OPTIMAL INVESTMENT STRATEGIES FOR CONTROLLING DRAWDOWNS |
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Authors: | Sanford J Grossman Zhongquan Zhou |
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Institution: | Department of Finance, the Wharton School, University of Pennsylvania, Philadelphia, PA |
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Abstract: | 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 t is the maximum level of wealth W attained on or before time t , then the constraint imposed on his portfolio choice is that Wtα M t, 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 t - α M t. 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 t. 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 t. It can be shown that at W t= M t, α M t is expected to grow at a faster rate than W t, 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 t is close to α M t. 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 t= M t). |
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Keywords: | drawdown portfolio insurance semimartingales stochastic control |
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