A one factor spot rate model for the New Zealand term structure of interest rates

Abstract

The term structure of interest rates describes the rates available now at which money can be borrowed or lent over future time intervals. In a one factor model, a single point on the term structure is modelled, generally the instantaneous spot rate. To date, a number of one factor models have been proposed in the literature. Some results for the model proposed by Vasicek (1977) are developed, and the model proposed by Cox, Ingersoll & Ross (1985) is briefly examined. Under certain assumptions, bond and interest rate contingent claim prices solve a second order partial differential equation. This equation can be solved, at least numerically, given the continuous and terminal payoffs from a contingent claim and the form of the instantaneous mean and standard deviation of the one factor model. A robust non-parametric estimation procedure, that uses the loess filter, is developed to estimate the instantaneous mean and variance terms of a one factor model. Estimates for simulated data are found to be consistent with theoretical values. A proxy for the instantaneous spot rate for New Zealand is obtained from the 90 day bank bill series, and the estimates obtained from this proxy motivate the introduction of two new parametric models. One of these models comprises a piecewise combination of models that either exist or are extensions of existing models. Estimates from different time intervals also suggest that the instantaneous variance function changes character over time. Maximum likelihood estimates are calculated for the parameters of the new models, in addition to those of already established models, and all models are compared using the Akaike information criterion. The new models are found to out-perform all others under this criterion. Leptokurtic errors were produced from all fitted models, which motivated a departure from one factor models. Modelling the daily changes in the instantaneous spot rate by a mixed Gaussian distribution leads to a considerable improvement in the likelihood. Using the new specification for the error for all models, the new models are again found to be superior.