Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування
The novel theory of investment portfolio optimization under uncertainty is presented based on fuzzy set theory and efficient forecasting methods. The direct problem of fuzzy portfolio optimization and dual problem are considered. In the direct problem structure of a portfolio is determined which pro...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
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| author | Zaychenko, Yuri Sydoruk, Inna |
| author_facet | Zaychenko, Yuri Sydoruk, Inna |
| author_institution_txt_mv | [
{
"author": "Yuri Zaychenko",
"institution": null
},
{
"author": "Inna Sydoruk",
"institution": null
}
] |
| author_sort | Zaychenko, Yuri |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2016-07-25T14:59:53Z |
| description | The novel theory of investment portfolio optimization under uncertainty is presented based on fuzzy set theory and efficient forecasting methods. The direct problem of fuzzy portfolio optimization and dual problem are considered. In the direct problem structure of a portfolio is determined which provides the maximum profitableness at the given risk level. In dual problem the portfolio structure is determined which provides the minimum risk level at the set level of critical profitableness. For estimation of stocks profitableness in future moment the application of forecasting method- Fuzzy Group Method of Data Handling (FGMDH) is suggested. This method enables to construct fuzzy forecasting models by experimental data almost automatically. The experimental investigations of the suggested theory were carried out and comparison with classical portfolio model was performed. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2016.1.09 |
| first_indexed | 2025-07-17T10:20:07Z |
| format | Article |
| fulltext |
© Yuri Zaychenko, Inna Sydoruk, 2015
Системні дослідження та інформаційні технології, 2016, № 1 85
TIДC
МЕТОДИ ОПТИМІЗАЦІЇ, ОПТИМАЛЬНЕ
УПРАВЛІННЯ І ТЕОРІЯ ІГОР
УДК 518.9
DOI: 10.20535/SRIT.2308-8893.2016.1.09
PROBLEM OF FUZZY PORTFOLIO OPTIMIZATION AND ITS
SOLUTION WITH APPLICATION OF FORECASTING METHODS
YURI ZAYCHENKO, INNA SYDORUK
The novel theory of investment portfolio optimization under uncertainty is presented
based on fuzzy set theory and efficient forecasting methods. The direct problem of
fuzzy portfolio optimization and dual problem are considered. In the direct problem
structure of a portfolio is determined which provides the maximum profitableness at
the given risk level. In dual problem the portfolio structure is determined which
provides the minimum risk level at the set level of critical profitableness. For
estimation of stocks profitableness in future moment the application of forecasting
method- Fuzzy Group Method of Data Handling (FGMDH) is suggested. This
method enables to construct fuzzy forecasting models by experimental data almost
automatically. The experimental investigations of the suggested theory were carried
out and comparison with classical portfolio model was performed.
INTRODUCTION
Historically, the first and the most common way to take account of uncertainty is
the use of probability theory. The beginning of modern investment theory was in
the article H. Markowitz, "Portfolio Selection", which was released in 1952. In
this article mathematical model of optimal portfolio of securities was first pro-
posed. Methods of constructing such portfolios under certain conditions are based
on theoretical and probabilistic formalization of the concept of profitability and
risk. For many years the classical theory of Markowitz was the main theoretical
tool for optimal investment portfolio construction, after which most of the novel
theories were only modifications of the basic theory. However, the global market
crisis of recent years has shown that the existing theory of investment portfolio
optimization and forecasting stock indices exhausted itself and a revision of the
basic theory of portfolio management is strongly needed.
New approach in the problem of investment portfolio construction under un-
certainty is connected with fuzzy sets theory. Fuzzy sets theory was created about
half a century ago in the fundamental work of Lotfi Zadeh [1]. By using fuzzy
numbers in the forecast parameters decision- making person was not required to
form probability estimates.
The application of fuzzy sets technique enabled to create a novel theory of
fuzzy portfolio optimization under uncertainty and risk deprived of drawbacks of
classical portfolio theory by Markovitz.
Yuri Zaychenko, Inna Sydoruk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 1 86
The main source of uncertainty is changing stock prices of securities at the
stock market as the decision on portfolio is based on current stock prices while
the implementation of portfolio is performed in future and portfolio profitableness
depends on future prices which are unknown at the moment of decision making.
