Heterogeneous Expectations and Bond Markets

Transcription

1 Heerogeneous Expecaions and Bond Markes Wei Xiong and Hongjun Yan January 2008 Absrac This paper presens a dynamic equilibrium model of bond markes in which wo groups of agens hold heerogeneous expecaions abou fuure economic condiions. The heerogeneous expecaions cause agens o ake speculaive posiions agains each oher and herefore generae endogenous relaive wealh flucuaion. The relaive wealh flucuaion amplifies asse price volailiy and conribues o he ime variaion in bond premia. Our model shows ha a modes amoun of heerogeneous expecaion can help explain several puzzling phenomena, including he excessive volailiy of bond yields, he failure of he expecaions hypohesis, and he abiliy of a en-shaped linear combinaion of forward raes o predic bond reurns. We are graeful o Nick Barberis, Markus Brunnermeier, Bernard Dumas, Nicolae Garleanu, Jon Ingersoll, Arvind Krishnamurhy, Owen Lamon, Debbie Lucas, Lin Peng, Monika Piazzesi, Chris Sims, Hyun Shin, Jeremy Sein, Sijn Van Nieuwerburgh, Neng Wang, Moo Yogo, and seminar paricipans a Bank of Ialy, Federal Reserve Bank of New York, Harvard Universiy, NBER Summer Insiue, New York Universiy, Norhwesern Universiy, Princeon Universiy, Universiy of Briish Columbia, Universiy of Chicago, Universiy of Illinois-Chicago, Wharon School, and Yale Universiy for heir helpful discussions and commens. Princeon Universiy and NBER. wxiong@princeon.edu. Yale Universiy. hongjun.yan@yale.edu.

2 1 Inroducion Following he seminal work of Cox, Ingersoll and Ross (1985, mos academic sudies in he economics and finance lieraure use represenaive-agen models o analyze yield curve dynamics. While his approach leads o racable parameric models of yield curve dynamics, represenaive-agen models ignore imporan ineracions among heerogeneous agens. In his paper, we analyze an equilibrium model of bond markes in which heerogeneous expecaions cause agens o rade wih each oher. We show ha he endogenous wealh flucuaions caused by agens rading can help resolve several challenges encounered by sandard represenaive-agen models, including he excessive volailiy of bond yields and he failure of he expecaions hypohesis. Moreover, our model also sheds ligh on he recen finding of Cochrane and Piazzesi (2005 ha a single en-shaped linear combinaion of forward raes predics excess bond reurns. We adop he sandard equilibrium framework of Cox, Ingersoll and Ross (1985 wih log-uiliy agens and a consan-reurn-o-scale risky invesmen echnology. Unlike heir model, we allow agens o hold heerogeneous expecaions of fuure economic condiions. 1 Specifically, we assume ha here are wo groups of agens using differen learning models o infer he value of an unobservable variable ha deermines fuure equilibrium shor raes. Consequenly, he wo groups of agens hold heerogeneous expecaions abou fuure ineres raes. A group is said o be more opimisic (pessimisic abou fuure shor raes if is expecaion of fuure shor raes is higher (lower han he oher group s. Noe ha when agens adop differen learning models, no only will heir beliefs differ from each oher, bu heir average belief can also deviae from he belief of an ouside objecive observer of he economy, whom we call an economerician. In order o isolae he effecs caused by he wo groups belief dispersion from hose caused by heir erroneous average 1 There is ample evidence supporing he exisence of heerogeneous expecaions among agens. Mankiw, Reis and Wolfers (2004 find ha he inerquarile range among professional economiss inflaion expecaions, as shown in he Livingson Survey and he Survey of Professional Forecasers, varies from above 2% in he early 1980s o around 0.5% in he early 2000s. Swanson (2005 finds ha in he Blue Chip Economic Indicaors survey of major U.S. corporaions and financial insiuions beween 1991 and 2004, he difference beween he 90h and 10h percenile forecass of nex-quarer real US GDP growh rae flucuaes beween 1.5% and 5%, and he 90h and 10h percenile forecass of four-quarer-behind 3-monh Treasury bill rae flucuaes beween 0.8% and 2.2%. 1

3 belief, we adop a novel specificaion in which he wo groups beliefs are divergen bu heir average is always idenical o he economerician s benchmark belief. Heerogeneous beliefs moivae agens o ake speculaive posiions agains each oher in bond markes, and marke clearing condiions deermine bond prices. We show ha he equilibrium bond price is a wealh-weighed average of bond prices in homogeneous economies, in each of which only one ype of agen is presen. Sandard represenaive-agen models have difficuly generaing he large bond yield volailiy and highly variable risk premia observed in acual daa because he aggregae consumpion is raher smooh. Our model shows ha he relaive wealh flucuaion caused by agens speculaive posiions amplifies bond yield volailiy and conribues o he ime variaion in bond premia. The inuiion is as follows. Agens who are opimisic abou fuure ineres raes would be on raes rising agains hose pessimisic agens. In equilibrium, bond prices aggregae agens heerogeneous beliefs and, in paricular, reflec heir wealh-weighed average belief. When agens wealh-weighed average belief abou he fuure shor raes is higher han he economerician s belief, hey would discoun bonds more heavily and he equilibrium bond prices would appear cheap o he economerician, i.e., he bond premium is high. Similarly, he bond premium is low when agens wealh-weighed average belief is lower han he economerician s belief. Thus, he bond premium varies wih he wo groups beliefs and wealh disribuion. Noe ha he wo groups wealh disribuion is endogenously deermined by heir rading. When a posiive shock his he marke, i favors opimisic agens and causes wealh o flow from pessimisic agens o opimisic agens, giving he opimisic belief a larger weigh in deermining bond prices. The relaive-wealh flucuaion hus amplifies he effec of he iniial news on bond prices and makes bond premia more variable. We provide a calibraion exercise o show ha even wih a modes amoun of belief dispersion, he volailiy amplificaion effec of agens relaive wealh flucuaion is significan enough o explain he excess volailiy puzzle documened by Shiller (1979, Gurkaynak, Sack and Swanson (2005, and Piazzesi and Schneider (2006. These sudies find ha longerm yields appear o be oo volaile relaive o he levels implied by sandard represenaive- 2

