Ambiguiy, Nominal Bond Yields, and Real Bond Yields Guihai Zhao a,1 a Bank of Canada, 234 Wellingon Sree, Oawa, ON K1A 0G9 Absrac The lieraure relies on inflaion non-neuraliy o generae upward sloping nominal yield curves (PS 2007). We develop a model ha can generae upward sloping nominal and real yield curves wihou relying on inflaion non-neuraliy, bu insead using ambiguiy abou inflaion and growh o produce his resul. This can help resolve he puzzle abou why upward sloping yield curves have persised despie posiive inflaion shocks become good news for fuure growh in he pos 1990s period. The model is also consisen wih he recen empirical findings on he erm srucure of equiy reurns. Keywords: Ambiguiy, Term Srucure, Regime Break, Bond Yields, Equiy Yields JEL Classificaion: G00, G12, E43 1. Inroducion To generae upward sloping nominal yield curve, equilibrium bond-pricing models rely on inflaion as bad news for fuure growh and he assumpion ha agens prefer early resoluion of uncerainy, for example, Piazzesi and Schneider (2007) (henceforh PS 2007) and Bansal and Shaliasovich (2013). The inuiion is ha a posiive surprise in inflaion is bad news for fuure growh, and a he same ime, long erm bonds pay off less han shor erm bonds. Therefore, long erm bonds command a erm spread over shor erm bonds. Recen developmens in he bond marke lieraure have shown ha he correlaion beween consumpion growh and inflaion has swiched from negaive o I am graeful o Larry Epsein, Simon Gilchris, and Francois Gourio for heir coninuing advice and suppor on his projec. I appreciae he helpful commens of Marin Eichenbaum, Bruno Feunou, Anonio Diez de los Rios, and Jonahan Wimer. All errors are my own. Email address: GZhao@bank-banque-canada.ca (Guihai Zhao) 1 Fax: +1 613 782 7535
posiive afer lae 90s, which can successfully explain he changes in correlaion beween U.S. Treasury bond reurns and sock reurns. 2 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 he nominal yield curve slopes up afer lae 90s. This paper provides an alernaive approach o undersand upward sloping nominal yield curves in boh environmens. 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 Unied Saes. 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, i is much harder for an equilibrium model of bond pricing o capure also real bond yields. In fac, excep for Wacher (2006), he previously menioned models generae a downward sloping real yield curve. Finally, he recen empirical findings on he erm srucure of equiy reurns pose some serious challenges o equilibrium models (see below for he deails). 3 This paper develops a consumpion-based asse pricing model ha helps o explain he preceding feaures in he daa, by posiing ha 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. There is incomplee confidence ha any single disribuion accuraely describes he environmen and hus here is uncerainy abou he disribuion. Specifically, i is assumed ha here is ambiguiy abou boh real growh and inflaion processes. Using forecas dispersion as an empirical measure for size of ambiguiy (or confidence), we find ha, before he lae 90s, he size of ambiguiy for long horizon inflaion is bigger han hese for shor horizons, and he erm srucure of ambiguiy is reversed aferwards. However, he erm srucure of 2 For example, Burkhard and Hasselof (2012); David and Veronesi (2013); Song (2014); Campbell e al. (2016). 3 See, e.g., Van Binsbergen and Koijen (2017) for a survey 2
ambiguiy for real oupu growh is always downward sloping. In equilibrium, ambiguiy averse agens evaluae fuure prospecs under he wors-case measure. Given he fac ha inflaion is bad news for fuure growh before he lae 90s, and good news aferwards, he equilibrium mean growh and inflaion is upward sloping for boh subperiods, which generaes upward sloping nominal and real yield curves in boh environmens. There are wo main ingrediens in he model. Firs, deparing from he raional expecaion hypohesis, he model assumes ha invesors are ambiguiy averse and have he recursive muliple-priors (or maxmin) preference wih consan relaive risk aversion (CRRA) uiliy funcion (Epsein and Schneider (2003)). Invesors in his economy have in mind a benchmark or reference measure of he economy s dynamics ha represens he bes esimae of he sochasic process. However, hey 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. Second, under he reference measure, real growh and inflaion are described by a simple sae space model as in Piazzesi and Schneider (2007). The se of alernaive measures for real growh/inflaion is generaed by a se of differen mean real growh/inflaion around is reference mean value. We use he Blue Chip Financial Forecas (BCFF) survey o characerize he properies of ambiguiy yields for US real oupu growh and consumer price index (CPI) from 1985 o 2016. Moivaed by he fac ha inflaion forecas dispersion has swiched from upward sloping o downward sloping afer he lae 90s, we model inflaion ambiguiy as a random walk wih posiive drif in he firs subperiod and wih negaive drif in he second subperiod. Ambiguiy abou real growh is modeled as a random walk wih negaive drif for boh subperiods because real oupu growh forecas dispersion has been consisen and always downward sloping. We assume ha he model has 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 3
