A High-Moment Trapezoidal Fuzzy Random Portfolio Model with Background Risk

Xiong DENG, Yanli LIU

Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (1) : 1-28.

PDF(545 KB)
PDF(545 KB)
Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (1) : 1-28. DOI: 10.21078/JSSI-2018-001-28
 

A High-Moment Trapezoidal Fuzzy Random Portfolio Model with Background Risk

Author information +
History +

Abstract

In most exiting portfolio selection models, security returns are assumed to have random or fuzzy distributions. However, uncertainties exist in actual financial markets. Markets are associated not only with inherent risk, but also with background risk that results from the differences among individual investors. This paper investigated the compliance of stock yields to the fuzzy-natured high-order moments of random numbers in order to develop a high-moment trapezoidal fuzzy random portfolio risk model based on variance, skewness, and kurtosis. Data obtained from the Shanghai Stock Exchange and Shenzhen Stock Exchange was used to assess the influence on the proposed model of both background risk and the maximum level of satisfaction of the portfolio. The empirical results demonstrated that the differences between the maximum and minimum variance, skewness, and kurtosis values of the portfolio were positively correlated with the variance of the background risk.

Key words

background risk / fuzzy random number / kurtosis / portfolio selection / satisfaction / skewness

Cite this article

Download Citations
Xiong DENG , Yanli LIU. A High-Moment Trapezoidal Fuzzy Random Portfolio Model with Background Risk. Journal of Systems Science and Information, 2018, 6(1): 1-28 https://doi.org/10.21078/JSSI-2018-001-28

1 Introduction

Classical portfolio theory is based on complete markets. However, real markets are usually incomplete. Researchers have conducted numerous theoretical and empirical studies concerning incomplete markets in order to investigate the decision-making behaviors of individual investors. Several factors can influence these decision-making behaviors, including cost, background risk, and preferential differences. Background risk, which cannot be expressed in terms of money or other forms of wealth, is influenced by the source of income, health status[1], real estate ownership[2] and necessary living expenses of consumers[3]. Since none of these factors are certain in financial markets, the statistical properties associated with background risk are not influenced by the allocation of wealth to risky stocks or risk-free bonds.
A substantial number of studies have focused on background risk. Baptista[4] investigated optimal delegated portfolio management with background risk. In addition, Alghalith[5] introduced an incomplete-market dynamic investment model with background risk. Heaton and Lucas[6] investigated the influence of background risk resulting from sources such as labor and entrepreneurial income on portfolio selection problems. Tsanakas[7] developed a distortion-type risk measurement in order to evaluate the risk associated with any uncertain position within a portfolio with a fixed level of background risk. Eichner and Wagener[8] analyzed the effects of the mean, variance, and covariance of assets as well as the tempering effects of background risk on portfolio composition. Georgescu[9] combined probabilistic and possibilistic to develop investment models that account for background risk. Jiang, et al.[10] investigated the influence of background risk on an investor's portfolio choice and hedging behavior. Li, et al.[11] developed a fuzzy portfolio selection model with background risk based on possibilistic return and possibilistic risk. Huang, et al.[12] developed an uncertain portfolio selection model with background risk based on uncertainty theory. Xu, et al.[13] developed a fuzzy model for portfolio selection problems that accounts for the vagueness of investor preferences.
For many types of problems, including portfolio selection problems, an optimal decision must be based on several criteria. Multi-objective programming is a mathematical methodology used to solve these types of problems. Since objectives are often contradictory, their optimum values cannot always be expressed simultaneously, the application of multi-objective optimization to portfolio optimization has attracted increased attention in recent years. Li, et al.[14] developed a multi-objective portfolio selection model with fuzzy random returns for investors by analyzing return, risk, and liquidity. Aouni[15] presented a detailed discussion and current review of two essential facets of financial decision support: Portfolio selection and corporate performance evaluation. Several multi-objective methods exist, including weighted programming[16], constraint programming[17], compromise programming[18], composite programming[19], the reference point method[20], and goal programming[21].
In addition, researchers have conducted empirical studies concerning negative skewness and excessive kurtosis. Negative skewness is associated with a higher probability of negative returns in markets that are more likely to experience decreases in asset pricing. In contrast, markets with excessive kurtosis are associated with a higher probability of extreme values than markets with normal distributions. Despite their importance, third or higher moments of portfolio returns are often neglected in portfolio construction problems because of computational difficulties. Kim, et al.[22] developed a robust mean-variance approach that could be used to control portfolio skewness and kurtosis without the introduction of high-moment terms. In another study, Jaaman, et al.[23] introduced and empirically tested a four moment mean-conditional-value-at-risk-skewness-kurtosis model. Hossain, et al.[24] developed a modeling approach for multiple objective portfolio selection problems based on the weighted possibilistic moments of trapezoidal fuzzy numbers. Briec, et al.[25] developed a nonparametric efficiency measurement approach for static portfolio selection problems in a mean-variance-skewness-kurtosis space. Hall, et al.[26] analyzed the higher moments, skewness, and kurtosis of portfolios in order to identify whether these factors could be used to improve portfolio construction and diagnose model misspecifications in portfolio returns. Brinkmann, et al.[27] presented measurements for the higher-order dependencies among assets in order to estimate the covariance, co-skewness, and co-kurtosis matrices of asset returns using only the forward-looking information of an options market. Furthermore, McLaughlin, et al.[28] investigated the role of higher moments during portfolio construction.
This paper presents a new proposed model as follows. Section 2 introduces theories regarding fuzzy random numbers. Section 3 explains the deduction of the variance, skewness, and kurtosis formulas of a fuzzy random variable as well as a fuzzy random portfolio selection model with background risk. Section 4 shows the development of the proposed multi-objective high-moment trapezoidal fuzzy random portfolio model based on background risk. Section 5 explains how data from the Shanghai Stock Exchange and Shenzhen Stock Exchange were utilized to provide an empirical analysis of the proposed model. Conclusions regarding this work are presented in Section 6.

2 Theories Regarding Fuzzy Random Numbers

The proposed approach to developing a multi-objective high-moment trapezoidal fuzzy random portfolio model is based on the following definitions.
Definition 1 (see [29]) Let ξ be a fuzzy variable with the membership function μ, and let r be a real number. The possibility of a fuzzy event {ξr} is defined as follows:
Pos{ξr}=supμ(t)(tr).
(1)
Definition 2 (see [29]) The credibility measure Cr of {ξr} is defined as shown
Cr{ξr}=12(1+Pos{ξr}Pos{ξ<r}).
(2)
Thus, Cr is a self-dual set function, in that
Cr{ξr}+Cr{ξ<r}=1,
(3)
for every r.
Definition 3 (see [29]) The expected value of a fuzzy variable {ξr} is also defined as follows, if at least one of the two integrals is finite
E[ξ]= 0+Cr{ξr}dr0Cr{ξ<r}dr.
(4)
Definition 4 (see [29]) Let ξ be a fuzzy variable with a finite expected value e. The variance of ξ is defined as follows:
V[ξ]= E[(ξe)2].
(5)
A fuzzy random variable ξ is said to be trapezoidal if, for each ω, ξω is a fuzzy variable denoted by (X1(ω),X2(ω),X3(ω),X4(ω)), where Xi,i=1,2,3,4 is a random variable defined on the probability space Ω. The randomness of ξ is determined by a random variable Xi,i=1,2,3,4.
Definition 5 (see [30]) The expected value operator of a fuzzy random variable ξ is defined as follows, if at least one of the two integrals is finite.
E[ξ]= 0+Pr{ωΩ|E[ξr]}dr0Pr{ωΩ|E[ξ<r]}dr.
(6)

