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Fixed-Income Portfolio Analytics pp 151–192Cite as

Modelling Yield Curves

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Abstract

Yield-curve models fall into two main categories: static and dynamic. Static yield-curve models—addressed in detail in the previous chapter—involve fitting a mathematical function to the yield-tenor relationship. This engineering type exercise—involving relatively little or no economic intuition—requires determination of the parameters of a mathematical function to a collection of bonds at a single point in time.

Essentially, all models are wrong, but some are useful.

Box and Draper 1987

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Notes

  1. 1.

    These two approaches are not independent. To describe the evolution of the yield curve over time, one will need to observe current and historical yield curves. This is turn will involve repeatedly performing the curve-fitting exercise for a large collection of different points in time. Fitted curves are, therefore, an essential input into any dynamic yield-curve model.

  2. 2.

    Typically one requires graduate-level mathematics and statistics to read, understand, and implement the models found in the finance literature.

  3. 3.

    Users also often simultaneously demand for greater model granularity and realism thereby further heightening the complexity. This is part of the natural tension between understandability and usability of any model.

  4. 4.

    Each of these yield curves was initially fit with a kernel regression and then smoothed out with a quartic regression approach. The consequence are quite smooth, but robust estimates of the US Treasury yield curve at each point in time.

  5. 5.

    This characteristic is termed curvature.

  6. 6.

    While arbitrage opportunities do occasionally exist, they typically disappear rapidly. When building a model of yield-curve dynamics, therefore, it is generally a good idea to construct it so that arbitrage is avoided.

  7. 7.

    These are both notions of unconditional volatility. Loosely speaking, this means volatility estimated over a long time period, without reference to the given time period. Volatility may also be measured conditionality, which involves explicitly taking into account current market conditions. When we consider the measurement of risk, we will have much more to say on this idea.

  8. 8.

    Imagine that you purchase a nominal bond and expect inflation to remain about 2 % over its life. If inflation unexpectedly jumps to 5 %, the value of your bond is eroded. The longer the tenor of the bond, the greater the erosion of value. Inflation risk is not the only risk for which bond holders must be compensated. Liquidity and market-specific factors are also important.

  9. 9.

    Whether interest rates revert to a long-term mean is a matter of some debate. Practically and economically, it appears logically obvious, but it is empirically hard to prove. See Ball and Torous [5].

  10. 10.

    For a more technical review of the stylized facts about yields, see Leippold and Wu [60], Fisher [35], or Bolder et al. [10].

  11. 11.

    Temperature, pressure, and entropy are common state variables used in a physical setting.

  12. 12.

    Similarly, the humidity index is a measure that addresses the fact that during the summer it feels hotter when the relative humidity is high. Combining temperature and relative humidity, according to a pre-determined formula, leads to another unobservable weather-related state variable.

  13. 13.

    The reason is simple: more can happen over a longer time horizon than over a shorter period.

  14. 14.

    The first prediction is termed a point estimate, whereas the second prediction includes a confidence interval.

  15. 15.

    This is not always the case, but it is rarely true with time series variables.

  16. 16.

    In some periods, the overall level of interest rates is low, the yield curve is very steep, or short- and long-term rates are very close in value. These situations typically persist for significant periods of time and typically change only gradually.

  17. 17.

    This idea is certainly not unique to yield-curve modelling, it is a foundational idea in the study of stochastic processes.

  18. 18.

    This random part has many names: random noise, shocks, innovations, or diffusion. Each name tries to capture the notion of unpredictability in the movement of our state variables.

  19. 19.

    One embarks on a sequence of mathematical computations and, generally after some heavy lifting, arrives with a potentially very complex mapping. It is also possible to write down choices of state variables and their dynamics for which no arbitrage-free mapping exists.

  20. 20.

    See, for example, Christensen et al. [19], Diebold and Rudebusch [29], Bolder [8] and Bolder and Liu [9].

  21. 21.

    We will not resolve this question in this document. The point, however, is that there are reasonable arguments for each choice of mapping.

