Ambiguity, Nominal Bond Yields, and Real Bond Yields

Transcription

1 Ambiguiy, Nominal Bond Yields, and Real Bond Yields Guihai Zhao a,1 a Bank of Canada, 234 Wellingon Sree, Oawa, ON K1A 0G9 Absrac Equilibrium bond-pricing models rely on inflaion being bad news for fuure growh o generae upward-sloping nominal yield curves. We develop a model ha can generae upward-sloping nominal and real yield curves by insead using ambiguiy abou inflaion and growh. Ambiguiy can help resolve he puzzling fac ha upward-sloping yield curves have persised despie posiive inflaion shocks changing from negaive o posiive news abou growh in he las weny years. The expecaions hypohesis roughly holds under invesors wors case beliefs. The difference beween he wors case scenario and he rue disribuion makes realized excess reurns on long erm bonds predicable. Keywords: Ambiguiy, Term Srucure, Regime Break, Bond Yields, Expecaions Hypohesis, Bond Reurn Predicabiliy, Equiy Yields JEL Classificaion: G00, G12, E43 1. Inroducion To be consisen wih he fac ha he nominal yield curves are upward sloping, equilibrium bond-pricing models rely on inflaion as bad news for fuure growh and he assumpion ha agens prefer early resoluion of uncerainy; see, for example, Piazzesi and Schneider (2007) (henceforh PS 2007) and Bansal and Shaliasovich (2013). The I am graeful o Larry Epsein, Simon Gilchris, and François Gourio for heir coninuing advice and suppor on his projec. I appreciae he helpful commens of Jason Allen, Tony Berrada, Marin Eichenbaum, Bruno Feunou, Anonio Diez de los Rios, Carolin Pflueger, Marcel Rindisbacher, Jonahan Wimer, Min Wei, and conference paricipans a BoC-FRBSF-SFU Fixed Income Conference 2017, Fed Board Conference on Risk, Uncerainy and Volailiy 2018, CICF 2018, SITE 2018, and NFA All errors are my own. address: GZhao@bankofcanada.ca (Guihai Zhao) 1 Fax:

2 inuiion is ha a posiive surprise o inflaion lowers fuure consumpion growh, and a he same ime, decreases he real payoff of long-erm nominal bonds. Therefore, long-erm nominal bonds are risky and command a erm spread over shor-erm bonds. However, in he curren macroeconomic environmen where inflaion is good news for fuure growh, hese models also imply a downward-sloping nominal yield curve, which is in conras o he fac ha in he daa he nominal yield curve coninues o slope up afer he lae 1990s. 2 This paper provides an alernaive approach o undersand upward-sloping nominal yield curves in boh environmens. An imporan relaed fac is excess bond reurn predicabiliy. Agains he expecaions hypohesis (EH), Fama and Bliss (1987), Campbell and Shiller (1991), Dai and Singleon (2002), and Cochrane and Piazzesi (2005) provide evidence for bond reurn predicabiliy using yield spreads and forward raes as predicors. Ohers, however, show ha he failure of he EH is due o expecaional errors (Froo (1989); Piazzesi e al. (2015)), and he resuls are in favor of EH using subjecive expecaions from survey. The mixed resuls call for a heory ha can separae subjecive belief from he perspecive of he economerician, and shor rae expecaions should be he mos imporan driver of upward sloping yield curves under subjecive belief. We will show ha hese resuls can be reconciled if invesors have ambiguiy abou inflaion and growh. From he perspecive of equilibrium asse pricing models, anoher puzzling fac is ha he erm srucure of Treasury inflaion-proeced securiies (TIPS) is upward sloping in he U.S. In he weny-year hisory of TIPS daa, he observed slope has never been significanly negaive. Campbell (1986) shows ha real bonds have a negaive real erm premium if consumpion growh follows a persisen process. While i has been difficul o accoun for he nominal bond yield curve and bond reurn predicabiliy, i is much harder for an equilibrium model of bond pricing o also capure real bond yields. These 2 Recen developmens in he bond marke lieraure have shown ha he correlaion beween consumpion growh and inflaion has swiched from negaive o posiive afer he lae 1990s, which can explain he changes in correlaion beween U.S. Treasury bond reurns and sock reurns. See, for example, Burkhard and Hasselof (2012); David and Veronesi (2013); Campbell e al. (2016); Song (2017). 2

3 hree facs (upward sloping nominal and real yield curves, and he EH) are closely relaed boh in heory and empirically. Ideally, an equilibrium model should be able o explain hem joinly, and his is he firs model ha offers a join explanaion. This paper develops a consumpion-based asse pricing model where invesors have limied informaion abou he sochasic environmen and hence face boh risk and ambiguiy. Risk refers o he siuaion where here is a probabiliy law o guide choice. However, ambiguiy averse agens (wih recursive muliple priors, or maxmin, preferences by Epsein and Schneider (2003)) lack he confidence o assign probabiliies o all relevan evens. Insead, hey ac as if hey evaluae fuure prospecs using a worscase probabiliy drawn from a se of muliple disribuions. The main resul is ha he shor rae expecaions are upward sloping under invesors wors-case equilibrium beliefs, which generaes upward-sloping nominal and real yield curves even wih a consan relaive risk aversion (CRRA) uiliy, and he EH roughly holds under heir subjecive beliefs. However, he sochasic processes evolve over ime under he rue disribuion ha is differen from heir wors-case beliefs. This difference makes realized bond yields o be lower han expeced under heir wors-case beliefs, and herefore excess reurns on long-erm bonds are predicable. Invesors in his economy have in mind a benchmark or reference measure of he economy s dynamics ha represens he bes poin esimae of he sochasic process. As in PS 2007, under he reference measure, real growh and inflaion are described by a sae space model. However, invesors are concerned ha he reference measure is misspecified and believe ha he rue measure is acually wihin a se of alernaive measures ha are saisically close o he reference disribuion. As argued by Ilu and Schneider (2014), a loss of confidence is capured by an increase in ha se. I could be riggered, for example, by conflicing news ha expers now disagree more abou he fuure. Conversely, an increase in confidence is capured by a shrinkage of he se of beliefs. For example, as agens observe expers moving oward consensus on he course of he economy, hey move closer oward hinking in erms of a single probabiliy. We assume ha here is ambiguiy abou boh real growh and inflaion disribuion, and he 3

