Оптимальна диверсифікація портфеля акцій за ринкових обмежень
The problem of optimal portfolio diversification is considered. Based on mathematical models of the dynamics of the market value formation of a single share and an optimal stock portfolio, the structure of the optimal portfolio is determined. Such models are built in a class of ordinary differential...
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| author | Kulian, Victor R. Korobova, M. V. Yunkova, Olena O. |
| author_facet | Kulian, Victor R. Korobova, M. V. Yunkova, Olena O. |
| author_sort | Kulian, Victor R. |
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| description | The problem of optimal portfolio diversification is considered. Based on mathematical models of the dynamics of the market value formation of a single share and an optimal stock portfolio, the structure of the optimal portfolio is determined. Such models are built in a class of ordinary differential equations. One of the problems of optimal investing is optimizing the expected return of the stock portfolio for the desired level of risk. Another problem is the choice of the stock portfolio with the same expected return, but with a smaller risk. For this purpose, we use a set of acceptable and effective portfolios. This sequence of steps of the algorithm allows consistently solve two optimization problems. The problem of portfolio diversification consists of the problem of determining the moments of time and the necessity to perform such a diversification. In the article, we constructed an algorithm for determining these points of time, based on the solution of an optimal control problem. The application of this algorithm enables to select an optimal risk portfolio at a certain level of its expected profitability. It uses an efficient and acceptable set of investment portfolios. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2020.1.08 |
| first_indexed | 2025-07-17T10:26:45Z |
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V. Kulian, M. Korobova, O. Yunkova, 2020
90 ISSN 1681–6048 System Research & Information Technologies, 2020, № 1
UDC 519.925.51
DOI: 10.20535/SRIT.2308-8893.2020.1.08
OPTIMAL STOCK PORTFOLIO DIVERSIFICATION
UNDER MARKET CONSTRAINTS
V. KULIAN, M. KOROBOVA, O. YUNKOVA
Abstract. The problem of optimal portfolio diversification is considered. Based on
mathematical models of the dynamics of the market value formation of a single
share and an optimal stock portfolio, the structure of the optimal portfolio is deter-
mined. Such models are built in a class of ordinary differential equations. One of the
problems of optimal investing is optimizing the expected return of the stock portfo-
lio for the desired level of risk. Another problem is the choice of the stock portfolio
with the same expected return, but with a smaller risk. For this purpose, we use a set
of acceptable and effective portfolios. This sequence of steps of the algorithm allows
consistently solve two optimization problems. The problem of portfolio diversifica-
tion consists of the problem of determining the moments of time and the necessity to
perform such a diversification. In the article, we constructed an algorithm for deter-
mining these points of time, based on the solution of an optimal control problem.
The application of this algorithm enables to select an optimal risk portfolio at a cer-
tain level of its expected profitability. It uses an efficient and acceptable set of in-
vestment portfolios.
Keywords: optimal portfolio, diversification, admissible set, effective set.
THE GOAL OF THE WORK
In this paper an attempt to construct new fundamental approaches for solving
portfolio investment problems, based on the application of dynamic systems
mathematical modeling and the admissible and effective set of portfolio is made.
Main purpose of this paper is to apply mathematical modeling methods and
management theory to solve dynamic investment management problems, to ex-
plore asset and liability portfolio management strategies, and in general financial
instruments in a dynamic case. We develop analytical methods and computational
procedures for solving the problem of two-criterion optimization of a portfolio of
risky securities. The problems are presented in the formulation of H. Markowitz [1]
in the presence of quantitative and qualitative instrumental market constraints on
the structure of the portfolio. It’s important to say, that an alternative approach to
solving the problem of investment optimization and which has proved to be quite
effective is to use the apparatus of fuzzy sets [6,7].
MATHEMATICAL FORMALIZATION AND CONSTRUCTION OF MODELS
Dynamic Asset and Liability Management (ALM) [1] models have found the
most successful use in long-term financial planning, where the need for multiple
decision-making is determined by the essence of the process. Examples of ALM
work include implemented models for pension funds, insurance companies, in-
vestment companies, banks, university funds [1].
Optimal stock portfolio diversification under market constraints
Системні дослідження та інформаційні технології, 2020, № 1 91
The general scheme of active management of a stock portfolio, as a rule,
consists of the following steps:
1. Calculation of the price of an individual share on the basis of static de-
terministic models.
