Does a central clearing counterparty reduce liquidity needs?
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Abstract
This study investigates whether and how central clearing influences the overall liquidity needs in a network of financial obligations. Utilizing the approach of flow network theory, we show that the effect of adding a central clearing counterparty (CCP) is decomposed into two effects: central routing, and central netting effects. Each effect can produce different liquidity needs according to different liquidity scenarios. The analysis indicates that adding a CCP in times of financial distress successfully reduces the overall liquidity needs if and only if the netting efficiency of the CCP is sufficiently high. Furthermore, once the economy is no longer in financial distress, higher netting efficiency of the CCP could conversely increase the overall liquidity needs. The results have implications for the effectiveness of CCPs in mitigating systemic risk in times of financial distress, and their operating costs once the distress has passed.
1 Introduction
In the recent financial crisis, the collapse of financial institutions, such as Bear Stearns and Lehman Brothers, demonstrated that the interconnected feature of bilateral exposure among financial institutions could lead to market disruption. In order to cope with this apparent vulnerability, the G20 leaders agreed at the 2009 Pittsburgh Summit that standardized overthecounter (OTC) derivatives should be cleared through central clearing counterparties (CCPs). Central clearing is expected to help mitigate counterparty credit risk by removing the direct risk exposure between counterparties, thereby reducing the systemic risk of the “domino” of defaults and relevant firesales.^{1}
However, the success and relevant cost of central clearing is never evident. For example, CCPs typically require margin themselves in order to bear the counterparty risk arising from cleared derivative transactions. In times of financial distress, margin requirements by CCPs could trigger firesales.^{2} Here, we should note that multilateral netting by CCPs could have already reduced relevant exposure, and have contributed to reduce the required margin compared to settlements without CCPs. The effect of CCPs on overall liquidity needs is not apparent until the two aspects are investigated in a consolidated manner.
We further argue about the operating cost of CCPs once the economy is no longer in financial distress. CCPs could affect overall liquidity needs in times of nondistress, and larger liquidity needs tend to imply larger costs, since liquidity is essentially scarce resource. In times of nondistress, when financial institutions are not in a rush to obtain liquidity, contracted trades would need less liquidity, or similarly, liquidity would circulate more efficiently among relevant financial institutions even without CCPs. Consequently, CCPs and their multilateral netting could have different effects on overall liquidity needs compared with those in times of financial distress.
In order to assess the effects of CCPs and discuss further improvements of the clearing mechanism, it is important to understand the essential nature of CCPs regarding how they affect overall liquidity needs in times of both financial distress and nondistress. This study develops a stylized model to probe this issue. Our focus is on how the introduction of a CCP could alter the interconnected feature of the relevant network of financial obligations, and how the change of network topology could affect overall liquidity needs. From the perspective of network topology, the introduction of a CCP serves as an additional entity itself in the relevant network, while its multilateral offsetting serves to eliminate relevant obligations. We explicitly show that the effect of a CCP is decomposed into two effects: the central routing effect, and the central netting effect, such that the total effect is essentially the addition of the two effects.
The effect of a CCP is examined on the basis of two polar liquidity scenarios. One is assumed as a situation in times of financial distress, by which liquidity circulates least efficiently. The other is assumed as a benchmark situation in times of nondistress, by which liquidity circulates most efficiently. We refer to the former situation as the bad environment and the latter as the good environment. We show that in the bad environment, the central routing effect is always negative, but the central netting effect is always positive. A negative central routing effect means that adding a CCP certainly increases the overall liquidity needs if there is no financial obligation to be offset by the CCP. A positive central netting effect means that larger offset amounts lead to smaller liquidity needs. This implies that the total effect of a CCP is positive in the bad environment when the offset amount is sufficiently large. By contrast, we show that it is possible for the central netting effect in the good environment to be negative, whereby, although counterintuitive, a larger offset amount could lead to larger liquidity needs. This is because eliminating financial obligations could effectively separate a connected network into multiple disconnected networks, thereby inhibiting the same liquidity from circulating through the whole network. We observe a tradeoff of multilateral offsetting regarding the overall liquidity needs. It is possible for a CCP to reduce liquidity needs during times of financial distress, thereby reducing the risk of firesales and relevant systemic risk of the “domino” of defaults. However, it could conversely increase overall liquidity needs during times of nondistress. For our benchmark situations, in order for the total effect of a CCP to be always positive in the bad environment, more than twothirds of the relevant trading needs to be offset. Since the overall liquidity needs in the good environment could become larger in proportion to the offset amount, the tradeoff could become serious.
There are two policy implications of this research. First, when central clearing is used, the expected offset efficiency for the relevant security should be examined with sufficient care, since insufficient netting efficiency could have an adverse effect. Second, possible severe tradeoff associated with multilateral netting suggests conditional utilization of a CCP. Since our analysis shows that multilateral netting could be costly at the time of nondistress but helps mitigate liquidity needs at the time of financial distress, a CCP could be used as an emergency scheme. Although it is out of the scope of this study to argue about the whole cost of such an emergency scheme, our analysis indicates there is possible merit in resolving the underlying tradeoff regarding overall liquidity needs.
1.1 Relevant literature and contributions
The role of CCPs has been examined largely focusing on how much relevant exposure is reduced through CCPs’ multilateral nettings, supposing that smaller amount of exposure to each counterparty implies smaller counterparty risk. This study departs from the literature by examining the roles of CCPs in overall liquidity needs.
Focusing on the effect on exposure, Duffie and Zhu (2011) examine central clearing in derivative markets and point out the possible disadvantage of central clearing arrangements compared with bilateral clearing arrangements when central clearing is provided only within each class of derivatives. The roles of CCPs in derivative markets are debated in Bliss and Kaufman (2006), Bliss and Steigerwald (2006), and Pirrong (2009). Jackson and Manning (2007) argue about the effects of CCPs in relation to “tiering,” which refers to the ratio between the number of indirect and direct members of CCPs. The authors argue there is incentive for a “tiered” structure. Galbiati and Soramäki (2012) analyze the implications of “tiering” in terms of network topology. In view of relevant network topologies, they examine the tree structure, while more general structures matter in our liquidity context.
Several recent studies have examined the role of CCPs in affecting overall liquidity needs, in the context of how CCPs set their margins. Murphy et al. (2014) investigate the procyclical nature of various margin models, proposing quantitative measures of procyclicality. Abruzzo and Park (2016) empirically analyze the marginsetting behavior of CCPs, and find that margininduced procyclicality is a concern during recessions, but not during times of expansion. Miglietta et al. (2015) quantify the impact on the cost of funding in repo markets of the initial margins applied by CCPs. These studies have helped clarify the possible negative effects of CCPs on overall liquidity needs, but they have not explicitly shown how the existence of CCPs could have negative effects compared with cases without CCPs.
This study provides a stylized model that enables us to compare a situation with a CCP with that without a CCP. Our focus is on the effects of CCPs on overall liquidity needs by affecting the relevant network topology. For this purpose, we utilize liquidity problems defined in the network, which are formally represented as problems of flow network. The model and liquidity problems used in this study are based on Hayakawa (2016), who investigates settlement efficiency of gross settlements in view of network topology.^{3} The present study serves as an application of Hayakawa (2016) to examine the role of CCPs, utilizing several basic results of the study.
Section 2 presents our model. Section 3 illustrates our analysis in a less formal manner. Section 4 provides an overview of the results. Section 5 shows our formal analysis. Section 6 concludes. The appendix includes proofs of the relevant results.
2 Model
There are a finite number of financial institutions that have obligations among themselves. Each obligation is formed by the trade of either of two types of securities. One type is called the target security, and the other the nontarget security. The target security is traded either with a CCP or bilaterally, while the nontarget security is traded bilaterally. We incorporate obligations for the nontarget security in order to analyze externality of the CCP in the relevant settlements. As becomes clearer soon in this section, both types of security are assumed to be of a bond or equity type in the sense that the amount of obligation formed by each trade is fixed. The simplification serves to clarify the nature of a CCP, and has implications for the debate about CCPs in derivative contracts and relevant margin issues, which are discussed in Sect. 2.4, and further in the analysis section.

