Modeling Yields at the Zero Lower Bound: Are Shadow Rates the Solution?

Transcription

1 FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES Modeling Yields a he Zero Lower Bound: Are Shadow Raes he Soluion? Jens H. E. Chrisensen, Federal Reserve Bank of San Francisco Glenn D. Rudebusch, Federal Reserve Bank of San Francisco December 2013 Working Paper hp:// The views in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem.

2 Modeling Yields a he Zero Lower Bound: Are Shadow Raes he Soluion? Jens H. E. Chrisensen and Glenn D. Rudebusch Federal Reserve Bank of San Francisco 101 Marke Sree, Mailsop 1130 San Francisco, CA Absrac Recen U.S. Treasury yields have been consrained o some exen by he zero lower bound (ZLB) on nominal ineres raes. In modeling hese yields, we compare he performance of a sandard affine Gaussian dynamic erm srucure model (DTSM), which ignores he ZLB, and a shadowrae DTSM, which respecs he ZLB. We find ha he sandard affine model is likely o exhibi declines in fi and forecas performance wih very low ineres raes. In conras, he shadow-rae model miigaes ZLB problems significanly and we documen superior performance for his model class in he mos recen period. JEL Classificaion: G12, E43, E52, E58. Keywords: erm srucure modeling, zero lower bound, moneary policy. We hank conference paricipans a he FRBSF Workshop on Term Srucure Modeling a he Zero Lower Bound especially Don Kim for helpful commens. The views in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem. We hank Lauren Ford for excellen research assisance. This version: December 17, 2013.

3 1 Inroducion Wih nominal yields on governmen deb in several counries having fallen very near heir zero lower bound (ZLB), undersanding how o model he erm srucure of ineres raes when some of hose ineres raes are near he ZLB is an issue ha commands aenion boh for bond porfolio pricing and risk managemen and for macroeconomic and moneary policy analysis. The iming of he ZLB period in he Unied Saes can be seen in Figure 1. 1 The sar of he ZLB period is commonly daed o December 16, 2008, when he Federal Open Marke Commiee (FOMC) lowered is arge policy rae he overnigh federal funds rae o a range of 0 o 1/4 percen. The key empirical quesion of he paper is o exrac reliable marke-based measures of expecaions for fuure moneary policy when nominal ineres raes are near he ZLB. Unforunaely, he workhorse represenaion in finance for bond pricing he affine Gaussian dynamic erm srucure model ignores he ZLB and places posiive probabiliies on negaive ineres raes as we will show. This counerfacual oucome resuls from ignoring he exisence of a readily available currency for ransacions. For in he real world, an invesor always has he opion of holding cash, and he zero nominal yield of cash will dominae any securiy wih a negaive yield. 2 Insead, o handle he problem of near-zero yields, we rely on he shadow-rae arbirage-free Nelson-Siegel (AFNS) model class inroduced in Chrisensen and Rudebusch (2013). 3 These are laen-facor models where he sae variables have sandard Gaussian dynamics, bu he shor rae is given an inerpreaion of a shadow rae in he spiri of Black (1995) o accoun for he effec on bond pricing from he exisence of he opion o hold currency. As a consequence, he models respec he ZLB. Furhermore, due o he Gaussian dynamics, hese shadow-rae models are as flexible and empirically racable as regular AFNS models. In he empirical analysis, we compare he resuls from his new shadow-rae AFNS model o hose obained from a regular AFNS model esimaed on he same sample. We find ha shadow-rae models can provide beer fi as measured by in-sample merics such as he RMSEs of fied yields and he likelihood values. Sill, i is eviden from hese in-sample resuls ha a sandard hree-facor Gaussian dynamic erm srucure model like our Gaussian hree-facor AFNS model has enough flexibiliy o fi he cross-secion of yields fairly well a each poin in ime even when he shorer end of he yield curve is flaened ou a he ZLB. However, i is no he case ha he Gaussian model can accoun for all aspecs of he erm srucure a he ZLB. Indeed, we show ha our esimaed hree-facor Gaussian model clearly fails along wo dimensions. Firs, despie fiing he yield curve, 1 The daa are nominal U.S. Treasury zero-coupon yields and described laer in he paper. 2 Acually, he ZLB can be a somewha sof floor. The non-negligible coss of ransacing in and holding large amouns of currency have allowed governmen bond yields o push slighly below zero in a few counries, noably in Denmark recenly. To capure a lower bound on bond yields ha depends on insiuional fricions, we could replace he lower bound of zero wih some appropriae, possibly ime-varying, negaive epsilon. 3 See Diebold and Rudebusch (2013) for a comprehensive presenaion of relaed applicaions of he AFNS model. 1

4 Rae in percen FOMC Dec Ten year yield One year yield FOMC Aug Figure 1: Treasury Yields Since One- and en-year weekly U.S. Treasury zero-coupon bond yields from January 7, 2005, o December 28, he model canno capure he dynamics of yields a he ZLB. One sark indicaion of his is he high probabiliy he model assigns o negaive fuure shor raes obviously a poor predicion. Second, he sandard model misses he compression of yield volailiy ha occurs a he ZLB as expeced fuure shor raes are pinned near zero, longer-erm raes flucuae less. The shadow-rae model, even wihou incorporaing sochasic volailiy, can capure his effec. In erms of forecasing fuure shor raes, we firs esablish ha he regular AFNS model is compeiive over he normal period from 1995 o Thus, his model could have been expeced o coninue o perform well in he mos recen period, if only i had no been for he problems associaed wih he ZLB. Second, we show ha during he mos recen period he shadow-rae model sands ou in erms of forecasing fuure shor raes in addiion o performing on par wih he regular model during he normal period. Third, he deerioraion in shor rae forecass implies ha he regular model delivers exaggeraed esimaes of he policy expecaions embedded in he yield curve in recen years. In urn, his leads us o conclude ha is erm premium esimaes are arificially low during ha period. 4 As a consequence of hese findings combined, we recommend o use a shadow-rae modeling approach no only when yields are as low as hey were owards he end of our sample, bu in general. 4 Ichiue and Ueno (2013) also compare sandard and shadow-rae Gaussian models for U.S. Treasury daa and find deerioraion in he performance of heir sandard model during he mos recen period. However, hey only sudy wo-facor models. 2

