Optimal Dynamic Trading with Leverage Constraints

We solve for the optimal dynamic trading strategy of an investor who faces two constraints. The first constraint is a limitation on his ability to borrow for the purpose of investing in a risky asset, i.e., the market value of his investments in the risky asset X, must be less than an exogenously given function of his wealth X(W). The second constraint is the requirement that the investor’s wealth be non-negative at all times, i.e., Wt>O. We assume that the investor has constant relative risk aversion A, and the value of the risky asset follows a diffusion with drift m+r (where r is the risk free rate) and per unit time variance s2. In the absence of the constraints, X &Mac186; (m/s2)*W/A. We prove that in the presence of the above constraints the optimal investment is X &Mac186; Min[(m/s2)*W/a, X(W)]. The coefficient a is not in general equal to A, and represents the extent to which the investor alters his strategy even when the constraints are not binding because of the possibility that the constraints will become binding in the future.