# Sharpe ratio

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In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) is a way to examine the performance of an investment by adjusting for its risk. The ratio measures 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 F. Sharpe.[1]

## Definition

Since its revision by the original author, William Sharpe, in 1994,[2] the ex-ante Sharpe ratio is defined as:

${\displaystyle S_{a}={\frac {E[R_{a}-R_{b}]}{\sigma _{a}}}={\frac {E[R_{a}-R_{b}]}{\sqrt {\mathrm {var} [R_{a}-R_{b}]}}},}$

where ${\displaystyle R_{a}}$ is the asset return, ${\displaystyle R_{b}}$ is the return on a benchmark asset, such as the risk free rate of return or the return of an index such as the S&P 500. ${\displaystyle E[R_{a}-R_{b}]}$ is the expected value of the excess of the asset return over the benchmark return, and ${\displaystyle {\sigma }}$ is the standard deviation of the asset excess return.

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 Sharpe ratio is similar to the Information ratio but, whereas the Sharpe ratio is the 'excess' return of an asset over the return of a risk free asset divided by the variability or standard deviation of returns, the information ratio is the active return to the most relevant benchmark index divided by the standard deviation of the 'active' return or tracking error.

## Use in finance

The Sharpe ratio characterizes how well the return of an asset compensates the investor for the risk taken. When comparing two assets versus a common benchmark, the one with a higher Sharpe ratio provides better return for the same risk (or, equivalently, the same return for lower risk). However, like any other 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.

## Tests

Several statistical tests of the Sharpe ratio have been proposed. These include those proposed by Jobson & Korkie[3] and Gibbons, Ross & Shanken.[4]

## 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.[5] 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:

${\displaystyle S={\frac {E[R-R_{f}]}{\sqrt {\mathrm {var} [R]}}}.}$

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:

${\displaystyle S={\frac {E[R-R_{b}]}{\sqrt {\mathrm {var} [R-R_{b}]}}}.}$

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

${\displaystyle {\sqrt {\mathrm {var} [R-R_{f}]}}={\sqrt {\mathrm {var} [R]}}.}$

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.[6]

## 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; 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 the old definition) will be${\displaystyle {\frac {R_{a}-R_{f}}{\sigma _{a}}}={\frac {0.15}{0.10}}=1.5}$

Example 2

For an example of calculating the more commonly used ex-post Sharpe ratio - which uses realized rather than expected returns - based on the contemporary definition, consider the following table of weekly returns.

Date Asset Return S&P 500 total return Excess Return
7/6/2012 -0.0050000 -0.0048419 -0.0001581
7/13/2012 0.0010000 0.0017234 -0.0007234
7/20/2012 0.0050000 0.0046110 0.0003890

We assume that the asset is something like a large-cap U.S. equity fund which would logically be benchmarked against the S&P 500. The mean of the excess returns is -0.0001642 and the (sample) standard deviation is 0.0005562248, so the Sharpe ratio is -0.0001642/0.0005562248, or -0.2951444.

Example 3

Suppose that you currently have \$250,000 invested in a portfolio with an expected return of 12% and a volatility of 10%. The efficient (tangent) portfolio has an expected return of 17% and a volatility of 12%. The risk-free rate of interest is 5%. What is the Sharpe ratio?

A lot of the information is extraneous, so we can ignore it. The Sharpe ratio is: ${\displaystyle {\frac {0.12-0.05}{0.1}}=70\%}$

## Strengths and weaknesses

The main complaint is this ratio relies on the notions that risk equals volatility and that volatility is bad. Simple logic will tell you that the more you reduce volatility, the less likely you are to be able to capture higher returns. But the bigger problem for the Sharpe ratio is that it treats all volatility the same. Basically, the ratio penalizes strategies that have upside volatility (i.e., big positive returns), but those that developed other risk adjusted ratios just don’t think big positive returns should be viewed as a negative thing. 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 systematic risk of a portfolio, the Sharpe ratio observes both systematic 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 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.[7]

Bailey and López de Prado (2012)[8] 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[9] 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.

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. 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.)[10]

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.[11]

## References

1. ^ a b Sharpe, W. F. (1966). "Mutual Fund Performance". Journal of Business. 39 (S1): 119–138. doi:10.1086/294846.
2. ^ Sharpe, William F. (1994). "The Sharpe Ratio". The Journal of Portfolio Management. 21 (1): 49–58. doi:10.3905/jpm.1994.409501. Retrieved June 12, 2012.
3. ^ Jobson JD; Korkie B (September 1981). "Performance hypothesis testing with the Sharpe and Treynor measures". The Journal of Finance. 36 (4): 888–908. JSTOR 2327554.
4. ^ Gibbons M; Ross S; Shanken J (September 1989). "A test of the efficiency of a given portfolio". Econometrica. 57 (5): 1121–1152. doi:10.2307/1913625. JSTOR 1913625.
5. ^ Roy, Arthur D. (July 1952). "Safety First and the Holding of Assets". Econometrica. 20 (3): 431–450. doi:10.2307/1907413. JSTOR 1907413.
6. ^ 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.
7. ^ "Understanding The Sharpe Ratio". Retrieved March 14, 2011.
8. ^ Bayley, D. and M. López de Prado (2012): "The Sharpe Ratio Efficient Frontier", Journal of Risk, 15(2), pp.3-44. Available at http://ssrn.com/abstract=1821643
9. ^ 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 http://ssrn.com/abstract=2003638
10. ^ Wilmott, Paul (2007). Paul Wilmott introduces Quantitative Finance (Second ed.). Wiley. pp. 429–432. ISBN 978-0-470-31958-1.
11. ^ Lo, Andrew W. (July–August 2002). "The Statistics of Sharpe Ratios". Financial Analysts Journal. 58 (4).