  22. 22.

    This chapter will be no exception.

  23. 23.

    PCA was by no means developed for yield-curve applications, but is a powerful and more than 100-year-old technique used by statisticians and engineers for understanding data and reducing dimensionality.

  24. 24.

    See Jolliffe [54] for much more detail on this technique.

  25. 25.

    This result is also very robust: it has been repeatedly reproduced using different markets and time periods.

  26. 26.

    For those who have read the technical description of PCA, these are the eigenvectors associated with the three largest eigenvalues of Ω.

  27. 27.

    It is also occasionally called butterfly. The reason is that looking at the factor loading—and with a bit of imagination—one can see two wings at short and long tenors and body in the intermediate part of the curve.

  28. 28.

    Should we wish to avoid the statistical computations of the PCA factors, we could easily have proxied them with our straightforward linear combinations of observed yields.

  29. 29.

    Using independent regression equations is entirely reasonable given the uncorrelated (i.e., orthogonal) nature of the state variables. If one opted for the linear combination of yields, then a vector auto-regression might be preferable.

  30. 30.

    The reason will become obvious once you see the complexity of the next example.

  31. 31.

    Better measures examine the capacity of the model to fit the observed historical data out of sample. This essentially amounts to predicting future interest-rate outcomes. As the thinking goes, if a model can predict future outcomes relatively well, then it is probably a good model. See Duffee [32] for more on this idea.

  32. 32.

    How is this done? One take the state variables for each period and applies the mapping Eq. (6.17) to it. This provides an estimated yield curve, which is immediately compared to the actual yield curve.

  33. 33.

    This is not to say that it is going to win any awards.

  34. 34.

    Vasicek [78] is a landmark finance paper that thoroughly warrants reading. The Black–Scholes idea was to create a replicating portfolio for an option. Vasicek [78] noticed that a zero-coupon bond is merely a kind of contingent claim (i.e., option) on an interest rate and adjusted the overall framework accordingly.

  35. 35.

    The curious, and masochistic reader, is referred to James and Webber [52], Brigo and Mercurio [14] or Bolder [7] and the excellent references they contain.

  36. 36.

    Like other non-existent theoretical concepts such the notion of perfect competition in microeconomics, it is quite useful.

  37. 37.

    If the instantaneous short rate actually existed, then an investment of K units of currency in a bank account, B, over [t, T] would return,

    $$\displaystyle\begin{array}{rcl} B(t,T) = \mathit{Ke}^{\int _{t}^{T}r(u)\mathit{du}}.& & {}\end{array}$$
    (6.19)

    The cumulative return on one’s bank account is merely the integral of the instantaneous short rate over the period. Conversely, if you wish to determine the discount rate over a given period, one need only use the negative of the integral in Eq. (6.19): \(e^{-\int _{t}^{T}r(u)\mathit{du}}\). This may seem a bit crazy, but it avoids the unwieldy geometric sums typically employed for computing fixed-income returns.

  38. 38.

    An option is a good example; the value of a commodity option on corn, for example, depends importantly on the underlying value of corn.

  39. 39.

    This short expression warrants significant explanation, but it is quite technical. \(\mathbb{Q}\), for example, is termed the equivalent martingale measure induced by using the money-market account as the numeraire asset. The still interested reader is referred to a broad mathematical finance literature. Some good starting points include Panjer et al. [71], Karatzas and Shreve [57], Duffie [31], Neftci [67], Musiela and Rutkowski [66] and Bjork [6].

  40. 40.

    Recall that we need three factors to fully explain the variance of yield-curve movements.

  41. 41.

    See Bolder [8] for the gory details.

  42. 42.

    To complete the story, affine is an old-fashioned mathematical term used to describe a linear function. Given the simple linear form of the mapping in Eq. (6.24), this is the genesis of the name affine model.

  43. 43.

    There is fortunately a burgeoning literature seeking to ease the estimation of these models through the use of linear regressions. See Diez [26] for more details.

