Multiperiod mean-variance customer constrained portfolio optimization for finite discrete-time markov chains

The multi-period formulation aims at selecting an optimal investment strategy in a time-horizon able to maximize the final wealth while minimize the risk and determine the exit time. This paper is dedicated to solve the multi-period mean-variance customer constrained Markowitz’s portfolio optimization problem employing the extraproximal method restricted to a finite discrete time, ergodic and controllable Markov chains for finite time horizon. The extraproximal method can be considered as a natural generalization of the convex programming approximation methods that largely simplifies the mathematical analysis and the economic interpretation of such model settings. We show that the multi-period mean-variance optimal portfolio can be decomposed in terms of coupled nonlinear programming problems implementing the Lagrange principle, each having a clear economic interpretation. This decomposition is a multi-period representation of single-period mean variance customer portfolio which naturally extends the basic economic intuition of the static Markowitz’s model (where the investment horizon is practically never known at the beginning of initial investment decisions). This implies that the corresponding multi-period mean-variance customer portfolio is determined for a system of equations in proximal format. Each equation in this system is an optimization mean-variance problem which is solved using an iterating projection gradient method. Iterating these steps, we obtain a new quick procedure which leads to a simple and logically justified computational realization: at each iteration of the extraproximal method the functional of the mean-variance portfolio converges to an equilibrium point. We provide conditions for the existence of a unique solution to the portfolio problem by employing a regularized Lagrange function. We present the convergence proof of the method and all the details needed to implement the extraproximal method in an efficient and numerically stable way. Empirical results are finally provided to illustrate the suitability and practical performance of the model and the derived explicit portfolio strategy.