SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES. Abstract

Transcription

1 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES KRISTOFFER P. NIMARK Absrac A racable equilibrium erm srucure model populaed wih raional bu heerogeneously informed raders is developed and esimaed. Traders ake on speculaive posiions o exploi wha hey perceive o be inaccurae marke expecaions abou fuure bond prices. Yield dynamics due o speculaion are (i) saisically disinc from classical erm srucure componens due o risk premia and expecaions abou fuure shor raes and mus be orhogonal o public informaion available o raders in real ime and (ii) quaniaively imporan, poenially accouning for a subsanial fracion of he variaion of long mauriy US bond yields. Dae: June 26, 215, Economics Deparmen, Cornell Universiy. pkn8@cornell.edu webpage: The auhor hanks David Backus, Francisco Barillas, Toni Braun, Bernard Dumas, Michael Gallmeyer, Jerome Henry, Bryce Lile, Elmar Merens, Francisco Peneranda, Ponus Rendahl, Helen Rey, Ken Singleon, Thomas Sargen, Chris Telmer and seminar and conference paricipans a New York Universiy, Bank of Norway, Sveriges Riksbank, EABCN conference on Uncerainy Over he Business Cycle, ESSIM 29, 3rd European Workshop on General Equilibrium Theory, Universiy of Alicane, SED 29 and NORMAC 29, Federal Reserve Bank of New York, Cornell Universiy, UPF Finance Lunch, Bank of England, Queen Mary College, UC Berkeley, Federal Reserve Bank of San Francisco, 21 NBER Summer Insiue, UC San Diego and he Wesern Finance Associaion 211 for useful commens and suggesions. 1

2 2 KRISTOFFER P. NIMARK 1. Inroducion A fundamenal quesion in finance is wha he economic forces are ha explain variaion in asse prices and reurns. This paper demonsraes ha allowing for heerogeneous informaion ses among raional raders inroduces a speculaive componen in bond yields ha is absen in models in which all raders share he same informaion. The speculaive erm is saisically disinc from boh risk premia and erms reflecing expecaions abou fuure risk free shor raes and poenially empirically imporan. Many bonds, and US reasury bonds in paricular, are raded in very liquid secondary markes. In such a marke, he price an individual rader will pay for a long mauriy bond depends on how much he hinks oher raders will pay for he same bond in he fuure. If raders have access o differen informaion, his price may differ from wha an individual rader would be willing o pay for he bond if he had o hold i unil mauriy. The possibiliy of reselling a bond hen changes is equilibrium price as raders ake speculaive posiions in order o exploi wha hey perceive o be marke mispercepions abou fuure bond prices. This paper presens and srucurally esimaes an equilibrium model of he erm srucure of ineres raes ha is populaed wih raders ha engage in his ype of speculaive behavior. In he model, individual raders can idenify bonds ha, condiional on heir own informaion ses, have a posiive expeced excess reurn. In he absence of arbirage, expeced reurns in excess of he risk free rae mus be compensaion for risk. Traders will hold more of he bonds wih a higher expeced reurn in heir porfolios. In equilibrium, he increased riskiness of a less balanced porfolio is exacly off-se by he higher expeced reurn. We show formally ha heerogeneous informaion inroduces a source of ime varying expeced excess reurns ha, unlike he excess reurns documened by for insance Fama and Bliss (1987) and Campbell and Shiller (1991), canno be prediced condiional on pas bond yields. When aggregaed, he speculaive behaviour of individual raders inroduces new dynamics o bond prices. We demonsrae ha when raders have heerogeneous informaion ses, bond yields are parly deermined by a speculaive componen ha reflecs raders expecaions abou he error in he average, or marke, expecaions of fuure risk-free ineres raes. Since i is no possible for individual raders o predic he errors ha oher raders make based on informaion available o everybody, he speculaive componen in bond prices mus be orhogonal o publicly available informaion. Heerogeneous informaion hus inroduces a hird erm in bond yields ha is saisically disinc from he classical componens of yield curve decomposiions, i.e. erms due o risk premia and erms reflecing expecaions abou fuure risk-free shor raes. Despie he fac ha he speculaive componen in bond yields mus be orhogonal o public informaion, i is possible o quanify is imporance using only publicly available daa on bond yields. This is so because we as economericians have access o he full sample of daa and he speculaive erm is orhogonal only o public informaion available o raders in real ime. Tha is, we can use public informaion available in period + 1, + 2,... and so on, o back ou an esimae of he speculaive erm in period. The esimaed model suggess ha speculaive dynamics are quaniaively imporan and can explain a subsanial fracion of he variaion in US bond yields.