4 se of alernaive measures is generaed by a se of differen mean real growh/inflaion raes around is reference mean value. We use he Blue Chip Financial Forecas (BCFF) survey o characerize he properies of ambiguiy yields for U.S. real oupu growh and he consumer price index (CPI) from 1985 o Using BCFF forecas dispersion as an empirical measure for he size of ambiguiy (or confidence), we find ha, before he lae 1990s, he size of ambiguiy for long horizon inflaion is bigger han hose for shor horizons, and he erm srucure of ambiguiy is reversed aferwards. The bigger long-erm inflaion forecas dispersion before he lae 1990s can be inerpreed as, for example, inflaion scares (Goodfriend and King, 2005) creaed during he moneary policy experimenaion period, or he shifing endpoins (Kozicki and Tinsley, 2001) of long-erm inflaion expecaions, or sronger disagreemen among expers. Agens in our model respond o his long run inflaion uncerainy from inflaion scares, or shifing endpoins, or disagreemen by aribuing his uncerainy esimaed by economericians o some deerminisic variaions in long-erm inflaion, which corresponding o a bigger size of ambiguiy. Currenly, when he long-erm inflaion expecaions are well anchored, invesors have less ambiguiy regarding longer horizon inflaion. However, for real oupu growh, he erm srucure of ambiguiy is always downward sloping, which suggess ha invesors do no have a shifing endpoins problem and have less ambiguiy abou long run real growh han for shor run real growh. We assume an unexpeced discree regime shif mainly due o changes in inflaion paerns and moneary policy wih he firs subperiod as he inflaion fighing period of Volcker and Greenspan and he second subperiod as he recen period of low inflaion and increased cenral bank ransparency. 3 Moivaed by he fac ha inflaion forecas dispersion has swiched from upward sloping o downward sloping afer he lae 1990s, we model inflaion ambiguiy as a random walk wih posiive drif in he firs subperiod and wih negaive drif in he second subperiod. Given ha real oupu growh forecas dispersion has been consisenly 3 See, for example, Campbell e al. (2014) and Zhao (2017) for a similar regime break. The resuls are robus o differen regime break poins. 4

5 downward sloping, we assume ha ambiguiy abou real growh is a random walk wih negaive drif in boh periods. In equilibrium, ambiguiy averse agens evaluae fuure prospecs under he wors-case measure. Given he erm srucure of ambiguiy for inflaion and real growh, we show ha, in equilibrium, he wors-case growh and inflaion expecaions are upward sloping for boh subperiods, which generaes upward-sloping nominal and real yield curves in boh environmens. In equilibrium, he values of bonds and dividend srips can be solved as funcions of he ambiguiy processes. For he whole period, ambiguiy averse agens make decisions using he lower bound of he se of alernaive mean oupu growh he wors-case measure which is upward sloping because of he downward-sloping dispersion yields for oupu forecass. Thus he real bond yield curve is always upward sloping. During he firs subperiod, when inflaion expecaion is negaively associaed wih he wors-case expeced real oupu growh, he wors-case mean inflaion is he upper bound, which is upward sloping because he dispersion is bigger for a longer horizon. This implies ha invesors subjecive nominal shor rae expecaion is upward sloping, which generaes an upward-sloping nominal yield curve. During he second subperiod, inflaion expecaion becomes posiively associaed wih he wors-case expeced real oupu growh, and he wors-case mean inflaion becomes he lower bound. However, a he same ime, he inflaion forecas dispersion urns o be downward sloping, which again implies an upward-sloping mean inflaion in equilibrium. Therefore he model generaes upwardsloping nominal yield curves in boh subperiods, bu wih a differen mechanism. The model-implied bond yield volailiy is also consisen wih daa across periods. Many sudies have documened ha excess reurns on long-erm bonds are predicable. However, using survey expecaions as subjecive beliefs, a small lieraure argues ha he failure of he EH is due o expecaional errors. For example, Piazzesi e al. (2015) show ha he expeced excess reurns on long-erm bonds consis of wo pars: he expeced subjecive bond premium and he difference beween subjecive and saisical fuure ineres rae expecaions, and hey find he second par is significan. In our model, yields for long-erm bonds are roughly equal o he average of expeced 5

6 fuure shor raes under he equilibrium wors-case belief. Thus, consisen wih Froo (1989) and Piazzesi e al. (2015), he EH roughly holds under he subjecive equilibrium belief. However, invesors ambiguiy arises from uncerainy in long-run inflaion or GDP growh expecaions does no maerialize when he ime arrives, hus here is no rend in he one-sep-ahead ambiguiy process, which is used o calculae realized yields. This difference makes realized bond yields o be lower han expeced under invesors worscase beliefs. This difference and curren yield spreads/forward raes are boh driven by he rend componens in he ambiguiy process. Hence, consisen wih he empirical evidence, he realized excess bond reurns are predicable in he model. This is he firs paper ha separaes subjecive belief from he perspecive of he economerician and provides a heoreical framework ha is consisen wih boh of hese findings. Even hough he model focuses primarily on bond yields, i has imporan implicaions for he erm srucure of dividend srips as well. The empirical findings on equiy yields are differen across counries. Using dividend fuure conracs for he S&P500, Van Binsbergen and Koijen (2017) show ha dividend fuure reurns are slighly upward sloping and he volailiy of equiy yields is downward sloping, and he marke reurns are no significanly differen from individual dividend spo reurns. This model is consisen wih hese findings. Forecas dispersion is also widely used as measure for disagreemen or uncerainy (volailiy) in he lieraure. For example, Ehling e al. (2018) show ha bigger inflaion forecas dispersion, corresponding o sronger inflaion disagreemen, is associaed wih higher nominal yields. While higher uncerainy (volailiy) ypically lowers yields hrough he precauionary savings channel. In his model, he impac of higher inflaion forecas dispersion (as a measure for he size of ambiguiy) on nominal yields is fundamenally differen from disagreemen or volailiy, i is associaed wih higher nominal yields before he lae 1990s (upper bound as he wors case belief), and i is associaed wih lower nominal yields aferwards (lower bound as he wors case belief). We es he impac of inflaion forecas dispersion on nominal yields in he daa for hese wo subperiods, and find he coefficien is posiive in he firs subperiod and negaive in he second 6

