Internationally affine term structure models
Next Document The Spanish Review of Financial Economics. 2011;09:31-4
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.
Full Text1. 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 Rios, 2009), these results can be used to estimate ATSMs in a multi-country setting, and to study the exchange rate forecasting ability of such models.2. An affine expected rate of depreciation
The analysis is similar to that in (Backus et al., 2001) and (Brandt and Santa-Clara, 2002). It is based on a two-country world where assets can be denominated in either domestic currency j=1 (i.e., “dollars”) or foreign currency j=2 (i.e., “pounds”). In particular, consider, based on a no-arbitrage argument, the existence of a (strictly positive) stochastic discount factor (SDF), , for each country. This SDF prices any traded asset denominated in currency j through the following relationship:is just the gross h-period return on the asset.
In this set-up, the law of one price implies that any foreign asset must be correctly priced by both the domestic and the foreign SDFs which, under complete markets, translates into the fact that the exchange rate St (“currency 1″ per unit of “currency 2″) is uniquely determined by the ratio of the two pricing kernels:
Therefore, one can obtain the law of motion of the (log) exchange rate, st=logSt, using Itô’s lemma on the stochastic processes of . To this end, assume the following dynamics of the domestic and foreign SDF:is the instantaneous interest rate (also known as short rate) in country j; Wt is an n-dimensional vector of independent Brownian motions that describes the shocks in this economy; and is an n-dimensional vector that is usually called the market price of risk because it describes how the SDF responds to the shocks given by Wt. In general, the short rates and the prices of risk are functions of time, t, and a Markovian n-dimensional vector, xt, that describes completely the state of the global economy. The law of motion of these state variables, xt, is given by a diffusion such as:
Using Itô’s lemma on and subtracting, one gets:
While the conditions needed to have bond yields in affine form can be found in (Duffie and Kan, 1996), the following proposition summarizes the conditions needed to get an expected rate of depreciation that is affine in the set of state variables given by xt.
Proposition 1 If the drift of the process that the log exchange rate s t follows is affine in a set of state variables x t , that is , (6) with and , and x t follows an affine diffusion under the physical measure: (7) where Φ and Σ are n × n matrices , θ is an n-vector , V ( x t ) is a diagonal n × n matrix with i-th typical element , W t is an n-dimensional vector of independent Brownian motions, and all the eigenvalues of Φ are positive to guarantee the stationarity of the process; then, the expected rate of depreciation h-periods ahead is a (known) affine function of the state vector x t : (8) where the coefficients and have the following expressions:
Proof First note that the expected rate of depreciation satisfies then take expectations with respect to the integral form of (7) : and use that along with the fact that E t x t+h = θ + e −Φh ( x t − θ ) and that Φ is invertible in order to obtain the desired result. □
The result in this proposition is novel because (to the best of our knowledge) the literature on continuous-time multi-country affine models has focused almost entirely on Euler approximations to the expected rate of depreciation h-periods ahead. For example, (Hodrick and Vassalou, 2002),(Leippold and Wu, 2007) and (Ahn, 2004) use an Euler approximation of the law of motion of the (log) exchange rate to obtain a formulae for the expected rate of depreciation that is valid only for an arbitrary small period h. Yet Eq. has the advantage of being exact and, hence, any model parameter estimates based on this result will not be subject to discretization biases. Similarly, (Backus et al., 2001) only provides an expression for the one-period ahead expected rate of depreciation (h=1) and, thus, this proposition generalizes their results to the case of an arbitrary choice of h. For example, (Diez de los Rios, 2009) exploits Eq. to estimate a two-country ATSM and analyze its forecasting ability when predicting exchange rates up to one year ahead.
