LO 10.7: Describe principal component analysis and explain how it is applied to

Regression analysis focuses on yield changes among a small number of bonds. Empirical approaches, such as principal components analysis (PCA), take a different approach by providing a single empirical description of term structure behavior, which can be applied across all bonds.

PCA attempts to explain all factor exposures using a small number of uncorrelated exposures which do an adequate job of capturing risk.

For example, if we consider the set of swap rates from 1 to 30 years, at annual maturities, the PCA approach creates 30 interest rate factors or components, and each factor describes a change in each of the 30 rates. This is in contrast to regression analysis, which looks at variances of rates and their pairwise correlations.

PCA sets up the 30 factors with the following properties:
1. The sum of the variances of the 30 principal components (PCs) equals the sum of the variances of the individual rates. The PCs thus capture the volatility of the set of rates.
2. The PCs are not correlated with each other.
3. Each PC is chosen to contain the highest possible variance, given the earlier PCs.

The advantage of this approach is that we only really need to describe the volatility and structure of the first three PCs since the sum of the variances of the first three PCs is a good approximation of the sum of the variances of all rates. Thus, the PCA approach creates three factors that capture similar data as a comprehensive matrix containing variances and covariances of all interest rate factors. Changes in 30 rates can now be expressed with changes in three factors, which is a much simpler approach.