Optimal Portfolio Policy for a Multi - Period Mean Variance Investors

We study the optimal portfolio policy for a multi period mean - variance investor facing multiple risky assets subject to proportional transaction costs, market impact or quadratic transaction costs. We demonstrate analytically that, in the presence of proportional transaction costs, the optimal strategy for the multiperiod investor is to trade in the first period to the boundary of a no - trade region shaped as a parallelogram, and not to trade thereafter. For the case with market impact costs, the optimal portfolio policy is to trade to the boundary of a state dependent rebalancing region. In addition, the rebalancing region converges to the Markowitz portfolio as the investment horizon grows large. We contribute to the literature by characterizing the no - trade region for a multiperiod investor facing proportional transaction costs, and studying the analytical properties of the optimal trading strategy for the model with impact costs. Finally, our contribution is to study numerically the utility losses associated with ignoring transaction costs and investing myopically, as well as how these utility losses depend on relevant parameters. We find that the losses associated with either ignoring transaction costs or behaving myopically can be large. Moreover, the losses from ignoring transaction costs increase in the level of transaction costs, and decrease with the investment horizon, whereas the losses from behaving myopically increase with the investment horizon and are concave unimodal on the level of transaction costs. Our work is related to mean - variance utility and proportional transaction costs. For the case with multiple risky assets, the optimal portfolio policy is characterized by a no - trade region shaped as a parallelogram.