7 subperiod. I could be he case ha inflaion forecas dispersion conains informaion for boh disagreemen and ambiguiy. However, he resul of he second subperiod suggess ha ambiguiy has a dominaing impac. Relaed lieraure The recen developmens in equilibrium bond-pricing models are dominaed by several groups. The firs group relies on inflaion as bad news for fuure growh o generaes upward-sloping nominal curve. For example, PS 2007 show ha inflaion is bad news for fuure consumpion growh using early daa, and he model wih Epsein and Zin (1989) preference can generae an upward-sloping nominal yield curve. In a similar vein, Wacher (2006) generaes upward-sloping nominal using an exernal habis model (Campbell and Cochrane (1999)), where innovaions o consumpion and inflaion growh are negaively correlaed. Taking inflaion as bad news for fuure growh, Bansal and Shaliasovich (2013) show ha a long-run risks model wih ime-varying volailiy of expeced consumpion growh and inflaion can accoun for bond reurn predicabiliy. Song (2017) finds ha he U.S. economy enered a posiive correlaion (beween consumpion growh and he inflaion arge) regime following he lae 1990s and has largely remained in ha regime hereafer. He argues ha if agens evaluae long-erm bonds using an uncondiional probabiliy of swiching from a posiive correlaion regime o a negaive one, he long-run risks model (Bansal and Yaron (2004)) generaes an upward-sloping nominal yield curve. 4 Given ha inflaion has swiched from bad o good news for fuure growh, and suppose ha he probabiliy of swiching back o a negaive correlaion regime is small, hese models imply a negaive inflaion risk premium and herefore imply a downward-sloping nominal yield curve. The second group focuses on real yield curve. Wacher (2006) shows ha habis model 4 In he posiive correlaion regime of he curren period, he condiional probabiliy of swiching back o a negaive correlaion regime is close o zero, while he uncondiional probabiliy is abou 2/3 because he economy has been in he negaive correlaion regime mos periods before he lae 1990s. Due o he downward-sloping real yield curve in he model, he model-implied nominal yield curve slope is only 1/3 of he daa. If agens form expecaion based on recen experiences raher han uncondiional probabiliy as in Malmendier and Nagel (2016), he model will implies a downward-sloping nominal yield curve. 7

8 can generae an upward-sloping real yield curve. Albuquerque e al. (2016) show ha he risk o ime preference implies a posiive real erm premium and can generae an upward-sloping real yield curve. They also show ha, in addiion o he erm srucure of U.S. TIPS, he erm srucure of U.K. Gils is also upward-sloping on average. Berrada e al. (2018) show he risk o belief updaing implies a posiive real erm premium when agens have beleifs-dependen risk aversion. The upward-sloping real yield curve in hese models is due o an upward-sloping real erm premium. This paper differs from hese previous sudies along some imporan dimensions. Firs, we provide a new undersanding of he upward-sloping nominal yield curve for wo environmens where inflaion can be bad or good news for fuure growh. Second, his paper provides an alernaive mechanism o generae an upward-sloping real yield curve for boh he pre- and pos-2000s. Mos imporanly, insead of relying upward sloping inflaion risk premium and real erm premium as in he previous sudies, he upward sloping nominal and real curves are mainly driven by upward sloping nominal and real shor rae expecaions in his paper, which is consisen wih findings in Froo (1989) and Piazzesi e al. (2015) ha he EH roughly holds under subjecive belief. The paper also reveals he source for he predicabiliy of he realized excess bond reurns ha found in he lieraure (Fama and Bliss (1987); Campbell and Shiller (1991); Dai and Singleon (2002); Cochrane and Piazzesi (2005)). Anoher group of lieraure invesigaes he role of ambiguiy in he bond-pricing models. For example, Ulrich (2013) argues ha, even wih log uiliy, ambiguiy abou rend inflaion can help generae an upward-sloping erm premium for nominal bonds if inflaion shocks make he size of ambiguiy bigger. Gagliardini e al. (2009) find ha ambiguiy premium can help explain he failure of he EH in he daa. Insead of focusing only on one dimension of he daa, his is he firs paper ha offers a join explanaion for hree mos imporan (and closely relaed) facs in he bond markes: upward-sloping nominal and real curves, and he mixed resuls of he EH. Moreover, in his paper, he channel hrough which ambiguiy works is fundamenally differen from hese previous sudies. We emphasize he imporance of shor rae expecaions under he wors-case 8

9 beliefs, oher han ambiguiy premium which implies ha he EH is violaed even under subjecive beliefs. This paper is relaed o a number of papers ha have sudied he implicaions of ambiguiy and robusness for finance and macroeconomics (see he survey by Epsein and Schneider (2010) and he references herein). Ilu and Schneider (2014) show how ime-varying ambiguiy abou produciviy generaes business cycle flucuaions. Using forecas dispersion daa, Zhao (2017) shows ha ambiguiy abou consumpion growh is driven by pas inflaion and argues ha bond risk changes are due o he ime-varying impac of inflaion on ambiguiy. This paper conribues o he ambiguiy lieraure by firs showing a differen erm srucure of ambiguiy for inflaion and oupu growh over wo subperiods, and hen using he recursive muliple-priors preference o link ambiguiy yields wih real and nominal bond yields and he equiy yields. The paper coninues as follows. Secion 2 oulines he model and solves i analyically. Secion 3 discusses he resuls of he empirical analysis. Secion 4 provides concluding commens. 2. The model In a pure exchange economy, idenical ambiguiy averse invesors maximize heir uiliy over endowmen/oupu processes. Oupu growh and inflaion are given exogenously. Equilibrium prices adjus such ha he agen is happy o consume he endowmen Economy dynamics Under reference measure P, oupu growh and inflaion follow a sae space model, while dividend growh is leveraged oupu growh: 5 We use oupu growh as he endowmen process because he non-durable good and service survey is no available in he BCFF. Using he Philadelphia Fed s Survey of Professional Forecasers (SPF), Zhao (2017) shows ha he dispersion for consumpion growh and oupu growh are highly correlaed. 9

