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 & Swi National Bank Thi verion: October, 2016 Abtract Why do market participant obtain fund through a repurchae agreement (repo) than by elling the aet pot? What determine the repo haircut? What i the role played by re-ue of the collateral old to the lender? To anwer thee quetion, we characterize the propertie of the repurchae agreement (repo) traded in equilibrium. We how that a repo allow invetor to borrow againt their aet holding while inuring both borrower and lender againt future market price rik. Repo on afer aet command a lower haircut and a higher liquidity premium relative to rikier aet. If collateral i carce, haircut may alo be negative. We how that trader benefit from re-uing the collateral old in a repo. Firt, re-ue allow the economy to utain more borrowing with the ame quantity of aet, thu generating a collateral multiplier effect. Second, with We thank audience at the Bank of Canada, EBI Olo, 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

2 collateral re-ue, lender might alo chooe to re-pledge the aet to third partie. We characterize the condition under which intermediation arie a an equilibrium choice of trader. Thee finding are helpful to rationalize chain of trade oberved on the repo market. 1 Introduction Gorton and Metrick (2012) make the cae that 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 o gain the option to ue the collateral during the length of the forward contract. Thi practice i known a re-ue or re-hypothecation. 2 from economit and regulator alike. Thi pecial feature of repo ha attracted a lot of attention Repo are extenively ued by market maker and dealer bank a well a 1 See Acharya (2010) A Cae for Reforming the Repo Market and (FRBNY 2010) 2 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. 2

3 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. They are cloely linked to market liquidity and o they are important to undertand from the viewpoint of Finance. Major central bank around the globe ue repo to teer the hort term nominal interet rate. The U.S. Federal Reerve recently introduced revere repo to better control hort term rate. Repo thu became eential to the conduct of monetary policy. Finally, firm 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 repo contract are trying to olve? 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 preent a imple model where we derive the propertie of the equilibrium repo. Trader prefer thee contract to pot trade and to collateralized contract that do not allow for re-ue. The model ha three period and two type of invetor, a natural borrower and a natural lender, both rik-avere, who lack the technology to commit to future promie. 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 then reflected in the 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 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 may find it optimal to default if the value of collateral fall below the promied repayment. We aume the punihment for default i the lo of the aet pledged a collateral together 3

4 with a penalty that may depend on the borrower characteritic and reflect hi creditworthine. To avoid default, the promied repayment hould not be too high relative to the market value of the aet. Both a borrowing and a hedging motive determine the repo contract traded in equilibrium. When the market value of the aet i low, the borrower cannot promie to repay much, a he would otherwie default, thu limiting hi borrowing capacity. In contrat, when the market value of the aet i high, the repurchae price of the equilibrium contract i contant thu hedging both invetor. Uing thi equilibrium contract we derive comparative tatic for haircut and liquidity premia. Haircut increae with counterparty rik, a a rikier invetor 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 invetor 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 however, invetor 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 tranaction. In our model, invetor alway chooe to re-ue the aet pledged a collateral whenever they have thi option. To fix idea uppoe the collateral i perfectly afe, ell for $100 in the firt period, and alo pay $100 in the econd period. Suppoe the borrower ha one unit of collateral and can promie to repay ay up to $110 per unit of the aet. So he obtain $110 in a firt round of repo with the lender. The lender can then re-ue ome of the collateral by elling it back to the borrower. The latter ue ome of hi $110 to purchae that amount of collateral and can 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. In thi imple example, the haircut i negative, but a we how in the text, a imilar reult obtain when haircut are poitive. Overall, re-ue ha a multiplier effect ince a borrower can pledge more income per unit of aet in thoe tate where he i contrained. We how that thi collateral multiplier effect depend on the 4

5 recoure nature of repo tranaction. There i no uch multiplier when the only punihment for default i the lo of collateral. 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. 3 Our model rationalize intermediation with difference in counterparty quality and ability to re-deploy the collateral. In our example, the hedge fund delegate borrowing to the dealer bank if the later i more creditworthy. 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 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 cot and benefit of the exemption from automatic tay for repo collateral. Lender eay acce to the borrower collateral may be privately optimal but collectively harmful in the preence of fire ale, a point alo made by Infante (2013) and Kuong (2015). 3 In the US, Direct Repo TM provide thi ervice 5

