Systemic Risk and Optimal Fee Structure for Central ...

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Kou, Steven, Xianhua Peng, Chris Heyde. 2013. External risk measures and basel accords. Mathematics of Operations. Research 38(2) 393–417. Rosenlund ...
Systemic Risk and Optimal Fee Structure for Central Clearing Counterparty under Partial Netting Zhenyu Cuia , Qi Fengb , Ruimeng Huc , Bin Zoud∗ a School

of Business, Stevens Institute of Technology, [email protected]

b Department c Department

of Mathematics, University of Connecticut, [email protected]

of Statistics and Applied Probability, University of California Santa Barbara, [email protected] d Department

of Mathematics, University of Connecticut, [email protected]

Abstract We propose a novel central clearing counterparty (CCP) design for a financial network with multilateral clearing, where the participation rate of individual banks depends on the volume-based fee charged by the CCP. We introduce a general demand function for the individual banks’ participation rate, and seek the optimal fee that maximizes the net worth of the CCP. The optimal fee is explicitly solved for the case of a quadratic demand function. We show that partial participation of banks in the CCP at the optimal fee rate reduces banks’ aggregate shortfall in the financial network and also reduces the overall systemic risk. This result justifies the alignment of interests of the profitable aspect and the regulatory aspect of the CCP. Furthermore, we carry out numerical examples to verify the theoretical results. Key Words: Clearing Fee; Systemic Risk; Financial Network; Optimization; Shortfall JEL: D85, C61, G21

1

Introduction

The recent 2007-2009 financial crisis has demonstrated the negative impacts of the failure of key financial institutions on the global markets and the subsequent cascading contagion effects throughout the financial system (see Financial Crisis Inquiry Commission (2011), Haldane and May (2011), Asmild and Zhu (2016)). Thus it is important to understand the nature of systemic risk and design financial systems (e.g. design the interbank liabilities clearing mechanism) to manage it. Financial institutions are increasingly and tightly connected together at an unprecedented scale and the complex dynamics of the inter-connectedness aggregate their idiosyncratic risks within the financial system. In consequence, failures of individual institutions, due to excessive risk taking, may quickly propagate throughout the entire financial network and cause cascading disasters in a systemic way. Prominent examples include the 2007-2009 financial crisis and the ongoing European debt crisis. In the aftermath of the 2007-2009 financial crisis, central to the new regulatory approach for financial stability is the implementation of central clearing for the over-the-counter (OTC) derivatives. A central clearing counterparty (CCP hereafter) stands as an intermediary between OTC derivatives counterparties (see Duffie and Zhu (2011)). In particular, the case of Lehman Brothers in 2008 exemplifies the importance of a CCP in facilitating efficient default management and clearance amid a financial crisis. One particular CCP, LCH.Clearnet, managed Lehman Brothers’ interest rate swaps worth of 9 trillion US dollars, and all open positions of Lehman Brothers were netted, hedged and auctioned off by LCH.Clearnet. See WSJ (2008) for more information on this case study of central clearing. ∗ Corresponding Author: Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs, Connecticut 06269-1009 USA. Email: [email protected]. Phone: (+1) 860-486-3921. Fax: (+1) 860-486-4238.

1

There is also an emerging field of academic literature studying the impact of a CCP or multiple CCPs on the systemic risk. The Dodd-Frank Wall Street Reform and Consumer Protection Act was introduced in 2010 (see Acharya et al. (2010)) with the aim to reform the over-the-counter (OTC) derivatives markets. Similar to the proposals by the Basel Committee for Banking Supervision, the Dodd-Frank act proposes that the majority of the OTC derivatives should be centrally cleared. In the academic literature, the risks of a financial system with the presence of a CCP or multiple CCPs have been studied in Duffie and Zhu (2011), Capponi et al. (2015), Cont and Kokholm (2014), Amini et al. (2015a), Duffie et al. (2015), Glasserman et al. (2015), Glasserman and Young (2015), Amini et al. (2016), Feinstein (2017), etc. In this paper, we incorporate a for-profit CCP into the financial network, which charges a volumebased fee for providing clearing service and putting its own capital at risk. This is a reasonable and practical assumption, and has been adopted in the literature, see for instance Amini et al. (2015a). In the previous literature, a fixed volume-based fee rate is assumed (see Amini et al. (2015a)). In this paper, a novel design is to explicitly take into account the tradeoff between the incentive/profitability of the CCP as a central bank who is profit-seeking (taking into consideration the interests of external shareholders of the CCP), and the acceptability/responsibility of the CCP as a regulator who seeks to reduce the systemic risk for the general welfare of the financial system and society. Instead of assuming a fixed percentage fee rate as in Amini et al. (2015a), we treat the fee rate charged by the CCP as a decision variable. We set an upper bound for the possible range of the fee rate. If the CCP charges a fee higher than this upper bound, banks in the financial network will not choose the CCP to clear their liabilities, i.e. the participation percentage is zero. This situation can naturally happen in the setup of multiple CCPs, where banks can choose an appropriate CCP to clear their interbank liabilities. Another practical feature we incorporate into the model is that banks are not obligated to participate fully in the central clearing process. Instead, under our setting, all the banks in the financial network can choose the proportion/percentage of their interbank liabilities to be centrally cleared through the CCP, while the remaining proportion will be cleared in a bilateral way, e.g. as in Eisenberg and Noe (2001). We further assume that the banks’ participation rates in the CCP depend solely on the clearing fee through a decreasing demand function. In other words, we take into account the natural trade-off between the per unit profit and the clearing volume of the CCP. In the gaming between the CCP and the banks, we study the case in which the banks’ participation in the CCP is observable or known to, but cannot be controlled by the CCP. We then formulate an optimization problem to determine the optimal fee rate the CCP should charge in order to maximize its net worth. We show that the optimal fee always exists and is unique if the demand function satisfies some technical conditions. An explicit example of a quadratic demand function is fully analyzed with optimal fee rate explicitly obtained. We next focus on the comparative studies on the clearing payment, bank shortfalls, and systemic risk between the partial multilateral clearance (with CCP) and the full bilateral clearance (without CCP). We show that the partial multilateral clearance under the optimal clearing fee always increases the clearing payment and reduces the shortfall for all participating banks and the systemic risk. Our results are clearly in favor of the introduction of a CCP, and we allow the practical assumptions that on one hand, the CCP is profit seeking, and on the other hand, the CCP is serving as a regulator aiming to reduce the overall systemic risk. Thus the two aspects of the CCP, the incentive and the acceptability, have aligned interests under our CCP design. To verify the theoretical findings, we consider the top 10 banks in Britain as a financial network, and based on the available data from the Bankscope database, simulate their liability matrix using the R package systemicrisk (see Gandy and Veraart (2016)). We then compute the optimal clearing fee that maximizes the CCP’s net worth and compare the shortfall of individual banks with and without the CCP in the network. The numerical examples show that, under the partial clearance with the CCP and the optimal clearing fee, banks in default impose strictly less shortfall to the system, as compared to the case when there is no CCP in the system. Our numerical results also confirm that when the CCP charges at the optimal fee, the systemic risk is indeed reduced. The most important objective of our paper is to solve the optimal fee problem for a profit-seeking 2

