Heterogeneous Beliefs and Short-term Credit Booms

Transcription

1 Heterogeneous Beiefs an Sort-term Creit Booms Ziguo He y Wei Xiong z Marc 2 Abstract We stuy te nancing of specuative asset-market booms in a stanar framework wit eterogeneous beiefs an sort-saes constraints. Cas-constraine optimists use teir asset oings as coatera to raise ebt nancing from ess optimistic creitors. Troug state-contingent re nancing, sort-term ebt aows te optimists to reuce ebt payment in upper states wic tey assign iger probabiities to, but at te expense of greater roover risk if te asset funamenta eteriorates at te ebt maturity. In contrast, ong-term ebt aows te optimists to ege teir nancing cost in ownturns. Our moe ienti es istinctive e ects of initia an future beief ispersion in riving a sort-term creit boom, wic can in turn fue an asset-market boom. Our moe igigts te important e ects of agents ebt-maturity an everage coices on asset market equiibrium. Keywors: Asset bubbe, Debt maturity, Leverage, Roover risk PRELIMINARY. We tank Ap Simsek an seminar participants at Nortwestern University, Stockom Scoo of Economics, an University of Minnesota for epfu comments. y University of Cicago, Boot Scoo of Business. Emai: ziguo.e@cicagoboot.eu. z Princeton University an NBER. Emai: wxiong@princeton.eu.

2 Introuction Stanar economic teories empasize agents consumption an portfoio coices as te key rivers of te asset market equiibrium. Te recent creit crisis of painte a i erent picture, in wic te nancing of market participants investment positions, suc as everage an ebt maturity, emerge at te center of te boom-an-crisis cyce. In te aftermat of te crisis, many commentators, e.g., Arian an Sin (29), Brunnermeier (29), Gorton an Metrick (29), an Krisnamurty (2), observe tat uring te ousing-market boom before te crisis many nancia institutions use arge everages to nance investment positions in mortgage backe securities. Furtermore, as we iscuss in Section 4, te arge everages were accompanie by a sortening of ebt maturity ue to te ramaticay increase use of sort-term ebt, suc as overnigt repos an asset-backe commercia paper. Te i cuties of tese nancia institutions in roing over teir sortterm ebt after te ousing market starte to ecine in 27 a eventuay e to a nancia market metown. Te signi cant roes paye by te sort-term creit boom in nancing te initia ousing-market boom an in triggering te ater nancia crisis igigt te important e ects of market participants nancing coices on asset-market ynamics. To te extent tat sort-term ebt exposes te borrowers to roover risk, teir nancing ecisions appear non-trivia an eserve a systematic anaysis. In tis paper, we eveop a ynamic moe to anayze te interactions between investors nancing coices an asset market ynamics. In particuar, we focus on te roe of ebt maturity, jointy wit tat of everage. We bui on te framework recenty propose by Geanakopos (29), wo nicey emonstrates tat investors everage cyce can irecty a ect te asset-market cyce. Tis framework is iea for our purpose as it invoves optimists wo o more optimistic beiefs about te funamenta vaue of an asset using te asset as coatera to borrow from not so optimistic creitors to nance teir asset purcases. Tis framework aso nests a wiey use asset-market setting for anayzing asset overvauation an bubbes as joint e ects of eterogeneous beiefs an sort-saes constraints, e.g., Mier (977), Harrison an Kreps (978), Morris (996), Cen, Hong, an Stein (22), an Sceinkman an Xiong (23). In tis setting, sort-saes constraints cause te equiibrium asset prices Interestingy, sort-term creit booms were aso present in severa oter istorica asset-market booman-crisis episoes. See Wite (99) an Eicengreen an Mitcener (23) for te stock market boom an cras of 929, an Rorik an Veasco (999) an Reinart an Rogo (29) for te emerging-market ebt crises in 99s.

3 to bias towar te beiefs of te optimists an, as a resut, iger beief ispersion can ea to iger asset prices. Speci cay, our moe as two perios an a risky asset wose funamenta vaue is unobservabe. We consier two groups of risk-neutra agents oing eterogeneous an state-contingent beiefs, wic originate from teir eterogeneous prior beiefs an earning processes, about te asset funamenta. If te optimists ave su cient funs, tey wou acquire a te asset an bi up te asset price to teir optimistic vauation. If te optimists ave insu cient funs, ten tey ave to use teir asset oings as coatera to raise ebt nancing from te pessimists wo ave excess funs. Like Geanakopos (29) an Simsek (29), we restrict te optimists to stanar non-contingent ebt contracts, wic are wiey use in practice. 2 Di erent from tem, we focus on te optimists ebt maturity coice in raising nancing from te pessimistic creitors. Te optimists nancing coices, wic incue bot maturity an everage, etermine te creit tey can obtain an tus te price tey can o er for te asset. In tis way, te asset market equiibrium is jointy etermine wit te creit market equiibrium. A key avantage of sort-term ebt is tat state-contingent re nancing aows te optimists to structure state-contingent ebt payo s to reuce nancing cost an increase everage. Because te possibe asset funamenta ecine uring a sort perio is sma, te optimists are abe to raise a arge everage at te risk-free rate espite te creitors not so optimistic beief. Tus, sort-term ebt provies a powerfu everage too to fue te optimists specuative incentives. Te ownsie of sort-term borrowing is tat if te asset funamenta ecines uring te initia perio, te borrower wi ave to promise a iger ebt payment to obtain re nancing or even to ose te coaterize asset in woe. If e sti os te more optimistic view about te asset funamenta, is greater promise (or te asset if forfeite) is uner-vaue by te creitor. Suc uner-vauation represents te so-cae roover risk, wic as been recognize as a key trigger of sort-term ebt crises. 3 On te oter an, by ocking in te nancing for a onger perio, ong-term ebt can act as a eging evice against roover risk uring ownturns. Tus, te optimists initia specuative incentive 2 Non-contigent ebt contract is sown to be optima in te costy state veri cation moe of Townsen (979), te monitoring moe of Diamon (984), an te contingent future nancing moe of Boton an Scarfstein (99). In tese moes, te unobservabiity of cas ows is important for te ebt contract to be optima. 3 See Acarya, Gae, an Yorumazer (29) an He an Xiong (29a, 29b). In contrast to tese moes, te uner-vauation (or te so-cae resae iscount) in our moe is enogenousy etermine by te eterogeneous beiefs between te borrowers an creitors. 2

