Clearing networks

Journal of Economic Behavior & Organization 83 (2012) 609–626

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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo

Clearing networks夽,夽 Marco Galbiati a,b,∗ , Kimmo Soramäki c a b c

Bank of England, United Kingdom ECB, Germany Aalto University School of Science and Technology, Helsinki, Finland

a r t i c l e

i n f o

Available online 26 June 2012

JEL classification: E58 G01 G18 Keywords: Central counterparty CCP Clearing Settlement Network analysis

a b s t r a c t In several financial markets, counterparty risk is reallocated away from traders via ‘novation’, a step of the clearing process. By novation, a third party steps into a bilateral contract, guaranteeing performance of both legs of the trade. Central counterparties (CCPs) are entities whose special purpose is novating trades, relieving market participants from counterparty risk. However, in most cases, the CCP is not the sole novator: the CCP novates contracts between its clearing members, which in turn novate trades for other (typically smaller) participants and so on. This paper develops an abstract model of such hierarchical clearing networks. Novation is modelled here as a function which transforms (trading) exposures into (cleared) exposures. By using Monte Carlo simulations, we study how such function is affected by the clearing network’s topology, drawing conclusions on the risks faced by the CCP, and on the system’s margin requirements. Crown Copyright © 2012 Published by Elsevier B.V. All rights reserved.

1. Introduction The recent financial crisis prompted the notion that important parts of the financial infrastructure may need reforms. Not because the infrastructures experienced failures (on the contrary, they held up remarkably well during the turmoil), but because different or new infrastructures could have mitigated some of the problems which fuelled the crisis, among which were lack of information and lack of trust (Haldane, 2009). A main reason why well-designed financial infrastructures can be socially beneficial is that they can allow a better allocation of risks. Market participants typically seek market risk in the form of profit opportunities. Instead, counterparty risk is most often considered an unwanted feature of a market, a friction that participants are unwilling, and sometimes unprepared, to manage.1 Indeed, during the crisis several key markets (e.g. money markets) ceased to function precisely due counterparty risk, threatening the viability of the whole system.

夽 The views expressed in this paper are those of the authors, and not necessarily those of the BoE nor ECB. 夽 Most of the practical knowledge about CCPs behind this paper comes from Matthew Dive, who patiently took us through the infinite intricacies of clearing. We also thank Simon Debbage, Claire Halsall, Matthew Vital, Anne Wetherilt for suggestions. We owe interesting discussion and comments to Charles Kahn, Richard Sowers, Katzeteru Tao and to participants to the 7th Bank of Finland simulation Seminar (Helsinki, August 2009). The views expressed here are those of the authors; they do not necessarily reflect the views of their parent institutions. Kimmo Soramäki gratefully acknowledges a grant from Säästöpankkien TutkimussäätiBank. ∗ Corresponding author at: Bank of England, United Kingdom. E-mail address: [email protected] (M. Galbiati). A possible reason for this is that counterparty risk is more difficult to measure and control, as the events associated with it are more sudden and ‘lumpier’. 1

0167-2681/$ – see front matter. Crown Copyright © 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jebo.2012.05.013

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Table 1 Examples of network structures.

Euro CCP LCH EquityClear LSE Eurex LCH SwapClear ICE Clear Europe LCH RepoClear

GCMs

Indirect memb.

Tiering

25 37 58 22 50 42

>550 >530 235 80 157 80

>22 >14 4.1 3.6 3.1 1.9

It is then natural that central counterparties (CCPs) are playing a main role in the current debate on the infrastructures reform: CCPs are entities whose main function is to novate contracts between the trading parties, becoming the ‘seller to every buyer, and buyer to every seller.’ By doing so, CCPs relieve their clients of counterparty risk, which they themselves manage by, e.g. calling margins and collecting default funds to mutualize possible losses.2 While particularly topical at the moment, CCPs have been long at the centre of attention of Central Banks and regulators. The Committee for Payment and Settlement Systems (CPSS), working under the auspices of the Bank for International Settlements (BIS), published its first fact-finding on risks in securities settlement in 1998.3 Three years later, a set of recommendations for securities settlement systems was developed by the International Organization of Securities Commissioners (IOSCO).4 Another 3 years later, a new CPSS/ISOCO joint report defined recommendations for central counterparties. Here, the focus was on major risks that CCPs face,5 rather than on their systemic risk-reducing function. Most recently, both U.S. and European regulators have voiced their views that CCPs should play a more prominent role in securities clearing and settlement and that they should be prudently regulated: ‘First, we propose to require that all standardized derivative contracts be cleared through well-regulated central counterparties and executed either on regulated exchanges or regulated electronic trade execution systems.’ – Secretary Timothy F. Geithner before the House Financial Services and Agriculture Committees, Joint Hearing on Regulation of OTC Derivatives, 10 July 2009. and ‘Risks arising from connections [. . .] can also be addressed through the more widespread adoption of CCP clearing. [. . .] the authorities may need to be more active in facilitating [. . .] the transition from bilateral to central clearing in markets where CCP clearing is warranted but product standardization is not yet sufficiently advanced.’ – Bank of England FSR, June 2009. Finally, the G20 leaders agreed in Pittsburgh that all standardized over-the-counter derivative contracts should be traded on exchanges where appropriate, and cleared through CCPs by the end of 2012.6 The recently established Financial Stability Board is working towards this remit, via CPSS and IOSCO working groups. Much of the debate has revolved around the question of whether a CCP should be introduced or not, for several specific markets. Yet, the choice of CCP vs. non-CCP does not exhaust the range of possible alternatives, because a system cleared by a CCP can be organized in very different architectures. In the simplest case, all market participants directly connect to the CCP. In other, by far more frequent, cases the CCP clears for a restricted number of institutions – the General Clearing Members, or GCMs – which in turn clear for other participants, and so on in a hierarchy of tiers. Table 1 gives an idea of how varied clearing networks can be. At one extreme there is Euro CCP, which appears to be very ‘tiered’.7 At the other extreme, LCH RepoClear, which appears much ‘flatter’, as each GCMs serves about two indirect members on average. CCPs like ICE Clear Europe and Eurex are in-between. Fig. 2 (later on) graphically illustrates some possible clearing networks. Such differences in the clearing structure have only marginally been considered in the policy debate, and never studied in the literature. This paper looks at these issues for the first time, arguing that the question ‘which structure should there be around a CCP?’ is as relevant as the typical question ‘should a CCP be introduced at all?’ This is the first paper to place special emphasis on clearing arrangements as networks.8 However, it is by no means the first paper looking into the economics of clearing. The literature on this is too broad to be summarized here. However, some studies focused on the causes and the implications of CCP membership – hence, on tiering. They are thus most relevant here.

