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\nCentral Counterparty Default Waterfalls and
\nSystemic Loss
\nMark Paddrik
\nOffice of Financial Research
\nmark.paddrik@ofr.treasury.gov
\nSimpson Zhang
\nOffice of the Comptroller of the Currency
\nsimpson.zhang@occ.treas.gov
\n20-04 | June 18, 2020
\nCentral Counterparty Default
\nWaterfalls and Systemic Loss \u2217
\nMark Paddrik\u2020
\nSimpson Zhang\u2021
\nJune 2020
\nAbstract
\nCentral counterparty default waterfalls act as last lines of defense in over-the-counter markets
\nby managing and allocating resources to cover defaults of clearing members and clients. However,
\ncentral counterparties face competing objectives in setting up their default waterfalls. In this
\npaper we evaluate the trade-offs between default waterfall resiliency and central clearing, using
\na unique and comprehensive dataset containing all U.S. cleared and bilateral credit default swap
\npositions. We evaluate the resiliency of different default waterfall designs, accounting for the
\ninterconnectedness of payments in the system, the presence of client clearing obligations for
\nmembers, and the distribution of losses among market participants.
\nKeywords: central counterparty, systemic risk, default waterfall, financial networks, credit
\ndefault swaps
\nJEL Classification Numbers: G10, G23, G28, L14
\n\u2217We thank Celso Brunetti, Roy Cheruvelil, Ben Craig, St\u00b4ephane Cr\u00b4epey, Peter Curley, Selman Erol, Katherine Gleason, Stephen Kane, David Li, Sriram Rajan, Stacey Schreft, Stathis Tompaidis, Robert Wasserman, Jessie
\nJiaxu Wang, H. Peyton Young, John Zitko and participants of the Fifth Network Science and Economics Conference, Canadian Economic Association Annual Meeting, SIAM Conference on Financial Mathematics & Engineering:
\nCCP Symposium, Western Finance Association Conference, Society for Economic Dynamics Meeting, 19th Annual
\nFDIC\/JFSR Bank Research Conference, and 2019 OFR\/FRBC Financial Stability Conference for their valuable comments. We would like to thank Elizabeth McKee for her excellent Helpance in aggregating CCP filings and analyzing
\nCCP manuals. Additionally, we would like to thank OFR\u2019s High Performance Computing, Data, and Legal teams
\nfor collecting and organizing the data necessary to make this project ( help with nursing paper writing from experts with MSN & DNP degrees) possible. The views expressed in this paper do
\nnot necessarily reflect the views of the Office of the Comptroller of the Currency, the Office of Financial Research,
\nthe U.S. Department of the Treasury, or any federal agency and do not establish supervisory policy, requirements, or
\nexpectations.
\n\u2020Office of Financial Research, U.S. Department of the Treasury, 714 14th St NW, Washington, DC 20220; phone:
\n202-927-8511; email: Mark.Paddrik@ofr.treasury.gov.
\n\u2021Office of the Comptroller of the Currency, 400 7th St SW, Washington, DC 20219; phone: 202-649-6288; email:
\nSimpson.Zhang@occ.treas.gov.
\n1
\nCentral counterparty (CCP) clearing in over-the-counter (OTC) financial markets has grown
\nsubstantially since the 2007-09 financial crisis, from nearly nonexistent in 2007 to more than 70
\npercent of new interest rate derivatives and index credit default swaps volume in the United States
\nin 2019 (Financial Stability Oversight Council (2019)). Financial regulators have encouraged this
\ngrowth in cleared products as a way to reduce the financial stability risks posed by large counterparty failures. CCPs help to ensure the continuity of payments within these markets and reduce the
\npotential losses that taxpayers suffer (Financial Stability Board (2017)). They do so by performing
\nrisk management and maintaining default waterfalls, financial resources that cover losses generated
\nby counterparty default.
\nAlthough default waterfalls are critical to a CCP\u2019s risk management, there is little consensus
\non the optimal structure of default waterfalls globally. After the introduction of central clearing to
\nnumerous previously non-cleared markets, many new CCPs were created with a tremendous degree
\nof variation in how they source default waterfall resources. Such variations reflect the conflicting
\nobjectives that the default waterfall must serve. As a CCP\u2019s default waterfall is its last line of
\ndefense in times of stress, it is important that the waterfall be resilient against market shocks.
\nBut it is also necessary for the waterfall to account for the incentives of participants, as requiring
\nlarge contributions can be costly for clearing members and discourages clearing through the CCP
\n(Ghamami and Glasserman (2017)). Lower rates of central clearing could in turn decrease financial
\nsystem resilience.
\nAssessing the systemic risk implications of CCP default waterfall designs is difficult for financial
\nregulators, CCPs, and market participants alike due to the historical rarity of CCP member defaults
\nand the complex interrelationship of payments within both cleared and non-cleared portions of
\nderivatives markets (Duffie (2015); Cont (2015)).1 Spillover effects can arise through both the
\ncleared and non-cleared network of exposures and from fire sales of illiquid collateral, causing
\ngreater losses in a crisis. As firms see a limited view of the overall market, they face difficulties
\nin accurately evaluating the extent of these spillover effects and determining the cost-benefit of
\na particular default waterfall structure (Cox and Steigerwald (2017)). Although groups such as
\nthe International Swaps and Derivatives Association (ISDA) and standard-setting bodies such as
\n1Kroszner (1999), Cox (2015), and Bignon and Vuillemey (2020) examine in depth a few historical examples of
\nlarge clearing member default at derivatives CCPs.
\n2
\nthe Committee on Payments and Market Infrastructures (CPMI) and International Organization
\nof Securities Commissions (IOSCO) have written reports that qualitatively discuss the merits of
\ndifferent default waterfall designs (Elliott et al. (2014); ISDA (2013); CPMI-IOSCO (2014)), there
\nhas been limited theoretical modeling of these mechanisms or empirical testing using market data.
\nIn this paper we evaluate the merits of existing CCP default waterfall designs through a structural modelling approach that can account for these complexities. Unlike the theoretical works
\nof Biais et al. (2012), Amini et al. (2015) and Wang et al. (2020) that have investigated optimal
\ncounterparty risk exposures through trade-offs in CCP risk-sharing, we take the network of exposures as given and focus on how the CCP\u2019s default waterfall influences financial system loss. We
\ncalibrate this model using a unique and comprehensive dataset on U.S. credit default swap (CDS)
\nCCP transactions to assess the resilience of the system against large market shocks. We also assess counterfactual default waterfall structures and determine their impacts on CCP stability and
\noverall financial system resilience.
\nWe examine the impact of the CCP\u2019s default waterfall and recovery mechanisms on both total
\nand individual losses suffered from variation margin owed by (and to) the CCP members of a major
\nCDS central counterparty. We also incorporate the hundreds of clients that clear through CCP
\nmembers, which the members are responsible for in the event of their default, and examine the
\nimpact of client defaults on systemic resiliency. client clearing losses have been highly significant
\nhistorically. For instance, clients were heavily responsible for the default of the CCP Caisse de
\nLiquidation des Affaires en Marchandise in 1974 (Bignon and Vuillemey (2020)). As recently as
\nMarch 2020, a large client of the CME clearing member ABN Amro defaulted and caused the
\nmember an estimated $200 million in losses (Mourselas and Smith (2020)).