4 an te subsequent roover risk jointy etermine weter sort-term or ong-term ebt is esirabe. Our moe igigts te istinctive roes of beief ispersion at i erent times. A iger initia beief ispersion about te asset funamenta over te rst perio creates a greater specuative incentive for te optimists an tus makes sort-term ebt more esirabe, wie a iger beief ispersion on te interim ate after te funamenta eteriorates increases te optimists roover risk an iscourages te use of sort-term ebt. Te traeo between te initia an future beief ispersion impies tat in a ynamic environment wit timevarying beiefs, te intuitive argument mae by Geanakopos (29) tat optimists aways prefer to max out risk-free sort-term borrowing can be sarpene. Our moe ienti es te conitions for suc intuition to o an not to o. Our moe suggests tat te emergence of a sort-term creit boom re ects excessive eterogeneous beiefs between agents. To te extent tat sort-term ebt reuces te borrowers nancia stabiity, our moe suggests tat te increasing sort-term everages uring a sort-term creit boom re ects te borrowers optimism about te future asset funamenta, as we as te creitors concerns about te borrowers abiity to repay in te ong term. Furtermore, as te roover risk increases wit te beief ispersion between te sort-term borrowers an te creitors in te future own state, te eavy use of sort-term ebt invoves not ony a arge initia beief ispersion, but aso te borrowers expectation tat beiefs wi soon converge, or, at east, not to iverge furter in te future own state. Our moe aso sows tat if cosen by te optimists, sort-term ebt can fue an assetmarket boom by aowing tem to raise arge everages at ow interest rates. It can aso act as te brige from boom to crisis as te roover risk ampi es te ownturn after te asset funamenta eteriorates. Troug te nancing-cost canne, our moe sows tat te common perception tat in te presence of sort-saes constraints iger beief ispersion eas to iger asset prices may not aways o. Tis is because te optimists nancing becomes more costy as teir beiefs iverge furter from te creitors. In particuar, our moe igigts te istinctive e ects of initia an future beief ispersion. To te extent tat te optimists can use sortterm ebt to mitigate te increase nancing cost cause by an increase in te initia beief ispersion, te equiibrium asset price remains increasing wit te initia beief ispersion uner a reasonabe conition in our moe. However, an increase of beief ispersion in te 3

5 future own state raises te roover risk an tus iscourages te use of sort-term ebt. Tis e ect exacerbates te nancing-cost e ect an causes te equiibrium asset price to ecrease wit te beief ispersion in te future own state. Te existing empirica stuies of asset-price e ects of eterogeneous beiefs, e.g., Cen, Hong, an Stein (22) an Dieter, Maoy, an Scerbina (22), commony treat investors beief ispersion in i erent time orizons as quaitativey simiar an focus on tat in sort term. In contrast, our moe suggests istinctive e ects between sort-term an ong-term beief ispersion. Te empasis of our moe i ers from tose focusing on te tigtening of creit uring crises (e.g., Brunnermeier an Peersen (29)) an tose on te sortening of ebt maturity uring crises (e.g., He an Xiong (29a) an Brunnermeier an Oemke (29)). Instea, our moe ienti es te environment tat fosters sort-term creit booms, wic ten to initiay fue asset bubbes an ater trigger nancia crises. Our moe is reate to te iterature tat stuies te pervasive use of sort-term ebt by banks an nancia rms. Te existing iterature as empasize severa avantages of sortterm ebt. First, sort-term ebt is a natura soution to a variety of agency probems insie a rm, e.g., Caomiris an Kan (99) an Diamon an Rajan (2). By coosing sortterm nancing, creitors keep te option to pu out if tey iscover tat rm managers are pursuing vaue-estroying projects. Secon, te sort commitment perio aso makes sortterm ebt ess information sensitive an tus ess expose to averse-seection probems, e.g., Gorton an Pennacci (99). Wie tese teories impy tat rms reguary use certain amounts of sort-term ebt, tey o not expain te increasing use of sort-term ebt uring asset-market booms. Our moe compements Garmaise (2), wo stuies te security-esign probem of a cas-constraine rm facing investors wit eterogeneous beiefs. His moe contrasts te optima security esign uner eterogeneous beiefs to tat uner rationa expectations, wie our moe focuses on te roe paye by ebt structure in fueing asset-market specuation riven by eterogeneous beiefs. Lanier an Tesmar (28) erives a moe in wic optimism eas entrepreneurs to use sort-term ebt. Teir moe assumes tat te entrepreneurs assign zero probabiity to te future own state, in wic sort-term ebt wi force tem give up te contro of teir rms. As a resut, te moe oes not consier roover risk an te traeo between te borrowers specuative incentives an roover risk, wic is te focus of our moe. 4

6 Tis traeo is anaogous to tat consiere by Diamon (99), wo anayzes ebt maturity coice for borrowers wit private information about teir future creit rating. In is moe, borrwers face a traeo between a preference for sort maturity ue to expecting teir creit rating to improve, against iquiity risk ue to te oss of teir private rents tat cannot be assigne to te creitors. Our moe ties bot sies of te traeo te borrowers specuative incentives an roover risk to te eterogeneous beiefs between te borrowers an creitors, wic in turn aows us to ink te borrowers nancing coices to te asset market equiibrium. Te paper is organize as foows. Section 2 presents a baseine moe wit two groups of agents oing exogenousy speci e beiefs. Section 3 extens te baseine moe wit earning an tree groups of agents. We iscuss te impications of te moe in Section 4, an concue in Section 5. A tecnica proofs are provie in te Appenix. 2 Te Moe 2. Asset an Agents Consier a moe wit tree ates an two perios. Te ate is inexe by t = ; ; 2: Tere is a ong-term risky asset, wic we interpret eiter as a ouse or a mortgage backe security. Te asset pays a na payo on ate 2. Te na payo is etermine by te na reaization of a pubicy observabe binomia tree. Figure iustrates te tree. Te tree can go eiter up or own from t = to t = an from t = to t = 2: Te tree as four possibe pats, wic we enote by uu, u, u, an (ere, u stans for up an stans for own ), an tree possibe na noes (pats u an u ea to te same na noe). We normaize te na payo of te risky asset at te en of pat uu as ; at te en of pats u an u as ; an at te en of pats as 2 ; were 2 (; ) : We enote te asset payo by e 2 ; ; 2 : Te probabiity of te tree going up in eac perio is unobservabe. Suppose tat tere are two groups of risk-neutra agents, wo i er in teir beiefs about tese probabiities. In tis section, we exogenousy specify two sets of beiefs for te agents. Utimatey, te i erence in te agents beiefs is riven by teir prior beiefs an earning processes, an we wi exten te moe wit earning in Section 3. Tere are tree intermeiate noes on te tree, one on ate an two on ate (u an epening on weter te tree goes up or own in te rst perio). We coect tese 5