5 Finally, we should menion wo alernaive frameworks o modeling yields near he ZLB ha guaranee posiive ineres raes: sochasic-volailiy models wih square-roo processes and Gaussian quadraic models. Boh of hese approaches suffer from he heoreical weakness ha hey rea he ZLB as a reflecing barrier and no as an absorbing one as in he shadow-rae model. Empirically, of course, he recen prolonged period of very low ineres raes seem more consisen wih an absorbing sae. In addiion, Dai and Singleon (2002) disparage he fi of sochasic-volailiy models, while Kim and Singleon (2012) compare quadraic and shadow-rae empirical represenaions and find a sligh preference for he laer. Sill, we consider all hree modeling approaches o be worhy of furher invesigaion, bu we view he shadow-rae model o be of paricular ineres because away from he ZLB i reduces exacly o he sandard Gaussian affine model, which is by far he mos popular dynamic erm srucure model. Therefore, he enire voluminous lieraure on affine Gaussian models remains compleely applicable and relevan when given a modes shadow-rae weak o handle he ZLB. The res of he paper is srucured as follows. Secion 2 describes Gaussian models in general as well as a specific empirical Gaussian model ha we consider, while Secion 3 deails our shadow-rae model. Secion 4 conains our empirical findings and discusses he implicaions for assessing policy expecaions and erm premiums in he curren low-yield environmen. Secion 5 concludes. Two appendices conain addiional echnical deails. 2 A Sandard Gaussian Term Srucure Model In his secion, we provide an overview of he affine Gaussian erm srucure model, which ignores he ZLB, and describe an empirical example of his model. 2.1 The General Model Le P (τ) be he price of a zero-coupon bond a ime ha pays $1, a mauriy + τ. Under sandard assumpions, his price is given by P (τ) = E P [ M+τ ], M wherehesochasic discounfacor, M, denoes hevaluea ime 0 of aclaim a afuuredae, and he superscrip P refers o he acual, or real-world, probabiliy measure underlying he dynamics of M. (As we will discuss in he nex secion, here is no resricion in his sandard seing o consrain P (τ) from rising above is par value; ha is, he ZLB is ignored.) We follow he usual reduced-form empirical finance approach ha models bond prices wih unobservable (or laen) facors, here denoed as X, and he assumpion of no residual arbirage 3

6 opporuniies. We assume ha X follows an affine Gaussian process wih consan volailiy, wih dynamics in coninuous ime given by he soluion o he following sochasic differenial equaion (SDE): dx = K P (θ P X )d+σdw P, where K P is an n n mean-reversion marix, θ P is an n 1 vecor of mean levels, Σ is an n n volailiy marix, and W P is an n-dimensional Brownian moion. The dynamics of he sochasic discoun funcion are given by dm = r M d+γ M dw P, and he insananeous risk-free rae, r, is assumed affine in he sae variables r = δ 0 +δ 1 X, where δ 0 R and δ 1 R n. The risk premiums, Γ, are also affine Γ = γ 0 +γ 1 X, where γ 0 R n and γ 1 R n n. Duffie and Kan (1996) show ha hese assumpions imply ha zero-coupon yields are also affine in X : y (τ) = 1 τ A(τ) 1 τ B(τ) X, where A(τ) and B(τ) are given as soluions o he following sysem of ordinary differenial equaions db(τ) dτ da(τ) dτ = δ 1 (K P +Σγ 1 ) B(τ), B(0) = 0, = δ 0 +B(τ) (K P θ P Σγ 0 )+ 1 2 n j=1 [ Σ B(τ)B(τ) Σ ], A(0) = 0. j,j Thus, he A(τ) and B(τ) funcions are calculaed as if he dynamics of he sae variables had a consan drif erm equal o K P θ P Σγ 0 insead of he acual K P θ P and a mean-reversion marix equal o K P + Σγ 1 as opposed o he acual K P. The probabiliy measure wih hese alernaive dynamics is frequenly referred o as he risk-neural, or Q, probabiliy measure since he expeced reurn on any asse under his measure is equal o he risk-free rae r ha a risk-neural invesor would demand. The difference is deermined by he risk premium Γ and reflecs invesors aversion o he risks embodied in X. 4

7 Finally, we define he erm premium as TP (τ) = y (τ) 1 τ +τ E P [r s]ds. (1) Tha is, he erm premium is he difference in expeced reurn beween a buy and hold sraegy for a τ-year Treasury bond and an insananeous rollover sraegy a he risk-free rae r. 2.2 An Empirical Affine Model A wide variey of Gaussian erm srucure models have been esimaed. Here, we describe an empirical represenaion from he lieraure ha uses high-frequency observaions on U.S. yields from a sample ha includes he recen ZLB period. I improves he economeric idenificaion of he laen facors, which faciliaes model esimaion. 5 The Gaussian erm srucure model we consider is an updae of he one used by Chrisensen and Rudebusch (2012). This CR model is an arbirage-free Nelson-Siegel (AFNS) represenaion wih hree laen sae variables, X = (L,S,C ). These are described by he following sysem of SDEs under he risk-neural Q-measure: 6 dl ds dc = λ λ 0 0 λ θ Q 1 θ Q 2 θ Q 3 where Σ is he consan covariance (or volailiy) marix. L S C In addiion, he insananeous risk-free rae is defined by d+σ dw L,Q dw S,Q dw C,Q, λ > 0, (2) r = L +S. (3) This specificaion implies ha zero-coupon bond yields are given by ( 1 e λτ ) ( 1 e λτ y (τ) = L + S + λτ λτ e λτ) C A(τ), (4) τ where he facor loadings in he yield funcion mach he level, slope, and curvaure loadings inroduced in Nelson and Siegel (1987). The final yield-adjusmen erm, A(τ)/τ, capures convexiy effecs due o Jensen s inequaliy. The model is compleed wih a risk premium specificaion ha connecs he facor dynamics o 5 Difficulies in esimaing Gaussian erm srucure models are discussed in Chrisensen e al. (2011), who propose using a Nelson-Siegel srucure o avoid hem. 6 Two deails regarding his specificaion are discussed in Chrisensen e al. (2011). Firs, wih a uni roo in he level facor under he pricing measure, he model is no arbirage-free wih an unbounded horizon; herefore, as is ofen done in heoreical discussions, we impose an arbirary maximum horizon. Second, we idenify his class of models by fixing he θ Q means under he Q-measure a zero wihou loss of generaliy. 5