  44. 44.

    Typically, the ability to find a global minimum for a non-linear optimization problem depends on the characteristics of one’s objective function. Rarely does the objective function used in the estimation of affine yield-curve models exhibit these characteristics.

  45. 45.

    Again, the interested reader is referred to Bolder [8] for more detail.

  46. 46.

    This is further supported by the fact that the parameters on these two factors—δ 1 and δ 2 from Eq. (6.22)—are similar in magnitude, but have the opposite sign.

  47. 47.

    At some tenors, the estimated curve exactly fits the observed curve across the entire data sample. The estimation algorithm permits the model to exactly fit one tenor for each state variable. With three state variables, this implies that three yield tenors are fitted perfectly.

  48. 48.

    There are many different flavours of affine term-structure models. Dai and Singleton [22, 23] provide, in their seminal papers, a definitive specification of the various types of affine model. Duffie [33], and Duffie et al. [34], and Cox et al. [20, 21] are also excellent references.

  49. 49.

    We’ve been almost criminally brief and imprecise in the development, but hopefully there is enough here to form an idea about the approach.

  50. 50.

    See Hurn et al. [51] for much more detail.

  51. 51.

    It was used extensively by central bankers and even extended by Svensson [77], because it provided a sensible, parsimonious description of the yield curve.

  52. 52.

    λ, given its non-linear form, is typically fixed and forgotten about.

  53. 53.

    By all rights, of course, this should be called the Diebold–Li model for all of their hard work and cleverness. Nevertheless, life is not always fair, and this model is predominately termed the Nelson–Siegel model.

  54. 54.

    These are flexible models used extensively in practice. Judge [55], Harvey [40] and Hamilton [39] are excellent references on this topic (among other things).

  55. 55.

    It is also possible to adjust the Diebold–Li model to preclude arbitrage opportunities. See Diebold and Rudebusch [29] for more details.

References

  1. P.A. Abken, Innovations in the modeling of the term structure of interest rates. Fed. Reserv. Bank Atlanta Econ. Rev. 10, 2–27 (1990)

    Google Scholar 

  2. S.H. Babbs, K.B. Nowman, Kalman filtering of generalized vasicek term structure models. J. Financ. Quant. Anal. 34, 115–130 (1999)

    Article  Google Scholar 

  3. D. Backus, S. Foresi, C. Telmer, Discrete-Time Models of Bond Pricing. Graduate School of Industrial Administration, Carnegie Mellon University (1998)

    Book  Google Scholar 

  4. D. Backus, C. Telmer, L. Wu, Design and Estimation of Affine Yield Models. Graduate School of Industrial Administration, Carnegie Mellon University (1999)

    Google Scholar 

  5. C.A. Ball, W.N. Torous, Unit roots and the estimation of interest rate dynamics. J. Empir. Financ. 3, 215–238 (1996)

    Article  Google Scholar 

  6. T. Björk, Arbitrage Theory in Continuous Time, 1st edn. (Oxford University Press, Oxford, 1998)

    Book  Google Scholar 

  7. D.J. Bolder, Affine term-structure models: theory and implementation. Bank of Canada: Working Paper 2001-15 (2001)

    Google Scholar 

  8. D.J. Bolder, Modelling term-structure dynamics for portfolio analysis: a practitioner’s perspective. Bank of Canada: Working Paper 2006-48 (2006)

    Google Scholar 

  9. D.J. Bolder, S. Liu, Examining simple joint macroeconomic and term-structure models: a practitioner’s perspective. Bank of Canada: Working Paper 2007-49 (2007)

    Google Scholar 

  10. D.J. Bolder, G. Johnson, A. Metzler, An empirical analysis of the canadian term structure of zero-coupon interest rates. Bank of Canada: Working Paper 2004-48 (2004)