2 CCPs are better known as “clearing houses” (CHs). However while all CCPs are CHs, not all CHs are CCPs, as not all CHs offer novation, but sometimes only other clearing services. 3 ‘OTC derivatives: Settlement Procedures and Counterparty Risk Management,’ BIS/CPSS report. 4 ‘Recommendations for Securities Settlement Systems’ – CPSS/IOSCO report, 2001. 5 ‘Recommendations for Central Counterparties’ – CPSS/IOSCO report, 2004. 6 See http://www.pittsburghsummit.gov/documents/organization/129866.pdf. 7 Tiering is defined as the ratio between the number of indirect members and the number of GCMs, i.e. direct members. 8 Jackson and Manning (2007) and Duffie and Zhu (2009) are most close to our work, as discussed later.

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Tiering is often a consequence of choices on the part of the CCP, which may prefer to offer services to a restricted circle of high-credit quality counterparties. However, other participants of a clearing system may also benefit from a tiered arrangement. Jackson and Manning (2007) show that, if market participants differ in their credit quality and the CCP is not able to tailor margin requirements to individual default probabilities, then high credit quality agents will favour tiered arrangements. Loosely speaking, in a non-tiered arrangement high credit quality agents end up subsidizing agents of lower credit quality, as margins need to be set to a high level, to cover for the more likely defaults of the ‘bad’ members. So, while access criteria are typically decided by CCPs, their tightness could reflect the interest of those market participants with a comparatively high credit standing. On the other hand, Moser (2002) shows that there are also reasons why participants with a high credit reputation may favour open access policies: broadening participation may increase market depth and liquidity, increasing the ‘performance’ of the market. Like in Jackson and Manning (2007), broader participation may require higher collateral costs (to guard against more likely defaults by less solid counterparties), thus generating a tradeoff performance vs. costs.9 As for the consequences of different clearing arrangements, Jackson and Manning (2007) is again a key reference, focusing on: (i) the cost of posting collateral and (ii) the magnitude of replacement costs/the distribution of these costs among participants. A CCP is found to decrease overall costs as compared to a bilateral clearing arrangement, thanks to the ‘netting effect’, whereby net exposures are made smaller by multilateral clearing. Duffie and Zhu (2009) show that, paradoxically, CCP clearing may be less netting efficient than bilateral clearing. This may occur if there are different classes of contracts and central clearing is applied only to one of them. If one class of contracts is cleared centrally, this is carved out from the process of netting across classes, which instead takes place under bilateral clearing. So, if the netting within CCP-cleared contracts does not compensate the loss of netting across different classes, overall netting efficiency may decrease. Pirrong (2009) considers the effects of CCP netting on the distribution of losses following a default. The crux of the argument is that netting assigns priority to certain claims over others. For example: counterparty A netting against B is equivalent to A paying to B, and then immediately receiving back its money, thus obtaining priority on B’s assets (inclusive of A’s original payment). Pirrong’s point is that ‘to the extent that other creditors are financial institutions whose failure could pose systemic risks, it is not necessarily true that it is better to shift the burden of default from CCP members to other creditors’. The main novelties of the present paper are two. First, we present a general model of clearing, designed to study the mechanics of any clearing network. Second, we offer a numerical application of this model, to study the statistical properties of exposures under a broad spectrum of clearing networks. The numerical application allows us to consider not only average exposures, but also other important statistics – in particular, the incidence of “rare but possible” events. This paper limits itself to studying the mechanics of clearing, in the same spirit at as Duffie and Zhu (2009) and Jackson and Manning (2007). That is, the paper leaves aside the issue of defaults, and thus programmatically differs from e.g. Pirrong (2009). This also means that, while modelling margins, the paper does not explain how these latter are used, nor whether they are in any sense sufficient. Also, the clearing network is taken as given, and is considered to be independent of trading – while in reality the clearing structure and the trading behaviour do affect each other. By modelling networks, this paper may be broadly seen as part of the recent literature on the ‘topology’ of financial system including, among other studies: (i) analyses of financial infrastructures (Soramäki et al., 2007; Becher et al., 2008; Bech et al., 2008 and references therein, Galbiati and Giansante, 2010), (ii) analyses on interbank markets (Boss et al., 2004; Bech and Atalay, 2008; Rørdam and Bech, 2008; Iori et al., 2008; Akram and Christophersen, 2010; Heijmans et al., 2011; Wetherilt et al., 2009); sector and country level analysis (Castrén and Kavonius, 2009; Garrett et al., 2011). The paper proceeds as follows: Section 2 describes our model, Section 3 covers the results and Section 4 provides the conclusions. 2. Model 2.1. Original exposures We consider one future contract, traded on a market populated by N ‘counterparties’. The details of the contract are immaterial: all we say is that it produces exposures between participants, summarized in a N × N matrix T. Tij represents the nominal position of trader i against trader j. By referring to a future contract, we mean that whenever Tij = / 0, i and j are both exposed. So e.g. Tij < 0 should not be read as ‘i has a negative exposure (possibly is not exposed) to j’ but: ‘i is exposed to j, j is exposed to i, and the size of these exposures is given by the absolute value of Tij ’ (we can imagine that T12 = x means ‘1 is long x contracts against 2’, but the actual being long or short is irrelevant here).

