Correlation Upper Bound Mystery
maximum possible correlation between variables is a medium quant interview question on Covariance, reported to have been seen at Citadel, Jane Street and Squarepoint Capital.
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This covariance and correlation question is about how pairwise linear relationships between random variables must fit together consistently. You are given two correlations and asked to pin down the possible range of a third one, under the global structure imposed by the joint correlation matrix. It turns a seemingly local quant question into a global compatibility check for random variables.
It trains your intuition for correlation matrices, variance-covariance structure, and positive semi-definiteness, all of which are central in quant prep. You practice translating qualitative constraints about dependence into sharp quantitative bounds, and you get used to reading feasibility directly from matrix conditions.
This matters for quant interviews because real-world risk models, portfolio construction, factor models, and statistical arbitrage all rely on internally consistent covariance estimates. Interviewers use this kind of question to see if you understand when a set of correlations can actually come from a realizable multivariate model.
What it tests
Whenever you are given several pairwise correlations among random variables and asked to find an extreme value (maximum or minimum) for another correlation, the key structural fact is that the entire correlation matrix must be positive semi-definite. This means all its principal minors, including the determinant of the full matrix, must be non-negative. This requirement encodes the geometric reality that not all sets of correlations are compatible: the correlations must fit together in a way that could arise from actual random variables, which is only possible if the matrix is positive semi-definite. The determinant condition, in particular, gives a quadratic (or higher degree for more variables) in the unknown correlation, whose feasible range is precisely where the matrix remains valid. This is why the determinant bounds the possible values for the unknown correlation, and why the extreme values are found at the roots of this quadratic.
Practise this question with written feedback, or hear it in a spoken mock interview.
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