10 g +1 = µ c + x c, + σ c ε c,+1 π +1 = µ π + x π, + σ π ε π,+1 x c,+1 = ρ c x c, + σc x ε c,+1 + σcπε x π,+1 (1) x π,+1 = ρ π x π, + σπε x π,+1 d +1 = ζ d g +1 + µ d + σ d ε d,+1 where g +1 and d +1 are he growh rae of oupu and dividends respecively, and π is inflaion. The expeced growh and inflaion are denoed by x c, and x π,. As argued in PS 2007, he sae space represenaion for z +1 = ( g +1, π +1 ) T does a good job in capuring he dynamics of inflaion, especially he high order auocorrelaions. For simpliciy, we assume ha he correlaion beween growh and inflaion is capured by σcπ. x All shocks are i.i.d normal and orhogonal o each oher. To model dividends and oupu separaely, we follow Ju and Miao (2012), where he parameer ζ d > 0 can be inerpreed as he leverage raio on expeced oupu growh, as in Abel (1999); ogeher wih he parameer σ d, his allows us o calibrae he correlaion of dividend growh wih consumpion growh. The parameer µ d helps mach he expeced growh rae of dividends. The above sae space sysem for inflaion and oupu growh represens economerician s bes poin esimae from he daa. However, invesors are concerned ha his reference measure is misspecified and ha he rue measure is acually wihin a se of alernaive measures ha are saisically close o he reference measure Ambiguiy abou inflaion and oupu growh The early ambiguiy lieraure focuses on eiher he real economy, for example, ambiguiy abou consumpion growh/tfp growh, or on he nominal side, for example, ambiguiy abou inflaion. However, due o he very differen paerns of he observed forecas dispersion for inflaion and oupu growh, in his paper, we assume ha invesors are ambiguous abou boh inflaion and oupu growh. The se of alernaive measures is generaed by a se of differen mean oupu growh (inflaion) raes around 10

11 he reference mean value µ c + x c, (µ π + x π, ). 6 Specifically, under alernaive measure p µ, oupu growh and inflaion are as follows: g +1 = µ c, + x c, + σ c ε c,+1 π +1 = µ π, + x π, + σ π ε π,+1 (2) where µ c, A c, = [µ c a c,, µ c + a c, ] and µ π, A π, = [µ π a π,, µ π + a π, ] wih boh a c, and a π, being posiive. Each rajecory of µ will yield an alernaive measure p µ for he join process. A larger a c, (a π, ) implies ha invesors are less confiden abou he reference disribuion. Ilu and Schneider (2014) describe he source of ambiguiy in deail, and we consider inflaion process as an example. From he one-sep-ahead inflaion process in equaion (2), every ransiory innovaion o inflaion π +1 x π, consiss of wo componens: σ π ε π,+1 is an i.i.d. normal sequence of innovaions wih mean zero and variance σ π 2 whereas µ π, is a deerminisic sequence. I is assumed ha he empirical momens of µ π, converge o hose of an i.i.d. normal sochasic process ha has mean zero and variance σπ 2 σ π 2 > 0 and ha i is independen of ε π,+1. To he economerician who observes he inflaion daa, he ransiory innovaion hus look like a realizaion of an i.i.d. process wih mean zero and variance σπ. 2 Therefore he deerminisic sequence µ π, canno be learned, even wih a large amoun of daa. For wo-sep-ahead (more han one-sep-ahead in general) inflaion process π +2 = µ π+1 + ρ π x π, + σ π x ε x π,+1 + σ π ε π,+2, he innovaions o inflaion π +2 ρ π x π, conains boh ransiory shocks σ π ε π,+2 and shocks o expeced inflaion σ π x ε x π,+1. Therefore, he size of ambiguiy for long versus shor horizon inflaion could poenially be differen. In he following secion, we specify how ambiguiy changes over ime and model he differen erm srucure of ambiguiy. 6 One requiremen for he alernaive measures is ha hey mus be equivalen o he reference measure P (i.e., hey pu posiive probabiliies on he same evens as P ). 11

12 2.3. Term srucure of ambiguiy To measure ambiguiy empirically, we follow he lieraure and use he forecas dispersion from he BCFF survey. 7 As argued in Ilu and Schneider (2014), he reason is ha invesors sample expers opinions and aggregae hem when making decisions. Thus large disagreemen among expers (as conflicing news) makes invesors less confiden in heir probabiliy assessmens, which corresponds o a bigger size of ambiguiy. We use BCFF forecas dispersion for GDP growh and CPI inflaion from 1985 o The BCFF survey conains forecass for shor-erm and long-erm horizons from he same paricipans, and he dispersion is calculaed as he difference beween he op 10 average and boom 10 average of he individual forecass in levels. Figure 1 shows he one-quarer-ahead and six-years-ahead forecas dispersion for CPI inflaion from 1985 o I is clear ha six-years-ahead dispersion is bigger han one-quarer-ahead dispersion before he lae 1990s, and he relaionship is reversed aferwards. Several approaches can be applied here as inerpreaion of his change. For example, Goodfriend and King (2005) argue ha inflaion scares were creaed during he moneary policy experimenaion of he lae 1970s and early 1980s, and invesors were unsure abou long run inflaion scenarios before he lae 1990s. Kozicki and Tinsley (2001) show he imporance of shifing endpoins from long run inflaion expecaions in he same period. Alernaively, bigger long run inflaion forecas dispersion can be inerpreed as sronger disagreemen among expers. Agens in our model respond o his long run inflaion uncerainy from inflaion scares, or shifing endpoins, or disagreemen by reaing his uncerainy esimaed by economericians as variaion from he deerminisic componen µ π, in equaion (2), herefore heir percepion of ambiguiy for long run inflaion is bigger. Afer he lae 1990s, when he long-erm inflaion expecaions are 7 See, for example, Anderson, Ghysels, and Juergens (2009), Ilu and Schneider (2014), Drechsler (2013), and Zhao (2017). 8 There are wo reasons why we use he BCFF insead of oher surveys such as he Philadelphia Fed s SPF. The firs one is ha he number of forecasers are more sable for he BCFF, which means he forecas dispersion is more accurae. The second is ha he BCFF provides monhly survey resuls, which gives us more daa poins. 12

