Transparency and collateral: central versus bilateral clearing

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Transparency and collateral: central versus bilateral clearing Gaetano Antinolfi, Francesca Carapella, and Francesco Carli 2018-017 Please cite this paper as: Antinolfi, Gaetano, Francesca Carapella, and Francesco Carli (2018). “Transparency and collateral: central versus bilateral clearing,” Finance and Economics Discussion Series 2018-017. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2018.017. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Transparency and collateral: central versus bilateral clearing∗ Gaetano Antinol(cid:28),†Francesca Carapella,‡and Francesco Carli(cid:159) February 28, 2018 Abstract Bilateral(cid:28)nancialcontractstypicallyrequireanassessmentofcounterpartyrisk. Central clearing of these (cid:28)nancial contracts allows market participants to mutualize their counterparty risk, but this insurance may weaken incentives to acquire and to reveal information about such risk. When considering this trade-o(cid:27), participants would choose central clearing if information acquisition is incentive compatible. If it is not, they may preferbilateralclearing,whenthischoicepreventsstrategicdefaultwhileeconomizingon costly collateral. In either case, participants independently choose the e(cid:30)cient clearing arrangement. Consequently, central clearing can be socially ine(cid:30)cient under certain circumstances. These results stand in contrast to those in Acharya and Bisin (2014), who (cid:28)nd that central clearing is always the optimal clearing arrangement. Keywords: Limited commitment, central counterparties, collateral JEL classi(cid:28)cation: G10, G14, G20, G23 ∗WeareverygratefultoNedPrescottforhisthoughtfuldiscussion,GarthBaughman,FlorianHeider,Marie Hoerova,CyrilMonnet,BorghanNarajabad,GuillaumeRocheteau,andWilliamRoberdsfortheircomments and suggestions. We also thank participants in the Spring 2013 Midwest Macroeconomics Meetings; 2013 SummerMeetingsoftheEconometricSociety;2013SocietyforAdvancementofEconomicTheoryConference; 2014ChicagoFedMoneyworkshop;SystemCommitteeMeetingonFinancialStructureandRegulationatthe Dallas Fed; First African Search and Matching Workshop; and seminar participants at the Federal Reserve Board, the Federal Reserve Bank of Atlanta, Bank of Portugal, Universities of Auckland, Bern, Birmingham, Cat(cid:243)lica-Lisbon, Porto, Tilburg, and UC Irvine. The views expressed in this paper are those of the authors and do not necessarily re(cid:29)ect those of the Board of Governors or the Federal Reserve System. †Washington University in St. Louis, gaetano@wustl.edu. ‡Federal Reserve Board of Governors, Francesca.Carapella@frb.gov. (cid:159)Deakin University, fcarli@deakin.edu.au. 1

1 Introduction Two important aspects characterize modern (cid:28)nancial contracting. One is that (cid:28)nancial institutions trade a variety of products bilaterally, such as over-the-counter (OTC) derivatives, repurchase agreements, and reserves held at the central bank.1 The second is the di(cid:30)culty in evaluating the risk that a counterparty will not ful(cid:28)ll its future obligations. To mitigate this risk by appropriately choosing contractual terms, such as prices and collateral, information about the exposure of a counterparty to various risks is necessary. This information, however, often lays within the walls of a bilateral relationship due to the high degree of specialization in understanding and pricing risks speci(cid:28)c not only to a certain (cid:28)nancial product, but to the interaction between the counterparties across other (cid:28)nancial markets. The recent (cid:28)nancial crisis has highlighted the systemic importance of this information.2 Both academic researchers and policy makers argued that during the crisis asymmetric information and lack of transparency in over-the-counter markets contributed to uncertainty over the risks that certain institutions posed, causing runs and exacerbating (cid:28)nancial distress.3 Consequently, particular attention has been devoted to the role of clearing institutions and to their potential in improving transparency in (cid:28)nancial markets.4 Mandatory clearing via a central counterparty (CCP), de(cid:28)ned below, has been at the center of (cid:28)nancial reforms both in the US and in Europe. However, the consequences of these reforms on the incentives of (cid:28)nancialmarketparticipantstoacquireinformationabouteachotherarenotwellunderstood. In this paper, we address the question of potential tradeo(cid:27)s between bilateral and central clearing with respect to market transparency. We develop a model where information about a counterparty is soft in the sense that it can be veri(cid:28)ed only by agents within the bilateral transaction. This assumption captures the idea that soft information is often related to 1See Krishnamurthy et al. (2014), http://www.newyorkfed.org/banking/tpr_infr_reform_data.html (2014), http://www.newyorkfed.org/markets/gsds/search.html (2014); and for the Federal Funds market Afonso and Lagos (2012a), Afonso and Lagos (2012b), Bech and Atalay (2010). 2Among many, see Caballero and Simsek (2009), Zawadowski (2011), and Zawadowski (2013). 3See Acharya and Bisin (2014), Pirrong (2009), and Powell (November, 21st 2013), Du(cid:30)e et al. (2010), Jackson and Miller (2013). 4See Acharya and Bisin (2014) on transparency, but also Biais et al. (2016), and Koeppl (2012) among others. 2

signi(cid:28)cant synergies across di(cid:27)erent projects and trades which are observable only to the agents involved in such a class of activities. Thus, soft information cannot be easily and publicly veri(cid:28)ed by a third party, or it is di(cid:30)cult to summarize and aggregate.5 In our economy, clearing arrangements a(cid:27)ect equilibrium outcomes, including incentives to acquire information about counterparties. Trading is bilateral and subject to two frictions: limited pledgeability of a counterparty’s future income, and private information about the degreeofpledgeabilityofincome. Costlymonitoringrevealstheextenttowhichacounterparty’s income is pleadgeable. This information, however, is not available to a third party, such as a clearing institution, which has to induce truthful reporting about the monitoring activity and its outcome by choosing contractual terms appropriately. When monitoring does not take place, counterparty pledgeability types cannot be part of contractual terms, and pooling contracts are the only feasible contracts. In this case, information is not available to (cid:28)nancial market participants, and in particular to clearing institutions; such lack of information may disrupt the provision of central clearing services. Because di(cid:27)erent clearing arrangements provide di(cid:27)erent incentives, the optimal clearing arrangement depends on the structure of (cid:28)nancial assets traded and the information set of marketparticipants. ThechoiceofclearingarrangementisalwaysconstrainedParetooptimal, and as a consequence any restriction on the contract traded or on the clearing arrangement reduceswelfare,despitetheabsenceofexternalitiesorsystemicriskconsiderations. Ourmodel is novel in this respect: it shows that crucial information acquired in a bilateral relationship may be lost when clearing services are transferred to a central counterparty, and it shows what characteristics of assets and trades are more likely to be associated with bilateral and central clearing arrangements. Clearing is the process of transmitting, reconciling and con(cid:28)rming payment orders or instructions to transfer securities prior to settlement. Clearing is bilateral when it takes place via traders’ respective clearing banks: under this arrangement each trader bears the risk that her bilateral counterparty may default. Traders manage this risk by requiring collateral to be 5See Stein (2002), Petersen (2004), Hauswald and Marquez (2006), Mian (2003). 3

posted. Central clearing is done by a third party, namely a central counterparty (CCP), that transforms the nature of the risk exposure of the two parties in a trade. A CCP is an entity that interposes itself between two counterparties, becoming the buyer to every seller and the seller to every buyer for the speci(cid:28)ed set of contracts.6 The substitution of the CCP as the sole counterparty for each of the two original traders in a bilateral exchange is called novation. Through novation of the original contract, the CCP observes all contracts traded by institutions for which it performs clearing services in a speci(cid:28)ed (cid:28)nancial market. Both all and speci(cid:28)ed are important components of this de(cid:28)nition: the(cid:28)rstoneimpliesthat,withinaspeci(cid:28)cmarket,theCCPhasinformationaboutthenetwork of trades across its members, which may not be available to the bilateral counterparties. The second implies that the CCP may lack information about its members, if that information is learned outside the speci(cid:28)ed set of contracts which the CCP clears, such as soft information. Previous research on CCPs, for example Acharya and Bisin (2014), has focused on the (cid:28)rst component, recognizing the potential welfare bene(cid:28)ts of CCP clearing. Instead, we focus on the second component and characterize the conditions under which CCP clearing might reduce welfare relative to bilateral clearing. The tradeo(cid:27) between bilateral and central clearing arises from i) two dimensions of risk against which traders value insurance, namely uncertain counterparty’s income and pledgeability type, and ii) private information about a counterparty’s pledgeability type, which introduces an adverse selection problem. The severity of the adverse selection problem interacts with the value of insurance in di(cid:27)erent ways in each clearing arrangement. With bilateral clearing, counterparty risk is managed through collateral requirements, which are costly in terms of foregone investment opportunities. Costly monitoring provides the information about the counterparty’s income necessary to tailor collateral requirements to the counterparty’s pledgeability type. With CCP clearing, uncertainty about a counterparty’s income is managed through loss mutualisation across members, as in Acharya and Bisin (2014), Koeppl and Monnet (2010), 6See Capital requirements for bank exposures to central counterparties and BIS glossary of terms used in payments and settlement systems, 2003. 4

and Biais et al. (2016). Loss mutualisation enables the CCP to diversify counterparty risk and save on collateral requirements. However, the ability of the CCP to pool risk across its members interacts in an important way with the supply of information about pledgeability types. When the CCP can induce each member to monitor a counterparty and truthfully reveal her type, it can implement separating contracts that make central clearing Pareto superior to bilateral clearing. We call an allocation that satis(cid:28)es these conditions incentivefeasible.7 When incentive feasible allocations do not exist, there is a trade-o(cid:27) between bilateral and centralclearing. CCPclearingnaturallymaintainstheabilitytoprovideinsurancebypooling risk over idiosyncratic shocks to income. Without the information generated by monitoring, however, the CCP cannot tailor contractstothequalityofthecounterpartiesinatrade,resultingineitherexcessiveorinsu(cid:30)cient collateral. Withbilateralclearingonlyinsuranceviacollateralrequirementsisfeasible. Thisis costly, but it is exactly this cost that preserves incentives to monitor. Intuitively, monitoring produces information useful in customizing collateral requirements to the type of counterparty and, when collateral is costly, this information is very valuable. If monitoring is not too costly, traders prefer bilateral clearing. The insurance provided by the CCP is not su(cid:30)cient to compensate for the loss of information about a counterparty’s type. Note that this result is not related to the common idea that CCPs may generate moral hazard and increase risk by providing insurance. In our economy the amount of risk is (cid:28)xed. Rather, it is due to the lack of incentives to acquire and transmit information about counterparties, which may result from the activity of the CCP. Related to the trade-o(cid:27) we analyze, certain practitioners and analysts have expressed concernsaboutrecentreformsofclearingarrangements. Gregory(2014),Section1.5,discusses possible dangers of introducing mandatory central counterparty clearing: (cid:16)A third potential problem[ofCCPclearing]isrelatedtolossmutualizationthatCCPsusewherebyanylossesin excess of a member’s own (cid:28)nancial resources are generally mutualized across all the surviving 7Because monitoring and truth-telling are incentive feasible, then the CCP tailors collateral requirements to counterparty types, and is able to implement transfers that make every participant better o(cid:27). 5

members. The impact of such mechanism is to homogenize the underlying credit risk such that all CCP members are more or less equal. ... Many (cid:28)rms trading derivatives (e.g. large banks and hedge funds) specialize precisely in understanding risks and pricing, and hence are likelytohavebetterinformationthanCCPsespeciallyformorecomplexderivatives.(cid:17) Indeed, (cid:16)One of the last futures exchanges to adopt a CCP was the London Metal Exchange in 1986 (again with regulatory pressure being a key factor).(cid:17) (Gregory (2014), Section 2.1.5.) The results and assumptions of our model are moreover consistent with the empirical evidence in Bignon and Vuillemey (2016). First, we assume that the CCP cannot directly monitor ultimate investors. Bignon and Vuillemey (2016) (cid:28)nd evidence of this information asymmetryinthefailureoftheCaisse de Liquidation des A(cid:27)aires et Marchandises (CLAM,a CCPclearingsugarfutures)inParisin1974, asretail investors were unsophisticated and nondiversi(cid:28)ed, did not have enough liquid (cid:28)nancial resources and that CLAM could not directly monitor ultimate investors.8 Second, we show the existence of equilibria where lenders do not have incentives to acquire information about their counterparties and/or pass it on to the CCP.Inequilibrium, then, theCCPisunabletochargemember-speci(cid:28)cmargins. Bignonand Vuillemey (2016) show that CLAM kept margins at a constant level across members, which was not su(cid:30)cient to ensure stable clearing and ended with the failure of a large CCP member and eventually of the CCP itself. Thepaperisorganizedasfollows: theremainderofthissectionprovidesaliteraturereview, Section 2 describes the model, Sections 3 and 4 characterize the contract with bilateral and central clearing respectively, and Section 5 characterizes the optimal contract and clearing arrangement chosen by traders. Section 6 concludes. 1.1 Related Literature Our paper relates to the literature that studies how changes in (cid:28)nancial market infrastructure in(cid:29)uence the exposure of market participants to default as well as market liquidity risk. Part of this literature has focused on the bene(cid:28)ts of CCP clearing. Carapella and Mills 8Bignon and Vuillemey (2016) go even further, theorizing risk-shifting behavior on the part of the CCP once it realized it was close to bankruptcy. 6

(2011) focus on netting and highlight a liquidity enhancing role for CCPs, which reduce trading costs and facilitate socially desirable transactions that would not occur with bilateral clearing. Koeppl and Monnet (2010) focus on novation and counterparty risk insurance: in their framework CCP clearing is the e(cid:30)cient arrangement for centralized trading platforms, and it improves on bilateral clearing for OTC trades by providing a better allocation of default risk. Acharya and Bisin (2014) focus on information dissemination and stress the welfare enhancing e(cid:27)ect of central clearing on transparency: CCP clearing can correct for an externality introduced by the non-observability of trading positions, when the exposure to third parties can cause a counterparty to default. Monnet and Nellen (2012) focus on twosided limited commitment and show that a CCP can improve on a segregation technology (de(cid:28)ned as a vault for collateral assets) through novation and mutualization. We di(cid:27)er from these papers as in our model the provision of clearing services by a CCP is endogenously limited, and central clearing may not be desirable. Du(cid:30)e and Zhu (2011) also show that introducing a CCP that clears a class of derivatives may lead to an increase in average exposure to counterparty default. However, their mechanism is very di(cid:27)erent from ours, as their focus is on netting. The authors show that when a CCP is dedicated to clear only one class of derivatives, the bene(cid:28)ts of bilateral netting between pairs of counterparties across di(cid:27)erent assets may be larger than the bene(cid:28)ts of multilateral netting among many clearing participants but within a single class of assets. In our model, we focus on novation and mutualization of losses as the key features of central clearing. In this respect, our paper is closer to Koeppl (2012), Biais et al. (2012), and Biais et al. (2016). In these papers moral hazard limits the provision of insurance. Biais et al. (2012) and Biais et al. (2016) show that central clearing can provide insurance against counterparty risk, but must be designed to preserve risk-prevention incentives. As a result traders end up bearing some of their idiosyncratic risk. Koeppl (2012) considers an environment with moral hazard, where collateral can serve either as an incentive device or as an insurance device. When a CCP cannot observe the degree of moral hazard and opts to use collateral as an incentive device, central clearing can have the unintended consequence of forcing collateral 7

to increase for all contracts, reducing market liquidity, and adversely a(cid:27)ecting market discipline. In our environment the CCP provides insurance via loss mutualisation as well, but, via novation, it interacts with adverse selection and costly monitoring. This interaction a(cid:27)ects traders’ incentives to acquire socially valuable information about their trading partners, and transmit it to the CCP. This mechanism is similar to what Pirrong (2009) suggests: information asymmetries between the CCP and its clearing members may result in an increase in counterparty risk at the CCP, especially for complex products traded by large and opaque (cid:28)nancial institutions. Ourpaperisalsorelatedtotheliteratureonpaymentsystems,inparticulartoKoeppletal. (2012), who study the e(cid:30)ciency of a clearing and settlement system in an environment with information asymmetry between the clearing institution and traders. In our model, trading is subject to an information asymmetry as well: traders can costly acquire soft information about their counterparty while the clearing institution cannot. However, the focus of our paper is the endogenous e(cid:27)ect of this information asymmetry on the credit risk faced by the clearing institution. In this respect our paper complements the one by Koeppl et al. (2012) by characterizing how central clearing can a(cid:27)ect transparency and risk management in (cid:28)nancial markets. Finally, it is worth highlighting that both our results and the economic mechanism at the core of our analysis are consistent with some empirical (cid:28)ndings on central clearing for credit default swaps. Although they cannot measure monitoring and transparency directly, Loon and Zhong (2014) (cid:28)nd that trading volume increase when credit default swaps are cleared centrally. This is an equilibrium outcome of our model, despite transparency may decrease with central clearing. 2 The Model Time is discrete and consists of two periods, t = 1,2. The economy is populated by two types of agents: a unit measure of lenders and a unit measure of borrowers. Lenders and borrowers have di(cid:27)erent preferences, and have access to di(cid:27)erent technologies. 8