4 agen models. We also show ha heerogeneous expecaions can help explain he failure of he classic expecaions hypohesis in he daa. The expecaions hypohesis suggess ha when he yield spread (long erm bond yield minus he shor rae is posiive, he long erm bond yield is expeced o rise (or he long erm bond price is expeced o fall, because, oherwise, an agen canno be indifferen abou invesing in he long bond or he shor rae. However, his hypohesis has been rejeced by many empirical sudies. To menion one here, Campbell and Shiller (1991 find ha when he yield spread is posiive, he long erm bond yield ends o fall raher han rise. This paern is a naural implicaion in our model: Suppose he wealhweighed average belief abou he fuure shor raes is higher han he economerician s belief. On he one hand, i implies ha agens discoun long erm bonds more heavily, which leads o higher long erm bond yields and so larger yield spreads; on he oher, i also implies ha he long erm bond prices appear cheap from he economerician s poin of view, i.e., he long erm bond prices are expeced o rise and bond yields are expeced o fall. Taken ogeher, a high wealh-weighed average belief implies boh large yield spreads and falling long erm bond yields in he fuure. Indeed, we show in our simulaions ha a reasonable amoun of belief dispersion is able o generae regression resuls similar o hose of Campbell and Shiller. Our model can also shed ligh on he recen finding of Cochrane and Piazzesi (2005 ha a single en-shaped linear combinaion of forward raes predics excess reurns on woo five-year bonds. As elaboraed laer in he paper, his en-shaped facor racks agens wealh-weighed belief: he higher he weighed average belief abou he fuure shor raes, he bigger value of he en-shaped facor. Moreover, a higher belief abou he fuure shor raes also makes bond prices cheap from he economerician s poin of view, and hus predics higher fuure bond reurns. As a resul, he en-shaped facor predics bond premia. Our simulaion confirms ha a reasonable amoun of belief dispersion is able o generae bond reurn predicabiliy resuls comparable o hose of Cochrane and Piazzesi. Our paper complemens he growing lieraure on equilibrium effecs of heerogeneous beliefs. Deemple and Murhy (1994 are he firs o demonsrae ha equilibrium prices have a wealh-weighed average srucure. More recenly, Basak (2000, Dumas, Kurshev and 3

5 Uppal (2005, Jouini and Napp (2005, Buraschi and Jilsov (2006, David (2007, and Li (2007 provide equilibrium models o sudy he effecs of heerogeneous beliefs on a variey of issues, including asse price volailiy, ineres raes, equiy premium, and he opion implied volailiy. Our model differs from hese models in wo aspecs. Firs, our model specificaion allows us o isolae belief-dispersion effecs from oher learning-relaed effecs ha also arise in hese earlier models, such as effecs caused by under-esimaion of risk and by erroneous average beliefs. Second, and more imporanly, our model provides new implicaions of heerogeneous beliefs on bond yield movemen and bond reurn predicabiliy. Our model also differs from he lieraure ha sudies he effec of invesor preference heerogeneiy on asse prices, e.g., Dumas (1989, Wang (1996, and Chan and Kogan (2002. In paricular, Wang analyzes he effec of preference heerogeneiy on he yield curve. In anoher relaed sudy, Vayanos and Vila (2007 analyze he effec of he difference in invesors preferred habias on bond markes. In conras o hese sudies, our model generaes new implicaions based on invesors belief dispersion. The res of he paper is organized as follows. Secion 2 presens he model. Secion 3 discusses he effec of heerogeneous expecaions on bond marke dynamics. Finally, Secion 4 concludes. We provide all he echnical proofs in he Appendix. 2 The Model We adop he equilibrium framework of Cox, Ingersoll and Ross (1985 wih log-uiliy agens and a consan-reurn-o-scale risky invesmen echnology. Unlike heir model, ours assumes ha agens canno direcly observe a random variable ha deermines fuure reurns of he risky echnology, and ha agens can only esimae is value. There are wo groups of agens holding heerogeneous expecaions regarding his variable. Because of he belief dispersion, agens speculae in capial markes. We sudy a compeiive equilibrium in which each agen opimizes consumpion and invesmen decisions based on his own expecaion. Marke clearing condiions deermine he equilibrium shor rae and asse prices. 4

6 2.1 The economy We consider an economy wih only one consan-reurn-o-scale echnology. The reurn of he echnology follows a diffusion process: di I = f d + σ I dz I (, (1 where f is he expeced insananeous reurn, σ I a volailiy parameer, and Z I ( a sandard Brownian moion process. The expeced insananeous reurn from he risky echnology, f, follows anoher linear diffusion process: df = λ f (f l d + σ f dz f (, (2 where λ f is a consan governing he mean revering speed of f, l a moving long-run mean of he risky echnology s expeced reurn, σ f a volailiy parameer, and Z f ( a sandard Brownian moion process independen of Z I (. The long-run mean l is unobservable and follows an Ornsein-Uhlenbeck process: dl = λ l (l ld + σ l dz l (, (3 where λ l is a parameer governing he mean-revering speed of l, l he long-run mean of l, σ l a volailiy parameer, and Z l ( a sandard Brownian moion process independen of Z I ( and Z f (. As we show laer, boh variables f and l affec he dynamics of shor-erm and long-erm ineres raes. Since he risky echnology represens an alernaive invesmen o invesing in he shor erm bond, he echnology s expeced insananeous reurn f, afer adjused for risk, deermines he shor rae. l is he level o which f mean-revers and so i affecs he fuure shor raes. In realiy, invesors ofen face grea uncerainy regarding he fuure shor raes and need o form expecaions in order o rade long-erm bonds. Inroducing he variable l and making i unobservable provides a convenien way of modelling agens expecaions. The cenral par of our analysis is o show ha agens heerogeneous expecaions can generae significan effecs on bond markes. 2 2 For simpliciy, his paper focuses on agens disagreemen abou he real side of he economy. In an 5