13 Figure 1: Term srucure of ambiguiy/dispersion for inflaion The dispersion is for one-quarer-ahead and six-years-ahead inflaion forecass from he BCFF from 1985 o Onequarer-ahead forecass are monhly and six-years-ahead forecass are semiannually. well anchored, agens aribue less long-erm uncerainy esimaed by economericians o he deerminisic componen and have less ambiguiy regarding long-erm inflaion. Figure 2 plos he long and shor horizon forecas dispersion for real GDP growh, and i shows ha long horizon dispersion is smaller han shor horizon dispersion for mos periods (excep for a few periods around 1992). This suggess ha invesors do no have a shifing endpoins problem for long-erm real growh and have less ambiguiy abou long run real growh han for shor run real growh. Table 1 shows quaniaively ha he erm srucure of inflaion forecas dispersion has swiched from upward sloping o downward sloping afer he lae 1990s. However, we sill observe a significan amoun of dispersion for even six-years-ahead inflaion forecass in he second subperiod. For real GDP growh, he erm srucure of forecas dispersion is consisenly downward sloping across he wo subperiods, and similar o inflaion, we observe a significan amoun of dispersion for six-years-ahead forecass in boh subperiods. Moivaed by he observed erm srucure of ambiguiy, we model a c, and a π, as a 13

14 Figure 2: Term srucure of ambiguiy/dispersion for GDP The dispersion is for one-quarer-ahead and six-years-ahead GDP forecass from he BCFF from 1985 o One-quarerahead forecass are monhly and six-years-ahead forecass are semiannually Inflaion_Disp_Q Inflaion_Disp_Q Inflaion_Disp_Q Inflaion_Disp_Q Inflaion_Disp_Q Inflaion_Disp_6Y GDP_Disp_Q GDP_Disp_Q GDP_Disp_Q GDP_Disp_Q GDP_Disp_Q GDP_Disp_6Y Table 1: Term srucure of dispersion Table 1 repors he erm srucure of forecas dispersion for inflaion and oupu in wo subperiods. Inflaion_Disp_Q1 refers o one-quarer-ahead inflaion forecas dispersion, Inflaion_Disp_6Y refers o six-years-ahead inflaion forecas dispersion, similarly for oher variables. One-quarer o five-quarers-ahead forecass are monhly and six-years-ahead forecass are semiannually. Survey daa are from he BCFF, and dispersions are in annual percenages. 14

15 random walk wih drif ha are specified in he following way: a c,+1 = µ a c + a c, + σ ac ε ac,+1 + σ ac a ε a,+1 a π,+1 = µ a π + a π, + σ aπ a ε a,+1 (3) where µ a c and µ a π are he drif parameers, which can be posiive or negaive. Given he high correlaion beween inflaion and GDP growh dispersion in he daa, boh a c, and a π, are driven by a common exogenous shock ε a,+1, where he coefficiens σ ac a and σ aπ a capure he correlaion beween hem. ε ac,+1 is an a c, specific shock ha capures he difference of hese wo. 9 Given he fac ha, saring from around 1999, inflaion ambiguiy has swiched from upward sloping o downward sloping and inflaion shocks have swiched from bad news o good news for fuure growh, we assume ha he model has an unexpeced discree regime shif a he end of 1999 (for a deailed discussion, see Secion 4). This is also consisen wih he lieraure for regime breaks; for example, Campbell e al. (2014) argue ha he firs subperiod is he inflaion fighing period of Volcker and Greenspan and he second subperiod is he recen period of low inflaion and increased cenral bank ransparency. Therefore µ a π is posiive for he firs subperiod (dispersion is bigger for a longer horizon) and negaive for he second subperiod (dispersion is smaller for a longer horizon). µ a c is negaive for boh subperiods. Source of ambiguiy - decomposiion Equaion (3) models ambiguiy abou inflaion and real growh in a parsimonious way, which can be furher decomposed ino wo pars. The firs par (denoed by a 1c, or a 1π, ) is a random walk wih no drif (or a saionary process, for example i.i.d normal process), which represens agens ambiguiy from ransiory shocks in equaion (2). The second par (denoed by a 2c, or a 2π, ) is he rend componen, which represens agens 9 We can modify he process by allowing for oupu growh shocks and inflaion shocks. However, due o he CRRA uiliy, we show in an earlier version ha hese shocks have very small effecs on erm premium and yields. 15

16 ambiguiy from he shocks o he expeced inflaion and growh. Since he one-sep-ahead process conains only he ransiory shocks, a 1c, (= a c, ) and a 1π, (= a π, ) are he oal size of ambiguiy for one-quaer-ahead inflaion and real growh. Our specificaion of a 1c, or a 1π, as a random walk wih no drif (or a i.i.d normal process) implies ha here is no rend in he realized one-sep-ahead ambiguiy/dispersion, which is consisen wih he daa (Figure 1 and Figure 2). However, innovaions in he wo-sepahead (more han one-sep-ahead in general) process consis of boh ransiory shocks and shocks o expeced inflaion and growh, hence he oal size of ambiguiy a c,+1 = a 1c,+1 + a 2c,+1 and a π,+1 = a 1π,+1 +a 2π,+1, where a 2c,+1 and a 2π,+1 are he rend componens in he oal size of ambiguiy. For example, one decomposiion could be (again, ake inflaion ambiguiy as an example), a 1π,+1 = c + σ a ε a,+1 and a 2π,+1 = µ + a 2π,. When invesors percepion of ambiguiy for long run inflaion is bigger due o long run uncerainy arises from inflaion scares, or shifing endpoins, or sronger disagreemen, µ is posiive. While in he second subperiod, agens aribue less long-erm uncerainy esimaed by economericians o ambiguiy, and µ is negaive. I is also possible ha, in he second subperiod, shocks o he expeced and unexpeced inflaion and growh are negaively correlaed, which lowers he long run uncerainy iself (see, for example, he dividend growh process in Leau and Wacher (2007)). A each poin in ime, due o he rend componen in ambiguiy from he uncerainy in expeced inflaion and growh, agens perceive long fuure o be more or less ambiguous han for shor horizons. However, when he ime arrives, he second par of ambiguiy conaining he rend componen does no maerialize (expeced inflaion and growh evolve over ime under he rue disribuion), and he realized one-sep-ahead ambiguiy does no become bigger or smaller on average. Only when agens evaluae fuure prospecs ha are more han one sep ahead, does he second par maer. As shown in Secion 3, bond prices are solved under he wors-case disribuion where he EH roughly holds, and he upward-sloping nominal and real curves are mosly due o he rend componen in ambiguiy. Whereas he realized yields are calculaed using he realized one-sep-ahead ambiguiy ha conains no rends, and his difference makes 16

