A dominance maximization approach to portfolio selection

Portfolio selection is a typical decision problem under uncertainty, where the drivers of uncertainty are the asset returns. Generally, the aim is to choose the fractions of a given capital invested in each asset, such that the resulting portfolio return satisfies specific criteria. Starting with the seminal work of Markowitz [1952, 1959], the key problem in asset allocation is to select a portfolio with appropriate features in terms of gain and risk. Among academics and practitioners, different measures of gain (expected return, median return, absolute or relative wealth, etc.) and risk (variance, semivariance, MAD, CVaR, etc.) have been proposed. From a mathematical viewpoint, the synthetic indices that represent gain and risk (or other features) are modeled by functions of n real variables to be optimized simultaneously. This optimization phase represents the first step of the gain-risk analysis [see, e.g., Blay and Markowitz, 2013, Elton et al, 2009, Markowitz, 1959], where the efficient portfolios, namely the Pareto optimal solutions, are detected. Multiobjective optimization problems arise naturally when the decision process involves several (potentially) conflicting goals that must be taken into account simultaneously. Solving multiobjective programs usually consists in computing the Pareto optimal solution that best suites the decision maker; this is in general carried out by scalarization. Just with respect to the latter approach, it is hardly possible here to even summarize the huge amount of solution methods that have been proposed for multiobjective optimization. We only mention that solution methods are commonly grouped into four main categories: no-preference, a posteriori, a priori and interactive methods. We refer the interested reader to the fundamental Miettinen [2012] for both theoretical bases and review of the literature on multiobjective optimization. Among the Pareto optimal portfolios, one can adopt preference criteria with respect to risk (or gain). Generally, these criteria are introduced in the second step of the gain-risk analysis, where the risk aversion characteristics of the decision maker are specified and by which the optimal choice between efficient portfolios is made. Once the preferences of a decision maker has been modeled by a utility function u(·), the optimal portfolio must be the efficient portfolio with (exact or approximate) maximum expected utility [see Blay and Markowitz, 2013, Carleo et al, 2017, Markowitz, 2014, and references therein]. One of the main limitations of expected utility maximization is the subjective specification of a utility function. A way to overcome this issue is to use exact and approximate stochastic dominance (SD) relations [see Bruni et al, 2012, 2015, Roman and Mitra, 2009, and references therein]. Similarly to Ballestero and Romero [1996], in this paper we provide an alternative framework to the expected utility approach, resulting in a no-preference strategy that requires the solution of a nonlinear nonconvex single-objective reformulation of the original multiobjective problem. More precisely, we develop here a portfolio selection method based only on the probabilistic features of the asset returns. To easily describe the rationale behind this no-preference approach, let us consider a portfolio selection problem with two objectives, a (convex) measure of risk ρP (x) and one (concave) that represents gain γP (x), where x ∈ Rn is the vector of portfolio weights. Each feasible solution x identifies, in the objectives space (in this case the risk-gain plane), a rectangle defined by a reference point (e.g., the so-called nadir vector or the worst values of the objectives on the feasible region) and the point (ρP (x), γP (x)). Clearly, x dominates all the feasible points whose objective values belong to the rectangle. In the light of this consideration, we aim at computing a Pareto optimal solution that maximizes the area of the corresponding rectangle. This simple idea draws inspiration from the hypervolume paradigm (see, e.g., Auger et al [2009], Fleischer [2003] and Zitzler and Thiele [1998]), shares some conceptual similarities with other approaches (e.g., GUESS Buchanan [1997] and the method of the global criterion, see Miettinen [2012]), and can be classified as a particular instance of the single-objective approach that consists in maximizing the (weighted) geometric mean of the difference between the objective functions and the components of the reference point (see Audet et al [2008] and Lootsma et al [1995]). The distinctive and significant features of the resulting approach are further investigated: computing a (global) solution of the reformulation is not sensitive to the scaling of the objective functions and provides one with a (global) Pareto optimum of the original problem that dominates the most (in the sense of the area of the induced rectangle) with respect to the objectives space. Interestingly, this Pareto optimum has the nice property that any other feasible point (including all the other Pareto optima), for which any objective is improved (e.g., entailing a larger gain or a smaller risk w.r.t. the ones provided by our approach) by a factor, is such that the other objective must worsen by at least the same factor. As for the solvability issue, in general, one can not expect to solve globally a nonconvex single-objective program (like the one arising in our framework) in order to recover global Pareto optimal solutions. Surprisingly, as for our reformulation, any stationary point with a nontrivial positive value of the corresponding area turns out to be global optimal. Hence, one can employ any standard nonlinear algorithm, such as the projected gradient one, in order to make our approach practically viable. We provide some numerical results showing the significance of the computed Pareto optima also with respect to the solutions provided by other classical approaches. Moreover, as further development, we envisage that the peculiar nature of the approach makes it fit particularly well into the noncooperative scenario of multi-portfolio selection.