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 "M"<sub>t</sub> is the maximum level of wealth W attained on or before time "t", then the constraint imposed on his portfolio choice is that W<sub>t</sub>α"M"<sub>t</sub>, 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""W"<sub>t</sub> - α"M"<sub>t</sub>. 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 α"M"<sub>t</sub>. 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 ="M"<sub>t</sub>. It can be shown that at "W"<sub>t</sub>="M"<sub>t</sub>, α"M"<sub>t</sub> is expected to grow at a faster rate than "W"<sub>t</sub>, 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 "W"<sub>t</sub> is close to α"M"<sub>t</sub>. 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 "W"<sub>t</sub>="M"<sub>t</sub>). Copyright 1993 Blackwell Publishers.
