Immunization of multiple liabilities. (English) Zbl 0666.62101

It is well known that a sufficient condition for the immunization of multiple liabilities is the separate immunization of each liability outflow. This paper proves that this is also a necessary condition. It also discusses how one constructs immunized portfolios by means of linear programming.


62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
90C90 Applications of mathematical programming
Full Text: DOI


[1] Bierwag, G.O., Duration analysis: managing interest rate risk, (1987), Ballinger Cambridge, MA
[2] Bierwag, G.O.; Kaufman, G.G.; Toevs, A., Immunization strategies for funding multiple liabilities, Journal of financial and quantitative analysis, 18, 113-123, (1983)
[3] Bierwag, G.O.; Kaufman, G.G.; Toevs, A., Recent developments in bond portfolio immunization strategies, (), 105-157
[4] Bierwag, G.O.; Kaufman, G.G.; Latta, C.M.; Roberts, G.S., Duration: response to critics, Journal of portfolio management, 48-52, (1987), Winter
[5] Blackwell, D., Comparison of experiments, (), 93-102
[6] Boyle, P.P., Immunization under stochastic models of the term structure, Journal of the institute of actuaries, 105, 177-187, (1978)
[7] Bratteli, O.; Robinson, D.W., Operator algebras and quantum statistical mechanics, 1, (1979), Springer Verlag New York · Zbl 0421.46048
[8] Brumelle, S.L.; Vickson, R.G., A unified approach to stochastic dominance, (), 101-113
[9] Bühlmann, H.; Gagliardi, B.; Gerber, H.U.; Straub, E., Some inequalities for stop-loss premiums, ASTIN bulletin, 9, 75-83, (1977)
[10] Fishburn, P.C.; Vickson, R.G., Theoretical foundations of stochastic dominance, (), 39-113
[11] Fisher, L.; Weil, R.L., Coping with the risk of interest-rate fluctuations: returns to bondholders from naïve and optimal strategies, Journal of business, 44, 408-431, (1971)
[12] Fong, H.G.; Fabozzi, F.J., Fixed income portfolio management, (1985), Dow Jones — Irwin Homewood, IL
[13] Fong, H.G.; Vasicek, O., Returned maximization for immunized portfolios, (), 227-238
[14] Fong, H.G.; Vasicek, O., A risk miminizing strategy for multiple liability immunization, (1983), Unpublished manuscript
[15] Fong, H.G.; Vasicek, O., A risk minimizing strategy for portfolio immunization, Journal of finance, 39, 1541-1546, (1984)
[16] Gerber, H.U., On the computation of stop-loss premiums, Mitteilungen der vereinigung schweizerischer versicherungsmathematiker, 77, 47-58, (1977) · Zbl 0381.62090
[17] Gerber, H.U.; Jones, D.A., Some practical considerations in connection with the calculation of stop-loss premiums, Transactions of the society of actuaries, 28, 215-231, (1976), Discussion 233-235
[18] Goovaerts, M.J.; De Vylder, F.; Haezendonck, J., Ordering of risks: A review, Insurance: mathematics and economics, 1, 131-161, (1982) · Zbl 0492.62090
[19] Goovaerts, M.J.; De Vylder, F.; Haezendonck, J., Insurance premiums: theory and applications, (1984), North-Holland Amsterdam · Zbl 0532.62082
[20] Granito, M.R., Review of duration analysis: managing interest rate risk, Journal of finance, 43, 264-267, (1988)
[21] ()
[22] Karamata, J., Sur une inégalité relative aux fonctions convexes, Publications mathématiques de l’université de belgrade, 1, 145-148, (1932) · JFM 58.0211.01
[23] Karlin, S.J.; Studden, W.J., Tchebycheff systems: with applications in analysis and statistics, (1966), Wiley New York · Zbl 0153.38902
[24] Kocherlakota, R.; Rosenbloom, E.S.; Shiu, E.S.W., Algorithms for cash-flow matching, Transactions of the society of actuaries, 40, (1988), forthcoming
[25] Le Cam, L., Asymptotic methods in statistical decision theory, (1986), Springer Verlag New York · Zbl 0605.62002
[26] Lidstone, G.J., On the approximate calculation of the values of increasing annuities and assurances, Journal of the institute of actuaries, 31, 68-72, (1893)
[27] Macaulay, F.R., Some theoretical problems suggested by the movements of interest rates, bond yields and stock prices in the united states Since 1856, (1938), National Bureau of Economic Research New York
[28] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (1979), Academic Press New York · Zbl 0437.26007
[29] Milgrom, P.R., Measuring the interest rate risk, Transactions of the society of actuaries, 37, 241-257, (1985), Discussion 259-302
[30] Phelps, R.R., Lectures on Choquet’s theorem, (1966), Van Nostrand Princeton, NJ · Zbl 0135.36203
[31] ()
[32] Promislow, S.D., Comparing risks, (), 79-93
[33] Promislow, S.D., Measurement of equity, Transactions of the society of actuaries, 39, 215-237, (1987), Discussion 239-256
[34] Redington, F.M., Review of the principles of life-office valuations, Journal of the institute of actuaries, 78, 286-315, (1952), Discussion 316-340
[35] Redington, F.M.; Redington, F.M., A ramble through the actuarial countryside: the collected papers, essays & speeches of Frank mitchell redington, MA, (1986), Institute of Actuaries Students’ Society London, Reprinted in
[36] Rothschild, M.; Stiglitz, J.E., Increasing risk: I. A definition, Journal of economic theory, 2, 225-243, (1970)
[37] Sherman, S., On a theorem of Hardy, Littlewood, Pólya, and blackwell, Proceedings of the national Academy of sciences of the united states of America, 37, 826-831, (1951) · Zbl 0044.27801
[38] Shiu, E.S.W., A generalization of Redington’s theory of immunization, Actuarial research clearing house, 69-81, (1986), 1986.2
[39] Shiu, E.S.W., Immunization — the matching of assets and liabilities, (), 145-156
[40] Shiu, E.S.W., On the fisher—weil immunization theorem, Insurance: mathematics and economics, 6, 259-266, (1987) · Zbl 0633.62109
[41] Strassen, V., The existence of probability measures with given marginals, Annals of mathematical statistics, 36, 423-439, (1965) · Zbl 0135.18701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.