What do eigenvalues/eigenvectors of the yield/forward rates covariance matrices mean?

I have 5 bonds (with maturities 1,2,3,4,5 years) which I calculated the yield curve for 10 days. I also calculated the forward rates from the yield rates. Now I've been told to calculate the covariance of the yield rates and the forward rates, as well as their eigenvalues/eigenvectors.

I'm assuming the covariance of the yield rates tell me how the bonds with different maturities move along with each other over time? But I'm not sure what the forward rates tell me.

Answer

The PCA analysis does not really tell you what the bonds do but it tells you how the rates move together. The variations of $n$ rates (i.e. 1 y, 2y, ...) are split up in (at first) abstract factors like

$$\Delta R_i = \sum_{j=1}^n e_{i,j} f_j$$ where $\Delta R_i$ is the change in the rate $i$ and $f_j$ is factor $j$ and $e_{i,j}$ is the (factor loading=) influence of factor $j$ to the rate $i$. The factors coming from PCA are uncorrelated and ordered by the size of their variance (largest first). Then it turns out that usually all rates have $e_{i,1}$, the influence of the first factor, with the same sign. This means that a change in this factor is in the same direction for all rates. The $e_{i,2}$ have a different sign for short terms as opposed to longer terms. Thus the second factor influences short and long rates differently - this is interpreted as steepening/flattening factor. For the third one often sees a curvature pattern (same sign for short and long and another sign for the middle terms).

Mathematical detail: the factor loadings are the eigenvectors of the covariance matrix of $\Delta R_i,i=1,\ldots,n$ and the variances of the factors are the squared eigenvalues.

Looking at the total variance explained by these it often turns out that $n$ rates can be described by the loadings to these 3 factors and the variances of these factors.

You can do this with spot rates and with forward rates. It would be interesting how a PCA of spot and forward rates together looks.

Note that such a reduction of dimensionality is an approximation and as always - take care doing it.

One of the first Google hits points to an article with more mathematical details:PRINCIPAL COMPONENT ANALYSIS by GRAEME WEST.

Problems of the interpretation are described here inPotential PCA interpretation problems for volatility smile dynamics by Reiswich and Tompkins.