8 he dynamics under he real-world P-measure. 7 The maximally flexible specificaion of he AFNS model has P-dynamics given by 8 dl ds dc = κ P 11 κ P 12 κ P 13 κ P 21 κ P 22 κ P 23 κ P 31 κ P 32 κ P 33 θ P 1 θ P 2 θ P 3 L S C d+ σ σ 21 σ 22 0 σ 31 σ 32 σ 33 dw L,P dw S,P dw C,P. (5) Using boh in- and ou-of-sample performance measures, CR wen hrough a careful empirical analysis o jusify various zero-value resricions on he K P marix. Imposing hese resricions resuls in he following dynamic sysem for he P-dynamics: dl ds dc = κ P 21 κ P 22 κ P κ P 33 0 θ P 2 θ P 3 L S C d+σ dw L,P dw S,P dw C,P, (6) where he covariance marix Σ is assumed diagonal and consan. Noe ha in his specificaion, he Nelson-Siegel level facor is resriced o be an independen uni-roo process under boh probabiliy measures. 9 As discussed in CR, his resricion helps improve forecas performance independen of he specificaion of he remaining elemens of K P. Because ineres raes are highly persisen, empirical auoregressive models, including DTSMs, suffer from subsanial small-sample esimaion bias. Specifically, model esimaes will generally be biased oward a dynamic sysem ha displays much less persisence han he rue process (so esimaes of he real-world mean-reversion marix, K P, are upward biased). Furhermore, if he degree of ineres rae persisence is underesimaed, fuure shor raes would be expeced o rever o heir mean oo quickly causing heir expeced longer-erm averages o be oo sable. Therefore, he bias in he esimaed dynamics disors he decomposiion of yields and conaminaes esimaes of long-mauriy erm premia. As described in deail in Bauer e al. (2012), bias-correced K P esimaes are ypically very close o a uni-roo process, so we view he imposiion of he uni-roo resricion as a simple shorcu o overcome small-sample esimaion bias. We re-esimaed his CR model over a larger sample of weekly nominal U.S. Treasury zerocoupon yields from January 4, 1985, unil December 28, 2012, for eigh mauriies: hree monhs, six monhs, one year, wo years, hree years, five years, seven years, and en years. 10 The model 7 I is imporan o noe ha here are no resricions on he dynamic drif componens under he empirical P- measure beyond he requiremen of consan volailiy. To faciliae empirical implemenaion, we use he essenially affine risk premium inroduced in Duffee (2002). 8 As noed in Chrisensen e al. (2011), he unconsrained AFNS model has a sign resricion and hree parameers less han he sandard canonical hree-facor Gaussian DTSM. 9 Due o he uni-roo propery of he firs facor, we can arbirarily fix is mean a θ P 1 = The yield daa include hree- and six-monh Treasury bill yields from he H.15 series from he Federal Reserve 6

9 K P K P,1 K P,2 K P,3 θ P Σ K1, P σ (0.0001) K2, P σ (0.1319) (0.1171) (0.0857) (0.0270) (0.0002) K3, P σ (0.1607) (0.0073) (0.0004) Table 1: Parameer Esimaes for he CR Model. The esimaed parameers of he K P marix, θ P vecor, and diagonal Σ marix are shown for he CR model. The esimaed value of λ is (0.0023). The numbers in parenheses are he esimaed parameer sandard deviaions. The maximum log likelihood value is 66, parameer esimaes are repored in Table 1. As in CR, we esed he significance of he four parameer resricions imposed on K P in he CR model relaive o he unresriced AFNS model. 11 As before, we found ha he four parameer resricions are no rejeced by he daa; hus, he CR model appears flexible enough o capure he relevan informaion in he daa compared wih an unresriced model. 2.3 Negaive Shor-Rae Projecions in Sandard Models Before urning o he descripion of he shadow rae model, i is useful o reinforce he basic moivaion for our analysis by examining shor rae forecass from he esimaed CR model. Wih regard o shor rae forecass, any sandard affine Gaussian dynamic erm srucure model may place posiive probabiliies on fuure negaive ineres raes. Accordingly, Figure 2 shows he probabiliy obained from he CR model ha he shor rae hree monhs ou will be negaive. Prior o 2008 he probabiliies of fuure negaive ineres raes are negligible excep for a brief period in 2003 and 2004 when he Fed s policy rae emporarily sood a one percen. However, near he ZLB since lae 2008 he model is ypically predicing subsanial likelihoods of impossible realizaions. 3 A Shadow-Rae Model In his secion, we describe an opion-based approach o he shadow-rae model and esimae a shadow-rae analog o he CR model wih U.S. daa. Board as well as off-he-run Treasury zero-coupon yields for he remaining mauriies from he Gürkaynak e al. (2007) daabase, which is available a hp:// 11 Tha is, a es of he join hypohesis κ P 12 = κ P 13 = κ P 31 = κ P 32 = 0 using a sandard likelihood raio es. 7

10 Probabiliy FOMC 12/ CR model Figure 2: Probabiliy of Negaive Shor Raes Since Illusraion of he condiional probabiliy of negaive shor raes hree monhs ahead from he CR model. 3.1 The Opion-Based Approach o he Shadow-Rae Model The concep of a shadow ineres rae as a modeling ool o accoun for he ZLB can be aribued o Black (1995). He noed ha he observed nominal shor rae will be nonnegaive because currency is a readily available asse o invesors ha carries a nominal ineres rae of zero. Therefore, he exisence of currency ses a zero lower bound on yields. To accoun for his ZLB, Black posulaed as a modeling ool a shadow shor rae, s, ha is unconsrained by he ZLB. The usual observed insananeous risk-free rae, r, which is used for discouning cash flows when valuing securiies, is hen given by he greaer of he shadow rae or zero: r = max{0,s }. (7) Accordingly, as s falls below zero, he observed r simply remains a he zero bound. While Black (1995) described circumsances under which he zero bound on nominal yields migh be relevan, he did no provide specifics for implemenaion. Gorovoi and Linesky (2004) derive bond price formulas for he case of one-facor Gaussian and square-roo shadow-rae models. 12 Unforunaely, heir resuls do no exend o mulidimensional models. Insead, he small se of previous 12 Ueno, Baba, and Sakurai (2006) use hese formulas when calibraing a one-facor Gaussian model o a sample of Japanese governmen bond yields. 8

