Covariance Matrix

The covariance matrix is the core input for portfolio optimization: it shows how asset returns co-move. Diagonal elements are variances (each asset's standalone volatility), while off-diagonal elements are covariances (how pairs move together). For example, stocks and bonds often have low or negative covariance — when stocks fall, bonds may rise — enabling diversification. Diversification benefit: A portfolio of two 20%-volatility assets with 0.5 correlation has ~17.3% volatility, less than either asset alone. The matrix is symmetric (Cov(A,B) = Cov(B,A)) and typically estimated from historical returns, but estimation error is huge—small sample changes can radically alter results. Advanced methods (shrinkage, factor models) improve stability.