Also notice that this proposition states that one can obtain an affine expected rate of depreciation when both the short rates, , and the instantaneous variances of the pricing kernels, , are affine in xt (which guarantees that the drift of the log exchange rate, st, is affine); and, at the same time, the process that xt follows must be an affine diffusion under the physical measure. Note, however, that these conditions are restrictive with respect to the general class of ATSMs. For example, it is possible to obtain affine bond yields without assuming a model where the instantaneous variance of the SDF is affine in xt (see (Duffee, 2002, Cheridito et al., 2007)) or without the condition that the state vector must follow an affine diffusion under the physical measure (see (Duarte, 2004)).3. Examples
This section presents additional details on the two main families of ATSMs that belong to the internationally affine class.3.1. Affine models of currency pricing
In this subsection, we focus on a multi-country version of the (Dai and Singleton, 2000) standard formulation of the ATSMs that nests most of the work on international term structure modelling.1 These models can be considered as multivariate extensions of the (Cox et al., 1985) model, and they are characterized by the following set of assumptions:
2. dxt=Φ(θ−xt)dt+Σ1/2V(xt)1/2dWt, where Φ and Σ are n×n matrices, θ is an n-vector, V(xt) is a diagonal n×n matrix with i-th typical element , and Wt is an n-dimensional vector of independent Brownian motions.2
Under these assumptions, one can show that bond yields satisfy:is the yield on an h-period zero-coupon bond in country j, and the coefficients and solve a system of ordinary differential equations whose details can be found in (Duffie and Kan, 1996) or (Piazzesi, 2009).
Notice that this model satisfies the conditions in , and therefore the expected rate of depreciation h-periods ahead is also an affine function of the state vector xt. Such a formulation is also known as a “completely affine” specification (Duffee, 2002), because it has an instantaneous variance of the SDFs, , that is affine in the set of factors xt. The fact that Et[st+h−st] is also affine adds a new meaning to the term “completely affine specification.” The problem is that such a specification has been found to be empirically restrictive. For example, (Duffee, 2002) finds that this parameterization produces forecasts of future Treasury yields that are beaten by a random walk specification4; and (Backus et al., 2001) point out that this model constrains the relationship between interest rates and the risk premium in such a way that the ability of the model to capture the forward premium puzzle is severely limited. In the next section, we analyze a more flexible family of dynamic term structure models that has been found empirically more plausible.3.2. Quadratic models of currency pricing
The quadratic term structure model was introduced by (Ahn et al., 2002) and (Leippold and Wu, 2003) in order to accommodate rich nonlinear and time-varying dynamics in bond yields. In particular, these models are characterized by the following set of assumptions5:
2. dxt=Φ(θ−xt)dt+Σ1/2dWt, where Φ and Σ are n×n matrices, θ is an n-vector; and Wt is an n-dimensional vector of independent Brownian motions.
It can be shown that in this framework bond yields have a quadratic form:, , and , solve a system of ordinary differential equations. Still, it is possible to view any quadratic model as being affine in the original set of variables and their respective squares and cross-products. To do so, just express as: with , , and Dn is the duplication matrix.6
Similarly, it can be shown that the expected rate of depreciation is also affine in this augmented set of factors. To do so, first note that the drift of the (log) exchange rate process can be expressed as:and . Finally, it can be shown that if one applies Itô’s lemma on then the joint process for xt and zt is an affine diffusion (see Appendix B in (Cheng and Scaillet, 2002)). In particular, the law of motion of the augmented set of factors satisfies: , and and being the Moore–Penrose inverse of matrix Dn: . In addition, the diffusion term satisfies , which implies a volatility matrix whose elements are affine in xt and (and, therefore, affine in xt and zt). Therefore, the quadratic model also satisfies the conditions given in if one interprets this model as being affine in an augmented set of state variables.
It is also interesting to note that this quadratic framework also nests the Gaussian essentially affine specification used in (Dai and Singleton, 2002) and (Duffee, 2002) when for j=1, 2. Such a model combines Gaussian state variables and an affine market price of risk, and it has been shown to be flexible enough to explain the rejection of the expectations hypothesis of the term structure of interest rates as well as to produce forecasts of future Treasury yields that beat the random walk specification. In this case, bond yields are affine in the set of state variables, while the expected rate of depreciation is quadratic.74. Conclusions
This note presents a set of conditions that extends the tractability of the single-country ATSM to the multi-country case. In particular, the main contribution of the present paper is to provide conditions to obtain an expected rate of depreciation that is affine on the set of state variables. As shown in (Diez de los Rios, 2009), this result can be exploited to estimate ATSMs in a multi-country setting, and to study the exchange rate forecasting ability of such models. Finally, two main families of dynamic term structure models are shown to satisfy these conditions.
☆ The views expressed in this paper are those of the author and do not necessarily reflect those of the Bank of Canada.