9 Baer et al. (2002) find that collateral is de facto the primary risk-management tool and collateral requirements are the same for most members. Hence, introduction of new, riskier members increases collateral costs for all members.

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Fig. 1. Novation and clearing.

Trades are first bilaterally netted, so T is negative-symmetric: Tij = − Tji for all i, j.

2.2. Clearing networks The N counterparties are arranged in a directed network, where the link i → j means that i clears through j. We consider networks with a tree structure and three types of nodes:

(a) one CCP, with no ‘parent’ but with one or more ‘children’ of type (b); (b) two or more GCMs, ‘children’ of the CCP, and possibly ‘parents’ of nodes of type (c); (c) one or more clients, each with one GCM as a ‘parent’, and no ‘children’. For any node i, we denote its parent (clearer) by C(i). So e.g. C(C(i)) is the CCP, for any i. The network is thus fully described by the N × 1 vector C. We call  the set of GCMs.

2.3. Novation and clearing Novation is the replacement of exposures between non-adjacent nodes in the clearing network, with other exposures according to a precise rule. Clearing, on the other hand, consists in applying novation iteratively, until no further novation is possible. A formal definition of the clearing process will be given in a moment; Fig. 1 illustrates it graphically. In Fig. 1 left, i and j are reciprocally exposed. As they are non-adjacent in the clearing network, there are two cases: either i and j use the same clearer, or not. In the first case, exposure i ↔ j is novated by C(i), i.e. is replaced with two exposures of the same size as i ↔ j but of opposite signs, in such a way that all three involved parties maintain their original market positions. It can be noted that the replaced i ↔ j is ‘off’ the clearing network, while the new exposures i ↔ C(i) and C(i) ↔ j are ‘along’ the network (i.e. between adjacent nodes). In the second case, i and j have different clearers. Here, first C(i) novates i ↔ j into i ↔ C(i) and C(i) ↔ j (dotted). The latter exposure is then novated by C(j) into C(i) ↔ C(j) and C(j) ↔ (j). Note that the order of novation does not matter: C(j) may have been the first novator and C(i) the second. It can also be noted that one of the new exposures, C(i) ↔ C(j), is between non-adjacent nodes. The clearing process thus continues (not shown in Fig. 1), with C(i)’s clearer, the CCP, further novating

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such exposure like in case 1. Eventually, all exposures are along the network so no further novation is possible. At this point, the clearing process is complete. Formally, the clearing process is described by the following algorithm: {INITIALIZE T

Step 0 :



Step 1 :

IF Tij = 0 for each ∀i, j : j = / C(i)

THEN END

ELSE

Step 2

⎧ BE: (i, j) any pair such that j = / C(i) ⎪ ⎪ ⎪ ⎨ SET: T = T = 0 ij

Step 2 :

(1)

ji

VARY: TiC(i) = Tij = −TC(i)j ⎪ ⎪ ⎪ ⎩ GO TO Step 1

It is easy to see that the algorithm stops after a finite number of steps, and that the resulting matrix of exposures does not depend on the precise sequence of chosen pairs (i, j), while of course it depends on the network C. We can therefore see the clearing process as a function transforming matrices trading exposures into matrices of cleared exposures. We call this the clearing function. For practicity, we use T∗ to indicate both the function, and its value (computed at T). We now show how to obtain T∗ directly from T, without going through the definitory algorithm. This is useful to understand the determinants of cleared exposures. Given a GCM k, we denote the group of its clients by gk : gk = k ∪ {i : C(i) = k} Accordingly, we define k’s internalized exposures as Ik = {Tij : (i, j) ∈ gk × gk }

(2)

i.e. trades within k’s group. So, k’s non-internalized trades are NI k = {Tij : i ∈ gk ⇔ j ∈ / gk } Recalling that cleared exposures are only between participants and their respective clearers, a moment of reflection reveals that

T∗ =

⎧  ⎪ T.,. for i = CCP and j any GCM ⎪ ⎪ ⎪ ⎨ T .,. ∈NI k ⎪ ⎪ ⎪ ⎪ ⎩

Tjk

for i any GCM and j = C −1 (i)

(3)

k

0

otherwise

In plain English, the cleared exposure between: (i) the CCP and a GCM j is given by the sum of j’s non-internalized trades, (ii) a GCM i and any of its clients j is given by j’s trades; (iii) any other pair is zero. To verify (i), note that k’s internalized trades either (a) are never novated (those between k and its clients) or (b) are novated by k and then novated no further (the trades between k’s clients). Thus, internalized trades never ‘reach’ the CCP. Note also that client-GCM exposures are unaffected by the network topology, as they only depends on the client’s trades. This model of clearing, where exposures are added together, reflects the accounting practice of account pooling, whereby a clearer pools (i.e. sums) exposures from its own proprietary trading with exposures inherited from its clients. Alternative accounting methods exist, but we focus on account pooling for its simplicity and wide prevalence.10 2.4. Margins Clearers (CCP and GCMs) collect margins from the parties they clear for. For simplicity, here we assume that margins equal the absolute value of net exposures (scaling these by a constant factor would be immaterial). That is, the margin due by i to C(i) is: ∗ mi = |TiC(i) |

(4)

10 Under account segregation, clearers may keep separate proprietary and client trades, to clear them separately with the CCP. Modelling this is cumbersome, but conceptually simple: one needs to keep track of several Ts, one for proprietary, the other(s) for client exposures.

