Does Macaulay Duration Provide the Most Cost-Effective Immunization Method a Theoretical Approach
In the following, we offer a theoretical approach that attempts to explain (Comments 1-3) why and when the Macaulay duration concept happens to be a good approximation of a bond's price sensitivity. We are concerned with the basic immunization problem with a single liability to be discharged at a future time q. Our idea is to divide the class K of all shifts a(t) of a term structure of interest rates s(t) into many classes and then to find a sufficient and necessary condition a given bond portfolio, dependent on a class of shifts, must satisfy to secure immunization at time q against all shifts a(t) from that class. For this purpose, we introduce the notions of dedicated duration and dedicated convexity. For each class of shifts, we show how to choose from a bond market under consideration a portfolio with maximal dedicated convexity among all immunizing portfolios. We demonstrate that the portfolio yields the maximal unanticipated rate of return and appears to be uniquely determined as a barbell strategy (portfolio) built up with 2 zero-coupon bearing bonds with maximal and respective minimal dedicated durations. Finally, an open problem addressed to researchers performing empirical studies is formulated. (original abstract)
- Bansal, R., Zhou, H., 2002. Term Structure of Interest Rates with Regime Shifts. Journal of Finance. 57(2002), pp.1997-2043.
- Elton, E.J., and Gruber, M.J., 1995. Modern Portfolio Theory and Investment Analysis. New York: Wiley.
- Fisher, L., and Weil, R., 1971. Coping with the Risk of Interest Rate Fluctuations: Returns to Bondholders from Naive and Optimal Strategies. Journal of Business, 44(1971), pp.408-431.
- Jorion, P. and Khoury, S., 1996. Financial Risk Management: Domestic and International Dimensions. Cambridge, MA: Blackwell Publishers.
- Litterman, R. and Scheinkman, J., 1991. Common Factors Affecting Bond Returns. Journal of Fixed Income, 54(1991), pp.54-61.
- Maculay, R., 1938. Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the U.S. Since 1856. National Bureau of Economic Research, New York.
- Redington, F., 1952. Review of the Principle of Life Office Valuations. Journal of the Institute of Actuaries, 3(1952), pp.286-315.
- Rządkowski, G. and Zaremba, L., 2000. New Formulas for Immunizing Durations. The Journal of Derivatives, 8(2000), pp.28-36.
- Rządkowski, G. and Zaremba L., 2010. Shifts of the Term Structure of Interest Rates Against Which a Given Portfolio Is Preimmunized. Control and Cybernetics, 39(2010), pp.857-867.
- Zaremba, L., 1998. Construction of a k-immunization Strategy with the Highest Convexity. Control & Cybernetics, 27(1998), pp.135-144.
- Zaremba, L. and Smoleński, W., 2000. How to Find a Bond Portfolio with the Highest Convexity in a Class of Fixed Duration Portfolios. Bulletin of the Polish Academy of Sciences, 48(2000), pp.279-286.
- Zaremba, L. and Rządkowski, G., 2016. Determination of Continuous Shifts in the Term Structure of Interest Rates Against Which Bond Portfolio B is Immunized. Control & Cybernetics, No. 4 (in the publishing process).
- Zheng, H., Thomas, L. and Allen, D., 2002. The Duration Derby: A Comparison of Duration Based Strategies in Asset Liability Management. Proceedings of 41st IEEE Conference on Decision and Control, Las Vegas, Nevada USA.
- Zheng, H., 2007. Macaulay Durations for Nonparallel Shifts. Annals Operation Research, 151 (2007), pp.179-191.