CCP and study the effects on aggregate shortfalls and systemic risk under the optimal clearing fee scheme. Hence, we do not consider the setup of a CCP where the contribution to a guarantee fund is required to join the CCP nor the liquidation effects of some external assets owned by the banks. Interested readers may refer to Cifuentes et al. (2005), Amini et al. (2015a), Amini et al. (2015b), Feinstein (2017), etc, for the impacts of these two factors on the model. We comment that the inclusion of a guarantee fund into our proposed framework can be done in a straightforward way, and if the liquidation price of the external assets owned by the banks is fully characterized by some random variable, as in Amini et al. (2015a), then adding illiquid external assets into the model is also straightforward. The key contributions of our paper are three-fold. 1. We set up a general framework for a financial network with partial clearance through the CCP. Our framework incorporates the classical network model (no CCP), as in Eisenberg and Noe (2001), and some recent models with full participation in the CCP, see Amini et al. (2015a). 2. We propose a novel CCP design where the banks can choose the percentage of their interbank liabilities to be cleared through the CCP, and where the CCP optimally chooses the clearing fee after observing banks’ participation percentage, which is determined by a demand function. 3. We prove that the aggregate shortfall of the financial system is always reduced when the CCP charges at the optimal fee rate. In particular, we provide rigorous justifications that the incentive (i.e. maximizing net worth of CCP) and the acceptability (i.e. reducing systemic risk) of the CCP as regulators are consistent. It is important to distinguish our work from several recent papers in the literature. Amini et al. (2015a) considers a multilateral network with a CCP that charges a fixed and pre-determined proportional fee. They assume that all the liabilities of banks are cleared through the CCP. In our analysis, the clearing fee charged by the CCP is a control variable and the banks choose the proportion of their liabilities to be centrally cleared through their own demand functions. In Amini et al. (2015b), they argue that only part of the interbank liabilities in the financial system arising from the standardized contracts can be centrally cleared through the CCP. That is they study a similar partial clearance scheme through the CCP as we do; however, the proportion of partial clearance in their paper is pre-determined by the nature of the contracts while such proportion in our work can be implicitly controlled through selecting the clearing fee by the CCP. In addition, Amini et al. (2015b) assumes that the CCP does not charge any clearing fee nor puts its own equity at risk. These two assumptions may not hold in the settings of a CCP in real world. In this paper, we take into consideration these two practical aspects of the CCP, and formulate an optimization problem to explicitly determine the optimal fee rate the CCP should charge. The rest of the paper is organized as follows. Section 2 models the interbank liabilities in the financial network, and presents the two-step clearance mechanism in our setting, namely, partial multilateral clearance with the CCP and partial bilateral clearance without the CCP. The optimal clearing fee is derived in Section 3 for a general demand function and explicitly calculated in a special example of a quadratic demand function. We study the impact of the partial clearance on the clearing payment and the shortfall of banks in Section 4 and on the systemic risk in Section 5, respectively. We conduct numerical studies in Section 6. Section 7 concludes the paper.

