Centralized Netting in Financial Networks - WordPress.com

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PRESSPRESSJID: 8, 2018;6:59JBF [m5G;January ]

Journal of Banking and (2018) 1–16 Finance 000

Contents lists available at ScienceDirect

Journal Banking andof Finance

journal homepage: www.elsevier.com/locate/jbf

Centralized netting financialin networksR

Rodney Garratt

a

, Peter Zimmerman

b

,∗

a

University California,of Santa Barbara, United States

b

Saïd Business School, University of Oxford, Kingdom United

a r t i c l e i n f o

Article history:

Received 30 2016 September

Accepted 20 2017 December

Available online xxx

Keywords:

Centralized netting

Central clearingExposures

Networks

a b s t r a c t

We consider how centralizedthe introduction of netting in financial networks affects total netted expo-

sures between therecounterparties. In casessome is netting increasesa centralizedtrade-off: the expec-

tation of variance. ofnet exposures, but reduces the We show that the set networks for which expected

net a trade-off can exposures decreases is strict subset of those for which the variance decreases, so the

only onebe in direction. network structures, introducing netting is beneficialFor some centralized never

to dealers sufficient is inunless weight placed on reductions variance. This may explain why, in the ab-

sence traders derivatives developof regulation, in a network do not central clearing. Our results can be

used to estimate requirements in networks.margin and counterparty risk financial We also provide tech-

niques theto evaluate robustness of our behavioral ofresults to responses theto introduction centralized

netting.

© 2017 Elsevier AllB.V. rights reserved.

1. Introduction

Centralized netting in a financial network is the novation of ex-

posures ato single counterparty. This can reduce the aggregate

level of exposures thein system by netting offsetting claims, and

so decrease systemic risk collateraland requirements. Examples of

centralized includenetting the introduction a central of counter-

party (CCP) to over-the-counter derivatives markets, triparty repo,

and innetting payments networks.

1

Improvements are likelymost

when all are exposures simultaneously netted.

2

Introducing cen-

tralized a subset exposures the effectnetting in of has of improv-

ing netting amongst those exposures, because they are now no-

vated to the same counterparty. itBut disrupts bilateral netting

sets amongst those exposures that are not novated. This paper ex-

R

An earlier of paper ofversion this appeared as Federal Reserve Bank New York

staff paper 717. The theviews expressed in this paper ofare those authors and do

not the thenecessarily reflect those of Federal Reserve Bank of orNew York Federal

Reserve System. The authors wish to

thank Evangelos MarcoBenos, Galbiati, Lavan

Mahadeva, Mark seminar CEMLA, Manning and participants England,at the Bank of

IBEFA/ASSA, International Finance and Banking Society, Reserve Bank of Australia,

Saïd EconomicBusiness School Societyand the for the Advancement of Theory for

useful comments and andsuggestions, Christopher Lewis for invaluable mathemat-

ical are ours.advice. Any errors∗

Corresponding author. E-mail addresses: [email protected] (R. Garratt), [email protected]

(P. Zimmerman).

1

For example, CLS (Continuous Linked Settlement) nets foreign exchange trans-

actions in payments systems.

2

This casemay not thebe when the theagents strategically reform network in

response introduction centralizedto the of netting. See Section 6.

amines how this trade-off depends on ofthe structure the net-

work.

We focus on of (CCP)the introduction a central counterparty

to a derivatives network. Duffie and Zhu (2011) show that, awhen

single asset class centrally cleared,is bilateral exposures between

agents may increase — as a result of reduced netting opportunities

across loss net-pairs of counterparties — resulting in an overall in

ting efficiency. derive theThey conditions, in terms of number of

asset classes number dealers,and the of for whether or not cen-

tralized the anetting in form of CCP is beneficial.

In Duffie and Zhu’s framework, theall agents in network are as-

sumed tradeto with one another. expressingBy the magnitude of

links inbetween agents each asasset class a theyrandom variable,

can then calculate respectexpected exposures with to this distri-

bution, and examine how this changes before and after the intro-

duction of centralized netting. clearA advantage approachof this

is it is notthat necessary to observe the actual exposures;network

instead it relates the question of whether or not the introduction

of orcentralized asset class netting in a single is beneficial not

to easily observable parameters. However, one drawback theirto

analysis is the aassumption of completely-connected network,

with all agents trading onewith another. Typically, real-world fi-

nancial arenetworks not completely connected. Empirical studies

show that there typically number well-connected counter-is a of

parties coupled with a poorly-connectedlarger number of more

counterparties.

In this netting thatpaper we develop centralizeda model of ex-

tends the framework Zhuanalytic developed by Duffie and to more

https://doi.org/10.1016/j.jbankfin.2017.12.008

0378-4266/© 2017 Elsevier B.V. All rights reserved.

Please cite this article as: R. Garratt, P. Zimmerman, Centralized netting in financial networks, Journal of Banking and Finance (2018),

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2 R. Garratt, P. Zimmerman / Journal of Banking and Finance 000 (2018) 1–16

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JID: 8, 2018;6:59JBF [m5G;January ]

general network models. develop analytical doWe results that not

depend anyon assumptions about the the net-precise structure of

work. In addition, centralizedwe examine the effect of netting on

the thevariance, as well as expectation, of Ournetted exposures.

rationale is ifthat, the agents have some aversion to volatility or

extreme outcomes, then they are likely to benefit from a lower

variance of net exposures, a as well as lower mean. As in Duffie

and Zhu, ofwe show that the introduction centralized netting is

more morelikely beneficial when areto be there agents, whenand

there asset classes.are fewer

Our first key result is ifthat, givenfor any network, introducing

centralized then alsonetting reduces expected exposures, it must

reduce the exposures.variance of However, the converse implica-

tion networksis not true. ofThis means that the set for which the

expectation of ofexposures a subsetis reduced is those for which

the variance is reduced.

We derive general expressions to determine cases when cen-

tralized the exposures.netting reduces expectation and variance of

These expressions are in terms of the network distribution theand

number asset classes. centralizedof As the benefits of netting are

decreasing in inthe number of asset classes the network, we can

determine the critical number of asset classes above which cen-

tralized netting does not deliver benefits. This critical number is

higher for forthe variance than the exposures.expectation of

These full first results require information about the structure

of Ourthe network. second assumption.result weakens this We es-

tablish upper lowerand bounds number asseton the maximum of

classes centralized Theserequired for netting deliver benefits. to

do degreenot depend on knowing the distribution theof network.

This helpsecond result could a judge thedecision maker to bene-

fits isof of central the clearing when full structure the network

unknown. We show that the ZhuDuffie and network structure

achieves isthe upper bound, suggesting that their criterion ex-

treme and beingwill lead clearingto central introduced too often.

