Annualizing Simple Returns and Volatility from Daily Data
Summary
The document considers how to calculate annualized simple returns and volatility for a rolling Sharpe ratio when historical observations are daily log returns. Converting each log return to a simple return makes the product of gross daily returns telescope: over a full period, it gives the cumulative price ratio. The response distinguishes that cumulative return from an average annual return, which requires taking a geometric root over the number of years in the observation period. A geometric average over daily observations instead describes an average daily growth rate.
For volatility, the response says daily simple-return standard deviation is commonly annualized by multiplying by the square root of the number of trading days, while noting that log returns are more commonly used and that the difference may be small. This is practical guidance rather than a derivation of conditions under which square-root scaling holds. The document does not provide data or a worked numerical example, and it cautions that annualized volatility should be compared with an annual return.
Key ideas
- Multiplying gross simple returns across consecutive days telescopes to the cumulative price ratio.
- A cumulative return over a year is different from an average annual return across multiple years.
- An average annual return can be expressed as the geometric growth rate over the full observation period.
- Daily return standard deviation is often scaled by the square root of trading days to annualize volatility.
- Annualized volatility should be compared with returns expressed on an annual basis.
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Full text
# How to calculate annualized simple returns and annualized simple standard deviation given historical daily data
# How to calculate annualized simple returns and annualized simple standard deviation given historical daily data
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.
## Answer by Richi Wa (score 0, accepted)
https://quant.stackexchange.com/a/43181
to answer the question about the annual return: First your product can only start with $i=1$ because otherwise you have negative indices. If this is corrected: with your definitions and so on, doesn't this simply reduce to $\log P_252 - \log P_1$ which is the log return of the last year? What you might try to express is the average annual return which makes sense if you have more than one year. E.g. if $R_2$ i the return over two years then the $R^a$ that satisfies $$ (1+R^a)^2 = 1 + R_2 $$ is the average annual return. With your formula this could be an average daily return though.
For the standard deviation you usually use log-returns (depending whether you agreed on something else or there is a law background that prescribes to use something else). The difference will be negligible. More important: with the square-root you get an annualized volatility and should compare it to an annual return.
EDIT: to answer the comment: By your definition: $$ \prod_{j = 0}^{252 - 1} ( 1 + R_{n - j} ) = \prod_{j = 0}^{252 - 1} e^{r_{n-j}} $$ and $$ e^{r_{n-j}} = \exp \left( \log P_{n-j} - \log P_{n-j-1} \right). $$ Thus the above product of telescope form. For $j=1$ you get $$ \exp \left( \log P_{n-1} - \log P_{n-2} \right) = P_{n-1} / P_{n-2} $$ and for $j=2$ it is $$ P_{n-2}/ P_{n-3} $$ thus $P_{n-2}$ cancels. For $j = 252$ you get $P_{n-252}/P_{n-253}$. Thus if all indices are correct then all terms except the first and the last cancel.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.