Research ArticleA Possibilistic Portfolio Model with Fuzzy Liquidity Constraint

Yunyun Sui 1 Jiangshan Hu1 and Fang Ma2

1School of Mathematics and Information Science Weifang University Weifang 261061 China2School of Science Shenyang University of Technology Shenyang 110023 China

Correspondence should be addressed to Yunyun Sui suiyunyun1231163com

Received 2 March 2020 Revised 27 June 2020 Accepted 6 July 2020 Published 29 July 2020

Academic Editor Zhile Yang

Copyright copy 2020 Yunyun Sui et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Investors are concerned about the reliability and safety of their capital especially its liquidity when investing (is paper sets up apossibilistic portfolio selection model with liquidity constraint In this model the asset return and liquidity are fuzzy variableswhich follow the normal possibility distributions Liquidity is measured as the turnover rate of the asset On the basis of possibilitytheory we transform the model into a quadratic programming problem to obtain its solution We illustrate that in the process ofinvestment investors canmake better use of capital by choosing their investment portfolios according to their expected return andasset liquidity

1 Introduction

Recently portfolio selection has received extensive atten-tions [1ndash4] (e portfolio selection model studies how toallocate investment funds among different assets to guar-antee profits and disperse investment risk Markowitz [5]proposed a mean-variance model (MVM) for portfolio se-lection which played an important role in the developmentof modern portfolio selection theory (e MVM uses meanand variance to describe respectively the expected returnand risk of a portfolio (e basic rule is the investorsrsquo trade-off between expected return and risk

In the MVM using the variance of return of aportfolio as a risk measure has some limitations andcomputational difficulties to construct large-scale port-folio In order to overcome the difficulty some re-searchers extended the MVM in various ways Examplesare the semivariance model [6] mean absolute deviationsmodel [7] semiabsolute deviation model [8] Value-at-Risk (VaR) model [9] Conditional Value-at-Risk (CVaR)model [10] and so on But whether there is a risk measurethat is best for all portfolios is still an open problem [11](e main reason is that each measure performs the best inits domain but not when considered in another measuredomain [12]

(e MVM combines probability theory with optimiza-tion techniques to model investment behavior under un-certainty However Black and Litterman [13] showed thatthe solutions of the MVM are very sensitive to perturbationsof input parameters (is means that a small uncertainty inthe parameters can make the usual optimal solution prac-tically meaningless (erefore it is necessary to developmodels that are immune as far as possible to data uncer-tainty Robust optimization is a good tool to solve thisproblem [14] In order to systematically counter the sensi-tivity of the optimal portfolio to statistical and modelingerrors in the estimation of relevant market parametersGoldfarb and Iyengar [15] established a robust portfolioselection problem (ey introduced ldquouncertainty structuresrdquofor the market parameters and transformed the robustportfolio selection problem corresponding to these uncer-tainty structures into a second-order cone program Piriet al [16] studied the robust models of the mean-Condi-tional Value-at-Risk (M-CVaR) portfolio selection problemin which the mean return risk is estimated in both the in-terval and ellipsoidal uncertainty sets (e correspondingrobust models are a linear programming problem and aquadratic programming problem respectively Kara et al[17] established a robust optimization problem based on thedata In order to obtain the robust optimal solution of the

HindawiComplexityVolume 2020 Article ID 3703017 10 pageshttpsdoiorg10115520203703017

portfolio they used the method of Robust ConditionalValue-at-Risk under Parallelepiped Uncertainty an evalu-ation and a numerical finding of the robust optimal port-folio allocation Kang and Li [18] proposed a robust mean-risk optimization problem with multiple risk measuresunder ambiguous distribution in which variance value ofrisk (VaR) and conditional value of risk (CVaR) were si-multaneously used as a triple-risk measure And a closed-form expression of the portfolio was obtained Khanjarpa-nah and Pishvaee [19] established a robust flexible portfoliooptimization model and extended the model by introducingan improved robust flexible method Khodamoradi et al[20] established a robust cardinality constraints mean-var-iance optimization model and proved that the model wasequivalent to a mixed integer second-order cone program(rough numerical examples they found that in both ro-bust and nonrobust models short selling and risk-neutralinterest rates would increase sharpe ratios Robust optimi-zation solves the effect of parameter uncertainty on theoptimal solution well In addition in real financial marketsinvestorsrsquo behaviors are influenced by such uncertaintyfactors as politics psychology and social cognition Fuzzyset theory [21] is one of the effective methods to deal withsuch uncertainty Many scholars have tried to employ fuzzyvariables to manage portfolio selection problem and buildmany fuzzy portfolio models [22ndash24] Wang and Zhu [25]introduced the fuzzy portfolio selection models In 2012Tsaur [26] constructed a fuzzy portfolio model with theparameters of fuzzy-input return rates and fuzzy-outputproportions Zhou et al [27] proposed the concept of fuzzysemientropy (ey used the semientropy to quantify thedownside risk and set up two mean-semientropy portfolioselection models To obtain the optimal solution they usedthe genetic algorithm Possibility theory is an importanttheory of fuzzy sets which was first proposed by Zadeh [28]and developed by Dubois and Prade [29] In possibilitytheory the relationship between fuzzy variables and pos-sibility distributions is the same as that between randomvariables and probability distributions in probability theoryTanaka et al [30] firstly proposed the possibilistic portfolioselection model In this model the fuzzy variables wereconsidered to follow the exponential possibility distribu-tions In 1999 Tanaka and Guo [31] believed that the upperand lower possibility distributions can be used to reflectexpertsrsquo knowledge in the portfolio selection modelCarlsson and Fuller [32] introduced the notions of lower andupper possibilistic mean values as well as the notations forcrisp possibilistic mean value and variance of continuouspossibility distributions Based on that Carlsson et al [33]assumed that the return of securities is trapezoidal fuzzyvariables and found the optimal portfolio model with thehighest utility score by scoring the utility Zhang and Nie[34] extended the possibilistic mean and variance conceptsproposed by Carlsson and Fuller and presented the notionsof upper and lower possibilistic variances and covariances offuzzy numbers (en in 2007 Zhang [35] proposed apossibilistic mean-standard deviation model in which theinvestment proportion has lower bound constrained Zhanget al [36] proposed two kinds of portfolio selection models

based on upper and lower possibilistic means and variancesand the notions of upper and lower possibilistic efficientportfolios In order to better integrate an uncertain decisionenvironment with vagueness and ambiguity in 2009 Zhanget al [37] proposed a possibilistic mean-variance portfolioselection model based on the definitions of the possibilisticreturn and possibilistic risk In this model the returns ofassets were fuzzy variables with LR-type possibility distri-butions Based on the definitions of the upper and lowerpossibilistic mean Zhang et al [38] defined the interval-valued possibilistic mean and possibilistic variance Based onthese a new portfolio selection model with the maximumutility was established Li et al [39] developed a possibilisticportfolio model under VaR and risk-free investment con-straints Liu and Zhang [40] defined the product of multiplefuzzy numbersrsquo possibilistic mean and variance and thenbased on these definitions they developed three multiperiodfuzzy portfolio optimization models To solve these modelsthey applied a novel fuzzy programming approach-basedself-adaptive differential evolution algorithm

Several modifications to the basic MVM have beensuggested in the literature to consider more realistic factorssuch as liquidity budget and lowerupper bound con-straints while deciding the allocation of money amongassets Liquidity refers to the ability to transact a largenumber of shares at prices that do not vary substantiallyfrom past prices unless new information enters the market

(ere are various ways of measuring liquidity amongwhich trading volume number of trades transactionamount turnover rate and velocity of circulation arecommonly used (e turnover rate was introduced by Dataret al [41] (e turnover rate is the total amount of tradedshares divided by the total net asset value of the fund over aparticular period Marshalla and Young [42] argued thatliquidity is the most important factor in a portfolio In ourmodel turnover rate is controlled through fuzzy chanceconstraint Fuzzy portfolio selection with chance constraintdifferent assumptions and estimation methods has beendiscussed in the literature (see [43ndash46]) Barak et al [47]developed a mean-variance-skewness fuzzy portfolio modelwith cardinality constraint and considered the fuzzy chanceconstraint to measure portfolio liquidity Furthermore theydesigned a genetic algorithm to solve the model

In real financial markets short-time and institutionalinvestors hope to not only reach the expected rate ofreturn but also ensure that the liquidity of the portfolioshould not be lower than the expected value Moreover inaddition to return risk and liquidity the thresholdconstraint is also the major concern for researchers andpractitioners because in order to manage the portfoliomore effectively it is necessary to limit the upper andlower bounds (threshold constraints) of the capitalinvested in each asset (e core of this study was inspiredby Li et al [39] and Barak et al [47] and has been sub-sequently further promoted and developed by other re-searchers Numerous studies have been done aboutpossibilistic portfolio selection but a few papers considerliquidity constraint and regard asset liquidity as fuzzyvariables obey the possibility distribution

2 Complexity

Our goal is to analyze the return-risk trade-off with li-quidity constraint and threshold constraints under an un-certain market environment To increase the applicability ofthe model the return rate of assets is expressed as a fuzzyvariable which is associated with a normal possibility dis-tribution Liquidity is measured by turnover rate and is alsorepresented by a fuzzy variable associated with a normalpossibility distribution (us we established a possibilisticportfolio selection model with fuzzy chance constraints Byusing the possibility theory we transformed the chance-constrained model into a deterministic mathematical modeland obtained the solution for the model

(e rest of this paper is organized as follows In Section2 we present some basic concepts regarding the possibilitytheory and the notions of the possibilistic mean andvariance of a fuzzy number At the same time in thissection we recall the notion of normal possibility distri-bution and introduce a theorem about it In Section 3 wepropose a possibilistic portfolio model with liquidityconstraint and threshold constraints We suppose theexpected rate of return and the turnover rate of the assetsare both normally distributed fuzzy variables and thenprovide a solution for the model by a theorem Section 4provides a numerical example to illustrate the proposedapproach Section 5 concludes and provides directions forfuture research

2 Preliminaries

In this section we review some definitions and propertiesLet ξ be a fuzzy variable with membership function μ

and let r be a real number (en the possibility of ξ isdefined as follows

Pos ξ ge r supxger

μ(x) (1)

Definition 1 (e upper possibilistic mean value of 1113957A withα-level set Aα [a(α) b(α)] is defined as

Mlowast(1113957A)

111393810 Pos[1113957Age b(α)]b(α)dα

111393810 Pos[1113957Age b(α)]dα

211139461

0αb(α)dα (2)

where Pos denotes the possibility measure

Definition 2 (e lower possibilistic mean value of 1113957A isdefined as

Mlowast(1113957A)

111393810 Pos[1113957Ale a(α)]a(α)dα

111393810 Pos[1113957Ale a(α)]dα

211139461

0αa(α)dα (3)

Definition 3 (e possibilistic mean value of 1113957A is defined as

M(1113957A) Mlowast(

1113957A) + Mlowast(1113957A)

2 1113946

1

0α(a(α) + b(α))dα (4)

