When Return-Difference and Regression Residuals Measure Tracking Error

Summary

This exchange compares two ways to calculate tracking error: the standard deviation of a fund’s returns minus its benchmark returns, and the standard deviation of residuals from a regression. The response says they can represent the same quantity when the regression model defines the benchmark appropriately. In the example discussed, the benchmark is a multi-factor model, so the residuals are interpreted as returns unexplained by the model’s systematic factors.

Under that interpretation, residual volatility measures idiosyncratic risk, and the associated alpha and information ratio describe return relative to that residual risk. The equivalence therefore depends on the model specification and what is treated as systematic; it is not a blanket claim that any regression residual volatility equals benchmark-relative tracking error. The exchange provides a conceptual explanation rather than worked calculations. It does not address practical choices such as return frequency, regression estimation details, or how results change when the benchmark and regression factors differ.

Key ideas

• Tracking error can be measured as the volatility of fund returns relative to benchmark returns.
• Regression residual volatility can align with that measure when the regression specifies the benchmark model.
• In a factor model, residuals represent returns left unexplained by the included systematic factors.
• The interpretation depends on model specification and the factors treated as systematic.
• Residual volatility is connected to idiosyncratic risk and the interpretation of the information ratio.

Tags

# Understanding how to calculate tracking error


# Understanding how to calculate tracking error












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?

## Answer by zuiqo (score 1)

https://quant.stackexchange.com/a/15156

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.