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Annualizing Log Returns and Volatility

Article Quant Q&A · Author: galmani
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Summary

The document explains how to compare log returns measured over different periods. Because log returns add across time, daily log returns can be summed to obtain the cumulative log return for the observed period. When only an average daily log return is available, annualization uses a factor based on the chosen number of periods per year. A common convention for trading days is mentioned, while the response emphasizes that the factor should match the use case. For a partial-year observation, it suggests scaling in proportion to the year length and the period length.

It distinguishes return scaling from volatility scaling: volatility is described as growing with the square root of time, rather than linearly. The answer points to Brownian motion as an explanation for that relationship, but provides no derivation. Annualization conventions depend on the calendar and application, so comparisons require consistent period definitions. The document offers general guidance rather than a calculation of the example return or a treatment of compounding beyond log-return additivity.

Key ideas

  • Log returns over successive periods can be added to obtain the total log return.
  • An average daily log return can be annualized with a factor reflecting the periods per year.
  • The appropriate annualization factor depends on the use case and period definition.
  • Volatility scales with the square root of time, so return and volatility annualization differ.

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StatisticsRisk management
Full text
# How to annualize log returns?


# How to annualize log returns?












I have daily log return from 01.01.2011 to 10.28.2011 and I'd like to compare the total return of that 10 months period (which is of -7.093%) to annual log returns of previous years. I know it's really a simple question but I want to be sure not to make any mistake. Thanks in advance.

## Answer by matt (score 6)

https://quant.stackexchange.com/a/3001

Log returns are additive. Just add the daily returns together. If you only have one average daily return you annualize simply by multiplying with an annualization factor. Often 252 is used but it depends on your specific use case. In your case you may want to multiply by (days per year / days in 10 months period) Make sure you do not apply the same to volatility. Volatility scales with the square root of time. If you want to know why then Google for properties of Brownian motion.

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