The correlation between interest rates varies across maturities. But very short-term rates might not be correlated with short-term and longer-term rates. The convenient methodology for simulating interest rate variations is to use the principal components analysis, or PCA. The reader should refer to the factor model chapter (Chapter 32) for the conceptual framework. As a reminder, PCA serves for explaining a set of variables using orthogonal factors, which are much easier to handle than usual (correlated) factors. The factors are called principal components. The number of factors that contribute significantly to the variance explained decreases with the correlations between variables. Hence PCA allows explaining the variance of multiple correlated variables with a smaller number of factors. It is notably successful for interest rates because they are highly correlated.

Let F be the set of variables to be explained, interest rates, the generic form of factor model is a linear relation between the explained variables and the factors X:

The notations are those of Chapter 32 on factor models. The PCA uses independent variables as factors. The independent variables are constructed as linear functions of the observable factors used in the standard regression technique. The concepts behind PCA are detailed in Appendix 1 of Chapter 32. The generic form of the PCA models is:

A major application of PCA is the modeling of the term structure, because it was observed that very few factors, usually no more than three, suffice for explaining most of the variance of interest rates. The technique is commonly used in market VaR, but serves as well for providing interest rate simulations in ALM with other than simple interest rate scenarios, such as parallel shifts or standard changes of the steepness of the term structure of rates.

Because PCA uses as independent factors combinations of standard regression variables, one can wonder what principal components mean. In the case of interest rates, the interpretation, as we will illustrate, is fairly simple.

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