# Internationally affine term structure models

Next Document The Spanish Review of Financial Economics. 2011;09:31-4

- Português: [Resumo]
- English: [Abstract - Full text - PDF]

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## Abstract

This note provides the conditions needed to obtain a multi-country term structure model where both bond yields for each country and the expected rate of depreciation (over any arbitrary period of time) are known affine functions of the set of state variables. In addition, two main families of dynamic term structure models are shown to satisfy these conditions.

**Keywords**: Term structure. Interest rates. Exchange rates.

## Full Text

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1. Introduction

The affine term structure model (ATSM), originally proposed by (Duffie and Kan, 1996), is widely regarded as the cornerstone of modern fixed income theory thanks to its main advantage: tractability. In particular, an ATSM provides analytical expressions for bond yields that are affine functions of some state vector. As noted by (Piazzesi, 2009), tractability is important because otherwise one would need to compute yields with Monte Carlo methods or solution methods for partial differential equations, which could be especially costly from a computational point of view when model parameters are estimated using data on bond yields.

This note presents a set of conditions that extends the tractability of the single-country ATSM to the multi-country case in the context of international term structure models as those in (Backus et al., 2001), (Brandt and Santa-Clara, 2002) and (Brennan and Xia, 2006) among others. In particular, this note focuses on internationally affine term structure models where not only bond yields in each one of the countries are known affine functions of a set of state variables, but also the expected rate of depreciation satisfies this property. The main contribution of the present paper is to provide conditions to obtain an expected rate of depreciation (over any arbitrary period of time) that is affine on the set of state variables (Section 2).

Two main families of ATSMs are shown to satisfy these conditions (Section 3). The first subgroup is the so-called completely affine term structure model introduced in (Dai and Singleton, 2000). However, such a specification has been found empirically restrictive. We overcome this issue by showing that the more flexible class of quadratic-Gaussian term structure models introduced in (Ahn et al., 2002) and (Leippold and Wu, 2003) can also deliver an affine expected rate of depreciation when interpreted as being affine in the original set of variables and their respective squares and cross-products. As shown in (Diez de los...

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