A Theory of Repurchase Agreements, Collateral Re-use, and Repo Intermediation

Transcription

1 A Theory of Repurchae Agreement, Collateral Re-ue, and Repo Intermediation Piero Gottardi European Univerity Intitute Vincent Maurin Stockholm School of Economic Cyril Monnet Univerity of Bern, SZ Gerzenee Thi verion: November, 2017 Abtract We how that repurchae agreement (repo) arie a the intrument of choice to borrow in a competitive model with limited commitment. The repo contract traded in equilibrium provide inurance againt fluctuation in the aet price in tate where collateral value i high and maximize borrowing capacity when it i low. Haircut increae both with counterparty rik and aet rik. In equilibrium, lender chooe to re-ue collateral. Thi increae the circulation of the aet and generate a collateral multiplier effect. Finally, we how that intermediation by dealer may endogenouly arie in equilibrium, with chain of repo among trader. We thank audience at the Third African Search & Matching Workhop, Bank of Canada, EBI Olo, SED 2016, EFA Olo 2016, London FTG 2016 Meeting, the 2015 Money, Banking, and Liquidity Summer Workhop at the St Loui Fed, The Philadelphia Fed, the Sverige Rikbank, Surrey, Eex, CORE and Univerity of Roma Tor Vergata for very helpful comment. 1

2 1 Introduction Gorton and Metrick (2012) argue that the financial panic of tarted with a run on the market for repurchae agreement (repo). Lender dratically increaed the haircut requeted for ome type of collateral, or topped lending altogether. Thi view wa very influential in haping our undertanding of the crii. 1 repo more deeply a well a call for regulation quickly followed. 2 Many attempt to undertand The very idea that a run on repo could lead to a financial market meltdown peak to their importance for money market. Overall, repo market activity i enormou. Recent urvey etimate outtanding volume at 5.4 trillion in Europe while $3.8 trillion to $5.5 trillion are traded in the US, depending on calculation. 3 The main market participant are large dealer bank and other financial intitution who ue repo for funding, to finance ecurity purchae, or imply to obtain a afe return on idle cah. For thee reaon, repo market have important implication for market liquidity, a Brunnermeier and Pederen (2009) illutrate. Dealer bank alo play a major role a repo intermediarie between cah provider and cah borrower. Finally, mot major central bank implement monetary policy uing repo, thu contributing to the ize and liquidity of thee market. Repo are imple financial intrument ued to lend cah againt collateral. Repo allow to carry out leveraged purchae of aet, which are pledged a collateral to obtain cah, or to borrow ecuritie. Technically, a repo contract i the ale of an aet combined with a forward contract that require the original eller to repurchae the aet at a future date for a pre-pecified (repurchae) price. The eller take a haircut defined a the difference between the elling price in a repo and the aet pot market price. Beide the haircut, a repo differ from a equence of pot trade becaue the eller commit to buying back the aet at a pre-et repurchae price. While a repo look very much like a imple collateralized loan, it ha two additional and important feature. It i a recoure loan and the borrower ell the collateral rather than merely pledging it. The lender thu acquire the legal title to the aet old and o the poibility to re-ue the collateral 1 Subequent tudie by Krihnamurty et al. (2014) and Copeland et al. (2014b) have qualified thi finding by howing that the run wa pecific to the - large - bilateral egment of the repo market. 2 See for example Acharya and Öncü (2013) and FRBNY (2010). 3 The number for Europe i from the International Capital Market Aociation (ICMA, 2016). The two figure for the US are from Copeland et al. (2014a) and Copeland et al. (2012) where the latter add revere repo. Thee number are only etimate becaue many repo contract are traded over the counter and thu difficult to account for. 2

3 before the forward contract with the eller mature. Repo, a well a the practice to re-ue the collateral, known a re-ue or re-hypothecation, have attracted a lot of attention from economit and regulator alike. 4 But a proper undertanding of the determinant and the conequence of repurchae agreement i miing. Why do invetor with aet in their portfolio chooe repo to raie fund? Why do dealer bank fund the purchae of riky ecuritie by imultaneouly elling them in a repo? Which economic force determine haircut? What are the conequence of collateral re-ue? Finally, why would borrower trade through dealer rather than directly with lender? To undertand repo market and their potential contribution to ytemic rik, a theory of repo hould anwer thee quetion while accounting for the baic feature of repo contract. In thi paper we analyze a imple competitive economy where ome invetor have ome funding need, but are unable to commit to future payment. To face their need, they can ue the aet they own, by either elling them in the pot market or in repo ale. Repo ale are characterized a loan contract exhibiting the key feature of repo decribed above. We how that in equilibrium invetor trade repo rather than pot. Furthermore, invetor value the option to re-ue collateral, that ditinguihe repo from tandard collateralized loan. In equilibrium they ue thi option a it allow to expand their borrowing capacity through a multiplier effect. Collateral re-ue alo hape the tructure of the repo market: intermediation by afer counterpartie, who ue repo to fund their purchae of aet, may endogenouly arie. The baeline verion of the model feature two type of rik avere invetor, a cah 4 Aghion and Bolton (1992) argue that ecuritie are characterized by cah-flow right but alo control right. Collateralized loan grant neither cah-flow right nor control right over the collateral to the lender unle the counterpartie ign an agreement for thi purpoe. A a ale of the aet, a repo automatically give the lender full control right over the ecurity a well a over it cah-flow. Re-ue right follow directly from ownerhip right. A Comotto (2014) explain, there i a ubtle difference between US and EU law however. Under EU law, a repo i a tranfer of the ecurity title to the lender. However, a repo in the US fall under New York law which i the predominant juridiction in the US. Under the law of New York, the tranfer of title to collateral i not legally robut. In the event of a repo eller becoming inolvent, there i a material rik that the right of the buyer to liquidate collateral could be uccefully challenged in court. Conequently, the tranfer of collateral in the US take the form of the eller giving the buyer (1) a pledge, in which the collateral i tranferred into the control of the buyer or hi invetor, and (2) the right to re-ue the collateral at any time during the term of the repo, in other word, a right of re-hypothecation. The right of re-ue of the pledged collateral (...) give US repo the ame legal effect a a tranfer of title of collateral. To conclude, although there are legal difference between re-ue and rehypothecation, they are economically equivalent (ee e.g. Singh, 2011) and we treat them a uch in our analyi. 3