7 2.2 Heerogeneous expecaions The economics and finance lieraure has widely adoped he Bayesian inference framework o model agens learning processes abou unobservable economic variables, such as produciviy of he economy and profiabiliy of a specific firm. One line of he lieraure, e.g., Harris and Raviv (1993, Deemple and Murhy (1994, Morris (1996 and Basak (2000, assumes ha agens hold heerogeneous prior beliefs abou unobservable economic variables. In hese models, agens coninue o disagree wih each oher even afer hey updae heir beliefs using idenical informaion, bu he difference in heir beliefs deerminisically converges o zero. In anoher srand of he lieraure, e.g., Scheinkman and Xiong (2003, Dumas, Kurshev and Uppal (2005, Buraschi and Jilsov (2006 and David (2007, heerogeneous beliefs arise from agens differen prior knowledge abou he informaiveness of signals and he dynamics of unobservable economic variables. In suppor of his approach, Kurz (1994 argues ha nonsaionariy of economic sysems and limied daa make i difficul for raional agens o idenify he correc model of he economy from alernaive ones. More recenly, Acemoglu, Chernozhukov, and Yildiz (2007 show ha when agens are uncerain abou a random variable and abou he informaiveness of a source of signal regarding he random variable, even an infinie sequence of signals from his same source does no lead agens heerogeneous prior beliefs abou he random variable o converge. This is because agens have o updae beliefs abou wo sources of uncerainy using one sequence of signals. Finally, behavioral biases such as overconfidence could also preven agens from efficienly learning abou he informaiveness of heir signals. Following his approach, we analyze wo groups of invesors who hold differen prior knowledge abou he informaiveness of a flow of signals on he long-run mean of risky echnology reurns (l. In paricular, we assume ha he signals are no informaive abou l, bu he heerogeneous prior knowledge leads he wo groups o reac differenly o he earlier version, we exended our model o incorporae he moneary side of he economy by specifying wo addiional random processes, wih a similar srucure o ha of (f, l. More specifically, one observable process deermines he shor-erm inflaion rae and he oher unobservable process deermines he long-run mean o which he shor-erm inflaion rae mean-revers. By allowing agens o hold heerogeneous beliefs abou he value of his unobservable long-run mean of inflaion rae, we can show ha agens disagreemen abou he moneary side of he economy has an effec on bond markes similar o heir disagreemen abou he real side. Since his analysis does no presen addiional concepual insigh, we do no repor i in his paper. 6

8 signal flow and herefore o possess heerogeneous expecaions. This approach is racable and generaes a saionary process for he difference in agens beliefs Benchmark belief We will evaluae he effecs of agens heerogeneous beliefs on asse price dynamics from he view poin of an ouside observer, an economerician. Hence, we firs derive he belief of he economerician. Since he economerician undersands ha he signals are no informaive, his informaion se a ime is {f τ } τ=0. We assume ha he economerician s prior belief abou l 0 has a Gaussian disribuion. Since heir informaion flow also follows Gaussian processes, heir poserior beliefs abou l mus likewise be Gaussian. According o he sandard resuls in linear filering, e.g., Theorem 12.7 of Lipser and Shiryaev (1977, he economerician s belief variance converges o a saionary level a an exponenial rae. For our analysis, we will focus on he seady sae, in which he belief variance has already reached is saionary level, denoed by γ, which is he posiive roo o he following quadraic equaion of γ: λ 2 f σ 2 f γ 2 + 2λ l γ σ 2 l = 0. (4 We denoe he economerician s poserior disribuion abou l a ime by where ˆl R l {f τ, S(τ} τ=0 (ˆlR N, γ, is he mean of he poserior disribuion. Applying Theorem 12.7 of Lipser and Shiryaev (1977, we obain ha dˆl R = λ l (ˆl R ld + λ f σ γdẑr f (, (5 f where dẑr f = 1 [df + λ f (f σ ˆl ] R d f (6 is he informaion shock in df. Ẑ R f poin of view. is a sandard Brownian moion from he economerician s 7

9 2.2.2 Heerogeneous beliefs We assume ha all agens observe he following process: ds = dz S (, where Z S ( is a sandard Brownian moion independen of all he Brownian moions inroduced earlier. Tha is, S( is pure noise. However, agens in he economy believe ha S( is parially correlaed wih he fundamenal shock o l and hus conains useful informaion. There are wo groups of agens, group A and group B. They have differen inerpreaions of S( and so possess heerogeneous expecaions. Before we formally inroduce he beliefs of he wo groups, i is useful o noe ha, in general, his heerogeneous economy could differ from a homogeneous economy hrough several channels. Firs, disagreemen induces speculaive rading beween he wo groups, herefore generaing endogenous relaive wealh flucuaion. Second, he average belief of he wo groups could differ from he economerician s belief, and he erroneous average belief would affec equilibrium asse price dynamics. Third, since each agen in he economy feels he has learned from he signal abou l, his poserior belief variance would be smaller han he economerician s and his leads o an underesimaion of he risk in he economy. While he effecs generaed by he second and hird channels are ineresing in heir own righ, we are primarily ineresed in he firs channel. In order o isolae he impac from he firs channel, we adop a specificaion ha shus down he oher wo channels. To shu down he second channel, we assume ha he wo groups have exacly he opposie inerpreaions of S. Agens in group A believe ha he signal generaing process is ds = φ S dz l ( + 1 φ 2 S dz S (, where he parameer φ S [0, 1 measures he perceived correlaion beween he signal ds and he fundamenal shock dz l (. Symmerically, agens in group B believe ha he signal has a correlaion of φ S wih he fundamenal shock dz l (: ds = φ S dz l ( + 1 φ 2 S dz S (. Since he wo groups of agens have opposie percepions abou he correlaion beween ds and dz l (, hey would hold differen beliefs and, as will be shown laer, he average of 8

