Best Method of Annualizing Standard Deviation?

[Anon9001]

[Topic is now in Investing - Theory, News & General. Thank you to the member who reported it and explained what was wrong - mod mkc]

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%

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

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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%

St.Dev of Shiller SP Data 1871-2022:

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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%

St.Dev of Amazon Returns 1997-2022:

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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%

St.Dev of MSCI EM Index 1987-2022:

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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%

[japh]

What do you use this information for>?

[Anon9001]

What do you use this information for>?

Good Question. I prefer using Annual Standard Deviation if the Asset is having 20 Years of Historical Data but for Assets which are only 5-10 years old calculating Annualized Standard Deviation should be more accurate than Annual Standard Deviation due to the small sample size so I would like to find out which method of Annualizing Standard Deviation is close to approximating Annual Standard Deviation due to this.

[jmk]

I am struck by the magnitude of the difference.

[bh1]

If you correctly compute the standard deviation of almost all financial data, the result will tend to infinity as you increase the number of data points. This drops out of the combination of the generalized central limit theorem (mathematics) and the efficient market hypothesis (arbitrage).

[toddthebod]

I question the value of calculating summary statistics of any sort on the returns of an individual company's stock. What does it tell you to calculate the mean annual return for Amazon who once returned 1,000% in a single year?

[Anon9001]

I question the value of calculating summary statistics of any sort on the returns of an individual company's stock. What does it tell you to calculate the mean annual return for Amazon who once returned 1,000% in a single year?

These are offered as examples to see how well the Wiki method of annualizing Standard Deviation does in comparison to the simple method that everyone uses to annualize Standard Deviation. I dont see any value in looking at the Past St.Dev of Amazon to get the Future St.Dev of Amazon as I personally believe Amazon's Future St.Dev will be much lower as Amazon is much larger company now compared to late 90s.

[international001]

Browsing to the article a bit...

Assume you have a normal distribution of independent returns for every month. The annual distribution of the sum would be std_annual=std_montly*sqrt(12)

However, when we look at annual returns we are not looking at sum of returns, but a multiplication of returns. That's the big difference

The impact is more important when you look at the average:

https://www.kitces.com/blog/volatility- ... t-returns/

i.e. for the same annual return, you always prefer something with lower volatility. It will grow more average over time. Another reason to have index funds

[BJJ_GUY]

Browsing to the article a bit...

Assume you have a normal distribution of independent returns for every month. The annual distribution of the sum would be std_annual=std_montly*sqrt(12)

However, when we look at annual returns we are not looking at sum of returns, but a multiplication of returns. That's the big difference

The impact is more important when you look at the average:

https://www.kitces.com/blog/volatility- ... t-returns/

i.e. for the same annual return, you always prefer something with lower volatility. It will grow more average over time. Another reason to have index funds

Can you clarify the point you are trying to make? What does it meant to grow average over time, and why is this statement supportive evidence for index funds?

As it relates to the use of arithmetic mean or the geometric mean: You use the arithmetic mean when doing forward looking estimations/modeling, and you use geometric returns when showing a historical return.

[international001]

I was just trying to point that the computation for standard deviation of geometric-returns(X) it gets more complicated than standard deviation of arithmetic-returns(X). That's the point of the post. Sorry if it was obvious to everybody else, but I had never even think about it.

But doing average of geometric-returns(X) <= average of arithmetic-returns(X). This is something well known (look at the Kitces post).
When stdX = standard deviation of X = volatility of the security is equal to 0%, then average of geometric-returns(X)= average of arithmetic-returns(X)
The lower the stdX, then you maximize average of geometric-returns(X), that it's what you really want to maximize. Index fund have the same average of arithmetic-returns that an individual stock, but since its stdX is lower, the average of geometric-returns(X) is higher.

[edgeagg]

If you correctly compute the standard deviation of almost all financial data, the result will tend to infinity as you increase the number of data points. This drops out of the combination of the generalized central limit theorem (mathematics) and the efficient market hypothesis (arbitrage).

The distribution of returns is well known to be long tailed. Consequently, measures of variance are meaningless as you indicate

[edgeagg]

If you correctly compute the standard deviation of almost all financial data, the result will tend to infinity as you increase the number of data points. This drops out of the combination of the generalized central limit theorem (mathematics) and the efficient market hypothesis (arbitrage).

The distribution of returns is well known to be long tailed. Consequently, measures of variance are meaningless as you indicate