Skip to main content
SpringerLink
Log in

An Econometric Model of the Term Structure of Interest Rates Under Regime-Switching Risk

  • Chapter
  • First Online:
Hidden Markov Models in Finance

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 209))

Abstract

This paper develops and estimates a continuous-time model of the term structure of interests under regime shifts. The model uses an analytically simple representation of Markov regime shifts that elucidates the effects of regime shifts on the yield curve and gives a clear interpretation of regime-switching risk premiums. The model falls within the broad class of essentially affine models with a closed form solution of the yield curve, yet it is flexible enough to accommodate priced regime-switching risk, time-varying transition probabilities, regime-dependent mean reversion coefficients as well as stochastic volatilities within each regime. A two-factor version of the model is implemented using Efficient Method of Moments. Empirical results show that the model can account for many salient features of the yield curve in the U.S.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In the context of continuous-time models, Landen [40] also uses a marked point process to represent Markov regime shifts in her model of the term structure of interest rates. However, we use a different construction of the mark space that simplifies the corresponding random measure. Other approaches to regime shifts include Hidden Markov Models (e.g. Elliott et al. [22]) and the Conditional Markov Chain models (e.g. Yin and Zhang [52]). An application of Hidden Markov Models to the term structure of interest rates can be found in Elliott and Mamon [23]. Bielecki and Rutkowski [8, 9] are examples of the application of conditional Markov Chain models to the term structure of interest rates.

  2. 2.

    See Last and Brandt [41] for detailed discussion of marked point process, stochastic intensity kernel and related results.

  3. 3.

    This simply means that m(t, A) −γ m (t, A) is a martingale.

  4. 4.

    This is analogous to the representation of Markov regime shifts as an AR(1) process in discrete-time models.

  5. 5.

    Absence of arbitrage is sufficient for the existence of the pricing kernel under certain technical conditions, as pointed out by Harrison and Kreps [38].

  6. 6.

    It is possible that \(\frac{1} {f} \frac{\partial f} {\partial X^{\prime}}\) depends on X(t) and S(t) as well. Nonetheless in the broad class of affine models, \(\frac{1} {f} \frac{\partial f} {\partial X^{\prime}}\) is a constant that depends only on the bond’s maturity.

  7. 7.

    If ψ 1 is regime-dependent, an analytical solution of the yield curve is in general unavailable. Bansal and Zhou [5] and Wu and Zeng [50] assume that ψ 1 depends on regimes and obtain the term structure of interest rates under log-linear approximation.

  8. 8.

    In order to obtain a closed form solution of the term structure of interest rates, we need to restrict σ 1 to be constant across regimes.

  9. 9.

    In the regime switching model of Bansal and Zhou [5], the risk-neutral mean reversion coefficient is allowed to shift across regimes. But the term structure of interest rates can only be solved analytically under log linear approximation.

  10. 10.

    In the more restrictive CIR models, \([\lambda ^{\prime}_{0,D}(S_{t-}) + X_{t}^{{\prime}}\lambda _{1,D} + X_{t}^{{\prime}}\varTheta _{1}(S)]\) is further restricted to be proportional to the variance of the state variables.

  11. 11.

    In Ang and Bekeart [4], however, the market price of regime-switching risk is not explicitly defined. λ S (u, S, X) can be derived from the specification of the pricing kernel.

  12. 12.

    Of course, A(τ, S) also depends on the factor loading B(τ) through the differential equation (3.18).

  13. 13.

    See Siu [47] for a discussion of the pricing of regime-switching risk in equity market.

  14. 14.

    Hamilton [37] and Filardo [29] are examples of regime-switching models of business cycles with time-varying transitions probabilities. In regime-switching models of interest rates, time-varying transition probabilities are assumed in Gray [35], Boudoukh et al. [10] and more recently Dai et al. [17] among others.

  15. 15.

    To make our study comparable, we consider the roughly same sample period as that in Bansal and Zhou [5].

  16. 16.

    Continuously compounded bond returns are \(H_{t+\varDelta t} \equiv -(\tau -\varDelta t)R_{t+\varDelta t}(\tau -\varDelta t) +\tau R_{t}(\tau )\), where R t (τ) is the yield on a τ-year bond at time t. Since we don’t have data on R t+Δ t (τ −Δ t), we approximate it by R t+Δ t (τ) for τ ≫ Δ t, where Δ t = 1 month. Also note that Corr(E t (H t+Δ t ), BC t ) = Corr(H t+Δ t , BC t ) under rational expectations.

  17. 17.

    Bansal and Zhou [5] and Bansal et al. [7] are excellent examples of applying EMM to estimate the term structure model under regime shifts. Dai and Singleton [14] also provides extensive discussions of estimating affine term structure models using EMM procedure.

  18. 18.

    As for model selection for regime switching models (or general Hidden Markov models), Scott [45] gave an excellent review on limitations of various criteria, including BIC and AIC.

