FINANCE RESEARCH SEMINAR SUPPORTED BY UNIGESTION

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1 FINANCE RESEARCH SEMINAR SUPPORTED BY UNIGESTION A Theory of Repurchae Agreement, Collateral Re-ue, and Repo Intermediation Prof. Piero GOTTARDI European Univerity Intitute Abtract Thi paper characterize repurchae agreement (repo) a equilibrium contract tarting from firt principle. We how that a repo allow the borrower to augment it conumption today while hedging both agent againt future market price rik. A a reult, afer aet will command a lower haircut and a higher liquidity premium relative to rikier aet. Haircut may alo be negative. When lender can re-ue the aet they receive in a repo, we how that there exit a collateral multiplier effect and borrowing increae. In addition, with collateral re-ue, lender might alo re-pledge the aet to third partie. In the model, intermediation arie a an equilibrium choice of trader and trutworthy agent play a role a intermediary. Thee finding are helpful to rationalize chain of trade oberved on the repo market. Friday, September 23, 2016, 10:30-12:00 Room 126, Extranef building at the Univerity of Lauanne

2 A Theory of Repurchae Agreement, Collateral Re-ue, and Repo Intermediation Piero Gottardi European Univerity Intitute Vincent Maurin European Univerity Intitute Cyril Monnet Univerity of Bern, SZ Gerzenee & Swi National Bank Thi verion: September, 2016 [Preliminary and Incomplete] Abtract Thi paper characterize repurchae agreement (repo) a equilibrium contract tarting from firt principle. We how that a repo allow the borrower to augment it conumption today while hedging both agent againt future market price rik. A a reult, afer aet will command a lower haircut and a higher liquidity premium relative to rikier aet. If collateral i very carce, haircut may alo be negative. We how that trader benefit from re-uing the collateral old in a repo. Firt, re-ue generate collateral multiplier effect a the economy can utain more borrowing with a imilar quantity of aet. Second, with collateral re-ue, lender might alo We thank audience at the Bank of Canada, the Third African Search & Matching Workhop, the 2015 Money, Banking, and Liquidity Summer Workhop at the St Loui Fed, The Philadelphia Fed, the Sverige Rikbank, Surrey and Univerity degli Studi di Roma Tor Vergata for generou comment. The view expreed in thi paper do not necearily reflect the view of the Swi National Bank. 1

3 re-pledge the aet to third partie. In the model, intermediation arie a an equilibrium choice of trader and trutworthy agent play a role a intermediary. Thee finding are helpful to rationalize chain of trade oberved on the repo market. 1 Introduction According to Gorton and Metrick (2012), the financial panic of tarted with a run on the market for repurchae agreement (repo). Their paper wa very influential in haping our undertanding of the crii. It wa quickly followed by many attempt to undertand repo market more deeply, both empirically and theoretically a well a call to regulate thee market. 1 A repo i the ale of an aet combined with a forward contract that require the original eller to repurchae the aet at a given price. Repo are different from imple collateralized loan in (at leat) one important way. A repo lender obtain the legal title to the pledged collateral and can thu ue the collateral during the length of the forward contract. Thi practice i known a re-ue or re-hypothecation. With tandard collateralized loan, borrower mut agree to grant the lender imilar right 2. Thi pecial feature of repo ha attracted a lot of attention from economit and regulator alike. 1 See Acharya (2010) A Cae for Reforming the Repo Market and (FRBNY 2010) 2 Aghion and Bolton (1992) argue that ecuritie are characterize 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 agent, 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 conlude, athough 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. 2

4 Repo are extenively ued by market maker and dealer bank a well a other financial intitution a a ource of funding, to acquire ecuritie that are on pecial, or imply to obtain a afe return on idle cah. A uch, they are cloely linked to market liquidity and o they are important to undertand from the viewpoint of Finance. The Federal Reerve Bank, the central bank in the United State, and other central bank ue repo to teer the hort term nominal interet rate. The Fed newly introduced revere repo are conidered an effective tool to increae the money market rate when there are large exce reerve. Repo thu became eential to the conduct of monetary policy. Finally, firm alo rent capital and ue collateralized borrowing and ome form of repo to finance their activitie or hedge expoure (notably interet rate rik, ee BIS, 1999). Thi affect real activitie, and o repo are alo an important funding intrument for the macroeconomy. Mot exiting reearch paper tudy pecific apect of the repo market, e.g. exemption from automatic tay, fire ale, etc., taking the repo contract and mot of it idioyncraie a given. Thee theorie leave many fundamental quetion unanwered, uch a why are repo different from collateralized loan? What i the nature of the economic problem olved by the repo contract? To anwer thee quetion, to undertand the repo market and the effect of regulation, one cannot preume the exitence or the deign of repo contract. In thi paper we characterize a imple economic environment where repo emerge a the funding intrument of choice. More preciely, we borrow technique from ecurity deign to derive the equilibrium collateralized contract. The interpretation a a repo contract i natural ince the borrower ultimately ell an aet pot combined with a promie to re-purchae at an agreed price. The model ha three period and two type of agent, a natural borrower and a natural lender, both rik-avere. The borrower i endowed with an aet that yield an uncertain payoff in the lat period. The payoff realization become known in the econd period and i reflected in the econd period price of the aet. To increae hi conumption in the firt period, the borrower could ell the aet to the lender in the pot market. However thi trade will expoe both partie to price rik in the econd period. Intead, the borrower can obtain reource from the lender by elling the aet combined with a forward contract promiing to repurchae the 3

