Investment strategies - fit your desire risk

level and the best plausible profit expectation.

Krzysztof Urbanowicz
Faculty of Physics Warsaw University of Technology
Koszykowa 75, PL00-662 Warsaw, Poland
Email: urbanow@pks.mpg.de

Abstract
Temporal linear correlations, which are not visible in average, can leads to changes
in time series probability distribution function. In the case of stock market data the
visible pivot point of the time series is zero. The unbalance of the gaussianity of
left and right side of zero point can lead to some statistical forecasting of stochastic
systems. Here we show the principle of objective value minimization which helps us
to correct gaussianity estimation. The experimental testing of investments strategies
based on this knowledge are presented. The positive eciency of strategies can be
as the proof of correctness of the objective value theory.
Key words: information entropy, stochastic processes, probability distribution,
stock market data
PACS: 05.45.Tp, 89.65.Gh

Introduction

Prediction the future is always most desire property of analysis, nevertheless

which kind of analysis we perform. When we would like to earn some money
on some investment, we try to catch the future in dierent ways. Sometime
we use knowledge and experience, sometimes computers show the results for
us, and many times we use intuition. In all the cases we can describe it and
put it in some schemes which rule the prices on the market. However number
of used schemes to forecast the future is large, one can say that too large to
capture it in some reasonable way. Nevertheless there are some dominant ways
and well-known knowledge which are used by many investors which generate
some little predictable future. Sometimes the panic-like behavior increases
the power of some sell/buy trigger and therefore enlarges its impact for selfrealized prediction. Every market is made by buyers and sellers and prices

Preprint submitted to Elsevier

2 February 2012

are just the eect of their behavior, so established knowledge on forecasting

is unstable. Usage of some prediction method by many investors will decrease
its eciency. On the other hand it generates the self made prophecy, but
prophecy only for those who are the rst, because the most of the people
make a use of the prediction too late. This is the sense of minority games
in which the only who are in minority wins, it is true also for stock market
games. When we would like to speculate on the stock market, then we try
to be on the minor site when we do invest and close the position when most
people follow us. In speculation we have to be rst so we take much of the risk,
because on the beginning well-established triggers are ease to misunderstand.
The well-known knowledge is overused make it many times useless or generates
self made prediction which of course should not be employed by reasonable
investor because it is too risky. There is for sure unpublished knowledge which
makes money for big investors or others, which can be on the beginning stable
(not altering in time or make too small prot to cover costs).
On the beginning we should tell what is known about stock market data for
sure that is statistic and known property. Most of the people working on stock
market are almost sure that the data are likely to be unpredictable, so that it is
dicult to have universal in time and space method to collect money. We can
say also that there are the ways of making the prot for very simple reason,
because there are people that do it and got rich on stock market investments.
From Physics point of view the data that we are dealing with are likely to be
stochastic, not linearly correlated on scales larger than an hour. Randomness
is dominant in such data but we think that there are deterministic component
which can be reveal by some method but complex, nonlinear method. Lets go
to the practice on investing, what most people know earn money.
1) Diversication.
The dominant way in optimization of portfolio is just diversication. Most of
papers on the investment strategies are somehow related to this issue. Diversication is the simplest way to decrease the risk with having the prot on
the same level. The assumption for ecient diversication is no correlation
among investments. Sometimes it is dicult to manage if we for example are
specialist in investments on banks only. On stock market many rms from the
same branch are highly correlated, so investor have to look at stocks from very
dierent branch to be consistent with eective diversication, or even should
make money not only on stock market but also on FOREX or other. This is
because it is known that highly develop Bessa shows much correlations in all of
the stocks so the risk is high. Diversication is the only one investment strategy which can help investors without doing prediction. Rests of the methods
are just prediction methods and prediction together with diversication.
2) Markowitz method. Putting together prediction and diversication with
2