10 heir beliefs coincides wih he belief of he economerician. This specificaion also includes he benchmark case wih one raional agen as a special case of φ S = 0. Noe also ha i is difficul for hese agens o deec he inconsisency beween he perceived signal generaing process and he acual process for wo reasons. Firs, in boh group-a and group-b agens minds, he perceived signal generaing process has he same quadraic variaion as he acual process. Second, agens canno direcly measure he correlaion beween ds and dz l ( since dz l ( is unobservable. To urn off he hird channel, we furher assume ha agens in boh groups also perceive a higher volailiy in he unobservable process of l : dl = λ l (l ld + kσ l dz l (, where k = 1 > 1. 1 φ 2 S Tha is, agens exaggerae he volailiy by a facor of k. Noe ha because agens canno observe l, hey canno deec heir exaggeraion by quadraic variaion. Moreover, we choose he magniude of k so ha, as we show nex, i exacly offses he underesimaion of he uncerainy in l. We now derive group-a and group-b agens belief abou l. Agens informaion se a ime includes {f τ, S(τ} τ=0. According o Theorem 12.7 of Lipser and Shiryaev (1977, agens poserior variance a he seady sae is he posiive roo o he following quadraic equaion of γ: λ 2 f σ 2 f γ 2 + 2λ l γ ( 1 φ 2 S k 2 σ 2 l = 0. (7 Since ( 1 φ 2 S k 2 = 1, equaions (4 and (7 imply ha agens in boh groups have he same poserior variance as he economerician. We denoe group-i agens poserior disribuion abou l a ime by l {f τ, S(τ} τ=0 (ˆli N, γ, i {A, B}, where ˆl i is he mean of group-i agens poserior disribuion. We will refer o ˆl i as heir 9

11 belief hereafer. Theorem 12.7 of Lipser and Shiryaev (1977 implies dˆl i = λ l (ˆl i ld + λ f σ γdẑi f ( + f φ S σ l 1 φ 2 S ds φ Sσ l ds 1 φ 2 S if i = A if i = B (8 where dẑi f = 1 [df + λ f (f σ ˆl ] d i f (9 is he informaion shock in df o group-i agens. group-i agens poin of view. Ẑ i f is a sandard Brownian moion from As shown in (8, due o he difference in heir prior knowledge abou he correlaion beween ds and dz l (, group-a agens reac posiively o ds, while group-b agens reac negaively. Thus, he wo groups hold heerogeneous beliefs abou l and he difference in heir beliefs flucuaes wih he signal flow. Furhermore, he wo groups reacions o ds have opposie signs bu he same magniude. As a resul, if he average of he wo groups prior beliefs abou l 0 is he same as he economerician s prior belief, hen in he fuure heir average belief abou l would always keep rack of he economerician s belief. We formally sae hese properies of agens beliefs in he following proposiion and provide he proof in he Appendix. Proposiion 1 The difference in he wo groups beliefs has he following process: ( d (ˆlA ˆl B = λ l + λ2 f σf 2 γ (ˆlA ˆl B d + 2φ Sσ l ds, 1 φ 2 S which flucuaes wih ds and mean-revers o zero. Furhermore, if 1 2 (ˆl A 0 + ˆl B 0 = ˆl R 0, hen he average of he wo groups beliefs abou l always racks he economerician s belief: 1 2 (ˆl A + ˆl B = ˆl R. This proposiion resuls from he symmeric learning processes of he wo groups. Unlike oher models of heerogeneous beliefs in he lieraure, his model specificaion allows us o isolae belief-dispersion effecs from oher learning relaed effecs, such as under-esimaion of risk and erroneous average belief. 10

12 2.3 Capial markes equilibrium The difference in agens beliefs causes speculaive rading among hemselves. Agens who are more opimisic abou l would be on ineres raes rising agains more pessimisic agens. Noe ha, from each group s perspecive, here are hree sources of random shocks. For group-i agens, he shocks are dz I, dẑi f, and ds. Thus, he markes are complee if agens can rade a risk-free asse and hree risky asses ha span hese hree sources of random shocks. In realiy, bond markes offer many securiies, such as bonds wih differen mauriies, for agens o consruc heir bes and o complee he markes. As a resul, we analyze agens invesmen and consumpion decisions, as well as heir valuaion of financial securiies, in a complee-markes equilibrium. We inroduce a zero-ne-supply risk-free asse and wo zero-ne-supply risky financial securiies in he capial markes, in addiion o he risky producion echnology. 3 A ime, he risk-free asse offers a shor rae r, which is endogenously deermined in he equilibrium. The wo risky financial securiies offer he following reurn processes: dp f p f = µ f (d + df, (10 dp S p S = µ S (d + ds. (11 We refer o hese securiies as securiy f and securiy S, respecively. Like fuures conracs, hese securiies are coninuously marked o he flucuaions of df and ds, respecively. Since agens hold differen views abou he underlying innovaion processes of hese securiies, hey disagree abou heir expeced reurns. As a resul, some agens wan o ake long posiions, while ohers wan o ake shor posiions. Through rading, he conrac erms µ f ( and µ S ( are coninuously deermined so ha he aggregae demand for each of he securiies is zero all he ime. We could also view hese financial securiies as synheic posiions consruced by dynamically rading bonds. We choose o inroduce hese securiies insead of specific bonds o simplify noaions, and our specific choice of securiies does no affec he equilibrium in complee markes. 3 We also allow agens o shor-sell he risky echnology. This can be implemened by offering a derivaive conrac on he reurn of he echnology. The marke clearing condiions, however, require ha agens in aggregae hold a long posiion in he risky echnology. 11

13 We assume ha all agens have a logarihmic preference. Group-i agens (i {A, B} maximize, according o heir beliefs, heir lifeime uiliy from consumpion by choosing heir consumpion c i, and he fracion of heir wealh invesed in he risky echnology and wo financial securiies (x i I, xi f, xi S : where E i max E i {c i,xi I,xi f,xi S } 0 e β u(c i d, is he expecaion operaor under heir probabiliy measure, β is heir imepreference parameer, and is heir uiliy funcion from consumpion. u(c i = log(c i We solve each agen s consumpion and invesmen problem using he sandard dynamic programming approach developed by Meron (1971. The resuls for logarihmic uiliy are well known: Each agen consumes wealh a a consan rae equal o his ime preference parameer (c i risk-reurn radeoff. = βw i and invess in risky asses according o he asses insananeous In a compeiive equilibrium, each agen chooses opimal consumpion and invesmen sraegies in accordance wih his expecaions and all markes clear. Marke clearing condiions ensure: 1 he aggregae invesmen o he risk-free asse is zero; 2 he aggregae invesmen o each of he risky securiies f and S is also zero; and 3 he aggregae invesmen o he risky echnology is equal o he oal wealh in he economy. We formally derive he equilibrium in Appendix A.2 and summarize hree imporan properies of he equilibrium in he following heorem. Theorem 1 The equilibrium has he following properies: 1. The shor rae is r = f σ 2 I. (12 2. Define he wealh raio of group-a and group-b agens by η W A W B follows a diffusion process: d ln (η = λ f σ f (ˆlA ˆl B. Is logarihm dẑr f (. (13 12