5 aet in period 2. Unlike in an outright ale, a contant repurchae price in a repo hedge market price rik. Under limited commitment however, the borrower might not honor hi promie. Indeed, he may find it optimal to default if the value of collateral fall below the promied repayment 3. We aume that in addition to the lo of the collateral, a defaulting borrower incur a cot commenurate with the ize of default. To avoid thi wateful default, the repurchae price of the repo contract hould thu lie below a multiplier of the aet price proportional to thi default cot parameter. In high tate of the world or when the aet i abundant, thi contraint doe not bind and the repurchae price i contant. In low tate of the world however, the aet pay very little and the borrower exhaut hi borrowing capacity : the repurchae price increae with the pot market price. Uing thi equilibrium contract we derive comparative tatic for haircut and liquidity premia. Haircut increae with counterparty rik a a rikier agent can promie le income per unit of aet pledged. More riky collateral command a higher haircut and a lower liquidity premium. Compared to a afe aet, a riky ecurity pay le in bad time and more in good time. Since agent are contrained in bad time, thi i preciely when collateral i valuable. Hence the liquidity premium i higher for the afe aet. In good time, agent do not exploit the higher value of the rikier collateral ince the repurchae price become contant. Hence, compared to the afe aet, le of the riky aet payoff i pledged and the haircut i larger. In Section 4, we introduce collateral re-ue. In a repo, the lender indeed acquire ownerhip of the aet ued a collateral in the repo tranaction. In our model, a lender might re-ue a fraction of the aet he receive a collateral. We how that agent trictly prefer to re-ue a it increae the borrowing capacity of the repo eller. To fix idea, uppoe the collateral i perfectly afe and pay $100 in the econd period. The net interet rate i 0 o that $100 i alo the price of the aet in period 1. With the extra cot for default, a borrower can promie to repay more than $100 per unit of the aet in period 2, ay $110. The lender can the re-ue ome of the collateral by elling it back to the borrower. The latter can 3 In practice, even in the abence of outright default, trader opportunitically delay the ettlement of tranaction, a documented by Fleming and Garbade (2005). In our model, the repurchae price can be made tate-contingent which rule out default in equilibrium. The tatecontigency omehow mimic margin adjutment in actual tranaction. 4

6 now pledge another $110 per unit. With one round of re-ue, the borrower netted an extra $10 per unit. Thee trade can be repeated until no collateral may be re-ued. Overall, re-ue ha a multiplier effect ince a borrower can pledge more income with the ame quantity of the aet in thoe tate where he i contrained 4. Without the non-pecuniary penalty, thi extra borrowing capacity diappear and and re-ue doe not affect the equilibrium allocation, a reult in line with Maurin (2015). Overall, the model implie that collateral re-ue hould be more prevalent for aet that command low haircut and when the lender trading partner have low counterparty rik. Finally, Section 5 dicue the implication of collateral re-ue for the repo market tructure. We argue that ome participant naturally emerge a intermediarie when they can re-ue collateral. In practice, dealer bank indeed make for a ignificant hare of thi market by intermediating between natural borrower (ay hedge fund) and lender (ay money market fund or MMF). Thi might eem puzzling if direct trading platform are available for both partie to bypa the dealer bank 5. Our model rationalize intermediation with difference in trutworthine and ability to re-deploy the collateral. In our example, the hedge fund delegate borrowing to the dealer bank if the later i more trutworthy. Although there are larger gain from trade with the MMF, the hedge fund prefer borrowing from the dealer bank if he i more efficient at re-uing collateral. Indeed, through re-ue, one unit pledged to the dealer bank can then upport more borrowing in the chain of tranaction. Our model thu provide an endogenou theory for repo intermediation baed on fundamental heterogeneity between trader. Relation to the literature Gorton and Metrick (2012) argue that the recent crii tarted with a run on repo whereby funding dropped dramatically for many financial intitution. Subequent tudie by Krihnamurty et al. (2014) and Copeland et al. (2014) have qualified thi finding by howing that the run wa pecific to the - large - bilateral egment of the repo market. Recent theoretical work indeed highlighted ome 4 Our tripped down example uggeted that re-ue only work when haircut are negative. However, borrower only want to pledge more income in low payoff tate where they are contrained. In good tate, they might till want to pledge le income than the future value of the aet. The haircut average over tate and might thu be poitive. 5 In the US, Direct Repo TM provide thi ervice 5

7 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) emphaize the trade-off from the exemption from automatic tay for repo collateral. Lender eay acce to the borrower collateral may be privately optimal but collectively nefariou in the preence of fire ale, a point alo made by Infante (2013) and Kuong (2015). Thee paper uually take repurchae agreement a given while we want to undertand their emergence a a funding intrument. One natural quetion i to ak why borrower do not imply ell the collateral to lender? Mot paper including our highlight the role of the commitment to the repurchae price. In Narajabad and Monnet (2012), Tomura (2013) and Parlatore (2015), it allow lender to avoid earch friction in the pot market when reelling the aet. In contrat, our model i fully competitive but aet payoff are riky. A a reult, repo are eential becaue the repurchae price provide hedging againt price rik. Bigio (2015) and Madion (2016) emphaize aymmetry of information about the quality of the aet. There, the commitment to repurchae inulate uninformed buyer from the information-enitive part of the aet cah flow. The repo contract thu reemble a leaing agreement a in Hendel and Lizzeri (2002) or the optimal debt financing arrangement of DeMarzo and Duffie (1999), both of which mitigate advere election. Our model ha ymmetric information but aet payoff are random. With uncertainty, agent may alo want to pledge le than the future value of the cah flow when it i expected to be high (the hedging component). Beide the different economic motivation, thee work eentially identify repo with tandard collateralized loan. We account for the ale of collateral in a repo by conidering re-ue. In addition, our theory rationalize haircut ince borrower chooe repo when they could obtain more cah in the pot market 6. To derive the repo contract, we follow Geanakoplo (1996), Araújo et al. (2000) and Geanakoplo and Zame (2014) where collateralized promie traded by agent are elected in equilibrium. Our model differ from their a we allow for an extra non-pecuniary penalty for default in the pirit of Dubey et al. (2005). While our reult on the deign of repo contract carry through without thi penalty, it i 6 In particular, we do not need tranaction cot a uggeted by Duffie (1996). 6