Our additional relate specificresults to network structures. The

previous literature net-has demonstrated that real-world financial

works can be accurately networkmodeled scale-freeas models

(e.g. (Soramäki et al., 2007; 2004;Inaoka et al., Garlaschelli al.,et

2005)) or by core-periphery models (e.g. (Craig and von Peter,

2014; 2014; 2012Langfield et al., Markose, )). For scale-free net-

works, we find netthat expected exposures always increase when

a single asset is novated to a CCP, of ofregardless the size the net-

work. Therefore, for the introduction central aof clearing to be op-

timal policy, dealer desireagents or mustpolicymakers have some

to reduce the exposures. Moreover, thevariance of net for asymp-

totic network we find is novatedthat, awhen single asset to a CCP,

expected net exposures always increase and netthe variance of ex-

posures the trade-off betweenalways decreases. Hence, mean and

variance is a thedefinite feature of limiting scale-free network. In

contrast, networks for core-periphery we find that, givenfor any

number asset classes, there minimumof is a size of the network

above which the introduction a theof CCP reduces both mean

and netvariance of exposures. Thus, core-for sufficiently large

periphery networks, centralized netting is benefi-unambiguously

cial.

Our offindings predictcan be applied to the impact introduc-

ing a CCP on margin requirements, since aggregate margin needs

are related to the distribution netted exposures.of

3

Our model

can also be applied to any network where bilateralthe issue of

vs centralized under consideration.netting is For example, it could

be used ofto consider the effect multilateral netting in the inter-

bank bilateralmarket, the benefits of vs triparty repo, or the effect

on payment system exposures aof introducing netting mechanism

(such as CLS or a liquidity-saving mechanism).

The techniques we develop canin paperthis be used to evalu-

ate novating sin-the efficiency gains of multiple asset classes to a

gle CCP or of ofdifferent configurations CCPs that dif-each handle

ferent sets of asset classes. efficient oneTrivially, if it is to novate

asset class to a CCP, then it will always be efficient to novate an-

other. This follows from the efficiencyfact that gains to novation

are higher when the number of asset classes is lower and novating

an asset effectively thereduces number of asset classes by 1. The

logical extension this idea that thatof is the well-known result the

unconstrained everythingfirst-best putis to through a single CCP.

Relatedly, show thatwe (1) merging will alwaysCCPs improve effi-

ciency, again with the ultimate conclusion athat single CCP is best,

and (2) it is novatemost efficient to asset classes to the largest ex-

isting CCP.

We also consider how ifthe mayresults change the agents

strategically reform the the introductionnetwork in response to of

centralized netting. circumstances, We show that, in certain cen-

tralized maynetting be more efficient there assetwhen are more

classes, rather than fewer. This can happen if the net-post-clearing

work is perhapsless connected, as agents raisinga result of their

bilateral limits as risk a the infrastructure,result of new system

and needingthus to form fewer links. Bilateral netting becomes

more moreefficient there connections,when are fewer so it is

likely will beneficial when that the CCP be it disrupts a smaller

fraction theof bilateral links.

The final two sections discuss how our results can be used in

practice, policy. Theand the implications of our findings for issue

of clearing networks comeis likely to to the Europe thefore in in

near as requiringfuture the CommissionEuropean is considering

the clearing of euro-denominated swaps to be carried out within

the theEuropean Union. Most such clearing is done in United

Kingdom, announced Europeanwhich has its intention leaveto the

Union in March 2019. analysts Our model allows to forecast the

disruption networks result.to netting clearingin that may

2. A thereview of relevant literature

The framework developed in Duffie and Zhu (2011) has been

utilized by other authors in investigateorder to specific problems.

Heath et al. (2013) is perhaps paper inthe most similar to our

that thanthey aexamine network other the completely-connected

structure struc-of Zhu.Duffie and They core-peripheryassume a

ture and use a computational approach to compare the effect of

various clearing arrangements on expected netted exposures.

Anderson et al. (2013) al. (2013) and Cox et apply the Duffie

and Zhu framework the issue to explore policy of interoperabil-

ity between CCPs. They examine whether a regulator can reduce

expected netted exposures tradesby mandating to be novated to

a local CCP, CCPwhich linkcan to a global that clears a range

of Both ofproducts. papers retain the assumption a homogeneous

link and hetero-network, though Cox, Garvin Kelly allow for some

geneity dealerbetween agents in the exposures (butmagnitude of

not in the existence of links).

Cont (2014)and Kokholm extend the ZhuDuffie and frame-

work by relaxing the exposures betweenassumption of normal

counterparties and show, using a simulation approach, that Duffie

and Zhu’s conclusions are sensitive distributional to different as-

sumptions. However, they the homogeneousretain network as-

sumption that Duffie and is inZhu use. This contrast to our pa-

per, which anduses more general realistic network structures.

4

3

See, for example, Sidanius and Žikeš (2012). More recent papers including

Duffie al. (2015)et (2014) and Campbell use actual bilateral exposure data to an-

alyze the issue empirically.

4

While our framework explicitly permits use exposures,the of non-normal some

distributional results.assumption is innecessary order to obtain analytical

Please cite this article as: R. Garratt, P. Zimmerman, Centralized netting Bankingin financial networks, Journal of and Finance (2018),

,

,

,

,

,

,

,

,


R. Garratt, (2018)P. Zimmerman / Journal of Banking and Finance 000 1–16 3

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A recent working paper by Menkveld (2015) considers the Duffie

and Zhu framework the theand looks at mean and variance of

aggregate exposure of CCP. the His isgoal to examine the ability

of CCPthe to survive simultaneous losses thein centrally cleared

asset. contrast, lookWe, in at the aimpact of introducing CCP

on ofthe mean and variance exposures across all asset classes

and is our focus on the aggregate exposures of all counterparties

across theall asset classes, not just those in centrally cleared asset

class.

Our paper makes innovations two key to the ZhuDuffie and

model which, to our knowledge, have not been considered before.

First, we provide an analytical generalization of the model so that

it appliedcan be to any network. lookSecond, we at how in-the

troduction of CCP ofa affects the variance counterparty exposures

for alternative network structures as aswell the mean.

In the literature, abroader there are variety of papers which use

a network derivativesapproach to tofocus on issues relating OTC

networks other than netting efficiency. Markose (2012) finds that

the empirical OTC derivatives network aggregated prod-over all

ucts can well-described core-peripherybe by a modified model

and whichderives summary statistics to identify institutions carry

the quantitygreatest of systemic risk. Heath et al. (2016) use the

same data set the to show that empirical network structure has

the potential to generate stability risks. Borovkova and Lalaoui El

Mouttalibi (2013) use ofa simulation model the effectapproach to

the introduction a aof CCP on default cascades in network. They

examine both networks,homogeneous and core-periphery and find

that homogeneous networks moreare resilient.

