Sharpe ratio
In
It was named after William F. Sharpe,[1] who developed it in 1966.
Definition
Since its revision by the original author, William Sharpe, in 1994,[2] the ex-ante Sharpe ratio is defined as:
where is the asset return, is the
The ex-post Sharpe ratio uses the same equation as the one above but with realized returns of the asset and benchmark rather than expected returns; see the second example below.
The information ratio is a generalization of the Sharpe ratio that uses as benchmark some other, typically risky index rather than using risk-free returns.
Use in finance
The Sharpe ratio seeks to characterize how well the return of an asset compensates the investor for the risk taken. When comparing two assets, the one with a higher Sharpe ratio appears to provide better return for the same risk, which is usually attractive to investors.[3]
However, financial assets are often
Even in less extreme cases, a reliable empirical estimate of Sharpe ratio still requires the collection of return data over sufficient period for all aspects of the strategy returns to be observed. For example, data must be taken over decades if the algorithm sells an insurance that involves a high liability payout once every 5–10 years, and a high-frequency trading algorithm may only require a week of data if each trade occurs every 50 milliseconds, with care taken toward risk from unexpected but rare results that such testing did not capture (see flash crash).
Additionally, when examining the investment performance of assets with smoothing of returns (such as
Sharpe ratios, along with Treynor ratios and Jensen's alphas, are often used to rank the performance of portfolio or mutual fund managers. Berkshire Hathaway had a Sharpe ratio of 0.76 for the period 1976 to 2011, higher than any other stock or mutual fund with a history of more than 30 years. The stock market[specify] had a Sharpe ratio of 0.39 for the same period.[5]
Tests
Several statistical tests of the Sharpe ratio have been proposed. These include those proposed by Jobson & Korkie[6] and Gibbons, Ross & Shanken.[7]
History
In 1952, Arthur 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, or MAR) and σ is standard deviation of returns.[8] This ratio is just the Sharpe ratio, only using minimum acceptable return instead of the risk-free rate in the numerator, and using standard deviation of returns instead of standard deviation of excess returns in the denominator. Roy's ratio is also related to the Sortino ratio, which also uses MAR in the numerator, but uses a different standard deviation (semi/downside deviation) in the denominator.
In 1966, William F. Sharpe developed what is now known as the Sharpe ratio.[1] Sharpe originally called it the "reward-to-variability" ratio before it began being called the Sharpe ratio by later academics and financial operators. The definition was:
Sharpe's 1994 revision acknowledged that the basis of comparison should be an applicable benchmark, which changes with time. After this revision, the definition is:
Note, if Rf is a constant risk-free return throughout the period,
The (original) Sharpe ratio has often been challenged with regard to its appropriateness as a fund performance measure during periods of declining markets.[9]
Examples
Example 1
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. We estimate the risk of the asset, defined as standard deviation of the asset's
Example 2
An investor has a portfolio with an expected return of 12% and a standard deviation of 10%. The rate of interest is 5%, and is risk-free.
The Sharpe ratio is:
Strengths and weaknesses
A negative Sharpe ratio means the portfolio has underperformed its benchmark. All other things being equal, an investor typically prefers a higher positive Sharpe ratio as it has either higher returns or lower volatility. However, a negative Sharpe ratio can be made higher by either increasing returns (a good thing) or increasing volatility (a bad thing). Thus, for negative values the Sharpe ratio does not correspond well to typical investor utility functions.
The Sharpe ratio is convenient because it can be calculated purely from any observed series of returns without need for additional information surrounding the source of profitability. However, this makes it vulnerable to manipulation if opportunities exist for smoothing or discretionary pricing of illiquid assets. Statistics such as the
While the
The returns measured can be of any frequency (i.e. daily, weekly, monthly or annually), as long as they are
For Brownian walk, Sharpe ratio is a dimensional quantity and has units , because the excess return and the volatility are proportional to and correspondingly. Kelly criterion is a dimensionless quantity, and, indeed, Kelly fraction is the numerical fraction of wealth suggested for the investment.