3 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 3 A necessary condiion for raders o have any relevan privae informaion abou fuure bond yields is ha bond prices do no perfecly reveal he sae of he economy. Recen saisical evidence suppors his view. In a few closely relaed papers, Joslin, Priebsch and Singleon (214) and Duffee (211) presen evidence suggesing ha he facors ha can be found by invering yields are no sufficien o opimally predic fuure bond reurns. They find ha while he usual level, slope and curvaure facors explain virually all of he cross secional variaion in yields, addiional facors are needed o forecas excess reurns. Ludvigson and Ng (29) provide more evidence ha curren bond yields are no sufficien o opimally forecas bond reurns. They show ha compared o using only yield daa, drawing on a very large panel of macroeconomic daa helps predic excess reurns. Saed anoher way, hese saisical models all sugges ha linear combinaions of curren bond yields are no sufficien o predic fuure bond yields opimally. In addiion o he empirical evidence cied above, we also have a priori reasons o believe ha bond prices should no reveal all informaion relevan o predicing fuure bond reurns. Grossman and Sigliz (198) argued ha if i is cosly o gaher informaion and prices are observed coslessly, prices canno fully reveal all informaion relevan for predicing fuure reurns. For he bond marke, he mos imporan variable o forecas is he shor ineres rae. In mos developed counries, he shor ineres rae is se by a cenral bank ha responds o macroeconomic developmens. If i is cosly o gaher informaion abou he macro economy, Grossman and Sigliz s argumen implies ha bond prices canno reveal all informaion relevan o predic bond reurns. In pracice, here is a vas amoun of financial and macro economic daa available ha could in principle help raders o predic fuure bond yields. If prices do no reveal all he informaion ha is relevan for predicing bond reurns i becomes more probable ha differen raders will use differen subses of he available informaion. Here, we model his by endowing raders wih parly privae informaion ha hey can exploi when rading. This se up also accords well wih he casual observaion ha a leas one moive for rade in asses is possession of informaion ha is no, or a leas is no believed o be, already refleced in prices. Formally, our se-up is similar o he informaion srucure in Diamond and Verrecchia (1981), Admai (1985), Singleon (1987), Allen, Morris and Shin (26) and Bacchea and van Wincoop (26). One implicaion of heerogeneous informaion ses is ha differen raders have differen expecaions abou fuure bond yields. This provides us wih anoher way of gauging he plausibiliy of his assumpion. While bond raders expecaions are unobservable, he average cross-secional dispersion of responses of one-quarer ahead Federal Funds Rae in he Survey of Professional Forecasers is abou 4 basis poins in he sample. There exiss a very large heoreical lieraure ha sudies asse pricing wih heerogeneously informed agens. Hellwig (198), Diamond and Verecchia (1981), Admai (1985) and Singleon (1987) are some of he early references. More recen examples include papers by Allen, Morris and Shin (26), Kasa, Walker and Whieman (214), Bacchea and van Wincoop (26, 27), Cespa and Vives (212) and Makarov and Rychkov (212). These papers eiher presen purely heoreical models or models calibraed o explain some feaure of he daa. The model presened here is esimaed direcly using likelihood based mehods.

4 4 KRISTOFFER P. NIMARK To he bes of my knowledge, his is he firs paper o empirically quanify he imporance of heerogeneous informaion ses for asse prices and reurns. 2. A Bond Pricing Model This secion presens an equilibrium bond pricing model. Traders are risk averse, raional, and ex ane idenical bu may observe differen signals relevan for predicing fuure bond prices. They choose a porfolio of risky bonds in order o maximize nex-period wealh. Traders ha have observed signals ha make hem more opimisic abou he reurn of a given bond will hold relaively more of ha bond in heir porfolio and in equilibrium, he increased riskiness of a less balanced porfolio is exacly offse by a higher expeced reurn. The equilibrium price of a bond is a funcion of he average expecaions of he price of he same bond in he nex period, discouned by he risk-free shor ineres rae. Bond prices are also affeced by supply shocks ha prevens equilibrium prices from revealing he average expecaion of fuure bond prices. The model is relaively racable and in he secion following his one i will be used o draw ou he consequences for erm srucure dynamics of relaxing he assumpion ha raders all have access o he same informaion Demand for long mauriy bonds. Time is discree and indexed by. As in Allen, Morris and Shin (26) here are overlapping generaions of agens who each live for wo periods. Each generaion consiss of a coninuum of households wih uni mass. Each household is endowed wih one uni of wealh ha i invess when young. When old, households unwind heir asse posiions and use he proceeds o consume. Unlike in he model of Allen e al, he owners of wealh, i.e. he households, do no rade asses hemselves. Insead, a coninuum of raders, indexed by j (, 1), rade on behalf of he households, wih households diversifying heir funds across he coninuum of raders. While no modeled explicily here, his se up can be moivaed as a perfecly compeiive limi case of he muual funds model of Garcia and Vanden (29) ha allows uninformed households o benefi from muual funds privae informaion, while diversifying away idiosyncraic risk associaed wih individual funds. More imporanly, he assumpion ha he ownership of he asses is separaed from he privaely informed raders keeps he model racable by absracing from informaion induced wealh heerogeneiy. 1 The formal srucure of he model is as follows. Trader j invess one uni of wealh in period on behalf of households born in period. In period +1 rader j unwinds he posiion of he now old generaion of households who hen use he proceeds o consume. Traders are infiniely lived and perform he same service for he nex generaion of households. There are wo ypes of asses: a risk-free one period bond wih (log) reurn r and risky zero-coupon bonds of mauriies 2, 3,..., n periods. Trader j chooses a vecor of porfolio weighs α j in order o maximize he expeced log of wealh under managemen W j +1 in period + 1. Tha is, rader j solves he problem max E [ log W j α j +1 Ω j ] (2.1) 1 Xiong and Yan (21) presen a calibraed difference-in-beliefs model ha hey use o analyze he ineracion of beliefs and wealh dynamics and how ha affec he erm srucure of ineres raes.

5 subjec o SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 5 W j +1 = 1 + r p,j (2.2) where Ω j denoes rader j s informaion se and r p,j is he log reurn of he porfolio chosen by rader j in period. All raders observe he shor risk-free rae r as well as he price of all bonds. In equilibrium, log reurns of individual bonds will be normally disribued. However, he log reurn on a porfolio of asses wih individual log normal reurns is no normally disribued. Following Campbell and Viceira (22a, 22b) we herefore use a second order Taylor expansion o approximae he log excess porfolio reurn as r p,j r = α j rx α j diag [ ] Σ j 1 rx, 2 α j Σ j rx,α j (2.3) where rx +1 is a vecor of period + 1 excess reurns on bonds defined as p 1 +1 p 2 r p 2 +1 p 3 r rx +1. p+1 n 1 p n r (2.4) and p n is he log price of a bond wih n periods o mauriy. The marix Σ j rx, is he covariance of log bond reurns condiional on rader j s informaion se. In equilibrium, condiional reurns will be normally disribued, ime invarian and wih a common condiional covariance across all raders. We can hus suppress he ime subscrips and rader indices on he condiional reurn covariance marix and wrie Σ rx insead of Σ j rx, for all and j. Maximizing he expeced log wealh (2.2) wih respec o α j hen gives he opimal porfolio weighs α j = Σ 1 rx E [ ] rx +1 Ω j 1 + rx diag [Σ rx ]. (2.5) 2 Σ 1 The higher reurn a rader expecs o earn on a bond, he more will he hold of i in his porfolio. However, risk aversion prevens he mos opimisic rader from demanding all of he available supply. Since each rader j has one uni of wealh o inves, inegraing he porfolio weighs (2.5) across raders yields he aggregae demand for bonds Bond supply. The vecor of bond supply s is sochasic and disribued according o s = µ + Σ 1 rx v : v N (, V V ) (2.6) To simplify noaion, he vecor of supply shocks v are normalized by he inverse of he condiional variance of bond prices Σ 1 rx. The supply shocks v play a similar role here as he noise raders in Admai (1985). Tha is, hey preven equilibrium prices from fully revealing he informaion held by oher raders. While here may be some uncerainy abou he oal number of bonds ousanding, a more appealing inerpreaion of he supply shocks is in erms of effecive supply, as argued by Easley and O Hara (24). They define he floa of an asse as he acual number of asses available for rade in a given period.

