Immunization of bond portfolios: A new general framework

Alberto Bueno-Guerrero; Manuel Moreno; Javier F. Navas; Alberto Bueno-Guerrero; Manuel Moreno; Javier F. Navas

doi:10.3934/QFE.2026007

Quantitative Finance and Economics

2026, Volume 10, Issue 1: 130-161. doi: 10.3934/QFE.2026007

Previous Article Next Article

Research article

Immunization of bond portfolios: A new general framework

1.
Department of Economics, IES Francisco Ayala, Avda. Francisco Ayala, s/n, 18014 Granada, Spain
2.
Department of Economic Analysis and Finance, University of Castilla-La Mancha, Cobertizo San Pedro Mártir s/n, 45071 Toledo, Spain
3.
Department of Financial Economics and Accounting, Universidad Pablo de Olavide, Ctra. de Utrera, Km. 1, 41013 Sevilla, Spain

Received: 25 July 2025 Revised: 06 November 2025 Accepted: 03 December 2025 Published: 13 March 2026
JEL Codes: C61, G11, G12

We present a new general setting for the classical immunization problem under which we recover and generalize many of the results in the literature related to immunization of bond portfolios. We also propose a new duration vector adapted to our framework that allows us to obtain immunized portfolios by duration matching. We introduce the concept of second-best portfolio that is the portfolio that produces the minimum loss in the worst-case scenario. We obtain explicit expressions for this class of portfolios when short positions are allowed or forbidden. We also show that, when immunization cannot be achieved, the second-best portfolio is the closest to the duration matching situation. Finally, we carry out a numerical illustration of our theoretical results, showing the effectiveness of the duration measures and the second-best portfolios.
- Immunization,
- duration,
- direction X,
- arbitrage,
- Gateaux differential,
- Kuhn-Tucker conditions
Citation: Alberto Bueno-Guerrero, Manuel Moreno, Javier F. Navas. Immunization of bond portfolios: A new general framework[J]. Quantitative Finance and Economics, 2026, 10(1): 130-161. doi: 10.3934/QFE.2026007

Related Papers:

Abstract

We present a new general setting for the classical immunization problem under which we recover and generalize many of the results in the literature related to immunization of bond portfolios. We also propose a new duration vector adapted to our framework that allows us to obtain immunized portfolios by duration matching. We introduce the concept of second-best portfolio that is the portfolio that produces the minimum loss in the worst-case scenario. We obtain explicit expressions for this class of portfolios when short positions are allowed or forbidden. We also show that, when immunization cannot be achieved, the second-best portfolio is the closest to the duration matching situation. Finally, we carry out a numerical illustration of our theoretical results, showing the effectiveness of the duration measures and the second-best portfolios.

