Financing Speculative Booms

Transcription

1 Financing Specuative Booms Ziguo He y Wei Xiong z Marc 2 Abstract Tis paper stuies te nancing of specuative asset-market booms in a stanar framework wit eterogeneous beiefs an sort-saes constraints. Casconstraine 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. Our moe ienti es istinctive e ects of initia an future beief ispersion in riving a sort-term creit boom, an sows tat it can initiay fue an asset-market boom an ten exacerbate te ownturn wen asset funamenta eteriorates. Keywors: Sort-term creit boom, Asset bubbe, Roover risk, Debt maturity PRELIMINARY. y University of Cicago, Boot Scoo of Business. Emai: ziguo.e@cicagoboot.eu. z Princeton University an NBER. Emai: wxiong@princeton.eu.

2 Introuction Te notion tat specuation eaing to bot booms an crises rests on an inerent instabiity of creit as a ong istory in economics. As summarize by Kineberger (2), a ost of cassica economists incuing Irving Fiser, Henry Simons, an Hyman Minsky empasize te roe of ebt contracte to everage te acquisition of specuative assets for future resae an te roe of ebt structures in causing nancia i cuties. Tere is aso growing evience of pronounce cyces of creit expansion an contraction tat accompany te boom-an-crisis cyces of asset markets: Wite (99) an Eicengreen an Mitcener (23) for te stock market boom an cras of 929; Arian an Sin (29), Brunnermeier (29), Gorton an Metrick (29), an Krisnamurty (2) for te recent creit crisis in 27-28; Rorik an Veasco (999) an Reinart an Rogo (29) for te emerging-market ebt crises in 99s. In particuar, tere is a saient pattern in te increasing use of sort-term creit in te booming perios of tese boom-an-crisis episoes. Motivate by te importance of creit in tese episoes, tis paper eveops a ynamic moe to anayze te inerent instabiity of creit an its roe in fueing specuative booms an in riving ebt crises. Our moe buis on te stanar asset-market framework tat combines bot 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 framework, sort-saes constraints cause te equiibrium asset prices to bias towar te beiefs of te optimists. However, existing moes ten to ignore ow optimists nance teir specuative positions. In tis regar, we foow Geanakopos (29), wo a ong avocate to incorporate te asset s coatera vaue into stanar asset-market equiibrium moes. His most recent paper provies a moe wit eterogeneous beiefs to igigt te important roe paye by everage cyces in riving asset market cyces. Our moe i ers from Geanakopos not ony o we focus on te optimists everage coice, but aso on te roe of ebt structure (e.g. ong term vs. sort term) in a ecting teir nancing cost an everage coice. Te enogenous everage an maturity coice jointy etermine te equiibrium asset price ynamics. Speci cay, our moe as two perios an a risky asset wose funamenta vaue is unobservabe an uctuates over time. 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

3 funs, tey wou acquire a te asset an bi up te asset price to teir optimistic vauation in settings were te optimists aways o te most optimistic beief (e.g., Mier (977)), or to be even iger tan teir optimistic vauation in settings were oter agents beiefs may rise above te optimists in te future (e.g., Harrison an Kreps (978)). 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. Te nancing cost irecty a ects te creit te optimists wi use an te price tey can o er for te asset. As a resut, te asset market equiibrium is jointy etermine wit te creit market equiibrium. In our moe, we restrict te optimists to stanar non-contingent ebt contracts, wic are wiey use in practice. Despite tis restriction, te optimists neverteess face a non-trivia coice of ebt structure to raise nancing from te pessimistic creitors. To tease out te nancing probem, it is usefu to consier te rst-best aocation of te asset payo s between te optimists an pessimists. Since te optimists assign iger probabiities to te upper states (i.e., states iger tan a certain treso) an ower probabiities to te ower states, te rst-best aocation is to assign te asset payo s in te upper states to te optimists, an tose in te ower states to te pessimists. However, te rst best payo aocation is infeasibe if agents ony ave access to non-state contingent ebt nancing. Te stanar non-contingent ong-term ebt, wic stipuates a monotone payo structure, is especiay ine cient in tis regar. More speci cay, te ong-term ebt contract requires te optimistic borrower to make te promise payment in fu in te upper states, wic e vaues igy, but not so igy by te creitor. Tis makes te creit rater costy to te borrower. Inee, our anaysis sows tat te nancing cost inirecty pus te equiibrium asset price towar te pessimistic creitors beiefs an can even overturn te stanar resut tat eterogeneous beiefs ea to an overvauation of assets. Tis outcome ecoes a recent paper by Simsek (29), wo aso sows tat nancing cost can severey constrain optimists from biing up asset prices in a static setting. A key insigt of our moe is tat state-contingent re nancing of sort-term ebt aows te optimists to structure state-contingent ebt payo s to reuce te nancing cost. Suppose an optimistic borrower initiay uses a sort-term ebt contract tat matures on te interim ate. If te asset funamenta improves wen te ebt matures, te borrower wi obtain a 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. 2