6 6 KRISTOFFER P. NIMARK 2.3. Equilibrium bond prices. Equaing aggregae demand α j dj wih supply s and solving for he log price p n gives p n = 1 2 σ2 n r + E [ p n 1 +1 Ω] j dj Σ n rx µ v n (2.7) where 1 2 σ2 n and v n are he relevan elemens of 1diag [Σ 2 rx] and v respecively and Σ n rx is he n h row of Σ rx. The price of an n period bond in period hus depends on he average expecaion in period of he price of an n 1 period bond in period The erm srucure of ineres raes and higher order expecaions. The log price of a one-period risk-free bond is he inverse of he shor ineres rae, i.e. p 1 = r. (2.8) Taking his as he saring poin we can apply (2.7) recursively o find he price of long mauriy bonds. The log price of a wo period bond is hen given by p 2 = 1 2 σ2 2 r E [ r +1 Ω] j dj Σ 2 rx µ v 2 (2.9) i.e. p 2 is a funcion of he average firs order expecaions abou he nex period risk free rae r. Coninuing wih he same logic, he price of a hree period bond is he average period expecaion of he price of a wo period bond in period + 1, discouned by he shor rae r. Leading (2.9) by one period and subsiuing ino (2.7) wih n = 3 gives p 3 = 1 ( ) σ σ3 2 Σ 2 p µ Σ 3 pµ (2.1) r E [ r +1 Ω] j dj [ [ ] ] E E r +2 Ω j +1 dj Ω j dj v 3. The expression (2.1) demonsraes ha he period price of a hree period bond is a funcion no only of he average expecaion of fuure risk-free ineres raes bu also of higher order expecaions. Tha is, he price parly depends on he average expecaion in period of he average expecaion in period + 1 of he risk-free rae in period + 2. In general, second and higher order expecaions do no coincide wih firs order expecaions when raders have heerogenous informaion ses. The price of a 3 period bond will hen deviae from he consensus value of he bond, i.e. he price he bond would have if i refleced only he average (firs order) period expecaion abou risk-free ineres raes in period + 1 and + 2. Higher order expecaions will maer for he price of all bonds of mauriy n > 2. Recursive forward subsiuion of (2.7) can be used o find a general expression for he price of an n

7 period bond as SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 7 p n = n ( ) 1 n 1 2 σ2 i Σ i rxµ r (k) :+k vn (2.11) i=2 where we used he more compac noaion [ [ r (k) :+k E E... E k= [ ] r +k Ω j +k 1 dj... Ω j +1 ] ] dj Ω j dj (2.12) for a k order expecaion of r +k. The price of an n-period bond hus depends on average expecaions of fuure shor raes of order up o n 1. As usual, he yield y n of an n period bond can be compued as y n = n 1 p n Uncondiional bond yields. Traders form model consisen expecaions which implies ha he uncondiional mean of he higher order expecaions of he risk-free rae in he bond price equaion (2.11) coincide wih he rue uncondiional mean. The uncondiional yield of an n period bond is hus given by E [y n ] = E [r ] + n 1 n i=2 ( Σ i rxµ 1 ) 2 σ2 i (2.13) The erm E [r ] in (2.13) reflecs how he average risk-free shor rae affecs long mauriy yields. The second erm, n 1 n i=2 Σi rxµ, capures boh risk-premia via he covariances in Σ i rx and supply effecs from he vecor µ. Risk premia will be high if he condiional variances are large or if condiional excess reurns are posiively correlaed. The average supply of bonds may increase or decrease bond yields depending on wheher he condiional reurns are posiively or negaively correlaed. The las componen on he righ hand side of (2.13) is a Jensen s inequaliy erm due o he log ransformaion. Uncondiional yields depend on he condiional covariance of bond reurns and will hus be influenced by raders informaion ses. However, he uncondiional yields are known o he raders in he model and do no influence heir filering problem. 3. Heerogeneous informaion, excess reurns and speculaion In his secion we derive he main heoreical implicaions of relaxing he assumpion ha all raders have access o he same informaion. Firs, we will demonsrae ha heerogeneous informaion inroduces rader-specific risk premia. We prove formally ha, unlike classical bond risk premia, risk premia due o informaion heerogeneiy mus be orhogonal o publicly available informaion. Second, we define he speculaive porfolio as he componen of a rader s porfolio held in order o exploi wha he perceives o be inaccurae marke expecaions abou nex period bond prices. Third, we derive he speculaive componen of bond yields and prove ha, jus like he rader specific componen in risk-premia, i mus be orhogonal o publicly available informaion.