Definition 4 (e possibilistic variance of 1113957A is defined as

Var(1113957A) 11139461

0Pos[1113957Ale a(α)]

a(α) + b(α)

2minus a(α)1113890 1113891

2

1113888 1113889dα

+ 11139461

0Pos[1113957Age b(α)]

a(α) + b(α)

2minus b(α)1113890 1113891

2

1113888 1113889dα

11139461

0α

a(α) + b(α)

2minus a(α)1113890 1113891

2

1113888 1113889dα

+ 11139461

0α

a(α) + b(α)

2minus b(α)1113890 1113891

2

1113888 1113889dα

12

11139461

0α[b(α) minus a(α)]

2dα

(5)

Definition 5 (e possibilistic covariance between fuzzynumbers 1113957A and 1113957B is defined as

Cov(1113957A 1113957B) 12

11139461

0α b1(α) minus a1(α)( 1113857 b2(α) minus a2(α)( 1113857dα

(6)

Lemma 1 (see[25]) Let α β isin R and let 1113957A and 1113957B be fuzzynumbers en

(1) M(α1113957A + β1113957B) αM(1113957A) + βM(1113957B)

(2) Var(α1113957A + β1113957B) α2Var(1113957A) + β2Var(1113957B)

+2|αβ|Cov(1113957A 1113957B)

Definition 6 (e fuzzy variable ξ is obeying the normalpossibility distribution if its membership function is

μξ(x) exp minusx minus μσ

1113874 11138752

1113896 1113897 (7)

(en it can be written as ξ sim FN(μ σ2)

Example 1 If the fuzzy variable ξ sim FN(μ σ2) then itsλ-level set is

(ξ)k μ minus σln λminus1

1113968 μ + σ

ln λminus1

11139681113876 1113877 λ isin (0 1) (8)

In fact from (4) we obtain

M(ξ) Mlowast(ξ) + Mlowast(ξ)

2 1113946

1

0λ(a(λ) + b(λ))dλ

11139461

0λ μ + σ

lnλminus1

1113968+ μ minus σ

lnλminus1

11139681113874 1113875dλ

11139461

02μλdλ μ

(9)

It follows from (5) that

Complexity 3

Var(ξ) 12

11139461

0λ μ + σ

lnλminus1

1113968minus μ + σ

lnλminus1

11139681113874 1113875

2dλ

12

11139461

04λσ2 lnλminus1dλ

2σ2 11139461

0λ lnλminus1dλ

2σ214

12σ2

(10)

Next we show the distribution of 1113936nk1 λkξk

Theorem 1 If ξk k 1 2 n are n normally distributedfuzzy variables expressed as ξk sim FN(μk σ2k) andλk k 1 2 n are n real numbers then

1113944

n

k1λkξk sim FN 1113944

n

k1λkμk

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠ (11)

Proof According to Lemma 1 and (9) one has

M 1113944n

k1λkξk

⎛⎝ ⎞⎠ 1113944n

k1λkM ξk( 1113857 1113944

n

k1λkμk (12)

From (6) we deduce

Cov ξi ξj1113872 1113873 12

11139461

0α2σ i

lnαminus1

radic2σj

lnαminus1

radicdα

2σiσj 11139461

0α lnαminus1dα

2σiσj

14

12σiσj i j 1 2 n

(13)

It follows from Lemma 1 and (10) that

Var 1113944n

k1λkξk

⎛⎝ ⎞⎠ 1113944n

k1λ2kVar ξk( 1113857 + 2 1113944

n

igtj1λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868Cov ξi ξj1113872 1113873

1113944n

k1λ2k12σ2k + 2 1113944

n

igtj1

12σiσj λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868

12

1113944

n

k1λ2kσ

2k + 2 1113944

n

igtj1σiσj λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

(14)

(us

1113944

n

k1λkξk sim FN 1113944

n

k1λkμk

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠ (15)

(is completes the proof

3 Model Foundation

31 Possibilistic Portfolio Model with Fuzzy Liquidity Con-straint and Risk-Free Investment Suppose that there are n

risky assets and one risk-free asset available for investmentLet 1113957rk be the return rate of asset k k 1 2 n which is afuzzy number Let xk represent the proportion invested inasset k k 1 2 n and let rf be the return of the risk-freeasset (us the return 1113957R on the portfolio can be written as

1113957R 1113944n

k1xk1113957rk + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ (16)

Obviously 1113957R is a fuzzy numberTo establish a new model we need the following values(e possibilistic mean of the portfolio return 1113957R is given by

M(1113957R) 1113944n

k1xkM 1113957rk( 1113857 + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ (17)

(e possibilistic variance of 1113957R is written as

Var(1113957R) 1113944n

k1x2kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixj

11138681113868111386811138681113868

11138681113868111386811138681113868Cov 1113957ri 1113957rj1113872 1113873

1113944n

k1x2kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

(18)

Now we can establish the following possibilistic port-folio selection model with fuzzy liquidity constraint

(PL1)

minVar(1113957R) 1113944

n

k1x2

kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

st 1113944

n

k1xkM 1113957rk( 1113857 + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ge μ

Pos 1113944

n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ge 1 minus α

1113944

n

k1xk le 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

where μ is the underestimated expected return rate and l0 is thepredetermined value dk and gk represent respectively thelower and the upper bounds on investment in asset k k

1 2 n 1113957lk is the turnover rate of asset k k 1 2 n

which reflects the liquidity of the asset α reflects the sensitivityof the investor If the value of α is close to 0 then the investor issensitive to liquidity of the portfolio Otherwise the investor isnot sensitive to the portfoliorsquos liquidity

In this section we suppose that the return rate of assetk k 1 2 n is a normally distributed fuzzy variabledenoted as 1113957rk sim FN(μk σ2k) and its membership function is

A1113957rk(x) exp minus

x minus μk

σk

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭ (20)

4 Complexity

(e turnover rate of asset k k 1 2 n is also anormally distributed fuzzy variable denoted as1113957lk sim FN(ak b2k) So its membership function is

A1113957lk(x) exp minus

x minus ak

bk

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭ (21)

From (eorem 1 we obtain

1113957R sim FN 1113944n

k1xkμk + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

1113944

n

k1

1113957lkxk sim FN 1113944n

k1xkak

12

1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

(22)

(en themembership function of 1113936nk1

1113957lkxk is defined by

A(u) exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ (23)

Furthermore

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ supugel0

exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭

exp minusl0 minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ for 1113944n

k1xkak lt l0

1 for 1113944n

k1xkak ge l0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

From (19) and (24) we have

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

(25)

From (22) and (25) (PL1) can be transformed into

(NPL1)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xk μk minus rf1113872 1113873 + rf ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk le 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

32 PossibilisticPortfolioModelwithoutRisk-Free InvestmentSimilar to the previous section we propose a portfolio se-lection model without risk-free investment as follows

(PL2)

minVar(1113957R) 1113944n

k1x2

kVar 1113957rk( 1113857 + 2 1113944n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

st 1113944n

k1xkM 1113957rk( 1113857ge μ

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ge 1 minus α

1113944

n

k1xk 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

If the return rate of asset k k 1 2 n is a normallydistributed fuzzy variable denoted as 1113957rk sim FN(μk σ2k) andthe turnover rate of asset k k 1 2 n is also a normallydistributed fuzzy variable denoted as 1113957lk sim FN(ak b2k) thenthe model (PL2) can be transformed into

(NPL2)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xkμk ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk 1

0ledk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

(NPL1) and (NPL2) are quadratic programmingproblems and can be solved by MATLAB Lingo etc

4 Numerical Example

In this section we give a real portfolio example to illustrateour approach In this example we selected eight stocks fromthe Shanghai Stock Exchange We collected data on monthlyreturns and turnover rate for each of the eight stocks over theperiod of January 2006 to December 2006 from the RESSETFinancial Research Database We use SPSS to generate thefrequency distributions of the monthly returns and turnoverrates

Table 1 shows the transaction codes and the frequencydistributions of the monthly returns while Table 2 shows thetransaction codes and the frequency distributions of themonthly turnover rates

Complexity 5

Let the risk-free asset be a bank deposit We use thethree-month deposit interest rates as the return on risk-freeassets So we get the return on risk-free assets rf 28 Ifthe possibility of the portfoliorsquos turnover rate must be morethan 0002 (in more than 90 times) that is l0 02()

and α 10 then the lower bound of investment ratio xk

must be d 002 0 01 0 002 01 01 005 and the upperbound g 03 02 03 02 02 04 04 03 By solving themodel (NPL1) and (NPL2) the possibilistic efficientportfolios for the different micros are obtained as shown inTables 3 and 4

From these two tables we can see that the risk increasesas μ increases We also can see that when the value of μ is low(eg μ 3) the proportion that invests is low For ex-ample when the value of μ is equal to 3 the investmentproportion with risk-free is the lower bound but the in-vestment proportion without risk-free is 002 00702 01 002 01 02098 and 03 Figure 1 shows these two efficientportfolios From Figure 1 we can see that when the risk-freeinvestment is included in the portfolio the risk is lower atthe same value ofμ And from Figure 1 we also can draw theconclusion that the risk of the portfolios can be spread bydiversification (e impact of liquidity constraint on theportfolio is shown in Figure 2 Figure 2 shows some pos-sibilistic efficient portfolios with and without liquidityconstraints

It can be seen from Figure 2 that under the same ex-pected return those investors who have requirements forfinancial liquidity need to take on greater risks In otherwords if investors want to keep the risks they take constantthe demand for financial liquidity will reduce the expectedreturn of the portfolio In addition the impact of liquidityfloor l0 on the portfolio is shown in Figure 3 Figure 3 showssome possibilistic efficient portfolios with different l0 (e

impact of l0 on the portfolio depends on the slope and theintercept of the l0 line and the position of the possibilisticefficient frontier

As shown in Figure 3 when the position of the possi-bilistic efficient frontier and the slope of the l0 line remainunchanged the intercept changes of the l0 line will affect theoptimal solution of the model As l0moves from ldquoardquo point toldquobrdquo point the optimal solution of the model would notchange But when l0 moves from ldquoardquo point to ldquocrdquo point thischange would affect the optimal solution of the model

Analogous to the above if the turnover rate of theportfolio must be more than 0002 ie l0 02() and theexpected rate of return must reach 8 or more then thelower and upper bound of investment ratio xk should bed 002 0 01 0 002 01 01 005 and g 03 02 0302 02 04 04 03 respectively By solving the model(NPL1) with different investor sensitivity parameter valueswe obtain the possibilistic efficient portfolios in Table 5 (erelationship between investor sensitivity and the portfoliorisk is shown in Figure 4

From Figure 4 we can see that an increase in investorsensitivity is associated with a decrease in the portfolio riskand when investor sensitivity reaches a certain threshold theportfolio risk no longer reduces (at is to say in this case ifinvestors seek at least 8 of the profits regardless of theinvestorsrsquo preference for liquidity the minimum risk of theportfolio is 05126 (or 5126)