\nThis work makes several contributions to the literature on central clearing and risk sharing.
\nFirst, we quantify how losses are allocated across a financial payment system. We do so by providing
\na comprehensive measure of systemic loss and examining how losses are influenced by the scale of
\nmarket shocks and the liquidity of collateral. We find substantive spillover effects due to network
\ncontagion that dramatically elevate losses from large shocks. Though asset fire sales can also
\nintensify the level of systemic losses, in line with Duarte and Eisenbach (2018), we find that current
\ncollateral standards are high enough to prevent major liquidation losses.
\nSecond, we analyze how different waterfall structures that vary in the proportion of losses
\n3
\nallocated to individual CCP members vs. the shared collective of all CCP members affect the
\nquantity of capital needed to sustain the CCP against market shocks. We compare our results with
\na unique data collection on the default waterfall designs of more than 60 global derivatives CCPs.
\nThis allows us to analyze how global variations in funded resources, including initial margins,
\nCCP capital, and guarantee funds, influence expected losses for market participants and the CCP\u2019s
\nresilience under stress. These data provide us with a measure of the preferences of CCPs and
\nclearing members in selecting a default waterfall\u2019s resource allocation.
\nFinally, we consider the CCP members\u2019 responses to changes in waterfall structure. As more
\ndefault waterfall resources are required, clearing members may become more hesitant to participate
\nin central clearing due to the heightened costs. We estimate the resilience of the default waterfall if
\nparticipation in central clearing is also reduced. We find that the changes in central clearing participation can have a large impact on the resilience provided by requiring more waterfall resources. In
\nthe case of more minor market shocks, requiring more waterfall resources will not counterbalance
\nthe consequences of decreased central clearing participation. This result highlights the downward
\npressure on waterfall resources that a CCP faces in periods of market calm, which is in line with
\nthe downward trends in CCP waterfall resources in the decade following the 2008 financial crisis.
\nPrevious papers that examined CCP default waterfall designs include Capponi et al. (2017),
\nwhich examines the CCP\u2019s role in attracting less risky membership and the consequences for risk
\nsharing from allocating risk to themselves in the default waterfall. In contrast, we consider the aggregate loss to firms and counterparty externalities, similar to Acharya and Bisin (2014), Ghamami
\n(2015) and Ghamami and Glasserman (2017), by incorporating the impacts of several layers of
\nthe waterfall on client clearing and non-cleared positions into our analysis. Importantly, previous
\npapers in this literature have not considered the full network implications of the CCP\u2019s default
\nwaterfall, nor have they had access to the detailed transaction-level market data that we use.
\nOther papers such as Huang (2019) have studied the conflicting objectives CCPs may have in
\ndetermining their waterfall structures. As commercial enterprises with profit-making incentives,
\nCCPs compete for the clearing business of members and their client positions (Glasserman et al.
\n(2016)). This is likely to drive how much and where waterfall resources are allocated, as collecting
\ncollateral has direct short-term costs for participants and could disincentivize participation in central clearing. Our paper is complementary to these other works and helps determine the magnitude
\n4
\nof the stability benefits provided by default waterfall resources, which must be balanced against
\nthe costs imposed on CCP members.
\nThe rest of this paper is divided into the following sections. Section 1 provides a background
\non CCP default waterfalls and how they have been implemented. Section 2 describes our CCP
\npayments model and how the waterfall is incorporated into it. Section 3 describes how to compute
\nsystemic losses and individual firm losses using the model. Sections 4 presents an empirical test of
\nCCP waterfall resiliency using U.S. CDS market data. Section 5 presents counterfactual analysis
\nthat considers the impact of various segments of the default waterfall. Section 6 concludes.
\n1 CCP Default Waterfall Structure
\nA CCP\u2019s recovery plan to deal with clearing member or client defaults is known as its default
\nwaterfall. The default waterfall provides a detailed list of resources that the CCP will use in attempting to recoup losses from clearing member defaults. While the exact rules of default waterfalls
\nvary across CCPs, their overall structures are similar and follow from standard industry guidelines
\n(ISDA (2013), ISDA (2015)). The stages of a typical default waterfall are depicted in Figure 1.
\n1.1 Default Waterfall Resources and Mechanisms
\nThe first several stages of the default waterfall are present in nearly all CCPs, and they involve
\nwidely used mechanisms. These stages are known as the funded waterfall stages because their
\nresources are contributed before the shock occurs. Thus the amount available to use is independent
\nof the shock. Since these are the first stages to be used, there is more historical precedent for them
\nthan for the final stages, and they are thus better understood and tested.
\nThe first stage of the default waterfall is the initial margin (IM) of the defaulting clearing
\nmember. IM is held at the CCP in case a clearing member defaults. IM can be used when the
\nclearing member does not fulfill its payment obligations. The IM amount is usually set at a certain
\nValue-at-Risk (VaR) level, such as 99 percent, but may also have additional components, like
\nconcentration and liquidity (Capponi et al. (2020)). IM is also collected for non-centrally cleared
\ntransactions. However, the margin period of risk (MPOR) used in IM calculations typically differ,
\nwith derivative CCPs typically using a 5-day MPOR while bilateral trades typically use a 10-day
\n5
\nFigure 1. Stages of CCP Default Waterfall
\nCCP Capital
\nGuarantee Fund Contribution of Defaulting M
\nIM of Defaulting M IM of Defaulting Client
\nMember Defaults Client Defaults
\nGuarantee Fund Contribution of Surviving Ms
\nFunded Resources
\nAssessments
\nVM Gains Haircutting
\nEnd-of-Waterfall Mechanisms
\nNote: The chart depicts the series of resources and mechanisms in the waterfall which will be accessed if previous
\nones are insufficient to cover total default losses in the event of a clearing member (M) or client default. The solid
\narrows depict the most common set of waterfall resource contingencies. A defaulting clearing member\u2019s, or client\u2019s,
\nobligation is first covered by their initial margin (IM). Positions of defaulting clients are the responsibility of the
\nassociated clearing member, who has to cover any shortfalls in variation margin (VM) payments owed for those
\npositions. If the clearing member cannot fulfill this obligation, the clearing member may be put into default. If the
\nclearing member\u2019s IM is insufficient to cover its obligations, the resources of the following stages will be used.
\nSource: Authors\u2019 creation.
\nMPOR.
\nThe second stage of the default waterfall is the guarantee fund contribution of the defaulting
\nclearing member. Guarantee fund contributions are collected from all clearing members and held
\nat the CCP. A clearing member\u2019s contribution is usually proportional to its VaR, and is thus also
\nproportional to its IM. The CCP\u2019s total guarantee fund amount is typically sized according to the
\n\u201cCover 2\u201d rule, which states that the guarantee fund should cover the default of the two largest
\nclearing members of the CCP. However, alternative risk-based rules can also be used. We will
\nempirically test the resilience offered by variations in the ratio of guarantee funds to IM. The
\n6
\nguarantee fund is more versatile than IM because it can be used to cover the losses of any clearing
\nmember, but this versatility also opens up non-defaulting clearing members to losses.
