P - Xw

Each principal component is a time series of the transformed X variables, and the full T x m matrix of principal components, which has Pm as its with column, may be written

P = XW. (6.1)

To see that this procedure leads to uncorrelated components, note that

P P = W X XW = rWWA.

However, W is an orthogonal matrix, that is W = W-1 and so PP = . Since this is a diagonal matrix the columns of P are uncorrelated, and the variance of the mth principal component is Xm.

Since the variance of each principal component is determined by its corresponding eigenvalue, the proportion of the total variation in X that is explained by the mth principal component is Xnl/(sum of eigenvalues). However, the sum of the eigenvalues is k, the number of variables in the system.4 Therefore the proportion of variation explained by the first n principal components together is

J2x,/k.

Because of the choice of column labelling in W the principal components have been ordered so that Pt belongs to the first and largest eigenvalue Xu P2 belongs to the second largest eigenvalue X2, and so on. In a highly correlated system the first eigenvalue will be much larger than the others, so the first principal component alone will explain a large part of the variation.

Since W = W~, equation (6.1) is equivalent to X = PW, that is,

X,- = P, + wi2P2 + .. + wlkPk (6.2)

Thus each vector of input data may be written as a linear combination of the principal components. This is the principal components representation of the original variables that lies at the core of PCA models. Often only the first few principal components are used to represent each of the input variables, because they are sufficient to explain most of the variation in the system. However, even without this dimension reduction, calculations of covariance for the original variables are greatly facilitated by the representation of these variables by (6.2); the principal components are orthogonal so their unconditional covariance matrix is diagonal.

4 To see why, note that the sum of the eigenvalues is the trace of , the diagonal matrix of eigenvalues of V. However, the trace of equals the trace of V (because trace is invariant under similarity transforms), and because V has Is along its diagonal, the trace of V is the number of variables in the system.

6.2 Application to Term Structures

This section shows how, in any highly correlated system, the first principal component captures an approximately parallel shift in all variables. Term structures are special because they impose an ordering on the system that provides an intuitive interpretation of all the principal components, not just the first. After interpreting the principal components of a single yield curve, we shall consider how PCA should be applied to modelling multiple yield curves. Finally, an empirical example of PCA is given for modelling futures prices on the same underlying asset.

Term structures are special because they impose an ordering on the system that provides an intuitive

interpretation of all the principal components, not just the first

6.2.1 The Trend, Tilt and Convexity Components of a Single Yield Curve

Figure 6.1 shows monthly data on US semi-annualized zero-coupon rates using monthly data from 1944 to 1992.5 There are three different programs on the CD that analyze these data; each one illustrates a slightly different approach to programming PCA.

The data for PCA must be stationary, so that the unconditional correlation matrix XX can be calculated (§3.1) but the yields in Figure 6.1 are not stationary. Therefore, the input to PCA should be the first differences of the yields. For ease of exposition we shall refer to these first differences as the bond returns.6 Other transformations, such as taking deviations from a trend, do not generally make financial data stationary (§11.1.5).

The return correlation matrix XX for the US data is shown in Table 6.1. It exhibits the typical behaviour of the yield curve: correlations tend to decrease with the spread, and the 1-month rate and long rate have lower correlations with other rates. The trace of this matrix (the sum of the diagonal elements) is the number of variables in the system, that is, 14.

The results of a PCA are shown in Tables 6.2a and 6.2b. From Table 6.2a, the largest eigenvalue is 11.01, so the proportion of total variation that it explains is 11.01/14, or 78.6%. The second largest eigenvalue, 1.632, explains a further 1.632/14=11.7% and the third largest eigenvalue (0.4963) explains another 3.5%) of the total variation. Thus 93.8% of the total variation in the zero-coupon bond returns is explained by the linear model with just three principal components, that is,

Xt = wnP + wi2P2 + w;3P3

for the ith maturity bond, where X, is the standardized return (with zero mean and unit variance).

5 Copyright Thomas S. Coleman, Lawrence Fisher, Roger G. Ibbotson, U.S Treasury Yield Curves 1926-1992". Ibbotson Associates, Chicago. See Coleman et at. (1992).

6The returns on zero-coupon bonds are often represented by the process / = r dt + 1 dr. where dr is the difference in rates.

-1-1-1-,-,-,-1-1-,-r-1

1/31/44 10/31/47 7/31/51 4/29/55 1/30/59 10/31/62 7/29/66 4/30/70 1/31/74 10/31/77 7/31/81 4/30/85 1/31/89 10/30/92

Figure 6.1 US zero-coupon yields from January 1944 to December 1992. Table 6.1 Correlation matrix for US zero-coupon bonds

1 mth 3 mth 6 mth 9 mth 12 mth 18 mth 2 yr 3 yr 4 yr 5 yr 7 yr 10 yr 15 yr long

An upward shift in the first principal component therefore induces a roughly parallel shift in the yield curve

The factor weights wn, wa and wi3 are given in Table 6.2b. Note that the correlations are quite high, and this is reflected in the similarity of the weights on the first principal component wn, except perhaps for the very short and very long maturities that have lower correlation with the rest of the system. An upward shift in the first principal component therefore induces a roughly parallel shift in the yield curve, as shown in Figure 6.2a. For this reason the first principal component is called the trend component of the yield curve, and in this example 78.6% of the total variation in the yield curve over the sample period can be attributed to (roughly) parallel shifts.