8 crucial for the reult in Section 4 and 5 related to collateral re-ue. In the econd part of the paper, we indeed account for the tranfer of the legal title to the collateral to the lender, opening the poibility for re-ue. Singh and Aitken (2010) and Singh (2011) argue that collateral re-ue or rehypothecation lubricate tranaction in the financial ytem 7. However rehypothecation may entail rik for collateral pledger a explained by Monnet (2011). While Bottazzi et al. (2012) or Andolfatto et al. (2014) abtract from the limited commitment problem of the collateral receiver, Maurin (2015) how that re-ue rik eriouly mitigate the benefit from circulation. In our model indeed, re-ue relaxe collateral contraint only thank to the extra penalty for default for borrower beide the collateral lo. Aet re-ue then play a role imilar to pyramiding (ee Gottardi and Kubler, 2015). One difference i that lender re-ue the collateral backing the debt rather than the debt itelf a collateral. We tre the role of collateral re-ue in explaining repo market intermediation a in Infante (2015) and Muley (2015). Unlike thee paper, intermediation arie endogenouly in our model a trutworthy agent re-ue the collateral from riky counterpartie to borrow on their behalf. In an empirical paper, Ia and Jarnecic (2016) indeed uggeted that the fee baed view of repo intermediation whereby dealer gain from difference in haircut doe not tand in the data. The tructure of the paper i a follow. We preent the model and the complete market benchmark in Section 2. We analyze the optimal repo contract, including propertie for haircut, liquidity premium, and repo rate in Section 3. In Section 4, we allow for collateral re-ue and tudy intermediation in Section 5. Finally, Section 6 conclude. 2 The Model 2.1 Setting The economy lat three date, t = 1, 2, 3. There are two type of agent i = 1, 2 and only one good each period. Both agent have endowment ω in all but the 7 Fuhrer et al. (2015) etimate an average 5% re-ue rate in the Swi repo market over

9 lat period. Agent 1 i alo endowed with a unit of an aet while agent 2 ha none. Thi aet pay dividend in date 3. The dividend i ditributed according to a cumulative ditribution function F () with upport S = [, ] and with mean E[] = 1. In date 2, the realization of in date 3 i known to all agent. Thi i an eay way to model price rik at date 2. Preference from conumption profile (c 1, c 2, c 3 ) for agent 1 and 2 are: U 1 (c 1, c 2, c 3 ) = c 1 + v(c 2 ) + c 3 U 2 (c 1, c 2, c 3 ) = c 1 + u(c 2 ) + βc 3 where β < 1, u(.) and v(.) are repectively trictly concave and concave function. We aume u (ω) > v (ω) and u (2ω) < v (0), o that there are gain from tranferring reource from agent 1 to agent 2 in date 2 and the optimal allocation i interior. Thee preference contain two important element. Firt, a β < 1, agent 2 value le conumption in date 3 o that agent 1 i the natural holder of the aet in that period. Second, agent with concave utility function dilike conumption variability in period 2. While they may want to engage in borrowing and lending, agent are not able to fully commit to future promied payment. Borrower can pledge the aet a collateral in a repurchae agreement to alleviate thi friction. In Section 2.3, we define contract and tudy agent incentive to default formally. For clarity, we introduce and dicu the ingredient of our repo model here. When facing a default, a creditor can eize the aet ued a collateral, which he can hold or ell in the pot market. In addition, he recover a fraction α [0, 1] of the hortfall, that i the difference between the promied repayment and the market value of the collateral. Finally, a defaulting agent i incur a non-pecuniary cot equal to a fraction π i [0, 1] of the contractual repayment, meaured in conumption unit. Our aumption about default cot match everal feature of repo contract. Firt, in a repo, the lender get poeion of the collateral and may thu ell it when the borrower default 8. Second, repo are recoure-loan. Under the mot popular mater agreement decribed in ICMA (2013), an agent may claim the hortfall 8 While a repo i not characterized a a ale in the US, the exemption from automatic tay for repo collateral give imilar right for the lender. See alo footnote 2 on thi point. 8

10 to a defaulting counterparty in a cloe-out proce. Our partial recovery rate α capture the monetary value of delay or other impediment in recouping thi hortfall. Finally, the non-pecuniary component proxie for legal and reputation cot or loe from future market excluion 9. We allow the parameter π to depend on the identity of the borrower. The functional form will enure that price are linear function of trade. The lat building block of our model of repo i the ability for the lender to re-ue collateral. Again, thi follow naturally from the tranfer of ownerhip of the aet ued a collateral. We aume that lender 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 10. Our model highlight ome key feature of repo: they are collateralized loan, with recoure and the lender get poeion of the collateral. The environment i a imple et-up where thee feature will play out. There are two rik-avere agent and one wihe to borrow fund from the other. Limited commitment implie that the aet mut be ued to tranfer fund acro time. Unlike a combination of pot trade, a repo allow to hedge price rik thank to the repurchae price. However, it expoe trader to default rik. When the aet i carce, the ability to re-ue collateral prove valuable becaue borrower can increae leverage. Throughout the paper, market are competitive and agent are price-taker. 2.2 Perfect commitment A a benchmark, we olve for equilibrium when agent can perfectly commit to future promie. Market are thu complete and the equilibrium allocation i efficient. A a reult, marginal rate of ubtitution are equalized unle one agent i at a corner. We gue that thi i the cae between the firt and the econd 9 We thu depart from mot model of collateralized lending a la Geanakoplo (1996) which aume α = π = 0. A we argued, our aumption that α > 0 i natural for repo which are recoure loan. In addition, the non-pecuniary punihment (π > 0) i neceary to explain re-ue a we how in Propoition 6 and 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. In practice, the bulk of traded repo have hort maturity, limiting the cope for re-ue. 9

