A generalization of principal component analysis for non-observable term structures in emerging markets

Principal Component Analysis (PCA) has been traditionally used for identifying the most important factors driving term structures of interest rates movements. Once one maps the term structure dynamics, it can be used in many applications. For instance, portfolio allocation, Asset/Liability models, and risk management, are some of its possible uses. This approach presents very simple implementation algorithm, whenever a time series of the term structure is disposable. Nevertheless, in markets where there is no database for discount bond yields available, this approach cannot be applied. In this article, we exploit properties of an orthogonal decomposition of the term structure to sequentially estimate along time, term structures of interest rates in emerging markets. The methodology, named Legendre Dynamic Model (LDM), consists in building the dynamics of the term structure by using Legendre Polynomials to drive its movements. We propose applying LDM to obtain time series for term structures of interest rates and to study their behavior through the behavior of the Legendre Coefficients levels and first differences properly normalized (Legendre factors). Under the hypothesis of stationarity and serial independence of the Legendre factors, we show that there is asymptotic equivalence between LDM and PCA, concluding that LDM captures PCA as a particular case. As a numerical example, we apply our technique to Brazilian Brady and Global Bond Markets, briefly study the time series characteristics of their term structures, and identify the intensity of the most important basic movements of these term structures.