When can you immunize a bond portfolio?

Abstract

This paper presents a condition equivalent to the existence of a Riskless Shadow Asset that guarantees a minimum return when the asset prices are convex functions of interest rates or other state variables. We apply this lemma to immunize default free and option free coupon bonds and reach three main conclusions. First, we give a solution to an old puzzle: why do simple duration matching portfolios work well in empirical studies of immunization even though they are derived in a model inconsistent with equilibrium and shifts on the term structure of interest rates are not parallel, as as sumed? Second, we establish a clear distinction between the concepts of immunized and maxmin portfolios. Third, we develop a framework that includes the main results of this literature as special cases. Next, we present a new strategy of immunization that consists in matching duration and minimizing a new linear dispersion measure of immunization risk. Ó 1998 Elsevier Science B.V. All rights reserved.JEL classi®cation: G11

Hedging Interest Rate Risk by Optimization in Banach Spaces

“…A theoretical justification is provided in Refs. [2][3][4][5]. These papers show that the bond price convexity implies that the duration strategies are close to maximin strategies; therefore, they are robust against different changes on the TSIR and not only against parallel shifts.…”

“…We have taken m = 5 years and K is the set of shocks introduced in Ref. 5 and analyzed in Proposition 2.1b. We have selected the parameter λ 2 = 0.05 = 5% and it may be proved that the final result does not depend on λ 1 .…”

Immunization using a stochastic-process independent multi-factor model: The Portuguese experience

“…A second argument, proposed by Soto (2001), can be found in the interest rate risk dispersion measures of Nawalkha and Chambers (1996) and Balbás and Ibáñ ez (1998). In these models, the attempt to reduce immunization risk is attained by min- Duration-matching strategies Maturity-Bullet Two-bond portfolio combining the bond chosen for the maturity strategy and a bond with duration longer than, but closest to, the horizon date.…”

“…Non-parallel shifts were proposed byBierwag (1977),Khang (1979) andBabbel (1983) or, in an equilibrium setting, byCox et al (1979),Ingersoll et al (1978),Brennan and Schwartz (1983),Nelson and Schaefer (1983) andWu (2000), among others. SeeBravo (2001) for a detailed analysis of interest rate risk models.2 SeeNawalkha and Chambers (1996),Balbás and Ibáñ ez (1998) andBalbás et al (2002) for alternative definitions of interest rate risk dispersion measures.3 In these models, the direction of interest rate shifts can be set on an a priori basis, or can be based on real data. In the latter case, the historical movements in the term structure of interest rates are used to identify a limited number of state variables, observable or not, which govern the yield curve.…”

On risk minimizing strategies for default-free bond portfolio immunization

“…They have proposed multiple-risk measure models (e.g. Fong and Vasicek, 1984, Balbás andIbáñez, 1998) or singlerisk measure models (e.g. Chambers, 1996, Kałuszka andKondratiuk-Janyska, 2004).…”