Annualizing Volatility from Log and Simple Returns
Summary
The document compares annualized return and volatility calculations based on logarithmic returns and simple returns. For monthly log returns, it describes scaling the mean by the number of periods in a year and scaling standard deviation by the square root of that number. It asks whether volatility needs conversion to percentage units and whether software that uses simple returns is wrong.
The response treats the two return definitions as conventions with different advantages, emphasizing that analysts should state which one they use. For small return intervals, the difference between log and simple returns is generally second order and may be modest; over longer intervals, such as a month, it can be more noticeable. The exchange does not fully answer the percentage-unit question or discuss assumptions behind annualization, such as stable, uncorrelated returns. Consequently, the scaling rules are useful conventions, not guarantees that annualized figures will match realized outcomes.
Key ideas
- Log returns and simple returns are distinct conventions for measuring performance and volatility.
- Annualized standard deviation is commonly scaled by the square root of the number of periods per year.
- The difference between return conventions tends to be smaller for short intervals with modest returns.
- State the return convention clearly to prevent comparisons from mixing definitions.
- Annualization relies on assumptions and does not guarantee realized annual performance.
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Full text
# The use of volatility from log returns and raw return
# The use of volatility from log returns and raw return
As far as I know, we usually use log returns( $ln\frac{p_{t+1}}{p_{t}}$ ) in quantitative finance.
For example, let's say we have lots of monthly log returns data, $R_m$.
Then, we can get the mean of monthly log return, $\mu_{month}=mean(R_m)$ and volatility of log return $\sigma_{month}=std(R_{m})$
From $\mu, \sigma$, we can calculate annualized return $\mu_{annual} = 12*\mu_{month}$ and annualized volatility $\sigma_{annual}=\sqrt{12}*\sigma_{month}$.
I would like to ask questions are below.
- $\mu_{annual}$ is log return. So, If I want to express it as a percentage, $e^{\mu_{annual}}-1$(%). I understand this. But, what about volatility? Do I have to change the unit? I mean, Can I say annual volatility is $\sigma_{annual}$(%) without changing the unit?
- I found that some people calculate annualized return and annualized volatility from the raw return not from the log return. Is it correct?. For instance, zipline also uses raw returns for getting annualized volatility(link). Could we say this is wrong?
But, as far as I know, we should use log returns, not raw returns.
- A Matter of Sale Returns and Volatility
- Volatility from Wikipedia
## Answer by Ezy (score 4)
https://quant.stackexchange.com/a/54484
There is no right or wrong, just those 2 conventions are different, each one with its pros/cons.
In general what is more important is to be clear about conventions used to avoid miscommunication and mistakes.
Now if you calculate returns over an interval where the magnitudes are meant to be small then mathematically speaking the difference between raw return and log return wont be material on average since the difference will be second order.
It is not the case here since you aggregate monthly return to get to annual return. Over the course of a month there could be observable difference between raw and log return of a security. If you were to look at daily or intraday returns that would not be as much the case.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.