Sharpe ratio
Encyclopedia
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:
where
is the asset return, 
is the return on a benchmark asset, such as the risk free rate of return, 
is the expected value
of the excess of the asset return over the benchmark return, and
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
is a constant risk free return throughout the period,
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
against the same benchmark with return 
, 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 ratio
s and Jensen's alpha
s, are often used to rank the performance of portfolio or mutual fund
managers.
In 1966, William Forsyth Sharpe
developed what is now known as the Sharpe ratio. 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
assuming a constant 
.
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.
(
and 
).
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.
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.
López de Prado (2008) 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.
Risk premium
A risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset, in order to induce an individual to hold the risky asset rather than the risk-free asset...
) per unit of deviation in an investment asset or a trading strategy, typically referred to as risk
Financial risk
Financial risk an umbrella term for multiple types of risk associated with financing, including financial transactions that include company loans in risk of default. Risk is a term often used to imply downside risk, meaning the uncertainty of a return and the potential for financial loss...
(and is a deviation risk measure
Deviation risk measure
In financial mathematics, a deviation risk measure is a function to quantify financial risk in a different method than a general risk measure...
), named after William Forsyth Sharpe
William Forsyth Sharpe
William Forsyth Sharpe is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences....
. Since its revision by the original author in 1994, it is defined as:
where
is the asset return, is the return on a benchmark asset, such as the risk free rate of return, is the expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the excess of the asset return over the benchmark return, and
is the standard deviationStandard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
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
Information ratio
The Information ratio is a measure of the risk-adjusted return of a financial security . It is also known as Appraisal ratio and is defined as expected active return divided by tracking error, where active return is the difference between the return of the security and the return of a selected...
utilizes standard deviation of excess return over the benchmark, i.e. the tracking error, as the denominator.) Note, if
is a constant risk free return throughout the period,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
against the same benchmark with return , 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 ratio
Treynor ratio
The Treynor ratio , named after Jack L. Treynor, is a measurement of the returns earned in excess of that which could have been earned on an investment that has no diversifiable risk , per each unit of market risk assumed.The Treynor ratio relates...
s and Jensen's alpha
Jensen's alpha
In finance, Jensen's alpha is used to determine the abnormal return of a security or portfolio of securities over the theoretical expected return....
s, are often used to rank the performance of portfolio or mutual fund
Mutual fund
A mutual fund is a professionally managed type of collective investment scheme that pools money from many investors to buy stocks, bonds, short-term money market instruments, and/or other securities.- Overview :...
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. 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
William Forsyth Sharpe
William Forsyth Sharpe is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences....
developed what is now known as the Sharpe ratio. 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
assuming a constant .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.
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(
and ).As a guide post, one could substitute in the longer term return of the S&P500
S&P 500
The S&P 500 is a free-float capitalization-weighted index published since 1957 of the prices of 500 large-cap common stocks actively traded in the United States. The stocks included in the S&P 500 are those of large publicly held companies that trade on either of the two largest American stock...
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 ratioBias ratio (finance)
The bias ratio is an indicator used in finance to analyze the returns of investment portfolios, and in performing due diligence.The bias ratio is a concrete metric that detects valuation bias or deliberate price manipulation of portfolio assets by a manager of a hedge fund, mutual fund or similar...
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
Treynor ratio
The Treynor ratio , named after Jack L. Treynor, is a measurement of the returns earned in excess of that which could have been earned on an investment that has no diversifiable risk , per each unit of market risk assumed.The Treynor ratio relates...
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 kurtosisKurtosis riskKurtosis risk in statistics and decision theory denotes the fact that observations are spread in a wider fashion than the normal distribution entails...
, fatter tailsFat tailA fat-tailed distribution is a probability distribution that has the property, along with the heavy-tailed distributions, that they exhibit extremely large skewness or kurtosis. This comparison is often made relative to the ubiquitous normal distribution, which itself is an example of an...
and higher peaks, or skewnessSkewness riskSkewness risk in financial modeling denotes that observations are not spread symmetrically around an average value. As a result, the average and the median can be different...
on the distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
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.
López de Prado (2008) 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
Modigliani Risk-Adjusted Performance
Modigliani risk-adjusted performance or M2 or M2 or Modigliani–Modigliani measure or RAP is a measure of the risk-adjusted returns of some investment portfolio. It measures the returns of the portfolio, adjusted for the deviation of the portfolio , relative to that of some benchmark...
measure, which is in units of percent return – universally understandable by virtually all investors.
See also
- Bias ratio (finance)Bias ratio (finance)The bias ratio is an indicator used in finance to analyze the returns of investment portfolios, and in performing due diligence.The bias ratio is a concrete metric that detects valuation bias or deliberate price manipulation of portfolio assets by a manager of a hedge fund, mutual fund or similar...
- Calmar ratioCalmar RatioCalmar ratio is a performance measurement used to evaluate Commodity Trading Advisors and hedge funds. It was created by Terry W. Young and first published in 1991 in the trade journal Futures....
- Capital asset pricing modelCapital asset pricing modelIn finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
- Coefficient of variationCoefficient of variationIn probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution. It is also known as unitized risk or the variation coefficient. The absolute value of the CV is sometimes known as relative standard deviation , which is...
- Hansen-Jagannathan bound
- Information ratioInformation ratioThe Information ratio is a measure of the risk-adjusted return of a financial security . It is also known as Appraisal ratio and is defined as expected active return divided by tracking error, where active return is the difference between the return of the security and the return of a selected...
- Jensen's alphaJensen's alphaIn finance, Jensen's alpha is used to determine the abnormal return of a security or portfolio of securities over the theoretical expected return....
- Modern portfolio theoryModern portfolio theoryModern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
- Modigliani Risk-Adjusted PerformanceModigliani Risk-Adjusted PerformanceModigliani risk-adjusted performance or M2 or M2 or Modigliani–Modigliani measure or RAP is a measure of the risk-adjusted returns of some investment portfolio. It measures the returns of the portfolio, adjusted for the deviation of the portfolio , relative to that of some benchmark...
- Risk adjusted return on capitalRisk adjusted return on capitalRisk adjusted return on capital is a risk-based profitability measurement framework for analysing risk-adjusted financial performance and providing a consistent view of profitability across businesses. The concept was developed by Bankers Trust and principal designer Dan Borge in the late 1970s...
- Roy's safety-first criterionRoy's safety-first criterionRoy's safety-first criterion is a risk management technique that allows an investor to select one portfolio rather than another based on the criterion that the probability of the portfolio's return falling below a minimum desired threshold is minimized....
- Sortino ratioSortino ratioThe Sortino ratio measures the risk-adjusted return of an investment asset, portfolio or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target, or required rate of return, while the Sharpe ratio penalizes both upside and downside...
- Treynor ratioTreynor ratioThe Treynor ratio , named after Jack L. Treynor, is a measurement of the returns earned in excess of that which could have been earned on an investment that has no diversifiable risk , per each unit of market risk assumed.The Treynor ratio relates...
- Upside potential ratioUpside potential ratioThe Upside-Potential Ratio is a measure of a return of an investment asset relative to the minimal acceptable return. The measurement allows a firm or individual to choose investments which have had relatively good upside performance, per unit of downside risk....
- Generalized Sharpe Ratio
Further reading
- 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