Why CAPM Beta Does Not Need Annualization

Summary

The document addresses whether a beta estimated from daily market and stock returns should be scaled to an annual value for CAPM analysis. Its central point is that beta is the ratio of covariance to variance, or equivalently correlation multiplied by the ratio of the two return standard deviations. The correlation has no time scale, and when both standard deviations use the same return interval, their annualization factors cancel. Beta therefore does not need to be multiplied by the square root of the number of trading days.

The response gives a concise scaling argument rather than a worked calculation or empirical comparison. It also does not address how to annualize the market return or align a yearly risk-free rate with daily data. The result assumes consistent return intervals and a conventional covariance-based beta estimate; it does not resolve other modeling or estimation choices in a CAPM analysis.

Key ideas

• Beta is covariance divided by the market return variance.
• Equivalently, beta is correlation multiplied by the ratio of the asset and market return standard deviations.
• When both standard deviations are measured over the same interval, their annualization factors cancel.
• Do not annualize a daily beta by multiplying it by the square root of the trading days per year.

Tags

# Finding an annual beta from daily data


# Finding an annual beta from daily data












In computing a CAPM analysis I have daily data for the market and security, but I have yearly per annum figures for the risk free rate. As such I am trying to annualise my market returns and my beta to be used in the calculation.

I have 10 years of daily data for both the market and the stock, I have calculated the daily geometric average of the market and have annualised it by putting it to the pwoer of 252 (the number of trading days per year). I then found the daily covariance and the daily variance to calculate a daily beta. I then annualise the daily beta by multiplying by sqrt(252).

I'm not confident in this method and it gives me srtange results, please help any advice is greatly appreciated :)

## Answer by Newquant (score 0)

https://quant.stackexchange.com/a/80762

Another way to write Beta is: $$ \rho_{ij}\frac{\sigma_i}{\sigma_j} $$

Correlation is a dimensionless variable, and has no scale/annualisation factor. The two standard deviations in the fraction both scale with sqrt(T), so must cancel out, meaning beta is time invariant.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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