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Construct the original active returns
Asset minus benchmark returns are −1/0.5/1/1.5/4 percentage points. Their mean is 1.2%, and their centered squared sum is 13.3. Sample tracking error over these supplied periods is √(13.3/4)≈1.823458 percentage points.
Construct the fitted residuals
Residuals around the line 1.2%+2.1X are 0/0.4/−0.2/−0.8/0.6 percentage points. Their squared sum is 1.2, a different risk object from the active-return centered sum.
Compare the named scales
| Quantity | Squared-sum divisor | Percentage points |
|---|---|---|
| Sample tracking error | 13.3 / 4 | 1.823458 approximately |
| Sample dispersion of fitted residuals | 1.2 / 4 | 0.547723 approximately |
| Residual standard-error estimate | 1.2 / 3 | 0.632456 approximately |
Retain the benchmark exposure meaning
Subtracting one benchmark return assumes unit benchmark exposure for the active-return series. Fitting a line estimates a different slope and intercept. Neither scale should be relabeled as the other without defining the transformation and denominator.