I have daily log returns of my asset that run over several years and I would like to calculate a time series of the Rolling Sharpe Ratio.
This Sharpe Ratio asks specifically for:
- Annualized simple returns;
- And annualized standard deviation of simple returns.
This is not standard procedure, and I'm confused. My questions are about how to calculate the annualized simple returns and annualized standard deviation of simple returns.
Is it correct to calculate the annualized returns given historical daily data as follows: Given $r_{n} = \ln (P_{n} / P_{n-1})$ so that my daily returns are $ R_{n} = e^{r_{n}} - 1$; will my annualized returns be the geometric mean:
$$ 1 + R_{\text{annualized}} = \left( \prod_{j = 0}^{252 - 1} ( 1 + R_{n - j} ) \right)^{1/252}?$$
About the annualized standard deviation. I calculate the daily standard deviation of simple returns $R_{n}$ and multiply by $\sqrt{252}$? I know that this works with log returns because these are normally distributed, but do simple returns work in the same way?
I apologize if this question is too basic. Any reference recommendation is highly appreciated.