Transparency and collateral: the design of CCPs' loss allocation rules

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Transparency and collateral: the design of CCPs’ loss allocation rules Gaetano Antinolfi, Francesca Carapella and Francesco Carli 2019-058 Please cite this paper as: Antinolfi, Gaetano, Francesca Carapella and Francesco Carli (2019). “Transparency and collateral: the design of CCPs’ loss allocation rules,” Finance and Economics Discussion Series 2019-058. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2019.058. 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: the design of CCPs’ loss allocation rules ∗ Gaetano Antinol(cid:28), Francesca Carapella, and Francesco Carli(cid:159) † ‡ July 23, 2019 Abstract This paper adopts a mechanism design approach to study optimal clearing arrangements for bilateral (cid:28)nancial contracts in which an assessment of counterparty risk is crucial for e(cid:30)ciency. The economy is populated by two types of agents: a borrower and lender. The borrower is subject to limited commitment andholdsprivateinformationabouttheseverityofsuchlackofcommitment. The WeareverygratefultoNedPrescottforhisthoughtfuldiscussionandtoGuillaumeRocheteaufor ∗ his valuable input. We also thank Garth Baughman, Florian Heider, Marie Hoerova, Cyril Monnet, BorghanNarajabad,andWilliamRoberdsfortheircommentsandsuggestions,andparticipantsinthe Spring2013MidwestMacroeconomicsMeetings;2013EuropeanSummerMeetingsoftheEconometric Society; 2013 Society for Advancement of Economic Theory Conference; 2014 Chicago Fed Money workshop;SystemCommitteeMeetingonFinancialStructureandRegulationattheDallasFed;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. All errors are our own. The views expressed in this paperaresolelytheresponsibilityoftheauthors,andshouldnotbeinterpretedasre(cid:29)ectingtheviews 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 (Corresponding Author). ‡ (cid:159)Deakin University, fcarli@deakin.edu.au. 1

lender can acquire information at a cost about the commitment of the borrower, whicha(cid:27)ectstheassessmentofcounterpartyrisk. Whentruthfulrevelationbythe borrower is not incentive compatible, the mechanism designer optimally trades o(cid:27) the value of information about the lack of commitment of the borrower with the cost of incentivizing the lender to acquire such information. Central clearing of these (cid:28)nancial contracts through a central counterparty (CCP) allows lenders to mutualize their counterparty risks, but this insurance may weaken incentives to acquire and reveal information about such risks. If information acquisition is incentive compatible, then lenders choose central clearing. If it is not, they may prefer bilateral clearing to prevent strategic default by borrowers and to economizeoncostlycollateral. Centralclearingisanalyzedunderdi(cid:27)erentinstitutional features observed in (cid:28)nancial markets, which place di(cid:27)erent restrictions on the contract space in the mechanism design problem. The interaction between the costly information acquisition and the limited commitment friction di(cid:27)ers signi(cid:28)cantlyineachclearingarrangementandineachsetofrestrictions. Thisresultsin novel lessons about the desirability of central versus bilateral clearing depending on traders’ characteristics and the institutional features de(cid:28)ning the operation of the CCP. Keywords: Limited commitment, central counterparties, collateral JEL classi(cid:28)cation: G10, G14, G20, G23 1 Introduction An important aspect of modern (cid:28)nancial contracting is that (cid:28)nancial institutions trade avarietyofproductsbilaterally,suchasover-the-counter(OTC)derivatives,repurchase 2

1 agreements,andreservesheldatthecentralbank. Informationabouttheexposureofa counterparty to various risks is necessary to select appropriate contractual terms, such as prices and collateral, in order to control the risk that a counterparty will not ful(cid:28)ll its future obligations. This information, however, is often con(cid:28)ned within a bilateral relationship because of the high degree of specialization in understanding and pricing risks speci(cid:28)c to a certain (cid:28)nancial product, and because of the interaction between the counterparties across other (cid:28)nancial markets. The (cid:28)nancial crisis of 2008 has highlighted the systemic importance of such infor- 2 mation. Both academic researchers and policymakers 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 3 exacerbating (cid:28)nancial distress. Consequently, particular attention has been devoted to the role of clearing institutions and to their potential in improving transparency in 4 (cid:28)nancial markets. Mandatory clearing via a central counterparty (CCP), de(cid:28)ned below, hasbeenatthecenterof(cid:28)nancialreformsbothintheUSandinEurope. However, the consequences of these reforms on the incentives of (cid:28)nancial market participants to acquire information about each other are not well understood. In this paper, we address the question of potential tradeo(cid:27)s between bilateral and central clearing with respect to market transparency. We adopt a mechanism design approach and develop a model of (cid:28)nancial contracting 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 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 FederalFundsmarketAfonsoandLagos(2012a),AfonsoandLagos(2012b),BechandAtalay(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 (2013) among others. 3

related to signi(cid:28)cant synergies across di(cid:27)erent projects and trades which are observable only to the agents involved in those activities. Thus, soft information cannot be easily 5 and publicly veri(cid:28)ed by a third party, or it is di(cid:30)cult to summarize and aggregate. In our economy, trading is bilateral and subject to two frictions: limited pledgeability of a counterparty’s future income, and private information about the degree of pledgeability of income, which we call an agent’s pledgeability type. Costly monitoring reveals the extent to which a counterparty’s income is pledgeable. 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 contractualtermsappropriately. Whenmonitoringdoesn’ttakeplace,thepledgeability type of a counterparty cannot be part of contractual terms. In this case, information is not available to (cid:28)nancial market participants, and in particular to clearing institutions; such lack of information results in ine(cid:30)cient collateral by the CCP and, possibly, in strategic defaults by some of its members. 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 posted. Central clearing is done by a third party, namely a central counterparty, 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 6 contracts. The substitution of the CCP as the sole counterparty for each of the two original traders in a bilateral transaction is called novation. Novation, however, doesn’t 5See Stein (2002), Petersen (2004), Hauswald and Marquez (2006), Mian (2003). 6See Capital requirements for bank exposures to central counterparties (2012), and BIS glossary of terms used in payments and settlement systems (2003). 4

eliminate counterparty risk: the CCP needs to use proper risk management tools and loss allocation mechanisms to guarantee that it has enough resources to perform the 7 obligations stemming from novation, despite possible default by some of its members. Through novation, 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)rst one implies that, within a speci(cid:28)c market, the CCP has information about the network of trades across its members, which may not be available to the bilateral counterparties. The second implies that theCCPmaylackinformationaboutitsmembers, ifthatinformationislearnedoutside the speci(cid:28)ed set of contracts which the CCP clears. 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 andcharacterizetheconditionsunderwhichCCPclearingmightreducewelfarerelative to bilateral clearing. In our economy, clearing arrangements and risk management tools adopted by the CCP a(cid:27)ect equilibrium outcomes, including incentives to acquire information about counterparties. Our model 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 which operates under commonly used loss allocation methods. We model the institutional features of loss allocation methods adopted by modern CCPs as restrictions on the contract space of the CCP. The tradeo(cid:27) between bilateral and central clearing arises from i) two dimensions of risk against which traders value insurance, namely a counterparty’s uncertain income and pledgeability type, ii) private information about a counterparty’s pledgeability type, which introduces an adverse selection problem, iii) and the risk management 7See,amongothers,https://www.chicagofed.org/publications/chicago-fed-letter/2017/ 389 (2017) 5

tools and loss allocation methods adopted by the CCP. 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 necessary to tailor collateral requirements to the counterparty’s pledgeability type. With CCP clearing, uncertainty about a counterparty’s income is managed through speci(cid:28)c loss allocation methods, which de(cid:28)ne the (cid:28)nancial resources that are used to absorb the losses caused by a default. We consider three institutional arrangements currently adopted by CCPs to absorb losses in excess of defaulting members’ margins. In the (cid:28)rst case, the CCP allocates losses pro-rata among surviving members, as in Acharya and Bisin (2014) and Koeppl and Monnet (2010). Loss mutualization enables the CCP to diversify counterparty risk and save on collateral requirements. The second institutional arrangement consists in partial tear-up of unmatched contracts. Under this loss allocation method, which is adopted under extreme default scenarios, the CCP canterminatecontractsbyoriginalcounterparties,thusde-factoundoingnovation. The third institutional arrangement is a pro-rata loss allocation method with due diligence, which is routine operation of modern CCPs, and is equivalent to the (cid:28)rst arrangement but with disciplinary actions. According to this method, the CCP must honor the contracts it clears, unless it detects lack of due diligence by its members, in which case 8 it adopts disciplinary actions. The loss allocation method adopted by the CCP implies a degree of insurance against counterparty risk which 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 8SeeIceClearCreditClearingRules(2018),ICEClearDisclosureFramework(2018),andNational Securities Clearing Corporation - Rules and Procedures (2018). 6

contracts that make central clearing Pareto superior to bilateral clearing. We call an 9 allocation that satis(cid:28)es these conditions incentive-feasible. An incentive-feasible allocation always exists if the CCP adopts, in equilibrium, a partial tear-up loss allocation method. When the CCP doesn’t or cannot rely on a partial tear-up loss allocation method, incentive-feasible allocations may not exist and there is a trade-o(cid:27) between bilateral and central clearing. CCP clearing naturally maintains the ability to provide insurance by pooling risk over idiosyncratic shocks to income. Without the information generated by monitoring, however, the CCP cannot tailor contracts to counterparty types in a trade, resulting in either excessive or insuf- (cid:28)cient collateral. With bilateral clearing, insurance requires to post collateral. This is costly, but it is exactly this cost that preserves incentives to monitor. Intuitively, monitoring produces information useful in tailoring collateral requirements to the type of counterparty and, when collateral is costly, this information is valuable. If monitoring is not too costly, traders prefer bilateral clearing. The insurance provided by the CCP may not be 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. As we discuss in the next section, our results and the economic mechanism at the core of our analysis are consistent with empirical (cid:28)ndings for certain (cid:28)nancial markets, with concerns of practitioners and regulators, and with (cid:28)ndings from recent work based on network analysis. 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 9Because 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 weakly better o(cid:27). 7

traded and the information set of market participants and clearing institutions. In this respect, our model shows what characteristics of assets and trades are optimally associated with bilateral and central clearing arrangements. Speci(cid:28)cally, for a region of the parameter space, our model implies that (cid:28)nancial institutions with high opportunity cost of collateral, such as dealers and hedge funds, should prefer to clear their trades bilaterally, whereas institutions with a low opportunity cost of collateral, such 10 as money market funds (MMMF), are more likely to rely on CCPs. The paper is organized as follows: the remainder of this section provides a literature review, Section2describesthemodel, Sections3optimalcontractswithoutinformation acquisition, Section 4 optimal contracts with information acquisition, and Section 5 comparative statics. 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 (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 10That the opportunity cost of collateral is larger for dealers than MMMF is re(cid:29)ected in higher returns produced by the former. That MMMF have taken up central clearing wherever possible is re(cid:29)ected,forexample,intheincreaseintherateofreposclearedatFixedIncomeClearingCorporation (FICC) between 2015 and 2017, which have tended to replace reverse repos with the Fed (see BIS Quarterly Review (2017)). 8

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 two-sided 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 by the loss allocation method adopted. 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. Two models of central clearing related to ours are Biais et al. (2016) and Koeppl (2013). In both papers, there is asymmetric information between buyers and sellers of (cid:28)nancial assets in the form of moral hazard, and collateral plays both the role of insuring counterparties and aligning incentives. Moral hazard generates the potential for excessive risk taking, which is controlled by margin requirements. In Biais et al. (2016), the risk pooling activity associated with central clearing allows the central counterpartytosetmarginsmoree(cid:30)cientlythaninbilateraltrade, andcentralclearing always dominates. In Koeppl (2013), margin requirements play a similar role as in Biais et al. (2016) in controlling moral hazard, but also, as in our case, generate a tradeo(cid:27) between bilateral and central clearing. In Koeppl (2013), a CCP has the 9

objective of minimizing the risk associated with trades. Moral hazard in risk-taking is potentially ampli(cid:28)ed by collusion between buyers and sellers, and the resulting margin requirements can have a negative impact on market liquidity (as measured by the probability of (cid:28)nding a counterparty). Thus, the tradeo(cid:27) between bilateral and central clearing lies in their relative impact on market liquidity. 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. Our paper is also related to the literature on payment systems, in particular to Koeppl et al. (2012), who study the e(cid:30)ciency of a clearing and settlementsysteminanenvironmentwithinformationasymmetrybetweentheclearing 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. An extensive analysis of central counterparties is provided by Pirrong (2009), which describes aspects of central clearing that are reminiscent of our formal model, noting that central clearing always involves a mechanism to redistribute losses in case of default. This redistribution may be a(cid:27)ected by asymmetric information problems, which are likely to be relatively more severe when central clearing involves members whose balance sheets are opaque as a result of trading positions outside the products that are 10

centrally cleared. These balance sheet risks and the asymmetric information problems associated with them a(cid:27)ect more severely centrally cleared markets than bilaterally cleared markets. A conclusion of Pirrong’s discussion is that "...a CCP prices default risk as if all members are homogeneous, when in fact they are not necessarily so. Although this imposed homogeneity can contribute to liquidity, it misprices balance sheet risks and tends to encourage trading by less creditworthy (cid:28)rms. Thus, a variety of considerations suggests that the cost of evaluating and pricing balance sheet risks are lower in bilateral OTC markets than centrally cleared ones, especially when intermediaries are complex (cid:28)rms engaged in information-intensive intermediation." (Pirrong (2009), page 50.) Ourresultsformalizeconcernsexpressedbypractitionersandanalystsaboutregulatory reforms of clearing arrangements. Gregory (2014), Section 1.5, discusses possible dangers of introducing mandatory central counterparty clearing: (cid:16)A third potential problem [of CCP clearing] is related to loss mutualization that CCPs use whereby any losses in excess of a member’s own (cid:28)nancial resources are generally mutualized across all the surviving 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 likely to have better information than CCPs especially for more complex derivatives.(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.) Our results and the economic mechanism at the core of our analysis are consistent with 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 11

an equilibrium outcome of our model, despite transparency may decrease with central clearing. The results and assumptions of our model are also 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 asymmetry in the failure of the Caisse de Liquidation des A(cid:27)aires et Marchandises (CLAM, a CCP clearing sugar futures) in Paris in 1974, as (cid:16)retail investors were unsophisticated and non-diversi(cid:28)ed, did not have enough liquid (cid:28)nancial 11 resources(cid:17) and that CLAM could not (cid:16)directly monitor ultimate investors(cid:17). 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. In equilibrium, then, the CCP is unable to charge member-speci(cid:28)c margins. Bignon and 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. Finally, our results about the existence of a trade-o(cid:27) between bilateral and central clearing based on the value of insurance provided by CCPs, resonate with those in Garratt and Zimmerman (2015). They study netting in a network model of trades and (cid:28)nd that CCPs (cid:16)can improve netting e(cid:30)ciency only if agents have some degree of risk aversion that allows them to trade o(cid:27) the reduced variance against the higher expected netted exposures.(cid:17) They further hypothesize that (cid:16)This may explain why, in the absence of regulation, traders in a derivatives network may not develop a CCP themselves,(cid:17) which is also consistent with our results about bilateral clearing being preferred in some (cid:28)nancial markets. 11BignonandVuillemey(2016)goevenfurther, theorizingrisk-shifting behavioronthepartofthe CCP once it realized it was close to bankruptcy. 12

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. There are two goods: a consumption good and a capital good. 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. The capital good can be invested at time t = 1 and transformed 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 and assume that the law of large numbers holds so that the aggregate return on investment is pθ. 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 1 and time 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, where i denotes − a borrower’s type. 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 we will refer to them as high-pledgeabilitiy borrowers, while a measure 1 q can pledge a fraction λL < λH, and we will refer to them as low- − 13

pledgeabilitiy borrowers. The type λi is private information of the borrower, but can be learned by a lender by exerting monitoring e(cid:27)ort. The preferences of a lender are de(cid:28)ned over second period consumption x , and 2 time-1 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 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-or- leave-it (TIOLI) o(cid:27)er, which also speci(cid:28)es a clearing and settlement 12 arrangement. In the second period, settlement takes place either bilaterally or trough a CCP, according to the lenders’ choice. Feasible contracts di(cid:27)er depending on the clearing arrangement initially chosen. In the next sections, we de(cid:28)ne and characterize optimal contracts with bilateral and central clearing. Our analysis includes the equilibrium characterization of economies with central clearing under di(cid:27)erent assumptions about the contract space, each corresponding to di(cid:27)erent possibilities for the CCPs’ operation, rules, and procedures. Speci(cid:28)cally, we analyze 1) a benchmark case, where potential CCP losses are allocated 12When the commitment constraint is binding, the assumption of a TIOLI is without loss of generality because of transferable utility. 14

pro-rata among its members; 2) a partial tear-up case, which is adopted by CCPs under extreme default scenarios; and 3) a case with pro-rata allocation of losses and due 13 diligence, describing the routine operation of a CCP with disciplinary actions. We model the benchmark case by assuming that, after novation, the (cid:28)nancial obligation between the original counterparties is eliminated and the CCP allocates losses pro-rata among its members. Novation is, in fact, a counterparty substitution: from the bilateral counterparty to the CCP, which (cid:16)stands in between buyers and sellers and guarantees the performance of trades ...[and]... is legally obliged to perform on the 14 contracts it clears.(cid:17) The assumption that the (cid:28)nancial obligation between original counterparties is eliminated is meant to model the legal obligation of the CCP to perform on cleared contracts. The pro-rata loss allocation method implies that any losses which the CCP experiences are mutualized across its members. Hence, no information about a member’s default is used to allocate losses ex-post. In the context of our 15 model, this is achieved by allowing the CCP to pool idiosyncratic risk. We model the partial tear-up case by assuming that the CCP can o(cid:27)er its members fully state contingent contracts. In extreme default scenarios (cid:16)loss allocation methods [that] go beyond the idea of simply using default funds on a pro-rata basis(cid:17) can be implemented. One such method gives the CCP an (cid:16)option [...] to (cid:16)tear-up(cid:17) unmatched contracts with surviving clearing members. [...] The aim of the tear-up is to return the CCP to a matched book by terminating the other side of a defaulter’s trades (or at least those that cannot be auctioned). All other contracts (possibly the majority of the 16 total contracts cleared) could remain untouched.(cid:17) This loss allocation method, in the 13See Gregory (2014) section 10.3. 14See Gregory (2014), section 8.3, and ?: (cid:16)CCPs are best seen as commitment mechanisms that assure the performance of (cid:28)nancial contract obligations. How they perform that function sets them apart from other infrastructures, intermediaries and (cid:28)nancial institutions.(cid:17) 15A possible interpretation of this (cid:28)rst case is that the contracts submitted for central clearing are liquid, and one side of the contract can sell his position to a third party, implying that the initial link between the lender and the borrower is destroyed. 16See Gregory (2014), pg 187-192. 15

context of our model, requires the CCP to terminate its contract with a lender upon default (exogenous or strategic) of her original borrower. To gain perspective into the preference of CCPs (and regulators) to avoid partial tear-ups, it is instructive to note that such loss allocation method was not adopted in the two stressful scenarios that involved CCPs in the past few decades. In September 2008, the default of Lehman did not trigger any partial tear up but was resolved worldwide with auctions and transfers of clients’ accounts resulting in closing out all of Lehman’s positions without 17 a(cid:27)ecting CCPs ability to perform on their obligations. Ten years later, in September 2018, despite the default of a major member wiping out (cid:16)2/3 of the default fund(cid:17) of Nasdaq Clearing, the CCP did not adopt a partial tear-up. Rather, Nasdaq Clearing i) contributed its Junion Capital fund, and ii) recapitalized the Default Fund using additional contributions from its clearing members (cid:21) in other words, a pro-rata loss 18 allocation method was adopted. We model the case of pro-rata with due diligence by assuming that the CCP must honor the contracts it clears, unless it detects lack of due diligence by its members. These assumptions are equivalent to a CCP operating with a pro-rata loss allocation method with the ability to impose sanctions for violations of its rules or for prohibited 19 conduct. The pro-rata loss allocation method implies that any losses which the CCP experiences are mutualized across its members. In the context of our model, this is achieved by allowing the CCP to pool idiosyncratic risk, similarly to Koeppl and Monnet (2010) and Biais et al. (2016). The ability to impose sanctions for violations 17See Central Counterparty Default Management and the Collapse of Lehman Brothers", CCP12, The Global Association of Central Counterparties (2009). Also, see Faruqui et al. (2018), and the speech by Sir Jon Cunli(cid:27)e, Deputy Governor for Financial Stability of the Bank of England, at the FIA International Derivatives Expo 2018, London (5 June 2018). 18See https://business.nasdaq.com/updates-on-the-Nasdaq-Clearing-Member-Default/index.html. See Elliott (2013) for a discussion of the disadvantages of a pro-rata loss allocation method with respect to the incentives of the CCP members to participate in a default management process. 19See Gregory (2014) sections 10.1.1-3 on the pro-rata loss allocation method. Also see Ice Clear Credit Clearing Rules (2018) and ICE Clear Disclosure Framework (2018): articles 701 2, rule 609. − 16