4 better term in re nancing an tus be abe to keep a greater fraction of te asset payo s in te subsequent upper states to imsef. Tis bene t, wic can be terme as specuative incentive, motivates te use of sort-term ebt. However, tere is aso an opposing force. If te asset funamenta eteriorates, e wi ave to promise a iger ebt payment to obtain re nancing or even to ose te coaterize asset in woe. As e sti os a 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 increasingy recognize as a key trigger of sort-term ebt crises. 2 In our moe, te optimists initia specuative incentive an te subsequent roover risk jointy etermine weter sort-term ebt is esirabe. In particuar, 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 after te funamenta eteriorates on te interim ate increases te borrower s roover risk an iscourages te use of sort-term ebt. Te traeo between te initia an future beief ispersion enrices te two-state setting consiere by Geanakopos (29), wo suggests tat te iger accumuative beief ispersion over ong-run (wic aso os in our moe) sou aways make sort-term ebt more esirabe. Our moe sows tat sort-term ebt aows te optimists to substantiay reuce teir nancing cost over a broa region were te ig cost of ong-term ebt nancing wou ave constraine teir capacity to bi up te asset prices. Our moe tus expains te syncronization of sort-term creit booms an asset price booms, a penomenon commony observe in various boom-an-crisis episoes (see Section 4 for tree exampes). Furtermore, sort-term ebt aso acts as te brige from booms to crises. As te asset funamenta eteriorates, te optimists wi face i cuties in roing over teir sort-term ebt an may even be force to turn over teir asset to te pessimistic creitors at a substantia iscount. Taken togeter, te inerent instabiity of sort-term ebt nancing initiay fues te asset market boom create by te rise of eterogeneous beiefs between optimists an pessimists, an ten exacerbates te ownturn wen te asset funamenta eteriorates. 2 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. 3

5 Te ong ebate on weter te commony observe creit expansions tat accompanie various asset-market booms were riven by suppy socks, suc as expansionary monetary poicies an externa capita in ows, or by internay generate eman for creit is sti unsette. 3 i erent episoes. Te suppy- an eman-riven factors ave probaby paye i erent roes in Our moe eps unerstaning te eman-riven creit expansions, especiay te expansions of sort-term creit uring asset-market booms. Tis saient penomenon cannot be easiy expaine by any suppy-sie teory. Te empasis of our moe aso 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)). In particuar, our moe suggests tat te concurrent sort-term creit boom an asset-price boom are a potentiay usefu preictor for future crises. Our moe aso provies insigts regaring reguating sort-term everages. 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 (29). 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. Finay, 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. Te paper is organize as foows. Section 2 presents a baseine moe wit two groups 3 Kineberger (2) provies a etaie account on i erent views about te riving force of creit expansions. See Eicengreen an Mitcener (23) for an anaysis of te argument tat excessive suppy of creit a fuee te stock market boom before te 929 market cras, an Wite (99) for te argument tat te eman for creit to buy stocks a pue funs into te market as te cost of creit a risen in sync wit te use of creit. 4

6 Figure : Timeine. 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 nay 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. 5

7 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 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 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 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 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 : 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 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 : 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. 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. 8

10 2.2.2 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 : A key insigt of our moe is tat statecontingent 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 : E min F S; ; e i = F S. 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) 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 9

11 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. 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 E ed i : (6)

12 Tis expression iustrates te traeo in te optimist s ebt coice. 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 E ed in V ed. 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 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. Moverover, Proposition os for any given ebt face vaue an is not restricte just to te face vaue use in te equiibrium. In tis sense, te insigt conveye by te proposition is more genera tan te speci c (binomia-tree) setting consiere in our paper.

