Network Implementation of Investment Management
This paper presents simulation of selection elements of portfolio applying classical Markowitz’s portfolio analysis theory using parallel computational environment – cellular automata. Simulation and its results for parallel computational environment are presented. Рассмотрено моделирование выбора э...
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2007
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| Cite this: | Network Implementation of Investment Management / A. Katkow, A. Ulfik // Электронное моделирование. — 2007. — Т. 29, № 4. — С. 7-17. — Бібліогр.: 9 назв. — англ. |
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| citation_txt | Network Implementation of Investment Management / A. Katkow, A. Ulfik // Электронное моделирование. — 2007. — Т. 29, № 4. — С. 7-17. — Бібліогр.: 9 назв. — англ. |
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| description | This paper presents simulation of selection elements of portfolio applying classical Markowitz’s portfolio analysis theory using parallel computational environment – cellular automata. Simulation and its results for parallel computational environment are presented.
Рассмотрено моделирование выбора элементов портфеля на основе классической теории анализа портфеля заказов Марковитца с использованием параллельной вычислительной среды — клеточного автомата. Приведены результаты моделирования в параллельной вычислительной среде.
Розглянуто моделювання вибору елементів портфоліо на базі класичної теорії аналізу портфеля заказів Марковитця з використанням паралельного обчислювального середовища — клітинного автомата. Наведено результати моделювання у паралельному обчислювальному середовищі.
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| first_indexed | 2025-11-24T06:04:39Z |
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A. Katkow, A. Ulfik
Faculty of Management
Czestochowa University of Technology
(E-mail: ulfik@zim.pcz.pl)
Network Implementation of Investment Management
This paper presents simulation of selection elements of portfolio applying classical Markowitz’s
portfolio analysis theory using parallel computational environment – cellular automata. Simula-
tion and its results for parallel computational environment are presented.
Ðàññìîòðåíî ìîäåëèðîâàíèå âûáîðà ýëåìåíòîâ ïîðòôåëÿ íà îñíîâå êëàññè÷åñêîé òåîðèè
àíàëèçà ïîðòôåëÿ çàêàçîâ Ìàðêîâèòöà ñ èñïîëüçîâàíèåì ïàðàëëåëüíîé âû÷èñëèòåëüíîé
ñðåäû — êëåòî÷íîãî àâòîìàòà. Ïðèâåäåíû ðåçóëüòàòû ìîäåëèðîâàíèÿ â ïàðàëëåëüíîé
âû÷èñëèòåëüíîé ñðåäå.
K e y w o r d s: simulation, portfolio selection, cellular automata
The use of computers in portfolio analysis is necessary because of computational
complexity. Imitating occurrences in nature lets us have very good results. The
most popular usage of such processes are genetic algorithms and neural net-
works. Another one is a cellular automaton. Cellular automata were created in
fortieth of the last century to emulate processes occurring in nature. Soon after
this invention, cellular automata occurred very interesting and useful. First phys-
icists with big success used them to simulation of complicated issues. Today cel-
lular automata are being used also in mathematic, mechanic, economy, graphic
(for example creating textures and fractals), sociology (for example spreading
epidemics or crowd behavior), computer games and many other fields. In spite
of this, there are still some researches done to find new application for cellular
automata. One of them can be problem of selection of portfolio in stock market.
Portfolio analysis. H. Markowitz in 1952 had published his first paper
about portfolio selection [1]. It has started a real revolution for capital markets
and created completely new technique of making investment decisions today,
called portfolio analysis. Although this theory was showing how to choose the
best stocks to get the highest income with lowest level of risk, it was very hard to
apply it in practice because of computational complexity. After few years an-
other scientist W. Sharp has simplified this model and made portfolio analysis
able to apply in practice [2]. He has added to this theory two coefficients giving
investors a clear hint which stocks are giving better results than a general ten-
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 7
dency on considered market. As factor reflecting market tendency he has sug-
gested stock exchange index. On stock market in Poland investors most com-
monly use Warsaw Stock Exchange Index.
