Abstract
The mean-variance (MV) portfolio is typically formulated as a quadratic programming (QP) problem that linearly combines the conflicting objectives of minimizing the risk and maximizing the expected return through a risk aversion profile parameter. In this formulation, the two objectives are expressed in different units, an issue that could definitely hamper obtaining a more competitive set of portfolio weights. For example, a modification in the scale in which returns are expressed (by one or percent) in the MV portfolio, implies a modification in the solution of the problem. Motivated by this issue, a novel mean squared variance (MSV) portfolio is proposed in this paper. The associated optimization problem of the proposed strategy is very similar to the Markowitz optimization, with the exception of the portfolio mean, which is presented in squared form in our formulation. The resulting portfolio model is a non-convex QP problem, which has been reformulated as a mixed-integer linear programming (MILP) problem. The reformulation of the initial non-convex QP problem into an MILP allows for future researchers and practitioners to obtain the global solution of the problem via the use of current state-of-the-art MILP solvers. Additionally, a novel purely data-driven method for determining the optimal value of the hyper-parameter that is associated with the MV and MSV approaches is also proposed in this paper. The MSV portfolio has been empirically tested on eight portfolio time series problems with three different estimation windows (composing a total of 24 datasets), showing very competitive performance in most of the problems.
Highlights
Markowitz (1952) [1,2] proposed the well-known mean-variance (MV) portfolio model under the assumption that a rational investor aims at maximizing returns and minimizing risks
The resulting portfolio model is a non-convex quadratic programming (QP) problem, which has been reformulated as a mixed-integer linear programming (MILP) problem to reach, in this way, the global solution of the problem
To compare the out-of-sample performance of the mean squared variance (MSV) portfolio with the performance provided by state-of-the-art MV-based strategies (Section 4.1); and, to analyse the diversification levels produced by the proposed MSV portfolio and the MV portfolio in problems with different dimensions (Section 4.2)
Summary
Markowitz (1952) [1,2] proposed the well-known mean-variance (MV) portfolio model under the assumption that a rational investor aims at maximizing returns and minimizing risks. Speaking, the MV portfolio framework is a bi-objective optimization problem, where an efficient frontier is composed by all combination assets that are not dominated by any other in expected return and risk simultaneously. The Sharpe ratio assesses the two objectives in the same unit level, unlike the third formulation of the optimization problem, which evaluates the objectives in different units [24] Motivated by this fact, a novel portfolio strategy, which is denoted as mean squared variance (MSV), which calculates the two objectives of the minimization function of the problem in the same unit is proposed in this paper.
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