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This assumption makes our margins more similar to initial margins, than to variation margins.11 As our model has no defaults nor price movements, this simplifying assumption may be reasonable. We concentrate on tree networks; hence, clients only pay margins, as they do not novate contracts. GCMs, instead, pay margins to the CCP, and receive margins from clients. Finally, the CCP only receives margins, being the ultimate clearer. This implies that margins flow through the network from bottom up. In addition, a GCM may end up retaining part of its clients’ margins. Indeed, a GCM collects margins for all client trades, but pays margins only for the non-internalized ones. Margin needs can be defined under two assumptions: 1. Re-hypothecation. GCMs are allowed to ‘recycle’ the margins received from clients to pay margin to the CCP. In this case, the margin need of a participant is the net of margin due minus margin received. 2. No re-hypothecation. GCMs have to pay margins before receiving margins. Here, the margin need of a participant equals the margin due. Formally, the margin need of i is:

i =

⎧ ⎪ ⎨ ⎪ ⎩

⎧ ⎨

max



0, mi −

⎩

mk

k∈gi

⎫ ⎬ ⎭

mi

rehypotecation no rehypotecat.

where gi is the group of clients of i (if i is a client, we trivially have gi = i and i = mi ). Note that re-hypothecation is unrelated to both account pooling and internalization (Section 2.3). Pooling and internalization determine the size of the cleared exposures. Re-hypothecation instead determines how the corresponding margins calls can be met. We define systemic margin need as the value of margin payments required by the system as a whole:

R =



⎧ ⎨

max

i∈

NR =



mi ,

⎩

i =

i∈N





mk

k∈gi \i

mi

⎫ ⎬ ⎭

rehypotecation

(5)

no rehypotecat.

(6)

i∈N

where we remind,  is the set of GCMs. Under re-hypothecation, the systemic margin need is the sum of the margin needs by each the margin  group as a whole.

For group gi , this equals 

need of the GCM i plus the margin needs of its clients: max

0, mi −



k∈gi \i

mk

+



k∈gi \i

mk = max

mi ,



k∈gi \i

mk . Without re-hypothecation, the systemic margin need is

simply the sum of all individual margin needs. Summing up: margin needs depend on margins due m. . Margins due depend on T∗ (Eq. (3)), which in turn depends on C and on T. So, a clearing network C defines two margin need functions: one for the case with rehypotecation (R), the other for the case without (NR). Each such function maps trading exposures (T) into scalars. 2.5. Analysis of the model Sections 2.3 and 2.4 led to the definition of: (a) the clearing function T∗ ; (b) the margin need functions R and NR. Our aim is to study the properties of (a) and (b), depending on the network C. We do so using Monte Carlo simulations. We fix the number of participants N and proceed in four steps: 1. 2. 3. 4.

construct all possible clearing networks; randomly create many Ts, representing ‘days’; compute T∗ , R and NR for each network C and T; compute statistics of these functions across Ts, putting them in relation with network properties of the corresponding Cs. To perform step 4, we classify networks along two dimensions: tiering and concentration, as illustrated in Fig. 2.

11 Initial margins are collected as initial fee. Variation margins are called through time, marking contracts to market prices and settling any variation. Margin calls are also adjusted on the basis of perceived creditworthiness of counterparties.

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Fig. 2. Clearing networks.

The left networks display lower tiering than the right networks, as they have four GCMs instead of two. The bottom networks are less concentrated than the top networks, as their clients there equally split among GCMs. Formally, we define the tiering level simply as the number of clients: Tiering = N − 1 − | | as there is 1 CCP, and a number | | of GCMs. Concentration is measured by a Gini coefficient: | |   k

Concentration = 2

k=1

| |

− Sk



k

where Sk = s and in turn si = |gi − 1|/(N − 1 − | |) (relative size of i’s group), with groups gi s ordered by size (i.e. i < j i=1 i for |gi | < |gj |). Unfortunately, the mapping f(C) ≡ (tiering(C), concentration(C)) is not injective, i.e. different networks may feature the same tiering/concentration levels. However, any two networks with identical tiering and concentration turn out to produce extremely similar exposures and margin needs. Later on, we will therefore take average values across such ‘duplicate’ networks, and associate this average with corresponding given tiering/concentration pair. Summing up: we identify each network with a pair of tiering/concentration ‘coordinates’, and we present our results accordingly. 3. Results Given the key role of the CCP, we concentrate on its exposures, looking at its: 1. 2. 3. 4.

total expected exposure (expected across Ts, i.e. across ‘days’); single expected exposure (the above, averaged across GCMs); extreme exposures (largest across GCMs, and across days); The first two indicators tell us about ‘average’ risks. The third is instead most relevant if the CCP has to ensure against extreme events, as typically specified by the regulators. Results on the exposures of the GCMs (which include exposures to their clients) are summarized in Appendix A. We then consider margin needs looking at:

1. expected margin needs (expected across different realizations of the trading matrixes T, which can also be interpreted as ‘days’); 2. extreme margin needs (95th percentile across Ts).

