Optimal Dynamic Trading with Leverage Constraints

We solve for the optimal dynamic trading strategy of an investor who faces a leverage constraint, i.e., a limitation on his ability to borrow for the purpose of investing in a risky asset. We assume that the investor has constant relative risk aversion, and that the value of the risky asset follows a geometric Brownian motion. In the absence of the leverage constraint, the optimal strategy involves investing a fixed proportion of wealth in the risky asset. We prove that, in the presence of the leverage constraint, the optimal investment also involves investing a fixed proportion of wealth in the risky asset when the leverage constraint is not binding. However, the two proportions are different, reflecting the extent to which the investor alters his strategy even when the leverage constraint is not binding because of the possibility that the leverage constraint will become binding in the future.