    Google Scholar 

  11. G.E.P. Box, N.R. Draper, Empirical Model-Building and Response Surfaces (Wiley, New York, 1987)

    Google Scholar 

  12. A. Brace, D. Gatarek, M. Musiela, The market model of interest rate dynamics. Math. Financ. 7, 127–155 (1997)

    Article  Google Scholar 

  13. M.J. Brennan, E.S. Schwartz, A continuous time approach to the pricing of bonds. J. Bank. Financ. 3, 133–155 (1979)

    Article  Google Scholar 

  14. D. Brigo, F. Mercurio, Interest-Rate Models: Theory and Practice, 1st edn. ( Springer, Berlin, 2001)

    Google Scholar 

  15. K.C. Chan, G.A. Karolyi, F.A. Longstaff, A.B. Sanders, An empirical comparison of alternative models of the short-term interest rate. J. Financ. XLVII, 1209–1227 (1992)

    Google Scholar 

  16. H. Chen, Estimation of a projection-pursuit type regression model. Ann. Stat. 19(1), 142–157 (1991)

    Article  Google Scholar 

  17. R.-R. Chen, A two-factor preference-free model for interest rate sensitive claims equilibrium model of the term structure of interest rates. J. Futur. Mark. 15, 345–372 (1995)

    Article  Google Scholar 

  18. R.-R. Chen, L. Scott, Maximum likelihood estimation for a multifactor equilibrium model of the term structure of interest rates. J. Fixed Income 3, 14–31 (1993)

    Article  Google Scholar 

  19. J.H.E. Christensen, F.X. Diebold, G.D. Rudebusch, The affine arbitrage-free class of Nelson–Siegel term structure models. J. Econ. 164, 4–20 (2011)

    Article  Google Scholar 

  20. J.C. Cox, J.E. Ingersoll, S.A. Ross, An intertemporal general equilibrium model of asset prices. Econometrica 53, 363–384 (1985)

    Article  Google Scholar 

  21. J.C. Cox, J.E. Ingersoll, S.A. Ross, A theory of term structure of interest rates. Econometrica 53, 385–407 (1985)

    Article  Google Scholar 

  22. Q. Dai, K.J. Singleton, Specification analysis of affine term structure models. J. Financ. 55, 1943–1978 (2000)

    Article  Google Scholar 

  23. Q. Dai, K.J. Singleton, Expectation puzzles, time-varying risk premia, and dynamics models of the term structure. J. Financ. Econ. 63, 415–441 (2002)

    Article  Google Scholar 

  24. Q. Dai, A. Le, K.J. Singleton, Discrete-time dynamic term structure models with generalized market prices of risk. University of Stanford Working Paper (2006)

    Google Scholar 

  25. F. de Jong, Time-series and cross-section information in affine term structure models. Department of Financial Management, University of Amsterdam (1998)

    Google Scholar 

  26. A.D. de los Rios, A new linear estimator for gaussian dynamic term structure models. Bank of Canada Working Paper No. 13–10 (2013)

    Google Scholar 

  27. F.X. Diebold, L. Ji, C. Li, A three-factor yield curve model: non-affine structure, systemic risk sources, and generalized duration. University of Pennsylvania Working Paper (2004)

    Google Scholar 

  28. F.X. Diebold, C. Li, Forecasting the term structure of government bond yields. University of Pennsylvania Working Paper (2003)

    Google Scholar 

  29. F.X. Diebold, G.D. Rudebusch, Yield Curve Modeling and Forecasting: The Dynamic Nelson–Siegel Approach (Princeton University Press, Princeton, 2013)

    Book  Google Scholar 

  30. J.-C. Duan, J.-G. Simonato, Estimating and testing exponential-affine term structure models by kalman filter. Technical report, Centre universitaire de recherche et analyze des organizations (CIRANO) (1995)

