Hi,
I did a beta matrix on bloomberg but strangely some pairs show beta values both below 1. Logically that should not be possible, no?
You mean a specific product had a beta <1 of itself?
It should be possible, perhaps even common. In a one factor model, beta is equal to the correlation times the ratio of the volatilities. If you have a correlation of 0, then you will get a beta of 0 (which is less than one) for both pairs (by pair, I assume you mean use one as the index for the other as one member of the pair and then you compare that with the situation where the other variable is used as the index). So for a correlation low enough, it should be possible to get a beta less than one for both directions.
If the volatilities are similar, the correlation doesn’t even have to be that low. For example, if the volatilities are identical, then any correlation less than one (which would be pretty much everything) will give you betas less than one in both directions.
Your intuition is probably based on a geometric reflection that suggests if the slope of y plotted against x is less than one, the slope of x plotted against y must be greater than one. But the noise (or unexplained variance) mucks up that intuition. Your intuition would be right if there were no noise in the relationship between X and Y (I.e. If the correlation = 1)
For multifactor models, the calculation is more complicated, but the logic is basically the same. So yes, it should be possible (unless I’ve misunderstood your question).
Thinking about this a little more, in a single factor model, if you multiply the betas of each pair, you should get the correlation squared, or the percentage of variance explained. So although some betas may be larger than one, the product of the betas should never be larger than one, and tell you something about how much noise there is in the correlation.
I haven’t thought through the multi factor model. I suspect that there are interactions that may not give such a clean result.