Computing the Nondominated Surface in Tri-Criterion Portfolio Selection

Abstract

Computing the nondominated set of a multiple objective mathematical program has long been a topic in multiple criteria decision making. In this paper, motivated by the desire to extend Markowitz portfolio selection to an additional linear criterion (dividends, liquidity, sustainability, etc.), we demonstrate an exact method for computing the nondominated set of a tri-criterion program that is all linear except for the fact that one of its objectives is to minimize a convex quadratic function. With the nondominated set of the resulting quad-lin-lin program being a surface composed of curved platelets, a multiparametric algorithm is devised for computing the platelets so that they can be graphed precisely. In this way, graphs of the tri-criterion nondominated surface can be displayed so that, as in traditional portfolio selection, a most preferred portfolio can be selected while in full view of all other contenders for optimality. Finally, by giving an example for socially responsible investors, we demonstrate that our algorithm can outperform standard portfolio strategies for multicriterial decision makers.

Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds

“…Whereas, since Markowitz (1956), it has been possible to compute a nondominated frontier as in Figure 1, it has not been until recently, namely by Hirschberger et al (2013), that it has been possible to compute a tri-criterion nondominated surface as in Figure 2. Using the CIOS (Custom Investment Objective Solver)…”

“…code from Hirschberger et al (2013) to solve for the tri-criterion nondominated surface, we are able to obtain from its output, for each paraboloidic platelet, (a) the platelet's 2-dimensional polyhedron of (λ µ , λ ν ) vectors in parameter space and (b) the platelet's polyhedron of efficient portfolio composition vectors in decision space. Recall that a point is efficient if and only if its image in criterion space is nondominated (where a point is nondominated if and only if it is impossible to move from it to another without at least deteriorating one criterion).…”

“…Pixelate the polyhedron in (λ µ , λ ν ) parameter space associated with each polyhedron of efficient portfolio composition vectors identified in the above step. Utilizing the relationship in Hirschberger et al (2013) that enables us to specify the x ∈ E for each point in (λ µ , λ ν ) parameter space, we are able to pixelate all incident polyhedra in E.…”

“…However, recent studies suggest that in many situations a more complex decision model may be at work (Abdelaziz et al, 2007;Ballestero et al, 2012;Bollen, 2007;Dorfleitner & Utz, 2012;Hallerbach et al, 2004;Steuer et al, 2007;. Toward that end, in Hirschberger et al (2013), a methodology is developed for extending the portfolio selection model of Markowitz to include a third criterion. Whether the third criterion is a financial or, as in this paper, a non-financial one, this causes the nondominated frontier 1 in two-dimensional space to become a nondominated surface in three-dimensional space, and that paper describes an algorithm for computing tri-criterion nondominated surfaces exactly.…”

“…Here, our paper relates to , who use an ε-constraint method (see Haimes et al, 1971) to generate a discretized representation of a nondominated surface. Because accurate knowledge of the nondominated surface is important in the inverse optimization, we use the approach of Hirschberger et al (2013) which enables us to compute the entire nondominated surface analytically, rather than being consigned to working only with a collection of dispersed dots.…”