Economics Letters 110 (2011) 1–3
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Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Afﬁne Nelson–Siegel model Rodrigo A. Alfaro ⁎ Central Bank of Chile, Agustinas 1180, Santiago, Chile
a r t i c l e
i n f o
Article history: Received 11 September 2009 Received in revised form 26 August 2010 Accepted 4 October 2010 Available online 11 October 2010
a b s t r a c t We introduce a discrete-time version of the dynamic yield curve model proposed by Diebold and Li (2006) which is based on Nelson and Siegel (1987). As in Christensen et al. (2010) we found an afﬁne process that matches the model. © 2010 Elsevier B.V. All rights reserved.
JEL classiﬁcation: E43 G12 Keywords: Nelson–Siegel Model Yield curve Euler equation Afﬁne models
1. Introduction The Nelson and Siegel (1987) (NS hereafter) model is very popular among practitioners and central banking analysts, but it was considered static and not related with the class of afﬁne models proposed by Vasicek (1977), and Cox et al. (1985), among others. Diebold and Li (2006) propose a dynamic version of the model (DNS) and Christensen et al. (2010) show that DNS belongs to the afﬁne class proposed in the literature. Thus, the DNS model is supported by both practical and theoretical grounds. In this paper we provide the same results for DNS but using a discrete-time setting. To that aim we use the Euler equation as the main tool for pricing, following the notation developed in Campbell et al. (1997, chapter 11). We believe that the discrete-time framework proposed here is more suitable for a broader audience than the one used in continuous-time.
n-period interest rate can be obtained taking the average over the sample:
1 n 1−e−αn 1−e−αn −αn : + λ3 −e ∫ f ðiÞdi = λ1 + λ2 αn αn n 0
The discrete-time version considered in this paper is based on the fact that e− α = 1 − α + O(α2), which means that when α is small e− α ≈ 1 − α ≡ ϕ, implying that 0 b ϕ b 1.1 In our discrete-time setting the short-term interest rate has maturity n = 1, and its dynamics is: z1, t + i = λ1t + λ2tϕi + λ3t(1 − ϕ) iϕi − 1. The exponent of the last component is adjusted to match the short-term interest rate in discrete-time. Also, we consider the Diebold and Li (2006) approach for time-variant factors. Taking the average over the sample period we get the following structure for the n maturity discount rate:
2. Discrete version of DNS model The original NS model is also set in continuous time. The authors consider that the instantaneous forward rate is a function of time, and that it is the solution of a second order differential equation which can be expressed as follows: f(i) = λ1 + λ2e− αi + αλ3ie− αi. Using this, the
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znt = λ1t +
n n λ2t 1−ϕ λ 1−ϕ n−1 : + 3t −nϕ n 1−ϕ n 1−ϕ
Note that znt is the continuously compounded interest rate, meaning that znt = log(1 + Znt). Also, the price of the discount bond
^ For example, Diebold and Li (2006) report α≈0:06 and thus ϕ ≈ 0.94.