7 Figure : Timeine. intermeiate noes in te foowing set: f; u; g : At eac of te noes, eac agent as a beief about te probabiity of te tree going up in te foowing perio. We coect eac agent s beiefs in te foowing set: f i ; i u; i g ; were i 2 f; g inicates te agent s type. Trougout tis section, we assume tat te -type agents are aways more optimistic tan te -type agents across a te intermeiate noes (ere, te superscript an stans for ig an ow.) Tat is, n > n for any n 2 f; u; g : Base on te reative orer, we ca te -type agents optimists an te -type pessimists. In particuar, we empasize tat te beief ispersion between te optimists an pessimists is not constant. Staning at t = ; te i erence between an represents te initia beief ispersion between te two groups about te asset funamenta from ate to, wie te i erence between an represents te future beief ispersion about te asset funamenta from te ate- state to ate 2: As we wi sow ater, tese two types of beief ispersion pay istinctive roes in etermining te optima ebt maturity coice. We summarize te na asset payo s at te en of te four possibe tree pats an te optimists an pessimists beief about eac of te pats in Tabe. Note tat te optimists assign a iger probabiity to pat uu an a ower probabiity to pat. But is beiefs about te mie pats u an u can be iger or ower tan tose of te pessimists, wic we wi speci cay iscuss ater. We normaize te tota suppy of te asset to be one unit. Tere are 2 (; ) units of 6

8 Tabe : Asset Payo an Agent s Beief across Di erent Pats Tree Pats uu u u Asset payo 2 Optimists beief u u Pessimists beief u u optimists, wo are omogeneous. On ate ; eac optimist is initiay enowe wit unit of te risky asset an c oars of cas. Given te optimists optimism, it is natura for tem to purcase te rest of te asset ( unit) from te pessimists. Foowing Mier (977), Harrison an Kreps (978), Morris (996), Cen, Hong, an Stein (22), an Sceinkman an Xiong (23), we assume tat sort-saes of te asset are not aowe. As a resut, te pessimists cannot specuate on te asset price faing in te future an wi sit on te sieine. Te focus of our anaysis is on te nancing of te optimists asset purcases. Since tey may not ave su cient cas, tey may nee to borrow from te pessimists wo sit on te sieine wit cas. As te pessimists beiefs a ect te cost of nancing to te optimists, teir beiefs can inirecty a ect te equiibrium asset price.for simpicity, we assume tat bot te risk-free interest rate an te agents scount rate are zero, an tat te pessimists on te sieine wi aways ave su cient cas. Terefore, in equiibrium tey aways eman zero expecte return in nancing te optimists. 2.2 Coateriize Debt Financing Like Geanakopos (29), we assume tat te optimists use teir asset oings as coatera to obtain ebt nancing. We focus on non-contingent ebt contracts. A non-contingent ebt contract speci es a constant ebt payment (face vaue) at maturity uness te borrower efauts. Non-contingent ebt contracts are wiey use in practice. Townsen (979) expains its popuarity base on te cost of verifying te state of te wor. Tat is, non-contingent ebt contracts circumvent te cost of verifying te vaue of te coatera as ong as te borrower makes te promise payment. Diamon (984) an Boton an Scarfstein (99) aso erive te optimaity of non-contingent ebt base on unobservabiity of cas ows. In 7

9 Tabe 2: Asset Payo an Debt Payment across Di erent Pats Tree Pat uu u u Asset payo 2 Long-term ebt face vaue F L 2 2 ; F L F L F L 2 Sort-term ebt face vaue F S 2 2 ; K F S F S F S; F S 2 Sort-term ebt face vaue F S 2 [K ; ] F S F S 2 tis moe, we wi restrict te optimists to use ony non-contingent ebt. We wi rst iscuss ong-term ebt contracts, an ten sort-term ones. We wi restrict our attention to contracts wit face vaues in 2 ; : We wi sow in Lemma 2 in Section tat tis is witout oss of generaity in te equiibrium Long-term Debt Consier a ong-term ebt contract, wic is coateraize by one unit of te asset. Te contract matures on ate 2 an as a face vaue of F L 2 2 ;. Te ebt payment is ed L (F L ) = min F L ; e : Depening on te four possibe pats of te tree, te asset payo an ebt payment are iste in Tabe 2. Given te ebt payment, a pessimistic creitor is wiing to provie te foowing creit on ate : ffff i C L (F L ) = E edl = FL + 2 ; () were E i n [] enotes te conitiona expectation of a type-i agent on noe n 2 f; u; g : On te oter an, from te optimistic borrower s perspective, te expecte cost of using tis ebt contract is E i edl = FL + 2 : (2) Te i erence between () an (2) igigts a key feature of our moe te borrower an creitor use i erent probabiities in assessing te cost an vaue of a ebt contract. 8