\nThe next stage is the CCP\u2019s own capital contribution.2 This is commonly referred to as \u201cskin in
\nthe game\u201d and is intended to reduce moral hazard on the part of the CCP. CCP capital contributions
\nare typically small relative to the total IM or the guarantee fund, and are generally one to three times
\nsmaller in relative magnitude, as we will discuss in the following section. The final stage of funded
\nresources is the loss mutualization of surviving clearing members\u2019 guarantee fund contributions.
\nThe guarantee fund covers the defaulted payments pro rata across each of the clearing members.
\nIn the event that the funded resources are entirely deployed, a few different end-of-waterfall
\nmechanisms can be implemented either to raise fresh funds, via assessments, or reduce obligations,
\nvia variation margin gains haircutting (VMGH). Assessments allow the CCP to request additional
\nfunds from non-defaulting clearing members, whereas VMGH allows the CCP to temporarily reduce
\nthe VM payments on its obligations. These mechanisms have rarely been used in practice and may
\nhave alterations made to them to further support the CCP. For the purpose of focus, we will not
\nanalyze these mechanisms in the main body of the text, but we do discuss and analyze them in
\nAppendix D.3
\n1.2 Empirical Comparison of CCP Default Waterfall Resources
\nThough CCPs use the same types of default waterfall resources and mechanisms globally, the
\namount of resources collected at each stage varies significantly in practice. To highlight the differences empirically, we collected a unique data sample of default waterfall resources from the
\nPrinciples for Financial Market Infrastructures (PFMI) filings of 60 global CCPs from the fourth
\nquarter of 2017. Our data show a large degree of heterogeneity in how resources are allocated
\nalong the waterfall. Particularly, there are major differences in the amount of resources CCPs have
\navailable through their IM, guarantee funds, and CCP capital contributions.
\nSuch differences are significant because the relative levels of IM and guarantee fund can have a
\n2This stage may come in one or two parts depending on the CCP. Some CCPs allow for a second part that comes
\nafter the guarantee fund stage. We use one part in our analysis for simplicity, but having a second part would not
\nmaterially change our model.
\n3We compare their assessment rules, specifically the cap on assessments as a function of the guarantee fund
\ncontribution sizes, and whether their rules permit usage of variation margin gains haircuts (VMGH) and initial
\nmargin haircuts (IMH). Each choice of rules can have a significant effect on the size of the shock the CCP can
\nwithstand.
\n7
\ntremendous impact on the resilience of the CCP, its clearing members, and the overall market. We
\nempirically test the magnitude of these effects below in Section 5 using positions-level data from a
\nmajor U.S. CCP. Additionally, though for the sake of focus we do not analyze CCP end-of-waterfall
\nmechanisms in the main body of the text, we do discuss and empirically analyze them in Appendix
\nD.
\nThe CCP data collection is summarized in the following sequence of tables. Table I shows the
\naverage percentage of resources for different stages of the default waterfall across CCPs grouped
\nby asset class. Commodity CCPs make the highest percentage of capital contributions, whereas
\nInterest Rate CCPs make the lowest percentage. Credit CCPs have relatively high levels of IM
\nand guarantee funds but low levels of capital contribution relative to other CCP asset classes. This
\nwaterfall structure shifts the losses from the CCP onto the clearing members. The table also shows
\nthe maximum assessment limit as a percentage of total guarantee funds.4 These values are due to
\ncaps on assessments set by each CCP as a function of guarantee fund size.
\nTable I. Waterfall Resources by CCP Asset Class
\nInterest Rate Currency Commodity Credit Equity
\nNumber of CCPs 13 12 16 6 13
\nFunded Resources
\nInitial Margin 79.2 73.6 77.2 77.9 81.1
\nGuarantee Fund 19.2 21.8 13.7 20.1 13.4
\nCCP Capital 1.6 4.6 9.1 2.0 5.5
\nEnd-of-Waterfall Resources
\nAssessments 86.5 96.9 75.8 60.2 124.9
\nNote: The table presents the mean percentage of funded resources collected at each stage, and the maximum assessment a CCP can make on its clearing members relative to the guarantee fund size, grouped by the asset class a CCP
\nclears. Looking across CCP types, initial margin makes up the majority of resources collected, ranging from 70 to 81
\npercent, followed by the guarantee fund with 13 to 22 percent. The CCP\u2019s contribution is minimal, ranging from 1
\nto 9 percent. More generally we find that no particular asset class appears to have any unique preference in assigning
\nresources.
\nSource: CCPView Clarus Financial Technology; authors\u2019 analysis.
\nTable II shows a similar summary grouped by the location of the CCP. There does not seem
\nto be a global consensus on the optimal waterfall design for minimizing systemic risk or ensuring
\nincentive compatibility for CCP members. The ratios vary dramatically across regions. Asian and
\nEuropean CCPs have significantly lower percentages of IM than North American CCPs. European
\nCCPs have larger levels of guarantee funds, while Asian CCPs have larger CCP capital. These
\n4Some CCPs allow for a greater assessment amount if there are multiple clearing member defaults versus a single
\nclearing member default.
\n8
\ndifferences across regions can have an important impact on the CCP\u2019s resilience under periods of
\nmarket stress. Our empirical analysis in Section 5 shows that CCPs with higher IM relative to
\nguarantee funds and capital are less resilient to market shocks.
\nTable II. Waterfall Resources by Jurisdictional Region
\nAsia Europe North America Oceania South America
\nNumber of CCPs 27 20 12 2 2
\nFunded Resources
\nInitial Margin 69.2 74.0 85.2 90.1 97.7
\nGuarantee Fund 18.7 25.3 13.5 2.2 2.2
\nCCP Capital 12.2 0.7 1.3 7.7 0.1
\nEnd-of-Waterfall Resources
\nAssessments 75.5 122.3 77.5 300.0 73.6
\nNote: The table presents the mean percentage of funded resources collected at each stage, and the maximum assessment a CCP can make on its clearing members relative the guarantee fund size, grouped by the continental
\njurisdiction a CCP resides. Looking across CCP jurisdictions, we see wide variation in funded resources and assessments, suggestive of jurisdictional regulatory preferences influencing CCP\u2019s default waterfall allocations.
\nSource: CCPView Clarus Financial Technology; authors\u2019 analysis.
\nAn additional dimension to the waterfall structure is the liquidity of the collateral resources
\nheld by the CCP and used in the event of default. Although intra-firm payments are made in
\ncash, holding IM and guarantee fund collateral in cash alone creates significant costs for clearing
\nmembers. As a result, other forms of collateral that pay higher interest rates are typically held, or
\nthe CCP may rely on credit lines in case of short-term delays in payments. Table III highlights the
\npercent of collateral and credit lines held by 30 CCPs as of the fourth quarter of 2017.