6 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? A firt trand of paper explain the exitence of repo uing earch friction (e.g. Narajabad and Monnet, 2012, Tomura, 2013, and Parlatore, 2015). 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 aymmetry of information about the quality of the aet to explain repo. There, the debt-like feature of a repo contract 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 repo exit in an environment with ymmetric information, where invetor trade contract on a Walraian market, but where the collateral ha uncertain payoff. Our theory alo rationalize haircut ince borrower chooe repo when they could obtain more cah in the pot market. 4 account for the ale of collateral in a repo by conidering re-ue. In addition, we To derive the repo contract, we follow the competitive approach of Geanakoplo (1996), Araújo et al. (2000) and Geanakoplo and Zame (2014) where collateralized promie traded by invetor are elected in equilibrium. Unlike thee paper where the only cot from default i the lo of the collateral, our model aim to capture the recoure nature of repo tranaction. We thu allow for a partial recovery of the hortfall and an extra 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, the recoure nature of repo i crucial to explain re-ue. Indeed, Maurin (2015) howed in a more general environment that the collateral multiplier effect diappear if loan are non-recoure. In the econd part of the paper, we 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 lubricate tranaction in the financial ytem. 5 However re-ue introduce the rik that the collateral taker doe not or cannot return the collateral a explained by Monnet (2011). Unlike In particular, we do not need tranaction cot a uggeted by Duffie (1996). 5 Fuhrer et al. (2015) etimate an average 5% re-ue rate in the Swi repo market over

7 Bottazzi et al. (2012) or Andolfatto et al. (2014), we thu account for the limited commitment problem of the collateral taker, when tudying the benefit of reue. Aet re-ue reemble pyramiding (ee Gottardi and Kubler, 2015) whereby a newly iued debt claim i ued a collateral. Collateral re-ue differ becaue of the two ided limited commitment problem which i abent with pyramiding. In addition, while pyramiding merely allow for a more efficient ue of collateral, re-ue ha a multiplier effect. 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 invetor 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 In thi ection we preent a imple environment where rik avere invetor have funding need to mooth their income. Thee invetor can trade ecuritie in a competitive financial market, but limited commitment require that borrowing i backed by ome collateral. invetor may trade the aet pot or imple collateralized contract without re-ue, but in equilibrium, they will chooe to trade repo. 2.1 Setting The economy lat three period, t = 1, 2, 3. There are two type of invetor i = 1, 2 and one conumption good each period. Both invetor have endowment ω in all but the lat period. Invetor 1 i alo endowed with a unit of an aet 7

8 while invetor 2 ha none. 6 Thi aet pay dividend in period 3. The dividend i ditributed according to a cumulative ditribution function G(.) with upport S = [, ] and with mean E[] = 1. The realization of become known to all invetor in period 2. A a conequence, price rik arie in period 2. Let c i t denote invetor i conumption in period t. Preference from conumption profile (c i 1, c i 2, c i 3) for invetor i = 1, 2 are: U 1 (c 1 1, c 1 2, c 1 3) = c v(c 1 2) + c 1 3 U 2 (c 2 1, c 2 2, c 2 3) = c u(c 2 2) + βc 2 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 invetor 1 to invetor 2 in date 2 and the optimal allocation i interior. Thee preference contain two important element. Firt, a β < 1, invetor 2 value le conumption in date 3, o invetor 1 i the natural holder of the aet in that period. Second, invetor with trictly concave utility function dilike conumption variability in period Arrow-Debreu equilibrium Here we how that the Arrow-Debreu equilibrium allocation (c 1, c 2 ) i not contingent on the realization of the aet dividend, a feature we ue extenively later on. In thi economy, (c 1, c 2 ) i characterized by equal marginal rate of ubtitution unle one invetor i at a corner. We gue and verify that thi i the cae between the firt and the econd period and 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, invetor 2 doe not conume in period 3 becaue he ha a lower marginal utility than invetor 1. The implicit price for period 2 and 3 6 Thi i for implicity only and we could eaily relax thi aumption, a none of the reult depend on it. 8