There are two goods: a consumption good and a capital good. The capital good can be invested at time t = 1 and transformed with a linear technology into time t = 2 consumption. Only borrowers have access to this technology. The technology is indivisible, takes one unit of capital good at t = 1, and returns θ(cid:101)units of consumption good at t = 2; θ(cid:101)is a random variable with support {0,θ}, whose realization is unknown at the time of investment. We de(cid:28)ne p = Prob(θ(cid:101)= θ) to be the probability of success of investment. In the (cid:28)rst period, lenders receive an endowment of one unit of capital, while borrowers receive an endowment of ω units of consumption good. The consumption good can be stored from t = 1 to t = 2 by both lenders and borrowers. Borrowers have preferences biased towards consumption in the (cid:28)rst period relative to lenders. Speci(cid:28)cally, borrowers’ preferences are de(cid:28)ned over t = 1 consumption c and time 1 t = 2 consumption c , and are represented by the utility function 2 U (c ,c ) = αc +c α > 1 1 2 1 2 Borrowers have limited commitment to repay: a borrower can repudiate a contract and, after default, consume a fraction 1−λi of the output realization. There are two types of borrowers, distinguished by the extent to which they can pledge their income. A measure q of borrowers can pledge a fraction λH of their income, and a measure 1−q can pledge a fraction λL, where λH > λL. The type λi is private information of the borrower, but can be learned by a lender before trading by exerting monitoring e(cid:27)ort. The preferences of a lender are de(cid:28)ned over second period consumption x , and time-1 2 monitoring e(cid:27)ort e, according to the utility function V(x ,e) = u(x )−γ ·e 2 2 where u is strictly increasing and strictly concave, and e ∈ {0,1}. We further assume that lim u(cid:48)(x) = +∞. x→0 The mismatch between endowments and preferences over consumption goods generates 9

incentives to trade: lenders have capital but they need borrowers to use their technology to transform it into consumption goods. Nevertheless, trade is subject to two frictions. First, there is limited commitment; second, each lender is randomly matched and can only contract with one borrower. Trade is bilateral. When a lender and a borrower are matched with each other, they enter into a relationship described by a contract. The lender provides the contract to the borrower as a take-it-orleave-it (TIOLI) o(cid:27)er, which also speci(cid:28)es a clearing and settlement arrangement. 9 In the second period, settlement takes place either bilaterally or trough a CCP, according tothelenders’choice. Feasiblecontractsdi(cid:27)erdependingontheclearingarrangementinitially chosen. In the next sections, we de(cid:28)ne and characterize optimal contracts with bilateral and central clearing. Labeling agents as lenders and borrowers, and modeling the contract between them as a loan is meant to capture the counterparty (credit) risk of a (cid:28)nancial relationship. In this respect, it should not be thought of as a restriction on the set of contracts analyzed in our model relative to the set of contracts which are bilaterally and centrally cleared in reality. A loan in our model is the analog of any (cid:28)nancial obligation with a component of counterparty risk, which we formalize as limited commitment to honor such obligation. Whether the obligation is a repayment for a loan obtained in the past (cid:21)as in a repurchase agreement or a bond(cid:21) or the transfer of an asset (cid:21)as in an option which is exercised by its holder(cid:21) the limited commitment to keep promises previously made is intrinsically the same. Limited commitment is the pivotal friction in the model, and it introduces interesting interactions between the clearing arrangement, the terms of the contract traded, and the information acquired about the counterparty. 9When the commitment constraint is binding, the assumption of a TIOLI is without loss of generality because of transferable utility. 10

3 Contracts with bilateral clearing When a lender and a borrower are matched at the beginning of t = 1, the lender chooses whether or not to verify her counterparty type by exerting e(cid:27)ort, a decision denoted respectively by e = 1 and e = 0. If e = 1, the lender learns her counterparty type λi, i ∈ {L,H}. The lender then can o(cid:27)er a contract that prevents the borrower from defaulting strategically in equilibrium. Therefore, a contract with bilateral clearing and monitoring is a list (xi , xi , ci, ci , ci ), where 2,h 2,l 1 2,h 2,l xi and ci are respectively the lender’s and the borrower’s consumption in time t and state t,s t,s s, when the borrower’s type is i. The contract is indexed by the borrower’s type i, and second-period consumption is indexed by the idiosyncratic state s ∈ {l,h}. If the borrower acceptsthecontract, thelendertransferstheunitofcapitaltotheborrower, andtheborrower transfersω−ci unitsofconsumptiongoodtothelender. Wecanthinkofthetransferω−ci as 1 1 collateral as it denotes the amount of consumption good stored by the lender to be consumed at t = 2. In this respect ω −ci is akin to margins in (cid:28)nancial transactions (or a house in a 1 mortgage) as it preserves the value of the lender’s investment by insuring the lender against the borrower’s default.10 The borrower then chooses to invest the unit of capital, while the lender chooses to store the consumption good ω −ci. In the second period, the borrower is 1 entitled to consumption ci , whereas the lender is entitled to xi . 2,s 2,s Di(cid:27)erently, if e = 0, the lender does not verify the counterparty’s type, which remains private information of the borrower. In this circumstance, the lender commits to a mechanism that speci(cid:28)es a menu of contracts. Without loss of generality, we assume that the lender commits to direct revelation mechanisms, that is, a contract is executed after the borrower truthfully announces her type. However, since type i is private information, we cannot conclude, as in the previous paragraph, that the contracts o(cid:27)ered by the lender will always prevent borrowers from defaulting in equilibrium. In other words, default may not be just an o(cid:27)-equilibrium event, and it is necessary that we specify contracts to account for this possibil- 10Notice that we are assuming one sided limited commitment, only on the side of the borrower. Therefore lenders always return the collateral to borrowers if θ˜=θ. Storage is veri(cid:28)able. 11

ity. Formally, a strategy for a borrower is a pair (mi,σi) ∈ {λL,λH}×{0,1}, where mi is the reporting strategy and σi is the default strategy: σi = 1 means that the borrower defaults in equilibrium. A mechanism is a list of contracts (Σi,x i,∆ , xi , ci, ci , ci ) , where Σi 2,h 2,l 1 2,h 2,l i={L,H} is the lender’s default recommendation (contingent to the idiosyncratic state s = h) to a borrower that reports her type to be λi. Σi = 1 means that the lender recommends the borrower todefaultinequilibrium. ∆representsthepublichistoryoftheborrower’sdefault/repayment decision. ∆ = 1 if the borrower defaults in equilibrium, and ∆ = 0 if the borrower repays. We say that a contract is incentive-compatible if a borrower’s best strategy (mi,σi) is to report truthfully her type, mi = λi, and then follow the default/repayment recommendation, σi = Σi. The timing is similar to the case with monitoring: after reporting the type and accepting the ensuing contract, the borrower receives one unit of capital and transfers ω−ci 1 units of consumption good to the lender. In the second period, after the shock realization is known, the borrower chooses whether to default (σi = 1) or to repay (σi = 0). In case of repayment, the borrower is entitled to consumption ci , and the lender to consumption xi . 2s 2,s In case of default, the borrower’s consumption is equal to (1−λi)θ, while the lender is left with λiθ+ω−ci. 1 3.1 The contract with information acquisition Lenders matched with a λH borrower solve a similar problem to the one that lenders matched with λL borrowers face, with λL replaced by λH. Let V denote the value to a lender of a match with a borrower of type λi, once the lender i has paid the cost γ and knows the borrower’s type. Then lenders choose contracts (xi , xi , 2,h 2,l ci, ci , ci ) to solve 1 2,h 2,l i∈{L,H} (Pi) V = max pu(xi )+(1−p)u(xi )−γ (1) i 2,h 2,l (xi ,xi ,ci,ci ,ci )∈(cid:60)5 2,h 2,l 1 2,h 2,l + s.t. αci +pci +(1−p)ci ≥ αω (2) 1 2,h 2,l ω ≥ ci ≥ 0 (3) 1 ci +xi ≤ ω−ci +θ (4) 2,h 2,h 1 12

ci +xi ≤ w−ci (5) 2,l 2,l 1 ci ≥ (1−λi)θ (6) 2,h Constraint (2) is the borrower’s participation constraint: the borrower can always refuse to trade, and consume the endowment ω in the (cid:28)rst period. Constraint (3) is time t = 1 feasibility of the consumption plan, and likewise (4) and (5) are time t = 2 feasibility in states h and l respectively. Constraint (6) is the borrower’s individual rationality constraint: the borrower can default and consume 1−λi units of consumption (in the low state θ(cid:101)= 0, and the limited commitment to repay is not relevant). It is easy to see that at a solution both second-period feasibility constraints (4) and (5) should bind. Solving for xi and xi and replacing their values in the objective function (1), 2,h 2,l we can solve for (ci, ci , ci ). 1 2,h 2,l Because α > 1, a lender’s expected consumption is larger when the borrower consumes her whole endowment ω in t = 1, and nothing in t = 2. However, such a contract violates the individual rationality constraint (6), and leaves the lender with no consumption in the second period when the output realization is low, as implied by constraint (5). Therefore, the lender will always store some of the borrower’s endowment from time t = 1 to time t = 2. Collateral then plays two roles. First, it provides insurance to the lender against the risk of the low-consumption state at t = 2 when s = l. Second, it provides the borrower incentives to repay at t = 2. It does so indirectly, by storing consumption goods up to t = 2. The larger this amount, the easier it is for the borrower to satisfy the limited commitment constraint (6). Lemma 1 With bilateral clearing, if the lender pays the monitoring cost γ then i) ci < ω, ii) 1 ci = 0 and xi > xi . 2,l 2,h 2,l Lemma 1 implies that the solution to the contract with bilateral clearing is such that i) collateral is always positive, and ii) insurance is incomplete. Counterparty risk, the risk that the counterparty may be unable or unwilling to settle her obligations, is managed by 13

requiring collateral to be posted. The collateral requirement ω−ci insures against this risk. 1 However collateral must be used e(cid:30)ciently, since it is costly. Therefore ci = 0 and insurance 2,l is incomplete. First, consider the case when the collateral endowment ω is scarse relative to the counterparty type λi, namely ω ≤ ω(λi) ≡ (1−λi)pθ . Then, in the next lemma we show that the α scarcity of collateral provides the borrower with some additional rents relative to her outside option, even though the optimal contract asks the borrower to post all the available collateral in t = 1. (1−λi)pθ Lemma 2 If ω < , the participation constraint (2) is slack. In addition, the limited α commitment constraint (6) is binding and c = 0. This is area 4 in Figure 1. 1 (1−λi)pθ Next, consider the case when ω > . Let µ and η be the multipliers associated with α (2) and (6) respectively. The (cid:28)rst order conditions for optimality are −pu(cid:48)(ω−ci +θ−ci )+pµ+η = 0 (7) 1 2,h −pu(cid:48)(ω−ci +θ−ci )−(1−p)u(cid:48)(ω−ci)+αµ ≤ 0 (8) 1 2,h 1 with equality if ci > 0. Together with the complementary slackness conditions 1 µ{αci +pci −αω} = 0 (9) 1 2,h and η{ci −(1−λi)θ} = 0 (10) 2,h they fully characterize the solution to the problem. Let λ∗ be the unique value satisfying (cid:16) (cid:17) u(cid:48) (1−λ∗)pθ α−p α = (11) 1−p u(cid:48) (cid:0) θ− α−p (1−λ∗)θ (cid:1) α Intuitively, λ∗ is the smallest value of λ such that the limited commitment constraint is slack. For any λ ≤ λ∗, the limited commitment constraint (6) is binding because the quality 14

of the counterparty is relatively low, which is equivalent to a high borrower’s temptation to default. Lemma 3 With bilateral clearing, if the lender pays the cost γ to monitor the borrower and (1−λi)pθ ω > , then the participation constraint (2) binds. Moreover, α a) If λi < λ∗, then ci = (1−λi)θ and ci = ω− (1−λi)pθ . 2,h 1 α b) If λi > λ∗, and ω < (1−λ∗)pθ , then ci = αω > (1−λi)θ and ci = 0. α 2,h p 1 c) If λi > λ∗, and ω ≥ (1−λ∗)pθ , then ci = ω− (1−λ∗)pθ and ci = (1−λ∗)θ > (1−λi)θ. α 1 α 2,h The solution to the problem (Pi) is shown in Figure 1. The partition of the state space depends on two key parameters: the borrower’s endowment, ω, and the borrower’s type λi, which indicates the borrower’s quality. The interaction of the two determines whether both the limited commitment and the participation constraint bind, or only one of them binds. The temptation to default 1−λi measures the severity of the commitment problem, so thatwhenλi isrelativelylowtheborrowerhasrelativelyhighincentivetodefault,thesolution to (Pi) must be such that the limited commitment constraint binds. (1−λi)pθ Wecanthendistinguishtwoscenarios: whenω isrelativelylow(ω ≤ ),thescarcity α of collateral limits the possibility of using it to provide incentives to repay, and the borrower earns some rents for this reason.11 The participation constraint is slack and the limited commitment constraint binds. Finally, because ω is relatively scarce, c = 0. This solution is 1 described in area 4 in Figure 1. Whenωisrelativelyhigh,theparticipationconstraintbinds: thisistrueforsolutionsa,b,c in Lemma 3, which correspond to areas 1,2,3 in Figure 1. Whether the limited commitment constraintbinds12 ornot13 dependsontheseverityofthecommitmentproblemwithrespectto λ∗: ifλi ≤ λ∗ thentheborrower’stemptationtodefaultisstrong,andthelimitedcommitment constraint binds. 11Because ω is low, any allocation satisfying the (2) at equality would violate the limited commitment constraint (6). In this sense, in order to give incentives to the borrower to repay at t=2, her utility is larger than her outside option αω, despite the lender makes a TIOLI o(cid:27)er. 12Solution 2a and area 3 in Figure 1. 13Solutions 2b,2c and areas 1,2 in Figure 1. 15

If λi > λ∗ then the borrower’s temptation to default is low, so the limited commitment constraint is slack. In this case, the more important role of collateral is the insurance against thelowrealizationofθ˜. Let (1−λ∗)pθ ,forλ∗ solving(11),bethelevelofcollateralthatprovides α the lender with the e(cid:30)cient level of insurance. Thus, we can distinguish two sub-cases: the (1−λ∗)pθ (cid:28)rst,(ω ≤ ),inwhichthescarcityofcollateraldoesnotallowforthee(cid:30)cientprovision α of insurance, and ci = 0; the second, (ω > (1−λ∗)pθ ), in which the borrower’s endowment is 1 α relatively abundant, the e(cid:30)cient level of insurance is provided, ci > 0, and collateral level 1 ω−ci = (1−λ∗)pθ > 0 is constant with respect to λi and ω. Because the commitment problem 1 α is not severe in both sub-cases, that is to say the limited commitment constraint is slack, then the threshold level of ω that separates the two is a function of λ∗ rather than the actual temptation to default, λi. !() (3) (1) (PC)binds (PC)binds (LC)slack (LC c ) 1 b > in 0 ds c1>0 !( ) ⇤ (4) (2) (PC)slack (PC)bind (LC)binds (LC)slack c1=0 c1=0 ⇤ Figure 1: Solution to bilateral problem with info acquisition. 3.2 Bilateral clearing without monitoring In Section 3 we de(cid:28)ned a mechanism with bilateral clearing and no monitoring as a list of contracts (Σ , xi∆, xi , c i,0 , ci , ci) . i 2,h 2,l 2,h 2,l 1 i=L,H Conditional on no monitoring, the lender chooses a mechanism with bilateral clearing to 16 1

solve the following problem: (cid:88) (cid:104) (cid:110) (cid:111) (cid:105) Vbil,e=0 =max q p Σiu(xi1)+(1−Σi)u(xi0) +(1−p)u(xi ) (12) i 2h 2h 2l i=L,H (cid:104) (cid:105) s.t. αci +p Σi(1−λi)θ+(1−Σi)ci +(1−p)ci ≥ αω (13) 1 2h 2l ω ≥ ci ≥ 0 (14) 1 xi0 +ci ≤ ω−ci +θ (15) 2h 2h 1 xi1 ≤ ω−ci +λiθ (16) 2h 1 xi +ci ≤ ω−ci (17) 2l 2l 1 (cid:110) (cid:104) (cid:105) (cid:111) (λi,Σi) ∈ argmax αcmˆ +p σˆ(1−λi)θ+(1−σˆ)cmˆ +(1−p)cmˆ (18) 1 2h 2l (mˆ,σˆ) Constraint (13) is borrower i’s participation constraint, for i ∈ {L,H}. Constraint (14) is time t = 1 feasibility, (15) and (16) are time t = 2 feasibility in states (s,∆) = (h,0) and (s,∆) = (h,1) respectively; (17) is the time t = 2 feasibility condition in state l. Finally, constraint(18)istheincentivecompatibilityconstraintforaborroweroftypeλi: thestrategy pair(λi,Σi)isincentivecompatibleifthereisnootherstrategypair(mˆ,σˆ)thatyieldsahigher payo(cid:27). Notice that a borrower can deviate by reporting a di(cid:27)erent type mˆ (cid:54)= λi, by choosing a di(cid:27)erent default strategy σˆ (cid:54)= Σi, or both. The solution to this problem is analogous to that of the problem with central clearing and no monitoring, characterized in the next section. However, we show in Lemma 17 in Section 5 that any contract with bilateral clearing and no monitoring is dominated by a contract with centralclearing. Therefore,becausethegoalofthepaperistocomparebilateralversuscentral clearing, characterizing mechanisms with bilateral clearing and no monitoring is irrelevant to the question we want to address. Finally, notice that the solution to the lender’s decision problem with bilateral clearing is the same solution to the problem of a social planner subject to the fricitions of 1) private information of the borrower’s type; 2) limited commitment on the borrower’s side at t = 2; and3)bilateralmatching. Thelastconstraintimpliesthatthesocialplannercanallocateonly 17

the resources which are available within the speci(cid:28)c lender-borrower match, but not across di(cid:27)erent matches. These are the same constraints which a lender faces in choosing a menu of contracts to o(cid:27)er the borrower, implying that the solution to the lender’s problem with bilateral clearing is constrained e(cid:30)cient. 4 Contracts with CCP clearing Withcentralclearing, borrowersandlenderssubmitthecontracttheyagreeupontotheCCP, which novates the contract. With novation, the original contract is suppressed and replaced by two contracts: one between the lender and the CCP, and one between the borrower and the CCP. The CCP takes the contract terms as given, but can require borrowers to post additional collateral, and lenders to contribute to a loss mutualization scheme. We model novation by assuming that the CCP commits to a mechanism at the beginning of t = 1, and that lenders and borrowers negotiate over these contracts. A contract speci(cid:28)es transfers between borrowers and the CCP and transfers between lenders and the CCP as a function of public information. Because no transfer between the borrower and the lender takes place in the second period, a mechanism with central clearing consists of two contracts: a contract between the lender and the CCP, and a contract between the borrower and the CCP. Contracts with central clearing may or may not prescribe monitoring by lenders. As in the environment with bilateral clearing, upon monitoring a lender learns the type λi of her counterparty. By assumption, this remains private information of the lender and the borrower. Asaresult, whendesigningacontractwithmonitoring, theCCPneedstotakeinto account the incentives that lenders have to monitor their counterparty and report truthfully the information they learn. Amechanismwithcentralclearingandmonitoringconsistsofcontractsforlenders,{X i,∆} 2 i=L,H and contracts for borrowers, {Ci,Ci } . Contracts are executed after the lender reports 1 2,s i=L,H to be matched with a borrower of type λi. The CCP promises to pay to the lender Xi1 if 2 the borrower defaults in equilibrium, and Xi0 if the borrower does not default. At t = 1 the 2 18