simplest way of prediction and risk evaluation. The standard way of portfolio
selection.
In late fties last century Markowitz found the analytical solution of ecient
portfolio selection. The standard Markowitz issue corresponds to simplest way
of risk and prot evaluation. Risk is found to be variance of the past data and
prot as the average from returns on the same region. In order to apply his
procedure to another type of risk and prot evaluation one has to exchange
in formulas below only m and D or C on correspond values. This is especially
valid when we would like to optimize portfolio to forecast future values of m
and D or C.
3) Fundamental analysis.
The Method of predictions can be selected on fundamentals and technical.
Fundamental evaluation are based on knowledge about condition of rms or
political or investor information about decision making process. Fundamental
analysis is rather long-term, which uses the disproportion between the market
value and the evaluation. There are twofold problems within:
a. Is it the evaluation properly made?
b. Market prices are not always showing the capital of a rm and future prot
from the capital, but what investors thought about this.
The second case is much visible when we calculate the capital of a rm and
compare to the capitalization on the stock market. It is well-known that the
second is much higher than the rst balance. The ratio of these two capitalizations is not universal and diers among stocks even in the same branch of
industry. In this case investor uses some intuitive knowledge about the proper
value.
4) Technical analysis.
The very beginning coecients which may help in forecasting are moments and
smoothing methods. These coecients are based on the statistical property
of price changes. Moments comes from the mathematics and are just rst
and second moment of time series so average and variance. In nance we
use names of trends and volatility to show the same but volatility uses the
second moment in many dierent ways but it is not as strictly dened as in
mathematics. Trends are believed by investors to be much stronger than on
beginning one can suppose. One can say that to know in which trend we are
(bullish or bearish) is most important. Wisdom is to know the trend and follow
it. The nest analysis about trend searching is Elliot Waves. Our expression
about this analysis is that waves which shows you trend are not always visible.
The correct investing structure in data can be seen from time to time and can
3

be easily misunderstand by non specialist. In the case of volatility (in technical

analysis is called moment) the established investing procedure is to buy when
volatility is low and sell when is high. It can be understood as not to enter
the market in the case of large risk and changes.
5) Capital Asset Pricing Model CAPM theory.
Many economists assume perfectly ecient market, market in which additional prot one can be gathered only when he take additional risk. This
procedure one can turn round and say that if someone would like to look for
more protable stock then he should search for more risk in the case of ecient portfolio. This is an essence of CAPM theory [14] which go further and
show more risky assets that is assets which are more correlated with a market
index. Correlations with average increase the risk because it do not allowed
for ecient diversication.
In the CAPM theory future prot is related to the covariance of the portfolio
i with market index:
xi xf = i + i (xm xf ) + i

(1)

where xi ,xf and xm is portfolio i, risk free and market return respectively.
The linear trend i and random noise i is selected to better t the data. The
relation prot=covariance could be explained in two ways:
1) Market portfolio is assumed to be most optimal portfolio one can get.
Larger correlation of returns from portfolio xi to the most optimal xm should
be awarded with higher prot in long time horizon.
2) The covariance i shows the level of decreased prot from ecient diversication. Higher risk connected to the ecient portfolio should collect more
prot in longer time. The most important risk factor related to i is connected
to collective fail of all investments in the portfolio, so that the risk of shortage
of money would force to close the position in the most unprotable way. In
long time horizon investment the collective panic-like behavior which would
force us to change the investment decision is related to the correlation i in
average. One can say that the risk which can bear by individual investor can
dier, so the risk should possess some deviations from the mean i . On the
other hand covariance i is collected from the past, but one never knows if it
will persist in the future. One more problem appears in CAPM modeling that
is risk related to the correlation to the market i is not only one risk factor
in ecient market, so one should look at some more correlations of the prot
portfolio to existing fundamental analysis parameters [18].
The fundamental law of nature tells that free energy in the relaxation pro4

cedure stays in the minimum [10]. This is related to the limited sources of
energy in nature. Nevertheless humans, because of desire, struggle with nature to possess more sources of the energy. Limited amount of energy as well
as information, money etc. are then minimized by nature laws but we would
like to have its unlimited sources. That is why we can say that the value which
are minimized express the real value of the corresponding source such a value
we call objective. In this paper we try to convince that the absolute value
of some source demonstrate not always the real value, the value which are
desired to maximize by humans, because what we would like to have more are
express by its objective value. Stock market is thought to be stochastic [13]
or mixture of deterministic and stochastic processes [48] but with prevailing
part of the stochastic one. Regarding last we apply the principle of objective
value minimization to the stock market data. We try to reveal the important
thing of the stock market analysis, i.e. which stock is most desirable to invest
bringing higher prot than can do average investors.