14 3. The ime price of an asse, which provides a single payoff X T a ime T, is given by P = ω A P A + ω B P B, where ω i is he wealh share of group-i agens, and P i is he price of he asse in a hypoheical economy, in which only group-i agens are presen. Theorem 1 shows ha he equilibrium shor rae is he expeced insananeous reurn of he risky echnology adjused for risk (equaion (12. This is because agens would demand a higher reurn from lending ou capial when he expeced reurn from he alernaive opion of invesing in he risky echnology is higher. Theorem 1 also shows ha when he wo groups hold differen beliefs, logarihm of heir wealh raio responds o he informaion shock dẑr f (. This is a resul of speculaion among he wo groups. For insance, if group-a agens hold a higher belief abou l (ˆl A > ˆl B, hey would be agains group-b agens on fuure ineres rae rising. Consequenly, a posiive informaion shock would favor group-a agens and cause wealh o flow from group B o group A. Also noe ha from he view poin of he economerician, he wo groups wealh raio does no converge o eiher zero or infiniy, i.e., no group is able o evenually dominae he economy in he long run. This is because neiher group has a superior learning model. 4 Finally, Theorem 1 shows ha he price of an asse is he wealh-weighed average of each group s valuaion of he asse in a corresponding homogeneous economy. This resul allows us o represen asse prices in a heerogeneous economy using prices in homogeneous economies. Thus, asse pricing is remarkably simple even in a complex environmen wih heerogeneous agens. While his price represenaion depends on agens logarihmic preference and linear risky echnology, i is independen of he specific informaion srucure in our model. Several earlier models, e.g., Deemple and Murhy (1994 and Basak (2000, provide similar price represenaions under differen informaion srucures. 2.4 Bond price in a homogeneous economy Theorem 1 allows us o express he price of a bond as he wealh-weighed average of each group s bond valuaion in a homogeneous economy. Thus, before analyzing he effecs of 4 See Kogan, e al. (2006 and Yan (2006 for more discussions on invesors survival in he long run. 13

15 agens heerogeneous expecaions on bond markes, we firs derive bond prices in homogeneous economies in he following proposiion 2, wih a proof in Appendix A.3. Proposiion 2 In a homogeneous economy wih only group-i agens, he price of a zerocoupon bond wih a face value of 1 and mauriy τ is deermined by ( B H τ, f, ˆl i = e a f (τf a l (τˆl i b(τ, where a f (τ = 1 λ f (1 e λ f τ, (14 a l (τ = 1 λ l (1 e λ lτ + b(τ = τ 0 [ λ l lal (s 1 2 σ2 f a2 f (s ( e λ f τ e λ lτ, (15 λ f λ l ( 1 1 φ 2 σl 2 2λ l γ S ] a 2 l (s λ f γa f (sa l (s σi 2 ds. Proposiion 2 implies ha he yield of a τ-year bond in a homogeneous economy, Y H ( τ, f, ˆl i = 1 τ log ( B H = a f (τ τ f + a l(τ ˆli τ + b(τ τ, is a linear funcion of wo fundamenal facors: f and ˆl i. This specific form belongs o he general affine srucure proposed by Duffie and Kan (1996. The loading on facor f, a f (τ/τ, has a value of 1 when he bond mauriy τ is zero and monoonically decreases o zero as he mauriy increases, suggesing ha shor-erm yields are more exposed o he flucuaions in f. The inuiion of his paern is as follows. f is he expeced insananeous reurn from he risky echnology, which serves as a close subsiue for invesing in shor-erm bonds. As a resul, f deermines he shor rae (r = f σ 2 I as in Theorem 1 and he flucuaion in f has a greaer impac on shor-erm yields. As bond mauriy increases, he impac of f becomes smaller. Agens belief abou l deermines heir expecaion of he fuure shor raes, because l is he level o which f mean-revers. The loading of he bond yield on ˆl i, a l (τ/τ, has a hump shape if l mean-revers (λ l > 0. 5 Since l describes he long run produciviy, as he 5 In he case where mean reversion is no presen (λ l = 0, he facor loading a l (τ/τ is a monoonically increasing funcion of bond mauriy. 14

16 bond mauriy increases from 0, he bond yield becomes more sensiive o he belief abou l, ha is, as he bond mauriy increases from 0 o an inermediae value, he loading a l (τ/τ increases. As he bond mauriy increases furher, he loading a l (τ/τ falls. This is caused by he mean reversion of l, which causes any shock o l o evenually die ou. This force causes he yields of very long-erm bonds o have low exposure o agens belief abou l. This hump shape in he bond yield s loading on ˆl i is imporan for undersanding laer resuls such as volailiy amplificaion and bond reurn predicabiliy. 2.5 Represenaive agen As is well known, one can consruc a represenaive agen o replicae he price dynamics in a complee-markes equilibrium wih heerogeneous agens. Does his mean ha we can simply focus on he represenaive agen s belief process and ignore he heerogeneiy beween agens? To undersand why he answer is no, we consruc a represenaive agen for our model. 6 If we resric he represenaive agen o having he same logarihmic preference as he group-a and group-b agens, we obain he same equilibrium as before by wising he represenaive agen s belief, as summarized in he following proposiion wih a proof in Appendix A.4. Proposiion 3 To replicae he compeiive equilibrium derived in Theorem 1, consider a represenaive agen who has he same logarihmic preference as agens in he heerogeneous economy. Then, a any poin of ime, he represenaive agen s belief abou l, ˆl N, has o be he wealh-weighed average belief of group-a and group-b agens: ˆlN = ω A ˆl A + ω B ˆl B. (16 I is imporan o sress ha he represenaive agen s belief mus equal he wealhweighed average belief, no only a one poin of ime, bu also a all fuure poins. Thus, over ime, he represenaive agen s belief would change in response no only o he belief flucuaion of each group, bu also o he relaive wealh flucuaion. According o Theorem 1, heerogeneous beliefs cause he wo groups o rade agains each oher and consequenly 6 See Jouini and Napp (2005 for a recen analysis of he exisence of a consensus belief for he represenaive agen in an exchange economy wih agens holding heerogeneous beliefs. 15