11 period 11. Let c i t denote agent i conumption in period t. We obtain the following equilibrium condition: u (c 2 2, ) = v (2ω c 2 2, ) (1) c 2 3, = 0 where we ued the reource contraint of period 2 to ubtitute for c 1 2, = 2ω c 2 2,. Intuitively, ince β < 1, agent 2 doe not conume in period 3 becaue he ha a lower marginal utility than agent 1. The implicit price for period 2 and 3 conumption are repectively u (c 2 2, ) and 1. To pin down the equilibrium allocation completely, we ue the budget contraint of agent 2 and obtain c 2 1, = ω u (c 2 2, )(c 2 2, ω). Thi expreion i poitive if : ω u (c 2 2, )(c 2 2, ω) (2) which we aume in the remainder of the text. In equilibrium, agent 1 borrow c 2 2, ω at a net interet rate r = 1/u (c 2 2, ) 1, uing unecured credit. Oberve that agent firt bet conumption (c 1 2,, c 2 2, ) in period 2 i determinitic although the aet payoff i already known. Indeed, rik avere agent prefer a mooth conumption profile. 2.3 Incomplete Market with Limited Commitment We now turn to the more intereting cae where agent face limited commitment. Oberve that pot trading i alway feaible, independently of the everity of the friction. To gain intuition about the benefit from uing repo, we how firt that agent cannot achieve the firt bet allocation by uing only pot trade, a it expoe them to price rik. 11 Conjecturing intead that marginal rate of ubtitution are equalized between the econd and the third period, we find a contradiction ince the reulting allocation i not budget feaible at the implied market price. 10

12 2.3.1 Spot Tranaction Suppoe agent can only trade the aet in a pot market. The pot market price in period 1 (rep. period 2 and tate ) i denoted p 1 (rep. p 2 ()). The price in period 2 indeed reflect the future known payoff of the aet. Let u denote a i 1 (rep. a i 2()) the aet holding of agent i after trading in period 1 (rep. period 2 and date ). The budget contraint of agent 2 in period 1 and 2 write c 2 1 = ω + p 1 a 2 1 c 2 2() = ω + p 2 ()(a 2 1 a 2 2()) Uing pot trade, agent 2 can implicitly lend to agent 1 if he buy the aet in period 1, that i a 2 1 > 0 and re-ell it in period 2, that i a 2 2() < a 2 1. We give a formal characterization of the equilibrium in the Appendix. Here, we tre our main point: a combination of pot trade can never finance the firt-bet allocation (1). Since agent 2 doe not want to conume in period 3 (β < 1), he would reell all the aet bought in period 1 o that a 2 2() = 0. Thi implie c 2 2() = ω +p 2 ()a 2 1. Agent 2 conumption mut then vary with becaue of price rik, while the firt bet conumption level c 2 2, i determinitic. Indeed, pot trade are too limited an intrument to tranfer wealth acro time. In particular, aet price rik generate undeirable conumption variability in period 2. A we will ee, the repo allow agent to commit to a repurchae price to hedge againt the aet payoff variability Trading in Spot and Repo Market In thi ection, we pecify the agent problem when they have acce to both pot and credit market. We naturally define a repo a the ale of an aet with a forward contract to buy it back. Compared to a combination of pot trade, a repo offer more flexibility a the repurchae price can hedge price rik. However, each agent i now expoed to the default rik of hi counterparty. A it will be clear, a repo eller effectively borrow with a loan collateralized by the aet old. Our model peak to repo pecifically becaue lender can re-ue a fraction of the aet pledged. Thi i a natural feature in a repo where the collateral i old. Definition 1. A repo contract i a price chedule f = {f()} S whereby the eller 11

13 agree to repurchae each unit old in period 1 at price f() in tate of period 2. When trading one unit of repo f, a eller i tranfer one unit of the aet to the buyer j. In exchange, he receive q ij (f) which i the price of the repo. We explain below how thi price may depend on trader type. Repo f i imilar to a tandard collateralized loan where the eller/borrower obtain q ij (f) per unit of aet pledged and promie to repay {f()} S. However, tandard model of collateralized borrowing do not account for collateral re-ue, a we do here. Borrower and Lender Default In a repo, the borrower promie to repay the lender who pledge to return the collateral. Hence, a dual limited commitment problem arie. To explicit each counterparty incentive to default, conider a trade of one unit of repo contract f between borrower i and lender j. Thi come without lo of generality becaue penaltie for default are linear in the amount traded. Borrower i prefer to repay rather than default if and only if: f() p 2 () + α(f() p 2 ()) + π i f() (3) The left hand ide i the repurchae price of the aet. For the borrower to repay, f() mut not exceed the total default cot. The firt component i the lo of the market value p 2 () of the collateral eized by the lender. The econd term α(f() p 2 ()) i the fraction of the hortfall recovered by the lender. The third component π i f() i the non-pecuniary cot for the borrower. We now turn to the lender incentive to return the aet 12. Oberve that he can only re-ue a fraction ν j of the collateral. We aume that he depoit or egregate the non re-uable fraction 1 ν j with a collateral cutodian. A a reult, he may only abcond with the re-uable fraction of the collateral 13. When 12 Technically, mot Mater Agreement characterize a fail and not outright default the event where the lender doe not return the collateral immediately. While our model doe not ditinguih the two event, lender alo incur penaltie when they fail. 13 It i eay to undertand why thi i optimal for him ex-ante. Firt, he i le likely to default ex-pot. Second, by definition, he would not derive ownerhip benefit from keeping the non re-uable collateral on hi balance heet. In the tri-party repo market, BNY Mellon and JP Morgan provide thee ervice. Our reult extend with ome modification to the cae where egregation i not available. Eentially, the no-default contraint of the lender might become binding for high value of, while it i not in our baeline pecification. 12

