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:

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

where R is the asset return, Rf is the return on a benchmark asset, such as the risk free rate of return, E[RRf] is the expected value of the excess of the asset return over the benchmark return, and σ is the standard deviation of the asset return. The risk-free interest rate is the Interest rate that it is assumed can be obtained by investing in Financial instruments with no Default risk In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values [1]

Note, if Rf is a constant risk free return throughout the period,

$\sqrt{\mathrm{var}[R-R_f]}=\sqrt{\mathrm{var}[R]}$.

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 Rf .

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 Rf, 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. The Treynor ratio is a measurement of the returns earned in excess of that which could have been earned on a riskless Investment (i In finance Jensen's alpha (or Jensen's Performance Index, ex-post alpha) is used to determine the excess return of a security or portfolio of securities A mutual fund is a professionally managed type of collective investments that pools money from many investors and Invests it in Stocks bonds,

This ratio was developed by William Forsyth Sharpe in 1966. William Forsyth Sharpe (born June 16, 1934) is the STANCO 25 Professor of Finance Emeritus at Stanford University 's Graduate School of Business [2] 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. [3]

## Examples

Suppose the asset has an expected return of 15% in excess of the risk free rate. 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 (R = 0. 15 and σ = 0. 10).

As a guide post, one could substitute in the longer term return of the S&P500 as 10%. The S&P 500 is a Stock market index containing the stocks of 500 Large-Cap Corporations all of which are from the United States. Assume the risk-free return is 3. 5%. 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. 40625 ((10%-3. 5%)/16%). 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

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