A dynamic programming approach to two-stage mean-variance portfolio selection in cointegrated vector autoregressive systems

In this paper we study the problem of optimal portfolio construction when the trading horizon consists of two consecutive decision intervals and rebalancing is permitted. It is assumed that the log-prices of the underlying assets are non-stationary, and specifically follow a discrete-time cointegrated vector autoregressive model. We extend the classical Markowitz mean-variance optimization approach to a multi-period setting, in which the new objective is to maximize the total expected return, subject to a constraint on the total allowable risk. In contrast to traditional approaches, we adopt a definition for risk which takes into account the non-zero correlations between the inter-stage returns. This portfolio optimization problem amounts to not only determining the relative proportions of the assets to hold during each stage, but also requires one to determine the degree of portfolio leverage to assume. Due to a fixed constraint on the standard deviation of the total return, the leverage decision is equivalent to deciding how to optimally partition the allowed variance, and thus variance can be viewed as a shared resource between the stages. We derive the optimal portfolio weights and variance scheduling scheme for a trading strategy based on a dynamic programming approach, which is utilized in order to make the problem computationally tractable. The performance of this method is compared to other trading strategies using both Monte Carlo simulations and real data, and promising results are obtained.