of low inflaion and increased cenral bank ransparency 4. One possible inerpreaion for he observed change in erm srucure of forecas dispersion is ha, as argued by Goodfriend and King (2005), inflaion scares were creaed during he moneary policy experimenaion of lae 1970s and early 1980s, and invesors are no sure abou fuure inflaion scenarios unil inflaion was fully under conrol afer lae 1990s. Currenly, invesors have less ambiguiy regarding longer horizon inflaion due o a clear undersanding of inflaion argeing and he low inflaion environmen. 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 yields is always upward sloping. During he firs subperiod, when he inflaion shocks are bad news for fuure real oupu growh, he wors-case mean inflaion is he upper bound, which is upward sloping because because he dispersion is bigger for longer horizon. This implies ha invesor s subjecive nominal ineres rae expecaion is upward sloping and similarly for he nominal yields. Whereas during he second subperiod, inflaion surprises become good news for fuure growh, and he wors-case mean inflaion is he lower bound now. 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 upward sloping nominal yield curves in boh subperiods, bu wih differen mechanism. The model implied bond yield volailiies are also consisen wih daa across periods. Piazzesi e al. (2015) show ha he expeced excess reurns on long 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, he wors-case measure as invesor s subjecive measure implies ha subjecive ineres rae expecaions are bigger for longer horizons. If he saisi- 4 See, for example, Campbell e al. (2014) and Zhao (Forhcoming) for a similar regime break. The resuls are robus o differen regime break poins. 4
cal fuure ineres rae expecaions is consan, our model would generae bigger excess reurns for long bonds, and we hus provide a poenial heoreical framework for heir empirical 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 for differen counries. Using dividend fuure conracs for he S&P500, Van Binsbergen and Koijen (2017) show ha he 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. Relaed Lieraure This paper is closely relaed o some recen developmens in erm srucure of bonds. Using Epsein and Zin (1989) preference, Piazzesi and Schneider (2007) show ha inflaion as bad news for fuure consumpion growh can generae upward sloping nominal yield curve. In a similar vein, Wacher (2006) generaes upward-sloping nominal and real yield curves in an exernal habis model of 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 volailiies of expeced consumpion growh and inflaion can accoun for bond reurn predicabiliy. Ulrich (2013) argues ha, even wih log uiliy, ambiguiy abou rend inflaion can help generae an upward sloping erm premium for nominal bond if inflaion shocks make he size of ambiguiy bigger. While all hese sudies focus on yield curve of he whole sample period (no regime swich), Song (2014) exends he long-run risks model of Bansal and Yaron (2004) by allowing regime swich in he correlaion beween consumpion growh and inflaion arge. He finds ha he U.S. economy has enered a posiive correlaion regime afer lae 90s and largely remained in ha regime hroughou he sample. He argue ha i will help o generae upward sloping nominal yield curve if, during he posiive correlaion regime of curren period, agen sill 5
evaluae long erm bonds using uncondiional probabiliy of swiching back o negaive correlaion regime for he whole sample period of 1963-2014 5. However, Malmendier and Nagel (2016) find ha individuals form expecaions abou fuure inflaion using heir experienced inflaion during heir lifeimes, which suggess ha i may be more reasonable for agen o use condiional probabiliy, which likely o generae an downward sloping yield curve. This paper differs from hese previous sudies along some imporan dimensions. Firs, given ha inflaion has swiched from bad news o good news for fuure growh afer he lae 90s, i is sill puzzling o generae upward sloping nominal yield curve from he perspecive of equilibrium bond-pricing models. This model provides an alernaive undersanding of upward sloping nominal yield curve for wo environmens where inflaion can be bad or good news for fuure growh. Second, in he weny year hisory of TIPS daa, he observed slope has never been maerially negaive. Wih he excepion of Wacher (2006), previous sudies ypically generae downward sloping real yield curve, and his paper provides a new mechanism o generae an upward sloping real yield curve for boh subperiods. We also show ha, in his model, he ambiguiy erm premium ha Ulrich (2013) uses o generae upward sloping nominal bond yields is quaniaively very small and is negaive in he second period where inflaion shocks make he size of inflaion ambiguiy smaller. The upward sloping feaure for bond yields is mainly driven by he erm srucure of ambiguiy. The model is consisen wih he empirical finding in Piazzesi e al. (2015) ha he difference beween subjecive and saisical fuure ineres rae expecaions is an imporan par of observed bond excess reurns. 