Sharpe ratio

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The Sharpe ratio is a measure of risk-adjusted performance of an investment asset, or a trading strategy. Since its revision by the original author made in 1994, it is defined as:

$S = \frac{E[R-R_f]}{\sigma} = \frac{E[R-R_f]}{\sqrt{Var[R-R_f]}}$ ,

where $R$ is the asset return, $R f$ is the return on a benchmark asset, such as the risk free rate of return, $E [R - R f]$ is the expected value of the excess of the asset return over the benchmark return, and $σ$ is the standard deviation of the excess return.

Note, if $R f$ is a constant risk free return throughout the period, $\sqrt{Var[R-R_f]}=\sqrt{Var[R]}$ . Sharpe´s 1994 revision acknowledged that the risk free rate changes with time, prior to this revision the definition was $S = \frac{E[R]-R_f}{\sigma}$ assuming a constant $R f$ .

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 $E [R]$ against the same benchmark with return $R f$ , 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. Sharpe originally called it the "reward-to-variability" ratio 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.

[edit] 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%. Finally, suppose the risk-free rate of return, $R f$ , is 4%. Then the Sharpe ratio will be 1.10 ( $R = 0.15$ , $R f = 0.04$ , and $σ = 0.10$ ).