(e impact of liquidity sensitivity α on the portfolio isshown in Figure 5 Figure 5 shows some possibilistic efficientportfolios with different α In Figure 5 when the position ofthe possibilistic efficient frontier and the intercept of the αline remain unchanged the slope changes of the α line willaffect the optimal solution of the model As α moves fromldquoardquo point to ldquobrdquo point the optimal solution of the model

Table 1 (e possibility distribution of returns

Stock code μk σk

600058 006 0167600028 009 0102600089 010 0207600115 004 0111600170 003 0049600495 010 0080600526 001 0086600662 003 0076

Table 2 (e possibility distribution of turnover rates ()

Stock code ak bk

600058 4070 31762600028 177 0802600089 3251 10681600115 975 4384600170 1486 8796600495 7385 51658600526 6547 16565600662 4014 19587

6 Complexity

ndash01

0

01

02

03

04

05

06

07

08

2 3 4 5 6 7 8 9

Without risk-free investmentWith risk-free investment

Figure 1 Variation of portfolio risks with μ

Table 3 Possibilistic efficient portfolios with risk-free investment for the different micros with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 0 01 0 002 01 01 005 00901 0395 002 0 01 0 002 02330 01 005 01441 052365 002 00717 01 0 002 03737 01 005 02654 073547 002 01521 01 0 002 03717 01 005 03287 0813875 002 02 01497 0 002 03482 01 005 04408 088798 002 02 02725 0 002 02929 01 005 06485 09554812 002 02 03 00197 002 02779 01 005 07252 09876

Table 4 Possibilistic efficient portfolios without risk-free investment for the different μs with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 00702 01 0 02 01 02098 03 03607 15 002 00760 01 0 02 01547 01493 03 03640 165 002 01554 01 0 02 02868 01 01378 04004 17 002 01906 01 0 02 03281 01 00613 04165 175 002 02 01817 0 01386 03097 01 005 05309 18 002 02 02853 0 00671 02776 01 005 06910 1812 002 02 03 003 0024 0276 01 005 07252 1

003

004

005

006

007

008

009

02 03 04 05 06 07 08 09

Expe

cted

rate

of r

etur

n

Deviation

Without liquidity constraintWith liquidity constraint

Figure 2 Some possibilistic efficient portfolios with and without liquidity constraint

Complexity 7

would not change But when α moves from ldquoardquo point to ldquocrdquopoint this change would affect the optimal solution of themodel

5 Conclusions and Directions forFuture Research

In this paper we proposed a possibilistic portfolio modelwhich is different from the MVM proposed by MarkowitzUnlike the MVM we measured the liquidity of asset as theturnover rate Besides we assumed that the expected rate ofreturns and turnover rates of assets are fuzzy variableswhich follow the normal possibility distribution We then

applied the possibility theory and fuzzy chance constraint toconsider the turnover rate and obtained the solution for themodel Furthermore we illustrated our proposed effectiveapproaches for the portfolio construction using numericalexamples In these examples we analyzed the risk betweenthe portfolios with and without risk-free constraints Wealso analyzed changes in the portfoliorsquos risk with respect to

0

001

002

003

004

005

006

007

008

009

0 01 02 03 04 05 06 07 08Deviation

Expe

cted

rate

of r

etur

n

Possibilistic efficient frontiera

bc

Figure 3 Some possibilistic efficient portfolios with different l0

Table 5 Possibilistic efficient portfolios with risk-free investment for the different α values with l0 02()and μ 8()

α () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

01 002 02 02725 0 002 02929 01 005 06485 0955402 002 02 02544 0 002 03110 01 005 06241 0956403 002 02 02315 0 002 03339 01 005 05942 0960504 002 02 02012 0 002 03642 01 005 05560 0955405 002 02 01654 0 002 04 01 005 05126 0955406 002 02 01654 0 002 04 01 005 05126 09554

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07

Risk

The sensitivity to liquidity

Figure 4 Variation of portfolio risk with investor sensitivity α 0

001

002

003

004

005

006

007

008

009

Expe

cted

rate

of r

etur

n

02 04 06 080Deviation

a

bc

Possibilistic efficientfrontier

Figure 5 Some possibilistic efficient portfolios with different α

8 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

Our goal is to analyze the return-risk trade-off with li-quidity constraint and threshold constraints under an un-certain market environment To increase the applicability ofthe model the return rate of assets is expressed as a fuzzyvariable which is associated with a normal possibility dis-tribution Liquidity is measured by turnover rate and is alsorepresented by a fuzzy variable associated with a normalpossibility distribution (us we established a possibilisticportfolio selection model with fuzzy chance constraints Byusing the possibility theory we transformed the chance-constrained model into a deterministic mathematical modeland obtained the solution for the model

(e rest of this paper is organized as follows In Section2 we present some basic concepts regarding the possibilitytheory and the notions of the possibilistic mean andvariance of a fuzzy number At the same time in thissection we recall the notion of normal possibility distri-bution and introduce a theorem about it In Section 3 wepropose a possibilistic portfolio model with liquidityconstraint and threshold constraints We suppose theexpected rate of return and the turnover rate of the assetsare both normally distributed fuzzy variables and thenprovide a solution for the model by a theorem Section 4provides a numerical example to illustrate the proposedapproach Section 5 concludes and provides directions forfuture research

2 Preliminaries

In this section we review some definitions and propertiesLet ξ be a fuzzy variable with membership function μ

and let r be a real number (en the possibility of ξ isdefined as follows

Pos ξ ge r supxger

μ(x) (1)

Definition 1 (e upper possibilistic mean value of 1113957A withα-level set Aα [a(α) b(α)] is defined as

Mlowast(1113957A)

111393810 Pos[1113957Age b(α)]b(α)dα

111393810 Pos[1113957Age b(α)]dα

211139461

0αb(α)dα (2)

where Pos denotes the possibility measure

Definition 2 (e lower possibilistic mean value of 1113957A isdefined as

Mlowast(1113957A)

111393810 Pos[1113957Ale a(α)]a(α)dα

111393810 Pos[1113957Ale a(α)]dα

211139461

0αa(α)dα (3)

Definition 3 (e possibilistic mean value of 1113957A is defined as

M(1113957A) Mlowast(

1113957A) + Mlowast(1113957A)

2 1113946

1

0α(a(α) + b(α))dα (4)

Definition 4 (e possibilistic variance of 1113957A is defined as

Var(1113957A) 11139461

0Pos[1113957Ale a(α)]

a(α) + b(α)

2minus a(α)1113890 1113891

2

1113888 1113889dα

+ 11139461

0Pos[1113957Age b(α)]

a(α) + b(α)

2minus b(α)1113890 1113891

2

1113888 1113889dα

11139461

0α

a(α) + b(α)

2minus a(α)1113890 1113891

2

1113888 1113889dα

+ 11139461

0α

a(α) + b(α)

2minus b(α)1113890 1113891

2

1113888 1113889dα

12

11139461

0α[b(α) minus a(α)]

2dα

(5)

Definition 5 (e possibilistic covariance between fuzzynumbers 1113957A and 1113957B is defined as

Cov(1113957A 1113957B) 12

11139461

0α b1(α) minus a1(α)( 1113857 b2(α) minus a2(α)( 1113857dα

(6)

Lemma 1 (see[25]) Let α β isin R and let 1113957A and 1113957B be fuzzynumbers en

(1) M(α1113957A + β1113957B) αM(1113957A) + βM(1113957B)

(2) Var(α1113957A + β1113957B) α2Var(1113957A) + β2Var(1113957B)

+2|αβ|Cov(1113957A 1113957B)

Definition 6 (e fuzzy variable ξ is obeying the normalpossibility distribution if its membership function is

μξ(x) exp minusx minus μσ

1113874 11138752

1113896 1113897 (7)

(en it can be written as ξ sim FN(μ σ2)

Example 1 If the fuzzy variable ξ sim FN(μ σ2) then itsλ-level set is

(ξ)k μ minus σln λminus1

1113968 μ + σ

ln λminus1

11139681113876 1113877 λ isin (0 1) (8)

In fact from (4) we obtain

M(ξ) Mlowast(ξ) + Mlowast(ξ)

2 1113946

1

0λ(a(λ) + b(λ))dλ

11139461

0λ μ + σ

lnλminus1

1113968+ μ minus σ

lnλminus1

11139681113874 1113875dλ

11139461

02μλdλ μ

(9)

It follows from (5) that

Complexity 3

Var(ξ) 12

11139461

0λ μ + σ

lnλminus1

1113968minus μ + σ

lnλminus1

11139681113874 1113875

2dλ

12

11139461

04λσ2 lnλminus1dλ

2σ2 11139461

0λ lnλminus1dλ

2σ214

12σ2

(10)

Next we show the distribution of 1113936nk1 λkξk

Theorem 1 If ξk k 1 2 n are n normally distributedfuzzy variables expressed as ξk sim FN(μk σ2k) andλk k 1 2 n are n real numbers then

1113944

n

k1λkξk sim FN 1113944

n

k1λkμk

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠ (11)

Proof According to Lemma 1 and (9) one has

M 1113944n

k1λkξk

⎛⎝ ⎞⎠ 1113944n

k1λkM ξk( 1113857 1113944

n

k1λkμk (12)

From (6) we deduce

Cov ξi ξj1113872 1113873 12

11139461

0α2σ i

lnαminus1

radic2σj

lnαminus1

radicdα

2σiσj 11139461

0α lnαminus1dα

2σiσj

14

12σiσj i j 1 2 n

(13)

It follows from Lemma 1 and (10) that

Var 1113944n

k1λkξk

⎛⎝ ⎞⎠ 1113944n

k1λ2kVar ξk( 1113857 + 2 1113944

n

igtj1λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868Cov ξi ξj1113872 1113873

1113944n

k1λ2k12σ2k + 2 1113944

n

igtj1

12σiσj λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868

12

1113944

n

k1λ2kσ

2k + 2 1113944

n

igtj1σiσj λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

(14)

(us

1113944

n

k1λkξk sim FN 1113944

n

k1λkμk

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠ (15)

(is completes the proof

3 Model Foundation

31 Possibilistic Portfolio Model with Fuzzy Liquidity Con-straint and Risk-Free Investment Suppose that there are n

risky assets and one risk-free asset available for investmentLet 1113957rk be the return rate of asset k k 1 2 n which is afuzzy number Let xk represent the proportion invested inasset k k 1 2 n and let rf be the return of the risk-freeasset (us the return 1113957R on the portfolio can be written as

1113957R 1113944n

k1xk1113957rk + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ (16)

Obviously 1113957R is a fuzzy numberTo establish a new model we need the following values(e possibilistic mean of the portfolio return 1113957R is given by

M(1113957R) 1113944n

k1xkM 1113957rk( 1113857 + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ (17)

(e possibilistic variance of 1113957R is written as

Var(1113957R) 1113944n

k1x2kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixj

11138681113868111386811138681113868

11138681113868111386811138681113868Cov 1113957ri 1113957rj1113872 1113873

1113944n

k1x2kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

(18)

Now we can establish the following possibilistic port-folio selection model with fuzzy liquidity constraint

(PL1)

minVar(1113957R) 1113944

n

k1x2

kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

st 1113944

n

k1xkM 1113957rk( 1113857 + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ge μ