10 In particuar, as te borrower is optimistic an assigns a iger probabiity to pat uu, te promise payment F L at te en of tis pat is more costy to te borrower tan vaue by te creitor. Tus, te rst-best aocation of asset payo s between te borrower an creitor wou be to assign a of te asset payo at te en of pat uu to te borrower. However, suc a non-monotonic aocation is infeasibe uner stanar non-contingent ebt contracts, wic stipuates monotonic payo s. Interestingy, as we wi sow next, te stanar non-contingent sort-term ebt troug re nancing can generate non-monotone ebt payments, wic is te main avantage of sort-term ebt over ong-term ebt Sort-term Debt We now consier a sort-term ebt contract coateraize by one unit of asset, an wit a promise payment F S 2 2 ; ue on ate. Di erent from ong-term ebt, sort-term ebt requires re nancing (or roover) at ate : As we wi prove in Propositions 5 an 6, no optimist saves cas in te equiibrium. Tis means tat it is impossibe for any optimist to se is asset oing to anoter optimist on ate. As a resut, we ony nee to consier te optimist s re nancing on ate if e uses sort-term ebt nancing. Moreover, if te optimist fais to ro over is ebt, e as to forfeit te asset to te creitor wo vaues it at a ower vaue. A key insigt of our moe is tat state-contingent re nancing of sort-term ebt makes it possibe for te borrower to reuce ebt payment at te en of pat uu by traing up payments at te en of some ower pats. Speci cay, in te upper interim state u; te borrower can aways get a new contract wit te same face vaue F S, because te asset payo is aways su cient to pay o te ebt regaress of te subsequent na state being or. However, in te ower interim state, te borrower wi get a worse term an may even ose te asset if te initiay promise payment is too arge. Speci cay, te face vaue of te new contract F S; nees to ensure tat te pessimistic ener s vauation of te new ebt contract is su cient for o setting te initiay promise payment F S : min F S; ; e i = F S. E Since te igest possibe ate-2 payment te borrower can promise is, te maximum amount of creit te borrower can raise in tis state is: K E min ; e i i = E e = + 2 < : (3) 9

11 Tis impies tat te borrower wi fai to re nance is sort-term ebt in te interim state if te initiay promise ate- ebt payment F S is iger tan K : Terefore, we ave te foowing two cases to consier:. If F S 2 2 ; K ; te initia sort-term ebt is riskess an te borrower can obtain a creit of C S (F S ) = F S on ate. In te ower interim state, te borrower can ro into a new sort-term ebt contract wit face vaue F S; : F S; = F S 2 F S : (4) Te borrower as to promise more as te vaue of te coatera as eteriorate. 2. If F S 2 (K ; ] ; te initia ebt contract is risky. In te ower interim state ; te payment ue excees te maximum amount of ebt te borrower can re nance from any pessimistic creitor using te asset as te coatera. Te borrower tus efauts an oses te asset to te creitor. Tis outcome is equivaent to te na ebt payment at te en of pats u an being an 2 ; respectivey. By using tis ebt contract, on ate te borrower can get a creit of: C S (F S ) = F S if F S 2 (K ; ] : Tabe 2 summarizes te na ebt payments across te four possibe tree pats for te two cases iscusse above. In bot cases, te state-contingent re nancing makes te na ebt payment non-monotonic wit respect to te na asset payo, i.e., te ebt payment at te en of pats uu an u is ower tan at te en of pat u. Tis rearrangement of ebt payment is potentiay vauabe to te borrower as e assigns a iger probabiity to pat uu tan te creitor. Note tat in orer to keep te ebt risk-free, te igest face vaue of ong-term ebt is 2 wie te igest face vaue of sort-term ebt is K > 2 : Tis means tat te borrower can obtain a iger everage from using sort-term ebt witout increasing te initia-perio borrowing cost. Of course, if te asset funamenta eteriorates to state on ate, te borrower as to re nance at a iger interest rate for te foowing perio because te ebt is now risky.

12 2.3 Te Optima Debt Contract To stuy te optima ebt contract use by an optimistic buyer, we take te asset price p as given. In igt of Tabe 2, we enote a ebt contract (eiter ong-term or sort-term) by a set of state-contingent ebt payment D. e i Furtermore, we enote C ed E ed as te ate- creit tat a borrower can obtain from a pessimist by using te ebt contract e D. Wat is te maximum unit of asset tat an optimist can a or on ate by using te ebt contract e D? He is initiay enowe wit c oars of cas an unit of te asset. Suppose tat e purcases aitiona x i units in te market. His tota purcasing power is c + ( + x i ) C ed ; te sum of is cas enowment an te creit e can raise by using is asset oing ( + x i units in tota) as coatera. Te buget contraint impies tat c + ( + x i ) C ed = x i p ) x i = c + C ed : (5) C ed An impicit assumption in tis cacuation is tat te optimist maxes out is purcasing power, a conjecture tat we wi verify in Propositions 5 an 6. For eac unit of asset, te optimists ate- expectation of te ate-2 cas ow after netting out te ebt payment i is E e e D : Terefore, te optimist s ate- vaue from using te contract D e (i.e., te expectation of te na weat) is V ed = ( + x i ) E e i D e = p c + p C ed p E e Tis expression iustrates te traeo in te optimist s ebt coice. E ed i : (6) On one an, by promising a coaterize ebt payment D e on eac unit of asset oing, te buyer can raise a creit of C ed c+p an tus estabis a arger initia position p C( D) e ; wic is te rst part in V ed : Tis term represents a everage e ect. On te oter an, te ebt payment reuces te asset payo to te buyer on ate 2. secon part E e ed in V ed. E Tis ebt-cost e ect is re ecte in te Te ebt contract contains two imensions: ebt maturity (ong-term or sort-term) an promise payment (i.e., te ebt face vaue). Bot are etermine by te traeo between te everage e ect an ebt-cost e ect. We wi rst anayze te agent s maturity coice, an ten te face-vaue coice.

13 2.3. Maturity Coice To erive te optima ebt maturity, we consier te foowing question: in orer to raise te same amount of creit at t = ; wic contract (i.e., ong-term or sort-term) entais te ower expecte cost? Equation (6) impies tat te one wit te ower cost ominates te oter. Te foowing key proposition sows tat te optima maturity coice is etermine by te initia an future beief ispersion between te optimists an pessimists. Proposition Consier two ebt contracts, one sort-term an te oter ong-term. Suppose tat bot contracts ave a face vaue in 2 ; an give te same ate- creit to an optimistic borrower. Ten, from te borrower s perspective on ate, te sort-term contract requires a (weaky) ower expecte cost if an ony if > : (7) Proposition sows tat weter te sort-term ebt contract ominates te ong-term ebt contract epens on te initia beief ratio between te optimists an pessimists ( = about te rst-perio funamenta), an te future beief ratio in te ower interim state ( = about te secon-perio funamenta.) Te sort-term contract is ominant if te initia beief ratio is su cienty arge, or if te future beief ratio is su cienty sma. To unerstan te intuition, a ebt contract not ony cannes te necessary nancing from a creitor to an optimistic borrower to purcase te asset, but aso represents an aocation of te asset payo s between te borrower an creitor. As we ave iscusse earier, a ong-term contract speci es a monotonic ebt payment wit respsect to te asset payo, wie a sort-term contract aows te borrower to reuce te ebt payments at te en of pats uu an u by traing up te payment at te en of pat u (te payment at te en of pat is maxe out.) Is tis traeo wortwie? It rst epens on te initia beief ratio =. If tis ratio becomes iger, te reuce future ebt payment after te upper interim state u becomes more vauabe to te borrower an te increase payment after te ower interim state becomes ess important. Conversey, te creitor ns te reuce payment after u ess important, wie te increase payment after more vauabe. As a resut, te sort-term ebt contract becomes more esirabe to bot of te borrower an creitor. Tis e ect re ects te two parties specuative incentives riven by teir initia beief ispersion. 2