1 See the models in Saá-Requejo (1993) , Frachot (1996) , Backus et al. (2001) , Hodrick and Vassalou (2002) , and Ahn (2004) .
2 Dai and Singleton (2000) provide a set of restrictions on the parameters of the model that guarantees that cannot take on negative values.
3 As noted earlier in the main text, this formulation rules out specifications of the price of risk such as those in Duffee (2002) and Cheridito et al. (2007) .
4 Duffee (2002) claims that this is because (i) the price of risk variability only comes from V(xt)1/2 and (ii) because the sign of cannot change as the elements of V(xt)1/2 are restricted to be nonnegative.
5 See Inci and Lu (2004) and Leippold and Wu (2007) for a quadratic model of currency pricing.
6 In particular, for a given n×n matrix Γ it can be shown that ; and given that is an n×n symmetric matrix then .
7 See Brennan and Xia (2006) , Dong (2006) and Diez de los Rios (2009) for the use of this model in an international set-up.
Received 30 July 2009
Accepted 3 May 2010
Bibliography1.Ahn DH. Common factors and local factors: implications for term structures and exchange rates. Journal of Financial and Quantitative Analysis. 2004;39:69-102.
2.Ahn DH, Dittmar RF, Gallant AR. Quadratic term structure models: theory and evidence. Review of Financial Studies. 2002;15:242-88.
3.Backus DK, Foresi S, Telmer CI. Affine term structure models and the forward premium anomaly. Journal of Finance. 2001;51:279-304.
4.Brandt MW, Santa-Clara P. Simulated likelihood estimation of diffussions with an application to exchange rate dynamics in incomplete markets. Journal of Financial Economics. 2002;63:161-210.
5.Brennan MJ, Xia Y. International capital markets and foreign exchange risk. Review of Financial Studies. 2006;19:753-95.
6.Cheng, P., Scaillet, O., 2002. Linear-Quadratic Jump-Diffusion Modeling with Application to Stochastic Volatility, FAME Research Paper Series No. 67.
7.Cheridito P, Filipovic D, Kimmel RL. Market price of risk specifications for affine models: theory and evidence. Journal of Financial Economics. 2007;83:123-70.
8.Cox J, Ingersoll J, Ross S. A theory of the term structure of interest rates. Econometrica. 1985;53:385-407.
9.Dai Q, Singleton KJ. Specification analysis of affine term structure models. Journal of Finance. 2000;55:1943-78.
10.Dai Q, Singleton KJ. Expectations puzzles, time-varying risk premia, and affine models of the term structure. Journal of Financial Economics. 2002;63:411-5.
11.Diez de los Rios A. Can affine term structure models help us predict exchange rates?. Journal of Money, Credit and Banking. 2009;41:755-66.
12.Dong S. Macro Variables Do Drive Exchange Rate Movements: Evidence from a No-Arbitrage Model. 2006.
13.Duarte J. Evaluating an alternative risk preference in affine term structure models. Review of Financial Studies. 2004;17:370-404.
14.Duffee GR. Term premia and interest rate forecasts in affine models. Journal of Finance. 2002;57:405-43.
15.Duffie D, Kan R. A yield-factor model of interest rates. Mathematical Finance. 1996;6:379-406.
16.Frachot A. A re-examination of the uncovered interest rate parity hypothesis. Journal of International Money and Finance. 1996;15:419-37.
17.Hodrick R, Vassalou M. Do we need multi-country models to explain exchange rate and interest rate and bond return dynamics?. Journal of Economic Dynamics and Control. 2002;26:1275-99.
18.Inci AC, Lu B. Exchange rates and interest rates: can term structure models explain currency movements?. Journal of Economic Dynamics and Control. 2004;28:1595-624.
19.Leippold M, Wu L. Design and estimation of quadratic term structure models. European Finance Review. 2003;7:47-73.
20.Leippold M, Wu L. Design and estimation of multi-currency quadratic models. Review of Finance. 2007;11:167-207.
21.Piazzesi M. Affine term structure models. pp. 691-766.
22.Saá-Requejo J. The Dynamics and the Term Structure of Risk Premia in Foreign Exchange Markets. 1993.
a Bank of Canada, Ottawa, Canada