Stage 0: Trade formation

Stage 1: CCP scheme/bilateral scheme

Stage 2: Offsetting under the CCP scheme^{4}

Stage 3: Settlements of obligations
2.1 Stage 0: Trade formation
At Stage 0, trades among the financial institutions are exogenously given. The contracts of the trades specify that all relevant obligations are finally settled at Stage 3. We express the obligations formed at Stage 0 utilizing a flow network representation.^{5}
The obligations are expressed with a network \(\mathcal {N}=\langle V,A,f \rangle \). V specifies a set of vertices, which corresponds to \(V1\) number of the financial institutions added with one hypothetical entity. The role of the hypothetical entity is stated shortly in this section. \(A=\{(v,v',n)v,v'\in V, n=1,2,\ldots \}\) specifies a set of arcs, where each arc \((v,v',k)\) shows that v has some amount of obligation to \(v'\), and k is used as an index that distinguishes among multiple arcs for \((v,v')\). The indices are usually not mentioned in order to avoid being notationally cumbersome. Then, \(f:A\rightarrow R_{+}\) expresses the amount of the relevant obligations. We assume that all the obligations for the trades of the target security are formed against the hypothetical entity. Using the introduced notations, \((\mathcal {N}= \langle V,A,f \rangle , v^*\in V)\) is exogenously given at Stage 0, which we call a financial system with hypothetical entity \(v^*\).
We confine ourselves to a balanced network throughout this study. We say a network \(\langle V,A,f \rangle \) is balanced when for each vertex \(v\in V\), the total amount of obligations owed by v and those owed to v are equal, that is, \(\sum _{v'\in V} f((v',v))= \sum _{v'\in V}f((v,v'))\) for every \(v\in V\). This is explicitly stated as Assumption 1. The assumption of balanced networks has dual roles. On the one hand, it implies market clearing regarding the trades of the target security, since the total amount of obligations owed to and by the hypothetical entity are supposed to be equal. On the other hand, a balanced network at Stage 0 derives a balanced network at later stages. There, we examine relevant liquidity needs based on each network, and the assumption of the balanced network serves to simplify our analysis.^{6}
Assumption 1
Given a financial system \((\mathcal {N}= \langle V,A,f \rangle , v^*\in V)\), network \(\mathcal {N}\) is balanced.
We further assume that each of the obligations owed to and by the hypothetical entity is the same amount, that is, the price of the target security is supposed to be fixed, and each trade is made in the same unit. The assumption is formally stated as follows.
Assumption 2
 i)
(obligations owed by \(v^*\)) \(f((v^*,v))=m\), for every \(v\in V\) such that \((v^*,v)\in A\), and
 ii)
(obligations owed to \(v^*\)) \(f((v',v^*))=m\), for every \(v'\in V\) such that \((v',v^*)\in A\).
Assumption 3
 i)
for every \(v\in V{\setminus } v^*\), the number of “buys” of the target security by v is equal to or less than the number of “sells” of the target security by the other financial institutions. Formally, \(\{(v,v^*,k)\in Ak=1,2,.., \} \le \{ (v^*,v',k)\in A k=1,2,.., v'\in V, v'\ne v \}\).
 ii)
for every \(v\in V{\setminus } v^*\), the number of “sells” of the target security by v is equal to or less than the number of “buys” of the target security by the other financial institutions. Formally, \(\{(v^*,v,k)\in Ak=1,2,.., \} \le \{ (v',v^*,k)\in A k=1,2,\ldots , v'\in V, v'\ne v \}\).
2.2 Stage 1: CCP scheme/bilateral scheme
At Stage 1, the hypothetical entity is materialized under either of the two schemes: the CCP scheme and the bilateral scheme. For the CCP scheme, the hypothetical entity is itself reinterpreted as a CCP for the target security. Thus, the network given at Stage 0 is unchanged. When the given network as Stage 0 is that shown in Fig. 1 with hypothetical entity \(v_c\), \(v_c\) is reinterpreted simply as the CCP under the CCP scheme at Stage 1. Note that the CCP does not offset obligations at Stage 1, but it will do so at Stage 2. In order to simplify relevant statements, we refer to a network derived under the CCP scheme at Stage 1 as a CCP network.
For the bilateral scheme, the hypothetical entity is made to vanish, and we examine all the possible bilateral trades for the target security. We refer to networks that are derived under the bilateral scheme as bilateral networks. Specifically, when the financial system shown in Fig. 1 is given at Stage 0 with hypothetical entity \(v_c\), we derive four bilateral networks, as shown in Fig. 3. Observe that in each bilateral network, each obligation \((v,v_c)\), \(v\in \{v_b, v_d,v_f\}\) that is previously owed to the hypothetical entity is paired with an obligation \((v_c,v')\), \(v'\in \{v_a, v_e,v_f\}\) that is previously owed by the hypothetical entity, and the pair of arcs is replaced with a new arc \((v,v')\). For the network shown on the upperleft in Fig. 3, the relevant arcs are paired such that \(\{(v_b,v_c),(v_c,v_f) \}, \{(v_f,v_c),(v_c,v_a) \},\{(v_d,v_c),(v_c,v_e) \} \), and each of the pairs is replaced by each of \((v_b,v_f)\), \((v_f,v_a)\), and \((v_d,v_e)\).
In general, for a given financial system \((\mathcal {N}= \langle V,A,f \rangle , v^*\in V)\), let \(A_{to}\subset A\) denote a set of obligations owed to the hypothetical entity, and \(A_{by}\subset A\) denote a set of obligations owed by the hypothetical entity. Note that \(A_{to}=A_{by}\).^{7} Consider all the possible onetoone matchings between \(A_{to}\) and \(A_{by}\). For each matching, let each pair of arcs \(\{ (v,v^*), (v^*,v')\}\), \(v,v'\in V\) be replaced with a new arc \((v,v')\). Here, we exclude any matching that includes a pair of arcs regarding the same vertex such that \(\{(v,v^*), (v^*,v)\}\), \(v\in V\).^{8} The reason for the exclusion of such matching is that the pair of obligations \(\{(v,v^*), (v^*,v)\}\) needs be replaced with vertex v, which effectively assumes offsetting between the paired obligations. We assume any bilateral or multilateral offsetting is not executed under the bilateral scheme.
2.3 Stage 2: Offsetting under the CCP scheme
2.4 Stage 3: Settlement of obligations
 Procedure to set \(\{p_v(s)\}_{v\in V}\)