References

[1]	Afik Z, Fooladi I, Jacoby G, et al. (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
[2]	Afik Z, Jacoby G, Wiener Z (2018) Duration and Globalization. J Fixed Income 28: 31–43. https://doi.org/10.3905/jfi.2018.28.2.031 doi: 10.3905/jfi.2018.28.2.031
[3]	Balbás A, Ibáñez A(1998) When Can You Immunize a Bond Portfolio? J Bank Financ 22: 1571–1595. https://doi.org/10.1016/S0378-4266(98)00070-3 doi: 10.1016/S0378-4266(98)00070-3
[4]	Balbás A, Ibáñez A, López S (2002) Dispersion Measures as Immunization Risk Measures. J Bank Financ 26: 1229–1244. https://doi.org/10.1016/S0378-4266(01)00168-6 doi: 10.1016/S0378-4266(01)00168-6
[5]	Balbás A, Montagut EH, Pérez-Fructuoso MJ (2004) Hedging bond portfolios versus infinitely many ranked factors of risk. Available form: https://e-archivo.uc3m.es/rest/api/core/bitstreams/d94d5038-e94e-45a0-83ae-f1dcfd267d64/content.
[6]	Balbás A, Romera R (2007) Hedging Interest Rate Risk by Optimization in Banach Spaces. J Optim Theory Appl 132: 175–191. https://doi.org/10.1007/s10957-006-9124-6 doi: 10.1007/s10957-006-9124-6
[7]	Bierwag GO (1977) Immunization, Duration and the Term Structure of Interest Rates. J Financ Quant Anal 12: 725–742. https://doi.org/10.2307/2330253 doi: 10.2307/2330253
[8]	Bierwag GO, Kaufman G (1977) Coping with the Risk of Interest Rate Fluctuations: A Note. J Bus 50: 364–370. http://www.jstor.org/stable/2352543
[9]	Bierwag GO, Khang C (1979) An Immunization Strategy is a Minimax Strategy. J Financ 34: 389–399. https://doi.org/10.2307/2326978 doi: 10.2307/2326978
[10]	Björk T, Kabanov Y, Runggaldier W (1997) Bond Market Structure in the Presence of Marked Point Processes. Math Financ 7: 211–239. https://doi.org/10.1111/1467-9965.00031 doi: 10.1111/1467-9965.00031
[11]	Bowden RJ (1997) Generalising Interest Rate Duration with Directional Derivatives: Direction X and Applications. Manage Sci 43: 586–595. https://doi.org/10.1287/mnsc.43.5.586 doi: 10.1287/mnsc.43.5.586
[12]	Bravo JM, Da Silva CMP (2006) Immunization using a Stochastic-Process Independent Multi-Factor Model: The Portuguese experience. J Bank Financ 30: 133–156. https://doi.org/10.1016/j.jbankfin.2005.01.006 doi: 10.1016/j.jbankfin.2005.01.006
[13]	Bueno-Guerrero A, Moreno M, Navas JF (2015) Stochastic String Models with Continuous Semimartingales. Physica A 433: 229–246. http://dx.doi.org/10.1016/j.physa.2015.03.070 doi: 10.1016/j.physa.2015.03.070
[14]	Bueno-Guerrero A, Moreno M, Navas JF (2016) The Stochastic String Model as a Unifying Theory of the Term Structure of Interest Rates. Physica A 461: 217–237. http://doi.org/10.1016/j.physa.2016.05.044 doi: 10.1016/j.physa.2016.05.044
[15]	Bueno-Guerrero A, Moreno M, Navas JF (2022) Bond Market Completeness under Stochastic Strings with Distribution-Valued Strategies. Quant Financ 22: 197–211. https://doi.org/10.1080/14697688.2021.2018483 doi: 10.1080/14697688.2021.2018483
[16]	Chambers DR, Carleton WT, McEnally RW (1988) Immunizing Default-Free Bond Portfolios with a Duration Vector. J Financ Quant Anal 23: 89–104. https://doi.org/10.2307/2331026 doi: 10.2307/2331026
[17]	Diebold FX, Ji L, Li C (2006) A Three Factor Yield Curve Model: Non-Affine Structure, Systematic Risk Sources, and Generalized Duration, In: L. R. Klein (ed.), Long-Run Growth and Short-Run Stabilization: Essay in Memory of Albert Ando, Cheltenham, U. K.: Edward Elgar, 240–274. https://doi.org/10.4337/9781781950500
[18]	Diebold FX, Li C (2006) Forecasting the Term Structure of Goverment Bond Yields. J Econom 130: 337–364. https://doi.org/10.1016/j.jeconom.2005.03.005 doi: 10.1016/j.jeconom.2005.03.005
[19]	Fisher L, Weil R (1971) Coping with the Risk of Interest Rate Fluctuations and Return to Bondholders from Naive and Optimal Strategies. J Bus 44: 408–431. https://www.jstor.org/stable/2352056
[20]	Fong HG, Vasicek OA (1984) A Risk Minimizing Strategy for Portfolio Immunization. J Finance 39: 1541–1546. https://doi.org/10.2307/2327743 doi: 10.2307/2327743
[21]	Fooladi IJ, Jacoby G, Jin L (2021) Real Duration and Inflation Duration: A Cross Country Perspective on a Multidimensional hedging Strategy. J Int Financ Markets Inst Money 70: 101265. https://doi.org/10.1016/j.intfin.2020.101265 doi: 10.1016/j.intfin.2020.101265
[22]	Goldstein RS (2000) The Term Structure of Interest Rates as a Random Field. Rev Financ Stud 13: 365–384. https://www.jstor.org/stable/2646029
[23]	Hicks JR (1939) Value and Capital, Oxford: Clarendon Press.
[24]	Hull J, White A (1990) Pricing Interest-Rate Derivative Securities. Rev Financ Stud 3: 573–592. https://doi.org/10.1093/rfs/3.4.573 doi: 10.1093/rfs/3.4.573
[25]	Ingersoll JE, Skelton J, Weil RL (1978) Duration Forty Years Later. J Financ Quant Anal 13: 627–650. https://doi.org/10.2307/2330468 doi: 10.2307/2330468
