Multi-Period Mean Expected-Shortfall Strategies: ‘Cut Your Losses and Ride Your Gains’

Abstract

Dynamic mean-variance (MV) optimal strategies are inherently contrarian. Following periods of strong equity returns, there is a tendency to de-risk the portfolio by shifting into risk-free investments. On the other hand, if the portfolio still has some equity exposure, the weight on equities will increase following stretches of poor equity returns. This is essentially due to using variance as a risk measure, which penalizes both upside and downside deviations relative to a satiation point. As an alternative, we propose a dynamic trading strategy based on an expected wealth (EW), expected shortfall (ES) objective function. ES is defined as the mean of the worst β fraction of the outcomes, hence the EW-ES objective directly targets left tail risk. We use stochastic control methods to determine the optimal trading strategy. Our numerical method allows us to impose realistic constraints: no leverage, no shorting, infrequent rebalancing. For 5 year investment horizons, this strategy generates an annualized alpha of 180 bps compared to a 60:40 stock-bond constant weight policy. Bootstrap resampling with historical data shows that these results are robust to parametric model misspecification. The optimal EW-ES strategy is generally a momentum-type policy, in contrast to the contrarian MV optimal strategy.