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 5 In he posiive correlaion regime of curren period, he condiional probabiliy of swiching back o 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 of periods before lae 90s. 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 even using he uncondiional probabiliy in Song (2014). 6
ime-varying ambiguiy abou produciviy generaes business cycle flucuaions. Using forecas dispersion daa, Zhao (Forhcoming) show ha ambiguiy abou consumpion growh is driven by pas inflaion and argue ha bond risk changes are due o ime-varying impac of inflaion on ambiguiy. This paper conribues o he ambiguiy lieraure by firs showing 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. From he perspecive of equilibrium models, his paper is he firs effor, o my knowledge, o joinly undersand real and nominal bond yield curves across differen subperiods. 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 endownmen 6. 2.1. Economy dynamics Under reference measure P, oupu growh and inflaion follow a sae space model, and dividend growh is leveraged oupu growh 6 We use oupu growh as he endowmen process because he non-durable good and service survey is no available in BCFF. Using he Philadelphia Fed s Survey of Professional Forecasers (SPF), Zhao (Forhcoming) shows ha he dispersion for consumpion growh and oupu growh are highly correlaed. 7
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 Piazzesi and Schneider (2007), he sae space represenaion for z +1 = ( g +1, π +1 ) T does a beer job a 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) 7, 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 one 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 he 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. 2.2. Ambiguiy abou Inflaion and Oupu Growh Early ambiguiy lieraure focuses on eiher he real economy, for example, ambiguiy abou consumpion growh / TFP growh, or on 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 7 The lieraure also repors several oher ways, See Campbell (1999); Cecchei, Lam, and Mark (1993); Bansal and Yaron (2004) 8
ambiguous abou boh inflaion and oupu growh. The se of alernaive measures is generaed by a se of differen mean oupu growh (inflaion) around he reference mean value µ c + x c, (µ π + x π, ) 8. Specifically, under alernaive measure p µ, oupu growh and inflaion follow: g +1 = µ c + µ c, + x c, + σ c ε c,+1 π +1 = µ π + µ π, + x π, + σ π ε π,+1 (2) where µ c, A c, = [ a c,, a c, ] and µ π, A π, = [ a π,, a π, ] wih boh a c, and a π, be 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 confidence abou he reference disribuion. In he following secion, we specify how ambiguiy changes over ime and model he differen erm srucure of ambiguiy. 2.3. Term srucure of ambiguiy The lieraure has been using forecas dispersion as a measure of ambiguiy 9. As argued in Ilu and Schneider (2014), he reason is ha invesors sample expers opinions and aggregae hem when making decisions. Thus sronger disagreemen among expers makes invesors less confiden in heir probabiliy assessmens for fuure prospecs. We follow he lieraure and use he BCFF survey o sudy he properies of ambiguiy yields for US real oupu growh and consumer price index (CPI) from 1985 o 2016 10. The dispersion is calculaed as he difference beween he op 10 average and boom 10 average of he individual forecass in levels. Figure 1 shows one quarer and five quarers ahead monhly forecas dispersion for 8 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 ). 9 See, for example, Anderson, Ghysels, and Juergens (2009), Ilu and Schneider (2014), and Drechsler (2013), and Zhao (Forhcoming). 10 There are wo reasons ha we use BCFF insead of oher survey such as he Philadelphia Fed s Survey of Professional Forecasers (SPF). The firs one is ha he number of forecasers are more sable for BCFF which means he forecas dispersion is more accurae. The second is ha BCFF provides monhly survey which gives us more daa poins. 9
