# Sharpe ratio

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, named after William Forsyth Sharpe. Since its revision made by the original author in 1994, it is defined as:

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

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 excess return.[1]

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

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

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. However like any mathematical model it relies on the data being correct. Pyramid schemes with a long duration of operation would typically provide a high Sharpe ratio when derived from reported returns but the inputs are false. When examining the investment performance of assets with smoothing of returns (such as With profits funds) the Sharpe ratio should be derived from the performance of the underlying assets rather than the fund returns.

Sharpe ratios, along with Treynor ratios and Jensen's alphas, are often used to rank the performance of portfolio or mutual fund managers.

## History

This ratio was developed by William Forsyth Sharpe in 1966.[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.

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 .

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 (RRf = 0.15 and σ = 0.10).

As a guide post, one could substitute in the longer term return of the S&P500 as 10%. 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.

## Strengths and Weaknesses

The Sharpe ratio has as its principle advantage that it is directly computable from any observed series of returns without need for additional information surrounding the source of profitability. Unfortunately, some authors are carelessly drawn to refer to the ratio as giving the level of 'risk adjusted returns' when the ratio gives only the volatility adjusted returns when interpreted properly. Other ratios such as the Bias ratio (finance) have recently been introduced into the literature to handle cases where the observed volatility may be an especially poor proxy for the risk inherent in a time-series of observed returns.

## References

1. ^ Sharpe, W. F. (1994). "The Sharpe Ratio". Journal of Portfolio Management 21 (1): 49–58.
2. ^ Sharpe, W. F. (1966). "Mutual Fund Performance". Journal of Business 39 (S1): 119–138. doi:10.1086/294846.
3. ^ Scholz, Hendrik (2007). "Refinements to the Sharpe ratio: Comparing alternatives for bear markets". Journal of Asset Management 7 (5): 347–357. doi:10.1057/palgrave.jam.2250040.