Jackson and Manning (2007) andand Galbiati So-

ramäki (2012) use different approaches examineto the desirability

of tiering — that is, restricting direct access the ato CCP to limited

set of counterparties. Song et al. (2014) extend the Galbiati and

Soramäki framework the effect to study of network structure on

the maximum theexposure risk of CCP itself and use extreme

value theory to obtain analytical results.

Our offinal results discuss the possibility agents strate-

gically reforming the the introduc-network in response to

tion of CCP.a To our knowledge, there is no existing lit-

erature which However,directly addresses this question. re-

cent work by Aldasoro (2016)et al. (2015), Chang Zhangand ,

Hollifield et al. (2017) al. (2016)and Van der Leij et provides some

insight into how networks form under strategic interaction.

3. A general network model of exposure netting

We assume thethat dealer network is not directly observable,

but the the distribution thenumber of nodes, degree and dis-

tribution the exposures of magnitude of bilateral is known. This

is a the realistic assumption for dealer networks, where regula-

tor and participants often oflack exact real-time knowledge bilat-

eral Thisexposures. is ina thegeneralization of assumption made

Duffie and Zhu (2011), the thein which exact structure of network

is fixed and known.

Let number nodesN be the of (i.e. market letparticipants) and

S be a the givenrandom variable denoting number of links any

node values non-negativehas. S takes on the integers. Define J

i

to

be the set with aof nodes which node i has link. The size of this

set given a the .is by realization of random variable S

Let number asset classes.K denote the of Links are undirected:

class k. When X

k

i j

0 athen node i has net exposure to node j in

asset class k, and when X

k

i j

0 thethen reverse is true. Each value

is generated with the same known distribution, independently of

one another and linkof the structure of the network.

5

First consider the situation without a CCP. Consider two linked

nodes i and j. Define Y K

i j

≡ max{ K

k=1

X

k

i j

0 the} to be value of node

i’s netted . exposuresexposure to node j Positive net in one asset

class can be partially or wholly offset by negative net exposures

in another asset class samewith the counterparty. and If i j are

not linked, net is netthen the exposure zero. The total exposure of

node equalsi

j J∈

i

YK

i j

.

Now define asthe ( )function f K the expected net exposure be-

tween givenany two linked nodes, that there are K asset classes:

f

K

≡ E

YK

i j

(1)

The expected total netted exposures a givenfor node i are:

N K

≡ E

j J∈

i

YK

i j

= E

E

j J∈

i

YK

i j

S

= E

[

S f

K

]

= E

[

S

]

f

K

(2)

where we have used the fact that each Y K

i j

is independent from one

another and from S.

Similarly, afterthe the betweenvariance of exposure two nodes

netting is:

g

K

≡ Var

YK

i j

(3)

and the the netted exposures thevariance of total of network is:

N K

≡ Var

j J∈

i

YK

i j

(4)

We can evaluate this expression using lawthe of total variance:

N K

= E

Var

j J∈

i

YK

i j

S

+ Var

E

j J∈

i

YK

i j

S

= E

[Sg K

]

+ Var

[S f K

]

= E

[

S

]

g K + Var

[

S

]

f K

2

(5)

3.1. Novating a single a CCPasset class to

Now we introduce a a lossCCP in single asset class. Without

of generality, reorder the so theasset classes that centrally cleared

asset class one The nodeis the labelled K. net exposure of a given

i becomes:

j J∈

i

Y

K−1

i j

+ max

j∈

∈

∈

∈

∈

∈

∈

∈

∈ J

J

J

J

J

J

J

J

J

i

i

i

i

i

i

i

i

i

XK

i j

0

Y

S

i CCP

(6)

where the the a node’s exposures thefirst term is sum of to other

nodes and is itsthe second term netted theexposure to CCP. We

can rewrite the second term as Y

Si CCP

with the S superscript arising

because the size of J

i

has distribution .S

6

Note that, a given , thefor realized value of S two terms in

(6) are independent: the first term is determined entirely by ex-

posures the 1 the secondarising from first K − assets, while is de-

each link is net inendowed with a thevector reflecting exposure

each agents. Thisasset class between the two dealer may be pos-

itive the direction the exposure.or negative, depending on of De-

fine X

ij

to be a -vector K of values (weights) on the betweenlink

nodes Thei and if itj, exists. k

th

element in each vector, denoted

X

k

i j

represents the between thenet exposure two nodes assetin

posures the 1 the secondarising from first K − assets, while is de-

termined entirely by exposures .arising from asset K

5

The constraint X

ki j

= −X

k

ji

does apply, our onbut as analysis focuses a represen-

tative node, we do not need to apply this constraint.

6

In Section 6, butwe consider the same setting allow the network structure S to

change as a result of ofthe introduction central clearing.


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Now, when there is a CCP, ofthe expected total net exposure

node i is:

N K

= E

E

j J∈

i

Y

K−1

i j

+ Y

Si CCP

S

= E

[

S f

K − 1

+ f S

]

= E

[

S

]

f

K − 1

+ E

[

f

S

]

(7)

and netthe thevariance of total exposure of node i is:

N K

= E

Var

j J∈

i

Y

K−1

i j

+ Y

Si CCP

S

+Var

E

j J∈

i

Y

K−1

i j

+ Y

Si CCP

S

= E

[

Sg K g S− +1

]

+ Var

[

S f K f S− +1

]

= E

[

S

]

g K − +1 E

[g S

]

+ Var

[

S f K f S− +1

]

(8)

Using that(2) (7)and , the change in expected net exposure results

from novating a asingle asset class to CCP is:

N K

− N K

= E

[

f

S

]

− E

[

S

]

f

K

− f

K − 1

(9)

Using that(5) (8)and , the change in variance results from novating

a asingle asset class to CCP is:

N K

− N K

= Var

[

S f K f S− +1

]

− Var

[

S

]

f K

2

+E

[g S

]

− E

[

S

]

g K g K− − 1

(10)

These introducing centralizedresults show that netting will change

both the exposures.mean and variance of total net Reduction in ei-

ther the the positivemean or variance is likely to be for users of

the system, since it means that counterparty risk — and total mar-

gin needs — are lower either in expectation or volatility. Therefore

we can say that:

centralized netting delivers netting benefits when both (9) and

(10) are negative;

netting is worsened when both expressions are positive; and

when the aexpressions have different signs, then there is

trade-off depending decision-makerson the weight place on

the the exposures.mean and variance of

The key focus of this identifypaper is to the extent to which

the effect the net-of centralized netting depends on underlying

work structure.