CPI inflaion from 1985 o 2016. I is clear ha five quarers ahead dispersion is bigger han one quarer dispersion before he lae 90s, and he relaionship has reversed aferward. One possible inerpreaion is ha, as argued by Goodfriend and King (2005), inflaion scares were creaed during he moneary policy experimenaion of lae 1970s and early 1980s, and invesors were no sure abou fuure inflaion scenarios unil inflaion was fully under conrol afer he lae 90s. Currenly, invesors have less ambiguiy regarding longer horizon inflaion due o a clear undersanding of inflaion argeing and he low inflaion environmen. Figure 2 plos long and shor horizon forecas dispersion for real GDP growh and i suggess ha, excep for 1985, longer horizon dispersion is smaller han shor horizon dispersion for mos of he periods. This paern is more clear when comparing one quarer ahead GDP forecas dispersion wih even longer horizon dispersion, for example, 6-o-10 years ahead forecass ha only available semiannually afer 1996. I seems ha, a leas afer 1986, here are no real GDP scares, which may be due o he fac ha invesors undersand GDP growh is always one key objecive for he Federal Reserve Bank and hey are less concerned abou long run growh. Table 1 shows quaniaively ha he erm srucure of inflaion forecas dispersion has swiched from upward sloping o downward sloping afer lae 90s. However, we sill observe a significan amoun of dispersion for even 6-o-10 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 as for inflaion, we observe a significan amoun of dispersion for 6-o-10 years ahead forecass in he boh subperiods. Moivaed by he observed erm srucure of ambiguiy, we model a c, and a π, as random walk wih drif ha are specified in he following way, a c,+1 = µ a c + a c, + σ ac c ε c,+1 + σ ac π ε π,+1 + σ ac a ε a,+1 a π,+1 = µ a π + a π, + σc aπ ε c,+1 + σπ aπ ε π,+1 + σa aπ ε a,+1 (3) where µ a c and µ a π are he drifs ha could be posiive and negaive; σ ac c and σ aπ c capure 10
Figure 1: Term srucure of ambiguiy/dispersion for Inflaion The dispersion is he one quarer ahead and five quarers ahead monhly inflaion forecass from BCFF from 1985 o 2016. Figure 2: Term srucure of ambiguiy/dispersion for GDP The dispersion is he one quarer ahead and five quarers ahead monhly inflaion forecass from BCFF from 1985 o 2016. 1985-1999 2000-2016 Inflaion_Disp_Q1 1.6 1.7 Inflaion_Disp_Q2 1.7 1.6 Inflaion_Disp_Q3 1.9 1.5 Inflaion_Disp_Q4 2.0 1.4 GDP_Disp_Q1 2.5 1.8 GDP_Disp_Q2 2.5 1.7 GDP_Disp_Q3 2.4 1.6 GDP_Disp_Q4 2.3 1.4 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 1 quarer ahead inflaion forecas dispersion, similarly for oher variables. Survey daa are from BCFF and dispersions are in percenage. 11
how oupu growh shocks affec ambiguiy; σπ ac and σπ aπ capure how ambiguiy is affeced by inflaion shocks; σ ac a shocks. and σ aπ a capure all oher shocks apar from inflaion and growh Given he fac ha inflaion ambiguiy has swiched from upward sloping o downward sloping, we assume ha he model has an unexpeced discree regime shif a he end of 1999. 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 longer horizon) and negaive for he second subperiod(dispersion is smaller for longer horizon). While µ a c is negaive for boh subperiods. Noe ha we focus on he average paern of bond and equiy yields in his paper, no heir hisorical movemens. A each poin of ime, he specificaion in equaion (3) represen agen s belief of how he size of ambiguiy evolves over ime when he makes decisions. The oal ambiguiy capured by a c,+1 or a π,+1 in equaion (3) can be hough of as wo pars: a consan par represening ambiguiy abou he ransiory shocks, and a random walk par ha capures ambiguiy abou he rend componen (or ambiguiy abou ambiguiy in he firs par). A each period, one observes he oal realized ambiguiy measured by one quarer ahead dispersion. Therefore, even hough he size of ambiguiy evolves following a random walk process in agen s mind, he observed realized ambiguiy is saionary (one quarer ahead dispersion). To infer he hisorical performance of he model, one can use hisorical one quarer ahead dispersion as a measure for size of amibguiy in he model. This specificaion of ambiguiy is consisen wih 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. 2.4. Magniude of Ambiguiy Given he specificaion for ambiguiy process, one naural quesion is wheher he size of ambiguiy is reasonable. We use he error deecion probabiliy approach suggesed 12
by Anderson, Hansen, and Sargen (2003) o provide a sense of magniude of he size of ambiguiy. This approach quanifies he saisical closeness of wo measures by calculaing he average error probabiliy in a Bayesian likelihood raio es of wo compeing models. Inuiively, measures ha are saisically close will be associaed wih large error probabiliies, bu measures ha are easy o disinguish imply low error probabiliies. Formally, le l be he log likelihood funcion of he wors-case measure relaive o he reference measure and P a be he alernaive wors-case measure. Then, he average probabiliy of a model deecion error in he corresponding likelihood raio es is ɛ = 0.5 P (l > 0)+0.5 P a (l < 0), where ɛ is jus a simple equally weighed average of he probabiliy of rejecing he reference model when i is rue (P (l > 0)) and he probabiliy of acceping he reference model when he wors case model is rue (P a (l < 0)). 2.5. Preference: Recursive Muliple Priors Piazzesi and Schneider (2007) show he imporance of Epsein and Zin (1989) preference o generae upward sloping nominal yields curve in a seing where inflaion shocks are bad news for fuure consumpion growh. 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 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. Agen evaluaes his expeced lifeime uiliy under 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 endowmen in his paper and he wors-case measure for oupu growh ha gives he minimum coninuaion 13
