Sharpe ratio

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The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return (or Risk Premium) per unit of risk in an investment asset or a trading strategy. Since its revision made by the original author in 1994, it is defined as :

<math>S = \frac{E[R-R_f]}{\sigma} = \frac{E[R-R_f]}{\sqrt{\mathrm{var}[R-R_f]}}</math>,

where <math>R</math> is the asset return, <math>R_f</math> is the return on a benchmark asset, such as the risk free rate of return, <math>E[R-R_f]</math> is the expected value of the excess of the asset return over the benchmark return, and <math>\sigma</math> is the standard deviation of the excess return (Sharpe 1994). Note, if <math>R_f</math> is a constant risk free return throughout the period,

<math>\sqrt{\mathrm{var}[R-R_f]}=\sqrt{\mathrm{var}[R]}</math>.

Sharpe's 1994 revision acknowledged that the risk free rate changes with time. Prior to this revision the definition was <math>S = \frac{E[R]-R_f}{\sigma}</math> assuming a constant <math>R_f</math>. The Sharpe ratio is used to characterize how well the return of an asset compensates the investor for the risk taken. When comparing two assets each with the expected return <math>E[R]</math> against the same benchmark with return <math>R_f</math>, the asset with the higher Sharpe ratio gives more return for the same risk. Investors are often advised to pick investments with high Sharpe ratios. Sharpe ratios, along with Treynor ratios and Jensen's alphas, are often used to rank the performance of portfolio or mutual fund managers. This ratio was developed by William Forsyth Sharpe in 1966. Sharpe originally called it the "reward-to-variability" ratio in before it began being called the Sharpe Ratio by later academics and financial professionals. Recently, the (original) Sharpe ratio has often been challenged with regard to its appropriateness as a fund performance measure during evaluation periods of declining markets (Scholz 2007).

1 Examples
2 References
3 See also
4 External links

Examples

Suppose the asset has an expected return of 15%. We typically do not know the asset will have this return; suppose we assess the risk of the asset, defined as standard deviation of the asset's excess return, as 10%. The risk-free return is constant. Then the Sharpe ratio (using a new definition) will be 1.5 (<math>R = 0.15 </math> and <math>\sigma = 0.10 </math>). As a guide post, one could substitute in the longer term return of the S&P500 as 10%. The risk-free return is constant. And the average standard deviation of the S&P500 is about ±16%. Doing the math, we get that the average, long-term Sharpe ratio of the US market is about 0.625. But we should note that if one were to calculate the ratio over, for example, three-year rolling periods, then the Sharpe ratio would vary dramatically.

References

Sharpe, W.F. (1966). Mutual Fund Performance. Journal of Business, 39, 119-138.
Sharpe, W.F. (1994). The Sharpe Ratio. Journal of Portfolio Management, 21, Fall, Issue 1 49-58.
Scholz, H. (2007). Refinements to the Sharpe ratio: Comparing alternatives for bear markets. Journal of Asset Management, Vol. 7, 347-357.