Optimization of the Fuzzy Investment Portfolio during the Time Period

The problem of portfolio optimization under uncertainty is considered. For its solution the application of fuzzy sets theory is suggested. Fuzzy portfolio optimization problem during the time period is stated, its model is provided, investigated, and algorithm of its solution presented. This problem includes two main criteria – portfolio profitableness and risk. A mathematical model of this problem is constructed and explored. For better estimation of stock profitableness, Fuzzy Group Method of Data Handling (FGMDH) is suggested. The experimental investigations of the suggested approach are carried out. The results with optimal portfolios based on forecasted profitableness are presented and its efficiency is evaluated.


I. INTRODUCTION
Portfolio is a purposefully formed set of investment assets (real or financial investments) owned by an individual or a legal person.The aim of this set is to implement the predeveloped strategy and to achieve the investment objectives.The main objective of portfolio investment is to create for a set of investment assets such investment conditions that are inaccessible from the position of a single asset and possible only when it is combined with the other.This includes achieving the optimal combination for investors of such investment characteristics as profitableness and risk level.
Portfolio analysis exists, perhaps, as long, as people think about making rational decisions connected with the use of limited resources.However, the occurrence moment of portfolio analysis started with a publication of pioneer work by Harry Markovitz (Markovitz's Portfolio Selection) in 1952.The model offered in that work, simple enough in essence, has allowed catching the basic features of the financial market, from the point of view of the investor, and has supplied the last with the tool for development of rational investment decisions.
The central problem in the Markovitz's theory is the portfolio choice, i. e., a set of operations.Thus, in the estimation, both separate operations and their portfolios are considered: profitableness and risk of operations and their portfolios.The risk thus receives a quantitative estimation.The account of mutual correlation dependences between profitableness of operations appears the essential moment in the theory.This account allows making effective diversification of portfolio, leading to essential decrease in risk of a portfolio in comparison with risk of the operations included in it.At last, the quantitative characteristic of the basic investment characteristics allows defining and solving a problem of a choice of an optimum portfolio in the form of a problem of quadratic optimization.
However, the worldwide market crises during the last 20 years have shown that existing theories of optimization of share portfolios and forecasting of share indices have exhausted themselves, and essential revision of share management methods is necessary.
Thus, in the light of obvious insufficiency of available scientific methods for management of financial assets, the development of a fundamentally new theory of management of the financial system functioning in the conditions of essential uncertainty is needed.The great assistance to this theory was rendered by the theory of the fuzzy sets, which have been developed about half a century ago in fundamental works of Lotfi Zadeh.
In previous studies, we have discovered the application of classical probabilistic method and fuzzy set theory.We have considered direct, dual and multi-criteria optimization portfolio problems.
The purpose of the present article is the research and experimental analysis of direct optimization problem during the time period.

II. PROBLEM STATEMENT
Let us consider a share portfolio from N components and its expected behavior at time interval [0,T].Each portfolio component is characterized by the financial profitableness i r , I = 1,N.
The holder of a share portfoliothe private investor, the investment company, mutual fundoperates the investments, being guided by certain reasons.On the one hand, the investor tries to maximize the profitableness.On the other hand, it fixes a maximum permissible risk of an inefficiency of the investments.We will assume the capital of the investor be equal to 1.The problem of optimization of a share portfolio consists in the finding of a vector of share price distribution of papers in a portfolio of the investor maximizing the income at the set risk level (obviously, that 1 1

Weaknesses of a Classical Markovitz's Model
In the process of practical application of Markovitz's model, its drawbacks were found out: Information Technology and Management Science ___________________________________________________________________________________________________________ 2014 / 17 67 1.The hypothesis about normality profitableness distributions in practice does not prove to be true.2. Stationarity of price processes also not always is in practice.3.At last, the risk of actives is considered a dispersion standard deviation of the prices of securities from the expected value, i.e., a decrease in profitableness of securities in relation to the expected value, and profitableness increase in relation to an average are estimated absolutely the same.Though for the proprietor of securities these events are absolutely different.
These weaknesses of Markovitz's theory define the necessity of the development of essentially new approach of definition of an optimum investment portfolio.