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Fig. 3. CCP total expected exposure.

We consider a network with 20 counterparties and one CCP (N = 21). Such N is large enough to produce a large variety of clearing networks, while at the same time keeping their number manageable. With we have precisely 626 topologically different tree networks, with 2–20 GCMs, and concentration levels in [0, 1]. For each network we draw 3000 matrices T, obtain statistics of cleared exposures and margin requirements. We thus obtain about12 626 data points in the tiering–concentration space, showing how exposures and margin requirements depend on the network characteristics. Further details on simulations are in Appendix A.

3.1. CCP total expected exposure We first look at the expected (across days) total exposure of the CCP:

 CTE = E



 ∗ |TCCP,k |

(7)

k∈

Our first result is: • Result 1: tiering and concentration decrease CTE.

This is illustrated by Fig. 3. The contour plot shows CTE as a function of tiering (horizontal axis) and concentration (vertical axis). Darker shades correspond to lower CTE. The figure is ‘blank’ in the bottom left portion, because low-tiering networks can only display a limited range of concentration levels.13 The reason behind Result 1 is that that tiering and concentration both enhance internalization and ‘netting’. The formal argument is presented in Appendix A.

12 As said above, different networks may feature the same tiering/concentration levels. Of our 626 network, only 579 produce distinct tiering/concentration combinations. 13 For example: with 18 GCMs, there are only 2 clients, which cannot be uniformly distributed across the GCMs. So concentration is necessarily high. For 2–20 GCMs, there are respectively 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 networks, that is ‘points’ along the concentration dimension.

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Fig. 4. CCP’s single expected exposure.

3.2. CCP’s single expected exposure We now consider the expected (again across days) CCP exposure against the single, average GCM: CSE =

CTE | |

(8)

We have the following: • Result 2: tiering increases CSE, concentration decreases it. This is illustrated by Fig. 4, where again darker shades correspond to lower CSE, and the figure is ‘blank’ in the bottom left portion. The second part of Result 2 immediately follows from Result 1: concentration reduces CTS, so it clearly reduces (| | is fixed when varying concentration). The effect of tiering on CSE is less trivial. A tiered system has few and hence large GCMs, capable of creating large exposures. However, large GCMs also internalize more trades, so it is not entirely trivial that the total effect is on CSE is positive. A more precise argument to explain Result 2 is given in Appendix A. 3.3. CCP’s extreme exposures Sections 3.3 and 3.3.2 looked at average exposures, but more significant risks may reside in extreme exposures. We consider ‘extreme’ exposures in two senses: extreme across days (i.e. Ts) and across GCMs. 3.3.1. Largest exposure across GCMs The largest exposure of the CCP against an individual GCM is: ∗ CME = max{TCCP,k } k∈

An analytical expression for CME seems very difficult to reach: CME is the 1st order statistic of a vector of non-independent, different random variables, and analytical expressions in these cases are notoriously difficult to obtain.14 Intuition, as often for order statistics, does not go far; however, it seems likely that the distribution of the CME, at least for extreme values, is driven by the largest GCM. Song et al. (2012) give an analytical expression for the lim p(CME ≥ L), which confirms this. More L→∞

extensive discussion is in Appendix A.

14 GCMs may have a different number of clients. So, their exposures with the CCP are random draws from different distributions. To see non-independence, recall that a CCP-cleared trade comprises 2 symmetric legs with different GCMs, which therefore introduce correlation between CCP exposures.

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Fig. 5. CCP largest exposure (tiering). (For interpretation of the references to colour in text, the reader is referred to the web version of the article.)

Given these difficulties, we just report the results of our Monte Carlo simulations, without attempting formal proofs. Fig. 5 shows the distribution of CME for three levels of tiering.15 Interestingly, the Gamma-distribution yields an extremely good fit, especially for high tiering.16 The content of this figure is summarized as: • Result 3: tiering (i) decreases CME’s average, (ii) increases CME’s variance and (iii) increases CME’s skewness.17 That is: tiering decreases the CCP’s largest exposure. However, it makes it less predictable, and increases the chances of it being very large. Fig. 6 shows the effects of concentration. Here, the pdfs correspond to three systems with four GCMs each, and clients distributed as follows: maximum concentration (8, 0, 0, 0), medium concentration (5, 2, 1, 0) and minimum concentration (2, 2, 2, 2). Concentration appears to decrease the expected value of the maximum exposure, to reduce its variance, and to makes the right tail of the distribution thinner. We summarize the content of Fig. 6 as: • Result 4: concentration (i) decreases CME’s average, (ii) decreases CME’s variance, and reduces the likelihood of extreme CMEs. In synthesis, concentration reduces the risks stemming from the CCP largest exposure across GCMs.

