Generalized Markowitz mean-variance principles for multi-period portfolio-selection problems

The multi-period portfolio-selection problem is formulated as a Markowitz mean-variance optimization problem. It is shown that the single-period Markowitz quadratic programming algorithm can be used to solve the multi-period asset-allocation problem with suitable modifications in the covariance and linear constraint matrices. It is assumed that the number of shares invested in risky assets is deterministic and the amount of money invested in the risk-free asset is random at future trading dates when short sales on assets are allowed. The general covariance matrix in the multi-period setup contains intertemporal correlations between assets, in addition to correlations between assets at all trading dates. Analytical solutions for the optimal trading strategy, which is linear in the risk-aversion parameter, are obtained. The efficient frontier is a straight line in the expected return/standard deviation of the portfolio space. When the dynamics of the risky assets follow geometric Brownian motion, it is shown that the time-zero allocations to the risky assets coincide with those obtained by Merton in the continuous-time framework. When short sales are not allowed on assets, the values of the portfolio at future trading dates may not be conserved. In the modified Markowitz mean-variance formulation, the value of the portfolio at future trading dates is conserved in the expected sense, and the optimal trading strategy is selected so that the deviation is minimized in the least-square sense. The efficient frontier is a parabola in the expected return/total variance space when short-sales are allowed and when the portfolio consists of risky assets only. When short sales are not allowed, it is shown that finding the optimal trading strategy is equivalent to solving the single-period Markowitz quadratic programming problem, with suitable modifications in the covariance and linear constraint matrices. By solving the Karush-Kuhn-Tucker conditions, analytical solutions are obtained for a two-period two-assets case, and the Markowitz mean-variance principle is illustrated by solving a test problem numerically using the revised simplex method.