Therefore in order to raise the reliability of decision concerning portfolio and cut
possible risk it’ s needed to forecast future prices of stocks. For this the applica-
tion of inductive modeling method, so-called Fuzzy Froup Method of Data Han-
dling (FGMDH) seems to be very perspective.
The goals of this work are to review the main results in fuzzy portfolio opti-
mization theory, to consider and analyze so-called direct and dual problem of
portfolio optimization, to estimate the application of FGMDH for stock prices
forecasting and to carry out experimental investigations for estimation of the effi-
ciency of the elaborated theory.
PROBLEM STATEMENT
Let us consider a share portfolio from N components and its expected behavior at
time interval ],0[ T . Each of a portfolio component Ni ,...,1= at the moment T is
characterized by it’s financial profitableness ir (evaluated at a point T as a rela-
tive increase in the price of the asset for this period) [2, 3]. The holder of a share
portfolio – the private investor, the investment company or mutual fund – oper-
ates the investments, being guided by certain reasons. On one hand, the investor
tries to maximize the profitableness. On the other hand, he fixes maximum per-
missible risk of an inefficiency of the investments.
Assume the capital of the investor be equal 1. The problem of share portfolio
optimization consists in a finding of a vector of share prices distribution in a port-
folio Nixx i ,1},{ == maximizing the income at the set risk level .
In process of practical application of Markovitz model its drawbacks were
detected:
The hypothesis about normality of profitableness distributions in practice
does not prove to be true.
Stationarity of price processes is not always confirmed in practice.
At last, the risk of stocks is considered as a dispersion i.e. a decrease in prof-
itableness of securities in relation to the expected value, and profitableness in-
crease in relation to an expected value are estimated in this model absolutely all
the same. While for the proprietor of securities these events are absolutely differ-
ent. These weaknesses of Markovitz theory determine necessity of essentially new
approach of definition of an optimum investment portfolio.
Let’s review the main principles and ideas of a fuzzy portfolio optimiza-
tion method.
The risk of a portfolio is not its volatility, but possibility that expected prof-
itableness of a portfolio will appear below some pre-established planned value.
Correlation of stock prices in a portfolio is not considered and not accounted.
Profitableness of each security is not random but a fuzzy number. Similarly,
restriction on extremely low level of profitableness can be both usual scalar and
fuzzy number of any kind.
Profitableness of a security on termination of ownership term is expected to
be equal r and lies in a settlement range.
Problem of fuzzy portfolio optimization and its solution with application of forecasting methods
Системні дослідження та інформаційні технології, 2016, № 1 87
For i-th security denote:
ir — the expected profitableness of the i-th security;
ir1 — the lower border of profitableness of the i-th security;
ir2 — the upper border of profitableness of the i-th security.
),,( 21 iiii rrrr = — profitableness of i-th security is a triangular fuzzy number.
Then profitableness of a portfolio:
,;;
1
2max
11
1min ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
==== ∑∑∑
===
N
i
ii
N
i
ii
N
i
ii rxrrxrrxrr
where ix is the weight of the i-th security in a portfolio ( its ratio), and
.,1,0,1
1
Nixx i
N
i
i =≥=∑
=
(1)
Critical level of profitableness of a portfolio at the moment of T may be
fuzzy triangular number );;( *
2
**
1
* rrrr = or non-fuzzy number.
To define structure of a portfolio which will provide the maximum profit-
ableness at the set risk level, it is required to solve the following problem [6]:
,const ,max | }{}{ opt =β→= rxx (2)
where r is a portfolio profitableness , β is a desired risk, vector x satisfies (1).
MATHEMATICAL MODEL OF FUZZY PORTFOLIO OPTIMIZATION
PROBLEM
Let us consider a risk estimation of portfolio investments. On fig. 1 membership
function of r and criterion value *r are shown.
Point with ordinate 1α is the crossing point of two membership functions.
Let us choose any level of membership α and define corresponding intervals
Fig. 1. Membership functions of r and *r
μrμr*
α
α1
Yuri Zaychenko, Inna Sydoruk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 1 88
],[ 21 rr and ],[ *
2
*
1 rr . At *
21 rr > , intervals are not crossed, the risk and inefficien-
cies level equal to zero. Level 1α is upper border of risk zone. At 10 α≤α≤ in-
tervals are crossed.