\nTable III. Funded Resource Collateral and Credit Lines
\nMean Median Std Dev Min Max
\nCollateral
\nSecured Cash Deposits 44.3 47.5 35.0 – 100.0
\nUnsecured Cash Deposits 14.6 1.8 31.0 – 100.0
\nRepo Lent Cash\/Securities 10.3 – 21.1 – 81.4
\nGovernment Securities 28.2 21.0 30.2 – 99.0
\nOther 2.6 – 13.5 – 74.2
\nUnsecured Credit Lines 8.1 – 23.4 – 121.8
\nNote: The table presents the percentage of collateral and liquidity resources held by 30 OTC derivative CCPs as
\nof the fourth quarter of 2017. The majority of CCP collateral holdings are in cash, repo, or government securities.
\nA small percentage of holdings are in other less liquid assets. Additionally some CCPs have unsecured credit lines,
\nwhich we present as a percentage of their total collateral holdings, that they may draw on in times of short-term
\nliquidity impairment.
\nSource: CCPView Clarus Financial Technology; authors\u2019 analysis.
\nIn the event of a default, non-cash collateral will need to be liquidated before payments can be
\n9
\nmade, likely at a reduced price in times of market stress. These liquidation costs and the ability
\nto raise cash in a timely fashion are additional elements of risk that need to be considered. As
\na result CCPs in general hold very high quality collateral, in cash or cash equivalents, as Table
\nIII highlights, and require haircuts. Though CCPs may take wider forms of collateral if needed,
\nthey generally encourage collateral delivered to be of high quality through applying steep haircuts
\nrelative to the general market.
\n2 Model of CCP Default Waterfall
\nCCP stress can arise via many forms. For instance, a substantial shock such as the Comprehensive Capital Analysis and Review (CCAR) shock can lead to a situation where some firms owe
\nmuch more VM than they expect to receive. These VM obligations are supposed to be satisfied
\nwithin a very short time \u2013 typically within a few hours \u2013 and can put severe stress on the firms\u2019
\ntrading desks. This stress may prevent a firm from fulfilling all its obligations to its counterparties.
\nThis, in turn, increases the stress on the firm\u2019s downstream counterparties, amplifying the impact
\nof the shock through the network of CDS exposures. The model assumes that payments are all
\nmade simultaneously as margin payments are made intra-day, though it can take several days or
\nweeks for some portfolio liquidations if necessary.
\nIn the next two sections, we introduce a network model to account for this type of contagion,
\nand we show how to incorporate the different components of the CCP waterfall into the model.
\n2.1 Basic Setup and Payments
\nThe setup is based on the framework of Glasserman and Young (2015), which in turn builds on
\nthe model of Eisenberg and Noe (2001). There are N + 1 agents in the market, with 0 indexing the
\nCCP and i \u2208 1, …, N indexing the non-CCP firms. Each pair of counterparties i, j can have a set of
\ncontracts between them, which we assume are in a single asset class, for instance CDS. Contracts
\ncan represent both bilateral and centrally cleared transactions.
\nThere are three types of non-CCP firms in our model: clearing members M, clients C, and
\nbilateral firms B. These firms differ in their use of central clearing. Clearing members may clear
\nthrough the CCP directly. Clients cannot clear through the CCP directly but must instead submit
\n10
\nFigure 2. Example Network of Cleared and Non-cleared Obligations
\nB1 M1
\nC1
\nCCP
\nM2
\nC2
\nB2
\nM3
\nC3
\nM4 B3
\nC4
\nB4
\nNote: Four firm types are depicted in the network: the CCP, clearing members (M), clients (C), and bilateral firms
\n(B). The links represent obligations, both direct (solid) and client clearing (dashed). These firms differ in their use
\nof central clearing. Clearing members may clear through the CCP directly. Clients cannot clear through the CCP
\ndirectly but must instead submit trades through a clearing member. Clearing members pass through the payments
\nreceived or owed from the client to the CCP, and clearing members must cover any shortfalls in these payments out
\nof their own funds.
\nSource: Authors\u2019 creation.
\ntrades through a clearing member. Clearing members pass through the payments received or owed
\nfrom the client to the CCP, and clearing members must cover any shortfalls in these payments out
\nof their own funds. We give the details of how this process works below. We denote Ck as the set
\nof client firms of a clearing member k. Also, we denote Mi as the set of clearing members of a
\nclient firm i.
\n5 Finally, some non-members are purely bilateral firms that do not participate in any
\ncentrally cleared transactions and instead engage only in bilateral transactions.
\nGiven a market shock, we can calculate the variation margin (VM) payment obligations for
\neach individual contract. We then net all the VM payments owed on contracts between every pair
\nof counterparties. This produces a net VM payment that needs to be made between this pair. We
\nexclude client clearing trades in this summation, and will instead consider them separately next.
\nWe write the net VM payment obligations as an obligation matrix P\u00af = (\u00afpij )i,j\u2208N0
\n, where \u00afpij is the
\nnet amount of VM owed by node i to node j in the aftermath of the shock. We denote the indices
\nof the first row and column of this matrix as 0 to account for the CCP. Note that if \u00afpij is positive
\n5
\nIn the data, we do not observe the identity of the clearing member that a client uses, but we assume that the
\nclient clears through the clearing member with which it has the most bilateral transactions.
\n11
\nthen \u00afpji is zero. Further, \u00afpii = 0 for all i.
\nGiven the market shock, firms may only be able to satisfy part of their payment obligations due
\nto shortfalls in funds. We denote the actual VM payments made between firms using the realized
\npayment matrix P = (pij ).
\nFigure 3. Example of Normal and Realized Payments
\nBj Ci Mk CCP0 Ml
\npij \u2264 p\u00afij pik \u2264 p\u00afik pk0 \u2264 p\u00afk0 p0l \u2264 p\u00af0l
\nq
\nc
\nik0 \u2264 q\u00afik0 q
\nm
\nik0 \u2264 q\u00afik0
\nNote: The example shows the types of obligations that could occur between four types of firms, both bilateral and
\nclient cleared. The figure helps to illustrate the dependencies that firms can have on one another, as clearing member
\nk\u2019s ability to pay CCP 0 may depend on client i\u2019s ability to fulfill its obligations. In the figure, if the realized payments
\nfrom client i, pij , pik, qc
\nik0
\n, qm
\nik0
\n, are less than its obligations, clearing member k and pure bilateral firm j will suffer
\ndirect losses. Additionally, the impact of client i\u2019s realized payments may lead to clearing member k reducing its
\npayments, causing potential losses to the CCP as well.
\nSource: Authors\u2019 creation.
\nThere are also client clearing transactions in the network that induce a set of corresponding
\nclient clearing obligations. We denote the net VM client clearing obligations with a set of two
\nmatrices Q\u00af = (\u00afqik0, q\u00af0ki)i\u2208C,k\u2208M. In this notation, \u00afqik0 denotes the net amount owed by client i to
\nthe CCP through clearing with member k, and \u00afq0ki denotes the net amount owed by the CCP to
\nclient i through clearing with member k.
