Equilibrium Collateral Constraints

Transcription

1 Equilibriu Collateral Constraints Cecilia Parlatore y New York University, Stern School of Business October 3, 4 bstract I stuy a ynaic oel of rs that nee to raise funs to invest in risky projects whose return is private inforation. Firs with an investent opportunity in the future value the asset ore than those without it since the asset allows the to increase the aount they can invest in prouctive projects. If the investent opportunities are persistent, borrowers toay will value the asset ore than leners. This, together with the asyetric inforation proble, iplies that collateral contracts are optial. The aount that can be borrowe against the assets is an equilibriu outcoe. Introuction Collateralize ebt is a wiely use for of nancing. Trillions of ollars are trae aily in ebt collateralize by iverse nancial assets, such as sale an repurchase agreeents (repos) an collateralize over-the-counter erivative traes. Many nancial institutions use collateralize ebt to raise funs that allow the to provie intereiation services. Soe of these institutions are private epository institutions, creit unions, ortgage real estate investent trusts, an security brokers an ealer. Most of these institutions are highly levere an they use the repo arket as a source of nancing. The assets that are "sol" using repos are nancial assets which coul also be sol without a repurchase agreeent in nancial arkets. These nancial assets are not prouctive assets an, in principle, their intrinsic value is the sae inepenently of the assets holer. Then, why o so any nancial institutions choose to use these nancial assets as collateral instea of selling the to raise funs? If we assue that the borrower values the asset ore than the lener, as is the case of a faily heirloo, the borrower will be better o by pawning the asset than by selling it since part of the value the borrower assigns to it will not be internalize by the arket. On the other han, the lener will be happy to receive the e-ail: ceciparlatore@nyu.eu, Webpage: y I a especially inebte to Ricaro agos an To Sargent for their avice in this project. I also thank Douglas Gale for very helpful coents an suggestions. I thank Saki Bigio, lberto Bisin, V.V. Chari, Gian uca Cleenti, Euaro Davila, To Keister, Eiliano Marabio Catan, ntoine Martin, Juan Pablo Nicolini, Michal Szkup an seinar participants at Wharton for their insightful rearks. See Gorton, G. an Metrick (a; b) an Copelan, Martin, an Walker ():

2 asset as collateral since it will create incentives for the borrower to repay the ebt, an in case of efault the lener keeps the collateral. However, in the case of nancial assets, there is no a-hoc reason why borrowers woul value the assets ore than leners since both borrowers an leners can sell the assets in the sae arket an receive the sae iviens. Yet, we observe nancial assets being collateralize. In this paper I evelop a oel in which borrowers an leners value the asset equally in autarky, but they assign i erent values to it in equilibriu. s a result of this enogenous i erence in valuations, collateralize ebt contracts ipleent the optial funing contract. Since the contract is an equilibriu outcoe, I can also characterize the aount that can be borrowe against an asset (i.e., its ebt capacity) an its eterinants. The oel is a iscrete-tie, in nite-horizon oel. There are two types of risk neutral agents, borrowers an leners, an one urable asset which pays iviens each perio. Borrowers can invest in risky projects but they nee external funs to o so. The return of the projects is private inforation of the agent who investe in the. In orer to raise funs, borrowers enter into a state contingent contract with leners. In equilibriu, rs value the asset ore than leners, an, therefore, choose to use collateral contracts. In y oel, borrowers have the investent opportunity before the asset s iviens are pai an, thus, cannot invest without external nancing. This tiing iplies a aturity isatch for the borrower between the nee of funs to invest an the availability of the iviens. eners o not have access to investent opportunities. Therefore, if left in autarky, both risk neutral borrowers an leners woul value the asset by the expecte iscounte su of iviens. However, once the agents are allowe to trae, the equilibriu features enogenous i erences in valuations an, thus, collateral contracts as optial. The ain assuptions that lea to this result are the aturity isatch between the tie in which the iviens are pai an the investent opportunity, the persistence in the role as borrowers an leners, an the asyetric inforation about the borrower s ability to repay. Suppose that an agent in this econoy will have an investent opportunity toorrow. Holing the asset toorrow will allow the agent to invest in the project either by selling the asset or by pleging it as collateral. Being able to raise funs against the asset in the funing arket will solve the aturity isatch proble between the tiing of the ivien realization an the investent opportunity. In turn, this iplies that agents with investent opportunities toorrow will value the asset ore than those without the. If the role as borrowers an leners is persistent, as it is the case in any collateralize ebt arkets, it follows that agents who are borrowers toay will value the asset toorrow ore than leners will. 3 This i erence in valuations, together with the asyetric inforation about the borrower s ability to repay, iplies that collateral contracts arise optially in equilibriu. The extra value borrowers assign to the asset on top of the expecte iscounte value of iviens can be ecopose in two preia for the borrower: a liquiity preiu an a collateral preiu. The liquiity Collateralize ebt contract can reain optial if savings are allowe in the oel. 3 This persistence is consistent with the observation that, in any collateralize ebt arkets, i erent types of institutions specialize in borrowing or lening. For exaple, in the repo arket, oney arket funs are usually leners whereas hege funs an specialty leners are usually borrowers.