References

  1. Aït-Sahalia, Y., Kimmel, R.: Estimating affine multifactor term structure models using closed-form likelihood expansions. J. Financ. Econ. 98, 113–144 (2010)

    Article  Google Scholar 

  2. Ang, A., Bekaert, G.: Regime switches in interest rates. J. Bus. Econ. Stat. 20(2), 163–182 (2002)

    Article  Google Scholar 

  3. Ang, A., Bekaert, G.: Short rate nonlinearities and regime switches. J. Econ. Dyn. Control 26(7–8), 1243–1274 (2002)

    Article  Google Scholar 

  4. Ang, A., Bekaert, G., Wei, M.: The term structure of real rates and expected inflation. J. Financ. 63, 797–849 (2008)

    Article  Google Scholar 

  5. Bansal, R., Zhou, H.: Term structure of interest rates with regime shifts. J. Financ. 57, 1997–2043 (2002)

    Article  Google Scholar 

  6. Bansal, R., Gallant, R., Tauchen, G.: Nonparametric estimation of structural models for high-frequency currency market data. J. Econom. 66, 251–287 (1995)

    Article  Google Scholar 

  7. Bansal, R., Tauchen, G., Zhou, H.: Regime-shifts, risk premiums in the term structure, and the business cycle. J. Bus. Econ. Stat. 22, 396–409 (2004)

    Article  Google Scholar 

  8. Bielecki, T., Rutkowski, M.: Multiple ratings model of defaultable term structure. Math. Financ. 10, 125–139 (2000)

    Article  Google Scholar 

  9. Bielecki, T., Rutkowski, M.: Modeling of the defaultable term structure: conditional Markov approach. Working paper, The Northeastern Illinois University (2001)

    Google Scholar 

  10. Boudoukh, J., Richardsona, M., Smithb, T., Whitelaw, R.F.: Regime shifts and bond returns. Working paper, Arizona State University (1999)

    Google Scholar 

  11. Cecchetti, S., Lam, P., Mark, N.: The equity premium and the risk-free rate: matching the moments. J. Monet. Econ. 31, 21–45 (1993)

    Article  Google Scholar 

  12. Chauvet, M., Potter, S.: Forecasting recessions using the yield curve. J. Forecast. 24, 77–103 (2005)

    Article  Google Scholar 

  13. Cox, J., Ingersoll, J., Ross, S.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Dai, Q., Singleton, K.: Term structure dynamics in theory and reality. Rev. Financ. Stud. 16, 631–678 (2003)

    Article  Google Scholar 

  17. Dai, Q., Singleton, K., Yang, W.: Regime shifts in a dynamic term structure model of the U.S. treasury bond yields. Rev. Financ. Stud. 20, 1669–1706 (2007)

    Google Scholar 

  18. Duarte, J.: Evaluating alternative risk preferences in affine term structure models. Rev. Financ. Stud. 17, 379–404 (2004)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  21. Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)

    Article  Google Scholar 

  22. Elliott, R.J. et al.: Hidden Markov Models: Estimation and Control. Springer, New York (1995)

    Google Scholar 

  23. Elliott, R.J., Mamon, R.S.: A complete yield curve descriptions of a Markov interest rate model. Int. J. Theor. Appl. Financ. 6, 317–326 (2001)

    Article  Google Scholar 

  24. Elliott R.J., Siu, T.K.: On Markov-modulated exponential-affine bond price formulae. Appl. Math. Financ. 16(1), 1–15 (2009)

    Article  Google Scholar 

  25. Estrella, A., Mishkin, F.S.: Predicting U.S. recessions: financial variables as leading indicators. Rev. Econ. Stat. 80, 45–61 (1998)

    Google Scholar 

  26. Evans, M.: Real risk, inflation risk, and the term structure. Econ. J. 113, 345–389 (2003)

    Article  Google Scholar 

  27. Fama, E.F., French, K.R.: Business conditions and expected returns on stocks and bonds. J. Financ. Econ. 25, 23–49 (1989)

    Article  Google Scholar 

  28. Ferland, R., Gauthier, G., Lalancette, S.: A regime-switching term structure model with observable state variables. Financ. Res. Lett. 7, 103–109 (2010)

    Article  Google Scholar 

  29. Filardo, A.: Business cycle phases and their transition dynamics. J. Bus. Econ. Stat. 12, 299–308 (1994)

    Google Scholar 

  30. Futami, H.: Regime switching term structure model under partial information. Int. J. Theor. Appl. Financ. 14, 265–294 (2011)

    Article  Google Scholar 

  31. Gallant, R., Tauchen, G.: Which moment to match? Econom. Theory 12, 657–681 (1996)

    Article  Google Scholar 

  32. Gallant, R., Tauchen, G.: Reprojecting partially observed system with application to interest rate diffusions. J. Am. Stat. Assoc. 93, 10–24 (1998)