R.A. Alfaro / Economics Letters 110 (2011) 1–3
is Bnt = (1 + Znt)− n. The proof of Eq. (1) is based on the following series: n−1
h i n−1 1−ϕn ϕ i n−1 n and ∑ iϕ = 1−nϕ + ðn−1Þϕ : 2 1−ϕ ð1−ϕÞ i=1
In particular the last term of Eq. (1) is obtained as follows: n−1
ð1−ϕÞ ∑ iϕ
n 1−ϕ n−1 i 1−ϕ n−1 −nϕ : ∑ iϕ = 1−ϕ ϕ i=1
The last term is identical to the one in the DNS model, and γn is zero if c2 = 0. For the DNS we consider 3 factors, one of which (λ3) is unobserved. This means λ1t = ϕ1λ1, t − 1 + e1t, λ2t = (1− ϕ2)λ3, t − 1 + ϕ2λ2, t − 1 + e2t, and λ3t = ϕ2λ3, t − 1 + e3t. As in the previous case, the (log) SDF is explained by observed factors mt + 1 = − λ1t − λ2t + e0, t + 1. Again, a natural guess for the (log) price is: bnt = Fn − G1nλ1t − G2nλ2t − G3nλ3t. Using previous results the n-period interest rate is:
znt = The discrete-time version of the model provides the same features as the continuous one: z1t = λ1t + λ2t, and lim n → ∞znt = λ1t. In other words it means that the ﬁrst factor is a long-term interest rate, also called ‘level effect’, and the second factor is the short-term interest rate (with n = 1) minus the long-term one, which is the negative of the so called ‘term spread’. The third factor is deﬁned as ‘curvature’ by Diebold and Li (2006). 3. Afﬁne version of DNS model We consider the Euler equation as the main tool for pricing assets. In particular if Mt is the stochastic discount factor (SDF), then a discount bond of maturity n can be priced at time t as follows: Bnt = Et (Bn − 1, t + 1Mt + 1). We consider a loglinear version of this equation, taking bnt = log(Bnt) and mt = log(Mt). Then we deﬁne znt = − bnt/n and bnt = Et(bn − 1, t + 1 + mt + 1); to simplify the Jensen's term is ignored in the latter.2 First, we consider that there is a unique factor that explains the (log) SDF in a linear relationship as mt + 1 = − λt + e0, t + 1, with λt + 1 = c + ϕλt + e1, t + 1, then we could guess that the (log) price of the discount bond is bnt = Fn − Gnλt. Taking γn ≡ − Fn/n the n-period interest rate can be written as follows:
znt = γn +
λt 1−ϕ ; n 1−ϕ
if |ϕ| b 1 and znt = γn + λt if ϕ = 1 (see Appendix A.1). Note that for n = 1, F1 is zero , then z1t = λt, which means that the factor is the short-term interest rate. This conclusion is identical to Vasicek (1977) who solves the non-arbitrage condition in a continuous-time setting. Balduzzi et al. (1998) add an unobserved factor, which implies that mt + 1 = − λ1t + e0, t + 1, as before, but now λ1t = (1 − ϕ1)λ2, t − 1 + ϕ1λ1, t − 1 + e1t where λ2t = c2 + ϕ2λ2, t − 1 + e2t. Again, we guess the (log) price is a linear function of the factors as bnt = Fn − G1nλ1t − G2nλ2t. Taking γn ≡ − Fn/n the n-period interest rate is (see Appendix A.2): znt = γn +
n λ1t 1−ϕn1 λ 1−ϕn2 ϕ −ϕn1 + 2t − 2 : n 1−ϕ1 n 1−ϕ2 ϕ2 −ϕ1
If ϕ2 = ϕ1 the limit of G2n is as follows lim G2n =
1−ϕn1 1−ϕn1 n−1 n−1 − lim nϕ2 = −nϕ1 ; ϕ2 →ϕ1 1−ϕ1 1−ϕ1
where L'Hopital's rule is used for the second term. Thus the n-period interest rate has the following structure: znt = γn +
λ1t 1−ϕn1 λ 1−ϕn1 n−1 + 2t −nϕ1 : n 1−ϕ1 n 1−ϕ1
n n n λ1t 1−ϕ1 λ 1−ϕ2 λ 1−ϕ2 n−1 + 2t + 3t −nϕ2 : n 1−ϕ1 n 1−ϕ2 n 1−ϕ2