2

Financial Network and Clearance Mechanism

We consider a financial network that consists of m interlinked banks1 in a single period setting. With slight abuse of letter m, we label those banks as 1, 2, · · · , m and denote M := {1, 2, · · · , m} as the set of all banks. There are two dates t = 0, 1, with the values at t = 0 being deterministic and the values at 1 The term “bank” in this paper refers to a broad collection of financial institutions, e.g., banks, credit unions, mortgage loan companies, insurance companies, pension funds, investment banks, etc.

3

t = 1 being random. At t = 0, the nominal interbank liabilities are represented by a matrix (Lij )i,j∈M , where Lij ≥ 0 denotes the nominal liabilities that bank i owes to bank j. We assume Lii = 0 for all i ∈ M, i.e., bank i does not owe to itself. At t = 1, these interbank liabilities have to be all settled in cash. P The total nominal liabilities of bank i sum P up to Li = m j=1 Lij , and the total liabilities that all m other banks owe to bank i are represented as j=1 Lji . In our model, the assets of bank i include: P γi ≥ 0 amount of cash and a total of m L j=1 ji receivables from other banks in the financial network. The assets and liabilities of bank i can be summarized through the following simplified balance sheet in Table 1. Assets P γi + m j=1 Lji

Liabilities P Li = m j=1 Lij

Table 1: Balance Sheet of Bank i The net exposure of bank i, after netting for all nominal liabilities, is given by Λi =

m X

Lji −

j=1

m X

Lij .

(1)

j=1

Bank i has either positive receivables from other banks if Λi ≥ 0 or positive payable (i.e. liabilities) to other banks if Λi < 0. In other words, the total receivables of bank i from other banks is Λ+ i and the total payable of bank i to other banks is Λ− . It is obvious that the following equality holds i m X

Λ+ i =

i=1

m X

Λ− i .

i=1

A CCP with initial capital γ0 > 0 is introduced into the financial network, and it is assumed that the CCP charges a volume-based proportional fee, denoted by f , for providing the clearing service. The banks in the financial network choose the proportion of their liabilities to be centrally cleared through the CCP, while the remaining interbank liabilities are to be cleared bilaterally without the involvement of the CCP. The following two subsections discuss in details this two-step clearance mechanism. Remark 1. By allowing partial clearance through the CCP, we generalize the work of Amini et al. (2015a), in which all interbank liabilities are cleared through the CCP, i.e., the full participation case. Note that our framework also incorporates the case of an interbank network without CCP , cf. Eisenberg and Noe (2001), which corresponds to setting the participation rate of banks to be zero.

2.1

First Step: Multilateral Clearance with the CCP

The liabilities cleared by the CCP are subject to proportional fees. We assume f ∈ [0, f¯], where f¯ ∈ (0, 1) is the highest level of the clearing fee such that all banks are willing to pay in order to participate in the central clearing system. Denote by di (f ) the proportion of the liabilities that bank i chooses to clear through the CCP and assume that it is a function of f . The following assumptions are imposed on di (f ) for all i ∈ M. Assumption 1. The function di (·), ∀ i ∈ M, satisfies the following conditions: • di (·) is a continuous and at least twice differentiable function mapping from [0, f¯] to [0, 1], where 0 < f¯ < 1; • di (·) is a strictly decreasing function of f , i.e., d0i (f ) < 0 for all f ∈ [0, f¯]; • di (0) = 1 and di (f¯) = 0. 4

Remark 2. In the model set-up, we consider the most general case regarding the demand function, namely, each bank may have different demand function di (f ), ∀ i ∈ M. However, in the main analysis of subsequent sections (Sections 3-5), we assume di (f ) = d(f ) for all i ∈ M, see Assumption 3. Remark 3. Note di (0) = 1 in Assumption 1 can be interpreted as follows. If the CCP does not charge any clearing fee, then all banks in the financial network will choose to clear all their liabilities through the CCP. That is because, with f = 0 (no clearing fees), the CCP provides γ0 as a buffer to all participating banks for free (i.e. an arbitrage opportunity to all banks). On the contrary, if the clearing fee f is too high, the cost of central clearing will outweigh the benefit of sharing the buffer γ0 in case of defaults, and consequently, banks do not participate in the CCP. Hence, we set f¯ to be a threshold beyond which all banks choose not to clear through the CCP, i.e., f¯ is the upper bound on the clearing fee that the CCP can charge to keep business. Denote the CCP as node 0 in the financial network, we now analyze the partial multilateral clearance with the CCP for a representative bank, say bank i. As a summary of notations, in general we shall b the nominal receivables or payable (depending on the sign of L), b and denote by L b ∗ its denote by L − bi be the net worth of bank i (the CCP, if i = 0), and C b and C b + are clearing counterpart. Let C i i introduced as node i’s shortfall and surplus, respectively. Under the clearing fee f , the proportion of bank i’s liabilities cleared by the CCP is di (f ). The quantity b 0i := (1 − f ) · di (f ) · Λ+ , L i with Λi defined by (1), is the nominal receivables of bank i from the CCP after taking into account the clearing fees. The quantity b i0 := di (f ) · Λ− L i is the nominal payable of bank i to the CCP (i.e., nominal liabilities). Assume that bank i has limited liabilities to the CCP, which are capped proportionally to its cash reserve by di (f ) · γi . (Note that (1 − di (f )) · γi is left for the bilateral clearing without CCP.) If we treat all the banks and the CCP involved in the multilateral clearance as a sub network, then bank i is endowed with cash γ bi := di (f ) · γi and its nominal cash flow with the CCP is captured by the pair b b (Li0 , L0i ), for all i ∈ M. Under the above assumption, we obtain the clearing payment of bank i to the CCP by b ∗i0 = di (f ) · (Λ− ∧ γi ), L i

for all i ∈ M,

(2)

and the shortfall imposed by bank i to the CCP is given by b− = L b i0 − L b ∗i0 = di (f ) · (Λ− − γi )+ , C i i

for all i ∈ M.