11 research on shadow-rae models has relied on numerical mehods for pricing. 13 However, in ligh of he compuaional burden of hese mehods, here have been only wo previous esimaions of mulifacor shadow-rae models: Ichiue and Ueno (2007) and Kim and Singleon (2012). Boh of hese sudies underake a full maximum likelihood esimaion of heir wo-facor Gaussian shadow-rae models on Japanese bond yield daa using he exended Kalman filer and numerical opimizaion. To overcome he curse of dimensionaliy ha limis numerical-based esimaion of shadow-rae models, Krippner (2013) suggesed an alernaive opion-based approach ha makes shadow-rae models almos as easy o esimae as he corresponding non-shadow-rae model. To illusrae his approach, consider wo bond-pricing siuaions: one wihou currency as an alernaive asse and he oher ha has a currency in circulaion ha has a consan nominal value and no ransacion coss. In he world wihou currency, he price of a shadow-rae zero-coupon bond, P (τ), may rade above par, ha is, is risk-neural expeced insananeous reurn equals he risk-free shadow shor rae, s, which may be negaive. In conras, in he world wih currency, he price a ime for a zero-coupon bond ha pays $1 when i maures in τ years is given by P (τ). This price will never rise above par, so nonnegaive yields will never be observed. Now consider he relaionship beween he wo bond prices a ime for he shores (say, overnigh) mauriy available, δ. In he presence of currency, invesors can eiher buy he zerocoupon bond a price P (δ) and receive one uni of currency he following day or jus hold he currency. As a consequence, his bond price, which would equal he shadow bond price, mus be capped a 1: P (δ) = min{1,p (δ)} = P (δ) max{p (δ) 1,0}. Tha is, he availabiliy of currency implies ha he overnigh claim has a value equal o he zerocoupon shadow bond price minus he value of a call opion on he zero-coupon shadow bond wih a srike price of 1. More generally, we can express he price of a bond in he presence of currency as he price of a shadow bond minus he call opion on values of he bond above par: P (τ) = P (τ) C A (τ,τ;1), (8) where C A (τ,τ;1) is he value of an American call opion a ime wih mauriy in τ years and srike price 1 wrien on he shadow bond mauring in τ years. In essence, in a world wih currency, he bond invesor has had o sell off he possible gain from he bond rising above par a any ime prior o mauriy. 13 Boh Kim and Singleon (2012) and Bomfim (2003) use finie-difference mehods o calculae bond prices, while Ichiue and Ueno (2007) employ ineres rae laices. 9

12 Unforunaely, analyically valuing his American opion is complicaed by he difficuly in deermining he early exercise premium. However, Krippner (2013) argues ha here is an analyically close approximaion based on racable European opions. Specifically, he argues ha he above discussion suggess ha he las incremenal forward rae of any bond will be nonnegaive due o he fuure availabiliy of currency in he immediae ime prior o is mauriy. As a consequence, he inroduces he following auxiliary bond price equaion P aux. (τ,τ +δ) = P (τ +δ) C E (τ,τ +δ;1), (9) where C E (τ,τ +δ;1) is he value of a European call opion a ime wih mauriy +τ and srike price 1 wrien on he shadow discoun bond mauring a + τ + δ. I should be sressed ha P aux. (τ,τ +δ) is no idenical o he bond price P (τ) in equaion (8) whose yield observes he zero lower bound. The key insigh of Krippneris ha he las incremenal forward rae of any bondwill benonnegaive due o he fuure availabiliy of currency in he immediae ime prior o is mauriy. In Equaion (9), his is obained by leing δ 0, which idenifies he corresponding nonnegaive insananeous forward rae: [ f (τ) = lim δ 0 ] δ lnpaux. (τ,τ +δ). (10) Now, he discoun bond prices whose yields observe he zero lower bound are approximaed by P app. (τ) = e +τ f (s)ds. (11) The auxiliary bond price drops ou of he calculaions, and we are lef wih formulas for he nonnegaive forward rae, f (τ), ha are solely deermined by he properies of he shadow rae process s. Specifically, Krippner (2013) shows ha f (τ) = f (τ)+z (τ), where f (τ) is he insananeous forward rae on he shadow bond, which may go negaive, while z (τ) is given by z (τ) = lim δ 0 [ δ C E ] (τ,τ +δ;1). (12) P (τ +δ) In addiion, i holds ha he observed insananeous risk-free rae respecs he nonnegaiviy equaion (7) as in he Black (1995) model. 10

13 Finally, yield-o-mauriy is defined he usual way as y (τ) = 1 τ = 1 τ +τ +τ = y (τ)+ 1 τ f (s)ds f (s)ds+ 1 τ +τ +τ lim δ 0[ δ lim δ 0[ δ C E (s,s+δ;1) P (s+δ) C E(s,s+δ;1) P (s+δ) ] ds. I follows ha bond yields consrained a he ZLB can be viewed as he sum of he yield on he unconsrained shadow bond, denoed y (τ), which is modeled using sandard ools, and an add-on correcion erm derived from he price formula for he opion wrien on he shadow bond ha provides an upward push o deliver he higher nonnegaive yields acually observed. I is imporan o sress ha since he observed discoun bond prices defined in equaion (11) differfromheauxiliary bondpricep aux. (τ,τ+δ) definedinequaion (9) andusedinheconsrucion of he nonnegaive forward rae in equaion (10), he Krippner (2013) framework should be viewed as no fully inernally consisen and simply an approximaion o an arbirage-free model. 14 Of course, away from he ZLB, wih a negligible call opion, he model will mach he sandard arbirage-free erm srucure represenaion. In addiion, he size of he approximaion error near he ZLB can be deermined via simulaion as we will demonsrae below. ] ds 3.2 The Shadow-Rae B-CR Model In heory, he opion-based shadow-rae resul is quie general and applies o any assumpions made abou he dynamics of he shadow-rae process. However, as implemenaion requires he calculaion of he limi in equaion (12), he opion-based shadow-rae models are limied pracically o he Gaussian model class. The AFNS class is well suied for his exension. 15 In he shadow-rae AFNS model, he affine shor rae equaion (3) is replaced by he nonnegaiviy consrain and he shadow risk-free rae, which is defined as he sum of level and slope as in he original AFNS model class: r = max{0,s },s = L +S. All oher elemens of he model remain he same. Namely, he dynamics of he sae variables used for pricing under he Q-measure remain as described in equaion (2), so he yield on he shadow 14 In paricular, here is no explici PDE ha bond prices mus saisfy, including boundary condiions, for he absence of arbirage as in Kim and Singleon (2012). 15 For deails of he derivaions, see Chrisensen and Rudebusch (2013). 11