\n6 Note again that if one of these obligations is positive
\nthen the other is zero. Because client clearing must involve a clearing member as an intermediary,
\nq\u00afik0 is fulfilled by having client i submit a payment to clearing member k, q
\nc
\nik0
\n, and that clearing
\nmember k then submitting a payment to the CCP, q
\nm
\nik0
\n. The reverse obligation \u00afq0ki is fulfilled by
\nthe CCP submitting a payment to clearing member k, q0ki, and clearing member k submitting a
\npayment to client i, q
\nm
\n0ki. A clearing member must pass through any payments that it receives in
\nthe course of client clearing, even if the clearing member is in default. Therefore q
\nm
\nik0 \u2265 q
\nc
\nik0
\nand
\nq
\nm
\n0ki \u2265 q0ki. We describe how the payment values are determined in equilibrium over the next few
\nsections.
\n2.1.a Initial Margin and Capital Buffer
\nIn most transactions, IM will be held to help cover deficiencies in VM payments. We denote
\nthe IM that i holds from counterparty k as zki. This IM value will always be positive unless the
\n6The CCP does not net the transactions of a client that clears through multiple different clearing members. The
\nclient is in effect treated as a different entity across each clearing member with which it clears.
\n12
\ncounterparty k is the CCP itself as the CCP does not post IM, z0i = 0 \u2200i. If counterparty k fails
\nto pay VM to i in a timely manner, the position will be closed out and the IM will be applied to
\nthe shortfall in VM payments. The value of IM held is calculated on a portfolio level using a VaR
\nestimate based on historical look back.7
\nFor client clearing transactions, IM will be held by the CCP for a transaction with client i
\nclearing through clearing member k. We denote the value of this IM by zik0. The IM is used to
\ncover shortfalls in payments if the client is unable to cover its obligations. The IM is then passed
\nthrough by the clearing member to the CCP to Help with the clearing member\u2019s passthrough
\nobligation if necessary. The passthrough payment of clearing member k must be weakly greater
\nthan the amount it receives, q
\nm
\nik0 \u2265 (q
\nc
\nik0 + zik0). The CCP does not post IM for the client clearing
\nobligations that it owes. For a given firm i, we denote the total amount of IM that it contributes
\nacross bilateral and client clearing obligations as zi
\n.
\nBeyond IM, firms are assumed to hold some quantity of assets on hand, which we term as capital
\nbuffer, bi > 0. This bi
\nis a function of the firm\u2019s risk management policies, non-CDS positions, or
\navailable cash. Firms will use their capital buffers to help meet their VM obligations.
\nGenerally collateral in the financial system can be treated as cash, particularly in the case of
\ncleared IM, as CCPs require IM to be in sovereign bonds or currency. As a result, if collateral zij
\nis seized by node j from node i, node i\u2019s payment obligation is reduced by exactly zij . Similarly
\nwe assume there is no liquidation cost for the capital buffer bi that is used to node i\u2019s payment
\nobligations. However, in the non-cleared derivatives market, a wide range of less liquid securities,
\nincluding corporate bonds, foreign-denominated bonds, and equities are accepted as collateral, and
\nthese types of securities are also used as collateral in repurchase agreements.
\nWe note that our model can be adjusted to incorporate the potential consequences of collateral
\nilliquidity. To do so, we assume that zij and bi are held in an asset with a normalized price \u03c0.
\nThe value of \u03c0 can range from zero to one, with zero indicating a full reduction in value and one
\nindicating no reduction in value (i.e. cash). In turn, \u03c0zij is the reduced value of the IM and \u03c0bi
\nthe reduced value of the capital buffer. Following the price-impact formulation used in Cifuentes
\net al. (2005) and Amini et al. (2016), we posit that \u03c0 is a strictly decreasing function G(1, \u2206) of
\n7We estimate VM and IM following the formulation of Luo (2005) and applying the framework adopted in
\nAppendix I of Paddrik et al. (2020).
\n13
\nthe total proportion of collateral asset liquidated \u2206, with the first argument of G indicating the
\ninitial price of 1. To be concrete, we set
\n\u03c0 = G(1, \u2206) = e
\n\u2212\u03bb\u2206 (1)
\nfor some \u03bb > 0 . A larger \u03bb corresponds to a less liquid asset. Though this setting makes the
\nsimplifying assumption that all collateral is held in a single illiquid asset, any overstatement can be
\noffset through a smaller value of \u03bb. We analyze the impact of collateral illiquidity in Appendix C,
\nand we show that the current collateral standards are sufficient to prevent major losses. To reduce
\nour notation in the subsequent sections we will present the case where IM and capital buffers are
\nheld in cash, \u03c0 = 1.
\n2.1.b Payments for Bilateral Firms
\nWe first derive payment equations for nodes other than the CCP. We start with the simplest
\ncase of a bilateral firm i, and we write out the payments pij that i makes to each counterparty j
\nconditional on the payments that it receives from each counterparty k, pki.
\nThe resources available to firm i are its capital buffer bi
\n, the payments from each counterparty
\nk, pki, and the IM that i holds from each counterparty k, zki. The total obligations of firm i to its
\ncounterparties are P
\nk6=i
\np\u00afik. If this exceeds the resources of firm i, it will not be able to fully pay
\nits counterparties. It will thus need to reduce its payments to its counterparties proportionally. We
\ndescribe next how this reduction is implemented.
\nFor a bilateral firm i \u2208 B, we define the stress at i, si
\n, to be the amount by which i\u2019s payment
\nobligations exceed the incoming payments from i\u2019s counterparties and IM held:
\nsi =
\n\uf8ee
\n\uf8ef
\n\uf8ef
\n\uf8ef
\n\uf8ef
\n\uf8ef
\n\uf8f0
\nX
\nk6=i,0
\np\u00afik
\n| {z }
\nObligations
\n\u2212
\n\uf8eb
\n\uf8ed
\nX
\nk6=i,0
\n((pki + zki) \u2227 p\u00afki) + bi
\n\uf8f6
\n\uf8f8
\n| {z }
\nResources
\n\uf8f9
\n\uf8fa
\n\uf8fa
\n\uf8fa
\n\uf8fa
\n\uf8fa
\n\uf8fb
\n+
\n. (2)
\nLet \u00afpi =
\nP
\nj6=i
\np\u00afij be the total payment obligations of i to all other nodes. In the following
\ndefinitions, we restrict attention to the nodes i in the system such that \u00afpi > 0. The others do not
\n7
\nIn general, x \u2227 y denotes the minimum of two real numbers x and y.
\n14
\nhave payment obligations and thus do not transmit payment shortfalls. In the CDS market that we
\nconsider, such firms would be buyers of CDS protection (not sellers) and under our CCAR shock
\nthey would not have VM obligations. Shortfall and payment equations are thus not necessary for
\nsuch nodes.
