Highest Achievable Covariance with Given Moments

This quant interview question is about the extreme behavior of covariance when only first and second moments are specified. In quant prep, it sits at the intersection of probability, statistics, and linear algebra, asking you to reason about how "tightly" two random variables can move together given fixed means and variances. It's a classic conceptual test in covariance and correlation, often appearing in quant interviews.

It trains your understanding of covariance bounds, correlation constraints, and how moment information restricts possible joint distributions. You must recognize what information is relevant, ignore what is not, and connect covariance to variance in a precise way. Conceptual clarity and comfort with random variables as vectors are central.

This matters for quant interviews because modeling dependence structures under incomplete information is a daily task in trading, risk, and derivatives pricing. Interviewers want to see that you can reason about feasible joint behaviors of assets or risk factors from limited summary statistics. A strong answer signals you understand both probabilistic structure and its implications for portfolio construction, hedging, and risk limits, which is exactly what front-office quant roles demand.

What it tests

For any two random variables, the covariance is fundamentally limited by the product of their standard deviations, a fact formalized by the Cauchy-Schwarz inequality. This arises because the correlation coefficient, which normalizes covariance, must always lie between -1 and 1. The reason is that the inner product of centered random variables cannot exceed the product of their lengths (standard deviations), just as the angle between two vectors cannot produce a cosine outside [-1,1]. This constraint ensures that no matter how strongly two variables are linearly related, their covariance cannot surpass the geometric mean of their individual variabilities. This bound is tight: it is achieved only when one variable is an exact affine function of the other.