7 - Principal Components: Empirical Results

Summary

We must improve our concrete experience of persons and things into the contemplation of general rules and principles; but without being grounded in individual facts […] we shall end as we began, in ignorance.

– William Hazlitt, On Reason and Imagination

THE PURPOSE OF THIS CHAPTER

In Chapter 6 we explained what principal components are, and why (and when) they are useful.

In this chapter we present some empirical results about the principal components obtained from the covariance matrix of changes in yields. The data we present clearly show the high degree of redundancy in the whole set of yieldchange data.

Readers are probably already familiar with the principal-component results obtained using nominal yields (or forward rates). However, in this chapter we present some results that are less widely discussed (such as the mean-reverting properties of principal components), but which are greatly relevant for some of the modelling approaches we present in Part VII. In the second part of the chapter we then present similar results for real rates and break-even inflation.

Beside being of intrinsic interest, all these results provide a motivation for themodelling choices discussed in Chapters 32 and 33, and a foundation for the discussion of the reversion-speed matrix in the real-world and the risk-neutral measures.

NOMINAL RATES

Descriptive Features

Most readers will probably already be familiar with the Principal-Component results for nominal yields, and therefore in this section wemove at a rather brisk space, and simply recall that the first three eigenvectors lend themselves to the usual interpretation – a ‘level’, ‘slope’ and ‘curvature’. Figure 7.1, which shows the first three eigenvectors (the first three principal components) obtained from orthogonalizing the covariance matrix of yield curve levels (upper panel L) or differences (lower panel D) for maturities out to 20 years, gives a justification for this interpretation. As well known, the first eigenvalue by itself explains more than 90% of the observed variance, the first two eigenvalues account for about 95%, and, depending on the currency and the time period under study, one can explain close to 98–99% of the yield curve variance using three principal components.