I think I figured out by accident the best method to annualize Standard Deviation of Monthly Returns. I was looking at the Wiki article of the Log-Normal Distribution and found under the Sidebar of the wiki page the Variance formula which allows you to convert Variance of Log Returns to Variance of Returns assuming that Returns follow a Log-Normal Distribution. It's

To get the Standard Deviation you just need to take the Square Root of this Formula. I got the Annualized Standard Deviation of Returns by using this formula on the Annualized Standard Deviation of Log Returns (

**Multiply St.Dev of Monthly Log Returns with SQRT(12)**) and the Annualized Mean (

**Multiply Mean of Monthly Log Returns with 12**) of Log Returns. I then compared the results with the usual incorrect assumption of multiplying the Standard Deviation of Returns with Square Root of 12 and Tobins method of Annualizing Standard Deviation with the Standard Deviation of Annual Returns to see which one is close to reality.

St.Dev of Apple Stock Returns from 1985-2022:

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```
St.Dev of Annual Returns= 63.3%
Simple Method of Annualizing St.Dev (Square Root of 12)= 42.8%
Wiki Method of Annualizing St.Dev= 61.2%
Tobins Method of Annualizing St.Dev= 47.7%
```

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```
St.Dev of Annual Returns= 47.3%
Simple Method Annualized St.Dev of Monthly Returns= 33.3%
Wiki Method Annualized St.Dev of Monthly Returns= 43.7%
Tobins Method Annualized St.Dev of Monthly Returns= 40.1%
```

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```
St.Dev of Annual Returns= 17.8%
Simple Method Annualized St.Dev of Monthly Returns= 14.2%
Wiki Method Annualized St.Dev of Monthly Returns= 15.3%
Tobins Method Annualized St.Dev of Monthly Returns= 15.0%
```

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```
St.Dev of Annual Returns= 199.6%
Simple Method Annualized St.Dev of Monthly Returns= 58.8%
Wiki Method Annualized St.Dev of Monthly Returns= 87.3%
Tobins Method Annualized St.Dev of Monthly Returns= 72.9%
```

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```
St.Dev of Annual Returns= 32.3%
Simple Method Annualized St.Dev of Monthly Returns= 22.2%
Wiki Method Annualized St.Dev of Monthly Returns= 25.7%
Tobins Method Annualized St.Dev of Monthly Returns= 24.4%
```