The yield curve is a high-dimensional object: a vector of rates at dozens of maturities, all moving simultaneously. Yet in practice, almost all the variation in daily curve movements can be described by just three independent directions — level, slope, and curvature. Principal component analysis (PCA) is the statistical tool that reveals this low-dimensionality from historical data, without any model assumption.
For the rates desk, PCA provides two things: (1) an empirical basis for multi-factor hedging — a 3-factor hedge is almost always sufficient to cover the residual risk left by a DV01 hedge; and (2) a validation that the Nelson-Siegel parameterisation is empirically grounded — its three shape factors align almost perfectly with the first three PCs of the forward curve.
Learning objectives. By the end of this note you should be able to:
- state the spectral decomposition theorem and identify the eigenvalues and eigenvectors of a covariance matrix;
- define the principal components of a random vector and list their key properties (uncorrelated, maximum-variance ordering);
- apply PCA to the forward curve: what the data matrix looks like, what the covariance matrix represents, and what the first three PCs mean;
- interpret the loading vectors of the first three PCs as level, slope, and curvature patterns;
- compute the percentage of variance explained by the first PCs;
- explain how PCA guides multi-factor hedging and why three factors typically suffice.
In this lesson
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- The spectral decomposition theorem
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- Principal components of a random vector
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- PCA of the forward curve in practice
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- Interpretation of the three factors
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- Percentage of variance explained
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- PCA and multi-factor hedging
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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