3 Variance, Skewness, and Kurtosis Formulas of a Trapezoidal Fuzzy Random Variable

In this section, the variance, skewness, and kurtosis formulas of a trapezoidal fuzzy random variable are deduced. These formulas will be applied to the portfolio selection and background risk problems in the next section.
Definition 6 (see [30]) Let ξ be a trapezoidal fuzzy random variable such that, for each ω,
ξ(ω)=(X(ω)α1,X(ω)α2,X(ω)β2,X(ω)β1)
(7)
is a trapezoidal fuzzy variable with the possibility distribution.
μξ(ω)(x)={1, X(ω)α2xX(ω)+β2,xX(ω)+α1α1α2, X(ω)α1xX(ω)α2,x+X(ω)+β1β1β2, X(ω)+β2xX(ω)+β1,0, otherwise,
where α1>α2>0, β1>β2>0, and X is a random variable with a finite expected value. Thus, the expected value of each ω can be expressed as shown:
E[ξ(ω)]= 14[4X(ω)β1β2+β1+β2].
(8)
In addition,
E[ξ]= 14[4Xβ1β2+β1+β2].
(9)
Proposition 1 Let ξ be a trapezoidal fuzzy random variable such that, for each ω,
ξ(ω)=(X(ω)α1,X(ω)α2,X(ω)+β2,X(ω)+β1)
(10)
and
m=E[X].
(11)
1) If X(ω)<mα1, then
E[ξ(ω)m]2=[X(ω)]22mX(ω)+(m2+ 13α12).
(12)
2) If mα1<X(ω)<mα2, then
E[ξ(ω)m]2=[X(ω)]2[2m+12(α1α2)]X(ω)+[m2+12(α1α2)m+16(α12+α22)]. 
(13)
3) If mα2<X(ω)<m, then
E[ξ(ω)m]2=[X(ω)]2(α2+2m)X(ω)+(m2+mα2+ 13α22).
(14)
4) If m<X(ω)<m+β2, then
E[ξ(ω)m]2=[X(ω)]2(β2+2m)X(ω)+(m2+mβ2+13β22).
(15)
5) If m+β2<X(ω)<m+β1, then
E[ξ(ω)m]2=[X(ω)]2[2m+12(β1β2)]X(ω)+[m2+12(β1β2)m+16(β12+β22)].
(16)
6) If X(ω)>m+β1, then
E[ξ(ω)m]2=[X(ω)]22mX(ω)+(m2+ 13β12).
(17)
Proof Only Assertion 2 is proven in this paper; the other assertions can be proven in a similar manner. For any r0,
Pos{[ξ(ω)m]2r}=Pos{[ξ(ω)m]r}Pos{[ξ(ω)m]r}.
If mα1<X(ω)<mα2, then
Pos{[ξ(ω)m]2r}=Pos{[ξ(ω)m]r}={1,0r<[X(ω)+m]2,rX(ω)+m+α1α1,   [X(ω)+m]2r<[X(ω)+m+α1]2,0,otherwise,
and
Pos{[ξ(ω)m]2<r}=Pos{[ξ(ω)m]<r}={1,r[X(ω)+m]2,r+X(ω)m+α2α2,   [X(ω)+mα2]2r<[X(ω)+m]2,0,otherwise.
Therefore,
Cr{[ξ(ω)m]2r}=12(Pos{[ξ(ω)m]r}Pos{[ξ(ω)m]<r})={1,0r<[X(ω)+mα2]2,rX(ω)+m+α22α2,   [X(ω)+mα2]2r<[X(ω)+m]2,rX(ω)+m+α12α1,[X(ω)+m]2r<[X(ω)+m+α1]2,0,otherwise.
In addition,
E[ξ(ω)m]2=0+Cr{[ξ(ω)m]2r}dr=[X(ω)]2[2m+12(α1α2)]X(ω)+[m2+12(α1α2)m+16(α12+α22)].
This concludes the proof of Proposition 1.
According to Proposition 1, the formula of the trapezoidal fuzzy variable can be obtained when XN(μ,σ2).
Theorem 1 Let ξ be a trapezoidal fuzzy random variable such that, for each ω,
ξ(ω)=(X(ω)α1,X(ω)α2,X(ω)+β2,X(ω)+β1)
and
m=E[X].
If XN(μ,σ2), the variance of ξ can be expressed as follows:
V[ξ]=(σ2+ 13α12)[1Φ(α1σ)]+σα12πexp( α122σ2)+[σ2+16(α12+α22)][Φ(α1σ)Φ(α2σ)]+(α1+α2)σ22πexp( α222σ2)(3α1α2)σ22πexp( α122σ2)+(σ2+13α22)[Φ(α2σ)12]+σα22π2πα2σexp(α222σ2)+(σ2+13β12)[1Φ(β1σ)]+σβ12πexp(β122σ2)+[σ2+16(β12+β22)][Φ(β1σ)Φ(β2σ)]+(β1+β2)σ22πexp( β222σ2)(3β1β2)σ22πexp( β122σ2)+(σ2+ 13β22)[Φ(β2σ)12]+σβ22π 2πβ2σexp( β222σ2).
(18)
Proof Since XN(μ,σ2), m=μ. According to Proposition 1:
1) If X(ω)<mα1, then
V1[ξ]=mα1E[ξ(ω)m]2 12πσexp( (xm)22σ2)dx =mα1[x22mx+(m2+13α12)] 12πσexp((xm)22σ2)dx=(σ2+13α12)[1Φ(α1σ)]+σα12πexp( α122σ2).
2) If mα1<X(ω)<mα2, then
V2[ξ]=mα1mα2E[ξ(ω)m]212πσexp((xm)22σ2)dx =mα1mα2{x2[2m+12(α1α2)]x+[m2+12(α1α2)m+ 16(α12+α22)]}12πσexp((xm)22σ2)dx=[σ2+16(α12+α22)][Φ(α1σ)Φ(α2σ)]+(α1+α2)σ2πexp(α222σ2)(3α1α2)σ2πexp( α122σ2).
3) If mα2<X(ω)<m, then
V3[ξ]=mα2mE[ξ(ω)m]212πσexp((xm)22σ2)dx=mα2m[x2(2m+α2)x+(m2+mα2+13α22)]12πσexp((xm)22σ2)dx=(σ2+13α22)[Φ(α2σ)12]+σα22π2πα2σexp(α222σ2).
4) If m<X(ω)<m+β2, then
V4[ξ]=mm+β2E[ξ(ω)m]212πσexp((xm)22σ2)dx=mm+β2[x2(2m+β2)x+(m2+mβ2+13β22)]12πσexp((xm)22σ2)dx=(σ2+13β22)[Φ(β2σ)12]+σβ22π2πβ2σexp(β222σ2).
