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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

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2 Answers 2

up vote 3 down vote accepted

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.

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2  
Just wanted to finish your logic to the complete answer: using the formula for beta above, one can see, that beta(A, relative to B) * beta(B, relative to A) = correlation between A and B. So one shouldn't expect them to be inverse of each over. In your example it just means, that the correlation between the returns is sqrt(0.48 * 0.74) ~ 0.59. –  LazyCat Dec 30 '13 at 16:21
    
Thanks for both insightful comments! I think it makes sense now :) –  user6859 Dec 31 '13 at 7:21

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.

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