9 conumption are repectively u (c 2 2, ) and 1. To pin down the equilibrium allocation completely, we ue the budget contraint of invetor 2 and derive hi period 1 conumption 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, invetor 1 borrow c 2 2, ω at a net interet rate r = 1/u (c 2 2, ) 1. In the following we refer for implicity to the Arrow-Debreu equilibrium allocation a the firt bet allocation. Oberve that conumption in period 2 (c 1 2,, c 2 2, ) i determinitic although the aet payoff i already known. Indeed, rik avere invetor prefer a mooth conumption profile. 2.3 Financial Market With Limited Commitment While invetor want to engage in borrowing and lending, they may not be able to fully commit to future promied payment. Hence the firt bet equilibrium allocation cannot be utained and borrowing poition mut be collateralized. invetor can trade their aet pot or they can trade financial ecuritie in zero net upply in a competitive market. Spot Trade The pot market price in period 1 i denoted p 1 and the price in period 2 and tate i p 2 () which reflect the future known payoff of the aet. Let u denote a i 1 (rep. a i 2()) the aet holding of invetor i after trading in period 1 (rep. period 2 and tate ). Uing pot trade, invetor 2 can implicitly lend to invetor 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. In the Appendix we how that a combination of pot trade alone can never utain the firt bet allocation becaue p 2 () i a function of the tate which generate undeirable conumption variability in period 2 for both invetor. Repo In addition to pot tranaction, invetor can alo trade in period 1 ecuritie that are eentially debt or promie to deliver the conumption good in period 2. We let f = {f()} S denote the payoff chedule for a generic ecurity. An invetor elling f promie to repay f() in tate of period 2 per unit of ecurity. 9

10 We allow for all poible value of f() o that the market for financial ecuritie i complete. Short poition are backed by collateral a otherwie invetor may default. Without lo of generality, an invetor mut pot one unit of collateral per unit of ecurity old. The aet i a financial claim and not a real aet, which make it poible for the lender to re-ue the collateral pledged. However, collateral re-ue introduce a double commitment problem. In what follow we pecify what happen to the collateral and the punihment for default to capture the main feature of repo contract. The ownerhip of the collateral i tranferred to the lender who i able to re-ue the collateral. Specifically, 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. 7 The other fraction 1 ν i i egregated. 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 invetor i incur a non-pecuniary cot equal to a fraction π i [0, 1] of the contractual repo payment, meaured in conumption unit. A pecified, the ecuritie match everal feature of repo contract. Firt, they are loan collateralized by financial aet. Second, the lender get poeion of the collateral ince it i old by the borrower. He may then ell it when the borrower default but alo re-ue the aet pledged during the lifetime of the tranaction. 8 Finally, repo are recoure-loan. Under the mot popular mater agreement decribed in ICMA (2013), an invetor can indeed claim the hortfall 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. The nonpecuniary component proxie for legal and reputation cot or loe from future market excluion. 9 The parameter π may depend on the identity of the borrower. 7 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. 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. 9 The functional form will enure that price are linear function of trade. We thu depart 10

11 We allow the repo repurchae price f() to be tate contingent, a natural feature in our environment. Thi might be viewed a unrealitic ince repo uually pecify a fixed repayment. But note that margin call or repricing of the term of trade during the lifetime of a repo are way in which contingencie can arie. 10 In Section 6, we alo dicu contract with fixed repayment to how that our main reult hold qualitatively. 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 term 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 term π i f() i the non-pecuniary cot for the borrower. Notice that default i only meaningful when π i + α < 1 and we concentrate on thi cae from now on. We now turn to the lender incentive to return the aet. 11 Recall 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. 12 When from mot model of collateralized lending a la Geanakoplo (1996) which aume α = π = 0. A we argued, our aumption eem natural for repo which are recoure loan. 10 When he face a margin call, a trader mut pledge more collateral to utain the ame level of borrowing. Thi i equivalent to reducing the amount borrowed per unit of aet pledged. 11 Technically, mot Mater Agreement characterize a a fail and not an outright default the event where the lender doe not return the collateral immediately. While our model doe not ditinguih between fail and default, lender alo incur penaltie when they fail. 12 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 11