(3)

Remark 4. If γi ≥ Λ− i for all i ∈ M, which means all banks have enough cash reserve to pay off their b − = 0) imposed to the CCP. However, such degenerated nominal liabilities, then there is no shortfall (C i case is not of our interest. Similarly, the case in which γi < Λ− i for all i ∈ M is trivial too, since in this case all banks default and the whole financial system will break down. Thus we assume hereafter that γi < Λ− i for at least one but not all i ∈ M. In addition, we assume that γj > 0 for some j ∈ M, i.e., some bank(s) have extra positive cash amount in their reserves. b 0 , is given by The total nominal liabilities of the CCP to the participating banks, denoted by L b0 = L

m X

b 0i = (1 − f ) L

i=1

m X

di (f ) · Λ+ i ,

(4)

i=1

b0 that can be used to pay off the liabilities is while its total assets A b0 = γ0 + A

m X

b ∗i0 = γ0 + L

i=1

m X i=1

5

 di (f ) · Λ− i ∧ γi .

(5)

Therefore, the clearing payment from the CCP to all participating banks is b ∗0 = A b0 ∧ L b0 . L b0 after multilateral clearance given by We immediately have the net worth of the CCP C b0 = A b0 − L b0 . C

(6)

b 0 instead of the actual payment L b ∗ so that the Notice that in (6), we use the nominal liabilities L 0 (possible) shortfall of the CCP is taken into consideration. We also notice that the CCP exposes b0 ≤ L b 0 and has positive surplus if and only if A b0 > L b0 , shortfall to the financial network if and only if A which directly implies the following lemma. Lemma 1. The shortfall of the CCP is given by b − = (A b0 − L b 0 )− = L b0 − L b∗ , C 0 0

(7)

and the surplus of the CCP is given by b + = (A b0 − L b 0 )+ = A b0 − L b ∗0 . C 0

(8)

We assume that all the liabilities cleared by the CCP have equal priority. Under this assumption, the clearing payment of the CCP to bank i is + b b ∗ = L0i · L b ∗ = di (f ) · Λi b∗ , L ·L 0i 0 0 m P b L0 dj (f ) · Λ+ j

(9)

j=1

b 0i /L b 0 is the ratio of bank i’s nominal receivables from the CCP to the CCP’s total nominal where L bi (the net worth of bank i in the multilateral clearance step) as liabilities. We then obtain C bi = di (f ) · γi + L b ∗0i − L b i0 . C We next illustrate the cash flow of bank i in the following two possible scenarios: 1. i ∈ M+ := {i ∈ M : Λi ≥ 0} The CCP charges the clearing fee of amount f · di (f ) · Λ+ i (this is reflected in the definition of ∗ ∗ b b L0 ), and returns the payment of amount L0i , represented by (9), to bank i. In this scenario, we have b + = di (f ) · γi + L b ∗ and C b − = 0. C 0i i i 2. i ∈ M− := {i ∈ M : Λi < 0} b ∗ , given by (2), to the CCP for its interbank liabilities to be centrally cleared, and Bank i pays L i0 b − , given by (3), to the CCP. In this scenario, we have imposes a shortfall C i

b + = di (f ) · (γi − Λ− )+ C i i

b − as in (3). and C i

Since the multilateral clearance through the CCP neither creates nor destroys extra value, we expect that the aggregate surplus of the banks and the CCP is equal to the summation of their initial cash value, as formulated in the lemma below. Lemma 2. In the multilateral clearance process, the aggregate surplus of the banks and the CCP equal the summation of their initial cash value, i.e., m X

b + = γ0 + C i

i=0

m X i=1

Proof. See proof in Appendix of Cui et al. (2017). 6

di (f ) · γi .

(10)

2.2

Second Step: Bilateral Clearance without the CCP

Recall that bank i only clears di (f ) proportion of its liabilities through the CCP (multilateral clearance) and handles the remaining 1−di (f ) proportion of its liabilities without going through the CCP (bilateral b L b ∗ , C, b etc. ), the notations clearance). To distinguish from the notations used in Subsection 2.1 ( L, e ij and L e i the nominal bilateral liabilities in this subsection come with the overhead e. We denote by L of bank i to bank j and the total nominal bilateral liabilities of bank i, respectively. Thus, we have  e ij = 1 − di (f ) · Lij L

and

ei = L

m X

e ij . L

j=1

e ij ) by Define the corresponding relative nominal liabilities matrix (Π  e ij  L e i > 0; , if L e ij = L ei Π  0, e i = 0. if L For all the interbank liabilities that are not cleared through the CCP P (i.e., in the bilateral clearance), ∗ ∗ e e e∗ we denote by Lij the clearing payment from bank i to bank j. Let Li = m j=1 Lij be the total clearance e ∗ = (L e∗ , · · · , L e ∗m )T be the clearing payment vector, where aT is the transpose payment of bank i and L 1 of vector a. Assume that all interbank liabilities in the bilateral clearance have equal priority, then the total capital received by bank i from its counterparties is given by m X j=1

e∗ = L ji

m X

eT · L e∗ . Π ij j

j=1

b∗ Since the clearing process through the CCP is taken place first, where bank i either receives L 0i ∗ b to the CCP, its cash reserve now becomes from the CCP or pays L i0 b∗ − L b ∗ = (1 − di (f )) · γi + C b+ , γ ei = γi + L 0i i0 i