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎨
⎧
≥−−
<≤<
−
−−−
><−
−+−
≥≥>
−
≥
=α
, if ),)((
, ; if ,
2
)())((
, , if ),(
2
)()(
, ,r if ,
2
)(
, if ,0
*
1212
*
1
*
2
*
222
*
11
2*
12
12
*
1
*
2
*
22
*
11
*
1
*
2
1
*
21
*
1
*
22
*
11
*
2
2
1
*
2
*
21
rrrrrr
rrrrrrrrrrr
rrrrrrrrrr
rrrrrr
rr
S
(3)
where αS are shaded areas of the phase space.
Since all realizations ),( *rr at set membership level )(αϕ are equally possible,
so the degree of inefficiencies risk )(αϕ is geometrical probability of event to drop
into any point ),( *rr in the zone of inefficient distribution of the capital [5]:
)()(
)(
12
*
1
*
2 rrrr
S
−−
=αϕ α .
Then total value of risk level of portfolio inefficiency is equal to:
.)(
1
0
ααϕ=β ∫
α
d
When the criterion of efficiency is defined as non-fuzzy level *r limiting
transition at **
1
*
2 rrr →→ we obtain:
⎪
⎪
⎩
⎪
⎪
⎨
⎧
>
∈α≤≤
−
−
<
=αϕ
. if ,1
,]1,0[ , if ,
)(
)(
, if ,0
)(
2
*
2
*
1
12
1
*
1
*
rr
rrr
rr
rr
rr
Fig. 2. Phase space ),( *rr
Problem of fuzzy portfolio optimization and its solution with application of forecasting methods
Системні дослідження та інформаційні технології, 2016, № 1 89
The most expected value risk degree of a portfolio is defined so [2]:
⎪
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎪
⎨
⎧
≥
<≤⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
α
α−
+−−
≤≤⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
α
α−
+
<
=β
, if 1,
, if ,)1(ln11)1(1
, if ,)1(ln11
, if ,0
max
*
max
*
1
1
1
*
min1
1
1
min
*
rr
rrraR
rrraR
rr
(4)
where
⎪
⎩
⎪
⎨
⎧
≥
<
−
−
=
,r if ,1
, if ,
max
*
max
*
minmax
min
*
r
rr
rr
rr
R (5)
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎨
⎧
≥
<<
−
−
=
<≤
−
−
<
=α
. if ,0
,~ if ,~
,~ if ,1
,~ if ,~
, if ,0
max
*
max
*
max
*
max
*
*
min
min
min
*
min
*
1
rr
rrr
rr
rr
rr
rrr
rr
rr
rr
Taking into account also that profitableness of a portfolio is equal to:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
==== ∑∑∑
===
N
i
ii
N
i
ii
N
i
ii rxrrxrrxrr
1
2max
11
1min ;; ,
where ),,( 21 iiii rrrr = is the profitableness of i-th security, we obtain the follow-
ing direct portfolio optimization problem [6]:
,max
1
→=∑
=
N
i
iirxr (6)
,const=β (7)
.,1,0,1
1
Nixx i
N
i
i =≥=∑
=
(8)
At a risk level variation β 3 cases are possible. Consider in detail each of
them.
• .0=β
From (3) it is evident, that this case is possible when .
1
1
* ∑
=
<
N
i
iirxr
Yuri Zaychenko, Inna Sydoruk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 1 90
Then we receive the following problem of linear programming:
,max
1
→=∑
=
N
i
iirxr (9)
,*
1
1 rrx
N
i
ii >∑
=
(10)
.,1,0,1
1
Nixx i
N
i
i =≥=∑
=
(11)
The solution of the problem (9)–(11) — vector Nixx i ,1},{ == deter-
mines a required structure of the optimum portfolio for the given risk level.
• 1=β .
From (3) it follows, that this case is possible when ∑
=
≥
N
i
iirxr
1
2
* . Then we
get the following problem
,max
1
→=∑
=
N
i
iirxr
.,1,0,1,
1
*
1
2 Nixxrrx i
N
i
i
N
i
ii =≥=≤ ∑∑
==
• 10 <β< .