5) If m+β2<X(ω)<m+β1, then
V5[ξ]=m+β2m+β1E[ξ(ω)m]212πσexp((xm)22σ2)dx=m+β2m+β1{x2[2m+12(β1β2)]x+[m2+12(β1β2)m+ 16(β12+β22)]}12πσexp((xm)22σ2)dx=[σ2+16(β12+β22)][Φ(β1σ)Φ(β2σ)]+(β1+β2)σ2πexp( β222σ2)(3β1β2)σ2πexp(β122σ2).
6) If X(ω)>m+β1, then
V6[ξ]=m+β1+E[ξ(ω)m]212πσexp((xm)22σ2)dx=m+β1+[x22mx+(m2+13β12)]12πσexp((xm)22σ2)dx=(σ2+13β12)[1Φ(β1σ)]+σβ12πexp(β122σ2).
Therefore,
V[ξ]=V1[ξ]+V2[ξ]+V3[ξ]+V4[ξ]+V5[ξ]+V6[ξ]=(σ2+13α12)[1Φ(α1σ)]+σα12πexp( α122σ2)+[σ2+16(α12+α22)][Φ(α1σ)Φ(α2σ)]+(α1+α2)σ22πexp(α222σ2)(3α1α2)σ22πexp(α122σ2)+(σ2+13α22)[Φ(α2σ) 12]+σα22π 2πα2σexp(α222σ2)+(σ2+13β12)[1Φ(β1σ)]+σβ12πexp( β122σ2)+[σ2+16(β12+β22)][Φ(β1σ)]Φ(β2σ)]+(β1+β2)σ22πexp( β222σ2)(3β1β2)σ22πexp( β122σ2)+(σ2+13β22)[Φ(β2σ)12]+σβ22π2πβ2σexp(β222σ2).
This concludes the proof of Theorem 1.
For narrative convenience, in this paper, the related symbols and formulas are defined as shown.
A1=i=1Nxiαi1+αb1,A2=i=1Nj=1Nxixjαi1αj1+i=1Nxiαi1αb1+αb12,A3= i=1Nj=1Nk=1Nxixjxkαi1αj1αk1+ i=1Nj=1Nxixjαi1αj1αb1+ i=1Nxiαi1αb12+αb13,A4=i=1Nj=1Nk=1Nl=1Nxixjxkxlαi1αj1αk1αl1+ i=1Nj=1Nk=1Nxixjxkαi1αj1αk1αb1+i=1Nj=1Nxixjαi1αj1αb12+i=1Nxiαi1αb13+αb14,B1= i=1Nxiαi2+αb2,B2= i=1Nj=1Nxixjαi2αj2+ i=1Nxiαi2αb2+αb22,B3= i=1Nj=1Nk=1Nxixjxkαi2αj2αk2+ i=1Nj=1Nxixjαi2αj2αb2+i=1Nxiαi2αb22+αb23,B4=i=1Nj=1Nk=1Nl=1Nxixjxkxlαi2αj2αk2αl2+ i=1Nj=1Nk=1Nxixjxkαi2αj2αk2αb2+ i=1Nj=1Nxixjαi2αj2αb22+ i=1Nxiαi2αb23+αb24,C1= i=1Nxiσi+σb,C2=i=1Nj=1Nxixjσiσj+ i=1Nxiσiσb+σb2,C3=i=1Nj=1Nk=1Nxixjxkσiσjσk+i=1Nj=1Nxixjσiσjσb+i=1Nxiσiσb2+σb3,C4=i=1Nj=1Nk=1Nl=1Nxixjxkxlσiσjσkσl+ i=1Nj=1Nk=1Nxixjxkσiσjσkσb+ i=1Nj=1Nxixjσiσjσb2+ i=1Nxiσiσb3+σb4,D1=i=1Nxiβi2+βb2,D2=i=1Nj=1Nxixjβi2βj2+i=1Nxiβi2βb2+βb22,D3=i=1Nj=1Nk=1Nxixjxkβi2βj2βk2+ i=1Nj=1Nxixjβi2βj2βb2+ i=1Nxiβi2βb22+βb23,D4=i=1Nj=1Nk=1Nl=1Nxixjxkxlβi2βj2βk2βl2+ i=1Nj=1Nk=1Nxixjxkβi2βj2βk2βb2+ i=1Nj=1Nxixjβi2βj2βb22+ i=1Nxiβi2βb23+βb24,E1=i=1Nxiβi1+βb1,E2=i=1Nj=1Nxixjβi1βj1+i=1Nxiβi1βb1+βb12,E3=i=1Nj=1Nk=1Nxixjxkβi1βj1βk1+ i=1Nj=1Nxixjβi1βj1βb1+i=1Nxiβi1βb12+βb13,E4=i=1Nj=1Nk=1Nl=1Nxixjxkxlβi1βj1βk1βl1+ i=1Nj=1Nk=1Nxixjxkβi1βj1βk1βb1+i=1Nj=1Nxixjβi1βj1βb12+i=1Nxiβi1βb13+βb14.
Corollary 1 Let xi be the investment proportions of security i, and let
ξi(ω)=(Xi(ω)αi1,Xi(ω)αi2,Xi(ω)+βi2,Xi(ω)+βi1)
and XiN(μi,σi2) be the trapezoidal fuzzy random returns of security i for i=1,2,,N. and let
ξb(ω)=(Xb(ω)αb1,Xb(ω)αb2,Xb(ω)+βb2,Xb(ω)+βb1)
and XbN(μb,σb2) be the trapezoidal fuzzy random return of the background asset risk.
Then the variance of the trapezoidal fuzzy random portfolio with background asset risk can be expressed as follows:
V[i=1Nxiξi(ω)+ξb(ω)]=(C2+13A2)[1Φ(A1C1)]+A1C12πexp(A22C2)+[C2+16(A2+B2)][Φ(A1C1)Φ(B1C1)]+(A1+B1)C122πexp(B22C2)(3A1B1)C122πexp(A22C2)+(C2+13B2)[Φ(B1C1)12]+B1C12π2πB1C1exp(B22C2)+(C2+13E2)[1Φ(E1C1)]+E1C12πexp(E22C2)+[C2+16(E2+D2)][Φ(E1C1)Φ(D1C1)]+(E1+D1)C122πexp(D22C2)(3E1D1)C122πexp(E22C2)+(C2+13D2)[Φ(D1C1)12]+D1C12π2πD1C1exp(D22C2).
(19)
According to the definition of the fuzzy random variable, the skewness can be defined as follows.
Definition 7 (see [30]) Let ξ be a fuzzy random variable with a finite expected value of e, the skewness of ξ can be defined as shown.
S[ξ]=E[(ξe)]3.
(20)
Proposition 2 Let ξ be a trapezoidal fuzzy random variable such that, for each ω,
ξ(ω)=(X(ω)α1,X(ω)α2,X(ω)+β2,X(ω)+β1)
(21)
and
m=E[X].
(22)
1) If X(ω)<mα1, then
E[ξ(ω)m]3=[X(ω)]3+3m[X(ω)]2(3m2+α12)X(ω)+m(m2+α12).
(23)
2) If mα1<X(ω)<mα2, then
E[ξ(ω)m]3=[X(ω)]3+[3m+34(α1α2)][X(ω)]2[3m2+32(α1α2)m+12(α12+α22)]X(ω)+[m3+34(α1α2)m2+12(α12+α22)m+18(α13α23)].
(24)
