The Sharpe Ratio

The Sharpe ratio is the ratio of reward to volatility . It's a popular way to look at the performance of an asset relative to its risk.

\mathrm{Sharpe\;Ratio} = \large \frac{r_{\mathrm{risky\;portfolio}}-r_{\mathrm{risk\;free}}}{\sigma_{\mathrm{excess\;return}}}

The numerator of the Sharpe ratio is called the excess return , differential return as well as the risk premium . It’s called “excess return” because this is the return in excess of the risk-free rate. It’s also called the “risk premium”, because this represents the premium that investors should be rewarded with for taking on risk.

The denominator is the volatility of the excess return.

How do you calculate this? The risk premium (which we’ll denote with D ) equals the portfolio return minus risk free rate over a period of time:

D_t = r_{\mathrm{portfolio},\; t} - r_{\mathrm{risk\;free},\; t}

Then, calculate the mean and standard deviation of D_t over the historical period from t = 1 to T :

D_{average} = \frac{1}{T}\sum_{t=1}^T D_t
\;\;\;\;\;\;\;\;\;\;\sigma_{D} = \sqrt{\frac{\sum_{t=1}^T(D_t- D_{average})^2}{T-1}}

\mathrm{Sharpe\;Ratio} = \frac{D_{average}}{\sigma_D}

As we saw previously, the Sharpe Ratio is the slope of the Capital Market Line .

The Sharpe Ratio allows us to compare stocks of different returns, because the Sharpe ratio adjusts the returns by their level of risk.

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Annualized Sharpe Ratio

Please keep in mind that the Sharpe Ratio depends on the time period over which it is measured, and it’s normally annualized. You annualize it in the same way you annualize volatility. (If you need a refresher on annualization, please refer to the video on annualization within the lesson on volatility) For example,

\mathrm{Sharpe\;Ratio} {\mathrm{year}} = \sqrt{252}\;\mathrm{Sharpe\;Ratio} {\mathrm{day}}

Let’s see where the square root of 252 trading days comes from by annualizing the risk premium in the numerator, and then annualizing the standard deviation in the denominator. Then we’ll put these two together as the annualized Sharpe Ratio.

To annualize daily risk premium (r_p - r_f) , we add the daily return 252 times, or more simply multiply by 252.
D_{year} = 252 \times D_{day}

To annualize the daily standard deviation, let’s first annualize the daily variance. To annualize daily variance, we add \sigma_D^2 252 times, or more simply multiply it by 252.
\sigma_{D, year}^2 = 252 \times \sigma_{D,day}^2

The standard deviation is the square root of the variance, which is
\sqrt[2]{252 \times \sigma_D^2} ,
or just \sqrt[2]{252} \times \sigma_D^2

In other words:
\sigma_{D, year} = \sqrt[2]{252} \times \sigma_{D,day}

If we combine the annualization factors of the numerator and denominator, this becomes:
\frac{252}{\sqrt[2]{252}}
which simplifies to \sqrt[2]{252}

So to convert the Sharpe ratio from daily to annual, we multiply by \sqrt[2]{252} .

Therefore:
\mathrm{Sharpe\;Ratio} {\mathrm{year}} = \sqrt{252}\;\mathrm{Sharpe\;Ratio} {\mathrm{day}}

More Information

This link gives more information on the Sharpe Ratio from its originator, William Sharpe, Professor of Finance, Emeritus at Stanford University's Graduate School of Business.