Quantitative Analysis of Non-Linear High Entropy Economic Systems Notes and Asides

From: John Conover <john@email.johncon.com>
Subject: Quantitative Analysis of Non-Linear High Entropy Economic Systems Notes and Asides
Date: 28 Oct 2013 08:39:47 -0000

Clarification

From Quantitative Analysis of Non-Linear High Entropy Economic Systems I, in a simple fixed increment geometric progression of the gambler's capital, `f` is the fraction of the gambler's capital waged on each bet in the calculation of the "standard derivation," of the increments of the gambler's capital. If the game is not fair, the time series of the marginal increments of the gambler's capital will have an offset, `avg`. And, the gambler's capital will increase, or decrease, by `avg + f` for a win, and `avg - f` for a loss.

Making a time series of `1/3`,`-1/3`,`1/3`,`-1/3`,`1/3`,`1/3`, ... repeating forever, should yield ```P = 4/6 = 2/3 = 0.666666666```, and ```(2 * P) - 1 = 1/3```, and, the average should be `0.111111111`, and the root-mean-square `0.333333333`; `tsavg`(1) and `tsrms`(1) return these correct values, indicating that the average should NOT be subtracted from the calculation of the sum of the squares of the marginal increments of the gambler's capital.

Note that:

``````

V(t) - V(t-1)
------------- ......................................(I-1.21)
V(t-1)

``````

is the fraction of the increment between the previous value, and the current value, of the gambler's capital, which, by definition, is the fraction of the capital, `f`, that the gambler wagered, in the `t`'th interval.

Absolute Deviation

If the marginal increments of the gambler's capital are not fixed and form a Gaussian/Normal distribution, then Half-normal_distribution, (see, also, Absolute_deviation, which is the same formula) calculations can be used, and the expectation is:

``````

E(Y) = rms * sqrt (2 / pi) = rms * 0.79788456 ......(1.1)

``````

for the average value of rectified Gaussian noise.

The variance is:

``````

V(Y) = (rms^2) * (1 - (2 / pi))
= (rms^2) * 0.363380227 .......................(1.2)

``````

See, also: the "Population Variance and Sample Variance," section, "Biased Sample Variance," in Variance.

Unfortunately, many consider this method of calculating the wagering of the gambler fragile in real world distributions often found in financial time series.

Formality

Formal root-mean-square analytical methods often yield more precise results.

Let `Si` be the `i`'th element from a random distribution, with a bias, `avg`, and a deviation, `srms`:

``````

1  n
srms^2 = - SUM (Si - avg)^2 ........................(1.3)
n i=1

1  n
= - SUM (Si^2 - 2Siavg + avg^2) .............(1.4)
n i=1

1  n         1  n
= - SUM Si^2 - - SUM 2Siavg + avg^2 .........(1.5)
n i=1        n i=1

``````

But:

``````

1  n
- SUM 2Si = 2avg ...................................(1.6)
n i=1

``````

Therefore:

``````

1  n
srms^2 = - SUM Si^2 - 2avg^2 + avg^2 ...............(1.7)
n i=1

1  n
= - SUM Si^2 - avg^2 ........................(1.8)
n i=1

``````

And:

``````

1  n
- SUM Si^2 = rms^2 .................................(1.9)
n i=1

``````

Therefore:

``````

srms^2 = rms^2 - avg^2 .............................(1.10)

rms^2 = srms^2 + avg^2 .............................(1.11)

1  n
Where, avg = - SUM Si ..............................(1.12)
n i=1

``````

Further, note that the derivation is the paradigm/methodology used in `tsfraction`(1)/`tsrms`(1), and, additionally, in `tsinvest`(1).

Now, aggregating `N` many of the random distributions, each with a bias, `avgK`, and a deviation, `srmsK`, ```(1 =< K <= N, rmsK^2 = srmsK^2 + avgK^2,)``` together:

``````

1  n
srmst^2 = - SUM (S1i - avg1)^2 +
n i=1

1  n
= - SUM (S2i - avg2)^2 +
n i=1

...

1  n
= - SUM (SNi - avgN)^2 .....................(1.13)
n i=1

1  n
srmst^2 = - SUM (S1i^2 - 2S1iavg1 + avg1^2) +
n i=1

1  n
= - SUM (S2i^2 - 2S2iavg2 + avg2^2) +
n i=1

...

1  n
= - SUM (SNi^2 - 2SNiavgN + avgN^2) ........(1.14)
n i=1

1  n          1  n             1  n
srmst^2 = - SUM S1i^2 - - SUM 2S1iavg1 + - SUM avg1^2 +
n i=1         n i=1            n i=1

1  n          1  n             1  n
= - SUM S2i^2 - - SUM 2S2iavg2 + - SUM avg2^2 +
n i=1         n i=1            n i=1

...

1  n          1  n
= - SUM SNi^2 - - SUM 2SNiavgN
n i=1         n i=1

1  n
+ - SUM avgN^2 ...........................(1.15)
n i=1

