# mean-variance-skewness model for portfolio

Innovative Applications of O.R.

Mean-variance-skewness model for portfolio selection with fuzzy returns

Xiang Li a ,Zhongfeng Qin b,*,Samarjit Kar c

a Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China

b

School of Economics and Management,Beihang University,Beijing 100191,China

c Department of Mathematics,National Institute of Technology,Durgapur 713209,India a r t i c l e i n f o Article history:Receive

d 21August 2008Accepted 4May 2009Availabl

e online 15May 2009Keywords:Portfolio selection Fuzzy variable Mean-variance-skewness model

Fuzzy programming

Credibility measure

a b s t r a c t

Numerous empirical studies show that portfolio returns are generally asymmetric,and investors would

prefer a portfolio return with larger degree of asymmetry when the mean value and variance are same.In

order to measure the asymmetry of fuzzy portfolio return,a concept of skewness is deﬁned as the third

central moment in this paper,and its mathematical properties are studied.As an extension of the fuzzy

mean-variance model,a mean-variance-skewness model is presented and the corresponding variations

are also considered.In order to solve the proposed models,a genetic algorithm integrating fuzzy simu-

lation is designed.Finally,several numerical examples are given to illustrate the modelling idea and

the effectiveness of the proposed algorithm.

Modern portfolio selection theory is derived from the seminal work of Markowitz [19,20]which considered trade-off between return and risk.Since then,numerous portfolio selection models are developed by considering the return and risk such as mean-variance mode and so on.Several researchers like Sharpe [26,27],Stone [28],Sengupta [25],Best and Grauer [3],etc.have done some articles by using various approximation scheme.

Most of the reasonable works on portfolio selection have been done based on only the ﬁrst two moments of return distributions.How-ever,there is a controversy over the issue of whether higher moments should be considered in portfolio selection.Many researchers (e.g.Arditti [1],Samuelson [24],Kraus and Litzenberger [10],Konno et al.[18],Konno and Suzuki [9],Liu et al.[17],Prakash et al.[21])argued that the higher moments cannot be neglected unless there are reasons to trust that the returns are symmetrically distributed (e.g.normal)or that higher moments are irrelevant to the investors’decisions.

Samuelson [24]also showed that higher moments are relevant for investors to make decisions in portfolio selection and almost all investors would prefer a portfolio with a larger third order moment if ﬁrst and second moments are same.Chunhachinda et al.[5],Mach-ado-Santos and Fernandes [18]provided evidence of skewness by using the data of stock markets.All the above discussions motivated us to add the third moment of return distribution of a portfolio selection into a general mean-variance model.

All the above literatures assume that the security returns are random variables.However,if there is not enough historical data,it is more reasonable to assume them as fuzzy variables.Fuzzy portfolio selection has been undertaken in the literature such as Parra et al.[2],Terol et al.[4],Tanaka and Guo [29,30]and Vercher et al.[31].In 2002,Liu and Liu [15]deﬁned the expected value and variance for measuring the portfolio return and the risk,respectively.Within the framework of credibility theory,several models for fuzzy portfolio selection were proposed such as,mean-semivariance model [7]and cross-entropy minimization model [23]and so forth.In addition,Qin and Li [22]con-sidered option pricing problem in fuzzy environment which is another hottest area in ﬁnance.

In fuzzy environment,investors also face to construct a portfolio selection from the potential securities with asymmetric returns.Sim-ilar to stochastic approaches,Huang [7]employed semivariance to describing asymmetry of fuzzy returns.Different from Huang’s ap-proach,we used skewness of fuzzy returns to characterize the corresponding asymmetry as alternative approach.The purpose of this paper is to establish and analyze fuzzy mean-variance-skewness models.

European Journal of Operational Research 202(2010)

239–247

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