The Affine Arbitrage-Free Class of Nelson-Siegel Term Structure Models† Jens H. E. Christensen Federal Reserve Bank of San Francisco [email protected] Francis X. Diebold University of Pennsylvania and NBER [email protected] Glenn D. Rudebusch Federal Reserve Bank of San Francisco [email protected] Abstract We derive the class of arbitrage-free affine dynamic term structure models that approxi- matethewidely-used Nelson-Siegel yield-curvespecification. Ourtheoretical analysis relates this new class of models to the canonical representation of the three-factor arbitrage-free affine model. Ourempirical analysis shows that imposing the Nelson-Siegel structure on the canonical representation of affine models greatly improves its empirical tractability; further- more,wefindthatimprovementsinpredictiveperformance areachievedfrom theimposition of absence of arbitrage. †TheviewsexpressedarethoseoftheauthorsanddonotnecessarilyreflecttheviewsofothersattheFederalRe- serveBankofSanFrancisco. GeorgStrasserprovidedexcellentresearchassistance. Wethankseminarparticipants atCopenhagen BusinessSchool, especiallyAndersBjerreTrolle,andatStanfordUniversityforcomments. Firstdraft: September 2007. Thisversion: May2008. 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, 2007). 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, 1 slope,andcurvature. SuchadynamicNelson-Siegel(DNS)modeliseasytoestimate,andDiebold 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 deriving the class of AFNS models, which are affine AF term structure models that maintain 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 models butcanbe easilyandrobustlyestimatedbecausethe Nelson-Siegelstructurehelpsidentifythe la- tentyield-curvefactors. Inparticular,empiricalimplementationoftheAFNSmodelsisfacilitated by the fact that zero-coupon bond prices have analytical solutions, which we provide. After deriving the new class of AFNS models, we examine their in-sample fit and out-of- sample forecast performance relative to standard DNS models. For both the DNS and the AFNS 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.1 Finally, and most importantly, we find that the parsimonious AFNS 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 AFNS class of models. Section 3 derives the relationship betweentheAFNSclassofmodelsandthecanonicalrepresentationofaffineAFmodelsasdetailed in Singleton (2006). For the four specific DNS and AFNS models used in our empirical analysis, Section 4 describes the estimation method, data, andin-sample fit, while Section 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 AFNS 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) 1Chua et al. (2008) also use a very parsimonious model with forward rates formulated as exponential-affine functionsofthestatevariablesandfindthatsuchamodelperformswellintermsofforecasting. 2 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 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 AFNS model Our derivation of the AFNS model starts from the standard continuous-time affine AF structure (DuffieandKan,1996).2 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)3 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.4 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 2Krippner(2006) derivesaspecialcaseoftheAFNSmodelwithconstant riskpremiums. 3Theaffinepropertyappliestobondprices;therefore,affinemodelsonlyimposestructureonthefactordynamics underthepricingmeasure. 4Stationarity 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 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    p 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 4 Given these pricing functions, for a three-factor affine model with X =(X1,X2,X3), the closest t t t t match to the Nelson-Siegel yield function would be 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 with ODEs for the B(t,T) functions that 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 In this case,the factor loadingsexactly matchthe Nelson-Siegelones,but there is anunavoidable additional term in the yield function −C(t,T), which only depends on the maturity of the bond. T−t As described in Proposition 1, there exists a unique class of affine AF models that satisfy the above 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              5 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 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. 