of the CCP’s rules and procedures implies that the net payment from the CCP to a member might di(cid:27)er from the payment speci(cid:28)ed in the contract submitted for central clearing. In the context of our model, the CCP detects lack of due diligence when it is able to identify whether a lender did not monitor her counterparty or did not report her counterparty’s type truthfully. When this happens we assume that the CCP can use any information about the original counterparty to punish such misbehavior by the lender. 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 madeisintrinsicallythesame. Limitedcommitmentisthepivotalfrictioninthemodel, and it introduces interesting interactions between the clearing arrangement, the terms of the contract traded, and the information acquired about the counterparty. 3 Optimal contracts without information acquisition In this section, we characterize optimal contracts without information acquisition. The goal of this section is twofold: 1) to introduce notation and the basic mechanics of our model; 2) to set a benchmark for contracts to which we will refer in subsequent sections when information acquisition will not occur in equilibrium. The de(cid:28)nition of a 17

contract di(cid:27)ers depending on the clearing arrangement chosen. With bilateral clearing, settlement involves only the original counterparties. Instead, when clearing is central, borrowers and lenders submit to the CCP the contract upon which they agree. The CCP then novates such 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 terms of the original contract as given, but can require borrowers to post collateral (i.e. margin), and lenders to contribute to a loss mutualization scheme (i.e. default or guarantee fund). 3.1 Bilateral Clearing without information acquisition Whenclearingisbilateral, lenderscommittoamechanismthatspeci(cid:28)esamenuofcontracts. Without loss of generality, we assume that lenders commit to direct revelation mechanisms, that is, a contract is executed after the borrower truthfully announces his type. Thus, a strategy for a borrower is a pair (mi,σi) λL,λH 0,1 , where ∈ { }×{ } mi is his reporting strategy and σi his default decision when the idiosyncratic state is s = h; σi = 1 means that the borrower defaults in equilibrium. 20 A mechanism with bilateral clearing is a menu of contracts (Σi, ci, ci , xi,∆) , 1 2,s 2,s i={L,H} where Σi is the lender’s default recommendation (contingent on the idiosyncratic state s = h) to a borrower that reports his type to be λi. We use the notation Σi = 1 to mean that a lender recommends her counterparty to default in equilibrium. Also, ∆ represents the public history of the borrower’s default/repayment decision, where ∆ = 1iftheborrowerdefaultsinequilibrium, and∆ = 0iftheborrowerrepays. Wesay that a contract is incentive-compatible if a borrower’s best strategy (mi,σi) is to report 20Tobemorespeci(cid:28)c,astrategyforaborrowerisatriple(mi,σi,σi),whereσi istype-iborrower’s l h s default strategy, when the idiosyncratic state is s. However, σi =0 is a (weakly) dominant strategy, l sowecanignoreitfromthede(cid:28)nitionoftheborrower’sfeasiblestrategiesandassumethataborrower always repays when s=l. 18

truthfully his type, mi = λi, and then follow the default/repayment recommendation, σi = Σi. After reporting his type and accepting the ensuing contract, a borrower receives one unit of capital and transfers ω ci units of consumption good to the lender. As − 1 an example of a (cid:28)nancial contract between the lender and the borrower, consider a repurchase agreement (repo): then we can think of the unit of capital transferred by the lender to the borrower at t = 1 as the starting leg of the repo, and of the payment xi by the borrower to the lender at t = 2 as the closing leg of the repo. 21 We can think 2 of the transferofω ci by theborrower to the lender att = 1as collateral, as it denotes − 1 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 mortgage) − 1 as it preserves the value of the lender’s investment by insuring the lender against the 22 borrower’s default. 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, after − 1 the shock realization is known, the lender is entitled to consumption xi , whereas the 2,s borrower is entitled to consumption ci and chooses whether to default (σi = 1) or to 2,s repay (σi = 0). When the realization of the borrower’s idiosyncratic state at t = 2 is low (s = l) the lender can use the additional resources from collateral for her own consumption. When the realization of the borrower’s idiosyncratic state at t = 2 is high (s = h) and the borrower repays (σi = 0) the lender returns the collateral to the borrower, who consumes it together with the return from his production technology 23 net of the payment to the lender for the closing leg of the repo. If the borrower 21See Garbade (2006). 22Notice 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. 23This example re(cid:29)ects the micro-foundations of a repo as derived in Antinol(cid:28) et al. (2015), where borrowers have access to an investment technology that needs lenders’ capital (and borrowers’ know how) to be operated, and where the possibility of borrowers’ default justi(cid:28)es lenders to require the transfer of collateral upfront from borrowers as a means to self insure. 19

defaults (σi = 1) the lender keeps the collateral and the borrower’s pledgeable income (ω ci +λiθ), whereas borrower’s consumption is equal to (1 λi)θ. − 1 − The optimal mechanism solves the following problem: (cid:88) (cid:104) (cid:110) (cid:111) (cid:105) (Pb) Vbil,e=0 =max q p Σiu(xi1)+(1 Σi)u(xi0) +(1 p)u(xi ) (1) 0 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 (2) 1 − − 2h − 2l ≥ ω ci 0 (3) ≥ 1 ≥ xi0 +ci ω ci +θ (4) 2h 2h ≤ − 1 xi1 ω ci +λiθ (5) 2h ≤ − 1 xi +ci ω ci (6) 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 ∈ 1 − − 2h − 2l (mˆ,σˆ) (7) Constraint (2) is borrower i’s participation constraint, for i L,H : the bor- ∈ { } rower can always refuse to trade, and consume the endowment ω in the (cid:28)rst period. Constraint (3) is time t = 1 feasibility, (4) and (5) are time t = 2 feasibility in states (s,∆) = (h,0) and (s,∆) = (h,1) respectively; (6) is time t = 2 feasibility condition in state l. Finally, constraint (7) is the incentive-compatibility constraint for a borrower of type λi: the strategy pair (λi,Σi) is incentive compatible if there is no other strategy pair (mˆ,σˆ) that yields a higher payo(cid:27). Notice that a borrower can deviate by reporting a di(cid:27)erent type mˆ = λi, by choosing a di(cid:27)erent default strategy σˆ = Σi, or both. (cid:54) (cid:54) InProposition3ofSection3.2weprovethatbilateralclearingisneveroptimalwhen borrowers’ type is private information and lenders have no technology to learn such type. Thus, we do not characterize the soloution to problem (Pb), because the goal of 0 the paper is to compare bilateral versus central clearing, and the characterization of 20

mechanisms with bilateral clearing and no monitoring is irrelevant to the question we want to address. 3.2 Central Clearing without information acquisition A key aspect of central clearing is novation, that is the legal act of erasing the original obligations between a borrower and a lender, and the CCP becoming the sole counterparty to each of them. 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 the contracts in such mechanism. Each contract speci(cid:28)es transfers between borrowers and the CCP and transfers between lenders and the CCP, while no transfer between the borrower and the lender takes place. Transfers are a function of i) public information and ii) the restrictions on the contracting space consistent with institutional arrangements adopted by CCPs in reality. Formally, a strategy for a type-i borrower is a pair (mi,σi) λL,λH 0,1 ∈ { } × { } that speci(cid:28)es a (cid:28)rst period announcement strategy mi λi,λH and a second period ∈ { } default decision σi 0,1 . Similarly to the bilateral clearing case, σi = 1 means that ∈ { } the borrower defaults in equilibrium. Let then ∆ 0,1 be the borrower’s observed ∈ { } default/repayment decision and s l,h denote his time t = 2 idiosyncratic state. ∈ { } De(cid:28)ne to be the set of all possible second-period histories for a borrower. Thus, 2 H an element of is a pair (s,∆), and = l,h 0,1 . A mechanism with central 2 2 H H { }×{ } clearing consists of contracts between the CCP and the lender, Xi(h ) , and { 2 2 } i=L,H between the CCP and the borrower, Σi, Ci, Ci , which are executed if the { 1 2,s} i=L,H borrower reports his type to be λi. Let h denote the public history of a borrower after 2 novation, and Σi the default decision that the CCP recommends to a borrower who reports his type to be λi. A mechanism is incentive compatible if it is the borrower’s bestresponsetoreporttruthfullyhistype, andthenfollowthedefaultrecommendation 21

Σi. In Lemma 2 we prove that optimal contracts between the CCP and lenders are independent of public history h . Thus, in this section we abstract from discussing the 2 di(cid:27)erences in the speci(cid:28)c institutional arrangements adopted by CCPs with respect to their loss allocation mechanism. We defer to Section 4.2 the discussion of these institutional arrangements and the analysis of the ensuing restrictions on the contracting space with central clearing. To simplify notation, we then write Xi (cid:0) (s,∆) (cid:1) = Xi,∆ . Referring to the example in 2 2,s Section3.1, ifthe(cid:28)nancialcontractisarepurchaseagreement, thecontractbetweenthe lender and the CCP involves a starting leg where the lender transfers his endowment of capital to the CCP at t = 1, and a closing leg where the CCP pays Xi,∆ to the lender, 2,s where (s,∆) denotes new contractible information about the borrower, observed by the CCP in t = 2. The contract between the borrower and the CCP involves a starting leg where the CCP transfers one unit of capital (received from the lender) to the borrower, and the borrower transfers ω Ci units of good to the CCP as a margin − 1 requirement. The closing leg of this contract involves the transfer of θ Ci units of − 2,s consumption good from the borrower to the CCP, of which θ Ci Xi,∆ are default − 2,s− 2,s 24 fund contributions from the borrower. Let mˆ and σˆ be de(cid:28)ned as in Section 3.1 for problem (Pb). The optimal mechanism 0 with central clearing and no monitoring solves (P ) VCCP,e=0 = max (cid:88) q (cid:110) p (cid:104) Σiu(X i,1 )+(1 Σi)u(X i,0 ) (cid:105) +(1 p)u(X i,0 ) (cid:111) (8) 0 i 2,h − 2,h − 2,l i s.t. αCi +p[Σi(1 λi)θ+(1 Σi)Ci ]+(1 p)Ci αω (9) 1 2h 2l − − − ≥ 0 Ci ω (10) 1 ≤ ≤ 24Clearly, if the realization of the borrower’s output shock is low (s=l) then this borrower makes no payment to the CCP at t=2. The CCP can use resources from margin requirements and default fund contributions of borrowers able to pay, in order to settle payments to lenders. 22

(cid:88) q (cid:110) p (cid:104) Σi (cid:110) X i,1 +(1 λi)θ (cid:111) +(1 Σi) (cid:110) X i,0 +Ci (cid:111)(cid:105) i 2,h − − 2,h 2h i +(1 p)(Ci +X i,0 ) (cid:111) pθ+ (cid:88) q ω Ci (11) − 2l 2,l ≤ 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 (12) 1 2h 2l ∈ − − − (mˆ,σˆ) Constraint(9)isborrower’siparticipationconstraint; (10)and(11)arerespectively time t = 1 and t = 2 feasibility constraints. Note that the feasibility constraint in t = 2 is de(cid:28)ned for the aggregate resources of the CCP in the second period, since the CCP becomes the buyer to every seller and the seller to every buyer. Also, note that CCP’s resources are constant, as there is no aggregate uncertainty. Constraint (12) is the incentive compatibility constraint of a borrower who must report his type truthfully, mi = λi, and then follow the default recommendation, σi = Σi. Lemma 1 Any contract preventing borrowers’ default, ΣH = ΣL = 0, must satisfy min CH , CL (1 λL)θ and αCH+pCH +(1 p)CH = αCL+pCL +(1 p)CL . { 2,h 2,h} ≥ − 1 2,h − 2,l 1 2,h − 2,l Lemma 1 is a direct consequence of the incentive-compatibility constraint (12), and the observation that a borrower may deviate by reporting a di(cid:27)erent type mˆ = λi, (cid:54) by choosing a di(cid:27)erent default strategy σˆ = Σi, or both. In other words, the limited (cid:54) commitment friction interacts with private information in a way that forces the CCP to treat both types of borrowers as if their constraints were the same as that of the worst type. This result has important consequences for the contract which the CCP can o(cid:27)er, as it prevents any separation of borrowers where information about a high pledgeabilitytypecanbeexploited, ifatthesametimethecontractpreventsborrowers’ default. Lemma 2 A solution to Problem (P ) is such that Xi,∆ = X−i,∆(cid:48) for all s,s(cid:48) l,h 0 2,s 2,s(cid:48) ∈ { } and ∆,∆(cid:48) 0,1 . ∈ { } 23

Lemma 2 proves that the CCP optimally ignores information about a borrower in the contract with the lender who was the original counterparty to that borrower: concavity of the utility function u( ) implies that a solution must satisfy Xi,∆ = Xi,∆(cid:48) , for all · 2,s 2,s(cid:48) s,s(cid:48) l,h and∆,∆(cid:48) 0,1 . Thus, anyrestrictiontothespaceofcontractsbetween ∈ { } ∈ { } the CCP and lenders is irrelevant. Comparing problem (P ) to problem (Pb) we can prove the following result: 0 0 Proposition 3 Without information acquisition, central clearing is the optimal clearing arrangement: the solution to (P ) dominates the solution to (Pb). 0 0 When lenders cannot learn the pledgeability type of their counterparty, they are no better than the CCP at evaluating the risk that a borrower will strategically default. Thus, the CCP can always replicate borrowers’ optimal contracts of Section 3.1, and, in addition, insure lenders against the idiosyncratic risk associated with the original counterparty. Despite borrowers’ expected repayments to lenders are the same as with bilateral clearing, counterparty risk vanishes with the CCP due to the provision of insurance. Hence, central clearing is always preferred to bilateral clearing. To characterize the solution to problem (P ), using Lemma 2 we simplify the nota- 0 tion and write Xi,∆ = X in (8) and in (11). Also, it is easy to see that the resource 2,s 2 constraint (11) is binding: (cid:88) (cid:104) (cid:105) (cid:88) X = pθ q Σip(1 λi)θ+(1 Σi)pCi +(1 p)Ci + q ω Ci . (13) 2 − i − − 2h − 2l i { − 1} i i Then the CCP chooses a collateral policy and a default recommendation to maximize time t = 2 revenues, subject to borrowers’ participation and incentive-compatibility constraints. Speci(cid:28)cally, because α > 1, the expected revenues of the CCP are larger when borrowers consumes all of their endowment ω in t = 1, and consume nothing in t = 2. However, such a contract would induce all borrowers to default in equilibrium. 24

In this environment, the CCP may provide the borrower incentives to repay at t = 2, by storing some of his endowment from time t = 1 to time t = 2. In particular, let ω(λ) be the smallest amount of collateral that a borrower with pledgeability λ needs to post in order to overcome his limited commitment problem: (1 λ)pθ ω(λ) = − (14) α Assumption 4 Assume that borrowers’ endowment ω is large enough: ω > ω(λL) ≡ (1−λL)pθ . α To ease exposition we maintain Assumption 4 throughout the paper. However, we 25 can relax this assumption and show that our main results holds true. Assumption 4 guarantees that a borrower’s participation constraint binds. In general, the interaction between the participation and the incentive constraints in problem (P ) can give rise to 0 solutions where the former is slack, for example if collateral is relatively scarce in the economyandthecommitment problemisrelativelysevere(i.e. ω isrelativelysmall and λisrelativelysmall). Thiswouldimplythattheconsumptionallocationoftheborrower at t = 2 exceeds the value of his outside option from simply consuming his endowment, which delivers utility αω. In this case the borrower would earn extra rents with respect to what is necessary to satisfy his participation constraint, as C > αω. Assumption 4 2h guaranteesthatthisdoesnothappen. Wecanthenproceedtocharacterizethesolution to problem (P ). 0 Proposition 5 Let Assumption 4 hold. With no information acquisition, the optimal contract with CCP clearing satis(cid:28)es CH = CL, and CH = CL . In particular, CH = 1 1 2,s 2,s 2,l 25The proof is available in Antinol(cid:28) et al. (2018) and upon request. 25

CL = 0, and 2,l    ω ω(λL) if q 1/α  (1 λL)θ if q 1/α C = − ≤ C = − ≤ 1 2,h  ω ω(λH) if q > 1  (1 λH)θ if q > 1 − α − α where ω(λ) is de(cid:28)ned in (14). If q < 1/α no borrower defaults in equilibrium (ΣL = 0, ΣH = 0); if q > 1/α low-pledgeability borrowers default in equilibrium (ΣL = 1, ΣH = 0). Proposition 5 shows that optimal contracts with central clearing do not separate high-pledgeability from low-pledgeability borrowers, with the only exception of the default recommendation in a region of the parameters’ space. Speci(cid:28)cally, without information acquisition, the CCP must choose between two classes of contracts: one in which no borrower defaults in t = 2, and one in which λH borrowers repay in t = 2 whereas λL borrowers default in equilibrium. In the (cid:28)rst scenario, the CCP o(cid:27)ers a pooling contract that treats all borrowers as if they were the worst possible type. As a result, λH borrowers end up posting excessive collateral with respect to what their type would require. In the second scenario, contracts which let λL borrowers post too little collateral and default in equilibrium are pooling over λH types. 26 Both excessive and insu(cid:30)cient collateral requirements play an important role in the decision of the CCP. Speci(cid:28)cally, on the one hand, higher collateral requirements increase the amount of resources available at t = 2 if they prevent λL borrowers from defaulting in equilibrium. On the other hand, higher collateral requirements reduce the amount of resources available at the CCP in t = 2 through a di(cid:27)erent channel: to leave the participation constraint una(cid:27)ected, if borrowers post an additional unit of collateral at t = 1 they must be compensated by α > 1 units of consumption at t = 2. 26Formally,thisisnotapoolingcontract,astherecommendeddefaultdecision,whichispartofthe optimal contracts, di(cid:27)ers for the two type of borrowers. However, these contracts are observationally equivalenttotheCCPo(cid:27)eringthesame(pooling)contractthatinducesdi(cid:27)erentdefaultstrategiesin equilibrium. 26

The resolution of this trade-o(cid:27) depends on the cost of collateral, α, and on the measure of λL borrowers, 1 q. In particular, when the population of λL types is relatively − large, i.e. q 1, the CCP bene(cid:28)ts from preventing the default of λL borrowers. Thus, ≤ α all borrowers post enough collateral to satisfy the limited commitment problem of λL types, namely ω(λL). If instead the population of λL types is relatively small, i.e. q > 1, it is too costly for the CCP to prevent the default of λL borrowers. Thus, the α CCP maximizes its resources by requiring all borrowers to post collateral to satisfy the limited commitment problem of λH types, namely ω(λH). Substituting the results from Proposition 5 into equation (8) we can (cid:28)nally write the value of central clearing with no information acquisition as:  (cid:16) (cid:17)   u ω(λL)+pθλL if q 1 VCCP,e=0 = ≤ α (15) (cid:16) (cid:17)   u ω(λH)+pθ[qλH +(1 q)λL] if q 1 − ≥ α This equation will be a useful benchmark in the following section. 4 Optimal contracts with information acquisition Next, we introduce the possibility for lenders to engage in costly monitoring, which reveals to her the type λi of her counterparty. By assumption, this remains private information of the lender and the borrower. As a result, when designing a contract with monitoring, the CCP needs to take into account the incentives that lenders have to monitor their counterparty and report truthfully the information they learn. 4.1 Bilateral clearing with information acquisition With bilateral clearing, when the lender monitors her counterparty and learns his type λi, i L,H , she will o(cid:27)er a contract that prevents the borrower from defaulting ∈ { } 27

strategically in equilibrium. Therefore, a contract with bilateral clearing and information acquisition is a list (xi , ci, ci ), where xi and ci are respectively the lender’s 2,s 1 2,s t,s t,s and the borrower’s consumption in time t and state 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 . ∈ { } Let V denote the value to a lender of a match with a borrower of type λi, once i the lender has paid the cost γ and knows the borrower’s type. Then lenders choose contracts (xi , ci, ci ) to solve 2,s 1 2,s i∈{L,H} (Pi) V = max pu(xi )+(1 p)u(xi ) γ (16) i (xi ,xi ,ci,ci ,ci )∈(cid:60)5 2,h − 2,l − 2,h 2,l 1 2,h 2,l + s.t. αci +pci +(1 p)ci αω (17) 1 2,h − 2,l ≥ ω ci 0 (18) ≥ 1 ≥ ci +xi ω ci +θ (19) 2,h 2,h ≤ − 1 ci +xi w ci (20) 2,l 2,l ≤ − 1 ci (1 λi)θ (21) 2,h ≥ − Constraint (17) is the borrower’s participation constraint, (18) is time t = 1 feasibility of the consumption plan, and likewise (19) and (20) are time t = 2 feasibility in states h and l respectively. Constraint (21) is the borrower’s limited commitment constraint: the borrower can default and consume 1 λi units of consumption (in the − low state θ(cid:101)= 0, and limited commitment to repay is not relevant). Itiseasytoseethatatasolutionbothsecond-periodfeasibilityconstraints(19)and (20) should bind. Solving for xi and xi and replacing their values in the objective 2,h 2,l function (16), we can solve for (ci, ci , ci ). 1 2,h 2,l Similarly to Section 3.2, because α > 1, a lender’s expected consumption is larger 28

when the borrower consumes his whole endowment ω in t = 1, and nothing in t = 2. However, such a contract violates the limited commitment constraint (21), and leaves the lender with no consumption in the second period when the output realization is low, as implied by constraint (20). Therefore, the lender will always store some of the borrower’s endowment from time t = 1 to time t = 2. Collateral with bilateral clearing 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 (21). To characterize the solution to (Pi) let λ∗ be the unique value satisfying (cid:16) (cid:17) u(cid:48) (1−λ∗)pθ α p α 1 − p = u(cid:48) (cid:0) θ α−p(1 λ∗)θ (cid:1) . (22) − − α − Intuitively,λ∗ isthesmallestvalueofλsuchthatthelimitedcommitmentconstraint isslack. Foranyλ λ∗, thelimitedcommitmentconstraint(21)isbindingbecausethe ≤ quality of the counterparty is relatively low, which is equivalent to a high borrower’s temptation to default. In the rest of the paper, we make the following assumption: Assumption 6 AssumethatthecommitmentproblemofλL borrowersissevereenough: (cid:16) (cid:17) u(cid:48) (1−λL)pθ α p α 1 − p > u(cid:48) (cid:0) θ α−p(1 λL)θ (cid:1) . − − α − Assumption 6 guarantees that, with bilateral clearing, information about the quality of counterparties has positive value: in the Appendix we show that when Assumption 6 is violated, information about the quality of a counterparty has no value and no trade-o(cid:27) 29

exists between bilateral and central clearing. Under this additional assumption we can characterize the solution to problem (Pi). 27 Lemma 7 Let ω(λ) be de(cid:28)ned in (14) and λ∗ in (22) and let Assumptions 4 and 6 hold. Then, optimal contracts with bilateral clearing and monitoring satisfy ci = 0, 2,l xi = θ ci +ω ci, xi = ω ci, cL = (1 λL)θ, cL = ω ω(λL), and 2,h − 2,h − 1 2,l − 1 2,h − 1 − (cid:18) (cid:19) u(cid:48) (1−λH)pθ (1) cH = ω ω(λ∗), cH = (1 λ∗)θ, if α > α−p. 1 − 2,h − u(cid:48)(θ−α−p(1−λH)θ) 1−p α (cid:18) (cid:19) u(cid:48) (1−λH)pθ (2) cH = ω ω(λH), cH = (1 λH)θ, if α−p > α . 1 − 2,h − 1−p u(cid:48)(θ−α−p(1−λH)θ) α To understand the intuition behind Lemma 7, recall that the limited commitment constraint of a λL borrower binds by Assumption 6. Thus, collateral always provides λL borrowers with incentives to repay at t = 2. Di(cid:27)erently, the limited commitment constraint (21) of a λH borrower may or may not bind. If λH > λ∗, which corresponds to Case (1) in Lemma 7, λH borrowers’ temptation to default is low and the limited commitment constraint is slack. In this case, the more important role of collateral is insurance against the low realization of θ ˜ . If instead λH < λ∗, which corresponds to Case (2) in Lemma 7, the limited commitment constraint of λH borrowers binds and collateral provides them with incentives to repay at t = 2. Lemma 7 also shows that collateral is the only tool that lenders can use to manage counterparty risk when clearing is bilateral. Because collateral is costly, lenders optimally choose to bear part of counterparty risk, namely lenders’ consumption is higher in the state of nature where her counterparty can repay. Corollary 8 A solution to problem (Pi) is such that insurance is incomplete, xi > 2,h xi . 2,l 27We characterize the entire set of solutions to the bilateral problem with informatino acquisition in Appendix 6.1. 30

