical values for `avg` and `rms` are `0.0004` and `0.02`, respectively, for daily data. `P` would then have a value of `0.51`, and `g` a value of `1.0002` per day, which is about `5%` per calendar year, for `253` business days in a calendar year.

### Example I, GE Equity Price

As an emperical exercise to demonstrate the usage of Equation (1.24) and Equation (1.20), the equity price history for General Electric, (ticker symbol, GE,) for the last 10,103 days, (January 4, 1962, to February 15, 2002,) was downloaded from Yahoo!'s Historical Prices database, and the `csv2tsinvest` program used to convert the database to a time series. The mean of the marginal increments of the price, `avg`, and the root-mean-square of the price, `rms`, was found to be `0.000499`, and `0.015145`, respectively, using the `tsfraction`, `tsavg`, and, `tsrms` programs. On January 4, 1962, the initial value of the equity was `\$0.761300`, adjusted for splits. On February 15, 2002, the value was `\$37.11`.

From Equation (1.24):

``````

0.000499
-------- + 1
0.015145
P = ------------ = 0.516474084 .....................(1.29)
2

``````

And, from Equation (1.20):

``````

0.516474084
g = (1 + 0.015145)            *

(1 - 0.516474084)
(1 - 0.015145)

= 1.00779357225 * 0.992648138

= 1.00038441336 ..................................(1.30)

`````` Figure I

Figure I is a plot of the GE's equity price, from January 4, 1962, to February 15, 2002, and the equation ```0.761300 * 1.00038441336^t```. Additionally, the `tsinvestsim` program was used to generate a time series with the same characteristics, (`P = 0.516474084`, `I = 0.761300`, and ```f = 0.015145```,) using a uniformly distributed random number generator, to produce a binomial frequency distribution with 10103 elements to simulate a normal/Gaussian distribution, which is overlayed, also. Figure II

Figure II is the frequency distribution of the marginal increments of GE's equity price, from January 4, 1962, to February 15, 2002, and a least squares fit of the distribution to a Normal/Gaussian bell curve distribution. The binomial frequency distribution of the marginal increments of the time series simulated with the `tsinvestsim` program shown in Figure I is also displayed. The data in Figure II was made with the `tsnormal` program. The comb effect in the binomial frequency distribution was caused by aliasing between the binomial elements in the frequency distribution and the histogram element size of the plot, and could be removed by choosing a larger histogram element size. Of passing interest is the leptokurtosis in the frequency distribution of the marginal increments of GE's equity price, which is explored further in Example III. Figure III

Figure III is a plot of the likelihood, `P`, of an up movement in GE's equity price as computed by the `tsshannonwindow` program for a 100 day moving window, from May 5, 1981 to February 15, 2002-4,900 business days for clarity. The first time series presents likelihood, `P`, computed by Equation (1.24) for the moving window. The second by counting the number of up movements in the window. Figure IV

Figure IV is a plot of the reconstructed GE equity price time series, as computed by the `tsgainwindow` program for a 100 day moving window, from January 6, 1962 to March 6, 2002, overlayed on the original GE equity price. The `tsgainwindow` program uses Equation (1.24) and Equation (1.20) to derive the statistics of the time series for the moving window, which is then used for the reconstruction:

``````

tsgainwindow -w 100 ge.jan.02.1962-mar.06.2002 | tsmath -l | tsintegrate | \
tsmath -e | tsmath -m 0.78 > reconstruction.ge.jan.02.1962-mar.06.2002

``````

where "```tsmath -l | tsintegrate | tsmath -e | tsmath -m 0.78```" implements the geometric reconstruction by taking the natural logarithm of the series, integrating it, exponentiating it, and scaling it by a factor of 0.78, which was the initial value of the original time series.

### Example II, Is the Equity Price Growth of the DJIA Component Companies Optimally Maximal?

If the component companies in the DJIA are the premier companies in the US exchanges, then their equity price characteristics should be near optimally maximal, i.e., consistent with Equation (1.26).

Symbol Name `avg` `rms` `rms^2`

AA

Alcoa

0.000510

0.020365

0.000415

AXP

American Express

0.000538

0.021724

0.000472

BA

Boeing

0.000493

0.024889

0.000619

C

CitiGroup

0.000405

0.022257

0.000495

CAT

Caterpillar

0.000423

0.019186

0.000368

DD

Du Pont

0.000331

0.017290

0.000299

DIS

Disney

0.000919

0.025126

0.000631

EK

Eastman Kodak

0.000338

0.028620

0.000819

GE

General Electric

0.000512

0.015452

0.000239

GM

General Motors

0.000257

0.017700

0.000313

HD

Home Depot

0.000818

0.027398

0.000751

HON

Honeywell

0.000438

0.020541

0.000422

HWP

Hewlett-Packard

0.001378

0.051051

0.002606

IBM

0.000321

0.016877

0.000285

INTC

Intel

0.001406

0.028905

0.000835

IP

International Paper

0.000359

0.018274

0.000334

JNJ

Johnson & Johnson

0.000703

0.016049

0.000258

JPM

JP Morgan Bank

0.000375

0.023279

0.000542

KO

Coca Cola

0.000534

0.016808

0.000283

MCD

McDonalds

0.000506

0.020492

0.000420

MMM

Minnesota Mining and Manufacturing (3M)