3 preiu for the borrowers arises fro the being able to sell the asset an use the funs to invest in the risky projects (fro solving the aturity isatch entione above). The collateral preiu for the borrowers is the aitional value they can obtain by using the asset as collateral instea of selling it. By solving the oel in close for, I a able to characterize the asset s ebt capacity copletely. I show that when all borrowers can invest in the sae type of projects, i.e., all projects have the sae correlation with the asset s future iviens, an increase in this correlation increases the asset s ebt capacity. Given the asyetric inforation proble, the borrower has incentives to lie when the return of the projects is high. Therefore, an asset that has a higher value when the borrower has incentives to lie akes it easier to otivate the borrower to tell the truth, an, hence, is better collateral. I also n that when the returns of the projects are positively correlate with future iviens, a ean preserving sprea of the istribution of the projects returns ecreases the asset s ebt capacity. Finally, when borrowers can invest in i erent types of projects, two regies ay arise in equilibriu: one in which all borrowers use the asset as collateral an one in which soe borrowers use the asset as collateral an other borrowers choose to sell the asset to raise funs. In this case, the e ect of changes in the correlation structure on the overall aount intereiate in the econoy epens on the istribution of project types across borrowers an on the regie of the econoy. There is a large literature that analyzes collateral contracts. My paper is closest to acker () an Rapini (5). acker () shows that collateralize ebt is the optial nancing contract when the borrower values the collateral goo ore than the lener. He shows this in a two-goo, two-perio oel with two agents, a risk-averse borrower an a risk-neutral lener that takes the i erence in valuations as given. In a siilar environent with non-pecuniary costs of efault, Rapini (5) stuies how efault varies with aggregate incoe when iniviual incoe is privately observe by the agents. The optial risk sharing contract allows for efault which, given the efault penalties assue, only occur when the realization of incoe is low. These efault penalties, which are oele as transfers of agent-speci c goos only value by the agent who is enowe with it, can be interprete as collateral which is only value by the original holer. s in these two papers, the friction that gives rise to collateral contracts in equilibriu in y oel is asyetric inforation between the borrower an the lener. In particular, the aount of repayent goo the borrower has when he has to repay the loan is private inforation of the borrower. In contrast, y oel is in in nite horizon, all agents are risk neutral, an, ost iportantly, the i erence in the arginal value of the collateral goo for the lener an the borrower is an equilibriu outcoe: in equilibriu, the borrower values the collateral goo ore than the lener an, therefore, collateral contracts are optial. s Barro (976) shows, enforceent frictions can also give rise to collateral contracts. Kocherlakota () analyzes optial repayent contracts in the presence of collateral when there are enforceent frictions. Collateral contracts are optial in Kocherlakota s oel, provie that leners are assue to be less willing than borrowers to substitute consuption for collateral goos. In his setup, collateral is use to force the borrower to share with the lener the returns on the project in which the borrower investe. Following Kiyotaki an Moore (997) there is a large literature that refers to these enforceent frictions 3

4 to analyze the e ects of collateral constraints on aggregate uctuations. For exaple, Jerann an Quarini () n that a tightening of the rs collateral constraints contribute signi cantly to the 8 9 recession an to the ownturns in 99 9 an : In these papers, at the tie of repayent, the lener can recover only a fraction of the collateral value. This fraction is stochastic an epens on (unspeci e) arket conitions. Moreover, urable assets are use not only as collateral for loans, but also as inputs for prouction (an, thus, selling the is not an option). In the general equilibriu literature, Geanakoplos (3a; 3b) ; an Geanakoplos an Zae (7) stuy the use of urable assets as collateral. They show that when collateral contracts are assue, the set of trae assets will be eterine enogenously. raujo, Orrillo an Pascoa (994) an raujo, Fajaro an Pascoa (5) assue collateral contracts an ake collateral enogenous by allowing each seller of assets to x the level of collateral or the bunle use as collateral. In all these papers, collateral contracts are assue. Finally, in a search oel with bilateral traing, Monnet an Narajaba () show that agents prefer to conuct repurchase agreeents than asset sales when they face substantial uncertainty about the value of holing the asset in the future. In contrast to the papers entione above, in the oel presente in this paper, the asset use as collateral is not an input in prouction an it can be sol to raise funs. s explaine above, given the persistence in the roles as borrowers an leners, the asset is use as collateral as an optial response to asyetric inforation about the resources available to the borrower at the tie of repayent. The aount that can be borrowe against the asset is eterine in equilibriu, an, thus, changes in the collateral constraints face by borrowers re ect changes in the econoy s funaentals. By characterizing collateral constraints in equilibriu, y oel provies a link between the aount that can be borrowe against the assets an the funaentals in the econoy beyon the value of the asset. It also allows e to ientify collateral an liquiity preia which a ect the asset s value an its ebt capacity. The rest of the paper is organize as follows. Section presents the oel. Section 3 e nes an characterizes equilibriu. Coparative statics results are presente in section 4. Section 5 extens the oel in section, allowing for heterogeneity aong borrowers. Section 6 conclues. Moel Tie is iscrete, starts at t, an goes on forever. Each perio t is ivie in two subperios, orning an afternoon. There are two types of consuption, non-storable goo, orning an afternoon speci c. There is one asset that lasts forever an which is in xe supply k. k units of the asset yiel t k units of (afternoon) consuption goo as ivien at the en of each afternoon t. The ivien t is known at the beginning of orning t: There are two types of in nitely-live, risk-neutral agents, borrowers an leners. There is a easure of each type an all agents have the sae iscount factor (; ). Each agent starts his life with an enowent of the asset. Each subperio leners are enowe with a large aount of consuption 4