13 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. re ects te two parties specuative incentives riven by teir initia beief ispersion. Tis e ect 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. Wen tis ratio becomes arge, te roover-risk e ect becomes severe an tus makes it more costy for te borrower to use sort-term ebt. Proposition re ects te traeo between te specuative-incentive e ect an te rooverrisk e ect, an ties tis traeo to te ynamics of te agents eterogeneous beiefs. A cose ook at te pat u makes tis connection crysta cear. Te state-contingent ebt re nancing aows te borrower to trae up te ebt payment at te en of pat u for ower payments at te en of uu an u: Te borrower assigns a probabiity of to te pat u, wie te creitor assigns : Suppose tat tere is ony initia beief ispersion, i.e., > an =. Ten, te borrower tinks tis pat is ess ikey tan te creitor. As a resut, teir specuative incentives make te sort-term ebt esirabe. 2

14 However, if tere is ony future beief ispersion, i.e., > an =, te borrower ns tis pat more ikey tan te creitor. As a resut, te sort-term ebt is more costy to te borrower because of te roover risk. 4 As te optimists o a more optimistic beief tan te pessimists about te asset funamenta in eac perio, te accumuative beief ispersion between tem increases wit time orizon. Geanakopos (29) argues tat tis property of beief ispersion is su cient to ensure te ominance of sort-term ebt over ong-term ebt in a setting wit ony two states. Proposition sows tat tis argument may be speci c to is moe. In contrast, in our more genera framework, te resut may reverse: sort-term ebt is more esirabe precisey wen te initia isagreement about te rst-perio funamenta is ig an future isagreement about te secon-perio funamenta is ow Optima Debt Face Vaue To erive te optima ebt face vaue, we rst caracterize its feasibe range: i Lemma 2 If E e p E e i ; 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 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 4 Conition (7) impies tat even if bot agents argree on te probabiity of te pat u, = sti pays a roe. Tis is because wen =, te borrower sti gains by saving on te pats uu an u; wic e tinks are more ikey to occur tan te ener. Tis expains te appearance of te ratio = on te LHS of (7). 3

15 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 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. 4

16 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 + + E i e juu; u; u + + E e j i 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 Date Te amount of asset purcase by an iniviua optimistic buyer using a ebt contract e D is given by Equation (5). Propositions, 3, an 4 jointy etermine te optima contract ed (p ) base on te buyer s an creitor s eterogeneous beiefs an te asset price 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 5

17 Figure 2: Te equiibrium wit ong-term ebt nancing. 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 6

18 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 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 ) : () 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. In aition, we prove tat te buyers ave no incentive to keep cas on ate. 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: 7

19 Figure 3: Te equiibrium wit sort-term ebt nancing. -LD: If C L 2 > ( ) P M c, ten p = c+c L( 2 ) an a te buyers use te same ong-term ebt contract wit face vaue 2 ; -LD2: If C L 2 ( ) P M c C L (), ten p = P M an eac buyer is ini erent between te ong-term ebt contracts wit face vaues of an 2, wit te fraction of buyers using te former contract given in (); -LD3: If C L () < ( ) P M c, ten p = c+c L() an a te buyers use te same ong-term ebt contract wit face vaue : 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. 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 ) an a te buyers use te same sort-term ebt contract wit face vaue 2 ; 8

20 -SD2: If C S 2 ( ) P H c C S (K ) ; ten p = P H an eac buyer is ini erent between te sort-term ebt contracts wit face vaues of K an 2, 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 ) -SD3: If ( ) P L c < C S (K ) < ( ) P H c; ten p = c+c S(K ) an a te buyers use te same sort-term ebt contract wit face vaue K ; -SD4: If C S (K ) ( ) P L c C S () ; ten p = P L an eac buyer is ini erent between te sort-term ebt contracts wit face vaues of an K, 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 ) -SD5: If C S () < ( ) P L c; ten p = c+c S() an a te buyers use te same sortterm ebt contract wit face vaue : Date On ate, tere are two states: u an : In te upper state u, eac asset oer s nancia conition is strong. Regaress of te possibe equiibrium ebt contract e as taken on ate, e faces no efaut risk going forwar. If e as use a sort-term ebt contract, e can aways re nance te maturing ebt wit a new one. Tus, any potentia buyer of te asset as to pay a price equa to te optimistic asset oer s vauation: i p u = E e u = u + u : () In te ower state, te equiibrium epens on te ebt contracts use by te asset oers on ate. As sown by Propositions 5 an 6, te optimists o not save any cas on ate. As a resut, if any asset oer runs into nancia istress after te asset funamenta eteriorates in state, te price of te asset is etermine by te pessimists, instea of te optimists. We etermine te equiibrium asset price in tis state by te saow cost of a potentia buyer, wo nees to buy out te stake of te current asset oer in te asset as we as te stake of is creitor. Te foowing proposition summarizes te equiibrium on ate. Proposition 7 On ate, in te upper state u; eac asset oer faces no efaut risk going forwar an te asset price is given in (). In te ower state, te equiibrium epens ; ; 9