8 8 KRISTOFFER P. NIMARK 3.1. Heerogenous informaion and expeced excess reurns. The holding period reurn on a zero-coupon bond depends on how is price changes over ime. To he exen ha differen raders have differen expecaions abou fuure bond prices, hey will also have differen expecaions abou bond reurns. In our model, his can be seen mos clearly from he definiion of he realized excess reurn on an n period bond rx n +1 p n 1 +1 p n r. (3.1) The excess reurn ha rader j expecs o earn on an n period bond is hus given by E [ ] [ rx n +1 Ω j = E p n 1 +1 Ω] j p n r. (3.2) since curren bond prices and shor raes are direcly observed by all raders. By subsiuing ou he curren bond price p n using he expression (2.7), he excess reurn ha rader j expecs o earn on an n period bond can be expressed as a sum of a rader specific and a common componen E [ ] [ ] [ ] rx n +1 Ω j = E p n 1 +1 Ω j E p+1 n 1 Ω j dj 1 2 σ2 n + Σ n rxµ + v n } {{ } } {{ } rader specific common In equilibrium, a posiive expeced excess reurn can only be earned as compensaion for risk. Since individual porfolios are deermined by expeced excess reurns and because raders are risk-averse, a rader who is more opimisic han he average rader abou he reurn of an n period bond will hold more of i in his porfolio and have a larger condiional porfolio reurn variance. The risk ha a more opimisic rader is compensaed for is hus he risk associaed wih holding a porfolio wih a higher condiional variance of reurns. In he absence of informaion heerogeneiy, he expeced excess reurn would be deermined by he consans 1 2 σ2 n + Σ n rxµ and he supply shock v n. There is hus a ime-varying componen in risk premia ha is common o all raders. However, he componen of excess reurn ha is due o informaion heerogeneiy is saisically disinc from he common componen since i mus be orhogonal o public informaion in real ime. Before proving his saemen formally, we firs define he relevan informaion se. Definiion 1. The public informaion se Ω a ime is he inersecion of he period informaion ses of all raders Ω Ω j. (3.3) j (,1) Proposiion 1. The rader specific componen in he expeced excess reurn E [ ] [ ] rx n +1 Ω j E rx n +1 Ω j dj (3.4) is orhogonal o public informaion in real ime. Proof. For any random variable X, he law of ieraed expecaions (e.g. Brockwell and Davis 26) saes ha E (E [X Ω ] Ω) = E (X Ω) (3.5)

9 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 9 if and only if Ω Ω. Take expecaions of he lef hand side of (3.4) wih respec o he public informaion se (3.3) and use ha Ω Ω j o ge [( E E [ ] [ ] ) ] rx n +1 Ω j E rx n +1 Ω j dj Ω = E [ ] rx n +1 rx n +1 Ω (3.6) = (3.7) which complees he proof. In wo influenial papers, Fama and Bliss (1987) and Campbell and Shiller (1991) argued ha excess reurns on bonds can be prediced using curren yields. One implicaion of Proposiion 1 is hus ha he rader specific componen in expeced excess reurn is saisically disinc from he classic predicable excess reurns documened in hese papers The speculaive porfolio. Traders ha have differen reurn expecaions will hold differen porfolios. We define he speculaive componen of rader j s porfolio as he bonds rader j holds because he believes average reurn expecaions are inaccurae. Tha is, he speculaive componen in rader j s porfolio is he difference beween rader j s acual porfolio and he porfolio rader j believes he average rader holds and i is given by ( ) [( α j E α i di Ω j = Σ 1 rx E rx +1 E ( ) ] ) rx +1 Ω i di Ω j. (3.8) The speculaive componen in rader j s porfolio is hus he (covariance weighed) difference beween rader j s expeced reurns and he reurns ha rader j believes he average rader expecs o earn on bonds. If all oher raders shared rader j s expecaions, bond prices would adjus unil all raders, including rader j, would hold he average porfolio. Trader j hus owns some bonds only because he believes ha he average, or marke, expecaions abou bond reurns are incorrec. Below, we will use he esimaed model o quanify how he speculaive porfolio of he average rader reacs o he shocks ha drive bond yields Speculaion, bond prices and public informaion. When aggregaed, he speculaive behaviour of individual raders affecs he demand for bonds, and in exension, bond prices. Above, we defined he speculaive porfolio in erms of differences in one-period reurn expecaions which depend on he expeced nex period price. Of course, he nex period price may will also be parly deermined by speculaive behavior, and expecaions abou he price furher ino he fuure, and so on. In order o accoun for he oal effec of speculaion on a bond s price, i is helpful o firs define a useful couner-facual price The consensus price. Following Allen, Morris and Shin (26) we define he consensus price p n of an n-period bond as he price ha would reflec he average opinion of he fundamenal value of he asse properly discouned. The consensus price is hus he counerfacual price a bond would have, if by chance, all raders happened o share he average rader s period expecaions abou he risk-free ineres raes beween period and + n 1 and his fac was common knowledge. I can be found by replacing he higher order

10 1 KRISTOFFER P. NIMARK expecaions of he risk-free rae in (2.11) wih he average rader s firs order expecaions p n 1 n ( σ 2 2 i Σ i rxµ ) n 1 E [ r +k Ω] j dj v n. (3.9) i=2 We use he couner-facual consensus price p n o define he speculaive componen in acual bond prices The speculaive componen in bond prices. The speculaive componen in bond prices is he difference beween he acual price and he couner-facual consensus price. Taking he difference beween (2.11) and (3.9), we ge n 1 ( p n p n = k= k= E [ ] r +k Ω j (k) dj r :+k ). (3.1) The speculaive componen in an n-period bond price can hus be expressed as he difference beween firs and higher order expecaions abou fuure shor ineres raes. 2 I is sraighforward o show ha he speculaive componen in bond prices mus be orhogonal o public informaion. Proposiion 2. The speculaive erm p n p n is orhogonal o public informaion in real ime, i.e. E [p n p n Ω p ] = (3.11) Proof. In he Appendix. While he formal proof of Proposiion 2 is given in he Appendix, he logic is simple and inuiive. The speculaive componen (3.1) consiss of higher order expecaions errors abou he risk-free ineres rae, ha is, predicions abou oher raders predicion errors. By definiion, he public informaion se is available o all raders. Clearly, i is no possible for an individual rader o predic he errors ha oher raders are making by using informaion ha is available also o hem. The speculaive componen in a bond s price mus herefore be orhogonal o public informaion available in real ime. Allen, Morris and Shin (26) argue ha wih privaely informed raders, asse prices may display drif, i.e. slow adjusmen o shocks wih several small price changes in he same direcion. While his is rue if one condiions on he acual value of he fundamenal, Proposiion 2 demonsraes ha here should be no discernible drif caused by privae informaion ha can be idenified simply by observing prices or oher informaion ha is publicly available in real ime. Tha he speculaive componen in bond prices mus be orhogonal o public informaion available is a consequence of ha raders form raional model consisen expecaions. This also makes he speculaive componen derived here differen from he speculaive componen in he difference-in-beliefs model of Xiong and Yan (21). In heir model, raders are boundedly raional and do no condiion on bond prices when hey form expecaions abou 2 In a differen conex, Bacchea and Wincoop (26) shows ha a similar erm (which hey label he higher order wedge ) can be expressed as an average expecaion error of he innovaions o he fundamenal process in heir model.