Although insurance is incomplete, lenders can customize collateral requirements to the speci(cid:28)c risk of their counterparty, after acquiring information about his pledgeability type. Such information may be valuable and, considering central clearing without information acquisition as a benchmark, as characterized in Section 3.2, we can prove the following: Proposition 9 Let Assumption 4 and Assumption 6 hold. Then, if (cid:34) (cid:35) 1 (cid:16) (cid:17) pu min λHθ+ω(λH),λ∗θ+ω(λ∗) +(1 p)u(max ω(λH),ω(λ∗) ) α { } − { } (cid:34) (cid:35) α 1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) + − pu λLθ+ω(λL) +(1 p)u ω(λL) > u ω(λL)+λLpθ , (23) α − there exists an interval (q,q) and a function γ(q) : (q,q) such that bilateral + → (cid:60) clearing with information acquisition is preferred to central clearing without information acquisition, if and only if q (q,q) and γ < γ(q). ∈ Proposition 9 can be understood in termsof the value ofinformationunder bilateral clearing. In particular, we can interpret bilateral clearing with monitoring as a lottery bil = (pq,p(1 q),(1 p)q,(1 p)(1 q) over outcomes (xH , xH, xL , xL ), whereas L − − − − 2,h 2,l 2,h 2,l central clearing results in a degenerate lottery over XCCP. The threshold γ(q) can be 2 rewritten as γ = u(E RP ) u(XCCP) (24) Lbil − Lbil − 2 where E and RP are, respectively, the expected value and the risk-premium of Lbil Lbil the lottery bil. 28 When Assumption 4 and Assumption 6 are satis(cid:28)ed, XCCP < E L 2 Lbil and a trade-o(cid:27) between bilateral and central clearing may exist, provided that lenders are not too risk-averse and the population of borrowers is su(cid:30)ciently heterogeneous. 28The risk premium of a lottery is a measure of how many resources, in expectation, an agent is willing to give up to get rid of uncertainty: RP =E CE . L L− L 31

In fact, central clearing has the advantage of providing insurance by pooling risk over idiosyncratic uncertainty. However, the CCP requires all traders to post the same amount of collateral. Thus, central clearing has the limitation of requiring a fraction of the borrowers’ population to post either excessive or insu(cid:30)cient collateral necessary to provide incentives to repay. On the one hand, the value of insurance provided by the CCP is smaller the less risk-averse lenders are. On the other hand, the bene(cid:28)ts from collateral customization are larger if the population of borrowers contains both highpledgeability and low-pledgeability types, hence the necessary and su(cid:30)cient condition is q (q,q). ∈ Risk-aversionmaychangethedesirabilityofbilateralvs. centralclearingthroughits e(cid:27)ectonγ. Inparticular,fromequation(24),risk-aversionhasane(cid:27)ectonγ throughits e(cid:27)ect on i) the risk-premium RP , ii) expected consumption under optimal contracts Lbil with bilateral clearing, and iii) the utility function u( ). In the next lemma we provide · su(cid:30)cient conditions which guarantee that an increase in risk-aversion results in central clearing becoming relatively more desirable. Lemma 10 Let u(x) and v(x) be v.N-M utility functions. Consider two economies: economy A populated by lenders with utility u( ) and economy B populated by lenders · with utility v( ). Suppose there exists a concave function ρ : such that v(x) = + + · (cid:60) → (cid:60) (cid:18) (cid:19) v(cid:48) (1−λL)pθ ρ(u(x)) and ρ(a) ρ(b) ζ a b , for ζ 1. Also, suppose that α−p > α | − | ≤ | − | ≤ 1−p v(cid:48)(θ−α−p(1−λL)θ) α (the equivalent of Assumption 6 for economy B). Then, if lenders in economy B prefer bilateral clearing with information acquisition to central clearing with no information acquisition, then also lenders in economy A prefer bilateral clearing with information acquisition to central clearing with no information acquisition. Lemma 10 can be understood through the di(cid:27)erent e(cid:27)ects of an increase in riskaversion. First, more risk-averse lenders are willing to give up more resources to avoid 32

uncertainty over consumption, i.e. RP increases with risk-aversion. Second, optimal Lbil contractswithbilateralclearingmaychangewithrisk-aversion, andsodoesE . Ifthe Lbil Arrow-Pratt coe(cid:30)cient of absolute risk-aversion increases uniformly with risk-aversion, the demand of collateral for insurance becomes larger and expected consumption under bilateral clearing decreases, i.e. E decreases. 29 Finally, risk-aversion a(cid:27)ects the Lbil desirability of bilateral vs. central clearing through the form of the utility function: that is the di(cid:27)erence u(E RP ) u(XCCP) is not in general scale invariant, Lbil − Lbil − 2 but depends on the function u( ). In Lemma 10 we show that when we consider · transformation in risk-aversion that result in a concave contraction of the original 30 utility function, central clearing becomes relatively more desirable with risk-aversion. 4.2 Central clearing with information acquisition The optimality of bilateral clearing in Proposition 9 hinges on lenders’ ability to acquire information about the quality of their counterparty and to customize collateral accordingly. A natural question is whether a central counterparty could aggregate such information. Speci(cid:28)cally, can lenders credibly acquire information about counterparty quality and transfer it to the CCP, which afterwards could customize collateral requirements to borrowers’ pledgeability types. The answer to this question depends on the consequences of novation and the risk management adopted by the CCP. Similarly to Section 3.2, we model novation by assuming that the CCP commits to a mechanism at the beginning of t = 1, and that borrowers and lenders negotiate over 29Assuming that the Arrow-Pratt coe(cid:30)cient increases uniformly is equivalent to assume a concave transformation of the original utility function: v(x)=ρ(u(x)) for ρ() concave. · 30Asanexampleofsuchanincreaseinrisk-aversion,considerthefunctionu: de(cid:28)nedas + + (cid:60) →(cid:60) u(x)=3x1 3, and the transformation ρ: + + , ρ(y)=2(1+y)2 1 2. Then, the more risk-averse (cid:60) (cid:0)→(cid:60)(cid:1) − lender has utility v : , v(x) = g u(x) . It is easy to check that u(0) = v(0) = 0. Also, + + (cid:60) → (cid:60) u(0) = v (0) = . Moreover, g (y) > 0 > g (y), and g (0) = 1. Then, g() is a concave contraction (cid:48) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48) ∞ · that de(cid:28)nes a new utility function, v(), that satis(cid:28)es all the assumptions that we made in the paper · about the utility function u(). · 33

these contracts. A contract speci(cid:28)es transfers between the CCP and lenders and between the CCP and borrowers. Transfers are a function of i) public information and ii) the restrictions on the contracting space consistent with the institutional arrangements adopted by CCPs in reality. A strategy for a lender is a pair (e, m) 0,1 λH,λL of e(cid:27)ort e and message ∈ { }×{ } reported to the CCP m; a strategy for a borrower is a default decision σ (λ,m) : s λL,λH 2 l,h 0,1 . For example, σ (λL,λH) is the default decision of a λLh { } ×{ } → { } borrower at the node corresponding to lender’s (cid:28)rst-period message m = λH, when the idiosyncratic state is s = h. Note that strategic default is relevant only when the idiosyncratic state is s = h, and easily σ (λ,m) = 0 when realization of the s idiosyncratic state s = l. Using the same notation as in Section 3.2, let h = (s,∆) be the public history of a 2 borrower after novation, where ∆ 0,1 is his observed repayment/default decision ∈ { } and s l,h his idiosyncratic state at t = 2. Let denote the set of all such 2 ∈ { } H ˆ histories, and de(cid:28)ne to be the set of all second-period histories of a borrower 2 2 H ⊆ H upon which the contract between the CCP and the original lender can be contingent ˆ ˆ on. Finally, let h be an element of this set. A mechanism with central clearing 2 2 ∈ H and monitoring consists of contracts between the CCP and lenders, Xm(h ˆ ) , and { 2 2 } contracts between borrowers and the CCP, Cm, Cm . { 1 2,s} ˆ ˆ We consider three speci(cid:28)cations for : 1) a benchamrk case where = ; 2) 2 2 H H {∅} ˆ ˆ = (s,∆) : s l,h ,∆ 0,1 ; and 3) = ∆ : ∆ 0,1 . The three cases 2 2 H { ∈ { } ∈ { }} H { ∈ { }} correspond to di(cid:27)erent assumptions about the CCP rules and procedures. Speci(cid:28)cally, we focus on the rules de(cid:28)ning the loss allocation methods, member’s due diligence, and prohibited conduct. These rules impose a constraint on the contract the CCP itself can o(cid:27)er. Consistently with our description in Section 2, case 1) describes a scenario where 34

the CCP allocates losses pro-rata among its members. Thus, the contract between a lender and the CCP must be independent of the second-period history of the original ˆ borrower. Therefore, in this case we write = . Case 2) corresponds to a scenario 2 H ∅ where the CCP adopts a partial tear-up loss allocation method, by which the CCP can select and terminate certain contracts upon default of a member. In practice, and as described in Section 2, CCPs adopts this loss allocation method under extreme default scenarios, and typically terminate contracts by original counterparty. In the context of ourmodelthisisequivalenttoimposingnorestrictionsonsecondperiodrepaymentsby ˆ the CCP to lenders, thus = (s,∆) l,h 0,1 . Case 3) describes a scenario 2 H { ∈ { }×{ }} where the CCP adopts a pro-rata loss allocation method with due-diligence, which are 31 key features of standard loss mutualization schemes. Pro-rata allocation of losses implies that even if the original counterparty of a lender defaults, the CCP cannot refuse to perform on her obligation and pay the lender. In our model, this requires that payments to a lender are independent of her borrower’s default. Due diligence, however, implies that the CCP can assess fees and impose sanctions for the violation 32 of its Rules and Procedures. In our model, because strategic default by a borrower signals that his lender did not monitor him or did not report his type truthfully, the CCP is able to identify a violation of its rules and punish the lender. Thus, a pro ˆ rata loss allocation method with due diligence requires = ∆ 0,1 and allows 2 H { ∈ { }} the CCP to choose the punishment for a lender who did not report truthfully her counterparty type. We assume that the CCP chooses the harshest feasible punishment 33 as this sets a lower bound on the measure of economies for which our results hold. Due to non-negativity of consumption, the CCP allocates zero consumption to a lender 31See National Securities Clearing Corporation - Rules and Procedures (2018), and UBS Response to the Committee on Payment and Settlement Systems (2012) 32SeeNationalSecuritiesClearingCorporation-RulesandProcedures(2018),andIceClearCredit Clearing Rules (2018). 33A more lenient punishment would results in a larger set of economies for which our results hold. 35

34 who misbehaved. Let wi(h ˆ ) = u (cid:0) Xi(h ˆ ) (cid:1) be the lender’s payo(cid:27) when she reports m = λi in t = 1 2 2 2 and, in addition to this message, the contract between the CCP and the lender is contingent on history h ˆ . De(cid:28)ne next π(h ˆ σ(λi,m)) to be the probability distribution 2 2 | over histories h ˆ conditional on the default strategy of a borrower of type λi, when the 2 lender’s message in t = 1 is m. Let π( σ(λi,m)) = 1. 35 The CCP contracts which ∅| induce monitoring and result in borrowers’ separation solve the following maximization problem: (cid:34) (cid:35) (cid:88) (cid:88) (cid:16) (cid:17) (P ) max q π hˆ σ(λi,λi) wi(hˆ ) γ (25) 1 i 2 2 | − i∈{L,H} hˆ 2∈Hˆ 2 s.t. αCi +pCi +(1 p)Ci αω (26) 1 2,h 2,l − ≥ Ci (1 λi)θ (27) 2,h ≥ − (cid:88) ω q Ci 0 (28) i 1 − ≥ i∈{L,H} (cid:34) (cid:35) (cid:88) (cid:88) (cid:16) (cid:17) q π hˆ σ(λi,λi) u−1(wi(hˆ )) i 2 2 | i∈{L,H} hˆ 2∈Hˆ 2 (cid:88) (cid:110) (cid:111) + q [Ci +pCi +(1 p)Ci ] ω+pθ (29) i 1 2,h 2,l − ≤ i∈{L,H} (cid:88) (cid:16) (cid:17) (cid:88) (cid:16) (cid:17) π hˆ σ(λi,λi) wi(hˆ ) π hˆ σ(λi,λ−i) w−i(hˆ ) (30) 2 2 2 2 | ≥ | hˆ 2∈Hˆ 2 hˆ 2∈Hˆ 2 (cid:34) (cid:35) (cid:88) (cid:88) (cid:16) (cid:17) γ + q π hˆ σ(λi,λi) wi(hˆ ) i 2 2 − | i∈{L,H} hˆ 2∈Hˆ 2   (cid:34) (cid:35)  (cid:88) (cid:88) (cid:16) (cid:17)  max q π hˆ σ(λi,λj) wj(hˆ ) (31) i 2 2 ≥ j∈{L,H} |  i∈{L,H} hˆ 2∈Hˆ 2 σ (λi,λj) = 0 i(cid:27) C j (1 λi)θ (32) s 2,h ≥ − 34In principle, the CCP could adopt a pro-rata loss allocation method without due diligence. However notice that this would be de-facto equivalent to the case where original counterparties are not ex-post identi(cid:28)able which we analyze in the benchmark case. 35Note that in the de(cid:28)nition of σ(λ,λi), the lender may know λ if she monitored her counterparty, but it can be as well private information of the borrower if monitoring did not occur. 36

whereweusethenotationqH = q andqL = 1 q. Onecaneasilyseethatσ (λi,λj) = 0 l − for i,j L,H , because borrowers have no incentive to default in state s = l. Con- ∈ { } straint (26) is borrower i’s participation constraint and (27) his limited commitment constraint. Equations (28) and (29) are time t = 1 and t = 2 feasibility constraints, and (30) and (31) are, respectively, the ex-post and ex-ante incentive-compatibility constraint for lenders. Speci(cid:28)cally, when (30) is satis(cid:28)ed lenders prefer, after monitoring, to report truthfully their counterparty’s type. When constraint (31) is satis(cid:28)ed a lender prefers, ex-ante, to monitor her counterparty (and then report her type truthfully) rather than not to monitor and report that her counterparty is either a highpledgeability or a low-pledgeability type. Constraint (32) de(cid:28)nes optimal borrowers’ default decision. ˆ 4.2.1 Contracts with pro-rata loss allocation method: = 2 H ∅ This section characterizes equilibrium contracts with central clearing when the CCP ˆ allocates losses pro-rata among its members, resulting in = . In this case, the 2 H ∅ contract between the CCP and a lender is independent of any information available at t = 2 after novation, such as information about the lender’s original counterparty. With ˆ = we can write wi(h ˆ ) = wi. The incentive-compatibility constraint (31) 2 2 H ∅ thus becomes: γ +qwH +(1 q)wL max wH,wL (33) − − ≥ { } The left-hand side of (33) is the payo(cid:27) for a lender after monitoring and truthtelling. The right-hand side of (33) is the payo(cid:27) of a lender who does not monitor her counterparty and reports the type associated with the larger repayment. One can easily see that there exists no incentive-compatible contract: for any non-negative (wH,wL) constraint (33) is violated. We summarize this result in the next proposition. 37

ˆ Lemma 11 For any γ > 0, if = there exists no solution to problem (P ). 2 1 H ∅ A consequence of Lemma 11 is that lenders cannot credibly transfer information about their counterparty to the CCP. Thus, the CCP must adopt a homogeneous collateral policy that treats all borrowers either as a low-pledgeability or as a high-pledgeability type. Thus, bilateral clearing is optimal for the same conditions as in Proposition 9. ˆ Corollary 12 For any γ > 0, if = bilateral clearing is optimal when the assump- 2 H ∅ tions in Proposition 9 are satis(cid:28)ed. ˆ 4.2.2 Contracts with partial tear-up: = = (s,∆) l,h 0,1 2 2 H H { ∈ { }×{ }} This section characterizes equilibrium contracts with central clearing that can be contingent not only the (cid:28)rst period message by the lender, but also on second period ˆ histories: = . As discussed in Section 2 we refer to these contracts as associated 2 2 H H 36 with a partial tear-up allocation of losses. In this case, contracts between a lender and the CCP can depend on the message reported by lender to the CCP at t = 1, the realization of the borrower’s returns, and the borrowers’ default decision. Hence, we write wi(h ˆ ) = wi,∆. 2 s Because the objective of the CCP is to customize collateral requirements to borrowers’ types, it is optimal to use the information available to lenders after monitoring, and punish lenders who misreport the type of their counterparty. This is feasible for the CCP when a borrower strategically defaults: if at t = 1 a lender reports that her counterparty is a high-pledgeability type, and at t = 2 the borrower defaults when s = h, this default signals the misbehavior of the lender. Indeed, from constraint (32), strategic default of a borrower (∆ = 1) can only be the consequence of inappropriate collateral requirement. Also, by constraint (27), collateral requirements can only be 36See Gregory (2014) section 10.3. 38

insu(cid:30)cient because the lender did not monitor or report truthfully her counterparty type. Hence, it is feasible for the CCP to identify a lender who misreported her counterparty type, and punish her. Constraint (31) implies that the optimal punishment is the hardest feasible one: in fact choosing wi,1 as low as it is feasible relaxes the incenh tive constraint (31). By non negativity of consumption, the lowest feasible punishment is wi,1 = 0. h Using this result, we can prove the following proposition. ˆ Proposition 13 Let Assumption 4 and Assumption 6 hold, and let = in prob- 2 2 H H lem (P ). If γ < γ, for γ de(cid:28)ned in Proposition 9, there exist contracts (wi,0,Ci,Ci ) 1 s 1 2,s feasible and incentive-compatible in problem (P ), such that 1 (cid:40) (cid:41) (cid:40) (cid:41) (cid:88) (cid:104) (cid:88) (cid:105) (cid:88) (cid:104) (cid:88) (cid:105) q p wi,0 q p u(xi∗ ) i s s ≥ i s 2,s i∈{L,H} s=l,h i∈{L,H} s=l,h where xi∗ is lenders’ consumption in the optimal contract with bilateral clearing and 2,s monitoring of Lemma 7. Proposition 13 proves that the CCP can replicate any contracts with bilateral clearing when it can make payments contingent on the message by the lender at t = 1 and ontheborrower’shistoryatt = 2. Asaresult, wheninformationacquisitionisvaluable under bilateral clearing, such information can be acquired and credibly transmitted to the CCP. ˆ Intuitively, when = , the set of contingencies spans the set of states of na- 2 2 H H ture and any bilateral contract becomes feasible with central clearing, including the optimal bilateral contract with information acquisition. Importantly, Proposition 13 shows that such contracts are also incentive compatible for the parameter space where information acquisition is valuable with bilateral clearing. That is, when γ < γ, bilateral contracts with information acquisition satisfy incentive-compatibility constraints 39