0.000378

0.014766

0.000218

MO

Philip Morris

0.000770

0.017934

0.000322

MRK

Merck

0.000583

0.015963

0.000255

MSFT

Microsoft

0.001593

0.026458

0.000700

PG

Procter and Gamble

0.000515

0.015394

0.000237

SBC

SBC Communications

0.000597

0.016261

0.000264

T

AT&T

0.000113

0.022534

0.000508

UTX

United Technologies

0.000588

0.018409

0.000339

WMT

WalMart Stores

0.001507

0.034505

0.001191

XOM

Exxon Mobil

0.000479

0.013938

0.000194

TABLE I.

TABLE I contains the average, `avg`, the root-mean-square, `rms`, and the root-mean-square, squared, `rms^2`, of the marginal increments for the equity price time series for each of the component companies in the DJIA. The daily close equity price history for each component company in the DJIA was downloaded from Yahoo!'s Historical Prices database, and the `csv2tsinvest` program used to convert the database to a time series. The `tsfraction`, `tsavg`, and, `tsrms` programs were used to calculate the average, `avg`, and, the root-mean-square, `rms`, of the marginal increments of each time series. The `tsmath` program was used to square the `rms` values. The time series files represented from between January 2, 1962 and July 9, 1986 for the various component companies, to March 6, 2002. Figure V

Figure V is a parametric scatter plot of the average v.s. the root-mean-square, squared, `avg` and `rms^2`, respectively, of the component companies of the DJIA. The figure is a pictographic representation of the data in TABLE I.

It does, indeed, appear that the equity price growth of most of the DJIA component companies is near optimally maximal, although slightly risk adverse.

### Example III, Leptokurtosis In the Increments of the DJIA

The historical time series of the DJIA's 27949 daily closes, from January 2, 1900 through March 6, 2002, inclusive, was obtained from Yahoo!'s database of equity Historical Prices, (ticker symbol ^DJI,) in csv format. The csv format was converted to a Unix database format using the `csv2tsinvest` program, filename `djia1900-2002`, and files of the marginal increments of the DJIA made:

``````

tsfraction djia1900-2002 | tsnormal -t > djia.1900.jan.02-2002.mar.06
tsfraction djia1900-2002 | tsnormal -t -f > djia.1900.jan.02-2002.mar.06.histogram

``````

From Figure IV, the average and root-mean-square values, ```avg = 0.0004```, and, `rms = 0.02`, respectively, seem reasonable approximations for the median values for the marginal increments of the equities in the DJIA. Additionally, the instantaneous values of the equities in the DJIA seem to have a, from Section II, a log-normal distribution with a range that is, approximately, an order of magnitude and a half. As a simple simulation of the DJIA, the `tsinvestsim` program can be used with the `data` file:

``````

tsinvestsim -n 1000 data 27949 | tsinvest -i -j | cut -f1 | \
tsfraction | tsnormal -t > tsinvestsim.jan.02.1900-mar.06.2002
tsinvestsim -n 1000 data 27949 | tsinvest -i -j | cut -f1 | \
tsfraction | tsnormal -t -f > tsinvestsim.jan.02.1900-mar.06.2002.histogram

``````

and the files plotted: Figure VI

Figure VI is the frequency distribution of the marginal increments of the DJIA's value, from January 2, 1900, to March 6, 2002, and a least squares fit of the distribution to a Normal/Gaussian bell curve distribution. The binomial frequency distribution of the marginal increments of the time series simulated with the `tsinvestsim` program. The data in Figure VI was made with the `tsnormal` program.