14 the lender default, the borrower get the 1 ν j unit of egregated collateral back. He alo recover a fraction α of the hortfall p 2 () f() (1 ν j )p 2 (), ymmetrically with the cae of a borrower default. Hence, the lender prefer to return the re-uable collateral rather than default if and only if ν j p 2 () f() + α(ν j p 2 () f()) + π j f() (4) The left hand ide i the cot of returning the re-uable unit of collateral at market value 14. The right hand ide i the total cot of defaulting. The lender then foregoe the payment f() from the borrower. He alo loe the fraction α of the hortfall ν j p() f() recovered by the borrower. Finally, he incur the non-pecuniary cot π j f(). Our model ha ubtle implication for the cot and benefit of default. Firt, the non-pecuniary punihment generate a deadweight lo. Thi hould encourage agent to trade default-free contract. However, becaue loan are recoure, borrower can indirectly pledge the endowment ω through the recovery payment when they default. To illutrate thi trade-off in a tark way, uppoe that the aet i worthle, that i = 0 for all. From (3), a borrower would default on any repo. Thi doe not mean that credit market hut down however. Indeed, uppoe he ell contract f uch that f() = 100 for all. The deadweight cot i 100π. However, the lender would recover α(100 p 2 ()) = 100α. The borrower effectively pledged 100α through default. Intuitively, when π i high and α i low, agent will avoid contract with equilibrium default. We how in the Proof of Propoition 3 that focuing on default-free contract come without lo of generality when the following condition hold: πv (ω) α(u (ω) v (ω)) (5) Intuitively, repo contract inducing default are dominated if the marginal cot of default πv (ω) exceed the marginal benefit α(u (ω) v (ω)) through the pe- 14 A lender might re-ue collateral and not have in on hi balance heet when he mut return it to the lender. However, oberve that he can 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

15 cuniary tranfer. We can now define the et of no-default repo contract F ij between two agent i and j a a function of the period 2 pot market price p 2 = {p 2 ()} S. To implify notation, we let θ i := π i /(1 α). Tranforming equation (3) and (4), we obtain the et of no-default repo. F ij (p 2 ) = { f [, ], ν j p 2 () f() p } 2() 1 + θ j 1 θ i Since agent i i le likely to default when θ i i high, we intepret thi parameter a a meaure of creditworthine. Oberve that the et F ij (p 2 ) i convex. Second, price and quantitie are linear function of quantity traded. In addition, we normalized all contract by unit of aet pledged. Hence, for any combination of multiple contract old by i, there exit an equivalent trade of a ingle repo contract. In the following, we thu call without ambiguity f 12 and f 21 the equilibrium contract. Agent optimization problem. We can now write the agent optimization problem. We let b ij (rep l ij ) denote the amount agent i borrow (rep. lend) with j uing equilibrium contract f ij (rep f ji ). We call q ij denote the price of the equilibrium contract f ij traded by agent i and j. When indexing a contract, the ubcript ij reflect the equilibrium choice of repo by agent i and j. The ubcript ij alo indexe the price to the extent that ome repo contract might have different price when traded by different pair of agent becaue of heteroegeneou incentive to default. For implicity, we write (6) 14

16 q ij := q ij (f ij ). max E [ U i (c i 1, c i 2(), c i 3()) ] (7) a i t,bij,l ij ubject to c i 1 = ω + p 1 (a i 0 a i 1) + q ij b ij q ji l ij (8) c i 2() = ω + p 2 ()(a i 1 a i 2()) f ij ()b ij + f ji l ij (9) c i 3() = a i 2() (10) a i 1 + ν j l ij b ij (11) b ij 0 (12) l ij 0 (13) At date 1, agent i ha reource ω+p 1 a i 0 and chooe aet holding a i 1, lending l ij and borrowing b ij. Given thee deciion, hi reource at date 2 i the endowment ω and the value of hi aet holding p 2 ()a i 1 a well the net value of the repo poition f ij ()l ij f()b ij. Equation (11) i the collateral contraint of agent i. A borrower (an agent for which b > 0) mut hold one aet per unit of repo contract old. He can buy thee aet either in the pot market (a 1 > 0) or in the repo market (l > 0). In the latter cae, however, only a fraction ν j of the aet purchaed can be re-ued. The collateral contraint alo how that a lender can take a hort poition on the pot market. Let indeed b = 0 and l > 0. Then, it can be that a 1 < 0 if ν > 0. With re-ue, a lender acquire ownerhip of the aet pledged by the lender and can then ell it. The only difference with a regular ale i the commitment to return the aet to the agent who initially old it. Definition 2. Repo equilibrium An equilibrium i a ytem of pot price p 1 and p 2 = {p 2 ()} S, a pair of repo contract (f 12, f 21 ) F 12 (p 2 ) F 21 (p 2 ) and their price q 12 and q 21, and allocation {c i t(), a i t, l ij, b ij } i=1,2.j i t=1..3, S uch that 1. {c i t(), a i t, l ij, b ij } j i t=1..3, S olve agent i = 1, 2 problem (7)-(13). 2. Market clear, that i a a 2 1 = a and b ij = l ji for i = 1, 2 and j i 3. For any contract f {f 12, f 21 }, there exit a price q( f) uch that agent do not trade thi contract. 15