The principle of objective value minimization

Let us dene x as the value of the studied variable. The information entropy
for stochastic process is given by the formula [1,9]
H=

P (x) ln P (x)dx.

(2)

It is common knowledge that in the equilibrium the entropy has its maximal
value [1,2,10]. One can think about certain function w(x) of x that we assign
as the objective value. We would like to show some properties of this function.
Let us create the functional which maximize the information entropy with
constraints of the given average objective value [1]:
S=

P (x) ln P (x)dx +

w(x)P (x)dx +

P (x)dx = max.

(3)

The last factor in Eq. (3) gives the normalization of all probability to 1. We
assume that for properly dened the objective value w(x), one should have
w(0) = 0. Resolving functional (3), i.e. calculate derivatives that should vanish
S
= 0,
P (x)

(4)

we get the following formula for the probability distribution

P (x) = e1+ ew(x) .

(5)
5

Table 1
Examples of objective values w(x) and corresponding the probability distribution
P (x).
Objective value

P (x)

w(x) = |x|

P (x) e|x|

w(x) = x2
)
(
w(x) = ln c+|x|
c

Probability distribution
exponential

x2

P (x) e
)
(
c
P (x) c+x

gaussian
power-law

The table 1 presents the examples of objective values w(x) and corresponding
to it the probability distribution P (x). Lets change a little functional (3) in
order to minimized objective value with the constraint of given value of the
information entropy
S =

w(x)P (x)dx

P (x) ln P (x)dx +

P (x)dx = min.

(6)

The above functional is minimized after calculating the the derivative that
should vanish, because
2S
> 0.
P (x)2

(7)

The minimization of the functional S gives the same dependence P (x) versus
x as maximization of the functional S. Using the fact that probability distribution P (x) include all the information about the stochastic process one
can say that: the principle of maximum entropy express in the principle of
minimum of the objective value.
Proposition In the equilibrium, for stochastic not correlated process, the
average objective value is minimized for constant entropy
w(x) =

w(xn ) = min,

(8)

where xn is a realization of stochastic process in the step n. The proof can be

made from the minimization of functional (6).
One can conclude from the above that objective value must be increasing function of absolute value |x|. The next conclusion is the following that having the
probability density function (pdf) of certain absolute value one can calculate
its objective value.
The objective value is a state function of a given system. One can say that it
is a parameter of the system that is minimized. The energy in the canonical
6

system known from statistical mechanics can serve as a objective value. Very
fast one can conclude that distribution of energy is P (E) eE what agrees
with statistical mechanics [10]. One can calculate from the above principle
Maxwell-Bolzmann distribution of velocities. Minimization of the energy of
the whole system which consist of particles without interaction can be regard
as a minimization of the energy of every particles alone, so minimized should
2
be E v 2 . The above gives the Maxwell-Boltzmann distribution P (v) ev
[10].