2. Analysis of static stock portfolio.
3. Modeling of asset dynamics.
4. Modeling the dynamics of the asset portfolio.
5. Integrated management and diversification of assets portfolio.
In order to move from the problem of static to dynamic consider Sharp's
market model
,ind SMr
where r — the market price of the stock; indSM — stock market index; —
some basic value for r . The equation describes the only general principles for
determining market value of a stock. From the above correlation it can be seen
that the market value of the stock is formed by the integral influence of the stock
market index and the random component. The ratio can be formally regarded as a
characteristic of the impact of the market index on the formation of one share
market value. Given that such processes occur over time, we can write down
.)()()( ind tSMttr
It would be a great simplification, when modeling such a complex process,
to consider function as the primary and the only one that generates value. The
analysis of the factors influencing the dynamics of the process gives grounds to
argue that such significant factors can also be attributed to the correlation between
securities and inflation. So let's write it in the form
,))(,),(),(()()( ind ttrtItSMtrttr iijii .,1, nji
Given the above, we write the right part in the form
,))()())()((()()(
1
2ind1 ttrtrtItSMtrttr i
n
j
ijiii
,)()())()((
)()(
1
2ind1 trtrtItSM
t
trttr
j
n
j
iji
ii
.,1 ni
Modern securities trading technologies and conditions of formation of mar-
ket value of shares on the stock market make it possible to take the next step
),()())()((
1
2ind1 trtrtItSM
dt
dr
j
n
j
iji
i
,,1 ni
where 21, — model parameters.
In the case ij , ni ,1 , is also a vector of model parameters that takes into
account correlations between stocks. Here n — the number of shares with which
it correlates i — share.
In the time interval ],[ 10 ttt equation that describes the return of the stock
portfolio pr , looks like
i
iip trtxtr ,)()()(
V. Kulian, M. Korobova, O. Yunkova
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 92
where ix — part of shares i — kind in the portfolio; ir — expected return on
shares of i — kind. Differentiating both parts by t , we will get
i
i
i
i
i
p
dt
tdr
tx
dt
tdx
tr
dt
dr
.
)(
)(
)(
)((
Let us consider that for ji there are relationships
i j
j
i j
iiii
i
i
i
ii tr
f
trtxtrtx
tr
f
trtx ,
)(
)()()()(
)(
)()(
i j
j
i j
iiii
i
i i
ii
txdt
tdx
trtxtrtx
dt
tdx
tx
trtx
.
)(
1)(
)()()()(
)(
)(
)()(
Function f is the right part of the mathematical model of stock price forma-
tion [3]
.))(),(()())()(( 32ind1 trtxtrtItSM
dt
dr
i
i
Then the dynamic equation of pricing of the stock portfolio will look like
)(2
)(
tr
dt
tdr
p
p
i j j
j
j
j
ii txdt
tdx
tr
f
trtx ,
)(
1)(
)(
)()( .ji
The last ratio, assuming the above, describes the dynamics of the behavior of
a portfolio of risky securities. His more detailed analysis points to two important
features that characterize the market value of the portfolio and that it depends on
the dynamics of both the expected return on the shares and the change in the
structure of the portfolio.
For solving and analyzing applied portfolio investment problems there are
wide range of approaches [2], [3]. A significant part of them involves the active
use of technical analysis methods, which make it possible to determine the market
value of the stock in the future. Such rules for constructing the forecast, due to the
well-developed mathematical formalizations and approaches and relatively not
complicated practical implementation, are actively developing and effectively
applied not only in the stock market. An application of fundamental analysis ana-
lytical methods allows for us to answer the question: why the market value of a
stock in the future will be just such? At present, due to the complexity of mathe-
matical models in the study of the market pricing processes of stock market as-
sets, the methods of fundamental analysis have not yet found effective develop-
ment and constructive application. Principles for analyzing processes based on the
development and application of mathematical modeling methods [2–6] are obvi-
ously the most promising and devote much attention to the research.
MATHEMATICAL FORMULATION OF THE PROBLEM
The mathematical problem of constructing the optimal dynamics of the portfolio
of shares in the most general formulation of H. Markowitz has the form [1]
Optimal stock portfolio diversification under market constraints
Системні дослідження та інформаційні технології, 2020, № 1 93
.
.,1,0
,1
,min
,max
T
T
T
nix
xI
Vxx
xr
i
x
x
(1)
Here T — is the sign of transposition.