(30) and (31). This happens for two reasons: (cid:28)rst, constraint (30) is always satis(cid:28)ed by constraint (31), and can then be ignored. In other words, lenders’ best deviation is always ex-ante, before monitoring, rather than lying after acquiring information about the counterparty type. Hence, the CCP needs to consider only two kinds of deviations, which are considered in the right-hand side of constraint (31): 1) not monitoring and reporting a high-pledgeability counterparty; 2) not monitoring and reporting a low-pledgeability counterparty. Second, these two deviations induce lotteries over time t = 2 consumption for lenders that are inferior to the one induced by optimal contracts in Proposition 5, which are pooling contracts with central clearing. As a result, bilateral contracts with information acquisition satisfy the incentive-compatibility constraint (31), because in economies with γ < γ¯ they are preferred to contracts with central clearing and no information acquisition, as shown in Proposition 9. In conclusion, Proposition 13 shows that central clearing is always preferred to bilateral clearing when CCP contracts are state contingent. Corollary 14 With partial tear-up allocation of losses, central clearing is always the optimal clearing arrangement. Notice that the results in Proposition 13, and its implications in Corollary 14, are non trivial. One might be induced to believe that the CCP is solving the same problem of as that of an unconstrained social planner in the case of a partial tear-up loss allocation method. This is, however, incorrect as the CCP is not endowed with the monitoringtechnologywithwhichlendersareendowed. Rather, theCCPhastoinduce lenders to exert e(cid:27)ort and monitor their borrowers. Using the replicability results in Proposition 13, the CCP can o(cid:27)er lenders fully state contingent contracts. This, in turn, allows the CCP to reduce the provision of ex ante insurance to lenders typical of a pro-rata loss allocation method, and to favor the provision of incentives to acquire 40

information about borrowers. For our result it is su(cid:30)cient that these incentives are just enough to implement the bilateral contract. ˆ 4.2.3 Contracts with pro-rata loss allocation and due diligence: = ∆ 2 H { ∈ 0,1 { }} This section characterizes equilibrium contracts where the CCP adopts a pro-rata loss allocation method and due diligence. In practice, CCPs impose sanctions for violation oftheirrules, andassess(cid:28)nesifamembershows(cid:16)prohibitedconduct(cid:17) orconductwhich is inconsistent with (cid:16)just and equitable principles of trade.(cid:17) Violations of the rules of a CCP include failure to provide information regarding the businesses and operations of the member and its risk management practices, or reporting its (cid:28)nancial or operational condition. Additionally, members may need to submit information to the CCP (cid:16)as the Corporation from time to time may reasonably require.(cid:17) In this respect, lenders’ costlyinformationacquisitioninourmodelcapturestheideathatmembersmightlearn their counteparties’ (cid:28)nancial and operation condition by, for example, monitoring their counterparties. Formally, we allow contracts between the CCP and lenders to depend on observed lack of due diligence, which can be revealed by the strategic default of a lender’s counterparty. Contracts, however, are not contingent on the realization of a borrower’s idiosyncraticreturn,astheCCPislegallyobligedtohonorthecontractsitclears. Todo so it may have to absorb losses, in which case it uses the resources contributed pro-rata ˆ ˆ to the default fund by surviving members. Hence, , as = ∆ 0,1 , 2 2 2 H ⊂ H H { ∈ { }} and we write wi(h ˆ ) = wi,∆. 2 We solve problem (P ) in two steps. In the (cid:28)rst step the CCP determines the con- 1 tracts o(cid:27)ered to borrowers, Ci,Ci ,Ci , that maximizes time t = 2 resources. { 1 2,h 2,l} i=L,H It is optimal to solve for Ci,Ci ,Ci that maximize time t = 2 resources { 1 2,h 2,l} i=L,H 41

because it relaxes constraints (29), (30), and (31), while satisfying (26), (27), and (28). Then, in the second step, the CCP determines the contracts it o(cid:27)ers to lenders, wi,∆ , given resources available. i=L,H { } In the next lemma we characterize optimal contracts o(cid:27)ered to borrowers. Lemma 15 Let ω(λ) be de(cid:28)ned in (14). Optimal contracts o(cid:27)ered to borrowers, Ci, { 1 Ci , Ci , satisfy Ci = (1 λi)θ, Ci = 0, Ci = ω ω(λi). 2,h 2,l} i=L,H 2h − 2l 1 − To gain intuition for Lemma 15, note that, when contracts are cleared centrally, there is no need for collateral for insurance purposes, because the CCP can fully insure lenders by pooling borrowers idiosyncratic risk. Hence, the CCP’s objective is to minimize collateral requirements, and the limited commitment constraint of both types of borrowers is binding. Recall that Assumption 4 guarantees that the participation constraint is always binding. Then the consumption allocation is determined as a residual from the borrowers’ participation constraint in t = 1. Next, we can characterize optimal contracts between lenders and the CCP: 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) − − which maps any value of γ 0 to the minimum aggregate resources (i.e. t = 2 ≥ consumption goods) consistent with the existence of a solution to the CCP problem. Further, de(cid:28)ne the threshold γˆ as the unique solution to (cid:88) (cid:104) (cid:105) φ(γˆ) = q λipθ+ω(λi) i (34) i∈{L,H} for ω(λ) de(cid:28)ned in (14). Thus, γˆ denotes the largest value of γ such that a solution to problem (P ) with pro-rata allocation of losses and due diligence exists. For economies 1 42

where such a solution exists we can prove the following result. Proposition 16 With pro-rata allocation of losses and due diligence, a solution to problem (P ) exists and is unique if and only if γ γˆ. If γ γˆ, wH,0∗ = wL,0∗ + γ, 1 ≤ ≤ q where wL,0∗ solves (cid:18) (cid:19) γ (cid:88) (cid:104) (cid:105) qu−1 wL,0∗ + +(1 q)u−1(wL,0∗ ) = q λipθ+ω(λi) i q − i∈{L,H} Finally, if γ γˆ, then ≤   (cid:40) (cid:41) (cid:40) (cid:41)  (cid:88)  (cid:88) (cid:104) (cid:88) (cid:105) max u(X∗), q wi,0∗ γ q p u(xi∗ ) γ 2 i − ≥ i s 2,s −   i∈{L,H} i∈{L,H} s=l,h where xi∗ is lenders’ consumption in the optimal contract with bilateral clearing and 2,s monitoring of Lemma 7, and X∗ is lenders consumption in (13) for the optimal contract 2 with CCP clearing and no monitoring in Proposition 5. The (cid:28)rst part of Proposition 16 shows that a solution to the CCP problem (P ) with 1 pro-rataallocationoflossesandduediligence, ifitexists, issuchthatlenders’incentivecompatibility constraint (31) binds for ˆi = L. The second part of Proposition 16 proves that information is more valuable with central clearing, because the CCP can providefullinsuranceagainsttheidiosyncraticreturnriskandpartialinsuranceagainst the counterparty-type risk. More precisely, in the proof of Proposition 16 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, 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 43

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. Following a similar argument, we prove the following result for economies that do not satisfy the conditions in Proposition 16. Proposition 17 Let Assumption 4 and Assumption 6 hold. Also, let γ be de(cid:28)ned in Proposition 9 and γˆ in (34). With pro-rata allocation of losses and due diligence, bilateral clearing (with monitoring) is the optimal clearing arrangement if and only if γ (γˆ,γ). ∈ The conditions in Proposition 17 are necessary and su(cid:30)cient for optimality of bilateral clearing with pro-rata allocation of losses and due diligence. Central clearing has the advantage of providing insurance by pooling risk over idiosyncratic risks and, as a result, has the potential to economize on the use of collateral necessary to insure against idiosyncratic risk in bilateral clearing. However, 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 with 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)cient collateral necessary to provide incentives to repay. When γ (γˆ,γ), the bene(cid:28)ts of central ∈ clearing do not compensate for its costs: the insurance against uncertain returns does not compensate lenders for the distortion in the use of collateral due to the lack of information about the counterparty quality. Thus lenders choose to clear contracts 37 bilaterally and acquire information about their borrowers. 37Notice that the assumption that monitoring activity is costly for lenders is necessary for the 44

5 Implications for Collateral and Default The goal of this section is to illustrate the implications of our model for the collateral policies chosen with each clearing arrangement and for the associated default rates in equilibrium. We combine results from the previous sections and analyze the role of some primitives of the model for equilibrium outcomes. We focus on economies where there exists a trade-o(cid:27) between bilateral and central clearing, since this is the main focus of our paper. Therefore, we restrict our analysis to economies where the CCP operates with a pro-rata loss allocation method with due diligence (these are the economies of Section 4.2.3), and where γ > γˆ, with γˆ de(cid:28)ned in (34). In these economies information acquisition is not incentive-compatible with central clearing, and, as a consequence, bilateral clearing might be preferred. Table 1 summarizes Assumethatg > gˆ,soinformationacquisitionisnotincentive-compatiblewithCCP results about default rates and collateral requirements. clearing. CCP Bilateral Parameteres collateral default collateral default lL lH lL lH lL lH lL lH lH < l w(lL) sL = 0 sH = 0 w(lL) w(lH) sL = 0 sH = 0 ⇤ q 1  a lH > l w(lL) sL = 0 sH = 0 w(lL) w(l ) sL = 0 sH = 0 ⇤ ⇤ lH > l w(lH) sL = 1 sH = 0 w(lL) w(l ) sL = 0 sH = 0 ⇤ ⇤ q 1 a lH < l w(lH) sL = 1 sH = 0 w(lL) w(lH) sL = 0 sH = 0 ⇤ Table 1: Collateral policy and strategic default strategies under bilateral clearing with monitoring and central clearing with no information acquisition, assuming that γ > γˆ, where γˆ is de(cid:28)ned in (34) and λ∗ in (22). In the economies we consider, collateral requirements with bilateral clearing are existence of a trade-o(cid:27) between bilateral and central clearing. More speci(cid:28)cally, in a simpler set-up wherelendersalwayshavetheinformationaboutthequalityoftheircounterparty,whichisequivalent to γ =0, information transmission is always incentive-compatible (0=γ <γˆ) and central clearing is always the optimal arrangement. 45

tailored to borrowers’ type. However, because information acquisition is not incentive compatible with central clearing (γ > γˆ), the CCP must then choose a homogenoues collateral policy to maximize resources at t = 2, trading-o(cid:27) collateral costs with the cost of default. Equilibrium default measures the opportunity cost of collateral: lowering collateral requirements and tailoring them to λH borrowers comes at the cost of equilibrium default of λL types. In economies where the measure of λH borrowers is small (q 1), average collateral with central clearing is larger, as λH borrowers ≤ α post more collateral than their type would require. As a consequence, default does not occur in equilibrium, both with central and bilateral clearing. Di(cid:27)erently, in economies where the measure of λH borrowers is large, the CCP saves resources by requiring less collateral and allowing default by λL borrowers. Thus, average collateral is lower than with bilateral clearing, but equilibrium default is larger. Lemma 18 summarizes these results. Lemma 18 Suppose the CCP adopts a pro-rata allocation of losses method with due diligence. Let Assumption 4 and Assumption 6 hold, and assume that γ > γˆ, for γˆ de(cid:28)ned in (34). If q 1, average collateral requirements are lower with bilateral ≤ α clearing, whereas average defaults are the same under the two clearing arrangements. If instead q 1, average collateral requirements are larger and average defaults are ≥ α smaller with bilateral clearing than they are with central clearing. Using the results from Lemma 18, we then investigate the e(cid:27)ects of an increase in the cost of collateral, α. The next lemma shows that when the measure of λH borrowers is small, and under some additional assumptions, an increase in the cost of collateral results in a larger set of economies where bilateral clearing is preferred to central clearing. 46

(cid:18) (cid:19) u(cid:48) (1−λH)pθ Lemma 19 Suppose that q 1, Assumption 4 holds, and α−p > α . ≤ α 1−p u(cid:48)(θ−α−p(1−λH)θ) α Also, assume that lenders are not prudent, i.e. u(cid:48)(cid:48)(cid:48)(x) 0. Then dγ > 0 and dγˆ < 0. ≤ dα dα Lemma 19 provides su(cid:30)cient conditions for an increase in the cost of collateral to result in a larger set of economies where bilateral clearing is preferred. When q < 1, the CCP adopts the homogeneous collateral policy that treats all borrowers as α low-pledgeability types. Therefore, high-pledgeability borrowers are required to post excessive collateral with respect to what their type would require. Bilateral clearing, on the other hand, features information acquisition, which allows lenders to tailor collateral requirements to the type of their counterparty. Thus, when q < 1, bilateral α clearing has the advantage of economizing over the average collateral requirement. As a result, ceteris paribus, an increase in the cost of collateral strengthens the relative advantage of bilateral clearing. Formally, an increase in α works as a negative income e(cid:27)ect that lowers lenders’ expected consumption at t = 2, because more resources must be allocated to satisfy borrowers’ participation constraint. Due to the distortion inducedbythehomogeneouscollateralpolicyoftheCCP,onaverage, thismechanismis stronger with central clearing. However, the fact that average collateral requirements are larger with central clearing is not enough to reach the conclusion that bilateral clearing is preferred for a larger set of economies when α increases. The reason is that, with bilateral clearing, lenders’ consumption depends also on the return of their counterparty’s technology. Thus, with bilateral clearing, the e(cid:27)ect of an increase in the cost of collateral must be weighted by the marginal utility of consumption at di(cid:27)erent levels of consumption. The assumption that lenders are not prudent in Lemma 19, i.e. u(cid:48)(cid:48)(cid:48) 0, is su(cid:30)cient to guarantee that this second-order e(cid:27)ect on lenders’ payo(cid:27)s ≤ works in the same direction as the (cid:28)rst-order e(cid:27)ect on average collateral requirement. Thus, undertheassumptionthatlendersarenotprudentwecanconcludethatbilateral 47

clearing is preferred to central clearing for a larger set of economies. Lemma 20 shows that the opposite result holds when we consider economies that do not fall under the assumptions of Lemma 19. (cid:18) (cid:19) u(cid:48) (1−λH)pθ Lemma 20 Suppose that q > 1, Assumption 4 holds, α−p > α , and α 1−p u(cid:48)(θ−α−p(1−λH)θ) α γ > γˆ for γˆ de(cid:28)ned in (34). Also, assume that lenders prudent enough, i.e. u(cid:48)(cid:48)(cid:48)(x) > 0 and (cid:18) (1 λH)pθ (cid:19) (cid:18) (1 λH)pθ (cid:19) (cid:18) (1 λL)pθ (cid:19) pu(cid:48) − +λHθ +(1 p)u(cid:48) − > u(cid:48) − +λLpθ α − α α Then, and increase in the cost of collateral α makes central clearing more desirable, i.e. dγ < 0. dα Lemma 20 considers economies where the optimal contract with central clearing adopts the homogeneous collateral policy that treats all borrowers as λH types. In this scenario, undersomeadditionalassumptions, anincreaseinthecostofcollateralresults in a larger set of economies where central clearing is preferred. The reasoning mirrors the arguments in Lemma 19. With central clearing, low-pledgeability borrowers are required to post too little collateral with respect to what their type would require, and default. Inaneconomywherecentralclearingispreferred,thebene(cid:28)tfromeconomizing oncollateralcompensateslendersforthecostofλL borrowersdefaultinginequilibrium. In this scenario, an increase in the cost of collateral must strengthen, ceteris paribus, this e(cid:27)ect. As in the proof of Lemma 19, this is not enough to reach the conclusion that central clearing is preferred for a larger set of economies, because an increase in α has a second-order e(cid:27)ect on the payo(cid:27) under bilateral clearing due to the interaction between collateral and uncertain returns. The assumption that lenders are prudent, i.e. u(cid:48)(cid:48)(cid:48) 0, is su(cid:30)cient conclude that this second order e(cid:27)ect works in the same direction ≥ 48

as the direct (cid:28)rst order e(cid:27)ect on economizing on collateral by letting λL borrowers default in equilibrium. 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 38 money market funds. 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 39 (e.g. General Collateral Finance Repo Service (GCF Repo) and triparty settlement). Analogously, our results are consistent with central clearing arising endogenously in markets where participants 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 40 riskiness in a counterparty’s set of activities. 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 heterogeneous quality of trading counterparties. Using a mechanism design approach, we considertheinstitutionalarrangementsofmodernCCPsandmodelthemasconstraints on the contract space of the CCP. 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 38At least under normal circumstances, disregarding events as money market funds breaking the buck. 39As an example, for evidence related to the US repo market see the O(cid:30)ce of Financial Research BriefPaperno. 17-04,Bene(cid:28)tsandRisksofCentralClearingintheRepoMarket. Seealsofootnote10 in the introduction for more details on MMMF and central clearing. 40As an example, recall that the (cid:28)rst central counterparties originated next to grain and co(cid:27)ee exchanges,wherefarmers andbakers tradedfutures. Amongmany,forreferencesseeKroszner(2006), and Gregory (2014). 49

with the risk pooling activity of the CCP, and under loss allocation methods commonly adopted by CCPs for their routine operations. 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 potential 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 limits inherent in di(cid:27)erent loss allocation mechanisms and the importance of the risk of the underlying assets and of the degree of heterogeneity of market participants in determining whether CCP’s can perform their risk sharing function e(cid:27)ectively. 50

Appendix Proof of Lemma 1 Consider problem (8) - (12) where the recommended default decision ΣH = 0 and ΣL = 0. Thesetwoconditionsrequirein(12)thatCH (1 λH)θ andCL (1 λL)θ 2,h ≥ − 2,h ≥ − respectively. Constraint (12) for λH-borrowers can be rewritten as CH (1 λH)θ (35) 2,h ≥ − αCH +pCH +(1 p)CH αCL +pCL +(1 p)CL (36) 1 2,h − 2,l ≥ 1 2,h − 2,l whereas constraint (12) for λL-borrowers becomes CL (1 λL)θ (37) 2,h ≥ − αCL +p(1 λL)θ+(1 p)CL αCH +pmax (1 λL)θ,CH +(1 p)CH (38) 1 − − 2,l ≥ 1 { − 2,h} − 2,l The optimal contract should satisfy CH (1 λL)θ. Then (35) can be ignored. 2,h ≥ − Furthermore both (38) and (36) bind. Combine (38) with (36): 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 (38) 2,h ≥ − and (36) bind. 51

Proof of Lemma 2 The conclusion follows directly from concavity of the utility function u( ) and linearity · of constraint (11) in Xi,∆ . 2,s Proof of Proposition 3 Proof. Let (Σi∗, ci∗, ci∗ , xi,∆∗) be the solution to problem (1)-(7): 1 2,s 2,s i={L,H} (cid:88) (cid:104) (cid:110) (cid:111) (cid:105) Vbil,e=0 = q p Σi∗u(xi1∗)+(1 Σi)u(xi0∗) +(1 p)u(xi∗) i 2h − 2h − 2l i=L,H De(cid:28)nethenthefollowingcontractswithCCPclearingforproblem(8)-(12): X ˆi,∆ 2 i=L,H { } and Σ ˆi, C ˆi, C ˆi where Σ ˆ = Σi∗, as well as C ˆi = ci∗, C ˆi = ci∗, and { 1 2s} i=L,H i 1 1 2s 2s (cid:32) (cid:33) (cid:88) (cid:104) (cid:110) (cid:111) (cid:105) X ˆi,∆ = u−1 q p Σi∗u(xi1∗)+(1 Σi∗)u(xi0∗) +(1 p)u(xi∗) 2 i 2h − 2h − 2l i=L,H (cid:88) (cid:104) (cid:110) (cid:111) (cid:105) < q p Σi∗xi1∗ +(1 Σi∗)xi0∗ +(1 p)xi∗ (39) i 2h − 2h − 2l i=L,H (cid:88) (cid:110) (cid:104) (cid:105)(cid:111) = pθ+ω C ˆi +p Σ ˆi(1 λi)θ+(1 Σ ˆi)C ˆi +(1 p)C ˆi − 1 − − 2h − 2l i=L,H wheretheinequalityfollowsfromconcavityofu( )(convexityofu−1( ))andtheequality · · in the last line follows from constraints (4), (5), and (6). By construction, constraints (9) and (10) are satis(cid:28)ed by (2) and (3). Constraint (11) is satis(cid:28)ed by (39) and, also by construction, constraint (12) is satis(cid:28)ed by (7). Then, the contracts X ˆi,∆ and Σ ˆi, C ˆi, C ˆi are feasible for problem { 2 } i=L,H { 1 2s} i=L,H (8)-(12). By optimality of X ˆi,∆ and Σ ˆi, C ˆi, C ˆi it needs to be { 2 } i=L,H { 1 2s} i=L,H VCCP,e=0 u(Xi,∆) = Vbil,e=0 2 ≥ 52

meaning that when borrowers pledgeability type is private information, central clearing is the optimal clearing arrangement: the solution to (8)-(12) dominates the solution to (1)-(7). The last expression shows that (8)-(12) Proof of Proposition 5 Proof. We prove the proposition in four steps: in the (cid:28)rst step, we show that we can ignore contracts that recommend ΣH = 1. In the second step we characterize optimal contracts where no borrower defaults in equilibrium, ΣH = ΣL = 0. In the third step we characterize optimal contracts where λL borrowers default in equilibrium, ΣH = 0 and ΣL = 1. In the fourth and last step we compare the two classes of contracts and we determine the optimal contracts. Step 1. Without loss of generality, in problem (8)-(12) we can ignore all contracts that recommend the strategy ΣH = 1. Therefore, λH borrowers never default in equilibrium. Suppose by contradiction that the optimal contracts X and Σi, Ci, Ci 2 { 1 2s} i=L,H recommend ΣH = 1. 41 Then, by (12) 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 (40) 1 − − 2,l ≥ 1 2,h − 2,l Weneedtoconsidertwocasesseparately: i)thecasewhentherecommendeddefault strategy to a λL borrower is ΣL = 0 and ii) when the recommended strategy is ΣL = 1. 41We are using the property that Xi,∆ is the same for all i and all ∆. 2 53

