Principal Component Analysis
PCA finds orthogonal directions of maximum variance in a covariance (or correlation) matrix — the yield-curve level/slope/butterfly and many equity ‘statistical factors’ are PCA.
Definition
Eigen-decompose Σ = V Λ V'. The first columns of V are the PCs. They are statistical, not structural: rotate the universe and they rotate. Explained-variance ratios are in-sample.
Why it matters
It compresses a curve or a cross-section into two or three risk factors you can actually limit. It does not tell you the economic names of those factors without a story.
Case
Treasury PCA: PC1 ≈ parallel, PC2 ≈ slope, PC3 ≈ curvature. In a twisty QT regime PC3’s share rises — the 3-factor story is not eternal.
How to read it
Use a correlation matrix if scales differ. Check stability out of sample. A PC that flips sign every quarter is not a factor; it is noise.