The content of this two-criterion task (1) is to determine the optimal invest-
ment strategy, which involves maximizing the expected profitability and minimiz-
ing the risk at the same time. According to H. Markowitz, the criteria in the task
are controversial, that is, improving the outcome of one of them leads to deterio-
ration beyond others. In practice, this means that increasing the profitability of
a portfolio corresponds to an increase in its riskiness. There are different ap-
proaches to solving the problem (1), but they are more academic in nature and
difficult to apply to real investment in securities. A step that can bring the prob-
lem formulation (1) closer to the practical investment needs divide this two-
criterion problem into two one-criterion ones. The first of them involves risk op-
timization at a predetermined level of expected profitability at the chosen time
point, and the second is the optimization of the expected profitability for the in-
vestor-defined "optimal" portfolio risk level. In some cases, such mathematical
statements of nonlinear programming problems allow for analytical solutions [2],
but do not consider the essential features. They consist in the fact that at each step
the solution of the problem of diversification of the portfolio of portfolios must be
taken into account as budgetary and instrumental constraints. The constraints
make possible to analyze the availability of the required quantity and quality of
financial instruments on the market
),()( tXtxi .,1 ni (2)
Here )(tX — limited set of admissible portfolios. The mathematical formu-
lation of the problem of optimizing the risk of an investment portfolio at a time
point determined by the level of its expected profitability is
.
.],[,,1),()(
,],[,,1,0
,1)(
,min)()(
,)()()(
0
0
T
T
T
TttnitXtx
Tttnix
TxI
TVxTx
TrTxTr
i
i
x
P
(3)
On an example of investing in a stock, we consider the problem of optimiz-
ing portfolio risk for a given level of its expected return (3), while taking into ac-
count the constraints (2).
V. Kulian, M. Korobova, O. Yunkova
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 94
THE PROBLEM OF OPTIMAL PORTFOLIO DIVERSIFICATION
The optimal diversification procedure consists of successive steps, each of which
makes the decision on the best one, taking into account the limitations of the
structure of the stock portfolio.
The first step in solving the problem is to identify the times at which it is ap-
propriate to diversify the portfolio. For this purpose, we will use Algorithm 2 [3]
to solve the problem of determining the optimal moments of stock portfolio diver-
sification. The algorithm procedure allows to determine consecutive time points
that divide the study interval into separate intervals, each of them builds optimal
trajectories of acceptable portfolios in the presence of market constraints on the
portfolio structure. The mathematical problem of optimizing a stock portfolio
based on a programme trajectory and possible algorithms for solving it are de-
scribed in [3]. A graph showing an example of the interval breakdown and corre-
sponding trajectories at each time interval is given on fig. 1.
,pr
2
1
4
3
6
5
7
0
t0 t1 t2 t3 t4 t5 t
Fig. 1. Stock portfolio trajectories that are optimal over time
On the basis of modeling the dynamics of one share market value and the
portfolio of shares, the task of optimal management of different types of shares in
the portfolio is solving. The result of this task is trajectories 0–1, 2–3, 4–5, 6–7.
The next step is to build an optimal diversification procedure for the
investment portfolio at each individual interval. General task of H. Markowitz of
risky assets portfolio optimization (1) involves consideration of another criterion
– riskiness. Let's use this in the future by the set of admissible and effective
portfolios. At the same time, one of the main tasks of portfolio investment is the
problem of optimal diversification of such portfolio.
The risk optimization procedure for optimal portfolio profitability is to select
at each step the permissible portfolios that lie on the appropriate line. This line
connects the point, that corresponds to the optimal market value of the portfolio
with the point, belonging to the effective set. This line is parallel to the axis of
riskiness of portfolios. The feature of this choice of optimal portfolio is that in this
line, according to the definition, each of the portfolios corresponds to the same
expected profitability, but the risk decreases in the direction of the axis. This
property of the admissible set of investment portfolios allows from one side to
take into account the limitations (2) and on the other —– to determine the
portfolio of “optimal” expected returns with less risk. For 1t time moment
admissible and effective sets will looks like as on a fig. 2.
In fig. 1 points 1 and 2 are defined for the time moment 1t . Point 1 is the op-
timal portfolio for the moment, calculated on the basis of solving the problem of
optimal portfolio management structure using one of the algorithms described
Optimal stock portfolio diversification under market constraints
Системні дослідження та інформаційні технології, 2020, № 1 95
in [3]. In order to obtain the initial optimal portfolio value for the next time mo-
ment 2t , we apply a valid and effective set.