11 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 11 fuure bond yields. To an ouside economerician, he speculaive componen in ha model looks like classical risk premia, i.e. i makes excess reurns predicable based on curren bond yields Decomposing bond prices. There exiss a very large empirical erm srucure lieraure ha implicily or explicily decomposes long-erm ineres raes ino expecaions abou fuure risk-free shor ineres raes and risk-premia, e.g. Cochrane and Piazzesi (28) and Joslin, Singleon and Zhu (211). The premise for hese ype of wo-way decomposiions is ha risk premia and expecaions abou fuure risk free ineres raes are sufficien o compleely accoun for he yield-o-mauriy of a bond. However, heerogeneous informaion inroduces a hird componen o bond yields due o speculaive behaviour by raders. Add and subrac he consensus price (3.9) from he righ hand side of he price of an n-period bond (2.11) and rearrange o ge n 1 p n = E [ r +k Ω] j dj (3.12) k= } {{ } Average 1 s order shor rae expecaions ( n 1 + k= E [ ] r +k Ω j (k) dj r :+k ) } {{ } Speculaive componen + 1 n ( σ 2 2 i Σ i rxµ ) v n i=2 } {{ } Common risk premia The price of a long-mauriy bond can hus be expressed as he sum of average firs order expecaions abou fuure risk-free shor raes, a speculaive componen due o higher order predicion errors and a risk-premia componen common o all raders. From Proposiion 2 we know ha he speculaive componen mus be orhogonal o public informaion in real ime. The speculaive componen is hus saisically disinc from boh common risk premia and firs order expecaions abou fuure risk-free raes. In a model wih perfec or common informaion, he speculaive componen would be zero a all imes and bond prices would hen be a funcion only of common shor rae expecaions and risk premia. The speculaive componen would also be zero if here were no secondary markes for rading bonds. In he absence of secondary markes, bonds can only be purchased when hey are issued and mus hen be held unil mauriy. In such a seing, he expecaion of oher raders expecaions would no maer for he equilibrium price, since he price of a zero coupon bond a mauriy is simply is face value, which is known o all raders and does no depend on he expecaions of oher raders. The new dynamics inroduced o he erm srucure by heerogeneous informaion ses are hus dependen on he fac ha long mauriy bonds can be raded in secondary markes. This ends he heoreical par of he paper. Before urning o he daa, we can summarize our findings so far. Wih heerogeneous informaion ses, individual raders can idenify and ake advanage of predicable excess reurns ha would be absen in a model wih only

12 12 KRISTOFFER P. NIMARK common informaion. We also demonsraed ha he new bond price dynamics inroduced by speculaive behavior mus be orhogonal o public informaion. This has an ineresing empirical implicaion: Speculaive dynamics canno be deeced using public daa in real ime. However, as economericians we can use public informaion from periods + s : s > o exrac an esimae of he speculaive componen in bond yields in period. To do so, we need o specify explici processes for he risk-free shor rae, bond supply and raders informaion ses. 4. Empirical Specificaion Above, bond prices were derived as funcions of higher order expecaions of fuure shor raes. In order o have an operaional model ha we can use o quanify he implicaions of heerogenous informaion, we here specify explici processes for he shor rae, he supply of long mauriy bonds and he informaion ses of he raders. In his secion we also describe how he model can be solved and esimaed The shor rae and he exogenous facors. The shor ineres rae r is an affine funcion of a vecor of exogenous facors x where he facors follow he vecor auoregressive process r = δ + δ x x (4.1) x = F x 1 + Cu : u N(, I). (4.2) We will normalize he shor rae and facor processes by assuming ha δ x is a vecor of ones, F is a diagonal marix wih he i h diagonal elemen denoed f i and C is a lower riangular marix wih he c ij in he i h row and j h column. Normalizing F and C o be diagonal and lower riangular do no resric he dynamics of r. In he esimaed model, x is a four dimensional vecor. This gives a sufficienly high dimensional laen sae o make he filering problem of raders non-rivial, while keeping he model compuaionally racable Parameerizing bond supply. The bond supply disribuion (2.6) is parameerized as follows. The mean supply vecor µ has a ypical elemen µ n given by µλ n. The parameer λ governs how he average supply of bonds changes wih mauriy n. Wih λ > 1, supply increases wih mauriy, and conversely, λ < 1 implies ha average supply decreases wih mauriy. The marix V, i.e. he square roo of he covariance of he supply shocks v, is diagonal wih he n h diagonal elemen given by σn 1 This parameerizaion implies ha he sandard deviaion of he direc effec of supply shocks on bond yields is consan across mauriies and ha supply shocks are independen across mauriies. These resricions are imposed in order o economize on he number of free parameers bu preliminary esimaes suggess ha hey imply very small coss in erms of fi.