3.3.2. Extreme-event exposures The other meaning of ‘large exposures’ is ‘in extreme market circumstances’. To study these, we concentrate on the tail of pdfs. Fig. 7, with the distribution of CTE (defined in Eq. (7)) for a given concentration and four different tiering levels,18 reveals that tiering has a beneficial effect, summarized as follows:

15 Fig. 5 is obtained for N = 13. The concentration level is kept at zero: in the three each of the 12, 4 or 2 GCMs (depending on the network) has respectively 0, 2 or 5 clients. These results are robust to changes in N (see Appendix A). 16 Song et al. (2012 – preliminary at the moment when this paper went to press) give the following analytical expression for the CME’s limiting distribution: 2 lim p(CME ≥ L) e−ˇL , where ˇ = 2/(N2 − 4|g∗ |) and |g∗ | is the largest GCM’s group size. This result, stating that the decay rate ˇ increases with the size of L→∞

the largest GCM, would actually imply that, in Fig. 5, both the 4- and the 12-GCM pdfs eventually fall below the 2-GCM pdf. Our Monte Carlo results show that, if true, this must be happening only for extremely large CMEs. Hence, if there really are exceptions Result 3, these have only theoretical relevance. 17 If the original exposures come from a distribution with finite support, this is simply shown: the maximum CCP-GCM exposure is finite, and it is an increasing function of the size of the GCM. 18 In Fig. 7 concentration equals to zero, i.e. all GCMs have the same size.

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Fig. 6. CCP largest exposure (concentration).

• Result 5: tiering reduces the likelihood of large total CCP exposures. It also reduces the variance of the total CCP exposure. Fig. 7 also confirms a finding of Fig. 3, i.e. that average falls with higher tiering. The same results obtain keeping tiering constant while increasing concentration: concentration reduces risks for the CCP.

Fig. 7. Total CCP exposure (CTE) – distribution across days.

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Fig. 8. Expected margin needs – re-hypothecation.

3.4. Margins We now move from exposures to margin needs. For both cases with and without re-hypothecation, we consider margin needs under average and extreme market circumstances. These results may shed light on whether (and which) clearing systems may produce liquidity stresses on their participants.

3.4.1. Expected margin needs – re-hypothecation Here we study the function R defined in Eq. (5). Fig. 8 is a contour plot of E[R] against tiering and concentration – where the expected value is taken, as usual, across Ts, i.e. across ‘days’. The content of Fig. 8 is summarized as follows: • Tiering first decreases, then increases margin needs. The effects of concentration are less clear cut. However, in most of the cases margin needs increase with it. A heuristic explanation of the pattern of Fig. 8 is given in Appendix A.

3.4.2. Expected margin needs – no re-hypothecation Margin needs in this case equal the sum of all margins paid in the system, i.e. the sum of all cleared exposures T∗ . Fig. 9 plots E[NR] (expected value taken across days) against tiering and concentration. Its content is reassumed as: • Without re-hypothecation, average margin needs depend non-monotonically on tiering and decrease with concentration. Without giving rigorous proof, Appendix A explains the pattern of Fig. 9. Predictably, margin needs are higher when re-hypothecation is not allowed. The size of this saving depends on the model’s parameters.19 However, such savings are higher when tiering is high, and when concentration is low. Indeed, when concentration is high (or tiering is low), large GCMs receive margin payments in excess of what they pay to the CCP. This entails a ‘waste’ of margins in terms of recycling, as part of the margins paid by clients are held by the GCMs and not recycled. Our main results on average margin needs are summarized as follows: • Result 6: the effects of tiering and concentration on margin needs are not univocal. They can be very different whether re-hypothecation is allowed or not. If re-hypothecation is not allowed, concentration may help save on margin needs. On the other hand, re-hypothecation is most powerful as margin-saving device when tiering is high and concentration is low.

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Fig. 9. Expected margin needs – no re-hypothecation.

Fig. 10. Extreme margin needs.

3.4.3. Extreme margin needs Having looked at average events, we now consider extreme conditions. Fig. 10 shows how clearing/concentration affects the 95th percentile of margin needs, when re-hypothecation is not allowed. Fig. 10 is very similar to Fig. 9. Although this similarity was not granted, it is un-surprising, as both figures are about the same random variable: Fig. 10 represents the 95th centile, Fig. 9 represents the average.

19

In particular, on the distribution from which Ts are drawn.

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In our model, margin needs without re-hypothecation equal the sum of all exposures in the system. So Fig. 10 also shows the total amount of resources needed, if each i is required to provide margin covering 100 • Result 7. (i) Systems with low concentration may have to put up comparatively large precautionary resources; (ii) Systems with an intermediate level of tiering may have to put up large precautionary resources. So, an increase in tiering may first increase and then decreases the needed amount of precautionary resources. 4. Conclusions This paper models clearing systems as networks whose function is to transform exposures. By looking at the mechanics of such networks, we study how their topology affects the resulting exposures and margin requirements. Our first set of findings, on the exposures faced by the CCP, are summarized in Table 2. These results suggest that the effect of tiering is rather complex. On one hand, tiering reduces the CCP total exposure, and the CCP’s average largest exposure across GCMs. On the other hand, it increases the typical CCP exposure towards a single GCM. Most importantly, tiering increases the likelihood of the largest CCP exposure being very large. The 5th CPSS-IOSCO Recommendation for CCPs states: ‘A CCP should maintain sufficient financial resources to withstand [. . .] a default by the participant to which it has the largest exposure in extreme but plausible market conditions [. . .]’ Our findings suggest that tiering increases precisely such exposure, it in extreme market conditions (while also increasing the variance of such exposure, making it less predictable). Hence, tiering increases the risk on which the 5th Recommendation focuses on. Equivalently, application of the 5th Recommendation implies that CCPs of a tiered systems should maintain larger precautionary resources. The effects of concentration on the CCP are instead beneficial, and very clear-cut. Concentration decreases risks for the CCP in all exposure-related aspects: it decreases the expected single exposure, the total exposure, the largest expected exposure, the variance such exposures and the skewness of their distribution (the right tail). This suggest that, for a given total volume being cleared, a CCP faces reduced risks when this volume is unequally divided by GCMs, as when some GCMs are ‘large’, and others are ‘small’. Our second set of results is on the amount of margin (collateral) absorbed by a clearing network. Analogous to liquidity in payment systems, collateral is the essential ‘lubricant’ for clearing systems. Looking at how much of it is needed is important from an efficiency point of view, but also from a ‘risk’ perspective. High margin requirements can put excessive strain on participants. Conversely, if a system can operate safely using less margin, the freed up collateral can be employed elsewhere with obvious benefits. An obvious fact is that a system requires more margins without re-hypothecation. But how tiering and concentration affect margin needs? An interesting finding here is that the relationship between tiering (or concentration) and margins (i) is non monotonic and (ii) crucially depends on whether re-hypothecation is allowed or not. In a system with re-hypothecation, concentration increases margin requirements, while tiering first decreases, and then increases them. If re-hypothecation is not allowed, exactly the opposite is true: concentration reduces margin requirements, while tiering first increases margin requirements, and then decreases them. This is summarized in Table 3. Table 2 Effect of network characteristics on exposures. CCP exposure