borrower transfers ω−Ci units of the consumption good to the CCP. The CCP promises to 1 pay the borrower Ci in the low state (s = l) at t = 2, and Ci0 in the high state (s = h) at 2,l 2,h t = 2. We assume that repayments to the lender are independent of the idiosyncratic state s, because the initial link between the lender and the borrower is suppressed upon novation. A strategy for a lender is a monitoring and reporting decision (e,m ) ∈ {0,1}×{λH,λL}; L a strategy for a borrower is a default decision function σi ∈ {0,1}. A mechanism with central clearing and no monitoring consists of contracts {X i,∆} 2 i=L,H and{Σi,Ci,Ci } , whichareexecutediftheborrowerreportshertypetobeλi. Σi isthe 1 2s i=L,H default decision that the CCP recommends to a borrower who reports her type to be λi; ∆ is the public history of the borrower’s default/repayment decision. As with bilateral clearing, a strategy for a borrower is a pair (mi,σi) ∈ {λL,λH}×{0,1}. A mechanism is incentive compatible if it is the borrower’s best response to report truthfully her type, and then follow the recommendation Σi. 4.1 Central clearing with monitoring and borrowers’ separation Assuming that the monitoring decision as well as its outcome are not observable, contracts must induce lenders to monitor their counterparty and report truthfully the information they learn. With such contracts, the CCP acquires full information about borrowers’ types, so it can design contracts that prevent borrowers’ default in equilibrium. In order to induce monitoring and truth-telling, the CCP can punish a lender whose original counterparty defaults in i,1 equilibrium. The worst punishment is to choose X = 0. This is optimal, because it relaxes 2 the incentive constraint for monitoring and truth-telling, without compromising the provision of insurance. To simplify the notation, let Xi = X i,0 . Also, let VFI denote the ex-ante value to the 2 2 lender before the borrower’s type is known. Then the CCP chooses contracts (XH,XL) and 2 2 {Ci,Ci } to solve the following maximization problem: 1 2,s i=L,H (P0FI) VFI = max qu (cid:0) XH(cid:1) +(1−q)u (cid:0) XL(cid:1) −γ 2 2 s.t. αCi +pCi +(1−p)Ci ≥αω, ∀i (19) 1 2h 2l 19

Ci ≥ (cid:0) 1−λi(cid:1) θ, ∀i (20) 2h 0≤Ci ≤ω, ∀i (21) 1 qXH +(1−q)XL+qpCH +(1−q)pCL + 2 2 2h 2h +q(1−p)CH +(1−q)(1−p)CL ≤ω−qCH −(1−q)CL+pθ (22) 2l 2l 1 1 −γ+qu (cid:0) XH(cid:1) +(1−q)u (cid:0) XL(cid:1) ≥max (cid:110) u (cid:0) XL(cid:1) , (23) 2 2 2 (q+(1−q)(1−p))u (cid:0) XH(cid:1) +(1−q)p[σL(cid:48) u(0)+(1−σL(cid:48) )u(XH)] (cid:111) 2 2 σL(cid:48) =argmax (cid:8) (1−σˆ)CH +σˆ (cid:0) 1−λL(cid:1) θ (cid:9) (24) 2h σˆ 0,1 ∈{ } Constraint (19) is the borrowers’ participation constraint, (20) is the borrowers’ limited commitment constraint, and (21) is t = 1 feasibility. Since the clearing process is channeled through the CCP, (22) de(cid:28)nes t = 2 feasibility. Note that t = 2 feasibility is not de(cid:28)ned for di(cid:27)erent realizations of borrowers’ idiosyncratic state, as there is no aggregate uncertainty. Constraint (23) is the incentive compatibility condition for the lenders; we apply a max operator to the right-hand side of the constraint because lenders can deviate in two ways. First, they can choose not to monitor their counterparty and select the contract designed for λL types. In this case, (20) implies that all borrowers repay, so that lenders always consume XL. Alternatively, lenders may choose not to monitor their counterparty and select the 2 contract designed for λH types; such a deviation is detected by the CCP only if the borrower is a λL type who defaults in equilibrium. Constraint (24) de(cid:28)nes the o(cid:27)-equilibrium optimal default strategy of a λL borrower who is entitled to consumption CH . 2,h In Appendix 7.4, we prove that we need to characterize only contracts that satisfy CH < 2,h (1 − λL)θ. Then, we replace σL(cid:48) = 1 in constraints (23) and (24). Notice that the incentive compatibility constraint (23) generates a non-convex set of feasible allocations. To this end, de(cid:28)ne wH = u (cid:0) XH(cid:1) , wL = u (cid:0) XL(cid:1) , and rewrite (P0FI) with the CCP choosing 2 2 (cid:8) wi,Ci,Ci (cid:9) to maximize lenders’ ex-ante utility: 1 2s i=H,L,s=h,l (PˆFI) VFI = max qwH +(1−q)wL−γ s.t. αCi +pCi +(1−p)Ci ≥αω, ∀i (25) 1 2h 2l 20

Ci ≥ (cid:0) 1−λi(cid:1) θ, ∀i (26) 2h 0≤Ci ≤ω, ∀i (27) 1 qu 1(cid:0) wH(cid:1) +(1−q)u 1(cid:0) wL(cid:1) +qpCH +(1−q)pCL + − − 2h 2h +q(1−p)CH +(1−q)(1−p)CL ≤ω−qCH −(1−q)CL+pθ (28) 2l 2l 1 1 −γ+qwH +(1−q)wL ≥ (cid:110) (cid:111) max wL,[q+(1−q)(1−p)]wH +(1−q)pu(0) (29) One can solve problem (PˆFI) in two steps. In the (cid:28)rst step, the CCP determines the contracts o(cid:27)ered to borrowers, {Ci,Ci } , to provide the maximal amount of resources 1 2s i=H,L,s=h,l in the second period. We denote such resources by Ω; they consist of the amount of consumption good stored by the CCP from t = 1 to t = 2 and of all t = 2 borrowers’ net payments. The contracts {Ci,Ci } must be feasible: they should satisfy the participation and 1 2s i=H,L,s=h,l the limited commitment constraints of the borrowers. Thus, contracts {Ci,Ci } 1 2s i=H,L,s=h,l solve the following problem: (Pˆb FI ) Ω = max (cid:104) ω−qCH −(1−q)CL (cid:105) +pθ 1 1 {Ci,Ci ,Ci } 1 2h 2l −q[pCH +(1−p)CH]−(1−q)[pCL +(1−p)CL] 2,h 2,l 2h 2l s.t. αCi +pCi +(1−p)Ci ≥ αω 1 2h 2l ω ≥ Ci ≥ 0 1 Ci ≥ (1−λi)θ 2,h In the second step, the CCP determines the contracts it o(cid:27)ers to lenders, for a given amount of resources Ω. Such contracts should persuade lenders to monitor their counterparty and report truthfully the information that they learn; thus they solve (Pˆa FI ) max qwH +(1−q)wL−γ Ω {wH,wL}∈(cid:60)2 + s.t. qu−1(cid:0) wH(cid:1) +(1−q)u−1(cid:0) wL(cid:1) ≤ Ω 21

−γ +qwH +(1−q)wL ≥ (cid:110) (cid:111) max wL,(q+(1−q)(1−p))wH +(1−q)pu(0) (30) Assume without loss of generality that u(0) = 0. We can prove the following: Lemma 4 (Ci,Ci ,Ci ,wi) solve the problem (PˆFI) if and only if (Ci, Ci , Ci ) 1 2h 2l i=L,H 1 2h 2l i=L,H solve (PˆbFI) and, letting Ω∗ denote the value of the objective in (PˆbFI) at its solution, (wH,wL) solve (Pˆa FI ). Ω∗ Theincentivecompatibility(30)hasamaxoperatorontheright-handsidebecauselenders have two feasible deviations: they can i) not monitor and report a λL type or ii) not monitor and report a λH type. In the next lemma we show that in problem (Pˆa FI ) we can restrict Ω∗ our attention to the space of utilities (wH,wL) where the best deviation for the lender is the (cid:28)rst. This is the space of utilities (wH,wL) that satisfy wL ≥ [q+(1−q)(1−p)]wH. Lemma 5 Let Ω ∈ (cid:60) . For any (w ,w ) ∈ (cid:60)2 such that + H L + qu−1(cid:0) wH(cid:1) +(1−q)u−1(cid:0) wL(cid:1) ≤ Ω (31) [q+(1−q)(1−p)]wH = max (cid:0) wL,(q+(1−q)(1−p))wH(cid:1) (32) −γ +qwH +(1−q)wL ≥ (q+(1−q)(1−p))wH (33) there exist (wH(cid:48) ,wL(cid:48) ) ∈ (cid:60)2 such that + (cid:16) (cid:17) (cid:16) (cid:17) qu−1 wH(cid:48) +(1−q)u−1 wL(cid:48) ≤ Ω (34) (cid:16) (cid:17) wL(cid:48) = max wL(cid:48) ,(q+(1−q)(1−p))wH(cid:48) (35) −γ +qwH(cid:48) +(1−q)wL(cid:48) ≥ wL(cid:48) (36) and qwH(cid:48) +(1−q)wL(cid:48) > qwH +(1−q)wL (37) 22

Lemma 5 follows from convexity of the function u−1(·) and the ine(cid:30)ciency that the incentivecompatibilityconstraint(30)createsindi(cid:27)erentregionsofthepayo(cid:27)s’space(wH,wL). Technically,di(cid:27)erentpayo(cid:27)s(wH,wL)areinducedbylotteriesoverdi(cid:27)erentoutcomes,(u−1(wH), u−1(wL);q,1−q). If we hold constant the amount of resources, which is equal to qu−1(wH)+ (1 − q)u−1(wL), we keep constant the expected cost of these lotteries. Consider then any payo(cid:27)s (wH,wL) such that the right-hand side of (30) equals to [q+(1−q)(1−p)]wH. We can reduce wH and increase wL so that the lottery over outcomes that is induced by the new payo(cid:27)s is a mean-preserving contraction of the lottery over outcomes that is induced by the original payo(cid:27)s. Because of risk aversion, the new lottery must be strictly preferred to the original one. Since we can continue this process until the the right-hand side of (30) equals wL, the result follows. As a corollary of Lemma 5, we can rewrite problem (PˆaFI) as follows: Ω (Pˆa FI )(cid:48) max qwH +(1−q)wL−γ Ω {wH,wL}∈(cid:60)2 + s.t. qu−1(cid:0) wH(cid:1) +(1−q)u−1(cid:0) wL(cid:1) ≤ Ω (38) −γ +qwH +(1−q)wL ≥ wL (39) wL−[q+(1−q)(1−p)]wH ≥ 0 (40) Lemma 6 A solution to problem (Pˆa FI )(cid:48) exists (and is unique) if and only if Ω ≥ Ωˆ, for Ωˆ Ω which solves (cid:18) (cid:19) (cid:18) (cid:19) γ γ[q+(1−q)(1−p)] Ωˆ = qu−1 +(1−q)u−1 (41) pq(1−q) pq(1−q) Moreover, at the solution, equations (38) and (39) hold with equality. Lemma 6 characterizes the optimal contract between lenders and the CCP, given available revenues Ω. Under the optimal contract, lenders matched with λH borrowers enjoy higher consumption than lenders matched with λL borrowers. Ex-ante, this contract induces lenders 23

to monitor their counterparty, anticipating that this might be a λH borrower. Ex-post this contract induces lenders matched with λL borrowers to truthfully reveal their counterparty’s type. In fact the punishment that lenders would incur if caught lying, that is if their original counterparty defaults, is large enough to deter misreporting. The remaining question concerns the consumption allocation implied by the optimal contract. The answer is provided in the following lemma. Lemma 7 A solution to problem (Pˆb FI ) is such that Ci = (1−λi)θ, Ci = 0, 2h 2l (cid:26) p(1−λi)θ (cid:27) Ci = max 0,ω− . 1 α And    q(p(1−λH)θ)+(1−q)(p(1−λL)θ) if ω ≤ p(1−λH)θ   α   (cid:16) (cid:17) Ω = pθ+ω− q ω+(α−1) p(1−λH)θ +(1−q)(p(1−λL)θ) if p(1−λH)θ < ω ≤ p(1−λL)θ  α α α    (cid:16) (cid:17) (cid:16) (cid:17)   q ω+(α−1) p(1−λH)θ +(1−q) ω+(α−1) p(1−λL)θ if ω > p(1−λL)θ α α α (42) To gain intuition for Lemma 7, note that, when contracts are cleared centrally, there is no need of collateral for insurance purposes, because the CCP can fully insure lenders by pooling risk. Hence, the objective in problem (Pˆb FI ) is to minimize collateral requirements. The limited commitment constraint always binds, and consumption is determined residually from the borrowers’ participation constraint in the (cid:28)rst period. When collateral is abundant, borrowers’t = 1consumptionispinneddownbythebindingparticipationconstraint,whereas when collateral is scarce borrowers’ t = 1 consumption equals zero. WecancombineLemma6andLemma7tocharacterizethesolutiontotheproblem(PˆFI). First, de(cid:28)ne the function (cid:18) (cid:19) (cid:18) (cid:20) (cid:21)(cid:19) γ 1−p(1−q) φ(γ) = qu−1 +(1−q)u−1 γ pq(1−q) pq(1−q) 24

which maps any value of γ ≥ 0 to the minimum aggregate resources (i.e. t = 2 consumption goods)consistentwiththeexistenceofasolutiontotheCCPfullinformationproblem(P0FI). Further, de(cid:28)ne the threshold γˆ(ω) as the unique solution to (cid:20) (cid:26) (1−λH)pθ (cid:27) (cid:21) φ(γˆ(ω)) = q min ω, +λHpθ α (43) (cid:20) (cid:26) (1−λL)pθ (cid:27) (cid:21) +(1−q) min ω, +λLpθ α Thus, γˆ(ω) denotes the largest value of γ which, for a given value of ω, is such that a solution to the CCP full information problem (P0FI) exists. Proposition 8 A solution to problem (P0FI) exists and is unique if and only if γ ≤ γˆ(ω). Then Ω is given by equation (42) and VFI = wL, for wL solving (cid:18) (cid:19) γ qu−1 wL+ +(1−q)u−1(wL) = Ω. q Proof. The conclusion follows combining Lemma 4, Lemma 5, Lemma 6, and Lemma 7. Intuitively, when the CCP wants to implement contracts with monitoring, it needs to take into account that a lender may deviate by choosing not to monitor her counterparty, while announcing that monitoring occurred and that the counterparty’s pledgeability type is either i) low, or ii) high. Constraint (39) guarantees that lenders will not undertake deviation i), requiring the CCP to reward members facing a high-quality counterparty relative to those facing a low-quality counterparty. Constraint (40) guarantees that lenders will not take deviation ii), requiring the CCP to ensure that the members who face a low-quality counterparty do not get penalized excessively relative to those facing a high-quality counterparty. In other words, the CCP needs to make payments that are far enough between lenders matched with di(cid:27)erentborrowertypes, butalsolarge enough tosustainthecostofmonitoring. Importantly, 25

these two conditions can be jointly satis(cid:28)ed only if the cost of monitoring is low relative to the resources available to the CCP, namely if γ ≤ γˆ(ω), for γˆ(ω) de(cid:28)ned in (43). 4.2 Central clearing without monitoring The CCP may prefer to o(cid:27)er contracts that do not require to monitor borrowers. Such contracts are chosen to solve: (cid:88) VCCP,e=0 =max q [Σiu(Xi,1)+(1−Σi)u(Xi,0)] (44) i 2 2 i s.t. αCi +p[Σi(1−λi)θ+(1−Σi)Ci ]+(1−p)Ci ≥αω (45) 1 2h 2l 0≤Ci ≤ω (46) 1 (cid:88) (cid:104) (cid:110) (cid:111) (cid:110) (cid:111) q Σi Xi,1+p(1−λi)θ +(1−Σi) Xi,0+pCi i 2 2 2h i (cid:105) (cid:88) +(1−p)Ci ≤pθ+ q {ω−Ci} (47) 2l i 1 i (cid:110) (cid:104) (cid:105) (cid:111) (λi,Σi)∈argmax αcmˆ +p σˆ(1−λi)θ+(1−σˆ)Cmˆ +(1−p)Cmˆ (48) 1 2h 2l (mˆ,σˆ) Concavity of the utility function u(·) implies that it is optimal to choose X H,1 =X H,0 = 2 2 L,1 L,0 i,∆ X = X . Therefore we simplify the notation and write X = X in (44) and in (47). 2 2 2 2 In addition, we ignore contracts such that good type borrowers default in equilibrium, as the next lemma shows that they are not optimal. Lemma 9 Without loss of generality, we can ignore all contracts that recommend the strategy ΣH = 1. Therefore, λH borrowers never default in equilibrium. According to Lemma 9, we have to consider only two classes of contracts: contracts in which no borrower defaults in t = 2, that is ΣH = ΣL = 0, and contract in which only λH borrowers repay in t = 2, whereas λL borrowers default in equilibrium, that is ΣH = 0 and ΣL = 1. 26