Objective portfolio selection

Random Matrix Theory can be used to describe correlations eigenvalues spectrum of random data with the same probability distributions [2]. Correlations
calculated from time series with dierent probability distributions will show
lower values than in the case of the same Pdf. In correlation calculations the
absolute value has its important impact so the smooth transormation of the
data can lead to change the correlation. In nancial interest is to relate correlations with risk but the aim of analysis should be the maximal correlation
so the real risk. Let us imagine such a case that we are dealing with two
time series transformed from one original that both possess dierent probability distribution. Let the rst time series has Gauss distribution and second
Couchy one. Correlation calculated between these two time series is apparent from one and can be evaluate as an integrate of the join part of these
two probabilities. When we would like to calculate some property of two time
series with dierent probability distributions we have to renormalized them
to the distributions with the same properties. We use the Objective Value
Theory to do this. Objective Value can be calculated from data by looking on
the probability distribution function (Pdf). The ObV correspond to each data
point representing its objectiveness, the value which in average is minimized
in isothermal processes. ObV is extensive, so one can use the mean value as
the reference value. Pdf renormalization process can be explained as follows.
We calculate the objective value of each point and on this level we normalize
average ObV of each time series to the mean ObV of all data. Next we calculate inverse function to objective value using only one set of parameters in
Pdf description. Now we possess normalized all time series that the statistical
properties calculated among these time series are nd to be extreme (maximal
or minimal). Apply the renormalization procedure to our example of time series with Couchy and other with Gauss distribution will give us transformation
of these two dierent distributions to the universal one so the join part will
give us unity. The validation of the theory will be calculations of correlation
7

showing larger values in the case of renormalization procedure.

(

x = w

w(x)
SD(x)
w(x)

(9)

where SD(x) is standard deviation od x. The ObV as well as the normalization

procedure is calculated for left and right side of Pdf separately.
The covariance matrix of normalized time series one can use to evaluate risk in
standard portfolio optimization [?]. The ObV w(x) can be a good measure
of risk related to the stock. The parameter w(x) for Gauss distribution is
just variance of original data but for any other data described by power law
is larger than variance and it depends from power exponent. Smaller power
exponent benet in much larger ObV than variance of original data. The
extreme shortfall of prices will increase ObV with respect to variance. The
Markowitz optimization problem can be replaced by the following functional:
wp (x) + mp + (

P (x)dx 1) = min,

(10)

where wp (x) is measure for risk and mp is expected prot from portfolio. Here
we use our forecasting method by unbalance in gaussianity:
mp =

pi Fi

(11)

Statistical forecasting of stochastic systems

The existence of some method of forecasting of stock market data is rather

controversial. Many scientists claim that there should not be any reasonable
method because we are dealing with ecient market (CAPM theory). This
is only the assumption, very reasonable assumption, but the strict proof one
never saw. We would like to show here some, the same, very reasonable theory
that could help to statistical forecasting of stochastic systems.
Let us consider time series of returns {xi }, which was made as follows:
xi = ln

Pi
,
Pi1

(12)

where Pi is the price at time i.

From the point of view of investors very important pivot point in returns
time series is just zero. In all cases of visualization of changes in prices the
8

0.7

Left side: Gauss

Right side: Leptocurtic

distribution

0.6

0.5

Pdf

0.4

0.3

0.2

0.1

0.0

-6

-4

-2

Fig. 1. Academic example of distribution behaviour with normal distribution on the

left side and leptokurtic distribution on the right.

green color reects growth and red falls of prices. That is why we can separate
our return time series on bullish and bearish which reects another type of
subsystem responsible for action in investors brain. We would like to convince
that there should exist two separate parts which are acting separately only
when in the stock market is hossa and second when is bessa. This is because
we sell or buy so zero is the threshold in which we earn or loose. Let us then
separate our time series to positive and negative values and put a notation as
follows:

x+
i : xi > 0 and xi : xi < 0 for i=1,...,N

(13)

All the time we would like to assign zero as the pivot point in both time series.
+
Let us consider that there exist some asymmetry between {x
i } and {xi }
probability distribution function Pdf (see example on Fig. 1). This unbalance
will lead us to do some forecasting basing on gaussianity of left and right Pdf
(our meaning of gaussianity we explain in details in section 5).

Gaussianity in a simple word would mean a level of maturity of the distribution. Gaussianity for Gaussian distribution is maximal. This is well known
from statistic that sum of many not correlated increments with nite variance will give us a Gaussian distribution, which further does not change in
its gure after adding additional data. Gauss distribution is the attractor of
convolutions of many distributions with nite second moment, that is why it
is so ubiquitous. When we are dealing in stock market data with situation
represented in Fig. 1 then our method says that we should BUY for three
9

reasons:
1) If the distribution will persist then average of the right side is higher than
average of the left side.
 