13 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES Traders informaion ses. All raders observe a vecor of public signals conaining he curren shor rae r and bond yields of mauriy 2,3,...,n colleced in he vecor y. Heerogeneous informaion is inroduced hrough rader-specific signals abou he laen facors x. The vecor of privae signals z j observed by rader j is specified as z j = x + Qζ j : ζ j N (, I 4 ) (4.3) where Q is a diagonal marix wih he i h diagonal elemen denoed q i. Each elemen in he signal vecor is hus he sum of he rue facor and an idiosyncraic noise componen. The noise is uncorrelaed across signals and ime. The vecor s j defined as s j = [ ] z j r y (4.4) hen conains all he signals ha rader j observes in period. Trader j s informaion se in period also includes all previous signals Ω j = { } s j, Ω j 1 (4.5) and raders hus condiion heir expecaions on he enire hisory of observed signals The law of moion of sae. When raders have heerogeneous informaion ses, i becomes opimal for hem o form expecaions abou oher raders expecaions. Naural represenaions of he sae in his class of models end o be infinie. 3 The model is solved using he mehod proposed in Nimark (211) which delivers a law of moion for he (finie dimensional) sae X of he form X = MX 1 + Ne. (4.6) ] The sae vecor X is given by he hierarchy of higher order expecaions of he exogenous facors x [ X x () x (1) x (k) (4.7) where he k order expecaions is defined recursively as [ ] x (k) E x (k 1) Ω j dj saring from x () = x. The soluion mehod in Nimark (211) uses ha he impac of higher order expecaions on bond prices decreases fas enough in he order of expecaion, and ha he variance of higher order expecaions is bounded by he variance of he rue facors. Togeher, hese facs imply ha he equilibrium represenaion can be approximaed wih a sae vecor ha conains only a finie number of higher order expecaions of he facors. The ineger k is he maximum order of expecaion considered and can be chosen o achieve an arbirarily close approximaion in he limi as k. In he esimaed model, k = 4. The vecor e conains all he aggregae shocks ha affec he exended sae X and includes boh he facor shocks u and he supply shocks v. The supply shocks do no direcly affec he facors x bu hey do affec raders (higher order) expecaions abou x since raders use bond yields o exrac informaion abou x. 3 See Townsend (1983), Sargen (1991) and Makarov and Rychkov (212).

14 14 KRISTOFFER P. NIMARK Common knowledge of he model among raders is used o pin down he law of moion for X, ha is, o find M and N in (4.6). The logic is as follows: As usual in raional expecaions models, firs order expecaions x (1) are opimal, i.e. model consisen esimaes of he acual facors x. The knowledge ha oher raders have model consisen expecaions allow raders o rea average firs order expecaions as a sochasic process wih known properies when hey form second order expecaions. Common knowledge of he model hus implies ha second order expecaions x (2) are opimal esimaes of x (1) given he law of moion for x (1). Imposing his srucure on all orders of expecaions allows us o find he law of moion for he complee hierarchy of expecaions as funcions of he srucural parameers of he model. The Appendix describes how o find he law of moion for he sae in pracice. The sae vecor X is high dimensional, bu his by iself does no increase our degrees of freedom in erms of fiing bond yields. In fac, because he endogenous sae variables x (k) are raional expecaions of he lower order expecaions in x (k 1), he marices M and N in he law of moion (4.6) are compleely pinned down by he parameers of he process governing he rue exogenous facors x and he precision of raders informaion ses Bond prices and he sae. For a given law of moion (4.6), bond prices can be derived using he average expecaion operaor H : R k+1 R k+1 ha annihilaes he lowes order expecaion of a hierarchy so ha x (1) x (2) ṭ. x (k+1) = H x () x (1) ṭ. x (k) (4.8) and where x (k) = : k > k. The average (firs order) expecaion abou he sae in period is hus given by HX. The average expecaion in period of wha he sae will be in period + 1 is hus given by MHX. Combing he operaor H ha increases he order of expecaions by one sep wih he marix M from he law of moion (4.6) ha moves expecaions one sep forward in ime, allows us o compue he k order expecaion of he shor rae in period + k as r (k) :+k = [ δ x ] (MH) n 1 X. (4.9) Subsiuing (4.9) ino he bond pricing equaion (2.11) hen gives p n = 1 n ( σ 2 2 i Σ i rxµ ) nδ (4.1) i=2 n 1 [ δx ] (MH) s X v n s= The marix M governs he acual dynamics of r while bonds are priced as if X was observed by all raders and followed a process governed by MH. The marices M and MH

15 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 15 are hus analogous o he physical and risk neural dynamics in a sandard no-arbirage framework, hough he inerpreaion is differen The esimaed sae space sysem. The sae equaion (4.6) and he bond price equaion (4.1) can be combined ino a sae space sysem of he form X = MX 1 + Ne (4.11) y = A + BX + Rv. (4.12) Combining he fac ha y n = n 1 p n wih (4.1) implies ha he rows of A and B ha correspond o he n period bond yield in he measuremen equaion are given by A n = n 1 1 n ( σ 2 2 i Σ i pµ ) + δ (4.13) i=2 n 1 [ B n = n 1 δx ] (MH) s X. (4.14) s= The vecor of parameers o be esimaed is denoed θ {F, C, Q, δ, µ, λ, σ v } and consiss of a oal of 22 parameers. Evaluaing he log likelihood funcion for he sae space sysem (4.11) - (4.12) allows us o form a poserior esimae for θ. The yields used for esimaion are he 1-, 2-, 3-, 4- and 5-year ineres raes on US Treasuries aken from he CRSP daa base. The sample period runs from July 1952 o January 213 and conains 727 monhly observaions. We use uniform priors on all model parameers. To ake ino accoun he evidence from he Survey of Professional Forecasers cied in he inroducion, an informaive prior is used on he model implied forecas dispersion. The prior disribuion of he sandard deviaion of he cross-secional forecas dispersion is cenered around 2 basis poins wih a sandard deviaion of 5 basis poins. This ensures ha a low poserior probabiliy is associaed wih parameerizaions ha imply eiher couner-facually small or implausibly large degrees of forecas dispersion among he raders in he model. The poserior parameer disribuions was generaed from 2 draws from an Adapive Meropolis algorihm (see Haario, Saksman and Tamminen 21), iniialized from a parameer vecor found by maximizing he poserior using he simulaed annealing maximizer of Goffe (1996). The resuls repored in he nex secion are based on he las 1 draws. 5. Empirical resuls Table 1 repors he poserior esimaes of he model parameers. The mode θ is he parameer vecor from he Markov chain ha achieves he highes poserior likelihood. All parameers appear o be well-idenified. The model fis uncondiional yields well. A he poserior mode, he uncondiional riskfree shor rae is 5.9 per cen and he uncondiional 5-year yield is 6.25 compared o he respecive sample means of 5.24 and 5.73 per cen. The models hus slighly over-predics uncondiional yields bu is able o generae an upward sloping yield curve. The upward slope is driven enirely by he covariance srucure of condiional reurns. In fac, he poserior

