PCA, factor extraction, and yield-curve geometry

Principal Component Analysis is what happens when you let the SVD loose on a centred data matrix. It is the single most-used dimension-reduction tool in finance, used to extract yield-curve factors, build covariance estimators, isolate sector exposures from a return panel, and identify common shocks across countries.

The PCA recipe

1. Start with data matrix X (T observations × n features). Centre: Xᶜ = X - 1·x̄ᵀ.
2. Compute the sample covariance Σ̂ = (1/(T-1)) XᶜᵀXᶜ. (Or use SVD of Xᶜ directly — numerically preferred.)
3. Eigendecompose Σ̂ = QΛQᵀ. Sort eigenvalues in descending order.
4. The k-th principal component is Xᶜ · qₖ — a length-T time series of scores for the k-th PC.
5. Variance explained by PC k: λₖ / tr(Λ). Cumulative through PC k: Σᵢ≤ₖ λᵢ / tr(Λ).

Yield-curve PCA — the classical application

Run PCA on daily changes in Kenyan Treasury yields across maturities 3M, 6M, 1Y, 2Y, 5Y, 10Y, 15Y, 20Y. The first three principal components, almost universally, capture:

1. Level — first PC, ~80-90% of variance. All maturities move together. This is the Fed/CBK policy news factor.
2. Slope — second PC, ~5-10% of variance. Short rates move opposite to long rates. This is the steepening/flattening factor.
3. Curvature — third PC, ~1-3% of variance. Middle maturities move opposite to short and long ends. The 'belly' factor.

PCA for risk factor extraction

On equity returns, the first PC of the NSE-20 index components is almost identical to the equally-weighted market portfolio — that's the market factor. The next 2-5 PCs typically pick up sector clusters (banking, telecoms, manufacturing) and behave like statistical factor models. This is the basis for PCA-based covariance estimators used at quant equity shops.

PCA pitfalls in finance

• Scaling matters. PCA on raw prices vs returns vs standardised returns gives entirely different decompositions. Default to standardised returns unless you have a reason.
• PCA finds variance, not signal. The first PC may dominate purely because of the largest-vol stock, not because it represents shared risk.
• PCA is unstable under sample additions when eigenvalues are close. Successive months can flip the sign of PC3 and rotate PC2 by 90°.
• PCs are orthogonal in sample but their economic interpretation can rotate. Sparse PCA and constrained variants are designed to fix this.

Implementation

import numpy as np
from numpy.linalg import svd

# X: T x n centred returns
U, s, Vt = svd(X, full_matrices=False)
# Principal components (scores)
PC = U * s          # T x n_components
# Loadings (eigenvectors)
loadings = Vt.T     # n x n_components
# Variance explained
var_explained = s**2 / (X.shape[0] - 1)
frac_explained = var_explained / var_explained.sum()