 As a side bar, Figure VI is interesting-the simulation was made using a binomial process, which produces normal/Gaussian distributions. The evolution of a geometric progression of a random variable with a normal/Gaussian distribution is a log-normal distribution. What happens is that the price distribution of the 30 equities in the DJIA evolves into a log-normal distribution, with one equity dominating the index value at all times; its price fluctuations dominate the fluctuations in the index, too. The contribution of the marginal increments of the other 29 equities contribute less to the index's fluctuations. So, it would be expected that an index, made up of the sum of equities from a geometric process, with price fluctuations that have a normal/Gaussian distribution, would have a bell curve that is pushed up around the mean with fat tails-exactly what is shown in Figure VI. (As a modeling exercise, slightly exponentiating normal/Gaussian distributed increments will simulate leptokurtosis, too.) Most economic time series exhibit such phenomena, and because of the importance, log-normal distributions will be discussed in Section II in some detail. Note that leptokurosis means higher risk; the tails contain too much of the distribution and the root-mean-square, `rms`, of the marginal increments can be misleading in calculating risk. The simple portfolio management strategy, from Section V, exploits this characteristic to maximize portfolio gain, while at the same time, minimizing risk, and the equities in the DJIA are used as a demonstration by eliminating the leptokurtosis in the increments of the value of the portfolio. In some sense, the long term objective of portfolio management is to minimize leptokurtosis in the increments of the portfolio's value; portfolios with leptokurtotic distributions of the increments gain fabulous wealth quite quickly, only to to lose it all when the risk asserts itself-a portfolio with leptokurtotic distribution of the increments can not remain a fugitive from the laws of probability forever. (Its why investing in indices is probably not such a good idea, and why the dot-com valuations came down so fast, too-the dot-com companies were notorious for having leptokurtotic distributed price fluctuations.) (Just so there is no confusion, Figure VI does not represent a log-normal distribution-it represents the sum of geometric processes that evolves into a log-normal distribution. Equity prices evolve into log-normal distributions, with, approximately, normal/Gaussian distributed increments; indices are the sum of equity prices, and have non-Gaussian distributed increments, i.e., distributions with fat tails, or leptokurtosis. As a final qualification, the amount of leptokurtosis is not stable, or constant over time-the long term evolution of indices with log-normal distributions of the constituent parts is for one part to totally dominate the index, and, in the long term, the marginal increments of the index will have the normal/Gaussian distribution of that part.)

It has been suggested that the leptokurtosis in economic time series is created by the fractal dimension of the data not being 2- which is the value for a simple Brownian motion fractal-and persistence, (i.e., a slight tendency for the next movement in the time series to be like the last.) But there can be other reasoning, too.

``````

tsmath -l djia1900-2002 | tslsq -o | tsroot -l | \
tsscalederivative > djia.1900.jan.02-2002.mar.06
tsinvestsim -n 1000 data 28048 | tsmath -l | tslsq -o | tsroot -l | \
tsscalederivative > tsinvestsim.jan.02.1900-mar.06.2002

``````

where the `data` file is the same as used for Figure VI, above. Figure VII

Figure VII is a plot of the persistence, (sometimes called the Hurst Exponent from rescaled range analysis, Range/Scale,) of the DJIA, and a simulation with similar statistics. Of interest is the variability of the persistence-it is far from constant over time, which would be expected in a well behaved fractal. Some observations:

1. The persistence of the DJIA at ```e^0.693147 = 1``` day is `0.573078`, and then falls rapidly-this is a contradiction with the Efficient Market Hypothesis, (EMH), which predicates that the market responds to information instantaneously; the contradiction is not serious for most analysis, and the approximation used in the EMH is probably reasonable for general purposes, (all it says is that an instantaneous market response to new information means within a few days.)

2. At about `e^2.2 = 9` days, the persistence drops to just over 0.5.

3. And then the persistence raises to about 0.6 through the rest of a calendar year, (of about 253 trading days in a calendar year, or at `ln (253) = 5.5`, when it, quite quickly, goes anti-persistent.

Although such variability in persistence could be attributed to Non-Linear Dynamical System, (NLDS,) dynamics, it is more likely a structural artifact. For example, the anomaly could be created by a downtrend, followed by an uptrend about 30 trading days later, on average, that happens, on average, about once a year, (in addition to the short term few day inefficiency.) Looking at the time series of the DJIA, October and November frequently have about a month and a half downward spike, which also happens to correspond with October 31, the time that mutual fund capital and income gains taxation liabilities are executed at the end of the fiscal year, as mandated under US Federal law, (usually, resulting in mutual fund companies holding a fire sale on equities they hold with bad pro forma.) Note, also, that many NLDS methodologies would indicate that structural phenomena are system dynamics-leading to inappropriate interpretations of the data.

However, using an average persistence of `0.57` for the DJIA, (and its constituent equities, too,) can adequately model the NLDS/structural phenomena. This means that the root-mean-square of the marginal increments, `rms`, would not be added as `((1 / n) * sum increment^2)^0.5`, but as ```((1 / n) * sum increment^(1 / 0.57))^0.57```. (For an example of the consequences of assuming that risk is represented as the root-mean-square of the marginal increments, see the section on the DJIA, from Section III.

Of passing interest is the fact that, as shown in Figure VII, the simulation showed substantial leptokurtosis, but no persistence.

### Appendix I, Useful Approximations

As a approximation for non-linear high entropy economic system's daily data, from Equation (1.20), the exponential marginal gain per unit time, `g`, will typically be about `1.0002`, and the ```ln (1.0002)``` will be about `0.0002`. The average of the time series' marginal increments, `avg`, that has exponential marginal gain per unit time, ```g = 1.0002``` will be about `0.0004`. So, as an approximation, divide the `avg` by two, and that's the `ln (g)`; add unity to get `g`.

If a non-linear high entropy economic system is assummed to be operating near optimality-a reasonable assumption for most economic systems-then substituting Equation (1.26) into Equation (1.24):

``````

avg
---- + 1
rms         rms + 1   sqrt (avg) + 1
--------  = ------- = -------------- ...............(1.31)
2           2            2

``````
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--

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

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