From (3) it is evident, that this case is possible when ≤≤∑
=
*
1
1 rrx
N
i
ii
∑
=
≤
N
i
iirx
1
, or when ∑∑
==
<≤
N
i
ii
N
i
ii rxrrx
1
2
*
1
.
а) Let be ∑∑
==
≤≤
N
i
ii
N
i
ii rxrrx
1
*
1
1 .Then using (4)–(5) the problem (6)–(8) is re-
duced to the following nonlinear programming problem :
,max
1
→=∑
=
N
i
iirxr (12)
β=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
−
−
⎟
⎠
⎞
⎜
⎝
⎛
−+⎟
⎠
⎞
⎜
⎝
⎛
−
− ∑∑
∑
∑∑
∑∑
==
=
==
==
N
i
ii
N
i
ii
N
i
iiN
i
ii
N
i
iiN
i
ii
N
i
ii rxrx
rrx
rrxrxr
rxrx
1
1
1
*
1*
11
1
*
1
1
1
2
ln1 , (13)
,*
1
1 rrx
N
i
ii ≤∑
=
(14)
,*
1
rrx
N
i
ii >∑
=
(15)
Problem of fuzzy portfolio optimization and its solution with application of forecasting methods
Системні дослідження та інформаційні технології, 2016, № 1 91
Nixx i
N
i
i ,1,0,1
1
=≥=∑
=
. (16)
b) Let *
2
1 1
N N
i i i i
i i
x r r x r
= =
≤ <∑ ∑ . Then the problem (6)–(8) is reduced to the
following nonlinear programming problem:
,max
1
→=∑
=
N
i
iirxr (17)
×
−∑∑
==
N
i
ii
N
i
ii rxrx
1
1
1
2
1
,ln
1
1
1
2
1
*
1
*
1
1
* β=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
−
−
⎟
⎠
⎞
⎜
⎝
⎛
−−⎟
⎠
⎞
⎜
⎝
⎛
−×
∑∑
∑
∑∑
==
=
==
N
i
ii
N
i
ii
N
i
iiN
i
ii
N
i
ii
rxrx
rxr
rxrrxr (18)
,*
1
2 rrx
N
i
ii >∑
=
(19)
,~ *
1
rrx
N
i
ii ≤∑
=
(20)
.,1,0,1
1
Nixx i
N
i
i =≥=∑
=
(21)
The R-algorithm of minimization of not differentiated functions was sug-
gested to find the solution of problems (12)–(16) and (17)–(21).
Let both problems: (12)–(16) and (20)–(24) be solvable. Then to the struc-
ture of a required optimum portfolio will correspond such vector },{ ixx =
Ni ,1= — the solution one of the problems (12)–(16), (17)–(21) whose the crite-
rion function value will be greater.
IV. ANALYSIS AND COMPARISON OF EXPERIMENTAL RESULTS
OBTAINED BY MARKOVITZ AND FUZZY PORTFOLIO MODELS
For the comparative analysis of investigated methods of a share portfolio optimi-
zation real data on share prices of the companies RAO» EES (EERS2) and Gaz-
prom (GASP), were taken from February, 2000 till May, 2006 [6, 7].
In Markovitz model expected profitableness of a share is calculated as a
mean }{rMm = and risk of an asset is considered as a dispersion of the profit-
ableness value ( ) ][ 22 rmM −=σ i.e. level of variability of expected incomes.
In the fuzzy-sets model obtained from a situation at the share market we
conclude:
Yuri Zaychenko, Inna Sydoruk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 1 92
• the profitableness of EERS2 shares lies in a settlement corridor
,]9,3:0,1[− the most expected value of profitableness is 2,1 % ;
• the profitableness of GASP shares lies in a settlement corridor [–4,1: 5,7],
the most expected value of profitableness is 4,8 % .
Let critical profitableness of a portfolio be 3,5 % i.e. portfolio investments
which bring the income below 3,5 %, are considered as the inefficient.