\nWe can use these VM payment obligations to define the relative liability of node i to node j as
\naij = \u00afpij\/p\u00afi
\n. (3)
\nWe assume that the stress of firm i is transmitted to i\u2019s counterparties pro rata the size of its
\npayment obligations. We can now define the payment functions for bilateral firms. Given any input
\nvector p \u2208 R
\n2n+2 such that 0 \u2264 pij \u2264 p\u00afij for all 0 \u2264 i, j \u2264 n, the payment functions pij (p) for a
\nbilateral firm i are given by:
\npij = \u00afpij \u2212 aijsi
\n. (4)
\n2.1.c Payments for Client Firms
\nNow we define the payments for client firms. Client firms have similar payment functions as
\nbilateral firms, but they also have client clearing transactions with the CCP. Recall that for these
\ntransactions, clients need to form a contract with a clearing member, which then intermediates
\ntransactions with the CCP. If the client owes money to the CCP, the client pays the clearing
\nmember , which then passes the payment onto the CCP. If the client is owed money by the CCP,
\nthe CCP will pay the clearing member, which then passes the payment onto the client. Clients also
\npost IM to cover shortfalls in their payments. However, the CCP does not post IM for the client.
\nThe total resources of a client firm i are given by the sum of its payments received from bilateral
\nand client clearing transactions along with its capital buffer. The total obligations of client i are
\ngiven by the sum of its bilateral and client clearing obligations. If this amount exceeds the resources
\nof firm i, it will not be able to pay its counterparties in full and will instead pass on partial payments.
\nFor a client firm i \u2208 C, we define the stress at i, si
\n, to be the amount by which i\u2019s payment
\nobligations exceed the incoming payments from i\u2019s counterparties and IM held:
\n15
\nsi =
\n\uf8ee
\n\uf8ef
\n\uf8ef
\n\uf8ef
\n\uf8ef
\n\uf8ef
\n\uf8f0
\nX
\nk6=i,0
\np\u00afik +
\nX
\nk\u2208Mi
\nq\u00afik0
\n| {z }
\nObligations
\n\u2212
\n\uf8eb
\n\uf8ed
\nX
\nk6=i,0
\n((pki + zki) \u2227 p\u00afki) + X
\nk\u2208Mi
\nq
\nm
\n0ki + bi
\n\uf8f6
\n\uf8f8
\n| {z }
\nResources
\n\uf8f9
\n\uf8fa
\n\uf8fa
\n\uf8fa
\n\uf8fa
\n\uf8fa
\n\uf8fb
\n+
\n. (5)
\nUnlike bilateral firms, clients will also have client clearing payment obligations. When the client
\nfaces stress, we assume that these payments are reduced pro rata the total payment obligations
\nowed by the client across all types of transactions.
\nDenote \u00afp
\ng
\ni =
\nP
\nk6=i
\np\u00afik +
\nP
\nk\u2208Mi
\nq\u00afik0 as the sum over all payment obligations for client i. We
\nuse these combined payment obligations to derive client i\u2019s relative payment liability to different
\nfirms.
\naik = \u00afpik\/p\u00af
\ng
\ni
\n\u2200k 6= i, (6)
\na
\nc
\nik0 = \u00afqik0\/p\u00af
\ng
\ni
\n\u2200k \u2208 Mi
\n. (7)
\nThe stress of client i is transmitted to i\u2019s counterparties pro rata the size of its combined
\npayment obligations. Given any vector p \u2208 R
\n2n+2 such that 0 \u2264 pik \u2264 p\u00afik for all 0 \u2264 i, k \u2264 n, the
\npayment functions pik(p), qc
\nik0
\n(p) for a client firm i are given by:
\npik = \u00afpik \u2212 aiksi
\n, (8)
\nq
\nc
\nik0 = \u00afqik0 \u2212 a
\nc
\nik0
\nsi
\n. (9)
\n2.1.d Payments for Clearing Members
\nNow we define the payments for clearing members. Clearing members may engage in bilateral
\ntrades, centrally cleared trades, and client clearing trades. Each of these three types of trades will
\nentail different obligations and resources for the clearing member. The total obligations of clearing
\nmember k to its counterparties are given by P
\ni6=k
\np\u00afki +
\nP
\ni\u2208Ck
\n(\u00afq0ki + \u00afqik0). If this amount exceeds
\nthe resources of clearing member k, it will not be able to pay its counterparties in full.
\nMember k\u2019s stress is given by:
\n16
\nsk =
\n X
\ni6=k
\np\u00afki +
\nX
\ni\u2208Ck
\n(\u00afq0ki + \u00afqik0)
\n| {z }
\nObligations
\n\u2212
\n\uf8eb
\n\uf8ed
\nX
\ni6=k
\n((pik + zik) \u2227 p\u00afik) + X
\ni\u2208Ck
\n((q
\nc
\nik0 + zik0) \u2227 q\u00afik0) + q0ki) + bk
\n\uf8f6
\n\uf8f8
\n| {z }
\nResources
\n +
\n. (10)
\nThe way that a clearing member k passes on stress is special because for centrally cleared
\ntransactions it must pass through at least the amount that was given to it by a client i for the
\nCCP, ((q
\nc
\nik0 + zik0) \u2227 q\u00afik0), or the CCP for a client i, q0ki. If these amounts are sufficient to cover
\nthe original obligations, \u00afqik0 and \u00afq0ki respectively, then the clearing member will make the payment
\nin full. However, if the amount received by the clearing member is less than the obligation, the
\nclearing member must cover the remainder out of its own funds or be in default. If the clearing
\nmember is under stress, then any remaining obligations will be cut pro rata based on the clearing
\nmember\u2019s stress.
\nDefine \u00afp
\nr
\nk =
\nP
\ni6=k
\np\u00afki +
\nP
\ni\u2208Mk
\n(\u00afq0ki \u2212 q0ki + [\u00afqik0 \u2212 q
\nc
\nik0 \u2212 zik0]
\n+) as the sum of the remaining
\npayments over all entities. Because the passthrough payments must always be made, we deduct
\nthem in determining the clearing member\u2019s relative liability.
\naki = \u00afpik\/p\u00af
\nr
\ni \u2200k 6= i, (11)
\na
\nm
\n0ki = (\u00afq0ki \u2212 q0ki)\/p\u00af
\nr
\ni \u2200i \u2208 Ck, (12)
\na
\nm
\nik0 = [\u00afqik0 \u2212 q
\nc
\nik0 \u2212 zik0]
\n+\/p\u00af
\nr
\ni \u2200i \u2208 Ck. (13)
\nThe stress of clearing member k is transmitted to k\u2019s counterparties pro rata the size of its
\nremaining payment obligations. We can now define the payment functions for clearing member k.
\npki = \u00afpki \u2212 akisk \u2200i 6= k, (14)
\nq
\nm
\n0ki = \u00afq0ki \u2212 a
\nm
\n0kisk \u2200i \u2208 Ck, (15)
\nq
\nm
\nik0 = \u00afqik0 \u2212 a
\nm
\nik0
\nsk \u2200i \u2208 Ck (16)
\nWe have defined the payment functions for all entities other than the CCP. We will define the
\nCCP\u2019s equilibrium payment functions and received payments in the next section. In equilibrium,
\n17
\nthe payments received by a firm and the payments made by a firm must be balanced. Firms that
\nare under stress will cut their payments pro rata, while unstressed firms will make their payments
\nin full. The equilibrium payment vector accounts for contagion effects from one firm failing to pay
\nits counterparties and propagating stress further down the network. Such contagion effects have
\nthe potential to be very large, as shown in Paddrik et al. (2020).