In line 1–2, fig. 2, depending on the formulation of the problem, choose the
optimal value of the initial value for the next interval portfolio value. It should be
noted that one of the tasks may involve determining the optimal value of the
expected return for a given level of risk, and the other – on the contrary. Thus, the
built-in algorithm of diversification of the optimal stock portfolio makes it possible
to take into account the instrumental and quantitative constraints in the construction
of the optimal stock portfolio that emerges on the market at any given time.
We will apply this procedure to build an optimal stock portfolio for the
following time intervals.
For 2t time moment admissible and effective sets will looks like as on a fig. 3.
Admissible
set
2V
2H
C
1H
*
D
1
Pr
*
Pr
E
Fig. 2. Solving the problem of optimizing the stock portfolio risk for 1t time moment
Fig. 3. Solving the problem of optimizing the stock portfolio risk for 2t time moment
Admissible
set
4V
4H
C
3
*
D
3
Pr
*
Pr
E
V. Kulian, M. Korobova, O. Yunkova
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 96
For 3t time moment admissible and effective sets will looks like as on a fig. 4.
Let's turn to the second task in the general statement of H. Markowitz about
optimization of risk optimal for the expected profitability of the portfolio of
shares. To do this, we use the sets of admissible and effective portfolios that cor-
respond to the selected set of shares [1, 2].
If a certain portfolio is at a point 5, that is, for which there is no risk of re-
ducing according to the above rule, then we define the “optimal portfolio” by
moving it from point 5 to point 6V that is an element of an effective portfolio of
portfolios. In fact, this means the definition of a portfolio of stocks with a higher
expected return. At the same time, such a procedure allows constructively to take
into account the existing limitations when diversifying the portfolio.
Another mathematical statement of the problem of optimizing the expected
return on the investment portfolio at a time point determined by the level of its
risk, is such
.
.],[,,1),()(x
,],[,,1,0
,1)(
,)()(
,max)()(
0i
0
TttnitXt
Tttnix
TxI
TVxTx
TxTr
i
T
T
x
T
The procedure for optimizing the expected return on a portfolio for a certain
level of its risk is to select a procedure for optimizing the expected return of a
portfolio for a certain level of its risk is to select at each step the admissible port-
folios that lie of the straight line 5–6 (fig. 4). The line connects the points 5 that
corresponds to the optimal expected return on the calculated portfolio and the
point 6 belonging to the effective set. This line is parallel to the axis of market
value. The feature of this choice of optimal portfolio is that in this line, according
Fig. 4. Solving the problem of optimizing the stock portfolio risk for 3t time moment
Admissible
set
6V
6H
C
5
*
D
Pr
*
Pr
5
Optimal stock portfolio diversification under market constraints
Системні дослідження та інформаційні технології, 2020, № 1 97
to the definition, each of the portfolios is responsible for the same risk, but the
market value increases.
This property of the admissible set of investment portfolios, as in the
previous case, allows one to take into account the restrictions (2), on the other
hand, to define a portfolio with “optimal” risk and higher expected returns.
As defined above, portfolio is at a point 5, that is, for which there is no
possibility to increase the expected yield, according to the above rule, then we
determine the “optimal portfolio” by moving it from point 5 to point 6H that is an
element of an effective portfolio of portfolios. In fact, this means reducing the risk
profile of the stock portfolio.
The above examples of admissible and effective sets are based on statistical
information on the market value of shares in the FMTS and UM markets. Such
information is concise and is provided to describe the computational capabilities
of the stock portfolio diversification algorithm.
CONCLUSION
In this study, new mathematical statements of optimization of stock portfolio
structure are presented and methods of their solution are developed. Mathematical
problems formulated on the basis of models of the dynamics of market value of
one share and portfolio of shares. That gives an opportunity to solve the problem
of optimal diversification of the portfolio of investments, taking into account of
quantitative and qualitative market restrictions on the structure of the portfolio.
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Received 13.02.2020
From the Editorial Board: the article corresponds completely to submitted manuscript.
