Control Bands for Tracking Constant Portfolio Allocations with Fixed and Proportional Transaction Costs

The vast majority of research related to optimal asset allocation strategies in the presence of transaction costs, requires formulation of highly sophisticated numerical schemes for the estimation of no-transaction bands; moreover, the optimization objectives examined are far less compared to the number of works that assume frictionless trading. In this article, we point out that an investor may alternatively try to track a constant allocation strategy as derived under the frictionless markets hypothesis and any optimization objective, by applying a loss function that reflects his/her risk preferences. We focus in the two-asset case (one riskless and one risky) and assume a fixed cost per transaction plus a cost proportional to the change in the risky fraction process. Using a recently proposed transformation of the risky fraction process by Nagai (2005), we derive optimal rebalancing policies for the quadratic loss case, using two alternative methods. First, we calculate no transaction bands for investors who choose the boundaries of the bands and their optimal rebalancing actions so that they minimize long run cost per unit time. The latter is defined as the expected cost per transaction cycle (opportunity cost/tracking error plus transaction cost) divided by the expected cycle time. In the second case, the objective is to minimize the expected discounted squared tracking error plus discounted transaction costs over an infinite horizon. On that purpose, similar to Suzuki and Pliska (2004), we use impulse control theory in a continuous-time, dynamic setting and characterize the optimal strategy in terms of a quasi-variational inequality. For both formulations, we derive explicit solutions, which we use to perform sensitivity analysis for the control bands with respect to the market parameters and the magnitude of the transaction costs.