time series - How to quickly estimate a lower bound on correlation for a large number of stocks?

I would like to find stock pairs that exhibit low correlation. If the correlation between A and B is 0.9 and the correlation between A and C is 0.9 is there a minimum possible correlation for B and C? I'd like to save on search time so if I know that it is mathematically impossible for B and C to have a correlation below some arbitrary level based on A to B and A to C's correlations I obviously wouldn't have to waste time calculating the correlation of B and C.

Is there such a "law"? If not, what are other methods of decreasing the search time?

Answer

Yes, there is such a rule and it is not too hard to grasp. Consider the 3-element correlation matrix

$$\left(\begin{matrix} 1 & r & \rho \\ r & 1 & c \\ \rho & c & 1 \end{matrix}\right)$$

which must be positive semidefinite. In simpler terms, that means all its eigenvalues must be nonnegative.

Assuming that $\rho$ and $r$ are known positive values, we find that the eigenvalues of this matrix go negative when

$$\begin{equation} c<\rho r-\sqrt{1-\rho ^2+\rho ^2 r^2-r^2}. \end{equation}$$

Therefore the right hand side of this expression is the lower bound for the AC correlation $c$ that you seek, with $\rho$ being the AB correlation and $r$ being the BC correlation.