The Effect of Additive Rate Shocks on Duration and Immunization: Examining the Theory

This paper examines the Macaulay, Hicks, and continuous time formulations of duration by analyzing the term structure dynamics that are necessary to derive these mathematical models. The results presented help to explain the true effects of interest rate risk upon the values of fixed-income securities, as well as provide a rationale for the development of duration models in a general equilibrium framework. In particular, we show that the Hicksian formulation of duration is derived from a uniform infinitesimal additive shock to all yields to maturity on a flat yield curve, while the Macaulay duration is derived from a uniform infinitesimal additive shock to all spot rates. In addition, the Macaulay duration is shown to be inconsistent with a uniform infinitesimal additive shock to any single period spot or forward rate spanning a period less than the maturity of the security. The continuous time representation of duration is examined to show the immunizing condition and how this result is related to the Macaulay and Hicksian durations through the phenomenon of post-shift convexity of investment value. The appendices provide more detailed proofs of the critical concepts discussed in this paper.