Note that, the case 1 the thein where K = CCP clears all of as-

sets that dealersthe trade with one another. In this case, we would

always expect the introduction aof CCP to improve netting, as it

does not disrupt any existing bilateral This of the netting sets. is

confirmed by observing that N 1

≤ N 1

and N 1

≤ N 1

.

7

3.2. Assigning a distribution to the bilateral exposures

To obtain tractable theresults for cases where K 1, we need to

assume a distribution the exposures betweenfor bilateral dealers.

This will enable writeus to down expressions for f and ing equa-

tions (9) and (10). (We follow Duffie and Zhu, 2011) assumeand

that each bilateral of the exposures X

k

i j

is and independent iden-

the theformula for sum of independent normal random variables,

we can write the function f as:

f

=

∞

0

1

√

√

√

√

√

√

√

√

√2

ye−

y

2

2

2

dy =

2

(11)

and:

g

=

∞

0

1

√

√

√

√

√

√

√

√

√2

y

2

e−

y

2

2

2

dy −

2

2

=

1

2

∞

−∞

1

√

√

√

√

√

√

√

√

√2

y

2

e−

y

2

2

2

dy −

2

2

2

2

2

2

2

2

2

2

=

2

− 12

(12)

We substitute these into and(9) (10) to show that a CCP re-

duces the exposuresexpectation of net if and only if:

√

√

√

√

√

√

√

√

√

K

K

K

K

K

K

K

K

K +

+

+

+

+

+

+

+

+

K − 1

E[

E[

E[

E[

E[

E[

E[

E[

E[S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]

and net if andintroducing a theCCP reduces variance of exposures

onl

l

l

l

l

l

l

l

ly

y

y

y

y

y

y

y

y if:

if:

if:

if:

if:

if:

if:

if:

if:

Var[S

K − 1 +

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

S K S] Var[ ] (13)

Rewriting firstthe variance term in (13) in terms of expectations

operators, canwe express this as:condition

2

K − 1

Var Var[ ]S − [

√

√

√

√

√

√

√

√

√

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[S

3

3

3

3

3

3

3

3

3

2

] − E[S]E[

√

√

√

√

√

√

√

√

√

S]†

when the denominator is non-zero.

The only denominator zero of ( ) † is if and if isS a constant,

which implies isthat the numerator also case,zero. thisIn the de-

gree node constant of every is and known with certainty, so the

variance of netted exposures introductiondoes not change upon of

centralized netting. callWe such a network network”;a “trivial the

Duffie and Zhu network network.is an example of such a Trivial

networks are not realistic real-worldrepresentations of networks,

which exhibittypically some heterogeneity their distri-in degree

butions.

3.2.1. Effect of number of classes Kchanging the asset

Proposition 1. As the number of classes Kasset increases, there is

less benefit from the introduction of centralized truenetting. This is

whether the benefit is inmeasured terms of lower expectations or

lower variance of netted exposures.

Proof. We can express this more precisely as follows: for any

number asset classesof K, let

K

be the set of networks for

which centralized netting meanreduces the of netted exposures;

i.e. S∈

K

if and ifonly S satisfies ( ). Then for all K

K, K

is a

subset of

K

. theThe same is true for variance. The result follows

immediately that left-hand sidesfrom the fact the of both expres-

sions ( ) ( ) .and † are increasing in K

tically distributed normally with 0mean and variance

2

.

8

Using

7

To show this, note that f

0 = 0 = g 0 and that the max{ · , 0} function is sub-additive.

8

We can generalize slightly: the results follow that will also apply asymptoti-

cally as ) so as(i.e. N, K→∞ long the distribution of the exposures has zero mean

and finite follows lawvariance. This from the of large numbers. In Section 7.1, we

show that using

Student’s t-distribution instead distributionof the normal does

not the change results substantively. findingsThis is in line with the of Cont and

Kokholm (2014).

The

intuition behind Proposition 1 is that, theas K increases,

CCP ofclears a lower proportion the withdealers’ activity one an-

other, so the with the benefit of netting CCP is reduced.

9

This is

consistent with the findings of Duffie and Zhu.

For a given network structure, we can define K ∗

as the criti-

cal decreasevalue of K below which netexpected exposures upon

9

See Section 6 for an examination of of Propositionthe robustness 1 to a behav-

ioral response introductionto the of a CCP by market participants.


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introduction aof CCP, and whichabove they increase. K

†

is the cor-

responding critical value variance.for Then:

K ∗ ≡1

4

E[

E[

E[

E[

E[

E[

E[

E[

E[S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]

+

+

+

+

+

+

+

+

+

E[

√

√

√

√

√

√

√

√

√

S]

E[S]

2

(14)

K

† ≡1

4

Var Var[ ]S − [

√

√

√

√

√

√

√

√

√

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[S

3

3

3

3

3

3

3

3

3

2

] − E[S]E[

√

√

√

√

√

√

√

√

√

S]

2

+ 1 (15)

In other words, expressions ( ) ( ) and † are equivalent respec-to,

tively, K K ∗

and K K

†

. the theThese expressions are focus of

analysis moving forward. They relate impactthe of introducing a

CCP of onto the number asset classes K the left-hand side, and to

the distribution thedegree S on right-hand side.

Next we present a the betweengeneral result on relationship K ∗

and K

†

that holds non-trivialfor any finite network network(i.e. a

which has in itssome variance degree distribution).

Proposition 2. K

†

K ∗

for all finite non-trivial networks.

Proof. See the Appendix.

An

implication of Proposition 2 is that the more the decision-

maker cares asset classesabout the thevariance, wider range of

K for which the CCP delivers netting benefits. Relative theto ex-

isting examinesliterature, which only conditions clear-for central

ing to reduce expected netted exposures, this proposition suggests

that clearing beneficial, longcentral is more likely to be so as the

decision-maker places non-zeroa weight on variance.

10

3.2.2. Bounds on K ∗

and K

†

So thisfar in paper, havewe followed the related literature in

treating degreethe distribution theS as known, even though exact

structure of the not.network is For example, expressions (14) and

(15) relate K ∗

and K

†

to moments of S, which are assumed to be

known. However, in reality a decision-maker — such as an associ-

ation a tryingof dealers or regulator to decide whether or not to

mandate clearingcentral in a trading network — may lack knowl-

edge even about the distribution thedegree of network. For exam-

ple, a may thedecision-maker only know number of asset classes

K and an upper bound on the thesize of network N. aIn such case,

a maydecision-maker be interested in bounds on K ∗

and K

†

which

can decision.help to inform a

Proposition 3 (Bounds on ∗

†

). Among all networks S with

P 1 1S N≤ − = K ∗

lies 1between and N

N

N

N

N

N

N

N

N

2

4 1N−

and both bounds can

be achieved. When K

†

is itdefined, lies 1 Nbetween and − 1 and both

bounds can be achieved.


