How good is the description of term structure movements using PCA?

The movements of the term structure of interest rates can be explained quite well using only a few basic factors, like a parallel movement of the curve (shift), a flattening or steepening or of the curve (twist) and a reshaping of the curve (butterfly). The factors, which can be calculated as the eigenvectors of the Gramian matrix of interest rate movements, provide a very powerful framework, since they accelerate the calculation of the historical VaR for interest rate derivatives (which is a very time consuming task, if a full valuation approach is used) significantly.
The following questions occur:

  • How good is the explaination of interest rate movements using only a few factors?
  • How many factors do we have to use to get a good approximation?
  • How stable are the principal components over time?
  • Does the shape of the principal components depend on the currency?
  • How large is the approximation error of a historical VaR calculation using PCA compared to a full valuation approach?
Performing a principal component analysis (PCA) on the weekly changes of historical EUR yield curve data (between 2000 and 2007) given on 16 curve points (1W,3M,6M,9M,1Y,2Y,3Y,4Y, 5Y,7Y,10Y,15Y, 20Y,25Y, 30Y,50Y), the principal components have the following form

As expected, the first three factors describe a shift, twist and butterfly movement of the curve. 
In my next blog(s), I will try to find the answers to the questions above....