4.2.1 The optimal contract without default: pooling over λL Without monitoring, the optimal contract that guarantees no defaults in equilibrium is a pooling contract that ignores borrowers’ heterogeneity and treats them all as if they were the worst borrower type. More speci(cid:28)cally, consider the following modi(cid:28)ed problem: (PL) VCCP,λL = max u(X ) (49) 2 s.t. αC +pC +(1−p)C ≥ αω (50) 1 2h 2l C ≥ (cid:0) 1−λL(cid:1) θ (51) 2h 0 ≤ C ≤ ω (52) 1 X +pC +(1−p)C ≤ ω−C +pθ (53) 2 2h 2l 1 Constraint (50) is the participation constraint, and (51) is the limited commitment constraint of λL borrowers. Equations (52) and (53) are t = 1 and t = 2 resource constraints, where in (53) no borrower defaults in equilibrium. Lemma 10 A solution to (PL) is such that: i) C < ω; ii) (53) always binds; iii) C = 0. 1 2,l Lemma 11 Let (X ,Ci,Ci ,Ci ) be the solution to (P0), with ΣH = 0 and ΣL = 0. Then, 2 1 2h 2l CH = CL, CH = CL , and (X , CL, CL , CL ) solve problem (PL). 1 1 2,l 2,l 2 1 2,h 2,l According to Lemma 11, problem (PL) characterizes the optimal contract where no borrower defaults in equilibrium. This contract resembles the one with monitoring and bilateral clearing when the borrower type is λL. The only di(cid:27)erence between the two contracts is the resourceconstraint,which,withcentralclearing,permitsriskpoolingoverinvestmentreturns, θ˜. As in the previous analysis, α > 1 implies C = 0. 2,l It is easy to see that (51) binds. Then C is determined residually from the participation 1 constraint (50). The next lemma summarizes these results. Lemma 12 The solution to problem (PL) is such that the limited commitment constraint (cid:110) (cid:111) always binds, C = (1−λL)θ, and C = max 0,ω− (1−λL)pθ . 2,h 1 α 27

Therefore we can rewrite VCCP,λL as  (cid:18) (cid:19)  u (1−λL)pθ +pθλL if ω≥ (1−λL)pθ VCCP,λL = α α  u (cid:0) ω+pθλL(cid:1) if ω< (1−λL)pθ α 4.2.2 The Contract in which λL borrowers default: pooling over λH The optimal contract that induces λL borrowers to default in equilibrium has an intuitive interpretation: itsoutcomeisequivalenttotheCCPignoringtheheterogeneityacrossborrowers and treat them all as if they were λH type borrowers. More speci(cid:28)cally, consider the following modi(cid:28)ed problem: (PH) VCCP,λH = max u(X ) (54) 2 (X2,C1,C 2h ,C 2l ) s.t. αC +pC +(1−p)C ≥ αω (55) 1 2h 2l (1−λL)θ ≥ C ≥ (cid:0) 1−λH(cid:1) θ (56) 2h 0 ≤ C ≤ ω (57) 1 X +qpC +(1−q)p (cid:0) 1−λL(cid:1) θ+q(1−p)C + 2 2h 2l +(1−q)pC ≤ ω−C +pθ (58) 2l 1 Constraint (55) is the participation constraint and (56) is the limited commitment constraint of λH borrowers. Equations (57) and (58) are t = 1 and t = 2 resource constraints. Note that constraint (58) assumes that λL borrowers always default in equilibrium. Lemma 13 A solution to (PH) is such that: i) (57), is always slack and C < ω; ii) (58), 1 always binds; iii) C = 0. 2,l Lemma 14 Let (X ,Ci,Ci ,Ci ) denote the solution to problem (P0), with ΣH = 0 and 2 1 2h 2l ΣL = 1. Then CH = CL, CH = CL , and (X ,CH,CH ,CH) solve problem (PH). 1 1 2,l 2,l 2 1 2,h 2,l According to Lemma 14 problem (PH) characterizes optimal contracts that induce λL borrowers to default in equilibrium. We characterize such optimal contracts in the next lemma. 28

Lemma 15 The solution to problem (PH) is such that (1−λL)pθ 1. If ω ≥ α (a) If q ≥ 1, then C = (1−λH)θ, C = ω− (1−λH)pθ . α 2h 1 α (b) If q < 1, then C = ω− (1−λL)pθ , C = (1−λL)θ α 1 α 2h (1−λL)pθ (1−λH)pθ 2. > ω ≥ α α (a) If q ≥ 1,then C = (1−λH)θ, C = ω− (1−λH)pθ . α 2h 1 α (b) If q < 1, then C = 0, C = αω α 1 2h p 3. If ω < (1−λH)pθ , then C = 0, C = (1−λH)θ α 1 2h Therefore we can rewrite VCCP,λH as   u u (cid:0) (cid:16) ω (1 + −λ p H θ ) [ p q θ λH + + pθ ( [q 1 λ − H q + )λ ( L 1 ] (cid:1) −q)λL] (cid:17) i i f f ω q≥ < 1 (1− an λ α H d ) ω pθ ≥ (1−λH)pθ VCCP,λH = α α α  u u (cid:16) (cid:16) ( ( 1 1 − − λ λ α H L) ) p p θ θ + + p p θ θ λ [q L λ (cid:17) H+(1−q)λL]+(1−qα) (cid:104) ω− (1−λ α H)pθ (cid:105)(cid:17) i i f f q q < < α 1 1 a a n n d d ω (1− ≥ λ α H (1 ) − pθ λL ≤ )pθ ω< (1−λ α L)pθ α α α Lemma 15 shows that the properties of the pooling contract over λH hinge on two key parameters: the fraction q of high-pledgeability borrowers and the cost of collateral 1. The α relative size of these parameters governs the e(cid:27)ect that collateral has on the total amount of resources available to the CCP at t = 2. The reason is that reducing the homogeneous collateralrequirementacrossborrowersatt = 1hastwoopposinge(cid:27)ectswhentheCCP(optimally)acceptsthatλL borrowersdefaultinequilibrium. First, becauseborrowers’preferences are biased toward t = 1 consumption, reducing collateral requirements has the potential of increasing the amount of resources available to the CCP at t = 2. This increase could occur because the reduction in collateral and corresponding increase in consumption at t = 1 leads toamorethanproportionalreductioninconsumptionatt = 2foraconstantlevelofexpected utility, as α > 1. However, a second indirect e(cid:27)ect of reducing collateral requirements for all borrowers by o(cid:27)ering a pooling contract over λH is the reduction in the aggregate resources available to the CCP at t = 2, because λL borrowers default in equilibrium. Intuitively, the 29

(cid:28)rst e(cid:27)ect is stronger than the second e(cid:27)ect if and only if the fraction of λH borrowers is large enough, i.e. q > 1. This last condition can also be rewritten as q(α−1) > 1−q, where the α left-hand-side is the bene(cid:28)t of lower collateral requirements, weighted by the fraction of λH borrowers, and the right-hand-side is the cost of lower collateral requirements, weighted by the fraction of λL borrowers. When q ≥ 1, maximizing the amount of resources available at t = 2 is equivalent to α minimizing the collateral requirement in t = 1. Then, the limited commitment constraint of a λH borrower binds, C = (1−λH)θ, and consumption in the (cid:28)rst period is determined 2,h residually from the participation constraint (55) and the feasibility condition C ≥ 0. 1 When q < 1, maximizing the amount of resources available at t = 2 is equivalent to min- α imizing the resources consumed by λL borrowers due to their defaults. This is accomplished by choosing the largest feasible collateral requirement, up to the point where the e(cid:27)ect of λL borrowers’ default on t = 2 resources is minimized. When ω < (1−λL)pθ , collateral is α scarce and borrowers are asked to post their entire endowment as collateral, which results (1−λL)pθ in C = 0. When ω ≥ , the contract chosen by the CCP e(cid:27)ectively replicates the 1 α allocation of a pooling contract over λL borrowers. As a result, the consumption allocation of λL borrowers is such that their limited commitment and participation constraints hold at equality. In this case λL borrowers are treated exactly as they would be in a full information contract, therefore they do not earn any information rents. 4.2.3 Equilibrium contracts with central clearing and no information acquisition The results discussed throughout Section 4.2 show that contracts without monitoring can impose costs to lenders in terms of ine(cid:30)cient collateral requirements. The following lemma characterizes the optimal contract with central clearing when there is no monitoring activity. Lemma 16 The optimal contract with central clearing and no information acquisition is (i) Pooling over λH if q ≥ 1, or if q < 1 and ω < (1−λL)pθ ; α α α (ii) Pooling over λL if q < 1 and ω ≥ (1−λL)pθ . α α 30

When lenders acquire no information about borrowers’ types, optimal contracts ignore heterogeneity in borrowers’ default risk. More precisely, if all borrowers are treated as if they were λL types, λH borrowers end up posting an excessive amount of collateral. This policy is costly for lenders because it requires them to forgo a larger amount of consumption good in t = 2 to satisfy borrowers’ participation constraint. When the population of λH borrowers is relatively large, i.e. q ≥ 1, the policy of treating all borrowers as λL types is not e(cid:30)cient for α the CCP, which thus chooses to let λL borrowers default in equilibrium. If instead all borrowers are treated as if they were λH types, λL borrowers post too little collateral and default in equilibrium at t = 2. This policy also imposes costs on lenders, because the defaults of λL borrowers reduce the amount of consumption good available to the CCP at t = 2. When the population of λL types is relatively large, and collateral is abundant relative to the commitment problem of λL borrowers, i.e. q < 1 and ω ≥ (1−λL)pθ , α α it is ine(cid:30)cient for the CCP to let λL borrowers default in equilibrium. Lemma 16 then implies that:    VCCP,λL if ω ≥ (1−λL)pθ and q < 1 VCCP,e=0 = α α (59)   VCCP,λH otherwise. whereVCCP,λH isde(cid:28)nedinSection4.2.2, andVCCP,λL isde(cid:28)nedinSection4.2.1. Finally, as with bilateral clearing, notice that the solution to the decision problem with central clearing is the same solution to the problem of a social planner subject to the frictions of 1) private information of the borrower’s type; 2) limited commitment on the borrower’s side at t = 2; 3) lenders’ private information on their use of the information acquisition technology. As a consequence, the optimal contract with central clearing is constrained e(cid:30)cient. The decision problem with central clearing is not constrained by the resources available within the speci(cid:28)c lender-borrower match, as it is with bilateral clearing, and the CCP can reallocate resources across di(cid:27)erent matches. However, the decision problem with central clearing is subject to the additional constraint arising from the lenders’ information acquisition decision. 31

5 Optimal Clearing In the previous sections, we characterized feasible contracts under di(cid:27)erent clearing arrangements. In this section, we determine lenders’ choice of clearing arrangement, and refer to it as the optimal clearing arrangement. First, weprovethatbilateralclearingisoptimalonlyiflendersmonitortheircounterparty. More precisely, we prove that contracts with bilateral clearing and no information acquisition arenotoptimal. ThereasonisthataCCPcanalwaysreplicatesuchcontracts,andinaddition it can provide insurance against idiosyncratic risks. Lemma 17 formalizes this result. Lemma 17 The optimal contract with CCP clearing and no monitoring, i.e. the solution to (44), dominates the optimal contract with bilateral clearing and no monitoring, i.e. the solution to (12). Second, in Lemma 18 we prove that for γˆ(w) de(cid:28)ned in (43), if γ ≤ γˆ(w) central clearing is the optimal arrangement. Lemma 18 If γ ≤ γˆ(w) de(cid:28)ned in (43), then the contract with bilateral clearing and monitoring is dominated either by the contract with CCP clearing and pooling over λL or by the contract with CCP clearing and monitoring. In the proof of Lemma 18, we show that lenders would prefer the contract with bilateral clearing and monitoring over the contract with central clearing and pooling over λL only if, given the monitoring cost γ, the value of facing a λH counterparty is signi(cid:28)cantly higher than the value of facing a λL counterparty. However, if this is the case and γ ≤ γˆ(w), a CCP can replicate such bilateral contracts and obtain enough resources at t = 2 to induce lenders to monitor their counterparties and report truthfully their type. Further, the CCP can transfer some resources from lenders facing a λH counterparty to lenders facing a λL counterparty, without violating lenders’ incentive compatibility constraints. As a result, central clearing improves on bilateral clearing by providing insurance against the risk of facing a counterparty type λL. 32

Third, in Lemma 19 we prove that bilateral clearing is optimal only if λH is su(cid:30)ciently large and λL is su(cid:30)ciently small. (cid:110) (cid:111) Lemma 19 If λL ≥ λ L ≡ max λ∗,1− αω , the contract with central clearing and pooling pθ over λL dominates the contract with information acquisition and bilateral clearing. If λH < λH = 1 − αω or λH > λH > λ∗, the contract with central clearing and pooling over λH pθ dominates the contract with information acquisition and bilateral clearing. Corollary 20 Bilateral clearing is never optimal if either of these conditions hold: (cid:110) (cid:111) i) λL ≥ λ L ≡ max λ∗, 1− αω , or pθ ii) λH < λH = 1− αω, or pθ iii) λH > λH > λ∗, or iv) γ ≤ γˆ(w). Corollary 20 provides su(cid:30)cient conditions for central clearing to be optimal. These su(cid:30)cientconditionscanbeunderstoodintermsofthevalueofinformationunderdi(cid:27)erentclearing arrangements. Speci(cid:28)cally, Corollary 20 states that central clearing is optimal if either information about the counterparty type has no value with bilateral clearing, cases i)-iii), or if the value of information about counterparty type is larger with central clearing than with bilateral clearing (case iv)). In case i), with bilateral clearing, the limited commitment constraint is slack for both borrowers’ types. Thus lenders do not need any information about their counterparty and prefer central clearing, which provides insurance against uncertain investment risk. Similarly, in the economies described by cases ii) and iii) information about the counterparty type has no value, although for di(cid:27)erent reasons. These are economies where, with bilateral clearing, optimal contracts require borrowers to post the maximum feasible amount of collateral (i.e. c = 0) regardless of their type. Thus, even if a lender knew the 1 type of her counterparty, she could not require λL-borrowers to post more collateral than 33

she already posted.14 As a consequence, lenders prefer central clearing because it provides insurance against uncertain investment risk. Finally, in economies where the monitoring cost is relatively small (case iv)), the CCP can induce monitoring by lenders, and information is more valuable with central clearing because the CCP can provide full insurance against the idiosyncratic return risk, and partial insurance against the counterparty-type risk. 5.1 Optimal bilateral clearing: the CCP contract where full information is not implementable (γ > γˆ(ω)). In the rest of the analysis, we consider parameter con(cid:28)gurations that do not satisfy any of the conditions of Corollary 20. Then, a trade-o(cid:27) between bilateral and central clearing arises. Central clearing has the advantage of providing insurance by pooling risk over idiosyncraticuncertaintyand,asaresult,thepotentialtoeconomizeontheuseofcollateralnecessary to insure against idiosyncratic risk. However, since γ > γˆ(w), monitoring is not incentive feasible for the CCP. Without the information generated by monitoring, the CCP must o(cid:27)er contracts that require all traders to post the same amount of collateral, which is associated either to a low-pledgeability or a high-pledgeability counterparty. Thus, central clearing has the limitation of requiring a fraction of the borrowers’ population to post either excessive or insu(cid:30)cientcollaterallevelsnecessarytoprovideincentivestorepay. Ontheotherhand, bilateralclearinghasthedisadvantageofcallingforlargercollateralrequirementstoinsureagainst idiosyncratic risk, but the bene(cid:28)t of preserving the incentives to monitor a counterparty, as long as the monitoring cost γ is not too large, and allow collateral requirements to be tailored to the type of counterparty. These insights are formalized in the following proposition. Proposition 21 Let Y ⊆ R3 ×(0,1)4 and y = (ω,α,θ,p,q,λH,λL) denote an element of + (cid:110) (cid:111) Y. Suppose that λL < λ L ≡ max λ∗,1− αω , λH ≥ λH ≡ 1 − αω, and λH ≤ λ∗. Let pθ pθ γˆ(ω) : Y → R be the map de(cid:28)ned for any vector y in (43), and γ : Y → R map any vector + + 14Economies described by case ii) correspond to area 4 in (cid:28)gure 1, where ω is so small that both types of borrowers are required to post their entire endowment as collateral. Economies described by case iv)) correspondtoarea2in(cid:28)gure1foraλH borrower,andeitherarea2orthepartofarea4suchthatω<ω(λ∗) for a λL borrower. The collateral good is not very abundant and, even if the limited commitment constraint is slack for a λH borrower, collateral requirement is at its maximum even for such borrower. 34

y to a value of monitoring cost: (cid:20) (cid:21) pθ φ(γˆ(ω)) = q (1−λH)+pθλH +(1−q)[Q +pθλL] (60) 2 α γ = qA+(1−q)B−C (61) where (cid:40) ω if ω < pθ (1−λL) α Q = (62) 2 pθ (1−λL) otherwise α and (cid:16) (cid:17) (cid:16) (cid:17) (cid:40) pu pθ (1−λ∗)+λ∗θ +(1−p)u pθ (1−λ∗) if λH ≥ λ∗ α α A = (63) (cid:16) (cid:17) (cid:16) (cid:17) pu pθ (1−λH)+λHθ +(1−p)u pθ (1−λH) otherwise α α (cid:16) (cid:17) (cid:16) (cid:17) (cid:40) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) if ω > pθ (1−λL) α α α B = pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) if pθ (1−λH) < ω < pθ (1−λL) α α (64) (cid:16) (cid:17) u pθ (cid:0) 1−λL(cid:1) +pθλL if ω > pθ (1−λL) and q < 1 (cid:40) α α α (cid:16) (cid:17) C = u pθ (cid:0) 1−λH(cid:1) +pθ(qλH +(1−q)λL) if q > 1 α α u (cid:0) ω(1−αq)+pθ[1−(1−q)(1−λL)] (cid:1) if pθ (1−λH) < ω < pθ (1−λL) and q ≤ 1. α α α (65) Then bilateral clearing with information acquisition is the optimal clearing arrangement if and only if γ ∈ (γˆ(ω),γ). Proposition 21 proves that lenders prefer bilateral clearing for intermediate values of the monitoring cost γ. The reason is that γˆ(ω) is the lower bound on the cost of monitoring, γ, such that the CCP can only o(cid:27)er pooling contracts. Since these are the only contracts that entail a trade-o(cid:27) with bilateral clearing, then γ > γˆ(ω) is necessary for the optimality of bilateral clearing. Similarly, γ is the largest value of γ such that the value of tailoring collateral requirements to the severity of the limited commitment friction, net of the cost of 35