Expected profitableness of the optimum portfolios received by Markovitz
model, is higher, than profitableness of optimum portfolios, received by the
fuzzy-set model because in Markovitz model the calculation of expected share
profitableness is based on indicators for the preceeding periods and the situation
in the share market at the moment of decision-making is not accountedby the in-
vestor. As profitableness of shares EERS2 and GASP in 2000- 2005 years was
much more higher than at the present moment, Markovitz model gives unfairly
high estimate.
In the fuzzy-set model the profitableness of each asset is a fuzzy number. Its
expected value is calculated not from statistical data, but by analysis of the market
at the moment of decision- making by the investor. Thus, in the considered case,
the expected profitableness of a portfolio is not too high.
The structures of an optimum portfolio which we get as a result of use of
both methods for the same risk levels are quite different too. To find out the rea-
son of this we consider following dependences obtained for both models (Fig. 3)
Dependence of expected profitableness on risk degree of the portfolio is presented
in Fig. 3.
The dependencies “optimal profitableness-risk” received by the above speci-
fied methods, are practically opposite. The reason of such result is the various
understanding of a portfolio risk.
In the fuzzy-set method the risk is recognized as a situation when expected
profitableness of a portfolio falls below the critical level, so with decrease of ex-
pected profitableness, risk of the real portfolio profitableness to be less thanthe
critical value, increases .
In Markovitz model the risk is considered as the degree of expected income
variability of a portfolio, in both cases of smaller and greater income that contra-
Fig. 3. Dependences of expected profitableness on degree of risk of the portfolio: a —
Fuzzy portfolio model; b — Markovitz model
x
y
a б
Problem of fuzzy portfolio optimization and its solution with application of forecasting methods
Системні дослідження та інформаційні технології, 2016, № 1 93
dicts common sense. The various understanding of portfolio risk level is also the
reason of difference of a portfolio structure, received by different methods.
From the point of view of the fuzzy-set approach, the greater is the portion
of GASP shares in a portfolio, the less is the risk of that efficiency of share in-
vestments will appear below the critical level which is in our case 3,5 %.
From the point of view of Markovitz model, average mean deviation from
average value for GASP shares is great enough, therefore with growth of this
share portion the risk of a portfolio increases. It leads to that portion of highly
profitable assets in the share portfolio received by Markovitz model, is unfairly
small.
According to Markovitz model, thanks to correlation between assets it is
possible to receive a portfolio with a risk level less than volatility of the least risk
security.
In this research after investing 96 % of the capital in EERS2 shares and 4 %
in GASP shares, the investor received portfolio with expected profitableness of
2,4 % and degree of risk 0,19. However investments with expected profitableness
of 2,4 % in our fuzzy-set model are considered as the inefficient. If to set critical
value of expected portfolio profitableness equal to 2,4% the risk of inefficient in-
vestments will decrease, too.
DUAL PORTFOLIO OPTIMIZATION PROBLEM
Now consider the portfolio optimization problem dual to the problem (6)–(8)
[2, 10]:
),(min xβ (22)
under conditions
,*
set
1
rrrxr
N
i
ii =≥=∑
=
(23)
.,1,0,1
1
Nixx i
N
i
i =≥=∑
=
(24)
In the paper [3], it was proved that the risk function )(xβ is convex where
,)(
)(
)(ln)()()( xD
xC
xBxBxAx ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=β
where
∑∑
==
−=−=
N
i
ii
N
i
ii rrxxBrxrxA
1
*
1
1
* ;)(;)(
.)(
1 1
1∑ ∑
= =
−=
N
i
N
i
iiii rxrxxC
So the dual portfolio problem (22)–(24) is convex programming problem.
Taking into account that constraints (23) are linear compose Lagrangian function:
Yuri Zaychenko, Inna Sydoruk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 1 94
.1))(),,(
11
*
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−μ+−λ+β=μλ ∑∑
==
N
i
i
N
i
ii xrxrxxL
The optimality conditions by Kuhn–Tucker are such [3]:
,,1,0)( Nir
x
x
x
L
i
ii
=≥μ+λ−
∂
β∂
=
∂
∂
,0*
1
≤+−=
λ∂
∂ ∑
=
rrxL N
i
ii
∑
=
=−=
μ∂
∂ N
i
ixL
1
,01
and conditions of complementary slackness
,,1,0,,0,0 *
1
NixrrxLx
x
L
i
N
i
iii
i
=≥λ=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−λ=
λ∂
∂
=
∂
∂ ∑
=
where 0≥λ and μ are indefinite Lagrange multipliers.