3) If mα2<X(ω)<m, then
E[ξ(ω)m]3=[X(ω)]3+(3m+32α2)[X(ω)]2(3m2+3α2m+α22)X(ω)+(m3+32α2m2+α22m+14α23).
(25)
4) If m<X(ω)<m+β2, then
E[ξ(ω)m]3=[X(ω)]3+(3m+32β2)[X(ω)]2+(3m23β2m+β22)X(ω)+(m3+32β2m2β22m+14β23).
(26)
5) If m+β2<X(ω)<m+β1, then
E[ξ(ω)m]3=[X(ω)]3+[3m+34(β1β2)][X(ω)]2+[3m232(β1β2)m+12(β12+β22)]X(ω)+[m3+34(β1β2)m212(β12+β22)m+18(β13β23)].
(27)
6) If X(ω)>m+β1, then
E[ξ(ω)m]3=[X(ω)]33m[X(ω)]2+(3m2+β12)X(ω)m(m2+β12).
(28)
Proof Only Assertion 2 is proven in this paper; the other assertions can be proven in a similar manner. For any r0,
Pos{[ξ(ω)m]3r}=Pos{[ξ(ω)m]r3}.
If mα1<X(ω)<mα2, then
Pos{[ξ(ω)m]3r}=Pos{[ξ(ω)m]r3}={1, 0r<[X(ω)+m]3,r3X(ω)+m+α1α1,   [X(ω)+m]3r<[X(ω)+m+α1]3,0, otherwise,
and
Pos{[ξ(ω)m]3<r}=Pos{[ξ(ω)m]<r3}={1, r[X(ω)+m]3,r3+X(ω)m+α2α2,   [X(ω)+mα2]3r<[X(ω)+m]3,0, otherwise.
Therefore,
Cr{[ξ(ω)m]3r}=12(Pos{[ξ(ω)m]r3}Pos{[ξ(ω)m]<r3})={1, 0r<[X(ω)+mα2]3,r3X(ω)+m+α22α2,   [X(ω)+mα2]3r<[X(ω)+m]3,r3X(ω)+m+α12α1, [X(ω)+m]3r<[X(ω)+m+α1]3,0, otherwise.
In addition,
E[ξ(ω)m]3=0+Cr{[ξ(ω)m]3r}dr=[X(ω)]3+[3m+34(α1α2)][X(ω)]2[3m2+32(α1α2)m+12(α12+α22)]X(ω)+[m3+34(α1α2)m2+ 12(α12+α22)m+ 18(α13α23)].
This concludes the proof of Proposition 2.
According to Proposition 2, the formula of the trapezoidal fuzzy variable can be obtained when XN(μ,σ2).
Theorem 2 Let ξ be a trapezoidal fuzzy random variable such that, for each ω,
ξ(ω)=(X(ω)α1,X(ω)α2,X(ω)+β2,X(ω)+β1)
and
m=E[X].
If XN(μ,σ2), the skewness of ξ can be expressed as follows:
S[ξ]=2π(2σ2+13α12)σ+[34(α1α2)σ2+18(α13α23)][Φ(α1σ)Φ(α2σ)]+(12α12+34α1α2114α22)σ2πexp(α222σ2)(2σ2+94α1234α1α2+12α22)σ2πexp(α122σ2)+σ[12π(2σ2+α22)+(32σ2+14α22)][Φ(α2σ)12]+2π(2σ2+13β12)σ+[34(β1β2)σ2+18(β13β23)][Φ(β1σ)Φ(β2σ)]+(12β12+34β1β2114β22)σ2πexp(β222σ2)(2σ2+94β1234β1α2+12β22)σ2πexp(β122σ2)+σ[12π(2σ2+β22)+(32σ2+14β22)][Φ(β2σ)12].
(29)
Proof SinceXN(μ,σ2), m=μ. According to Proposition 1:
1) If X(ω)<mα1, then
S1[ξ]=mα1E[ξ(ω)m]312πσexp((xm)22σ2)dx=mα1[x3+3mx2(3m2+α12)x+m(m2+α12)]12πσexp((xm)22σ2)dx=2π(2σ2+α12)σ.
2) If mα1<X(ω)<mα2, then
S2[ξ]=mα1mα2E[ξ(ω)m]312πσexp((xm)22σ2)dx=mα1mα2{x3[3m+34(α1α2)]x2[3m2+32(α1α2)+12(α12+α22)]x+[m3+34(α1α2)+12(α12+α22)+18(α13α23)]}12πσexp((xm)22σ2)dx=[σ2+16(α12+α22)][Φ(α1σ)Φ(α2σ)]+(2σ2+12α12+34α1α2+34α22)σ2πexp(α222σ2)(2σ2+94α1234α1α2+12α22)σ2πexp( α122σ2).
3) If mα2<X(ω)<m, then
S3[ξ]=mα2mE[ξ(ω)m]312πσexp((xm)22σ2)dx=mα2m[x3+(3m+32α2)x2(3m2+3α2m+α22)x]+[m3+32α2m2+α22m+14α23]12πσexp((xm)22σ2)dx =(2σ2+α22)σ2π+(32σ2+α22)σ[Φ(α2σ)12]+(2σ2+72α22)σ2πexp(α222σ2).
4) If m<X(ω)<m+β2, then
S4[ξ]=mm+β2E[ξ(ω)m]312πσexp((xm)22σ2)dx=mm+β2[x3+(3m+32β2)x2+(3m23β2m+β22)x+(m3+32β2m2β22m+14β23)]12πσexp((xm)22σ2)dx =(2σ2+β22)σ2π+(32σ2+β22)σ[Φ(β2σ)12]+(2σ2+72β22)σ2πexp(β222σ2).
5) If m+β2<X(ω)<m+β1, then
S5[ξ]=m+β2m+β1E[ξ(ω)m]312πσexp((xm)22σ2)dx=m+β2m+β1{x3+[3m+34(β1β2)]x2+[3m232(β1β2)+12(β12+β22)]x+[m3+34(α1β2)12(β12+α22)+18(β13β23)]}12πσexp((xm)22σ2)dx=[σ2+16(β12+β22)][Φ(β1σ)Φ(β2σ)]+(2σ2+12β12+34β1β2+34β22)σ2πexp(β222σ2)(2σ2+94β1234β1β2+12β22)σ2πexp(β122σ2).
6) If X(ω)>m+β1, then
S6[ξ]=m+β1+E[ξ(ω)m]312πσexp((xm)22σ2)dx=m+β1+[x33mx2+(3m2+β12)x(m2+β12)m]12πσexp((xm)22σ2)dx =2π(2σ2+β12)σ.
Therefore,
S[ξ]=S1[ξ]+S2[ξ]+S3[ξ]+S4[ξ]+S5[ξ]+S6[ξ]=2π(2σ2+13α12)σ+[34(α1α2)σ2+18(α13α23)][Φ(α1σ)Φ(α2σ)]+(12α12+34α1α2114α22)σ2πexp(α222σ2)(2σ2+94α1234α1α2+12α22)σ2πexp(α122σ2)+σ[12π(2σ2+α22)+(32σ2+14α22)][Φ(α2σ)12]+2π(2σ2+13β12)σ+[34(β1β2)σ2+18(β13β23)][Φ(β1σ)Φ(β2σ)]+(12β12+34β1β2114β22)σ2πexp(β222σ2)(2σ2+94β1234β1α2+12β22)σ2πexp(β122σ2)+σ[12π(2σ2+β22)+(32σ2+14β22)][Φ(β2σ)12].