 
 +
 
xi  > x
i 

(14)

2) Gaussianity of the left side distribution Glef t is larger than gaussianity of

the right side Gright , so a parameter of unbalance
F =

Glef t Gright
max(Glef t , Gright )

(15)

will show us a future. It can be understood twofold:

a. Maximal value of F is 1, so as minimal -1, so unbalance after approaching
some large value should go back to zero because mean value is just this point.
b. Not stable leptokurtic distributions can be understood as generated by
temporal correlations which are not visible in average. These temporal correlations will lead to extreme or zero values when correlate data are added, so
if xi > 0 then is more probable that xi+1 > 0. It generate a more frequent
appearance of x+
i which gives shorter correlation lag in real time. On the end
we will nished in higher gaussianity of right side distribution and positive
trend.
3) Gauss is mature form of all distributions with nite second moment (what
is always the case of stock market data). In other words gaussianity should
increase in time when there are no external forces in the market. The increase
of gaussianity can happen after more frequent appearance of not correlated
data. In our example more frequent positive returns means statistically positive trend.

Gaussianity evaluation

For calculations of gaussianity we use tted Pdf to histogram by the maximum

likelihood principle. The distribution function, which we would like to t to
the data, should be the one which can emulate power-law distributions as
well as Gaussian. We took the Student distribution which is well known as
a power-law distribution with convergence to Gaussian for innite power [2].
10

The Student function look as follows:

PSt (, x) =

(( + 1)/2)
(/2)(1 + x2 /)(+1)/2

(16)

where (z) = 0 tz1 et dt. Now we would like to t our Student distribution
to data. Let us note as xi clean time series which correspond to Student distribution Eq. (16) best tted to data, so we would like to minimize (
xi xi )2 .
In order to do so we use Maximum likelihood principle. We use equation on
the conditional probability which we would like to maximize:
P (xi |
xi ) =

P (
xi and xi )
= P (xi ),
P (
xi )

(17)

where xi and xi are independent from each other. The lack of dependence
should be true because we are dealing with stochastic data. The method of
maximum likelihood try to nd maximum sum of logarithmus of probability
given by Eq. (17), so to nd we should resolve the following equality:

ln(PSt (, xi ))
=0

(18)

We solve this equation numerically with constraint that variance of the data
should follow 2 = /( 2) and mean equal zero. In the case of power
nding one should preliminary normalized time series by division with average
absolute value of x:
xi
xi =
i = 1, ..., N
(19)
|xi |
The objective value of all stocks one can present as
(

((x x)/ |x|)2

w(x) = ln 1 +
.

(20)

The gaussianity is calculated as the ratio of statistical Shannon entropy of

returns distribution and objective value w(x) as the consequence of objective
value minimization theory:
G=

P (x) ln P (x)
.
w(x)

(21)

The gaussinianity can be further expand in order to release entropy and ObV
evaluation from data. Complex calculations from data sometimes lead to large
11

uncertainty. We can say that entropy as well as objective value (ObV) is

calculated only from Pdf which are dependent from power exponent. Next
we will show the approximation which relate gaussianity only from power
exponent calculated from Student distribution. In the case of functional
w(x) S(w(x)) +

P (x)dx = min,

(22)

we get probability distribution as follows

(

1
P (x) = exp 1 + + w(x)

(23)

Parameter can be assigned from Pdf normalization and it is:

(

1
1 = ln exp
w(x) dx

(24)

The entropy S in further calculations comes as:

S=

1
1
P (x) ln P (x)dx = w(x) ln exp
w(x)

(25)

The parameter of our aim gaussianity implicate as the formula:

1 ln exp (w(x)/) dx
G=

w(x)

(26)

In the case when we t to data Student distribution we have the following

ObV:
(

w(x) = 0.5 ( + 1) ln 1 + x2 / ,

(27)

where variance of x is one. In the simply approximation w(x) can be replaced

as follows
w(x) = ( + 1) ln (1 + x/)