This problem may be solved by standard methods of convex programming,
for example method of feasible directions or method of penalty functions.
THE APPLICATION OF FGMDH FOR STOCK PRICES FORECASTING
AND EXPERIMENTAL INVESTIGATIONS
The profitableness of leading companies at NYSE in the period from 03.09.2013
to 17.01.2014 were used as the input data in experimental investigations. The
companies included: Canon Inc. (CAJ), McDonald's Corporation (MCD), Pep-
siCo, Inc (PEP), The Procter & Gamble Company (PG), SAP AG (SAP).
For forecasting we have used Fuzzy GMDH method [5, 2] with triangular
membership functions, linear partial descriptions, training sample of 70 and fore-
casting for 1 step. The next profitableness values on date 17.01.2014 were ob-
tained (table 1).
T a b l e 1 . The profitableness of shares on date 17.01.2014, %
Profitableness
Companies
Real value Low bound Forecasted
value
Upper
bound
MAPE
test
sample
MSE
test
sample
CAJ –1,270 –1,484 –1,246 –1,008 2,2068 0,0295
MCD –0,105 –0,347 –0,118 0,111 2,5943 0,0091
PEP 0,206 0,001 0,242 0,483 3,0179 0,0177
PG 0,162 0,041 0,170 0,299 1,6251 0,0197
SAP 0,843 0,675 0,867 1,059 2,3065 0,0164
Let the critical profitableness level be 0,7%. Varying the risk level we obtain
the following results at the end of 2-nd week (17.01.2014) for triangular MF. The
results are presented in the tables 2, 3 and the Fig. 4.
Problem of fuzzy portfolio optimization and its solution with application of forecasting methods
Системні дослідження та інформаційні технології, 2016, № 1 95
T a b l e 2 . Distribution of components of the optimal portfolio for triangular
MF with critical level %7,0*=r
CAJ MCD PEP PG SAP
0,05482 0,00196 0,0027 0,00234 0,93818
0,06145 0,00113 0,00606 0,0039 0,92746
0,0698 0,00577 0,00235 0,00219 0,91989
0,06871 0,00228 0,0057 0,00244 0,92087
0,07567 0,00569 0,00106 0,00094 0,91664
0,07553 0,00002 0,0029 0,00208 0,91947
0,06774 0,00121 0,006 0,00234 0,92271
0,0764 0,001 0,00612 0,00464 0,91184
0,09072 0,00849 0,00655 0,0039 0,89034
T a b l e 3 . Parameters of the optimal portfolio for triangular MF with critical
level %7,0*=r
Low bound Expected
profitableness Upper bound Risk level
0,55133 0,74591 0,94049 0,2
0,53462 0,72954 0,92446 0,25
0,51544 0,71084 0,90624 0,3
0,51894 0,71431 0,90968 0,35
0,5045 0,70018 0,89587 0,4
0,50877 0,70425 0,89973 0,45
0,522 0,71731 0,91262 0,5
0,50197 0,69752 0,89308 0,55
0,46358 0,66014 0,8567 0,6
As we can see on fig. 4, the dependence profitableness — risk has descend-
ing type, the greater risk the lesser is profitableness which is opposite to classical
probabilistic method by Markovitz . It may be explained so that at fuzzy approach
by risk is meant the situation when the expected profitableness happens to be less
than the given criteria level. When the expected profitableness decreases, the risk
grows.
Fig. 4. Dependence of the expected portfolio profitableness versus risk level
for triangular MF
Yuri Zaychenko, Inna Sydoruk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 1 96
The profitableness of the real portfolio is 0,7056%. This value falls in calcu-
lated corridor of profitableness for optimal portfolio [0,5346, 0,7295, 0,9245]
built with application of forecasting method FGMDH, indicating the high accu-
racy of the forecast.
Now consider the same portfolio using Gaussian MF (table 4, fig. 5).
