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.
G. Roberts: deceased.
Notes
- 1.
- 2.
- 3.
This discussion of immunization begins by assuming default and option free securities in order to separate interest rate risk from other risks.
- 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.
- 6.
- 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.
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.
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.
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.
- 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.
In case of deflation, the principal amount cannot be indexed to a level lower than its unadjusted principal.
- 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.
Afik, Z., G. Jacoby, and Z. Wiener. 2018. Duration and globalization. The Journal of Fixed Income 28 (2): 31–43.
Balbás, A., and A. Ibánez. 1998. When can you immunize a bond portfolio? Journal of Banking & Finance 22: 1571–1595.
Balbás, A., A. Ibánez, and S. López. 2002. Dispersion measures as immunization risk measures. Journal of Banking & Finance 26: 1229–1244.
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.
Bierwag, G.O. 1977. Immunization, duration, and the term structure of interest rates. Journal of Financial and Quantitative Analysis 12 (5): 725–741.
———. 1987a. Duration analysis, managing interest rate risk. Cambridge, MA: Ballinger Publishing.
———. 1987b. Bond returns, discrete stochastic processes, and duration. Journal of Financial Research 10 (3): 191–209.
Bierwag, G.O., and I. Fooladi. 2006. Duration analysis: An historical perspective. Journal of Applied Finance 16 (2): 144–160.
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.
———. 1988. Duration of non-default free securities. Financial Analysts Journal 44 (4): 39–46.
———. 1996. Managing interest rate risk with duration gaps to achieve multiple targets. Journal of Financial Engineering 5: 53–73.
Bierwag, G.O., and C. Khang. 1979. An immunization strategy is a minimax strategy. Journal of Finance 34 (2): 389–399.
Bierwag, G.O., and G.S. Roberts. 1990. Single factor duration models: Canadian tests. Journal of Financial Research 13: 23–38.
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.
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.
———. (1982b). Empirical tests of alternative single factor duration models. Paper presented at Western Finance Association, Portland, Oregon (June).
———. 1983. Immunization strategies for funding multiple liabilities. Journal of Financial and Quantitative Analysis 18: 113–124.
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.
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.
———. 1993b. Designing and immunization portfolio: Is M-squared the key? Journal of Banking & Finance 17: 1147–1170.
———. 2000. Risk management with duration: Potential and limitations. Canadian Journal of Administrative Sciences 17 (2): 126–142.
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.
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.
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.
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.
Chance, D.M. 1990. Default risk and the duration of zero coupon bonds. Journal of Finance 45 (1): 265–274.
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.
Cornett, M.M., and A. Saunders. 2008. Financial institutions management. 6th ed. Boston: Irwin McGraw Hill, Chapters 8, 9 and 18.
Diaz, A., M.O. González, El Navarro, and F.S. Skinner. 2009. An evaluation of contingent immunization. Journal of Banking & Finance 33: 1874–1883.
Dym, S.I. 1991. Measuring the risk of foreign bonds. Journal of Portfolio Management 17 (2): 56–61.
Dym, S. 1992. Global and local components of foreign bond risk. Financial Analysts Journal 48 (2): 83–91.
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.
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.
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.
Fong, H.G., and O.A. Vasicek. 1984. A risk minimizing strategy for portfolio immunization. Journal of Finance 39 (5): 1541–1546.
Fooladi, I., and G. Roberts. 1992. Portfolio immunization: Canadian tests. Journal of Economics and Business 44 (1): 3–17.
———. 2004. Macrohedging for financial institutions: Beyond duration. Journal of Applied Finance 14 (1): 11–19.
Fooladi, I., G. Roberts, and F. Skinner. 1997. Duration for bonds with default risk. Journal of Banking and Finance 21 (1): 1–16.
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.
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.
Hicks, J.R. 1939. Value and capital. 2nd ed. Oxford: Clarendon Press. 1946.
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.
Ingersoll, J.E., J. Skelton, and R.L. Weil. 1978. Duration forty years later. Journal of Financial and Quantitative Analysis 13 (4): 621–650.
Jacoby, G. 2003. A duration model for defaultable bonds. Journal of Financial Research 26: 129–146.
Jacoby, G., and G. Roberts. 2003. Default- and call-adjusted duration for corporate bonds. Journal of Banking & Finance 27: 2297–2321.
Jacoby, G., and I. Shiller. 2008. Duration and pricing of TIPS. Journal of Fixed Income 18 (2): 71–84.
Kaufman, G.G. 1984. Measuring and managing interest rate risk: A primer. Economic Perspectives 8: 16–29.
Leibowitz, M.L., and A. Weinberger. 1981. The uses of contingent immunization. Journal of Portfolio Management 8 (1): 51–55.
———. 1982. Contingent immunization, part I: Risk control procedures. Financial Analysts Journal 38 (6): 17–32.
———. 1983. Contingent immunization, part II: Problem areas. Financial Analysts Journal 39 (1): 35–50.
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.
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.
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.
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.
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.
Prisman, E. 1986. Immunization as a maxmin strategy: A new look. Journal of Banking & Finance 10 (4): 491–509.
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.
Redington, F.M. 1952. Review of the principle of life-office valuations. Journal of the Institute of Actuaries 78: 286–340.
Samuelson, P.A. 1945. The effects of interest rate increases on the banking system. American Economic Review 35 (1): 16–27.
Soto, G.G. 2001. Immunization derived from a polynomial duration vector in the Spanish bond market. Journal of Banking & Finance 25: 1037–1057.
Soto, G.M. 2004. Duration models and IRR management: A question of dimensions? Journal of Banking & Finance 28: 1089–1110.
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.
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.
Zion, D., and B. Carache. 2002. The magic of pension accounting. Boston: Credit Suisse First.
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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Afik, Z., Fooladi, I., Jacoby, G., Roberts, G. (2021). Duration Concepts, Analysis, and Applications. In: Lee, CF., Lee, A.C. (eds) Encyclopedia of Finance. Springer, Cham. https://doi.org/10.1007/978-3-030-73443-5_14-1
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