Suppose (cid:28)rst that optimal contracts recommend ΣL = 0: from (12) 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 (40) we obtain a contradiction. Therefore, it is not possible for the contracts to recommend ΣL = 0. Suppose then optimal contracts recommend ΣL = 1. De(cid:28)ne a new contract (C ˜i,C ˜i ), X ˜ as X ˜ = X , C ˜H = (1 λH)θ, { 1 2,s 2 } 2 2 2,h − C ˜i = Ci if either i = H and s = h, C ˜i = C for i = L,H. Let such a contract 2,s 2,s (cid:54) (cid:54) 1 1 recommend Σ ˜H = 0, Σ ˜L = 1. It is easy to check that all constraints in problem (8) - (12) are satis(cid:28)ed, and as X ˜i = Xi, the new contract is payo(cid:27) equivalent to the original 2 2 (optimal) one, which concludes the proof of Step 1. According to Step 1, 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. Step 2. A solution to problem (8)-(12) with ΣH = 0 and ΣL = 0 is such that (1 λL)pθ CH = CL = 0, CH = CL = ω − , 2,l 2,l 1 1 − α CH = CL = (1 λL)θ X = ω(λL)+pθλL 2h 2,h − 2 Step 2.1: From Lemma 1 we know that, given ΣH = 0 and ΣL = 0, constraint (12) requires min CH , CL (1 λL)θ and αCH +pCH +(1 p)CH = αCL +pCL + { 2,h 2,h} ≥ − 1 2,h − 2,l 1 2,h (1 p)CL . − 2,l 54

Step 2.2: W.l.o.g we can ignore the participation constraint (9) of the λH borrower. The result follows immediately from the previous step. Step 2.3: We have that CL = 0. 2,l Suppose not: CL > 0. Then it must be CL = ω. If not we could reduce CL by (cid:15), 2,l 1 2,l increase CL by (1−p)(cid:15) , and increase X by (1 q)(1 p)(cid:15)[1 1] > 0. The new contract 1 α 2 − − − α would be feasible and expected utility would increase. Then CL = ω, and therefore as 1 CL > 0 and CL (1 λL)θ, the participation constraint (9) of λL borrowers can be 2,l 2,h ≥ − ignored as well. Moreover, it must be CH = (1 λL)θ, otherwise we could reduce CL 2,h − 2,l by (cid:15) and CH by (1−p)(cid:15) and increase X by p(cid:15). The new contract would still satisfy all 2,h p 2 constraints and the expected utility would increase. Similarly it should be CH = 0. If 2,l not we could reduce CL and CH by (cid:15), and increase X by (1 p)(cid:15). Finally, it should 2,l 2,l 2 − be CH = 0, otherwise we could reduce CL by (cid:15), reduce CH by (1−p)(cid:15) and increase X 1 2,l 1 α 2 by (1 p)(cid:15)[1 +(1 q)]. Combing CH = CH = CL = 0, CH = (1 λL)θ, CL = ω, we − α − 1 2,l 2,l 1,h − 1 obtain that the binding (12) becomes 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 ≥ − Step 2.4: We have that CH = 0. 2,l Suppose not: suppose CH > 0. Then it should be CH = ω, otherwise we could educe 2,l 1 CH by (cid:15), increase CH by (1−p)(cid:15) , and increase X by q(1 p)(cid:15)[1 1] > 0. Moreover 2,l 1 α 2 − − α the participation constraint (9) 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 1 2,h − 55

participation constraint (9) of λL borrowers and the binding (12) 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 (9) of λL borrowers and (12) give 1 2,h αω = αCL +pCL = αω +pCH > αω 1 2,h 2,h which is a contradiction. Step 2.5: 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], 2,h − 2,h 1 α 2 i − α and the expected utility would increase. Step 2.6: CH = CL. 1 1 Follows from (12) holding with equality. Step 2.7: We have that Ci = ω (1−λL)pθ . 1 − α Suppose by contradiction that Ci > ω (1−λL)pθ > 0, where the last inequality (> 0) 1 − α comes from Assumption 4. Then, we could modify the allocation by de(cid:28)ning allocation ˆ ˆ de(cid:28)ned by C = C ε for ε > 0 arbitrarily small, and X = X +αε. This allocation 1 1 2 2 − is still in the constraint set of problem problem (8)-(12), and contradicts optimality of 56

Ci > ω (1−λL)pθ . 1 − α Step 3. A solution to problem (8)-(12) with ΣH = 0 and ΣL = 1 is such that CH = CL = 0, CH = CL, where 2,l 2,l 1 1     (1 λH)θ if q 1   ω ω(λH) if q 1 Ci = − ≥ α Ci = − ≥ α 2,h 1   (1 λL)θ if q < 1   ω ω(λL) if q < 1 − α − α Consider problem (8) - (12). In (12), the recommended default decision ΣH = 0 and ΣL = 1 require CH (1 λH)θ and CL < (1 λL)θ respectively. Constraint 2,h ≥ − 2,h − (12) for λH-borrowers can be rewritten as CH (1 λH)θ (41) 2,h ≥ − αCH +pCH +(1 p)CH αCL +pmax (1 λH)θ,CL +(1 p)CL (42) 1 2,h − 2,l ≥ 1 { − 2,h} − 2,l whereas constraint (12) for λL-borrowers becomes CL (1 λL)θ (43) 2,h ≤ − αCL +p(1 λL)θ+(1 p)CL αCH +pmax (1 λL)θ,CH +(1 p)CH (44) 1 − − 2,l ≥ 1 { − 2,h} − 2,l Step 3.1: W.l.o.g. we can choose CL = (1 λH)θ, and ignore constraint (43). 2,h − This choice satis(cid:28)es (43) and relaxes (44) as much as possible. Since ΣL = 1 is the recommended (i.e. incentive compatible) deafult choice, CL does not appear in any 2,h other constraint. This means that we can assume CL = (1 λH)θ. 2,h − Step 3.2: We can ignore the participation constraint of λL-borrowers. 57

From (44) 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 3.3: 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 2,h 2,l 2 } 2,h (1 λL)θ > (1 λH)θ. Then we can ignore (41). Moreover it needs to be that − − CH = ω: if not, the CCP could reduce CH by (cid:15), increase CH by p(cid:15), and increase X 1 2,h 1 α 2 by α−1qp(cid:15) > 0. All constraints are still satis(cid:28)ed but the expected utility of lenders α increases. But then, since CH = ω, we can also ignore the participation constraint of 1 λH-borrowers. Fromconstraint(42), wecanignore(44). Thereforetheonlyconstraints left are (42), the resource constraint (10) for i = L, and the second-period resource constraint (11). Note that (42) should bind or the CCP could reduce CH and increase 2,h X accordingly, without violating any constraint: 2 αω +pCH +(1 p)CH = αCL +p(1 λH)θ+(1 p)CL (45) 2,h − 2,l 1 − − 2,l From this expression and (10) 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 2,l 2,l 2 − to be CL > CH 0. Then it has to be CH = 0, otherwise we could decrease both 2,l 2,l ≥ 2,l CL andCH by (cid:15), and increase X by (1 p)(cid:15). Replacing CH = 0 we obtain that 2,l 2,l 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 1 1 2,l increase CL by p(cid:15), and increase X by α−1(1 q)p(cid:15) > 0. All constraints are still 1 α 2 α − satis(cid:28)ed but the expected utility of lenders increases. Moreover it needs to be that 58

CL = 0. If not, the CCP could reduce CH by (cid:15), CL by p (cid:15), and increase X by p(cid:15). 2,l 1 2,l 1−p 2 All constraints are still satis(cid:28)ed but the expected utility of lenders strictly increase. But then, equation (45) becomes (1 λL)pθ < pCH = p(1 λH)θ − 2,h − which is not possible. This proves that it must be that CH (1 λL)θ. Replacing 2,h ≤ − this value in (44), the latter becomes αCL +(1 p)CL αCH +(1 p)CH 1 − 2,l ≥ 1 − 2,l Step 3.4: Attheoptimalsolution,equation(44)holdswithequality: αCL+(1 p)CL = 1 − 2,l αCH +(1 p)CH. 1 − 2,l Suppose not: suppose that (44) is slack. The only active constraints are then the resource constraint in t = 1, (10) the resource constraint in t = 2, (11), and the incentive compatibility constraints (41) and (42). But then it should easily be that it CL = CL = 0. As a result, (44) can only hold if CH = CH = 0, and equation (44) 1 2,l 1 2,l holds with equality. Step 3.5: Constraint (42) can be ignored. Use the fact that (44) binds and (43), 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 3.6: 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 2,l ≥ 2,l ≥ 59

strict inequality. If CH = CL > 0, then we could decrease both by (cid:15), increase X by 2,l 2,l 2 (1 p)(cid:15), satisfying all the relevant constraints and increasing the expected utility. If − instead CH > CL = 0, it has to be 0 CH < CL. But then we could reduce CH by 2,l 2,l ≤ 1 1 2,l (cid:15), reduce CL by (1−p)(cid:15) , and increase X by (1 p)(cid:15)[q + 1−q]. All constraints would be 1 α 2 − α satis(cid:28)ed, and the expected utility would increase. Step 3.7: It is optimal to choose CH = CL. 1 1 It follows immediately by the binding (44) once we replace CH = CL = 0. 2,l 2,l αCH = αCL 1 1 Step 3.8: The feasibility constraint (10) can be ignored. By Assumption 4, ω > ω(λL) > ω(λH). Suppose (cid:28)rst by contradiction that C = ω. 1 Then, the participation constraint (9) can be ignored for both type of borrowers. Then, we could decrease slightly C and increase slightly X , to satisfy (11). Since CH = CL, 1 2 1 1 constraint (12) is una(cid:27)ected by such modi(cid:28)cation. Suppose next that C = 0. Then, 1 it means that CH > (1−λL)pθ > (1−λH)pθ for the participation constraint (9) to hold 2,h α α for i = H. But then, we can decrease CH and CL slightly by the same amount and 2,h 2,h increase X to leave (11) unchanged and to induce the same borrowers’ default choice. 2 This increases lenders’ consumption, and contradicts that C = 0 can be optimal. 1 Step 3.9: The participation constraint of a λH type binds: αC +pCH = αω. 1 2h Suppose that the participation constraint is slack. Then easily C = (1 λH)θ and 2h − C = 0. This can be a solution if ω < (1−λH)pθ . 1 α Step 3.10: Rewrite the residual problem" 60

(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 − ≥ ≥ − If q 1, the objective is decrasing in C , so the solution is C = (1 λH)θ. This ≥ α 2h 2h − can be a solution only if ω (1−λH)pθ . If q < 1, then the solution is increasing in C , ≥ α α 2h (cid:110) (cid:111) so the solution is C = min (1 λL)θ, αω = (1 λL)θ, where the last inequality 2h − p − follows by Assumption 4. Step 4. The optimal contract induces no borrower to default in equilibrium, ΣH = ΣL = 0, if q < 1, and induces λL borrowers to default in equilibrium, ΣH = 0, ΣL = 1, α if q 1. ≥ α This follows just by comparing the payo(cid:27)s of the two contracts. Proof of Lemma 7 Proof. Consider problem (16)-(21). By Assumption 4, ω > ω(λL) > ω(λH). Replace xi and xi from the binding constraints (19) and (20), and rewrite the problem 2,h 2,l (cid:16) (cid:17) (cid:16) (cid:17) (Pi) V = max pu θ ci +ω ci +(1 p)u ω ci ci γ i (ci,ci ,ci )∈(cid:60)3 − 2,h − 1 − − 1 − 2,l − 1 2,h 2,l + s.t. αci +pci +(1 p)ci αω (17) 1 2,h − 2,l ≥ ω ci 0 (18) ≥ 1 ≥ 61

ci (1 λi)θ (21) 2,h ≥ − Step 1. We have that ci < ω. 1 Suppose by contradiction that c = ω. Then, because of (21), 1 αci +pci +(1 p)ci αω +p(1 λi)θ > αω 1 2,h − 2,l ≥ − and the participation constraint (17) is slack. But then, we can slightly decrease ci 1 without violating any constraint and increasing the objective function, which proves that c = ω is not possible. 1 Step 2. We have that ci = 0. 2,l Next, we show that second period borrowers’ consumption in the low state equals zero, i.e. ci = 0. To prove this, (cid:28)rst notice that it must be that xi xi . If not, i.e. 2,l 2,h ≥ 2,l if xi < xi , combining equations (19) and (20) (with equality) we obtain 2,h 2,l 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 2,h 2,h ci by p (cid:15), and reduce xi by the same amount. All constraints would be satis(cid:28)ed, 2,l 1−p 2,l and by 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 2,h ≥ 2,l 2,l that xi = xi . If not, i.e. if xi > xi , the lender could increase ci by (cid:15), reduce x 2,h 2,l 2,h 2,l 2,h 2,h by the same amount, reduce ci by p (cid:15), and increase x by the same amount. All 2,l 1−p 2,l constraints would be satis(cid:28)ed, and by concavity of u( ) the lender would increase her · 62

expected utility. Since xi = xi , combining (19) and (20) (with equality) we obtain 2,h 2,l 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 2,h 2,l 1 α x and x by the same amount α−1(cid:15). All constraints would be satis(cid:28)ed and the 2,h 2,l α lender expected revenues would increase. Therefore it can not be that ci > 0, and we 2,l conclude that it should be that ci = 0. 2,l Step 3. The participation constraint (17) binds. Suppose by contradiction that the participation constraint (17) is slack. Then, the limited commitment constraint (21) should bind, i.e. ci = (1 λi)θ. Indeed, both 2,h − constraints (17) and (21) can not be slack, because then we could decrease ci without 2,h violating any constraint. Then, since ω > ω(λL), it must be that ci > 0. Indeed, if 1 we had ci = 0, then αci + pci = p(1 λi)θ < αω and the participation constraint 1 1 2,h − (17) would be violated. But then, if ci > 0 and (17) is slack, we could just decrease 1 ci slightly, and increase lenders’ consumption, which proves that the participation 1 constraint (17) should bind. Step 4. A solution to problem (16)-(21) is such that ci = (1 λi)θ, ci = ω (1−λi)pθ 2,h − 1 − α ifλ λ∗, whereasci = (1 λ∗)θ, ci = ω (1−λ∗)pθ ifλ λ∗, forλ∗ de(cid:28)ned in equation ≤ 2,h − 1 − α ≤ (22). Ignore (18) and replace ci = αω−ci 1 in the objective function and in (21), and rewrite 2,h p 63

the residual problem as (cid:18) (cid:19) α p (cid:16) (cid:17) (Pi) V = max pu θ (ω ci) − +(1 p)u ω ci γ i (ci 1 )∈(cid:60)+ − − 1 p − − 1 − (1 λi)pθ ci ω − (21) 1 ≤ − α Ignore constraint (21): the (cid:28)rst order condition for optimality is (cid:18) (cid:19) α p (α p)u(cid:48) θ (ω ci) − (1 p)u(cid:48)(ω ci) (46) − − − 1 p ≤ − − 1 with equality if ci > 0. Notice that the left-hand side is decreasing in ci and the 1 1 right-hand side is increasing in ci. 1 Suppose (cid:28)rst that ci = 0: equation (46) requires 1 (cid:18) (cid:19) α p (α p)u(cid:48) θ (ω) − (1 p)u(cid:48)(ω) − − p ≤ − Since ω > (1−λL)pθ by Assumption 4, α (cid:18) (1 λL)θ (cid:19) (cid:18) α p (cid:19) (cid:18) (1 λL)pθ (cid:19) (α p)u(cid:48) θ − [α p] < (α p)u(cid:48) θ (ω) − (1 p)u(cid:48)(ω) < (1 p)u(cid:48) − − − α − − − p ≤ − − α which violates Assumption 6. Then, there exists a unique ci∗ > 0 that solves (46). 1 Given this c∗, the solution to problem (16)-(21) depends on λi: either ci = c∗ and (21) 1 1 1 holds for c∗, αω−c∗ 1 (1 λi), or (21) binds, αω−ci 1 = (1 λi). Notice that (21) is 1 pθ ≥ − pθ − decreasing in ci; thus, there exists unique λ∗ such that the solution is c∗ if and only if 1 1 λ λ∗, whereas the solution is pinned down by the binding constraint (21) if λ < λ∗. ≥ Speci(cid:28)cally, λ∗ solves (46) for ω ci = (1−λ∗)pθ : − 1 α (cid:18) (1 λ∗)pθ (cid:19) (cid:18) (1 λ∗)pθ (cid:19) (α p)u(cid:48) θ − [α p] = (1 p)u(cid:48) − − − α − − α 64

It is easy to see that such λ∗ is de(cid:28)ned in equation (22). De(cid:28)ne then the function (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 NoticethatF(0) = 1 < α−p andF(cid:48)(λ) > 0. Thus, thereexistsauniqueλ∗ (0,1)such 1−p ∈ that F(λ∗) = α−p. Then, we can conclude that ci = c∗ if λi λ∗, and ci = ω (1−λi)pθ 1−p 1 1 ≥ 1 − α if λi < λ∗. Step 5. We have cL = (1 λL)θ and cL = ω (1−λL)pθ . 2,h − 1 − α It follows from Step 4 and Assumption 6. Step 6. We have that (cid:18) (cid:19) u(cid:48) (1−λH)pθ (1) cH = ω ω(λ∗), cH = (1 λ∗)θ, if α > α−p. 1 − 2,h − u(cid:48)(θ−α−p(1−λH)θ) 1−p α (cid:18) (cid:19) u(cid:48) (1−λH)pθ (2) cH = ω ω(λH), cH = (1 λH)θ, if α−p > α . 1 − 2,h − 1−p u(cid:48)(θ−α−p(1−λH)θ) α It follows from Step 4. Proof of Proposition 9 Proof. Let ω(λ) be de(cid:28)ned in equation (14). Consider the payo(cid:27) ensuing the optimal contract with central clearing in equation (15), VCCP,e=0 = u(X ), where 2    ω(λL)+pθλL if q 1 X = ≤ α (47) 2   ω(λH)+pθ[qλH +(1 q)λL] if q 1 − ≥ α 65

and the payo(cid:27) associated with the optimal contracts with bilateral clearing and screening (cid:88) (cid:110) (cid:88) (cid:111) Vbil,e=1 = q p u(xi ) γ i s 2,s − i=L,H s=l,h where xH = min λHθ+ω(λH),λ∗θ+ω(λ∗) , xH = max ω(λH),ω(λ∗) , 2,h { } 2,l { } xL = λLθ+ω(λL), xL = ω(λL) (48) 2,h 2,l whereλ∗ isde(cid:28)nedin(22), andfromLemma7weknowthatλHθ+ω(λH) = min λHθ+ (cid:18) (cid:19) { u(cid:48) (1−λH)pθ ω(λH),λ∗θ+ω(λ∗) if and only if α−p > α . Notice that xi as well as λ∗ } 1−p u(cid:48)(θ−α−p(1−λH)θ) 2,s α are independent of q. Easily, bilateral clearing with information acquisition is preferred to central clearing if and only if Vbil,e=1 VCCP,e=0. Let then γ(q) : [0,1] ≥ → (cid:60) (cid:88) (cid:110) (cid:88) (cid:111) γ(q) = q p u(xi ) u(X ) (49) i s 2,s − 2 i=L,H s=l,h for X de(cid:28)ned in (47) and xi de(cid:28)ned in (48). 2 2,s Step 1. The function γ(q) de(cid:28)ned in (49) satis(cid:28)es γ(0) < 0 and γ(1) < 0. Consider then such function: at q = 0 we have (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:16) (cid:17) γ(0) = pu λLθ+ω(λL) +(1 p))u ω(λL) u ω(λL)+pθλL < 0 − − where the inequality comes from concavity of u( ). Consider next the same function at · 66