A

decision-maker can compare the observed K to the upper

bounds, which isare both knownfunctions theof quantity .N If K

larger than upperthe bound on K ∗

(K

†

), then decision-makerthe

knows that increase mean (variance)introducing a CCP will the

For both K∗

and K

†

, the lower bound onlyis achieved if and if

S takes values only on 0 aand 1, which is trivial case thesince

network consists only disconnectedof pairs and singletons. An im-

mediate characterizes uppercorollary to Proposition 3 the bound

on K∗

:

Corollary 1. The upper bound on K ∗

is achieved if if and only we

have a complete, homogeneous network.

As we demonstrate in the next section, this means thatresult

the homogeneous network of Duffie and Zhu (2011) achieves the

highest expected netting clearing,efficiency centralunder com-

pared to all networks of the same size. To put it another way, using

the Zhudecision criterion in Duffie and is almost certain to lead to

central acceptedclearing being too often.

4. Examination of different network structures

In this impactsection thewe consider how of CCPintroducing a

is affected Theby the theunderlying structure of network. struc-

ture of on ofthe network is reflected the right-hand sides expres-

sions for K∗

and K

†

via variablethe distribution theof random S.

As a network object,is a very general we restrict our analysis

to specific identifiedcases in the literature.existing First we apply

our Duffieresults to and Zhu’s network and recover their results.

Then we apply our analysis andto core-periphery fat-tailed net-

works, in order the effect theto examine on results of the presence

of large counterparties, are presentwell-connected which typically

in real-world financial networks.

4.1. Homogeneous Zhunetwork of Duffie and

Duffie and Zhu (2011) assume the network is completely con-

nected, so every agent Thishas full degree. means that S = N − 1

with certainty, and so K ∗

= N

N

N

N

N

N

N

N

N

2

4 1N−

which corresponds to their

Eq. (6). Zhu aThe Duffie and network is trivial network as defined

earlier, and has nointroducing a CCP effect on variance. The ben-

efit examiningof CCP justthe can consideredbe by the effect on

the mean.

A con-slight generalization of Duffie and Zhu’s completely

nected network is the Erdos-Rényi

random network, where links

are formed independently probability thiswith some p. In net-

work, the homogeneous but maynodes are still ex ante there be

random variations theirin realized link patterns. As Erdan os-Rényi

network grows in size, K ∗

and K

†

grow without limit.

11

However,

the Erdos-Rényi

homogeneous network is not pa-the focus of our

per, it is not fitas a good to real-world financial networks (as

shown, example,for in Markose (2012) and Craig and von Pe-

ter (2014)). beenInstead which havewe focus on two models

shown to be more networkrealistic: fat-tailed and core-periphery

structures.

4.2. networksFat-tailed

knows that increase mean (variance)introducing a CCP will the

of netted exposures. The upper bound for K

†

is at as highleast

as that for K ∗

when 2, N≥ which means that when the decision-

maker that clearing increasesis sure central the nettedvariance of

exposures, thethen she can be sure that it increases mean too.

There intermediatemay be cases decision-makerwhere a is sure

that increase mean there would be an in the but cannot be sure

about the avariance; in such cases, decision-maker who puts more

weight likelyon onthe effect variance is more to introduce central

clearing.

10

In Section 7.1, we follow (Cont and 2014Kokholm, ) effectin examining the of

introducing a Value-at-Risk anCCP on (VaR), alternative target statistic.

Many real-world financial networks have degree distributions

with a num-significant excess kurtosis. There is likely to be small

ber of highly-connected thinknodes can(we of these as the major

dealers), with the majority of nodes connections.having few One

popular this model for property is a ‘fat-tailed network’: that is,

a network with a distribution asymptoticallydegree which obeys

a power law P S s s= ∼ −

for some real-valued parameter 1.

Fatter tails are associated valueswith lower of .

In this section we will focus on scale-free networks, which are

a particular class of fat-tailed networks that have been shown to

11

This is a consequence of Proposition 5 — see Section 4.3.2.


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Table 1

Asymptotic behavior of K ∗

and K

†

in a fat-tailed network.

K ∗ K

†

1 2≤ Tends to positive infinity Tends to positive infinity

2

≤ 3 Tends to finite limit Tends to positive infinity

3 Tends to finite limit Tends to finite limit

arise many applications, including finance.in real-world in

12

Fo-

cusing classon this is instructive because they aarise according to

simple and intuitive theprocess, as explained in following section.

4.2.1. networksHow do scale-free arise?

Barabási and Albert (1999) show that scale-free networks can

be formed via growth and preferential attachment. As time goes

on, nodesnew join the with thenetwork and tend to form links

nodes which are better-connected. a modelThis is realistic of how

a derivatives networktrading may develop. Over time we would

expect new dealers market to enter as the grows. And there are

several with thereasons why these dealers are more likely to trade

agents are which already better-connected, such as name recog-

nition, relationship market,an existing in another or economies

of scale allowing better-connected agents to offer attractivemore

terms. Barabási and Albert show that networks formed through

this process have fat tails with 3.exponent =

Barabási onlyand Albert’s general solution applies when the

number nodesof becomes but the avery large, assumption of large

network is not necessarily purposes.realistic for our For example

Duffie and which isZhu aconsider network of size 12, the number

of ofentities that, the timeat writing their paper, had partnered

with create aICE Trust to CCP for clearing credit default swaps.

13

In order modelto networks settle of finite size, we need to on

a Barabási-Albert specific form of network. We use the scale-free

network formation process described in Dorogovtsev et al. (2001);

henceforth DMS network. major we refer to this as the The ad-

vantage of the DMS network is that it has an exact solution for

a network of any size. The DMS network generating process is as

follows:

1. Begin at time 2 with 3t = nodes. Each has two links connecting

to one another.

2. Each time a theperiod, new node is added to network and con-

nects nodes. nodes,to two existing To determine which choose

an existing at (each link random with equal probability). The

new whichnode then nodesconnects to the two share that ex-

isting link. Repeat.

3. This anprocess generates undirected DMS network. We now

need each to determine the value of link. ifWe assume that

two dealers have link in have linka one asset class, then they a

in pair andall eachK asset classes. For of connected nodes i j,

E[S

m

] ∼

N−1

Z

sm−

ds (16)

where Z is some constant.

