Smooth estimation of yield curves by Laguerre functions

This paper is concerned with the estimation of the nominal yield curve by means of two complementary approaches. One approach models the yield curve directly while the other focuses on a model of the forward rate from which a description of the yield curve may be developed by integration of the forward rate specification. This latter approach may be broadly interpreted as a generalisation of the widely used parametric functional form proposed by Nelson and Siegel (1987) for yield curve estimation. Nelson and Siegel describe the yield curve in terms of three linear factors commonly referred to as the level factor, the slope factor and the curvature factor, together with a nonlinear factor, say λ, which represents a time-scale. In a recent paper, Diebold and Li (2005) use the Nelson-Siegel method to fit the yield curve for US bonds. In their application, the value of λ (the parameter representing the time-scale) is fixed, leaving only the level, slope and curvature factors with which to capture the behaviour of US yields. Although this approach has the advantage of simplicity in terms of implementation, it is likely to be suboptimal if the fit of the yield curve is sensitive to the choice of the time-scale parameter. In this article, a limited empirical calibration exercise is performed using gilts yield curves published by the Bank of England. The data set comprises weekly yield curves for maturities from 1 to 19 years at 6monthly intervals for the time period January 1985 to December 2004. In this exercise, the 3-factor NelsonSiegel form with λ assigned the value suggested by Diebold and Li on theoretical grounds proved inadequate to capture the variation in the shape of the UK yield curve. In fact, there was no value of λ for which the Nelson-Siegel specification provided an adequate fit to the shape of the UK yield curve, a finding which suggests that this model may not be flexible enough to describe commonly occurring patterns in observed yields. This article presents two possible ways in which the Nelson-Siegel model for fitting yield curves may be generalised. These generalisations are based on the Fourier-Laguerre representation of a continuous function in (0,∞), and enjoy the advantage that they retain the overall structure of the Nelson-Siegel model and its ease of implementation. Both models describe the yield curve as a sum of linear factors multiplying nonlinear functions of maturity based on Laguerre functions, all of which use a common value for λ. Just as for the Nelson-Siegel model of yields, these generalised models may be implemented within a simple least-squares framework. Both the yield-based and forward rate based models are applied to the UK Gilts data for various numbers of factors. Both 3-factor models exhibit the same poor quality of fit as that experienced by the NelsonSiegel model. Both variants of the generalised 4factor model performed equally well when fitted to UK yield curve, and both proved superior to the fit of the Nelson-Siegel model. Rather interestingly, the move to five factors did not significantly improve the quality of the fit. A matter of concern when using the Nelson-Siegel model to fit yield curves is the sensitivity of the fit to the value of the time-scale parameter λ. The value of λ used in the Diebold and Li (2005) investigation of the yield curve for US bonds was unsuitable for the UK yield curve. In particular, the sensitivity of the Nelson-Siegel model to the value of λ was such that the factors of the model could not compensate for an inappropriate choice of λ. On the other hand, the improved fit to the UK yield curve achieved by the generalised models with four factors is, in addition, achieved with less sensitivity to the value of λ. This sensitivity to the value of λ is further reduced with five factors although, as commented previously, the quality of the fit is not improved significantly. Finally, although the optimal fit achieved by the generalised model of yields is indistinguishable from that based on the forward rate, the sensitivity of this fit to the value of λ is greater for the former than the latter. It is conjectured that this difference stems from the different behaviour of both models at long maturities, and suggests that the model based on forward rates is to be preferred.