Thanks to the very active feedback on Parts 1 to 4 of our series on Nelson-Siegel yield curve smoothing versus spline technologies, we took another tour through the most recent literature on the subject. We’ve found a gem of a summary by Jeffrey R. Greco of the University of Chicago that we reproduce in full with Jeff’s permission in this post. We have also found a large number of articles on why technique x for yield curve smoothing is better than technique y, with no mathematical proof of why one is better than the other. We show how to do that in this post.
Judging the “best” yield curve smoothing technique between two alternatives is not like evaluating a ballet performance or the finalists for Miss Universe or Mr. Universe. Determining the better of two alternative approaches without articulating what “best” means is a waste of ink and paper. In the original Adams and van Deventer paper [1994, as corrected in van Deventer and Imai, 1996], the definition of smoothness was defined mathematically and the resultant best smoothing approach (for forward rates) was derived, not argued, using a proof by Oldrich Vasicek. Jeffrey R. Greco’s overview of smoothing, reproduced as Appendix A, discusses how to do this for both continuous and discrete definitions of smoothness.
When the data contains obvious errors or inconsistencies so that a perfect fit to the data is either impossible or undesirable, Prof. Greco’s summary explains how to define a mathematical function that trades off smoothness of the forward rate curve for errors in pricing. This approach can be used for any definition of “best,” such as the following:
- Smoothest continuous forward rates given an arbitrary constraint on the left hand side (shortest maturity) or right hand side (longest maturity) of the yield curve
- Smoothest discrete forward rates
- Smoothest continuous zero coupon yields
- Smoothest discrete zero yields
- Smoothest continuous zero coupon bond prices
- Smoothest discrete zero coupon bond prices
- Minimal pricing error (often this criterion does not produce a unique answer, especially if there are bid-offered spreads or errors in the data)
- Combination of minimal pricing error and one of the other criterions, as Professor Greco illustrates in Appendix A.
Given the criterion for “best,” there can be no debate about whether Method A is better than Method B, unless the criterion does not provide for a unique solution. We know from Adams and van Deventer (1994), as corrected in van Deventer and Imai (1996), that there is a unique forward curve that is maximum smoothness for any given set of constraints on the left hand side and right hand side of the curve. ANY alternative, say Method X, subjected to the same constraints, will be less smooth, and this fact can be confirmed by calculating the smoothness for Method X and the maximum smoothness curve. Method X will have a smoothness statistic that is less smooth.
For any other definition of “best,” the procedure is the same. Derive the “best” curve by that criterion. Then examine the mathematical value of the criterion for Method Y and confirm that method Y is inferior by the given definition of best.
Any other form of argument about whether Method A or Method B is better is the financial equivalent of judging the Miss Universe or Mr. Universe contest. We can do better than that.
Professor Jeffrey R. Greco demonstrates exactly how to do this in his lecture notes from his class in fixed income derivatives from the Financial Mathematics Program at the University of Chicago. In my view, this is the best exposition of the principals behind yield curve smoothing in the literature, and we believe Prof. Greco’s notes should be studied by all serious yield curve analysts. Prof. Greco’s notes are displayed with his permission in Appendix A. Appendix B contains references from our first post on Nelson-Siegel versus spline methods of yield curve smoothing.
Donald R. van Deventer
Honolulu, September 8, 2009
Click here for the full text of Professor Greco’s lecture notes on yield curve smoothing
Professor Jeffrey R. Greco
Lecture Notes on Fixed Income Derivatives
Financial Mathematics Program
University of Chicago
Key References on Yield Curve Smoothing
Kenneth J. Adams and Donald R. van Deventer, 1994, Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness, The Journal of Fixed Income, June 1994, 52-62.
Bank for International Settlements, Monetary and Economic Department, “Zero Coupon Yield Curves: Technical Documentation,” 2005.
Mark Buono, Russell B. Gregory-Allen, and Uzi Yaari, 1992, The Efficacy of Term Structure Estimation Techniques: A Monte Carlo Study, The Journal of Fixed Income 1, 52-59.
Damir Filipovic, “A Note on the Nelson-Siegel Family,” Mathematical Finance, October, 1999, pp. 349-359.
F. B. Hildebrand, 1987, Introduction to Numerical Analysis (Dover Publications Inc., New York).
J. Huston McCulloch, 1975, The Tax Adjusted Yield Curve, Journal of Finance 30, 811-29.
Charles R. Nelson and Andrew F. Siegel, “Parsimonious Modeling of Yield Curves” The Journal of Business, Vol. 60, No. 4. (Oct., 1987), pp. 473-489.
P. M. Penter, 1989, Splines and Variational Methods (John Wiley & Sons, New York).
H. R. Schwartz, 1989, Numerical Methods: A Comprehensive Introduction (John Wiley & Sons, New York).
Gary S. Shea, 1985, Term Structure Estimation with Exponential Splines, Journal of Finance 40, 319-325.
Donald R. van Deventer and Kenji Imai, Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance, and Investment Management, Irwin Professional Publishing, Chicago, 1997.
Donald R. van Deventer, Kenji Imai, and Mark Mesler, Advanced Financial Risk Management, John Wiley & Sons, 2004. Translated into modern Chinese and published by China Renmin University Press, Beijing, 2007. See especially chapters 8 and 18.
Oldrich A. Vasicek and H. Gifford Fong, 1982, Term Structure Modeling Using Exponential Splines, Journal of Finance 37, 339-56.