Optimal Investment Strategies For Controlling Drawdowns

We analyze the optimal risky investment policy for an investor who, at each point in time, wants to lose no more than a fixed percentage of the maximum value his wealth has achieved up to that time. In particular, if Mt is the maximum level of wealth W attained on or before time t, then the constraint imposed on his portfolio choice is that WtαMt, where α is an exogenous number betweenα O and 1. We show that, for constant relative risk aversion utility functions, the optimal policy involves an investment in risky assets at time t in proportion to the “surplus”Wt ‐ αMt. the optimal policy may appear similar to the constant‐proportion portfolio insurance policy analyzed in Black and Perold (1987) and Grossman and Vila (1989). However, in those papers, the investor keeps his wealth above a nonstochastic floor F instead of a stochastic floor αMt. the stochastic character of the floor studied here has interesting effects on the investment policy in states of nature when wealth is at an all‐time high; i.e., when Wt =Mt. It can be shown that at Wt=Mt, αMt is expected to grow at a faster rate than Wt, and therefore the investment in the risky asset can be expected to fall. We also show that the investment in the risky asset can be expected to rise when Wt is close to αMt. We conjecture that in an equilibrium model the stochastic character of the floor creates “resistance” levels as the market approaches an all‐time high (because of the reluctance of investors to take more risk when Wt=Mt).