Time horizon of estimation period CAPM beta

When calculating CAPM beta, it is done by rolling regressions. If it is only the beta we want to obtain, am I correct to assume that we can estimate rolling correlations and stds, and use this to calcuate the betas as corr*stockSTD/mktSTD ?

Next thing I wonder is what is the time horizon used in CAPM? More specifically, how long is the rolling windows? Is there any newer articles stating more optimal time horizons for the std and corr? Daily data as obviously best, but for 1 year, 3 years or 5 years?

Thanks :)

Yes, you may use rolling correlations and standard deviations to get rolling beta estimates.

The time horizon used is your investment decision horizon: in other words, how often might you adjust your portfolio? If you revisit your portfolio's investments every year, then the time horizon should be a year. This also affects the risk-free rate you use when calculating excess returns for the stock and market index.

Daily data is not necessarily the best for estimating betas. Daily data is slightly polluted by bid-ask bounce, so that added source of noise will tend to bias beta estimates to be lower than estimates produced using weekly, monthly, or longer-term data.

The tougher question is how much data you should use for estimation: one year back? three years? more? The answer gets very complicated (see the statistical work of Allan Timmerman on optimal windows for mean vs variance estimation). Barring tackling that question with a complicated method, you can do as many people do and use three years prior.

Finally, I would be remiss if I did not point out that the CAPM betas will vary across the business cycle and suggest the presence of alpha more often than we would expect from economic theory. This is because your model is too simple: you should use a multi-factor model. Typically, adding in other factors like small vs large firms (e.g. outperformance of Russell 2000 vs the S&P 500), changes in the level and slope of the yield curve, momentum, inflation, and a credit spread (e.g. IBoxx IG - UST 10Y) will yield coefficients that are more stable across the business cycle; and, the alpha will be much closer to 0.

• "The time horizon used is your investment decision horizon", you say. So if the portfolio is rebalanced monthly, you use monthly time horizon? Does not make sense? We are testing the betting against beta strategy, and are trying to calculate their ex ante betas different from the original methodology. Do you have any suggustions to a more "optimal" ex anta beta in this manner? May 8 '21 at 13:31
• If you would rebalance monthly, then it would make sense to use monthly data -- and you definitely need to use a monthly risk-free rate wen calculating excess returns. May 10 '21 at 15:38
• But the methodology uses daily data, so everything is calculated used daily excess returns. Later it is added cumulatively to get the monthly excess returns. What i wonder is how the window i should use to calculate the betas? Does any articles research on optimal window (i.e. 250 days, 750 days..) for correlations and standard deviations? May 11 '21 at 9:14
• You could use daily log-returns with a daily fraction of the monthly risk-free rate subtracted to get excess returns; however, as I noted, using daily data induces a noise (error-in-variables) bias. Just modify the methodology to use monthly data. May 11 '21 at 21:03
• Modifying the methodology is what we do, so it is absolutely relevant to use monthly as you state. But only if this is more optimal, or at least can be argued to be more optimal. Trading costs of asset pricing anomalies. In the BAB article it is stated: Whenever possible, we use daily data, rather than monthly data, as the accuracy of covariance estimation improves with the sample frequency (Merton, 1980). Do you think your noise argument is trumphs this, and do you have an article that states it (we need to refer to something) ? Thank you for taking your time!! May 13 '21 at 7:13