6 - Principal Components: Theory

Summary

Parlare oscuramente lo sa fare ognumo, ma chiaro pochissimi.

– Galileo Galilei

THE PURPOSE OF THIS CHAPTER

Principal Component Analysis (PCA) is a well-known technique used to reduce the dimensionality of multidimensional problems. When it comes to termstructure modelling, it is well known that the principal components obtained by orthogonalizing the covariance matrix (of yields or forward rates) afford a synthetic yet ‘rich’ description of the yield curve. Principal components are also loved by practitioners because they lend themselves to an appealing intuitive interpretation as the ‘level’, ‘slope’ and ‘curvature’ of the yield curve.

Beside being of intrinsic descriptive interest, principal components (or their proxies) are also important because several affine modelling approaches use them (or their proxies) as their state variables. Furthermore, as we shall see in Part VI, one principal component (the ‘slope’) by itself explains a lot about excess returns.

In this chapter we therefore explain what principal components are. In the next, we present what is known empirically about them (looking at the covariance matrix both of levels and of differences). We explore important issues such as their persistence, and we highlight how this feature is linked to the quasi-unit-root nature of rates. We look both at nominal and at real yields.

WHAT ARE PRINCIPAL COMPONENTS?

The Axis Rotation

There so many excellent introductions to Principal Component Analysis that we feel that it is really unnecessary to reinvent this particular wheel and give a self-contained presentation here. However, it may still be useful to try and explain the intuition behind Principal Component Analysis and to discuss the two distinct functions that this mathematical technique fulfills – both in general, and in term-structuremodelling applications. Unfortunately, these distinct roles are frequently confused.

The first reason why principal components are used is that they allow a significant reduction of redundancy. This reduction is obtained by forming combinations of the original variables to create new variables that behave in a simpler and more ‘informative’ manner. (What ‘redundancy’ and ‘informative’ mean will become apparent in the following).