\(\cdot \)Let \(\langle V,A,f \rangle \) and \(s:A\rightarrow \{1,2,..,A\}\) be given.

\(\cdot \)Let \(k=0,1,2,\ldots ,\) show the current order for the relevant settlements.

\(\cdot \)Let \(p^h_v(k)\) indicate the current liquidity holding.

\(\cdot \)Let \(p^d_v(k)\) indicate the current liquidity needs.

 Initialization

\(\cdot \) Set \(p^h_v(0)=0\) and \(p^d_v(0)=0\), for every \(v\in V\).

\(\cdot \) Set \(k=1\).

 Main Procedure
 \(\cdot \) Take \(a=(v,v')\in A\) such that \(s(a)=k\).
 i)For the payer v, set \(p^d_v(k)\) and \(p^h_v(k)\) as follows.

\(\cdot \; p^d_v(k)= max(f(a) p^h_v(k1),0)\),

\(\cdot \; p^h_v(k) = max(p^h_v(k1) f(a), 0)\).

 ii)For the receiver \(v'\), set \(p^d_{v'}(k)\) and \(p^h_{v'}(k)\) as follows.

\(\cdot \; p^d_{v'}(k)=0\),

\(\cdot \; p^h_{v'}(k)=p^h_{v'}(k1)+f(a)\).

 iii)For the other \(v''\in V{\setminus } \{v,v'\}\), set \(p^d_{v''}(k)=0\) and \(p^h_{v''}(k)=p^h_{v''}(k1)\).

\(\cdot \) Update the current order as \(k:=k+1\).

\(\cdot \) If \(k>A\), proceed to the finalization; otherwise, repeat the main procedure with the updated k.

 i)

 Finalization

\(\cdot \) For each \(v\in V\), set \(p_v(s)=\sum _{k=1}^{A} p^d_v(k)\).