value is generaed by likelihood wih he wors mean a c, each period 11. For he worscase inflaion measure, i depends on he correlaion beween inflaion and oupu growh. Consisen wih Piazzesi and Schneider (2007), inflaion and oupu growh are negaively associaed in he firs subperiod, hus he wors-case inflaion measure is generaed by likelihood wih he highes mean inflaion +a π,. However, inflaion and oupu growh are posiively correlaed in he second subperiod, and he he wors-case inflaion measure is generaed by likelihood wih he lowes mean inflaion a π,. Thus, in equilibrium, he min operaor in he preference can be replaced by he wors-case measure. 2.6. 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 1ε +1 + σ a 2ε 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 earlier specificaion in secion 2. Noe ha equaion (5) describes he wors-case measure in equilibrium. In he following wo subsecions, we will solve bond yields and equiy yields using vecor forms. 2.6.1. Bond Price Since he represenaive agen evaluaes 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 discoun facor can be wrien as m $,+1 = logβ γ g +1 π c,+1 = logβ v z +1 (6) 11 See also Epsein and Wang (1994) for a proof. 14
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) Then we subsiue p (n) and p (n 1) +1 in he 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 periods bond a ime and selling i as an n 1 periods 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 C (n 1) σ a 1σ 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 implies ha expeced growh and inflaion in he fuure do no affec bond yields. The channel hrough which ambiguiy affec 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 longer horizon). To solve he price and yields for real bonds, one can jus replace v wih v = (γ, 0). 2.6.2. Sock Price Equiy price and reurns can be solved using real sochasic discoun facor m,+1 = logβ γ g +1. For any asse j wih real payoff, he firs-order condiion yields he 15
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 claim, 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), one can solve A 0, A 1, and A 2 wih A 1 = ζ d γ 1 k 1 ρ c way. and A 2 = ζ d γ 1 k 1. For he price of individual dividends (or dividend srips), one can solve i in a similar 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) Similar as for 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 16
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). The logic is same for bond yields ha average x z, is zero and 1 n A(n) 2 is consan. I worh o menion 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 volailiies explicily. To be consisen wih our empirical finding ha he slope of yield curve for inflaion ambiguiy has swiched from posiive o negaive, he whole sample, 1985.Q1 o 2016.Q4, is broken ino wo subperiods consisen wih major shifs in moneary policy. Because he earlies available daa for BCFF forecas dispersion is 1985.Q1, our firs subperiod 1985.Q1 o 1999.Q4, covers par of Fed chairmanships of Paul Volcker and Alan Greenspan. The second subperiod, 2000.Q1 o 2016.Q4, covers he laer par of Greenspan s chairmanship and he earlier par of Ben Bernanke s chairmanship. 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. We assume ha ransiions from one regime o anoher are srucural breaks, compleely unanicipaed by invesors. 17
3.1. Daa We use quarerly US daa on oupu growh, inflaion, ineres raes, and forecas dispersion from 1985.Q1 o 2016.Q4. Real oupu growh and PCI inflaion are from he Bureau of Economic Analysis. The forecas dispersion (1 quarer ahead and 5 quarers ahead) for real oupu growh and PCI inflaion are from Blue Chip Financial Forecas survey. The end-of-quarer yields for hree monh and one o en year bonds are from he daily daase consruced by Gürkaynak e al. (2007)(GSW 2007). The TIPS yields are from U.S. deparmen of he reasury via he Fed daabase a he S. Louis Federal Reserve, which are available from 2003 o 2017. For he 1 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. 3.2. Esimaion and calibraion The sae space sysem for oupu growh and inflaion is esimaed using maximum likelihood separaely for each subperiod. 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 Piazzesi and Schneider (2007), inflaion shocks were bad news for fuure growh in he firs subperiod, however, hey urned o be good news 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 ac c correlaion beween inflaion dispersion and oupu growh, σ ac π is chosen o mach correlaion beween inflaion dispersion and inflaion, σ ac a is chosen o mach is chosen o mach inflaion dispersion volailiy. Similarly for σc aπ, σπ aπ, and σa aπ. Table 2 shows ha hese values are quaniaive 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 volail- 18
