How Excess Returns Affect Beta, Correlation, and Benchmark Analysis
Summary
The document considers how subtracting a return average or a benchmark return changes beta, correlation, and covariance calculated from daily log returns. Its answer says that subtracting the same average from each series does not alter these measures. Subtracting the benchmark return from a security’s return changes beta by one, while the resulting correlation and covariance are described as less meaningful for the original comparison.
It also addresses comparisons against several correlated benchmarks. The response cautions that trying to isolate a security’s association with each benchmark can invite causal interpretations that correlation alone cannot support. As a possible approach, it proposes principal component analysis of benchmark returns, followed by regression of the security against the resulting independent components for attribution. This is a brief conceptual answer rather than a worked calculation; it does not specify implementation choices, assumptions, or tests of the approach.
Key ideas
- Subtracting the same average from both return series leaves beta, correlation, and covariance unchanged.
- Subtracting benchmark returns from security returns changes the beta interpretation and can make related correlation measures unsuitable for the original comparison.
- Correlated benchmarks make it difficult to interpret a security’s separate association with each one.
- Principal component analysis can form independent benchmark return components for use in a regression-based attribution.
Tags
Full text
# Questions about beta, correlation, and covariance # Questions about beta, correlation, and covariance Currently, I calculate beta, correlation, and covariance measures using daily log normal returns of Security A and Benchmark A. What would it mean if I were to use daily log normal excess returns in these tests? In such a case, I have two hypothetical definitions of excess returns: 1. daily return - average return 2. daily return of security A - daily return of benchmark A Is there an advantage to doing this? If I am comparing security A to benchmarks A, B, and C, where there is some correlation between the benchmarks, how could I "clean out" this correlation, so that when I ran the comparison, the correlation I calculate between sec A and the benchmarks is more clear. (E.g. Security A correlates 0.5 to Benchmark A, 0.3 to B, 0.7 to C independently of the benchmarks' correlation to each other). Note: adding covariance to the list of measurements since it plays a role in beta and port. variance. ## Answer by demully (score 1) https://quant.stackexchange.com/a/47288 I'm a bit confused by this. If "excess returns" = daily - average returns, this shouldn't change any of the beta, correlation or covariance outcomes. If "excess" = security - market, then the beta should be 1 lower. Correlation and covariance should be commensurately lower, but essentially meaningless. About "clean" correlations, this is a really tough one. IE easy to ask, but it rapidly gets complicated. Imagine you correlated Apple to the S&P and the NASDAQ: and correlation is ~80% to both indices, that are ~90% correlated to each other. "Cleaning" this to try to say that it's really one more than the other is essentially trying to infer causation from correlation, which is a huge minefield ;-) But if you insist, you would need to do a Principal Component Analysis of your 3 indices. This would take their returns, and constructs the 3 cross-market signals that explain the returns and are independent of each other. Regressing your security against these three PCs gives you (theoretically) a beta and performance attribution to these different cross-market signals.
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