Isaac Scientific Publishing
Journal of Advances in Economics and Finance
JAEF > Volume 1, Number 1, November 2016

Maximally Smooth Forward Rate Curves for Coupon Bearing Bonds

Download PDF  (830 KB)PP. 28-43,  Pub. Date:December 23, 2016
DOI: 10.22606/jaef.2016.11003

Hussain Abusaaq, Paul M. Beaumont, Yaniv Jerassy-Etzion
Saudi Arabian Monetary Agency, Riyadh, Saudi Arabia; Department of Economics, Florida State Univeristy, Tallahassee, Florida 32306, United States; School of Economics and Business Administration, Ruppin Academic Center, Emek Hefer, Israel
We present a fast and accurate algorithm to compute the maximally smooth instantaneous forward rate curve and the associated spot rate curve for the term structure of interest rates for coupon bearing bonds. The method produces zero pricing errors, constrains initial and terminal conditions of the spot and forward curves and produces the maximally smooth forward rate curve among the class of polynomial spline functions. The algorithm is simple enough to be quickly explained to traders and clients and flexible enough to be easily modified for different types of fixed income security markets. We illustrate the algorithm using on-the-run U.S. Treasury bonds for periods where the yield curve has a normal shape and also when it is inverted.
Term structure of interest rates, yield curve, coupon stripping, curve interpolation, maximally smooth curves
  • [1]  K. Adams and D. van Deventer, “Fitting yield curves and forward rate curves with maximum smoothness,”The Journal of Fixed Income, vol. 4, no. 1, pp. 52–62, 1994.
  • [2]  K. G. Lim and Q. Xiao, “Computing maximum smoothness forward rate curves,” Journal of Statistics and Computing, vol. 12(3), pp. 275–279, 2002.
  • [3]  P. Diament, “Semi-empirical smooth fit to the treasury yield curve,” The Journal of Fixed Income, vol. 3, no.1, pp.55–70, 1993. [Online]. Available:
  • [4]  P. S. Hagan and G. West, “Interpolation methods for curve construction,” Applied Mathematical Finance, vol.13 (2), pp. 89–129, 2006.
  • [5]  K. Adams, “Smooth interpolation of zero curves,” ALGO Research Quarterly, vol. 4, no. 1/2, pp. 11–22, March/June 2001.
  • [6]  F. A. Lutz, “The structure of interest rates,” The Quarterly Journal of Economics, vol. 55 (1), pp. 36–63,1940.
  • [7]  J. M. Culbertson, “The term structure of interest rates,” The Quarterly Journal of Economics, vol. 71, no. 4, pp. 485–517, 1957.
  • [8]  R. E. Lucas Jr., “Asset prices in an exchange economy.” Econometrica, vol. 46, no. 6, pp. 1429–1445, November 1978.
  • [9]  J. C. Cox, J. E. Ingersoll Jr., and S. A. Ross, “A re-examination of traditional hypotheses about the term structure of interest rates,” The Journal of Finance, vol. Vol. 36, No. 4, pp. 769–799, 1981.
  • [10]  J. C. Cox, J. Ingersoll Jr., Jonathan E., and S. A. Ross, “An intertemporal general equilibrium model of asset prices,” Econometrica, vol. 53, no. 2, pp. 363–384, 1985.
  • [11]  O. A. Vasicek, “An equilibrium characterization of the term structure,” Journal of Financial Economics, vol. 40, pp. 319–325, 1977.
  • [12]  M. J. Brennan and E. S. Schwartz, “A continuous-time approach to the pricing of bonds,” Journal of Banking and Finance, vol. 3, pp. 135–155, 1979.
  • [13]  T. C. Langetieg, “A multivariate model of the term structure,” The Journal of Finance, vol. 35, no. 1, pp.71–97, 1980.
  • [14]  T. S. Y. Ho and S.-B. Lee, “Term structure movements and pricing interest rate contingent claims,” The Journal of Finance, vol. 41, no. 5, pp. 1011–1029, 1986.
  • [15]  C. R. Nelson and A. F. Siegel, “Parsimonious modeling of yield curves,” The Journal of Business, vol. 60 (4), pp. 473–489, 1987.
  • [16]  F. A. Longstaff, “Time varying term premia and traditional hypotheses about the term structure,” The Journal of Finance, vol. 45, no. 4, pp. 1307–1314, 1990.
  • [17]  L. E. O. Svensson, “Estimating forward interest rates with the extended nelson & siegel method,” Sveriges Riksbank Quarterly Review, vol. 3, pp. 13–26, 1995.
  • [18]  F. X. Diebold and C. Li, “Forecasting the term structure of government bond yields,” Journal of Econometrics,vol. 130, pp. 337–364, 2006.
  • [19]  D. Heath, R. Jarrow, and A. Morton, “Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation,” Econometrica, vol. 60 (1), pp. 77–107, 1992.
