How Portfolio Beta Relates to Tracking Error

Summary

The document distinguishes two definitions of tracking error and explains how each relates to a portfolio’s beta against its benchmark. When tracking error is the volatility of active returns—the portfolio return minus the benchmark return—a beta of one is associated with the lowest tracking error, while moving away from one tends to increase it. This follows because beta differences add systematic variation to the return spread.

A different convention uses the standard error of estimate from a regression of portfolio returns on benchmark returns. That measure is not affected in the same way by beta, because the regression adjusts for the portfolio’s beta. The discussion is conceptual and provides no formula for the tracking-error decomposition or empirical example. The practical conclusion depends on which definition is used and does not imply that beta alone determines tracking error; other sources of active-return variation still matter.

Key ideas

• Under the active-return definition, tracking error measures the volatility of portfolio returns minus benchmark returns.
• A beta of one tends to minimize active-return tracking error, while departures from one can increase it.
• Regression residual volatility is a separate measure that adjusts for the portfolio’s beta.
• The relationship depends on the tracking-error definition, and beta alone does not capture every source of active risk.

Tags

# Relationship between Tracking Error and Beta to benchmark


# Relationship between Tracking Error and Beta to benchmark












Analyzing an indexed portfolio, can we say there is any relationship between ex-ante TE and Beta to benchmark?

Tracking error is the volatility of the difference in returns between the portfolio and the benchmark.

Beta can be calculated as correl(portfolio, bmk) * ( vol portfolio / vol bmk).

I am trying to assess if a change in Beta would be systematically matched by a change in TE.

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/45739

If you define tracking error as the volatility of the difference in returns between the portfolio and the benchmark, then the Beta of a portfolio needs to be 1 to have the best TE and deviations from 1 will cause an increase in TE over this optimal value.

If you define TE, as a few people do, as the SEE of a regression of portfolio returns on the benchmark, then this kind of TE is not affected by the beta of the portfolio. (The regression automatically adjusts for a different beta).

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