14 discoun bond mainains he popular Nelson and Siegel (1987) facor loading srucure ( ) ( 1 e λτ 1 e λτ y (τ) = L + S + λτ λτ e λτ ) where A(τ)/τ is he same mauriy-dependen yield-adjusmen erm. The corresponding insananeous shadow forward rae is given by C A(τ), (13) τ f (τ) = T lnp (τ) = L +e λτ S +λτe λτ C +A f (τ), (14) where he yield-adjusmen erm in he insananeous forward rae funcion is given by A f (τ) = A(τ) τ = 1 2 σ2 11τ 2 1 ( 1 e 2 (σ2 21 +σ22) 2 λτ λ ) (σ2 31 +σ2 32 +σ2 33 ) [ 1 λ 2 2 λ 2e λτ 2 λ τe λτ + 1 λ 2e 2λτ + 2 λ τe 2λτ +τ 2 e 2λτ] σ 11 σ 21 τ 1 e λτ λ σ 11 σ 31 [ 1 λ τ 1 λ τe λτ τ 2 e λτ] (σ 21 σ 31 +σ 22 σ 32 )[ 1 λ 2 2 λ 2e λτ 1 λ τe λτ + 1 λ 2e 2λτ + 1 λ τe 2λτ]. Krippner (2013) provides a formula for he zero lower bound insananeous forward rae, f (τ), ha applies o any Gaussian model ( f (τ) ) 1 f (τ) = f (τ)φ +ω(τ) exp ω(τ) 2π ( 1 2 [ f (τ) ] 2 ), ω(τ) where Φ( ) is he cumulaive probabiliy funcion for he sandard normal disribuion, f (τ) is he shadow forward rae, and ω(τ) is relaed o he condiional variance, v(τ,τ + δ), appearing in he shadow bond opion price formula as follows ω(τ) 2 = 1 2 lim 2 v(τ,τ +δ) δ 0 δ 2. 12

15 K P K P,1 K P,2 K P,3 θ P Σ K1, P σ (0.0001) K2, P σ (0.1533) (0.1334) (0.0908) (0.0343) (0.0002) K3, P σ (0.1252) (0.0085) (0.0004) Table 2: Parameer Esimaes for he B-CR Model. The esimaed parameers of he K P marix, θ P vecor, and diagonal Σ marix are shown for he B-CR model. The esimaed value of λ is (0.0027). The numbers in parenheses are he esimaed parameer sandard deviaions. The maximum log likelihood value is 66, Wihin he shadow-rae AFNS model, ω(τ) akes he following form ω(τ) 2 = σ11τ 2 +(σ21 2 +σ22) 2 1 e 2λτ 2λ [ 1 e +(σ31 2 +σ32 2 +σ33) 2 2λτ 1 4λ 2 τe 2λτ 1 2 λτ2 e 2λτ] 1 e λτ [ +2σ 11 σ 21 +2σ 11 σ 31 τe λτ + 1 e λτ ] λ λ +(σ 21 σ 31 +σ 22 σ 32 ) [ τe 2λτ + 1 e 2λτ ]. 2λ Therefore, he zero-coupon bond yields ha observe he zero lower bound, denoed y (τ), are easily calculaed as y (τ) = 1 τ +τ [ ( f (s) ) 1 f (s)φ +ω(s) exp ω(s) 2π ( 1 2 [ f (s) ] 2 ) ] ds. (15) ω(s) As in he affine AFNS model, he shadow-rae AFNS model is compleed by specifying he price of risk using he essenially affine risk premium specificaion inroduced by Duffee (2002), so he realworld dynamics of he sae variables can be expressed as equaion (5). Again, in an unresriced case, boh K P and θ P are allowed o vary freely relaive o heir counerpars under he Q-measure. However, we focus on he case wih he same K P and θ P resricions as in he CR model on he assumpion ha ouside of he ZLB period, he shadow-rae model would properly collapse o he sandard CR form. We label his shadow-rae model he B-CR model. We esimae he B-CR model from January 4, 1985, unil December 28, 2012, for eigh mauriies: hree monhs, six monhs, one year, wo years, hree years, five years, seven years, and en years Due o he nonlinear measuremen equaion for he yields in he shadow-rae AFNS model, esimaion is based on he sandard exended Kalman filer as described in Chrisensen and Rudebusch(2013). We also esimaed unresriced and independen facor shadow-rae AFNS models and obained similar resuls o hose repored below. 13