16 16 KRISTOFFER P. NIMARK mode of λ is.94. A value of λ smaller han 1 implies ha he average supply of bonds is decreasing in mauriy which by iself would generae a downward sloping yield curve. The supply shocks have a sandard deviaion of 83 basis poins. This is larger han he pricing errors usually found using yields-only affine no-arbirage models. However, he supply shocks are no formally equivalen o he pricing errors of sandard facor models such as hose in he model of Joslin, Singleon and Zhu (211). Firs, he supply shocks change raders required compensaion for risk and capures variaion in common risk premia. The supply shocks are hus priced facors and no pricing errors. Second, raders use he observed bond yields o exrac informaion abou he sae X. Because he supply shocks affec bond yields, supply shocks affec raders (higher order) expecaions abou he persisen facors x. So while supply shocks are independen across ime and mauriies, a supply shock o a single mauriy bond has persisen effecs on he enire cross-secion of bond yields. The supply shocks hus have quie differen observable implicaions compared o classical whie noise pricing errors. Tha a single supply shock have persisen effecs on yields across all mauriies also means ha he model is no subjec o Hamilon and Wu s (211) criique ha classical measuremen errors in affine erm srucure models can be saisically rejeced. A he poserior mode, he model implied cross-secional dispersion of forecass across he raders is approximaely 1 basis poins. While by iself, his level of dispersion is neiher oo large nor oo small o appear a priori unreasonable, i is somewha lower han he prior and also lower han wha is found in survey daa. Tha he poserior dispersion is lower han he prior and he dispersion measured in surveys sugges ha condiional on he model, here is a rade-off beween fiing bond yields and he cross-secional dispersion. One possible inerpreaion of his resul is ha raders in realiy are beer and more uniformly informed han survey respondens and his may be inferable from bond yield dynamics.

17 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 17 Table 1 Poserior Parameer Esimaes 1964:1-27:12 θ Mode θ Prior dis. Poserior 2.5%-97.5% Shor rae process δ.59 U (, ) ρ 1.99 U (,.999) ρ 2.96 U (,.999) ρ 3.7 U (,.999) ρ 4.22 U (,.999) c 1.12 U (, ) c 2.37 U (, ) c 3.12 U (, ) c 4.1 U (, ) c U (, ) (.17) (.13) c U (, ) (.2) (.9) c U (, ) (.49) (.43) c U (, ) c U (, ) (.21) (.19) c U (, ) Noise in privae signals q 1.11 U (, ) q 2.15 U (, ) q 3.27 U (, ) q 4.14 U (, ) Bond supply µ.5 U (, ) λ.94 U (, ) σ v.86 U (, ) Yields and speculaion. We can use he esimaed model o inspec he join responses of yields, he speculaive componen and he speculaive porfolio o innovaions o he exogenous facors. The op row of Figure 1 illusraes he response of he 1-, 3- and 5-year bond yields. The middle row illusraes he response of he speculaive componen defined as (3.1) in he 1-, 3- and 5-year bond yields. The boom row illusraes he speculaive porfolio in 1-, 3- and 5-year bonds of he average rader defined as (3.8). Facors are ordered according o persisence wih he impulse response funcions o he mos persisen facor in firs column. The signs of he innovaions are normalized so ha he iniial impac on yields is posiive. A few paerns sand ou. Firs, shor bond yields responds more o innovaions han long bond yields. The magniude of he response of he speculaive erm is uniformly larger in long mauriy bonds han in shorer mauriies. Condiional on he innovaion, he speculaive erm responds in he same direcion across all mauriies. However, while an innovaion o he firs facor imply a posiive response of he speculaive componen, innovaions o

18 18 KRISTOFFER P. NIMARK he remaining facors imply negaive responses. Afer an innovaion o he mos persisen facor, he average rader hus believes ha oher raders overesimae fuure shor ineres raes, while afer an innovaion o a less persisen facors, he average rader believes ha oher raders underesimae fuure shor ineres raes. Inspecing he boom row of Figure 1 shows ha he speculaive porfolio responses are sronger for 3-year bonds han for 1- and 5-year bonds. So while he speculaive erm is larges for long mauriy bonds, he speculaive posiion of he average raders is larger in medium mauriy bonds. This may seem counerinuiive. However, he speculaive porfolio (3.8) depends only on he difference beween firs and second order expecaions of he one-period reurn while he speculaive componen depends on he accumulaed difference beween firs and higher order expecaions of fuure shor raes over he enire life of he bond. The speculaive porfolio is a couner-facual hough experimen, and does no represen an acual change in bond demand in equilibrium. By definiion, he average rader holds he average porfolio, which is unaffeced by innovaions o x so here are no acual changes in he average porfolio in response o he innovaions ploed in Figure 1. However, he average rader is generally unaware of being he average rader. From his subjecive perspecive, he number of bonds ha he holds because he hinks ha average reurn expecaions are incorrec do in fac respond o facor innovaions. Quaniaively, he speculaive posiions are large. This is due parly o he low degree of risk-aversion implied by he logarihmic preferences in expeced wealh. Anoher, and more imporan, explanaion o he large speculaive posiions is he srong correlaion beween reurns on bonds of differen mauriies. A large long posiion in a paricular mauriy bond inended o exploi an expeced posiive excess reurn can hen be hedged effecively by aking a large off-seing shor posiions Hisorical decomposiion of bond yields. Proposiion 2 above esablished ha he speculaive erm in he price of a bond can be expressed as a higher order predicion error ha is orhogonal o public informaion. Neverheless, as economericians, we can quanify his erm using public bond price daa since he period higher order predicion error is only orhogonal o informaion known o all raders up o period. Since we can ex pos use he full sample and exploi informaion for +s : s > o back ou informaion abou he higher order predicion error in period, we can form an esimae of he speculaive componen. The procedure is as follows. For a given parameer vecor θ, he Kalman simulaion smooher can be used o draw from he smoohed sae disribuion p ( X T y T, θ ) (e.g. Durbin and Koopman 22). To consruc he poserior disribuion of he sae X T, draw repeaedly from he poserior parameer disribuion p ( θ y ) T and for each draw of θ generae a draw from he condiional sae disribuion p ( X T y T, θ ). Since he speculaive erm (3.1) can be expressed as a linear funcion of he sae X, he simulaed disribuion of he sae can be used o compue he implied poserior disribuion of he speculaive componen in bond yields. Agens average firs order expecaions of fuure risk-free raes are also linear funcions of he sae. Once we have a poserior disribuion of he sae we can hus consruc a poserior disribuion of he decomposiion (3.12) and quanify how much he erms due o

