NBER WORKING PAPER SERIES THE AFFINE ARBITRAGE-FREE CLASS OF NELSON-SIEGEL ...

38 Pages·2007·0.33 MB·German

NBER WORKING PAPER SERIES
THE AFFINE ARBITRAGE-FREE CLASS OF
NELSON-SIEGEL TERM STRUCTURE MODELS
Jens H. E. Christensen
Francis X. Diebold
Glenn D. Rudebusch
Working Paper 13611
http://www.nber.org/papers/w13611
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
November 2007
The views expressed are those of the authors and do not necessarily reflect the views of others at the
Federal Reserve Bank of San Francisco. Georg Strasser provided excellent research assistance. The
views expressed herein are those of the author(s) and do not necessarily reflect the views of the National
Bureau of Economic Research.
© 2007 by Jens H. E. Christensen, Francis X. Diebold, and Glenn D. Rudebusch. All rights reserved.
Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided
that full credit, including © notice, is given to the source.
The Affine Arbitrage-Free Class of Nelson-Siegel Term Structure Models
Jens H. E. Christensen, Francis X. Diebold, and Glenn D. Rudebusch
NBER Working Paper No. 13611
November 2007
JEL No. G1,G12
ABSTRACT
We derive the class of arbitrage-free affine dynamic term structure models that approximate the widely-used
Nelson-Siegel yield-curve specification. Our theoretical analysis relates this new class of models to
the canonical representation of the three-factor arbitrage-free affine model. Our empirical analysis
shows that imposing the Nelson-Siegel structure on this canonical representation greatly improves
its empirical tractability; furthermore, we find that improvements in predictive performance are achieved
from the imposition of absence of arbitrage.
Jens H. E. Christensen Glenn D. Rudebusch
Federal Reserve Bank of San Francisco Federal Reserve Bank of San Francisco
Economic Research, MS 1130 Economic Research, MS 1130
101 Market Street 101 Market Street
San Francisco, CA 94105-9967 San Francisco, CA 94105-9967
[email protected] [email protected]
Francis X. Diebold
Department of Economics
University of Pennsylvania
3718 Locust Walk
Philadelphia, PA 19104-6297
and NBER
[email protected]
1 Introduction
Understanding the dynamic evolution of interest rates and the yield curve is important for many
diverse tasks, such as pricing long-lived assets and their financial derivatives, managing financial
risk, allocating portfolios, conducting monetary policy, purchasing capital goods, and structuring
fiscal debt. To investigate yield-curve dynamics, researchers have produced a vast literature and
a wide variety of models. However, many of these models have tended to be either theoretically
rigorous but empirically disappointing or empirically appealing but not well grounded in theory.
Inthis paper,weintroduce ahybridmodelofthe yieldcurvethatdisplaystheoreticalconsistency
as well as empirical tractability and fit.
Sincenominalbondstradeindeepandwell-organizedmarkets,thetheoreticalrestrictionsthat
ruleoutopportunitiesforrisklessarbitrageacrossmaturitiesandovertimeholdapowerfulappeal,
and they provide the foundation for a large finance literature on arbitrage-free (AF) models that
started with Vasiˇcek (1977) and Cox, Ingersoll, and Ross (1985). These models specify the risk-
neutral evolution of the underlying yield-curve factors as well as the dynamics of risk premiums.
Following Duffie and Kan (1996), the affine versions of these models are particularly popular
becauseyieldsareconvenientlinearfunctions ofunderlyinglatentfactors(state variablesthatare
unobserved by the econometrician) with parameters, or “factor loadings,” that can be calculated