    Google Scholar 

  31. D. Duffie, Dynamic Asset Pricing Theory, 2nd edn. (Princeton University Press, Princeton, 1996)

    Google Scholar 

  32. G.R. Duffee, Term premia and interest rate forecasts in affine models. J. Financ. 57, 405–443 (2002)

    Article  Google Scholar 

  33. D. Duffie, R. Kan, A yield-factor model of interest rates. Math. Financ. 6, 379–406 (1996)

    Article  Google Scholar 

  34. D. Duffie, D. Filipovic, W. Schachermayer, Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003)

    Article  Google Scholar 

  35. M. Fisher, Forces that shape the yield curve: parts 1 and 2. U.S. Federal Reserve Board Working Paper (2001)

    Google Scholar 

  36. B. Flesaker, L. Hughston, Positive interest. Risk 9, 46–49 (1996)

    Google Scholar 

  37. A.L.J. Geyer, S. Pichler, A state-space approach to estimate and test multifactor cox-ingersoll-ross models of the term structure of interest rates. Department of Operations Research, University of Economics Vienna (1998)

    Google Scholar 

  38. P. Hall, On projection pursuit. Ann. Stat. 19(1), 142–157 (1989)

    Google Scholar 

  39. J.D. Hamilton, Time Series Analysis, Chapter 22 (Princeton University Press, Princeton, 1994)

    Google Scholar 

  40. A.C. Harvey, Forecasting, Structural Time Series Models and the Kalman Filter, 1st edn. (Cambridge University Press, Cambridge, 1990)

    Book  Google Scholar 

  41. D. Heath, R. Jarrow, A. Morton, Bond pricing and the term structure of interest rates: a discrete time approximation. J. Financ. Quant. Anal. 25, 419–440 (1990)

    Article  Google Scholar 

  42. D. Heath, R. Jarrow, A. Morton, Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60, 77–105 (1992)

    Article  Google Scholar 

  43. T.S.Y. Ho, S.-B. Lee, Term structure movements and pricing contingent claims. J. Financ. XLI, 1011–1029 (1986)

    Google Scholar 

  44. P. Honoré, Maximum likelihood estimation of non-linear continuous term structure models. Technical report, The Aarhus School of Business (1995)

    Google Scholar 

  45. P. Hördahl, O. Tristani, D. Vestin, A joint econometric model of macroeconomic and term structure dynamics. European Central Bank Working Paper No. 405 (2004)

    Google Scholar 

  46. P.J. Huber, Projection pursuit. Ann. Stat. 13(2), 435–475 (1985)

    Article  Google Scholar 

  47. J. Hull, A. White, Pricing interest-rate derivative securities. Rev. Financ. Stud. 3, 573–592 (1990)

    Article  Google Scholar 

  48. J. Hull, A. White, One-factor interest rate models and valuation of interest rate derivative securities. J. Financ. Quant. Anal. 28, 235–254 (1993)

    Article  Google Scholar 

  49. J. Hull, A. White, Numerical procedures for implementing term structure models I: single-factor models. J. Deriv. 2, 7–16 (1994)

    Article  Google Scholar 

  50. J. Hull, A. White, Numerical procedures for implementing term structure models II: two-factor models. J. Deriv. 2, 37–48 (1994)

    Article  Google Scholar 

  51. A.S. Hurn, K.A. Lindsay, V. Pavlov, Smooth estimation of yield curves by laguerre functions. Queensland University of Technology Working Paper (2005)

    Google Scholar 

  52. J. James, N. Webber, Interest-Rate Modelling (Wiley, Chichester, 2000)

    Google Scholar 

  53. A. Jeffrey, O. Linton, T. Nguyen, Flexible term structure estimation: which method is preferred. Yale International Centre for Finance: Discussion Paper No. ICF-00-25 (2000)

    Google Scholar 

  54. I.T. Jollliffe, Principal Component Analysis (Springer, New York, 2002)

    Google Scholar 

  55. G.G. Judge, W.E. Griffiths, R.C. Hill, H. Lütkepohl, T.-C. Lee, The Theory and Practice of Econometrics, 2nd edn. (Wiley, New York, 1985)