monitoring, exceeds the value of insurance against uncertain returns. When γ ∈ (γˆ(ω),γ), the insurance over uncertain returns provided by the CCP does not compensate lenders for the ine(cid:30)cient use of collateral due to the lack of information over the counterparty quality. Thus lenders choose to clear contracts bilaterally and acquire information about their borrowers. Naturally, the bounds on γ depend on the parameters of the model; among them, the degree of risk-aversion, the opportunity cost of collateral, and the degree of heterogeneity of the population of borrowers play an important role. Proposition 21 also imples that for any valueofλH andω,γ < 0eitherwhenq isarbitrarilyclosetounity(alargepresenceofλH-type borrowers) or when q is arbitrarily close to 0 (a large presence of λL-type borrowers). When q is close to unity, it is very likely for a borrower to be a λH type. In these cases, it is optimal to save on the monitoring cost and clear the contract centrally. On the other hand, when q is close to 0, it is very likely for a lender to face a λL-type borrower, so it does not pay o(cid:27) to monitoraborrowerandclearthecontractbilaterally. Inthesecases,learningthecounterparty type is not valuable, and central clearing still allows to pool investment risk. Thus, we expect bilateral clearing to emerge only if there is su(cid:30)cient uncertainty over counterparty types. On the other hand, bilateral clearing is preferred for a larger set of parameters when α, the opportunity cost of collateral, increases. This increase causes the threshold γˆ(ω) to (weakly) decrease, implying that monitoring and borrowers’ separation under central clearing is not implementable for a larger set of parameters.15 Moreover, for su(cid:30)ciently large values of p and q, the threshold γ is also weakly increasing in α: as a result, the set of monitoring costs satisfying the assumptions of Proposition 21 is larger.16 Similarly, the degree of risk-aversion plays an important role: in general, the threshold γ is smaller the higher is the degree of risk aversion. Intuitively, theadvantageoftheCCPinpoolingriskoveruncertainreturnsislarger 15Thethresholdγˆ(ω)isde(cid:28)nedbyequation(43): anincreaseinαcausestherighthandsideof(43)toweakly decrease, and, since the function φ(γ) is increasing in γ, then γˆ(ω) must (weakly) decrease as a consequence. As a consequence the set of γ values for which γ ≤ γˆ(ω) becomes smaller. As shown in the proof of Lemma 18, this implies that monitoring and borrowers’ separation are no longer feasible under central clearing. 16That γ increases in α can be seen from equations (63)-(65). Consider for simplicity an increase in α su(cid:30)cientlylargetosatisfyq> 1. ThenC decreasesand,aslongaspissu(cid:30)cientlylargethenAincreases,as α λ∗ (weakly) increases. With q su(cid:30)ciently large the change in A dominates the change in B, resulting in an overall increase in γ. If the increase in λ∗, however, is large enough that λH <λ∗, then A decreases, but due to strict concavity of u, any change in A is dominated by a change in C for p and q su(cid:30)ciently large. 36

the more risk-averse the lenders are. Intuitively, large values of α can be associated with (cid:28)nancial institutions such as hedge funds or broker-dealers, whose opportunity cost of collateral is higher than, say, that of money market funds.17 In this respect, our results are broadly consistent with evidence of dealers and hedge funds clearing a substantial share of their trades bilaterally, whereas money market funds are more likely to rely on (cid:28)nancial market infrastructure (e.g. General Collateral Finance Repo Service (GCF Repo) and triparty settlement).18 Analogously, our resultsareconsistentwithcentralclearingarisingendogenouslyinmarketswhereparticipants are homogenous in terms of their business type (in the model, q close to 1 or 0), when we interpret the pledgeability parameter λ as the riskiness in a counterparty’s set of activities.19 6 Conclusions This paper characterizes optimal clearing arrangements for (cid:28)nancial transactions in a model where insurance is valuable because of uncertain returns to investment and heterogenous quality of trading counterparties. The contribution of the analysis is the identi(cid:28)cation of a trade-o(cid:27) between clearing bilaterally and channeling clearing services through a CCP. This trade-o(cid:27) arises when incentives to monitor bilateral trades are incompatible with the risk pooling activity of the CCP. Thus, even though the motivation for central clearing might arise from reasons outside the model, such as systemic risk consequences of opaque bilateral positions, the consequence of mandatory CCP clearing is a loss of information across markets due to decreased incentives to monitor trading partners. This result should not of course lead to the conclusion that CCP’s are not useful in sharing risk in markets. It rather highlights the importance of the risk of the underlying assets and the degree of heterogeneity of market participants in determining whether CCP’s can perform their risk sharing function e(cid:27)ectively. 17At least under normal circumstances, disregarding events as money market funds breaking the buck. 18As an example, for evidence related to the US repo market see the O(cid:30)ce of Financial Research Brief Paper no. 17-04, Bene(cid:28)ts and Risks of Central Clearing in the Repo Market. 19As an example, recall that the (cid:28)rst central counterparties originated next to grain and co(cid:27)ee exchanges, where farmers and bakers traded futures. Among many, for references see Kroszner (2006), and Gregory (2014). 37

7 Appendix 7.1 Proof of Lemma 1 Proof. First, we show that the optimal contract requires positive collateral, meaning that ω−ci > 0. Suppose by contradiction that constraint (3) binds, i.e. ci = ω. From the limited 1 1 commitment constraint (6) we know that ci ≥ (1−λi)θ > 0; therefore the participation 2,h (2) is slack. But then, the lender could decrease ci: all constraints would still be satis(cid:28)ed, 1 and her expected utility would increase. This is a contradiction and proves that it must be that ci < ω. Then we conclude that the optimal contract requires positive collateral and 1 constraint (3) is slack. Next, weshowthatsecondperiodborrowers’consumptioninthelowstateequalszero, i.e. ci = 0. To prove this, (cid:28)rst notice that it must be that xi ≥ xi . If not, i.e. if xi < xi , 2,l 2,h 2,l 2,h 2,l combining equations (4) and (5) (with equality) we obtain ci = ci +θ+(xi −xi ) > ci +θ > (1−λi)θ 2,h 2,l 2,l 2,h 2,l Then, the lender could reduce ci by (cid:15), increase xi by the same amount, increase ci 2,h 2,h 2,l by p (cid:15), and reduce xi by the same amount. All constraints would be satis(cid:28)ed, and by 1−p 2,l concavity of u(·) the lender would increase her expected utility. Now that we established that xi ≥ xi , suppose by contradiction that ci > 0. Then it should be that xi = xi . If not, 2,h 2,l 2,l 2,h 2,l i.e. if xi > xi , the lender could increase ci by (cid:15), reduce x by the same amount, reduce 2,h 2,l 2,h 2,h ci by p (cid:15), and increase x by the same amount. All constraints would be satis(cid:28)ed, and by 2,l 1−p 2,l concavity of u(·) the lender would increase her expected utility. Since xi = xi , combining 2,h 2,l (4) and (5) (with equality) we obtain ci = ci +θ > (1−λi)θ 2,h 2,l But then the lender could reduce ci and ci by (cid:15), increase c by (cid:15), and increase both x 2,h 2,l 1 α 2,h and x by the same amount α−1(cid:15). All constraints would be satis(cid:28)ed and the lender expected 2,l α 38

revenues would increase. Therefore it can not be that ci > 0, and we conclude that it should 2,l be that ci = 0. 2,l Finally, we show that insurance is incomplete, meaning that xi > xi . Suppose by 2,h 2,l contradiction that xi = xi = x. Combining (4) and (5) (with equality) we obtain 2,h 2,l ci = θ > (1−λi)θ 2,h Then the lender can decrease ci by (cid:15), increase ci by p(cid:15) , decrease x by p(cid:15) and increase x 2,h 1 α 2,l α 2,h α−p by (cid:15). For (cid:15) su(cid:30)ciently small, the lender’s expected utility can be rewritten as p (cid:18) (cid:19) α−p (cid:16) p(cid:15)(cid:17) pu x+ (cid:15) +(1−p)u x− α α (cid:20) (cid:21) α−p (cid:104) p(cid:15)(cid:105) ≈ p u(x)+u(cid:48)(x) (cid:15) +(1−p) u(x)−u(cid:48)(x) α α (cid:20) (cid:21) α−1 = u(x)+u(cid:48)(x) p(cid:15) > u(x) α Thereforethelendercouldincreaseherexpectedutility,whichprovesthattheoriginalcontract could not be optimal, and concludes the proof. 7.2 Proof of Lemma 2 Proof. Itiseasytoseethatboththeparticipationconstraint(2)andthelimitedcommitment constraint (6) can not be slack: if this was the case, the lender could increase her revenues just by decreasing ci . 2,h Suppose then that ω < (1−λi)pθ . Because ci ≥ (1−λi)θ and ci ≥ 0, the participation α 2,h 1 constraint (2) is slack. Since both (2) and (6) can not be slack, it must be that (6) binds: ci = (1−λi)θ. Easily, ci = 0: if not, the lender could decrease ci, satisfy all constraints, 2,h 1 1 and increase her expected utility. 39

7.3 Proof of Lemma 3 (1−λi)pθ Proof. First, we show that when ω > , the participation constraint (2) always binds. α (1−λi)pθ Suppose by contradiction the participation constraint (2) is slack when ω > . Then, α since both constraints can not be slack, the limited commitment constraint (6) should bind, i.e. ci = (1−λi)θ. Then, since ω > (1−λi)pθ , it must be that ci > 0. If instead we had 2,h α 1 ci = 0, then αci +pci = p(1−λi)θ < αω and the participation constraint (2) would be 1 1 2,h (1−λi)pθ violated. Then if ω > , the participation constraint (2) always binds. α Next, we show that equation (11) de(cid:28)nes a unique threshold λ∗. De(cid:28)ne the function F(λ) as (cid:16) (cid:17) u(cid:48) (1−λ)pθ α F(λ) = (cid:16) (cid:17) u(cid:48) θ+(1−λ) pθ (1− α) α p Easily F(0) = 1 < α−p and F(cid:48)(λ) > 0. Therefore, if a λ∗ exists, this is unique. A necessary 1−p and su(cid:30)cient condition for λ∗ to exist is that α−p < F(1) = u(cid:48)(0) . 1−p u(cid:48)(0) Next, given the unique threshold λ∗ de(cid:28)ned by (11), we show that if λi < λ∗, the limited liability constraint (6) binds. Suppose not: λi < λ∗ and (6) is slack. Then ci > (1−λi)θ > 2,h (1 − λ∗)θ. Therefore η = 0 in (10); moreover we know from above that the participation constraint (2) binds. Solving (2) for ci we obtain ci = α(ω−c ), combined with the slack 2,h 2,h p 1 limited commitment constraint (6) gives ω−ci > (1−λi)pθ > (1−λ∗)pθ . From (7), as η = 0 we 1 α α have (cid:18) (cid:20) (cid:21)(cid:19) α µ = u(cid:48)(ω−ci +θ−ci ) = u(cid:48) θ−(ω−c ) 1− 1 2,h 1 p Replaced in (8), we obtain (cid:18) (cid:20) (cid:21)(cid:19) α 0 ≥ (α−p)u(cid:48) θ−(ω−c ) 1− −(1−p)u(cid:48)(ω−c ) 1 1 p (cid:18) (1−λ∗)pθ (cid:20) α−p (cid:21)(cid:19) (1−λ∗)pθ > (α−p)u(cid:48) θ− −(1−p)u(cid:48)( ) = 0 α p α which is a contradiction. Then, we conclude that if λi < λ∗, the limited commitment con- 40

straint (6) should bind. The consumption of the lender is (1−λi)pθ xi = λiθ+ 2,h α (1−λi)pθ xi = 2,l α Next, we show that if λi > λ∗, the limited commitment constraint (6) is slack. Suppose by contradiction that λi > λ∗ and the limited commitment constraint (6) binds. Then, ci = (1−λi)θ and, as the participation constraint (2) binds as well, ci = ω− (1−λi)pθ > 0. 2,h 1 α From (8) we have (cid:16) (cid:17) (cid:16) (cid:17) pu(cid:48) λiθ+ (1−λi)pθ (1−p)u(cid:48) (1−λi)pθ α α µ = + α α which replaced in (7) gives p (cid:20) (cid:18) (1−λi)pθ (cid:19) (cid:18) (1−λi)pθ (cid:19)(cid:21) η = (α−p)u(cid:48) λiθ+ −(1−p)u(cid:48) < 0 α α α where the inequality follows since λi > λ∗. Therefore, if λi > λ∗, the limited commitment constraint (6) is slack. Finally, we have to determine for λi > λ∗ whether ci > 0 or ci = 0. Since (6) is slack, 1 1 therefore η = 0, and (2) binds, therefore ci = α(ω−ci 1 ) , condition (7) gives 2,h p (cid:18) (cid:19) α−p µ = u(cid:48) θ−(ω−ci) 1 p replaced in (8) gives (cid:18) (cid:19) α−p (α−p)u(cid:48) θ−(ω−ci) −(1−p)u(cid:48)(ω−ci) ≤ 0 1 1 p with equality if ci > 0. Then, by the de(cid:28)nition of λ∗ in (11), it is clear that ci > 0 if and 1 1 only if ω > (1−λ∗)pθ , and ci = 0 if ω < (1−λ∗)pθ . This concludes the proof of Lemma 2. α 1 α 41

7.4 If CH > (1−λL)θ in problem (P0FI), then central clearing with screening 2h can not be be optimal Proof. Let (XH,XL), (C ,i,Ci ) be the solution to problem (P0FI) and suppose 2 2 1 2,s i=L,H,s=h,l CH > (1 − λL)θ. Consider now the contract with central clearing, no monitoring, and 2h pooling over λL de(cid:28)ned as Xˆ = qXH +(1−q)XL, Cˆ = qCH +(1−q)CL , and Cˆ = 2 2 2 2,s 2,s 2,s 1 qCH +(1−q)CL. Easily such constraints (50)-(53) in problem (PL). Concavity of u(·) gives 1 1 u(Xˆ2) ≥ qu(XH)+(1−q)u(XL) = VFI+γ, so it is strictly better than the original contract 2 2 with monitoring. 7.5 Proof of Lemma 4 Proof. Firstweshowtheonlyifdirection. Supposethat(Ci,Ci ,Ci ,wi) isthesolution 1 2h 2l i=L,H to problem (PˆFI), but either (Ci,Ci ,Ci ) does not solve (PˆbFI), or for Ω∗ the solution 1 2h 2l i=L,H to (PˆbFI), (w ,w ) solve (Pˆa FI ). H L Ω∗ If (Ci,Ci ,Ci ) does not solve (PˆbFI), let (Ci(cid:48) ,Ci(cid:48) ,Ci(cid:48) ) be the solution to 1 2h 2l i=L,H 1 2h 2l i=L,H (PˆbFI). From problem (PˆbFI), it must be that for some i, either Ci(cid:48) < Ci, or Ci(cid:48) < C , 1 1 2h 2h or Ci(cid:48) < Ci Suppose w.l.o.g. that CH(cid:48) < CH. Then, in problem (PˆFI) consider a new 2l 2l 1 1 contract (Ci(cid:48)(cid:48) ,Ci(cid:48)(cid:48) ,Ci(cid:48)(cid:48) ,wi(cid:48)(cid:48) ) where Ci(cid:48)(cid:48) = Ci , Ci(cid:48)(cid:48) = Ci , Ci(cid:48)(cid:48) = Ci − (cid:15). If wL > 1 2h 2l i=L,H 2h 2h 2l 2l 1 1 [q + (1 − q)(1 − p)]wH, then choose wH(cid:48)(cid:48) to solve u−1(wH(cid:48)(cid:48) ) = u−1(wH) + (cid:15); if instead wL < [q+(1−q)(1−p)]wH, choose wL(cid:48)(cid:48) to solve u−1(wL(cid:48)(cid:48) ) = u−1(wL)+ q (cid:15). In both cases, 1−q it is easy to show that (Ci(cid:48)(cid:48) ,Ci(cid:48)(cid:48) ,Ci(cid:48)(cid:48) ,wi(cid:48)(cid:48) ) satis(cid:28)es constraints (25)-(29) in problem 1 2h 2l i=L,H (PˆFI), and qwH(cid:48)(cid:48) +(1−q)wL(cid:48)(cid:48) > qwH+(1−q)wL, that contradicts optimality of the original contract in problem (PˆFI). If instead wL = [q+(1−q)(1−p)]wL, then choose wH(cid:48)(cid:48) and wL(cid:48)(cid:48) to solve u−1(wH(cid:48)(cid:48) ) = u−1(wH)+q(cid:15), and u−1(wL(cid:48)(cid:48) ) = u−1(wL)+q(cid:15). It is easy to show that wL(cid:48)(cid:48) > [q+(1−q)(1−p)]wH(cid:48)(cid:48) , that (Ci(cid:48)(cid:48) ,Ci(cid:48)(cid:48) ,Ci(cid:48)(cid:48) ,wi(cid:48)(cid:48) ) satis(cid:28)es constraints (25)-(29) in 1 2h 2l i=L,H problem(PˆFI),andqwH(cid:48)(cid:48) +(1−q)wL(cid:48)(cid:48) > qwH+(1−q)wL,whichcontradictsagainoptimality of the original contract in problem (PˆFI). If instead (Ci,Ci ,Ci ,wi) solve problem (PˆFI), but for Ω∗ the solution to (PˆbFI), 1 2h 2l i=L,H (w ,w ) does not solve (Pˆa FI ), let (wH(cid:48) ,wL(cid:48) ) solve (Pˆa FI ). It is straightforward to show H L Ω∗ Ω∗ 42

that (Ci,Ci ,Ci ,wi(cid:48) ) satis(cid:28)es constraints (25)-(29) in problem (PˆFI), and qwH(cid:48) +(1− 1 2h 2l i=L,H q)wL(cid:48) > qwH +(1−q)wL, which contradicts optimality of the original contract in problem (PˆFI). Next, we show the if direction. Let (Ci,Ci ,Ci ) solve (PˆbFI), and for Ω∗ the solu- 1 2h 2l i=L,H tion to (PˆbFI), (wH,wL) solve (Pˆa FI ). Suppose by contradiction that (Ci,Ci ,Ci ,wi) Ω∗ 1 2h 2l i=L,H does not solve problem (PˆFI). Let (Ci(cid:48) ,Ci(cid:48) ,Ci(cid:48) ,wi(cid:48) ) be the solution to (PˆFI). Then easily 1 2h 2l it must be that either Ci(cid:48) (cid:54)= Ci, or Ci(cid:48) (cid:54)= Ci , or Ci(cid:48) (cid:54)= Ci : if not it must be wH = wH(cid:48) and 1 1 2h 2h 2l 2l wL(cid:48) = wL by comparing (Pˆa FI ) with (PˆFI). By de(cid:28)nition of problem (PˆbFI), then it should Ω∗ bethateitherCi(cid:48) > Ci, orCi(cid:48) > Ci , orCi(cid:48) > Ci . Suppose,w.l.o.g. thatCH(cid:48) > CH. Then, 1 1 2h 2h 2l 2l 1 1 following the same argument as in the only if part, we can prove that (Ci(cid:48) ,Ci(cid:48) ,Ci(cid:48) ,wi(cid:48) ) can 1 2h 2l not be the solution to (PˆFI), which is a contradiction. 7.6 Proof of Lemma 5 Proof. Let (wH,wL) ∈ (cid:60)2 satisfy equations (31), (32), and (33). De(cid:28)ne X as + qu−1(cid:0) wH(cid:1) +(1−q)u−1(cid:0) wL(cid:1) = X and (wH(cid:48) ,wL(cid:48) ) as the unique solution to [q+(1−q)(1−p)]wH(cid:48) = wL(cid:48) (cid:16) (cid:17) (cid:16) (cid:17) qu−1 wH(cid:48) +(1−q)u−1 wL(cid:48) = X We want to show that (wH(cid:48) ,wL(cid:48) ) satisfy equations (34), (35), (36), and (37). Notice that equation (34) and equation (35) are satis(cid:28)ed by construction. Now, suppose by contradiction that equation (36) is violated. Therefore γ wH(cid:48) < wL(cid:48) + q wL(cid:48) = [q+(1−q)(1−p)]wH(cid:48) 43