from a simple system of differential equations.
Unfortunately, the canonical affine AF models can exhibit poor empirical time series perfor-
mance, especially when forecasting future yields (Duffee, 2002). In addition, the estimation of
these models is known to be problematic, in large part because of the existence of numerous
model likelihood maxima that have essentially identical fit to the data but very different impli-
cations for economic behavior (Kim and Orphanides, 2005). These empirical problems appear
to reflect an underlying model over-parameterization, and as a solution, many researchers (e.g.,
Duffee, 2002, and Dai and Singleton, 2002) simply restrict to zero those parameters with small
t-statistics in a first round of estimation. The resulting more parsimonious structure is typically
somewhat easier to estimate and has a more robust economic interpretation (fewer troublesome
likelihood maxima). However,these additional restrictions on model structure are arbitrary from
both a theoretical and a statistical perspective. Furthermore, their arbitrary application, along
with the computational burden of estimation, effectively precludes thorough simulation studies
of the finite-sample properties of the estimators of the canonical affine model, thus, complicating
model validation. In part to overcome such problems, this paper considers the application of a
new, arguably less arbitrary,structure to the affine AF class of models.
Our new AF model structure is based on the workhorseyield-curve representationintroduced
byNelsonandSiegel(1987). TheNelson-Siegelmodelisaflexiblecurvethatprovidesaremarkably
good fit to the cross section of yields in many countries, and it is very popular among financial
market practitioners and central banks (e.g., Svensson, 1995,Bank for International Settlements,
2005, and Gu¨rkaynak, Sack, and Wright, 2006). Moreover, Diebold and Li (2006) develop a
dynamic version of this model and show that it corresponds exactly to a modern factor model,
with yields that are affine in three latent factors, which have a standard interpretation of level,
slope,andcurvature. SuchadynamicNelson-Siegel(DNS)modeliseasytoestimate,andDiebold
1
and Li (2006) show that it forecasts the yield curve quite well. Unfortunately, despite its good
empiricalperformance, the DNS model does not impose the desirable theoreticalrestrictions that
rule out opportunities for riskless arbitrage (e.g., Filipovi´c, 1999, and Diebold, Piazzesi, and
Rudebusch, 2005).
In this paper, we show how to reconcile the Nelson-Siegel model with the absence of arbitrage
by derivingthe classofAFDNS models,whichareaffine AF termstructuremodels thatmaintain
the Nelson-Siegel factor-loading structure. These models combine the best of both yield-curve
modeling traditions. They maintain the AF theoretical restrictions of the canonical affine mod-
els but can be easily and robustly estimated because the Nelson-Siegel structure helps identify
the latent yield-curve factors. In particular, empirical implementation of the AFDNS models is
facilitated by the fact that zero-couponbond prices have analytical solutions, which we provide.
After deriving the new class of AFDNS models, we examine their in-sample fit and out-of-
sampleforecastperformancerelativetostandardDNSmodels. ForboththeDNSandtheAFDNS
models,we estimate parsimoniousandflexible versions(with independent factorsandmorerichly
parameterized correlated factors, respectively). We find that the flexible versions of both models
arepreferredforin-samplefit; however,the parsimoniousversionsexhibitsignificantlybetterout-
of-sample forecast performance. Finally, and most importantly, we find that the parsimonious