5 goo, e in the orning an ea in the afternoon. Each orning t borrowers receive a large enowent of (orning) consuption goo, an each afternoon t rs can invest in short-ter projects on which they can ake pro ts. Projects are risky: one unit of (afternoon) consuption goo investe in the risky projects at the beginning of afternoon t by borrower j yiels a rano payo j t in (afternoon) consuption goo at the en of the perio. j t f ; H g : The return of the projects is i.i.. across borrowers an tie. There is an unerlying unobservable i.i.. aggregate state! t (! ;! ) that eterines the probability of success of the risky projects. 4 et p n i : Pr j t i j! t! n, an p i : Pr (! t! ) p i +Pr (! t! ) p i Pr j t i, i ; H; 8j; :8t : I assue that E j t > > ; 8j; :8t, i.e., that the projects are pro table in expectation but that they incur losses if the low state is realize, an that > E(). The return of the projects is private inforation known by the r who investe in the, an it is potentially correlate with the ivien pai by the asset the following perio. 5 The joint istribution of returns in perio t an iviens in perio t + is stationary, i.e., where E t+ j j t i X n; Pr! t! n j j t i E ( t+ j! t! n ) : i for i ; H; 8j; 8t ; Pr! t! n j j t i pn i Pr (!! n) : p i The ivien is pai after the investent in the projects is ae. Therefore, iviens cannot be investe by the borrower. This tiing iplies that rs nee to borrow to ake risky loans, an creates a aturity isatch between the nee of funs (liquiity) an the availability of funs. 6 Moreover, this iplies that if agents were left in autarky, they woul all value the asset as the iscounte su of expecte iviens. Each orning, after the ivien level t is known, the asset arket opens. Borrowers an leners eet ranoly. Traes ae in this arket are bilateral an the ters of the transaction are eterine through Nash bargaining. t the beginning of each afternoon, the funing arket opens: each borrower is ranoly paire with a lener an there is a bilateral funing contract in which all bargaining power is given to the borrower. The tiing is illustrate in gure.. sset Market t the beginning of each orning, each borrower is ranoly atche with a lener in the asset arket. The ters of trae, quantity an price, are eterine through Nash bargaining. The bargaining power of borrowers is equal to (; ]. 7 Since atches are rano both in the asset arket an in the loan arket, the aggregate state of the econoy is (; F ; F B ), where F an F B are the istribution of assets in the hans of leners in the orning, an the istribution of assets in the hans of borrowers in 4 I assue the law of large nuber, an thus the fraction of successful investens in loans will also epen on the state realize. 5 One can think of the asset as ortgage backe security or as an asset backe security backe by loans ae by banks. 6 The results still survive if borrowers were able to save in consuption goo between the orning an afternoon. 7 If both agents value the asset the sae in equilibriu an the optial ebt contract is ineterinate. 5

6 Figure : Tiing the orning, respectively. I assue that borrowers have enough consuption goo each orning to buy all assets fro the non-proucer with who they eet. By aking this assuption, the oel abstracts fro borrowing constraints in the asset arket. Since the ain focus of the paper is to unerstan collateralize loan contracts, consiering such borrowing constraints, though interesting an realistic, akes it harer to isentangle the forces at work without aing uch to the analysis. et k T (k B ; k ; ) ; an P (k B ; k ; ) be the quantity an price that result fro the encounter of a borrower with k B units of the asset an a lener with k units of the asset. Then, the value of a borrower in the orning, before entering the asset arket, is Z VB (k B ; ) V a B k B + k T (k B ; k ; ) ; P (k B ; k ; ) F (k ) () where V B a (k B; ) R VB a (k B; k ; ) F a (k ) ; an VB a (k B; k ; ) is the value of a borrower with assets k B who enters a loan contract with a lener with assets k in the afternoon: The value of a lener in the orning, before entering the asset arket, is Z V (k ; ) P (k B ; k ; ) + k k T (k B ; k ; ) + E V k k T (k B ; k ; ) ; FB (k B ) () Therefore, since the ters of trae are eterine by Nash bargaining, P (k B ; k ; ) an k T (k B ; k ; ) solve ax P k E V k ; V k k ; P ;k V a B (k B + k ; ) V a B (k B ; ) P (3). Funing Market Every afternoon, a bilateral funing arket opens. Each borrower is atche ranoly with a lener an the ters of the loan contract are eterine by the borrower. 8 oan contracts are one-subperio contracts 8 The shape of the contract is the sae if all bargaining power is assigne to the lener. 6

7 an they consist of a loan aount in ters of (afternoon) consuption goo, q, an contingent repayents in ters of (afternoon) consuption, r i, an in ters of assets, t i, i ; H: 9 Given a ivien level, a contract (q; r ; r H ; t ; t H ) is feasible at tie t if q e a r i k Bt + i q i ; H t i k Bt i ; H where k Bt is the aount of assets hel by the borrower in afternoon t. Since t is only observe by the borrower, for contracts to be creible, they ust be incentive copatible. contract (q; r ; r H ; t ; t H ) is incentive copatible if r + E V B k B t ; j t H rh + E V B k B t H ; j t H (4) an whenever r H (k B + q) where V B k B r + E V B k B t ; j t rh + E V B k B t H ; j t t; e ; R V B k B t; k ; e ; F (k ) The rst constraint states that, in the high state, it is always at least as goo for the borrower to tell the truth an report that the high state has occurre than to lie an report a low realization of. The secon constraint states the analogous for the low state with the restriction that (5) is only active when lying in the low state is feasible, i.e., when there are enough resources in the low state to atch the contingent repayent in ters of goos in the high state, r H : I assue that the aount of the loan in equilibriu is never restricte by the aount of consuption goo owne by leners, i.e., that q < e a : et V a B (k B; k ; ) be the value of a borrower with assets k B who is atche with a lener with assets k in the loan arket. Then, V a B (k B; k ; ) sup E () q r H p r + k P (6) q;r ;r H ;t H ;t + E VB k B t H ; j H + p E VB k B t ; j (5) s.t. (q; r ; r H ; t ; t H ) is feasible an IC E V k ; q + r H + E V k + t H ; j H +p r + p E V k + t ; j (7) H (; ) (8) 9 Uner certaing paraetric assuptions, one sub-perio contracts ipleent long-ter contracts. If this was not the case, the incentive copatibity constraints woul not bin, the size of the loan woul be e cient an ebt woul be riskless. 7

