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A note on “Portfolio selection under possibilistic mean-variance utility and a SMO algorithm”

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Highlights

  • The corrections of all the mistakes in papers should be published.

  • The mistake has brought no problems in the considered paper and/or in other ones.

  • The mistake can put in difficulties researchers developing new models based on it.

  • References have been updated.

  • Careful revision of the paper, correcting typos and inaccuracies.

Abstract

In a paper published in this journal – Zhang W.-G., Zhang X.-L., Xiao W.-L. (2009), Portfolio selection under possibilistic mean-variance utility and a SMO algorithm. European Journal of Operational Research, 197(2), 693-700 –, the Authors investigate a fuzzy approach to the portfolio selection problem in which the stock returns are represented in terms of trapezoidal fuzzy numbers. In this note, we show that the expression provided for the possibilistic covariance is not consistent with the definition of possibilistic covariance given in the paper itself, and we derive the right expression for such a covariance.

Introduction

In an interesting paper published in this journal – Zhang W.-G., Zhang X.-L., Xiao W.-L. (2009), Portfolio selection under possibilistic mean-variance utility and a SMO algorithm. European Journal of Operational Research, 197(2), 693-700 –, the Authors investigate a fuzzy approach to the portfolio selection problem in which the stock returns are represented in terms of trapezoidal fuzzy numbers, also called possibilistic numbers. In short, first they take into consideration novel and recent definitions of possibilistic mean, variance and covariance related to these numbers, then they apply such definitions for building a new portfolio selection model, finally they solve some numerical examples of this model by using a sequential minimal optimization algorithm.

In this note, we show that the expression provided in Zhang, Zhang, and Xiao (2009) for the possibilistic covariance is not consistent with the definition of possibilistic covariance given in the paper itself. In particular, we derive the right expression for such a covariance.

The remainder of the paper is organized as follows. In the next section we briefly recall the notions and definitions we need trough the note. In Section 3 we derive the right expression for the possibilistic covariance. Finally, in Section 4 we give some closing remarks.

Section snippets

Notions and definitions

In the approach to the fuzzy portfolio selection considered in Zhang et al. (2009), the stock returns are represented in terms of trapezoidal fuzzy numbers. In detail, a fuzzy number A=(a,b,α,β) is called trapezoidal with tolerance interval [a, b], in which a,bR with a ≤ b, with left width α ≥ 0 and with right width b ≥ 0 if its membership function A(γ) has the form

A(γ)={1aγαifaα<γa1ifa<γb1γbβifb<γb+β0otherwise.

In Fig. 1, the membership of a generic trapezoidal fuzzy number is

The right expression of the possibilistic covariance between stock returns

As known, in portfolio selection models an important role is played by the covariances given their impact on the contraction of the risk of the portfolio return.

In Section 2 of Zhang et al. (2009), the Authors propose the following new definition of possibilistic covariance between trapezoidal fuzzy numbers: given two trapezoidal fuzzy numbers A=[a1,b1,α1,β1] and B=[a2,b2,α2,β2] having, respectively, the γ-level sets [A]γ=[a1(γ),a2(γ)]=[a1(1γ)α1,b1+(1γ)β1] and [B]γ=[b1(γ),b2(γ)]=[a2(1γ)α2,b

Some final remarks

With reference to Proposition 1, we provide three final remarks.

  • – Considering the expressions of the possibilistic variance and covariance provided in Section 3 of Zhang et al. (2009), respectivelyVar(ri)=(biai2+αi+βi6)2+(αi+βi)272

    and formula 2, it is immediate to verify that Var(ri) and Cov(ri, rj) are inconsistent to each other as Var(ri)Cov(ri,ri). In fact, from formula 2 one getsCov(ri,ri)=(αi+βi)224.

    which is clearly different from formula 3 (for instance, note that quantity “biai” is

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