mean–variance–skewness fuzzy portfolio selection model based on intuitionistic fuzzy optimization

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Procedia Engineering

Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 2062 – 2066 www.elsevier.com/locate/procedia

Advanced in Control Engineeringand Information Science

mean–variance–skewness fuzzy portfolio selection model based on intuitionistic fuzzy optimization Guohua Chen a*，Zhijun Luo, Xiaolian Liao, Xing Yu, Lian Yang Department of mathematics Hunan Institute of Humanities Science and Technology, Loudi 417000, China

Abstract mean–variance–skewness model for optimal portfolio selection in intuitionistic fuzzy environment is proposed, Firstly, membership and non-membership functions of object and constrain functions were defined. Secondly, intuitionistic fuzzy programming model was presented based on intuitionistic fuzzy "min-max" operator. Then, intuitionistic fuzzy programming was resolved by Matlab software.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011] Keywords: Mean–variance–skewness model; Fuzzy portfolio selection; Matlab software; Intuitionistic fuzzy optimization;

1. Introduction The mean–variance model originally introduced by Markowitz[1] plays an important and critical role in modern portfolio theory. Markowitz’s portfolio model is a bi-criteria optimization problem where a reasonable trade-off between return and risk is considered—minimizing risk for a given level of expected return, or equivalently, maximizing expected return for a given level of risk. Since Markowitz’s pioneering work[1] was published, With the continuous effort of various researchers, Markowitz’s seminal work has been widely extended. Extended models mainly include the mean semi-variance model [2], mean absolute deviation model[3–5], mean target model[6] and some frictional models.

* Corresponding author. Tel.: +86-15173813065; fax: +86-0738-8325585. E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.08.385

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Guohua Chen et al. Procedia Engineering 15 00 (2011) 2062 – 2066 Guohua Chen ,et/al/ Procedia Engineering (2011) 000–000

The higher moments are irrelevant to the investor’s decision. As a result, in some recent studies, such as [7–9], the concept of mean–variance trade-off has been extended to include the skewness of return in portfolio selection. In other words, a mean–variance–skewness trade-off model for portfolio selection has been generated. One problem with the mean–variance–skewness trade-off model for portfolio selection is that it is not easy to find a trade-off between the three objectives because this is a nonsmooth multi-objective optimization problem. Until now, many methods used to tackle this problem have been restricted to goal programming or linear programming techniques. Recently, Liu and Liu developed the credibility theory including the credibility measure, pessimistic value and expected value as fuzzy ranking methods[10]. Lin and Yin[11] discussed the fuzzy quadratic assignment problem with penalty, and formulated as expected value model, chance-constrained programming and dependent-chance programming according to various decision criteria. Li et al[12] presented mean–variance–skewness based on credibility measure. And a genetic algorithm integrating fuzzy simulation is designed. fuzzy set theory has been widely developed and various modifications and generalizations have appeared. One of them is the concept of intuitionistic fuzzy (IF) sets [13]. They consider not only the degree of membership to a given set, but also the degree of rejection such that the sum of both values is less than 1[14]. Applying this concept it is possible to reformulate the optimization problem by using degrees of rejection of constraints and values of the objective which are non-admissible. The degrees of acceptance and of rejection can be arbitrary (the sum of both have to be less than or equal to 1). a mean– variance–skewness model portfolio selection model based on intuitionistic fuzzy optimization is proposed in the paper.. 2. Problem formulation 2.1. Formulation of the mean–variance–skewness model Previous studies[7–9] revealed that maximizing the skewness of return could efficiently improve performance of the traditional Markowitz mean–variance portfolio model. That is, we can obtain a better portfolio by maximizing expected return and skewness of return and minimizing the variance of return simultaneously. Typically, the mean–variance–skewness model can be represented by n n ⎧ T max R ( x ) X R x R ( R Rit / p ) = = = ∑ ∑ i i ⎪ i =1 i =1 ⎪ n n n ⎪ 2 2 T σ min V ( x ) X VX x + xi x jσ ij (i ≠ j ) = = ∑ ∑∑ i i ⎪ i =1 i =1 j =1 ( P1 ) ⎨ ⎪ n n ⎛ n n ⎞ ⎪max S ( x) = E ( X T ( R − R ))3 = ∑ xi3 si3 +3∑ ⎜ ∑ xi2 x j siij + ∑ xi x 2 sijj ⎟(i ≠ j ) j ⎪ i =1 i =1 ⎝ j =1 j =1 ⎠ ⎪ T subject to X I 1 = ⎩ Where σ i2 and σ ij are the variance and covariance of the excess returns based on various forecasting techniques i whereas si3 , siij , and sijj are the skewness and coskewness of the excess returns