This concludes the proof of Theorem 2.
Corollary 2 Let xi be the investment proportions of security i, and let
ξi(ω)=(Xi(ω)αi1,Xi(ω)αi2,Xi(ω)+βi2,Xi(ω)+βi1)
and XiN(μi,σi2) be the trapezoidal fuzzy random returns of security i for i=1,2,,N. And let
ξb(ω)=(Xb(ω)αb1,Xb(ω)αb2,Xb(ω)+βb2,Xb(ω)+βb1),
and XbN(μb,σb2) be the trapezoidal fuzzy random return of the background asset risk. Then the skewness of the trapezoidal fuzzy random portfolio with background asset risk can be expressed as follows:
S[i=1Nxiξi(ω)+ξb(ω)]=2π(2C2+A2)C1+[34(A1B1)+18(A3B3)][Φ(A1C1)Φ(B1C1)]+(12A2+34A1B1114B2)C12πexp(B22C2)(2C2+94A234A1B1+12B2)C12πexp(A22C2)+[2π(2C2+B2)C1+(32C2+14B2)C1][Φ(B1C1)12]+2π(2C2+E2)C1+[34(E1D1)+18(E3D3)][Φ(E1C1)Φ(D1C1)]+(12E2+34E1D1114D2)C12πexp(D22C2)(2C2+94E234E1D1+12D2)C12πexp(E22C2)+[2π(2C2+D2)C1+(32C2+14D2)C1][Φ(D1C1)12].
(30)
According to the definition of the fuzzy random variable, the kurtosis can be defined as follows.
Definition 8 (see [30]) Let ξ be a fuzzy random variable with a finite expected value of e, the skewness of ξ can be defined as
K[ξ]=E[(ξe)]4.
(31)
Proposition 3 Let ξ be a trapezoidal fuzzy random variable such that, for each ω,
ξ(ω)=(X(ω)α1,X(ω)α2,X(ω)+β2,X(ω)+β1)
(32)
and
m=E[X].
(33)
1) If X(ω)<mα1, then
E[ξ(ω)m]4=[X(ω)]44m[X(ω)]3+2(α12+3m2)[X(ω)]24m(m2+α12)X(ω)+(m4+m2α12+15α14).
(34)
2) If mα1<X(ω)<mα2, then
E[ξ(ω)m]4=[X(ω)]4[4m+(α1α2)][X(ω)]3+[6m2+3(α1α2)m+(α12+α22)][X(ω)]2[4m3+3(α1α2)m2+2(α12+α22)m+12(α13α23)]X(ω)+[m4+(α1α2)m3+(α12+α22)m2+12(α13α23)+15(α14+α24)].
(35)
3) If mα2<X(ω)<m, then
E[ξ(ω)m]4=[X(ω)]42(2m+α2)[X(ω)]3+2(3m2+3α2m+2α22)[X(ω)]2(4m3+4α2m2+6α22m+α23)[X(ω)]+(m4+2α2m3+2α22m2+α23m+15α24).
(36)
4) If m<X(ω)<m+β2, then
E[ξ(ω)m]4=[X(ω)]42(2mβ2)[X(ω)]3+2(3m23β2m+2β22)[X(ω)]2(4m3+4β2m26β22mβ23)[X(ω)]+(m42β2m3+2β22m2β23m+15β24).
(37)
5) If m+β2<X(ω)<m+β1, then
E[ξ(ω)m]4=[X(ω)]4[4m+(β1β2)][X(ω)]3+[6m23(β1β2)m+(β12+β22)][X(ω)]2[4m33(β1β2)m2+2(β12+β22)m12(β13β23)]X(ω)+[m4(β1β2)m3+(β12+β22)m212(β13β23)m+15(β14+β24)].
(38)
6) If X(ω)>m+β1, then
E[ξ(ω)m]4=[X(ω)]44m[X(ω)]3+2(β12+3m2)[X(ω)]24m(m2+β12)[X(ω)]+(m4+m2β12+15β14).
(39)
Proof Only Assertion 2 is proven in this paper; the other assertions can be proven in a similar manner. For any r0,
Pos{[ξ(ω)m]4r}=Pos{[ξ(ω)m]r4}Pos{[ξ(ω)m]r4}.
If mα1<X(ω)<mα2, then
Pos{[ξ(ω)m]4r}=Pos{[ξ(ω)m]r4}={1, 0r<[X(ω)+m]4,r4X(ω)+m+α1α1,   [X(ω)+m]4r<[X(ω)+m+α1]4,0, otherwise,
and
Pos{[ξ(ω)m]4<r}=Pos{[ξ(ω)m]<r4}={1, r[X(ω)+m]4,r4+X(ω)m+α2α2,   [X(ω)+mα2]4r<[X(ω)+m]4,0, otherwise.
Therefore,
Cr{[ξ(ω)m]4r}=12(Pos{[ξ(ω)m]r4}Pos{[ξ(ω)m]<r4})={1, 0r<[X(ω)+mα2]4,r4X(ω)+m+α22α2,   [X(ω)+mα2]4r<[X(ω)+m]4,r4X(ω)+m+α12α1,   [X(ω)+m]4r<[X(ω)+m+α1]4,0, otherwise.
In addition,
E[ξ(ω)m]4=[X(ω)]4[4m+(α1α2)][X(ω)]3+[6m2+3(α1α2)m+(α12+α22)][X(ω)]2[4m3+3(α1α2)m2+2(α12+α22)m+12(α13α23)]X(ω)+[m4+(α1α2)m3+(α12+α22)m2+12(α13α23)+15(α14+α24)].
This concludes the proof of Proposition 3.
According to Proposition 3, the formula of the trapezoidal fuzzy variable can be obtained when XN(μ,σ2).
Theorem 3 Let ξ be a trapezoidal fuzzy random variable such that, for each ω,
ξ(ω)=(X(ω)α1,X(ω)α2,X(ω)+β2,X(ω)+β1)
and
m=E[X].
If XN(μ,σ2), the kurtosis of ξ can be expressed as follows:
K[ξ]=(3σ4+2α12σ2+15α14)[1Φ(α1σ)]+3(α12+σ2)α1σ2πexp(α122σ2)+[3σ4+(α12+α22)σ2+110(α14+α24)][Φ(α1σ)Φ(α2σ)]+[(2α1+α2)σ2+(α1+α2)α1α2+12(α13+α23)]σ2πexp(α222σ2)[(5α12α2)σ2+2α1α22+32(α13α23)]σ2πexp(α122σ2)+(3σ4+2α22σ2+15α24)[Φ(α2σ)12]+(α24+4σ4)α2σ2π(6α24+7σ4)α2σ2πexp(α222σ2)+(3σ4+2β12σ2+15β14)[1Φ(β1σ)]+3(β12+σ2)β1σ2πexp(β122σ2)+[3σ4+(β12+β22)σ2+110(β14+β24)][Φ(β1σ)Φ(β2σ)]+[(2β1+β2)σ2+(β1+β2)β1β2+12(β13+β23)]σ2πexp(β222σ2)[(5β12β2)σ2+2β1β22+32(β13β23)]σ2πexp(β122σ2)+(3σ4+2β22σ2+15β24)[Φ(β2σ)12]+(β24+4σ4)β2σ2π(6β24+7σ4)β2σ2πexp(β222σ2).