17 cause heir relaive wealh o flucuae wih informaion shock o he marke. Thus, alhough he wo groups beliefs are symmerically disribued around he economerician s belief, hrough he relaive wealh flucuaion channel, heir belief dispersion could sill affec he represenaive agen s belief and herefore equilibrium asse prices. We analyze hese effecs in he nex secion. 3 Effecs of Heerogeneous Expecaions Combining Proposiion 2 wih Theorem 1, we can express he price of a τ-year zero-coupon bond a ime as ( B = ω A B H τ, f, ˆl ( A + ω B B H τ, f, ˆl B, (17 where ω A and ω B are he wo groups wealh shares in he economy, and B (τ, H f, ˆl i, given in Proposiion 2, is he bond price in a homogeneous economy in which only group-i agens are presen. The implied bond yield in his heerogeneous economy is Y (τ = a f (τ f + b(τ 1 ] [ω τ τ τ log A e a l(τˆl A + ω B e a l(τˆl B. (18 Noe ha Y is no a linear funcion of agens beliefs ˆl A and ˆl B. Tha is, bond yields in his heerogeneous economy have a non-affine srucure. This srucure derives from he marke aggregaion of agens heerogeneous valuaions of he bond. 3.1 Trading volume Heerogeneous expecaions cause agens o ake speculaive posiions agains each oher in bond markes. These speculaive posiions can cause flucuaions in agens wealh shares upon he arrival of new informaion. Agens hen rade wih each oher o rebalance heir posiions. Inuiively, when belief dispersion increases, he size of heir speculaive posiions becomes larger. This in urn leads o a higher volailiy of agens wealh and herefore a larger rading volume in he bond markes. We use he volailiy of one group s posiion changes as a measure of rading volume. This measure corresponds o he convenional volume measure in a discree-ime se up. We summarize he effec of agens belief dispersion on rading volume in Proposiion 4, and provide a formal derivaion and furher discussion on our volume measure in Appendix A.5. 16

18 Proposiion 4 Trading volume (he flucuaion in agens speculaive posiions increases wih he belief dispersion beween he wo groups of invesors. There is now a growing lieraure analyzing rading volume caused by heerogeneous beliefs, e.g., Harris and Raviv (1993 and Scheinkman and Xiong (2003. While hese models demonsrae ha heerogeneous beliefs lead o rading, rading ypically occurs when agens beliefs flip, ha is, ˆl A ˆl B changes is sign. Thus, rading volume of his ype only increases wih he frequency wih which agens beliefs flip. Our model adds o his lieraure by showing ha even wihou agens beliefs flipping, he wealh flucuaion caused by heir speculaive posiions already leads o rading. 3.2 Volailiy amplificaion The wealh flucuaion caused by agens speculaive posiions no only leads o rading in bond markes, bu also amplifies bond yield volailiy. Loosely speaking, bond yields are deermined by agens wealh-weighed average belief abou fuure ineres raes (equaion (18. Since agens who are more opimisic abou fuure raes be on hese raes rising agains more pessimisic agens, any posiive news abou fuure raes would cause wealh o flow from pessimisic agens o opimisic agens, making he opimisic belief carry a greaer weigh in bond yields. The relaive-wealh flucuaion hus amplifies he impac of he iniial news on bond yields. As a resul, a higher belief dispersion increases he relaive-wealh flucuaion and so increases he bond yield volailiy. We summarize his inuiion in he following proposiion, and provide a formal proof in Appendix A.6. Proposiion 5 Bond yield volailiy increases wih belief dispersion. This volailiy amplificaion mechanism can help explain he excess volailiy puzzle for bond yields. Shiller (1979 shows ha he observed bond yield volailiy exceeds he upper limis implied by he expecaions hypohesis and he observed persisence in shor raes. Gurkaynak, Sack and Swanson (2005 also documen ha bond yields exhibi excess sensiiviy o paricular shocks, such as macroeconomic announcemens. Furhermore, Piazzesi and Schneider (2006 find ha by esimaing a represenaive-agen asse pricing model wih recursive uiliy preferences and exogenous consumpion growh and inflaion, 17

19 he model predics less volailiy for long yields relaive o shor yields. Relaing o his lieraure, Proposiion 5 shows ha exending sandard represenaive-agen models wih heerogeneous expecaions can help accoun for he observed high bond yield volailiy. In Secion 3.4, we provide a calibraion exercise o illusrae he magniude of his mechanism. 3.3 Time-varying risk premium Flucuaions in agens belief dispersion and relaive wealh also cause risk premia o vary over ime. To examine he ime variaion in risk premia, we firs derive he dynamics of he sochasic discoun facor, wih a proof in Appendix A.7. Proposiion 6 From he view poin of he economerician, he sochasic discoun facor has he following process: dm M = (f σ 2 I d σ I dz I λ f σ f ( ˆlR B ωˆl i i dẑr f, (19 i=a where ˆl R is he economerician s belief abou l, and dẑr f he informaion shock defined in equaion (6. Proposiion 6 shows ha from he view poin of he economerician he marke price of risk (risk premium per uni of risk for he aggregae producion shock dz I is σ I, while he marke price of risk for he informaion shock dẑr f is proporional o ˆl R B i=a ωi ˆl i, he difference beween he economerician s belief abou l and he wealh-weighed average belief. In he benchmark case where agens are homogeneous and have he same belief as he economerician (ˆl i = ˆl R, he risk premium for he informaion shock dẑr f is zero and he marke only offers a consan price of risk for he exposure o he aggregae producion shock dz I. When he wo groups beliefs are divergen, however, here is a non-zero risk premium for he informaion shock dẑr f. Moreover, his premium varies over ime depending on he wealh flucuaion among agens. The inuiion is as follows. Suppose he wo groups wealh-weighed average belief abou l is above he economerician s belief. Then, relaive o he economerician, agens are more opimisic abou he rise of f in he fuure, and so more opimisic abou asses ha are 18