16 The esimaed B-CR model parameers are repored in Table 2. In comparing he esimaed B-CR and CR model parameers, we noe ha none of he parameers are saisically significanly differen from he corresponding parameer in he oher model wih he excepion of λ, which is saisically, bu no economically, differen across he wo models. Hence, for parsimoniously specified Gaussian models, we conjecure ha he differences in he esimaed parameers beween wo oherwise idenical models ha are only disinguished by one being a sandard model and he oher being a shadow-rae model would end o be small. 3.3 How Good is he Opion-Based Approximaion? As noed in Secion 3.1, Krippner (2013) does no provide a formal derivaion of arbirage-free pricing relaionships for he opion-based approach. Therefore, in his subsecion, we analyze how closely he opion-based bond pricing from he esimaed B-CR model maches an arbirage-free bond pricing ha is obained from he same model using Black s (1995) approach based on Mone Carlo simulaions. The simulaion-based shadow yield curve is obained from 50,000 en-year long facor pahs generaed using he esimaed Q-dynamics of he sae variables in he B-CR model, which, ignoring he nonnegaiviy equaion (7), are used o consruc 50,000 pahs for he shadow shor rae. These are convered ino a corresponding number of shadow discoun bond pahs and averaged for each mauriy before he resuling shadow discoun bond prices are convered ino yields. The simulaion-based yield curve is obained from he same underlying 50,000 Mone Carlo facor pahs, bu a each poin in ime in he simulaion, he resuling shor rae is consrained by he nonnegaiviy equaion (7) as in Black (1995). The shadow-rae curve from he B-CR model can also be calculaed analyically via he usual affine pricing relaionships, which ignore he ZLB. Thus, any difference beween hese wo curves is simply numerical error ha reflecs he finie number of simulaions. To documen ha he close mach beween he opion-based and he simulaion-based yield curves is no limied o any specific dae where he ZLB of nominal yields is likely o have maered, we underake his simulaion exercise for he las observaion dae in each year since Table 3 repors he resuling shadow yield curve differences and yield curve differences for various mauriies on hese 7 daes. Noe ha he errors for he shadow yield curves solely reflec simulaion error as he model-implied shadow yield curve is idenical o he analyical arbirage-free curve ha would prevail wihou currency in circulaion. These simulaion errors in Table 3 are ypically very small in absolue value, and hey increase only slowly wih mauriy. Their average absolue value shown in he boom row is less han one basis poin even a a en-year mauriy. This implies ha using simulaions wih a large number of draws (N = 50,000) arguably delivers enough accuracy for he 17 Of course, away from he ZLB, wih a negligible call opion, he model will mach he sandard arbirage-free erm srucure represenaion. 14

17 Daes 12/29/06 12/28/07 12/26/08 12/31/09 12/31/10 12/30/11 12/28/12 Mauriy in monhs Shadow yields Yields Shadow yields Yields Shadow yields Yields Shadow yields Yields Shadow yields Yields Shadow yields Yields Shadow yields Yields Average Shadow yields abs. diff. Yields Table 3: Approximaion Errors in Yields for Shadow-Rae Model. A each dae, he able repors differences beween he analyical shadow yield curve obained from he opion-based esimaes of he B-CR model and he shadow yield curve obained from 50,000 simulaions of he esimaed facor dynamics under he Q-measure in ha model. The able also repors for each dae he corresponding differences beween he fied yield curve obained from he B-CR model and he yield curve obained via simulaion of he esimaed B-CR model wih imposiion of he ZLB. The boom wo rows give averages of he absolue differences across he 7 daes. All numbers are measured in basis poins. ype of inference we wan o make here. Given his calibraion of he size of he numerical errors involved in he simulaion, we can now assess he more ineresing size of he approximaion error in he opion-based approach o valuing yields in he presence of he ZLB. In Table 3, he errors of he fied B-CR model yield curves relaive o he simulaed resuls are only slighly larger han hose repored for he shadow yield curve. In paricular, for mauriies up o five years, he errors end o be less han 1 basis poin, so he opionbased approximaion error adds very lile if anyhing o he numerical simulaion error. A he en-year mauriy, he approximaion errors are undersandably larger, bu even he larges errors a he en-year mauriy do no exceed 4 basis poins in absolue value and he average absolue value is less han 2 basis poins. Overall, he opion-based approximaion errors in our hree-facor seing appear relaively small. Indeed, hey are smaller han he fied errors o be repored in Table 4. Tha is, for he B-CR model analyzed here, he gain from using a numerical esimaion approach insead of he opion-based approximaion would in all likelihood be negligible. 15

18 Rae in basis poins Five year Treasury yield Ten year Treasury yield Bear Searns rescue Mar. 24, 2008 FOMC Dec FOMC Aug Figure 3: Value of Opion o Hold Currency. We show ime-series plos of he value of he opion o hold currency embedded in he Treasury yield curve as esimaed in real ime by he B-CR model. The daa cover he period from January 6, 1995, o December 28, Measuring he Effec of he ZLB To provide evidence ha we should anicipae o see a leas some difference across he regular and shadow-rae models, we urn our focus o he value of he opion o hold currency, which we define as he difference beween he yields ha observe he zero lower bound and he comparable lower shadow discoun bond yields ha do no. Figure 3 shows hese yield spreads a he fiveand en-year mauriy based on real-ime rolling weekly re-esimaions of he B-CR model saring in 1995 hrough Beyond a few very emporary small spikes, he opion was of economically insignifican value prior o he failure of Lehman Brohers in he fall of However, despie he zero shor rae since 2008, i is no really unil Augus 2011 ha he opion obains significan susained value. A he end of our sample, he yield spread is 80 and 60 basis poins a he five- and en-year mauriy, respecively. Opion values a hose levels sugges ha i should maer for model performance wheher a model accouns for he ZLB of nominal yields. 18 Consisen wih our series for he 2003-period, Bomfim (2003) in his calibraion of a wo-facor shadow-rae model o U.S. ineres rae swap daa repors a probabiliy of hiing he zero-boundary wihin he nex wo years equaling 3.6 percen as of January Thus, i appears ha here was never any maerial risk of reaching he ZLB during he period of low ineres raes. 16

19 Full sample Mauriy in monhs All RMSE yields CR B-CR Normal period (Jan. 6, 1995-Dec. 12, 2008) Mauriy in monhs All RMSE yields CR B-CR ZLB period (Dec. 19, 2008-Dec. 28, 2012) RMSE Mauriy in monhs All yields CR B-CR Table 4: Summary Saisics of he Fied Errors. The roo mean squared fied errors (RMSEs) for he CR and B-CR model are shown. All numbers are measured in basis poins. The daa covers he period from January 6, 1985, o December 28, Comparing Affine and Shadow-Rae Models In his secion, we compare he empirical affine and shadow-rae models across a variey of dimensions, including in-sample fi, volailiy dynamics, and ou-of-sample forecas performance. 4.1 In-Sample Fi and Volailiy The summary saisics of he model fi are repored in Table 4 and indicae a very similar fi of he wo models in he normal period up unil he end of However, since hen we see a noable advanage o he shadow-rae model ha is also refleced in he likelihood values. Sill, we conclude from his in-sample analysis ha i is no in he model parameers nor in he model fi ha he shadow-rae model really disinguishes iself from is regular cousin. A serious limiaion of sandard Gaussian models is he assumpion of consan yield volailiy, which is paricularly unrealisic when periods of normal volailiy are combined wih periods in which yields are grealy consrained in heir movemens near he ZLB. A shadow-rae model approach can miigae his failing significanly. In he CR model, where zero-coupon yields are affine funcions of he sae variables, model- 17