4 poor invetor (natural borrower) and a cah rich invetor (natural lender). The borrower own an aet, whoe future payoff i uncertain. A large variety of poible repo contract, characterized by different value of the repurchae price, are available for trade. Due to borrower inability to commit, they may find it optimal to default on thee contract. The punihment for default i the lo of the aet old in the repo together with a penalty reflecting the borrower creditworthine. Hence there i a maximal amount that the borrower can credibly promie to repay, that depend on the future market value of the aet. Thi amount determine hi borrowing capacity. The recoure nature of repo contract implie that the borrowing capacity may exceed the future pot market price of the aet. Rik-avere invetor value the ability to borrow but dilike fluctuation in the future value of the aet price. We how that both a hedging and a borrowing motive determine the repurchae price of the repo contract that invetor chooe to trade in equilibrium. In the tate where the market value of the aet i low, the ability to borrow i limited. There the borrowing motive prevail and the repurchae price equal the borrowing capacity. In the other tate, where the aet price i high, the borrowing capacity i alo high and the hedging motive implie that the repurchae price i et at a contant level, below the borrowing capacity. Thee motive explain why invetor prefer repo contract over pot trade. The combination of a pot ale and a future repurchae of the aet in the pot market fully expoe invetor to the fluctuation in the future aet price. Moreover, a already noticed, a repo allow to pledge more income than the future value of the aet. We derive comparative tatic propertie for equilibrium haircut and liquidity premia. Haircut increae when counterparty quality decreae, becaue rikier borrower can credibly promie to repay lower amount, or when collateral i more abundant. We alo how that rikier aet command higher haircut and lower liquidity premia, ince higher rik entail a wore ditribution of collateral value acro tate relative to collateral need. Next, we analyze the benefit of collateral re-ue. In equilibrium lender reell in the pot market the collateral acquired via repo. Borrower in turn purchae thee additional amount of the aet o a to pledge them again in repo ale to lender. Thee trade increae the borrowing capacity of invetor. We find that the iteration of thee tranaction generate a collateral multiplier effect. The benefit of collateral re-ue are clear when haircut on the repo traded in equilibrium are negative, ince re-ue allow to increae the fund borrower can get for a given amount of the aet. We how that 4

5 re-ue i alo beneficial when haircut are poitive, although the reaon i different and more ubtle in thi cae, a we explain later. Since re-ue generate a multiplier effect, the benefit are larger when the aet i carce. Even though re-ue relaxe the borrowing contraint, it may till increae the liquidity premium of the aet ued a collateral. Thi effect arie becaue the propertie of the repo contract traded in equilibrium are alo affected by re-ue. In addition, our paper hed light on the way in which dealer bank ue repo to lever up and fund their activitie. Dealer bank obtain fund to purchae aet by uing thee aet a collateral in (tri-party or bilateral) repo. A a reult, they only need to tap into their cah holding to pay the repo haircut. Since haircut are uually mall, dealer bank can be highly levered. Dealer leverage i cloely related to their role in channeling fund between different invetor. In fact, dealer bank make for a ignificant hare of the repo market by intermediating between cah poor invetor, e.g. hedge fund, and cah rich invetor, e.g. inurance companie, or money market fund. To account for thee trading pattern, we extend the model by introducing a third type of invetor, to whom we refer a dealer. Dealer have limited cah and no aet, but a higher counterparty quality. We how that in thi environment, under ome condition we identify, dealer emerge in equilibrium a intermediarie between natural borrower and natural lender. Even though they could trade directly, natural borrower (ay, hedge fund) prefer to ell the aet in the pot market to dealer, who then pledge it a collateral in a repo with natural lender (ay, inurance companie). The emergence of dealer bank a leveraged intermediarie hinge on their uperior counterparty quality. Finally, we how that with collateral re-ue intermediation may alo occur via a chain of repo trade. In a repo chain, a natural borrower enter a repo with a dealer bank who in turn enter another repo with the natural lender. Intermediation via a chain of repo can arie when the dealer bank ha both a higher counterparty quality than the natural borrower and i better able at re-deploying collateral than the natural lender. Then, through re-ue, one unit pledged to the dealer bank can indeed upport more borrowing in the chain of tranaction. Thi explain why a natural borrower chooe to trade with dealer even when there are larger gain from trade with natural lender. 5

6 Relation to the literature Recent theoretical work highlighted ome feature of repo contract a ource of funding fragility. A a hort-term debt intrument to finance long-term aet, Zhang (2014) and Martin et al. (2014) how that repo are ubject to roll-over rik. Antinolfi et al. (2015) how that the benefit of an exemption from automatic tay 5 granted to repo may be harmful for ocial welfare in the preence of fire ale, a point alo made by Infante (2013) and Kuong (2015). Thee paper uually take the trade of repurchae agreement and their pecific feature a given while we want to undertand their emergence a a funding intrument. One natural quetion i why borrower do not imply ell the collateral to lender? A firt trand of paper explain the exitence of repo uing tranaction cot (e.g. Duffie, 1996) or earch friction (e.g. Narajabad and Monnet, 2012, Tomura, 2016, and Parlatore, 2016). Bundling the ale and the repurchae of the aet in one tranaction lower earch cot or mitigate bargaining inefficiencie. Bigio (2015) and Madion (2016) emphaize the role of informational aymmetrie regarding the quality of the aet to explain repo: their collateralized debt feature reduce advere election between the informed eller and the uninformed buyer a in DeMarzo and Duffie (1999) or Hendel and Lizzeri (2002). We how that invetor chooe to trade repo in an environment with ymmetric information, where market are Walraian, but where collateral ha uncertain payoff. One limitation of the work mentioned above i that the borrower chooe to ell repo if he can obtain more cah than in a pot ale of the aet, that i if the haircut i negative. Our analyi rationalize the ue of repo with poitive haircut when invetor are rik-avere. In addition, we account for the poible re-ue of collateral in repo by howing it benefit. To derive the equilibrium repo contract, we follow the competitive approach of Geanakoplo (1996), Araújo et al. (2000), and Geanakoplo and Zame (2014) where the propertie of the collateralized promie traded by invetor are elected in equilibrium. Unlike thee paper where the only cot of default i the lo of the collateral, our model aim to capture the recoure nature of repo tranaction. We thu allow for additional penaltie for default, ome of them non-pecuniary in the pirit of Dubey et al. (2005). While our reult on the characterization of repo contract traded in equilibrium remain valid alo in the abence of thee additional penaltie, the recoure nature of repo i crucial to 5 A hown by Eifeldt and Rampini (2009) for leae, uch benefit i in term of eaier repoeion of collateral in a default event. 6