20 posiively exposed o dẑr f (i.e., hose prices are posiively correlaed wih f. From he economerician s poin of view, hose asses appear expensive and have low risk premia. Similarly, hose asses would have high risk premia if he wealh-weighed average belief is below he economerician s belief. As a resul, he wealh flucuaion affecs he difference beween he wealh-weighed average belief and he economerician s belief and so conribues o he ime variaion in he risk premium. In he nex subsecion, we provide a calibraion exercise o show ha a modes amoun of belief dispersion can generae sufficien ime variaion in he risk premium o explain he failure of he expecaions hypohesis, and ha he ime variaion of he risk premium is relaed o a en-shaped linear combinaion of forward raes. 3.4 Calibraion This secion illusraes he impac of agens heerogeneous expecaions on bond markes by simulaing he heerogeneous economy based on a se of calibraed model parameers. Theorem 1 implies ha he shor rae follows dr = λ f [r (l σ 2 I ]d + σ f dz f. The shor rae mean-revers o a ime-varying long-run mean l σi 2. Balduzzi, Das and Foresi (1998 esimae wo-facor ineres rae models wih his srucure and find ha he long-run mean of he shor rae moves slowly wih a mean-reversion parameer of 0.07 in heir sample. Since he mean-reversion parameer of his long-run mean process corresponds o λ l, we choose λ l o be 0.07, which implies ha i akes ln(2/λ l = 9.9 years for he effec of a shock o he long-run mean of he shor rae o die ou by half. Balduzzi, Das and Foresi also show ha he mean-reversion parameer of he shor rae (λ f in our model ranges from 0.2 o 3 in differen sample periods. We choose a value of 1 for λ f and his implies ha i akes ln(2/λ f = 0.69 year for he difference beween he shor rae and is long-run mean o die ou by half. 7 7 These wo mean-reversion parameers affec he magniude of agens belief dispersion effec. Inuiively, a larger λ l parameer causes l o rever faser o is long run mean l, herefore making agens belief dispersion abou l less imporan for bond prices; a larger λ f parameer causes f o rever faser o l, herefore making agens belief dispersion abou l more imporan for bond prices. 19

21 We choose σ f = 1.25% o mach he shor rae volailiy in he daa, and se σ l = 1.2% so ha he volailiy of l is 0.35% per monh, he middle poin of he range from 0.1% o 0.6% esimaed by Balduzzi, Das, and Foresi (1998. Furhermore, since σ I measures agens aggregae consumpion volailiy (Theorem 1, we choose σ I = 2% o mach he aggregae consumpion volailiy in he daa. 8 Parameer φ S direcly affecs he amoun of belief dispersion beween he wo groups. We choose φ S = 0.75 o generae some modes belief dispersion: In our simulaed daa, he average dispersion beween he wo groups, ˆl A ˆl B, is only 1.70%. This amoun is raher modes compared wih he ypical dispersions in surveys of fuure inflaion and GDP growh raes (see foonoe 1. We choose he following iniial condiions for our simulaion. The wo groups have an equal wealh share a = 0, i.e., ω A 0 = ωb 0 = 0.5; Boh f 0 and l 0 sar from heir seady sae value l, which we se a 5%; And, he wo groups also share an idenical prior belief abou l 0 equal o he seady value l: ˆl A 0 = ˆl B 0 summarized below: λ l = 0.07, λ f = 1, σ f = 1.25%, σ l = 1.2%, σ I = 2%, = l. All he model parameers are φ S = 0.75, ω A 0 = ω B 0 = 0.5, f 0 = l 0 = ˆl A 0 = ˆl B 0 = l = 5%. (20 Based on hese model parameers, we simulae he economy for 50 years a daily frequency and exrac bond yields and forward raes for various mauriies a he end of each monh. The lengh of 50 years roughly maches he sample duraion used in mos empirical sudies of he yield curve. The simulaion is repeaed 10,000 imes Yield volailiy curve Figure 1 plos he monhly bond yield volailiy, defined as he sandard deviaion of yield changes, for differen mauriies from zero o 10 years. The upper solid line corresponds o he yield volailiy in he heerogeneous economy. The wo dashed lines around he volailiy curve provide he 95h and 5h percenile of he volailiy esimaes across he 10,000 simulaed pahs. As he mauriy increases from zero o hree years, he yield volailiy increases from 36 o above 41 basis poins per monh. As he mauriy furher increases, he yield volailiy 8 One could also choose σ I o mach he volailiy of he aggregae producion. This would have lile or no impac on he volailiy amplificaion effec and he bond-yield regression resuls. 20

22 hen sars o fall slighly. The magniude and shape of his volailiy curve is similar o hose esimaed in Dai and Singleon ( Heerogeneous Case Monhly Yield Volailiy Homogeneous Case Bond Mauriy Figure 1: The erm srucure of bond yield volailiy. Using parameers specified in equaion (20, he economy is simulaed for 50 years o calculae bond yield volailiy, defined as he sandard deviaion of yield changes, for zero coupon bonds wih mauriies ranging from zero o 10 years. The simulaion is ieraed 10,000 imes and he figure plos he average (solid line, 95h and 5h percenile (dashed lines of he esimaed volailiy across he 10,000 pahs on bond mauriy. Similar simulaions are also performed on a homogeneous economy wih a represenaive agen holding he equal weighed average belief of he wo groups in he heerogeneous economy. The plos a he boom of he figure correspond o he average (solid line, 95h and 5h percenile (dashed lines of he esimaed volailiy across he 10,000 pahs in he homogeneous economy. To illusrae he volailiy amplificaion effec discussed in Secion 3.2, we compue he volailiy curve in a hypoheical homogeneous economy in which all agens hold he equal weighed average belief of he wo groups in he above simulaed heerogeneous economy (his average belief coincides wih he economerician s belief, as shown in Proposiion 1. Noe ha he average belief reflecs he changes in he wo groups beliefs, bu no heir relaive wealh flucuaion. As a resul, he volailiy curve in he homogeneous economy does no capure he volailiy amplificaion effec caused by he wo groups relaive wealh 21