17 excess reurns on long-erm bonds predicable. Noe ha we focus on he average paern of bond and equiy yields in his paper. To infer he hisorical performance of he model, we can use hisorical one-quarer-ahead dispersion as a measure for he size of ambiguiy (only he firs par) in he model. This specificaion of ambiguiy is consisen wih a recen finding ha he esimaed ambiguiy is very persisen; for example, Dew-Becker and Bidder (2016) esimae he ambiguiy shocks have a half-life of 70 years Preference: Recursive muliple priors PS 2007 show he imporance of he Epsein and Zin (1989) preference o generae an upward-sloping nominal yield curve. To illusrae he key role of ambiguiy yields, we assume invesors have recursive muliple priors preference axiomaized by Epsein and Schneider (2003), bu wih CRRA uiliy funcion (invesors are indifferen beween early or lae resoluion of uncerainy): V (C ) = min p P E p (U (C ) + βv +1 (C +1 )) (4) 1 1 γ where U (C ) = C1 γ, γ is he coefficien of risk aversion, and β reflecs he invesor s ime preference. The wors-case belief The agen evaluaes his expeced lifeime uiliy under he subjecive belief p P, and he se of one-sep-ahead beliefs P consiss of he measures p µ generaed in Secion 2.2. Because invesors are ambiguiy averse, hey ac pessimisically and evaluae fuure prospecs under he wors-case measure. We use oupu growh as he endowmen, and he wors-case measure for oupu growh associaed wih he minimum uiliy is generaed by he disribuion wih a c, (he wors mean a each period). 11 For he wors-case inflaion measure, i depends on he correlaion beween inflaion expecaions 10 Resuls in he model rely mainly on he second par ambiguiy. We can change he firs par ambiguiy o a saionary process and he main resuls sill hold. 11 See Epsein and Wang (1994) for a proof. 17

18 and wors-case expeced real oupu growh. Using he boom 10 average of individual GDP growh forecass from he BCFF survey as he wors-case expeced real growh, we find i is negaively associaed wih inflaion expecaions in he firs subperiod and posiively associaed wih inflaion expecaions in he second subperiod. The paern is he same for all differen measures of inflaion expecaion from he BCFF survey: op 10 average, median, and boom 10 average of he individual inflaion forecass (he correlaions are 0.61, 0.52, and 0.39, respecively, for he firs subperiod, and 0.18, 0.38, and 0.46, respecively, for he second subperiod). Thus he wors-case inflaion measure is generaed by disribuion wih he highes mean inflaion +a π, in he firs subperiod and he lowes mean inflaion a π, in he second subperiod. In equilibrium, he min operaor in he preference can be replaced by he wors-case measure Asse markes To solve he model, we firs rewrie he economy dynamics in vecor forms: z +1 = φ a a + µ z + x z, + σ z ε +1 x z,+1 = ρ x x z, + σ x ε +1 (5) a +1 = µ a + a + σ a ε a +1 where z = ( g, π ) T, x = (x c,, x π, ) T, and a = (a c,, a π, ) T. All oher parameers are in vecor forms ha are consisen wih he earlier specificaion in Secion 2. Noe ha equaion (5) describes he wors-case measure in equilibrium. φ a represens he equilibrium choice of he upper or lower bound, equal o 1 or +1. In he following wo subsecions, we will solve bond yields and equiy yields using vecor forms Bond price Since he represenaive agen forms expecaions under he wors-case measure when making porfolio choices, he Euler equaion holds under he wors-case measure. Given he CRRA uiliy funcion, he log nominal pricing kernel or he nominal sochasic 18

19 discoun facor can be wrien as m $,+1 = logβ γ g +1 π c,+1 = logβ v z +1 (6) where v = (γ, 1). The ime- price of a zero-coupon bond ha pays one uni of consumpion n periods from now is denoed P (n), and i saisfies he recursion P (n) = E p o [M $,+1P (n 1) +1 ] (7) wih he iniial condiion ha P (0) = 1 and E p o is he expecaion operaor for he worscase measure. Given he linear Gaussian framework, we assume ha p (n) a linear funcion of a and x : = log(p (n) ) is p (n) = A (n) B (n) x C (n) a. (8) When we subsiue p (n) and p (n 1) +1 in he Euler equaion (7), he soluion coefficiens in he pricing equaion can be solved wih B (n) = B (n 1) ρ x + v = v ( n 1 i=o (ρ x ) i), C (n) = C (n 1) + v φ a = v φ a n, and A (n) is given in he appendix. The log holding period reurn from buying an n period bond a ime and selling i as an n 1 period bond a ime + 1 is defined as r n,+1 = p (n 1) +1 p (n), and he subjecive excess reurn is er n,+1 = Cov ( rn,+1, m $,+1) = B (n 1) σ x σ z v. As we can see from he soluion, he yield parameer for ambiguiy is consan over horizons n, and he average x z, is zero, implying ha, on average, expeced growh and inflaion do no affec long-erm bond yields. The channel hrough which ambiguiy affecs bond yields is he expeced fuure ineres rae embedded in A (n) (due o he rend componen µ a, A (n) /n is bigger for a longer horizon). To solve he price and yields for real bonds, we can jus replace v wih v = (γ, 0). 19

20 Sock price Equiy price and reurns can be solved using he real sochasic discoun facor m,+1 = logβ γ g +1. For any asse j wih a real payoff, he firs-order condiion yields he following asse pricing Euler condiion: E p o [exp(m,+1 + r j,+1 )] = 1 (9) where E p o is he expecaion operaor for he wors-case measure, and r j,+1 is he log of he gross reurn on asse j. To solve he marke reurn, i is assumed ha he log price-dividend raio for dividend claims, z, is linear in a c, and x c, : z = A 0 + A 1 x c, + A 2 a c,. (10) The log marke reurn is given by he Campbell and Shiller (1988) approximaion r m,+1 = k 0 + k 1 z +1 + d +1 z (11) where k 0 and k 1 are log linearizaion consans, which will be discussed wih more deail in he appendix. By subsiuing (10) and (11) ino he Euler equaion (9), we can solve A 0, A 1, and A 2 wih A 1 = ζ d γ 1 k 1 ρ c and A 2 = ζ d γ 1 k 1. For he price of individual dividends (or dividend srips), we can solve i in a similar way. Le P,n denoe he price of a dividend a ime ha is paid n periods in he fuure. Le D +1 denoe he realized dividend in period + 1. The price of he firs D dividend srip is given by P,1 = E p o [M,+1 D +1 ] = D E p o [M +1,+1 D ], and he recursion P,n = E p o [M,+1 P +1,n 1 ] allows us o compue he remaining dividend srip prices. Given he linear Gaussian framework, we assume ha he log dividend srip prices, scaled by he curren dividend, are also affine in he sae variables: pd (n) = A (n) 0 + A n 1x c, + A (n) 2 a c, (12) 20