(40)
Proof Since XN(μ,σ2), m=μ. According to Proposition 3:
1) If X(ω)<mα1, then
K1[ξ]=mα1E[ξ(ω)m]412πσexp((xm)22σ2)dx =mα1[x44mx3+2(α12+3m2)x24m(m2+α12)x+(m4+m2α12+15α14)]12πσexp((xm)22σ2)dx=(3σ4+2α12σ2+15α14)[1Φ(α1σ)]+3(α12+σ2)α1σ2πexp(α122σ2).
2) If mα1<X(ω)<mα2, then
K2[ξ]=mα1E[ξ(ω)m]412πσexp((xm)22σ2)dx =mα1{x4[4m+(α1α2)]x3+[6m2+3(α1α2)m+(α12+α22)]x2[4m3+3(α1α2)m2+2(α12+α22)m+12(α13α23)]x+[m4+(α1α2)m3+(α12+α22)m2+12(α13α23)+15(α14+α24)]}12πσexp((xm)22σ2)dx =[3σ4+(α12+α22)σ2+110(α14+α24)][Φ(α1σ)Φ(α2σ)]+[(2α1+α2)σ2+(α1+α2)α1α2+12(α13+α23)]σ2πexp(α222σ2)[(5α12α2)σ2+2α1α22+32(α13α23)]σ2πexp(α122σ2).
3) If mα2<X(ω)<m, then
K3[ξ]=mα2mE[ξ(ω)m]412πσexp((xm)22σ2)dx=mα2m[x42(2m+α2)x3+2(3m2+3α2m+2α22)x2(4m3+4α2m2+6α22m+α23)x+(m4+2α2m3+2α22m2+α23m+15α24)]12πσexp((xm)22σ2)dx=(3σ4+2α22σ2+15α24)[Φ(α2σ)12]+(α24+4σ4)α2σ2π(6α24+7σ4)α2σ2πexp(α222σ2).
4) If m<X(ω)<m+β2, then
K4[ξ]=mm+β2E[ξ(ω)m]412πσexp((xm)22σ2)dx=mm+β2[x42(2m+β2)x3+2(3m2+3β2m+2β22)x2(4m3+4β2m2+6β22m+β23)x+(m4+2β2m3+2β22m2+β23m+15β24)]12πσexp((xm)22σ2)dx=(3σ4+2β22σ2+15β24)[Φ(β2σ)12]+(β24+4σ4)β2σ2π(6β24+7σ4)β2σ2πexp(β222σ2).
5) If m+β2<X(ω)<m+β1, then
K5[ξ]=m+β2m+β1E[ξ(ω)m]412πσexp((xm)22σ2)dx=m+β2m+β1{x4[4m+(β1β2)]x3+[6m2+3(β1β2)m+(β12+β22)]x2[4m3+3(β1β2)m2+2(β12+β22)m+12(β13β23)]x+[m4+(β1β2)m3+(β12+β22)m2+12(β13β23)+15(β14+β24)]}12πσexp((xm)22σ2)dx=[3σ4+(β12+β22)σ2+110(β14+β24)][Φ(β1σ)Φ(β2σ)]+[(2β1+β2)σ2+(β1+β2)β1β2+12(β13+β23)]σ2πexp(β222σ2)[(5β12β2)σ2+2β1β22+32(β13β23)]σ2πexp(β122σ2).
6) If X(ω)>m+β1, then
K6[ξ]=m+β1+E[ξ(ω)m]412πσexp((xm)22σ2)dx=m+β1+[x44mx3+2(β12+3m2)x24m(m2+β12)x+(m4+m2β12+15β14)]12πσexp((xm)22σ2)dx =(3σ4+2β12σ2+15β14)[1Φ(β1σ)]+3(β12+σ2)β1σ2πexp(β122σ2).
Therefore,
K[ξ]=K1[ξ]+K2[ξ]+K3[ξ]+K4[ξ]+K5[ξ]+K6[ξ]=(3σ4+2α12σ2+15α14)[1Φ(α1σ)]+3(α12+σ2)α1σ2πexp(α122σ2)+[3σ4+(α12+α22)σ2+110(α14+α24)][Φ(α1σ)Φ(α2σ)]+[(2α1+α2)σ2+(α1+α2)α1α2+12(α13+α23)]σ2πexp(α222σ2)[(5α12α2)σ2+2α1α22+32(α13α23)]σ2πexp(α122σ2)+(3σ4+2α22σ2+15α24)[Φ(α2σ)12]+(α24+4σ4)α2σ2π(6α24+7σ4)α2σ2πexp(α222σ2)+(3σ4+2β12σ2+15β14)[1Φ(β1σ)]+3(β12+σ2)β1σ2πexp(β122σ2)+[3σ4+(β12+β22)σ2+110(β14+β24)][Φ(β1σ)Φ(β2σ)]+[(2β1+β2)σ2+(β1+β2)β1β2+12(β13+β23)]σ2πexp(β222σ2)[(5β12β2)σ2+2β1β22+32(β13β23)]σ2πexp(β122σ2)+(3σ4+2β22σ2+15β24)[Φ(β2σ)12]+(β24+4σ4)β2σ2π(6β24+7σ4)β2σ2πexp(β222σ2).
(41)
This concludes the proof of Theorem 3.
Corollary 3 Let xi be the investment proportions of security i, and let
ξi(ω)=(Xi(ω)αi1,Xi(ω)αi2,Xi(ω)+βi2,Xi(ω)+βi1)
and XiN(μi,σi2) be the trapezoidal fuzzy random returns of security i for i=1,2,,N. And let
ξb(ω)=(Xb(ω)αb1,Xb(ω)αb2,Xb(ω)+βb2,Xb(ω)+βb1)
and XbN(μb,σb2) be the trapezoidal fuzzy random return of the background asset risk. Then the skewness of the trapezoidal fuzzy random portfolio with background asset risk can be expressed as follows:
K[i=1Nxiξi(ω)+ξb(ω)]=(3C4+2A2C2+15A4)[1Φ(A1C1)]+3(A2+C2)A1C12πexp(A22C2)+[3C4+(A2+B2)C2+110(A4+B4)][Φ(A1C1)Φ(B1C1)]+[(2A1+B1)C2+(A1+B1)A1B1+12(A3+B3)]C12πexp(B22C2)[(5A12B1)C2+2A1B2+32(A3B3)]C12πexp(A22C2)+(3C4+2B2C2+15B4)[Φ(B1C1)12]+(B4+4C4)B1C12π(6B4+7C4)B1C12πexp(B22C2)+(3C4+2E2C2+15E4)[1Φ(E1C1)]+3(E2+C2)E1C12πexp(E22C2)+[3C4+(E2+D2)C2+110(E4+D4)][Φ(E1C1)Φ(D1C1)]+[(2E1+D1)C2+(E1+D1)E1D1+12(E3+D3)]C12πexp(D22C2)[(5E12D1)C2+2E1D2+32(E3D3)]C12πexp(E22C2)+(3C4+2D2C2+15D4)[Φ(D1C1)12]+(D4+4C4)D1C12π(6D4+7C4)D1C12πexp(D22C2).
(42)