19 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES 19 1 x s facor 3 x nd facor 6 x rd facor 4 x h facor Bond yields Speculaive componen 4 x x x x Speculaive porfolio year bond 3 year bond 5 year bond Figure 1. Impulse response funcion of 1,3 and 5 year yields (op row), he speculaive componen (middle row) and he speculaive porfolio (boom row) o innovaions o he exogenous facors (by columns). average firs order expecaions abou fuure risk free raes, common risk premia and he speculaion each conribued o bond yields over he sample period. Figure 2 and Figure 3 illusrae his decomposiion for 1- and 5-year bond yields. As in sandard models, mos of he variaion in bond yields is explained by variaion in expeced fuure risk-free raes. Risk premia are posiive on average and mos volaile around he early 198s for boh he 1- and 5-year yield. The speculaive erms are posiively correlaed across mauriies and more volaile in he 5-year bond han in he 1-year bond. The larges variaions in he speculaive erm occur around 198 when he speculaive componen for he 5-year bond conribues 12 negaive basis poins o he 5-year yield. This period coincides wih he so-called Volcker disinflaion when he hen Federal Reserve chairman Paul Volcker raised ineres raes sharply o bring inflaion under conrol, causing a recession (see for insance he accoun in Goodfriend and King 25). A negaive speculaive componen indicaes ha raders hough oher raders underesimaed fuure shor raes. Saed differenly, he episode around 198 during which

20 2 KRISTOFFER P. NIMARK 2 1 Average Shor Rae Expecaions Term in 1-year Yield 2.5% Median 97.5% Common Risk Premia in 1-year Yield Speculaive erm in 1 year yield Figure 2. Hisorical decomposiion of 5 year yield, median (solid) and 95% probabiliy inerval (doed). he speculaive componen is large and negaive was a period when individual raders perhaps believed ha oher raders aached oo much credibiliy o chairman Volcker s disinflaion policy and were individually more scepical abou he probabiliy of his evenual success. In absolue erms, he speculaive erm was larges in he early 198s. However, as a fracion of he oal yield, speculaion appears o have been more imporan in he las decade. In 211, he mode of he esimae of he speculaive erm reached 4 basis poins a a ime when 5-year yields were around 3 per cen Speculaion and he expecaions hypohesis. One way o hink of he well-known failure of he expecaions hypohesis is in erms of a yield decomposiion: If expecaions of fuure shor-raes are no sufficien o explain he variaion in bond yields, he expecaions

21 SPECULATION AND THE TERM STRUCTURE OF INTEREST RATES Average Shor Rae Expecaions Term in 5-year Yield 2.5% Median 97.5% Common Risk Premia in 5-year Yield Speculaive erm in 5 year yield Figure 3. Hisorical decomposiion of 5 year yield, median (solid) and 95% probabiliy inerval (doed). hypohesis fails. In his sense, he speculaive componen help o explain he failure of he expecaions hypohesis since i provides a second wedge, in addiion o classical risk-premia, beween bond yields and expecaions of fuure shor raes. A second way o hink abou he failure of he expecaions hypohesis is in erms of excess reurns being predicable, which is anoher way of saing ha expecaions of shorraes are no enough o explain bond yields. In his sense, he speculaive componen does no help explaining he well-documened empirical regulariy ha fuure excess reurns are predicable based on he curren yield curve (as well as many oher variables). This is so because he speculaive componen mus be orhogonal o publicly available informaion in real ime.

22 22 KRISTOFFER P. NIMARK % Median 2.5% Monhs o mauriy Figure 4. Relaive sandard deviaion of speculaive erm and yields across mauriies. Median (solid) and 95% probabiliy inerval (doed). Singleon (26) poins ou ha violaions of he expecaion hypohesis in US daa are mos pronounced when he period is included in sample. Risk-premia based explanaions of his episode emphasize ha he early 198s was a period when raders demanded eiher more compensaion o hold a given amoun of risk because of he recession, or when he amoun of risk was perceived o be higher han usual because of more volaile ineres raes. This is also he case for our model, hough i absracs from persisen variaion in common risk premia, which may be a source of misspecificaion. One concern migh be ha because of he resricive way ha common risk premia is inroduced, he model here simply relabels some of wha in realiy is risk-premia as speculaion. However, while he early 198s are associaed wih large movemens in boh risk-premia and speculaion, he wo are no observaionally equivalen. The fac ha speculaive dynamics mus be orhogonal o public informaion in real ime makes i economerically disinc from oher sources of ime variaion in bond yields. This is exploied in Barillas and Nimark (214) who use an affine no-arbirage model ha allows for heerogeneous informaion o separaely idenify risk premia and speculaion over he same sample period. Tha model ness a sandard hree facor affine model and aribues a similar quaniaive imporance o he speculaive componen o wha is found here. 4 4 The affine no-arbirage model in Barillas and Nimark (214) imposes less economic srucure han he equilibrium model presened here and is empirically more flexible. However, in ha paper, he porfolio choice of raders is no modeled explicily.