# Sharpe ratio

In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment (e.g., a security or portfolio) compared to a risk-free asset, after adjusting for its risk. It is defined as the difference between the returns of the investment and the risk-free return, divided by the standard deviation of the investment (i.e., its volatility). It represents the additional amount of return that an investor receives per unit of increase in risk.

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:

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

where ${\displaystyle R_{a}}$ is the asset return, ${\displaystyle R_{b}}$ is the risk-free return (such as a U.S. Treasury security). ${\displaystyle E[R_{a}-R_{b}]}$ is the expected value of the excess of the asset return over the benchmark return, and ${\displaystyle {\sigma _{a}}}$ 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 information ratio is similar to the Sharpe ratio, the main difference being that the Sharpe ratio uses a risk-free return as benchmark whereas the information ratio uses a risky index as benchmark (such as the S&P500).

## 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 and enough data is given that we observe all risks that the algorithm or strategy is actually taking. Ponzi schemes with a long duration of operation would typically provide a high Sharpe ratio when derived from reported returns, but eventually the fund will run dry and implode all existing investments when there are no more incoming investors willing to participate in the scheme and keep it going. Similarly, selling very low-strike put options may appear to have a very high Sharpe ratios over the time-span of even years, because low-strike puts act like insurance. On the contrary to the perceived Sharpe ratio, selling puts is a high-risk endeavor that's unsuitable for low-risk accounts due to their maximal potential loss. If the underlying security ever crashes to zero or defaults and investors want to redeem their puts for the entire equity valuation, all of the since-obtained profits and much of the underlying investment could be wiped out.

Thus the data for the Sharpe ratio must be taken over a long enough time-span to integrate all aspects of the strategy to a high confidence interval. 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 with-profits funds), the Sharpe ratio should be derived from the performance of the underlying assets rather than the fund returns (Such a model would invalidate the aforementioned Ponzi scheme, as desired).

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 had a Sharpe ratio of 0.39 for the same period.[3]

## Tests

Several statistical tests of the Sharpe ratio have been proposed. These include those proposed by Jobson & Korkie[4] and Gibbons, Ross & Shanken.[5]

## 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.[6] 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.[7]

## 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 someone currently is invested in a portfolio with an expected return of 12% and a volatility of 10%. The risk-free rate of interest is 5%. What is the Sharpe ratio?

The Sharpe ratio is: ${\displaystyle {\frac {0.12-0.05}{0.1}}=0.7}$

## Strengths and weaknesses

A negative Sharpe ratio means the portfolio has underperformed its benchmark. All other things being equal, an investor wants to increase a positive Sharpe ratio, by increasing returns and decreasing volatility. However, a negative Sharpe ratio can be brought closer to zero by either increasing returns (a good thing) or increasing volatility (a bad thing). Thus, for negative returns, the Sharpe ratio is not a particularly useful tool of analysis.[citation needed]

The Sharpe ratio's principal advantage is 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.[citation needed]

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.[8]

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.)[9]

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.[10]

### Drawback as fund selection criteria

Bailey and López de Prado (2012)[11] 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[12] 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.

Goetzmann, Ingersoll, Spiegel, and Welch (2002) determined that the best strategy to maximize a portfolio's Sharpe ratio, when both securities and options contracts on these securities are available for investment, is a portfolio of one out-of-the-money call and one out-of-the-money put. This portfolio generates an immediate positive payoff, has a large probability of generating modestly high returns, and has a small probability of generating huge losses. Shah (2014) observed that such a portfolio is not suitable for many investors, but fund sponsors who select fund managers primarily based on the Sharpe ratio will give incentives for fund managers to adopt such a strategy.[13]

## 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. ^ http://docs.lhpedersen.com/BuffettsAlpha.pdf
4. ^ Jobson JD; Korkie B (September 1981). "Performance hypothesis testing with the Sharpe and Treynor measures". The Journal of Finance. 36 (4): 888–908. doi:10.1111/j.1540-6261.1981.tb04891.x. JSTOR 2327554.
5. ^ Gibbons M; Ross S; Shanken J (September 1989). "A test of the efficiency of a given portfolio". Econometrica. 57 (5): 1121–1152. CiteSeerX 10.1.1.557.1995. doi:10.2307/1913625. JSTOR 1913625.
6. ^ Roy, Arthur D. (July 1952). "Safety First and the Holding of Assets". Econometrica. 20 (3): 431–450. doi:10.2307/1907413. JSTOR 1907413.
7. ^ 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.
8. ^ "Understanding The Sharpe Ratio". Retrieved March 14, 2011.
9. ^ Wilmott, Paul (2007). Paul Wilmott introduces Quantitative Finance (Second ed.). Wiley. pp. 429–432. ISBN 978-0-470-31958-1.
10. ^ Lo, Andrew W. (July–August 2002). "The Statistics of Sharpe Ratios". Financial Analysts Journal. 58 (4).
11. ^ 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
12. ^ 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
13. ^ Shah, Sunit N. (2014), The Principal-Agent Problem in Finance, CFA Institute, p. 14