Stock market in Poland is currently strongly developed. Apart from stock,
investors can invest money in bonds, futures contracts and also other financial
instruments. The main aim for investors is to estimate correctly future value of
securities and then choose those with the biggest profit to buy as well as to find
the best moment to sell. Problem is securities with high level of expected income
are characterized by high level of risk. Investors can reduce this risk by investing
in more than one security.
According to theory of portfolio analysis we can predict future values of
stock basing on historical quotation. From this data a rate of return is counted, in-
terpreted as expected profit and standard deviation witch is measure of disper-
sion and is interpreted as risk connected with this expected profit. Investments
are usually interested in shares with large profit and low risk level. Portfolio
analysis studies how these values will change if we will invest in more than one
share. This theory also shows how to choose assets during constructing of port-
folio to diversifcate risk which means, that risk of portfolio would be lower than
risk of shares that are this portfolio’s components.
Rate of return and risk. In classical Markowitz’s model of portfolio selec-
tion the most important profiles of assets are rate of return and standard devia-
tion. These two values have to be calculated for all shares taken by investor into
his consideration. Positive value of return rate is interpreted as expected profit
and negative value — as expected loss. This quantity is count in period of time Rt
with following formula:
R
P P D
P
t
t t t
t
�
� �
�
�
1
1
,
where Pt — the price of securities in period t; Pt �1 — the price of securities in pe-
riod t –1; Dt — paid-out dividend in period t.
Return rate is appointed for every period of time t, there for it becomes func-
tion of time. The real value of income depends on many factors and investor can
not be sure that he will get calculated profit. This is the reason that we use ex-
pression: expected rate of return. Expected rate of return from securities R is
counted with following formula:
R
R
N
t
t
N
�
�
�
1
,
where Rt — the rate of return in period t; N — number of all analyzed rates of
return.
A. Katkow, A. Ulfik
8 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
Defined in this way level of profit or loss always is accompanying the in-
vestment risk. Risk in portfolio analysis is calculated using statistics. Standard
deviation S is interpreted as amount of risk. Below is shown formula to calculate
variation of stock’s rate of return
S
n
R Rt
t
n
2 2
1
1
1
�
�
�
�
�( ) ,
where S2
— the variation of rate of return; n — number of all analyzed rate of return.
Both values: expected income and risk can be shown on a graph. It is called
map of risk and income. This graph makes possible retrieval companies with
possibly lowest risk and largest income. In the centre of this graph investors put
«market tendency» often understood as those variables counted for stock index.
In Poland investors use Warsaw Stock Exchange Index called WIG. In Table 1
shown return rates and standard deviation of WIG calculated on base of different
history periods.
Fig. 1 represents map of risk and income for counted all companies’ quota-
tion from polish stock market in different time periods. Fig. 1, a represents re-
sults of computation with one year history, Fig. 1, b — with two years and the
last with five years of history of quotation. On all figures axis of rate of return —
R and standard deviation — S cross in coordinates of WIG market as grey dot.
Too long history can include information that don’t have any influence on
current situation [3, 4]. Because of that there is no point of calculating rate of re-
turn and standard deviation on very long quotation history. On Fig. 1, a—e
shown above maps of risk and income based on long history are less legible then
first one — based on one year history. As we consider longer quotation history,
we will find stocks with high level of rate of return interpreted as level of ex-
pected income and high level of standard deviation interpreted as risk. Those
stocks most probably will not repeat its history in future and there for its consi-
deration by investors is pointless. Fig. 1, a—e are showing that the shorter quo-
tations’ history is, the better prognosis we get.
Expected rate of return and standard deviation are two basic characteristics
of individual securities. When we consider more than one security there is one
Network Implementation of Investment Management
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 9
Time period, year Number of quotations R S
1 246 0,001395 0,013586
2 498 0,001532 0,011759
3 751 0,001172 0,010756
4 1003 0,000515 0,032951
5 1254 �0,000057 0,029594
Table 1. Rate of return and standard deviation of WIG depending on quantity
of historical quotation
more important value. It is the coefficient of correlation between two securities. It
defines connection of rates of return of two stocks. The coefficient of correlation of
rates of return of stock 1 and 2 �
12
can be calculated by following formula:
�
12
1 1 2 2
1
1 2
�
� �
�
�( ) ( )
,
R R R R
S S
i i
i
n
where R1i — rate of return in period I; R1 — expected rate of return from first
stock; S1 — standard deviation of rate of return first stock; R2i, R2, S2 — the same
for second stock; n — number of all analyzed rate of return.