Sae Space Model µ c µ π ρ c ρ π σ c σ π σc x σπ x σcπ x 85.Q1-99.Q4 0.86 0.74 0.92 0.76 0.46 0.27-0.03 0.10-0.07 00.Q1-16.Q4 0.45 0.57 0.49-0.19 0.55 0.42 0.21 0.17 0.03 Ambiguiy µ a c µ a π σ ac c σ ac π σ ac a σ aπ c σ aπ π σ aπ a a 0,c a 0,π 85.Q1-99.Q4-0.0067 0.0012-0.008 0.0067 0.0126-0.0035 0.005 0.0076 0.32 0.095 00.Q1-16.Q4-0.0067-0.005-0.0045-0.0014 0.0105-0.0007-0.0004 0.0183 0.238 0.143 Oher β γ ζ d µ d (P1) µ d (P2) σ d (P1) σ d (P2) 1.004 2 3-2.40-1.17 2.70 2.55 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 percenage. iy, our resuls are quaniaive close o he exreme case where here is no uncerainy in ambiguiy process. The rend componen µ a and he iniial value a 0 are joinly calibraed o mach boh error deecion probabiliy and bond yields. a 0,c is chosen o mach he level of real ineres rae, µ a c is chosen o mach he slope of real yields, a 0,π is chosen o mach he level of 3 monh T-bill, and µ a π is chosen o mach he slope of nominal yields. A he same ime error deecion probabiliies for boh oupu and inflaion are a leas 5% in each subperiods (5% and 15% for oupu growh in period 1&2, 5% and 13% for inflaion in period 1&2). Noe ha he calibraed rend µ a and iniial value a 0 are very close o heir values in dispersion daa. For oher parameers, we follow he lieraure and se risk aversion as 2, 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 leverage raio, σ d can be calibraed o mach he sandard deviaion of dividend growh in he daa. Finally, we follow Piazzesi and Schneider (2007) and se he ime preference β = 1.004 12. 12 Higher ime preference helps o lower bond yield level. One can also se β o be smaller han 1, bu hen need o eiher decrease risk aversion parameer, or change he level of ambiguiy o mach he bond yield level. 19
3.3. Bond yields and volailiy 3.3.1. Real bond Using TIPS daa from U.S. deparmen of he reasury from 2003 o 2016, 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 significan negaive. The volailiy of real yields is smaller for 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 process 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, which is upward sloping. As a resul, he fuure ineres rae is higher for longer horizons. The model implied real yields are repored in Table 3, which is 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 longer horizon (due o he persisence in expeced oupu growh ρ c ), and (2) shocks from ambiguiy where he weigh is consan. Therefore our model implied volailiy is consisen wih daa and 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, and he volailiy also rapidly declines o almos zero (due o small ρ c and no ambiguiy shocks in he long end of he yield curve). 20
Real Bond 00.Q1-16.Q4 1Q 5Y 7Y 10Y Daa Model Model - No Ambiguiy Yield 0.53 0.81 1.05 Sd 1.13 1.03 0.93 Yield 0.08 0.53 0.74 1.05 Sd 1.33 0.4 0.38 0.37 Yield 1.98 1.93 1.93 1.93 Sd 1.25 0.12 0.09 0.06 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 5 years, 7 years, and 10 years o mauriy from 2003 and 2016. TIPS yields are 3.3.2. Nominal bond There is a large body of lieraure modeling nominal bond yields which are usually focused on he whole sample period. And mos sudies use inflaion non-neuraliy esablished in Piazzesi and Schneider (2007) o generae upward sloping yield curves. However, from he perspecive of equilibrium asse-pricing models, i is sill puzzling if we spli he whole sample ino differen subperiods. For example, before he lae 90s, a posiive surprise in inflaion is bad news for fuure growh, and a he same ime, long erm bonds pay off less han shor erm bonds. Therefore, long erm bonds command a erm spread over shor erm bonds. However, inflaion shocks have become good news for fuure growh afer lae 1990s, and a he same ime, we sill observe an upward sloping nominal yields 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 inflaion in longer horizons, ogeher wih he fac ha inflaion shocks are bad news for fuure oupu growh, 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. Whereas during he second subperiod, inflaion surprises become good news for fuure growh, and he wors-case mean inflaion is he lower bound now. A he same ime, invesors have less ambiguiy abou inflaion in longer horizons, which again implies an upward sloping 21