As N→∞:

E[S

m

] ∼

O N − 1

m− +1

ifm − 1

O N mlog − 1 if = − 1

O m1 if − 1

(17)

and so E[S

m

] a 1.has finite limit if and only if m −

We can apply this expressionsresult to our for K ∗

and K

†

,

(14) (15)and . Table 1 below summarizes the asymptotics for vari-

ous values of .

When 3, the right-hand sides of both expressions (14) and

(15) tend to finite limits. This willmeans that there be some crit-

ical values K ∗

and K

†

beyond which the introduction aof CCP will

increase the exposures matterexpectation and variance of net no

how large the network But converseis. when 1 2 the≤ is true:

for any willgiven ,number of asset classes K there be some crit-

ical size network beyond which andthe expectation variance of

net exposures with the introduction adecline of CCP. This suggests

that deliver dealersa CCP is more likely to benefits to in networks

with intuitivefatter tails. This makes sense, because such networks

have a large number of major dealers (well-connected nodes), and

so the bilateral netting dealer thanwith major is less efficient in

a network thosewhere arethere fewer of them (i.e. with thinner

tails).

For real-world networks typically lies 3.between 2 and

14

This

corresponds to the case , a intermediate in Table 1 implying po-

tential trade-off between a exposureshigher expected value of net

and isa the lower variance when number of asset classes larger

than the asymptotic limit of K ∗

.

Proposition 4. For the infinite DMS network, novating a single asset

class to a CCP increases expected netted exposures and decreases the

variance of any number of classes Knetted exposures, for asset ≥ 2.


The

increase in expected exposures thethat results from intro-

duction of CCPa ( N K

− N K

) a .has finite upper bound as N→∞

However, inthe reduction variance ( N K

− N K

) withoutis bound.

This as agents means long that—so preference functions put any

weight any on volatility—then, for given ,K there must be some

critical introducingsize of the network above which a CCP is ben-

eficial dealerfor the agents.

4.2.3. Finite analysis of the scale-free network

The distribution a given givenof S for t is in Dorogovtsev,

Mendes (see their ):and Samukhin Eq. 8

in pair andall eachK asset classes. For of connected nodes i j,

we generate the exposuresnet X

k

i j

k = 1 K as K iid normal

random variables meanwith 0 .and standard deviation

Carried out for t steps, this sizeproduces a network of N =

t + 1 a withwhich tends towards scale-free network exponent

= 3 becomes theas t large. The left-hand panel of Fig. 1 shows

realization processof a DMS for t = 100.

4.2.2. Asymptotic analysis of fat-tailed and scale-free networks

We can use of the definition fat tails to approximate the mo-

ments of ofthe distribution thedegree as size the network N→∞:

12

See, for example, (Soramäki et etal., 2007; Inaoka al., 2004) and

Garlaschelli al.et (2005).

13

Sizes of other real-world CCP networks can be found in inTable 1 Galbiati and

Soramäki (2012) (2013)and footnote 12 in Cox et al. .

P

t

s =

t

t + 1

s − 1

2 3t −

P

t−1

s − 1 +

1 − s

2 3t −

P

t−1 s

+

+

+

+

+

+

+

+

+

1

t + 1

1

[s=2]

(18)

for t≥3, initialwith condition P

2

s = 1

[s=2]

. 1(Here,

[ ]·

denotes

the identity function which if intakes the 1 value the condition

the subscript is true, and zero otherwise.)

Fig. 2 shows the effect a aof introducing CCP for range of val-

ues of ofK and t. distributionThe S is determined by Eq. (18) for

14

For example, Soramäki et etal. Inaoka(2007) and al. (2004) both estimate

= 2 1 for the interbank payment networks in the US and Japan respectively.

Garlaschelli al.et (2005) estimate

lies between share-2.2 3.0and for networks of

holdings in the US and Italy. For Barabási-Albert scale-free networks,

has an

asymptotic limit of 3.


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Fig. 1. Left-hand panel: Scale-free network of size t = 100 generated using the DMS process. Right-hand panel: Core-periphery network of size N = 100 generated using a

Bernoulli distribution with parameters z = 0 25 p = 2 c

0

= 0.

Fig. 2. The effect of the introduction of a CCP in a DMS scale-free network, for a

range of ofvalues (K, t). The chart is calculated up to t = 249 — that is, N = 250.

Values of K ∗

are black grayrepresented by the frontier between the and areas. Val-

ues of K

†

are grayrepresented by the frontier between the and white areas. Illus-

trative shownreal-world values of N, K are by the cross markers (see footnote

15).

the . net-corresponding value of t This has been calculated for DMS

works with members. theup to 250 The upper frontier of black

area in the figure corresponds to K ∗

, thewhile upper frontier of

the gray area corresponds to K

†

.

K∗

tends monotonically upperupward with an bound of ap-

proximately 1.73 (see the theproof of Proposition 4 in Appendix).

This willconfirms the introduction athat of CCP increase ex-

pected net exposures for any non-trivial oneDMS network (i.e.

where 2).K≥ In contrast, K

†

(shown by the the grayfrontier of

4.3. networksCore-periphery

Core-periphery networks have inbeen presented the recent lit-

erature as an alternative scale-freeto networks as a model of real-

world financial linkages. These networks are characterized by a

partition of the anodes into two sets: heavily-connected set of

“core” nodes, with a connected setalong sparsely of “peripheral”nodes.

Borgatti Everett and (1999) present a modelgeneral to allow

for the detection of core-periphery networks: they assume that

in such networks all are another, whilecore nodes linked to one

there nodes. They thenare peripheralno links between present a

statistic to test for correlations between core-such an idealized

periphery network and isthe actual modeldata. Their agnostic

about the distribution betweenof links core and periphery nodes;

this is because their aimed empiricalspecification is at verification

of ofthe a model structure, rather than general a core-periphery

network.

Langfield et al. (2014) (2014)and Craig and von Peter use

the Borgatti–Everett approach to identify core-periphery struc-

tures in andthe UK German interbank markets respectively.

Wetherilt et al. (2010) and Markose (2012) use alternative methods

to show that, respectively, UK the money market and the global

network derivatives of OTC can be characterized as having core-

periphery structures.

4.3.1. networksHow do core-periphery arise?

Van der Leij et al. (2016) show that core-periphery structures

can arise as stablethe outcome of a process of strategic network

formation between heterogeneous theiragents. In model, there are

“big” (core) banks and banks, “small” (peripheral) and the payofffrom

forming a link with a big bank is greater than a link with

a small bank. Chang Zhangand (2015) demonstrate that, when

where 2).K≥ In contrast, K

(shown by the the grayfrontier of

area) increases without limit as ∼ O log N − 1

2

as predicted

by the asymptotic analysis. This largemeans that, for sufficiently

networks, the introduction a a theof CCP will cause reduction in

variance of exposures.