T a b l e 4 . Parameters of the optimal portfolio for Gaussian MF with critical
level %7,0*=r
Low bound Expected
profitableness Upper bound Risk level
0,6833 0,87551 1,06772 0,2
0,66972 0,86178 1,05384 0,25
0,66955 0,86161 1,05368 0,3
0,66468 0,85682 1,04896 0,35
0,64944 0,8415 1,03356 0,4
0,65975 0,85185 1,04394 0,45
0,63439 0,8266 1,0188 0,5
0,63184 0,82389 1,01594 0,55
0,62452 0,81666 1,0088 0,6
The profitableness of the real portfolio is 0,8316%. This value falls in calcu-
lated corridor of profitableness for optimal portfolio [0,6833; 0,8756; 1,0677].
In the above results the optimal portfolio corresponds to the first row of
tables. As it can be seen from these data, the profitableness obtained using Gaus-
sian and bell-shaped MF is higher than the profitableness obtained using triangu-
lar MF.
The optimal portfolio obtained with different MF actually have the same
structure, the main portion falls on the company SAP shares, due to high rates of
return as compared with other companies.
Let’s consider the results obtained by solving the dual problem using trian-
gular MF. In this case, the investor sets the rate of return, and the problem is to
minimize the risk.
The optimal portfolio is presented in tables 5, 6 and Fig. 6.
Fig. 5. Dependence of expected portfolio profitableness versus risk for Gaussian MF
Problem of fuzzy portfolio optimization and its solution with application of forecasting methods
Системні дослідження та інформаційні технології, 2016, № 1 97
T a b l e 5 . Distribution of components of the optimal portfolio (dual task)
CAJ MCD PEP PG SAP
0,01627 0,02083 0,02226 0,02231 0,91833
0,01112 0,02085 0,02391 0,02383 0,92029
0,00333 0,01992 0,02517 0,02476 0,92682
0,0021 0,01579 0,02457 0,02344 0,9341
0,00004 0,00921 0,02423 0,02135 0,94517
0,00224 0,00144 0,01825 0,01095 0,96712
0,00044 0,00682 0,02508 0,02058 0,94708
0,0011 0,00917 0,02448 0,02039 0,94486
0,00294 0,01206 0,02533 0,02154 0,93813
T a b l e 6 . Parameters of the optimal portfolio (dual task)
Low bound Expected
profitableness Upper bound Risk level Critical rate
of return
0,58944 0,78264 0,97584 0,00025 0,6
0,59846 0,79141 0,98437 0,01468 0,65
0,61478 0,80735 0,99991 0,04973 0,7
0,6229 0,81531 1,00772 0,13347 0,75
0,63606 0,82822 1,02037 0,26399 0,8
0,64945 0,84181 1,03417 0,49937 0,85
0,63712 0,82933 1,02153 0,72631 0,86
0,63382 0,82612 1,01843 0,8333 0,87
0,62559 0,81805 1,01052 0,91214 0,88
From these results one can readily see that the curve “dependence risk —
given critical level of profitability” has a ascending character, because with the
growth of the critical value of profitability increases the probability that the ex-
pected return would be lower than a given critical value.
CONCLUSION
1. The problem of optimization of the investment portfolio under uncertainty
is considered in this paper. We suggested and explored the fuzzy-set approach for
solving the direct and dual portfolio optimization problems. In the direct problem
we used triangular, bell-shaped and Gaussian membership functions. The results
of solving the tasks were presented.
Fig. 6. Dependence of the risk level on a given critical return
Yuri Zaychenko, Inna Sydoruk
ISSN 1681–6048 System Research & Information Technologies, 2016, № 1 98
2. The optimal portfolios for the five assets at NYSE stock market were
constructed and analyzed.
3. The problem of stock prices forecasting for portfolio optimization was al-
so investigated. The fuzzy GMDH was proposed for its solution.. The fuzzy
GMDH allows to construct forecasting model using experimental data automati-
cally without participation of an expert. Besides, it may work under uncertainty
conditions with fuzzy input data or data given as intervals. The fuzzy GMDH was
applied for stocks profitableness forecasting at NYSE stock market in the problem
of fuzzy portfolio optimization. The application of fuzzy GMDH enabled to de-
crease risk of the wrong decisions and to raise the groundness of decisions con-
cerning portfolio content.