Pos 1113944

n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ge 1 minus α

1113944

n

k1xk le 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

where μ is the underestimated expected return rate and l0 is thepredetermined value dk and gk represent respectively thelower and the upper bounds on investment in asset k k

1 2 n 1113957lk is the turnover rate of asset k k 1 2 n

which reflects the liquidity of the asset α reflects the sensitivityof the investor If the value of α is close to 0 then the investor issensitive to liquidity of the portfolio Otherwise the investor isnot sensitive to the portfoliorsquos liquidity

In this section we suppose that the return rate of assetk k 1 2 n is a normally distributed fuzzy variabledenoted as 1113957rk sim FN(μk σ2k) and its membership function is

A1113957rk(x) exp minus

x minus μk

σk

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭ (20)

4 Complexity

(e turnover rate of asset k k 1 2 n is also anormally distributed fuzzy variable denoted as1113957lk sim FN(ak b2k) So its membership function is

A1113957lk(x) exp minus

x minus ak

bk

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭ (21)

From (eorem 1 we obtain

1113957R sim FN 1113944n

k1xkμk + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

1113944

n

k1

1113957lkxk sim FN 1113944n

k1xkak

12

1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

(22)

(en themembership function of 1113936nk1

1113957lkxk is defined by

A(u) exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ (23)

Furthermore

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ supugel0

exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭

exp minusl0 minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ for 1113944n

k1xkak lt l0

1 for 1113944n

k1xkak ge l0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

From (19) and (24) we have

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

(25)

From (22) and (25) (PL1) can be transformed into

(NPL1)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xk μk minus rf1113872 1113873 + rf ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk le 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

32 PossibilisticPortfolioModelwithoutRisk-Free InvestmentSimilar to the previous section we propose a portfolio se-lection model without risk-free investment as follows

(PL2)

minVar(1113957R) 1113944n

k1x2

kVar 1113957rk( 1113857 + 2 1113944n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

st 1113944n

k1xkM 1113957rk( 1113857ge μ

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ge 1 minus α

1113944

n

k1xk 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

If the return rate of asset k k 1 2 n is a normallydistributed fuzzy variable denoted as 1113957rk sim FN(μk σ2k) andthe turnover rate of asset k k 1 2 n is also a normallydistributed fuzzy variable denoted as 1113957lk sim FN(ak b2k) thenthe model (PL2) can be transformed into

(NPL2)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xkμk ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk 1

0ledk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

(NPL1) and (NPL2) are quadratic programmingproblems and can be solved by MATLAB Lingo etc

4 Numerical Example

In this section we give a real portfolio example to illustrateour approach In this example we selected eight stocks fromthe Shanghai Stock Exchange We collected data on monthlyreturns and turnover rate for each of the eight stocks over theperiod of January 2006 to December 2006 from the RESSETFinancial Research Database We use SPSS to generate thefrequency distributions of the monthly returns and turnoverrates

Table 1 shows the transaction codes and the frequencydistributions of the monthly returns while Table 2 shows thetransaction codes and the frequency distributions of themonthly turnover rates

Complexity 5

Let the risk-free asset be a bank deposit We use thethree-month deposit interest rates as the return on risk-freeassets So we get the return on risk-free assets rf 28 Ifthe possibility of the portfoliorsquos turnover rate must be morethan 0002 (in more than 90 times) that is l0 02()

and α 10 then the lower bound of investment ratio xk

must be d 002 0 01 0 002 01 01 005 and the upperbound g 03 02 03 02 02 04 04 03 By solving themodel (NPL1) and (NPL2) the possibilistic efficientportfolios for the different micros are obtained as shown inTables 3 and 4

From these two tables we can see that the risk increasesas μ increases We also can see that when the value of μ is low(eg μ 3) the proportion that invests is low For ex-ample when the value of μ is equal to 3 the investmentproportion with risk-free is the lower bound but the in-vestment proportion without risk-free is 002 00702 01 002 01 02098 and 03 Figure 1 shows these two efficientportfolios From Figure 1 we can see that when the risk-freeinvestment is included in the portfolio the risk is lower atthe same value ofμ And from Figure 1 we also can draw theconclusion that the risk of the portfolios can be spread bydiversification (e impact of liquidity constraint on theportfolio is shown in Figure 2 Figure 2 shows some pos-sibilistic efficient portfolios with and without liquidityconstraints

It can be seen from Figure 2 that under the same ex-pected return those investors who have requirements forfinancial liquidity need to take on greater risks In otherwords if investors want to keep the risks they take constantthe demand for financial liquidity will reduce the expectedreturn of the portfolio In addition the impact of liquidityfloor l0 on the portfolio is shown in Figure 3 Figure 3 showssome possibilistic efficient portfolios with different l0 (e

impact of l0 on the portfolio depends on the slope and theintercept of the l0 line and the position of the possibilisticefficient frontier

As shown in Figure 3 when the position of the possi-bilistic efficient frontier and the slope of the l0 line remainunchanged the intercept changes of the l0 line will affect theoptimal solution of the model As l0moves from ldquoardquo point toldquobrdquo point the optimal solution of the model would notchange But when l0 moves from ldquoardquo point to ldquocrdquo point thischange would affect the optimal solution of the model

Analogous to the above if the turnover rate of theportfolio must be more than 0002 ie l0 02() and theexpected rate of return must reach 8 or more then thelower and upper bound of investment ratio xk should bed 002 0 01 0 002 01 01 005 and g 03 02 0302 02 04 04 03 respectively By solving the model(NPL1) with different investor sensitivity parameter valueswe obtain the possibilistic efficient portfolios in Table 5 (erelationship between investor sensitivity and the portfoliorisk is shown in Figure 4

From Figure 4 we can see that an increase in investorsensitivity is associated with a decrease in the portfolio riskand when investor sensitivity reaches a certain threshold theportfolio risk no longer reduces (at is to say in this case ifinvestors seek at least 8 of the profits regardless of theinvestorsrsquo preference for liquidity the minimum risk of theportfolio is 05126 (or 5126)

(e impact of liquidity sensitivity α on the portfolio isshown in Figure 5 Figure 5 shows some possibilistic efficientportfolios with different α In Figure 5 when the position ofthe possibilistic efficient frontier and the intercept of the αline remain unchanged the slope changes of the α line willaffect the optimal solution of the model As α moves fromldquoardquo point to ldquobrdquo point the optimal solution of the model

Table 1 (e possibility distribution of returns

Stock code μk σk

600058 006 0167600028 009 0102600089 010 0207600115 004 0111600170 003 0049600495 010 0080600526 001 0086600662 003 0076

Table 2 (e possibility distribution of turnover rates ()

Stock code ak bk

600058 4070 31762600028 177 0802600089 3251 10681600115 975 4384600170 1486 8796600495 7385 51658600526 6547 16565600662 4014 19587

6 Complexity

ndash01

0

01

02

03

04

05

06

07

08

2 3 4 5 6 7 8 9

Without risk-free investmentWith risk-free investment

Figure 1 Variation of portfolio risks with μ

Table 3 Possibilistic efficient portfolios with risk-free investment for the different micros with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 0 01 0 002 01 01 005 00901 0395 002 0 01 0 002 02330 01 005 01441 052365 002 00717 01 0 002 03737 01 005 02654 073547 002 01521 01 0 002 03717 01 005 03287 0813875 002 02 01497 0 002 03482 01 005 04408 088798 002 02 02725 0 002 02929 01 005 06485 09554812 002 02 03 00197 002 02779 01 005 07252 09876

Table 4 Possibilistic efficient portfolios without risk-free investment for the different μs with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 00702 01 0 02 01 02098 03 03607 15 002 00760 01 0 02 01547 01493 03 03640 165 002 01554 01 0 02 02868 01 01378 04004 17 002 01906 01 0 02 03281 01 00613 04165 175 002 02 01817 0 01386 03097 01 005 05309 18 002 02 02853 0 00671 02776 01 005 06910 1812 002 02 03 003 0024 0276 01 005 07252 1

003

004

005

006

007

008

009

02 03 04 05 06 07 08 09

Expe

cted

rate

of r

etur

n

Deviation

Without liquidity constraintWith liquidity constraint

Figure 2 Some possibilistic efficient portfolios with and without liquidity constraint

Complexity 7

would not change But when α moves from ldquoardquo point to ldquocrdquopoint this change would affect the optimal solution of themodel

5 Conclusions and Directions forFuture Research

In this paper we proposed a possibilistic portfolio modelwhich is different from the MVM proposed by MarkowitzUnlike the MVM we measured the liquidity of asset as theturnover rate Besides we assumed that the expected rate ofreturns and turnover rates of assets are fuzzy variableswhich follow the normal possibility distribution We then

applied the possibility theory and fuzzy chance constraint toconsider the turnover rate and obtained the solution for themodel Furthermore we illustrated our proposed effectiveapproaches for the portfolio construction using numericalexamples In these examples we analyzed the risk betweenthe portfolios with and without risk-free constraints Wealso analyzed changes in the portfoliorsquos risk with respect to

0

001

002

003

004

005

006

007

008

009

0 01 02 03 04 05 06 07 08Deviation

Expe

cted

rate

of r

etur

n

Possibilistic efficient frontiera

bc

Figure 3 Some possibilistic efficient portfolios with different l0

Table 5 Possibilistic efficient portfolios with risk-free investment for the different α values with l0 02()and μ 8()

α () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

01 002 02 02725 0 002 02929 01 005 06485 0955402 002 02 02544 0 002 03110 01 005 06241 0956403 002 02 02315 0 002 03339 01 005 05942 0960504 002 02 02012 0 002 03642 01 005 05560 0955405 002 02 01654 0 002 04 01 005 05126 0955406 002 02 01654 0 002 04 01 005 05126 09554

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07

Risk

The sensitivity to liquidity

Figure 4 Variation of portfolio risk with investor sensitivity α 0

001

002

003

004

005

006

007

008

009

Expe

cted

rate

of r

etur

n

02 04 06 080Deviation

a

bc

Possibilistic efficientfrontier

Figure 5 Some possibilistic efficient portfolios with different α

8 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

Var(ξ) 12

11139461

0λ μ + σ

lnλminus1

1113968minus μ + σ

lnλminus1

11139681113874 1113875

2dλ

12

11139461

04λσ2 lnλminus1dλ

2σ2 11139461

0λ lnλminus1dλ

2σ214

12σ2

(10)

Next we show the distribution of 1113936nk1 λkξk

Theorem 1 If ξk k 1 2 n are n normally distributedfuzzy variables expressed as ξk sim FN(μk σ2k) andλk k 1 2 n are n real numbers then

1113944

n

k1λkξk sim FN 1113944

n

k1λkμk

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠ (11)

Proof According to Lemma 1 and (9) one has

M 1113944n

k1λkξk

⎛⎝ ⎞⎠ 1113944n

k1λkM ξk( 1113857 1113944

n

k1λkμk (12)

