How does Principal Components Analysis build a factor model?

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AcadiFi · Accepted Answer

A statistical factor model applies Principal Components Analysis (PCA) to a covariance matrix of asset returns and treats the top K principal components as factors. Each PC is a linear combination of assets that maximizes variance explained; successive PCs are orthogonal. Example: Prairie Alpha runs PCA on 500 US stocks over 5 years of daily returns. The first PC explains about 38% of variance — its loadings are nearly equal across all stocks, so PC1 is essentially a market factor. PC2 explains 6% and has positive loadings on small-caps, negative on large-caps — a size factor. PC3 explains 4% with value-versus-growth character. The top 5 PCs typically capture 55-65% of variance. Interpretation challenges: PCA gives you mathematical factors that need economic naming. PC4 might be a sector rotation factor; PC7 might be a liquidity factor; often they are pure noise at higher components. Rotation (varimax) or post-hoc labeling is subjective. ```mermaid flowchart LR A[Return
Matrix] --> B[Covariance
Matrix] B --> C[Eigen
Decompose] C --> D[Top K
PCs] D --> E[Statistical
Factors] ``` Statistical models are best when you need a risk model but don't want to impose economic priors. They are commonly used in risk budgeting, portfolio optimization under estimation error, and hedging illiquid portfolios against systematic risk. Limitations: factors are unstable across samples, and out-of-sample they may not price returns.

How does Principal Components Analysis build a factor model?

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