q = 1: (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:16) (cid:17) γ(1) = pu min λHθ+ω(λH),λ∗θ+ω(λ∗) +(1 p))u max ω(λH),ω(λ∗) u ω(λH)+pθλH { } − { } − (cid:32) (cid:33) (cid:16) (cid:17) < u pmin λHθ+ω(λH),λ∗θ+ω(λ∗) +(1 p)max ω(λH),ω(λ∗) u ω(λH)+pθλH { } − { } − (cid:32) (cid:33) (cid:16) (cid:17) u p[λHθ+ω(λH)]+(1 p)ω(λH) u ω(λH)+pθλH = 0 ≤ − − where the (cid:28)rst inequality comes from concavity of u( ), the second one from and the · de(cid:28)nition of λ∗. Thus, we also have that γ(1) < 0 as well. Step 2. The function γ(q) de(cid:28)ned in (49) is monotone increasing for q < 1. α Consider the (cid:28)rst derivative of the function γ(q) de(cid:28)ned in (49), for q < 1: α ∂γ(q) (cid:16) (cid:17) (cid:16) (cid:17) = pu min λHθ+ω(λH),λ∗θ+ω(λ∗) +(1 p))u max ω(λH),ω(λ∗) ∂q { } − { (cid:16) (cid:17) (cid:16) (cid:17) pu λLθ+ω(λL) +(1 p))u ω(λL) > 0 − − where the inequality comes from the fact that the contract xL = λLθ +ω(λL), xL = 2,h 2,l ω(λL) is feasible, but not optimal, for the problem of a lender that faces a λH borrower. Then, the function γ(q) de(cid:28)ned in (49) is monotone increasing for q < 1. α Step 3. The function γ(q) de(cid:28)ned in (49) is convex for q > 1. α Consider now the (cid:28)rst and second derivatives of the function γ(q) de(cid:28)ned in (49), for q > 1: α ∂γ(q) (cid:16) (cid:17) (cid:16) (cid:17) = pu min λHθ+ω(λH),λ∗θ+ω(λ∗) +(1 p))u max ω(λH),ω(λ∗) ∂q { } − { (cid:16) (cid:17) (cid:16) (cid:17) pu λLθ+ω(λL) +(1 p))u ω(λL) u(cid:48)(X )pθ(λH λL) 2 − − − − 67

and ∂2γ(q) = u(cid:48)(cid:48)(X )[pθ(λH λL)]2 > 0 ∂q2 − 2 − where the inequality comes from concavity of u( ), thus u(cid:48)(cid:48)( ) < 0 and u(cid:48)(cid:48)( ) > 0. · · − · Step 4. The function γ(q) > 0 for some q [0,1] if and only if γ(1) > 0. Moreover, ∈ α if γ(1) > 0, then there exist q,q [0,1], where q < q, such that γ(q) > 0 for γ [q,q]. α ∈ ∈ The conclusion follows from Steps 1,2, and 3. Suppose (cid:28)rst that γ(1) > 0. Since γ(q) α is strictly increasing for q < 1 and q(0) < 0, by the intermediate value theorem there α should exists a unique q (0, 1) such that γ(q) = 0. Also, since γ(1) < 0 and γ(q) is ∈ α convex for γ < 1, given that γ(1) > 0 it must be that for q > 1 the function γ(q) is α α α initially decreasing, crosses the horizontal axes for a unique q where γ(q) = 0, and then stays negative. Then, we proved the if direction: if γ(1) > 0, there exist q,q [0,1] α ∈ such that γ(q) > 0 for γ [q,q]. ∈ Next, we prove the only if direction. Suppose that γ(q) > 0 for some q (0,1), but ∈ γ(1) < 0. Since the function γ(q) is strictly increasing for q < 1, then γ(q) < 0 for all α α q 1, and it must be that γ(q) > 0 for some q > 1. But then, since γ(1) < 0, and ≤ α α α the function γ(1) is continuous, there should exist an interval [q(cid:48),q(cid:48)(cid:48)] (1,1] such that α ⊂ α γ(q) > 0 and γ(cid:48)(q) > 0 for q [q(cid:48),q(cid:48)(cid:48)]. But then, since the function γ(q) is convex, it ∈ must be that γ(cid:48)(q) > 0 also for all q > q(cid:48)(cid:48), and therefore γ(1) > γ(q(cid:48)(cid:48)) > 0, which is a contradiction. Step 5. Replacing q = 1 in (49) we obtain the condition in (23). Also, the function α γ(q) : (q,q) is theγ(q) de(cid:28)ned in (49), restricting the domain to the interval + → (cid:60) (q,q) 68

Proof of Lemma 10 Proof. Let (xiu ,xiu) be the optimal contract in the economy populated by lenders 2,h 2,l with utility u( ) and (xiv ,xiv ) be the optimal contract in the economy populated by · 2,h 2,l lenders with utility v( ). Also, let X be the optimal contract with central clearing. 2 · Note that X is the same with utility u( ) and with utility v( ). Let then λ∗ and λ∗ be 2 · · u v the threshold λ∗ de(cid:28)ned in (22) when the utility function is u( ) and v( ) respectively. · · Notice that (cid:16) (cid:17) (cid:16) (cid:17) u(cid:48) (1−λ∗ u )pθ v(cid:48) (1−λ∗ v )pθ α α p α u(cid:48) (cid:0) θ α−p(1 λ∗)θ (cid:1) = 1 − p = v(cid:48) (cid:0) θ α−p(1 λ∗)θ (cid:1) − α − u − − α − v (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) (cid:16) (cid:17) ρ(cid:48) v (1−λ∗ v )pθ u(cid:48) (1−λ∗ v )pθ u(cid:48) (1−λ∗ v )pθ α α α = > ρ(cid:48) (cid:0) v (cid:0) θ α−p(1 λ∗)θ (cid:1)(cid:1) u(cid:48) (cid:0) θ α−p(1 λ∗)θ (cid:1) u(cid:48) (cid:0) θ α−p(1 λ∗)θ (cid:1) − α − v − α − v − α − v (cid:18) (cid:19) v(cid:48) (1−λL)pθ and therefore λ∗ > λ∗. Then, since we assumed that α−p > α , using the u v 1−p v(cid:48)(θ−α−p(1−λL)θ) α same argument as above we can easily prove that (cid:16) (cid:17) (cid:16) (cid:17) v(cid:48) (1−λL)pθ u(cid:48) (1−λL)pθ α p α α 1 − p > v(cid:48) (cid:0) θ α−p(1 λL)θ (cid:1) > u(cid:48) (cid:0) θ α−p(1 λL)θ (cid:1) − − α − − α − and therefore the limited commitment constraint of λL borrowers binds under both lenders’ utility functions. Hence, (1 λL)pθ xLu = xLv = λLθ+ − 2,h 2,h α (1 λL)pθ xLu = xLv = − 2,l 2,l α Also, regarding xHu and xHv there are three possible cases. 2,s 2,s (cid:18) (cid:19) (cid:18) (cid:19) v(cid:48) (1−λH)pθ u(cid:48) (1−λH)pθ Case 1: α−p > α > α . 1−p v(cid:48)(θ−α−p(1−λH)θ) u(cid:48)(θ−α−p(1−λH)θ) α α 69

In this case xHu = xHv = λHθ + (1−λH)pθ and xHu = xHv = (1−λH)pθ . Then, since 2,h 2,h α 2,l 2,l α lenders in economy B prefer bilateral clearing, it must be that γ < γ , where v (cid:104) (cid:105) (cid:104) (cid:105) γ = q pv(xHv)+(1 p)v(xHv) +(1 q) pv(xLv)+(1 p)v(xLv) v(X ) v 2,h − 2,l − 2,h − 2,l − 2 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:16) (cid:17) = q pρ u(xHv) +(1 p)ρ u(xHv) +(1 q) pρ u(xLv) +(1 p)ρ u(xLv) ρ u(X ) 2,h − 2,l − 2,h − 2,l − 2 (cid:32) (cid:33) (cid:104) (cid:105) (cid:104) (cid:105) (cid:16) (cid:17) ρ q pu(xHv)+(1 p)u(xHv) +(1 q) pu(xLv)+(1 p)u(xLv) ρ u(X ) ≤ 2,h − 2,l − 2,h − 2,l − 2 (cid:104) (cid:105) (cid:104) (cid:105) q pu(xHv)+(1 p)u(xHv) +(1 q) pu(xLv)+(1 p)u(xLv) u(X ) ≤ 2,h − 2,l − 2,h − 2,l − 2 (cid:104) (cid:105) (cid:104) (cid:105) = q pu(xHu)+(1 p)u(xHu) +(1 q) pu(xLu)+(1 p)u(xLu) u(X ) = γ 2,h − 2,l − 2,h − 2,l − 2 u where the (cid:28)rst inequality follows from concavity, and the second one from the contraction property of ρ( ). Thus, when γ < γ , then γ < γ , and so agents in economy A as v u · well prefer bilateral clearing over central clearing. (cid:18) (cid:19) (cid:18) (cid:19) v(cid:48) (1−λH)pθ u(cid:48) (1−λH)pθ Case 2: α > α−p > α . v(cid:48)(θ−α−p(1−λH)θ) 1−p u(cid:48)(θ−α−p(1−λH)θ) α α In this case xHu = λHθ + (1−λH)pθ and xHu = (1−λH)pθ , xHv = λ∗θ + (1−λ∗ v )pθ , and 2,h α 2,l α 2,h v α xHv = (1−λ∗ v )pθ . In this case, since λ∗ λH, thus the contract (xHv,xHv) is feasible 2,l α v ≤ 2,h 2,l and incentive-compatible for the problem that lenders with utility u( ) face. But since · such lenders prefer the contract (xHu,xHu), it must be that 2,h 2,l pu(xHu)+(1 p)u(xHu) > pu(xHv)+(1 p)u(xHv) 2,h − 2,l 2,h − 2,l Now, since lenders in economy B prefer bilateral clearing, it must be that γ < γ , v 70

where (cid:104) (cid:105) (cid:104) (cid:105) γ = q pv(xHv)+(1 p)v(xHv) +(1 q) pv(xLv)+(1 p)v(xLv) v(X ) v 2,h − 2,l − 2,h − 2,l − 2 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:16) (cid:17) = q pρ u(xHv) +(1 p)ρ u(xHv) +(1 q) pρ u(xLv) +(1 p)ρ u(xLv) ρ u(X ) 2,h − 2,l − 2,h − 2,l − 2 (cid:32) (cid:33) (cid:104) (cid:105) (cid:104) (cid:105) (cid:16) (cid:17) ρ q pu(xHv)+(1 p)u(xHv) +(1 q) pu(xLv)+(1 p)u(xLv) ρ u(X ) ≤ 2,h − 2,l − 2,h − 2,l − 2 (cid:104) (cid:105) (cid:104) (cid:105) q pu(xHv)+(1 p)u(xHv) +(1 q) pu(xLv)+(1 p)u(xLv) u(X ) ≤ 2,h − 2,l − 2,h − 2,l − 2 (cid:104) (cid:105) (cid:104) (cid:105) < q pu(xHu)+(1 p)u(xHu) +(1 q) pu(xLu)+(1 p)u(xLu) u(X ) = γ 2,h − 2,l − 2,h − 2,l − 2 u where the (cid:28)rst inequality follows from concavity, the second one from the contraction property of ρ( ), and the third one from the previous argument. Thus, when γ < γ , v · then γ < γ , and so agents in economy A as well prefer bilateral clearing over central u clearing. (cid:18) (cid:19) (cid:18) (cid:19) v(cid:48) (1−λH)pθ u(cid:48) (1−λH)pθ Case 3: α > α > α−p. v(cid:48)(θ−α−p(1−λH)θ) u(cid:48)(θ−α−p(1−λH)θ) 1−p α α In this case xHu = λ∗θ + (1−λ∗ u )pθ and xHu = (1−λ∗ u )pθ , xHv = λ∗θ + (1−λ∗ v )pθ , and 2,h u α 2,l α 2,h v α xHv = (1−λ∗ v )pθ . We proved earlier that λ∗ < λ∗. Then, thus the contract (xHv,xHv) 2,l α v u 2,h 2,l is feasible and incentive-compatible for the problem that lenders with utility u( ) face. · But since such lenders prefer the contract (xHu,xHu), it must be that 2,h 2,l pu(xHu)+(1 p)u(xHu) > pu(xHv)+(1 p)u(xHv) 2,h − 2,l 2,h − 2,l FollowingthesameargumentasinCase2, wecanprovethatwhenγ < γ , thenγ < γ . v u So, when agents in economy B prefer bilateral with information acquisition to central clearingwithoutinformationacquisition, alsoagentsineconomyApreferbilateralwith 71

information acquisition to central clearing without information acquisition. Proof of Proposition 13 Proof. Let (xi∗ , ci∗, ci∗ ) be the optimal contract with bilateral clearing and screening 2,s 1 2,s in Lemma 7, for i L,H and s l,h . Also, let γ be de(cid:28)ned in Proposition 9. ∈ { } ∈ { } ˆ Rewrite problem (P ) for = (s,∆) : s l,h ,∆ 0,1 : 1 2 H { ∈ { } ∈ { }} (cid:104) (cid:88) (cid:105) (cid:104) (cid:88) (cid:105) (P ) max q p wH,0 +(1 q) p wL,0 γ (25) 1 s s s s − − s=l,h s=l,h s.t. αCi +pCi +(1 p)Ci αω (26) 1 2,h 2,l − ≥ Ci (1 λi)θ (27) 2,h ≥ − (cid:88) ω q Ci 0 (28) i 1 − ≥ i∈{L,H} (cid:40) (cid:41) (cid:88) (cid:88) q p u−1(wi,0)+Ci +pCi +(1 p)Ci ω+pθ (29) i s s 1 2,h 2,l − ≤ i∈{L,H} s∈{l,h} (cid:88) (cid:88) (cid:104) (cid:105) p wi,0 p σ (λi,λ−i)w−i,1+[1 σ (λi,λ−i)]w−i,0 (30) s s s s s s s ≥ − s∈{l,h} s∈{l,h}   (cid:88)  (cid:88)  γ + q p wi,0 i s s −   i∈{L,H} s∈{l,h}    max  (cid:88) q i (cid:88) p s (cid:104) σ s (λi,λ ˆi)w ˆ s i,1+[1 σ s (λi,λ ˆi)]w ˆ s i,0 (cid:105)   ≥ˆi∈{L,H} −  i∈{L,H} s∈{l,h} (31) σ (λi,λj) = 0 i(cid:27) C j (1 λi)θ (32) s 2,h ≥ − where we use the notation p = p, p = 1 p, qH = q and qL = 1 q. Consider h l − − the contracts (wˆi,∆, C ˆi, C ˆi ) which, on the equilibrium replicate the optimal contracts s 1 2,s in Lemma 7, and that assign maximum punishment for lenders in the o(cid:27)-equilibrium 72

(when the original borrower counterparty strategically defaults, ∆ = 1): wˆi,0 = u−1(xi∗ ), wˆi,1 = 0, s 2,s s C ˆi = ci∗, C ˆi = ci∗ . 1 1 2,s 2,s Step 1. The contract (wˆi,∆, C ˆi, C ˆi ) is feasible and satis(cid:28)es borrowers’ individually s 1 2,s rationality: constraints (26)-(29) are satis(cid:28)ed. Constraint (26) is directly satis(cid:28)ed by (17). Similarly (27) is directly satis(cid:28)ed by (21). Also, (28) is satis(cid:28)ed by (18) (cid:88) (cid:88) ω q C ˆi = ω q ci∗ 0 − i 1 − i 1 ≥ i∈{L,H} i∈{L,H} and (29) is satis(cid:28)ed by (19) and (20):   (cid:88)  (cid:88) (cid:104) (cid:105) q p u−1(wˆH,0)+C ˆi i s s 2,s   i∈{L,H} s∈{l,h} (cid:40) (cid:41) (cid:88) (cid:88) (cid:2) (cid:3) (cid:88) (cid:110) (cid:104) (cid:105) (cid:104) (cid:105)(cid:111) = q p xi∗ +ci∗ q p ω ci∗ +θ +(1 p) ω ci∗ i s 2,s 2,s ≤ i − 1 − − 1 i∈{L,H} s=l,h i∈{L,H} (cid:88) (cid:88) = pθ+ω q ci∗ = pθ+ω q C ˆi − i 1 − i 1 i∈{L,H} i∈{L,H} Step 2. Only ex-ante incentive-compatibility matters: constraint (30) is satis(cid:28)ed by 42 (31). We prove this by contraposition: we prove that when (30) is not satis(cid:28)ed, then (31) 42This is true in general and not only for the replication contract. 73

is violated as well. Assume that (30) is violated for ˜i, ˜i L,H , where ˜i =˜i: − ∈ { } − (cid:54) pw ˜i,0 +(1 p)w ˜i,0 < (cid:88) p (cid:110) σ (λ ˜i,λ−˜i)w−˜i,1 +[1 σ (λ ˜i,λ−˜i)]w−˜i,0 (cid:111) h − l s s s − s s s∈{l,h} Fromthede(cid:28)nitionofthemaxoperator, theexpressionabove, thefactthatσ (λi,λi) = s 0, and the fact that γ > 0, we obtain    ˆi∈ m {L a ,H x }   (cid:88) q i (cid:88) p s (cid:104) σ s (λi,λ ˆi)w ˆ s i,1 +[1 − σ s (λi,λ ˆi)]w ˆ s i,0 (cid:105)    i∈{L,H} s∈{l,h}   (cid:88) q  (cid:88) p (cid:104) σ (λi,λ−˜i)w−˜i,1 +[1 σ (λi,λ−˜i)]w−˜i,0 (cid:105) ≥ i s s s − s s   i∈{L,H} s∈{l,h} = q (cid:104) pw−˜i,0 +(1 p)w−˜i,0 (cid:105) +q (cid:88) p (cid:110) σ (λ ˜i,λ−˜i)w−˜i,1 +[1 σ (λ ˜i,λ−˜i)]w−˜i,0 (cid:111) −˜i h − l ˜i s s s − s s s∈{l,h} (cid:104) (cid:105) (cid:104) (cid:105) > q pw−˜i,0 +(1 p)w−˜i,0 +q pw ˜i,0 +(1 p)w ˜i,0 −˜i h − l ˜i h − l     (cid:88)  (cid:88)  (cid:88)  (cid:88)  = q p wi,0 > γ + q p wi,0 i s s − i s s     i∈{L,H} s∈{l,h} i∈{L,H} s∈{l,h} which proves that equation (31) is also violated. Step 3. If γ < γ, for γ de(cid:28)ned in Proposition 9, the contract (wˆi,∆, C ˆi, C ˆi ) is s 1 2,s incentive compatible: constraint (31) is satis(cid:28)ed. For (xi∗ , ci∗, ci∗ ) the optimal contract with bilateral clearing and screening in 2,s 1 2,s Lemma 7, consider the following contract (X ˜i,∆ , C ˜i, C ˜i , Σ ˜ ) in (8)-(12): 2 1 2s i C ˜i = cH∗, C ˜i = cH∗, Σ ˜i = 1 i(cid:27) cH∗ < (1 λi)θ, 2s 2,h 1 1 2,h − (cid:88) (cid:110) (cid:104) (cid:105) (cid:111) X ˜i,1 = X ˜i,0 X ˜ = q p (1 Σ ˜i)xH∗ +Σ ˜iλHθ +(1 p)xH∗ 2 2 ≡ 2 i − 2,h − 2,l i∈{L,H} 74

It is easy to show that such contract satis(cid:28)es constraints (9)-(12). Thus, by optimality, it must be that   (cid:88) (cid:110) (cid:104) (cid:105) (cid:111) VCCP,e=0 ≥ u(X ˜ 2 ) = u q i p (1 − Σ ˜i)xH 2,h ∗ +Σ ˜iλHθ +(1 − p)xH 2,l ∗  i∈{L,H} (cid:88) (cid:110) (cid:16) (cid:104) (cid:105) (cid:17)(cid:111) q u p (1 Σ ˜i)xH∗ +Σ ˜iλHθ +(1 p)xH∗ ≥ i − 2,h − 2,l i∈{L,H} (50) (cid:88) q (cid:110) pu (cid:16) (1 Σ ˜i)xH∗ +Σ ˜iλHθ (cid:17) +(1 p)u (cid:0) xH∗ (cid:1) (cid:111) ≥ i − 2,h − 2,l i∈{L,H} (cid:88) q (cid:110) p (cid:104) Σ ˜iu(λHθ)+(1 Σ ˜i)u (cid:0) xH∗ (cid:1) (cid:105) +(1 p)u (cid:0) xH∗ (cid:1) (cid:111) ≥ i − 2,h − 2,l i∈{L,H} Similarly, consider the following contract (X ¯i,∆ , C ¯i, C ¯i , Σ ¯ ) in (8)-(12): 2 1 2s i C ¯i = cL∗, C ˜i = cL∗, Σ ¯i = 1 i(cid:27) cL∗ < (1 λi)θ, 2s 2,h 1 1 2,h − (cid:88) (cid:110) (cid:104) (cid:105) (cid:111) X ¯i,1 = X ¯i,0 X ¯ = q p (1 Σ ¯i)xH∗ +Σ ¯iλLθ +(1 p)xL∗ 2 2 ≡ 2 i − 2,h − 2,l i∈{L,H} It is also easy to show that such contract satis(cid:28)es constraints (9)-(12). Thus, by optimality, it must be that   (cid:88) (cid:110) (cid:104) (cid:105) (cid:111) VCCP,e=0 ≥ u(X ¯ 2 ) = u q i p (1 − Σ ¯i)xL 2, ∗ h +Σ ¯iλLθ +(1 − p)xL 2, ∗ l  i∈{L,H} (cid:88) (cid:110) (cid:16) (cid:104) (cid:105) (cid:17)(cid:111) q u p (1 Σ ¯i)xL∗ +Σ ¯iλLθ +(1 p)xL∗ ≥ i − 2,h − 2,l i∈{L,H} (51) (cid:88) q (cid:110) pu (cid:16) (1 Σ ¯i)xL∗ +Σ ¯iλLθ (cid:17) +(1 p)u (cid:0) xL∗ (cid:1) (cid:111) ≥ i − 2,h − 2,l i∈{L,H} (cid:88) q (cid:110) p (cid:104) Σ ¯iu(λLθ)+(1 Σ ¯i)u (cid:0) xL∗ (cid:1) (cid:105) +(1 p)u (cid:0) xL∗ (cid:1) (cid:111) ≥ i − 2,h − 2,l i∈{L,H} 75

