On the Dybvig-Ingersoll-Ross Theorem

TL;DR

This paper refines the Dybvig-Ingersoll-Ross theorem by identifying the reciprocal of maturity as the key factor limiting the dominance of long-term rates over time, under weaker market assumptions.

Contribution

It provides a more precise version of the DIR theorem with a weaker viability assumption, enhancing understanding of long-term rate behavior in arbitrage-free models.

Findings

Long-term rates cannot fall in arbitrage-free models.

The reciprocal of maturity date bounds rate dominance.

Weaker market assumptions still uphold the theorem.

Abstract

The Dybvig-Ingersoll-Ross (DIR) theorem states that, in arbitrage-free term structure models, long-term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long-term rates at earlier dates can dominate long-term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.