Effect

Total Average Likelihood of extreme realizations Single Average Max (across GCMs) Average Likelihood of extreme realizations

Tiering

Concent.

Negative Negative

Negative Negative

Positive

Negative

Negative Positive

Negative Negative

Table 3 Effect of network characteristics on margins. Margin requirements

Rehypotecation Non-rehypotecation

Effect Tiering

Concent.

Neg/pos Pos/neg

Positive Negative

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Finally, we assume that each participant is required to hold enough resources to withstand ‘extreme market events’, and look at how this prudential buffer depends on tiering and concentration. Concentration is found to have a beneficial effect, decreasing the size of the required buffer. Tiering instead has a non-monotonic effect: the needed buffer is lowest for extreme tiering levels, and reaches a maximum for intermediate levels. Needless to say, all our results depend on the assumption made. A key assumption is the single account hypothesis. In the model each GCM holds a unique account at the CCP, where proprietary and client trade are pooled together. A second important assumption is the statistical independence of pre-clearing exposures. This could be easily removed in the numerical exercise, but it would then be difficult to decide how this assumption should be removed. If traders are not ex-ante identical, then the way they are distributed among GCMs would mostly likely matter. In other words, tiering and concentration would be insufficient to describe a clearing network, and we could not express our results in terms of these two simple dimensions. A third important assumption is that margins are a simple (linear) function of exposures. This is a reasonable approximation for initial margins, but probably less so for variation margins. Less crucial is instead the choice of the system size, as shown by robustness checks in Appendix A. Last but not least, we have assumed that trading (T) does not depend on the clearing network (C). In reality, exposures generated from trading may well depend on the clearing infrastructure which supports them. Appendix A. A.1. Generation of random Ts / j are iid random draws from a Normal (0,1). We draw our instances of T in the simplest way: all Tij : i = This seems a reasonable assumption for exposures in a centrally cleared market, where traders do not distinguish between counterparties, and indeed, trading platforms ensure anonymity and ‘fungibility’ of contracts. This assumption is more questionable for an OTC market, where preferential trading partnerships may emerge. These would give rise to, e.g. sparse matrices T, or at least some asymmetry in the trading exposures. A.2. GCM exposures GCMs can differ in the number of their clients. So, for given tiering and concentration levels, it is impossible to represent in a compact way the cleared exposures of all GCMs. However, cleared exposures of a single GCM have a simple analytical form. A.2.1. Cleared exposures towards the CCP Consider a GCM i, with a group of S members. It is easy to see that |NIi | = NS − S2 . Hence, i’s exposure to the CCP is the sum of NS − S2 normal random variables, so: ∗ Ti,CCP ∼N(0, NS − S 2 ) ∗ Hence, the expected absolute value of Ti,CCP scales with NS − S2 , so, it first increases, then decreases in S. Intuitively, the reason is the following. At first, acquiring clients creates more non-internalized exposures, so i’s exposure to the CCP grows. But after a while internalization kicks in; |NIi | starts to fall, and so does i’s exposure.

A.2.2. Cleared exposures towards clients Consider a GCM i. Its cleared exposure to a client j is a random variable ∗ Ti,j ∼N(0, N − 1)

as, indeed, it is the sum of all the exposures inherited from i. More difficult to determine is i’s largest exposure across clients. Formally, if i has S clients, its largest exposure is the 1st order statistic of a vector of S normally distributed random variables, which may be correlated. Indeed, if both j and k clear at i, the exposures (i, j) and (i, k) are sums of normal r.v.s. with a term in common, albeit of opposite sign, i.e. the exposure (j, k). Such correlation makes it very difficult to obtain an analytical expression for the maximum exposure’s distribution. A.3. Proofs and explanations of some results To simplify notation, given a set of exposures A, we adopt the following shorthand:

 T ∈A

T = ˙A

(9)

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Fig. 11. Non-internalized trades when i becomes a GCM.