It is easy to show that the two conditions can hold only if wL(cid:48) < q+(1−q)(1−p) , therefore pq(1−q) wH(cid:48) = wL(cid:48) < γ . Since u−1 is increasing, by the de(cid:28)nition of wH(cid:48) and wL(cid:48) we q+(1−q)(1−p) pq(1−q) have (cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) γ q+(1−q)(1−p) X = qu−1 wH(cid:48) +(1−q)u−1 wL(cid:48) < qu−1 +(1−q)u−1 γ pq(1−q) pq(1−q) (66) It is easy to show that equations (32) and (33) can hold only if wH ≥ γ and wL ≥ pq(1−q) q+(1−q)(1−p) γ. Then, since u−1 is increasing, from the de(cid:28)nition of X we have pq(1−q) (cid:18) (cid:19) (cid:18) (cid:19) γ q+(1−q)(1−p) X ≥ qu−1 +(1−q)u−1 γ pq(1−q) pq(1−q) that contradicts equation (66). Therefore equation (36) can not be violated. Finally notice that we can rewrite (cid:32) (cid:90) wL(cid:48) (cid:20) (cid:18) X −(1−q)u−1(s) (cid:19) 1−q 1 (cid:21) (cid:33) qwH(cid:48) +(1−q)wL(cid:48) = q wH + −u(cid:48) ds q q u(cid:48)(s) wL (cid:32) (cid:90) wL(cid:48) (cid:33) +(1−q) wL+ 1ds wL (cid:90) wL(cid:48)  u(cid:48) (cid:16) X−(1−q)u−1(s) (cid:17) q = qwH +(1−q)wL+(1−q) 1− ds u(cid:48)(s) wL > qwH +(1−q)wL X−(1−q)u−1(s) wherethelastinequalityfollowsfromconcavityofutogetherwiththefactthat > q s for all s ∈ [wL,wL(cid:48) ]. Therefore equation (37) is as well satis(cid:28)ed. 7.7 Proof of Lemma 6 Proof. The smallest values of wH and wL that jointly satisfy (39) and (40) are wH = γ pq(1−q) and wL = γ[q+(1−q)(1−p)] . Then constraint (38) can be satis(cid:28)ed jointly with (39) and (40) pq(1−q) only if Ω ≥ Ωˆ as de(cid:28)ned above. 44

Easily,whenΩ ≥ Ωˆ both(39)and(40)havetobind. If(39)doesnotbind,wecanincrease wH and wL by (cid:15) and all constraints are still satis(cid:28)ed. If (40) is not binding, we can construct a mean-preserving contraction on u−1(wH) and u−1(wL) so that (39) is una(cid:27)ected, but by convexity of u−1(·) the objective function strictly increases. 7.8 Proof of Lemma 7 Proof. The solution is the consequence of linearity of the objective function, and α > 1. 7.9 Proof of Lemma 9 Proof. Supposebycontradictionthattheoptimalcontracts{(Ci,Ci ,Ci ),X }recommend 1 2,h 2,l 2 ΣH = 1. Then, by (48) it must be that λH-borrowers prefer the strategy (mˆ,σˆ) = (λH,1) to the strategy (mˆ,σˆ) = (λL,0): αCH +p(1−λH)θ+(1−p)CH ≥ αCL+pCL +(1−p)CL (67) 1 2,l 1 2,h 2,l Suppose (cid:28)rst that the contracts recommend ΣL = 0: from (48) λL-borrowers need to prefer the strategy (mˆ,σˆ) = (λL,0) over the strategy (mˆ,σˆ) = (λL,1): αCL+CL +(1−p)CL ≥ αCH +p(1−λL)θ+(1−p)CH 1 2,h 2,l 1 2,l Combining this expression with (67) we obtain a contradiction. Therefore, it is not possible for the contracts to recommend ΣL = 0. Suppose then the the optimal contracts {(Ci,Ci ,Ci ), X } recommend ΣL = 1. De(cid:28)ne 1 2,h 2,l 2 a new contract {(C˜i,C˜i ,C˜i ), X˜ } as X˜ = X , C˜H = (1 − λH)θ, C˜i = Ci if either 1 2,h 2,l 2 2 2 2,h 2,s 2,s i (cid:54)= H and s (cid:54)= h, C˜i = C for i = L,H. Let such a contract recommend Σ˜H = 0, Σ˜L = 1. 1 1 It is easy to check that all constraints in problem (44) - (48) are satis(cid:28)ed, and as X did not 2 change the new contract is payo(cid:27) equivalent to the original (optimal) one, which concludes the proof. 45

7.10 Proof of Lemma 10 Proof. i) Suppose not. Then C = ω and the participation constraint (50) must be slack if 1 C satis(cid:28)es the limited commitment constraint (51). Consider then the allocation de(cid:28)ned 2,h by Cˆ = C −ε for ε > 0 arbitrarily small, and Xˆ = X +αε. This allocation is still in the 1 1 2 2 constraint set of problem (PL) and yields higher value of the objective. ii) Supposenot. Considerthentheallocationde(cid:28)nedbyXˆ = X +ε, forε > 0arbitrarily 2 2 small so that the resource constraint at t = 2, (53), is still satis(cid:28)ed. This allocation is still in the constraint set of problem (PL) and yields higher value of the objective. iii) Suppose not. Then C > 0: consider the allocation de(cid:28)ned by Cˆ = C − ε, 2,l 2,l 2,l Cˆ = C + ε andXˆ = X +ε(1−1). Becauseα > 1thenXˆ > X . Thereforethisallocation 1 1 α 2 2 α 2 2 is still in the constraint set to problem (PL) and yields higher value of the objective. 7.11 Proof of Lemma 11 Proof. Consider problem (44) - (48). In (48), the recommended default decision ΣH = 0 and ΣL = 0 require CH ≥ (1−λH)θ and CL ≥ (1−λL)θ respectively. Constraint (48) for 2,h 2,h λH-borrowers can be rewritten as CH ≥ (1−λH)θ (68) 2,h αCH +pCH +(1−p)CH ≥ αCL+pCL +(1−p)CL (69) 1 2,h 2,l 1 2,h 2,l whereas constraint (48) for λL-borrowers becomes CL ≥ (1−λL)θ (70) 2,h αCL+p(1−λL)θ+(1−p)CL ≥ αCH +pmax{(1−λL)θ,CH }+(1−p)CH (71) 1 2,l 1 2,h 2,l Step 1: The optimal contract should satisfy CH ≥ (1 − λL)θ. Then (68) can be ignored. 2,h Furthermore both (71) and (69) bind. Combine (71) with (69): 46

αCH +pCH +(1−p)CH ≥ αCL+pCL +(1−p)CL 1 2,h 2,l 1 2,h 2,l ≥ αCH +pmax{(1−λL)θ,CH }+(1−p)CH 1 2,h 2,l ≥ αCH +pCH +(1−p)CH 1 2,h 2,l Then all weak inequalities have to hold with equality, CH ≥ (1−λL)θ, and both (71) and 2,h (69) bind. Step 3: W.l.o.g we can ignore the participation constraint (45) of the λH borrower. It follows immediately from the previous step. Step 4: We have CL = 0. 2,l Suppose not: CL > 0. Then it must be CL = ω. If not we could reduce CL by (cid:15), increase 2,l 1 2,l CL by (1−p)(cid:15) , and increase X by (1−q)(1−p)(cid:15)[1− 1] > 0. The new contract would be 1 α 2 α feasible and expected utility would increase. Then CL = ω, and therefore as CL > 0 and 1 2,l CL ≥ (1−λL)θ, the participation constraint (45) of λL borrowers can be ignored as well. 2,h Moreover, it must be CH = (1 − λL)θ, otherwise we could reduce CL by (cid:15) and CH by 2,h 2,l 2,h (1−p)(cid:15) and increase X by p(cid:15). The new contract would still satisfy all constraints and the p 2 expected utility would increase. Similarly it should be CH = 0. If not we could reduce CL 2,l 2,l and CH by (cid:15), and increase X by (1−p)(cid:15). Finally, it should be CH = 0, otherwise we could 2,l 2 1 reduce CL by (cid:15), reduce CH by (1−p)(cid:15) and increase X by (1−p)(cid:15)[1 +(1−q)]. Combing 2,l 1 α 2 α CH = CH = CL = 0, CH = (1−λL)θ, CL = ω, we obtain that the binding (48) becomes 1 2,l 2,l 1,h 1 αω+pCH +(1−p)CL = (1−λL)θ 2,h 2,l which can never be satis(cid:28)ed for CL > 0 and CL ≥ (1−λL)θ, which is a contradiction. 2,l 2,h 47

Step 5: We have CH = 0. 2,l Suppose not: suppose CH > 0. Then it should be CH = ω, otherwise we could educe CH by 2,l 1 2,l (cid:15), increase CH by (1−p)(cid:15) , and increase X by q(1−p)(cid:15)[1− 1] > 0. Moreover the participation 1 α 2 α constraint (45) of λL borrowers should bind: if not following the same arguments of the previous step it should be CL = 0 and CL = (1−λL)θ. But the participation constraint (45) 1 2,h of λL borrowers and the binding (48) would give (1−λL)pθ = αCL+pCL +(1−p)CL = αCH +pCH +(1−p)CH 1 2,h 2,l 1 2,h 2,l = αω+pCH +(1−p)CL > (1−λL)pθ 2,h 2,l which is a contradiction. Then it should be αCL+pCL = αω 1 2,h This implies that CL < ω, as CL > 0. Then (45) of λL borrowers and (48) give 1 2,h αω = αCL+pCL = αω+pCH > αω 1 2,h 2,h which is a contradiction. Step 6: CH = CL = (1−λL)θ. 2,h 2,h Suppose Ci > (1−λL)θ. Reduce Ci by (cid:15), increase Ci by p(cid:15) and X by q p(cid:15)[1− 1], and 2,h 2,h 1 α 2 i α the expected utility would increase. Step 7: CH = CL. 1 1 Follows from (48) holding with equality. Step 8: The optimal contract that induces no borrower to strategically default in equilib- (cid:110) (cid:111) rium is Ci = 0, Ci = (1−λL)θ, Ci = min 0,ω− (1−λL) , and (CL CL,CL ,X ) solve 2,l 2,h 1 α 2,h 1 2,l 2 48

problem (PL). The conclusion follows from comparing the residual problem with (PL). 7.12 Proof of Lemma 12 Proof. From linearity of the objective, (51) always binds. Then whether (50) binds on not (1−λL)pθ depends on whether ω ≥ or not. α 7.13 Proof of Lemma 13 Proof. The proof is identical to the one of Lemma 10. 7.14 Proof of Lemma 14 Proof. Consider problem (44) - (48). In (48), the recommended default decision ΣH = 0 and ΣL = 1 require CH ≥ (1−λH)θ and CL < (1−λL)θ respectively. Constraint (48) for 2,h 2,h λH-borrowers can be rewritten as CH ≥ (1−λH)θ (72) 2,h αCH +pCH +(1−p)CH ≥ αCL+pmax{(1−λH)θ,CL }+(1−p)CL (73) 1 2,h 2,l 1 2,h 2,l whereas constraint (48) for λL-borrowers becomes CL ≤ (1−λL)θ (74) 2,h αCL+p(1−λL)θ+(1−p)CL ≥ αCH +pmax{(1−λL)θ,CH }+(1−p)CH (75) 1 2,l 1 2,h 2,l Step 1: W.l.o.g. we can choose CL = (1−λH)θ, and ignore constraint (74). 2,h This choice satis(cid:28)es (74) and relaxes (75) as much as possible. Since ΣL = 1 is the recommended (i.e. incentive compatible) deafult choice, CL does not appear in any other 2,h constraint. This means that we can assume CL = (1−λH)θ. 2,h Step 2: We can ignore the participation constraint of λL-borrowers. 49

From (75) and the participation constraint of λH-borrowers, αCL+p(1−λL)θ+(1−p)CL ≥ αCH +pCH +(1−p)CH ≥ αω 1 2,l 1 2,h 2,l Step 3: The optimal contract requires CH ≤ (1−λL)θ. 2,h Suppose by contradiction the optimal contracts {(Ci,Ci ,Ci ), X } satis(cid:28)es CH > (1 − 1 2,h 2,l 2 2,h λL)θ > (1−λH)θ. Then we can ignore (72). Moreover it needs to be that CH = ω: if not, 1 the CCP could reduce CH by (cid:15), increase CH by p (cid:15), and increase X by α−1qp(cid:15) > 0. All 2,h 1 α 2 α constraints are still satis(cid:28)ed but the expected utility of lenders increases. But then, since CH = ω, we can also ignore the participation constraint of λH-borrowers. From constraint 1 (73), we can ignore (75). Therefore the only constraints left are (73), the resource constraint (46) for i = L, and the second-period resource constraint (47). Note that (73) should bind or the CCP could reduce CH and increase X accordingly, without violating any constraint: 2,h 2 αω+pCH +(1−p)CH = αCL+p(1−λH)θ+(1−p)CL (76) 2,h 2,l 1 2,l From this expression and (46) it needs to be CL > CH ≥ 0. Then it has to be CH = 0, 2,l 2,l 2,l otherwise we could decrease both CL andCH by (cid:15), and increase X by (1−p)(cid:15) it needs to 2,l 2,l 2 be CL > CH ≥ 0. Then it has to be CH = 0, otherwise we could decrease both CL andCH 2,l 2,l 2,l 2,l 2,l by (cid:15), and increase X by (1−p)(cid:15). Replacing CH = 0 we obtain that 2 2,l (1−p)CL = α(ω−CL)+p[C −(1−λH)θ] > 0 2,l 1 2,h But then it has to be that CL = ω: if CL < ω, the CCP can decrease CL by (cid:15) and increase 1 1 2,l CL by p (cid:15), and increase X by α−1(1−q)p(cid:15) > 0. All constraints are still satis(cid:28)ed but the 1 α 2 α expected utility of lenders increases. Moreover it needs to be that CL = 0. If not, the CCP 2,l could reduce CH by (cid:15), CL by p (cid:15), and increase X by p(cid:15). All constraints are still satis(cid:28)ed 1 2,l 1−p 2 but the expected utility of lenders strictly increase. But then, equation (76) becomes (1−λL)pθ < pCH = p(1−λH)θ 2,h 50

which is not possible. This proves that it must be that CH ≤ (1−λL)θ.Replacing this value 2,h in (75), the latter becomes αCL+(1−p)CL ≥ αCH +(1−p)CH 1 2,l 1 2,l Step 4: At the optimal solution, equation (75) holds with equality: αCL + (1 − p)CL = 1 2,l αCH +(1−p)CH. 1 2,l Suppose not: suppose that (75) is slack. The only active constraints are then the resource constraint in t = 1, (46) the resource constraint in t = 2, (47), and the incentive compatibility constraints (72) and (73). But then it should easily be that it CL = CL = 0. As a result, 1 2,l (75) can only hold if CH = CH = 0, and equation (75) holds with equality. 1 2,l Step 5: Constraint (73) can be ignored. Use the fact that (75) binds and (74), we obtain αCH +pCH +(1−p)CH = αCL+pCH +(1−p)CL 1 2,h 2,l 1 2,h 2,l ≥ αCL+p(1−λH)θ+(1−p)CL 1 2,l Step 6: It is optimal to choose CH = CL = 0. 2,l 2,l Suppose not: suppose w.l.o.g. that CH ≥ CL ≥ 0 with one inequality holding has a strict 2,l 2,l inequality. If CH = CL > 0, then we could decrease both by (cid:15), increase X by (1−p)(cid:15), satis- 2,l 2,l 2 fyingalltherelevantconstraintsandincreasingtheexpectedutility. IfinsteadCH > CL = 0, 2,l 2,l it has to be 0 ≤ CH < CL. But then we could reduce CH by (cid:15), reduce CL by (1−p)(cid:15) , and 1 1 2,l 1 α increase X by (1−p)(cid:15)[q+ 1−q ]. All constraints would be satis(cid:28)ed, and the expected utility 2 α would increase. Step 7: It is optimal to choose CH = CL. 1 1 51