based on each forecasting method i , respectively. X is the proportion invested in various assets when the best trade-off is found. It is noted that, in this study, negative x represents a short sale. 2.2. Portfolio selection model based on fuzzy decision theory Due to the complexity of the financial system and stock market are difficult to forecast, and investors can not give accurate expectations in the return and risk and skewness of the three goals, it could be that the return and risk and skewness of the three fuzzy investors objectives, as investors believe that earnings and skewness over at quite a high level can be acceptable, and the bigger the better, under a certain level of risk is acceptable and the smaller the better, the fuzzy mean–variance–skewness model can be represented by :

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n n ~ ⎧ T max R ( x ) X R x R ( R Rit / p ) = = = ∑ ∑ i i ⎪ i =1 i =1 ⎪ n n n ~ ⎪ min V ( x) = X T VX = ∑ xi2σ i2 +∑∑ xi x jσ ij (i ≠ j ) ⎪ i =1 i =1 j =1 ( P2 ) ⎨ ⎪ ~ n n n ⎛ n ⎞ ⎪max S ( x) = E ( X T ( R − R ))3 = ∑ xi3 si3 +3∑ ⎜ ∑ xi2 x j siij + ∑ xi x 2 sijj ⎟(i ≠ j ) j ⎪ i =1 i =1 ⎝ j =1 j =1 ⎠ ⎪ T subject to X I 1 = ⎩ 3 mean–variance–skewness fuzzy portfolio selection model formulation based on intuitionistic fuzzy optimization It is transformed via Bellman-Zadeh's approach[15] to the following optimization problem: To maximize the degree of membership (acceptance) of the objective(s) and constraints to the respective fuzzy sets: ⎧ ⎪max μ ( x) x ∈ U , max μ ( x) x ∈ U , max μ ( x) x ∈ U R V S ⎪⎪ ( P3 ) ⎨ s.t 0 ≤ μ R ( x) ≤ 1, 0 ≤ μV ( x) ≤ 1, 0 ≤ μ S ( x) ≤ 1 ⎪ n ⎪ x j = 1, x j ≥ 0, j = 1, L , n ∑ ⎪⎩ j =1

where

μ R ( x), μV ( x), μ S ( x)

denotes degree of acceptance of x to the respective fuzzy sets.

In the case when the degree of rejection (non-membership) is defined simultaneously with the degree of acceptance (membership) and when both these degrees are not complementary to each other then IF sets can be used as a more general and full tool for describing this uncertainty [16]. It is possible to represent deeply existing nuances in problem formulation defining objective(s) and constraints (or part of them) by IF sets, i.e. by pairs of membership ( μ ( x) ) and rejection ( υ ( x) ) functions [5]. An problem is formulated as follows: To maximize the degree of acceptance of IF objective(s) and constraints and to minimize the degree of rejection of IF objective(s) and constraints: ⎧max μ R ( x) x ∈ U , max μV ( x) x ∈ U , max μ S ( x) x ∈ U ⎪ ⎪min γ R ( x) x ∈ U , max γ V ( x) x ∈ U , max γ S ( x) x ∈ U ⎪⎪ s.t 0 ≤ μ R ( x) ≤ 1, 0 ≤ μV ( x) ≤ 1, 0 ≤ μ S ( x) ≤ 1 ( P3 ) ⎨ 0 ≤ γ R ( x) ≤ 1, 0 ≤ γ V ( x) ≤ 1, 0 ≤ γ S ( x) ≤ 1 ⎪ n ⎪ x j = 1, x j ≥ 0, j = 1, L , n ⎪ ∑ j =1 ⎪⎩ Let U = {( x1 , x2 ,L , xn ) | x1 + x2 + L + xn = 1}

Definition The intuitionistic fuzzy set R = {< x, μ R ( x ), γ R ( x ) >| x ∈ U } ,

V = {< x, μV ( x), γ V ( x) >| x ∈ U} , S = {< x, μ S ( x), γ S ( x) >| x ∈ U} of R( x) , V ( x) S ( x) membership function μ R ( x), μV ( x), μ S ( x) and rejection function γ R ( x), γ V ( x), γ S ( x) as

follow:

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Guohua Chen et al. Procedia Engineering 15 00 (2011) 2062 – 2066 Guohua Chen ,et/al/ Procedia Engineering (2011) 000–000