23 flucuaion. The lower solid line in Figure 1 plos he volailiy curve in his homogeneous economy. The volailiy drops monoonically from 36 o 25 basis poins per monh as he bond mauriy increases from zero o 10 years. The difference beween he wo solid lines measures he volailiy amplificaion effec induced by wealh flucuaion. This effec is small a shor mauriies bu becomes subsanial when bond mauriy increases. For he 10 year bond, his amplificaion effec is 12 basis poins per monh, or roughly one hird of he oal bond yield volailiy. Why does he volailiy curve have a hump shape in he heerogeneous case, bu a monoonically decreasing shape in he homogeneous case? In he homogenous case, he bond yield is a linear combinaion of wo facors: Y (τ = a f (τ τ f + a l(τ ˆlR τ + b(τ τ. From our earlier discussion, he loading on random facor f, a f (τ τ, decreases monoonically wih τ, while he loading on random facor ˆl R, a l(τ τ, has a hump shape. The monoonically decreasing shape of he volailiy curve reflecs ha he conribuion of he firs facor o he bond yield volailiy dominaes he conribuion of he second facor. To simplify our discussion of he heerogenous case, we approximae equaion (18 by a linear form: Y (τ a f (τ f + a ( l(τ ω A τ τ ˆl A + ω B ˆl B + b(τ τ. Noe ha he second facor now becomes he wealh-weighed average belief ω A ˆl A + ω B ˆl B, which is more volaile han he second facor in he homogeneous economy, ˆl R. In oher words, he wealh flucuaion effec makes he second facor more volaile. The volailiy curve displays a hump shape when he wealh flucuaion effec is srong enough Campbell-Shiller bond yield regression This secion demonsraes ha he ime variaion in he risk premium in our model can help explain he failure of he expecaions hypohesis. The expecaions hypohesis posis ha an invesor in he bond marke should be indifferen abou he invesmen in a long-erm bond or in he shor rae over he same period. Despie is inuiive appeal, his predicion 22

24 is rejeced by many empirical sudies, e.g., Fama and Bliss (1987 and Campbell and Shiller (1991. In paricular, Campbell and Shiller (1991 run he following regression, Y (n Y (1 Y +1 (n 1 Y (n = α n + β n, (21 n 1 where Y (n is he n-monh yield a monh, α n is he regression consan, and β n is he regression coefficien. They show ha he expecaions hypohesis is equivalen o he following null hypohesis for regression (21: β n = 1. Inuiively, when he yield spread, Y (n Y (1, is posiive, he long erm bond yield is expeced o rise (or he long erm bond price is expeced o fall, because oherwise an agen canno be indifferen abou invesing in he long erm bond or he shor rae. The regression resuls in Panel A of Table 1 are colleced from Table 10.3 of Campbell, Lo and MacKinlay (1997, which uses 40 years of U.S. reasury bond yield daa from I shows ha β n sars wih a value of for 2-monh yield, and hen monoonically decreases as he bond mauriy increases. β n evenually akes a value of for 10-year yield. All hese coefficiens are significanly differen from 1 (he null, and he coefficien of 10-year yield is significanly negaive. Taken ogeher, hese regression resuls rejec he expecaions hypohesis: when he yield spread is posiive, he long erm bond yield ends o fall, raher han rise. This paern, however, is a naural implicaion of our model: Suppose he wealh-weighed average belief abou he fuure shor raes is higher han he economerician s belief. On he one hand, i implies ha agens discoun long erm bonds more heavily, which leads o higher long erm bond yields and so larger yield spreads; on he oher, i also implies ha he long erm bond prices appear cheap from he economerician s poin of view, i.e., he long erm bond prices are expeced o rise and bond yields are expeced o fall. Taken ogeher, a high wealh-weighed average belief implies boh large yield spreads and falling long erm bond yields in he fuure. To examine wheher his mechanism can explain he failure of he expecaions hypohesis, we simulae our economy 10,000 imes using he parameers summarized in (20. For each 23

25 n A. Resuls from Campbell-Lo-MacKinlay β n s.e. (0.191 (0.282 (0.442 (0.599 (1.004 (1.458 (2.076 B. Resuls from our simulaion of a heerogeneous economy β n s.e. (0.516 (0.541 (0.620 (0.785 (1.110 (1.632 (1.986 Table 1: The coefficiens of yield change regressions. This able repors he esimaes of β n in (21 and heir sandard errors for bond mauriies of n monhs. Panel A is aken from Table 10.3 of Campbell, Lo and MacKinlay (1997, which uses U.S. reasury bond yield daa from Panel B repors he mean and sandard deviaion of he esimaes of β n across he 10,000 simulaed pahs of he heerogeneous economy wih parameers from (20. simulaed pah, we run regression (21 using our simulaed bond yield daa. Panel B of Table 1 repors he average regression coefficiens and heir sandard errors. The average of he regression coefficiens decreases monoonically from o as bond mauriy increases from 2 monhs o 10 years, wih a similar rend and magniude o ha in Panel A. These coefficiens are also significanly lower han 0 based on he sandard errors across he 10,000 sample pahs. Noe ha he null hypohesis holds in a homogeneous economy wih each agen holding he same belief as he economerician. Therefore, exending a sandard asse pricing model wih modes heerogeneous expecaions offers a poenial explanaion for he failure of he expecaions hypohesis in he daa. The lieraure ofen aribues he failure of he expecaions hypohesis o ime-varying risk premia. Dai and Singleon (2002 find ha cerain classes of affine erm srucure models wih ime-varying risk premia are able o mach he aforemenioned bond yield regression resuls. However, he economic deerminans of he ime-varying risk premia sill remain elusive. Wacher (2006 argues for ime-varying risk preference of he represenaive agen, while our model proposes a new mechanism based on agens heerogeneous expecaions Cochrane-Piazzesi bond reurn regression Cochrane and Piazzesi (2005 find ha a single facor based on a en-shaped linear combinaion of forward raes predics excess reurns on bonds wih mauriy ranging from wo o 24