Stability of Descriptive Models for the Term Structure of Interest Rates with Application to German Market Data

This paper discusses the use of parametric models for the term structure of interest rates and their uses. The paper focuses initially on a potential problem which arises out of the use of certain models. In most cases the process of parameter estimation involves the minimization or maximization of a function (for example, least squares or maximum likelihood). In some cases this function can have a global minimum/maximum plus one or more local minima/maxima. As we progress through time this leads to a process under which parameter estimates and the fitted term structure can jump about in a way which is inconsistent with bond-price changes. Here a number of models are identified as susceptible to this sort of problem. However, under one of the descriptive models (the restricted-exponential model) it is proved that the likelihood and Bayesian posterior functions have unique maxima: both in a zero-coupon bond market and in a low-coupon bond market. A counterexample shows that this result can break down for larger-coupon bond markets. An alternative Bayesian estimator in combination with the restricted-exponential model is shown to be free from the problem of catastrophic jumps in all coupon-bond markets. This model has previously been applied to UK data (Cairns, 1998). Here, we consider its wider application in European bond markets. In particular, German market data from 1996 and 1997 is analysed using the restricted-exponential model. We find that the model gives a good description of the German market during this period.