21 Similar o he bond prices, we can firs compue pd (1) using pd (1) = log ( D E p o [M +1,+1 D ] ), and hen use he recursion pd (n) = log ( E p o [exp ( m,+1 + d +1 + pd (n 1) +1 ) ] ) o compue he remaining dividend srip prices. The soluion coefficiens in he pricing equaion (12) are A n 1 = A n 1 1 ρ c + ζ d γ = (ζ γ) ( n 1 i=o (ρ c ) i), A (n) 2 = A (n 1) 2 (ζ d γ) = n (ζ d γ), and A (n) 0 is given in he appendix. Dividend yield or equiy yield is defined as ey n = 1 n pd(n), which is downward sloping as 1 n A(n) 0 is downward sloping (due o he rend componen µ a c). 1 n A(n) 2 is consan. The logic is he same for bond yields where average x z, is zero and I is worh menioning ha alhough he ambiguiy averse agen acs pessimisically and prices asses under he wors-case measure, we are ineresed in expeced reurns under he reference model because i is he bes esimae of he daa generaing process based on hisorical daa, which are he counerpar of he observed expeced reurns. The wedge beween reference and wors-case mean growh makes he model-implied expeced reurn bigger (ambiguiy premium). Soluions are provided in he appendix. 3. Empirical findings Given he analyical soluions, in his secion we can calculae he nominal/real bond yields, dividend yields, and volailiy explicily. To be consisen wih our empirical finding ha he slope of he yield curve for inflaion ambiguiy has swiched from posiive o negaive, he whole sample, 1985.Q1 o 2017.Q4, is broken ino wo subperiods consisen wih major shifs in moneary policy. Because he earlies available daa for he BCFF forecas dispersion is 1985.Q1, our firs subperiod covers 1985.Q1 o 1999.Q4, par of he Fed chairmanships of Paul Volcker and Alan Greenspan. The second subperiod, 2000.Q1 o 2017.Q4, covers he laer par of Greenspan s chairmanship and he earlier par of Bernanke s chairmanship. We assume ha ransiions from one regime o anoher are srucural breaks, compleely unanicipaed by invesors. In Secion 4, we discuss he model implicaions of allowing a more gradual ransiion beween hese wo regimes. 21

22 3.1. Daa We use quarerly US daa on oupu growh, inflaion, ineres raes, and forecas dispersion from 1985.Q1 o 2017.Q4. Real oupu growh and CPI inflaion are from he Bureau of Economic Analysis. The forecas dispersion for real oupu growh and CPI inflaion are from he Blue Chip Financial Forecas survey. The end-of-quarer yields for one- o en-year bonds are from he daily daase consruced by Gürkaynak e al. (2007) (GSW 2007). The TIPS yields and end-of-quarer yields for hree-monh Treasury bills are from he U.S. Deparmen of he Treasury via he Fed daabase a he S. Louis Federal Reserve, which are available from 2003 o For he one-quarer real risk-free rae, we follow Beeler and Campbell (2012) and creae a proxy for he ex-ane risk-free rae by forecasing he ex-pos quarerly real reurn on hree-monh Treasury bills wih pas one-year inflaion and he mos recen available hree-monh nominal bill yield Esimaion and calibraion The sae space sysem for oupu growh and inflaion is esimaed using maximum likelihood separaely for each subperiod. The resuling parameer values are repored in Table 2. The correlaion beween oupu growh and inflaion is capured by σ x cπ, which is negaive for he firs subperiod and posiive for he second subperiod. Consisen wih PS 2007, inflaion shocks were bad news for fuure growh in he firs subperiod, however, hey urned o be good news in he second subperiod. A he same ime, worscase expeced real growh is negaively associaed wih inflaion expecaion in he firs subperiod and posiively associaed wih inflaion expecaion in he second subperiod. Thus, for ambiguiy averse invesors, he wors-case inflaion measure is he upper bound in he firs subperiod and is he lower bound in he second subperiod. The volailiy parameers in he ambiguiy process are calibraed o mach heir counerpars in dispersion daa. For example, wihin each subperiod, σ aπ a one-quarer-ahead inflaion forecas dispersion volailiy, σ ac is chosen o mach onequarer-ahead oupu growh forecas dispersion volailiy, and σ ac a is chosen o mach is chosen o mach he correlaion beween one-quarer-ahead wors-case inflaion and one-quarer-ahead 22

23 Sae Space Model µ c µ π ρ c ρ π σ c σ π σc x σπ x σcπ x 85.Q1 99.Q Q1 17.Q Ambiguiy µ a c µ a π σ ac σ ac a σ aπ a a 0,c a 0,π β 85.Q1 99.Q Q1 17.Q Oher γ ζ d µ d (P1) µ d (P2) σ d (P1) σ d (P2) Table 2: Configuraion of model parameers Table 2 repors oupu growh, dividend growh, inflaion, and ambiguiy processes parameers. All parameers are given in quarerly erms. Mean and sandard deviaion are in percenages. wors-case oupu growh. Table 2 shows ha hese values are quaniaively small, and we acually show in he following secion ha he impac of volailiy in he ambiguiy process on bond yields is negligible in his model. Given he small volailiy, our resuls are quaniaively close o he exreme case where here is no uncerainy in he ambiguiy process. The rend componen µ a and he iniial value a 0 are also calibraed o mach he daa in dispersion. For each subperiod, a c,0 and a π,0 are chosen o mach average onequarer-ahead dispersion values in he daa, µ a c and µ a π are chosen o mach he average difference beween six-years-ahead and one-quarer-ahead forecas dispersion (six-yearsahead minus one-quarer-ahead dispersion and hen divide by 24). For oher parameers, we follow he lieraure and se risk aversion as 3, and se leverage parameer ζ d = 3. µ d is chosen such ha he average rae of dividend growh is equal o he mean growh rae of dividends in he daa. Given he leverage raio, σ d can be calibraed o mach he sandard deviaion of dividend growh in he daa. Finally, ime preference β is calibraed o mach one-year nominal yields in he daa for each subperiod, which are close o he value in PS Higher ime preference helps o lower bond yield levels. We can also se β o be smaller han 1, bu hen we need o eiher decrease he risk aversion parameer or change he level of ambiguiy o mach he bond yield level. 23