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⎧ 0, ⎪ ⎪ R − Rmin , μ R ( x) = ⎨ ⎪ Rmax − Rmin ⎪ 1, ⎩ ⎧ 0, ⎪ ⎪ S − Smin , μ S ( x) = ⎨ ⎪ Smax − S min ⎪ 1, ⎩

⎧ 1, V ( x) ≤ Vmin ⎪ − V V ⎪ , Vmin < V ( x) ≤ Vmax Rmin < R ( x) ≤ Rmax μV ( x) = ⎨ max ⎪Vmax − Vmin ⎪ 0, R ( x) > Rmax V ( x) > Vmax ⎩ ⎧ 1, R ( x) ≤ Rmin S ( x) ≤ Smin ⎪ ⎪ Rmax − R , Rmin < R ( x) ≤ Rmax Smin < S ( x) ≤ Smax γ R ( x) = ⎨ ⎪α .Rmax − Rmin ⎪ 0, S ( x) > S max R ( x) > Rmax ⎩ R( x) ≤ Rmin

⎧ ⎧ 1, S ( x) ≤ S min 0, V ( x) ≤ Vmin ⎪ ⎪ ⎪ V − Vmin ⎪ S max − S , S min < S ( x) ≤ S max , Vmin < V ( x) ≤ Vmax γ S ( x) = ⎨ γ V ( x) = ⎨ ⎪α Vmax − Vmin ⎪α .Smax − S min ⎪ ⎪ 0, 1, V ( x) > Vmax S ( x) > S max ⎩ ⎩ Where Rmax and Rmin ,Vmax and Vmin , Smax and Smin is the minimum and maximum in the domain U,

a ≥ 1 ,Used to adjust the degree of hesitation, when a = 1 , degree of hesitation is 0, when a → +∞ , degree of hesitation tend to 1 − μ f ( x) , in general 1 ≤ a ≤ 3 .[17]

It can be transformed to the following system of equations[29]: α ≤ μ R ( x), α ≤ μV ( x), α ≤ μ S ( x)

β ≥ γ R ( x), β ≥ γ V ( x), β ≥ γ S ( x) α ≥ β , β ≥ 0, α + β ≤ 1 where α denotes the minimal acceptable degree of objective(s) and constraints and β denotes the maximal degree of rejection of objective(s) and constraints. Now the IFO problem can be transformed to the following crisp (non-fuzzy) optimization problem ：: max α − β ⎧ ⎪ s.t 0 ≤ μ ( x) ≤ 1, 0 ≤ μ ( x) ≤ 1, 0 ≤ μ ( x) ≤ 1 R V S ⎪ ⎪⎪ 0 ≤ γ R ( x) ≤ 1, 0 ≤ γ V ( x ) ≤ 1, 0 ≤ λS ( x) ≤ 1 ( P5 ) ⎨ α ≥ β,0 ≤ α + β ≤1 ⎪ n ⎪ x j = 1, x j ≥ 0, j = 1, L , n ⎪ ∑ ⎪⎩ j =1 3. Solving the Optimal Solution of ( P5 ) Based on Matlab software we use the Matlab optimization toolbox to solve (CPS), let solution vector x = [ x1 , L , xn , α , β ] , Algorithm is as follows: Step 1: Establishment of M file fun.m, definition the objective function: Function f=fun(X); F = α − β ; Step 2: Establishment of M file nonlcon.m, definition the constraints function G (x): Function G=nonlcon(X); G=[-μ R ( x);μ R ( x) − 1; -μV ( x);μV ( x) − 1; -μ S ( x);μ S ( x) − 1; ; −γ R ( x); γ R ( x) − 1; −γ V ( x); γ V ( x) − 1; −γ S ( x); γ S ( x) − 1] Step 3: Build the main program cps.m:

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x0 = [1/ n, L,1/ n,1/ 2,1/ 3] ; A = [0, L , 0, −α , β ; 0,L , 0, −α , − β ; 0,L , 0, α , β ] ; b=[0;0;1];Aeq=[1,…,1,0,0];beq=[1];VLB=[0,…,0];VUB=[]; [x,fval]=fmincon(‘fun’,x0,A,b,Aeq,beq,VLB,VUB, ‘nonlcon’) 4. Conclusion This study proposes decision models for mean–variance–skewness fuzzy portfolio selection based on intuitionistic fuzzy optimization and uses matlab software to solve the model.

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