This paragraph describes how linear algebra can help explain the way U.S. Treasury interest rates move over time. The key idea is that rates at different maturities often move together, so many rates can sometimes be summarised by a small number of underlying patterns.
1. What is the yield curve?
A Treasury yield curve shows the interest rates, or yields, on U.S. government debt for different time periods until repayment. A 1-month Treasury bill is a short maturity; a 30-year Treasury bond is a long maturity. The curve is the set of yields across these maturities on a particular date.
2. Turning yields into daily changes
The analysis uses daily observations from 2015 to 2025 for eight maturities, from 1 month to 30 years. Instead of analysing the yield levels directly, it calculates each maturity’s daily yield change: today’s yield minus the previous day’s yield. This focuses on how rates move from day to day.
Each day can be represented as a vector with eight entries—one yield change for each maturity. Stacking all the daily vectors creates a data matrix: rows represent days, and columns represent maturities.
3. Measuring how rates move together
The analysis calculates a covariance matrix from the yield changes. Its diagonal entries describe how much each maturity’s changes vary. Its off-diagonal entries describe how two maturities tend to move together: positive covariance means they often move in the same direction, while negative covariance means they often move in opposite directions.
4. Using eigenvalue decomposition and PCA
Principal Component Analysis (PCA) uses eigenvalue decomposition of the covariance matrix to find the most important patterns in the data.
- An eigenvector gives a pattern of weights across the eight maturities. It describes how the maturities combine to form a particular movement.
- The corresponding eigenvalue measures how much variation in the data is associated with that pattern.
The patterns are ranked by their eigenvalues. The first principal component (PC1) explains the most variation, the second (PC2) explains the most remaining variation, and so on. These components are mathematically designed to capture distinct, uncorrelated directions of variation in the dataset.
5. What the first three components represent
- PC1 — the level pattern: This represents a broad movement in yields, especially across medium- and long-term maturities. When its influence is positive, many yields may rise together; when negative, many may fall together. This is often called a level factor.
- PC2 — the steepening or flattening pattern: This captures a contrast between short-term and long-term rates. If short and long yields move differently, the slope of the yield curve changes. That is described as steepening or flattening, depending on which end rises or falls relative to the other.
- PC3 — the curvature pattern: This represents a change in the middle of the yield curve relative to its short and long ends. For example, medium-term yields may rise while both ends move less, or move in the opposite direction. This makes the curve bend or change curvature.
The exact signs of a principal component are not meaningful by themselves: an eigenvector and its negative describe the same pattern, with the direction reversed. So the important point is the relative movement across maturities, not whether a component is labelled positive or negative.
6. Interpreting the percentages
The first three components together explain about 90% of the observed variation in daily yield changes. Adding a fourth brings the total to roughly 95%. “Explained variation” means the share of the data’s overall movement represented by those components; it does not mean that the model predicts future yields with 90% accuracy.
Why this is useful
Although the dataset tracks eight separate maturities, their changes are not all independent. A few recurring patterns—broad level shifts, slope changes, and curvature changes—summarise most of their movement. PCA therefore provides a compact mathematical description of the yield curve and connects a data matrix, covariance, eigenvalue decomposition, and variance-based evaluation to meaningful patterns in financial markets.