14 Tere is aso te so-cae roover-risk e ect working against sort-term ebt. In te ower interim state, te borrower sti os te more optimistic beief going forwar. Tis means tat te increase future ebt payment is uner-vaue by te creitor. In oter wors, te borrower gives up some future asset payo s at a price ower tan is own vauation. In te extreme case, if e cannot obtain su cient re nancing to repay is maturing ebt obigation, e as to give up te asset in woe to te creitor wo oes not vaue te asset as muc as e oes. Tis uner-vauation of te increase ebt payment is etermine by te beief ratio between te borrower an creitor = in te interim state after ajusting for te probabiity tat tis state is reaize =. Wen te rigt-an sie of conition 7 becomes arge, te roover-risk e ect becomes severe an tus makes it more costy for te borrower to use sort-term ebt. Te traeo between te specuative-incentive e ect an roover-risk e ect impies tat sort-term ebt is not aways preferabe. Tis resut contrasts an intuitive argument mae by Geanakopos (29) tat te optimists aways coose to nance teir asset purcases using te most aggresive risk-free sort-term ebt contract (i.e., te contract wit a face vaue of K in our moe.) Tis argument ignores te possibe roover risk an oes not o if te future beief ispersion in state is su cienty arge Optima Debt Face Vaue To erive te optima ebt face vaue, we rst caracterize its feasibe range: i i Lemma 2 If E e p E e ; ten te optima ebt face vaue is insie 2 ;. Lemma 2 sows tat te optima ebt face vaue (for eiter ong-term or sort-term contract) ies insie te interva 2 ;. Te conition for tis resut te ate- asset price ies between te pessimists an optimists asset vauations is innocuous because it aways os in te equiibrium trougout tis section. Wen we exten te moe to incorporate earning in te next section an aow agents beiefs to ip on ate, te equiibrium asset price cou be iger tan te optimists asset vauation because of te asset owner s resae option. However, te feasibe range of te optima ebt face vaue erive in Lemma 2 sti os after we moify te conition to account for te resae option. Te intuition of Lemma 2 is as foows. If te optima ebt face vaue is ower tan 2 ; ten te ebt is risk free. Tus, increasing te face vaue by a sma amount oes not cange te risk of te ebt, an tus aows te borrower to increase is initia nancing by 3

15 at a cost exacty equa to. Since te asset price is ower tan is asset vauation, te increase creit aows im to take a arger asset position an terefore be better o. Tis sows tat te optima ebt face vaue cannot be smaer tan 2 : On te oter an, if te optima ebt face vaue is iger tan ; te borrower aways efauts on te ebt except at te en of te pat uu. Tis impies tat reucing te face vaue by a sma amount aows te borrower to save ebt payment at te en of pat uu, wic e vaues more tan te creitor. Of course, tis aso cuts own is initia asset position. In te proof provie in Appenix A.2, we sow tat as ong as te asset price is iger tan te pessimistic creitor s asset vauation, te borrower is better o by reucing te ebt face vaue. Tus, te optima ebt face vaue cannot be iger tan eiter. Te foowing proposition provies te borrower s optima ong-term ebt face vaue conitiona on ong-term ebt being more esirabe. Proposition 3 Suppose tat te conition in (7) oes not o. Tus, it is optima for te borrower to use ong-term ebt. De ne P M E e juu; u; u i + E e j i : Ten, te borrower s optima ebt face vaue is if p < P M ; is eiter or 2 if p = P M ; or is 2 if p > P M. Te iscrete asset payo impies tat varying te ong-term ebt face vaue F L between 2 an oes not cange te risk of te ebt. In oter wors, regaress of te vaue of F L in tis region, te borrower wi aways make te promise ebt payment F L at te en of pats uu, u, an u, an efaut an tus give up a te asset payo at te en of. Since te borrower is risk-neutra, e wi use te igest face vaue to maximize is position if te asset price is beow a critica eve P M, wic weigs is asset vauation an cost of nancing. More precisey, P M is a weigte average of te borrower s asset vauation in te upper (non-efaut) states fuu; u; ug an te creitor s vauation in te ower (efaut) state. If we interpret te ong-term ebt contract as a static contract tat spans two perios, ten Proposition 3 is anaogous to te resut of Simsek (29). If te borrower uses sort-term ebt, te efaut risk of te contract epens on weter te ebt face vaue is iger or ower tan K ; were K is te asset s maximum ebt capacity in te ower interim state : If te face vaue is between 2 an K, te borrower is aways 4

16 abe to re nance an te initia ebt contract is risk free, even toug te foow-up contract in te interim state is risky as te borrower wi efaut on ate 2 at te en of pat. If te face vaue is between K an, te borrower cannot get a new ebt contract to pay o te initia ebt in te state, an tus efaut on te ebt. In te same spirit to Proposition 3, te next proposition sows tat te borrower wi coose to use te igest face vaue insie te two regions 2 ; K an [K ; ] if te asset price is beow two tresos P H an P L, respectivey. Tese tresos re ect te borrower s asset vauation an te cost of using ebt in tese regions. Proposition 4 Suppose tat te conition in (7) os. Tus, it is optima for te borrower to use sort-term ebt. De ne P H + i E e + juu; u; u + i E e + j an wic satisfy P L E e juu; u i i + E e ju; ; P L < P M < P H : Ten, te borrower s optima sort-term ebt face vaue is if p < P L ; is eiter or K if p = P L ; is K if P L < p < P H ; is eiter K or 2 if p = P H ; or is 2 if p > P H : Te core of Propositions 3 an 4 is tat wen te asset price becomes ceaper reative to te buyer s own vauation (after ajusting for te nancing cost), e wi eman a greater position. To nance te greater position, e uses a iger ebt face vaue to obtain more creit. In te next subsection, we wi use tese two propositions to erive te joint equiibrium of te asset an creit markets. 2.4 Te Equiibrium of Asset an Creit Markets We now erive te equiibrium on ates an. On ate, te amount of asset purcase by an iniviua optimistic buyer using a ebt contract D e is given by Equation (5). Propositions, 3, an 4 jointy etermine te optima contract D e (p ) base on te buyer s an creitor s eterogeneous beiefs an te asset price 5