The cross markers illustrate values of N, K for four real-world

networks.

15

In some cases markers the are in whitethe region,

meaning that underlying— assuming the network DMShas a

structure — the case a moti-for introducing CCP would need to be

vated by considerations casesother than netting efficiency. In two

the markets are in the gray region, suggesting that the introduc-

tion of CCP ona would be beneficial weightif sufficient is placed

the netted exposures.reduced variance of

15

We have drawn these from wherepublished papers the values of N and K are

easiest deduce. to These are as follows:— point A (N = 12 K = 6): US derivatives

network Junein 2010 used by Duffie and (2011)Zhu ; B (point N =

176 K = 5): UK

interbank network in 2011 used by Langfield al.et (2014); C ( 5point N = 38 K = ):

global OTC derivatives network, used by Markose (2012), restricted to most active

participants; point D (N = 14 K

= 6): CDS network in March 2010 estimated from

DTCC Trade Information Warehouse Report.

agents need willdiffer thatin the amount they to trade, they self-

organise core-peripheryinto a structure, with the with theagents

lowest Chang Zhangtrading need forming the core. and associate

these nodes makers.core with market

Underlying this network assumptiongeneration process is the

that, for any given agent, links to core nodes are desirable, while

links to peripheral are are plausible whynodes not. There reasons

this may the case maybe for real-world financial networks. Agents

prefer to deal larger with players, whom with they are likely to

have in existing relationships other markets. Exposures to larger

players easiermay be to monitor. economies scaleAnd of may

mean that larger playersthese offer attractivemore trading terms.

Abstracting away from consideration of individual nodes’ op-

timal strategies, characterizewe can the a core-formation of

periphery network using simplethe following process:

1. Begin with c

0

core nodes, which are eachconnected to other.

2. eachAt step a new node is added. With probability z this new

node is labeled ‘core’. Otherwise, node the is labeled ‘periph-

eral’.


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3. all existingA withnew core node forms links of the core nodes

with certainty, and links pe-forms with the each of existing

ripheral periph-nodes according to some distribution . AW new

eral will existing peripheralnode never linksform with nodes,

but withwill form links existing core nodes according to the

distribution .W

This process, carried out for N − c

0

steps, will produce a net-

work of Borgatti-Everett ofsize N which meets the definition a

core-periphery The parametersnetwork. of the model are c

0

, z and

the distribution .W Borgatti and Everett allow any feasible distribu-

tion to determine core-periphery links; de-for example W could

pend on existing links in the network at a given point in time.

One simplenatural and way to model between thelinks core and

periphery eachis to assume that link occurs independently with

some fixed probability (0,p∈ 1) — that is, pro-the link formation

cess a assumption, thefollows Bernoulli distribution. Under this

number node canof links for any randomly chosen be expressed as

a plus a constant.mix of binomial distributions

16

The right-hand

panel of Fig. 1 shows a realization network-generatingof such a

process. Bernoulli core-peripheryWe will focus on the network in

the remainder of this section.

4.3.2. Asymptotic analysis of core-periphery networks

In order to make asymptotic Bernoulliinferences about the

core-periphery network, more we will state and prove a general

Proposition on the asymptotic limit of networks with ‘thin tails’,

which we define as follows.

Proposition 5. Suppose the degree distribution S of the network has

the following properties:

Var[ ]/E[ ]S S

2

tends to a finite as N ;limit → ∞

All of as N .the higher moments tend to zero →∞

Then K∗

and K

†

are O S as N .both ∼ (E[ ]) →∞

Proof. See the .Appendix

For

a network which meets the conditions in Proposition 5, K ∗

and K

†

will asincrease without bound the thesize of network

becomes large. This anysuggests that, for given number of asset

classes there minimumK, is a size of the network above which the

introduction a theof CCP would reduce both mean and variance of

net exposures.

Table 2

Parameter estimates for selected real-world core-periphery networks.

Global GermanOTC Dutch interbank interbank

Period 2009 Q4 Q4 Q22008 2003

N 204 100 1802

Estimated 0.10 0.15z 0.02

Estimated p 0.01 0.31 0.11

Fig. 3. The effect of the introduction core-peripheryof a CCP in a network with

z p c = 0 10 = 10

0

= = 0 for a range of values of (K, N) up to N 250. Values ofK ∗

are black grayrepresented by the frontier between the and areas. Values of K

†

are represented whiteby the thefrontier between gray and areas. Illustrative real-

world values of N, K are shown by the cross markers (see footnote

15).

c

0

. order derive solutions,In to numerical we need to choose feasi-

ble parameter empiricalvalues, so thewe turn to literature. Table 2

below parametersummarizes estimates selectedfrom three pa-

pers: the global OTC derivatives network from Markose (2012),

the Dutch interbank market from Van der Leij et al. (2016), and

the German interbank market from Craig and von Peter (2014).

17

The value of c

0

is impossible to observe and is likely to make lit-

tle difference for larger networks, so assume we c

0

= 0. We will

use z = 0 solutions10 p = 10 for our numerical — these parame-

18

net exposures.

Examples of network models meet thesewhich “thin-tailed”conditions

are the Duffie and Zhu network, the Erd os-Rényi

net-

work described in Section 4.1, theand Bernoulli core-periphery

network described these cases,above. allIn of E[S] (∼O N).

Proposition 5 does not apply to a network with a link forma-

tion network process W which generates fat tails; such a would

not inmeet conditionsthe stated the Proposition. For example, if

the acore and peripheral nodes form links according to Barabási–Albert

preferential attachment process, then such a network will

exhibit inasymptotic results similar to those derived Section 4.2.2.

Markose (2012) shows that the network of global OTC derivatives

exposures a can be modeled by core-periphery network with fat

tails in the the thedegree distribution. In this case, analytics of

fat-tailed network morewould be appropriate.

4.3.3. Finite analysis of core-peripherythe Bernoulli network

While the only parameter sizein is itsthe DMS network N, the

Bernoulli core-periphery parametersnetwork has four N, z, p and

16

Let C be the numbe nodes.r

r

r

r

r

r

r

r

r of

of

of

of

of

of

of

of

of core Then C ∼ c

0

+ Bin

n

n

n

n

n

n

n

n

cN

N

N

N

N

N

N

N

N −

−

−

−

−

−

−

−

−

0 z

and the condi-

tional distribution S

|C ∼

C

C

C

C

C

C

C

C

C

N

· [

C Bin N C p− 1 + − ] 1+ −

C

C

C

C

C

C

C

C

C

N

· Bin C p . Note that

the the Zhucase z = 1 corresponds to Duffie and network with equalsize to N.

use z = 0 solutions10 p = 10 for our numerical — these parame-

ter values representativeare fairly of Table 2.

