Yield Curve Modelling at the Bank of Canada
The primary objective of this paper is to produce a framework that could be used to construct a historical data base of zero-coupon and forward yield curves estimated from Government of Canada securities' prices. The secondary objective is to better understand the behaviour of a class of parametric yield curve models, specifically, the Nelson-Siegel and the Svensson methodologies. These models specify a functional form for the instantaneous forward interest rate, and the user must determine the function parameters that are consistent with market prices for government debt. The results of these models are compared with those of a yield curve model used by the Bank of Canada for the last 15 years. The Bank of Canada's existing model, based on an approach developed by Bell Canada, fits a so-called "par yield" curve to bond yields to maturity and subsequently extracts zero-coupon and "implied forward" rates. Given the pragmatic objectives of this research, the analysis focuses on the practical and deals with two key problems: the estimation problem (the choice of the best yield curve model and the optimization of its parameters); and the data problem (the selection of the appropriate set of market data). In the absence of a developed literature dealing with the practical side of parametric term structure estimation, this paper provides some guidance for those wishing to use parametric models under "real world" constraints.
In the analysis of the estimation problem, the data filtering criteria are held constant (this is the "benchmark" case). Three separate models, two alternative specifications of the objective function, and two global search algorithms are examined. Each of these nine alternatives is summarized in terms of goodness of fit, speed of estimation, and robustness of the results. The best alternative is the Svensson model using a price-error-based, log-likelihood objective function and a global search algorithm that estimates subsets of parameters in stages. This estimation approach is used to consider the data problem. The authors look at a number of alternative data filtering settings, which include a more severe or "tight" setting and an examination of the use of bonds and/or treasury bills to model the short-end of the term structure. Once again, the goodness of fit, robustness, and speed of estimation are used to compare these different filtering possibilities. In the final analysis, it is decided that the benchmark filtering setting offers the most balanced approach to the selection of data for the estimation of the term structure.
This work improves the understanding of this class of parametric models and will be used for the development of a historical data base of estimated term structures. In particular, a number of concerns about these models have been resolved by this analysis. For example, the authors believe that the log-likelihood specification of the objective function is an efficient approach to solving the estimation problem. In addition, the benchmark data filtering case performs well relative to other possible filtering scenarios. Indeed, this parametric class of models appears to be less sensitive to the data filtering than initially believed. However, some questions remain; specifically, the estimation algorithms could be improved. The authors are concerned that they do not consider enough of the domain of the objective function to determine the optimal set of starting parameters. Finally, although it was decided to employ the Svensson model, there are other functional forms that could be more stable or better describe the underlying data. These two remaining questions suggest that there are certainly more research issues to be explored in this area.