7 explain re-ue. Indeed, Maurin (2017) howed in a more general environment that the collateral multiplier effect diappear when loan are non-recoure. Collateral re-ue i dicued by Singh and Aitken (2010) and Singh (2011), who claim that it lubricate tranaction in the financial ytem. 6 At the ame time, re-ue generate the rik that the lender, who receive the collateral, doe not or cannot return it when due, a explained by Monnet (2011). Unlike Bottazzi et al. (2012) or Andolfatto et al. (2015), we account for the double commitment problem induced by re-ue. The increae in the circulation of collateral obtained with re-ue alo arie with pyramiding (ee Gottardi and Kubler, 2015), where collateralized debt claim are themelve ued a collateral. However, the mechanim i different: in pyramiding, no two ided commitment problem arie and the recoure nature of loan alo play no role. We tre the role of collateral re-ue in explaining the preence of intermediation in the repo market, a in Infante (2015) and Muley (2015). Unlike in thee paper, in our analyi intermediation arie endogenouly ince direct trade between borrower and lender i poible. The tructure of the paper i a follow. We preent the model and the et of contract available for trade in Section 2. We characterize the equilibrium and the propertie of repo contract traded in Section 3, where we alo derive the propertie of haircut and liquidity premia. In Section 4 we examine the effect of collateral re-ue and in Section 5 how that intermediation arie in equilibrium. Finally, Section 6 etablihe the robutne of our finding to alternative pecification of the repurchae price and Section 7 conclude. The proof are collected in the Appendix. 2 The Model In thi ection we preent a imple environment where rik avere invetor have funding need. To accommodate thee need, they can ell an aet in poitive net upply and take hort poition in a variety of ecuritie in zero net upply. Thee trade occur in a competitive financial market. Short poition are ubject to limited commitment and require collateral. Trade in thee ecuritie capture the main ingredient of repo contract. 6 Fuhrer et al. (2015) etimate an average 10% re-ue rate in the Swi repo market over

8 2.1 Setting The economy lat three period, t = 1, 2, 3. There i a unit ma of invetor of each type i = 1, 2 and one conumption good each period. All invetor have endowment ω in the firt two period and zero in the lat one. Invetor 1 i alo endowed with a unit of an aet while invetor 2 ha none. 7 Each unit of the aet pay dividend in period 3. The dividend i ditributed according to a cumulative ditribution function G(.) with upport S = [, ] and mean E[] = 1. The realization of become known to all invetor in period 2, one period before the dividend i paid. A a conequence, price rik arie in period 2. Let c i t denote invetor i conumption in period t. Invetor have preference over conumption profile c i = (c i 1, c i 2, c i 3) decribed by the following utility function, repectively for i = 1, 2 : U 1 (c 1 ) = c v(c 1 2) + c 1 3 U 2 (c 2 ) = c u(c 2 2) + βc 2 3 where β < 1, u(.) and v(.) are repectively trictly and weakly concave function. We aume u (ω) > v (ω) and u (2ω) < v (0). 8 The main role of thi pecification i to capture the fact that invetor 1 want to borrow hort-term in period 1. He want to borrow becaue hi intertemporal rate of ubtitution between period 1 and 2 i lower than that of invetor 2. Hi borrowing hould be hort-term becaue invetor 2 dicount period 3 cah flow more than invetor 1 (β < 1). In addition, the concavity of the invetor utility over date 2 conumption implie that they dilike variability in repayment term in period 2. We implify the analyi by auming that their utility i linear over conumption at the other date, but linearity play no eential role in our reult. 2.2 Arrow-Debreu equilibrium To illutrate the baic feature of thi economy, it i ueful to conider it Arrow-Debreu equilibrium allocation (c 1, c 2 ). Conumption at date 2 i determined by equating the 7 Thi i for implicity only and we could eaily relax thi aumption, a none of the reult depend on it. 8 Oberve that a pecial cae of the preference a pecified above i v(.) = δu(.), where invetor only differ with repect to their dicount factor. 8

9 invetor marginal rate of ubtitution between period 1 and period 2 while invetor 2 doe not conume in the lat period: 9 u (c 2 2, ) = v (2ω c 2 2, ) (1) c 2 3, = 0 where we ued the reource contraint in period 2 to ubtitute for c 1 2, = 2ω c 2 2,. The price for period 2 and 3 conumption are repectively u (c 2 2, ) and 1, with period 1 conumption a the numeraire. Conumption in period 1 i then obtained from the budget contraint. Thu for invetor 2 we have c 2 1, = ω u (c 2 2, )(c 2 2, ω) and we will aume that ω u (c 2 2, )(c 2 2, ω) (2) in the remainder of the text. In the Arrow-Debreu equilibrium, invetor 1 borrow u (c 2 2, )(c 2 2, ω) from type 2 invetor in period 1 and repay with a net interet rate r = 1/u (c 2 2, ) 1 in period 2. In the following we refer for implicity to thi equilibrium allocation a the firt bet allocation. Oberve that conumption in period 2 (c 1 2,, c 2 2, ) i determinitic even though the aet payoff i already known. Indeed, rik avere invetor prefer a mooth conumption profile. 2.3 Financial Market With Limited Commitment We aume invetor can buy or ell the aet each period in the pot market. They can alo take long and hort poition in financial ecuritie in the initial period 1, before the uncertainty i realized. Unlike in the Arrow-Debreu framework, agent are unable to fully commit to future promied payment. A we will ee, thi implie that borrowing poition mut be collateralized and the firt bet allocation cannot alway be utained. Spot Trade Let p 1 and p 2 () denote the period 1 and period 2 pot market price of the aet when the realized payoff i. We let a i 1 (rep. a i 2()) be the aet holding of invetor i after trading in period 1 (rep. period 2 and tate ). Note that pot trade could be 9 Intuitively, ince β < 1 invetor 2 ha a lower marginal utility for period 3 conumption utility than invetor 1. 9