III. FUZZY SET PORTFOLIO MODEL
Main principles and idea of the method are as follows:  The risk of a portfolio is not its volatility, but possibility that expected profitableness of a portfolio will appear below some pre-established planned value. Correlation of assets in a portfolio is not considered and not accounted. Profitableness of each asset is not a random fuzzy number.
Similarly, restriction on an extremely low level of profitableness can be both usual scalar and fuzzy number of any kind.Therefore, optimization of a portfolio in such a statement may mean in that specific case the requirement to maximize expected profitableness of a portfolio at a point of time T at the fixed risk level of a portfolio. Profitableness of a security on termination of ownership term is expected to be equal to r and is in a settlement range.
Denote for the i-th security: , , Then profitableness of a portfolio: where i x is a portion of i-th asset in portfolio, and Critical level of profitableness of a portfolio at the moment of T is r * .

IV. MATHEMATICAL MODEL OF A FUZZY OPTIMIZATION PROBLEM
To define the structure of a portfolio, which will provide the maximum profitableness at the set risk level, it is required to solve the following problem (1) -( 6): where r is profitableness,  is a desired risk, vector's components x satisfy (2).
As shown in ( 1) -( 6), the most expected value risk degree of a portfolio is defined: The profitableness of a portfolio is shown in (1).

V. MATHEMATICAL MODEL OF PORTFOLIO OPTIMIZATION
DURING THE PERIOD In this case we must define the structure of the portfolio, which provides the maximum average return for a given level of risk.Thus, we calculate the profitableness from (3) as: rthe expected profitableness of i-th security in time unit t, Tthe number of time units.We should find an optimal portfolio from the following problem: At a risk level variation  3 cases are possible.We will consider in detail each of them.1.

 
From (4) it is evident that this case is possible when The found result of the problem decision ( 10) -( 12) vector is a required structure of an optimum portfolio for the given risk level. 2.

 
From (4) it follows that this case is possible when From (4) it is evident that this case is possible when 11 Then using ( 4) -( 7) problem ( 7) -( 9) is reduced to the following problem of nonlinear programming: Then the problem ( 7) -( 9) is reduced to the following problem of nonlinear programming: The R-algorithm of minimization of not differentiated functions is applied to the decision of problems ( 16) -( 20) and ( 21) -(25).Vector is the required portfolio structure.

VI. EXPERIMENTAL INVESTIGATIONS AND RESULT ANALYSIS
As the input data closing prices of leading companies at the stock exchange NYSE, the authors used: Canon Inc.By using the Fuzzy GMDH method, the portfolio optimization system stops to be dependent on the factor of expert subjectivity.Let us see the results of application of the suggested approach to determining an optimal invest portfolio to the date 17 January 2014.Let the critical profitableness level set by trader 0.7 %.Varying the risk level we obtain the following results for triangular MF presented in Tables III, IV and Fig. 1.As we can see in Fig. 1, the dependence of profitableness-risk has descending type; the greater risk, the lesser profitableness is opposite to classical probabilistic methods.It may be explained by the fact that in the 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.The calculated corridor of profitableness for optimal portfolio is [0.55133; 0.74591; 0.94049].The main portfolio portion in this case goes to company SAP that can be explained by the high level of its profitableness in comparison with other companies.Now consider an optimal portfolio during 6 weeks (see Tables V, VI and Fig. 2).The calculated corridor of profitableness for optimal portfolio is [0.49289; 0.69582; 0.94049].
The main portfolio portion in this case goes to company PG that can be explained by the high level of its average profitableness in comparison with other companies.

VII. CONCLUSION
In this paper, the research in the field of portfolio management has been carried out.Fuzzy set theory has been used as a tool for getting an optimal portfolio.s a result of research, the mathematical model based on the fuzzy-set approach for finding of structure of the optimal investment portfolio has been received.On the basis of the theory of fuzzy sets, the algorithm of optimization of a share portfolio has been developed.As a result of research, the following conclusions have been made: 1.The dependence of profitableness-risk has a descending type; the greater risk, the lesser profitableness is opposite to classical probabilistic methods.2. Portfolios during the time period and at the end of period have different structure and characteristics that can be explained by the different calculations of profitableness.
3. For improving the accuracy of the suggested fuzzy portfolio model, the fuzzy GMDH method has been applied to profitableness forecasting.The experimental investigations have proved its high efficiency.
i rexpected profitableness of i-th security; 1 i rthe lower border of profitableness of i-th security; 2 i rthe upper border of profitableness of i-th security.

TABLE IV PARAMETERS
OF OPTIMAL PORTFOLIO WITH CRITICAL LEVEL R * =0.7% Fig. 1.Dependence of expected portfolio profitableness on risk level for triangular MF.