From (6) we deduce

Cov ξi ξj1113872 1113873 12

11139461

0α2σ i

lnαminus1

radic2σj

lnαminus1

radicdα

2σiσj 11139461

0α lnαminus1dα

2σiσj

14

12σiσj i j 1 2 n

(13)

It follows from Lemma 1 and (10) that

Var 1113944n

k1λkξk

⎛⎝ ⎞⎠ 1113944n

k1λ2kVar ξk( 1113857 + 2 1113944

n

igtj1λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868Cov ξi ξj1113872 1113873

1113944n

k1λ2k12σ2k + 2 1113944

n

igtj1

12σiσj λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868

12

1113944

n

k1λ2kσ

2k + 2 1113944

n

igtj1σiσj λiλj

11138681113868111386811138681113868

11138681113868111386811138681113868⎛⎝ ⎞⎠

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

(14)

(us

1113944

n

k1λkξk sim FN 1113944

n

k1λkμk

12

1113944

n

k1λk

11138681113868111386811138681113868111386811138681113868σk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠ (15)

(is completes the proof

3 Model Foundation

31 Possibilistic Portfolio Model with Fuzzy Liquidity Con-straint and Risk-Free Investment Suppose that there are n

risky assets and one risk-free asset available for investmentLet 1113957rk be the return rate of asset k k 1 2 n which is afuzzy number Let xk represent the proportion invested inasset k k 1 2 n and let rf be the return of the risk-freeasset (us the return 1113957R on the portfolio can be written as

1113957R 1113944n

k1xk1113957rk + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ (16)

Obviously 1113957R is a fuzzy numberTo establish a new model we need the following values(e possibilistic mean of the portfolio return 1113957R is given by

M(1113957R) 1113944n

k1xkM 1113957rk( 1113857 + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ (17)

(e possibilistic variance of 1113957R is written as

Var(1113957R) 1113944n

k1x2kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixj

11138681113868111386811138681113868

11138681113868111386811138681113868Cov 1113957ri 1113957rj1113872 1113873

1113944n

k1x2kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

(18)

Now we can establish the following possibilistic port-folio selection model with fuzzy liquidity constraint

(PL1)

minVar(1113957R) 1113944

n

k1x2

kVar 1113957rk( 1113857 + 2 1113944

n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

st 1113944

n

k1xkM 1113957rk( 1113857 + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠ge μ

Pos 1113944

n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ge 1 minus α

1113944

n

k1xk le 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

where μ is the underestimated expected return rate and l0 is thepredetermined value dk and gk represent respectively thelower and the upper bounds on investment in asset k k

1 2 n 1113957lk is the turnover rate of asset k k 1 2 n

which reflects the liquidity of the asset α reflects the sensitivityof the investor If the value of α is close to 0 then the investor issensitive to liquidity of the portfolio Otherwise the investor isnot sensitive to the portfoliorsquos liquidity

In this section we suppose that the return rate of assetk k 1 2 n is a normally distributed fuzzy variabledenoted as 1113957rk sim FN(μk σ2k) and its membership function is

A1113957rk(x) exp minus

x minus μk

σk

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭ (20)

4 Complexity

(e turnover rate of asset k k 1 2 n is also anormally distributed fuzzy variable denoted as1113957lk sim FN(ak b2k) So its membership function is

A1113957lk(x) exp minus

x minus ak

bk

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭ (21)

From (eorem 1 we obtain

1113957R sim FN 1113944n

k1xkμk + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

1113944

n

k1

1113957lkxk sim FN 1113944n

k1xkak

12

1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

(22)

(en themembership function of 1113936nk1

1113957lkxk is defined by

A(u) exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ (23)

Furthermore

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ supugel0

exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭

exp minusl0 minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ for 1113944n

k1xkak lt l0

1 for 1113944n

k1xkak ge l0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

From (19) and (24) we have

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

(25)

From (22) and (25) (PL1) can be transformed into

(NPL1)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xk μk minus rf1113872 1113873 + rf ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk le 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

32 PossibilisticPortfolioModelwithoutRisk-Free InvestmentSimilar to the previous section we propose a portfolio se-lection model without risk-free investment as follows

(PL2)

minVar(1113957R) 1113944n

k1x2

kVar 1113957rk( 1113857 + 2 1113944n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

st 1113944n

k1xkM 1113957rk( 1113857ge μ

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ge 1 minus α

1113944

n

k1xk 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

If the return rate of asset k k 1 2 n is a normallydistributed fuzzy variable denoted as 1113957rk sim FN(μk σ2k) andthe turnover rate of asset k k 1 2 n is also a normallydistributed fuzzy variable denoted as 1113957lk sim FN(ak b2k) thenthe model (PL2) can be transformed into

(NPL2)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xkμk ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk 1

0ledk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

(NPL1) and (NPL2) are quadratic programmingproblems and can be solved by MATLAB Lingo etc

4 Numerical Example

In this section we give a real portfolio example to illustrateour approach In this example we selected eight stocks fromthe Shanghai Stock Exchange We collected data on monthlyreturns and turnover rate for each of the eight stocks over theperiod of January 2006 to December 2006 from the RESSETFinancial Research Database We use SPSS to generate thefrequency distributions of the monthly returns and turnoverrates

Table 1 shows the transaction codes and the frequencydistributions of the monthly returns while Table 2 shows thetransaction codes and the frequency distributions of themonthly turnover rates

Complexity 5

Let the risk-free asset be a bank deposit We use thethree-month deposit interest rates as the return on risk-freeassets So we get the return on risk-free assets rf 28 Ifthe possibility of the portfoliorsquos turnover rate must be morethan 0002 (in more than 90 times) that is l0 02()

and α 10 then the lower bound of investment ratio xk

must be d 002 0 01 0 002 01 01 005 and the upperbound g 03 02 03 02 02 04 04 03 By solving themodel (NPL1) and (NPL2) the possibilistic efficientportfolios for the different micros are obtained as shown inTables 3 and 4

From these two tables we can see that the risk increasesas μ increases We also can see that when the value of μ is low(eg μ 3) the proportion that invests is low For ex-ample when the value of μ is equal to 3 the investmentproportion with risk-free is the lower bound but the in-vestment proportion without risk-free is 002 00702 01 002 01 02098 and 03 Figure 1 shows these two efficientportfolios From Figure 1 we can see that when the risk-freeinvestment is included in the portfolio the risk is lower atthe same value ofμ And from Figure 1 we also can draw theconclusion that the risk of the portfolios can be spread bydiversification (e impact of liquidity constraint on theportfolio is shown in Figure 2 Figure 2 shows some pos-sibilistic efficient portfolios with and without liquidityconstraints

It can be seen from Figure 2 that under the same ex-pected return those investors who have requirements forfinancial liquidity need to take on greater risks In otherwords if investors want to keep the risks they take constantthe demand for financial liquidity will reduce the expectedreturn of the portfolio In addition the impact of liquidityfloor l0 on the portfolio is shown in Figure 3 Figure 3 showssome possibilistic efficient portfolios with different l0 (e

impact of l0 on the portfolio depends on the slope and theintercept of the l0 line and the position of the possibilisticefficient frontier

As shown in Figure 3 when the position of the possi-bilistic efficient frontier and the slope of the l0 line remainunchanged the intercept changes of the l0 line will affect theoptimal solution of the model As l0moves from ldquoardquo point toldquobrdquo point the optimal solution of the model would notchange But when l0 moves from ldquoardquo point to ldquocrdquo point thischange would affect the optimal solution of the model

Analogous to the above if the turnover rate of theportfolio must be more than 0002 ie l0 02() and theexpected rate of return must reach 8 or more then thelower and upper bound of investment ratio xk should bed 002 0 01 0 002 01 01 005 and g 03 02 0302 02 04 04 03 respectively By solving the model(NPL1) with different investor sensitivity parameter valueswe obtain the possibilistic efficient portfolios in Table 5 (erelationship between investor sensitivity and the portfoliorisk is shown in Figure 4

From Figure 4 we can see that an increase in investorsensitivity is associated with a decrease in the portfolio riskand when investor sensitivity reaches a certain threshold theportfolio risk no longer reduces (at is to say in this case ifinvestors seek at least 8 of the profits regardless of theinvestorsrsquo preference for liquidity the minimum risk of theportfolio is 05126 (or 5126)

(e impact of liquidity sensitivity α on the portfolio isshown in Figure 5 Figure 5 shows some possibilistic efficientportfolios with different α In Figure 5 when the position ofthe possibilistic efficient frontier and the intercept of the αline remain unchanged the slope changes of the α line willaffect the optimal solution of the model As α moves fromldquoardquo point to ldquobrdquo point the optimal solution of the model

Table 1 (e possibility distribution of returns

Stock code μk σk

600058 006 0167600028 009 0102600089 010 0207600115 004 0111600170 003 0049600495 010 0080600526 001 0086600662 003 0076

Table 2 (e possibility distribution of turnover rates ()

Stock code ak bk

600058 4070 31762600028 177 0802600089 3251 10681600115 975 4384600170 1486 8796600495 7385 51658600526 6547 16565600662 4014 19587

6 Complexity

ndash01

0

01

02

03

04

05

06

07

08

2 3 4 5 6 7 8 9

Without risk-free investmentWith risk-free investment

Figure 1 Variation of portfolio risks with μ

Table 3 Possibilistic efficient portfolios with risk-free investment for the different micros with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 0 01 0 002 01 01 005 00901 0395 002 0 01 0 002 02330 01 005 01441 052365 002 00717 01 0 002 03737 01 005 02654 073547 002 01521 01 0 002 03717 01 005 03287 0813875 002 02 01497 0 002 03482 01 005 04408 088798 002 02 02725 0 002 02929 01 005 06485 09554812 002 02 03 00197 002 02779 01 005 07252 09876

Table 4 Possibilistic efficient portfolios without risk-free investment for the different μs with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 00702 01 0 02 01 02098 03 03607 15 002 00760 01 0 02 01547 01493 03 03640 165 002 01554 01 0 02 02868 01 01378 04004 17 002 01906 01 0 02 03281 01 00613 04165 175 002 02 01817 0 01386 03097 01 005 05309 18 002 02 02853 0 00671 02776 01 005 06910 1812 002 02 03 003 0024 0276 01 005 07252 1

003

004

005

006

007

008

009

02 03 04 05 06 07 08 09

Expe

cted

rate

of r

etur

n

Deviation

Without liquidity constraintWith liquidity constraint

Figure 2 Some possibilistic efficient portfolios with and without liquidity constraint

Complexity 7

would not change But when α moves from ldquoardquo point to ldquocrdquopoint this change would affect the optimal solution of themodel

5 Conclusions and Directions forFuture Research

In this paper we proposed a possibilistic portfolio modelwhich is different from the MVM proposed by MarkowitzUnlike the MVM we measured the liquidity of asset as theturnover rate Besides we assumed that the expected rate ofreturns and turnover rates of assets are fuzzy variableswhich follow the normal possibility distribution We then

applied the possibility theory and fuzzy chance constraint toconsider the turnover rate and obtained the solution for themodel Furthermore we illustrated our proposed effectiveapproaches for the portfolio construction using numericalexamples In these examples we analyzed the risk betweenthe portfolios with and without risk-free constraints Wealso analyzed changes in the portfoliorsquos risk with respect to