8 The borrower chooses a feasible an incentive copatible contract to axiize his expecte utility subject to the lener s participation constraint (7) an given the perceive law of otion for the aggregate state (8). (7) states that the lener has to be at least as goo participating in the contract as he woul be if he in t participate in it. The borrower invests all the loan aount, q, in the risky technology since E () >, an gets an expecte return E () q: He expects to repay r H + p r in ters of consuption goo an he gets iviens k P fro his stock of assets at the beginning of the afternoon. Finally, his expecte continuation value in the orning epens on the contingent transfers of assets t an t H that are part of the contract. By inspecting the constraints in the proble above, one can consierable siplify the borrower s proble. First, in any equilibriu, the participation constraint for leners (7) will hol with equality. If it i not, the borrower coul increase the loan aount an increase his expecte utility without violating any of the aitional constraints. Siilarly, as is usual in this kin of probles, the incentive copatibility constraint will bin in the high state (4) will hol with equality. Finally, in orer to axiize the size of the loan, the repayent in ters of goos in the low state, r, will be the axiu possible, i.e., r q + k B : eas 8, 9 an in the appenix foralize these arguents. Proposition The borrower s proble can be rewritten as VB a (k B; k ; ) sup (th ;t )[;k B ] k B + (E () ) E V k + t H ; j H + p E V k + t ; j (E () ) + E V B k B t H ; j H E V B k B t ; j H + p E V B k B t ; j + ph E V B k B t H ; j H E V k ; subject to q k B + E V k + t H ; j H + p E V k + t ; j + E V B k B t H ; V B k B t ; j H E V k ; q axf; E V k + t H ; j H + p E V k + t ; j p E V B k B t H ; V B k B t ; j H E V k NP ; g The proof follows fro leas 8, 9 an in the appenix. 3 Equilibriu De nition recursive equilibriu in this econoy is a pair of value functions for borrowers, in the orning an in the afternoon, V B (k B; ) an V a B (k B; k ; ), a value function for leners V (k ; ) ; price 8

9 an quantity functions in the bilateral asset arket, P (k B ; k ; ) an k T (k B ; k ; ), a loan contract (q (k B ; k ; ) ; r (k B ; k ; ) ; r H (k B ; k ; ) ; t (k B ; k ; ) ; t H (k B ; k ; )) an a law of otion for ; H (;!), such that () ; () ; (3), an (6) are satis e an the law of otion for satis es: Z F (k;! i ) p i H +p i Z F a (k t H (k B ; k; ; ) ;! ) FB a (k B ;! ) F a (k t (k B ; k; ; ) ;! ) FB a (k B ;! ) Z FB a (k;! ) FB k k T (k; k ; ; ) F (k ;! ) Z FB (k;! i ) p i H FB a (k + t H (k; k ; ; ) ;! ) F a (k ;! ) Z +p i FB a (k + t (k; k ; ; ) ;! ) F a (k ;! ) Z F a (k;! i ) F k + k T (k B ; k; ; ) ;! i F B (k B ;! i ) of istributions where! is the realize aggregate state the perio before. 3. ne Equilibriu I will focus on recursive a ne equilibria, i.e., in equilibria in which the value functions are a ne in asset holings. Within this class of equilibria, there is a unique equilibriu. I will use the guess an verify etho to show that such equilibriu exists an that it is unique. Suppose that V B a (k B ; ) VB a (k B ; k ; ) c B () k B + a B () V (k ; ) c () k + a () where c B () an c () are a ne in : Uner these assuption, the ters of trae in the asset arket are the solution to ax P k c k (cb () k P ) P ;k where E ( ) :Then, P ( ) c B () k + k + c k an k T (k B ; k ; ) arg ax k [ k P ;k NP ] c B () c k The apping that characterizes an equilibriu is not a contraction apping. Therefore I cannot show that the equilibriu is unique within a ore general class of value functions. 9

10 which iplies k T (k B ; k ; ) f k B ; k g : s it is usually the case with linear value functions, the quantity trae through Nash bargaining axiizes surplus an thus traes are e cient. Since the borrower can wait until the afternoon an sell the asset in the loan arket, by setting t t H k B, he will never choose to sell in the orning. Selling in the afternoon, allows the borrower to invest the procees of the sale in the risky projects which has a higher expecte return than consuing the goos in the orning. Therefore, the ters of trae in the asset arket are inepenent of the asset stock with which the borrower (buyer) enters the arket, k B, an k T (k B ; k ; ) k ; for all k B ; k, an. Given this structure, the value functions for leners an borrowers in the orning before entering the asset arket are, respectively, V (k ; ) P (k B ; k ; ) ( ) c B () + + c k an V B (k B ; ) c B () + c Z kf (k) + c B () k B : Thus, using the guesse functional for for the value functions gives an Therefore, c () c () ( ) c B () + + c " c ( ) cb + ( ) c B () + + ( ) c B + Finally, the value of a borrower who enters the loan arket with k B units of asset an eets a lener with k units of assets when the state is is, using proposition, V a B (k B ; k ; ) ax t H ;t [;k P ] (E () ) q + k B + ( c ( H ) t H + c ( ) t ) + c B ( H ) (k B t H ) + p c B ( ) (k B t ) +E c B ( ) Z + c kf (k) :!# : s.t. q k B + ( c ( H ) t H + p c ( ) t ) c B ( H ) (t H t ) (9) q ax f ( c ( H ) t H + p c ( ) t ) + p c B ( H ) (t H t ) ; g () r H q c ( H ) t H p c ( ) t + p c B ( H ) (t H t ) r q c ( H ) t H p c ( ) t + c B ( H ) (t H t ) OM for F Rewriting the proble in this way shows that the borrower will choose the sae contract that a planner who is subject to the asyetric inforation proble an put all weight on the borrower woul choose. Therefore, the equilibriu contract is constraine Pareto e cient.