12 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 benefit of defaulting and keeping the re-uable unit of collateral evaluated at market value. 13 The right hand ide i the cot of defaulting. The firt term i the foregone payment f() from the borrower. The lender alo loe the fraction α of the hortfall ν j p() f() which i recovered by the borrower. Finally, he incur the non-pecuniary cot π j f(). Our model ha everal implication for the cot and benefit of default. Firt, the non-pecuniary punihment generate a deadweight lo. Thi hould encourage invetor to trade default-free contract. However, invetor may want to trade default-prone contract becaue borrower can indirectly pledge the endowment ω through the recovery payment (if α > 0) when they default. We how in the Proof of Propoition 1 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 pecuniary tranfer with the recovery of the hortfall. We can now define the et of no-default repo contract F ij between two invetor 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 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. 13 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. 12

13 the et of no-default repo. F ij (p 2 ) = { f [, ], ν j p 2 () f() p } 2() 1 + θ j 1 θ i (6) Since invetor i i le likely to default when θ i i high, we interpret thi parameter a a meaure of creditworthine or counterparty quality. Oberve that the et F ij (p 2 ) i convex and that price 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. We denote q ij (f ij ) the price of the contract f ij F ij traded by invetor i and j. When indexing a contract, the ubcript ij reflect the equilibrium choice of repo by invetor i and j. 14 a the repo price. invetor optimization problem. For implicity, we write q ij := q(f ij ) and refer to q ij We can now write the invetor optimization problem. Given price, invetor are chooing which contract to trade and the volume of trade for that contract. We formalize the equilibrium choice of the repo contract below in Definition (2.3). We let b ij (rep. l ij ) denote the amount invetor i borrow (rep. lend) with j uing equilibrium contract f ij (rep. f ji ). max a i t,bij,l ij E [ U i (c i 1, c i 2(), c i 3()) ] (7) 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) 14 The ubcript ij alo indexe the price to the extent that invetor may have different re-ue abilitie. 13

14 At date 1, invetor 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 ji ()l ij f ij ()b ij. Equation (11) i the collateral contraint of invetor i. When invetor i borrow, that i b ij > 0, he mut hold one aet per unit of repo contract old. He can buy thee aet either in the pot market, that i a i 1 > 0 or in the repo market if l ij > 0. In the latter cae, however, only a fraction ν j of the aet purchaed can be re-ued. For later reference, it i important to note from the collateral contraint that a lender can take a hort poition on the pot market. Let indeed b ij = 0 and l ij > 0. Then, it can be that a i 1 < 0 if ν i > 0. With re-ue, a lender acquire ownerhip of the aet pledged by the lender and can then ell it to create a hort-poition. Indeed, when the repo mature, invetor 2 would then have the obligation to return an aet that he doe not hold anymore. The only difference with a regular ale i that the lender who acquired the aet in a repo i committed to return the aet to the borrower. Definition. 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 ), 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 invetor 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 ij ( f) uch that invetor do not trade thi contract. Point 1 and 2 are elf-explanatory. Point 3 formalize the optimality condition for the choice of contract. A repo contract can be part of an equilibrium if and only if invetor 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 invetor 1 (rep. invetor 2) to wih to ell (rep. to buy) thi contract. Hence, with our equilibrium definition, all contract are available to trade and invetor elect their preferred contract taking price a given. 14

