Description of term structure movements using PCA continued

In my last blog entry How good is the description of term structure movementsusing PCA a lot of open questions remained. Today I want to give first answers...
 - How good is the description of interest rate movements using only a few factors?
We assume, that we have a time series of  yield curves , where each of them is given on 16 curve points (1W,3M,6M,9M,1Y,2Y,3Y,4Y, 5Y,7Y,10Y,15Y, 20Y,25Y, 30Y,50Y). Calculating the principal components ei, based on daily interest rate movements, the increments of the yield curve dr =(dr1,…,dr16) can be exactly described using the formula

 where (.,.) is defined to be the inner product of two vectors. The following pictures show, how good an arbitrary chosen interest rate increment (blue curve) can be approximated using only 4 (left), 5 (middle), 6 (right) factors, i.e.

The table below shows, how many percent of the variance of daily, weekly and monthly historical interest rate movements can be described using only a few PCA factors:


Using a time series of daily EUR interest rate movements, the following picture shows the variation of the original data (left) and the remaining variation (right) after the filtration of the first four principal components. One can see, that on average about 1 basis point of the interest rate movements remain unexplained.


So, using a few principal components for the description of interest rate movements, leads to a good approximation of the original data. Furthermore, combinations of principal components produce  realistic yield curve scenarios, which can be used for the calculation of interest rate risk measures of instruments and portfolios.