Some results on Shift, Slope and Curvature

In this paper we systematize and develop some theoretical results about shift, slope and curvature for correlations matrices of interest rates. We provide a general investigation on the relations among some standard features of correlation models for interest rates and the existence of shift, slope and curvature. Our results show how their presence, excluding some peculiar behavior strictly related to low dimensions, can not be directly connected to the usual assumptions on the interest rates correlations structure.We provide some estimates of the distance between a shift and the vector, named pure shift, having all entries equal. We prove also that in a two-factor framework shift and slope are sufficient to justify the usual properties of correlations between rates.