2 3 2 j,j Zt Zt j=1Zt X (cid:0) (cid:1) 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. The existence of an AFNS model, as defined in this proposition, is not too surprising from a theoreticalperspective. FollowingTrolleandSchwartz,2007,thedynamicsofaforwardratecurve in a general m-dimensional Heath-Jarrow-Morton (HJM) model can always be represented by a finite-dimensional Markov process with time-homogeneous volatility structure if each volatility function is given by σ (t,T)=p (T −t)e−γi(T−t), i=1,...,m, i n,i where p (τ) is an n-order polynomial in τ. Since the forward rates in the DNS model satisfy n,i this requirement, there exists such an arbitrage-freethree-dimensional HJM model. However,the simplicity of the solution in the case of the Nelson-Siegel model presented in Proposition 1 is striking. Proposition 1 also has several interesting implications. First, the three state variables are Gaussian Ornstein-Uhlenbeck processes with a constant volatility matrix Σ.5 The instantaneous interest rate is the sum of level and slope factors (X1 and X2), while the curvature factor (X3) t t t is a truly latent factor in the sense that its sole role is as a stochastic time-varying mean for the slope factor under the Q-measure. Second, Proposition 1 only imposes structure on the 5Proposition1canbeextended toincludejumpsinthestatevariables. Aslongasthejumparrivalintensityis state-independent, theNelson-SiegelfactorloadingstructureintheyieldfunctionismaintainedsinceonlyC(t,T) isaffectedbytheinclusionofsuchjumps. SeeDuffie,Pan,andSingleton(2000)fortheneededmodificationofthe ODEsforC(t,T)inthiscase. 6 dynamics of the AFNS model under the Q-measure and is silent about the dynamics under the P-measure. Still, theobservationthatcurvatureisatrulylatentfactorgenerallyaccordswiththe 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 processundertheQ-measure,whichaccordswiththeusualfindingthatoneormoreoftheinterest rate factors are close to being nonstationary processes under the P-measure.6 Third, Proposition 1 provides insight into the nature of the parameter λ. Although in principle λ could vary over time, starting with Nelson and Siegel (1987), implementations of the Nelson-Siegel model have almost always fixed λ over the sample. In the AFNS 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-Siegelmodel,theAFNSmodelcontainsanadditionalmaturity-dependentterm−C(t,T) in T−t thefunctionforthezero-couponbondyields. Thenatureofthis“yield-adjustment”termiscrucial in assessing differences between the AFNS and DNS models, and we now turn to a theoretical analysis of this term. 2.3 The AFNS yield-adjustment term The only parameters in the system of ODEs for the AFNS 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 AFNS 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 AFNS 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) 6With 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 λ −λ   0 0 λ   andconsideraconvergingsequence εn>0,εn↓0,thenthereisaconverging sequence ofAFmodelswithalimit given bythe resultinProposition 1. Thus, bychoosing ε>0sufficiently small,we can obtain anAFmodel that isindistinguishablefromtheAFNSmodelinProposition1. 7 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(cid:0) (cid:1)j,j (T −t)2 1 1 1−e−λ(T−t) 1 1−e−2λ(T−t) = A +B − + 6 2λ2 λ3 T−t 4λ3 T −t h 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)− + 2λ2 λ2 4λ 4λ2 λ3 T −t 8λ3 T−t h i 1 1 1 1−e−λ(T−t) + D (T −t)+ e−λ(T−t)− 2λ λ2 λ3 T −t h i 3 1 1 3 1−e−λ(T−t) + E e−λ(T−t)+ (T −t)+ (T −t)e−λ(T−t)− λ2 2λ λ λ3 T−t h 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)− + , λ2 λ2 2λ2 λ3 T −t 4λ3 T−t h i where • 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 Thisresulthastwoimplications. First,thefactthatzero-couponbondyieldsintheAFNSclassof models aregivenby ananalyticalformulawillgreatlyfacilitateempiricalimplementationofthese models. Second, the nine underlying volatility parametersare not identified. Indeed, only the six terms A, B, C, D, E, and F can be identified; thus, the maximally flexible AFNS specification that can be identified has a triangular volatility matrix given by7 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 AFNS models relative to their corresponding DNS models. These models are introduced next. 7Thechoiceofupperorlowertriangularisirrelevantforthefitofthemodel. 8 2.4 Four specific Nelson-Siegel models In general, the DNS and AFNS models are silent about the P-dynamics, so there are an infinite 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 AFNS 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      The correlated-factor DNS model has factor P-dynamics described by a first-order vector au- toregression(VAR(1)) Lt−µL a11 a12 a13 Lt−1−µL ηt(L)  St−µS = a21 a22 a23  St−1−µS + ηt(S) ,  Ct−µC   a31 a32 a33  Ct−1−µC   ηt(C)                asinDiebold,Rudebusch,andAruoba(2006). Theinnovationsη (L),η (S),andη (C)areallowed t t t tobe correlatedwithaconditionalcovariancematrixgivenbyQ=qq′,wherethe Choleskyfactor q of the covariance matrix Q is q 0 0 11 q = q q 0 . 21 22  q31 q32 q33      In both of these DNS models, the measurement equation takes the form  yytt((..ττ21)) = 11.. 11−−λλee−−ττ..12λλττ12 11−−λλee−−ττ12λλττ12..−−ee−−λλττ12  LStt + εεtt((..ττ12)) ,  .   . . .   .   yt(τN)   1 1−λe−τNλτN 1−λe−τNλτN −e−λτN  Ct   εt(τN)        where the measurement errors ε (τ ) are assumed to be i.i.d. white noise. t i The corresponding AFNS models are formulated in continuous time and the relationship be- 9