It follows immediately by the binding (75) once we replace CH = CL = 0. 2,l 2,l αCH = αCL 1 1 Step 8: The optimal contract that induces λL types to strategically default in equilibrium is CL = (1−λH)θ, CH = CL = 0, CL = CH, and (CH ,CH,X ) that solve problem (PH). 2,h 2,l 2,l 1 1 2,h 1 2 Rewriting the problem for (CH ,CH,X ) with the relevant constraints, we obtain: 2,h 1 2 max u(X ) 2 (X2,C 1 H,C 2 H h ) s.t. αCH +pCH ≥ αω 1 2h (1−λL)θ ≥ CH ≥ (cid:0) 1−λH(cid:1) θ 2h 0 ≤ CH ≤ ω 1 X +qpCH +(1−q)p (cid:0) 1−λL(cid:1) θ ≤ ω−CH +pθ 2 2h 1 From Lemma 13 we have C = 0 in problem (PH), which completes the proof of Lemma 14. 2,l 7.15 Proof of Lemma 15 Proof. Because the resourse constraint in t = 2 binds and in t = 1 is slack, we can rewrite: VCCP,λH = max u (cid:2) ω−C +pθ−qpC −(1−q)p (cid:0) 1−λLθ (cid:1)(cid:3) 1 2h s.t. αC +pC +(1−p)C ≥ αω 1 2h 2l C ≥ (cid:0) 1−λH(cid:1) θ 2h C ≤ (cid:0) 1−λL(cid:1) θ 2h C ≥ 0 1 1. Suppose (cid:28)rst that the participation constraint binds. Then C = ω−pC 2h. Then in the 1 α 52

objective we have (cid:18) (cid:19) pC max u pθ+ 2h −qpC −(1−q)p(1−λL)θ 2h α pC 2h s.t. ω− ≥ 0 α (1−λL)θ ≥ C ≥ (1−λH)θ 2h (a) If q ≥ 1, the objective is decrasing in C , so the solution is C = (1−λH)θ. α 2h 2h (1−λH)pθ This can be a solution only if ω ≥ . α (cid:110) (cid:111) (b) Ifq < 1,thenthesolutionisincreasinginC ,sothesolutionisC = min (1−λL)θ, αω α 2h 2h p 2. Suppose now that the participation constraint is slack. Then easily C = (1−λH)θ 2h (1−λH)pθ and C = 0. This can be a solution if ω < . 1 α 7.16 Proof of Lemma 16 Proof. Let us consider when pooling over λL can be an equilibrium: VCCP,λL ≥ VCCP,λH. If p (cid:0) 1−λL(cid:1) θ ≤ αω and qα ≥ 1, VCCP,λL ≥ VCCP,λH if and only if: (cid:32)(cid:0) 1−λL(cid:1) pθ (cid:33) (cid:32)(cid:0) 1−λH(cid:1) pθ (cid:33) u +pθλL ≥ u +pθ (cid:2) qλH +(1−q)λL(cid:3) α α (cid:0) λH −λL(cid:1) pθ ≥ pθ (cid:2) qλH +(1−q)λL−λL(cid:3) α 1 ≥ αq which is a contradiction, so VCCP,λH ≥ VCCP,λL . If p (cid:0) 1−λL(cid:1) θ ≤ αω and qα < 1, VCCP,λH = VCCP,λL . Without loss of generality we say it is optimal to pool over λL. If p (cid:0) 1−λH(cid:1) θ ≤ αω < p (cid:0) 1−λL(cid:1) θ and q ≥ 1 then VCCP,λL ≥ VCCP,λH if and only if: α (cid:32)(cid:0) 1−λH(cid:1) pθ (cid:33) u (cid:0) ω+pθλL(cid:1) ≥ u +pθ (cid:2) qλH +(1−q)λL(cid:3) α 53

(cid:0) 1−λH(cid:1) pθ ω ≥ +pθq (cid:0) λH −λL(cid:1) α ≥ (cid:0) 1−λH(cid:1) pθ + pθ (cid:0) λH −λL(cid:1) = (1−λL)pθ α α α So it is never possible, and we have VCCP,λH ≥ VCCP,λL . If p (cid:0) 1−λH(cid:1) θ ≤ αω < p (cid:0) 1−λL(cid:1) θ and q < 1 then VCCP,λL ≥ VCCP,λH if and only if: α u (cid:0) ω+pθλL(cid:1) ≥ u (cid:18) (1−λH)pθ +pθ[qλH +(1−q)λL]+(1−qα) (cid:20) ω− (1−λH)pθ (cid:21)(cid:19) α α ω+pθλL ≥ ω(1−qα)+pθ−(1−q)(1−λL)pθ 0 ≥ pθ(1−λL)−αω > 0 Which is a contradiction, so it is never possible, and we have VCCP,λH ≥ VCCP,λL . If αω < p (cid:0) 1−λH(cid:1) θ then VCCP,λL ≥ VCCP,λH if and only if: u (cid:0) ω+pθλL(cid:1) ≥ u (cid:0) ω+pθ (cid:2) qλH +(1−q)λL(cid:3)(cid:1) but this is never possible. We can summarize the equilibrium CCP clearing conditional on no info acquisition: 1. ω < (1 − λ α L)pθ ==> pooling over λH   1<αq ==>pooling over λH 2. ω ≥ (1 − λ α L)pθ ==>  1≥αq ==>pooling over λL 7.17 Proof of Lemma 17 Proof. Let(Σi,x i,1 ,x i,0 ,xi ,ci,ci ,ci ) ,betheoptimalcontractwithbilateralclear- 2,h 2,h 2,l 1 2,h 2,l i=L,H i,1 i,0 ing and no information acquisition. De(cid:28)ne then the contracts with CCP clearing (X ,X ) 2 2 and (Σ ,Ci,Ci ) as Σ the same recommended default decision as in the contract with bilati 1 2,s i 54

eral clearing, as well as Ci = ci, Ci = ci , and 1 1 2s 2s   X 2 i,∆ = u−1  (cid:88) q i (cid:104) p (cid:110) Σiu(xi 2 1 h )+(1−Σi)u(xi 2 0 h ) (cid:111) +(1−p)u(xi 2l ) (cid:105)  i=L,H (cid:88)(cid:110) (cid:104) (cid:105)(cid:111) < pθ+ω− Ci +p Σi(1−λi)θ+(1−Σi)Ci +(1−p)Ci 1 2h 2l i Easily the contracts are incentive compatible for the same strategies as the contracts with bilateral clearing. Moreover all constraints are easily satis(cid:28)ed by concavity of u(·). Then it has to be VCCP,e=0 ≥ u(X i,∆ ) = Vbil,e=0 2 meaning that the contract with CCP clearing and no screening dominates the contract with bilateral clearing and no screening. 7.18 Proof of Lemma 18 Proof. Suppose not: suppose that the contract with bilateral clearing and screening dominatesboththecontractwithcentralclearingandscreeningandthecontractwithCCPclearing and pooling over λL. Let (xi , xi , ci, ci , ci ) be the optimal contracts with bilateral clearing, when the lender 2h 2l 1 2h 2l upon screening learns that her counterparty is of type i. Similarly, let (wi∗ , Ci∗ , Ci∗ , Ci∗ ) be 2h 2l 1 the optimal contract with CCP clearing and screening. Since the contract with bilateral clearing dominates the contract with CCP clearing and screening, we have qV +(1−q)V > VFI (77) H L Moreover, since the contract with bilateral clearing and screening dominates the contract with CCP clearing and pooling over λL, qV +(1−q)V ≥ VCCP,λL H L 55

De(cid:28)ne then the contract (X , Ci, Ci , Ci ) with Ci = cL, Ci = cL , and X = u−1(V +γ). 2 1 2h 2l 1 1 2,s 2,s 2 L Therefore VCCP,λL ≥ V +γ L Put together the two expressions: qV +(1−q)V ≥ VCCP,λL ≥ V +γ H L L γ ⇒ V ≥ V + H L q where V = pu(xH)+(1−p)u(xH)−γ H 2h 2l V = pu(xL )+(1−p)u(xL)−γ L 2h 2l Therefore γ pu(xH)+(1−p)u(xH) ≥ pu(xL )+(1−p)u(xL)+ 2h 2l 2h 2l q De(cid:28)nenowwH andwLasthelenders’utilitiesfrombilateralclearing,grossofthescreening cost γ: wH = pu(xH)+(1−p)u(xH) = V +γ 2h 2l H wL = pu(xL )+(1−p)u(xL) = V +γ 2h 2l L Concavity of u(·) gives us that u−1(wH) < pxH +(1−p)xH = ω−cH +pθ−pcH −(1−p)cH 2h 2l 1 2h 2l Similarly, u−1(wL) < pxL +(1−p)xL = ω−cL+pθ−pcL −(1−p)cL 2h 2l 1 2h 2l Consider then the contract with CCP clearing (wi, Ci , Ci , Ci), where wH and wL are 2h 2l 1 56

de(cid:28)ned above, Ci = ci, Ci = ci , Ci = ci . Easily if [q + (1 − q)(1 − p)]wH < wL, all 1 1 2h 2h 2l 2l constraints in the CCP problem (PˆFI) are automatically satis(cid:28)ed. Then, by de(cid:28)nition of optimality, it must be that VFI = qwH∗ +(1−q)wL∗ ≥ qwH +(1−q)wL = qV +(1−q)V H L thatcontradictsequation(77). Thenitmustbethat[q+(1−q)(1−p)]wH > wL,sotherelevant incentive constraint in the full information problem of the CCP is −γ +qwH +(1−q)wL ≥ [q+(1−q)(1−p)]wH. In this case, if the incentive constraint for screening is satis(cid:28)ed then the CCP solution always dominates the bilateral one because for any pair (wH,wL) such that [q+(1−q)(1−p)]wH > wL,theCCPcanalways(cid:28)ndanalternativepair(wH,wL)thatviolates [q+(1−q)(1−p)]wH > wL, satis(cid:28)es the incentive constraint −γ +qwH +(1−q)wL ≥ wL, and yields strictly higher utility to lenders. Then, it must be that the incentive compatibility constraint for screening is not satis(cid:28)ed by such a contract: it has to be [q+(1−q)(1−p)]wH > wL and −γ +qwH +(1−q)wL < [q+(1−q)(1−p)]wH. Consider then the solution to problem (PˆbFI): we know from Lemma 4 that (Ci∗ , Ci∗ , Ci∗ ) solve problem (PˆbFI). Moreover, by de(cid:28)nition of the maximization 1 2h 2l problem, it has to be that Ω∗ = pθ+ω−q[CH∗ +pCH∗ +(1−p)CH∗]−(1−q)[CL∗ +pCL∗ +(1−p)CL∗] 1 2h 2l 1 2h 2l ≥ pθ+ω−q[cH +pcH +(1−p)cH]−(1−q)[cL+pcL +(1−p)cL] 1 2h 2l 1 2h 2l = q[p(θ−cH +ω−cH)+(1−p)(ω−cH)]+(1−q)[p(θ−cL +ω−cL)+(1−p)(ω−cL)] 2h 1 2l 2h 1 2l = q[pxH +(1−p)xH]+(1−q)[pxL +(1−p)xL] 2h 2l 2h 2l > qu−1(wH)+(1−q)u−1(wL) De(cid:28)ne then δ = Ω−qu−1(wH)+(1−q)u−1(wL) 57

and de(cid:28)ne wH(cid:48) such that δ u−1(wH(cid:48) ) = u−1(wH)+ (78) q Since u−1(·) is increasing, wH(cid:48) ≥ wH. De(cid:28)ne now the operator  (cid:16) (cid:17) qu−1(wH(cid:48) )+(1−q)u−1(wL)−qu−1 y q+(1−p)(1−q) T(y) = u −y 1−q (cid:18) (cid:19) qu−1(wH(cid:48) )+(1−q)u−1(wL) NoticethatT(y)ismonotonedecreasinginy,thatfory = y ≡ [q+(1−p)(1−q)]u > q 0, it is T(y) = u(0)−y < 0 Furthermore, the two conditions wH(cid:48) ≥ wL + γ and wH(cid:48) ≥ wL − γ , imply that q 1−p (1−q)(1−p) wH(cid:48) ≥ wL . where the second inequality follows from wH(cid:48) ≥ wH > wL − γ , q+(1−p)(1−q) 1−p (1−q)(1−p) which results from the assumption that the incentive constraint is violated, −γ+qwH +(1− q)wL < [q+(1−q)(1−p)]wH, and from the de(cid:28)nition of wH(cid:48) that implies wH(cid:48) ≥ wH. Then for y = wL it is true that  (cid:16) (cid:17) qu−1(wH(cid:48) )+(1−q)u−1(wL)−qu−1 wL T(w L ) = u q+(1−p)(1−q) −wL 1−q ≥ u(u−1(wL))−wL = 0 By the intermediate value theorem, there must be a wL(cid:48)(cid:48) ≥ wL such that T(wL(cid:48)(cid:48) ) = 0. De(cid:28)ne then wL(cid:48)(cid:48) ∈ [wL,y) to be the value that satis(cid:28)es T(wL(cid:48)(cid:48) ) = 0, and then de(cid:28)ne wH(cid:48)(cid:48) as the solution to wL(cid:48)(cid:48) wH(cid:48)(cid:48) = q+(1−p)(1−q) Notice that wH(cid:48)(cid:48) ≤ wH(cid:48) , since wL(cid:48)(cid:48) ≥ wL. Consider then the contract (wH(cid:48)(cid:48) , wL(cid:48)(cid:48) , Ci∗ , Ci∗, Ci∗), where wH(cid:48)(cid:48) and wL(cid:48)(cid:48) are de(cid:28)ned 1 2h 2l above,andCi∗ ,Ci∗ ,Ci∗ solveproblem(PˆbFI). Noticethatthiscontractisfeasibleandsatisfy 1 2h 2l the limited commitment constraint in problem (PˆFI): participation, limited commitment 58

and feasibility constraints are easily satis(cid:28)ed by the de(cid:28)nition of Ci∗ , Ci∗, Ci∗. Moreover, 1 2h 2l by construction [1+(1−q)(1−p)]wH(cid:48)(cid:48) = wL(cid:48)(cid:48) . All is left to show is that this contract is incentive compatible. By construction, via the operator T qu−1(wH(cid:48)(cid:48) )+(1−q)u−1(wL(cid:48)(cid:48) ) = qu−1(wH(cid:48) )+(1−q)u−1(wL) = Ω∗ ≥ Ωˆ Replacing wH(cid:48)(cid:48) , wL(cid:48)(cid:48) and Ωˆ with their de(cid:28)nitions we can rewrite (cid:32) (cid:33) wL(cid:48)(cid:48) (cid:18) wˆL (cid:19) qu−1 +(1−q)u−1(wL(cid:48)(cid:48) ) ≥ qu−1 +(1−q)u−1(wˆL) q+(1−p)(1−q) q+(1−p)(1−q) Notice that this can hold if and only if wL(cid:48)(cid:48) ≥ wˆL and therefore wH(cid:48)(cid:48) ≥ wˆH. Moreover, recall that wˆH = wˆL+ γ . Therefore, for wL(cid:48)(cid:48) ≥ wˆL and wH(cid:48)(cid:48) ≥ wˆH, the following hold: q 1 wH(cid:48)(cid:48) = wˆH + (wL(cid:48)(cid:48) −wˆL) q+(1−q)(1−p) γ 1 = wˆL+ + (wL(cid:48)(cid:48) −wˆL) q q+(1−q)(1−p) γ 1 = wˆL+ + (wL(cid:48)(cid:48) −wˆL)+wL(cid:48)(cid:48) −wL(cid:48)(cid:48) q q+(1−q)(1−p) (cid:20) (cid:21) γ 1 γ = wL(cid:48)(cid:48) + +(wL(cid:48)(cid:48) −wˆL) −1 ≥ wL(cid:48)(cid:48) + q q+(1−q)(1−p) q that proves that the contract (wH(cid:48)(cid:48) , wL(cid:48)(cid:48) , Ci∗ , Ci∗, Ci∗) satis(cid:28)es as well the incentive com- 1 2h 2l patibility constraint. Then, by the de(cid:28)nition of optimality, it must be VFI ≥ qwH(cid:48)(cid:48) +(1−q)wL(cid:48)(cid:48) (cid:32) (cid:33) qu−1(wH(cid:48) )+(1−q)u−1(wL)−qu−1(wH(cid:48)(cid:48) ) = qwH(cid:48)(cid:48) +(1−q)u 1−q (cid:32) (cid:33) qu−1(wH)+δ+(1−q)u−1(wL)−qu−1(wH(cid:48)(cid:48) ) = qwH(cid:48)(cid:48) +(1−q)u 1−q (cid:32) (cid:33) Ω−qu−1(wH(cid:48)(cid:48) ) = qwH(cid:48)(cid:48) +(1−q)u 1−q (cid:32) (cid:90) wH(cid:48) (cid:33) (cid:32) (cid:90) wH(cid:48) (cid:20) (cid:18) Ω−qu−1(s) (cid:19) q 1 (cid:21) (cid:33) = q wH(cid:48) − 1ds +(1−q) wL+ u(cid:48) ds wH(cid:48)(cid:48) wH(cid:48)(cid:48) 1−q 1−qu(cid:48)(s) 59

