In finance, the Sharpe ratio is a way to examine the performance of an investment by adjusting
for its risk.
The ratio measures the excess return per unit of deviation in an investment asset or a trading
strategy, typically referred to as risk, named after William Forsyth Sharpe.
Definition Since its revision by the original author,
William Sharpe, in 1994, the ex-ante Sharpe ratio is defined as:
where is the asset return, is the return on a benchmark asset, such as the risk free rate
of return or an index such as the S&P 500.
is the expected value of the excess of the asset return over the benchmark return, and
is the standard deviation of this excess return.
This is often confused with the information ratio, in part because the newer definition
of the Sharpe ratio matches the definition of information ratio within the field of finance.
Outside of this field, information ratio is simply mean over the standard deviation of
a series of measurements.
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.
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.
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 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 and Gibbons, Ross & Shanken.
History In 1952, Arthur D. Roy suggested maximizing
the ratio "(m-d)/σ", where m is expected gross return, d is some "disaster level" and
σ is standard deviation of returns.
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.
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.
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,
Recently, the 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 will be 1.5.
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.
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 standard deviation is 0.0005562248,
so the Sharpe ratio is -0.0001642/0.0005562248, or -0.2951444.
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 systematic risk of a portfolio, the Sharpe ratio observes
both systematic and idiosyncratic risks.
The returns measured can be of any frequency, 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.
Bailey and López de Prado 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 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 accuracy of Sharpe ratio estimators hinges on the statistical properties of returns,
and these properties can vary considerably among strategies, portfolios, and over time.
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
Risk adjusted return on capital Roy's safety-first criterion
Sortino ratio Treynor ratio
Upside potential ratio V2 ratio
Z score Signal-to-noise ratio
References
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 0-471-26849-6 External links
The Sharpe ratio Generalized Sharpe Ratio
All Hail the Sharpe Ratio - Uses and abuses of the Sharpe Ratio