0

001

002

003

004

005

006

007

008

009

0 01 02 03 04 05 06 07 08Deviation

Expe

cted

rate

of r

etur

n

Possibilistic efficient frontiera

bc

Figure 3 Some possibilistic efficient portfolios with different l0

Table 5 Possibilistic efficient portfolios with risk-free investment for the different α values with l0 02()and μ 8()

α () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

01 002 02 02725 0 002 02929 01 005 06485 0955402 002 02 02544 0 002 03110 01 005 06241 0956403 002 02 02315 0 002 03339 01 005 05942 0960504 002 02 02012 0 002 03642 01 005 05560 0955405 002 02 01654 0 002 04 01 005 05126 0955406 002 02 01654 0 002 04 01 005 05126 09554

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07

Risk

The sensitivity to liquidity

Figure 4 Variation of portfolio risk with investor sensitivity α 0

001

002

003

004

005

006

007

008

009

Expe

cted

rate

of r

etur

n

02 04 06 080Deviation

a

bc

Possibilistic efficientfrontier

Figure 5 Some possibilistic efficient portfolios with different α

8 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

(e turnover rate of asset k k 1 2 n is also anormally distributed fuzzy variable denoted as1113957lk sim FN(ak b2k) So its membership function is

A1113957lk(x) exp minus

x minus ak

bk

1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭ (21)

From (eorem 1 we obtain

1113957R sim FN 1113944n

k1xkμk + rf 1 minus 1113944

n

k1xk

⎛⎝ ⎞⎠12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

1113944

n

k1

1113957lkxk sim FN 1113944n

k1xkak

12

1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

⎛⎝ ⎞⎠

(22)

(en themembership function of 1113936nk1

1113957lkxk is defined by

A(u) exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ (23)

Furthermore

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ supugel0

exp minusu minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭

exp minusl0 minus 1113936

nk1 xkak( 1113857

2

(12) 1113936nk1 xkbk( 1113857

2

⎧⎨

⎩

⎫⎬

⎭ for 1113944n

k1xkak lt l0

1 for 1113944n

k1xkak ge l0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

From (19) and (24) we have

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

(25)

From (22) and (25) (PL1) can be transformed into

(NPL1)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xk μk minus rf1113872 1113873 + rf ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk le 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

32 PossibilisticPortfolioModelwithoutRisk-Free InvestmentSimilar to the previous section we propose a portfolio se-lection model without risk-free investment as follows

(PL2)

minVar(1113957R) 1113944n

k1x2

kVar 1113957rk( 1113857 + 2 1113944n

igtj1xixjCov 1113957ri 1113957rj1113872 1113873

st 1113944n

k1xkM 1113957rk( 1113857ge μ

Pos 1113944n

k1

1113957lkxk ge l0⎡⎣ ⎤⎦ge 1 minus α

1113944

n

k1xk 1

0le dk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

If the return rate of asset k k 1 2 n is a normallydistributed fuzzy variable denoted as 1113957rk sim FN(μk σ2k) andthe turnover rate of asset k k 1 2 n is also a normallydistributed fuzzy variable denoted as 1113957lk sim FN(ak b2k) thenthe model (PL2) can be transformed into

(NPL2)

minVar(1113957R) 12

1113944

n

k1xkσk

⎛⎝ ⎞⎠

2

st 1113944n

k1xkμk ge μ

minus l0 minus 1113944n

k1akxk

⎛⎝ ⎞⎠

2

ge12ln(1 minus α) 1113944

n

k1xkbk

⎛⎝ ⎞⎠

2

1113944

n

k1xk 1

0ledk lexk legk k 1 2 n

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

(NPL1) and (NPL2) are quadratic programmingproblems and can be solved by MATLAB Lingo etc

4 Numerical Example

In this section we give a real portfolio example to illustrateour approach In this example we selected eight stocks fromthe Shanghai Stock Exchange We collected data on monthlyreturns and turnover rate for each of the eight stocks over theperiod of January 2006 to December 2006 from the RESSETFinancial Research Database We use SPSS to generate thefrequency distributions of the monthly returns and turnoverrates

Table 1 shows the transaction codes and the frequencydistributions of the monthly returns while Table 2 shows thetransaction codes and the frequency distributions of themonthly turnover rates

Complexity 5

Let the risk-free asset be a bank deposit We use thethree-month deposit interest rates as the return on risk-freeassets So we get the return on risk-free assets rf 28 Ifthe possibility of the portfoliorsquos turnover rate must be morethan 0002 (in more than 90 times) that is l0 02()

and α 10 then the lower bound of investment ratio xk

must be d 002 0 01 0 002 01 01 005 and the upperbound g 03 02 03 02 02 04 04 03 By solving themodel (NPL1) and (NPL2) the possibilistic efficientportfolios for the different micros are obtained as shown inTables 3 and 4

From these two tables we can see that the risk increasesas μ increases We also can see that when the value of μ is low(eg μ 3) the proportion that invests is low For ex-ample when the value of μ is equal to 3 the investmentproportion with risk-free is the lower bound but the in-vestment proportion without risk-free is 002 00702 01 002 01 02098 and 03 Figure 1 shows these two efficientportfolios From Figure 1 we can see that when the risk-freeinvestment is included in the portfolio the risk is lower atthe same value ofμ And from Figure 1 we also can draw theconclusion that the risk of the portfolios can be spread bydiversification (e impact of liquidity constraint on theportfolio is shown in Figure 2 Figure 2 shows some pos-sibilistic efficient portfolios with and without liquidityconstraints

It can be seen from Figure 2 that under the same ex-pected return those investors who have requirements forfinancial liquidity need to take on greater risks In otherwords if investors want to keep the risks they take constantthe demand for financial liquidity will reduce the expectedreturn of the portfolio In addition the impact of liquidityfloor l0 on the portfolio is shown in Figure 3 Figure 3 showssome possibilistic efficient portfolios with different l0 (e

impact of l0 on the portfolio depends on the slope and theintercept of the l0 line and the position of the possibilisticefficient frontier

As shown in Figure 3 when the position of the possi-bilistic efficient frontier and the slope of the l0 line remainunchanged the intercept changes of the l0 line will affect theoptimal solution of the model As l0moves from ldquoardquo point toldquobrdquo point the optimal solution of the model would notchange But when l0 moves from ldquoardquo point to ldquocrdquo point thischange would affect the optimal solution of the model

Analogous to the above if the turnover rate of theportfolio must be more than 0002 ie l0 02() and theexpected rate of return must reach 8 or more then thelower and upper bound of investment ratio xk should bed 002 0 01 0 002 01 01 005 and g 03 02 0302 02 04 04 03 respectively By solving the model(NPL1) with different investor sensitivity parameter valueswe obtain the possibilistic efficient portfolios in Table 5 (erelationship between investor sensitivity and the portfoliorisk is shown in Figure 4

From Figure 4 we can see that an increase in investorsensitivity is associated with a decrease in the portfolio riskand when investor sensitivity reaches a certain threshold theportfolio risk no longer reduces (at is to say in this case ifinvestors seek at least 8 of the profits regardless of theinvestorsrsquo preference for liquidity the minimum risk of theportfolio is 05126 (or 5126)

(e impact of liquidity sensitivity α on the portfolio isshown in Figure 5 Figure 5 shows some possibilistic efficientportfolios with different α In Figure 5 when the position ofthe possibilistic efficient frontier and the intercept of the αline remain unchanged the slope changes of the α line willaffect the optimal solution of the model As α moves fromldquoardquo point to ldquobrdquo point the optimal solution of the model

Table 1 (e possibility distribution of returns

Stock code μk σk

600058 006 0167600028 009 0102600089 010 0207600115 004 0111600170 003 0049600495 010 0080600526 001 0086600662 003 0076

Table 2 (e possibility distribution of turnover rates ()

Stock code ak bk

600058 4070 31762600028 177 0802600089 3251 10681600115 975 4384600170 1486 8796600495 7385 51658600526 6547 16565600662 4014 19587

6 Complexity

ndash01

0

01

02

03

04

05

06

07

08

2 3 4 5 6 7 8 9

Without risk-free investmentWith risk-free investment

Figure 1 Variation of portfolio risks with μ

Table 3 Possibilistic efficient portfolios with risk-free investment for the different micros with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 0 01 0 002 01 01 005 00901 0395 002 0 01 0 002 02330 01 005 01441 052365 002 00717 01 0 002 03737 01 005 02654 073547 002 01521 01 0 002 03717 01 005 03287 0813875 002 02 01497 0 002 03482 01 005 04408 088798 002 02 02725 0 002 02929 01 005 06485 09554812 002 02 03 00197 002 02779 01 005 07252 09876

Table 4 Possibilistic efficient portfolios without risk-free investment for the different μs with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 00702 01 0 02 01 02098 03 03607 15 002 00760 01 0 02 01547 01493 03 03640 165 002 01554 01 0 02 02868 01 01378 04004 17 002 01906 01 0 02 03281 01 00613 04165 175 002 02 01817 0 01386 03097 01 005 05309 18 002 02 02853 0 00671 02776 01 005 06910 1812 002 02 03 003 0024 0276 01 005 07252 1

003

004

005

006

007

008

009

02 03 04 05 06 07 08 09

Expe

cted

rate

of r

etur

n

Deviation

Without liquidity constraintWith liquidity constraint

Figure 2 Some possibilistic efficient portfolios with and without liquidity constraint

Complexity 7

would not change But when α moves from ldquoardquo point to ldquocrdquopoint this change would affect the optimal solution of themodel

5 Conclusions and Directions forFuture Research

In this paper we proposed a possibilistic portfolio modelwhich is different from the MVM proposed by MarkowitzUnlike the MVM we measured the liquidity of asset as theturnover rate Besides we assumed that the expected rate ofreturns and turnover rates of assets are fuzzy variableswhich follow the normal possibility distribution We then

applied the possibility theory and fuzzy chance constraint toconsider the turnover rate and obtained the solution for themodel Furthermore we illustrated our proposed effectiveapproaches for the portfolio construction using numericalexamples In these examples we analyzed the risk betweenthe portfolios with and without risk-free constraints Wealso analyzed changes in the portfoliorsquos risk with respect to

0

001

002

003

004

005

006

007

008

009

0 01 02 03 04 05 06 07 08Deviation

Expe

cted

rate

of r

etur

n

Possibilistic efficient frontiera

bc

Figure 3 Some possibilistic efficient portfolios with different l0

Table 5 Possibilistic efficient portfolios with risk-free investment for the different α values with l0 02()and μ 8()

α () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

01 002 02 02725 0 002 02929 01 005 06485 0955402 002 02 02544 0 002 03110 01 005 06241 0956403 002 02 02315 0 002 03339 01 005 05942 0960504 002 02 02012 0 002 03642 01 005 05560 0955405 002 02 01654 0 002 04 01 005 05126 0955406 002 02 01654 0 002 04 01 005 05126 09554