11 Given the a ne speci cation of the utility functions, the solution to the borrower s proble in the afternoon will be in corner solution. If the constraint () is ignore, there are four possible solutions: t t H, t an t H k B, t k B an t H, an t k B t H : If t t H then the constraints on q are satis e. If t < t H, () ight bin. In the appenix I show that () can t bin in equilibriu. Therefore, ignoring the constraints on q ; () ;an using the guesse functional for for V a P (k P ; k NP ; ) ; one can atch coe cients an get c B () (E () ) + (E () ) c B ( H ) a B () E + c B ( H ) c ( H H + p c P + p c B ( ) c B ( ) + c Z kf NP (k) Proposition 3 In the only a ne equilibriu, t k B an P : () The proof of this proposition can be foun in the appenix. 3. Ipleentation of the Optial oan Contract Using the results fro the previous section, the optial loan contract (q ; r ; r H ; t ; t H ) is given by q () ( + p c ( ) + c B ( H )) k B r () ( q () + ) k B r H () r () + c B ( H ) k B t k B, t H The optial loan contract only epens on the aggregate state through the current ivien level. Borrowers are collateral constraine: the axiu aount that the borrowers are able to borrow epens linearly on the aount of assets they have. The collateral constraint is an equilibriu outcoe an it epens on how uch the expecte holer values the asset. With probability p the return of the projects is low an the borrower transfers all his asset holings to the lener whose expecte iscounte value of one unit of asset is c ( ). With probability the return of the projects is high an the borrower keeps all his assets which he values c B ( H ) per unit. The optial loan contract can be ipleente using two i erent ebt contracts siultaneously: riskless ebt an collateralize ebt. This ipleentation is not unique. Below, I escribe the ipleentation with the axiu aount of riskless ebt, which is the one that inclues riskless ebt at interest rate. The axiu aount that can be repai inepenently of the realize state, risklessly, is r (). Therefore, the aount of riskless ebt in this ipleentation is r (). The reaining part of the loan aount

12 q () is repai in consuption goos only if the return of the project is high, whereas if it is low, the lener receives an asset transfer fro the borrower. I will refer to this fraction of the loan aount as collateralize ebt. Then, the aount of collateralize ebt in this ipleentation is q c : q () r () (p c ( ) + c B ( H )) k P which is inepenent of the ivien level. The interest rate on this loan can be copute as r H () r () c P ( H ) q c q c k b The expecte return on collateralize ebt is q c p (c B ( H ) c ( )) q c p (c B ( H ) c ( )) (p c ( ) + (c B ( H ))) : p c ( ) t + (r H r ) q c p c ( ) + c B ( H ) (p c ( ) + c B ( H )) : The axiu aount that can be borrowe against the asset, its ebt capacity, is given by D (p c ( ) + c B ( H )) : The asset s ebt capacity epens on the value of collateral for leners when there is efault (insurance) an on the value of collateral for borrowers when there isn t efault (incentives). Ceteris paribus, a higher value of collateral for leners increases the loan aount since they can recover ore when there is efault, while a higher value of collateral for borrowers ecreases the proucers incentives to lie an therefore allows the to borrow ore. 3.3 Preia The i erence between the agents valuation of the asset an the funaental value of the asset can be ecopose in several preia. The borrower values the asset ore than the expecte iscounte ivien strea for two reasons. The rst reason is that the asset serves as a liquiity transforation evice, it allows the borrower to solve the aturity isatch proble he faces. The extra value ue to this function is capture by the private liquiity preiu, which I e ne as the i erence between how uch the borrower woul value the asset if he chose to sell it to get funs an the funaental value of the asset. If the borrower chose to sell the asset in the afternoon, he woul get + c per unit of asset, which is the axiu aount the lener woul be willing to pay for it. With this funs, the borrower woul be able to invest in projects an he woul get the return on equity E() on the: Therefore, the borrower woul value each unit of asset E() + c, an the private liquiity preiu is e ne as

13 E () + c E () E () c The secon reason why the borrower values the asset ore than the lener is that he expects to use it as collateral the following perio. I e ne the private collateral preiu as the extra value a borrower gets fro using the asset as collateral instea of selling it to raise funs. Fro the characterization of the a ne equilibriu, a borrower who chooses to use the asset as collateral values it c B () per unit. Then, the private collateral preiu is : c B () E () + c E () (c B ( H ) c ( H )) : This preiu epens on the i erence in valuations for the borrower an the lener. If both agents value the asset the sae, the private collateral preiu is. In this case, the borrower woul be ini erent between selling the asset an pleging it as collateral. When the borrower values the asset ore than the lener, the private collateral preiu is positive an the borrower chooses to use the asset as collateral. Finally, the lener ay value the asset ore than the expecte iscounte su of its iviens if he has soe bargaining power in the asset arket. By being able to sell the asset to agents that value the asset ore than theselves, leners can extract soe of this extra value whenever their bargaining power is positive, i.e., <. This extra value is what I call a liquiity preiu an it is e ne as c () + ( ) E (c B ( H ) c ( H )) + + E() ( ) E() ( ) s I entione above, this preiu is when the borrowers have all bargaining power in the asset arket. : 4 Coparative Statics 4. Di erent correlation structures The ebt capacity of the asset epens on the correlation between future iviens an the success of the investent ae by borrowers. In this subsection, I will show that, keeping the (unconitional) expecte return of the asset,, xe, an increase in the correlation between the success of the investent ae by borrowers an the future iviens pai by the asset increases the asset s ebt capacity. 3

