Definitions
Covariance measures how two variables move together, in the product of their units. Correlation divides by both standard deviations to give a unit-free number between -1 and 1.
rho = Cov(X, Y) / (sigma_X sigma_Y)
Linear only
Correlation captures linear association and nothing else.
The standard counterexample: let X be symmetric about zero and Y = X squared. Y is completely determined by X, yet their correlation is zero. Perfect dependence, no correlation.
So zero correlation does not imply independence. Independence implies zero correlation, but not the reverse - except for jointly normal variables, where the two coincide. Interviewers ask this constantly and the joint-normal exception is the part candidates forget.
Portfolio intuition
For two assets with weights w and 1-w:
Var(portfolio) = w^2 sigma_1^2 + (1-w)^2 sigma_2^2 + 2w(1-w) rho sigma_1 sigma_2
The cross term is where diversification lives. With rho = 1 there is no benefit at all. With rho = -1 you can construct a zero-variance portfolio. Anything below 1 reduces risk relative to the weighted average.
That is the entire mathematical case for diversification and it should be immediate.
Correlation is unstable
The practical warning. Correlations estimated from historical data are noisy, and they are not constant - they rise sharply in crises.
Assets that appeared diversifying for years can move together precisely when it matters. Risk models calibrated on calm periods therefore understate crisis risk systematically, which is a failure that has recurred repeatedly.
Estimation
Correlation estimates need a lot of data. For n assets there are n(n-1)/2 pairwise correlations - 4,950 for 100 assets - and estimating all of them from limited history produces a matrix dominated by noise, often not even positive definite.
Shrinkage toward a structured target is the standard fix.
Common questions
"Can three assets be pairwise correlated at -0.9?" No - the correlation matrix must be positive semi-definite, and that configuration is not.
"Correlation of X and 2X + 3?" Exactly 1. Correlation is invariant to positive linear transformations.
Practise in statistics.