A.3.1. Proof of Result 1: tiering and concentration decrease CTE To prove that tiering decreases CTE, consider a network C, and a network C obtained from C by ‘promoting’ i, client of r, to GCM status - so C is more tiered than C . Recalling the definition of CTE (Eq. (7)) and Eq. (3) and using the shorthand defined in (9) we can write CTE in both networks as:

 CTE(C) = E abs(˙NI r ) +



 abs(˙NI t )

t∈ \r

and

 

CTE(C ) = E

abs(˙NI i ) + abs(˙NI r ) +



 abs(˙NI t )

t∈ \r

where the ‘prime’ indicates non-internalized payments in the network C . Because NI t = NI t for each t ∈  \ r, let us compare the different terms A = abs(˙NIr ) and B = abs(˙NI i ) + abs(˙NI r ). Fig. 11 makes clear that the sums in B contain, together, all the terms in A, plus some others (to be precise, those Tij : j ∈ gr ). Thus, because abs(x) + abs(y) ≥ abs(x + y), we have B ≥ A. Hence, CTE(C ) ≥ CTE(C). Let us now prove that concentration decreases CTE. Start from a network C and increase concentration by moving a client i from a correspondent k to a correspondent q with an equal or larger group of clients. It is obvious that |NI | ≤ |NI|, i.e. the number of exposures contributing to CTE(C ) is no larger than the number of those contributing to CTE(C). Hence, as expectation is taken, CTE(C ) ≤ CTE(C). A.3.2. Explanation of Result 2: tiering increases CSE To simplify, we instead show that tiering increases the ratio NI/| |, i.e. non-internalized trades per GCM. To see this, consider then a network where the N participants (ignore the CCP) are equally split across two GCMs. Then, NI = (N/2)2 = f(N). Increase the number of GCMs to four, splitting each group in two equal-sized groups. This gives f(N/2) = (N/4)2 new crossgroup exposures which, summed to the previous ones, gives a total of 2(N/4)2 non-internalized exposures. Hence, as splitting proceeds, NI grows as a geometric series whose kth term is N2 /2k+1 . So NI increases with | |, but at a decreasing rate; hence NI/| | falls with | |. A.3.3. Explanation of claim: ‘with re-hypothecation, tiering first decreases, then increases systemic margin needs’ – pp. 3.4.1 To intuitively see this, recall that R =



max{mi , ˙k∈gi \i mk }

i∈



When tiering is low, groups are small. Hence for each correspondent i we should expect mi > ˙k∈gi \i mk so R = i∈ mi . As tiering decreases, members move out of the top tier, so the summation in R loses terms and system margin needs fall. In the meantime, some GCMs gain clients so, at certain point mi = ˙k∈gi \i mk for some GCM i. At that point, as new clients join i, start to grow, max{mi , ˙k∈gi \i mk } starts to grow. That is, a term of the summation in R starts to grow, and so does R.

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Fig. 12. Largest CCP exposure - 100 members (compare with Fig. 5). (For interpretation of the references to colour in text, the reader is referred to the web version of the article.)

A.3.4. Explanation of claim: ‘without re-hypothecation, average margin needs depend non-monotonically on tiering (a) and decrease with concentration (b)’ – pp. 3.4.2 The reason behind (b) is the following. The margin needs of clients are unaffected by C. Hence, to assess the effect of concentration on margins we only need to consider the effect of concentration on the margins paid by the GCMs. These equal (a multiple of) the total exposure of the CCP. Fig. 3 and the discussion therein showed that the CCP total exposure decreases with concentration. Hence, so does the aggregate GCMs margin need, and the systemic margin need. The reason behind (a) is the following. As tiering increases, the total margins paid by the GCMs fall (see previous paragraph). On the other hand, the bottom tier becomes larger, so more margins are paid by the bottom tier to the GCMs. So, there are two countervailing effects on total margin requirements. It is not surprising that the tiering-margin relationship is non-monotonic, as one effect or the other prevails at opposite extremes of tiering. A.4. Analytical result on CME and robustness checks Song et al. (2012), in a work still preliminary when this paper went to press, give the following analytical expression for the CME’s limiting distribution: lim p(CME ≥ L) e−ˇL

2

L→∞

where ˇ = 2/(N2 − 4|g∗ |) and |g∗ | is the largest GCM’s group size. This result states that the probability of observing very large CMEs decays faster, when the ‘top’ GCM is larger. If confirmed, this would actually imply that extremely large CMEs are more likely in highly tiered systems. In Figs. 5 and 12, this would mean that the red curves eventually fall below all other curves. We could not observe this reversal in none of the systems we simulated; we estimate that, if this reversal really does take place, it can only be happening at events whose probability is smaller than 10−7 (for CMEs happening say 1 day every 30,000 years). Thus we conclude that, if our Result 3 admits exceptions, these are relevant only from a theoretical point of view. Fig. 12 illustrates large exposures for N = 101, showing that the substance of our results is unaffected by the system size (similar figures result from other values of N as well). References Akram, F., Christophersen, C., 2010. Interbank overnight interest rates – gains from systemic importance. Norges Bank Working Paper 11/2010. Baer, H., France, V.G., Moser, J.T., 2002. Opportunity cost and prudentiality: an analysis of collateral decisions in bilateral and multilateral settings. Federal Reserve Bank of Chicago Working Paper October 2001. Bech, M., Chapman, J.T.E., Garratt, R., 2008. Which Bank Is the ‘Central’ Bank? An Application of Markov Theory to the Canadian Large Value Transfer System. FRBNY Staff Report No. 356. Bech, M.L., Atalay, E., 2008. The Topology of the Federal Funds Market. FRBNY Staff Report No. 354. Rørdam, K.B., Bech, M., 2008. The topology of Danish interbank money flows. FRU Working Papers 2009/01. University of Copenhagen, Department of Economics, Finance Research Unit. Becher, C., Millard, S., Soramäki, K., 2008. The network topology of CHAPS sterling. Bank of England Working Paper No. 355. Boss, M., Elsinger, H., Thurner, S., Summer, M., 2004. Network topology of the interbank market. Quantitative Finance 4, 1–8.

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