Because γ < γ, from Proposition 9 we know that (cid:88) (cid:110) (cid:88) (cid:104) (cid:105)(cid:111) q p u(xi∗ ) γ VCCP,e=0 i s 2,s − ≥ i∈{L,H} s∈{l,h} Combining these expressions together with the contract (wˆi,∆, C ˆi, C ˆi ) we obtain s 1 2,s   (cid:88)  (cid:88)  (cid:88) (cid:110) (cid:88) (cid:104) (cid:105)(cid:111) γ + q p wˆi,0 = γ + q p u(xi∗ ) − i s s − i s 2,s   i∈{L,H} s∈{l,h} i∈{L,H} s∈{l,h} VCCP,e=0 ≥ (cid:40) max (cid:88) q (cid:110) p (cid:104) Σ ¯iu(λLθ)+(1 Σ ¯i)u (cid:0) xL∗ (cid:1) (cid:105) +(1 p)u (cid:0) xL∗ (cid:1) (cid:111) , ≥ i − 2,h − 2,l i∈{L,H} (cid:88) q (cid:110) p (cid:104) Σ ˜iu(λHθ)+(1 Σ ˜i)u (cid:0) xH∗ (cid:1) (cid:105) +(1 p)u (cid:0) xH∗ (cid:1) (cid:111) i − 2,h − 2,l i∈{L,H}    (cid:41) > ˆi∈ m {L a ,H x }   (cid:88) q i (cid:88) p s (cid:104) σ s (λi,λ ˆi)w ˆ s i,1 +[1 − σ s (λi,λ ˆi)]w ˆ s i,0 (cid:105)    i∈{L,H} s∈{l,h} where the last inequality follows from wˆi,1 < u(λLθ) < u(λH)θ, and the fact that the s de(cid:28)nition of Σ ¯i and Σ ˜i is equivalent to the de(cid:28)nition of σ (λi,λj) in (32), and the fact s that σ (λi,λj) = 0. l This proves that, if γ < γ, the incentive-compatibility constraint (31) is satis(cid:28)ed. Step 4. The contract (wˆi,∆, C ˆi, C ˆi ) is feasible, incentive-compatible, and s 1 2,s (cid:40) (cid:41) (cid:40) (cid:41) (cid:88) (cid:104) (cid:88) (cid:105) (cid:88) (cid:104) (cid:88) (cid:105) q p wˆi,0 = q p u(xi∗ ) i s s i s 2,s i∈{L,H} s=l,h i∈{L,H} s=l,h Feasibility and incentive-compatibility follows from the previous steps. The last equation is true by construction, and concludes the proof of Proposition 13. 76

The optimal contract with pro-rata allocation of losses and due diligence ˆ Rewrite the problem (P ) for the case where = ∆ 0,1 : 1 2 H { ∈ { }} max qwH,0 +(1 q)wL,0 γ (25) − − s.t. αCi +pCi +(1 p)Ci αω (26) 1 2,h − 2,l ≥ Ci (1 λi)θ (27) 2,h ≥ − (cid:88) ω q Ci 0 (28) − i 1 ≥ i∈{L,H} qu−1(wH,0)+(1 q)u−1(wL,0) − s (cid:88) (cid:110) (cid:111) + q [Ci +pCi +(1 p)Ci ] ω +pθ (29) i 1 2,h − 2,l ≤ i∈{L,H} (cid:88) (cid:104) (cid:105) wi,0 p σ (λi,λ−i)w−i,1 +[1 σ (λi,λ−i)]w−i,0 (30) s s s ≥ − s∈{l,h}    γ + (cid:88) q i wi,0 max  (cid:88) q i (cid:88) p s (cid:104) σ s (λi,λ ˆi)w ˆi,1 +[1 σ s (λi,λ ˆi)]w ˆi,0 (cid:105)   − ≥ˆi∈{L,H} −  i∈{L,H} i∈{L,H} s∈{l,h} (31) σ (λi,λj) = 0 i(cid:27) Cj (1 λi)θ (32) s 2,h ≥ − where we use the notation p = p, p = 1 p, qH = q, and qL = 1 q. It is easy to s l − − see that maximum punishment for lack of due diligence is optimal: wi,1 = 0. Replace wi,1 = 0, and simplify notation rewriting wi = wi,0, that is lenders’ promised utility when the (cid:28)rst period message was m = λi and the original borrower counterparty L does not strategically default in equilibrium. 77

Claim 1 Constraint (30) can be ignored. Proof. The proof is identical to Step 2 in the proof of Proposition 13. Claim 2: With pro-rata allocation of losses and due diligence, optimality of central clearing with information acquisition requires that CH < (1 λL)θ. 2,h − Proof. Let (wH,wL), (C ,i,Ci ) be the solution to problem (26)-(??), 2 2 1 2,s i=L,H,s=h,l and suppose CH (1 λL)θ. Consider now the contract with central clearing, 2h ≥ − no monitoring in problem (8)-(12) de(cid:28)ned as X ˆ = qu−1(XH) + (1 q)u−1(XL), 2 2 − 2 C ˆ = qCH + (1 q)CL , and C ˆ = qCH + (1 q)CL. Easily this contract in- 2,s 2,s − 2,s 1 1 − 1 duces strategies ΣL = ΣH = 0 in (12), and satis(cid:28)es (9)-(11). Concavity of u( ) gives · u(X ˆ2) qwH + (1 q)wL > qwH + (1 q)wL γ, so it is strictly better than the ≥ − − − original contract with information acquisition. Claim 3: The optimal contract satis(cid:28)es σ (λH,λL) = 1 and σ (λL,λH) = 0. h h Proof. The conclusion σ (λH,λL) = 1 follows from Claim 2 above. On the other h hand, the conclusion σ (λL,λH) = 0 follows easily from (27), since CL (1 λL)θ > h 2,h ≥ − (1 λH)θ. − Ignoring constraint (30) and replacing σ (λH,λL) = 1 and σ (λL,λH) = 0 in (31) h h and (32), we can rewrite problem (25)-(32) as follows: (P ˆFI) max qwH +(1 q)wL γ (25) − − s.t. αCi +pCi +(1 p)Ci αω (26) 1 2,h − 2,l ≥ 78

Ci (1 λi)θ (27) 2,h ≥ − (cid:88) ω q Ci 0 (28) − i 1 ≥ i∈{L,H} qu−1(wH)+(1 q)u−1(wL) − s (cid:88) (cid:110) (cid:111) + q [Ci +pCi +(1 p)Ci ] ω +pθ (29) i 1 2,h − 2,l ≤ i∈{L,H} (cid:110) (cid:104) (cid:105) (cid:111) γ +qwH +(1 q)wL max wL, q +(1 q)(1 p) wL (31) − − ≥ − − We can solve problem (P ˆFI) 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 { 1 2s} i=H,L,s=h,l of resources 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: { 1 2s} i=H,L,s=h,l they should satisfy the participation and the limited commitment constraints of the borrowers. Thus, contracts Ci,Ci solve the following problem: { 1 2s} i=H,L,s=h,l (cid:104) (cid:105) (P ˆ b FI ) Ω = max ω qCH (1 q)CL +pθ {Ci,Ci ,Ci } − 1 − − 1 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 79

counterparty and report truthfully the information that they learn; thus they solve (P ˆ a FI ) max qwH +(1 q)wL γ Ω {wH,wL}∈(cid:60)2 − − + (cid:0) (cid:1) (cid:0) (cid:1) s.t. qu−1 wH +(1 q)u−1 wL Ω − ≤ γ +qwH +(1 q)wL − − ≥ (cid:110) (cid:111) max wL,(q +(1 q)(1 p))wH (52) − − Claim 4: (Ci,Ci ,Ci ,wi) solve the problem (P ˆFI) if and only if (Ci, Ci , 1 2h 2l i=L,H 1 2h Ci ) solve (P ˆ bFI) and, letting Ω∗ denote the value of the objective in (P ˆ bFI) at 2l i=L,H its solution, (wH,wL) solve (P ˆ a FI ). Ω∗ Proof. First we show the only if direction. Suppose that (Ci,Ci ,Ci ,wi) is the 1 2h 2l i=L,H solution to problem (P ˆFI), but either (Ci,Ci ,Ci ) does not solve (P ˆ bFI), or for 1 2h 2l i=L,H Ω∗ the solution to (P ˆ bFI), (w ,w ) solve (P ˆ a FI ). H L Ω∗ If (Ci,Ci ,Ci ) does not solve (P ˆ bFI), 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 (P ˆ bFI). Fromproblem(P ˆ bFI),itmustbethatforsomei,eitherCi(cid:48) < Ci,orCi(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 (P ˆFI) consider a 2l 2l 1 1 new 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 1 2h 2l i=L,H 2h 2h 2l 2l 1 1 − wL > [q+(1 q)(1 p)]wH, thenchoosewH(cid:48)(cid:48) tosolveu−1(wH(cid:48)(cid:48)) = u−1(wH)+(cid:15); ifinstead − − 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 − − 1−q cases, 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 (26)-(31) in 1 2h 2l i=L,H problem (P ˆFI), 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 (P ˆFI). 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)) − − 1 2h 2l i=L,H satis(cid:28)es constraints (26)-(31) in problem (P ˆFI), and qwH(cid:48)(cid:48) +(1 q)wL(cid:48)(cid:48) > qwH +(1 − − 80

q)wL, which contradicts again optimality of the original contract in problem (P ˆFI). If instead (Ci,Ci ,Ci ,wi) solve problem (P ˆFI), but for Ω∗ the solution to 1 2h 2l i=L,H (P ˆ bFI), (w ,w ) does not solve (P ˆ a FI ), let (wH(cid:48),wL(cid:48)) solve (P ˆ a FI ). It is straight- H L Ω∗ Ω∗ forward to show that (Ci,Ci ,Ci ,wi(cid:48)) satis(cid:28)es constraints (26)-(31) in problem 1 2h 2l i=L,H (P ˆFI), and qwH(cid:48) +(1 q)wL(cid:48) > qwH +(1 q)wL, which contradicts optimality of the − − original contract in problem (P ˆFI). Next, we show the if direction. Let (Ci,Ci ,Ci ) solve (P ˆ bFI), and for Ω∗ the 1 2h 2l i=L,H solutionto(P ˆ bFI),(wH,wL)solve(P ˆ a FI ). Supposebycontradictionthat(Ci,Ci ,Ci ,wi) Ω∗ 1 2h 2l i=L,H does not solve problem (P ˆFI). Let (Ci(cid:48),Ci(cid:48) ,Ci(cid:48),wi(cid:48)) be the solution to (P ˆFI). Then 1 2h 2l easily it must be that either Ci(cid:48) = Ci, or Ci(cid:48) = Ci , or Ci(cid:48) = Ci : if not it must be 1 (cid:54) 1 2h (cid:54) 2h 2l (cid:54) 2l wH = wH(cid:48) and wL(cid:48) = wL by comparing (P ˆ a FI ) with (P ˆFI). By de(cid:28)nition of problem Ω∗ (P ˆ bFI), then it should be that either Ci(cid:48) > Ci, or Ci(cid:48) > Ci , or Ci(cid:48) > Ci . Sup- 1 1 2h 2h 2l 2l pose ,w.l.o.g. that CH(cid:48) > CH. Then, following the same argument as in the only if 1 1 part, we can prove that (Ci(cid:48),Ci(cid:48) ,Ci(cid:48),wi(cid:48)) can not be the solution to (P ˆFI), which is a 1 2h 2l contradiction. Claim 5: Let Ω . For any (w ,w ) 2 such that ∈ (cid:60) + H L ∈ (cid:60)+ (cid:0) (cid:1) (cid:0) (cid:1) qu−1 wH +(1 q)u−1 wL Ω (53) − ≤ (cid:0) (cid:1) [q +(1 q)(1 p)]wH = max wL,(q +(1 q)(1 p))wH (54) − − − − γ +qwH +(1 q)wL (q +(1 q)(1 p))wH (55) − − ≥ − − there exist (wH(cid:48),wL(cid:48)) 2 such that ∈ (cid:60)+ (cid:16) (cid:17) (cid:16) (cid:17) qu−1 wH(cid:48) +(1 q)u−1 wL(cid:48) Ω (56) − ≤ 81

(cid:16) (cid:17) wL(cid:48) = max wL(cid:48),(q +(1 q)(1 p))wH(cid:48) (57) − − γ +qwH(cid:48) +(1 q)wL(cid:48) wL(cid:48) (58) − − ≥ and qwH(cid:48) +(1 q)wL(cid:48) > qwH +(1 q)wL (59) − − Proof. Let (wH,wL) 2 satisfy equations (53), (54), and (55). De(cid:28)ne X as ∈ (cid:60)+ (cid:0) (cid:1) (cid:0) (cid:1) qu−1 wH +(1 q)u−1 wL = 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 (56), (57), (58), and (59). Notice that equation (56) and equation (57) are satis(cid:28)ed by construction. Now, suppose by contradiction that equation (58) is violated. Therefore γ wH(cid:48) < wL(cid:48) + q wL(cid:48) = [q +(1 q)(1 p)]wH(cid:48) − − 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) q+(1−q)(1−p) pq(1−q) 82

we 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) − − (60) It is easy to show that equations (54) and (55) can hold only if wH γ and ≥ pq(1−q) wL 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 (60). Therefore equation (58) 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:16) (cid:17) (cid:90) wL(cid:48) u(cid:48) X−(1−q)u−1(s) q = qwH +(1 q)wL +(1 q) 1 ds − − − u(cid:48)(s) wL > qwH +(1 q)wL − where the last inequality follows from concavity of u together with the fact that X−(1−q)u−1(s) > s for all s [wL,wL(cid:48)]. Therefore equation (59) is as well satis(cid:28)ed. q ∈ Proof of Lemma 15 Proof. Since from Claim 4 (Ci, Ci , Ci ) need to solve (P ˆ bFI), the conclusion 1 2h 2l i=L,H follows easily from linearity of the objective function in (P ˆ bFI) and the fact taht α > 1. 83

Sepci(cid:28)cally, from Assumption 4 the participation constraint should bind: αCi+pCi + 1 2,h (1 p)Ci = αω. From linearity of the objective function and α > 1 we obtain Ci = 0 − 2,l 2,l and the fact that the limited commitment constraint binds, Ci = (1 λi)θ. 2,h − Proof of Proposition 16 Proof. From Claim 5, we can simplify and rewrite problem (P ˆ aFI) as follows: Ω (P ˆ a FI )(cid:48) max qwH +(1 q)wL γ Ω {wH,wL}∈(cid:60)2 − − + (cid:0) (cid:1) (cid:0) (cid:1) s.t. qu−1 wH +(1 q)u−1 wL Ω (61) − ≤ γ +qwH +(1 q)wL wL (62) − − ≥ wL [q +(1 q)(1 p)]wH 0 (63) − − − ≥ Step 1: A solution to problem (P ˆ a FI )(cid:48) exists (and is unique) if and only if Ω Ω ˆ , Ω ≥ ˆ for Ω which solves (cid:18) (cid:19) (cid:18) (cid:19) γ γ[q +(1 q)(1 p)] Ω ˆ = qu−1 +(1 q)u−1 − − (64) pq(1 q) − pq(1 q) − − Moreover, at the solution, equations (61) and (62) hold with equality. Proof. The smallest values of wH and wL that jointly satisfy (62) and (63) are wH = γ and wL = γ[q+(1−q)(1−p)] . Then constraint (61) can be satis(cid:28)ed jointly with (62) pq(1−q) pq(1−q) ˆ and (63) only if Ω Ω as de(cid:28)ned above. ≥ ˆ Easily, when Ω Ω both (62) and (63) have to bind. If (62) does not bind, we can ≥ increase wH and wL by (cid:15) and all constraints are still satis(cid:28)ed. If (63) is not binding, we can construct a mean-preserving contraction on u−1(wH) and u−1(wL) so that (62) is una(cid:27)ected, but by convexity of u−1( ) the objective function strictly increases. (cid:3) · 84

ˆ FI Step 2: The solution to (Pb ) gives (cid:88) (cid:104) (cid:105) Ω = q λipθ+ω(λi) (65) i i∈{L,H} Proof. It follows easily from Lemma 15. (cid:3) Step 3: A solution to problem (P ˆFI) exists and is unique if and only if γ γˆ, for γˆ ≤ de(cid:28)ned in (34). Then, for Ω de(cid:28)ned in (65), qwH +(1 q)wL γ = wL, − − for wL solving (cid:18) (cid:19) γ qu−1 wL + +(1 q)u−1(wL) = Ω. q − Proof. The conclusion follows from Claim 4 and Claim 5 above, Lemma 15 and Step (cid:3) 1 and Step 2 in the proof of the current proposition. Step 4: If γ γˆ de(cid:28)ned in (34), then ≤   (cid:40) (cid:41) (cid:40) (cid:41)  (cid:88)  (cid:88) (cid:104) (cid:88) (cid:105) max u(X ), q wi,0∗ q p u(xi∗ ) 2 i ≥ i s 2,s   i∈{L,H} i∈{L,H} s=l,h where xi∗ is lenders’ consumption in the optimal contract with bilateral clearing and 2,s monitoring of Lemma 7, and X is lenders consumption in (13) for the optimal contract 2 with CCP clearing and no monitoring in Proposition 5. Proof. Suppose not: suppose that the optimal contract with bilateral clearing and screening dominates both the optimal contract with central clearing and screening and 85

the optimal contract with CCP clearing and no information acquisition:   (cid:40) (cid:41) (cid:40) (cid:41)  (cid:88)  (cid:88) (cid:104) (cid:88) (cid:105) max u(X∗), q wi,0∗ γ < q p u(xi∗ ) γ (66) 2 i − i s 2,s −   i∈{L,H} i∈{L,H} s=l,h Let (xi∗, xi∗, ci∗, ci∗, ci∗) be the optimal contracts with bilateral clearing, when the 2h 2l 1 2h 2l lender upon screening learns that her counterparty is of type i. Similarly, let (wi,0∗ , Ci∗ , Ci∗ , Ci∗) be the optimal contract with CCP clearing and screening. De(cid:28)ne wH 2h 2l 1 and wL as follows: wH = pu(xH∗)+(1 p)u(xH∗) 2h − 2l wL = pu(xL∗)+(1 p)u(xL∗) 2h − 2l and consider, in problem (P ˆFI), the contract with CCP clearing and screening (wi, Ci , Ci , Ci), where wH and wL are de(cid:28)ned above, Ci = ci∗, Ci∗ = ci∗, Ci∗ = ci∗. 2h 2l 1 1 1 2h 2h 2l 2l Step a: The contract (wi, Ci , Ci , Ci) satis(cid:28)es qwH +(1 q)wL γ wL. 2h 2l 1 − − ≥ Consider, in problem (8)-(12), the contract (X , Ci, Ci , Ci ) with Ci = cL∗, 2 1 2h 2l 1 1 Ci = cL∗, and X = u−1(pu(xL∗) + (1 p)u(xL∗)). Such contract is feasible in (8)- 2,s 2,s 2 2h − 2l (12), therefore it must be u(X∗) u(X ) = pu(xL∗)+(1 p)u(xL∗) (67) 2 ≥ 2 2h − 2l Moreover, since the contract with bilateral clearing and screening dominates the con- 86

tract with CCP clearing and no screening, (cid:104) (cid:105) (cid:104) (cid:105) q pu(xH∗)+(1 p)u(xH∗) +(1 q) pu(xL∗)+(1 p)u(xL∗) γ u(X∗) 2h − 2l − 2h − 2l − ≥ 2 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) wH wL Combining the last expression with (66) we obtain qwH +(1 q)wL γ u(X∗) pu(xL∗)+(1 p)u(xL∗) = wL − − ≥ 2 ≥ 2h − 2l γ wH wL + ⇒ ≥ q Step b: The contract (wi, Ci , Ci , Ci) must be that [q +(1 q)(1 p)]wH > wL. 2h 2l 1 − − From concavity of u( ) we have 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 Then the contract (wi, Ci , Ci , Ci) easily satis(cid:28)es (26)-(29). Suppose that we also 2h 2l 1 had [q + (1 q)(1 p)]wH wL. Then from step a, constraint (31) would also be − − ≤ satis(cid:28)ed in (P ˆFI). By de(cid:28)nition of optimality, it must be that (cid:40) (cid:41) (cid:88) q wi,0∗ = qwH,0∗ +(1 q)wL,0∗ qwH +(1 q)wL i − ≥ − i∈{L,H} (cid:104) (cid:105) (cid:104) (cid:105) = q pu(xH∗)+(1 p)u(xH∗) +(1 q) pu(xL∗)+(1 p)u(xL∗) 2h − 2l − 2h − 2l which contradicts equation (66). 87

