Question 14.3: Determine the first five principal components for the UK for...

Recall from Chapter 11 that each principal component is the combination of an eigenvalue which represents the volatility of a particular independent factor, and an eigenvector which contains the weights of this factor that contribute to the returns of each variable. Each eigenvector can be represented by a single line, showing the weights of each factor for each forward rate:

This shows that the dominant change of shape for the forward rate curve over this period is for rates at opposite ends of the curve to move in opposite direction – given a random number scaled by the first eigenvalue the movement of the one-year forward rate in respect of the first principal component would be around 0.5 times this random number, whilst the twenty-five year forward rate would move by −0.1 times the same random number. The second most important move is a move in the same direction, as evidenced by the fact that all values of the second eigenvector are above the horizontal axis. The third principal component produces changes in short and long forward rates that are in the opposite direction to those in the middle of the curve, whilst the fourth principal component produces still more involved changes. The first four eigenvalues are shown below:

These show that the first principal component is by far the most dominant. The dominance of the first principal component – and its shape – can be verified by looking at the correlations between the various excess returns.

The correlations between the terms shown above correspond largely with the magnitudes and signs in the first eigenvector, supporting the results of the PCA.