15 3 Repo market with no re-ue In thi ection, we characterize the equilibrium when invetor cannot re-ue collateral, that i ν 1 = ν 2 = 0. Then, a repo contract i a tandard collateralized loan taken by invetor 1. Since only the contract f 12 will be traded in equilibrium, we implify notation by etting f = f Equilibrium repo contract To gain intuition, remember 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 and given p 2 (), the maximum pledgeable income of invetor 1 i ap 2() Thi 1 θ 1. expreion obtain when invetor 1 ell all hi aet in a repo, that i b 12 = a, with the highet poible repurchae price p 2 ()/(1 θ 1 ). In low tate, thi income may fall hort of c 2 2, ω and the repurchae price hould indeed be et a high a poible becaue gain from trade are not exhauted. In high tate however, thi could raie invetor 2 conumption too much. There, the repurchae price f() hould be contant. We let be the threhold between thee two region. Formally, it i the olution to c 2 2, = ω + ap 2( ) (1 θ 1 ) = ω + a v (c 1 2, )(1 θ 1 ). (14) The econd equality follow from the obervation that p 2 () = /v (c 1 2()) in equilibrium, ince invetor 1 hold the aet into period 3. At invetor 1 can jut finance the firt-bet allocation if he pledge hi entire wealth. Oberve that i decreaing with a and θ 1. So it i eaier to achieve the firt bet allocation the larger the tock of aet and the more creditworthy invetor 1 i. Given the repo trade above, we can now determine the equilibrium p 2 () a the unique olution increaing in to ( p 2 ()v ω a p ) 2() = if < 1 θ 1 (15) p 2 ()v (c 1 2, ) = if 15

16 We have the following reult. Propoition 1. There i a unique equilibrium allocation where invetor trade repo contract f characterized a follow: 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 ( ) (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 ) ]. where p 2 i defined in (15). In equilibrium, invetor trictly prefer to trade repo over any combination of repo and pot trade in cae 1 and 2. They are indifferent between both in cae 3. The equilibrium contract reflect both invetor 1 deire to borrow in period 1 and the averion to the payoff rik in period 2. A we explained, invetor 1 can 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 invetor 1 i binding and the repurchae price f() i equal to the maximum pledgeable income for invetor 1. Thi borrowing motive explain why f() i increaing in for. However, when the collateral value i high, invetor 1 doe not want to increae the income pledged over the firt bet amount. Hence, the repurchae price become flat for. A a reult, conumption i contant thu hedging invetor againt the price rik for tate. We call thi the hedging motive. Given thi repo contract, we how in the Appendix that invetor do not want to trade any other contract. Figure 1 plot the equilibrium repo contract, in the cae v(x) = δx for δ (0, 1). It i intereting to emphaize why invetor prefer trading repo rather than pot. Suppoe indeed that invetor 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 16

17 f() δ(1 θ 1 ) /δ δ(1 θ 1 ) + Figure 1: Repo contract (v(x) = δx). repo contract ˆf with ˆf() = p 2 (). 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 ) with a repo. When the collateral value i high, the hedging motive inure invetor againt the price rik of pot trade with a flat repurchae price. We can aociate the equilibrium repurchae price to the repo rate r where: 1 + r = E[f()] q = E[f()] E[f()u (c 2 2())]. (17) When invetor are contrained (cae i) and ii) of Propoition 1), we have u (c 2 2()) > u (c 2 2, ) for [, ] o that 1 + r < 1 + r. invetor 2 would like to lend at the frictionle interet rate 1 + r. However, invetor 1 cannot increae borrowing ince he run out of collateral. The interet rate mut then fall for invetor 2 to be indifferent. 17