(11)

b ∗ and L b ∗ are defined respectively in (2) and (9). where L i0 0i Given the assumptions of absolute priority and limited liability (see (Eisenberg and Noe, 2001, Definition 1) for details), we obtain that the total clearing payment of bank i to its counterparties e ∗ solves must be the minimum of its total nominal liabilities and its total available assets, namely, L the following equation   m X e ∗i = L e i ∧ γ e Tij · L e ∗j  . Π L ei + j=1

by

e = (L e γ e → [0, L] e e1 , · · · , L e m )T and γ e L, e = (e e ) : [0, L] Denote L γ1 , · · · , γ em )T . Define a function Φ(·; Π,   e γ e ∧ γ e L, eT · l . e ) := L e+Π Φ(l; Π,

(12)

e ∗ is thus equivalent to finding a fixed point of Φ(l; Π, e γ e L, e ). Finding the clearing payment vector L The following lemma summarized from Eisenberg and Noe (2001) provides the existence and uniqueness e ∗. results to this fixed point problem, and therefore provides a characterization for L Lemma 3. ((Eisenberg and Noe, 2001, Theorems 1 and 2)) e γ e γ e L, e L, e ), there exists a solution to Φ(l; Π, e ) = l. If γ e > 0, i.e., γ For any given (Π, ei > 0 for all i ∈ M, then the solution l is unique.

7

e ∗ be the unique clearing payment vector ensured by Lemma 3, and L e ∗ be the corresponding Let L ij P e∗ clearing payment from bank i to bank j. In the bilateral clearance process, bank i receives m j=1 Lji e i to its counterparties. So we obtain the net worth of bank i after the two-step and is supposed to pay L clearing as     m m   X X ei = γ e∗ − L e i  = γi + L b∗ − L b∗ +  e∗ − L ei  . C ei +  L L (13) ji 0i i0 ji j=1

j=1

If a default happens in the bilateral clearance, the overall shortfall imposed by bank i is obtained as e− = L ei − L e ∗i , C i

(14)

where we have used the relation e ∗i = γ L ei +

m X

e Tij · L e ∗j = γ Π ei +

m X

j=1

e ∗ji < L ei . L

j=1

e + is given by On the contrary, if there is a surplus for bank i, then C i e+ = γ C ei + i

m X

e ∗ji − L e ∗i L

(15)

j=1

Similar to Lemma 2, we observe the same result for the aggregate surplus of all banks in the bilateral clearance in the following lemma. Lemma 4. In the bilateral clearance process, the aggregate surplus of the banks equal the summation of their initial cash value, i.e., m m X X e+ = C γ ei , (16) i i=1

i=1

where γ ei is given by (11). Remark 5. Combining (10), (11) and (16), we obtain b+ + C 0

m X

e+ = C i

m X

γi ,

i=0

i=1

which can be regarded as the conservation result on the surplus of the banks and the CCP in the two-step clearance. We end this section by summarizing all the model assumptions and will impose those assumptions in the sequel. Assumption 2. In the financial network model, we assume that the following conditions hold. 1. The multilateral clearance with the CCP takes place first. 2. Absolute priority and limited priority. In the multilateral clearance with the CCP, bank i pays either its liabilities di (f ) · Λ− i in full amount or its proportional cash reserve di (f ) · γi to the CCP; e i in full amount in the bilateral clearance without the CCP, bank i pays either its total liabilities L or its total available assets to the creditors, where i ∈ M. 3. Equal priority (proportionality). In both clearance steps, if there is a default, the creditors receive the payments in proportion to the size of their normal claim on the assets. 4. There are distressed banks in the network, i.e., γi < Λ− i for some but not all i ∈ M. 5. All banks have positive cash reserve, i.e., γi > 0 for all i ∈ M. 8

3

Optimal Clearing Fee

For a profit-oriented CCP, the objective is to find the optimal proportional clearing fee f ∗ that maxib0 , defined in (6), i.e., mizes its net worth C b0 . f ∗ = arg max C 0≤f ≤f¯

Assumption 3. We make the following assumption for the analysis in the rest of paper. for all i ∈ M,

di (f ) = d(f )

i.e., all the banks apply the same demand function to determine the proportion of their liabilities to be cleared through the CCP based on the fee charged. b 0 and A b0 (see (4) and (5)) into (6), we obtain Plugging the definitions of L "m # m m X X X + + − b0 = γ0 + C f · d(f ) · Λ − d(f ) · Λ − (Λ ∧ γi ) i

i

i=1

i=1

= γ0 + d(f )

i

i=1

m X   − + f · Λ+ , i − (Λi − γi ) i=1

where we have applied

3.1

Pm

+ i=1 Λi

=

− i=1 Λi

Pm

to derive the last equality above.

