Optimal Portfolio Revision with a Proportional Transaction Cost

The theory of the optimal investment decision is reexamined for a case in which the portfolio initialization problem is trivial but the portfolio adjustment decision presents some new and important problems. The investor is assumed to be a risk-averse expected utility maximizer whose portfolio consists of two risky assets. His objective is to maximize his expected utility at some time horizon, and there is a transaction cost for intermediate adjustment of the portfolio proportional to the value of assets traded. The solution for the optimum adjustment strategy consists of a set of pairs of control limits for the ratio of the amounts of the assets in the portfolio, one pair for each decision opportunity. The model falls naturally into the class of “cash balance” and dynamic portfolio selection models. The most important departure from both of these subclasses of models is the explicit consideration of more than one risky asset in the portfolio together with the transaction cost. The introduction of the transaction cost implies investor behavior which is systematically different from that implied by the no-transaction-cost dynamic portfolio selection models. The implied behavior is, however, quite similar in form to that implied by the cash-balance models, in which the objective is expected cost minimization.