Principal component analysis is widely used in fixed income risk management and pricing to estimate the effects of yield curve movements by transforming them into a reduced set of factors. When the analysis is done on spot rates, the factors have generally been described as level, steepness or slope, and curvature. But why? And is there a mathematical or economic reason for this observation?

Let’s now take a more critical look at the PCA decomposition of spot and forward rates. This analysis is based in part on a paper by Ilias Lekkos. In his paper, Lekkos presents a simple scenario. Divide the forward curve up into a collection of representative forwards, and then suppose that there is no correlation between each of these forwards (and hence the forward curve by extension), so that each is driven by an independent factor. If we simulated a bunch of forward curves out of this zero correlation factor structure, built the spot curves associated to each, and then performed a PCA on the spot curves, what would the PC loading structure look like?

Would there be correlation in the derived spot curves even though there was none present in the forward curves from which they were born?

The answer, it turns out, is a resounding yes.

I performed a similar analysis to Lekkos. In addition to replicating his findings I try to explain why the level, slope, and curvatures are important factors even if they appear to be a statistical construct of the no arbitrage averaging process that relates spots to forwards.

I started by taking 251 independent sets of 30 observations drawn from a uniform distribution between 0 and 1 to represent 251 independent one-year forward rates (any distribution would have given the same results). I then created a 30-year spot curve from these forwards for each of the 251 observations and applied principal component analysis to the z-scored returns of the spot rates. The results are reported in the following figures:

The first three factors explain about 87% of the total variance of the spot curves that are calculated from randomly generated forwards and the pattern of the factor loadings looks similar to what many expect the term structure to exhibit as drivers of its covariance: level, slope, and curvature.

To confirm this, I performed principal component analysis on one year of historical returns of the spot curve as of April 29, 2011. The spot curve here consists of 30 points with annual intervals. The results are summarized in the following figures:


The interpretation is almost identical between the two cases. So it appears that the factors that we observe in historical data are just a statistical construct of the no arbitrage averaging process that relates spots to forwards.

Next I repeated the same historical analysis but instead of using the spot curves I transformed the data into a forward curve with annual intervals by bootstrapping the spot rates into one-year forward rates. The results of the PCA on the one-year forward rate returns are presented below:

The first three factors still explain a large part of the variance but the pattern of their loadings is different. The results still exhibit curvature and slope but the level driver seems to have disappeared by the bootstrapping process.

Level and slope do have an economic interpretation because changes in those drivers for the spot curve imply changes in the forward curve. However, even if the first factor in the spot curve PCA looks like a level shift and explains the most variance, it does not seem to have the biggest impact on the forward curve. Its impact may have been exaggerated by the averaging process that defines the relationship between spots and forwards.

To illustrate this further, take a look at what happens to the spot and forward curves when the factor loadings for the first three factors are shifted. To do this, I took the PC loadings matrix I got from the PCA on the spot curve and applied shocks to the first three factors. Each of the factors was shocked by an amount equal to one unit of the explained variance for that factor. I then took a look at the resulting spot curve and the corresponding forward curve.

As we can see, the forward curve responds more to shifts in the second and third factor which look like a combination of slope and curvature (hump).

To complete the analysis I took a look at the forward curves generated by shifts to the forward PCA. Here again the results look like changes in slope and curvature. 

Forward rates are easier to interpret economically because they reflect future interest rate expectations. Most of their variation is probably due to slope and curvature changes. As for spot rates, a change in their slope and curvature seem to have a bigger impact on forward rates than a change in their level. The level factor is very likely the result of the mathematical relationship between spots and forwards.

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