2.4.1 Liquidity scenarios
Formally, the total liquidity needs in each environment is derived by each of the following minimization and maximization problems.
(Liquidity problem for the good environment) ^{10}
Given network \(\langle V,A,f \rangle \), take onetoone mapping \(s:A\rightarrow \{1,2,\ldots ,A\}\) and associated \(\{p_v(s)\}_{v\in V}\) such that
\(\min _{s} \sum _{v\in V}p_{v}(s)\).
(Liquidity problem for the bad environment)
Given network \(\langle V,A,f \rangle \), take onetoone mapping \(s:A\rightarrow \{1,2,\ldots ,A\}\) and associated \(\{p_v(s)\}_{v\in V}\) such that
\(\max _{s} \sum _{v\in V}p_{v}(s)\).
In other words, the liquidity problem for the good (bad) environment derives the minimum (maximum) total liquidity needs with respect to every possible order of settlements. For clarity of the problems, take the networks shown in Fig. 3 as inputs for each of the minimization and maximization problems. Then, Fig. 8 shows relevant settlements in the good environment, and Fig. 9 shows those in the bad environment.
3 Illustrative examples
We argue that the effect of a CCP needs be examined carefully in each liquidity scenario. We show that in the bad environment, a CCP increases overall liquidity needs if the netting efficiency is not sufficiently high. Still, higher netting efficiency contributes to decrease liquidity needs in the bad environment, and sufficiently high netting efficiency ensures that the CCP decreases the total liquidity needs compared to the corresponding bilateral settlements. However, when we consider the good environment, higher netting efficiency does not necessarily serve to decrease overall liquidity needs. Actually, it is possible that higher netting efficiency leads to larger overall liquidity needs. Thus, there is a possible tradeoff regarding the netting service provided by the CCP.
The tradeoff of the multilateral offsetting by the CCP is better understood by decomposing the total effect of the CCP into two types. One type is referred to as the central routing effect, and the other as the central netting effect. The base result shown in the analysis section is summarized as follows.
(Total effect) \(=\) (Central netting effect) \(+\) (Central routing effect).
The base result essentially states that the total effect is quantitatively decomposed into the two effects, such that the two effects are additive with each other.
3.1 Central routing effect
For financial system (A), there is no offsetting under the CCP scheme, which lets the netout CCP network be equal to the CCP network. Thus, the total effect of the CCP for financial system (A) equals the central routing effect. Furthermore, the central routing effect of the CCP for each financial system (B) and (C) is equal to that for financial system (A), from the observation that each netout CCP network for financial system (B) and (C) is essentially the same as the CCP network for financial system (A). This is formally shown in the analysis section.
The intuition for the results presented above is as follows. In the good environment, the CCP tends to provide additional routes for liquidity to circulate more efficiently. This is well illustrated in Fig. 14. For the bilateral network shown on the right of the figure, there are four mutually isolated cycles of obligations. When we turn to the netout CCP network shown on the left, we observe that the CCP serves to connect two of the previous isolated cycles to let them form one cycle. The same liquidity can now circulate throughout the united cycle. By contrast, in the bad environment, the CCP provides an additional stop for liquidity, which always increases the total liquidity needs. This is easier to observe in Fig. 11. Each of the bilateral and netout CCP networks consists of three mutually isolated cycles. The difference is that for the netout CCP network, there is one additional vertex for one of the cycles (the cycle at the center). This actually increases the liquidity needs for the relevant cycle from 30 to 40.
The central routing effect clarifies a negative aspect of adding a CCP during times of financial distress. For the case of derivative contracts, the CCP tends to demand liquidity in the form of margin. Suppose that the relevant derivatives are traded without any CCP; then, it is possible that the direct counterparty instead of the CCP demands margin, intending to ensure counterparty risk. Suppose that for some derivative trade, the margin demanded by the CCP is the same level as that demanded by the direct counterparty; then, there is no change in the amount of the required margin for the trade itself. However, during times of financial distress when margin requirement is prevalent regardless of the types of derivatives, adding a CCP indicates that the number of institutions that require margin effectively increases in total. This is interpreted as a negative externality of adding a CCP, which is well demonstrated by the central routing effect.
3.2 Central netting effect
Although we observe a negative aspect of the central netting effect in Figs. 15 and 16, it could conversely have a positive effect. We illustrate this point using financial systems (D) and (E), shown in Fig. 17. Note that (D) is the standard financial system for (E). Thus, the central routing effect for (E) is captured through (D). Figure 18 shows example bilateral networks for our examination of the central netting effect. We observe that the central netting effect is positive in the good environment. We observe three mutually separated cycles for (E) and two for (D). The liquidity needs are reduced by reducing one cycle.
Suppose each of isolated vertices \(v_f\) and \(v_g\) in (E) forms a cycle with additional vertices and arcs through the trades of the nontarget security, as is the case for financial system (C). Then, whether the central netting effect in the good environment is positive or negative depends on the amount of obligations for the nontarget security. Actually, we show that it is positive when the amount of obligations for the nontarget security is sufficiently small (financial system (E) is understood as an extreme case such that there is no relevant trade of the nontarget security.).
4 Overview of the results
We briefly overview the analysis and relevant results presented in the next section. In the analysis, first, the decomposition of the total effect of adding a CCP is formally presented. Although we illustrate the decomposition in Sect. 3 using example bilateral networks for given financial system, Theorem 1 ensures that the decomposition is well defined in the sense that the decomposition is applied to all the possible bilateral networks consistently. The decomposition serves as the analytical basis for showing our relevant results.
Theorem 2 shows the general results for the central routing effect both in the good and bad environment, and Theorem 4 shows the general results for the central netting effect in the bad environment. When we focus on the results in the bad environment, the combination of Theorems 2 and 4 implies that in order for a CCP to have a positive total effect during times of financial distress, it needs to provide sufficiently high netting efficiency.
For the quantitative aspect regarding how much netting efficiency is needed for a CCP to have a positive effect in the bad environment, we introduce a specific class of financial systems to capture the interconnected feature of real world networks of financial obligations. For the specific class, Theorem 3 shows the quantitative aspect of the central routing effect, while Theorem 5 shows the quantitative aspect of the central netting effect. The results for the combined total effect are summarized in Theorem 6. The theorem shows that the required netting efficiency is 66.6% for the specific class of financial systems, in order to ensure the total effect of a CCP to be positive in the bad environment . The theorem further explicitly shows the tradeoff of multilateral netting by a CCP in that higher netting efficiency leads to a larger negative effect in the good environment when each obligation settled by the CCP is relatively small.
5 Analysis
5.1 Decomposition of the effect of a CCP
Given financial system \((\mathcal {N},v^*)\) (\(\textcircled {1}\)), denote the network derived under the CCP scheme as \(\mathcal {N}^{net}\) (\(\textcircled {2}\)), which is referred to as the netout CCP network for \((\mathcal {N}, v^*)\). In an informal description, \(\mathcal {N}^{net}\) is derived from \((\mathcal {N},v^*)\) by offsetting all obligations regarding \(v^*\). Then, for the original financial system \((\mathcal {N},v^*)\), take the corresponding financial system \((\mathcal {N}^{net},v^*)\) (\(\textcircled {3}\)), which is referred to as a netout financial system. Note that the netout CCP network for the obtained netout financial system \((\mathcal {N}^{net},v^*)\) is the same \(\mathcal {N}^{net}\) (\(\textcircled {2}\)).^{12}
Then, for each bilateral network \(\mathcal {N}^B\) (\(\textcircled {4}\)) for the original financial system \((\mathcal {N},v^*)\) (\(\textcircled {1}\)), we take its corresponding bilateral network \(\mathcal {N}^{B_{net}}\) (\(\textcircled {5}\)) for the obtained financial system \((\mathcal {N}^{net},v^*)\) (\(\textcircled {3}\)). The procedure for taking corresponding \(\mathcal {N}^{B_{net}}\) is specified by (P1) provided below.
For our formal expression regarding the total liquidity needs in each environment, let \(x^{min}(\langle V,A,f \rangle )\) (\(x^{max}(\langle V,A,f\rangle )\)) denote the total liquidity needs in the good (bad) environment for given network \(\langle V,A,f\rangle \). Specifically, \(x^{min}(\langle V,A,f\rangle ) = \min _{s} \sum _{v\in V}p_{v}(s)\), and \(x^{max}(\langle V,A,f \rangle )= \min _{s} \sum _{v\in V}p_{v}(s)\), using the notations defined in Sect. 2.4.
The total effect of a CCP in the good environment is examined through a set of values \(\{x^{min}(\mathcal {N}^{B}) x^{min}(\mathcal {N}^{net})\}\) with respect to every possible \(\mathcal {N}^{B}\), and the effect in the bad environment is examined in exactly the same manner. For now, suppose we have somehow derived a corresponding \(\mathcal {N}^{B_{net}}\) for each \(\mathcal {N}^B\). We show our decomposition below, which states that for each \(\mathcal {N}^{B}\), the total effect a CCP is decomposed into two effects based on each corresponding \(\mathcal {N}^{B_{net}}\).

