New Formulas for Immunizing Durations

Abstract

The concept of duration has long been established as a standard tool for measuring and managing the interest sensitivity of a fixed-income instrument or portfolio. One important use of duration is in setting up an immunization strategy, such that the risk attached to a given future cash flow is immunized against a shift in the term structure by an offsetting hedge position that has the same duration. Earlier models in the literature developed duration-based strategies for immunizing against specific types of yield curve shifts, for example, an additive shift of a fixed number of basis points at every maturity. This article offers several new duration formulas that greatly extend the range of allowable yield curve shifts. Three theorems develop duration-based immunization techniques that cover, in the most general case, a portfolio of instruments exposed to any sum of a finite number of piecewise continuous functions. The results of previous duration models are shown to be special cases of this general formulation.