Arbitrage and Equilibrium Foundations of the Duration Risk Measure

Abstract

This paper provides arbitrage and equilibrium foundations of the traditional duration risk measure (see Macaulay [1938] and Hicks [1939]), by relating it to the Heath, Jarrow and Morton (HJM) [1992] term structure theory and Merton's intertemporal CAPM [1973]. Under the new approach the duration model is shown to be consistent with a subset of arbitrage-free forward rate processes of HJM, some of which preclude the occurrence of negative interest rates by allowing interest rate level dependent volatilities. Conditions are derived under which the convexity risk measure may or may not be priced. Finally, we demonstrate that when Merton's [1973] ICAPM is identified with the above HJM [1992] forward rate processes, the appropriate equilibrium measure of the systematic risk of a default-free security is its duration, and not its bondbeta as derived by Jarrow [1978], and others, under more restrictive assumptions. This paper addresses all of the arbitrage-based and equilibrium-based criticisms of the duration risk measure given by Ingersoll [1978], Sharpe [1983], and others.