(Total effect) \(=\) (Central Netting effect) \(+\) (Central Routing effect).

\(\cdot x^{min}(\mathcal {N^B})  x^{min}(\mathcal {N}^{net}) = ( x^{min}(\mathcal {N}^B)  x^{min}(\mathcal {N}^{B_{net}})) + (x^{min}(\mathcal {N}^{B_{net}})  x^{min}(\mathcal {N}^{net}))\).

\(\cdot x^{max}(\mathcal {N^B})  x^{max}(\mathcal {N}^{net}) = ( x^{max}(\mathcal {N}^B)  x^{max}(\mathcal {N}^{B_{net}})) + (x^{max}(\mathcal {N}^{B_{net}})  x^{max}(\mathcal {N}^{net}))\).
5.1.1 (P1) procedure and decomposition
We prepare for the statement of the (P1) procedure. For a financial system \((\mathcal {N}=\langle V,A,f\rangle ,v^*\in V)\), take a bilateral network \(\mathcal {N}^B\). We denote a set of obligations owed to the hypothetical entity as \(A_{to} \subset A\), and a set of obligations owed by the hypothetical entity as \(A_{by}\subset A\). Furthermore, we say \(\{ (v,v^*),(v^*,v)\}\) as an offsettable pair with respect to v. Let \(\mathcal {M}:A_{to}\rightarrow A_{by}\) denote a onetoone matching that yields the bilateral network \(\mathcal {N}^B\). When \(\{(v,v^*), (v^*,v')\}\) are matched in some matching, then we say \((v,v')\) is an arc derived by the matching. Given financial system \((\mathcal {N},v^*)\) and bilateral network \(\mathcal {N}^B\) derived by onetoone matching \(\mathcal {M}\), the following (P1) procedure yields the corresponding bilateral network \(\mathcal {N}^{B_{net}}\).

\(\cdot \) For financial system \((\mathcal {N},v^*)\), let \(f^m\) denote the unit price of the target security.

\(\cdot \) For \(\mathcal {N}^{B_{net}}=\langle V^{B_{net}}, A^{B_{net}}, f^{B_{net}}\rangle \), set \(\mathcal {N}^{B_{net}}=\mathcal {N}^B\).
 \(\cdot \) Take an offsettable pair \(\{ (v,v^*),(v^*,v)\}\) for \((\mathcal {N},v^*)\). Let \((v,v')\) and \((v'',v)\) be arcs derived by matching \(\mathcal {M}\), in which \((v,v^*)\) is matched with \((v^*,v')\), and \((v^*,v)\) is matched with \((v'',v^*)\).
 1.
Remove the pair of arcs \(\{(v,v'),(v'',v)\}\) from \(A^{B_{net}}\).
 2.
Then, if \(v'\ne v''\), add a new arc \((v'',v')\) to \(A^{B_{net}}\), and let \(f^{B_{net}} ((v'',v'))= f^m\).
 1.