AFDNS model outperforms its DNS counterpart in forecasting, which supports the imposition of
the AF restrictions.
We proceed as follows. Section 2 introduces the DNS model and derives the main theoretical
result of the paper, which defines the AFDNS class of models. Section 3 derives the relationship
between the AFDNS class of models and the canonical representation of affine AF models intro-
duced by Dai and Singleton (2000). For the four specific DNS and AFDNS models used in our
empirical analysis, Section 4 describes the estimation method, data, and in-sample fit, while Sec-
tion 5 examines out-of-sample forecastperformance. Section 6 concludes, and appendices contain
additional technical details.
2 Nelson-Siegel term structure models
In this section, we review the DNS model and introduce the AFDNS class of arbitrage-free affine
term structure models that maintain the Nelson-Siegel factor loading structure.
2.1 The dynamic Nelson-Siegel model
The original Nelson-Siegel model fits the yield curve with the simple functional form
1−e−λτ 1−e−λτ
y(τ)=β +β +β −e−λτ , (1)
0 1 λτ 2 λτ
(cid:16) (cid:17) (cid:16) (cid:17)
where y(τ) is the zero-coupon yield with τ years to maturity, and β , β , β , and λ are model
0 1 2
parameters.
As noted earlier, this representation is commonly used by financial market practitioners to
fit the yield curve at a point in time. Although for some purposes such a static representation
2
appearsuseful,adynamicversionisrequiredtounderstandtheevolutionofthebondmarketover
time. Therefore, Diebold and Li (2006) reinterpret the β coefficients as time-varying factors L ,
t
S , and C , so
t t
1−e−λτ 1−e−λτ
y (τ)=L +S +C −e−λτ . (2)
t t t t
λτ λτ
(cid:16) (cid:17) (cid:16) (cid:17)
Given their Nelson-Siegel factor loadings, these factors can be interpreted as level, slope, and
curvature. DieboldandLiassumeanautoregressivestructureforthesethree factors,whichyields
the DNS model—a fully dynamic Nelson-Siegel specification.
Empirically, the DNS model is very tractable and provides a good fit to the data; however,as
a theoretical matter, it does not require that the dynamic evolution of yields and the yield curve
at any point in time cohere such that arbitrage opportunities are precluded. Indeed, the results
of Filipovi´c (1999) imply that whatever stochastic dynamics are chosen for the DNS factors,
it is impossible to rule out arbitrage at the bond prices implicit in the resulting Nelson-Siegel
yield curve. Hence, the discounted prices of zero-coupon bonds in the DNS model are not semi-
martingale processes under the pricing or Q-measure. The next subsection shows how to remedy
this theoretical weakness.
2.2 The AFDNS model
OurderivationoftheAFDNSmodelstartsfromthestandardcontinuous-timeaffineAFstructure
(DuffieandKan,1996).1 Torepresentanaffinediffusionprocess,defineafilteredprobabilityspace
(Ω,F,(F ),Q), where the filtration (F ) = {F : t ≥ 0} satisfies the usual conditions (Williams,
t t t
1997). The state variable X is assumed to be a Markov process defined on a set M ⊂ Rn that
t
solves the following stochastic differential equation (SDE)2
dX =KQ(t)[θQ(t)−X ]dt+Σ(t)D(X ,t)dWQ, (3)
t t t t
where WQ is a standard Brownian motion in Rn, the information of which is contained in the
filtration(F ). ThedrifttermsθQ :[0,T]→RnandKQ :[0,T]→Rn×n arebounded,continuous
t
functions.3 Similarly, the volatility matrix Σ : [0,T] → Rn×n is assumed to be a bounded,
continuous function, while D : M ×[0,T] → Rn×n is assumed to have the following diagonal
structure
γ1(t)+δ1(t)X1+...+δ1(t)Xn ... 0
1 t n t
 p ... ... ... .
 0 ... γn(t)+δn(t)X1+...+δn(t)Xn 
 1 t n t 
 