|
| id | journaliasakpiua-article-209138 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:26:45Z |
| publishDate | 2020 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/58/2e36e6fea8719c622273df1d802f0a58.pdf |
| spelling | journaliasakpiua-article-2091382020-08-11T08:50:57Z Optimal stock portfolio diversification under market constraints Оптимальная диверсификация портфеля акций при рыночных ограничениях Оптимальна диверсифікація портфеля акцій за ринкових обмежень Kulian, Victor R. Korobova, M. V. Yunkova, Olena O. математична модель допустима множина ефективна множина диверсифікація портфеля акцій математическая модель допустимое множество эффективное множество диверсификация портфеля акций mathematical model acceptable set efficient set diversification of the investment portfolio The problem of optimal portfolio diversification is considered. Based on mathematical models of the dynamics of the market value formation of a single share and an optimal stock portfolio, the structure of the optimal portfolio is determined. Such models are built in a class of ordinary differential equations. One of the problems of optimal investing is optimizing the expected return of the stock portfolio for the desired level of risk. Another problem is the choice of the stock portfolio with the same expected return, but with a smaller risk. For this purpose, we use a set of acceptable and effective portfolios. This sequence of steps of the algorithm allows consistently solve two optimization problems. The problem of portfolio diversification consists of the problem of determining the moments of time and the necessity to perform such a diversification. In the article, we constructed an algorithm for determining these points of time, based on the solution of an optimal control problem. The application of this algorithm enables to select an optimal risk portfolio at a certain level of its expected profitability. It uses an efficient and acceptable set of investment portfolios. Посвящено построению новых и применению существующих методов математического моделирования при решении задачи оптимального инвестирования в рисковые ценные бумаги. Сформулированы новые постановки задач и разработаны методы траекторного моделирования динамики рыночной стоимости одной акции и портфеля акций. При решении задачи моделирования оптимальной траектории портфеля акций применены методы оптимального управления системой, параметрами управления в которой являются части акций разного вида в портфеле. Задача оптимального управления динамикой портфеля инвестиций сформулирована для критерия качества, который использует “программную траекторию”. Решена задача построения оптимального по прогнозируемой рыночной стоимости и/или риску портфеля акций. Рассмотрена задача о диверсификации построенного портфеля. Для ее решения применено допустимое и эффективное множества инвестиционных портфелей. Алгоритм решения задачи позволяет динамично учитывать инструментальные рыночные ограничения, которые формулируются при математической постановке задачи. Присвячено розробленню нових та застосуванню відомих методів математичного моделювання для розв’язання задачі оптимального інвестування у ризиковані цінні папери. Сформульовано нові постановки задач і побудовано методи траєкторного моделювання динаміки ринкової вартості однієї акції та портфеля акцій. Для розв’язання задачі моделювання оптимальної траєкторії портфеля акцій застосовано методи оптимального управління системою, у якій параметрами керування є частки акцій різних видів у портфелі. Задачу оптимального управління динамікою інвестиційного портфеля сформульовано для критерію, що використовує “програмну траєкторію”. Розв’язано задачу побудови оптимального за очікуваною ринковою вартістю та/або ризикованістю портфеля акцій. Розглянуто задачу про диверсифікацію побудованого портфеля інвестицій. Для її розв’язання застосовано допустиму та ефективну множину портфелів. Алгоритм розв’язання задачі дозволяє динамічно враховувати інструментальні ринкові обмеження, які задаються для математичного формулювання задачі. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2020-06-23 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/209138 10.20535/SRIT.2308-8893.2020.1.08 System research and information technologies; No. 1 (2020); 90-97 Системные исследования и информационные технологии; № 1 (2020); 90-97 Системні дослідження та інформаційні технології; № 1 (2020); 90-97 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/209138/209530 Copyright (c) 2021 System research and information technologies |
| spellingShingle | математична модель допустима множина ефективна множина диверсифікація портфеля акцій Kulian, Victor R. Korobova, M. V. Yunkova, Olena O. Оптимальна диверсифікація портфеля акцій за ринкових обмежень |
| title | Оптимальна диверсифікація портфеля акцій за ринкових обмежень |
| title_alt | Optimal stock portfolio diversification under market constraints Оптимальная диверсификация портфеля акций при рыночных ограничениях |
| title_full | Оптимальна диверсифікація портфеля акцій за ринкових обмежень |
| title_fullStr | Оптимальна диверсифікація портфеля акцій за ринкових обмежень |
| title_full_unstemmed | Оптимальна диверсифікація портфеля акцій за ринкових обмежень |
| title_short | Оптимальна диверсифікація портфеля акцій за ринкових обмежень |
| title_sort | оптимальна диверсифікація портфеля акцій за ринкових обмежень |
| topic | математична модель допустима множина ефективна множина диверсифікація портфеля акцій |
| topic_facet | математична модель допустима множина ефективна множина диверсифікація портфеля акцій математическая модель допустимое множество эффективное множество диверсификация портфеля акций mathematical model acceptable set efficient set diversification of the investment portfolio |
| url | https://journal.iasa.kpi.ua/article/view/209138 |
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