# Sharpe ratio

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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 deviation in an investment asset or a trading strategy, typically referred to as risk (and is a deviation risk measure), named after William Forsyth Sharpe. Since its revision 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 excess of the asset return. (This is often confused with the excess return over the benchmark return; the Sharpe ratio utilizes the asset standard deviation whereas the information ratio utilizes standard deviation of excess return over the benchmark, i.e. the tracking error, as the denominator.) 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, the higher the Sharpe ratio number the better. 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

In 1952, A. D. Roy suggested maximizing the ratio "(m-d)/σ", where m is expected gross return, d is some "disaster level" (a.k.a., minimum acceptable return) and σ is standard deviation of returns.[1] This ratio is just the Sharpe Ratio, only using minimum acceptable return instead of risk-free return in the numerator, and using standard deviation of returns instead of standard deviation of excess returns in the denominator.

In 1966, William Forsyth Sharpe developed what is now known as the Sharpe ratio.[2] Sharpe originally called it the "reward-to-variability" ratio before it began being called the Sharpe Ratio by later academics and financial operators.

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 if 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 0.4 ((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 could vary dramatically.

## Strengths and weaknesses

The Sharpe ratio has as its principal advantage that it is directly computable from any observed series of returns without need for additional information surrounding the source of profitability. Other ratios such as the bias ratio 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.

While the Treynor ratio works only with systemic risk of a portfolio, the Sharpe ratio observes both systemic and idiosyncratic risks.

The returns measured can be of any frequency (i.e. daily, weekly, monthly or annually), as long as they are normally distributed, as the returns can always be annualized. Herein lies the underlying weakness of the ratio - not all asset returns are normally distributed. Abnormalities like kurtosis, fatter tails and higher peaks, or skewness on the distribution can be a problematic for the ratio, as standard deviation doesn't have the same effectiveness when these problems exist. Sometimes it can be downright dangerous to use this formula when returns are not normally distributed. [4]

López de Prado (2008)[5] shows that Sharpe ratios tend to be "inflated" in the case of hedge funds with short track records.

Because it is a dimensionless ratio, laypeople find it difficult to interpret Sharpe Ratios of different investments. For example, how much better is an investment with a Sharpe Ratio of 0.5 than one with a Sharpe Ratio of -0.2? This weakness was well addressed by the development of the Modigliani Risk-Adjusted Performance measure, which is in units of percent return – universally understandable by virtually all investors.

## References

1. ^ Roy, Arthur D. (1952). "Safety First and the Holding of Assets". Econometrica 1952 (July): 431–450.
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.
4. ^ [1]
5. ^ López de Prado, M. (2008): "The Sharpe Ratio Efficient Frontier", Working paper, RCC at Harvard University http://ssrn.com/abstract=1821643

• Bacon Practical Portfolio Performance Measurement and Attribution 2nd Ed: Wiley, 2008. ISBN 978-0-470-05928-9
• Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0471268496

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