1Krippner(2006) derivesaspecialcaseoftheAFDNSmopdelwithconstantriskpremiums.
2Theaffinepropertyappliestobondprices;therefore,affinemodelsonlyimposestructureonthefactordynamics
underthepricingmeasure.
3Stationarity of the state variables is ensured if all the eigenvalues of KQ(t) are positive (if complex, the real
component should be positive), see Ahn, Dittmar, and Gallant (2002). However, stationarity is not a necessary
requirementfortheprocesstobewelldefined.
3
To simplify the notation, γ(t) and δ(t) are defined as
γ1(t) δ1(t) ... δ1(t)
1 n
γ(t)= ...  and δ(t)= ... ... ... ,
 γn(t)   δn(t) ... δn(t) 
   1 n 
   
where γ : [0,T] → Rn and δ : [0,T] → Rn×n are bounded, continuous functions. Given this
notation, the SDE of the state variables can be written as
γ1(t)+δ1(t)X ... 0
t
dXt =KQ(t)[θQ(t)−Xt]dt+Σ(t) p ... ... ... dWtQ,
 0 ... γn(t)+δn(t)X 
 t 
 
p
where δi(t) denotes the ith row of the δ(t)-matrix. Finally, the instantaneous risk-free rate is
assumed to be an affine function of the state variables
′
r =ρ (t)+ρ (t)X ,
t 0 1 t
where ρ :[0,T]→R and ρ :[0,T]→Rn are bounded, continuous functions.
0 1
Duffie and Kan (1996)prove that zero-couponbond prices in this framework are exponential-
affine functions of the state variables
T
P(t,T)=EQ exp − r du =exp B(t,T)′X +C(t,T) ,
t u t
Zt
(cid:2) (cid:0) (cid:1)(cid:3) (cid:0) (cid:1)
where B(t,T) and C(t,T) are the solutions to the following system of ordinary differential equa-
tions (ODEs)
n
dB(t,T) 1
= ρ +(KQ)′B(t,T)− (Σ′B(t,T)B(t,T)′Σ) (δj)′, B(T,T)=0 (4)
dt 1 2 j,j
j=1
X
n
dC(t,T) 1
= ρ −B(t,T)′KQθQ− (Σ′B(t,T)B(t,T)′Σ) γj, C(T,T)=0, (5)
dt 0 2 j,j
j=1
X
and the possible time-dependence of the parameters is suppressed in the notation. These pricing
functions imply that the zero-couponyields are given by
1 B(t,T)′ C(t,T)
y(t,T)=− logP(t,T)=− X − .
t
T −t T −t T −t
Therefore, for a three-factor affine model with X = (X1,X2,X3), the closest approximation to
t t t t
the Nelson-Siegel yield function is a yield function of the form
1−e−λ(T−t) 1−e−λ(T−t) C(t,T)
y(t,T)=X1+ X2+ −e−λ(T−t) X3− ,
t λ(T −t) t λ(T −t) t T −t
h i
4
so the ODEs for the B(t,T) functions have these solutions:
B1(t,T) = −(T −t),
1−e−λ(T−t)
B2(t,T) = − ,
λ
1−e−λ(T−t)
B3(t,T) = (T −t)e−λ(T−t)− .
λ
As described in the following proposition, there exists a unique class of affine AF models that
satisfy these ODEs.
Proposition 1:
Assume that the instantaneous risk-free rate is defined by
r =X1+X2.
t t t
In addition, assume that the state variables X = (X1,X2,X3) are described by the following
t t t t
system of SDEs under the risk-neutral Q-measure
dX1 0 0 0 θQ X1 dW1,Q
t 1 t t
 dX2 = 0 λ −λ  θQ − X2 dt+Σ dW2,Q , λ>0.
t 2 t t
 dXt3   0 0 λ  θ3Q   Xt3   dWt3,Q 
        
        
Then, zero-coupon bond prices are given by
T
P(t,T)=EQ exp − r du =exp B1(t,T)X1+B2(t,T)X2+B3(t,T)X3+C(t,T) ,
t u t t t
Zt
(cid:2) (cid:0) (cid:1)(cid:3) (cid:0) (cid:1)
whereB1(t,T),B2(t,T),B3(t,T),andC(t,T)arethe unique solutions tothe followingsystemof
ODEs:
dB1(t,T) 1 0 0 0 B1(t,T)
dt
 dB2(t,T) = 1 + 0 λ 0  B2(t,T)  (6)
dt
 dB3(t,T)   0   0 −λ λ  B3(t,T) 
 dt      
      