4 A Multi-Objective High-Moment Trapezoidal Fuzzy Random Portfolio Model with Background Risk

According to the definitions of the variance, skewness, and kurtosis of the trapezoidal random variable presented in Section 3, the multi-objective high-moment trapezoidal fuzzy random portfolio model with background risk can be expressed as follows.
(P-1){max  M[i=1Nxiξi(ω)+ξb(ω)] min  V[i=1Nxiξi(ω)+ξb(ω)] max  S[i=1Nxiξi(ω)+ξb(ω)] max  K[i=1Nxiξi(ω)+ξb(ω)] s.t.    0xi1,       i=1Nxi=1.  
Solutions for multi-objective decision-making problems usually cannot be obtained using multi-objective optimization models because of non-commensurability and contradiction among targets. These issues can be resolved by converting these multi-objective decision-making problems into single-objective decision-making problems. For this research, Zimmermann's[33] fuzzy programming method was used to convert the problem into a single-objective programming problem. An overview of Zimmermann's fuzzy programming method is presented as follows.
For a multi-objective optimization problem,
(P-2){max  φ(x)=[φ1(x),φ2(x),,φq(x)] min  ϕ(x)=[ϕ1(x),ϕ2(x),,ϕp(x)] s.t.    xX, 
where xX be represents the feasible region, φi (i=1,2,,q) denotes the income object functions, and ϕj (j=1,2,,p) denotes the cost object functions.
First, satisfaction functions are formulated in the following manner:
μ[φi(x)]=1exp{αi[φi(x)φiM(x)]},  i=1,2,,q,
(43)
μ[ϕj(x)]=1exp{αj[ϕj(x)ϕkM(x)]},  j=1,2,,p,
(44)
where
φiM(x)=12[maxφi(x)xX+minφi(x)xX],  i=1,2,,q,
(45)
ϕiM(x)=12[maxϕi(x)xX+minϕi(x)xX],  j=1,2,,p.
(46)
Then, the multi-objective optimization problem is converted into a corresponding equivalent single-objective programming problem as follows.
(P-3){max  λ s.t.    λ1exp{αi[φi(x)φiM(x)]},  i=1,2,,q,        λ1exp{αj[ϕj(x)ϕkM(x)]},  j=1,2,,p,         xX,  
where αi and αj represent the membership coefficient parameters. Higher values of these parameters are associated with lower values of their corresponding fuzzy degree of membership functions.
According to the fuzzy programming method presented above, the fuzzy programming method can be applied to model {P}-{1} to obtain the single-objective parameter programming model {P}-{4}, expressed as shown.
(P-4){max  λ s.t.    λM[i=1Nxiξi(ω)+ξb(ω)],        λV[i=1Nxiξi(ω)+ξb(ω)],        λS[i=1Nxiξi(ω)+ξb(ω)],        λK[i=1Nxiξi(ω)+ξb(ω)],         0xi1,         i=1Nxi=1,  
where M[i=1Nxiξi(ω)+ξb(ω)], S[i=1Nxiξi(ω)+ξb(ω)], and K[i=1Nxiξi(ω)+ξb(ω)] are the income objective functions, and V[i=1Nxiξi(ω)+ξb(ω)] is the cost objective function.

5 Empirical Analysis

In order to illustrate the effectiveness of the model presented in Section 4, eight stocks from the Shanghai Stock Exchange and Shenzhen Stock Exchange were selected randomly. The year-to-date return ratio and variance values of the eight stocks (2014) are displayed in Table 1.
Table 1 The year-to-date return ratio and variance of the eight stocks
Stock Year return ratio Variance
900941 0.0561 0.1844
900943 0.0558 0.2725
900945 0.1595 0.4949
900948 0.0583 0.4744
900949 0.1019 0.3422
900950 0.1002 0.4464
900956 0.0880 0.3478
900957 0.0773 0.3468
In addition, ten background risk values were assumed to exist. All ten of the background risk values possessed Year-to-date return ratio of approximately 0.0100, with standard deviations of 0.0000, 0.1000, 0.2000, 0.3000, 0.4000, 0.5000, 0.6000, 0.7000, 0.8000, and 0.9000 respectively. The model {({P}-4)} was also applied to the random return ratios of the eight stocks, yielding the following eight trapezoidal fuzzy random return ratios.
ξ1(ω)=(X1(ω)0.0112,X1(ω)0.0056,X1(ω)+0.0050,X1(ω)+0.0117),X1N(0.0558,0.0743),ξ2(ω)=(X2(ω)0.0112,X2(ω)0.0056,X2(ω)+0.0051,X2(ω)+0.0118),X2N(0.0561,0.0340),ξ3(ω)=(X3(ω)0.0319,X3(ω)0.0160,X3(ω)+0.0144,X3(ω)+0.0335),X3N(0.1595,0.2449),ξ4(ω)=(X4(ω)0.0117,X4(ω)0.0058,X4(ω)+0.0052,X4(ω)+0.0122),X4N(0.0583,0.2251),ξ5(ω)=(X5(ω)0.0204,X5(ω)0.0102,X5(ω)+0.0092,X5(ω)+0.0214),X5N(0.1019,0.1171),ξ6(ω)=(X6(ω)0.0200,X6(ω)0.0100,X6(ω)+0.0090,X6(ω)+0.0210),X6N(0.1002,0.1993),ξ7(ω)=(X7(ω)0.0176,X7(ω)0.0088,X7(ω)+0.0079,X7(ω)+0.0185),X7N(0.0880,0.0185),ξ8(ω)=(X8(ω)0.0155,X8(ω)0.0077,X8(ω)+0.0070,X8(ω)+0.0162),X8N(0.0773,0.1203).
Finally, the background risk was assumed to have ten trapezoidal fuzzy random return ratios as shown.
ξi(ω)=(Xi(ω)0.0020,Xi(ω)0.0010,Xi(ω)+0.0009,Xi(ω)+0.0021),  XiN(0.010,σbi2),
where i=1,2,,10, σb1=0.000, σb2=0.100, σb3=0.200, σb4=0.300, σb5=0.400, σb6=0.500, σb7=0.600, σb8=0.700, σb9=0.800, σb10=0.900.
A genetic algorithm was used to solve the model P-4. The results are shown in Figures 1~5.
Figure 1 Relationship between the standard deviation from background risk and portfolio return

Full size|PPT slide

Figure 2 Relationship between the standard deviation from background risk and portfolio variance

Full size|PPT slide

Figure 3 Relationship between the standard deviation from background risk and portfolio skewness

Full size|PPT slide

Figure 4 Relationship between the standard deviation from background risk and portfolio kurtosis

Full size|PPT slide

Figure 5 Relationship between the standard deviation from background risk and maximum portfolio satisfaction

Full size|PPT slide

As shown in Figure 5, the membership coefficients of the four objective functions to serials 1, 2, 3, and 4 were all equal to 5, 10, 20, and 40, respectively.
As shown in Figures 1~5, the return of the portfolio was not influenced by the variance of the background risk. However, the variance, skewness, and kurtosis of the portfolio were positively correlated with the variance of the background risk. The differences between the maximum and minimum variance, skewness, and kurtosis values of the portfolio were also positively correlated with the variance of the background risk. Furthermore, the relationship between the maximum satisfaction level of the portfolio and the variance of the background risk was volatile. There was a negative correlation between the maximum satisfaction level of the portfolio and the membership coefficients of the four objective functions. Thus, the maximum satisfaction level of the portfolio and the fuzzy degrees of the four objective functions were correlated positively.