\n2.2 CCP Default Waterfall
\nIn this section, we derive the CCP\u2019s payment function from the funded stages. Recall that the
\nCCP is indexed as node 0. The CCP\u2019s payment obligations are given by P
\nk\u2208M<\/p>\n
p\u00af0k +
\nP
\ni\u2208Ck
\nq\u00af0ki
\n.
\nThe resources that the CCP has available come from the CCP\u2019s default waterfall and the VM payments it receives. As described in the introduction, the funded layers of the CCP default waterfall
\nconsist of the following stages: IM of defaulting clearing members, guarantee fund contributions of
\ndefaulting clearing members, CCP capital, and guarantee fund contributions of surviving clearing
\nmembers.
\nIM is the first stage used in the default waterfall. It is used to cover shortfalls in the payments of
\nclearing member obligations, including a clearing member\u2019s client clearing obligations. zk0 denotes
\nthe IM collected by the CCP from clearing member k. Including IM, the CCP receives total
\nresources from clearing member k of:<\/p>\n
pk0 +
\nX
\ni\u2208Ck
\nq
\nm
\nik0 + zk0<\/p>\n
\u2227<\/p>\n
p\u00afk0 +
\nX
\ni\u2208Ck
\nq\u00afik0<\/p>\n
. (17)
\nThe next several stages of the default waterfall utilize the guarantee fund of the CCP. Funds
\nare first taken from contributions of the defaulting clearing members. If that is insufficient to cover
\nthe CCP stress, funds are next taken from the CCP\u2019s capital contribution b0, and then from the
\nremaining guarantee fund contributions of all the clearing members.
\nWe define some notation to describe the guarantee fund. Let \u03b3 be the total size of the CCP\u2019s
\nguarantee fund. We assume that each clearing member k contributes an amount \u03b3k to the guarantee
\nfund. Thus the total guarantee fund is given by the following equation:
\n\u03b3 =
\nX
\nk\u2208M
\n\u03b3k. (18)
\n18
\nThe guarantee fund contributions \u03b3k are computed by examining the riskiness of the portfolio
\nof each clearing member. We approximate the contribution \u03b3k by using the proportion of IM of
\nclearing member k held by the CCP , as zk0 is a reflection of the relative amount of risk clearing
\nmember k contributes to the total portfolio of the CCP.8
\n\u03b3k = \u03b3P
\nzk0
\nj\u2208M zj0
\n(19)
\nIn section 1.2 we will present a survey of the default waterfall sizes of real-world CCPs. There
\nexists a large degree of variation in guarantee fund sizing, as CCPs follow different rules such as
\nCover 1 or Cover 2 (CPMI-IOSCO (2012)). We examine the resiliency and trade-offs of these
\ndifferent rules in our empirical tests.
\nThe CCP also provides its own capital contribution to cover losses, which we denote by b0. This
\ncapital contribution is used after the guarantee fund of defaulting clearing members and before the
\nguarantee fund of non-defaulting clearing members.
\nThe CCP prefunds the first four stages of the default waterfall. The amount that is available is
\nthus fixed and independent of the market shock. In addition, implementing these stages does not
\ncreate additional stress for the network.9
\nThe remaining CCP stress after the funded waterfall layers, denoted s0, is given by the following
\nequation:
\ns0 \u2261
\n\uf8ee
\n\uf8f0
\nX
\nk\u2208M
\n\uf8eb
\n\uf8edp\u00af0k +
\nX
\ni\u2208Ck
\nq\u00af0ki \u2212 (pk0 +
\nX
\ni\u2208Ck
\nq
\nm
\nik0 + zk0) \u2227 (\u00afpk0 +
\nX
\ni\u2208Ck
\nq\u00afik0)
\n\uf8f6
\n\uf8f8 \u2212 \u03b3 \u2212 b0
\n\uf8f9
\n\uf8fb
\n+
\n. (20)
\nThe CCP\u2019s stress is zero if it can cover its payment obligations without using up its own capital
\nb0 and the guarantee fund contributions of all clearing members \u03b3. Only after the guarantee fund
\nis fully used up (in the fourth stage) will the CCP stress be greater than zero. After the guarantee
\nfund is used up, the CCP passes on this level of stress to its counterparties by cutting its payments
\npro rata. In Appendix D, we analyze the impact of including the end-of-waterfall mechanisms to
\n8
\n\u03b3 reflects the expected loss to the CCP if the two largest clearing members were to default, after accounting for
\nthe clearing members beyond their zk0. For the purposes of this model, we use public disclosures of \u03b3, discussed in
\nSection 4.1.
\n9Clearing members must eventually replenish the IM and guarantee fund contributions that are used, but this
\nwill usually come later. The PFMI regulations require that the IM and guarantee fund payments be replenished by
\nthe start of the next business day to ensure compliance with the cover two rule.
\n19
\ncover the remaining stress.
\nGiven the CCP\u2019s stress s0, we define the combined payment obligations for the CCP to derive
\nits payments. We let \u00afp
\ng
\n0 =
\nP
\ni6=0 p\u00afki +
\nP
\nk\u2208M q\u00af0ki.
\nWe use these combined payment obligations to derive the CCP\u2019s relative payment liability to
\ndifferent firms.
\na0k = \u00afp0k\/p\u00af
\ng
\n0
\n, (21)
\na0ki = \u00afq0ki\/p\u00af
\ng
\n0
\n. (22)
\np0k = \u00afp0k \u2212 a0ks0 \u2200k \u2208 M, (23)
\nq0ki = \u00afq0ki \u2212 a0kis0 \u2200k \u2208 M. (24)
\nSimilar to other firms, the CCP prorates its outgoing payments proportionally based on its level
\nof stress.
\n2.3 Existence of Payment Equilibrium
\nIn this section, we prove that a payment equilibrium will exist for any default waterfall design
\nin the financial system we have proposed. Compared with the classical network model of Eisenberg
\nand Noe (2001), our model features the additional complexities of IM, client clearing obligations,
\nand CCP default waterfall obligations. However, we show in the following theorem that a maximal
\npayment equilibrium still exists for our model. A maximal payment equilibrium refers to an equilibrium with the largest value of payments made by all the firms. We will focus on the analysis of
\nthe maximal payment equilibrium for our model in the empirical section of the paper. We state
\nour existence result in the following theorem.
\nTHEOREM 1. A maximal payment equilibrium exists for the financial clearing system.
\nProof. To prove the existence theorem, we build on the results obtained in Paddrik et al. (2020).
\nProposition 1 of that paper guarantees existence of a greatest fixed point for a function that is
\nupper-semicontinuous, monotone, and order-preserving on a bounded lattice. To apply this result
\nto our model, we start by showing that our payment functions satisfy these properties. The payment
\n20
\nfunctions in our model are given by Equation (4) for purely bilateral firm payments, Equations (8)-
\n(9) for client payments, Equations (14)-(16) for clearing members, and Equations (23)-(24) for the
\nCCP. We define the mapping function \u03a6(p, q) as the set of all the p and q payment functions for
\nall firms in our model.
\nFirst, we note that all of the payment functions are continuous, and thus upper-semicontinuous.