21 Figure 4: An iustration of te equiibrium e ect of using sort-term ebt. on te ebt contracts use by te asset oers, wic are given by i erent cases erive in Propositions 5 an 6. In cases LD an SD, a te asset oers use ebt contracts wit face vaue 2 ; an face no efaut risk going forwar. As a resut, te equiibrium asset price is etermine by teir vauation: E i e = + 2 : In te oter cases, at east some of te asset oers use aggresive ebt contracts wit face vaues of K or, an tese asset oers are now in istress. As a resut, te equiibrium asset price is etermine by te pessimistic creitors vauation: E i e = + 2 : 2.5 Te Roe of Sort-term Debt We now anayze te equiibrium e ect of sort-term ebt in nancing optimistic buyers asset purcases. To faciitate our anaysis, suppose tat sort-term ebt is initiay not avaiabe an asset buyers ave access ony to ong-term ebt. Figure 4 iustrates te canges in te equiibrium after te asset buyers ave access to sort-term ebt. Let us rst focus on Case A sown in te gure. In te absence of sort-term ebt, te asset buyers manage to purcase te asset at a price iger tan P M by using risk-free ong-term ebt. As we ave erive in Proposition 5, tis situation occurs ony wen te 2

22 asset buyers initia cas enowment is reasonaby ig so tat a moest amount of everage is enoug for te buyers to cear te asset market. Te price eve is aso ig so tat eac buyer is iscourage from using te more aggresive ong-term ebt contract wit face vaue to take a greater position. However, if te buyers are aowe to use sort-term ebt contracts, teir creit eman curve sifts up wen te asset price p is between te two critica eves P M an P H : Tis is because sort-term ebt aows te buyers to obtain te same creit at a ower cost uner te conition in Proposition. Tis sift causes buyers to use more creit an resuts in a iger equiibrium asset price. Figure 4 aows us to precisey ientify te conition for tis situation to occur an te exact canges in equiirbium price an creit usage. Proposition 8 Suppose tat te conition in (7) os, an tat ( ) P M c < C S (K ) : Ten, after te introuction of sort-term ebt, te equiibrium asset price on ate increases, an te price increase is supporte by te use of te sort-term ebt contract wit face vaue K by at east some of te asset buyers. To some extent te situation anayze in Proposition 8 is anaogous to te US ousing market in te pre-arm (ajustabe rate mortgage) era. Before te ARMs became popuar in te ast ecae, ome buyers a preominanty reie on 3-year xe rate mortgages to nance teir ome purcases. Historicay speaking, tese ong-term mortgage oans a ow efaut rates. Accoring to Mayer, Pence, an Serun (29), te sare of US mortgage oans tat were seriousy eiquent (9 ays or more past ue or in te process of forecosure) average ony.7 percent from 979 to 26. Tese xe-rate mortgages tus resembe te risk-free ong-term ebt contracts erive in case A of Figure 4. Te ARMs are e ectivey sort-term oans because tey typicay ave a xe teaser rate for 2 or 3 years, wic is sceue to rise by two or more percentage points after te initia perio ens. Te rate ike often forces te ome owners to re nance teir mortgages at te market rate, in te same spirit of roing over te sort-term ebt on te interim ate of our moe. If te ousing price rises wen te rate is reset, a ome owner faces no i cuty in re nancing is oan at te market rate. However, if te ousing price fas, te ome owner wi not be abe to re nance is oan an wi be straine by te argey increase mortgage payment. As a resut, e may be force to turn over te ome to is creitor. 2

23 Mayer, Pence, an Serun (29) ocument tat by mi-28, serious einquencies on ARMs a risen to over 29 percent, wie te simiar rate for xe-rate mortgages rose ony to 9 percent. 5 Financia innovations, suc as te securitization of subprime oans, were at east partiay responsibe for te popuarity of ARMs in te ast ecae. Tus te syncronous growt of ARMs an ousing prices uring te recent boom, togeter wit te arge spike in te einquency rate of ARMs uring te bust, matc squarey wit te situation escribe in Proposition 8. 6 Figure 4 aso iustrates anoter possibity, case B, in wic eac asset buyer as scarce cas enowment. As a resut, in te absence of sort-term ebt te equiibrium asset price remains beow P M espite te buyers aggresive use of ong-term ebt contract wit face vaue. Ten, te introuction of sort-term ebt aows te buyers to use a more moest sort-term ebt contract wit face vaue K. As a resut, te equiibrium asset price rops, accompanie by a reuction in creit use by te buyers. Proposition 9 Suppose tat te conition in (7) os an tat ( ) P M c > C S (K ) : Ten, after te introuction of sort-term ebt, te equiibrium asset price on ate ecreases an te price ecrease is accompanie by te sift to te more moest sort-term ebt contract wit face vaue K by at east some of te asset buyers. Taken togeter, Propositions 8 an 9 emonstrate tat introucing sort-term ebt cou eiter increase or ecrease te creit use by optimists in te equiibrium, epening on te severity of teir cas constraints. If tey are so severey constraine tat te asset prices remain ow even after tey ave use extremey aggressive ong-term ebt, ten introucing sort-term ebt wou actuay aow tem to reuce teir everages. Tis resut is intriguing from a teoretica perspective. However, tis situation is probaby ess reevant to te recent 5 Te ARMs were primariy use to nance te ome purcases of sub-prime an near-prime ouseos, wo were constraine from quaifying for te reguar xe-rate mortgages. Mian an Su (29) n tat from 22 to 25, areas wit isproportionatey arge sare of subprime borrowers in 996 a experience an unpreceente growt in subprime creit, espite te sarpy ecining reative income growt in tese areas. Interestingy, tese areas aso a signi canty iger mortgage einquency rates in 27 wen te ousing prices across US starte to ecine. 6 One cou aso attribute te syncronous growt of ARMs an ousing prices to a growing ivergence of beiefs among agents about te ousing-market funamentas. Tis argument is aso consistent wit our moe, wic we wi anayze in te next subsection. 22

