Smallest Correlation Among Seven Variables

This question is about understanding how a symmetric covariance structure constrains the possible correlations between multiple identically distributed random variables. It asks you to reason about when a joint distribution is mathematically feasible, not just about computing a single covariance or correlation. In quant prep, this sits at the intersection of probability, linear algebra, and multivariate statistics.

It trains your intuition for covariance matrices, positive semi-definiteness, and eigenvalues in a high-dimensional setting. You must see how a single parameter governing all pairwise relationships interacts with the dimension of the system, and how feasibility of a covariance matrix puts a sharp lower bound on that parameter.

This matters in quant interviews because modeling correlated assets, risk factors, or PnL components always involves valid covariance structures. Being able to detect impossible correlation assumptions is key for robust models, risk management, and simulation in trading and research.

What it tests

When dealing with a collection of identically distributed random variables with equal pairwise correlations, the structure of their covariance matrix is highly constrained. The requirement that any covariance matrix be positive semi-definite (so that all variances are non-negative) imposes limits on how negative the pairwise correlations can be. Specifically, for $n$ variables each with variance 1 and common correlation $\rho$, the smallest eigenvalue of the covariance matrix must be non-negative, which leads to a lower bound on $\rho$. This is because if the negative correlations are too strong, the joint variability implied by the covariance matrix would become impossible, violating the basic properties of variances and covariances. This principle ensures that the relationships among the variables remain physically and probabilistically feasible, no matter how many variables are involved.