The last step is to ﬁx ϕ1 = 1 and ϕ2 = ϕ for which an afﬁne process (Eq. (2)) matches the discrete-time version of DNS model presented in Eq. (1). 4. Conclusion We show that Dynamic Nelson–Siegel belongs to the afﬁne class processes. Our proof uses the Euler equation as a tool for pricing discount-bonds in a discrete-time framework. The result is obtained under the assumption that Jensen's term can be ignored. This is valid when dynamic factor disturbances have constant and small variances. If variances are constant but not negligible then an additional term should be added to the right hand side of Eq. (2), that term is timeinvariant and negative. Appendix A. Proofs A.1. One factor We note that Et(mt + 1) = − λt, and Et(bn − 1, t + 1) = Fn − 1 − Gn − 1Et (λt + 1) = Fn − 1 − Gn − 1(c + ϕλt), then using the loglinear version of the Euler equation we have: Fn − Gnλt = Fn − 1 − Gn − 1(c + ϕλt) − λt. The restrictions on the unknown coefﬁcients are Gn = 1 + ϕGn − 1, and Fn = Fn − 1 − cGn − 1. Also by construction b0, t + 1 = 0 then F0 = G0 = 0, using that we solve recursively for Gn = (1 − ϕn)/(1 − ϕ). Finally, for the case that ϕ = 1 we take the limit of Gn as follows lim ϕ → 1(1 − ϕn)/ (1 − ϕ) = lim ϕ → 1nϕn − 1 = n by L'Hopital's rule. A.2. Two factors The guess for the (log) price is: bnt = Fn − G1nλ1t − G2nλ2t. Given that only the ﬁrst factor is observed we have Et(mt + 1) = − λ1t, as before, meanwhile Et(bn − 1, t + 1) = Fn − 1 − G1, n − 1Et(λ1, t + 1) − G2, n − 1Et (λ2, t + 1) = Fn − 1 − G1, n − 1[(1 − ϕ1)λ2t + ϕ1λ1t] − G2, n − 1(c2 + ϕ2λ2t). The restrictions on coefﬁcients imply Fn =Fn − 1 − c2G2, n − 1, G1n = 1 + ϕ1G1, n − 1, and G2n = (1− ϕ1)G1, n − 1 + ϕ2G2, n − 1. Using the results of previous section we have G1n = (1− ϕn1)/(1− ϕ1). Replacing this into G2n gives us G2n = (1 − ϕn1 − 1) + ϕ2G2, n − 1, which is solved recursively. For that we consider that under n = 0 all coefﬁcients are zero, then G21 =0, G22 = 1 − ϕ1 + ϕ2G21 = 1 − ϕ1, G23 = 1 − ϕ21 + ϕ2G22 = 1 − ϕ21 + ϕ2(1 − ϕ1) = (1 + ϕ2) − (ϕ1ϕ2 + ϕ21) = (1 + ϕ2) − ϕ22 1 2 3 3 (ϕ 1ϕ− + ϕ21ϕ − 2 2 ) y G 24 = 1 − ϕ 1 + ϕ 2 G 23 = 1 − ϕ 1 + ϕ 2 [1 + ϕ 2 − (ϕ1ϕ2 + ϕ21)] = (1 + ϕ2 + ϕ22) − (ϕ1ϕ22 + ϕ21ϕ2 + ϕ31)=(1 + ϕ2 + ϕ22) − 1 2 −2 3 −3 ϕ32(ϕ1ϕ− 2 + ϕ1ϕ2 + ϕ1ϕ2 ), this means: n−2
G2n = ∑ ϕ2 −ϕ2 i=0
The Jensen's term is a positive expression that should be added next to the expected value in the loglinear version of the Euler equation. If factors have constant variances (homoscedastic) the term is proportional to those and time-invariant. Ignoring the Jensen's term implies assuming that variances are constant and small.
∑ ϕ1 ϕ2
i ϕ1 i=0 i = 0 ϕ2 n 1−ϕn2 ϕ −ϕn1 − 2 : = 1−ϕ2 ϕ2 −ϕ1 n−1
= ∑ ϕ2 −ϕ2
R.A. Alfaro / Economics Letters 110 (2011) 1–3
References Balduzzi, P., Das, S., Foresi, S., 1998. The central tendency: a second factor in bond yields. Review of Economics and Statistics 80, 62–72. Campbell, J., Lo, A., MacKinlay, A., 1997. The Econometrics of Financial Markets. Princeton University Press. Christensen, J., Diebold, F., Rudebusch, G., 2010. The Afﬁne Arbitrage-Free Class of Nelson–Siegel Term Structure Models. Manuscript University of Pennsylvania.
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