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07

Risk

The sensitivity to liquidity

Figure 4 Variation of portfolio risk with investor sensitivity α 0

001

002

003

004

005

006

007

008

009

Expe

cted

rate

of r

etur

n

02 04 06 080Deviation

a

bc

Possibilistic efficientfrontier

Figure 5 Some possibilistic efficient portfolios with different α

8 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

Let the risk-free asset be a bank deposit We use thethree-month deposit interest rates as the return on risk-freeassets So we get the return on risk-free assets rf 28 Ifthe possibility of the portfoliorsquos turnover rate must be morethan 0002 (in more than 90 times) that is l0 02()

and α 10 then the lower bound of investment ratio xk

must be d 002 0 01 0 002 01 01 005 and the upperbound g 03 02 03 02 02 04 04 03 By solving themodel (NPL1) and (NPL2) the possibilistic efficientportfolios for the different micros are obtained as shown inTables 3 and 4

From these two tables we can see that the risk increasesas μ increases We also can see that when the value of μ is low(eg μ 3) the proportion that invests is low For ex-ample when the value of μ is equal to 3 the investmentproportion with risk-free is the lower bound but the in-vestment proportion without risk-free is 002 00702 01 002 01 02098 and 03 Figure 1 shows these two efficientportfolios From Figure 1 we can see that when the risk-freeinvestment is included in the portfolio the risk is lower atthe same value ofμ And from Figure 1 we also can draw theconclusion that the risk of the portfolios can be spread bydiversification (e impact of liquidity constraint on theportfolio is shown in Figure 2 Figure 2 shows some pos-sibilistic efficient portfolios with and without liquidityconstraints

It can be seen from Figure 2 that under the same ex-pected return those investors who have requirements forfinancial liquidity need to take on greater risks In otherwords if investors want to keep the risks they take constantthe demand for financial liquidity will reduce the expectedreturn of the portfolio In addition the impact of liquidityfloor l0 on the portfolio is shown in Figure 3 Figure 3 showssome possibilistic efficient portfolios with different l0 (e

impact of l0 on the portfolio depends on the slope and theintercept of the l0 line and the position of the possibilisticefficient frontier

As shown in Figure 3 when the position of the possi-bilistic efficient frontier and the slope of the l0 line remainunchanged the intercept changes of the l0 line will affect theoptimal solution of the model As l0moves from ldquoardquo point toldquobrdquo point the optimal solution of the model would notchange But when l0 moves from ldquoardquo point to ldquocrdquo point thischange would affect the optimal solution of the model

Analogous to the above if the turnover rate of theportfolio must be more than 0002 ie l0 02() and theexpected rate of return must reach 8 or more then thelower and upper bound of investment ratio xk should bed 002 0 01 0 002 01 01 005 and g 03 02 0302 02 04 04 03 respectively By solving the model(NPL1) with different investor sensitivity parameter valueswe obtain the possibilistic efficient portfolios in Table 5 (erelationship between investor sensitivity and the portfoliorisk is shown in Figure 4

From Figure 4 we can see that an increase in investorsensitivity is associated with a decrease in the portfolio riskand when investor sensitivity reaches a certain threshold theportfolio risk no longer reduces (at is to say in this case ifinvestors seek at least 8 of the profits regardless of theinvestorsrsquo preference for liquidity the minimum risk of theportfolio is 05126 (or 5126)

(e impact of liquidity sensitivity α on the portfolio isshown in Figure 5 Figure 5 shows some possibilistic efficientportfolios with different α In Figure 5 when the position ofthe possibilistic efficient frontier and the intercept of the αline remain unchanged the slope changes of the α line willaffect the optimal solution of the model As α moves fromldquoardquo point to ldquobrdquo point the optimal solution of the model

Table 1 (e possibility distribution of returns

Stock code μk σk

600058 006 0167600028 009 0102600089 010 0207600115 004 0111600170 003 0049600495 010 0080600526 001 0086600662 003 0076

Table 2 (e possibility distribution of turnover rates ()

Stock code ak bk

600058 4070 31762600028 177 0802600089 3251 10681600115 975 4384600170 1486 8796600495 7385 51658600526 6547 16565600662 4014 19587

6 Complexity

ndash01

0

01

02

03

04

05

06

07

08

2 3 4 5 6 7 8 9

Without risk-free investmentWith risk-free investment

Figure 1 Variation of portfolio risks with μ

Table 3 Possibilistic efficient portfolios with risk-free investment for the different micros with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 0 01 0 002 01 01 005 00901 0395 002 0 01 0 002 02330 01 005 01441 052365 002 00717 01 0 002 03737 01 005 02654 073547 002 01521 01 0 002 03717 01 005 03287 0813875 002 02 01497 0 002 03482 01 005 04408 088798 002 02 02725 0 002 02929 01 005 06485 09554812 002 02 03 00197 002 02779 01 005 07252 09876

Table 4 Possibilistic efficient portfolios without risk-free investment for the different μs with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 00702 01 0 02 01 02098 03 03607 15 002 00760 01 0 02 01547 01493 03 03640 165 002 01554 01 0 02 02868 01 01378 04004 17 002 01906 01 0 02 03281 01 00613 04165 175 002 02 01817 0 01386 03097 01 005 05309 18 002 02 02853 0 00671 02776 01 005 06910 1812 002 02 03 003 0024 0276 01 005 07252 1

003

004

005

006

007

008

009

02 03 04 05 06 07 08 09

Expe

cted

rate

of r

etur

n

Deviation

Without liquidity constraintWith liquidity constraint

Figure 2 Some possibilistic efficient portfolios with and without liquidity constraint

Complexity 7

would not change But when α moves from ldquoardquo point to ldquocrdquopoint this change would affect the optimal solution of themodel

5 Conclusions and Directions forFuture Research

In this paper we proposed a possibilistic portfolio modelwhich is different from the MVM proposed by MarkowitzUnlike the MVM we measured the liquidity of asset as theturnover rate Besides we assumed that the expected rate ofreturns and turnover rates of assets are fuzzy variableswhich follow the normal possibility distribution We then

applied the possibility theory and fuzzy chance constraint toconsider the turnover rate and obtained the solution for themodel Furthermore we illustrated our proposed effectiveapproaches for the portfolio construction using numericalexamples In these examples we analyzed the risk betweenthe portfolios with and without risk-free constraints Wealso analyzed changes in the portfoliorsquos risk with respect to

0

001

002

003

004

005

006

007

008

009

0 01 02 03 04 05 06 07 08Deviation

Expe

cted

rate

of r

etur

n

Possibilistic efficient frontiera

bc

Figure 3 Some possibilistic efficient portfolios with different l0

Table 5 Possibilistic efficient portfolios with risk-free investment for the different α values with l0 02()and μ 8()

α () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

01 002 02 02725 0 002 02929 01 005 06485 0955402 002 02 02544 0 002 03110 01 005 06241 0956403 002 02 02315 0 002 03339 01 005 05942 0960504 002 02 02012 0 002 03642 01 005 05560 0955405 002 02 01654 0 002 04 01 005 05126 0955406 002 02 01654 0 002 04 01 005 05126 09554

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07

Risk

The sensitivity to liquidity

Figure 4 Variation of portfolio risk with investor sensitivity α 0

001

002

003

004

005

006

007

008

009

Expe

cted

rate

of r

etur

n

02 04 06 080Deviation

a

bc

Possibilistic efficientfrontier

Figure 5 Some possibilistic efficient portfolios with different α

8 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

ndash01

0

01

02

03

04

05

06

07

08

2 3 4 5 6 7 8 9

Without risk-free investmentWith risk-free investment

Figure 1 Variation of portfolio risks with μ

Table 3 Possibilistic efficient portfolios with risk-free investment for the different micros with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 0 01 0 002 01 01 005 00901 0395 002 0 01 0 002 02330 01 005 01441 052365 002 00717 01 0 002 03737 01 005 02654 073547 002 01521 01 0 002 03717 01 005 03287 0813875 002 02 01497 0 002 03482 01 005 04408 088798 002 02 02725 0 002 02929 01 005 06485 09554812 002 02 03 00197 002 02779 01 005 07252 09876

Table 4 Possibilistic efficient portfolios without risk-free investment for the different μs with l0 02() and α 10

μ () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

3 002 00702 01 0 02 01 02098 03 03607 15 002 00760 01 0 02 01547 01493 03 03640 165 002 01554 01 0 02 02868 01 01378 04004 17 002 01906 01 0 02 03281 01 00613 04165 175 002 02 01817 0 01386 03097 01 005 05309 18 002 02 02853 0 00671 02776 01 005 06910 1812 002 02 03 003 0024 0276 01 005 07252 1

003

004

005

006

007

008

009

02 03 04 05 06 07 08 09

Expe

cted

rate

of r

etur

n

Deviation

Without liquidity constraintWith liquidity constraint

Figure 2 Some possibilistic efficient portfolios with and without liquidity constraint

Complexity 7

would not change But when α moves from ldquoardquo point to ldquocrdquopoint this change would affect the optimal solution of themodel

5 Conclusions and Directions forFuture Research

In this paper we proposed a possibilistic portfolio modelwhich is different from the MVM proposed by MarkowitzUnlike the MVM we measured the liquidity of asset as theturnover rate Besides we assumed that the expected rate ofreturns and turnover rates of assets are fuzzy variableswhich follow the normal possibility distribution We then

applied the possibility theory and fuzzy chance constraint toconsider the turnover rate and obtained the solution for themodel Furthermore we illustrated our proposed effectiveapproaches for the portfolio construction using numericalexamples In these examples we analyzed the risk betweenthe portfolios with and without risk-free constraints Wealso analyzed changes in the portfoliorsquos risk with respect to

0

001

002

003

004

005

006

007

008

009

0 01 02 03 04 05 06 07 08Deviation

Expe

cted

rate

of r

etur

n

Possibilistic efficient frontiera

bc

Figure 3 Some possibilistic efficient portfolios with different l0

Table 5 Possibilistic efficient portfolios with risk-free investment for the different α values with l0 02()and μ 8()

α () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

01 002 02 02725 0 002 02929 01 005 06485 0955402 002 02 02544 0 002 03110 01 005 06241 0956403 002 02 02315 0 002 03339 01 005 05942 0960504 002 02 02012 0 002 03642 01 005 05560 0955405 002 02 01654 0 002 04 01 005 05126 0955406 002 02 01654 0 002 04 01 005 05126 09554

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07

Risk

The sensitivity to liquidity

Figure 4 Variation of portfolio risk with investor sensitivity α 0

001

002

003

004

005

006

007

008

009

Expe

cted

rate

of r

etur

n

02 04 06 080Deviation

a

bc

Possibilistic efficientfrontier

Figure 5 Some possibilistic efficient portfolios with different α

8 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

would not change But when α moves from ldquoardquo point to ldquocrdquopoint this change would affect the optimal solution of themodel