(28)

Next we put Eq. (28) to the formula on gaussianity Eq. (26) we get
G = ln ( ( + 2))

1
,
+1

(29)

12

where we put for parameter = 1. The forecasting procedure needs unbalance

of left and right side of Pdf so it is
F = G G+ = ln

+
( + 2)
+
+ (+ + 2) ( + 1)(+ + 1)

(30)

Investment strategy using only objective value

In order to omit entropy of Pdf calculations we can apply second method of

forecasting which uses only objective value to look for the leptokurtic distributions. The same as in previous section the objective value can be presented
very similar as in Eq. 20
)

((x x)/SD)2
w(x) = ln 1 +
,

(31)

where SD is the standard deviation of x. We divide w(x) for left side w (x)
and right side w+ (x) of the value of x with the respect of x. The inverse
function calculated from normalized objective value is as follows
x = w
1 (w) =

(ew/w 1).

(32)

Then one can create the normalized average value of left side R and right
side R+ of x i.e.

R/+ = SD w

w (x)
/+

w/+ (x)

(33)

The interpretation of R = R+ R as the forecasting is that it is normalized

mean return with respect of normalization using the objective value. If R
is larger than zero then statistically on the right hand side of Pdf of x the
distribution is more leptokurtic, so previous explanation in section 4 is valid
also to forecasting value R.

Numerical results for stock market data

We calculate the objective value for stocks from New York Stock Exchange
(NYSE), Warsaw Stock Exchange (WSE) and FOREX market. Results of
calculated of dierent stocks from New York Stock Exchange in years 1999
13

and 2000 are presented in Table 2. In Figs 2,3 there are plotted the Pdfs with
tted t-Student distribution for Ford and Bank of America respectively.
We use the forecasting described in section 6 in investment strategy (IS1) as
follows. In order to create create the portfolio with a help of R we do so that

pi = R/ j |pj |, where pi is the amount of money invested in stock i (negative

values are possible). In our approach we have to nd optimal size of window
Nwin taken in calculation of objective value. Our optimization method is quite
straightforward and it resembles a genetic algorithm. During the optimization
process we change the selection probability for actual values of optimized
parameters that is we increase the probability if the prot from portfolio is
positive and we decrease in the opposite case. The optimization process is
terminated when we reach a satisfactory mean value of a yearly prot from
past data (here it is 30%).
We begin our algorithm by generating randomly chosen stocks in the initial
portfolio. Then we randomly select a starting moment for our virtual investment. The next step is to optimize the parameters Nwin and risk value r:

pi = r R/ j |pj /r| using available data from the period prior to the selected starting point. Finally we invest in the portfolio but only if risk value
r is positive. The procedure was repeated over 1000 times and at the end we
calculated an average prot i.e. the eciency of the method. At Fig. 4 we
show a distribution of returns for our portfolio at Warsaw Stock Exchange.
We have calculated recommendations for windows 27 65 and 17 41 days
long for NYSE and WSE respectively. In the case of NYSE it was the period
October 1999 - December 2000 and for WSE on the period of July 2002 December 2003 (see Figs 5,6,7). For NYSE The annual return received in such
a way after commissions subtracting is around 15.2% for 11 stocks (7.5% for
60 stocks) and for WSE it was 14.6% (the commission level has been set to
0.25%). To omit articially large price changes that can be caused by such
eects as stock splitting, extreme returns larger than 0.4 have been rejected.
Looking at the results of investment strategy one can conclude that in the
case of hossa (WSE) and bessa (NYSE) we generate positive high prot.
The second forecasting procedure (IS2) which use the gaussianity (see sec. 4)
we applied to the FOREX market. FOREX market is global market of currency exchange, commodities as well as derivatives. In the FOREX there is
possibility to sell or buy given currency rate, e.g. EUR to USD (euro to American dollar). The common amount of currency handled on the FOREX is lot
(100000 of basing currency). The minimal value of lot in order to open position is 0.1. Nevertheless if investor open the position on one lot of currency
it should possess on the account only 1% from one lot so 1000. The cost of
opening the position is spread, so dierence between ask and bit oer. Another
cost of the investor is swap, which is constant cost subtracted at closing of
each day. Here swap is the dierence between costs of credit in one and second
14

currency. Swap can be positive but mostly is negative in order to prevent riskless opportunities of making the prot. In this strategy IS2 we use parameter
F in Eq. (15) to forecast future prices movement as follows:
RIS2 (Nf ) =