4. After analysis of the direct problem experiments it was detected that the
dependence “profitableness – risk” has descending type, the greater risk the
lesser is profitableness that is opposite to classical probabilistic methods.
5. The dependence “risk versus given critical level of profitability” has
ascending type, because as the value of the critical level of profitability increases
the probability that the expected return appears to be lower than a given critical
value also grows.
6. As the main result of this research the theory of fuzzy portfolio optimi-
zation under uncertainty was developed based on fuzzy set approach and fore-
casting method FGMDH.
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Received 12.11.2015
|
| id | journaliasakpiua-article-65695 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:20:07Z |
| publishDate | 2016 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/b2/74bd8f8df06d1f43cffdab9fd092fcb2.pdf |
| spelling | journaliasakpiua-article-656952016-07-25T14:59:53Z Problem of Fuzzy portfolio optimization and its solution with application of forecasting methods Проблема нечеткой портфельной оптимизации и ее решение с применением методов прогнозирования Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування Zaychenko, Yuri Sydoruk, Inna The novel theory of investment portfolio optimization under uncertainty is presented based on fuzzy set theory and efficient forecasting methods. The direct problem of fuzzy portfolio optimization and dual problem are considered. In the direct problem structure of a portfolio is determined which provides the maximum profitableness at the given risk level. In dual problem the portfolio structure is determined which provides the minimum risk level at the set level of critical profitableness. For estimation of stocks profitableness in future moment the application of forecasting method- Fuzzy Group Method of Data Handling (FGMDH) is suggested. This method enables to construct fuzzy forecasting models by experimental data almost automatically. The experimental investigations of the suggested theory were carried out and comparison with classical portfolio model was performed. Представлена новая теория портфельной оптимизации в условиях неопределенности, базирующая на применении теории нечетких множеств и эффективного метода прогнозирования. Рассмотрены прямая и двойственная задачи нечеткой портфельной оптимизации. В прямой задаче определяется структура оптимального портфеля ценных бумаг, обеспечивающая максимум ожидаемой доходности при ограничениях на риск, в двойственной задаче — структура портфеля, обеспечивающая минимум риска при установленном уровне критической доходности. Для определения доходности ценных бумаг предложен метод прогнозирования — нечеткий метод группового учета аргументов, позволяющий строить нечеткие прогнозные модели по экспериментальным данным автоматически при минимальном участии эксперта. Описаны экспериментальные исследования предложенной теории и проведено срав-нение с классической моделью портфельной оптимизации. Подано нову теорію портфельної оптимізації в умовах невизначеності, що ґрунтується на застосуванні теорії нечітких множин та ефективного методу прогнозування. Розглянуто пряму та двійкову задачі нечіткої портфельної оптимізації. У прямій задачі визначається структура оптимального портфеля цінних паперів, яка забезпечує максимум очікуваної дохідності при обмеженнях на ризик, у двійковій задачі – структура портфеля, який забезпечує мінімум ризику при обмеженнях на встановлений рівень критичної дохідності. Для визначення дохідності цінних паперів запропоновано метод прогнозування нечіткого методу врахування аргумента, що дозволить будувати нечіткі прогнозні моделі за експериментальними даними автоматично за незначної участі експерта. Описано експериментальні дослідження запропонованої теорії та проведено порівняння з класичною моделлю портфельної оптимізації. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2016-03-18 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/65695 10.20535/SRIT.2308-8893.2016.1.09 System research and information technologies; No. 1 (2016); 85-94 Системные исследования и информационные технологии; № 1 (2016); 85-94 Системні дослідження та інформаційні технології; № 1 (2016); 85-94 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/65695/60837 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Zaychenko, Yuri Sydoruk, Inna Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування |
| title | Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування |
| title_alt | Problem of Fuzzy portfolio optimization and its solution with application of forecasting methods Проблема нечеткой портфельной оптимизации и ее решение с применением методов прогнозирования |
| title_full | Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування |
| title_fullStr | Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування |
| title_full_unstemmed | Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування |
| title_short | Проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування |
| title_sort | проблема нечіткої портфельної оптимізації та її вирішення із застосуванням методів прогнозування |
| url | https://journal.iasa.kpi.ua/article/view/65695 |
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