17 p. Te tota measure of buyers in te economy is. Teir aggregate purcase P i x i sou equa to te tota asset enowe by te (pessimistic) seers : X x i = : (8) i If a te buyers use te same ebt contract e D (p ) to nance teir purcases, te market cearing conition impies tat c + C ed (p ) = p C ed (p ) C ed (p ) = ( ) p c. (9) Tis equation iustrates te intricate interaction between te asset price an te enogenousy etermine amount of creit to te asset buyers. In igt of Propositions 3 an 4, te buyers optima creit eman te term C ed (p ) on te eft an sie ecreases wit te asset price p. On te oter an, te creit avaiabe to te buyers nees to be su cient to support teir asset purcases (market cearing conition), i.e., in equiibrium te buyers aggregate cas sortfas sou equate teir creit eman. Te ineary increasing function on te rigt an sie gives te buyers cas sortfa te vaue of teir purcases ( sares mutipie by te price p ) minus teir cas enowments c Long-term Debt Equiibrium Figure 2 pots te two sies of (9) wen te conition in (7) fais an borrowers prefer to use ong-term ebt. As erive in Proposition 3, eac buyer s optima creit eman can take two possibe vaues, C L 2 or C L () : Te two-piece orizonta ine wit a ownwar jump at P M represents te creit eman C ed (p ), i.e., te eft an sie of (9). Te upwar soping curve represents te necessary creit neee to cear te asset market, te rigt an sie of (9). As te intercept of te asset-market cearing conition ( c) increases, we encounter tree possibe cases in equiibrium. First, if eac buyer s cas enowment c is ig, te optima creit eman curve an te asset-market cearing conition intersect at a point, were te equiibrium asset price p is iger tan P M an eac buyer emans a moest amount of creit C L 2. We abe tis case by case LD. In tis case, te buyers ampe 6

18 Figure 2: Te equiibrium wit ong-term ebt nancing. cas enowments aow tem to bi up te asset price to a ig eve witout using muc creit. Secon, if eac buyer s cas enowment c is ow, te two curves intersect at a point, were te equiibrium price p is ower tan P M an eac buyer emans a arge amount of creit C L (). We abe tis case by case LD3. In tis case, te buyers imite cas enowments constrain te price from rising ig even toug eac buyer uses an aggresive ebt contract. Te case LD2 occurs wen te upwar soping asset-market cearing conition passes te mie of te two orizonta eves of te buyers creit eman curve. In tis case, te equiibrium price is exacty P M an eac buyer is ini erent between using ong-term ebt contracts wit face vaues an 2 (Proposition 3). Ten, te asset market cearing conition (8) is fu e by ning a certain mix of buyers using tese two contracts. Denote by te fraction of buyers using te contract wit face vaue : Conition (8) is equivaent to wic impies c + C L () P M C L () + ( ) c + C L 2 P M C L 2 = ; = c+c L () P M C L () c+c L( 2 ) P M C L( 2 ) c+c L( 2 ) P M C L( 2 ) : () Te equiibrium on ate epens on te optimistic asset oers nancia conitions in te two possibe states: u an : In te upper state u; eac optimist as gaine on is 7

19 initia position an is now in a strong nancia conition. As a resut, eac optimist is free to buy or se some sare of te asset at a price equa to is own vauation: i p u = E e u = u + u : () In te ower state ; eac optimist su ers a oss from is initia position. In te LD case, because of te moest everage use by eac optimist from te ong-term ebt contract wit face vaue 2, is portfoio is sti sovent. At te margina, e can sti a or to buy a sma unit of asset at is own vauation i p = E e = + 2 : (2) if it is o ere in te market. He is aso wiing to se te asset at te same price. In te LD2 an LD3 cases, at east some of te optimists ave use te aggressive ong-term contract wit face vaue an teir portfoios are e ectivey uner water in state (i.e., regaress of weter te na state is or 2, te asset payo a goes to te creitor.) Teir nancia istress eas to a market freeze. On one an, te asset oers ave no incentive to se teir assets at any price beow because tey cannot get any bene t from suc a trae. Tis is exacty te so-cae ebt overang probem coine by Myers (977). Tis probem is aso wiey recognize as an important issue in te unerstan te ry up of asset market iquiity uring te recent creit crisis, e.g., Diamon an Rajan (29). On te oter an, ony pessimists ave cas an tey ony vaue te asset at a vaue of i e = + 2 < : Tis gap between te ower ask an igest bi prices impies E tat tere cannot be any trae in te equiibrium. For te sake of our iustration ater, we wi use a price p = (3) in tese scenarios, even toug tis is not a market price. Base on te market equiibrium iscusse above, it is irect to see tat tere is no incentive for any optimist to save cas on ate. Regaress of te ebt contracts use by oter optimists on ate an te state reaize on ate, saving cas on ate oes not ea to any possibe gain on ate. In te foowing proposition, we summarize te iscussion on te joint equiibrium of te asset an creit markets in wic te buyers ony use ong-term ebt contracts.. 8

