Why a New Ratio-Based Portfolio Rule Beats the Old Sharpe Maxim

A new paper from researchers at ETH Zurich and the University of Zurich shows that portfolio rules designed for ratio-type periodic evaluation – think Sharpe ratio, Sortino ratio, or Calmar ratio – outperform standard expected-utility maximization when the investor faces convex trading constraints like leverage limits or margin requirements.

The paper, posted to arXiv this week, models an investor who is evaluated not on terminal wealth but on a ratio of performance to risk over discrete evaluation periods. That matches how most institutional mandates actually work: a fund's Sharpe ratio over the last 12 months determines bonus pools, capital allocations, and client redemptions.

Under convex constraints – the kind that bind in practice, like a 2x leverage cap or a VaR limit – the optimal strategy under ratio evaluation differs sharply from the Merton-style solution that maximizes expected utility of terminal wealth. The ratio-optimal investor takes less leverage early in the evaluation period and more later, once the denominator (risk) is locked in. The utility-maximizer does the opposite.

The authors prove existence and uniqueness of the optimal policy under a general stochastic factor model, and they derive a verification theorem that lets them compute the policy numerically. For a simple one-factor model calibrated to S&P 500 and Treasury data, the ratio-optimal strategy delivers a 0.12 higher annualized Sharpe than the utility-maximizing strategy, with lower drawdowns.