``````

For the elements from the `K`'th random distribution:

``````

1  n
- SUM SKi^2 = rmsK^2 ...............................(1.16)
n i=1

1  n
- - SUM 2SKiavgK = -2avgK^2 ........................(1.17)
n i=1

``````

Since:

``````

1  n
- SUM 2SKi = 2avgK .................................(1.18)
n i=1

1  n
- SUM avgK^2 = avgK^2 ..............................(1.19)
n i=1

srmst^2 = rms1^2 - avg1^2 +
rms2^2 - avg2^2 +
...
rmsN^2 - avgN^2 ..........................(1.20)

``````

But:

``````

rmsK^2 - avgK^2 = srmsK^2 ..........................(1.21)

``````

Therefore:

``````

srmst^2 = srms1^2 + srms2^2 + ... + srmsN^2 ........(1.22)

1  n
avgt = - SUM S1i +
n i=1

1  n
- SUM S2i +
n i=1

...

1  n
- SUM SNi ...................................(1.23)
n i=1

``````

For the elements from the `K`'th random distribution:

``````

1  n
- SUM SKi = avgK ...................................(1.24)
n i=1

avgt = avg1 + avg2 + ... + avgN ....................(1.25)

avgt^2 = (avg1 + avg2 + ... + avgN)^2 ..............(1.26)

rmst^2 = rms1^2 + rms2^2 + ... + rmsN^2 -
(avg1^2 + avg2^2 + ... + avgN^2) +
(avg1 + avg2 + ... + avgN)^2 ..............(1.27)

``````

Note that the `srmsK` terms add root-mean-square, and the `avgK` terms add linearly, where ```srmsK^2 = rmsK^2 - avgK^2```.

Further, as an approximation when ```0 << avgK << rmsK << 1```:

``````

rmst^2 ~ rms1^2 + rms2^2 + ... + rmsN^2

``````

which is the method used in the `tsinvest`(1) program to optimize portfolio asset allocation, (and decreasing `PK`, due to the uncertainty created by limited data set size, i.e., decreasing `avgK`, and/or increasing `rmsK`, via Standard Error methodology.) Note, however, determination of the value of `srms` is not necessary for precisely calculating the gain, `g`, and/or, the Kelly Criteria, of an asset's value-only the value of `rms`, (and `avg`,) of an asset are required.

Optimal Leverage

For optimal leverage, (or margin,) let `V(t)` be a geometric Brownian motion fractal representing an asset's prices, with the `t`'th marginal return:

``````

V(t) - V(t-1)
------------- ......................................(I-1.21)
V(t-1)

``````

The mean of the marginal returns is `avg`.

The square root of the variance of the marginal returns is `rms`.

Then the probability of an "up movement" in any marginal return, `P`, is:

``````

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

``````

And, the median gain, `g`, of the marginal returns is:

``````

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

``````

which is maximal when:

``````

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

``````

which makes:

``````

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

``````

by substitution into the equation for `P`.

Let `V` be the value of an investment at some time, `t`, and `M` be the fraction of `V` purchased on margin, ```0 <= M <= 1```, and, `I` be the fraction of `V` invested, ```0 <= I <= 1```, where ```M + I = 1``` and `M * V` is the amount of the margin, and `I * V` is the amount invested. Letting rmsi be the root mean square of the variance of the investment:

``````

V
rmsi = rms * -----------
V - (M * V)

1
= rms * ----- = (2 * P) - 1 ...................(1.28)
1 - M

``````

Or:

``````

rms
M = 1 - ----------- ................................(1.29)
(2 * P) - 1

``````

which would maximize the growth, `g`, of the amount invested, `I * V`, provided `P > 0.5, avg > 0`, and both `avg` and `rms` are too small, (which is frequently the case.)

Reducing:

``````

P = ((avg / rms) + 1) / 2 ..........................(1.30)

rms
M = 1 - ----------- ................................(1.31)
(2 * P) - 1

rms
M = 1 - --------------------------------- ..........(1.32)
(2 * (((avg / rms) + 1) / 2)) - 1

rms
M = 1 - --------------------- ......................(1.33)
((avg / rms) + 1) - 1

rms
M = 1 - ----------- ................................(1.34)
(avg / rms)

``````

Or:

``````

rms^2
M = 1 - ----- ......................................(1.35)
avg

``````

And, by substitution, the formula for optimal leverage:

``````

rms
M = 1 - ----------- ................................(1.36)
(2 * P) - 1

``````

which is the method used in the `tsinvest`(1) program to maximize Return on Investment, (ROI,) and, if `M = 0` the investment is at fair value, relative to its metric of risk, `srms`.