10 a way for invetor 1 to meet hi borrowing need: he could ell the aet in period 1 to carry only a 1 1 < a into period 2 and then buy it back in period 2 to carry a 1 2() > a 1 1 into period 3. However, a combination of pot trade alone can never utain the firt bet allocation. Indeed, ince p 2 () i a function of the tate, uch trade generate undeirable conumption variability in period 2 for both invetor. 10 Repo In period 1 invetor can alo trade promie to deliver the conumption good in period 2. We pecify thee financial ecuritie o a to capture everal feature of repo contract, and we will refer to thee ecuritie a repo. There are in particular three feature of repo contract we want to match. Firt, a repo agreement i a ale of an aet combined with a forward repurchae of that aet. The repurchae price i effectively the amount to be repaid by the eller and the aet play a role a collateral. We thu model ecuritie a collateralized loan. Second, in a repo agreement the lender acquire ownerhip of the aet and can ell or re-pledge the aet before the repo mature. We then alo aume that the collateral backing the ecurity i tranferred to the lender who enjoy a right to re-ue it. Finally, repo are recoure loan and the non-defaulting counterparty can claim the payment of any remaining hortfall and other expene. In our environment, default on the ecuritie entail additional cot beyond the lo of the aet pledged a collateral. We now decribe in detail how we model each of thee feature. (i) Collateralized loan - We let f = {f()} S denote the payoff chedule for a generic ecurity. An invetor elling ecurity f promie to repay f() in tate of period 2 per unit of ecurity old. We allow for all poible value of f o that the market for financial ecuritie i complete. A we how below, the invetor inability to commit implie that hort poition mut be backed by the aet a collateral. Without lo of generality, we et the collateral requirement to one unit of aet per unit of ecurity old. We refer to the payoff chedule {f()} S a the repurchae price, which can be tate contingent. Repo uually pecify a contant repayment but margin call or repricing of the term of trade during the lifetime of a repo are way in which contingencie can arie. 11 In Section 6 we examine the cae where repo contract are retricted to have a contant repurchae price and how that the main qualitative propertie of our reult 10 See Online Appendix C.1 for the formal argument. 11 When a trader face a margin call, he mut pledge more collateral to utain the ame level of borrowing. Thi i equivalent to reducing the amount borrowed per unit of aet pledged. 10

11 till hold. (ii) Ownerhip tranfer - The aet ued a collateral i a financial claim. borrower tranfer to the lender both the aet ued a collateral and the ownerhip title to thi aet. The lender can then re-ue thi aet a he pleae. 12 Specifically, we aume that invetor i can re-ue a fraction ν i of the collateral he receive, where ν i [0, 1]. We interpret ν i a a meaure of the operational efficiency of a trader in re-deploying collateral for hi own trade. 13 (iii) Recoure loan - In a collateralized loan with re-ue there i a double commitment problem. The borrower can fail to pay back the lender, but the lender can alo fail to return the collateral. In the following, we decribe the punihment faced by invetor when they default on their obligation. Beide the lo of the collateral pledged or of the right to receive the repayment due, the defaulting party incur additional cot ince the other party can claim compenation. Thi capture the recoure nature of repo tranaction. We tart by pecifying the penalty for borrower and then move on to the cae of lender default. Borrower Default When the borrower default, the lender can retain or liquidate the collateral. practice, the lender typically need to ell the aet at a dicount below it fair market value, and we model the cot of liquidation a a linear function of the market value of the collateral, that i κp 2 () per unit of aet, for ome κ 0. Then, the lender can claim the hortfall he face in a default, equal to the difference (when poitive) between the payment due, f() and the market value of the collateral, p 2 (), net of the cot aociated with the liquidation of the collateral. Thi i in line with the recoure loan feature of repo and the proviion in the event of default decribed in tandard Repo Mater Agreement. 14 We aume that the lender i only able to collect a fraction α [0, 1] of thi hortfall from 12 Thi ditinguihe the ituation under conideration from that, for intance, of a mortgage loan where the aet ued a collateral i a phyical aet and the borrower retain ownerhip of the collateral 13 Singh (2011) dicue the role played by collateral dek at large dealer bank in channeling thee aet acro different buine line. Thee dek might not be available for le ophiticated repo market participant uch a money market mutual fund or penion fund. 14 See Appendix A for further detail on the contractual proviion in an event of default according to the two tandard Mater Agreement. US dealer motly ue the Mater Repo Agreement (MRA, 1996) with US dometic counterpartie while non-us repo typically ue the General Mater Repo Agreement (GMRA, 2011). The treatment of default event i imilar in the two cae. The full document can be found at mla-and-mfta/. The In 11