A. Katkow, A. Ulfik
10 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
�0,01
�0,005
0
0,005
0,01
R
0,02 0,04 0,06 0,08 0,1 0,12 S
�0,005
0
0,005
0,01
0,015
0,02
0,025
R
0,05 0,1 0,15 0,2 0,25 0,3 S
�0,005
0,005
0,01
0,015
0,02
0,025
R
0
0,05 0,1 0,15 0,2 0,25 0,3 S
�0,005
0,005
0,01
0,015
0,02
0,025
0,03
0,035
0,04
R
0
0,2 0,4 0,6 0,8 1 1,2 S
�0,01
0,00
0,01
0,02
0,03
0,04
0,05
R
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 S
Fig. 1. Map of risk and income for companies’
quotation of all companies from polish stock
market from a — 01.04.06 to 01.04.07; b —
01.04.05 to 01.04.07; c — 01.04.04 to
01.04.07; d — 01.04.03 to 01.04.07; e —
01.04.02 to 01.04.07
Effective portfolios. Portfolio is effective when rate of return is higher than
for any other portfolio with the same risk; level of risk is lowest from portfolios
with the same rate of return.
Problem of selection of effective portfolio is not simple in spite of existence
of theoretical solution. With choosing portfolio investor has to first choose secu-
rities in which he wants to invest, and then establish how the invested capital will
divided among securities.
Two element portfolios. Investors are seeking the possibility of investing
capital in stocks with high rate of return. However these securities very often
characterize with high level of risk. For investor will be interesting such invest-
ments, in which with growth of rate of return, the risk will go down. The portfo-
lio analysis gives us such possibility.
Portfolio is set of stock, which we have or we want to buy. The rate of return
from two component portfolio is the sum of rates of return individual values
multiplied by their parts in investment
R x R x Rp A A B B� �
where xA — the part of share A in portfolio; RA — the rate of return of stock A; xB,
RB — the same for stock B; 0 1� �x A ; 0 1� �xB ; x xA B� �1.
Calculations of risk for two elements investment portfolio Sp
2
are more com-
plicated. Variation of two component portfolio is define as:
S x S x S x x S S rp A A B B A B A B AB
2 2 2 2 2
2� � �
where rAB — the coefficient of correlation between the rate of return of stock A
and B; Sp — standard deviation of two component portfolio.
Many elements portfolios. For more than two elements of portfolio, formu-
las for rate of return and standard variation are as follow:
R x Rp i i
i
N
�
�
� ,
1
S x S x x S S rp i i i
j
N
i
N
j i j ij
i
N
2 2 2
111
2� �
���
��� .
Markowitz’s portfolio analysis gives investors the best portfolio, depending
on their expectations. Disadvantage of this theory is its numerical complexity.
Even thorough we can use very fast multi threads computers and other types of
parallel machines, we still can not find the best solution using classical methods
for finding the best solution. First of all we have to gather many data and with
use of this data, calculate expected rate of return and standard deviation. We
have to count this characteristics for all considered stock companies. We also
Network Implementation of Investment Management
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 11
need to have coefficients of correlation between all considered companies.
There are
n n( )�1
2
coefficients of correlation, so for 50 stocks we have to count
1225 coefficients of correlation, for 100 stocks – 4950 coefficients and for 265
stocks (on 01.04.2007 Warsaw Stock Exchange had 265 companies) 34 980 co-
efficients of correlation.
When we will count all necessary characteristics we have to choose portfo-
lio. Number of possible portfolios is 2 1
n
� , where n is number of stocks. For 5
stocks we can have 31 possible portfolios, for 10 — already 1023. For 50 stocks
we will have 1 125 899 906 842 623 possible portfolio and with 265 stocks this
number will reach 5,9 �10
79
. Additionally every portfolio can have different per-
centage of every stock that it includes. To solve this problem we not only need
very fast computer but also special technique that will allow to find the most ef-
fective portfolio and solution proposed in this paper is cellular automata.
