Why Beta Is Not the Reciprocal When You Swap Portfolio and Benchmark
Summary
The document explains why calculating beta in both directions between a portfolio and a benchmark does not generally produce reciprocal values. It defines beta as the covariance of the two return series divided by the variance of the variable used as the regression benchmark. Swapping the assets changes that denominator, so the resulting coefficient can differ in a way that does not match a simple inverse relationship.
It also expresses beta as correlation multiplied by the ratio of the portfolio’s volatility to the benchmark’s volatility. This separates beta’s dependence on co-movement from the relative scale of returns. The residual term in the regression captures variation not explained by the benchmark, so beta alone can be an incomplete description of risk when the series are weakly related. The discussion is conceptual and refers to weekly returns and a global equity benchmark denominated in Danish kroner; it does not analyze the underlying data or establish a broader risk conclusion. It notes that beta’s role in CAPM involves additional context.
Key ideas
- Regression beta is covariance divided by the variance of the chosen benchmark returns.
- Swapping portfolio and benchmark changes the variance in the denominator, so the two beta estimates need not be reciprocals.
- Beta can also be viewed as correlation multiplied by the ratio of the two return volatilities.
- Regression residuals represent portfolio variation that the benchmark does not explain.
- Beta alone may say little about total risk when portfolio and benchmark returns are weakly related.
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Full text
# How to interpret beta meaningfully?
# How to interpret beta meaningfully?
Although this is probably a basic question, this is probably also the right forum to post it in :)
I thought I understood beta, but know I am really confused...
The beta between my portfolio (weekly returns) and the benchmark (ACWI in Danish Kroner) is 0,48. So historically my portfolio has had half the volatility of the benchmark. Great.
If I turn the calculation around and look at the benchmark relative to my portfolio (I hope it makes sense) I get a beta of 0,74. So the benchmark has now been less volatile, than my portfolio. I can this be? I would expect the beta of the benchmark relative to my portfolio to be greater than 1...
Here is a link to the data (weekly) if needed:
http://www.market-trends.net/?attachment_id=4060
Kind regards
René
## Answer by Richi Wa (score 4, accepted)
https://quant.stackexchange.com/a/9829
I did not look at the data, but recall that beta is a parameter in the following equation:
$$ r_A = \alpha + \beta r_B + \epsilon $$ relating two returns (random variables, samples) $r_A$ and $r_B$. To calculate beta you peform $$ \beta = \frac{cov(r_A,r_B)}{var(r_B)}. $$ Thus if assets $A$ and $B$ exchange roles, then only the denominator changes. In your example the variance of your benchmark is smaller than the variance of your portfolio.
Futhermore note that the $\epsilon$ above models all volatility/risk that remains and that is not explained by $r_B$. If $r_B$ and $r_A$ are not too much related then the beta does not tell you too much about risk.
In the CAPM beta plays a more prominent role. But this is a slightly different story.
## Answer by assylias (score 1)
https://quant.stackexchange.com/a/12837
Another way to look at it is that:
$$\begin{align} \beta &= \frac{cov(R_p,R_M)}{var(R_M)}\\ &= \rho(R_p,R_M)\frac{\sigma(R_p)}{\sigma(R_M)} \end{align}$$
In other words, the beta is the product of the correlation between your portfolio and the market and the ratio of their volatility. You can then see why the inverse beta is not what you expected.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.