17 Point 1 and 2 are elf-explanatory. Point 3 i a natural requirement to characterize the repo contract traded in equilibrium. A repo contract can be part of an equilibrium if and only if agent do not wih to trade an alternative contract f. For example, if f F 12 (p 2 ), the implicit equilibrium price q( f) mut be too low (rep too high) for agent 1 (rep. agent 2) to wih to ell (rep. to buy) thi contract. Hence, with our equilibrium definition, all contract are available to trade and agent elect their preferred contract taking price a given State-Contingent Repo We allow the repurchae price to depend on market obervable, namely the payoff of the aet ued a collateral. Thi expand the pace of feaible contract but might be viewed a unrealitic. However, we argue that tate-contingency of f ultimately reproduce the effect of margin call or repricing on the term of trade during the lifetime of a repo. In practice, counterpartie quote an interet rate r for the tranaction. The non-tate contingent repurchae price then obtain a f = (1 + r)q(f) where q(f) it the repo ale price. Suppoe now that the borrower pledge one unit of aet. The expected value of the aet i 100 and the borrower i to repay 80. If the aet price fall to 90, the lender call a margin and require the borrower to pot more collateral 15. After a margin call, the borrower mut then pledge more aet to utain the ame level of borrowing. Thi i imilar to leaving the quantity of aet unchanged and reducing the amount borrowed. One may till wonder about the implication of impoing a contant repurchae price to our contract pace. Eentially, default can become valuable even when condition (5) hold, becaue it introduce tate-contingency in the payoff function. Uing contraint (3) in the lowet tate, borrower i could not pledge more than p 2 ()/(1 θ 1 ) without defaulting. Raiing the fixed promied payment entail default cot in low tate a before. However, it now ha the additional benefit of increaing the amount pledged in high tate. We dicu thi trade-off in more detail after Propoition (3). 15 See the? guide for technical detail 16

18 3 Equilibrium contract without re-ue In thi ection, we characterize the equilibrium when agent cannot re-ue collateral, that i ν 1 = ν 2 = 0. Then, a repo contract i a tandard collateralized loan. To gain intuition, remember that agent 1 want to borrow in period 1 by pledging to repay c 2 2, ω in period 2. Conider the following trade pattern. Agent 1 ell all hi aet in a repo, that i b 12 = a and doe not trade pot. The maximum per-unit payoff of the repo i p 2 ()/(1 θ 1 ). Hence, in period 2 and tate, uing hi budget contraint, agent 2 conumption mut atify c 2 2() ω + ap 2() 1 θ 1 In low tate, thi amount may fall hort of c 2 2,. The repurchae price hould then be et at it maximum value f() = p 2 ()/(1 θ 1 ) ince agent are contrained. In high tate however, thi could raie agent 2 conumption too much. There f() hould be contant. We thu define a the olution to c 2 2, = ω + ap 2( ) (1 θ) = ω + a v (c 1 2, )(1 θ). (14) Thi i the minimal tate where the firt-bet allocation can be financed. econd equality follow from the obervation that p 2 () = /v (c 1 2()) ince agent 1 i the natural holder of the aet into period 3. Oberve that i decreaing with a and θ. Therefore, it i eaier to achieve the firt bet level of conumption the larger the tock of aet and the more agent 1 i able to commit. We have the following reult. Propoition 3. Define p 2 () a the unique olution - increaing in - to The ( p 2 ()v ω a p ) 2() = 0 if < 1 θ 1 (15) p 2 () = /v (c 1 2, ) if There i a unique equilibrium allocation with repo contract f where: 1. If (a i low), f() = p 2 ()/(1 θ 1 ) for all S 17

19 2. If [, ] (a i intermediate), p 2 () for f() = 1 θ 1 p 2 ( ) (16) 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 ) ]. In equilibrium, agent trictly prefer to trade repo over any combination of repo and pot trade in cae 1 and 2. They are indifferent to uing a combination of both in cae 3. The equilibrium contract reflect the optimal ue of the collateral value. A we explained, agent 1 can indeed pledge at mot p 2 ()/(1 θ 1 ) per unit of aet in tate. Thi amount increae in together with the collateral value p 2 (). When the collateral value i low, for, the borrowing contraint of agent 1 i binding and the repurchae price f() i equal to thi maximal amount. However, when the collateral value i high, agent 1 doe not want to borrow above the firt bet amount. Hence, the repurchae price become flat for. We call thi the hedging motive. The proof of Propoition 1 in the Appendix formalize thi argument enuring that agent do not want to trade another contract p. Figure 1 plot the equilibrium repo contract, in the cae v(x) = x. It i intereting to emphaize why agent prefer trading repo rather than pot. Suppoe indeed that agent 1 ell 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 a repo contract ˆf with ˆf() = p 2 (). Thi alternative trade i dominated for two reaon. When the collateral value i low, agent 1 can increae the amount he pledge from p 2 () to p 2 ()/(1 θ 1 ) with a repo. More importantly, when the collateral value i high, the equilibrium repo limit the repayment to agent 2 to the firt bet level. Non-tate-contingent repo. When repurchae price may be tate-contingent, agent chooe to trade a default-free repo, for which inequality (3) hold. Indeed, under aumption (5), deadweight cot from defaulting alway exceed the implicit tranfer through the partial recovery of the hortfall. 18

