Short CommunicationA note on “Portfolio selection under possibilistic mean-variance utility and a SMO algorithm”
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 is called trapezoidal with tolerance interval [a, b], in which with a ≤ b, with left width α ≥ 0 and with right width b ≥ 0 if its membership function A(γ) has the form
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 and having, respectively, the γ-level sets and
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), respectively
and formula 2, it is immediate to verify that Var(ri) and Cov(ri, rj) are inconsistent to each other as . In fact, from formula 2 one gets
which is clearly different from formula 3 (for instance, note that quantity “” is
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