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Fitting the term structure of interest rates The term structure of interest rates defines the set of spot or zero-coupon rates that exist in a debt capital market, of default-free bonds, distinguished only by their term to maturity. In practice the term structure is defined as the array of discount factors on the same maturity term. Extracting the term structure from market interest rates has been the focus of extensive research, reflecting its importance in the field of finance. The term structure is used by market practitioners for valuation purposes and by central banks for forecasting purposes. The accurate fitting of the term structure is vital to the smooth functioning of the market. A number of approaches have been proposed with which to undertake this, and the method chosen is governed by the user’s requirements. Practitioners desire an approach that is accessible, straightforward to implement and as accurate as possible. In general there are two classes of curve fitting techniques; the parametric methods, so-called because they attempt to model the yield curve using a parametric function; and the spline methods.1 Parametric methods include the Nelson-Siegel model and a modification of this proposed by Svensson (1994, 1995), as well as models described by Wiseman (1994) and Bjork and Christensen (1997).2 James and Webber (2000) suggest that these methods produce a satisfactory overall shape for the term structure but are suitable only where good accuracy is not required.3 Market practitioners instead generally prefer an approach that gives a reasonable trade-off between accuracy and ease of implementation, an issue we explore in this article. The cubic spline process presents no conceptual problems, and is an approximation of the market discount function. McCulloch (1975) uses cubic splines and Beim (1992) states that this approach performs at least as satisfactorily as other methods.4 Although the basic approach can lead to unrealistic shapes for the forward curve (for example, see Vasicek and Fong (1982) and their suggested improvement on the approach using exponential splines), it is an accessible method and one that gives reasonable accuracy for the spot rate curve. Adams and Van Deventer (1994) illustrate using the technique to obtain maximum smoothness for forward curves (and an extension to quartic splines), while the basic technique has been improved as described by Fisher, Nychka and Zervos (1995), Waggoner (1997) and Anderson and Sleath (1999). These references are considered later. Splines are a non-parametric polynomial interpolation method.5 There is more than one way of fitting them. The simplest method is an ordinary least squares regression spline, but this approach produces wildly oscillating curves. The more satisfactory is a smoothing splines method. We consider the basic approach and how to implement it in this article.

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