mean inflaion in equilibrium. Therefore he model generaes upward sloping nominal yield curves in boh subperiods, bu wih differen mechanism. Table 4 repors nominal bond yields from daa and implied by he model for boh subperiods, and i is clear ha he model mach he daa very well. Anoher imporan difference in nominal yields is ha he average yield level has dropped dramaically from 6.14 for 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, 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 upper bound and lower bound of he inflaion dispersion provides anoher significan conribuion o drop in nominal yields. In a similar way o he real bond, nominal bond yield volailiy consiss of boh volailiy from he expeced growh x z,+1, which is decreasing over horizons, and volailiy from 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. Beside he small risk aversion parameer as one reason, one can also increase he ambiguiy volailiy in order o increase he bond yield volailiy 13. Wihou ambiguiy Since he nominal ineres rae is he sum of real ineres rae and expeced inflaion, and given he upward sloping real yield curve (due o ambiguiy abou oupu growh), i is naural o ask wheher inflaion ambiguiy maers for generaing upward sloping nominal yield curve. For his purpose, we shu down he ambiguiy for inflaion only 13 Currenly we mach he ambiguiy volailiy in he model o volailiy in he dispersion daa, one can also calibrae ambiguiy volailiy using yield volailiy. 22
Period 1 1985-1999 Daa Model Nominal Bond 1Y 2Y 3Y 4Y 5Y 10Y Yield 6.14 6.47 6.69 6.87 7.01 7.47 Sd 1.56 1.51 1.48 1.46 1.45 1.39 Yield 6.14 6.25 6.37 6.45 6.60 7.18 Sd 0.64 0.59 0.55 0.52 0.49 0.42 Model (No inflaion ambiguiy) Yield 5.75 5.86 5.96 6.07 6.18 6.70 Model (No ambiguiy) Period 2 2000-2016 Sd 0.75 0.71 0.68 0.65 0.63 0.57 Yield 8.23 8.23 8.23 8.23 8.23 8.23 Sd 0.52 0.45 0.40 0.35 0.31 0.19 Daa Yield 1.86 2.09 2.35 2.61 2.86 3.75 Model Model (No inflaion ambiguiy) Model (No ambiguiy) Sd 1.91 1.80 1.69 1.58 1.50 1.28 Yield 1.86 1.99 2.13 2.27 2.42 3.13 Sd 0.97 0.75 0.70 0.68 0.67 0.65 Yield 2.40 2.49 2.60 2.70 2.80 3.32 Sd 0.80 0.53 0.45 0.42 0.40 0.38 Yield 4.23 4.21 4.21 4.21 4.21 4.20 Sd 0.69 0.36 0.24 0.18 0.15 0.07 Table 4: Nominal Bond Yields and Volailiy This able presens daa and model implied nominal bond yields and volailiy for boh subperiods. The end-of-quarer yields for one o en year bonds are from he daily daase consruced by GSW 2007. 23
and provide he yields and volailiies in Table 4. There are wo main differences, (1) he slopes (10 year yield - 1 year yield) for boh subperiods are smaller wihou inflaion ambiguiy (1.04 vs. 0.95 for Period 1 and 1.27 vs. 0.92 for Period 2) which is due o he rend inflaion ambiguiy, and (2) he yield level is smaller/bigger for he firs/second subperiod because of invesors differen wors case inflaion choice. To check he overall effeciveness of ambiguiy for boh inflaion and oupu growh, we shu down he ambiguiy for boh inflaion and oupu growh. The resuls are provided in Table 4. As expeced, same as for real bond, he nominal yield curve is almos fla for boh subperiods, and volailieis also rapidly decline o almos zero. The yield levels are also higher for boh subperiods under he reference measure wih higher mean oupu growh (oupu growh ambiguiy dominaes inflaion ambiguiy in deermining yield level because of risk aversion parameer γ > 1). Wors case measure as subjecive belief In his model, ambiguiy averse agens evaluae fuure prospecs under he wors-case measure which is also heir subjecive measure. Piazzesi e al. (2015) show ha he expeced excess reurns on long bonds consis of wo pars: he expeced subjecive bond premium and he difference beween subjecive and saisical fuure ineres rae expecaions E (er n,+1 ) = E (er n,+1 ) + (n 1) ( ( ) ( )) E i n 1 +1 E i n 1 +1 wih he subjecive expecaion denoed by sar. In our model, he subjecive bond premium is solved in secion 2 as er n,+1 = B (n 1) σ x σ z v C (n 1) σ a 1σ z v where he firs erm is bond risk premium and he second erm is bond ambiguiy premium. Since he volailiy of ambiguiy process is very small (see parameer values in Table 2), also because invesors wih CRRA uiliy are indifferen beween early or lae resoluion of uncerainy, he bond premium is very small and quaniaively negligible in he model 14. However, he subjecive ineres rae expecaions are bigger for longer horizons in our model, if he saisical fuure ineres rae expecaion is consan, 14 In a model wih only ambiguiy abou rend inflaion, Ulrich (2013) argues ha he bond ambiguiy premium can help generae an upward sloping yield curve for nominal bond. However, he real yield curve is downward sloping in his model. 24
our model would generae bigger excess reurns for long bonds ha is consisen wih daa. 3.4. Slope, dispersion, and recession In his paper, we focus on he average nominal and real yield curves over he wo subperiods, no heir hisorical movemens. Therefore he rend componens in he wo ambiguiy processes are fixed wihin each subperiods. Neverheless, he hisorical slope of he Treasury yield curve has ofen been cied as a leading economic indicaor wih inversion of he curve being hough of as a signal of a recession (for example, Esrella and Hardouvelis (1991); Esrella and Mishkin (1996, 1998)). Wrigh (2006) finds ha using boh he level of he federal funds rae and he erm spread gives beer fi in forecasing forecasing recessions. Ang e al. (2006) argue ha he erm premium and expecaions hypohesis componens of he erm spread have quie differen saisical correlaions wih fuure growh. To shed ligh on he hisorical performance of he