12 the borrower. Thi partial recovery rate capture variou friction in recouping payment from unecured claim. 15 We alo poit that a defaulting borrower of type i incur an additional, non-pecuniary cot, equal to a fraction π i [0, 1] of the hortfall, meaured in conumption unit. Thi non-pecuniary component proxie for legal and reputation cot. It may thu depend on the borrower type and increae in the ize of the default. To analyze the borrower incentive to default, conider a trade of one unit 16 of repo contract f between borrower i and lender j. Borrower i prefer to repay rather than default in tate if and only if f() p 2 () + (α + π i ) max {f() p 2 ()(1 κ), 0} (3) The borrower will repay whenever the repurchae price f() doe not exceed the total default cot, given by the expreion on the right hand ide of (3). The firt term in that expreion i the market value p 2 () of the collateral eized by the lender. The econd term collect the fraction α of the hortfall recovered by the lender and the non-pecuniary cot π i max {f() p 2 ()(1 κ), 0} for the borrower. We ee from equation (3) that a borrower may only chooe to default if f() p 2 (). Hence (3) can be written a follow: Lender Default f() 1 (α + π i)(1 κ) 1 α π i p 2 () (4) We now dicu the punihment faced by a lender (of type j) when he fail to return the collateral. Recall that he can only re-ue a fraction ν j of the aet pledged. We aume that the non re-uable fraction 1 ν j i depoited or egregated with a collateral cutodian. 17 A a reult, the lender may only abcond with the re-uable fraction of the collateral. Abconding with the collateral i a default by the lender. 18 In thi 15 For intance, it might take time for the borrower to make thee payment. In addition, uch claim have a junior tatu if the borrower file for bankruptcy. 16 Thi i without lo of generality ince penaltie for default are linear in the amount traded, hence incentive to default do not depend on the ize of a poition. 17 It i eay to undertand why thi i optimal for the lender. He would not derive ownerhip benefit from keeping the non re-uable collateral on hi balance heet and egregation reduce hi incentive to default. In the tri-party repo market, BNY Mellon and JP Morgan provide thee ervice. If egregation i not available, incentive for the lender are clearly harder to utain. Thi can be een from equation (5) by taking ν j = Standard Repo Mater Agreement allow counterpartie to ditinguih between outright default by 12

13 event, the borrower get back the 1 ν j unit of egregated collateral and may alo claim any hortfall remaining after the cancellation of hi obligation to repay f(), equal to max {p 2 () f() (1 ν j )p 2 (), 0}. Like in a borrower default, we aume the borrower can only recover a fraction α of the claim and the lender incur a non-pecuniary cot equal to a fraction π j [0, 1] of the hortfall. Hence, the lender prefer to return the non-egregated collateral rather than default if and only if ν j p 2 () f() + (α + π j ) max {ν j p 2 () f(), 0} (5) The left hand ide of (5) i the benefit of defaulting given by the market value of the re-uable collateral held by the lender. 19 The expreion on the right hand ide i the cot of defaulting which include the foregone payment f() from the borrower, the fraction α of the hortfall max {ν j p 2 () f(), 0} recovered by the borrower, and the non-pecuniary cot π j max {ν j p 2 () f(), 0}. Oberve that condition (5) can be rewritten a follow: f() ν j p 2 () (6) that i, the lender prefer to return the collateral whenever the repurchae price f() exceed the value of the re-uable collateral he can abcond with. No-Default Repo The environment decribed extend the framework conidered in tandard model of collateralized lending, a for intance Geanakoplo (1996), to allow for recoure loan. When a borrower default, in addition to the lo of the collateral, he incur pecuniary and non-pecuniary cot. We aume thee cot are ufficiently low and the non-pecuniary the lender and failure to deliver the collateral on time. Late delivery of collateral i not necearily characterized a an event of default becaue finding the appropriate ecurity to deliver might take time in practice. We focu here on the firt one, in which cae the contraint impoed by the limited commitment of the lender are more relevant. 19 A lender might re-ue the collateral and not have it on hi balance heet when he mut return it to the lender. However, oberve that he can alway purchae the relevant quantity of the aet in the pot market to atify hi obligation. When he return the aet, the lender effectively cover a hort poition ν j. 13

14 cot i not too low compared to the recovery rate. Specifically, for every i: π i + α < 1 (7) α(u (ω) v (ω)) π i v (ω) (8) Aumption (7) implie that a borrower alway default if the loan i not collateralized. In our environment unecured borrowing i equivalent to a repo collateralized by an aet with zero value. Under (7) the no-default condition for the borrower, (3), never hold when p 2 () = 0. The econd property, condition (8), then enure that in equilibrium invetor prefer to trade default-free repo contract. When the recovery rate α i poitive, the recoure feature of loan in our environment implie that borrower could make higher payment to lender with contract inducing default. 20 However, by doing o, borrower incur a non pecuniary penalty which i a deadweight lo. A we will how in the proof of Propoition 1, under (8) uch deadweight lo alway outweigh the benefit of increaing the income pledged through default. To ummarize, invetor will ue the aet a collateral to utain borrower incentive, and chooe to trade ecuritie that do not induce default. We can now define the et of repo contract F ij that can be old by invetor i to invetor j o that no default occur. Thi et depend on the period 2 pot market price p 2 = {p 2 ()} S. To implify notation, we let θ i := (α + π i )κ/(1 (α + π i )(1 κ)). From condition (4) and (6) we obtain: F ij (p 2 ) = { f [, ], ν j p 2 () f() p } 2() 1 θ i (9) The upper bound of thi et, p 2() 1 θ i, contitute the borrowing capacity of invetor i per unit of aet held. It i increaing in θ i, which we can interpret a a meaure of the creditworthine or counterparty quality of invetor i. Oberve that the et F ij (p 2 ) i convex and that all contract have the ame collateral requirement given our normalization. Hence, for any combination of multiple contract old by i, there exit an equivalent trade of a ingle repo contract. We can thu focu our attention on equilibria where at mot one 20 It i eay to verify that, for f large enough, the actual payment to the lender after a borrower default, given by (1 α(1 κ))p 2 () + αf(), exceed the maximum amount a borrower can repay withtout defaulting, given by the right hand ide of (4). 14