    Google Scholar 

  56. I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, Berlin, 1991)

    Google Scholar 

  57. I. Karatzas, S.E. Shreve, Methods of Mathematical Finance (Springer, New York, 1998)

    Book  Google Scholar 

  58. T.C. Langetieg, A multivariate model of the term structure. J. Financ. XXXV, 71–97 (1980)

    Google Scholar 

  59. M. Leippold, L. Wu, Quadratic term-structure models. Swiss Institute of Banking and Finance Working Paper (2000)

    Google Scholar 

  60. M. Leippold, L. Wu, Empirical performance of quadratic term structure models. Swiss Institute of Banking and Finance Working Paper (2001)

    Google Scholar 

  61. O. Linton, E. Mammen, J. Nielsen, C. Tanggaard, Estimating yield curves by kernel smoothing methods. Yale International Centre for Finance: Discussion Paper (1999)

    Google Scholar 

  62. R. Litterman, J. Scheinkman, Common factors affecting bond returns. J. Fixed Income 1, 54–61 (1991)

    Article  Google Scholar 

  63. F.A. Longstaff, E.S. Schwartz, Interest rate volatility and the term structure: a two-factor general equilibrium model. J. Financ. XLVII, 1259–1282 (1992)

    Google Scholar 

  64. F.A. Longstaff, E.S. Schwartz, A two-factor interest rate model and contingent claims evaluation. J. Fixed Income 2, 16–23 (1992)

    Article  Google Scholar 

  65. J. Lund, Econometric analysis of continuous-time arbitrage-free models of the term structure of interest rates. Department of Finance, The Aarhus School of Business (1997)

    Google Scholar 

  66. M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling, 1st edn. (Springer, Berlin, 1998)

    Google Scholar 

  67. S.N. Neftci, An Introduction to the Mathematics of Financial Derivatives (Academic Press, San Diego, 1996)

    Google Scholar 

  68. C.R. Nelson, A.F. Siegel, Parsimonious modeling of yield curves. J. Bus. 60, 473–489 (1987)

    Article  Google Scholar 

  69. K.B. Nowman, Gaussian estimation of single-factor continuous time models of the term structure of interest rates. J. Financ. LII, 1695–1706 (1997)

    Google Scholar 

  70. B.K. Oksendal, Stochastic Differential Equations, 4th edn. (Springer, Berlin, 1995)

    Book  Google Scholar 

  71. H.H. Panjer, P.P. Boyle, S.H. Cox, D. Dufresne, H.U. Gerber, H.H. Mueller, H.W. Pedersen, S.R. Pliska, M. Sherris, E.S. Shiu, K.S. Tan, Financial Economics (The Actuarial Foundation, Schaumberg, 1998)

    Google Scholar 

  72. N.D. Pearson, T.-S. Sun, Exploiting the conditional density in estimating the term structure: an application to the cox-ingersoll-ross model. J. Financ. XLIX, 1279–1304 (1994)

    Google Scholar 

  73. S.F. Richard, An arbitrage model of the term structure of interest rates. J. Financ. Econ. 6, 33–57 (1978)

    Article  Google Scholar 

  74. L.C.G. Rogers, Which model for term-structure of interest rates should one use? IMA Vol. Math. Appl. 67, 93–115 (1995)

    Article  Google Scholar 

  75. L.C.G. Rogers, The potential approach to the term structure of interest rates and foreign exchange rates. Math. Financ. 7, 157–176 (1997)

    Article  Google Scholar 

  76. S.M. Schaefer, E.S. Schwartz, A two-factor model of the term structure: an approximate analytical solution. J. Financ. Quant. Anal. 4, 413–424 (1984)

    Article  Google Scholar 

  77. L.E.O. Svensson, Estimating and interpreting forward interest rates: Sweden 1992–1994. International Monetary Fund: Working Paper No. 114 (1994)

    Google Scholar 

  78. O. Vasicek, An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)

    Article  Google Scholar 

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    David Jamieson Bolder

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Bolder, D.J. (2015). Modelling Yield Curves. In: Fixed-Income Portfolio Analytics. Springer, Cham. https://doi.org/10.1007/978-3-319-12667-8_6

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