\(\cdot \) Repeat the main procedure until there is no offsettable pair.
For this specific example, the derived \(\mathcal {N}^{B_{net}}\) is easily confirmed as a bilateral network for \((\mathcal {N}^{net},v_c)\), where \(\mathcal {N}^{net}\) is the netout CCP network for the original financial system \((\mathcal {N},v_c)\). The first part of the following Theorem 1 ensures that this observation holds for each given bilateral network. The second part of the theorem shows consistency of the decomposition for the given financial system. The theorem shows that the decomposition is welldefined.
Theorem 1
Welldefined feature of the decomposition
 (i)
Take arbitrary bilateral network \(\mathcal {N}^B\) for financial system \((\mathcal {N},v^*)\). The (P1) procedure for \(\mathcal {N}^B\) uniquely yields a bilateral network for financial system \((\mathcal {N}^{net},v^*)\).
 (ii)
For any bilateral network \(\mathcal {N}^{B_{net}}\) for \((\mathcal {N}^{net},v^*)\), there is always a bilateral network \(\mathcal {N}^B\) for \((\mathcal {N},v^*)\) such that the (P1) procedure for \(\mathcal {N}^B\) yields \(\mathcal {N}^{B_{net}}\).
Proof
See Appendix 7.2. \(\square \)
5.2 Central routing effect
Theorem 2
 (i)
The central routing effect is always strictly negative in the bad environment.
 (ii)
The central routing effect is always weakly positive in the good environment.
 (i)
\(x^{max}(\mathcal {N}^{B_{net}})x^{max}(\mathcal {N}^{net})<0\).
 (ii)
\(x^{min}(\mathcal {N}^{B_{net}})x^{min}(\mathcal {N}^{net})\ge 0\).
Proof
See Appendix 7.3. \(\square \)
The theorem shows that the central routing effect works in different directions between the good and bad environments. In the good environment, additional CCP tends to provide additional routes for liquidity to circulate more efficiently. By contrast, in the bad environment, additional CCP serves as an additional stop for liquidity, which always increases liquidity needs.
We illustrate the intuition for the proof using Figs. 22 and 23. Given financial system \((\mathcal {N}^{net},v_c)\), which is shown in the upper part of the box in each figure, the same \(N^{net}\) shows the netout CCP network. In each of the two figures, bilateral network \(\mathcal {N}^{B_{net}}\) is shown in the lower part in the box. In the proof, we define a procedure to derive \(\mathcal {N}^{net}\) from arbitrary \(\mathcal {N}^{B_{net}}\). The procedure consists of two operations on the relevant network. As illustrated in the figures, the operations are referred to as arc separation and vertex contraction, for which the relevant concrete operations are described in each figure, and the definitions are formally stated in the appendix. In the relevant procedure, the arc separation serves to add additional vertices, while the vertex contraction reduces the added vertices by merging them into one vertex.
In order to observe the quantitative aspect of the central routing effect, we define a class of basic netout financial systems, which includes financial systems shown in the upper row in Fig. 24. We let a triangle refer to a cycle that is expressed with three different vertices \(\{v_a,v_b,v_c\}\) and three arcs \(\{(v_a,v_b), (v_b,v_c),(v_c,v_a)\}\).
Definition 1
Basic netout financial system
 (i)
it consists of triangles, and
 (ii)
\(v^*\) is included in every triangle, and every triangle is connected with each other only with vertex \(v^*\).
Theorem 3
The central routing effect: Basic netout financial system
 (i)
\(  J f^m \le x^{max}(\mathcal {N}^{B_{net}})x^{max}(\mathcal {N}^{net}) \le  f^m \).
 (ii)
\( 0 \le x^{min}(\mathcal {N}^{B_{net}})x^{min}(\mathcal {N}^{net}) \le (J1)f^m \).
Proof
See Appendix 7.4. \(\square \)
Figure 24 easily confirms each of the lower bound and upper bound of the abovementioned results. Observe that each network shown in the middle row in the figure consists of one cycle, while each network in the bottom row consists of J number of cycles with J as the number of the relevant triangles. In the bad environment, the smallest negative effect of the central routing effect is attained by networks in the middle row, while the largest negative effect is attained by those in the bottom row. This is because for each cycle, exactly one vertex is exempt from inputting additional liquidity in the bad environment, and thus, a larger number of cycles means that more vertices are exempt, given a fixed number of vertices in total. In the good environment, the middle row shows no effect of the central routing effect, while the bottom row shows the largest positive effect. It is easy to observe that a larger number of cycles means less efficient circulation of liquidity, since at least one vertex in each cycle needs to input liquidity.
Focusing on the bad environment, part (i) of Theorem 3 quantifies the range of the negative effect of the central routing effect for the relevant class. In Sect. 5.4, we argue about how much netting efficiency is required in order to cancel out the negative effect.
5.3 Central netting effect
The central netting effect is also clarified through the operations of arc separation and vertex contraction. We first illustrate this point using Fig. 25. The upper part of the box shown in the figure is the same as that in Fig. 20, which illustrates the (P1) procedure. Figure 25 illustrates that the (P1) procedure is replicated with the operations of the reverse of arc separation and the reverse of vertex contraction. From the opposite view, suppose that the (P1) procedure is derived \(\mathcal {N}^{B_{net}}\) from \(\mathcal {N}^{B}\), as shown in Fig. 20. Then, Fig. 25 illustrates that we conversely derive \(\mathcal {N}^{B}\) from \(\mathcal {N}^{B_{net}}\) by applying vertex contraction and arc separation in this sequence. The effect of the combination of arc separation and vertex contraction is already examined in Sect. 5.2.
The central netting effect is not essentially as simple as the central routing effect is, since cycle addition is also relevant. Still, the next theorem shows that the central netting effect is rather simply stated in the bad environment.
Theorem 4
Central netting effect in the bad environment
The central netting effect is always strictly positive in the bad environment.
Proof
See Appendix 7.5. \(\square \)
The intuition of the proof is simple. Regarding the combination of the operations of arc separation and vertex contraction, we have already confirmed that it strictly increases the total liquidity needs in the bad environment. In addition, cycle addition serves to generate additional liquidity needs, and thus, the total liquidity needs also strictly increase in the bad environment. Since the central netting effect is examined in the opposite direction, the central netting effect serves to strictly decrease the total liquidity needs in the bad environment.
So far, we have confirmed that, in the bad environment, the central routing effect is negative (part (i) of Theorem 2 in Sect. 5.2), while the central netting effect is positive (Theorem 4). According to this, larger netting efficiency tends to improve the total effect, but insufficient netting efficiency leads to a negative effect in total. In the next subsection, we argue the quantitative aspect with respect to the extent of netting efficiency required to cancel out the negative effect caused by the central routing effect.
For our formal statements of the relevant results, we prepare the terminology of isolated cycles. Given financial system \((\mathcal {N},v^*)\) with network \(\mathcal {N}=\langle V,A,f\rangle \), an isolated cycle is a cycle with a set of vertices \(V'\subset V{\setminus } v^*\) such that \(V'\) constitute a cycle in which each vertex \(v\in V'\) is not included in more than two arcs within A, thereby an isolated cycle consists of no more than one cycle, there is no more than one arc for any pair of vertices within \(V'\), and there is no other vertex \(v'\in V{\setminus } V'\) that constitutes an arc with a vertex \(v\in V'\). For a given balanced network, we refer to the weight of each isolated cycle as the weight of an arbitrary arc in each cycle, since any arc in the same isolated cycle has the same weight. Specifically, for financial system \((\mathcal {N}^{net},v_c)\) shown on the upperleft in Fig. 26, there are two isolated cycles, one with vertices \(\{v_f,v_g,v_h\}\), and the other with \(\{v_i,v_j,v_k\}\), in which the weight of each cycle is 10.
We compare a financial system that includes isolated cycles, with its corresponding financial system in which the isolated cycles are connected to the hypothetical entity. An example is shown in Fig. 26, in which each of the two isolated cycles in \((\mathcal {N}^{net},v_c)\) is connected to the hypothetical entity \(v_c\) in \((\mathcal {N},v_c)\). Specifically, let \(v^*\) denote the hypothetical entity, and take arbitrary vertex v for an isolated cycle. Then, add a pair of arcs \(\{(v,v^*),(v^*,v)\}\). Add a pair of arcs for every isolated cycle in this manner. The weight of each added arc is endowed by the given unit price of the targetsecurity.
The next theorem shows the central netting effect for a class of financial systems that include connected isolated cycles. The theorem reveals the quantitative aspect of the central netting effect both in the good and bad environment, which is further examined in combination with the central routing effect in the next subsection.
Theorem 5
Central netting effect: quantitative aspect
Given netout financial system \((\mathcal {N}^{net},v^*)\) in which there exist 2K number of isolated cycles \(\left\{ V_{1},V_{2},..,V_{2K}\right\} \) with integer \(K\ge 1\), and there is at least a pair of arcs \(\{(v,v^*),(v^*,v')\}\) such that \(v\ne v'\) and \(v,v'\notin \{V_k\}_{k=1,2,\ldots ,2K}\), take another financial system \((\mathcal {N},v^*)\) by connecting every isolated cycle to the hypothetical entity \(v^*\). Denote the unit price of the target security as \(f^m\), and denote the weight of each cycle indexed with \(k=1,2,..,2K\) as \(f^{k}\).
 (i)
\( Kf^m \le x^{max}(\mathcal {N}^{B})x^{max}(\mathcal {N}^{B_{net}}) \le 2Kf^m \), and
 (ii)
\(  \Sigma _1 ^{2K} \min (f^k , f^m) \le x^{min}(\mathcal {N}^{B})x^{min}(\mathcal {N}^{B_{net}}) \).
 (ii’)