20 Thi i not true if we contrain repurchae price to be contant acro tate, that i f() = f. A we explained before, the highet repurchae price without default i f nd = p 2 ()/(1 θ), the dahed red line on Figure 1. Suppoe that f nd i traded in equilibrium and agent 1 conider allocating one unit of collateral to repo f d intead where f d > f nd. There exit a threhold (f d ) [, ] o that the borrower default on repo f d in tate < (f d ) and repay otherwie. The effective payoff to the lender can be written p 2 () + α (f d p 2 ()) if < (f d ) ˆf d () = f d if (f d ) There are now three effect. In low tate < (f d ), defaulting entail a cot πf nd. In addition, through default, the net additional amount pledged in thoe tate i ˆf d () f nd. Aumption (5) tate that the net benefit from thee two firt effect i negative. However, agent alo benefit in high tate (f d ) where the net additional amount pledged i f d f nd. Thee gain are not preent if the contract can be made tate-contingent. Without tate-contingency in the contract pace, the trade-off between default and economy efficiency i no longer trivial. It i eay to realize that the equilibrium contract f hould then belong to [f nd, p 2 ( )/(1 θ 1 )]. The lower bound i the default-free contract while the upper bound enure the firt-bet level of conumption in tate. Intuitively, f i cloer to the upper bound if the non-pecuniary cot π i low, the recovery rate α i high and the ditribution G i kewed toward high tate. 3.1 Haircut, liquidity premium, and repo rate In thi ection, we derive the equilibrium propertie of the liquidity premium and repo haircut. We compare the haircut and liquidity premia of two aet with different rik profile. We alo invetigate the role of counterparty rik, a meaured by θ. We define the liquidity premium L a the difference between the pot price of the aet in period 1 and it holding value. We thu obtain L = p 1 E[] 19

21 f() 1 θ 1 1 θ 1 + f nd Figure 1: Repo contract (v(x) = x). The holding value E[] follow naturally from the preference of agent 1. The liquidity premium i alo the hadow price of the collateral contraint. Hence, whenever the aet i carce and agent are contrained, the aet bear a poitive liquidity premium. Uing the equilibrium characterization, we can relate the liquidity premium to the repo contract and the allocation: L = E[f()(u (c 2 2()) v (c 1 2())] The liquidity premium i poitive if there exit (low) tate where agent are contrained becaue they cannot increae borrowing. In thoe tate, u (c 2 2()) > v (c 1 2()) which implie L > 0. The repo haircut i the difference between the pot market price and the repo price. Indeed, it cot p 1 to obtain 1 unit of the aet, which can be pledged a collateral to borrow q. So to purchae 1 unit of the aet, an agent need p 1 q 20

22 which i the downpayment or haircut 16. H p 1 q = E[(p 2 () f())v (c 1 2())] (17) where the econd equality follow from the firt order condition of agent 1 with repect to pot and repo trade. Finally, the repo rate i 1 + r = E[f()] q = E[f()] E[f()u (c 2 2())]. (18) When agent are contrained (cae i) and ii) of Propoition 3), we have u (c 2 2()) > u (c 2 2, ) for [, ] o that 1 + r < 1 + r. Agent 2 would like to lend at the frictionle interet rate 1 + r. However, agent 1 cannot increae borrowing ince he run out of collateral. The interet rate mut then fall for agent 2 to be indifferent. Interetingly, r < r when the liquidity premium L i trictly poitive. Remember that a poitive liquidity premium preciely indicate collateral carcity. Net repo rate r can thu be negative for aet with large liquidity premium. Thi i conitent with market data a reported in ICMA (2013). 17 We now derive the haircut and liquidity premium for the optimal price chedule f. Corollary 4. The haircut and liquidity premium are: L = ˆ ( ) u ω + a p 2() 1 θ ( ) 1 df () 1 θ 1 v ω a p 2() 1 θ ˆ ( H = θ ˆ 1 df () + 1 θ 1 1 θ 1 ) df () when, where p 2 () i the period 2 pot market price defined in (15). When agent can reach the FB allocation in all tate, that i, the liquidity 16 An alternative but equivalent definition i (p 1 q)/q. 17 The ICMA (2013) report that The demand for ome aet can become o trong that the repo rate on that particular aet fall to zero or even goe negative. The repo market i the only financial market in which a negative rate of return i not an anomaly. (p.12) and in footnote 6 negative repo rate have been a frequent occurrence and can be deeply negative. Alo, ee Duffie (1996), or Vayano and Weill (2008). 21

23 premium i L = 0 and the haircut lie in the following range: H [ E[] ], E[] 1 θ 1 1 θ 1 A Figure 1 how, the borrowing and hedging motive have oppoite effect on the ize of the haircut. In the tate < where agent are contrained, the borrower ue the maximum pledgeable capacity p 2 ()/(1 θ 1 ) per unit while the aet price trade at p 2 (). From expreion (17), thi contribute negatively to the haircut. However, in tate, agent 1 doe not wih to borrow more than c 2 2, ω. Hence, he doe not ue the full collateral value of the aet. In particular, the repayment f() i flat while the aet value p 2 () increae with. Thi contribute poitively to the haircut. The overall ign of the haircut depend on the weight on both region in the ditribution of. Finally, oberve that the haircut i not pinned down when ince everal (contant) repurchae price f are poible in equilibrium. The liquidity premium capture the value of the aet a an intrument to borrow over and above it holding value. Thi premium i zero when agent are not contrained in any tate, that i a hown by the expreion above. When >, the liquidity premium i an average of the pledging capacity of the aet /(1 θ 1 ) multiplied by the wedge in marginal utilitie Counterparty rik We now perform a comparative tatic exercie varying θ a proxy for counterparty quality. Indeed, a higher θ implie a higher punihment from defaulting and thu a uperior ability to honor debt. Although there i no default in equilibrium, the equilibrium contract reflect default rik. Uing the expreion derived in Corollary 4, we obtain that haircut increae with counterparty rik, or: H 1 = θ 1 (1 θ 1 ) 2 ˆ df () 0 Indeed, a Figure 2 how, a higher θ 1 increae the amount a borrower can raie per unit of the aet pledged. Thi naturally lead to a decreae in the haircut, by 22