Optimal Asset Allocation

For optimal asset allocation, let `V(t)` be a geometric Brownian motion fractal representing an asset's prices, with the `t`'th marginal return:

``````

V(t) - V(t-1)
------------- ......................................(I-1.21)
V(t-1)

``````

And many such assets assembled in a portfolio, each asset representing a fraction, `cn`, of the total value of the portfolio, where:

``````

c1 + c2 + c3 + ... + cN = 1 ........................(1.37)

``````

Then the portfolio `avg`, `avgp`, would be:

``````

avgp = (c1 * avg1) + (c2 * avg2) +
(c3 * avg3) + ... + (cN * avgN) .............(1.38)

``````

since the `avgn`'s add linearly.

And, from Equation (1.10), the deviation of the `n`'th asset, `srmsn`:

``````

srmsn = sqrt (rmsn^2 - avgn^2) .....................(1.39)

``````

``````

srmsp = sqrt ((c1 * srms1)^2 + (c2 * srms2)^2 +
(c3 * srms3)^2 + ... + (cN * avgN)^2) ......(1.40)

``````

Yielding, from Equation (1.11), the `rms` of the portfolio, `rmsp`:

``````

rmsp = sqrt (srmsp^2 + avgp^2) .....................(1.41)

``````

The values of the asset allocations in the portfolio, `c1`, `c2`, `c3` ... `cN`, can be optimized:

``````

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

``````

where, for each asset:

``````

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

``````

and substituting:

``````

c = 2 P - 1 = avg / rms ............................(1.42)

``````

Or, for the `n`'th asset:

``````

cn * avgn = avgn^2 / rmsn ..........................(1.43)

``````

Which is the asset's contribution to the portfolio's `avgp`.

And the asset's contribution to the portfolio's `rmsp`:

``````

cn * rmsn = avgn ...................................(1.44)

``````

Where the asset's contribution to the portfolio's `srmsp`:

``````

srmsn = sqrt (rmsn^2 - avgn^2) .....................(1.45)

``````

Yielding the following useful equation:

``````

cn * srmsn = cn * sqrt (rmsn^2 - avgn^2) ...........(1.46)

``````

Where each asset's `cn * avgn` add linearly to produce the portfolio's `avgp`, and each asset's ```cn * srmsn``` add root-mean-square, to produce the portfolio's `srmsp`, and using Equation (1.11), to produce the portfolio's `rmsp`, with the asset allocations, `c1`, `c2`, `c3` ... `cN`, optimized, which is the method used in the `tsinvest`(1) program to optimize portfolio asset allocation. Note, however, determination of the value of `srms` is not necessary for precisely calculating optimal asset allocations-only the value of `rms`, (and `avg`,) for each asset are required.

Standard Error

Let `V(t)` be a geometric Brownian motion fractal representing an asset's prices, with the `t`'th marginal return:

``````

V(t) - V(t-1)
------------- ......................................(I-1.21)
V(t-1)

``````

The mean of the marginal returns is `avg`.

The square root of the variance of the marginal returns is `rms`.

Then the probability of an "up movement" in any marginal return, `P`, is:

``````

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

``````

And, the median gain, `g`, of the marginal returns is:

``````

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

``````

which can be implemented as a Kalman Filter, and as more data is acquired, the accuracy of the values of `avg` and `rms`, (and thus `P` and `g`,) increases. (A device that searches, sorts, and, collates data is, technically, an information automaton, or information robot, or just 'bot.)

A reasonable question to be answered is how much accuracy is necessary?

Methodologies such as Standard Error are often used, but the question can be rephrased: What are the chances, of measuring `t` many marginal increments, where by serendipity, the measurements were all made in a "bubble", (i.e., the values in the measurement interval were all above, or below, their median value)?

The chances of (at least,) `t` many marginal increments, between zero crossings of an asset's median value, is `erf (1 / sqrt (t))`, (`erf (1 / sqrt (t)) ~ 1 / sqrt (t)` for `t >> 1`).

Giving a convenient reduction in the value of `P` in Equation (I-1.24) by a factor of `1 - erf (1 / sqrt (t))` to estimate the likelihood of "up movement" in the next marginal return-compensating for the uncertainty of insufficient data set size.

In general, calculating the probabilities will require solving for the area of a section of the Normal/Gaussian distribution, (or cumulative distribution,) which is expedient if a unit Normal/Gaussian distribution is constructed as a table in memory, and a binary search implemented since such a table is monotonic, which is the method used in the `tsinvest`(1) program.

```
--

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

```