Annualizing Volatility When Log Returns May Be Autocorrelated

Summary

The document compares rolling volatility estimates from long-run U.S. stock market data. It considers the standard deviation of annual log returns, the monthly standard deviation scaled by the square root of twelve under a no-autocorrelation assumption, and an annualized estimate adjusted for autocorrelation. The author reports that the annualized measures still do not match the standard deviation calculated directly from annual returns and asks why.

This is a methodological question rather than a resolved analysis: the calculations are mentioned, but the spreadsheet, chart, and detailed results are not provided in the text. The comparison highlights that square-root-of-time scaling assumes uncorrelated returns, while serial dependence can change the variance accumulated over a year. Even with an autocorrelation adjustment, estimates may differ because the rolling windows and return aggregation may not align, and because annual and monthly samples have different statistical properties. The document does not establish which calculation is correct or identify a specific source of the discrepancy.

Key ideas

• Square-root-of-time scaling from monthly volatility assumes returns have no autocorrelation.
• Serial dependence can change the relationship between monthly and annual return variance.
• Rolling annual-return volatility and annualized monthly volatility may differ even when both use the same historical period.
• The document reports a discrepancy but provides no calculation details sufficient to diagnose it.

Tags

# How to get Annualized Volatility of Log Returns to match or be close to Annual Volatility of Log Returns?


# How to get Annualized Volatility of Log Returns to match or be close to Annual Volatility of Log Returns?












I done 30 Year Rolling Standard Deviation of Annual Log Returns of US Stock Market Data from Robert Shiller 1871-Present (ie I calculate standard deviation of annual returns from Jan 1871-Jan 1901,Feb 1871-Feb 1901,etc), Rolling Annualized Standard Deviation of Log Returns assuming no autocorrelation (ie I calculate standard deviation of monthly returns from Jan 1871-Jan 1901,Feb 1871-Feb 1901,etc and I multiply the standard deviation with the square root of 12 to annualize it) and Rolling Annualized Standard Deviation of Log Returns assuming autocorrelation (I found out how to scale volatility assuming there is autocorrelation from this video:"https://www.youtube.com/watch?v=_z-08wZUfBc", but please correct me if I am wrong in following his method).

What I found is that even when I am scaling volatiity assuming there is autocorrelation I still cant get annualized volatilty of log returns to match annual volatility of log returns. Could anyone tell me why this is the case?

Here is link to download excel sheet which is containing the calculations behind this chart.

EDIT: Fixed minor mistake with calculations.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.