24 ousing boom an oter istorica episoes. Tus, we wi focus on te situation ienti e in Proposition 8 in te foowing anaysis. 2.6 Heterogeneous Beiefs an Asset Price Cyces In tis subsection, we anayze te e ects of agents eterogeneous beiefs in riving te boom-an-bust cyce of asset prices. Te stanar resut of Mier (977) suggests tat eterogeneous beiefs cause asset overvauation in te presence of sort-saes constraints. Our anaysis wi focus on te interaction between eterogeneous beiefs an ebt nancing of optimistic buyers. For iustration, we use te foowing baseine parameter vaues: = :3; c = :5; = :4; = :7; = :3; u = :6; u = :4; = :6; = :4: (2) 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: In te iustration, 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 sort-term ebt. We aso iscuss te ebt contracts use by te asset buyers to nance teir purcases 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 vaue an instea take te foowing vaues: = :5 + an = :5 were canges from to :45 an rives te initia beief ispersion between te optimists an pessimists. Figure 5 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 23

25 Fraction of Optimists p 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 Initia beief ispersion Initia beief ispersion p p Pane C: Date Price Drop after Ba Sock Bot SD an LD LD ony Mier case Initia beief ispersion Pane D: Long term Contracts Use in LD Ony Equiibrium.8.6 θ 2 contract θ contract Initia beief ispersion Figure 5: Te equiibrium e ects of initia beief ispersion on asset an creit markets. 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 ase ine pots p wen te optimists ave access ony to ong-term ebt to nance teir asset purcases. Interestingy, tis price ecreases from :485 to :443 as increases from to :32 an ten stays at as increases furter. Te ramatic i erence between tis ine an te price uner te Mier setting origins from te optimists nancing cost. Te increase in beief ispersion not ony makes te optimists more optimistic, but aso makes te pessimists more pessimistic. As a resut, optimists face more costy nancing from te pessimists even toug teir own vauation of te asset is iger. Te nancing cost gives an inirect canne for te pessimists to a ect te equiibrium price espite te sort-saes constraints. In te iustrate LD ony equiibrium, te nancingcost e ect can even overturn te stanar Mier resut an cause te equiibrium price to ecrease wit agents beief ispersion. 24

26 Pane D provies a breakown of te ong-term ebt contracts use by te optimists in te LD ony equiibrium. Initiay, wen increases from to :32; tere is a mix of optimists using ong-term contracts wit face vaues of an 2 (e.g., te LD2 case in Proposition 5) an te fraction of optimists using te risk-free contract wit face vaue 2 rises from :9 to. Tis fraction stays at as increases furter (e.g., te LD case in Proposition 5). 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 sigty ecreases from its initia vaue at :485 as increases from to :5, ten monotonicay increases to :58 as increases furter to :23, an nay stays at at :58 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 roe of sort-term ebt in reucing te optimists nancing cost. 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 contract wit face vaue K rises monotonicay from to as rises from :5 to :23, an stays at as rises furter. Tis pane sows tat te increase of te equiibrium price wit in te region 2 [:5; :23] 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 substantiay ower tan tat in te stanar Mier setting, sort-term ebt aows te optimists to manage teir nancing cost more e ectivey an tus to preserve te stanar Mier resut on te equiibrium price increasing wit agents beief ispersion. Date- cras Pane C of Figure 5 pots te price rop 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 aso increases wit. However, in te LD ony equiibrium, 25