Browsing by Subject "Mean-variance problem"

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Open Access
A tri-objective reformulation for the dynamic mean-variance problem
(2024-07) Çolak, Muhammed Mustafa
The classical mean-variance problem aims to find a portfolio that minimizes a linear combination of the expectation and the variance of the terminal wealth. The dynamic version of the problem is known to be time-inconsistent in the classical sense, which makes the scalar dynamic programming approach inapplicable. By decomposing variance into two separate objectives, we introduce a tri-objective formulation in a discrete-time framework that generalizes the scalar problem and can reduce to the original setting. Using a less restrictive concept of time-consistency in a vector-valued sense, we show that the new formulation is time-consistent. Following the literature on set optimization, we develop a set-valued dynamic programming principle with the upper image of the vector-valued problem used as a value function. Finally, we reduce the generalized solutions of the formulation to the classical mean-variance problem using the minimal points of the three-dimensional upper images. We compute portfolios that are optimal for the initial mean-variance problem, and that remain time-consistent with respect to the tri-objective formulation.
Open Access
Dynamic mean-variance problem: recovering time-consistency
(2021-08) Düzoylum, Seyit Emre
As the foundation of modern portfolio theory, Markowitz’s mean-variance port-folio optimization problem is one of the fundamental problems of ﬁnancial math-ematics. The dynamic version of this problem in which a positive linear com-bination of the mean and variance objectives is minimized is known to be time-inconsistent, hence the classical dynamic programming approach is not applicable. Following the dynamic utility approach in the literature, we consider a less re-strictive notion of time-consistency, where the weights of the mean and variance are allowed to change over time. Precisely speaking, rather than considering a ﬁxed weight vector throughout the investment period, we consider an adapted weight process. Initially, we start by extending the well-known equivalence be-tween the dynamic mean-variance and the dynamic mean-second moment prob-lems in a general setting. Thereby, we utilize this equivalence to give a complete characterization of a time-consistent weight process, that is, a weight process which recovers the time-consistency of the mean-variance problem according to our deﬁnition. We formulate the mean-second moment problem as a biobjective optimization problem and develop a set-valued dynamic programming principle for the biobjective setup. Finally, we retrieve back to the dynamic mean-variance problem under the equivalence results that we establish and propose a backward-forward dynamic programming scheme by the methods of vector optimization. Consequently, we compute both the associated time-consistent weight process and the optimal solutions of the dynamic mean-variance problem.