  • [20]  A. Brace and M. Musiela, “A multifactor gauss markov imiplementatin of heath, jarrow, and morton,”Mathematical Finance, vol. 4, no. 3, pp. 259–283, 1994.
  • [21]  Q. Dai and K. J. Singleton, “Specification analysis of affine term structure models,” The Journal of Finance,vol. 55(5), pp. 1943–1978, 2000.
  • [22]  F. Audrino and E. De Giorgi, “Beta regimes for the yield curve,” Journal of Financial Econometrics, vol. 5,no. 3, pp. 456–490, 2007.
  • [23]  J. H. Christensen, F. X. Diebold, and G. D. Rudebusch, “An arbitrage-free generalized nelson–siegel term structure model,” The Econometrics Journal, vol. 12, no. 3, pp. C33–C64, 2009.
  • [24]  M. P. Laurini and L. K. Hotta, “Bayesian extensions to diebold-li term structure model,” International Review of Financial Analysis, vol. 19, no. 5, pp. 342–350, 2010.
  • [25]  J. H. Christensen, F. X. Diebold, and G. D. Rudebusch, “The affine arbitrage-free class of nelson–siegel term structure models,” Journal of Econometrics, vol. 164, no. 1, pp. 4–20, 2011.
  • [26]  C. Hevia, M. Gonzalez-Rozada, M. Sola, and F. Spagnolo, “Estimating and forecasting the yield curve using a markov switching dynamic nelson and siegel model,” Journal of Applied Econometrics, vol. 30, no. 6, pp.987–1009, 2015.
  • [27]  J. H. McCulloch, “The tax-adjusted yield curve,” The Journal of Finance, vol. 30 (3), pp. 811–830, 1975.
  • [28]  O. A. Vasicek and H. G. Fong, “Term structure modeling using exponential splines,” The Journal of Finance, vol. 37 (2), pp. 339–348, 1982.
  • [29]  G. S. Shea, “Interest rate term structure estimation with exponential splines: A note,” The Journal of Finance,vol. 40 (1), pp. 319–325, 1985.
  • [30]  T. S. Coleman, L. Fisher, and R. G. Ibbotson, “Estimating the term structure of interest rates from data that include the prices of coupon bonds,” The Journal of Fixed Income, vol. 2, no. 2, pp. 85–116, 1992. [Online]. Available:
  • [31]  V. Frishling and J. Yamamura, “Fitting a smooth forward rate curve to coupon instruments,”The Journal of Fixed Income, vol. 6, no. 2, pp. 97–103, 1996. [Online]. Available:
  • [32]  S. A. Mansi and J. H. Phillips, “Modeling the term structure from the on-the-run treasury yield curve,” The Journal of Financial Research, vol. 24 (4), pp. 545–564, 2001.
  • [33]  J. V. Jordan and S. A. Mansi, “Term structure estimation from on-the-run treasuries,” Journal of Banking & Finance, vol. 27, 2003.
  • [34]  M. Fisher, “Modeling the term structure of interest rates: An introduction,” Federal Reserve Bank of Atlanta Economic Review, vol. 89, no. 3, pp. 41–62, 2004.
  • [35]  J. Manzano and J. Blomvall, “Positive forward rates in the maximum smoothness framework,” Quantitative Finance, vol. 4, no. 2, pp. 221–232, APR 2004.
  • [36]  R. Gimeno and J. M. Nave, “Genetic algorithm estimation of interest rate term structure,” Banco de Espa?a,Tech. Rep. 0634, Dec. 2006.
  • [37]  F. Fernández-Rodríguez, “Interest rate term structure modeling using free-knot splines,” The Journal of Business, vol. 79, no. 6, pp. 3083–3099, 2006.
  • [38]  B. Jungbacker, S. J. Koopman, and M. Wel, “Smooth dynamic factor analysis with application to the us term structure of interest rates,” Journal of Applied Econometrics, vol. 29, no. 1, pp. 65–90, 2014.
  • [39]  B. G. Turan and A. k. Karagozoglu, “Pricing eurodollar futures options using the edt term structure model:The effects of yield curve smoothing,” The Journal of Futures Markets, vol. 20 (3), pp. 293–306, 2000.
  • [40]  F. E. Benth, S. Koekkebakker, and F. Ollmar, “Extracting and applying smooth forward curves from average-based commodity contracts with seasonal variation,” Journal of Derivatives, vol. 15, no. 1, pp. 52–66, 2007.
  • [41]  M. Ioannides, “A comparison of yield curve estimation techniques using UK data,” Journal of Banking & Finance, vol. 27, no. 1, pp. 1–26, JAN 2003.
  • [42]  P. Veronesi, Fixed Income Securities: Valuation, Risk, and Risk Management. Hoboken, New Jersey: Wiley, 2010.
  • [43]  P. Wilmott, Derivatives: The theory and practice of financial engineering. New York, New York: Wiley, 1998.
  • [44]  R. Chen and K. Du, “A generalised arbitrage-free Nelson-Siegel model: The impact of unspanned stochastic volatility,” Finance Research Letters, vol. 10, no. 1, pp. 41–48, March 2013.
Copyright © 2017 Isaac Scientific Publishing Co. All rights reserved.