The Long-Term Swap Rate and a General Analysis of Long-Term Interest Rates

TL;DR

This paper introduces the concept of the long-term swap rate, analyzes its relationship with other long-term interest rates, and examines their existence within specific term structure models, with applications to CoCo bonds.

Contribution

It defines the long-term swap rate as a fair rate for an infinite exchange swap and explores its connections with long-term yields and discount rates, including model existence analysis.

Findings

Introduced the long-term swap rate as an infinite-exchange overnight indexed swap

Analyzed relationships between long-term swap rate, yield, and simple rate

Demonstrated application to estimating components of CoCo bonds

Abstract

We introduce here for the first time the long-term swap rate, characterised as the fair rate of an overnight indexed swap with infinitely many exchanges. Furthermore we analyse the relationship between the long-term swap rate, the long-term yield, see Biagini et al. [2018], Biagini and H\"artel [2014], and El Karoui et al. [1997], and the long-term simple rate, considered in Brody and Hughston [2016] as long-term discounting rate. We finally investigate the existence of these long-term rates in two term structure methodologies, the Flesaker-Hughston model and the linear-rational model. A numerical example illustrates how our results can be used to estimate the non-optional component of a CoCo bond.