and
3
dC(t,T) 1
=−B(t,T)′KQθQ− Σ′B(t,T)B(t,T)′Σ , (7)
dt 2 j,j
j=1
X(cid:0) (cid:1)
with boundary conditions B1(T,T) = B2(T,T)= B3(T,T) = C(T,T) = 0. The unique solution
5
for this system of ODEs is:
B1(t,T) = −(T −t),
1−e−λ(T−t)
B2(t,T) = − ,
λ
1−e−λ(T−t)
B3(t,T) = (T −t)e−λ(T−t)− ,
λ
and
T T 1 3 T
C(t,T)=(KQθQ) B2(s,T)ds+(KQθQ) B3(s,T)ds+ Σ′B(s,T)B(s,T)′Σ ds.
2Z 3Z 2 Z j,j
t t Xj=1 t ` ´
Finally, zero-coupon bond yields are given by
1−e−λ(T−t) 1−e−λ(T−t) C(t,T)
y(t,T)=X1+ X2+ −e−λ(T−t) X3− .
t λ(T −t) t λ(T −t) t T −t
h i
Proof: See Appendix A.
This proposition defines the AFDNS model and has several interesting implications. First,
the three state variables are Gaussian Ornstein-Uhlenbeck processes with a constant volatility
matrix Σ.4 The instantaneous interest rate is the sum of level and slope factors (X1 and X2),
t t
while the curvature factor (X3) is a truly latent factor in the sense that its sole role is as a
t
stochastic time-varying mean for the slope factor under the Q-measure. Second, Proposition
1 only imposes structure on the dynamics of the AFDNS model under the Q-measure and is
silent about the dynamics under the P-measure. Still, the observation that curvature is a truly
latent factor generally accords with the empirical literature where it has been difficult to find
sensible interpretations of curvature under the P-measure (Diebold, Rudebusch, and Aruoba,
2006). Similarly, the level factor is a unit-root process under the Q-measure, which accords with
the usual finding that one or more of the interest rate factors are close to being nonstationary
processes under the P-measure.5 Third, Proposition 1 provides insight into the nature of the
parameter λ. Nelson and Siegel (1987) allowed λ to vary at each point in time, but subsequent
dynamicimplementationsoftheNelson-Siegelmodeltendedtofixλoverthesample(e.g.,Diebold
4Proposition1canbeextended toincludejumpsinthestatevariables. Aslongasthejumparrivalintensityis
state-independent, theNelson-SiegelfactorloadingstructureintheyieldfunctionismaintainedsinceonlyC(t,T)
isaffectedbytheinclusionofsuchjumps. SeeDuffie,Pan,andSingleton(2000)fortheneededmodificationofthe
ODEsforC(t,T)inthiscase.
5With the unit root in the level factor, as maturity increases, −C(t,T) → −∞, which implies that, strictly
T−t
speaking,thismodelisnotarbitrage-free. However,ifwemodifythemean-reversionmatrixKQ to
ε 0 0
KQ(ε)=0 0 λ −λ 1
0 0 λ
@ A
andconsideraconvergingsequence εn>0,εn↓0,thenthereisaconverging sequence ofAFmodelswithalimit
given bythe resultinProposition 1. Thus, bychoosing ε>0sufficiently small,we can obtain anAFmodel that
isindistinguishablefromtheAFDNSmodelinProposition1.
6
and Li, 2006). In the AFDNS model, λ is indeed a constant, namely, the mean-reversion rate of
the curvature and slope factors as well as the scale by which a deviation of the curvature factor
from its mean affects the mean of the slope factor. Fourth, relative to the Nelson-Siegel model,
the AFDNS model contains an additional maturity-dependent term −C(t,T) in the function for
T−t
the zero-coupon bond yields. The nature of this “yield-adjustment” term is crucial in assessing
differences between the AFDNS and DNS models, and we now turn to a theoretical analysis of
this term.
2.3 The AFDNS yield-adjustment term
The only parameters in the system of ODEs for the AFDNS B(t,T) functions are ρ and KQ,
1
i.e., the factor loadings of r and the mean-reversion structure for the state variables under the
t
Q-measure. The drift term θQ and the volatility matrix Σ do not appear in the ODEs but in
the yield-adjustment term −C(t,T). Therefore, in the AFDNS model, the choice of the volatility
T−t
matrix Σ affects both the P-dynamics and the yield function through the yield-adjustment term.
In contrast, the DNS model is silent about the real-world dynamics of the state variables, so the
choice of P-dynamics is irrelevant for the yield function.
As discussed in the next section, we identify the AFDNS models by fixing the mean levels of
the state variablesunder the Q-measureat 0, i.e., θQ =0. This implies that the yield-adjustment
term will have the following form:
C(t,T) 1 1 3 T
′ ′
− =− ΣB(s,T)B(s,T)Σ ds.
T −t 2T −t j,j
j=1Zt
X (cid:0) (cid:1)
Given a general volatility matrix
σ σ σ
11 12 13
Σ= σ σ σ ,
21 22 23
 σ31 σ32 σ33 
 