Cellular automata. Cellular automata (CA) are discrete models used
mainly in physic, mathematics and computability theory. They were created in
forties of last century by Stanis³aw Ulman and later developed by his colleague
also working at Los Alamos National Laboratory — John von Neumann. CA are
structures of the same cells putted into lattice. It is usually one, two or three-di-
mensional and involves large number cells (theoretically this number is infinite
but it is impossible to implement such model). Every cell has defined type and
starting value. It also has its algorithm called function of conversion. This func-
tion defines what will be future value depending on values of neighboring cells
in current time.
There are two types of neighborhoods in CA: von Neumann’s (Fig. 2, a) and
Moore’s (Fig. 2,b). Von Neumann’s neighborhood is made up of four cells adja-
cent vertically and horizontally. Eight cells surrounding given cell from right,
left, top, bottom and on a slant make up Moore’s neighborhood. In every itera-
tion current value of every cell is calculated on base of values of adjacent cells
from previous iteration [5].
A. Katkow, A. Ulfik
12 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
i�1, j i�1, j�1 i�1, j i�1, j+1
i, j�1 i, j i, j+1 i, j�1 i, j i, j+1
i+1, j i+1, j�1 i+1, j i+1, j+1
a b
Fig. 2. Two dimensional cellular automata with neighborhood von Neumann (a) and Moore (b)
The best known example of CA is game «Life» created by John Conway. In
this game every cell receives a starting state: it can be active or unactive. This
game simulates real environment where animals can be born or day when they
don’t have enough food. In this game there are settled rules of behavior of every
cell. If it is inactive and three other active cells surround it, it comes to life so be-
comes active. If cell is active and in its surrounding are two or three other active
cells, it stays active also. In every different case cell is inactive; it stays inactive
if such she was earlier or she days if earlier conditions are not fulfilled. So simple
rules leads to astounding solutions [6]. The final effect of CA can be systematize
as following:
CA achieves stable state, in which nothing does not it change (all cells stay
in determinate state);
state of CA changes cyclically after some quantity of iteration;
CA achieves chaotic state in which it is hard to find any order;
in CA we can find stable local configurations with long time of life.
The CA are already applied in many fields of science. In every issue it is
necessary to establish three basic parameters CA:
type of cells creating CA, which means to establish what kind of informa-
tion they have to contain;
starting value of every cell;
function of passage which is algorithm deciding what will be the state of
cells in current iteration on the base of values of neighboring cells in previ-
ous iteration.
It is purposeful to check how CA can simulate mechanisms occurring on the
stock market [7, 8] and how this simulation can be useful for potential capital
markets investors.
Simulation. To find the most effective portfolio we have to establish work
of CA and their three basic parameters: type of cells, starting value and function
of passage from one iteration to the other. In suggested solution all cells of cellu-
lar automata work collectively exchanging information between each other [9].
All cells of CA have one aim: choose the most effective portfolio so they
have to include information about composition of this portfolio and during itera-
tions improve it. To be able to choose portfolio, they also have to have access to
all input data. Every cell can be interpreted as artificial investor choosing the
most efficient portfolio. This artificial investor is working on his own choosing
the best result but at the same time he exchanges all information with other artifi-
cial investors which are in CA arranged in grid.
At the beginning of simulation all cells of CA get randomly chosen portfo-
lio. Then they calculate the basic characteristics of portfolio: rate of return and
standard deviation. Then every cell communicates with its neighbors with
Moore’s or von Neumann’s neighborhood (it is that moment when they work
Network Implementation of Investment Management
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 13
collectively). If in another cell there is portfolio with better characteristics then
cell gets this portfolio from its neighbor. If portfolios in adjoining cells have
worse characteristics, then cell stays with its original portfolio (in fallowing for-
mula this process is called Compare P[k]). In next step all cells work individu-
ally by looking for new, better portfolio. They get this portfolio as completely
new, randomly chosen one. If new portfolio is better it stays as current one if not,
then it is passed over (in fallowing formula this process is called Select P[k]).
This is repeated until all cells in CA get the same , most effective portfolio.
Implementation of simulation described above may be given by following
code.
