Using PCA to Assess a Parallel-Shift Yield Curve Model

Summary

The document explains how principal component analysis can help assess whether yield curve movements are well described by a parallel shift. Apply PCA to changes in yields across maturities, then inspect the first component’s loadings: a parallel-shift interpretation is plausible when the loadings have the same sign and broadly similar magnitudes across the curve. The component’s share of total variance indicates how much of the observed movement this single-factor description captures.

The answers describe PCA components as eigenvectors of the covariance matrix and their associated eigenvalues as measures of explained variance. They state that the first component often explains a large share of yield changes, but provide no analysis of the questioner’s data or specific results. A shift need not be perfectly uniform; short and long maturities can receive different weights while moving in the same direction. The conclusion is therefore conditional on both the loading pattern and the variance explained over the chosen sample period.

Key ideas

• Inspect the first principal component’s loadings across maturities to judge whether it resembles a parallel shift.
• Loadings with the same sign indicate that maturities tend to move in the same direction.
• The first component’s share of total variance measures how much movement the one-factor model captures.
• A parallel shift can have different weights at the front and back of the curve.
• The assessment describes the selected sample period and depends on correct PCA inputs.

Tags

# Principal component analysis for yield curve


# Principal component analysis for yield curve












I have Treasury yield data across 11 maturities for past 1 year. I have used a code in MATLAB for PCA on change in yield curve. Now, I have covariance matrix of daily/monthly yield curve changes, principal components and the fractions (individual and cumulative) explained by the principal components.

So with this data, how do I conclude if a parallel shift model is a good way to describe fluctuations of the yield curve over this time period ?

## Answer by Bram (score 2)

https://quant.stackexchange.com/a/39228

Based on factor loadings you should be able to tell if the first component is a parallel shift (if you did everything correctly it's highly like that it is). The variance explained by the factor then a measure of how good that model is. Note that a parallel shift normally actually isn't fully parallel, but instead has different weights on the front and back of the curve (but with all the same sign)

## Answer by Kiann (score 1)

https://quant.stackexchange.com/a/41844

your PCA's are effectively the eigen-values and the eigen-vectors of the covariance matrix.

The eigen-values (corresponding to each eigen-vector) show the proportion of (orthogonal moves) that is explanable by each eigen-vector. Usually, the first eigen-vector (i.e. 1st PCA) will have a eigen-value (relative to the sum of all eigen-values) that shows it explains >90% of the moves.

## Answer by Hritabrata Das (score 0)

https://quant.stackexchange.com/a/85754

Look if PC1 is truly parallel shift and then look for how much variance is explained by PC1, eigen value divided by total eigen values. That should give your answer

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.