\nTo show that the payment vectors are also monotone and order preserving, note that all the payment
\nfunctions are decreasing in the amount of stress at each entity. We therefore need to show that
\nlower payments received will increase the stress in each of these equations. For the non-CCP
\nequations, this is straightforward to see directly from the definitions of stresses: Equation (2) for
\npurely bilateral firms, Equation (5) for the clients, and Equation (10) for the clearing members.
\nStress at the CCP is given in Equation (20), which also shows that it is also a decreasing function of
\npayments. Since each individual payment function is upper-semicontinuous, monotone, and order
\npreserving, the function \u03a6(p, q) satisfies these properties as well.
\nSince the function \u03a6(p, q) is upper-semicontinuous, monotone and order preserving, we can
\napply Proposition 1 in Paddrik et al. (2020) to show that starting from \u00afp and \u00afq, the sequence of
\npayment vectors will converge to the greatest fixed point payment vector. Therefore a maximal
\npayment equilibrium will exist.
\nIn the empirical applications, we will compute the maximal equilibrium by choosing values of the
\ncapital buffers {b0, b1…, bn} and then recursively computing a fixed point of this system by taking
\nthe limit of the sequence (p
\n1
\n, q1
\n) = \u03a6(\u00afp, q\u00af),(p
\n2
\n, q2
\n) = \u03a6(p
\n1
\n, q1
\n), …,(p
\nn+1, qn+1) = \u03a6(p
\nn
\n, qn
\n),… The
\nresulting fixed point is a maximal equilibrium of the network model. In Appendix A, we describe
\nthis fictitious algorithm in more detail and how it is used to calculate the equilibrium payment
\nvector in the empirical settings. Finally, we note that both our existence result and our fictitious
\nalgorithm can be extended in a straightforward way to cover collateral illiquidity.10
\n10In the case where \u03c0zij and \u03c0bi are held in an illiquid asset, rather than cash, and the asset follows the priceimpact function given in Equation (1), finding the greatest fixed point for the clearing payments and the illiquid asset
\nvalue may require further iterations of the fictitious algorithm. However, a greatest fixed point is guaranteed to exist
\ngiven the monotonicity of \u03c0 by Tarski\u2019s fixed point theorem (1955), as applied in Cifuentes et al. (2005), Elliott et al.
\n(2014), Gofman (2017), and Amini et al. (2016).
\n21
\n3 Measuring Losses
\nA social planner concerned with financial system risk must select a desirable default waterfall
\ndesign by assessing the consequences of different waterfall structures on overall system losses and
\nhow those losses are distributed. We use systemic loss, defined as the total of bankruptcy losses
\nthat creditors suffer, which follows the interpretation applied in Eisenberg and Noe (2001) and
\nGlasserman and Young (2015). However, unlike the measures used in these works, which consider
\nonly the shortfall of payments, we also consider bankruptcy costs resulting from client clearing and
\nthe CCP\u2019s default waterfall. Clearing members are responsible for defaulted client obligations and
\nclearing members\u2019 default resources are used by the CCP to cover defaulting member obligations.
\nThus clearing members can become creditors to the defaulted firms as if they were directly receiving
\npayments.11 Systemic loss captures the overall level of payment disruption to the system felt across
\nall firm types.
\nWe next discuss how losses are calculated among the individual participant types. Then we
\ndescribe how total systemic losses can be derived in our model. This measure will allow us to later
\nempirically measure the social welfare implications of different default waterfall structures.
\n3.1 Losses for Different Types of Firms
\nWe now write out the total losses for the different types of firms. For each firm x, we denote
\nthe loss as lx. These loss equations will be used in the empirical section to compute the amount of
\nlosses suffered by individual market participants based on different shock sizes and default waterfall
\nstructures.12
\nFor a purely bilateral firm i, losses given a payment equilibrium are composed only of bilateral
\nlosses. These bilateral losses are equal to the difference in expected payments versus received
\npayments plus IM in bilateral transactions.
\nli =
\nX
\nk6=i
\n[\u00afpki \u2212 (pki + zki)]+
\n| {z }
\nbilateral
\n, \u2200i \u2208 B. (25)
\n11An example of this bankruptcy cost on clearing members can be seen in the default of a clearing member of
\nNasdaq OMX CCP in late 2018, which caused clearing members to become creditors to the order of \u20ac107 million.
\n12In Appendix B we provide a detailed description of the different types of losses in our model.
\n22
\nFor a client firm j, losses given a payment equilibrium are composed of bilateral plus client
\nclearing losses, which occur when clients do not receive their full obligations in client clearing
\ntransactions.
\nlj =
\nX
\nk6=j
\n[\u00afpkj \u2212 (pkj + zkj )]+
\n| {z }
\nbilateral
\n+
\nX
\nk\u2208Mj<\/p>\n
q\u00af0kj \u2212 q
\nm
\n0kj
\n| {z }
\nclient clearing
\n, \u2200j \u2208 C. (26)
\nFor a clearing member k, losses given a payment equilibrium are composed of bilateral and
\ndirect clearing, client clearing, and waterfall losses, which occur when waterfall contributions are
\nused to cover obligations of a separate defaulting clearing member. We denote the waterfall losses
\nby \u02c6\u03b3k, and we show how they can be computed in Appendix B.
\nlk =
\nX
\ni6=k
\n[\u00afpik \u2212 (pik + zik)]+
\n| {z }
\nbilateral and cleared
\n+
\nX
\ni\u2208Ck<\/p>\n
[\u00afqik0 \u2212 (q
\nc
\nik0 + zik0)]+ + \u00afq0ki \u2212 q0ki
\n| {z }
\nclient clearing
\n+ \u02c6\u03b3k
\n|{z}
\nwaterfall
\n,
\n\u2200k \u2208 M. (27)
\n3.2 Losses for the CCP
\nWe now describe the losses of the CCP, which may be different from the losses of other types
\nof firms. The CCP\u2019s losses can come from two channels. First, the CCP has a capital contribution
\nof b0 that it could lose if used in the default waterfall. We denote the amount used in equilibrium
\nby \u02c6b0, which is equal to the amount the CCP needs to cover in excess of received payments, IM,
\nand own default fund contributions of defaulting clearing members:
\n\u02c6b0 = min
\n\uf8eb
\n\uf8ed
\nX
\nk\u2208M
\n\uf8eb
\n\uf8edp\u00af0k +
\nX
\ni\u2208Ck
\nq\u00af0ki \u2212 (pk0 +
\nX
\ni\u2208Ck
\nq
\nm
\nik0 + zk0 + \u03b3k) \u2227 (\u00afpk0 +
\nX
\ni\u2208Ck
\nq\u00afik0)
\n\uf8f6
\n\uf8f8, b0
\n\uf8f6
\n\uf8f8 (28)
\nThe CCP can also suffer losses from payment shortfalls that exceed its total waterfall resources
\n(IM, guarantee fund, and capital contribution). Since the CCP has a balanced book, the remaining
\nshortfall in the payments it receives after using its default waterfall is equal to the equilibrium<\/p>\n","protected":false},"excerpt":{"rendered":"
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