\(  2K f^m \le x^{min}(\mathcal {N}^{B})x^{min}(\mathcal {N}^{B_{net}}) \le Kf^m\).
Proof
See Appendix 7.6. \(\square \)
The top row in Fig. 28 shows a financial system with four isolated cycles and a corresponding financial system with connected isolated cycles. The middle and bottom rows show relevant bilateral networks such that each network on the left is derived by the (P1) procedure for the network on the right in the same row.
Observe that for the bilateral networks in the middle row, cycle addition is not relevant for the relevant (P1) procedure, while only cycle addition is relevant for those in the bottom row. For result (i) of Theorem 5, we observe that the largest positive effect in the bad environment is attained for the bilateral networks shown in the middle row, while the smallest positive effect is attained for the bilateral networks in the bottom row. The intuition is that in the relevant (P1) procedure, liquidity needs reduced by the reverse of cycle addition is smaller than those reduced by corresponding combination of reverse vertex contraction and reverse arc separation.
For result (ii), the largest negative effect in the good environment is attained for the bilateral networks shown in the middle row. For the intuition, in the good environment, the total liquidity needs become larger as the number of cycles increases. Actually, for networks shown in the middle row, the number of cycles increases by 4 (from 1 to 5), while for networks in the bottom row, the number increases by 2 (from 4 to 6).
5.4 Netting efficiency of a CCP and total effect
To examine the total effect of a CCP, we examine a specific class of financial systems, in which a basic netout financial system (defined in Sect. 5.2) is combined with isolated cycles connected to the hypothetical entity. For example, financial system (C) in Fig. 26, (C’) in Fig. 27, and a financial system shown on the upper right in Fig. 28 are within the class. We refer to a financial system within the class as a (J, 2K) financial system with integers \(J\ge 2\) and \(K\ge 1\), in which J triangles constitute a basic netout financial system, and 2K isolated cycles are connected to the hypothetical entity. Both financial systems (C) and (C’) are (2, 2) financial systems, while the financial system shown on the upper right in Fig. 28 is a (2, 4) financial system.
We refer to netting efficiency for a given financial system, as the ratio of the amount of obligations that are eventually offset to the total amount of obligations with respect to the hypothetical entity. Netting efficiency for a (J, 2K) financial system is derived as \(\frac{2K}{J+2K}\). Thus, note that netting efficiency for the financial system shown on the upper right in Fig. 28 is twothirds, which is shown as the threshold value for the direction of the total effect of adding a CCP in the bad environment.
According to part (i) of the following Theorem 6, for the specified class, the netting efficiency of a CCP must be larger than twothirds in order for the total effect of a CCP to be always positive in the bad environment.
According to part (ii), the worst of the total effect of adding a CCP in the good environment is always negative, while it is possible for the best of the total effect to be positive. Furthermore, according to part (ii’), when netting efficiency is larger than twothirds and each obligation settled by the added CCP is relatively small, then the total effect of adding the CCP in the good environment is always negative.
Theorem 6
 (i)
\(x^{max}(\mathcal {N}^{B}) x^{max}(\mathcal {N}^{net}) \ge 0 \) if and only if \(J\le K\), and
 (ii)
\(  \Sigma _1 ^{2K} \min (f^k , f^m) \le x^{min}(\mathcal {N}^{B}) x^{min}(\mathcal {N}^{net}) \).
 (ii’)