    Article  Google Scholar 

  33. Gallant, R., Tauchen, G.: Efficient method of moments. Working paper, University of North Carolina (2001)

    Google Scholar 

  34. Garcia, R., Perron, P.: An analysis of the real interest rate under regime shifts. Rev. Econ. Stat. 78(1), 111–125 (1996)

    Article  Google Scholar 

  35. Gray, S.: Modeling the conditional distribution of interest rates as a regime-switching process. J. Financ. Econ. 42, 27–62 (1996)

    Article  Google Scholar 

  36. Hamilton, J.D.: Rational expectations econometric analysis of changes in regimes: an investigation of the term structure of interest rates. J. Econ. Dyn. Control 12, 385–423 (1988)

    Article  Google Scholar 

  37. Hamilton, J.D.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384 (1989)

    Article  Google Scholar 

  38. Harrison, M., Kreps, D.: Martingales and arbitrage in multiperiod security markets. J. Econ. Theory 20, 381–408 (1979)

    Article  Google Scholar 

  39. He, W., Wang, J.G., Yan, J.A.: Semimartingale theory and stochastic calculus. Science Press, Beijing/CRC, Boca Raton (1992)

    Google Scholar 

  40. Landen, C.: Bond pricing in a hidden Markov model of the short rate. Financ. Stoch. 4, 371–389 (2000)

    Article  Google Scholar 

  41. Last, G., Brandt, A.: Marked point processes on the real line. Springer, New York (1995)

    Google Scholar 

  42. Le, A., Dai, Q., Singleton, K.: Discrete-time dynamic term structure models with generalized market prices of risk. Rev. Financ. Stud. 23, 2184–2227 (2010)

    Article  Google Scholar 

  43. Lewis, K.K.: Was there a ‘Peso problem’ in the US term structure of interest rates: 1979–1982? Int. Econ. Rev. 32, 159–173 (1991)

    Article  Google Scholar 

  44. Protter, P.: Stochastic intergration and differential equations, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  45. Scott, S.: Bayesian methods for hidden Markov models: recursive computing in the 21st Century. J. Am. Stat. Assoc. 97, 337–351 (2002)

    Article  Google Scholar 

  46. Siu, T.K.: Bond pricing under a Markovian regime-switching jump-augmented Vasicek model via stochastic flows. Appl. Math. Comput. 216(11), 3184–3190 (2010)

    Article  Google Scholar 

  47. Siu, T.K.: Regime switching risk: to price or not to price? Int. J. Stoch. Anal. 2011, 1–14 (2011)

    Article  Google Scholar 

  48. Sola, M., Driffill, J.: Testing the term structure of interest rates using a vector autoregression with regime switching. J. Econ. Dyn. Control 18, 601–628 (1994)

    Article  Google Scholar 

  49. Timmermann, A.: Moments of Markov switching models. J. Econom. 96, 75–111 (2000)

    Article  Google Scholar 

  50. Wu, S., Zeng, Y.: A general equilibrium model of the term structure of interest rates under regime-siwthcing risk. Int. J. Theor. Appl. Financ. 8, 839–869 (2005)

    Article  Google Scholar 

  51. Xiang, J., Zhu, X.: A regime-switching NelsonCSiegel term structure model and interest rate forecasts. J. Financ. Econom. 11, 522–555 (2013)

    Article  Google Scholar 

  52. Yin, G.G., Zhang, Q.: Continuous-time Markov chains and applications: a singular perturbation approach. Springer, Berlin (1998)

    Book  Google Scholar 

Download references

Acknowledgements

We would like to thank seminar participants at Federal Reserve Bank of Kansas City, North American Summer Meetings of Econometric Society and Missouri Valley Economics Association Annual Meetings for helpful comments. A part of the research is done while the first author is visiting Federal Reserve Bank of Kansas City. He is grateful for their hospitalities. The second author gratefully acknowledges financial support of National Science Foundation under DMS-1228244 and University Missouri Research Board.

Author information

Authors and Affiliations

  1. Department of Economics, University of Kansas, 1450 Jayhawk Blvd, Lawrence, KS, 66045, USA

    Shu Wu

  2. Department of Mathematics and Statistics, University of Missouri at Kansas City, 5100 Rockhill Rd, Kansas City, MO, 64110, USA

    Yong Zeng

Authors
  1. Shu Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shu Wu .

Editor information

Editors and Affiliations

  1. Dept of Statistical & Actuarial Sciences, University of Western Ontario, London, Ontario, Canada

    Rogemar S. Mamon

  2. School of Mathematics, University of Adelaide, Adelaide, Australia

    Robert J. Elliott

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Wu, S., Zeng, Y. (2014). An Econometric Model of the Term Structure of Interest Rates Under Regime-Switching Risk. In: Mamon, R., Elliott, R. (eds) Hidden Markov Models in Finance. International Series in Operations Research & Management Science, vol 209. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7442-6_3

Download citation

Publish with us

Policies and ethics

Search

Navigation