14 Therefore, et f () : 8 [; (c B ( H ) + p c ( j (E () ( ) / ( ) ) p ) H / sign j ) ( ( + p )) ( ( + p )) : ) ( ( + p )) : Since by < H j E() >, f () > This result iplies that a higher H, increases the asset s ebt capacity an it akes it better collateral. The asset is worth ore to the borrower when he has ore incentives to efault, in the event of a successful investent, an therefore ecreases his incentives to lie. 4. Change in risk Without loss of generality, can be set to. Then, the variance of the project is given by V () ( H H ) + p ( H ) ( ) H E () H E () Therefore, one can get a ean preserving sprea by increasing H an setting E () H : Proposition 4 If H in, where in <, an increase in risk ecreases the ebt capacity of the j E() >. When the return of the projects an the future ivien level are su ciently positively correlate, an increase in the riskiness of the projects ecreases the ebt capacity of the asset. In this setup, an increase in risk ecreases the probability of success of the project an, therefore, increases the probability of efault. This shift in probabilities has a irect an an inirect e ect on the ebt capacity of the asset. The irect e ect is that now the ebt capacity of the asset puts ore weight on the lener s value of collateral in the event of efault at the cost of ecreasing the weight on borrower s value of collateral in the event of no efault. The sign of this irect e ect epens on whether the lener values the asset ore in the efault 4

15 state, than oes the borrower when he gets to keep it. The inirect e ect coes fro the change in the asset s valuation for the lener. n increase in the probability of efault ecreases the lener s valuation of the asset. When H, both e ects are negative. If H < ; the net e ect then epens on which of these e ects is larger. When H > in, the increase in the efault probability is the oinating e ect. 5 Multiple Project Types In this section I exten the benchark oel an introuce heterogeneity aong borrowers. Each borrower is characterize by the returns of the projects in which he is able to invest. Projects available to i erent borrowers i er in their correlation with the iviens pai by the asset but they share the sae success probability an unconitional expecte return. There are J types of loans an a fraction j of borrowers can invest in projects of type j, j ; :::; J. In this case, in an a ne equilibriu, the arginal value of assets for a leners is c () ( ) P ii ic B ()+ + c NP :The borrowers proble reains unchange, though now the asset value for leners epens on the average valuation aong borrowers. Depening on the paraeters of the oel, two i erent kins of regies ight arise. In one, all borrowers choose to use the asset as collateral. This is clearly the case when since the ultiple project type oel an the benchark oel give the sae contract for each agent. The existence of other types of borrowers only atters through the resale value of the asset in the asset arket. If the sellers on t get any surplus fro this sale, then the price of the asset in the asset arket will be equal to the expecte iscounte value of ivien an, thus, it woul be inepenent of the istribution of borrower types in the econoy. If < there ight be borrowers who choose to sell the asset at the beginning of the afternoon an invest those funs rather than pleging it as collateral. Since the expecte price at which leners can sell the asset in the asset arket epens on the borrowers average valuation of the asset, it ay be the case that this average valuation is high enough to otivate soe of the borrowers who value the asset the least to sell the asset to raise funs. Whether there are soe borrowers that on t use the asset as collateral or not epens on the value of. For high values of everyboy uses the asset as collateral in the only syetric equilibriu. For lower values of soe borrowers ay choose to sell the asset instea of using it as collateral. Reark 5 When or the asset s iviens are uncorrelate with the projects ae by borrowers (e.g. risk free assets) all borrowers choose to use the asset as collateral. In both cases, the asset will always be use as collateral by at least one type of borrower. Proposition 6 The borrowers with the highest arginal valuation of the asset will always use it as collateral. The proof of this proposition can be foun in the appenix. Borrowers whose projects have the highest positive correlation with the ivien pai by the asset value the asset the ost. The higher this correlation, 5

16 the larger the aount that can be borrowe against the asset since it is better at proviing incentives to solve the asyetric inforation proble. 5. Changes in the correlation structure In this subsection I present an exaple to illustrate how aggregate quantities respon to changes in the correlation structure an risk when there are ultiple types of projects in the econoy an all borrowers choose to use the asset as collateral. There are J types of borrowers. There is a population j of type j borrowers, j ; : et j be the return of the projects in which a borrower of type j can invest. The unconitional probability of success is equal for both types of projects but conitional on the aggregate state, the success probabilities are given by Pr H j! Pr H j! Pr H j! Pr H j! ; ( ) ; ( ) ; ; where Pr (! ) Pr (! ) :5. et i E ( t+ j! i ), i ;, where > : Then, the expecte value of the ivien conitional on the return of project j is E t+ j t H E t+ j t H + ( ) ; ( ) + : Given this correlation structure, an increase in, increases (ecreases) the correlation between the return of type (type ) projects an the future ivien level while keeping the unconitional probability of success,, an the unconitional expecte ivien,, constant. et D D + D be the average ebt capacity of the asset. This is the ebt capacity a representative borrower woul have in this econoy. Proposition 7 n increase in has the following e ects. If, the econoy s average ebt capacity reains unchange, type borrowers collateral constraint is relaxe an type borrowers collateral constraint is tightene. if >, the econoy s average ebt capacity increases, an type borrowers collateral constraint is relaxe. if <, the econoy s average ebt capacity ecreases, an type borrowers collateral constraint is tightene. 6