5 Conclusions and Directions forFuture Research

In this paper we proposed a possibilistic portfolio modelwhich is different from the MVM proposed by MarkowitzUnlike the MVM we measured the liquidity of asset as theturnover rate Besides we assumed that the expected rate ofreturns and turnover rates of assets are fuzzy variableswhich follow the normal possibility distribution We then

applied the possibility theory and fuzzy chance constraint toconsider the turnover rate and obtained the solution for themodel Furthermore we illustrated our proposed effectiveapproaches for the portfolio construction using numericalexamples In these examples we analyzed the risk betweenthe portfolios with and without risk-free constraints Wealso analyzed changes in the portfoliorsquos risk with respect to

0

001

002

003

004

005

006

007

008

009

0 01 02 03 04 05 06 07 08Deviation

Expe

cted

rate

of r

etur

n

Possibilistic efficient frontiera

bc

Figure 3 Some possibilistic efficient portfolios with different l0

Table 5 Possibilistic efficient portfolios with risk-free investment for the different α values with l0 02()and μ 8()

α () 600058 600028 600089 600115 600170 600495 600526 600662 Risk 11139368k1 xk

01 002 02 02725 0 002 02929 01 005 06485 0955402 002 02 02544 0 002 03110 01 005 06241 0956403 002 02 02315 0 002 03339 01 005 05942 0960504 002 02 02012 0 002 03642 01 005 05560 0955405 002 02 01654 0 002 04 01 005 05126 0955406 002 02 01654 0 002 04 01 005 05126 09554

0

01

02

03

04

05

06

07

0 01 02 03 04 05 06 07

Risk

The sensitivity to liquidity

Figure 4 Variation of portfolio risk with investor sensitivity α 0

001

002

003

004

005

006

007

008

009

Expe

cted

rate

of r

etur

n

02 04 06 080Deviation

a

bc

Possibilistic efficientfrontier

Figure 5 Some possibilistic efficient portfolios with different α

8 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

changes in investor sensitivity Finally we drew a conclusionthat our method can play a leading role in financial markets

Future research can extend our model in the followingways Firstly researchers can use other methods to solve theproblem and compare the results Secondly our fuzzyportfolio model can be extended to a multiperiod case(irdly our model can be considered in a fuzzy randomenvironment Finally with the development of behavioralfinance investor behavior has received more attention Webelieve that investor behavior can be considered in fuzzyportfolio theory

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

(is manuscript was presented in 2018 International Con-ference on Applied Finance Macroeconomic Performanceand Economic Growth

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was funded by the National Natural ScienceFoundation of China (11401438) the National Social ScienceFoundation Project of China (13CRK027) and the SocialScience Planning Project of Shandong Province (17CCXJ14)

References

[1] O S Fard and M Ramezanzadeh ldquoOn fuzzy portfolio se-lection problems a parametric representation approachrdquoComplexity vol 2017 Article ID 9317924 12 pages 2017

[2] W Gao and K Hong ldquo(e portfolio balanced risk indexmodel and analysis of examples of large-scale infrastructureprojectrdquo Complexity vol 2017 Article ID 5174613 13 pages2017

[3] A G Bernabeu J V Salcedo A Hilario D P Santamariaand J M Herrero ldquoComputing the mean-variance-sustain-ability nondominated surface by Ev-MOGArdquo Complexityvol 2019 Article ID 6095712 12 pages 2019

[4] J F Lavin M A Valle N S Magner and B M TabakldquoModeling overlapped mutual fundsrsquo portfolios a bipartitenetwork approachrdquo Complexity vol 2019 Article ID1565698 20 pages 2019

[5] H Markowitz ldquoPortfolio selectionrdquo e Journal of Financevol 7 no 1 pp 77ndash91 1952

[6] H Markowitz Portfolio Selection Efficient Diversification ofInvestments Yale University Press New York NY USA 1959

[7] H Konno and H Yamazaki ldquoMean-absolute deviationportfolio optimization model and its applications to tokyostock marketrdquo Management Science vol 37 no 5 pp 519ndash531 1991

[8] M G Speranza ldquoLinear programming model for portfoliooptimizationrdquo Finance vol 14 no 1 pp 107ndash123 1993

[9] P Jorion ldquoRisk2 measuring the risk in value at riskrdquo Fi-nancial Analysts Journal vol 52 no 6 pp 47ndash56 1996

[10] R Mansini W Ogryczak and M G Speranza ldquoConditionalvalue at risk and related linear programming models forportfolio optimizationrdquo Annals of Operations Researchvol 152 no 1 pp 227ndash256 2007

[11] P Byrne and S Lee ldquoDifferent risk measures differentportfolio compositionsrdquo Journal of Property Investment ampFinance vol 22 no 6 pp 501ndash511 2004

[12] H Kandasamy Portfolio Selection under Various Risk Mea-sures Clemson University ProQuest Dissertations and(esesGlobal Clemson University ProQuest Dissertations and(eses Global 2008

[13] F Black and R Litterman ldquoGlobal portfolio optimizationrdquoFinancial Analysts Journal vol 48 no 5 pp 28ndash43 1992

[14] D Bertsimas D B Brown and C Caramanis ldquo(eory andapplications of robust optimizationrdquo SIAM Review vol 53no 3 pp 464ndash501 2011

[15] D Goldfarb and G Iyengar ldquoRobust portfolio selectionproblemsrdquoMathematics of Operations Research vol 28 no 1pp 1ndash38 2003

[16] F Piri M Salahi and F Mehrdoust ldquoRobust mean-condi-tional value at risk portfolio optimizationrdquo InternationalJournal of Economic Sciences vol 3 no 1 pp 2ndash11 2014

[17] G Kara A Ozmen and G-W Weber ldquoStability advances inrobust portfolio optimization under parallelepiped uncer-taintyrdquo Central European Journal of Operations Researchvol 27 no 1 pp 241ndash261 2017

[18] Z Kang and Z Li ldquoAn exact solution to a robust portfoliochoice problem with multiple risk measures under ambiguousdistributionrdquo Mathematical Methods of Operations Researchvol 87 no 2 pp 169ndash195 2017

[19] H Khanjarpanah and M S Pishvaee ldquoA fuzzy robust pro-gramming approach to multi-objective portfolio optimisationproblem under uncertaintyrdquo International Journal of Math-ematics in Operational Research vol 12 no 1 pp 45ndash65 2018

[20] T Khodamoradi M Salahi and A R Najafi ldquoRobust CCMVmodel with short selling and risk-neutral interest raterdquoPhysica A Statistical Mechanics and Its Applications vol 547Article ID 124429 13 pages 2020

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[22] J Watada ldquoFuzzy portfolio selection and its applications todecision makingrdquo Tatra Mountains Mathematical Publica-tion vol 13 pp 219ndash248 1997

[23] S Ramaswamy ldquoPortfolio selection using fuzzy decisiontheoryrdquo Working Paper of Bank for International Settlementsvol 59 1998

[24] Y Fang K K Lai and S Y Wang Fuzzy Portfolio Opti-mization eory and Methods Springer London UK 2008

[25] S Wang and S Zhu ldquoOn fuzzy portfolio selection problemsrdquoFuzzy Optimization and Decision Making vol 1 no 4pp 361ndash377 2002

[26] R-C Tsaur ldquoFuzzy portfolio model with fuzzy-input returnrates and fuzzy-output proportionsrdquo International Journal ofSystems Science vol 46 no 3 pp 438ndash450 2013

[27] J Zhou X Li and W Pedrycz ldquoMean-semi-entropy modelsof fuzzy portfolio selectionrdquo IEEE Transactions on FuzzySystems vol 24 no 6 pp 1627ndash1636 2016

[28] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[29] D Dubois and H Prade ldquo(emean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1978

Complexity 9

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity

[30] H Tanaka H Nakayama and A Yanagimoto ldquoPossibilityportfolio selectionrdquo in Proceedings of the 1995 IEEE Inter-national Conference on Fuzzy Systems pp 77ndash91 PerthAustralia November 1995

[31] H Tanaka and P Guo ldquoPortfolio selection based on upperand lower exponential possibility distributionsrdquo EuropeanJournal of Operational Research vol 114 no 1 pp 115ndash1261999

[32] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Fuzzy Sets and Systems vol 122no 2 pp 315ndash326 2001

[33] C Carlsson R Fuller and P Majlender ldquoA possibilisticapproach to selecting portfolios with highest utility scorerdquoFuzzy Sets and Systems vol 131 no 1 pp 13ndash21 2002

[34] W G Zhang and Z K Nie ldquoOn possibilistic variance of fuzzynumbersrdquo Rough Sets Fuzzy Sets Data Mining and GranularComputing vol 2639 pp 398ndash402 2003

[35] W-G Zhang ldquoPossibilistic mean-standard deviation modelsto portfolio selection for bounded assetsrdquo Applied Mathe-matics and Computation vol 189 no 2 pp 1614ndash1623 2007

[36] W-G Zhang Y-L Wang Z-P Chen and Z-K NieldquoPossibilistic mean-variance models and efficient frontiers forportfolio selection problemrdquo Information Sciences vol 177no 13 pp 2787ndash2801 2007

[37] W-G Zhang W-L Xiao and Y-L Wang ldquoA fuzzy portfolioselection method based on possibilistic mean and variancerdquoSoft Computing vol 13 no 6 pp 627ndash633 2009

[38] W-G Zhang X-L Zhang and W-L Xiao ldquoPortfolio se-lection under possibilistic mean-variance utility and a SMOalgorithmrdquo European Journal of Operational Researchvol 197 no 2 pp 693ndash700 2009

[39] T Li W Zhang and W Xu ldquoFuzzy possibilistic portfolioselection model with VaR constraint and risk-free invest-mentrdquo Economic Modelling vol 31 pp 12ndash17 2013

[40] Y-J Liu and W-G Zhang ldquoPossibilistic moment models formulti-period portfolio selection with fuzzy returnsrdquo Com-putational Economics vol 53 no 4 pp 1657ndash1686 2018

[41] V T Datar N Y Naik and R Radcliffe ldquoLiquidity and stockreturns an alternative testrdquo Journal of Financial Marketsvol 1 no 2 pp 203ndash219 1998

[42] B R Marshall and M Young ldquoLiquidity and stock returns inpure order-driven markets evidence from the Australianstock marketrdquo International Review of Financial Analysisvol 12 no 2 pp 173ndash188 2003

[43] X Huang ldquoRisk curve and fuzzy portfolio selectionrdquo Com-puters and Mathematics with Applications vol 55 no 6pp 1102ndash1112 2007

[44] X Huang ldquoFuzzy chance-constrained portfolio selectionrdquoApplied Mathematics and Computation vol 177 no 2pp 500ndash507 2006

[45] P Li and B Liu ldquoEntropy of credibility distributions for fuzzyvariablesrdquo IEEE Transaction on Fuzzy System vol 16 no 1pp 123ndash129 2008

[46] L Li J Li Q Qin and S Cheng ldquoFuzzy chance-constrainedproject portfolio selection model based on credibility theoryrdquoAdvances in Intelligent Systems and Computing vol 277pp 731ndash743 2014

[47] S Barak M Abessi and M Modarres ldquoFuzzy turnover ratechance constraints portfolio modelrdquo European Journal ofOperational Research vol 228 no 1 pp 141ndash147 2013

10 Complexity