# Regarding correlation calculation

Hi. I can’t find an exact answer, though I think I know it. When calculating correlation to use in calculation of beta, we should be using the risk-adjusted returns (i.e., excess returns of Ri-Rf) instead of the non-excess returns, right? I ask this because in calculating various performance measures, we take care to specify Rf-Rf. Just want to verify that we take the correlation of the excess returns, and that the outputted correlation coefficient is what is then used in calculating beta.

Thank you!

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Because beta is the slope of a regression line, it doesn’t matter whether you use actual returns or returns in excess of the risk-free rate; the slope will be the same either way.

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Thanks for answering. In my case, I am calculating based on monthly investment returns and return (slightly) different slopes depending on whether or not I adjust for the risk-free rate (say, 3M T-Bills).

If I just make a silly example in Excel (pasted screenshot below), I can make random arrays of X, Y, and a smaller random Rf (just completely random numbers). If I take the slope of (X, Y), I end up getting 0.303545. However, if I take the slope of (X-Rf, Y-Rf), I get 0.299347. These would both show up as 0.30 in a report, but I would prefer to use a more accurate methodology. Note by the below that I also return a slightly different correlation. I’m inclined to believe I should use the correlation and beta of the risk-adjusted returns. Or, am I completely missing something? Please provide your thoughts – thanks a lot. The risk-free rate has to be a constant.  Varying it from one observation to the next is the source of the difference.

Simplify the complicated side; don't complify the simplicated side.

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Thanks again. The 3M T-bill rate changes on a monthly basis, which means the risk-free rate changes as well. At this point I am more asking for practical and application purposes instead of CFA purposes.

So, when calculating with that in mind, is it better to use the risk-adjusted returns?