General Demand Function

Define

m P

R :=

+ (Λ− i − γi )

i=1

m P i=1

.

(17)

Λ+ i

> 0, we have 0 ≤ R ≤ 1. According to Assumption 2, γi < Λ− = Since i for some but not all i ∈ M, hence we obtain 0 < R < 1. − i=1 Λi

Pm

Pm

+ i=1 Λi

+ Recall that Λ− i denotes the total payable of bank i to other banks and Λi denotes the total + receivables of bank i from other banks. Notice that (Λ− i − γi ) > 0 if and only if bank i has positive liabilities to other banks in the system and its own cash is not enough to pay off its total liabilities. + Thus we can interpret (Λ− i − γi ) as the measurement of bank i’s capability to pay off its liabilities m m P P without the help of the CCP. Λ+ Λ− i (= i ) is the aggregate measurement of total receivables (or i=1

i=1

total payable). Hence, the ratio R intuitively measures the overall ability of the system to meet its obligations or liabilities. In general, the smaller the value of R, the more likely the system is able to meet its obligations and stay solvent. Consider two extreme cases: if R = 0, which is equivalent to − γi ≥ Λ − i for all i ∈ M , all banks in the system have enough cash to meet their obligations, resulting in no default and a completely solvent system; if R = 1, i.e., γi = 0 for all i ∈ M− , all banks with positive liabilities in the system do not have any positive cash reserve, and apparently, this situation may be seen as a sign for a systemic breakdown. These two extreme cases also provide the intuitions behind the last two assumptions in Assumption 2, which together imply 0 < R < 1. The following proposition describes the optimal proportional clearing fee f ∗ under a general demand function d(·). Proposition 1. Under Assumptions 1-3, if the demand function is strictly convex (i.e., d00 (·) > 0), we obtain the following results. (i) If 0 < R < f¯ < 1 and d0 (f ) d(f ) 2 00 < 0 , (18) d (f ) d (f ) 9

then there exists an optimal proportional clearing fee f ∗ ∈ (0, f¯), which solves f∗ +

d(f ∗ ) = R, d0 (f ∗ )

(19)

where R < f ∗ < f¯ due to d0 (f ) < 0 and d(f ) > 0 for all f ∈ (0, f¯). (ii) If 0 < f¯ ≤ R < 1, then the optimal proportional clearing fee does not exist, because no bank will participate through the CCP. Proof. See proof in Appendix of Cui et al. (2017).

3.2

An Example: Quadratic Demand Function

In this subsection, we illustrate the previous findings using a specific demand function. There is a vast economics literature on theoretical demand models investigating the impact of prices on the consumer demand. Motivated from this strand of literature, we choose a quadratic demand function d(·) from the class of linear power models (see Song et al. (2008)):  d(f ) :=

2 f − f¯ , f¯

0 ≤ f ≤ f¯.

(20)

It is straightforward to check that d(f ) defined by (20) satisfies all the conditions imposed in Proposition 1. The corresponding optimal clearing fee is obtained in the following proposition. Proposition 2. If the demand function is of the quadratic form given by (20) and 0 < R < f¯ < 1, then the optimal proportional clearing fee f ∗ is f∗ =

1 ¯ 2 · f + · R, 3 3

where R is defined in (17). Proof. See proof in Appendix of Cui et al. (2017).

4

Impact of Partial Clearance on Banks

In this section, we are concerned about the impact of the partial clearance mechanism, as characterized in Section 2, on the clearing payment and the shortfall of individual banks in the financial network. To rule out the trivial cases, we assume the clearing fee f ∈ (0, f¯). The impact of bank i’s default on the financial network is two-fold. In the first clearing step b − , given by (3), to the (multilateral clearance with the CCP), bank i’s default imposes a shortfall C i CCP, which may cause financial stress on the CCP and propagate throughout the whole network. In the second clearing step (bilateral clearance without the CCP), bank i’s default directly imposes a e − , given by (14), to its counterparties. shortfall C i Thanks to the above analysis, we immediately have the following lemma. Lemma 5. Under the design of partial clearance with the CCP, the aggregate shortfall imposed by bank i to the financial network is given by    b− + C e − = d(f ) · Λ− − γi + + L ei − L e∗ . ACi− := C (21) i i i i Remark 6. For the simplicity of notations, we denote by ACi the aggregate net worth of bank i in the financial network under the design of partial clearance with the CCP (see two-step clearance scheme in b− + C e − . That means the shortfall imposed by bank i to Section 2). As denoted in Lemma 5, ACi− = C i i 10

the CCP in the multilateral clearance step is not carried over to the bilateral clearance step. However, by (11), bank i’s cash amount for the bilateral clearance γ ei is given by b+ , γ ei = (1 − d(f )) · γi + C i b + , is rolled over into the which indicates that the surplus of bank i from the multilateral clearance, C i cash reserve for the bilateral clearance. Therefore, we set e + with C e + given in (15). ACi+ := C i i In the seminal work of Eisenberg and Noe (2001), a CCP is not included in the modeling of the financial network. We incorporate a profit-pursuing CCP into the financial network and take into account the partial clearance for banks through the CCP. A natural question then arises: Does the design of partial clearance through the CCP reduce the shortfall of the participating banks imposed to the financial network? To answer the above question, we need to compute the shortfall of bank i without a CCP, and compare it to the one given in (21). If a CCP is not present in the financial network (equivalently, d(f ) = 0), then banks clear their interbank liabilities through the mechanism as in Subsection 2.2, see also Eisenberg and Noe (2001). To facilitate the analysis, we introduce the following notations when a CCP is not present in the network. In this case, the relative nominal liabilities matrix (Πij )i,j∈M is defined by   Lij , if L > 0; i Πij = Li 0, if L = 0. i