18 3.2 Haircut and liquidity premium In thi ection, we derive the equilibrium propertie of the liquidity premium and repo haircut. We define the liquidity premium L a the difference between the pot price of the aet in period 1 and it fundamental value. 15 We thu obtain L p 1 E[] The liquidity premium i alo the hadow price of the collateral contraint. It thu capture the value of the aet a an intrument to borrow 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 payoff of the repo contract and the marginal utilitie of the borrower and lender: L = E[f()(u (c 2 2()) v (c 1 2())] When, invetor are not contrained and c 2 2() = c 2 2, for all, and L = 0. When >, we have u (c 2 2()) > v (c 1 2()) for <, that i ome gain from trade are not realized in low tate becaue repo collateral i carce. The trictly poitive liquidity premium reflect the carcity: it i equal to the average of the repurchae price multiplied by the wedge in marginal utilitie. 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 invetor need p 1 q which i the downpayment or haircut. 16 H p 1 q = E[(p 2 () f())v (c 1 2())] (18) where the econd equality 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 tate < where 15 Formally, the fundamental value of the aet i it price in the Arrow-Debreu equilibrium. 16 An alternative but equivalent definition i (p 1 q)/q. 18

19 invetor are contrained, the borrower ue the maximum pledgeable capacity p 2 ()/(1 θ 1 ) per unit while the aet trade at price p 2 (). From expreion (18), thi contribute negatively to the haircut. However, in tate, invetor 1 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 Collateral carcity and counterparty quality In thi ection we derive comparative tatic for the liquidity premium and haircut relative to the carcity of collateral and the counterparty quality. Propoition 2. L i decreaing and H i increaing in the amount of collateral a. L = 0 whenever a i large enough that invetor can reach the FB allocation in all tate, that i. H decreae in counterparty quality θ 1 while the effect on L i ambiguou. When a increae, there i more aet to ue a collateral 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. The liquidity premium, which i the hadow price of collateral, geo down a more gain from trade are realized. Haircut increae with a becaue goe down a the quantity of aet a increae. Hence, there are le tate where the repurchae price contribute negatively to the haircut. A higher counterparty quality θ 1 decreae haircut ince the pledgeable capacity p 2 ()/(1 θ 1 ) increae. Intuitively, a better counterparty ha a higher ability to honor debt, which reduce the downpayment. Figure 2 illutrate the effect of an increae from θ 1L to θ 1H > θ 1L. The olid line repreenting the borrowing capacity hift to the left. 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. When it come to the liquidity premium L, counterparty quality θ 1 ha an ambiguou effect. Firt, an increae in θ 1 allow invetor 1 to borrow more 17 in tate 17 Although the effect i intuitive, the effect of an increae of θ 1 on the pledgeable amount 19

20 f() δ(1 θ 1H ) δ(1 θ 1L ) L δ(1 θ 1L ) = H δ(1 θ 1H ) H L Figure 2: Influence of θ, with θ H > θ L (v(x) = δx) <, which reduce the wedge u (c 2 2()) v (c 1 2()) in marginal utilitie. Thi effect, imilar to an increae in the aet available a, tend to reduce the liquidity premium. However, conditional on a value of, θ 1 determine the repurchae price chedule while a doe not. Indeed, on thoe tate where the invetor are contrained, the repo contract f() i equal to the maximum pledgeable capacity 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. Thu, counterparty quality θ 1 can have a non-monotonic impact on the liquidity premium L Aet rik Our model allow u to analyze the relationhip between haircut and liquidity premium of two aet with different rik profile. We characterize the equilibrium p 2 ()/(1 θ 1 ) i not traightforward. Remember indeed that the pot market price p 2 () i pinned down by the relationhip 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 on the pledgeable income i poitive, that i [p 2 ()/(1 θ 1 )] θ 1 > 0. 20

21 when invetor can trade both aet at the ame time rather than comparing quantitie acro equilibria with a ingle aet. 18 The extent to which invetor are contrained, that i the marginal utility wedge u (c 2 2()) v (c 1 2()) for any, i then the ame for both aet. A our exercie effectively control for market condition, we think it i more meaningful to bring it to the data. To make thing imple, we introduce two aet with perfectly correlated payoff but different rik to ignore the effect of rik haring on the tructure of the repo contract. A before, G[, ] 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[]. invetor 1 i endowed with a unit of aet A and b unit of aet B, while invetor 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. 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 3. 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 acro tate 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). The collateral i valuable when invetor are contrained, that i in low tate. 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 which mean that more income can be pledged compared to aet A. 18 For the ake of completene, we alo computed the comparative tatic related to a mean preerving pread. It implie a higher haircut, but the effect on the liquidity premium i indeterminate and depend on rik averion. 21

