Sharpe ratio

Sharpe ratio is a risk-adjusted return measure that evaluates the excess return generated by an investment relative to the risk-free rate in relation to the volatility of the investment's returns. It was developed by William F. Sharpe.

The Sharpe ratio is commonly used to compare the risk-adjusted performance of investment portfolios, funds, and securities.

Calculation

The Sharpe ratio is commonly calculated as:

$Sharpe Ratio = \frac{R_{p} - R_{f}}{σ (R_{p})}$

where:

$R_{p}$ = portfolio or investment return;
$R_{f}$ = risk-free rate of return;
$σ (R_{p})$ = standard deviation of portfolio or investment returns.

The numerator represents the investment's excess return over the risk-free rate, while the denominator represents the volatility of the investment's returns.

Example

Suppose an investment has an annual return of 12%, the risk-free rate is 4%, and the annualized standard deviation of returns is 10%.

$Sharpe Ratio = \frac{12 % - 4 %}{10 %} = 0.8$

The investment therefore has a Sharpe ratio of 0.8.

Interpretation

A higher Sharpe ratio generally indicates that an investment has generated greater excess return relative to the volatility undertaken.

A lower Sharpe ratio indicates that the investment generated less excess return relative to its volatility.

A negative Sharpe ratio indicates that the investment's return was below the risk-free rate over the measurement period.

The Sharpe ratio should generally be interpreted in the context of the investment strategy, benchmark, measurement period, and calculation methodology.

Annualized Sharpe ratio

When returns are measured at a frequency other than annually, the Sharpe ratio can be annualized using the appropriate scaling factor.

For periodic returns with a Sharpe ratio of $S R_{p}$ and $N$ periods per year:

$S R_{annual} = S R_{p} \sqrt{N}$

For example, when using monthly observations:

$S R_{annual} = S R_{monthly} \sqrt{12}$

The annualization convention assumes that the return and volatility measures are scaled consistently.

Excess return

The numerator of the Sharpe ratio represents excess return:

$Excess Return = R_{p} - R_{f}$

where the risk-free rate represents the return that could theoretically be earned from an investment with negligible credit risk over the relevant measurement period.

Volatility

The denominator of the Sharpe ratio is the standard deviation of investment returns:

$σ (R_{p}) = \sqrt{\frac{1}{n - 1} \sum_{t = 1}^{n} (R_{p, t} - {\bar{R}}_{p})^{2}}$

where:

$R_{p, t}$ = portfolio return during period $t$ ;
${\bar{R}}_{p}$ = average portfolio return;
$n$ = number of observations.

Standard deviation measures the dispersion of returns around their average.

Sharpe ratio and information ratio

The Sharpe ratio and Information ratio are related risk-adjusted performance measures but use different reference points.

The Sharpe ratio measures excess return relative to the risk-free rate:

$Sharpe Ratio = \frac{R_{p} - R_{f}}{σ (R_{p})}$

The information ratio measures active return relative to a benchmark:

$Information Ratio = \frac{R_{p} - R_{b}}{σ (R_{p} - R_{b})}$

where:

$R_{p}$ = portfolio return;
$R_{f}$ = risk-free rate;
$R_{b}$ = benchmark return.

Thus, the Sharpe ratio evaluates return relative to total portfolio volatility, while the information ratio evaluates active return relative to tracking error.

Sharpe ratio and Sortino ratio

The Sortino ratio is similar to the Sharpe ratio but focuses on downside volatility rather than total volatility.

The Sharpe ratio uses standard deviation of all returns:

$Sharpe Ratio = \frac{R_{p} - R_{f}}{σ (R_{p})}$

A simplified Sortino ratio is:

$Sortino Ratio = \frac{R_{p} - R_{t}}{σ_{d}}$

where $R_{t}$ is the target or minimum acceptable return and $σ_{d}$ represents downside deviation.

Portfolio management

The Sharpe ratio is commonly used in portfolio management to compare investments with different levels of risk.

It can be considered alongside:

A high Sharpe ratio does not necessarily indicate that an investment is appropriate for every investor because the measure does not incorporate all forms of investment risk or investor-specific objectives.

Limitations

The Sharpe ratio has several limitations.

These include:

reliance on historical returns and volatility;
sensitivity to the measurement period;
sensitivity to the choice of risk-free rate;
potential distortion from non-normal return distributions;
inability to distinguish between upside and downside volatility;
dependence on consistent return and volatility calculations.

The ratio is therefore most useful when investments are evaluated using consistent methodologies and comparable measurement periods.

Importance

The Sharpe ratio is one of the most widely used measures of risk-adjusted investment performance. It provides a standardized way to compare the excess returns of investments with the volatility associated with generating those returns and is commonly used in portfolio analysis, fund evaluation, and investment research.

Sharpe ratio

Contents

Calculation

Example

Interpretation

Annualized Sharpe ratio

Excess return

Volatility

Sharpe ratio and information ratio

Sharpe ratio and Sortino ratio

Portfolio management

Limitations

Importance

See also