15 contract i old by each agent and we ue f ij F ij (p 2 ) to denote the (unique) contract old by invetor i to invetor j. Invetor optimization problem. We can now write the optimization problem of an invetor of type i. Let q ij (f ij ) be the unit price of contract f ij. 21 The collection of thee repo price i q ij = {q ij (f ij ) f ij F ij (p 2 )}. Given the pot price and the price of the repo contract, invetor i chooe which contract to ell in F ij (p 2 ), which contract to buy in F ji (p 2 ), the volume of trade for the two contract a well a the trade of the aet in the pot market. Let b ij (rep. l ij ) denote the amount old (rep. bought) by invetor i to invetor j uing the choen contract f ij (rep. f ji ), that i the amount borrowed and lent. Thee contract mut be uch that invetor i doe not trictly benefit from trading any other exiting contract at the price he face. The quantitie traded of the two contract a well a the pot trade mut be a olution of the following problem: max E [ U i (c i 1, c i 2(), c i 3()) ] (10) a i 1,ai 2 (),bij,l ij ubject to c i 1 = ω + p 1 (a i 0 a i 1) + q ij (f ij )b ij q ji (f ji )l ij (11) c i 2() = ω + p 2 ()(a i 1 a i 2()) f ij ()b ij + f ji ()l ij (12) c i 3() = a i 2() (13) a i 1 + ν i l ij b ij (14) b ij 0 (15) l ij 0 (16) a i 2() 0 (17) Equation (11) i the budget contraint in period 1 for invetor i, where the reource available are ω + p 1 a i 0. Equation (12) i the budget contraint in period 2 for every realization of, with the reource available given by the endowment ω, the value of the invetor aet holding p 2 ()a i 1 and the net value of the repo poition f ji ()l ij f ij ()b ij. Equation (13) i the budget contraint in period 3. The collateral contraint of invetor i i pecified in (14). When invetor i ell contract f ij (that i b ij > 0), he mut pledge 21 Even in the abence of default the price may depend on the identitie of the agent trading the contract, to the extent that invetor may have different re-ue abilitie. 15

16 a collateral one unit of the aet per unit of repo contract old. He can atify thi requirement either by acquiring the aet in the pot market (that ia i 1 > 0), or in the repo market (that i l ij > 0). In the latter cae however, only a fraction ν i of the aet purchaed can be re-ued. It i important to oberve that, when invetor i buy but doe not ell a repo contract (that i l ij > 0 and b ij = 0), the collateral contraint may be atified with a i 1 < 0 if ν i > 0. Indeed, with re-ue agent i can ell in the pot market an aet that he acquired by purchaing a repo contract. When the repo mature, the invetor would then buy the aet to atify hi obligation to return it to the repo eller, thu covering hi hort poition. Hence (14) how that a lender can ue repo trade to take a hort poition in the aet. However, invetor cannot engage in naked hort ale of the aet. We can now define a competitive equilibrium (in hort a repo equilibrium) in the environment decribed: Definition. A repo equilibrium i a ytem of pot price p 1, p 2 = {p 2 ()} S, repo price q 12, q 21, a pair of repo contract (f 12, f 21 ) F 12 (p 2 ) F 21 (p 2 ) and an allocation {c i t(), a i 1, a i 2(), l ij, b ij } for i = 1, 2, j i, t = 1, 2, 3 and S uch that 1. {c i t(), a i 1, a i 2(), l ij, b ij } j i t=1..3, S olve problem (10) with contract (f ij,f ji ), j i, for invetor i = 1, Spot market clear: a a 2 1 = a and a 1 2() + a 2 2() = a for any. Repo market clear: b ij = l ji for i = 1, 2 and j i. 3. For every other contract f ij F ij (p 2 )\ {f ij } the price q ij ( f ij ) i uch that invetor i and j do not wih to trade thi contract, for j i = 1, 2. The equilibrium elect the repo contract that agent trade. Condition 3. enure that the market for other repo contract clear with a zero level of trade. 3 Repo market with no re-ue It i ueful to characterize firt the equilibrium when invetor cannot re-ue collateral, that i ν 1 = ν 2 = 0. We will how that in thi cae the only repo contract traded in 16

17 equilibrium i a contract old by invetor 1, who ha a funding need. In the remainder of thi ection, we imply refer to thi contract a f and to it price a q := q 12 (f). 3.1 Equilibrium repo contract To gain ome intuition, recall that, at the firt bet allocation, invetor 1 borrow in period 1 by promiing to repay c 2 2, ω in period 2. In a repo equilibrium, the maximum pledgeable income of invetor 1 in tate i ap 2 ()/(1 θ 1 ). Thi expreion i the amount invetor 1 can promie to repay when he ell all the aet uing the repo contract with a repurchae price equal to hi borrowing capacity p 2 ()/(1 θ 1 ). For low realization of p 2 (), thi payment may fall hort of c 2 2, ω. We will ee that in equilibrium invetor 1 ell all hi aet in a repo. At the choen contract, the repurchae price equal the invetor borrowing capacity in the tate where the value of the aet i low, while in the other tate it i independent of and lie below the borrowing capacity. In thoe tate, where p 2 () i relatively high, the pledgeable income allow to finance the firt bet allocation: the contant level of conumption c 2 2, in period 2 i then attained with a contant repurchae price. In thi equilibrium agent do not trade in the pot market in period 2. Hence all the aet i held by invetor 1 at the end of period 2. Invetor 1 conumption in period 2 i then: c 1 2() = ω af() and the equilibrium pot price i determined by the following firt order condition p 2 () = /v (c 1 2()) (18) A we aid above, f() i independent of in ome tate and equal to p 2 ()/(1 θ 1 ) in other tate. Thi, together with the above expreion, implie that p 2 () i trictly increaing in. Hence, there exit a threhold defined by the following equation c 2 2, = ω + ap 2( ) (1 θ 1 ) = ω + a v (c 1 2, )(1 θ 1 ). (19) uch that for all, the equilibrium conumption i equal to the firt bet conumption level (c 1 2,, c 2 2, ). For, the equilibrium conumption of invetor 1 i c 1 2() = ω 17