Input:
R[i] – array of rates of return of n stocks ( for i = 1, ..., n)
S[i] – array of standard deviations of n stocks (for i = 1, ..., n)
�[i][j] – array of coefficients of correlation between socks i and j (for i and
j=1, ..., n)
Output:
P[k] – array of portfolio for all cells
A. Katkow, A. Ulfik
14 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0,0131
0,0132
0,0133
0,0134
0,0135
0,0136
R
0,104 0,1045 0,105 0,1055 0,106 0,1065 0,107 S
b
a
WIG
00
0,002
0,004
0,006
0,008
0,01
0,012
0,014
R
0,02 0,04 0,06 0,08 0,1 S
Fig. 3. Map of risk and income with all portfolios from simulation shown above (a) and their rela-
tion to WIG (b)
Start
Random(P[k])
Repeat
Compare P[k]
Select P[k]
until P[i] = P[j] for all i and j
Stop
On 01.04.07 on Warsaw Stock Exchange there were 265 stock companies
that have enough long history, to be able to calculate their characteristics. For simu-
lation were chosen all quotation with at least 50 days history. Some of companies
had just started and had too less quotation. If we would have counted expected rate
of return and standard deviation would not realize in future. All characteristics were
counted on base of quotation since 01.04.06 till 01.04.07.
Simulation were done in application written in Borland C++ Builder. In
those simulation were used two-dimensional CA with 10 000 cells (100 rows
and 100 columns in grid). There were formed two component portfolios. The
aim of this simulation was achieving stable state by CA, in which all cells have
the same portfolio. This simulations was repeated several times with necessary
Network Implementation of Investment Management
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 15
Number of
simulation
Composition of
portfolio (%)
R S � ij
1 EPL96 + APT4 0,01339289 0,10421372 0,0191
2 EPL96 + ZAP4 0,01317766 0,10404359 0,0505
3 EPL98 + EDR2 0,01355884 0,10623490 0,0176
4 EPL99 + EFK1 0,01363012 0,10725951 0,0216
5 EPL99 + CSS1 0,01353179 0,10618511 0,0762
6 EPL98 + GRL2 0,01347554 0,10617583 0,1121
7 EPL99 + INK1 0,01364862 0,10724388 0,1263
8 EPL97 + CAR3 0,01344098 0,10512677 �0,0683
9 EPL99 + GRJ1 0,01362272 0,10723304 �0,0124
10 EPL98 + TLX2 0,01349710 0,10620266 0,1048
11 EPL97 + AGO3 0,01333366 0,10514575 �0,0061
12 EPL95 + SKA5 0,01326135 0,10406501 0,0671
13 EPL97 + AMB3 0,01334683 0,10512602 0,0332
14 EPL96 + ASL4 0,01328401 0,10433455 0,0125
15 EPL99 + PCG1 0,01362945 0,10730811 0,0516
Table 2. Efficient portfolios chosen by cellular automata and their characteristics with coef-
ficient of correlation between stock
number of iteration. In cells portfolios changed when new randomly chosen
portfolio was more effective then present one.
In simulation as main aim was assumption to achieve portfolio with maxi-
mum income and risk as less as it is possible. In Table 2 we can see results of 15
simulations. All results were unique and often did not appear in next execution of
simulation even though input data were the same. Because aim was to have maxi-
mum income in all portfolios CA has chosen stock EPL which had very high level of
expected rate of return. As other component were chosen different companies with
different characteristics. In Table 2 are shown those characteristics for all companies
chosen by CA in simulation results shown in Table 3 and WIG. Fig. 3, a presents
map of risk and income with portfolios chosen by CA during simulation; numbers
near dots are numbers of portfolios from Table 2. Fig. 3, b shows its relation to
WIG; WIG is marked as grey dot.
