When excess return is negative, the Sharpe ratio is also negative, which can be counterintuitive. The author presents a modification that provides a more useful ranking than does the traditional Sharpe ratio.
The Sharpe ratio is a measure of volatility-adjusted performance and is calculated by dividing excess return by the standard deviation of excess return. Excess return is defined as the return in excess of the risk-free rate of return—for example, the three-month T-bill rate. When portfolio performance is ranked by using the Sharpe measure, a higher value indicates better risk-adjusted performance. Although one might interpret excess return to be a positive number, it can be negative at times, such as during long market downturns. As mentioned by other authors and demonstrated by the author in this article, the reliability of the Sharpe ratio as an evaluating measure comes into question when excess return, and thus the calculated Sharpe ratio, is negative. The author illustrates the issue with a hypothetical example that compares two funds in which the second fund has a lower return and higher volatility than the first fund and thus has a larger (less negative) Sharpe ratio. He presents a modification to the Sharpe ratio to resolve this counterintuitive result that appears when excess return is negative.
The author’s solution is to add an exponent to the denominator of the Sharpe ratio (standard deviation of excess return). The exponent is excess return divided by the absolute value of excess return. Using this modification in the hypothetical example mentioned earlier results in an intuitive ranking, with the second fund now having a smaller (more negative) Sharpe ratio than the first fund does. The author presents graphical and statistical correlations of U.S. equity fund rankings with the two Sharpe measures and observes that correlations are tighter when he uses the modified Sharpe ratio. He concludes that when excess returns are negative, rankings produced by the Sharpe metric are improved by adding the exponent to the denominator.