18 ap 2 ()/(1 θ 1 ). Oberve that i decreaing with a and θ 1. Hence, when the quantity of the aet i large and/or invetor 1 i ufficiently creditworthy, lie below and the firt bet conumption i achieved in all tate. Subtituting for c 1 2() in equation (18) yield the following expreion for the equilibrium pot price in period 2 ( p 2 ()v ω a p ) 2() = if < 1 θ 1 (20) p 2 ()v (c 1 2, ) = if The reult i formally tated in the following: Propoition 1. Repo Equilibrium. There exit an equilibrium where invetor only trade a repo contract f with the following characteritic: 1. If (a i low), f() = p 2 ()/(1 θ 1 ) for all S 2. If [, ] (a i intermediate), p 2 () for f() = 1 θ 1 p 2 ( ) (21) for (1 θ 1 ) 3. If (a i high), f() = f for all S where f [ p 2( ) (1 θ 1 ), p 2( ) (1 θ 1 ) ]. where p 2 i defined in (20). The equilibrium allocation i alway unique; the pattern of trade i alo unique in cae 1. and 2., when θ 1 > 0. Two force are haping the equilibrium repo contract: invetor 1 deire to borrow in period 1 and the averion of both invetor to rik in their portfolio return in period 2. When the value of the aet i low, for, the maximum pledgeable income of invetor 1 i inufficient to exhaut all gain from trade o that thi invetor i borrowing contrained. In thee tate, the repurchae price i equal to thi borrowing capacity. Hence, f() i increaing in and i only determined by invetor 1 borrowing motive. On the other hand, when the collateral value i high, for >, the maximum pledgeable income exceed invetor 1 borrowing need. Hence, the repurchae price i contant for 18

19 f() δ(1 θ 1 ) /δ δ(1 θ 1 ) + Figure 1: Repo contract (v(x) = δx). and allow invetor to perfectly hedge againt the price rik in thoe tate. The repurchae price i thu pinned down by thi hedging motive. Figure 1 plot the equilibrium repo contract in cae 2. when v(x) = δx for ome δ (0, 1). Note that when *, there i a unique equilibrium repo contract and invetor 1 ell all hi aet uing thi repo. When the collateral i abundant o that <, invetor attain the firt bet allocation in equilibrium. In thi cae, everal repo contract with contant repurchae price or a combination of repo and pot trade allow to upport the equilibrium allocation. A we how in the proof of Propoition 1, when invetor trade the repo contract f they do not want to trade other repo contract nor to engage in pot trade. In addition, there i no other equilibrium where a different contract i traded. To gain ome intuition about the firt point, uppoe intead that invetor 1 ell (ome of) the aet pot in period 1 and buy it back at the pot market price p 2 () in period 2. Thi i formally equivalent to elling a repo contract ˆf with ˆf() = p 2 () for each. Thi alternative trade i dominated for two reaon. When the collateral value i low, invetor 1 can increae the amount he pledge from p 2 () to p 2 ()/(1 θ 1 ) by elling the equilibrium repo contract. When the collateral value i high, the pot trade expoe invetor to fluctuation in 19

20 conumption which they can avoid by trading the equilibrium repo contract. conideration apply to trade involving other repo contract. We can aociate the equilibrium repurchae price to a repo rate r given by: Similar 1 + r = E[f()] q = E[f()] E[f()u (c 2 2())]. (22) When invetor are contrained (cae 1. and 2. of Propoition 1), the borrowing rate i lower than in the firt bet allocation: 1+r < 1+r ince u (c 2 2()) > u (c 2 2, ) for [, ]. In the repo equilibrium, invetor 1 i borrowing contrained o the equilibrium interet rate mut fall to induce type 2 invetor to lend an amount compatible with market clearing. 3.2 Haircut and liquidity premium In thi ection we derive the propertie of the liquidity premium and the haircut in the repo equilibrium. We define the liquidity premium L a the difference between the pot price of the aet in period 1 and it fundamental value. Setting the fundamental value of the aet to be it price in the Arrow-Debreu equilibrium, the liquidity premium i: L p 1 E[] The liquidity premium i alo equal to the hadow price of the collateral contraint. It thu capture the value of the aet a an intrument that facilitate borrowing over and above it holding value. Hence, whenever the aet i carce and invetor are contrained, the aet bear a poitive liquidity premium. Uing the equilibrium characterization, we can relate the liquidity premium to the repurchae price of the equilibrium contract and the marginal utilitie of the borrower and the lender: L = E[f()(u (c 2 2()) v (c 1 2())] (23) When the repo collateral i abundant ( ), invetor are not contrained and c 2 2() = c 2 2, for all, o that L = 0. When the repo collateral i carce ( > ), we have u (c 2 2()) > v (c 1 2()) for <, that i ome gain from trade are not realized in low tate and L > 0. 20

21 The repo haircut i the difference between the pot market price of the aet and the repo price in period 1. One unit of the aet can be bought in the pot market at price p 1 and old at the equilibrium repo price q. So to purchae 1 unit of the aet, an invetor need p 1 q, which i the down payment or haircut: 22 H p 1 q = E[(p 2 () f())v (c 1 2())] (24) The econd equality in (24) follow from the firt order condition of invetor 1 with repect to pot and repo trade. A Figure 1 how, the borrowing and hedging motive have oppoite effect on the ize of the haircut. In the region <, where invetor 1 i contrained, the repurchae price i equal to hi borrowing capacity p 2 ()/(1 θ 1 ) while the aet trade at price p 2 (). From expreion (24) we ee that thi contribute negatively to the haircut. On the other hand, in tate the repurchae price f() i contant while the aet price p 2 () increae with. Thi contribute poitively to the haircut (more preciely, thi i true when f() a pecified in (21) i maller than p 2 ()). Thee two cae correpond repectively to the dotted and dahed region in Figure 1. The overall ign of the haircut depend on the probability ma attributed to the two region by the ditribution of. Finally, oberve that the haircut i not uniquely pinned down when ince everal (contant) repurchae price f are compatible with the unique equilibrium allocation Collateral carcity and counterparty quality In thi ection we tudy the impact of collateral carcity and counterparty quality on the level of the liquidity premium and the haircut. Propoition 2. L i decreaing and H i increaing in the amount of collateral a. H decreae in counterparty quality θ 1 while the effect on L i ambiguou. When a increae, more aet can be old in a repo. Invetor 1 can thu borrow more in tate <, which reduce the wedge u (c 2 2()) v (c 1 2()) in marginal utilitie. A more gain from trade are realized, the hadow price of collateral, that i L, goe down. Haircut increae with the quantity a of the aet becaue decline when a increae. 22 An alternative but equivalent definition i (p 1 q)/q. 21