In some settings, the Kelly criterion can be used to convert the Sharpe ratio into a rate of return. The Kelly criterion gives the ideal size of the investment, which when adjusted by the period and expected rate of return per unit, gives a rate of return.[11]
The accuracy of Sharpe ratio estimators hinges on the statistical properties of returns, and these properties can vary considerably among strategies, portfolios, and over time.[12]
Drawback as fund selection criteria
Bailey and López de Prado (2012)[13] show that Sharpe ratios tend to be overstated in the case of hedge funds with short track records. These authors propose a probabilistic version of the Sharpe ratio that takes into account the asymmetry and fat-tails of the returns' distribution. With regards to the selection of portfolio managers on the basis of their Sharpe ratios, these authors have proposed a Sharpe ratio indifference curve[14] This curve illustrates the fact that it is efficient to hire portfolio managers with low and even negative Sharpe ratios, as long as their correlation to the other portfolio managers is sufficiently low.
In recent years, many financial websites have promoted the idea that a Sharpe Ratio "greater than 1 is considered acceptable; a ratio higher than 2.0 is considered very good; and a ratio above 3.0 is excellent." While it is unclear where this rubric originated online, it makes little sense since the magnitude of the Sharpe ratio is sensitive to the time period over which the underlying returns are measured. This is because the nominator of the ratio (returns) scales in proportion to time; while the denominator of the ratio (standard deviation) scales in proportion to the square root of time. Most diversified indexes of equities, bonds, mortgages or commodities have annualized Sharpe ratios below 1, which suggests that a Sharpe ratio consistently above 2.0 or 3.0 is unrealistic.
See also
- Bias ratio
- Calmar ratio
- Capital asset pricing model
- Coefficient of variation
- Hansen–Jagannathan bound
- Information ratio
- Jensen's alpha
- List of financial performance measures
- Modern portfolio theory
- Omega ratio
- Risk adjusted return on capital
- Roy's safety-first criterion
- Signal-to-noise ratio
- Sortino ratio
- Sterling ratio
- Treynor ratio
- Upside potential ratio
- V2 ratio
- Z score
References
- ^ doi:10.1086/294846.
- S2CID 55394403. Retrieved June 12, 2012.
- ^ Gatfaoui, Hayette. "Sharpe Ratios and Their Fundamental Components: An Empirical Study". IESEG School of Management.
- JSTOR 1262669.
- ^ http://docs.lhpedersen.com/BuffettsAlpha.pdf [bare URL PDF]
- JSTOR 2327554.
- JSTOR 1913625.
- JSTOR 1907413.
- S2CID 154908707.
- ^ "Understanding The Sharpe Ratio". Retrieved March 14, 2011.
- ISBN 978-0-470-31958-1.
- ^ Lo, Andrew W. (July–August 2002). "The Statistics of Sharpe Ratios". Financial Analysts Journal. 58 (4).
- ^ Bailey, D. and M. López de Prado (2012): "The Sharpe Ratio Efficient Frontier", Journal of Risk, 15(2), pp.3-44. Available at https://ssrn.com/abstract=1821643
- ^ Bailey, D. and M. Lopez de Prado (2013): "The Strategy Approval Decision: A Sharpe Ratio Indifference Curve approach", Algorithmic Finance 2(1), pp. 99-109 Available at https://ssrn.com/abstract=2003638
- ^ Shah, Sunit N. (2014), The Principal-Agent Problem in Finance, CFA Institute, p. 14
Further reading
- Lo, Andrew W. "The statistics of Sharpe ratios." Financial analysts journal 58.4 (2002): 36-52 https://doi.org/10.2469/faj.v58.n4.2453
- 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 0-471-26849-6
- Steven E. Pav. The Sharpe Ratio: Statistics and Applications. CRC Press, 2022. ISBN 978-1-032-01930-7
- Goetzmann, William; Ingersoll, Jonathan; Spiegel, Matthew; Welch, Ivo (2002), Sharpening Sharpe Ratios (PDF), National Bureau of Economic Research.
- Shah, Sunit N. (2014), The Principal-Agent Problem in Finance, CFA Institute
External links
- The Sharpe ratio
- Generalized Sharpe Ratio
- All Hail the Sharpe Ratio - Uses and abuses of the Sharpe Ratio
- What is a good Sharpe Ratio? - Some example calculations of Sharpe ratios
- "A Comparison of Different Measures of Risk-adjusted Return". September 2013.
- What is a good Sharpe Ratio? - Some example calculations of Sharpe ratios
- Sharpe ratio in MS excel - Risk adjusted return calculations
- Calculating and Interpreting Sharpe Ratios online - Cloud calculator