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Duration Concepts, Analysis, and Applications

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Encyclopedia of Finance

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Abstract

We discuss duration and its development, placing particular emphasis on various applications. The survey begins by introducing duration and showing how traders and portfolio managers use this measure in speculative and hedging strategies. We then turn to convexity, a complication arising from relaxing the linearity assumption in duration. Next, we present immunization – a hedging strategy based on duration – and then examine stochastic process risk, foreign-exchange risk, and duration extensions that address these risks. We also examine the track record of duration and how the measure applies to financial futures. The discussion then turns to macrohedging the entire balance sheet of a financial institution. We develop a theoretical framework for duration gaps and apply it, in turn, to banks, life insurance companies, and defined benefit pension plans.

For The Encyclopedia of Finance, C.F. Lee, editor, Third Edition, Kluwer Academic Publishers.

Gordon Roberts: deceased.

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Notes

  1. 1.

    Bierwag (1977, 1987a) provide excellent reviews of duration analysis.

  2. 2.

    For more on market risk see J.P. Morgan (1994) and Cornett and Saunders (2008), Chap. 9.

  3. 3.

    This discussion of immunization begins by assuming default and option free securities in order to separate interest rate risk from other risks.

  4. 4.

    Bierwag and Kaufman (1977) maintain that, for the duration matched portfolios, these two effects (unexpected gains and losses resulting from interest rate shifts) cancel each other out, unless the stochastic process is not consistent with the equilibrium conditions, in which case the unexpected gain will be greater than the unexpected loss.

  5. 5.

    For more details on contingent immunization, see Bierwag (1987a) and Leibowitz and Weinberger (1981, 1982, 1983).

  6. 6.

    Detailed discussion of these modelling issues is in Bierwag (1977, 1987a).

  7. 7.

    For simplicity, the derivation assumes away any difference between rates of return for assets and liabilities, rA and rL. This may not be strictly true because interest earned on assets is higher than interest paid on liabilities. However, the essence of the argument is not affected by this assumption. In practice, duration gap implementation uses the average of the rates on assets and liabilities.

  8. 8.

    Following prior research, we choose the change in the market value of equity (E) as the target because maximizing equity value is most likely to be the goal of management and shareholders. Another possible target is E/A (the capital ratio) particularly in the case in which E/A is at the regulatory minimum of 8% and management wishes to immunize against any fall in the ratio. In practice, a financial institution likely targets equity along with other variables. Further discussion of this issue is in Kaufman (1984) and Bierwag and Kaufman (1996).

  9. 9.

    Some readers may be familiar with the concept of funding gap. Funding gap is defined as rate sensitive assets minus rate-sensitive liabilities. It differs from duration gap in two fundamental respects. First, funding gap relates interest rate shifts to the book value of net income; duration gap relates rate shifts to the market value of equity. Second, funding gap ignores the repricing of long-term assets when rates change. Because of these differences, a positive funding gap corresponds to a negative duration gap.

  10. 10.

    More generally, academic research supports the view that FI shares move in the direction expected by the theory of duration gaps. For an example, see Flannery and James (1984).

  11. 11.

    Early researchers on duration (Macaulay 1938; Redington 1952, for example) were actuaries who published their results in actuarial journals.

  12. 12.

    According to the U.S. Bureau of Labor Statistics, in 2019 only 16% of private-sector employees were enrolled in a defined benefit plan (see: www.bls.gov/ncs/ebs/factsheet/defined-benefit-frozen-plans.pdf)

  13. 13.

    In case of deflation, the principal amount cannot be indexed to a level lower than its unadjusted principal.

  14. 14.

    Note that the inflation-indexation schemes will affect an instrument’s value and elasticity. Fooladi et al. (2021) demonstrate that, when imperfect indexation implies under (over) protection against inflation, the instrument will have a positive (negative) expected-inflation duration, and its value will be negatively (positively) affected by inflation.

References

  • Acharya, V.V., and J.N. Carpenter. 2002. Corporate bond valuation and hedging with stochastic interest rates and endogenous bankruptcy. Review of Financial Studies 15: 1355–1383.

    Article  Google Scholar 

  • Afik, Z., G. Jacoby, and Z. Wiener. 2018. Duration and globalization. The Journal of Fixed Income 28 (2): 31–43.

    Article  Google Scholar 

  • Balbás, A., and A. Ibánez. 1998. When can you immunize a bond portfolio? Journal of Banking & Finance 22: 1571–1595.

    Article  Google Scholar 

  • Balbás, A., A. Ibánez, and S. López. 2002. Dispersion measures as immunization risk measures. Journal of Banking & Finance 26: 1229–1244.

