Sharpe Ratio
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In
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for its
risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
. It is defined as the difference between the returns of the investment and the
risk-free return The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ra ...
, divided by the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
of the investment returns. It represents the additional amount of return that an investor receives per unit of increase in risk. It was named after
William F. Sharpe William Forsyth Sharpe (born June 16, 1934) is an American economist. He 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. ...
, who developed it in 1966.


Definition

Since its revision by the original author, William Sharpe, in 1994, the '' ex-ante'' Sharpe ratio is defined as: : S_a = \frac = \frac, where R_a is the asset return, R_b is the
risk-free return The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ra ...
(such as a U.S. Treasury security). E
_a-R_b AR, Ar, or A&R may refer to: Arts, entertainment, and media Music * Artists and repertoire Periodicals * ''Absolute Return + Alpha'', a hedge fund publication *''The Adelaide Review'', an Australian arts magazine * ''American Renaissance'' ( ...
/math> is the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of the excess of the asset return over the benchmark return, and is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
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 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. However, financial assets are often not normally distributed, so that standard deviation does not capture all aspects of risk.
Ponzi schemes A Ponzi scheme (, ) is a form of fraud that lures investors and pays profits to earlier investors with funds from more recent investors. Named after Italian businessman Charles Ponzi, the scheme leads victims to believe that profits are coming ...
, for example, will have a high empirical Sharpe ratio until they fail. Similarly, a fund that sells low-strike
put options In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or at) a s ...
will have a high empirical Sharpe ratio until one of those puts is exercised, creating a large loss. In both cases, the empirical standard deviation before failure gives no real indication of the size of the risk being run. 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 High-frequency trading (HFT) is a type of algorithmic financial trading characterized by high speeds, high turnover rates, and high order-to-trade ratios that leverages high-frequency financial data and electronic trading tools. While there is no ...
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 A with-profits policy ( Commonwealth) or participating policy (U.S.) is an insurance contract that participates in the profits of a life insurance company. The company is often a mutual life insurance company, or had been one when it began its ...
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 ratio The Treynor reward to volatility model (sometimes called the reward-to-volatility ratio or Treynor measure), 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 ...
s and Jensen's alphas, are often used to rank the performance of portfolio or
mutual fund A mutual fund is a professionally managed investment fund that pools money from many investors to purchase securities. The term is typically used in the United States, Canada, and India, while similar structures across the globe include the SICAV i ...
managers.
Berkshire Hathaway Berkshire Hathaway Inc. () is an American Multinational corporation, multinational conglomerate (company), conglomerate holding company headquartered in Omaha, Nebraska, United States. Its main business and source of capital is insurance, from ...
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.


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" (a.k.a., minimum acceptable return, or MAR) 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. Roy's ratio is also related to the
Sortino ratio The 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 t ...
, which also uses MAR in the numerator, but uses a different standard deviation (semi/downside deviation) in the denominator. In 1966,
William F. Sharpe William Forsyth Sharpe (born June 16, 1934) is an American economist. He 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. The definition was: :S = \frac. 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: :S = \frac. Note, if ''R''f is a constant risk-free return throughout the period, :\sqrt=\sqrt. 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

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
excess return Alpha is a measure of the active return on an investment, the performance of that investment compared with a suitable market index. An alpha of 1% means the investment's return on investment over a selected period of time was 1% better than the ...
, as 10%. The risk-free return is constant. Then the Sharpe ratio (using the old definition) will be\frac=\frac=1.5 Example 2 Suppose that someone currently is invested in a portfolio with an expected return of 12% and a standard deviation of 10%. The risk-free rate of interest is 5%. What is the Sharpe ratio? The Sharpe ratio is: \frac = 0.7


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 bias ratio and first order autocorrelation are sometimes used to indicated the potential presence of these problems. While the
Treynor ratio The Treynor reward to volatility model (sometimes called the reward-to-volatility ratio or Treynor measure), 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 ...
considers only the
systematic risk In finance and economics, systematic risk (in economics often called aggregate risk or undiversifiable risk) is vulnerability to events which affect aggregate outcomes such as broad market returns, total economy-wide resource holdings, or aggreg ...
of a portfolio, the Sharpe ratio considers both systematic and
idiosyncratic risk An idiosyncrasy is an unusual feature of a person (though there are also other uses, see below). It can also mean an odd habit. The term is often used to express eccentricity or peculiarity. A synonym may be "quirk". Etymology The term "idiosyncr ...
s. Which one is more relevant will depend on the portfolio context. 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 In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
, fatter tails and higher peaks, or
skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal d ...
on the
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
can be problematic for the ratio, as standard deviation doesn't have the same effectiveness when these problems exist. 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 In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula that determines the optimal theoretical size for a bet. It is valid when the expected returns are known. The Kelly bet size is found by maximizing the expec ...
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. 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.


Drawback as fund selection criteria

Bailey and López de Prado (2012) 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. 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 selling one out-of-the-money call and selling 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.


See also

* Bias ratio *
Calmar ratio Calmar ratio (or Drawdown 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''. Young owned California Mana ...
*
Capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into accou ...
*
Coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as ...
* Hansen–Jagannathan bound * Information ratio * Jensen's alpha *
List of financial performance measures This article comprises a list of measures of financial performance. Return measures * Arithmetic return: average return of different observation periods * Geometric return: return depending only on start date and end date of one overall obser ...
*
Modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificatio ...
* Omega ratio *
Risk adjusted return on capital Risk-adjusted return on capital (RAROC) 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 ...
*
Roy's safety-first criterion Roy's safety-first criterion is a risk management technique, devised by A. D. Roy, 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 d ...
*
Signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
*
Sortino ratio The 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 t ...
*
Sterling ratio The Sterling ratio (SR) is a measure of the risk-adjusted return of an investment portfolio. While multiple definitions of the Sterling ratio exist, it measures return over average drawdown, versus the more commonly used max drawdown. While the ma ...
*
Treynor ratio The Treynor reward to volatility model (sometimes called the reward-to-volatility ratio or Treynor measure), 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 ...
*
Upside potential ratio The 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 do ...
*
V2 ratio The V2 ratio (V2R) is a measure of excess return per unit of exposure to loss of an investment asset, portfolio or strategy, compared to a given benchmark. The goal of the V2 ratio is to improve on existing and popular measures of risk-adjusted r ...
*
Z score In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...


References

* . *


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. * Bruce J. Feibel. ''Investment Performance Measurement''. New York: Wiley, 2003.


External links


The Sharpe ratio

Generalized Sharpe Ratio

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