24 3.3. Bond yields and volailiy Real bond yields Using TIPS daa from he U.S. Deparmen of he Treasury from 2003 o 2017, Table 3 repors he level and volailiy of real yields. Alhough here are less han weny years of TIPS daa, he observed slope has never been quaniaively significanly negaive. The volailiy of real yields is smaller for a longer horizon. Campbell (1986) argues ha, if consumpion growh is modeled as a persisen process where posiive shocks cause upward revisions in expeced fuure growh, a posiive consumpion shock causes real ineres raes o increase and bond prices o fall. In his case, real bonds hedge consumpion risk and have a negaive real erm premium. Thus, asse pricing models wih persisen consumpion growh processes are likely o be inconsisen wih he daa. In his model, for boh subperiods, invesors are less ambiguous abou longer horizon oupu growh. In equilibrium, ambiguiy averse agens choose he lower bound from he se of alernaive mean oupu growh raes, which are upward sloping. As a resul, he fuure ineres raes are higher for longer horizons. The model-implied real yields are repored in Table 3, which are upward sloping and consisen wih he daa. The volailiy in yields consiss of wo pars: (1) shocks from expeced growh where he weigh is smaller for a longer horizon (due o he persisence in expeced oupu growh ρ c ), and (2) shocks from ambiguiy where he weigh is consan. Therefore our modelimplied volailiy is consisen wih he daa and is downward sloping. However, due o our small risk aversion parameer, he size of volailiy is somewha smaller in magniude. To check he effeciveness of he mechanism described above, we shu down he ambiguiy for oupu growh and repor he resuls for real yield in Table 3 as well. As expeced, he real yield curve is almos fla now (he higher yield for one-quarer real bonds is due o he fac ha he real ineres rae provides hedges o growh risks. Bu wih CRRA uiliy, his effec only appears in shor horizons), and he volailiy also rapidly declines o almos zero (due o a small ρ c and no ambiguiy shocks in he long end of he yield curve). 24

25 Real Bond 00.Q1 17.Q4 1Q 5Y 7Y 10Y Daa Model Model (No ambiguiy) Yield Sd Yield Sd Yield Sd Table 3: Real bond yields and volailiy This able presens daa and model-implied real bond yields and volailiy for he second subperiod. available for five years, seven years, and en years o mauriy from 2003 and TIPS yields are Nominal bond yields There is a large body of finance lieraure modeling bond yields wihou disinguishing differences beween subperiods. Mos of hese sudies use inflaion non-neuraliy esablished in PS 2007 o generae upward-sloping yield curves. This mechanism requires ha agens prefer early resoluion of uncerainy, and a he same ime, inflaion is bad news for fuure growh. A posiive surprise o inflaion implies lower fuure growh and lower real payoff of long-erm bonds. Therefore, agens require excess reurns o hold long-erm bonds over shor-erm bonds. To undersand he changes in correlaion beween U.S. Treasury bond reurns and sock reurns, recen sudies have shown ha he correlaion beween consumpion growh and inflaion has swiched from negaive o posiive afer he lae 1990s. For example, Song (2017) esimaes a regime swich version of PS 2007 and finds ha he U.S. economy enered a posiive correlaion regime (beween inflaion and growh) afer he lae 1990s and largely remained in ha regime hroughou he sample. Our esimaion for he reference sae space model in Table 2 is also consisen wih hese findings. Given hese changes, he sandard approach implies a downward-sloping nominal yield curve for he curren period. However, we sill observe an upward-sloping nominal yield curve in he daa (as repored in Table 4), which implies ha we need o undersand nominal yields using a differen approach, a leas for he curren period. During he firs subperiod in his model, invesors have more ambiguiy abou infla- 25

26 ion in longer horizons. Togeher wih he fac ha he wors-case expeced growh is negaively associaed wih inflaion expecaion, ambiguiy averse invesors choose he upper inflaion bound o evaluae he fuure perspecive. This implies ha expeced inflaion in equilibrium is upward sloping, which generaes an upward-sloping nominal yield curve. During he second subperiod, he wors-case expeced growh is posiively associaed wih inflaion expecaion, and he wors-case mean inflaion becomes he lower bound. A he same ime, invesors have less ambiguiy abou inflaion in longer horizons, which again implies an upward-sloping mean inflaion in equilibrium. Therefore he model generaes upward-sloping nominal yield curves in boh subperiods, bu wih a differen mechanism. Table 4 repors nominal bond yields from he daa and implied by he model for boh subperiods, and i is clear ha he model maches he daa very well. Anoher imporan difference in nominal yields is ha he average yield level has dropped dramaically from 6.14 for a one-year nominal bond in he firs subperiod o 1.86 in he second subperiod. Par of he reason for his change is he decrease in mean oupu growh (from 0.86% quarerly o 0.45% quarerly) and decrease in mean inflaion (from 0.74% quarerly o 0.57% quarerly). They alone (including differences in ime preference for he wo subperiods), however, are far from providing a complee answer o he almos 70% drop in nominal yields. In his model, he wors-case mean inflaion in equilibrium is he upper bound in he firs subperiod and swiches o he lower bound in he second subperiod. Thus, he difference beween he upper bound and he lower bound of he inflaion dispersion provides anoher significan conribuion o he drop in nominal yields (accouning for 42% of he changes). In a similar way o he real bonds, nominal bond yield volailiy consiss of boh volailiy from he expeced growh x z,+1, which is decreasing over horizons, and volailiy from he ambiguiy process a +1, which is consan over horizons. Thus he model-implied volailiy shares he same paern of decreasing over horizons as in he daa. However, he size of volailiy is somewha smaller in magniude. Besides he small risk aversion parameer as one reason, we can also increase he ambiguiy volailiy in order o increase 26