 
the yield-adjustment term can be derived in analytical form (see Appendix B) as
C(t,T) 1 1 T 3
= Σ′B(s,T)B(s,T)′Σ ds
T −t 2T−tZt jX=1` ´j,j
(T −t)2 1 1 1−e−λ(T−t) 1 1−e−2λ(T−t)
= A +B − +
6 h2λ2 λ3 T−t 4λ3 T −t i
1 1 1 3 2 1−e−λ(T−t) 5 1−e−2λ(T−t)
+ C + e−λ(T−t)− (T −t)e−2λ(T−t)− e−2λ(T−t)− +
h2λ2 λ2 4λ 4λ2 λ3 T −t 8λ3 T−t i
1 1 1 1−e−λ(T−t)
+ D (T −t)+ e−λ(T−t)−
h2λ λ2 λ3 T −t i
3 1 1 3 1−e−λ(T−t)
+ E e−λ(T−t)+ (T −t)+ (T −t)e−λ(T−t)−
hλ2 2λ λ λ3 T−t i
1 1 1 3 1−e−λ(T−t) 3 1−e−2λ(T−t)
+ F + e−λ(T−t)− e−2λ(T−t)− + ,
hλ2 λ2 2λ2 λ3 T −t 4λ3 T−t i
where
7
• A=σ2 +σ2 +σ2 ,
11 12 13
• B =σ2 +σ2 +σ2 ,
21 22 23
• C =σ2 +σ2 +σ2 ,
31 32 33
• D =σ σ +σ σ +σ σ ,
11 21 12 22 13 23
• E =σ σ +σ σ +σ σ ,
11 31 12 32 13 33
• F =σ σ +σ σ +σ σ .
21 31 22 32 23 33
This result has two implications. First, the fact that zero-coupon bond yields in the AFDNS
class of models are given by an analyticalformula will greatlyfacilitate empiricalimplementation
of these models. Second, the nine underlying volatility parameters are not identified. Indeed,
only the six terms A, B, C, D, E, and F can be identified; thus, the maximally flexible AFDNS
specification that can be identified has a triangular volatility matrix given by6
σ 0 0
11
Σ= σ σ 0 .
21 22
 σ31 σ32 σ33 
 
 
In Section 4, we quantify the yield-adjustment term and examine how it affects the empirical
performance of two specific AFDNS models relative to their corresponding DNS models. These
models are introduced next.
2.4 Four specific Dynamic Nelson-Siegel models
In general,the DNS andAFDNS models are silentabout the P-dynamics,so there are aninfinite
number of possible specifications that could be used to match the data. However, for continuity
with the existing literature, our econometric analysis focuses on two specific versions of the DNS
model that have been estimated in recent studies, and, for consistency, we also examine the two
corresponding versions of the AFDNS model.
In the independent-factor DNS model, all three state variables are assumed to be independent
first-order autoregressions, as in Diebold and Li (2006). Using their notation, the state equation
is given by
Lt−µL a11 0 0 Lt−1−µL ηt(L)
 St−µS = 0 a22 0  St−1−µS + ηt(S) ,
 Ct−µC   0 0 a33  Ct−1−µC   ηt(C) 
      
      
where the error terms η (L), η (S), and η (C) have a conditional covariance matrix given by
t t t
q2 0 0
11
Q= 0 q2 0 .
22
 0 0 q323 
 
 
6Thechoiceofupperorlowertriangularisirrelevantforthefitofthemodel.
8