20 Figure 3: Te equiibrium wit sort-term ebt nancing. Proposition 5 Suppose tat te conition in (7) fais an te buyers use ong-term ebt contracts to nance teir asset purcases. Ten, tere is no incentive for any buyer to save cas an te equiibrium can be broken own into te foowing tree cases: -LD: If C L 2 > ( ) P M c, ten p = c+c L( 2 ) ; a te buyers use te same ongterm ebt contract wit face vaue 2 on ate, an te asset price in te two states on ate are given by equations () an (2); -LD2: If C L 2 ( ) P M c C L (), ten p = P M, eac buyer is ini erent between te ong-term ebt contracts wit face vaues of an 2 on ate wit te fraction of buyers using te former contract given in (), an te asset price in te two states on ate are given by equations () an (3); -LD3: If C L () < ( ) P M c, ten p = c+c L(), a te buyers use te same ong-term ebt contract wit face vaue on ate, an te asset price in te two states on ate are given by equations () an (3) Sort-term Debt Equiibrium If te conition in (7) os, te buyers wou prefer sort-term ebt. Figure 3 sows ve possibe cases for te equiibrium. Proposition 6 ists tese cases. Te ogic for tese cases is simiar to tat for Proposition 5. It is wort mention tat on ate if any optimist runs into 9

21 nancia istress in te ower state from using te aggressive sort-term ebt contracts wit face vaue of eiter or K ; e as to forefeit te asset to te creitor at a price equa to te creitor s vauation i p = E e = + 2 : (4) Furtermore, we can irecty verify tat te optimists ave no incentive to save cas on ate by comparing te margina vaues of saving cas an of estaising a greater asset position. Proposition 6 Suppose tat te conition in (7) os an te buyers use sort-term ebt contracts to nance teir asset purcases. Ten, tere is no incentive for any buyer to save cas an te equiibrium can be broken own into te foowing ve cases: -SD: If C S 2 > ( ) P H c; ten p = c+c S( 2 ), a te buyers use te same sortterm ebt contract wit face vaue 2 on ate, an te asset price in te two states on ate are given by equations () an (2); -SD2: If C S 2 ( ) P H c C S (K ) ; ten p = P H, eac buyer is ini erent between te sort-term ebt contracts wit face vaues of K an 2 on ate wit te fraction of buyers using te former contract as c+c S (K ) P H C S (K ) c+c S( 2 ) P H C S ( 2 ) c+c S ( 2 ) P H C S ( 2 ) in te two states on ate are given by equations () an (4); ; an te asset price -SD3: If ( ) P L c < C S (K ) < ( ) P H c; ten p = c+c S(K ), a te buyers use te same sort-term ebt contract wit face vaue K on ate, an te asset price in te two states on ate are given by equations () an (4); -SD4: If C S (K ) ( ) P L c C S () ; ten p = P L, eac buyer is ini erent between te sort-term ebt contracts wit face vaues of an K on ate wit te fraction of buyers using te former contract as c+c S () P L C S () c+c S (K ) P L C S (K ) c+c S (K ) P L C S (K ) two states on ate are given by equations () an (4);, an te asset price in te -SD5: If C S () < ( ) P L c; ten p = c+c S(), a te buyers use te same sort-term ebt contract wit face vaue on ate, an te asset price in te two states on ate are given by equations () an (4). 2

22 2.5 Heterogeneous Beiefs an Asset Price Dynamics In tis subsection, we anayze te e ects of agents eterogeneous beiefs in riving te boom-an-bust cyce of asset prices. In particuar, we focus on te i erent e ects of initia an future beief ispersion on te asset price ynamics troug teir i erent e ects on te optimists nancing coices. Te stanar resut of Mier (977) suggests tat a iger beief ispersion (in eiter te rst or secon perio) eas to iger asset prices in te presence of sort-saes constraints. However, nancing cost can aways signi cant eviation from tis resut. As we observe uring te perio aroun te recent creit crisis, te ebt contracts use by nancia institutions to nance teir investments in mortgage backe securities are mosty safe from te creitors perspectives. Tese contracts incue repo transactions an asset backe commercia paper. To tie our anaysis cose to tis observation, we focus on te region were if te optimists can ony use ong-term ebt, tey wi coose te risk-free ebt wit face vaue 2 (te LD case in Proposition 5). To faciitate te anaysis, we examine te equiibrium asset price ynamics wen ) ony LD (ong-term ebt) is avaiabe, an 2) bot LD an SD (ong-term an sort-term ebt) are avaiabe. Te i erence between tese two cases igigts te roe of ebt maturity. We use te foowing baseine parameter vaues: = :3; c = :6; = :4; = :7; = :3; u = :6; u = :4; = :6; = :4: (5) Tese numbers impy te foowing: Optimists consist of 3% of te popuation an eac is enowe wit :5 oar in cas. Te na asset payo can be ; :4, or :6: We et te objective probabiity of te tree going up eac perio be :5 an te optimists an pessimists beeifs be equay sprea aroun te objective probabiity. As earning is ikey to cause beief ispersion to ecrease over time, we make te beiefs of te optimists an pessimists on ate to be :7 an :3; an on ate in bot of te u an states to be :6 an :4: 2.5. Initia Beief Dispersion on Date We rst examine te e ect of te initia beief ispersion on ate. We et te vaues of an to eviate from teir baseine vaues an instea take te foowing ones: = :5 + an = :5 2

23 p Fraction of Optimists Fraction of Optimists Pane A: Date Equiibrium Price Bot SD an LD LD ony Mier case Obj price Pane B: Sort term Contracts Use in Equiibrium θ 2 contract K contract θ contract Sort run beief ispersion Sort run beief ispersion p p Pane C: Date Price Drop after Ba Sock Bot SD an LD LD ony Mier case Sort run beief ispersion Pane D: Long term Contracts Use in LD Ony Equiibrium θ 2 contract θ contract Sort run beief ispersion Figure 4: Te equiibrium e ects of initia beief ispersion on asset an creit markets. were canges from to :45 an rives te initia beief ispersion between te optimists an pessimists. Figure 4 iustrates te asset market an creit market equiibrium. Pane A pots te ate- asset price p wit respect to. Te orizonta otte ine at te :49 eve represents te asset s funamenta vaue by te objective probabiities. Te otte upwar soping ine represents te asset price in te Mier setting, were optimists aways ave su cient funs to execute teir purcases. As increases from to :45, te optimists become more optimistic an p increases from :548 to :74 (p is iger tan :49 at = because of te beief ispersion on ate ). LD-ony equiibrium Te orizonta ase ine at :485 pots p wen te optimists ave access ony to ong-term ebt to nance teir asset purcases. Interestingy, espite te wie range of beief ispersion between te optimists an pessimists, te equiibrium asset price is inepenent of te beief ispersion an is sigty beow te asset s funamenta vaue sown by te orizonta otte ine. Bot of tese caracteristics re ect te important e ects of optimists nancing cost. Pane D provies a breakown of te ong-term ebt 22

