# 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 ?

## 2 Answers

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)

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.