22 However, invetor 1 doe not wih to borrow over the firt bet level in thoe tate. Since le of the riky aet payoff i pledged, the haircut i larger. Oberve that without the hedging motive, aet rik would have no impact on the haircut. So far, repo are inditinguihable from tandard 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 invetor 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. Thi i a natural feature of a repo trade where the collateral i old to the lender. Many have dicued the conequence of re-ue in repo contract (ee Singh and Aitken, 2010). Our model allow to preciely characterize the benefit of re-ue and the effect on repo contract. The lender, invetor 2 i now able to re-ue collateral, that i ν 2 > 0 while for implicity we maintain ν 1 = 0 and we dicu the conequence of ν 1 > 0 in a remark, later in the text. To undertand the potential benefit, conider the equilibrium without re-ue. In the firt round, invetor 1 pledge all hi aet a collateral in a repo with the lender. At thi tage, invetor 2 (the lender) hold a unit of collateral. Allowing for re-ue free up a fraction ν 2 of thi collateral. Let u conider the following pattern of trade. After thi firt round, invetor 2 ell ɛ unit to invetor 1 (where ɛ i mall). Invetor 1 purchae thi aet and pledge it in a econd round of repo with the ame term. By definition of equilibrium, the marginal gain for invetor 1 from thi trade i null ince it i already feaible without re-ue. The marginal gain to invetor 2 i: U 2 ɛ = p 1 E[p 2 ()u (c 2 2())] q + E[f()u (c 2 2())] = θ 1 1 θ 1 p 2 () ( u (c 2 2()) v (c 1 2()) ) df () where we derive the econd equality in the Appendix. Hence, thi marginal gain i 22

23 trictly poitive when > (invetor are contrained) and θ 1 > 0. To undertand thi lat condition, notice that invetor 2 get p 2() 1 θ 1 ɛ in tate of period 2 from the econd round of repo, but ha to purchae ɛ unit of aet to return the full collateral on the firt repo. When invetor 2 reell ome collateral he receive from invetor 1, he effectively hort-ell the aet. So the net additional tranfer to invetor 2 in period 2 for all < where invetor are contrained i p 2 ()ɛ + p 2() 1 θ 1 ɛ = θ 1 1 θ 1 p 2 ()ɛ Thi tranfer i poitive and increae invetor 1 borrowing only if θ 1 > 0. In all other tate > gain from trade are exhauted, o the marginal impact of the net tranfer i null. Thee tep can be repeated over multiple round. At the end of the econd round, invetor 2 ha ɛ unit of the aet from which he can re-ue ν 2 ɛ and ell it to invetor 1. invetor 1 would then pledge an additional θ 1 1 θ 1 ν 2 ɛp 2 () in tate. After thi operation, invetor 2 ha (ν 2 ) 2 ɛ unit of re-uable aet. Iterating over thee round infinitely, the total pledgeable amount per unit of aet in tate obtain: M 12 p 2 () : = p 2() 1 θ 1 + = 1 1 ν 2 [ (ν 2 ) i θ 1 p 2 () 1 θ 1 i=1 1 1 θ 1 ν 2 ] p 2 () (19) where we call M 12 the borrowing multiplier, that i the pledgeable amount normalized for the value of one unit of the aet. The borrowing multiplier i trictly increaing in ν 2 a long a θ 1 > 0. The multiplier M 12 and the aet quantity held by invetor 1 determine hi borrowing capacity with invetor 2. Re-ue however induce two change to the original contract. Firt, it lower : the borrowing multiplier increae the number of tate where invetor can attain the firt bet ince more income can be pledged for a given quantity of collateral. Uing the multiplier, we can define (ν 2 ) a the minimal tate above which invetor 1 can pledge enough income to finance the firt bet allocation, that 23

