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...
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.
