Fixed-income portfolio managers might have expectations on the future development of certain yield curve key rates but not on the entire yield curve. As a result, it can be challenging to construct a sensible curve depending on only a small number of key rates. Principal component analysis (PCA) assigns weights to the most likely yield curve shifts (e.g., parallel shifts and tilts). This method makes it possible to construct most likely yield curves based on only a limited amount of predicted key rates. Because the method is flexible, more extreme shifts or strange deformations can also be incorporated.

Most portfolio managers have views on future economic developments without being aware of how to incorporate these views into their work in a consistent manner. Furthermore, most economic analyses and subjective views are focused on particular points on the yield curve, making it more difficult for a portfolio manager to use this information in less-described parts of the yield curve.

PCA is a solution method that is well known in physics, and hence, the authors use a physics optimization problem to introduce the concept. The intuition behind principal components can be compared with that behind springs: loose springs can easily be changed and will explain the major part of an entire shift, whereas stiffer springs are harder to change but might be required to get the overall effect correct. Changing the loose springs is the preferred method to use the least amount of energy possible.

In addition to the PCA framework being applied to yield curves, other asset classes or cross-currency interactions can be modeled (e.g., what happens to the yield of Currency X when Currency Y’s yield increases?). As a result, this research would be interesting not only to fixed-income investors but also to currency traders and other portfolio managers.

The authors introduce the concept by referring to a well-known problem in physics in which three lifeguards have to save a swimmer. The lifeguards can use a combination of running and swimming to reach the swimmer: the first lifeguard decides to start swimming immediately, the most direct path to the swimmer. The second lifeguard decides to run to the point closest to the swimmer before he starts swimming because he can run faster than he can swim, but by doing so, he makes his route longer. The third lifeguard chooses a solution somewhere in between. The last method turns out to be optimal.

This example illustrates important elements of PCA: although some changes might explain most of the yield curve changes (e.g., a parallel shift), a combination of changes yields the optimal result. The key rates on the yield curve are modeled to reflect changes in the yield curve, and PCA is used to reflect the most likely changes in the yield curve.

Thereafter, stylized and realistic examples are applied to test the effectiveness of the approach. When as many principal components as shocks in key rates are used, a unique yield curve can be derived. When more principal components than shocks in key rates are used, a combination of principal components can be derived and thus several yield curves can probably be derived.

Three realistic examples the authors offer concern a butterfly shift, a large shift, and a slope deformation. The authors show, in more challenging examples, that two principal components are not sufficient to create realistic yield curves but that using three principal components results in realistic shapes. Adding more principal components leads to even better results.

The authors introduce a mathematical and somewhat abstract topic in a helpful way: the example of the swimmer and lifeguards is very intuitive and allows the reader to become comfortable with the mathematics that follow. The issue that people’s expectations are concentrated in certain areas (e.g., key rates on a yield curve) is also very intuitive and will be familiar to many readers. The examples used in the article illustrate the method nicely. The research is a good introduction for financial professionals who want to become more familiar with PCA.