20 Rae in basis poins CR model B CR model Three monh realized volailiy of wo year yield Correlaion = 78.4% Figure 4: Three-Monh Condiional Volailiy of Two-Year Yield Since Illusraion of he hree-monh condiional volailiy of he wo-year yield implied by he esimaed CR and B-CR model. Also shown is he subsequen hree-monh realized volailiy of he wo-year yield based on daily daa. implied condiional prediced yield volailiies are given by he square roo of V P [y N T (τ)] = 1 τ 2B(τ) V P [X T ]B(τ), where T is he predicion period, τ is he yield mauriy, and V P [X T ] is he condiional covariance marix. 19 In he B-CR model, on he oher hand, zero-coupon yields are non-linear funcions of he sae variables and condiional prediced yield volailiies have o be generaed by sandard Mone Carlo simulaion. Figure 4 shows he implied hree-monh condiional yield volailiy of he wo-year yield from he CR and B-CR models. To evaluae he fi of hese prediced hree-monh-ahead condiional yield sandard deviaions, hey are compared o a sandard measure of realized volailiy based on he same daa used in he model esimaion, bu a daily frequency. The realized sandard deviaion of he daily changes in he ineres raes are generaed for he 91-day period ahead on a rolling basis. The realized variance measure is used by Andersen and Benzoni (2010), Collin-Dufresne e al. (2009), as well as Jacobs and Karoui (2009) in heir assessmens of sochasic volailiy models. This measure is 19 The condiional covariance marix is calculaed using he analyical soluions provided in Fisher and Gilles (1996). 18

21 fully nonparameric and has been shown o converge o he underlying realizaion of he condiional variance as he sampling frequency increases; see Andersen e al. (2003) for deails. The square roo of his measure reains hese properies. For each observaion dae he number of rading days N during he subsequen 91-day ime window is deermined and he realized sandard deviaion is calculaed as RV STD,τ = N y+n 2 (τ), where y +n (τ) is he change in yield y(τ) from rading day +(n 1) o rading day +n. 20 n=1 While he condiional yield volailiy from he CR model only change lile (due o updaing of is esimaed parameers), he condiional yield volailiy from he B-CR model closely maches he realized volailiy series Forecas Performance To exrac he erm premiums embedded in he Treasury yield curve is ulimaely an exercise in forecasing policy expecaions. Thus, o sudy bond invesors expecaions in real ime, we perform a rolling re-esimaion of he CR model and is shadow-rae equivalen on expanding samples adding one week of observaions each ime, a oal of 939 esimaions. As a resul, he end daes of he expanding samples run from January 6, 1995, o December 28, For each end dae during ha period, we projec he shor rae six monhs and one year ahead. Imporanly, he esimaes of hese objecs rely essenially only on informaion ha was available in real ime. For robusness, we include resuls from anoher esablished U.S. Treasury erm srucure model inroduced in Kim and Wrigh (2005, henceforh KW). I is a sandard laen hree-facor Gaussian erm srucure model of he kind described in Secion This model is updaed on an ongoing basis by he saff of he Federal Reserve Board. 23 However, we emphasize ha he forecass from his model are no real-ime forecass, bu based on he full sample esimae. As yields were boh economically and saisically far away from he zero lower bound prior o December 2008, i seems reasonable o disinguish beween model performance in he normal period prior o he policy rae reaching is effecive lower bound and he period afer i. For he 20 Noe ha oher measures of realized volailiy have been used in he lieraure, such as he realized mean absolue deviaion measure as well as fied GARCH esimaes. Collin-Dufresne e al. (2009) also use opion-implied volailiy as a measure of realized volailiy. 21 In heir analysis of Japanese governmen bond yields, Kim and Singleon (2012) also repor a close mach o yield volailiies for heir Gaussian shadow-rae model. 22 The KW model is esimaed using one-, wo-, four-, seven-, and en-year off-he-run Treasury zero-coupon yields from he Gürkaynak e al. (2007) daabase, as well as hree- and six-monh Treasury bill yields. To faciliae empirical implemenaion, model esimaion includes monhly daa on he six- and welve-monh-ahead forecass of he hreemonh T-bill yield from Blue Chip Financial Forecass and semiannual daa on he average expeced hree-monh T-bill yield six o eleven years hence from he same source. See Kim and Orphanides (2012) for deails. 23 The daa is available a 19

22 Full forecas period Six-monh forecas One-year forecas Mean RMSE Mean RMSE Random walk KW model CR model B-CR model Normal forecas period Six-monh forecas One-year forecas Mean RMSE Mean RMSE Random walk KW model CR model B-CR model ZLB forecas period Six-monh forecas One-year forecas Mean RMSE Mean RMSE Random walk KW model CR model B-CR model Table 5: Summary Saisics for Targe Federal Funds Rae Forecas Errors. Summary saisics of he forecas errors mean and roo mean-squared errors (RMSEs) of he arge overnigh federal funds rae six monhs and one year ahead. The forecass are weekly. The op panel covers he full forecas period ha sars on January6, 1995, and runs unil June 29, 2012, for he six-monh forecass (913 forecass) and unil December 30, 2011, for he one-year forecass (887 forecass). The middle panel coves he normal forecas period from January 6, 1995, o December 12, 2008, 728 forecass. The lower panel covers he zero lower bound forecas period ha sars on December 19, 2008, and runs unil June 29, 2012, for he six-monh forecass (185 forecass) and unil December 30, 2011, for he one-year forecass (159 forecass). All measuremens are expressed in basis poins. 13 years from 1995 hrough 2008 we should anicipae o be able o documen essenially idenical performance along mos dimensions, while we could hope o esablish some superior performance for he shadow-rae model in he years since 2009 in erms of forecasing fuure policy raes up o wo years ahead. The summary saisics for he forecas errors relaive o he subsequen realizaions of he arge overnigh federal fundsraese byhefomc arereporedintable5, which alsoconains heforecas errors obained using a random walk assumpion. We noe he srong forecas performance of he KW model relaive o he CR model during he normal period, 24 while i is equally obvious ha he KW model underperforms grossly during he ZLB period since December 19, As expeced, he 24 As noed earlier, his is no an enirely fair race as he KW model is no esimaed on a rolling real-ime basis unlike he oher models. 20