[26]	Jacoby G (2003) A Duration Model for Defaultable Bonds. J Financ Res 26: 129–146. https://doi.org/10.1111/1475-6803.00049 doi: 10.1111/1475-6803.00049
[27]	Kennedy DP (1994) The Term Structure of Interest Rates as a Gaussian Random Field. Math Financ 4: 247–258. https://doi.org/10.1111/j.1467-9965.1994.tb00094.x doi: 10.1111/j.1467-9965.1994.tb00094.x
[28]	Khang C (1979) Bond Immunization when Short-Term Interest Rates Fluctuate more than Long-Term Rates. J Financ Quant Anal 14: 1085–1090. https://doi.org/10.2307/2330309 doi: 10.2307/2330309
[29]	Lapshin V (2019) A Nonparametric Approach to Bond Portfolio Immunization. Mathematics 7: 1121. https://doi.org/10.3390/math7111121 doi: 10.3390/math7111121
[30]	Litterman RB, Scheikman J (1991) Common Factors Affecting Bond Returns. J Fixed Income 1: 54–61. https://doi.org/10.3905/jfi.1991.692347 doi: 10.3905/jfi.1991.692347
[31]	Luenberger DG (1969) Optimization by Vector Space Methods, Wiley, New York.
[32]	Macaulay FR (1938) Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the U.S. since 1856, Columbia University Press. https://doi.org/10.2307/2143533
[33]	Merton RC (1973) Theory of Rational Option Pricing. Bell J Econ Manage Sci 4: 141–183. https://doi.org/10.2307/3003143 doi: 10.2307/3003143
[34]	Moreno M (2007) Managing Interest Rate Risk under Non-Parallel Changes: An Application of a Two-Factor Model, In: G. Gregoriou (ed.), Advances in Risk Management. Palgrave-MacMillan, London, United Kingdom, 69-85. https://doi.org/10.1057/9780230625846
[35]	Munk C (1999) Stochastic Duration and Fast Coupon Bond Option Pricing in Multi-Factor Models. Rev Deriv Res 3: 157–181. https://doi.org/10.1023/A:1009654427422 doi: 10.1023/A:1009654427422
[36]	Musiela M (1993) Stochastic PDEs and Term Structure Models, Statistics Technical Reports, CSIR-002-93, SMS-017-93, University of South Wales.
[37]	Nawalkha SK, Chambers DR (1996) An Improved Immunization Strategy: M-Absolute. Financ Anal J 52: 69–76. https://doi.org/10.2469/faj.v52.n5.2026 doi: 10.2469/faj.v52.n5.2026
[38]	Nawalkha SK, Chambers DR (1997) The M-squared Vector Model: Derivation and Testing of Extensions to M-Squared. J Portf Manag 23: 92–98. https://doi.org/10.3905/jpm.23.2.92 doi: 10.3905/jpm.23.2.92
[39]	Nawalkha SK, Soto GM, Zhang J (2003) Generalized M-Vector Models for Hedging Interest Rate Risk. J Bank Financ 27: 1581–1604. https://doi.org/10.1016/S0378-4266(03)00089-X doi: 10.1016/S0378-4266(03)00089-X
[40]	Nelson CR, Siegel AF (1987) Parsimonious Modeling of Yield Curves. J Bus 60: 473–489. https://doi.org/10.1086/296409 doi: 10.1086/296409
[41]	Oliveira L, Nunes JPV, Malcato L (2014) The Performance of Deterministic and Stochastic Interest Rate Risk Measures: Another Question of Dimensions? Port Econ J 13: 141–165. https://doi.org/10.1007/s10258-014-0104-8 doi: 10.1007/s10258-014-0104-8
[42]	Ortobelli S, Vitali S, Cassader M, et al. (2018) Portfolio Selection Strategy for Fixed Income Markets with Immunization on Average. Ann Oper Res 260: 395–415. https://doi.org/10.1007/s10479-016-2182-8 doi: 10.1007/s10479-016-2182-8
[43]	Prisman EZ, Shores MR (1988) Duration Measures for Especific Term Structure Estimations and Applications to Bond Portfolio Immunization. J Bank Financ 12: 493–504. https://doi.org/10.1016/0378-4266(88)90011-8 doi: 10.1016/0378-4266(88)90011-8
[44]	Protter PE (2005) Stochastic Integration and Differential Equations, Second Edition, Springer. https://doi.org/10.1007/978-3-662-10061-5
[45]	Redington FM (1952) Review on the Principle of Life-Office Valuations. J Inst Act 78: 286–315.https://doi.org/10.1017/S0020268100052811 doi: 10.1017/S0020268100052811
[46]	Samuelson PA (1945) The Effect of Interest Rates Increases on the Banking System. Am Econ Rev 35: 16–27. https://www.jstor.org/stable/1810106
[47]	Santa-Clara P, Sornette D (2001) The Dynamics of the Forward Interest Rate Curve with Stochastic String Shocks. Rev Financ Stud 14: 149–185. https://doi.org/10.1093/rfs/14.1.149 doi: 10.1093/rfs/14.1.149
[48]	Soto GM (2001) Immunization Derived from a Polynomial Duration Vector in the Spanish Bond Market. J Bank Financ 25: 1037–1057. https://doi.org/10.1016/S0378-4266(00)00112-6 doi: 10.1016/S0378-4266(00)00112-6
[49]	Yan AX, Liu S, Wu C, et al. (2009) The Effects of Default and Call Risk on Bond Duration. J Bank Finan 33: 1700–1708. https://doi.org/10.1016/j.jbankfin.2009.04.004 doi: 10.1016/j.jbankfin.2009.04.004
[50]	Zhu W, Zhang CH, Liu Q, et al. (2018) Incorporating Convexity in Bond Portfolio Immunization using Multifactor model: A Semidefinite Programming Approach. J Oper Res Soc China 6: 3–23. https://doi.org/10.1007/s40305-018-0196-4 doi: 10.1007/s40305-018-0196-4

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)