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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?

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This Wilmott thread has a bit of detail on exactly this question. –  Tal Fishman Jun 8 '12 at 17:17

2 Answers 2

up vote 23 down vote accepted

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.

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Thanks, that is exactly what I was looking for! –  Joshua Chance Feb 15 '11 at 22:49
    
i think it has a name, sometimes, like "law of the triangle" or something similar. look at fx volatility, it has exactly the same problem all the times, since you correlate currency pairs, those consistency conditions appear naturally –  nicolas Mar 13 '12 at 7:52
    
What is the upper bound on that correlation ? –  Qbik Apr 24 '12 at 23:59
    
An upper bound, in the general case, can be obtained in the same way -- compute the determinant of the $3\times3$ matrix and solve for $c$: one of the roots is a lower bound, the other an upper bound. The formula is the same, except for the sign in front of the square root. –  Vincent Zoonekynd Jun 7 '12 at 8:07
    
If you expand the above, would it be possible to derive an analytic solution for simulating correlated data based on a correlation matrix (any size) with arbitrary correlation values? That is, for example, start with a $30 \times 30$ $\mathbf{R}$ matrix with balanced coefficients. Then apply Cholesky factorization and /or Iman and Conover's approach for simulation. I think what always happens with arbitrary corr values is that $\mathbf{R}$ is not positive definite. Given this, what rules are there for properties of $\mathbf{R}$ when simulating correlated data? –  LEP Dec 28 '13 at 21:47

The upper bound on BC correlation would be 1 for the example given. B=C would correlate to 1. If AB and Ac are different, I don't know off the top of my head.

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Is this supposed to be a joke? The upper bound for correlation is 1? –  chrisaycock Jun 7 '12 at 11:12

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