17 This proposition states that, when there are ultiple types of projects, the e ect of changes in the correlation structure on how uch can be borrowe in the econoy epens on the istribution of project types aong borrowers. When there are ultiple projects, an increase in the correlation between the return of the project in which a borrower invests an the future ivien has two e ects. On the one han, an increase in this correlation akes the asset ore valuable for the borrower aking it better collateral. On the other han, this change in correlation ecreases the value of the asset for other types of borrowers an thus a ects the resale value of the asset. The net e ect epens on the istribution of types of projects across borrowers. 6 Conclusion In this paper I showe that when the roles as borrowers an leners are persistent, an there is ex-post asyetric inforation about the borrower s ability to repay, collateralize ebt is the optial way for borrowers to raise funs. Borrowers woul rather o er their asset as collateral than sell it since they value it ore than leners. If they sol it, they woul get at ost the valuation of leners, whereas by o ering it as collateral they keep it when there is no efault. This i erence in arginal valuations of the asset between borrowers an leners is an equilibriu outcoe. In autarky, both borrowers an leners value the asset as the expecte iscounte su of the ivien strea. When the agents are able to trae, the borrower values the asset ore then the lener an they both value the asset (weakly) ore than its funaental value. The borrowers excess valuation can be ivie in two preia: a private liquiity preiu an a private collateral preiu. The rst coes fro the asset solving a aturity isatch for the borrower: the asset pays iviens in the future but the borrower has an investent opportunity toay. Being able to sell the asset provies the borrower with funs in the oent he nees the. The private collateral preiu is the extra value the borrower assigns to the asset ue to its role as collateral. Since the borrower will use collateral contracts in the future, the asset relaxes a borrowing constraint an thus has ore value toay. Since collateral contracts are optial in this setup, an the arginal valuations of both borrowers an leners are enogenous, the axiu aount that can be borrowe against the asset, its ebt capacity, is also an equilibriu outcoe. I a able to solve the oel in close for which, in turn, allows e to copute soe coparative statics with respect to the correlation structure, an the riskiness of the projects in which borrowers invest. I n that a higher correlation between the success of the projects an future iviens akes the asset better collateral since it is better at otivating the borrower to tell the truth when he has incentives to lie. If the return of the risky projects an the asset s iviens are su ciently positively correlate, an increase in the riskiness of the projects ecreases the asset s ebt capacity. I nally exten the oel to inclue heterogeneity in return of the projects aong borrowers. I show that changes in the correlation between the asset use as collateral an the return of the projects ae by borrowers have i erent e ects on the asset s ebt capacity epening on the istribution of types of 7

18 borrowers in the econoy. We know fro previous literature that changes in nancial constraints playe an iportant role in recent crises. Having a oel that characterizes these constraints as equilibriu outcoes is iportant both fro a positive an a norative point of view. In positive ters, it is interesting to see where the nancial shocks coe fro an how they interact with the funaentals of the econoy. Fro the norative sie, policies that ai at stabilizing the cycle an preventing nancial crises shoul take into account what rives changes in the nancing conitions face by nancial intereiaries, rs, an househols. This paper is an attept to eliver soe of the insights neee to unerstan collateralize ebt arkets better. 7 References Financial Syste Oversight Council annual report, Chapter 5, Financial Developents, raujo,., J. Orrillo an M. R. Pascoa.994. "Equilibriu with Default an Enogenous Collateral," Matheatical Finance :. raujo,., J. Fajaro, M. R. Páscoa. 5. "Enogenous collateral," Journal of Matheatical Econoics, Volue 4(4 5) : Barro, R "The oan Market, Collateral, an Rates of Interest," Journal of Money, Creit an Banking 8 (4) : : Bester, H "The Role of collateral in Creit Markets with Iperfect Inforation," European Econoic Review 3 : Bianchi, J.. "Overborrowing an Systeic Externalities in the Business Cycle," erican Econoic Review (7): Chan, Y. an G. Kanatas "syetric Valuations an the Role of Collateral in oan greeents," The Journal of Money, Creit, an Banking 7 (). Chan, Y. an.v.thakor "Collateral an Copetitive Equilibria with Moral Hazar an Private Inforation," Journal of Finance, Vol XII, No.. Copelan.,. Martin, an M. W. Walker.. The tri-party repo arket before the refors, Feeral Reserve Bank of New York Sta Report. Diaon, D Financial Intereiation an Delegate Monitoring, Review of Econoic Stuies 5 : Geanakoplos, J.. 3. "Proises Proises," Cowles Founation Paper No. 57. Geanakoplos, J.. 3. "iquiity, Default, an Crashes," Cowles Founation Paper No. 74. Geanakoplos, J., an W. Zae. 7. "Collateralize sset Markets," ieo. Gorton, G. an. Metrick.. "Haircuts," Feeral Reserve Bank of St. ouis Review 9(6) : Gorton, G. B. an. Metrick.. Securitize Banking an the Run on Repo. Journal of Financial Econoics 4 (3) : Jerann an Quarini (9), Bianchi (), Perri an Quarini (): 8

19 Gorton, G. an G. Oroñez.. "Collateral Crises," ieo. Jerann, Urban, an Vincenzo Quarini.. "Macroeconoic E ects of Financial Shocks," erican Econoic Review, () : Kiyotaki, N., an J. Moore "Creit Cycles, Journal of Political Econoy 5: 48. Kiyotaki, N., an J. Moore. 8. "iquiity, Business Cycles, an Monetary Policy," Unpublishe Manuscript. UR Kocherlakota, Narayana R... "Risky Collateral an Deposit Insurance," vances in Macroeconoics: Vol. : Iss, rticle. Krishnaurthy,.. 3. "Collateral Constraints an the pli cation Mechanis," Journal of Econoic Theory () : acker, J. M... "Collateralize Debt as the Optial Contract," Review of Econoic Dynaics 4 : agos, R. an G. Rocheteau. 9. "iquiity in sset Markets with Search Frictions," Econoetrica vol 77 () : i, Y, an Y. i.."iquiity, sset Prices, an Creit Constraints", ieo. Martin,., D. Skeie, an E. Von Thaen., revise January. "Repo Runs," Feeral Reserve Bank of New York Sta Reports N Monnet, Cyril an Borghan N. Narajaba.. "Why Rent When You Can Buy? Theory of Repurchase greeents", ieo. Perri an Quarini.. "International Recessions," NBER Working Papers 7: Rapini,.. 5. "Default an aggregate incoe," Journal of Econoic Theory :