    Article  Google Scholar 

  • Bertocchi, M., R. Giacometti, and S.A. Zenios. 2005. Risk factor analysis and portfolio immunization in the corporate bond market. European Journal of Operational Research 161: 348–363.

    Article  Google Scholar 

  • Bierwag, G.O. 1977. Immunization, duration, and the term structure of interest rates. Journal of Financial and Quantitative Analysis 12 (5): 725–741.

    Article  Google Scholar 

  • ———. 1987a. Duration analysis, managing interest rate risk. Cambridge, MA: Ballinger Publishing.

    Google Scholar 

  • ———. 1987b. Bond returns, discrete stochastic processes, and duration. Journal of Financial Research 10 (3): 191–209.

    Article  Google Scholar 

  • Bierwag, G.O., and I. Fooladi. 2006. Duration analysis: An historical perspective. Journal of Applied Finance 16 (2): 144–160.

    Google Scholar 

  • Bierwag, G.O., and G.G. Kaufman. 1977. Coping with the risk of interest-rate fluctuations: A note. Journal of Business 50 (3): 364–370.

    Article  Google Scholar 

  • ———. 1988. Duration of non-default free securities. Financial Analysts Journal 44 (4): 39–46.

    Article  Google Scholar 

  • ———. 1996. Managing interest rate risk with duration gaps to achieve multiple targets. Journal of Financial Engineering 5: 53–73.

    Google Scholar 

  • Bierwag, G.O., and C. Khang. 1979. An immunization strategy is a minimax strategy. Journal of Finance 34 (2): 389–399.

    Article  Google Scholar 

  • Bierwag, G.O., and G.S. Roberts. 1990. Single factor duration models: Canadian tests. Journal of Financial Research 13: 23–38.

    Article  Google Scholar 

  • Bierwag, G.O., G.G. Kaufman, R. Schweitzer, and A. Toevs. 1981. The art of risk management in bond portfolios. Journal of Portfolio Management 7: 27–36.

    Article  Google Scholar 

  • Bierwag, G.O., G.G. Kaufman, and A. Toevs. 1982a. Single factor duration models in a discrete general equilibrium framework. Journal of Finance 37: 325–338.

    Article  Google Scholar 

  • ———. (1982b). Empirical tests of alternative single factor duration models. Paper presented at Western Finance Association, Portland, Oregon (June).

    Google Scholar 

  • ———. 1983. Immunization strategies for funding multiple liabilities. Journal of Financial and Quantitative Analysis 18: 113–124.

    Article  Google Scholar 

  • Bierwag, G.O., G.G. Kaufman, and C. Latta. 1987. Bond portfolio immunization: Tests of maturity, one- and two-factor duration matching strategies. Financial Review 22 (2): 203–219.

    Article  Google Scholar 

  • Bierwag, G.O., I. Fooladi, and G. Roberts. 1993a. Designing and immunization portfolio: Is M-squared the key? Journal of Banking & Finance 17 (6): 1147–1170.

    Article  Google Scholar 

  • ———. 1993b. Designing and immunization portfolio: Is M-squared the key? Journal of Banking & Finance 17: 1147–1170.

    Article  Google Scholar 

  • ———. 2000. Risk management with duration: Potential and limitations. Canadian Journal of Administrative Sciences 17 (2): 126–142.

    Article  Google Scholar 

  • Bodie, Z. 1996. What the pension benefit guaranty corporation can learn from the Federal Savings and Loan Insurance Corporation. Journal of Financial Services Research 10: 83–100.

    Article  Google Scholar 

  • Bravo, J.M., and C.M.P. Da Silva. 2006. Immunization using a stochastic-process independent multi-factor model: The Portuguese experience. Journal of Banking & Finance 30: 133–156.

    Article  Google Scholar 

  • Brennan, M.J., and E.S. Schwartz. 1983. Duration, bond pricing, and portfolio management, in innovations in bond portfolio management, duration analysis and immunization. In Proceedings from the Ashland conference, 1981, ed. G.G. Kaufman, G.O. Bierwag, and A. Toevs. Greenwich: JAI Press.

    Google Scholar 

  • Chambers, D.R., W.T. Carleton, and R.W. McEnally. 1988. Immunizing default-free bond portfolios with a duration vector. Journal of Financial and Quantitative Analysis 23 (1): 89–104.

    Article  Google Scholar 

  • Chance, D.M. 1990. Default risk and the duration of zero coupon bonds. Journal of Finance 45 (1): 265–274.

