Assets 1 and 2 have correlation 0.8, and assets 1 and 3 have correlation 0.8. What is the full set of values the correlation between assets 2 and 3 can take?

Assets 1 and 2 have correlation 0.8, and assets 1 and 3 have correlation 0.8. What is the full set of values the correlation between assets 2 and 3 can take?

Approach: Impose that the 3 by 3 correlation matrix is positive semidefinite, which for this size reduces to its determinant being non-negative, and solve the resulting quadratic in the unknown correlation.

[0.28, 1]. The determinant of the 3 by 3 correlation matrix is 1 + 2 r12 r13 r23 - r12^2 - r13^2 - r23^2, and positive semidefinite requires it to be non-negative with all leading minors non-negative. Substituting r12 = r13 = 0.8 gives 1 + 1.28 r - 0.64 - 0.64 - r^2 >= 0, so the quadratic inequality is r^2 - 1.28 r + 0.28 <= 0. The roots are (1.28 plus or minus sqrt(1.6384 - 1.12))/2 = (1.28 plus or minus 0.72)/2, that is 0.28 and 1. So r23 lies in [0.28, 1] and cannot be negative. Geometrically the three assets are unit vectors, angles 1-2 and 1-3 are each arccos(0.8) = 36.87 degrees, and the angle between 2 and 3 is at most their sum, 73.74 degrees, whose cosine is 0.28.

Follow-up: For a general common correlation r12 = r13 = a, what is the lower bound on r23 as a function of a, and where does it turn negative?

Key concepts: positive semidefinite, correlation matrix, determinant condition, quadratic inequality.