As we can see on above figure all portfolios chosen by cellular automata had
better characteristics than characteristics of WIG. In Table 4 we can see how
prices of stock companies has changed in two months time. That prices of 4 com-
A. Katkow, A. Ulfik
16 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
Name R S
WIG 0,001395 0,013586
AGO 0,000312 0,027090
AMB 0,000750 0,022069
APT �0,003240 0,060476
ASL 0,002426 0,068524
CAR 0,003889 0,022262
CSS 0,003505 0,023912
EDR 0,004858 0,043015
EFK 0,003108 0,041203
EPL 0,013736 0,108290
GRJ 0,002367 0,020355
GRL 0,000693 0,020296
INK 0,004958 0,028927
PCG 0,003041 0,078690
SKA 0,001860 0,020722
TLX 0,001771 0,030702
ZAP �0,000230 0,016641
Table 3. Stock companies chosen by cel-
lular automata and their characteristics
Name
Price of stock Rate of return
in 2 months
period (%)on 01.04.07 on 31.05.07
WIG 57197,48 63064,92 10,26
AGO 46,82 45,75 �2,29
AMB 19,01 17,37 �8,63
APT 22,10 21,69 �1,86
ASL 5,15 5,15 0
CAR 62,25 120,50 93,57
CSS 45,00 43,00 �4,44
EDR 111,50 119,00 6,73
EFK 38,10 38,90 2,10
EPL 34,00 35,50 4,41
GRJ 67,25 71,35 6,10
GRL 39,10 41,50 6,14
INK 35,80 45,00 25,70
PCG 1,24 2,02 62,90
SKA 44,00 54,00 22,73
TLX 16,20 17,30 6,79
ZAP 65,70 103,60 57,69
Table 4. Stock companies chosen by cellular
automata
panies has drop down to lower level than it was month ago. But at the same time
we can see that all others has raided. Prices of 5 companies has raised much more
than quotation of WIG (bold font) and this increase was insignificant as for two
months time. Because of that cellular automata ca be useful tool for capital in-
vestors for choosing the best portfolio for their investments.
Ðîçãëÿíóòî ìîäåëþâàííÿ âèáîðó åëåìåíò³â ïîðòôîë³î íà áàç³ êëàñè÷íî¿ òåî𳿠àíàë³çó
ïîðòôåëÿ çàêàç³â Ìàðêîâèòöÿ ç âèêîðèñòàííÿì ïàðàëåëüíîãî îá÷èñëþâàëüíîãî ñåðåäî-
âèùà — êë³òèííîãî àâòîìàòà. Íàâåäåíî ðåçóëüòàòè ìîäåëþâàííÿ ó ïàðàëåëüíîìó îá÷èñ-
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Ïîñòóïèëà 14.06.07
Network Implementation of Investment Management
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 17
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| id | nasplib_isofts_kiev_ua-123456789-101780 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-11-24T06:04:39Z |
| publishDate | 2007 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Katkow, A. Ulfik, A. 2016-06-07T08:58:57Z 2016-06-07T08:58:57Z 2007 Network Implementation of Investment Management / A. Katkow, A. Ulfik // Электронное моделирование. — 2007. — Т. 29, № 4. — С. 7-17. — Бібліогр.: 9 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101780 This paper presents simulation of selection elements of portfolio applying classical Markowitz’s portfolio analysis theory using parallel computational environment – cellular automata. Simulation and its results for parallel computational environment are presented. Рассмотрено моделирование выбора элементов портфеля на основе классической теории анализа портфеля заказов Марковитца с использованием параллельной вычислительной среды — клеточного автомата. Приведены результаты моделирования в параллельной вычислительной среде. Розглянуто моделювання вибору елементів портфоліо на базі класичної теорії аналізу портфеля заказів Марковитця з використанням паралельного обчислювального середовища — клітинного автомата. Наведено результати моделювання у паралельному обчислювальному середовищі. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Network Implementation of Investment Management Сетевая реализация управления инвестициями Article published earlier |
| spellingShingle | Network Implementation of Investment Management Katkow, A. Ulfik, A. |
| title | Network Implementation of Investment Management |
| title_alt | Сетевая реализация управления инвестициями |
| title_full | Network Implementation of Investment Management |
| title_fullStr | Network Implementation of Investment Management |
| title_full_unstemmed | Network Implementation of Investment Management |
| title_short | Network Implementation of Investment Management |
| title_sort | network implementation of investment management |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101780 |
| work_keys_str_mv | AT katkowa networkimplementationofinvestmentmanagement AT ulfika networkimplementationofinvestmentmanagement AT katkowa setevaârealizaciâupravleniâinvesticiâmi AT ulfika setevaârealizaciâupravleniâinvesticiâmi |