22 f() δ(1 θ 1 ) δ(1 θ 1 ) δ(1 θ 1 ) = δ(1 θ 1 ) Figure 2: Influence of counterparty quality, with θ 1 > θ 1 (v(x) = δx) Hence, there are le tate where the repurchae price i equal to the borrowing capacity, which contribute negatively to the haircut (ee Figure 1). A higher counterparty quality θ 1 decreae haircut ince the borrowing capacity p 2 ()/(1 θ 1 ) increae. Intuitively, a better counterparty ha a higher ability to honor debt, which reduce the required down payment. Figure 2 illutrate the effect of an increae from θ 1 to θ 1 > θ 1. The olid line repreenting the borrowing capacity hift counterclockwie. Thi naturally lead to a decreae in the haircut, by increaing the ize of the region where f() > p 2 () while leaving the other region unchanged. The increae in the ize of the firt region correpond to the area with dener dot on Figure 2. When it come to the liquidity premium L, counterparty quality θ 1 ha an ambiguou effect. An increae in θ 1 increae the borrowing capacity in tate < and o allow invetor 1 to borrow more, 23 reducing the wedge u (c 2 2()) v (c 1 2()) in marginal utilitie. Thi effect, imilar to the one we found for an increae in the aet available, tend to 23 To ae properly the effect of θ 1 on the borrowing capacity p 2 ()/(1 θ 1 ), one hould account for the effect of θ 1 on the equilibrium value of p 2 (). The period 2 pot market price i indeed determined by p 2 ()v (ω ap 2 ()/(1 θ 1 )) = 0 for, o that p 2 () decreae with θ 1. However, one can eaily how that the net effect i poitive, that i [p 2 ()/(1 θ 1 )]/ θ 1 > 0. 22

23 reduce the liquidity premium. However, the increae in the borrowing capacity due to a higher θ 1 alo affect the propertie of the equilibrium repo contract in the tate <, ince f() = p 2 ()/(1 θ 1 ). A more income can be pledged when thi i mot valuable, the aet become a better borrowing intrument, which raie it price and o it liquidity premium Aet rik Our model alo allow u to compare haircut and liquidity premia for aet with different rik profile. To thi end, we extend the environment by introducing a econd aet. For implicity, we aume that the econd aet ha a perfectly correlated payoff with the firt aet but carrie higher rik. Hence there i no poibility of hedging poition in one aet with an oppoite poition in the other aet. Therefore the pattern of equilibrium trade a well a the propertie of repo contract are determined by the ame principle a before. The econd aet pay a mean preerving pread of the dividend of the firt aet dividend, ρ() = + σ( E[]), where σ > 0. Invetor 1 i till endowed with a unit of the firt aet and alo own b unit of the econd aet, while invetor 2 i not endowed with any of the aet. The et of available contract conit of all feaible repo uing any of the two aet. It i relatively traightforward to extend the equilibrium analyi of the previou ection to thi new environment. For each aet, the repurchae price of the equilibrium repo contract i equal to the borrowing capacity of that aet in all tate where the firt bet level of conumption cannot be reached and i contant otherwie. We then etablih the following reult. Propoition 3. The afer aet alway ha a higher liquidity premium and a lower haircut than the rikier aet. The key intuition behind the reult i that the mean preerving pread of the dividend induce a miallocation of collateral value acro tate. While the two aet have the ame expected payoff E[], the rikier aet pay relatively more in high tate (where there i upide rik) and le in low tate (downide rik). An aet i particularly 23

24 valuable a collateral in low tate where invetor 1 i borrowing contrained. Since the afer aet pay more in thee tate, it carrie a larger liquidity premium. Turning now to the haircut, the rikier aet ha a higher dividend in high tate, which enure a higher borrowing capacity in thee tate compared to the afer aet. However, invetor 1 doe not benefit by borrowing more in thoe tate where he attain the firt bet level of conumption. Thu, ince a maller fraction of the aet dividend i pledged in the equilibrium repo for the econd aet, the haircut i larger. Oberve that, without the hedging motive, the repurchae price would alway be equal to the borrowing capacity and o, by the previou argument, aet rik would have no impact on the haircut. In the analyi above, we compared haircut and liquidity premia for two aet with different rik when invetor can trade them both at the ame time, rather than examining how equilibrium price vary in the one aet economy when the dividend rik i modified. 24 An advantage of our approach i that the ame tochatic dicount factor are ued to price both aet. Hence the comparion effectively control for market condition and it implication can be brought to the data in a more meaningful way. So far, the tranfer of ownerhip of the collateral from the borrower to the lender did not play any role in the analyi. Indeed, with ν 2 = 0, the aet i immobile once pledged in a repo to invetor 2. The next two ection how that allowing for re-ue deliver new prediction. Firt, re-ue increae the borrowing capacity of invetor 1. Second, the poibility of re-uing collateral may lead to endogenou intermediation in equilibrium. 4 Re-ue and the collateral multiplier In thi ection, we analyze the impact of collateral re-ue on equilibrium contract and allocation. Variou author (ee for intance Singh and Aitken, 2010) have treed the importance of thi feature of a repo trade where the collateral i old to the lender. Our model allow to preciely characterize the benefit of re-ue and the effect on repo contract, in the preence of limited commitment. We will how that invetor alway want to re-ue collateral becaue it expand their borrowing capacity. The propertie of the repo contract traded in equilibrium then need to be uitably adjuted. In particular, 24 For completene we alo performed thi econd analyi, finding that a mean preerving pread implie a higher haircut while the effect on the liquidity premium i indeterminate and depend on rik averion. 24