On the analytical derivation of efficient sets in quad-and-higher criterion portfolio selection
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On the analytical derivation of efficient sets in quad-and-higher criterion portfolio selection Yue Qi1 · Ralph E. Steuer2 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract This paper provides results in the area of the analytical derivation of the efficient set of a mean-variance portfolio selection problem that has more than three criteria. By “analytical” we mean derived by formula as opposed to being computed by algorithm. By “more than three criteria”, we mean that beyond the mean and variance of regular portfolio selection, the problems addressed have two or more additional linear objectives. The additional objectives might include sustainability, dividend yield, liquidity, and R&D as extra objectives like these are being seen with greater frequency. While not all multiple criteria portfolio selection problems lend themselves to an analytical derivation, a certain class does and the problems in this class are covered by the mathematics of this paper. Keywords Multiple criteria portfolio selection · Analytical derivation · Minimum-variance surface · Nondominated set · Efficient set · Paraboloid
1 Introduction From Markowitz (1952) we have the mean-variance model of portfolio selection which can be expressed, in bi-criterion format, as min{z 1 = xT x} max{z 2 = μT x} s.t. 1T x = 1 x≥0
(1)
where x is a vector specifying the proportion of capital invested in each stock. Because this vector specifies a portfolio, x is often called a portfolio. In this model, because of the nonnegativity restriction on x, short selling is not allowed. With an n ×n covariance matrix of stock returns, the first objective (z 1 ) is portfolio variance. With μ a vector of expected stock returns, the second objective (z 2 ) is portfolio expected return.
B
Ralph E. Steuer [email protected]
1
Department of Financial Management, Business School Nankai University, Tianjin 300071, China
2
Terry College of Business, University of Georgia, Athens, GA 30602, USA
123
Annals of Operations Research
Let S ⊂ Rn be the feasible region in decision space. Here S = {x ∈ Rn | 1T x = 1, x ≥ 0}. Because the problem has more than one objective, the problem has a second representation of the feasible region, designated Z ⊂ Rk , called the feasible region in criterion space, where k is the number of objectives. Criterion space is the space of the objectives, and any z ∈ Z is a criterion vector. With a criterion vector resulting from the values of the objectives at a given point in S, here Z is given by Z = {z ∈ R2 | z 1 = xT x, z 2 = μT x, x ∈ S}. In this way, Z is the set of all images of the points in S, and S is the set of all inverse images of the criterion vectors in Z . With respect to Z , a criterion vector z¯ ∈ Z is nondominated iff there exists no z ∈ Z such that z 1 ≤ z¯ 1 , z 2 ≥ z¯ 2 with z = z¯ . The set of all nondominated criterion vectors is called the nondominated set and is designated N . Similarly, z¯ ∈ Z is weakly nondominated iff there exists no z ∈ Z such that z 1 < z¯ 1 , z 2 > z¯ 2 . The dif
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