Sharpe quotient

The Sharpe ratio , and the Sharpe ratio or Sharpe ratio called ( English Sharpe ratio ), is an economic indicator that a financial instrument the transfer return over the risk-free rate in relation to the volatility - a measure of the risk is - . Namesake is William F. Sharpe .

The Sharpe quotient can be used to compare the past performance of investments. The Sharpe quotient measures, so to speak, the excess return per unit of risk assumed. The measure of the risk is the volatility of the returns, whereby all returns are included in the calculation of the volatility (i.e. also those return values that are below the risk-free interest rate).

definition

The Sharpe quotient is defined as ${\ displaystyle S}$

{\ displaystyle S: = {\ frac {\ overline {D}} {\ sigma _ {D}}}}

,

where is defined as the average excess return of the investment with the return compared to the risk-free investment with return : ${\ displaystyle {\ overline {D}}}$ ${\ displaystyle D_ {t}: = R_ {t} ^ {a} -R_ {t} ^ {f}}$ ${\ displaystyle (R_ {t} ^ {a})}$ ${\ displaystyle (R_ {t} ^ {f})}$

{\ displaystyle {\ overline {D}}: = {\ frac {1} {T}} \ sum _ {t = 1} ^ {T} D_ {t} \ ,.}

The volatility is determined using the empirical standard deviation : ${\ displaystyle {\ sigma _ {D}}}$

${\ displaystyle \ sigma _ {D} = {\ sqrt {\ frac {\ sum _ {t = 1} ^ {T} \ left (D_ {t} - {\ overline {D}} \ right) ^ {2 }} {T-1}}} \ ,.}$

Annual returns and annual volatilities are usually used here. If monthly returns are used when calculating the Sharpe quotient, the monthly Sharpe quotient can be multiplied by annualized for comparison with the annual-based Sharpe quotient . ${\ displaystyle (R_ {t} ^ {a, f})}$ ${\ displaystyle {\ sqrt {12}}}$

As long as the Sharpe quotient is positive, the following applies: the higher the value of the Sharpe quotient, the better the performance of the invested investment in relation to the risk taken. If the Sharpe quotient is negative, the performance was worse than that of a risk-free investment.

The comparison of Sharpe quotients uses the sign to indicate whether underperformance or overperformance has been achieved, and enables an ordinal scaling for positive values: the higher the value, the better the performance. How high the risk was taken cannot be read from this key figure.

Examples

The risk-free interest rate is 3%. The achieved return of plant A is 4%, its volatility 1%. Plant B returned 5% with a volatility of 2%. The Sharpe quotient for both cases is 1.
In another example, the risk-free rate is also 3%. The achieved return of plant C is 2%, its volatility 1%. Plant D returned 1% with a volatility of 2%. The Sharpe quotient for both cases is −1, although Appendix D has a lower return with a higher risk.

Application and criticism

The Sharpe quotient is a measure in risk management (see performance (risk management) ). Like the similar Treynor quotient, it is a relative risk measure that is suitable for comparing the risk-adjusted earnings power of portfolios with a different systematic risk. The Jensen Alpha is an absolute benchmark for performance . Other measures include the sterling ratio or the information ratio .

A prerequisite for calculating the Sharpe quotient is the selection of the risk-free interest rate used as a benchmark. This must match the observation period of the investment. The use of a current (only effective in the future) interest rate is not permitted.

Sharpe quotients in the negative range are not meaningful, since a higher risk then leads to a better (less negative) Sharpe quotient (see example above).

Furthermore, the Sharpe quotient does not provide any information about the composition of the risk into systematic and unsystematic risk such as the Treynor quotient , although the unsystematic risk of an investor's overall portfolio can be reduced by further independent investments. It must also be noted that, depending on the investor's willingness to take risks, the risk compared to the return can be viewed as overweighted when forming the ratio, so that conservative bonds are overvalued.

While the Sharpe quotient measures the total risk of a portfolio, the Treynor quotient provides information about the systematic risk of the portfolio. The higher the diversification of the measured portfolio, the smaller the difference between the Sharpe quotient and the Treynor quotient divided by the market volatility.

In contrast to the Sharpe quotient, the Sortino quotient derived from it only takes into account those returns that are below an accepted lower limit (so-called downside volatility) when determining the risk component in the denominator.

literature

William F. Sharpe : Mutual Fund Performance . In Journal of Business , January 1966, pp. 119-138
William F. Sharpe: Adjusting for Risk in Portfolio Performance Measurement . In Journal of Portfolio Management , Winter 1975, pp. 29-34
William F. Sharpe: The Sharpe Ratio . In Journal of Portfolio Management , Fall 1994
Hendrik Scholz & Marco Wilkens: The market phase dependence of the Sharpe ratio - an empirical study for German equity funds . In Zeitschrift für Betriebswirtschaft , H. 12, 2006, pp. 1275-1302

Individual evidence

↑ Manfred Berger: Efficient price hedging of fixed-income securities with financial futures contracts . Springer, 2013 ( full text in the Google book search).
↑ Bernd R. Fischer: Performance Analysis in Practice: Performance Measures, Attribution Analysis, Global Investment Performance Standards . Walter de Gruyter GmbH & Co KG, 2010, p. 454-455 .

[1] Manfred Berger: Efficient price hedging of fixed-income securities with financial futures contracts . Springer, 2013 ( full text in the Google book search).

[2] Bernd R. Fischer: Performance Analysis in Practice: Performance Measures, Attribution Analysis, Global Investment Performance Standards . Walter de Gruyter GmbH & Co KG, 2010, p. 454-455 .