24 p() 1 θ H 1 θ L L 1 θ L = H 1 θ H H L Figure 2: Influence of θ, with θ H > θ L increaing the ize of the region where f() > p 2 () while leaving the other region unchanged. When it come to the liquidity premium L, counterparty quality θ 1 ha an ambiguou effect. Firt, remember that agent 1 can pledge at mot ap 2 ()/(1 θ 1 ) in tate. Hence, an increae in θ raie the pledgeable amount 18. Agent 1 can thu borrow more in tate <, which reduce the wedge u (c 2 2())/v (c 1 2()) 1. between marginal utilitie. Thi effect, imilar to an increae in the aet available a, tend to reduce the liquidity premium. However, θ 1 alo increae the lope of the repurchae price 1/(1 θ 1 ) on thoe tate where the agent are contrained. A more income can be pledged when thi i mot valuable, the aet become a better borrowing intrument, which raie it price. Oberve that thi econd effect doe not arie when we vary the aet upply a. Thu, counterparty quality θ 1 can have a non-monotonic impact on the liquidity premium L. 18 Thi argument abtract from the negative equilibrium impact of θ on the pot market price p 2 () which i pinned down by the relationhip p 2 ()v (ω ap 2 ()/(1 θ)) = 0 for. However, one can eaily how that the net effect i poitive, that i [p 2 ()/(1 θ)] θ > 0. 23

25 3.1.2 Aet rik We now want to compare haircut and liquidity premium a a function of aet rikine. For thi purpoe, we introduce two aet with different rik profile but perfectly correlated payoff 19. We compute the liquidity premium of the afer aet relative to the rikier, the haircut that both aet carry, and the repo rate. A before, F [, ] but there are now two aet i = A, B with payoff ρ i (): ρ i () = + α i ( E[]), where α B > α A = 0. With α A = 0, aet A i our benchmark aet. Since α B > 0, aet B ha the ame mean but a higher variance than aet A. Indeed V ar[ρ α ] = (1+α)V ar[]. We chooe to conider two aet with perfectly correlated payoff to ignore the effect of rik haring on the tructure of the repo contract. Agent 1 i endowed with a unit of aet A and b unit of aet B, while agent 2 doe not hold any of the aet. It i relatively traightforward to extend the equilibrium analyi of the previou ection to thi new economy with two aet. The et of available contract conit of feaible repo uing aet A and B. For each aet i = A, B, the repo contract f i ue the maximum pledgeable capacity up to the tate where the firt bet level of conumption can be reached. We then prove the following reult. Propoition 5. The afer aet A alway ha a higher liquidity premium and a lower haircut than the rikier aet B. The key intuition behind the reult i the miallocation of collateral value induced by a mean preerving pread. Aet A and B have the ame expected payoff. However, ince ρ B () ρ A () = α B ( E[]), the riky aet pay relatively more in high tate (upide rik) and le in low tate (downide rik). Since agent are contrained for low value of, thi i preciely when collateral i valuable. Since the afe aet A pay more in thee tate, it carrie a larger liquidity premium. We now turn to the haircut. In high tate, the rikier aet B ha a higher payoff 19 We can prove imilar reult, in the one aet cae, by conidering a mean preerving pread. However, we would then compare quantitie acro equilibrium rather than within an equilibrium a we do here. 24

26 which mean that more income can be pledged compared to aet A. However, agent 1 doe not wih to borrow over the firt bet level. Hence agent do not exploit the the higher collateral value of the riky aet in high tate, implying a larger haircut. Oberve that without thi hedging motive, aet rik would have no impact on the haircut. So far, repo are inditinguihable from collateralized loan. Indeed, with ν = 0, the aet i immobile once pledged in a repo.the next two ection how that allowing for re-ue deliver new prediction. Firt, re-ue increae the borrowing capacity of agent 1. Second, the poibility to re-ue collateral may lead to endogenou intermediation in equilibrium. 4 The multiplier effect of re-ue In thi ection, we analyze the impact of collateral re-ue on equilibrium contract and allocation. i old to the lender. Thi i a natural feature of a repo trade where the collateral Re-ue ha been very much under crutiny following the crii (ee Singh and Aitken, 2010) ince a default on re-ued collateral may affect everal agent along a credit chain. While we do not model the conequence of uch default cacade, we provide the foundation for thi analyi by highlighting the benefit of re-ue. The lender, agent 2 i now able to re-ue collateral, that i ν 2 > 0. To undertand the potential benefit, conider the equilibrium without re-ue. Agent 2 (the lender) hold collateral pledged by agent 1. Re-ue free up a fraction ν 2 of thi collateral. Suppoe agent 2 then ell ɛ unit where ɛ i mall to agent 1 at the equilibrium price p 1. The marginal gain for agent 1 i null ince buying the aet i feaible without re-ue. The marginal gain to agent 2 i U 2 ɛ = p 1 E[p 2 ()u (c 2 2())] = η 2 1 where η 2 1 i the hadow price of the aet for agent 2. Uing the equilibrium 25