The clearing payment from bank i to bank j is denoted by L∗ij , and the total payment made by bank i P ∗ ∗ ∗ ∗ T to its counterparties is L∗i = m j=1 Lij . The clearing payment vector L = (L1 , · · · , Lm ) is a solution to the fixed point problem for the function  Φ = Φ(l; Π, L, γ) = L ∧ γ + ΠT · l , where Π = (Πij )i,j∈M , L = (L1 , · · · , Lm )T , and γ = (γ1 , · · · , γm )T . Then, following from Lemma 3 and γ > 0 (γi > 0 for all i), the uniqueness of such L∗ is ensured. Finally, we obtain the shortfall of bank i to its creditors as Ci− = Li − L∗i . Theorem 1. If Assumptions 1-3 hold, then the design of partial clearance through the CCP reduces the shortfalls for all banks, i.e., ACi− − Ci− ≤ 0, ∀ i ∈ M. Proof. See proof in Appendix of Cui et al. (2017). Remark 7. In Theorem 1, positive reduction on the shortfall holds for some banks in the financial network. Lemma 6. We have e∗ , (1 − d(f )) · L∗i ≤ L i where the strict inequality holds for some i ∈ M. Proof. See proof in Appendix of Cui et al. (2017).

11

∀ i ∈ M,

5

Impact of Partial Clearance on Systemic Risk

Previously we have shown in Theorem 1 that partial clearance through the CCP reduces the shortfall for all banks, i.e., ACi− − Ci− ≤ 0 for all i ∈ M and strict inequality holds for some i ∈ M. In this section, we shall provide answers to an important question of practical interest: Does the design of partial clearance through the CCP reduce the systemic risk? Denote by C := (C0 , C1 , · · · , Cm )T the net worth in the financial network without CCP, where C0 := γ0 and Ci is the net worth of bank i in a full bilateral clearance setting without CCP. Recall the analysis in the previous section, m X Ci = γ i + L∗ji − Li , j=1

which can be interpreted as “available assets minus nominal liabilities”. Let AC := (AC0 , AC1 , · · · , ACm )T be the vector of net worth in the financial network with the b0 given by design of partial multilateral clearance setting with a CCP (as in Section 2). Here AC0 := C + + − − − e b e (6), ACi := Ci given by (15), and ACi := Ci + Ci given by (21). We essentially want to measure the systemic risk of the two financial systems, summarized by C and AC, and then investigate whether the systemic risk is reduced with AC. To this end, we follow the standard practice where an aggregation function Aα (·) and a scalar risk measure ρ(·) are used, see Chen et al. (2013), Brunnermeier and Cheridito (2014) and Feinstein et al. (2017) for further detailed discussions. Let x = (x0 , x1 , · · · , xm )T , we define the following convex aggregation function Aα : Rm+1 → R (cf. Amini et al. (2015a)), Aα (x) = α

m X

x− i − (1 − α)

i=0

m X

x+ i

(22)

i=0

for some parameter α ∈ [1/2, 1], which weighs the shortfall (bankruptcy costs) against the risk bearing capacity of the surplus in the network. We consider a coherent scalar risk measure ρ(·), which satisfies the properties of monotonicity, subadditivity, positive homogeneity, and translation invariance. For the definition of coherent risk measure and its properties, we refer to Artzner et al. (1999). A widely used coherent risk measure is Conditional-Value-at-Risk (CVaR), also called expected shortfall (ES) or expected tail loss (ETL). The Basel proposes using CVaR (a coherent risk measure) to replace Value-at-Risk (VaR, a non coherent risk measure) in Basel III, see Embrechts et al. (2014) for detailed studies. However, in both industry and academia, there is no agreement on the question: which is the best risk measure? Readers who are interested in this topic may refer to Kou et al. (2013) and Emmer et al. (2015) for a comprehensive survey and discussions. We then define the systemic risk measure by R(x) := ρ(Aα (x)). We obtain an upper bound for Aα (AC) − Aα (C) in the proposition below. Proposition 3. We have the following inequality  + b− = α L b0 − A b0 . Aα (AC) − Aα (C) ≤ α · C 0 Proof. See proof in Appendix of Cui et al. (2017).

