On the Level-Slope-Curvature Effect in Yield Curves and Eventual Total Positivity

Abstract

Principal components analysis has become widely used in a variety of fields. In finance and, more specifically, in the theory of interest rate derivative modeling, its use has been pioneered by Litterman and Scheinkman [J. Fixed Income, 1 (1991), pp. 54--61]. Their key finding was that a few components explain most of the variance of treasury zero-coupon rates and that the first three eigenvectors represent level, slope, and curvature (LSC) changes on the curve. This result has been, since then, observed in various markets. Over the years, there have been several attempts at modeling correlation matrices displaying the observed effects as well as trying to understand what properties of those matrices are responsible for them. Using recent results of the theory of total positiveness [O. Kushel, Matrices with Totally Positive Powers and Their Generalizations, 2014], we characterize these matrices and, as an application, we shed light on the critique to the methodology raised by Lekkos [J. Derivatives, 8 (2000), pp. 72--83].