Factor models from PCA

A covariance matrix with structure

Clipping and shrinkage clean a full matrix. A factor model builds one with structure instead: returns are driven by a few common factors plus asset-specific noise, $\mathbf r=B\mathbf f+\boldsymbol\varepsilon$. Here $B$ is the $N\times k$ matrix of loadings, $\mathbf f$ holds $k\ll N$ factor returns with covariance $\Lambda$, and $\boldsymbol\varepsilon$ are idiosyncratic returns, uncorrelated with each other and with the factors, with variances $d_i$. PCA gives a ready-made version: take the top $k$ principal components as the factors.

The first term is the systematic part: it depends on the portfolio only through its factor exposures $B^{T}\mathbf w$. The second is idiosyncratic risk. From PCA, $B$ holds the first $k$ eigenvectors scaled by $\sqrt{\lambda_j}$ (or the unscaled eigenvectors with $\Lambda=\operatorname{diag}(\lambda_j)$), and $d_i$ is whatever variance of asset $i$ the factors do not explain. The saving is large: instead of $N(N+1)/2$ covariances you estimate $Nk+N$ numbers. For $N=500$ and $k=5$ that is 3,000 instead of 125,250. The model matrix is positive definite whenever all $d_i>0$, even if $N>T$, and it embodies the lesson of clipping: only the top $k$ eigen-directions are trusted, the rest becomes diagonal noise.

• Structure: $\Sigma\approx B\Lambda B^{T}+D$, with a few factors and a diagonal idiosyncratic matrix.
• Savings: $Nk+N$ parameters instead of $N(N+1)/2$; positive definite if all $d_i>0$.
• Risk split: variance = factor exposure part + idiosyncratic part; diversification removes only the latter.

Lab · what's graded

• Counting parameters10 pts
• Idiosyncratic variance15 pts
• Share explained by the factor10 pts
• Portfolio exposure15 pts
• Portfolio variance20 pts
• When the model matrix inverts15 pts
• What diversification removes15 pts