    Article  Google Scholar 

  • Chow, E.H., W.Y. Lee, and M.E. Solt. 1997. The exchange-rate risk exposure of asset returns. Journal of Business 70 (1): 105–123.

    Article  Google Scholar 

  • Cornett, M.M., and A. Saunders. 2008. Financial institutions management. 6th ed. Boston: Irwin McGraw Hill, Chapters 8, 9 and 18.

    Google Scholar 

  • Diaz, A., M.O. González, El Navarro, and F.S. Skinner. 2009. An evaluation of contingent immunization. Journal of Banking & Finance 33: 1874–1883.

    Article  Google Scholar 

  • Dym, S.I. 1991. Measuring the risk of foreign bonds. Journal of Portfolio Management 17 (2): 56–61.

    Article  Google Scholar 

  • Dym, S. 1992. Global and local components of foreign bond risk. Financial Analysts Journal 48 (2): 83–91.

    Article  Google Scholar 

  • Fisher, L., and R.L. Weil. 1971. Coping with the risk of market-rate fluctuations: Returns to bondholders from naïve and optimal strategies. Journal of Business 44 (4): 408–431.

    Article  Google Scholar 

  • Flannery, M.J., and C.M. James. 1984. The effect of interest rate changes on the common stock returns of financial institutions. Journal of Finance 39: 1141–1153.

    Article  Google Scholar 

  • Fong, H.G., and O. Vasicek. 1983. Return maximization for immunized portfolios, in innovations in bond portfolio management, duration analysis and immunization. In The Ashland conference, 1981, ed. G.G. Kaufman, G.O. Bierwag, and A. Toevs. Greenwich: JAI Press.

    Google Scholar 

  • Fong, H.G., and O.A. Vasicek. 1984. A risk minimizing strategy for portfolio immunization. Journal of Finance 39 (5): 1541–1546.

    Article  Google Scholar 

  • Fooladi, I., and G. Roberts. 1992. Portfolio immunization: Canadian tests. Journal of Economics and Business 44 (1): 3–17.

    Article  Google Scholar 

  • ———. 2004. Macrohedging for financial institutions: Beyond duration. Journal of Applied Finance 14 (1): 11–19.

    Google Scholar 

  • Fooladi, I., G. Roberts, and F. Skinner. 1997. Duration for bonds with default risk. Journal of Banking and Finance 21 (1): 1–16.

    Article  Google Scholar 

  • Fooladi, I.J., G. Jacoby, G.S. Roberts, and Z. Wiener. 2006. Domestic elasticity of default-free foreign bonds. Journal of Applied Finance 16 (2): 174–182.

    Google Scholar 

  • Fooladi, I.J., G. Jacoby, and L. Jin. 2021. Real duration and inflation duration: A cross country perspective on a multidimensional hedging strategy. Journal of International Financial Markets, Institutions and Money 70: 101265.

    Article  Google Scholar 

  • Hicks, J.R. 1939. Value and capital. 2nd ed. Oxford: Clarendon Press. 1946.

    Google Scholar 

  • Hopewell, M.H., and G.G. Kaufman. 1973. Bond price volatility and term to maturity: A generalized respecification. American Economic Review 63 (4): 749–753.

    Google Scholar 

  • Ingersoll, J.E., J. Skelton, and R.L. Weil. 1978. Duration forty years later. Journal of Financial and Quantitative Analysis 13 (4): 621–650.

    Article  Google Scholar 

  • Jacoby, G. 2003. A duration model for defaultable bonds. Journal of Financial Research 26: 129–146.

    Article  Google Scholar 

  • Jacoby, G., and G. Roberts. 2003. Default- and call-adjusted duration for corporate bonds. Journal of Banking & Finance 27: 2297–2321.

    Article  Google Scholar 

  • Jacoby, G., and I. Shiller. 2008. Duration and pricing of TIPS. Journal of Fixed Income 18 (2): 71–84.

    Article  Google Scholar 

  • Kaufman, G.G. 1984. Measuring and managing interest rate risk: A primer. Economic Perspectives 8: 16–29.

    Google Scholar 

  • Leibowitz, M.L., and A. Weinberger. 1981. The uses of contingent immunization. Journal of Portfolio Management 8 (1): 51–55.

    Article  Google Scholar 

  • ———. 1982. Contingent immunization, part I: Risk control procedures. Financial Analysts Journal 38 (6): 17–32.

    Article  Google Scholar 

  • ———. 1983. Contingent immunization, part II: Problem areas. Financial Analysts Journal 39 (1): 35–50.

