The factor-residual model requires a set of orthogonal risk factors that explain the systematic variation in stock returns. PCA provides these automatically: applied to the matrix of stock returns, it finds the portfolio weights that explain the most variance with the fewest uncorrelated components. The resulting portfolios — eigenportfolios — are the statistical risk factors. The first captures the market-wide trend (approximately the equal-weighted index); subsequent ones capture sector rotation, growth-vs-value tilts, and other systematic patterns.
This note derives the eigenportfolio construction from the spectral decomposition of the return covariance matrix, interprets the leading eigenvectors, and shows how the principal component representation allows any stock's return to be expressed as a linear combination of eigenportfolio returns plus a residual. The full algebraic derivation of the eigenvalue problem is in the companion derivation note.
Learning objectives. By the end of this note you should be able to:
- set up the return matrix and compute the sample covariance matrix;
- state the spectral decomposition and interpret each matrix;
- define eigenportfolios as the columns of and compute their returns;
- compute the fraction of total variance explained by the leading eigenportfolios;
- express a stock's return as a linear combination of eigenportfolio returns and identify the residual.
In this lesson
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- Setting up the return matrix
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- Spectral decomposition and eigenportfolios
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- Variance explained and factor selection
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- Principal component representation of returns
Before you start
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