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Beta calculation: homogeneity of time series used

March 13th, 2018, 4:34 pm

Hi all,
excuse me, I noted (in a work of others) that for the computation of the beta (as in the CAPM framework), i.e. as "Covariance(asset i, market) / variance(market)", has been used a different lenght of the time series. I mean, for the numerator, if for the asset i for example only 5 years of historical data are available, the Covariance has been computed by using 5 years of historical data (both for asset i and for the market); while, if for the market instead are available 10 years of historical data, the denominator, i.e. the variance(market), has been computed by using 10 years of data.
I think this is a bit incorrect, since a common sense would suggest me to use the same lenght of (historical) data for both the numerator and denominator... but, do you know if there is some explicit reference for this? I mean, some article, paper, or even book for that?

Thanks a lot.
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Re: Beta calculation: homogeneity of time series used

March 19th, 2018, 6:55 pm

I doubt there is a reference. I agree with your common sense. 
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Re: Beta calculation: homogeneity of time series used

March 20th, 2018, 9:26 am

If you have reason to beleive that variance or covariance changes over time (i.e., the system is not perfectly IID), then you absolutely cannot use different time periods for the two terms.

If the system is truly IID, then maybe you can use different time periods (and might even get a better estimate in that the 10-year value of the variance would be less noisy than using a 5-year value) but I'd double check that with a Monte Carlo simulation.

In any case, the markets aren't IID so if this is market data, then using different periods for the different terms is a bad idea.

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