    Article  Google Scholar 

  • Macaulay, F. R. 1938. Some theoretical problems suggested by the movement of interest rates, bonds, yields, and stock prices in the United States since 1856. National Bureau of Economic Research. https://www.nber.org/books-and-chapters/some-theoretical-problems-suggested-movements-interest-rates-bond-yields-and-stock-prices-united

  • Morgan, J.P. 1994. Introduction to riskmetrics. New York: J.P. Morgan.

    Google Scholar 

  • Nawalkha, S.K., and G.M. Soto. 2009. Term structure estimation. Available at: https://ssrn.com/abstract=1096182

  • Nawalkha, S.K., and D.R. Chambers. 1997. The M-vector model: Derivation and testing of extensions to M-square. Journal of Portfolio Management 23 (2): 92–98.

    Article  Google Scholar 

  • Nawalkha, S.K., G.M. Soto, and J. Zhang. 2003. Generalized M-vector models for hedging interest rate risk. Journal of Banking & Finance 27: 1581–1604.

    Article  Google Scholar 

  • Oliveira, L., J.P.V. Nunes, and L. Malcato. 2014. The performance of deterministic and stochastic interest rate risk measures: Another question of dimensions? Portuguese Economic Journal 13: 141–165.

    Article  Google Scholar 

  • Ortobelli, S., S. Vitali, M. Cassader, and T. Tichy. 2018. Portfolio selection strategy for fixed income markets with immunization on average. Annals of Operations Research 260: 395–415.

    Article  Google Scholar 

  • Prisman, E. 1986. Immunization as a maxmin strategy: A new look. Journal of Banking & Finance 10 (4): 491–509.

    Article  Google Scholar 

  • Prisman, E.Z., and M.R. Shores. 1988. Duration measures for specific term structure estimations and applications to bond portfolio immunization. Journal of Banking & Finance 12 (3): 493–504.

    Article  Google Scholar 

  • Redington, F.M. 1952. Review of the principle of life-office valuations. Journal of the Institute of Actuaries 78: 286–340.

    Article  Google Scholar 

  • Samuelson, P.A. 1945. The effects of interest rate increases on the banking system. American Economic Review 35 (1): 16–27.

    Google Scholar 

  • Soto, G.G. 2001. Immunization derived from a polynomial duration vector in the Spanish bond market. Journal of Banking & Finance 25: 1037–1057.

    Article  Google Scholar 

  • Soto, G.M. 2004. Duration models and IRR management: A question of dimensions? Journal of Banking & Finance 28: 1089–1110.

    Article  Google Scholar 

  • Yan, A.X., S. Liu, C. Wu, and B. Anderson. 2009. The effects of default and call risk on bond duration. Journal of Banking & Finance 33 (9): 1700–1708.

    Article  Google Scholar 

  • Zhu, W., C.H. Zhang, Q. Liu, and S.S. Zhu. 2018. Incorporating convexity in bond portfolio immunization using multifactor model: A semidefinite programming approach. Journal of the Operations Research Society of China 6: 3–23.

    Article  Google Scholar 

  • Zion, D., and B. Carache. 2002. The magic of pension accounting. Boston: Credit Suisse First.

    Google Scholar 

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Acknowledgments

The authors gratefully acknowledge the support of the Social Sciences and Humanities Research Council of Canada. In addition, Iraj Fooladi received support from the Douglas C. Mackay Fund at Dalhousie.

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Authors and Affiliations

  1. Management of Service Organizations Department, Hadassah Academic College, Jerusalem, Israel

    Zvika Afik

  2. Rowe School of Business, Dalhousie University, Halifax, NS, Canada

    Iraj Fooladi

  3. I.H. Asper School of Business, The University of Manitoba, Winnipeg, MB, Canada

    Gady Jacoby

  4. Schulich School of Business, York University, Toronto, ON, Canada

    Gordon Roberts

Authors
  1. Zvika Afik
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  2. Iraj Fooladi
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  3. Gady Jacoby
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  4. Gordon Roberts
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Corresponding author

Correspondence to Iraj Fooladi .

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Editors and Affiliations

  1. Department of Finance & Economics, Rutgers University, Piscataway, NJ, USA

    Cheng-Few Lee

  2. Center for PBBEF Research, Phoenix, AZ, USA

    Alice C. Lee

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Afik, Z., Fooladi, I., Jacoby, G., Roberts, G. (2022). Duration Concepts, Analysis, and Applications. In: Lee, CF., Lee, A.C. (eds) Encyclopedia of Finance. Springer, Cham. https://doi.org/10.1007/978-3-030-91231-4_14

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