# 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.

1. $$\mu_{annual}$$ 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?

2. 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.

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.

• Thanks for your answer, What do you think about question no 1. ? changing unit of volatility. May 29 '20 at 3:14
• I don’t understand the first question
– Ezy
May 29 '20 at 3:15
• We can change the unit of the annualized return to the percentage(%). What about volatility? Is it the right way to change the unit in the same way of return like $e^{\sigma_{annual}} -1$ ? Jun 1 '20 at 2:25
• I still don’t get it. What is the alternative unit ?
– Ezy
Jun 1 '20 at 10:19
• I mean we usually use the percentage as a unit of return and volatility. If we calculated the volatility from log returns, the unit of the volatility is not the percentage. So, If I want to express the volatility as a percentage, I have to change the unit like $e^{\sigma_{annual}} - 1$. I would like to confirm my logic is right. Jun 2 '20 at 3:24