12

(23)

Remark 8. The result in Proposition 3 can be interpreted from the economic point of view as follows. The introduction of a CCP does not create any extra value for the financial system, and this observation immediately implies the conservation of the aggregate surplus in the financial system. Then Aα (AC) − Aα (C) is nothing but the difference between the aggregate shortfall under two clearance schemes scaled by a factor α. From Theorem 1, we know that the shortfall for any individual bank is reduced under the partial multilateral clearance with the CCP. This result helps us derive the following inequality b− − C − ) = α · C b − , since C0 = γ0 > 0. Aα (AC) − Aα (C) ≤ α(C 0 0 0 Theorem 2. If Assumptions 1-3 hold and R < f¯, where R is defined in (17), then for any choice of proportional clearing fee f ∈ (R, f¯), we have Aα (AC) − Aα (C) ≤ 0, and then R(AC) ≤ R(C). Consequently, if the demand function d(·) satisfies the conditions of Proposition 1 (i), then the systemic risk is reduced under the optimal clearing fee structure. Proof. See proof in Appendix of Cui et al. (2017). Remark 9. The systemic risk is reduced while the aggregate surplus is conserved under the optimal fee structure. This justifies that the reduction of systemic risk is achieved solely through the netting benefits with the introduction of a CCP. It also indicates that the two seemingly contradictive objectives of the CCP, i.e. the incentive and the acceptability, are consistent and aligned under the optimal fee structure. Remark 10. The above Theorem 2 shows that the design of partial clearance through the CCP reduces the systemic risk, measured by (23), and enhances the stability of a financial network. In Amini et al. (2015b), they show that partial clearance through the CCP increases bank shortfall and reduces aggregate bank surplus, and can be worse than no netting at all (i.e. full bilateral clearance without CCP). These two contradictory conclusions are mainly due to the completely different design of the CCP in the two papers. In the design of Amini et al. (2015b), the CCP does not charge any clearing fee nor provides any additional capital as a default buffer. In addition, banks participating in central clearing are required to contribute to a guarantee fund which is used to net against their own liabilities (acting as a margin fund) and help clear the CCP’s liabilities in the case of a default (acting as a default fund). Amini et al. (2015b) assumes that the sole liabilities receivers of the CCP are all participating banks. Under such assumptions, if the CCP defaults, the guarantee fund is fully depleted and furthermore, banks with receivables from the CCP need to bear the shortfall of the CCP, and thus further reduces individual banks’ assets.

6

Numerical Examples

In this section, we carry out numerical studies to confirm the theoretical findings obtained in Sections 3-5 (Assumptions 1-3 hold in this section). We pick the top 10 banks in Britain, ranked by total assets in 2015, and consider them as a financial network. From the Bureau Van Dijk Bankscope database2 , we list the information of those banks in Table 2: Column 1 is bank’s numeric label, Column 2 is bank’s name, Columns 3 and 4 are bank’s total assets and liabilities. The unit (except Column 1) in Table 2 and the subsequent tables is one billion pounds sterling. In our studies, we assume that 90% of the banks’ total liabilities are interbank liabilities and 90% of their total assets are assets in the financial system (i.e., interbank assets plus cash). Within the financial network, the total interbank liabilities are equal to the total interbank assets (with amount 5145 billion pounds). We compute the ratio of the total interbank assets over the total assets in the 2

Source link https://bankscope.bvdinfo.com

13

Bank 1 2 3 4 5 6 7 8 9 10 SUM

Name Barclays Bank Goldman Sachs Lloyds Bank Royal Bank of Scotland HSBC Bank Merrill Lynch Standard Chartered Bank Credit Suisse Bank of Scotland National Westminster Bank

Total Assets 1077 850 817 812 588 461 404 400 341 302 6052

Total Liabilities 1019 823 764 770 555 426 370 378 325 287 5717

Table 2: Balance Sheet of Top 10 Banks in Britain 5145 financial system by 90%×6052 ≈ 94%, and assume this ratio applies to all individual banks. With such an assumption, we obtain the columns in Table 3 by3

Column 2 (Interbank Liabilities)

= 90% × Total Liabilities (Column 4 from Table 2)

Column 3 (Interbank Assets)

= 94% × 90% × Total Assets (Column 3 from Table 2)

Column 4 (Net Exposure)

= Column 3 − Column 2

We then simulate a systematic shock which causes the cash reserve of each bank to decrease by 90%, i.e. Column 5 (Cash Reserve) = 10% × (90% × Total Assets − Column 3) Notice that we have γi > 0 for all i, so Assumption 2.5 is satisfied. The total cash reserve in the financial system after the shock is 30 billion pounds. Bank 1 2 3 4 5 6 7 8 9 10 SUM

Interbank Liabilities 917 741 688 693 500 383 333 340 293 258 5145

Interbank Assets 916 723 695 690 500 392 343 340 290 257 5145

Net Exposure Λi -1 -18 7 -3 0 9 10 0 -3 -2 0

Cash Reserve γi 5 4 4 4 3 2 2 2 2 2 30

Table 3: Balance Sheet of the Financial System In Table 3, we know the total interbank assets/liabilities of each bank in the financial system, but we do not know the accurate asset/liability data the Pm between any two banks. Namely, regarding Pm liability matrix L = (Lij ), we know the row sums j=1 Lij for all i and the column sums i=1 Lij for all j, but not the individual entries Lij for all i, j (except Lii = 0). To further continue our empirical studies, we apply the methodology and the R package4 from Gandy and Veraart (2016) to estimate the liability matrix L. The following command in R generates 5000 liability matrices: L