(cid:90) wH(cid:48)  u(cid:48) (cid:16) Ω−qu−1(s) (cid:17)  = qwH(cid:48) +(1−q)wL+q  1−q −1ds wH(cid:48)(cid:48) u(cid:48)(s) ≥ qwH(cid:48) +(1−q)wL ≥ qwH +(1−q)wL = qV +(1−q)V H L Ω−qu−1(s) where the (cid:28)rst inequality in the last line follows from the fact that < s for all 1−q s ∈ (wH(cid:48)(cid:48) ,wH(cid:48) ], and the inequality in the last line follows from the fact that wH(cid:48) ≥ wH, given the de(cid:28)nition in (78). But this contradicts (77). 7.19 Proof of Lemma 19 Proof. Suppose (cid:28)rst that λL ≥ λ L = λ∗. As λH > λL, λH > λ∗ as well. Moreover, λ∗ ≥ 1− αω gives us ω ≥ (1−λ∗)pθ . Then from Lemma 3, we have c = ω−c∗ 1 > (1−λL)θ, pθ α 2,h p and (cid:18) ω−c∗(cid:19) V = V = pu θ−(α−p) 1 +(1−p)u(ω−c∗)−γ H L 1 p (cid:16) (cid:17) Moreover, from Lemma 12, we have VCCP,λL = u (1−λL)pθ +λLpθ . Combining the two α expressions and using concavity of u(·), we have (cid:18) ω−c∗(cid:19) Vbil,e=0 = pu θ−(α−p) 1 +(1−p)u(ω−c∗)−γ 1 p (cid:16) (cid:17) (cid:18) (1−λL)pθ (cid:19) < u pθ−(α−1)(ω−c∗) < u pθ−(α−1) 1 α (cid:18) (1−λL)pθ (cid:19) = u +λLpθ = VCCP,λL α which proves that if λL ≥ λ L = λ∗, then the contract with central clearing and pooling over λL dominates the contract with information acquisition and bilateral clearing. Suppose now that λL ≥ λ L = 1 − αω > λ∗. Then, as λH > λL, we have λH > λ∗ as pθ well. Moreover, since 1− αω > λ∗, we have ω < (1−λ∗)pθ . Then from Lemma 3, we have pθ α αω = c > (1−λL)θ, so ω ≥ (1−λL)pθ , and p 2h α (cid:18) (cid:19) ω V = V = pu θ−(α−p) +(1−p)u(ω)−γ H L p 60

(cid:16) (cid:17) Moreover, from Lemma 12, we have VCCP,λL = u (1−λL)pθ +λLpθ . Combining the two α expressions and using concavity of u(·), we have (cid:18) (cid:19) ω Vbil,e=0 = pu θ−(α−p) +(1−p)u(ω)−γ p (cid:16) (cid:17) (cid:18) (1−λL)pθ (cid:19) < u pθ−(α−1)ω < u pθ−(α−1) α (cid:18) (1−λL)pθ (cid:19) = u +λLpθ = VCCP,λL α that proves that if λL ≥ λ L = 1 − αω > λ∗, then central clearing and pooling over λL pθ dominates information acquisition and bilateral clearing. So we have proven the (cid:28)rst half of (cid:110) (cid:111) the Lemma: if λL ≥ λ L ≡ max λ∗,1− αω , the contract with central clearing and pooling pθ over λL dominates the contract with information acquisition and bilateral clearing. Suppose now that λH < λH = 1− αω. Then as λL < λH, we have 1− αω ≥ λH > λL. pθ pθ Then from Lemma 3 we have V = p (cid:0) ω+λHθ (cid:1) +(1−p)u(ω)−γ H V = p (cid:0) ω+λLθ (cid:1) +(1−p)u(ω)−γ L From Lemma 15, we have that VCCP,λH = u (cid:0) ω+pθ[qλH +(1−q)λL] (cid:1) . Combining the two expressions and using concavity of u(·), we have Vbil,e=0 = qV +(1−q)V −γ H L = (1−p)u(ω)+p[qu(ω+λHθ)+(1−q)u(ω+λLθ)]−γ < u (cid:0) ω+pθ[qλH +(1−q)λL] (cid:1) = VCCP,λH that proves that if λH < λH = 1− αω, then central clearing and pooling over λH dominates pθ information acquisition and bilateral clearing. 61

Suppose (cid:28)nally that λH > λH > λ∗. Then from Lemma 3 we have (cid:20) (cid:18) (cid:19) (cid:21) α V = pu ω+θ− ω +(1−p)u(ω) −γ H p V = (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −γ L (cid:16) (cid:17) If q ≥ 1, from Lemma 15 we have VCCP,λH = u (1−λH)pθ +pθ[qλH +(1−q)λL] . Com- α α bining the two expressions, using concavity of u(·), λH > λH, and the fact that q ≥ 1, we α have Vbil,e=1 = qV +(1−q)V < u (cid:0) ω+qpθ−qαω+(1−q)pθλL(cid:1) −γ H L (cid:18) (1−λH)pθ (cid:20) (1−λH)pθ (cid:21) (cid:19) = u +pθ[qλH +(1−q)λL]+ ω− (1−qα) −γ α α (cid:18) (1−λH)pθ (cid:19) < u +pθ[qλH +(1−q)λL] −γ α = VCCP,λH −γ < VCCP,λH Ifinsteadq < 1,fromLemma15wehaveVCCP,λH = u(ω(1−αq)+pθ−(1−q)(1−λL)pθ). α Combining this with the payo(cid:27)s from bilateral clearing, using concavity of u(·), we get Vbil,e=1 = qV +(1−q)V < u (cid:0) ω+qpθ−qαω+(1−q)pθλL(cid:1) −γ H L = VCCP,λH −γ < VCCP,λH that completes the proof. 7.20 Proof of Proposition 21 Proof. Ifγ > γˆ(ω),withγˆ(ω) de(cid:28)nedconsistentlywith(43),thenthefullinformationcontract with CCP clearing is not implementable. CASE 1: CCP contract pools over λL. If q ≤ 1 and ω ≥ pθ (cid:0) 1−λL(cid:1) then the best contract with CCP clearing is the pooling α α contract over λL. Also, lemma 19 implies that only λL < λ L and λH > λH may be consistent 62

with bilateral clearing and information acquisition as an equilibrium outcome. Since ω ≥ pθ (cid:0) 1−λL(cid:1) > pθ (cid:0) 1−λH(cid:1) , then λ L = λ∗, and λH < λH is satis(cid:28)ed. α α Thus γˆ(ω) in this case is: (cid:20) (1−λH)pθ (cid:21) (cid:20) (1−λL)pθ (cid:21) φ(γˆ(ω)) = q +λHpθ +(1−q) +λLpθ (79) α α (cid:16) (cid:17) Lenderspayo(cid:27)withCCPclearingisVCCP,λL = u pθ (cid:0) 1−λL(cid:1) +pθλL . Lenders’payo(cid:27)with α bilateral clearing, since λL < λ∗, depends on whether (i) λL < λ∗ < λH or (ii) λL < λH < λ∗. (i) λL < λ∗ < λH. Claim 22 If ω ≥ (1−λL)pθ , λH > λ∗ > λL, αq ≤ 1, and γ ≥ γˆ(ω), the optimal contract α is such that: (a) bilateral clearing and information acquisition if γ ≤ γa. (b) CCP clearing and pooling over λL If γ > γa. Proof. Theexpectedpayo(cid:27)frombilateralclearing, usingω−c∗ = pθ (1−λ∗), tolenders 1 α is: (cid:20) (cid:18) (cid:19) (cid:21) α −γ +q pu ω−c∗+θ− (ω−c∗) +(1−p)u(ω−c∗) + 1 1 1 p (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) (80) α α Hence bilateral clearing is preferred to CCP clearing if γ < γa, where (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) pθ pθ γa =q pu (1−λ∗)+λ∗θ +(1−p)u (1−λ∗) α α (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) +(1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) α α (cid:18) (cid:19) −u pθ (cid:0) 1−λL(cid:1) +pθλL (81) α 63

(ii) λL < λH < λ∗. Claim 23 If ω ≥ (1−λL)pθ , λ∗ > λH > λL, αq ≤ 1, and γ ≥ γˆ(ω), the optimal contract α is such that: (a) bilateral clearing and information acquisition if γ ≤ γb. (b) CCP clearing and pooling over λL If γ > γb. Proof. The expected payo(cid:27) from bilateral clearing to lenders is: (cid:20) (cid:18) (1−λH)pθ (cid:19) (cid:18) (1−λH)pθ (cid:19)(cid:21) q pu +λHθ) +(1−p)u + α α (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) −γ α α Bilateral clearing is preferred if γ < γb where (cid:20) (cid:18) (1−λH)pθ (cid:19) (cid:18) (1−λH)pθ (cid:19)(cid:21) γb =q pu +λHθ) +(1−p)u + α α (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) α α (cid:18) (cid:19) −u pθ (cid:0) 1−λL(cid:1) +pθλL α CASE 2: CCP contract pools over λH. If either ω < pθ (cid:0) 1−λL(cid:1) or ω > pθ (cid:0) 1−λL(cid:1) and q > 1 then the best contract with CCP α α α clearing pools over λH. When ω < pθ (cid:0) 1−λL(cid:1) , we need to distinguish the two sub-cases: α (cid:28)rst ω < pθ (cid:0) 1−λH(cid:1) , second pθ (cid:0) 1−λH(cid:1) < ω < pθ (cid:0) 1−λL(cid:1) . α α α 1. ω < pθ (cid:0) 1−λH(cid:1) . α Lemma19showsthatinthiscasecentralclearingisalwayspreferredtobilateralclearing. 64

2. pθ (cid:0) 1−λH(cid:1) < ω ≤ (1−λL)pθ and q ≥ 1. α α α In this case, consistently with lemma 19, λL < λ L and λH > λH Also, γˆ(ω) is de(cid:28)ned as follows: φ(γˆ(ω)) = q (cid:20) (1−λH)pθ +λHpθ (cid:21) +(1−q) (cid:2) ω+λLpθ (cid:3) (82) α (cid:16) (cid:17) Lenders’payo(cid:27)withcentralclearingisVCCP,λH = u (1−λH)pθ +pθ[qλH +(1−q)λL] . α Lenders’ payo(cid:27) with bilateral clearing, since both λL < λ∗ and λ∗ < λL < 1− αω∗ are pθ feasible, depends on whether (i) λH ≥ λ∗ and ω > ω∗ = (1−λ∗)pθ , (ii) λH ≥ λ∗ and α ω ≤ ω∗, or (iii) λH < λ∗. (i) λH ≥ λ∗, ω > ω∗. Claim 24 If pθ (cid:0) 1−λL(cid:1) ≥ ω > pθ (cid:0) 1−λH(cid:1) , λH ≥ λ∗, ω ≥ ω∗, q ≥ 1, and α α α γ ≥ γˆ(ω), the optimal contract involves i. bilateral clearing and information acquisition if γ ≤ γc. ii. CCP clearing and pooling over λH If γ > γc. Proof. The expected payo(cid:27) from bilateral clearing to lenders is: (cid:20) (cid:18) (cid:19) (cid:21) α q pu ω−c∗+θ− (ω−c∗) +(1−p)u(ω−c∗) + 1 1 1 p (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −γ Bilateral clearing is preferred if γ < γb where (cid:20) (cid:18) (1−λ∗)pθ (cid:19) (cid:18) (1−λ∗)pθ (cid:19)(cid:21) γc =q pu +λ∗θ) +(1−p)u + α α (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −u (cid:18) (1−λH)pθ +pθ[qλH +(1−q)λL] (cid:19) α 65

(ii) λH ≥ λ∗, ω ≤ ω∗. This is true by Lemma 19. (iii) λH < λ∗. Claim 25 If pθ (cid:0) 1−λH(cid:1) ≥ ω > pθ (cid:0) 1−λH(cid:1) , λH < λ∗, q ≥ 1, and γ ≥ γˆb, the α α α optimal contract involves i. bilateral clearing and information acquisition if γ ≤ γd. ii. CCP clearing and pooling over λH if γ > γd. Proof. The expected payo(cid:27) from bilateral clearing to lenders is: (cid:20) (cid:18) (1−λH)pθ (cid:19) (cid:18) (1−λH)pθ (cid:19)(cid:21) q pu +λHθ +(1−p)u + α α (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −γ Bilateral clearing is preferred if γ < γd where (cid:20) (cid:18) (1−λH)pθ (cid:19) (cid:18) (1−λH)pθ (cid:19)(cid:21) γd =q pu +λHθ +(1−p)u + α α (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −u (cid:18) (1−λH)pθ +pθ[qλH +(1−q)λL] (cid:19) α 3. pθ (cid:0) 1−λH(cid:1) < ω ≤ (1−λL)pθ and q < 1. α α α Lenders’ payo(cid:27) with central clearing is then VCCP,λH = u(ω(1−αq)+pθ−(1−q)(1− λL)pθ). Lenders’ payo(cid:27) with bilateral clearing, as in the previous case, depends on whether (i) λH ≥ λ∗ and ω > ω∗ = (1−λ∗)pθ , (ii) λH ≥ λ∗ and ω ≤ ω∗, or (iii) λH < λ∗. α (i) λH ≥ λ∗, ω > ω∗. Claim 26 If pθ (cid:0) 1−λL(cid:1) ≥ ω > pθ (cid:0) 1−λH(cid:1) , λH ≥ λ∗, ω ≥ ω∗, q < 1, q ≥ 1, α α α α and γ ≥ γˆ(ω), the optimal contract involves 66

i. bilateral clearing and information acquisition if γ ≤ γe. ii. CCP clearing and pooling over λH If γ > γe. Proof. The expected payo(cid:27) from bilateral clearing to lenders is: (cid:20) (cid:18) (cid:19) (cid:21) α q pu ω−c∗+θ− (ω−c∗) +(1−p)u(ω−c∗) + 1 1 1 p (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −γ Bilateral clearing is preferred if γ < γe where (cid:20) (cid:18) (1−λ∗)pθ (cid:19) (cid:18) (1−λ∗)pθ (cid:19)(cid:21) γe =q pu +λ∗θ) +(1−p)u + α α (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −u(ω(1−αq)+pθ−(1−q)(1−λL)pθ) (ii) λH ≥ λ∗, ω ≤ ω∗. This case is ruled out by Lemma 19. (iii) λH < λ∗. Claim 27 If pθ (cid:0) 1−λH(cid:1) ≥ ω > pθ (cid:0) 1−λH(cid:1) , λH < λ∗, q < 1, and γ ≥ γˆ(ω), the α α α optimal contract involves i. bilateral clearing and information acquisition if γ ≤ γg. ii. CCP clearing and pooling over λH if γ > γg. Proof. The expected payo(cid:27) from bilateral clearing to lenders is: (cid:20) (cid:18) (1−λH)pθ (cid:19) (cid:18) (1−λH)pθ (cid:19)(cid:21) q pu +λHθ +(1−p)u + α α (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −γ 67

Bilateral clearing is preferred if γ < γg where (cid:20) (cid:18) (1−λH)pθ (cid:19) (cid:18) (1−λH)pθ (cid:19)(cid:21) γg =q pu +λHθ +(1−p)u + α α (1−q) (cid:2) pu (cid:0) ω+θλL(cid:1) +(1−p)u(ω) (cid:3) −u(ω(1−αq)+pθ−(1−q)(1−λL)pθ) 4. ω > pθ (cid:0) 1−λL(cid:1) and q ≥ 1. α α (cid:16) (cid:17) Lenders’payo(cid:27)withcentralclearingisthenVCCP,λH = u (1−λH)pθ +pθ[qλH +(1−q)λL] . α Lenders’ payo(cid:27) with bilateral clearing depends on whether (i) λH ≥ λ∗ and ω > ω∗, or (ii) λH < λ∗. In fact, in this case, λL > 1− αω, thus lemma 19 implies that we can pθ restrict to the case λL < λ∗. Also, in this case φ(γˆ(ω)) is de(cid:28)ned as follows: φ(γˆ(ω)) = q (cid:2) ω+λHpθ (cid:3) +(1−q) (cid:2) ω+λLpθ (cid:3) (83) (i) λH ≥ λ∗, ω > ω∗ Claim 28 If ω ≥ (1−λL)pθ , λH > λ∗ > λL, αq ≥ 1, and γ ≥ γˆ(ω), the optimal α contract involves i. bilateral clearing and information acquisition if γ ≤ γh. ii. CCP clearing and pooling over λH If γ > γh. Proof. Lenders’ expected payo(cid:27) from bilateral clearing is to lenders is: (cid:20) (cid:18) (cid:19) (cid:21) α q pu ω−c∗+θ− (ω−c∗) +(1−p)u(ω−c∗) + 1 1 1 p (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) −γ α α Then using the fact that ω−c∗ = (1−λ∗)pθ , we can rewrite that bilateral clearing 1 α 68

is preferred if γ ≤ γh de(cid:28)ned as (cid:20) (cid:18) (1−λ∗)pθ (cid:19) (cid:18) (1−λ∗)pθ (cid:19)(cid:21) γh = q pu +λ∗θ +(1−p)u + α α (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) − α α (cid:18) (cid:19) u pθ (cid:0) 1−λH(cid:1) +pθ (cid:2) qλH +(1−q)λL(cid:3) α (ii) λH < λ∗. Claim 29 If ω ≥ (1−λL)pθ , λH < λ∗, αq ≥ 1, and γ ≥ γˆ(ω), the optimal contract α involves i. bilateral clearing and information acquisition if γ ≤ γi. ii. CCP clearing and pooling over λH if γ > γi. Proof. Lenders’ expected payo(cid:27) from bilateral clearing is: (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) pθ pθ q pu (1−λH)+λHpθ +(1−p)u (1−λH) + α α (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) −γ α α Then bilateral clearing is preferred if γ ≤ γi, with γi de(cid:28)ned as follows: (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) pθ pθ γi = q pu (1−λH)+λHpθ +(1−p)u (1−λH) + α α (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (1−q) pu pθ (cid:0) 1−λL(cid:1) +θλL +(1−p)u pθ (cid:0) 1−λL(cid:1) − α α (cid:18) (cid:19) u pθ (cid:0) 1−λH(cid:1) +pθ (cid:2) qλH +(1−q)λL(cid:3) α 69

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