Principal Component Analysis Table 6.2a: Eigenvalues of correlation matrix

Component Eigenvalue Cumulative R2

Pi 11.01 0.786

P2 1.632 0.903

P3 0.4963 0.938

Table 6.2b: Eigenvectors of correlation matrix

The factor weights on the second principal component, w,2, are monotonically decreasing from 0.57207 on the 1-month rate to -0.48628 on the long rate. Thus an upward movement in the second principal component induces a change in slope of the yield curve, where short maturities move up but long maturities move down, as shown in Figure 6.2b. Therefore, the second principal component is called the tilt component, and in this example 11.7% of the total variation is attributed to changes in slope.

The factor weights on the third principal component, wa, are positive for the short rates, but decreasing and becoming negative for the medium-term rates, and then increasing and becoming positive again for the longer maturities. Therefore, the third principal component influences the convexity of the yield curve, and in this example 3.5% of the variation during the data period is due to changes in convexity (Figure 6.2c).

An upward movement in the second principal component induces a change in slope of the yield curve, where short maturities move up but long maturities move down

6.2.2 Modelling Multiple Yield Curves with PCA

It is often necessary to model more than one yield curve, for example when risk factors for fixed income portfolios include both government and corporate

Figure 6.2 Effect of (a) first, (b) second and (c) third principal component.

Jan-92 May-92 Sep-92 Jan-93 May-93 Sep-93 Jan-94 May-94 Sep-94 Jan-95

-1M 2M 6M-12M-2Y------3Y-4Y 5Y 7Y 10Y

Jan-92 May-92 Sep-92 Jan-93 May-93 Sep-93 Jan-94 May-94 Sep-94 Jan-95

-1M 2M 6M-12M-2Y----3Y-4Y 5Y 7Y 10Y

Figure 6.3 (a) US and (b) UK zero-coupon yields.

bond yields. In this case PCA can be a very useful computational tool because it allows a great reduction in dimensions. Moreover, when PCA is applied to more than one yield curve, the natural ordering in the system still gives a meaningful interpretation to the principal components.

To see this, consider an example where four principal components are used to represent a system of 20 yields. Contemporaneous data on both US and UK yields are illustrated in Figure 6.3. The data are daily from 1 January 1992 to 24 March 1995. There are 21 variables in the system: 10 maturities in the US yield curve and 11 maturities in the UK yield curve.

The results of PCA with four principal components are shown in Table 6.3. The largest eigenvalue of the correlation matrix is 14.97, so the first principal