Fourier estimation method applied to forward interest rates

Abstract

Principal component analysis (PCA) is a general method to analyse the factors of the term structure of interest rates. There are usually two or three factors. However, it is shown by Liu that when we apply PCA to forward rates, not spot rates, we need more factors to explain 95% of variability. In order to verify the robustness of this result, we introduce another method based on Fourier series, which is proposed by Malliavin and Mancino. The results reconfirm the observation of Liu with different data sets. In particular, the Fourier series method gives us similar results to PCA.

Affine term structure as multi-soliton

“…In real market, however, the term structure of spot rates behaves nicer. According to the series of studies by N.L.Liu and her collaborators [1][2][3], from the term structure of spot rates only two or three factors up to almost 99% are detected when applied a principal component analysis (or its variants), while that of forward rates exhibits more than 10, sometimes 15, or even more factors. Much more straightforward peculiarity is that the samples of the term structure of forward rates often have more humps than those of spot rates.…”

The Fourier estimation method with positive semi-definite estimators

“…A motivation of the present study is to make an implementation of the Fourier method easier when it is applied to "dynamic principal component analysis", an important application of the spot volatility estimation (see [1,2]). Due to the lack of symmetry of the matrices, its estimated eigenvalues are sometimes non-positive, or even worse, nonreal.…”