24 i: ω + am 12 p 2 ( (ν 2 )) = c 2 2,. The econd change in the tructure of the repo contract come from the hort poition that invetor 2 build when he re-ue the collateral (ee the dicuion preceding the definition of a repo equilibrium). To unwind hi hort poition, invetor 2 ha to purchae the aet in the pot market in period 2, which expoe him to price rik. The repo contract will eek to correct thi additional rik. A before, when < (ν 2 ) the borrowing motive dominate the hedging motive, o that the tructure of the contract doe not change. But for > (ν 2 ) the contract will reflect the cot for invetor 2 of unwinding hi hort poition ν 2 p 2 () a keeping f() contant in that range would make invetor 2 uffer the price rik. We can then introduce the candidate equilibrium repo contract f(; ν 2 ) where: p 2 () if < (ν 2 ) 1 θ f(, ν 2 ) = 1 (ν 2 ) (1 θ 1 )v (c 1 2, ) + ν 2( (ν 2 )) (20) if (ν v (c 1 2 ) 2, ) In general, when re-uing collateral (ν 2 > 0), the lender could default on hi promie to return the aet. However, contract (20) atifie the no-default contraint of the lender (4) for any value of ν 2 ; the payment from the repo contract f(, ν 2 ) i alway higher than the value of the re-uable collateral ν 2 p 2 (). The following Propoition etablihe that invetor trade thi contract in an equilibrium with re-ue: Propoition 4. Collateral Re-ue. Let ν 1 = 0, ν 2 (0, 1), θ 1 > 0, and (0) = > (the firt-bet allocation cannot be achieved without re-ue). There i a unique equilibrium allocation where invetor 1 borrow uing repo contract f(ν 2 ) defined in (20) and invetor 2 re-ell collateral in equilibrium. There exit ν < 1 uch that for ν 2 > ν invetor reach the firt-bet allocation. A we dicued before, when θ 1 > 0, re-ue trictly increae the amount invetor 1 can pledge to invetor 2. Thi i valuable when invetor are contrained and want to expand borrowing in low tate. From the expreion of M 12 in (19), 24

25 it i clear that for ν 2 high enough, the firt-bet allocation can even be financed in the lowet tate. One can obtain the expreion for ν by etting (ν 2 ) =. Some imple algebra in the Appendix how ν = (1 θ 1 ). Propoition (4) how that invetor alway want to re-ue collateral when they can, and thi reult hold independently of the ign of the haircut. If the haircut i negative, it i intuitive that re-ue i beneficial to the natural borrower, invetor 1. Indeed, by buying 1 unit of aet from invetor 2, invetor 1 can pledge it back in a repo which yield him a net gain of p 1 +p F = H in period 1. Thi increae the conumption of invetor 1 whenever H < 0. When H > 0 it may eem that invetor 1 loe from re-ue. But thi logic i incomplete ince invetor 1 may alo gain by tranferring conumption acro tate in period 2. If the haircut i poitive, an incremental amount of collateral re-ue decreae invetor 1 conumption in period 1, but it moothe hi conumption acro tate in period 2, a he conume more in the high tate and le in the low tate. We how that the econd effect alway dominate the firt when H > 0. ν 2. The liquidity premium L can exhibit non-monotonicity in the re-ue factor While re-ue relaxe the collateral contraint, it alo increae the amount pledgeable in tate where invetor are contrained. Thi lat effect make the aet more valuable and can increae the liquidity premium. Thee two effect are reminicent of the comparative tatic with repect to counterparty quality θ. Finally, our model predict that the benefit of re-ue are larger when collateral i mot carce (that i > ) and there i evidence that thi i indeed the cae (ee Fuhrer et al., 2015). Remark. Re-ue through repo v. pot ale (ν 1 > 0) Since invetor 2 i the natural lender, it eem that the re-ue capacity of the borrower ν 1 hould play no role. But recall that invetor 2 i free to re-ue a fraction of the collateral in a pot ale or in a repo where he would borrow from invetor 1. invetor 1 i willing to engage in a repo a a lender a long a he can re-ue a high enough fraction of the aet to increae hi borrowing. Propoition 4 25