20 8 ppenix 8. Borrower s Proble ea 8 Without loss of generality, (7) can be replace by in the borrower s proble. E V k ; q + r H + E V k + t H ; j H +p r + p E V k s + t ; j : () Proof. et V be the solution to the borrower s proble. et fv j g j be such that li j! V j V, where V j E () q j r Hj p r j + k B (3) + E V B k B t Hj ; j i t H + p E V B k B t j ; j i t for soe feasible an incentive copatible fq j ; r j ; r Hj ; t j ; t Hj g that satis es the participation constraint (7) : Suppose that for soe j, (q j ; r j ; r Hj ; t j ; t Hj ) is such that (7) is slack. Then, one coul increase q j an increase V j to Vj still satisfying all the other constraints. et Vj fv j g j if at (q j ; r j ; r Hj ; t j ; t Hj ) (7) hols with equality an V j an therefore, li V j li V j V : j! j! Therefore, we can replace (7) by () in the borrower s proble. j be a sequence ientical to otherwise. Then, by construction, V j V j ea 9 Without loss of generality, the incentive copatibility constraints can be replace by r + E V B k B t ; j i t H rh + E V B k B t H ; j i t H : in the borrower s proble. Proof. et V be the solution to the borrower s proble. et fv j g j be such that li j! V j V, where V j E () q j r Hj p r j + k P (4) + E V B k B t Hj ; j i t H + p E V B k B t j ; j i t for soe feasible an incentive copatible fq j ; r j ; r Hj ; t j ; t Hj g that satis es the participation constraint () : Suppose that for soe s, no incentive copatibility constraint bins. Then, there exists " s > such that E V B k B t Hs ; V B k B t s ; j r Hs r s + ( H ) " s r Hs r s + ( H ) " s E V B k B t Hs ; V B k B t s ; j H : Replace fq s ; r s ; r Hs ; t s ; t Hs g by fq s + " s + " ; r s + " s ; r Hs + H " s ; t j ; t Hj g where " > is such that the participation constraint bins. This contract still satis es all the constraints, but it attains a value V s > V s :

21 If r Hs > r s, (4) is the only relevant incentive copatibility constraint. For all s such that r Hs > r s an (4) is not bining, the previous arguent applies an a value V s > V s can be attaine. Now consier those s such that r H s r s < q s + k P : If (5) bins, one coul keep r Hs constant by increasing both r s an r Hs an by increasing q s to keep the participation constraint bining which woul result in an increase in the objective function. et this new value be V s. If for s, r Hs r s q s + k P, (5) oesn t bin unless (4) bins. Suppose (5) bins an (4) oesn t. Then, one coul increase r Hs still satisfying incentive copatibility an relaxing the participation constraint. Therefore, one coul increase q s which woul increase the objective function an give a value V s r ss. Therefore, once can construct a new sequence V j j, V j V j such that V j V j is the incentive copatibility constraint in the high state bins an V j V j if it oesn t. By construction, li V j li V j V : j! j! Therefore, without loss of generality one can concentrate on those sequences that are feasible in which (4) hols with equality, i.e., r Hj r j E V B k B t Hj ; V B k B t j ; j H for all j: (5) ea Without loss of generality, the feasibility constraints on contingent transfers in consuption goo can be replace by in the borrower s proble. r q + k B an r H Proof. By assuption q e a will not bin in a solution to the borrower s proble. Using lea 8, the participation constraint can be assue to hol with equality, an using this in the objective function one can see that the objective function is always increasing in the aount of the loan q. Using lea 9; the incentive copatibility constraint hols with equality which iplies that the upper boun for q is given by the axiu aount that can be repai in the low state, i.e., by r q + k P : where et V be a solution to the borrower s proble. et fv j g be a sequence such that li j! V j V an V j (E () ) q j + E V k + t Hj ; j i t H + p E V k + t j ; j i t + kp + E V B k B t Hj ; j i t H + p E V B k B t j ; j i t E V k ; (6) for soe feasible an incentive copatible that satis es (5) ; an (7) with equality, that is that, r j q E V k + t Hj ; j H + p E V k + t j ; j E V B k B t Hj ; V B k B t j ; j H + E V k ; (7) r Hj q E V k + t Hj ; j H + p E V k + t j ; j p E V B k B t j ; V B k B t Hj ; j H + E V k NP ; (8)

22 Then, for all j; the contract can be suarize by fq j ; t j ; t Hj g. The feasibility constraints for r an r H iply the following constraints q j k B + E V k + t Hj ; j H + p E V k + t j ; j ` + E V B k B t Hj ; V B k B t j ; j H E V k NP ; ; (9) q j E V k + t Hj ; j H + p E V k + t j ; j + E V B k B t Hj ; V B k B t j ; j H E V k ; ; () q j k P + E V k + t Hj ; j H + p E V k + t j ; j H + p E V B k B t j ; V B k B t Hj ; j H E V k ; () H q j E V k + t Hj ; j H + p E V k + t j ; j p E V B k B t Hj ; V B k B t j ; j H E V k ; : () Construct the following sequence V j : if fq j ; t j ; t Hj g is such that (9) hols with equality, set V j V j: If fq j ; t j ; t Hj g is such that (9) is slack let V j be the value attaine by the contract that satis es (9) with equality. Since the transfers in ters of consuption goo are e ne by (7) an (8), this contract is still incentive copatible an feasible. Moreover, q j > q j an V j > V j: Therefore, li V j li V j V j! j! an without loss of generality one can concentrate on the sequences fv j g as e ne above in (6), such that the loan quantities fq j g satisfy (9) with equality. Having this constraint hol with equality iplies r j q j + k P. Since q j always, this iplies that all contracts along this sequence satisfy r > which is the sae as satisfying () with strict inequality: Suppose that for soe j () hols with equality. This iplies r Hj H q j + k P an since r j q j + k P this woul iply that participation constraint is slack, an that the proucer is giving the non-proucer all the gains fro the project. Fro lea 8, there exists a feasible an incentive copatible contract that attains a higher value that contract j an therefore, without loss of generality we can ignore sequences in which for soe eleents j, () hols with equality. 8. Uniqueness ea () can t bin in equilibriu. Proof. Suppose () bins, then, If r H bins, k B + ( c ( H ) t H + p c ( ) t ) c B ( H ) q ( c ( H ) t H p c ( ) t ) ( ) + p k B t H t : (3)