6 Conclusions

This paper presented the variance, skewness, and kurtosis formulas of a trapezoidal fuzzy random number. Next, a multi-objective high-moment trapezoidal fuzzy random portfolio model with background risk was developed. Then, the portfolio model was transformed into a single objective optimization model using the membership function. The resulting model was subjected to an empirical analysis using data from the Shanghai Stock Exchange and Shenzhen Stock Exchange in order to identify the influence of background risk on the variance, skewness, and kurtosis of the portfolio model. In future studies, the proposed model will be used to investigate portfolio models that return fuzzy random numbers, double random numbers, double fuzzy numbers, and uncertain numbers. Constrained chance models, chance models, and expected value models could also be altered to account for background risk in future studies.

References

1
Fan E, Zhao R Y. Health status and portfolio choice: Causality or heterogeneity?. Journal of Banking and Finance, 2009, 33 (6): 1079- 1088.
2
Shum P, Faig M. What explains household stock holdings?. Journal of Banking and Finance, 2006, 30 (9): 2579- 2597.
3
Cardak B A, Wilkins R. The determinants of household risky asset holdings: Australian evidence on background risk and other factors. Journal of Banking and Finance, 2009, 33 (5): 850- 860.
4
Baptista A M. Optimal delegated portfolio management with background risk. Journal of Banking and Finance, 2008, 32 (6): 977- 985.
5
Alghalith M. The impact of background risk. Physica A: Statistical Mechanics and Its Applications, 2012, 391 (391): 6506- 6508.
6
Heaton J, Lucas D. Portfolio choice in the presence of background risk. The Economic Journal, 2000, 110 (460): 1- 26.
7
Tsanakas A. Risk measurement in the presence of background risk. Insurance Mathematics and Economics, 2008, 42 (2): 520- 528.
8
Eichner T, Wagener A. Tempering effects of (dependent) background risks: A mean-variance analysis of portfolio selection. Journal of Mathematical Economics, 2012, 48 (6): 422- 430.
9
Georgescu I. Combining probabilistic and possibilistic aspects of background risk. IEEE, International Symposium on Computational Intelligence and Informatics. IEEE, 2012, 225- 229.
10
Jiang C, Ma Y, An Y. An analysis of portfolio selection with background risk. Journal of Banking and Finance, 2009, 34 (12): 3055- 3060.
11
Li T, Zhang W, Xu W. A fuzzy portfolio selection model with background risk. Applied Mathematics and Computation, 2015, 256, 505- 513.
12
Huang X, Di H. Uncertain portfolio selection with background risk. Applied Mathematics and Computation, 2015, 276, 284- 296.
13
Xu W, Deng X, Li J. A New Fuzzy Portfolio Model Based on Background Risk Using MCFOA. International Journal of Fuzzy Systems, 2015, 17 (2): 246- 255.
14
Li J, Xu J. Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm. Information Sciences, 2013, 220 (1): 507- 521.
15
Aouni B. Comments on: Multicriteria decision systems for financial problems. Top An Official Journal of the Spanish Society of Statistics & Operations Research, 2013, 21 (2): 262- 264.
16
Ballestero E, Garcia-Bernabeu A, Hilario A. Portfolio Selection by Goal Programming Techniques. Socially Responsible Investment, 2015, 111- 129.
17
Cabot J, ClarisóR, Riera D. On the verification of UML/OCL class diagrams using constraint programming. Journal of Systems and Software, 2014, 93, 1- 23.
18
Ballestero E, Garcia-Bernabeu A. Compromise programming and utility functions. Socially Responsible Investment, 2015, 155- 175.
19
Lan G. Gradient sliding for composite optimization. Mathematical Programming, 2016, 159 (1): 201- 235.
20
Méndez-Rodríguez P, Pérez-Gladish B, Cabello J M, et al. Portfolio selection with SRI synthetic indicators: A reference point method approach. Socially Responsible Investment, 2015, 65- 69.
21
Jadidi O, Zolfaghari S, Cavalieri S. A new normalized goal programming model for multi-objective problems: A case of supplier selection and order allocation. International Journal of Production Economics, 2014, 148 (1): 158- 165.
22
Kim W C, Fabozzi F J, Cheridito P, et al. Controlling portfolio skewness and kurtosis without directly optimizing third and fourth moments. Economics Letters, 2014, 122 (2): 154- 158.
23
Jaaman S H, Weng H L, Isa Z. A new higher moment portfolio optimisation model with conditional value at risk. International Journal of Operational Research, 2014, 21 (4): 451.
24
Hossain S A, Bhattacharyya R. Portfolio Selection with Possibilistic Kurtosis. Facets of Uncertainties and Applications. Springer India, 2015, 303- 313.
25
Briec W, Kerstens K, Ignace V D W. Risk-loving and risk-averse preferences in mean-variance-skewness-kurtosis portfolio modeling: A common characterization using the shortage function. 2013.
26
Hall A D, Satchell S E. The anatomy of portfolio skewness and kurtosis. Journal of Asset Management, 2013, 14 (4): 228- 235.
27
Brinkmann F, Kempf A, Korn O. Forward-looking measures of higher-order dependencies with an application to portfolio selection. Cfr Working Papers, 2014.
28
McLaughlin K W. Higher moments in portfolio construction. Proceedings of the Northeast Business and Economics Association, 2013, 36 (34): 213- 220.
29
Liu B, Liu Y K. Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 2002, 10 (4): 445- 450.
30
Hao F F, Liu Y K. Mean-variance models for portfolio selection with fuzzy random returns. Journal of Applied Mathematics and Computing, 2009, 30 (1): 9- 38.
31
Liu Y, Wu X, Hao F. A new chance-variance optimization criterion for portfolio selection in uncertain decision systems. Expert Systems with Applications, 2012, 39 (7): 6514- 6526.
32
Qin Z, Xu L. Mean-variance adjusting model for portfolio selection problem with fuzzy random returns. International Joint Conference on Computational Sciences and Optimization. IEEE Computer Society, 2014, 83- 87.
33
Zimmermann H J. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1978, 1 (1): 45- 55.
PDF(545 KB)

264

Accesses

0

Citation

Detail

Sections
Recommended

/