model, one can ask he quesion ha wha would he model imply if we allow he rend componens in he ambiguiy processes o be ime varying. In he daa, he rend componens are measured by he difference beween long run and shor run (one quarer ahead) forecas dispersions. The bigger he difference, or he seeper he erm srucure of dispersion, he seeper he slope of nominal curve implied by he model. For real GDP growh, he 6-o-10 years ahead forecas dispersion vary very lile over ime. Since 6-o-10 years ahead forecas only available semiannually afer 1996, we assume long run forecas is consan and he difference is approximaed by one quarer ahead GDP forecas dispersion. Similarly for inflaion forecass afer lae 90s, he difference beween long run and shor run is approximaed by one quarer ahead inflaion forecas dispersion. For inflaion forecas before lae 90s, here is significan amoun of long run forecas dispersion variaions, he difference is calculaed as he difference beween five quarer ahead and one quarer ahead inflaion forecas dispersion. Using he model implied parameer values (γ = 2), we combine he slopes for erm srucure of inflaion and real GDP forecas dispersion 25
Figure 3: Slope, Dispersion, and Recession The slope is 10 year nominal yield minus 3 monh reasury rae. The dispersion is calculaed as a combinaion of slopes of erm srucure of inflaion and real GDP forecas dispersion. All daa are monhly from 1985 o 2016. over he whole sample period. Figure 3 shows he comovemens of model implied slope of he Treasury yield curve and he hisorical realized slopes, along wih NBER recessions as shaded areas. The model implied slopes are significanly correlaed wih slopes in he daa (he correlaion is 0.27), especially for hose periods around recessions. Before he recessions, shor run GDP dispersion (same for inflaion dispersion pos lae 90s) is geing smaller, invesors have less ambiguiy regarding shor run real GDP growh and inflaion, which implies a fla curve. However, he size of shor run ambiguiy become bigger enering ino he recessions, bu invesors sill believe he long run growh and inflaion have less uncerainy and he difference beween heir shor run and long run ambiguiy implies a seeper curve a his sage. Afer recessions, he size of shor run ambiguiy drops and he nominal curve becomes fla. The only difference for he wo subperiods is ha shor run and long run inflaion ambiguiy difference is driven by boh shor run and long run before lae 90s, and only by shor run afer lae 90s. 3.5. Equiy yields Recen empirical findings regarding he price and reurn for individual dividends (or dividend srips) pose some serious challenges o curren equilibrium asse pricing models 26
Model (Period 2) 1Q 1Y 5Y 7Y 10Y Marke Reurn 8.38 Dividend Spo Reurn 8.43 8.42 8.42 8.42 8.42 Dividend Fuure Reurn 2.76 2.78 2.80 2.81 2.82 Equiy Yield 7.74 7.69 7.47 7.36 7.20 Equiy Yield Volailiy 3.60 1.80 0.71 0.65 0.61 Table 5: Dividend Srip Reurn and Volailiy This able presens he model implied marke reurn, dividend spo reurns, dividend fuure reurns, and forward equiy yield volailiies for he second subperiod. To calculae reurns and volailiy for dividend srips, as well as for marke reurn, we se ime preference β = 0.99 in order o have sable approximaion for he Campbell and Shiller approximaion. (see Van Binsbergen and Koijen (2017) for a summary) 15. For example, using dividend fuure conracs for he S&P500 from 2002 o 2014, Van Binsbergen and Koijen (2017) show ha he dividend fuure reurns are slighly upward sloping and he volailiy of forward equiy yields is downward sloping. And he marke reurns are no significanly differen from individual dividend spo reurns. They argue ha leading asse pricing models are no able o mach hose feaures in daa. Noe ha heir empirical findings on equiy yields are differen for differen counries. Since our model is esimaed and calibraed using US daa, we will focus on he findings from S&P500. Table 5 repors he model s marke reurn, dividend spo reurn, dividend fuure reurn, and equiy yield volailiy. The closed form soluion for marke reurn is given in secion 2, and dividend spo reurn is defined as log (P +1,n 1 ) log (P,n ) = pd (n 1) +1 pd (n) + d +1. Dividend fuure reurn is dividend spo reurn less same horizon bond holding period reurn. Since agen faces he same size of one sep ahead ambiguiy, marke reurn and dividend spo reurn is very close in his model, however, dividend yields are downward sloping because long horizon dividends feaure less ambiguiy. Because he holding period reurn for real bonds is downward sloping in our model, he dividend fuure reurns is slighly upward sloping ha consisen wih Van Binsbergen and Koijen (2017). For he same reason as for he bond yields volailiy, he forward equiy yield volailiy is downward sloping ha is consisen wih daa. 15 Also, van Binsbergen e al. (2012) provide he firs direc measuremen of dividend srip prices using opions daa. van Binsbergen e al. (2013) exend his evidence using dividend fuures conrac. 27