Nf

Fi ,

(34)

i=1

2
where Fi is F calculated on window of the size Nwin
shifted i data back. The
2
parameter Nwin is equal 20000 minutes. Strategy consist of two parameters
RIS2 (Nf ). First one is slow changing parameter with Nf = 5000 minutes and
second fast changing with Nf = 90 minutes. In our calculations we use 15
minuts high, low and close price that is time series is: PiIS2 = Pihigh + Pilow +
2 Piclose . Time series is calculated as in Eq. (12). Investment strategy tells
to buy if RIS2 (5000) > RIS2 (90) and RIS2 (5000) > 0.2. In the opposite one
should sell that is RIS2 (5000) < RIS2 (90) and RIS2 (5000) < 0.2. After the
opening in 2-6 weeks the position should be close. The trigger to close buy
position is when RIS2 (5000) > RIS2 (90) and RIS2 (5000) < 0. In the case of
sell position it is respectively: RIS2 (5000) < RIS2 (90) and RIS2 (5000) > 0. In
Table 3 are presented prots of investment procedure IS2 using smallest value
of investment 0.1 lot. The average prot from past investments are positive
about 152 USD in two and half month (smallest deposit is about 100 USD,
- one per cent of 0.1 lot), what is the experimental proof of our theory used
in method IS2. One can see on the above that prot is high but it should be
tted to the risk aversion of the investor, because the price movements are
sometimes against the method so it appears the risk of shortage of money on
the account. Even if further method wins but it can be so that without us.
Each investor should answer the question how much money it can invest to
bear the risk which is inherit property of each stochastic system forecasting.

We placed the source les of method IS2 written in MetaQuotas Language 4

on the site http://www.urbanowicz.org.pl as well as on-line trades using this
method and details of testing.

Conclusions

We would like to emphasize that correctness of the theory lays on the temporal correlations which are not visible in the average. These correlations lead
to leptokurtic probability density distribution. With a help of our parameters
we see these correlations in the unbalanced left and right side of the returns
distribution. Such a correlation is more visible in FOREX when we use 15
minutes lags between the data. The theory based on the objective value help
us to more appropriate compute the unbalance in gaussianity, what is most
15

Fig. 2. Plot of the histogram of Ford stock counted in NYSE in years 1999 and 2000
and corresponding Student distributions with = 3.63, = 3.02 and + = 4.11.

important in our methods of stochastic systems forecasting. In the case of both

investment methods we gain, so we conclude that the theory is correct. In the
stock market data the possibility of cumulating of temporal correlations is
plausible and can be explained with some sociological knowledge of collective
behavior of investors. People when investing are incorporated in whole nancial world of information transfer, so temporal correlation can appear and we
reveal it with our parameters. Statistical forecasting can be possible of stock
market behavior, because it is not purely random system.

References

[1] J.Voit, The Statistical Mechanics of Financial Markets, (Springer-Verlag 2001).

[2] J.P. Bouchaud, M. Potters, Theory of financial risks - from statistical physics
to risk management, (Cambridge University Press 2000).
[3] R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics. Correlations
and Complexity in Finance, (Cambridge University Press 2000).
[4] E.E. Peters, Chaos and Order in the Capital Markets. A new view of cycle,
Price, and Market Volatility, (John Wiley & Sons 1997).