23 Rae in percen CR model B CR model Realized arge rae FOMC Aug. 9, Figure 5: Forecass of he Targe Overnigh Federal Funds Rae. Forecass of he arge overnigh federal funds rae one year ahead from he CR and B-CR model. Subsequen realizaions of he arge overnigh federal funds rae are included, so a dae, he figure shows forecass as of ime and he realizaion from plus one year. The forecas daa are weekly observaions from January 6, 1995, o December 28, CR and B-CR models exhibi very similar performance during he normal period, while he B-CR model sands ou in he mos recen ZLB period. 25 Figure 5 compares he forecass a he one-year horizon from he CR and B-CR models o he subsequen arge rae realizaions. For he CR model, he deerioraion in forecas performance is no really deecable unil afer he Augus 2011 FOMC meeing when explici forward guidance was firs inroduced. Since he CR model miigaes finie-sample bias in he esimaes of he meanreversion marix K P by imposing a uni-roo propery on he Nelson-Siegel level facor, i suggess ha he recen deerioraion for he CR model mus be caused by oher more fundamenal facors. 4.3 Decomposing Ten-Year Yields One imporan use for affine DTSMs has been o separae longer-erm yields ino a shor-rae expecaions componen and a erm premium. Here, we documen he differen decomposiions of 25 As anoher robusness check, we used he esimaed parameers of he CR model combined wih filering from he B-CR model o generae an alernaive se of shor rae forecass. The resuls are very close o hose shown in Table 5 for he B-CR model and hence no repored. 21

24 Rae in percen Ave. shor rae nex en years, CR model Ave. shor rae nex en years, B CR model FOMC Aug. 9, 2011 Rae in percen Ten year erm premium, CR model Ten year erm premium, B CR model FOMC Aug. 9, (a) Expeced shor rae. (b) Term premium. Figure 6: Ten-Year Expeced Shor Rae and Term Premium Panel (a) provides real-ime esimaes of he average policy rae expeced over he nex en years from he CR and B-CR model. Panel (b) shows he corresponding en-year erm premium series. The daa cover he period from January 6, 1995, o December 28, he en-year Treasury yield implied by he CR and B-CR model. To do so, we calculae, for each end dae during our rolling re-esimaion period, he average expeced pah for he overnigh rae, (1/τ) +τ E P[r s]ds, as well as he associaed erm premium assuming he wo componens sum o he fied bond yield, ŷ (τ). 26 Figure 6 shows he real-ime decomposiion of he en-year Treasury yield ino a policy expecaions componen and a erm premium componen according o he CR and B-CR models. Sudying he ime-series paerns in greaer deail, we firs noe he similar decomposiions from he CR and B-CR model unil December 2008 wih he noable excepion of he period when yields were low he las ime. Second, we see some smaller discrepancies across hese wo model decomposiions in he period beween December 2008 and Augus Finally, we poin ou he susained difference in he exraced policy expecaions and erm premiums in he period since Augus These resuls sugges ha a leas hrough lae 2011, he ZLB did no grealy effec he erm premium decomposiion. To provide a concree example of his, we repea he analysis in CR of he Treasury yield response o eigh key announcemens by he Fed regarding is firs large-scale asse purchase (LSAP) program. Table 6 shows he CR and B-CR model decomposiions of he 10-year U.S. Treasury yield on hese eigh daes and he oal changes. The yield decomposiions on hese daes are quie similar for boh of hese models, hough he B-CR model ascribes a bi more of he 26 The deails of hese calculaions for boh he CR and B-CR model are provided in Appendices A and B. 22

25 Announcemen dae Nov. 25, 2008 Dec. 1, 2008 Dec. 16, 2008 Jan. 28, 2009 Mar. 18, 2009 Aug. 12, 2009 Sep. 23, 2009 Nov. 4, 2009 Toal ne change Decomposiion from models Ten-year Model Avg. arge rae Ten-year Treasury nex en years erm premium Residual yield CR B-CR CR B-CR CR B-CR CR B-CR CR B-CR CR B-CR CR B-CR CR B-CR CR B-CR Table 6: Decomposiion of Responses of Ten-Year U.S. Treasury Yield. The decomposiion of responses of he en-year U.S. Treasury yield on eigh LSAP announcemen daes ino changes in (i) he average expeced arge rae over he nex en years, (ii) he en-year erm premium, and (iii) he unexplained residual based on he CR and B-CR model of U.S. Treasury yields. All changes are measured in basis poins. changes in yields o a signaling channel effec adjusing shor-rae expecaions. 4.4 Assessing Recen Shifs in Near-Term Moneary Policy Expecaions In his secion, we aemp o assess he exen o which he models are able o capure recen shifs in near-erm moneary policy expecaions. To do so, we compare he variaion in he models one- and wo-year shor rae forecass since 2007 o he raes on one- and wo-year federal funds fuures conracs as shown in Figure We noe ha he exisence of ime-varying risk premiums even in very shor-erm federal funds fuures conracs is well documened (see Piazzesi and Swanson 2008). However, he risk premiums in such shor-erm conracs are small relaive o he sizeable variaion over ime observed in Figure 7. As 27 The fuures daa are from Bloomberg. The one-year fuures rae is he weighed average of he raes on he 12- and 13-monh federal funds fuures conracs, while he wo-year fuures rae is he rae on he 24-monh federal funds fuures conrac hrough 2010, and he weighed average of he raes on he 24- and 25-monh conracs since hen. Absence of daa on he 24-monh conracs prior o 2007 deermines he sar dae for he analysis. 23