\( x^{min}(\mathcal {N}^{B}) x^{min}(\mathcal {N}^{net}) \le f^m \).
Proof
See Appendix 7.7. \(\square \)
The results are rather simply understood in light of our decomposition. We illustrate the intuition using Fig. 28. For financial system \((\mathcal {N},v_c)\) shown on the topright of the figure, observe that in the bad environment, the smallest positive central netting effect is attained for bilateral networks shown in the bottom row. Then, the largest negative central routing effect in the bad environment is attained for the bilateral networks between the bottomleft network and the netout CCP network shown on the topleft. Thus, for result (i), the largest negative total effect in the bad environment is derived by adding the relevant effects, in which the relevant values are explicitly shown in part (i) of Theorem 3 and part (i) of Theorem 5.
For result (ii) regarding the lower bound of the total effect (largest possible negative effect), in the good environment, the largest negative central netting effect is attained for the bilateral networks shown in the middle row, and the smallest positive central routing effect, which is zero, is again attained for the bilateral networks between the bottomleft network and the netout CCP network shown on the topleft. The relevant values are derived by combining part (ii) of Theorem 3 and part (ii) of Theorem 5.
For result (ii’) regarding the upper bound of the total effect, under the added conditions, the smallest negative central netting effect is attained for the bilateral networks shown in the bottom row, and the largest positive central routing effect is attained between the bottomleft network and the netout CCP network. The relevant values are derived by combining part (ii) of Theorem 3 and part (ii’) of Theorem 5.
5.5 Policy implications
Our analysis shows that even during times of financial distress, utilization of a CCP is not unconditionally successful in reducing overall liquidity needs. This is because the added CCP itself tends to demand liquidity, typically in the form of margin for derivative trades. Still, multilateral netting provided by the CCP tends to decrease the overall liquidity needs. Thus, adding a CCP reduces the liquidity needs in total as long as the netting efficiency is sufficiently large. Conversely, this indicates that a CCP should not be used if netting efficiency is expected to be sufficiently small. Although the threshold netting efficiency depends on each network topology, our analysis indicates that the threshold is not trivially low, suggesting twothirds as one benchmark value for a relevant policy.
Furthermore, our analysis indicates that operating a CCP after an economy is no longer in financial distress could be costly from the perspective of overall liquidity needs. Especially when an economy is far from financial distress, and liquidity could be circulated in a highly efficient manner, multilateral netting by the CCP could hinder the efficient circulation of liquidity. A negative effect of a CCP is more likely when each contracted amount for trades settled by the CCP is relatively small. Thus, our analysis indicates that operating a CCP that settles trades in relatively small amounts is rather costly even when the expected netting efficiency is sufficiently high.
In total, the analysis indicates the possible merit in the flexible utilization of a CCP only during times of financial distress. For trades that are settled by a CCP with high netting efficiency but each trade amount is relatively small, flexible utilization has an advantage over inflexible utilization of a CCP, in that the CCP operates regardless of the state of the economy. Nevertheless, this needs to be examined in combination with other relevant costs, which are not considered in this study.
5.6 Remark: specification of a CCP
In our analysis, we assume there is no difference between a CCP and other financial institutions regarding the manner in which liquidity is demanded. However, this is not necessarily the case in reality. Here, we take up two types of CCP specifications and argue about the implications for our results. Each specification can be expressed by adding constraints to our liquidity problems.
The first specification requires that the CCP never inputs its own liquidity to settle its obligations. This is a probable setting to reduce the liquidity cost born by the CCP. We call such a CCP “passive.” For a given financial system, for the CCP to be “passive” in the relevant settlements, the liquidity problems introduced in Sect. 2.4 are altered as follows.
(Liquidity problems with passive CCP)
Next, we further assume that CCPs must receive all the relevant payments before making any payments. This would be a reasonable specification to guarantee that CCPs are passive. We say that such a CCP is both passive and synchronous. For the CCP to be passive and synchronous in the relevant settlements, our liquidity problems are now altered as follows.
(Liquidity problems with passive and synchronous CCP)
We confirm that passive and synchronous CCP ends up exacerbating the negative effect in the good environment. This merely strengthens our relevant results regarding the disadvantage of adding a CCP, and accordingly suggests even larger merit of flexible utilization of a CCP.
Note that it is possible for the specifications of the CCP to affect the central routing effect but they never affect the central netting effect. This is generally true for arbitrary specification of the CCP, since the CCP is not explicitly relevant for the central netting effect. This observation shows another aspect of the analytical usefulness of our decomposition of the effect.
6 Concluding remarks
Utilization of CCPs for derivative trades has become a trend after the recent financial crisis. The possible disadvantages of CCPs need to be examined carefully in combination with their advantages. The existent literature has well clarified how CCPs possibly increase counterparty risk, by affecting relevant exposure. However, there has been little investigation on how CCPs possibly affect the overall liquidity needs. Such investigation is crucial, since significant liquidity needs could lead to liquidity shortage in times of financial distress, which could cause firesales and even “dominos” of bankruptcy. The difficulty of such investigation has its root in the dynamic nature of liquidity needs, which is essentially different from the static nature of corresponding exposure or obligations.
The presented model captures the static nature of the formed obligations through the concept of network. Then, the model captures the dynamic nature of liquidity needs through the relative order of payments regarding each network. This approach enables us to differentiate economic states regarding how much liquidity tends to be needed. We especially argue about liquidity needs during times of financial distress and when an economy is far from such distress. The effect of adding a CCP to the overall liquidity needs is shown to be decomposed into two effects: the central routing effect and the central netting effect. The decomposition is captured with respect to network topology, specifically through several operations in each network.
Our analysis primarily reveals the qualitative nature of each decomposed effect. Observing that the quantitative nature of the effect essentially depends on each network topology, the analysis also provides quantitative results for a specific class of networks. Further investigation into the quantitative aspects remains for future research.
Beyond the specific topic of CCPs, our analytical approach sheds new light on relevant liquidity issues. For example, liquidity is crucial in the debate about the propagation of loss of a financial institution through an interconnected network of contracts. However, most existent literature focuses on the effects on balance sheets of the relevant institutions, ignoring how much liquidity is available at each moment. This necessarily ignores bankruptcy arising from the liquidity problem, which is important in reality. Our model and analysis sets a foundation for examining dynamics of liquidity transfers, and have wide potential in the relevant applications.
Footnotes
 1.
In this respect, Brunnermeier and Pedersen (2009) argue there is a spiral nature between funding liquidity and market liquidity, whereby the initial loss could lead to firesales, which could further exacerbate the loss.
 2.
The possible procyclical feature associated with margin requirements of CCPs is pointed out in Domanski et al. (2015). Rennison et al. (2016) report a real world case that suggests the procylicality; on the day after Britain’s vote for Brexit, the five of the largest clearing houses demanded $27bn in additional collateral across derivatives products.
 3.
The original version of the models and liquidity problems are shown in Hayakawa (2014) with additional results, which constitutes a chapter of his doctoral thesis accepted in 2011.
 4.
We interchangeably use the word “offsetting” and “netting.”
 5.
For basic terminologies of flow networks, we obey the textbook usages. See, for example, Ahuja et al. (1993).
 6.
Our analysis on balanced networks could be extended to networks that are not balanced. For a nonbalanced network, we could derive a balanced network by “local changes,” such as adding arcs and/or adjusting relevant weights. Conversely, we derive the original nonbalanced network through the relevant “reverse” local changes on the balanced network. The amount of liquidity needs for the original nonbalanced network could be derived by examining the effects of those “reverse” local changes on the balanced network.
 7.
For a network \(\mathcal {N}\) shown in Fig. 1 with hypothetical entity \(v_c\), \(A_{to}=\{ (v_b,v_c),(v_d,v_c), (v_f,v_c) \}\), and \(A_{by}=\{ (v_c,v_a),(v_c,v_e), (v_c,v_f) \}\), with \(A_{to}=A_{by}=3\).
 8.
 9.
The World Bank (2013) reports that more than 80% of the surveyed payment systems had adopted realtime gross settlement (RTGS) systems. Several interbank payment systems incorporate offsetting mechanisms into their RTGS systems, which are referred to as the liquidity saving mechanism. From this perspective, this study assumes that settlements at Stage 3 are under an RTGS system without any liquidity saving mechanism.
 10.
 11.
Note that we refer to a cycle even when it consists of multiple cycles that are mutually connected.
 12.
The dotted line in Fig. 19 shows that \(\mathcal {N}^{net}\) is the same for \(\textcircled {2}\) and \(\textcircled {3}\).
 13.
Note that when there is more than one “passive” CCP, the overall required liquidity in the bad environment tends to be smaller. This means that adding more CCPs tends to have a smaller negative effect per CCP. Still, the smaller effect is not probable when the added CCP is not sufficiently close to the existent CCPs in the relevant network.
Notes
Acknowledgements
I would like to thank the editor and the anonymous referees for their careful reviews on an earlier version of this paper. This work was partly supported by JSPS KAKENHI Grant Number 16H02009, and by the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement n\(^{\circ }\) PIRSESGA2012317767.
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