16

Fig. 3. Plot of the histogram of Bank of America stock counted in NYSE in years
1999 and 2000 and corresponding Student distributions with = 3.28, = 2.98
and + = 4.08.
Table 2
Results of calculations of , and + parameter occurring in Student distribution
for 11 stocks counted on NYSE.
Stock

+
Apple

2.94

2.85

3.06

Bank of America

3.28

2.98

4.08

Boeing

3.8

4.07

3.24

Cisco

3.14

3.16

3.11

Compaq

3.05

2.67

3.99

Ford

3.63

3.02

4.11

General Electrics

3.52

3.76

3.34

General Motors

3.93

3.35

3.96

IBM

3.08

3.2

2.99

McDonalds

3.83

3.12

4.35

Texas Instruments

3.56

3.94

3.25

17

Fig. 4. Histogram of returns received by investment strategy IS1. The mean return
equals to 0.415% while the histogram dispersion is about 2.4%.

i K. Urbanowicz, European Physical Journal B 20,

[5] J.A. Holyst, M. Zebrowska
531-535 (2001).
[6] K. Urbanowicz and J.A. Holyst, Noise estimation by the use of neighboring
distance in Takens space and its application to the stock market data,
Proceedings of the Conference Complexity in science and society, International
Journal of Bifurcation and Chaos, arXiv:cond-mat/0412098;
[7] K. Urbanowicz and J.A. Holyst, Physica A 344, 284-288 (2004).
[8] K. Urbanowicz and J.A. Holyst, Application of noise level estimation for
portfolio optimization, Proceedings of the Third Nikkei Conference, Springer
(2005).
[9] C.E. Shannon, Bell System Technical Journal 27, 379-423 (1948).
[10] R.P. Feynman, Statistical Mechanics:A set of lectures, (Addison-Wesley 1993).
[11] http://www.smartquant.com/references.php
[12] M. Britten-Jones and S. M. Schafer, Non-linear Value-At-Risk, Europ. Fin.
Rev. 2: 161-187, 1999.
[13] M. Gili and E. Kellezi, A global optimization heuristic for portfolio choice with
VaR and Expected shortfall.
[14] J.H. Cochrane, New facts in nance, Economic Perspectives.

18

Fig. 5. The aggregated return for our investment strategy IS1 applied for 19 stocks
from the Warsaw Stock Exchange. The return corresponds to the mean annual
return 18% while the simple average return of 19 stocks was about 5.3% at the
same time period.
[15] V. Tola, F. Lillo, M. Gallagati, and R.N. Mantegna, Cluster analysis for
portfolio optimization, (2006)
[16] Krzysztof Urbanowicz, The principle of objective value minimization applied
to portfolio selection (2006).
[17] D. Polkow, T. Gebbie, How many bets are there?, arXiv:0601166 (2006).
[18] G.C. Lim, T.H. McCurdy,

19

Fig. 6. The aggregated return for our investment strategy IS1 applied for 11 stocks
from the New York Stock Exchange. The return corresponds to the mean annual
return 18.7% while the simple average method of all 11 stocks was about 45% at
the same time period.

20

Fig. 7. The aggregated return for our investment strategy IS1 applied for 60 stocks
from the New York Stock Exchange. The return corresponds to the mean annual
return 11% while the simple average return of all 60 stocks was about 10% at the
same time period.

21

Table 3
Results of testing the investment method IS2 on dierent currency rates in the
FOREX market from 2.06.2006 to 18.08.2006.
Currency rate size in lots No of investments Prot in USD
EURUSD

0.1

399.21

GBPUSD

0.1

677.27

USDCHF

0.1

86.31

CHFJPY

0.1

53.57

GBPCHF

0.1

-333.94

USDPLN

0.1

191.20

NZDUSD

0.1

173.14

GBPNZD

0.1

178.57

USDCAD

0.1

391.60

EURJPY

0.1

229.12

AUDJPY

0.1

393.70

EURCAD

0.1

176.61

AUDJPY

0.1

115.94

AUDNZD

0.1

127.22

AUDUSD

0.1

60.93

EURGBP

0.1

-74.03

EURCHF

0.1

-134.13

USDJPY

0.1

40.26

22