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I have come across two ways of calculating Tracking Error (TE) but i'm not sure if they are essentially the same.

The first way is to calculate the standard deviation of the difference between a fund's returns and a benchmark as shown here.

The second method is to run a regression and calculate the standard deviation of the error terms and shown here in section 8.

Many of the academic papers I have read use the latter. My question is, do these 2 methods yield the same answer?

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This appears to be the same thing, however, in the former case, the benchmark is the FF-Model. This means you assume the model stated in their eq. 9 is correct (as per your regression), and use the vol of the residuals as TE.

They go on and explain:

The volatility of the residuals in equation [9] is a measure of idiosyncratic (non-systematic) risk.20 Since alpha measures the return earned for taking on idiosyncratic risk, the information ratio measures of the amount of idiosyncratic return earned per unit of idiosyncratic risk exposure. For this reason, the information ratio is often interpreted as a measure of investment efficiency.

And thats exactly right: If you assume the FF-model, all factors of your regression are systematic. The remainder will be captured by the residuals, and is thus the remaining idiosyncratic risk.

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