Step c: The contract (wi, Ci , Ci , Ci) must be such that γ +qwH +(1 q)wL < 2h 2l 1 − − [q +(1 q)(1 p)]wH. − − If not, constraint (31) would be satis(cid:28)ed in (P ˆFI) and the CCP can always (cid:28)nd a pair (wH,wL) that violates [q + (1 q)(1 p)]wH > wL, satis(cid:28)es the incentive con- − − straint γ +qwH +(1 q)wL wL, and yields strictly higher utility to lenders than − − ≥ the optimal bilateral contract. Step d: Condition (66) must be violated. Consider then the solution to problem (P ˆ bFI): we know from Claim 4 that (Ci∗ , 1 Ci∗ , Ci∗) solve problem (P ˆ bFI). Moreover, by de(cid:28)nition of the maximization problem, 2h 2l 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) − − and de(cid:28)ne wH(cid:48) such that δ u−1(wH(cid:48)) = u−1(wH)+ (68) q 88

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 − − Notice that T(y) is monotone decreasing in y, that for y = y [q + (1 p)(1 ≡ − − q)]u (cid:16) qu−1(wH(cid:48) )+(1−q)u−1(wL) (cid:17) > 0, it is q T(y) = u(0) y < 0 − Furthermore, the two conditions wH(cid:48) wL + γ and wH(cid:48) wL γ , imply ≥ q ≥ 1−p − (1−q)(1−p) that wH(cid:48) wL . where the second inequality follows from wH(cid:48) wH > wL ≥ q+(1−p)(1−q) ≥ 1−p − γ , which results from the assumption that the incentive constraint is violated, (1−q)(1−p) 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 1 2h 2l de(cid:28)ned above, and Ci∗ , Ci∗ , Ci∗ solve problem (P ˆ bFI). Notice that this contract is 1 2h 2l 89

feasibleandsatisfythelimitedcommitmentconstraintinproblem(P ˆFI): participation, limited commitment and feasibility constraints are easily satis(cid:28)ed by the de(cid:28)nition of Ci∗ , Ci∗, Ci∗. Moreover, by construction [1+(1 q)(1 p)]wH(cid:48)(cid:48) = wL(cid:48)(cid:48) . All is left to 1 2h 2l − − 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∗ Ω ˆ − − ≥ Replacing wH(cid:48)(cid:48) , wL(cid:48)(cid:48) and Ω ˆ with their de(cid:28)nitions we can rewrite (cid:18) wL(cid:48)(cid:48) (cid:19) (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 1 2h 2l compatibility constraint. Then, by the de(cid:28)nition of optimality, it must be (cid:40) (cid:41) (cid:88) q wi,0∗ qwH(cid:48)(cid:48) +(1 q)wL(cid:48)(cid:48) i ≥ − i∈{L,H} (cid:18) qu−1(wH(cid:48))+(1 q)u−1(wL) qu−1(wH(cid:48)(cid:48)) (cid:19) = qwH(cid:48)(cid:48) +(1 q)u − − − 1 q − (cid:18) qu−1(wH)+δ +(1 q)u−1(wL) qu−1(wH(cid:48)(cid:48)) (cid:19) = qwH(cid:48)(cid:48) +(1 q)u − − − 1 q − 90

(cid:18) Ω qu−1(wH(cid:48)(cid:48)) (cid:19) = 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) − −  (cid:16) (cid:17)  (cid:90) wH(cid:48) u(cid:48) Ω−qu−1(s) = qwH(cid:48) +(1 q)wL +q  1−q 1ds − wH(cid:48)(cid:48) u(cid:48)(s) − (cid:40) (cid:41) (cid:88) (cid:104) (cid:88) (cid:105) qwH(cid:48) +(1 q)wL qwH +(1 q)wL = q p u(xi∗ ) ≥ − ≥ − i s 2,s i∈{L,H} s=l,h where the (cid:28)rst inequality in the last line follows from the fact that Ω−qu−1(s) < 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, ∈ ≥ (cid:3) given the de(cid:28)nition in (68). But this contradicts (66). Proof of Proposition 17 Proof. The conclusion follows from Proposition 9 and Proposition 16. Speci(cid:28)cally, when γ > γˆ, from Proposition 16, central clearing with information acquisition is not feasible. When γ < γ, then from Proposition 9, bilateral clearing with information acquisition is preferred to central clearing with no information acquisition. Proof of Lemma 19 Proof. To show that bilateral clearing is more desirable, we show that dγˆ < 0 and dα dγ > 0. Thus, if γ (γˆ,γ), then γ (γˆ(cid:48),γ(cid:48)), where γˆ(cid:48) and γ(cid:48) are the thresholds dα ∈ ∈ computed for α(cid:48) = α+(cid:15), for (cid:15) small. The conclusion dγˆ < 0 comes directly from the de(cid:28)ntion of γˆ in (34).On the other dα (cid:18) (cid:19) u(cid:48) (1−λH)pθ hand, since α−p > α , from Lemma 7 we have that xH = (1−λH)pθ +λHθ 1−p u(cid:48)(θ−α−p(1−λH)θ) 2,h α α 91

and xH = (1−λH)pθ . Thus, 2,l α dγ (1 λH)pθ (cid:20) (cid:18) (1 λH)pθ (cid:19) (cid:18) (1 λH)pθ (cid:19)(cid:21) = q − pu(cid:48) − +λHθ +(1 p)u(cid:48) − dα − α2 α − α (1 λL)pθ (cid:20) (cid:18) (1 λL)pθ (cid:19) (cid:18) (1 λL)pθ (cid:19)(cid:21) (1 q) − pu(cid:48) − +λLθ +(1 p)u(cid:48) − − − α2 α − α (cid:18) (1 λL)pθ (cid:19) (1 λL)pθ +u(cid:48) − +λLpθ − α α2 (1 λH)pθ (cid:18) (1 λH)pθ (cid:19) (1 λL)pθ (cid:18) (1 λL)pθ (cid:19) q − u(cid:48) − +λHpθ (1 q) − u(cid:48) − +λLpθ ≥ − α2 α − − α2 α (cid:18) (1 λL)pθ (cid:19) (1 λL)pθ +u(cid:48) − +λLpθ − α α2 (cid:16) (cid:17) u(cid:48) (1−λL)pθ +λLpθ pθ α > [ q(1 λH) (1 q)(1 λL)+(1 λL)] > 0 α2 − − − − − − where the (cid:28)rst inequality follows from u(cid:48)(cid:48)(cid:48)( ) 0, the second one from u(cid:48)(cid:48)( ) < 0, and · ≤ · the last one from λH > λL. Proof of Lemma 20 (cid:18) (cid:19) u(cid:48) (1−λH)pθ Since α−p > α , from Lemma 7 we have that xH = (1−λH)pθ +λHθ and 1−p u(cid:48)(θ−α−p(1−λH)θ) 2,h α α xH = (1−λH)pθ . Also, since q > 1, from equation (15) we have that X = (1−λH)pθ + 2,l α α 2 α pθ[qλH +(1 q)λL]. Then, − dγ (1 λH)pθ (cid:20) (cid:18) (1 λH)pθ (cid:19) (cid:18) (1 λH)pθ (cid:19)(cid:21) = q − pu(cid:48) − +λHθ +(1 p)u(cid:48) − dα − α2 α − α (1 λL)pθ (cid:20) (cid:18) (1 λL)pθ (cid:19) (cid:18) (1 λL)pθ (cid:19)(cid:21) (1 q) − pu(cid:48) − +λLθ +(1 p)u(cid:48) − − − α2 α − α (cid:18) (1 λH)pθ (cid:19) (1 λH)pθ +u(cid:48) − +pθ[qλH +(1 q)λL] − α − α2 (1 λH)pθ (cid:20) (cid:18) (1 λH)pθ (cid:19) (cid:18) (1 λH)pθ (cid:19)(cid:21) < q − pu(cid:48) − +λHθ +(1 p)u(cid:48) − − α2 α − α (1 λL)pθ (cid:20) (cid:18) (1 λL)pθ (cid:19) (cid:18) (1 λL)pθ (cid:19)(cid:21) (1 q) − pu(cid:48) − +λLθ +(1 p)u(cid:48) − − − α2 α − α 92

(cid:18) (1 λL)pθ (cid:19) (1 λH)pθ +u(cid:48) − +pθλL − α α2 (1 λH)pθ (cid:20) (cid:18) (1 λL)pθ (cid:19)(cid:21) < (1 q) − u(cid:48) − +pθλL − α2 α (1 λL)pθ (cid:20) (cid:18) (1 λL)pθ (cid:19) (cid:18) (1 λL)pθ (cid:19)(cid:21) (1 q) − pu(cid:48) − +λLθ +(1 p)u(cid:48) − − − α2 α − α 0 ≤ (cid:16) (cid:17) wherethe(cid:28)rstinequalitycomesfromu(cid:48)(cid:48)(x) < 0andq > 1,thusu(cid:48) (1−λH)pθ +pθ[qλH +(1 q)λL] α α − is maximized when q = 1, the second inequality comes from the assumption α (cid:18) (1 λH)pθ (cid:19) (cid:18) (1 λH)pθ (cid:19) (cid:18) (1 λL)pθ (cid:19) pu(cid:48) − +λHθ +(1 p)u(cid:48) − > u(cid:48) − +λLpθ , α − α α and the third inequality from prudence, i.e. u(cid:48)(cid:48)(cid:48) 0, and from λH > λL. ≥ 6.1 Bilateral clearing with information acquisition Consider the problem (8)-(12) when we do not impose the Assumption 4 and Assumption 6. Assumption 4 guaranteed that ω > (1−λL)pθ > (1−λH)pθ . Consider then (cid:28)rst α α a generic solution for the case when ω < (1−λi)pθ . The next lemma characterizes the α solution to this problem. Lemma 21 If ω < (1−λi)pθ , the participation constraint (17) is slack. In addition, the α limited commitment constraint (21) is binding and ci = 0. This is the area shaded in 1 yellow in Figure 1. Proof. It is easy to see that both the participation constraint (17) and the limited commitment constraint (21) can not be slack: if this was the case, the lender could increase her revenues just by decreasing ci . 2,h 93

Suppose then that ω < (1−λi)pθ . Because ci (1 λi)θ and ci 0, the partici- α 2,h ≥ − 1 ≥ pation constraint (17) is slack. Since both (17) and (21) can not be slack, it must be that (21) binds: ci = (1 λi)θ. Easily, ci = 0: if not, the lender could decrease ci, 2,h − 1 1 satisfy all constraints, and increase her expected utility. Next, consider the case when ω > (1−λi)pθ . Suppose we do not impose Assumption 6 α and let µ and η be the multipliers associated with (17) and (21) respectively. The (cid:28)rst order conditions for optimality are pu(cid:48)(ω ci +θ ci )+pµ+η = 0 (69) − − 1 − 2,h pu(cid:48)(ω ci +θ ci ) (1 p)u(cid:48)(ω ci)+αµ 0 (70) − − 1 − 2,h − − − 1 ≤ with equality if ci > 0. Together with the complementary slackness conditions 1 µ αci +pci αω = 0 (71) { 1 2,h − } and η ci (1 λi)θ = 0 (72) { 2,h − − } they fully characterize the solution to the problem. Lemma 22 If ω > (1−λi)pθ , then the participation constraint (17) binds. Moreover, if α we do not impose Assumption 6 a) If λi < λ∗, then ci = (1 λi)θ and ci = ω (1−λi)pθ . This is area a) in Figure 1. 2,h − 1 − α b) If λi > λ∗, and ω < (1−λ∗)pθ , then ci = αω > (1 λi)θ and ci = 0. This is area b) α 2,h p − 1 in Figure 1. 94

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 − − This is area 1) in Figure 1. Proof. We know from Lemma 7 that when ω > (1−λi)pθ , the participation constraint α (17) always binds. From the same lemma we also know that if λi < λ∗, the limited liability constraint (21) binds. This proves case (a). Also from Lemma 7, we know from that if λi > λ∗, the limited commitment constraint (21) is slack. Thus, when ω > (1−λi)pθ and λi > λ∗, the only thing left to determine is whether ci > 0 or ci = 0. α 1 1 The consumption of the lender is (1 λi)pθ xi = λiθ+ − 2,h α (1 λi)pθ xi = − 2,l α Since (21) is slack, therefore η = 0, and (17) binds, therefore ci = α(ω−ci 1 ) , condition 2,h p (69) gives (cid:18) (cid:19) α p µ = u(cid:48) θ (ω ci) − − − 1 p replaced in (70) gives (cid:18) (cid:19) α p (α p)u(cid:48) θ (ω ci) − (1 p)u(cid:48)(ω ci) 0 − − − 1 p − − − 1 ≤ with equality if ci > 0. Then, by the de(cid:28)nition of λ∗ in (22), it is clear that ci > 0 if 1 1 and only if ω > (1−λ∗)pθ , and ci = 0 if ω < (1−λ∗)pθ . This characterizes cases (b) and α 1 α (c), and concludes the proof. 95

!() (a) (1) (PC)binds (PC)binds (LC)slack (LC c ) 1 b > in 0 ds c1>0 !( ) ⇤ (c) (b) (PC)slack (PC)bind (LC)binds (LC)slack c1=0 c1=0 ⇤ Figure 1: Solution to bilateral problem with info acquisition. References Acharya, V. and A. Bisin(2014): (cid:16)Counterpartyriskexternality: Centralizedversus over-the-counter markets,(cid:17) Journal of Economic Theory, 149, 153(cid:21)182. Afonso, G. and R. Lagos (2012a): (cid:16)An empirical study of trade dynamics in the fed funds market,(cid:17) FRB of New York Sta(cid:27) Report. (cid:22)(cid:22)(cid:22) (2012b): (cid:16)Trade dynamics in the market for federal funds,(cid:17) FRB of New York Sta(cid:27) Report. Antinolfi, G., F. Carapella, and F. Carli (2018): (cid:16)Transparency and Collateral: Central Versus Bilateral Clearing,(cid:17) FEDS Working Paper No. 2018-017. Available at SSRN: https://ssrn.com/abstract=3137995 or http://dx.doi.org/10. 17016/FEDS.2018.017. 1 Antinolfi, G., F. Carapella, C. Kahn, A. Martin, D. C. Mills, and 96

E. Nosal (2015): (cid:16)Repos, (cid:28)re sales, and bankruptcy policy,(cid:17) Review of Economic Dynamics, 18, 21(cid:21)31. Bech, M. L. and E. Atalay (2010): (cid:16)The topology of the federal funds market,(cid:17) Physica A: Statistical Mechanics and its Applications, 389, 5223(cid:21)5246. Biais, B., F. Heider, and M. Hoerova (2016): (cid:16)Risk-Sharing or Risk-Taking? Counterparty Risk, Incentives, and Margins,(cid:17) The Journal of Finance. Bignon, V. and G. Vuillemey (2016): (cid:16)The failure of a clearinghouse: Empirical evidence,(cid:17) . BIS glossary of terms used in payments and settlement systems (2003): (cid:16)https://www.bis.org/cpmi/glossary_030301.pdf,(cid:17) . BIS Quarterly Review (2017): (cid:16)https://www.bis.org/publ/qtrpdf/r_ qt1712w.htm,(cid:17) . Caballero, R. J. and A. Simsek (2009): (cid:16)Complexity and (cid:28)nancial panics,(cid:17) Tech. rep., National Bureau of Economic Research. Capital requirements for bank exposures to central counterparties (2012): (cid:16)https://www.bis.org/publ/bcbs227.pdf,(cid:17) . Carapella, F. and D. Mills (2011): (cid:16)Information insensitive securities: the true bene(cid:28)ts of central counterparties,(cid:17) Tech. rep., mimeo, Board of Governors. Central Counterparty Default Management and the Collapse of Lehman Brothers", CCP12, The Global Association of Central Counterparties (2009): (cid:16)https://ccp12.org/wp-content/uploads/2017/05/ CCPDefaultManagementandtheCollapseofLehmanBrothers.pdf,(cid:17) . 97

Duffie, D. and H. Zhu (2011): (cid:16)Does a central clearing counterparty reduce counterparty risk?(cid:17) Review of Asset Pricing Studies, 1, 74(cid:21)95. Duffie, J. D., A. Li, and T. Lubke(2010): (cid:16)FederalReserveBankofNewYorkSta(cid:27) Reports Policy Perspectives on OTC Derivatives Market Infrastructure Sta(cid:27) Report No. 424,(cid:17) Rock Center for Corporate Governance at Stanford University Working Paper. Elliott, D. (2013): (cid:16)Central Counterparty Loss-Allocation Rules,(cid:17) Bank of England Financial Stability Paper No. 20. Faruqui, U., W. Huang, , and E. TakÆts (2018): (cid:16)Clearing risks in OTC derivatives markets: the CCP-bank nexus,(cid:17) BIS Quarterly Review. Garbade, K. D. (2006): (cid:16)The Evolution of Repo Contracting Conventions in the 1980s,(cid:17) New Yord Fed Economic Policy Review, 12. Garratt, R. and P. Zimmerman (2015): (cid:16)Does Central Clearing Reduce Counterparty Risk in Realistic Financial Networks?(cid:17) Federal Reserve Bank of New York, Sta(cid:27) Report. Gregory, J. (2014): Central Counterparties: Mandatory Central Clearing and Initial Margin Requirements for OTC Derivatives, John Wiley & Sons. Hauswald, R. and R. Marquez (2006): (cid:16)Competition and strategic information acquisition in credit markets,(cid:17) Review of Financial Studies, 19, 967(cid:21)1000. Ice Clear Credit Clearing Rules (2018): (cid:16)https://www.theice.com/ publicdocs/clear_credit/ICE_Clear_Credit_Rules.pdf,(cid:17) . ICE Clear Disclosure Framework (2018): (cid:16)https://www.theice.com/ publicdocs/clear_us/ICUS_DisclosureFramework.pdf,(cid:17) . 98

Jackson, J. K. and R. S. Miller(2013): (cid:16)ComparingG-20ReformoftheOver-the- Counter Derivatives Markets,(cid:17) Congressional Research Service, Report for Congress. Koeppl, T., C. Monnet, and T. Temzelides (2012): (cid:16)Optimal clearing arrangements for (cid:28)nancial trades,(cid:17) Journal of Financial Economics, 103, 189(cid:21)203. Koeppl, T. V. (2013): (cid:16)The limits of central counterparty clearing: Collusive moral hazard and market liquidity,(cid:17) Tech. rep., Queen’s Economics Department Working Paper. Koeppl, T. V. and C. Monnet (2010): (cid:16)The emergence and future of central counterparties,(cid:17) Tech. rep., Queen’s Economics Department Working Paper. Krishnamurthy, A., S. Nagel, and D. Orlov (2014): (cid:16)Sizing Up Repo,(cid:17) forthcoming in The Journal of Finance. Kroszner, R. S. (2006): (cid:16)Central counterparty clearing: History, innovation, and regulation,(cid:17) . Loon, Y. C. and Z. K. Zhong (2014): (cid:16)The impact of central clearing on counterparty risk, liquidity, and trading: Evidence from the credit default swap market,(cid:17) Journal of Financial Economics, 112, 91(cid:21)115. Mian, A. (2003): (cid:16)Foreign, private domestic, and government banks: New evidence from emerging markets,(cid:17) Journal of Banking and Finance, 27, 1219(cid:21)1410. Monnet, C. and T. Nellen (2012): (cid:16)Clearing with two-sided limited commitment,(cid:17) Tech. rep., mimeo, University of Berne. National Securities Clearing Corporation - Rules and Procedures (2018): (cid:16)http://www.dtcc.com/~/media/Files/Downloads/legal/rules/nscc_ rules.pdf,(cid:17) . 99

Petersen, M. A. (2004): (cid:16)Information: Hard and soft,(cid:17) Northwestern University Working Paper. Pirrong, C. (2009): (cid:16)The economics of clearing in derivatives markets: Netting, asymmetric information, and the sharing of default risks through a central counterparty,(cid:17) Tech. rep., Working paper. Powell, J. H. (November, 21st 2013): (cid:16)Speech at the Clearing House 2013 Annual Meeting, http://www.federalreserve.gov/newsevents/speech/ powell20131121a.htm,(cid:17) Tech. rep., The Waldorf Astoria, New York City. speech by Sir Jon Cunliffe, Deputy Governor for Financial Stability of the Bank of England, at the FIA International Derivatives Expo 2018, London (5 June 2018): (cid:16)https://www.bankofengland.co.uk/-/media/boe/files/speech/2018/ central-clearing-and-resolution-learning-some-of-the-lessons-of-lehmans-speech-by-jon-cunliffe,(cid:17) . Stein, J. C. (2002): (cid:16)Information production and capital allocation: Decentralized versus hierarchical (cid:28)rms,(cid:17) The Journal of Finance, 57, 1891(cid:21)1921. UBS Response to the Committee on Payment and Settlement Systems (2012): (cid:16)https://www.bis.org/cpmi/publ/comments/d103/ubs.pdf,(cid:17) . https://www.chicagofed.org/publications/chicago-fed-letter/2017/389 (2017): (cid:16)Central counterparty risk management: Beyond default risk,(cid:17) Tech. rep., Chicago Fed Letter, Number 389. http://www.newyorkfed.org/banking/tpr_infr_reform_data.html (2014): (cid:16)Triparty repo statistical data,(cid:17) Tech. rep., Federal Reserve Bank of New York. 100

http://www.newyorkfed.org/markets/gsds/search.html (2014): (cid:16)Primary Dealer statistics,(cid:17) Tech. rep., Federal Reserve Bank of New York. Zawadowski, A. (2011): (cid:16)Interwoven lending, uncertainty, and liquidity hoarding,(cid:17) Boston U. School of Management Research Paper. (cid:22)(cid:22)(cid:22) (2013): (cid:16)Entangled (cid:28)nancial systems,(cid:17) Review of Financial Studies, 26, 1291(cid:21) 1323. 101