24 contracts use by te optimists in te LD ony equiibrium. Trougout te wie range of vaue, te optimists aways use te same risk-free ong-term ebt contract wit face vaue 2 to nance teir asset purcases. Tis is because te aternative contract wit a iger face vaue is risky an more costy uner te speci e parameters. Te use of te risk-free ebt contract etermines te optimists purcasing power, wic can ony support a price eve ower tan te asset s funamenta vaue even toug te pessimists cannot sort se te asset. Equiibrium wit bot SD an LD avaiabe Te soi ine in Pane A pots p wen te optimists can coose between ong-term an sort-term ebt. p stays at at :485 as increases from to :5, ten monotonicay increases to :623 as increases furter to :3, an nay stays at at :623 as continues to rise. Tis pattern is ramaticay i erent from tat of te asset price in te LD ony equiibrium. Tis i erence igigts te important roes of nancing coices in a ecting te asset price ynamics. Pane B provies a breakown of te sort-term ebt contracts use by te optimists. Over te region 2 [; :5] ; te optimists o not use any sort-term ebt. Te reason is Proposition : Sort-term ebt is avantageous to ong-term ebt ony wen te specuative incentives cause by bot parties initia beief ispersion ominates te roover-risk e ect ue to teir future beief ispersion. Once rises above :5, te optimists start to use a mix of sort-term ebt contracts wit face vaues 2 an K. Te fraction of optimistis using te more aggresive K contract rises monotonicay from to as rises from :5 to :3, an stays at as rises furter. Tis pane sows tat te increase of te equiibrium price wit in te region 2 [:5; :3] is nance by te optimists increasing reiance on te sort-term ebt wit face vaue K. Taken togeter, even toug te asset price in te coatera equiibrium is ower tan tat in te stanar Mier setting, sort-term ebt aows te optimists to manage teir nancing cost more e ectivey an tus ensures tat te equiibrium price increases wit agents initia beief ispersion. Date- cras Pane C of Figure 4 pots te price cange on ate wen te ower state is reaize, i.e., p p, uner i erent settings. After te reaization of te negative sock, te asset price rops an te optimistic asset oers su er osses on teir positions. In te Mier setting, as te ate- price p monotonicay increases wit te initia beief ispersion, te price rop in state (i.e., jp p j) aso increases wit. In te LD 23

25 ony equiibrium, te optimists aways coose te same risk-free ong-term ebt contract to nance teir asset purcases. As a resut, te ate- price is inepenent of an so is te price rop in state : In contrast, te price rop in te setting wit bot LD an SD avaiabe is generay increasing wit. In fact, te sope of te price rop wit respect to in te LD-an-SD equiibrium is even steeper tan tat in te Mier setting in te mie region. Tis is because of te roover risk e ect. Wen te optimists nance teir purcases by te aggresive sort-term ebt wit face vaue K, tey are force to turn over teir asset to te pessimistic creitor after te negative sock. Tis ebt contract is sti risk-free to te creitor as te forefeite asset is just enoug to o set te ebt payment. However, te sift of te asset s margina investor from te optimists to te pessimists ampies te price impact of te negative funamenta sock. Tis e ect sows tat not ony can sort-term ebt fue te asset over-vauation on ate, but can aso exacerbate te ownturn after a negative sock Future Beief Dispersion on Date Next, we examine te e ects of te beief ispersion between te optimists an pessimists on ate. We wi focus on te ispersion in te ower state. Proposition suggests tat te beief ispersion in tis state introuces roover risk, wic iscourages optimists from using sort-term ebt. Speci cay, we eviate from te baseine parameters in (5) by specifying te foowing beiefs for te optimists an pessimists: = :5 + ; an = :5 were canges from to :45: Figure 5 iustrates te impact on te asset an creit markets. Pane A of Figure 5 pots p wit respect to. In te Mier setting, p is again increasing wit for te same reason as before an increase in makes te optimists more optimistic about te asset funamenta. In contrast, p ecreases wit in bot te equiibria wit LD an SD an wit LD ony. In te LD ony equiibrium, p ecreases wit wen is smaer tan :5, an is inepenent of is arger tan :5: Pane D furter sows tat wen is ess tan :5; te optimists use a mix of ong-term ebt contracts wit face vaues of an 2 : As increases in tis region, te fraction of optimists wo use te aggresive contract ecreases because te increase future ispersion makes tis contract more costy. Te optimists a switc to te risk-free 2 contract once rises above :5: 24

26 Fraction of Optimists Fraction of Optimists.7 Pane A: Date Equiibrium Price Pane B: Sort term Contracts Use in Equiibrium.65.8 p.6.55 Bot SD an LD LD ony Mier case Obj price.6.4 θ 2 contract K contract θ contract Long run beief ispersion Long run beief ispersion p p Pane C: Date Price Drop after Ba Sock Bot SD an LD LD ony Mier case Pane D: Long term Contracts Use in LD Ony Equiibrium.8.6 θ 2 contract θ contract Long run beief ispersion Long run beief ispersion Figure 5: Te equiibrium e ects of ong-run beief ispersion in state on te asset an creit markets. In te equiibrium wit bot LD an SD, p is iger tan tat in te LD ony equiibrium wen is ess tan :22 because in tis region te conition (7) in Proposition 7 os an sort-term ebt gives te optimists ceaper nancing tan ong-term ebt. Inee, Pane B sows tat te optimists use a mix of sort-term ebt contracts wit face vaues K an 2 : But, neverteess, p ecreases wit even in tis region. Tis is because an increases of raises te roover risk of tose optimists wo coose to use te K contract an causes teir fraction to reuce. Wen rises above :22; a te optimists switc to te 2 contract an p becomes insensitive to an ientica to tat in te LD ony equiibrium. Taken togeter, Figures 4 an 5 emonstrate substantia i erences in te equiibrium e ects of initia an future beief ispersion. Wie bot types of beief ispersion ten to increase te optimists beief an tus increase te equiibrium asset price, tey ave i erent e ects on te optimists nancing. In particuar, an increase in te future beief ispersion after te asset funamenta eteriorates increases te optimists roover risk an tus iscourages tem from using sort-term ebt, a usefu too in controing nancing 25