18

Fig. 3 shows the effect of introducing valuesa CCP for various

of N up to 250, for the 0parameters z = 10 p = 10 c

0

= 0. As pre-

dicted, K∗

and K

†

increase approximately linearly with .N There is

a gray intermediate area where K ∗

K K

†

— in such cases, intro-

ducing a exposures theCCP will increase expected net in network,

but the thereduce variance. The cross markers represent same four

real-world networks shown on Fig. 2.

19

Comparing can seeFigs. 2 and 3, we that network structure has

an important effect the aon benefits of introducing CCP. Compare

17

For the the global OTC network we use “inner core” as defined by Markose.

Each paper provides size the thenumber of nodes N, of the the core C and total

l

l

l

l

l

l

l

l

numbe r

r

r

r

r

r

r

r

r e ed of

of

of

of

of

of

of

of

of dir

dir

dir

dir

dir

dir

dir

dir

dir ct

ct

ct

ct

ct

ct

ct

ct

ct links L. We assume that c

0

= =0. We can then estimate z

C

C

C

C

C

C

C

C

C

N

and p = L

L

L

L

L

L

L

L

L−

−

−

−

−

−

−

−

−C

C

C

C

C

C

C

C

C C

C

C

C

C

C

C

C

C−

−

−

−

−

−

−

−

−12C N−C .

18

The theMarkose parameter mayvalues be bethought to best proxy, given that

we are interested in central derivativesclearing of OTC and not other forms of in-

terbank lending. However, it should be noted thethat Markose network exhibits fat

tails and so does not conform very well

to a Bernoulli core-periphery structure.

19

We use different used figures.networks in Table 2 to those to illustrate the

This confidenceis due to differing degrees of in our ability to estimate the relevant

parameters, given information the data and provided in the original papers. The

networks in Figs. 2 and 3 are those for

which estimate we can most confidently

N Kand . The networks in Table 2 are those for which we can most confidently

estimate N, z and p.


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point C thein two networks. This represents the deriva-global OTC

tive network explored in Markose (2012). the grayThis point lies in

region regionin and whiteFig. 2 the in Fig. 3. assume aIf we

DMS network networkstructure, then introducing a CCP to this in-

creases the netted exposures, butmean of reduces their variance.

Therefore a risk-averse maysufficiently policymaker be willing to

introduce a CCP. But, if we assume a core-periphery structure, then

introducing a the so aCCP increases both mean and variance, even

highly risk-averse introduce apolicymaker would not wish to CCP.

Clearly, the ought network structure to be an important factor in

policymakers” decisions about whether to mandate central clear-

ing.

As Fig. 3 shows, there un-are wherenetwork sizes N a CCP is

ambiguously bad for netting efficiency. For all N≤ 93, K ∗

2 and

so a exposuresCCP will never reduce average net for networks of

this size. And for all 29, N≤ K

†

2 so a and introducing CCP for

networks bothof this size increases the mean and variance of net

exposures.

Testing findother parameter values, we also that K ∗

and K

†

al-

ways increase with with .N and both z and p We can interpret

higher K∗

and K

†

as indicating that bringsthe introduction aof CCP

greater Thesenetting benefits. results arise because, when the net-

work relies (theon oflinks with a fractionsmall key nodes scale-

free nettingcase), bilateral substantively reduces net exposures,

and linkscentral clearing disrupts this. But when are spreadmore

amongst number nodes core-periphery a larger of key (the case

with central maylarge z or p), clearing bring greater netting ben-

efits.

20

This strongly suggests that knowledge of network structure —as

well as the size and the number of asset classes — is important

in benefitsassessing possiblethe of on effi-central clearing netting

ciency. degree distribution,Particularly important is the thetail of

which indetermines largestthe connectivity theof nodes the net-

work.

5. Generalization multipleto several CCPs and asset classes

Until now have we only considered the case the introduc-of

ting are:

N K

= − +E[S f K] A

M

m=1

E[ f a

m

S ] (19)

N K

= − +E[S g K] A

M

m=1

E[g a

m

S ]

+Var

S f K A− +

M

m=1

E[ f a

m

S ]

(20)

Note that in the case 1m = a

1

= =1 A 1 we recover Eqs.

(7) (8)and .

These equations provide us with a the net-way of comparing

ting efficiency under different constellations of CCPs, by writing

a given constellation a the setin terms of partition of of asset

classes. Let us assume theagain that underlying bilateral exposures

X

k

i j

are iid normally distributed (0,∼N

2

) for some parameter .

We can use the expressions (11) (12)and to show that introducing

a given constellation theCCP reduces expectation and variance of

exposures if and only if, respectively:

1

A

√

√

√

√

√

√

√

√

√

K

K

K

K

K

K

K

K

K +

+

+

+

+

+

+

+

+

K A−

M

M

M

M

M

M

M

M

M

m=1

√

√

√

√

√

√

√

√

√

a

m

E[

E[

E[

E[

E[

E[

E[

E[

E[S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]

2

K A−

A SVar[ ] −

M

m=1

1

1

1

1

1

1

1

1

√

√

√

√

√

√

√

√

√

a

m

2

Var[

√

√

√

√

√

√

√

√

√

S]

M

m=1

1

1

1

1

1

1

1

1

√

√

√

√

√

√

√

√

√

a

m

E[S

3

3

3

3

3

3

3

3

3

2

] − E[S]E[

√

√

√

√

√

√

√

√

√

S]

†

Expressions (

) (and †

) ( ) (are generalizations of and †), respec-

tively. As before, all areof the K terms on ofthe left-hand sides the

expressions, which inare increasing K. a given con-Therefore CCP

stellation less efficient numberwill tend to be when the total of

asset classes is higher. asThe sameintuition theis before: when A

is fixed and K increases, then the CCPs collectively clear a smaller

proportion of the network, fromand benefitsso the they bring

netting within are likely asset classes less to be greater than the

cost from disruption of existing bilateral netting sets across asset

tion of centralized one asset class.netting in In reality, a single

CCP ormay clear more than one asset class, there may severalbe

CCPs all different Thisclearing asset classes. may affect netting effi-

ciency: for example Duffie and Zhu (2011) show that merging mul-

tiple CCPs into a single CCP will always improve netting efficiency.

In this briefly showsection, we how our results can be generalized

to a with severalsetting CCPs.

Suppose that, as before, there are agentsN dealer in the OTC

derivatives network, and K asset classes. M CCPs are introduced,

indexed by m = 1 . aM CCP m clears number of asset classes

a

m

. the so the setWe relabel asset classes that asset classes in

{1 a

1

} {are cleared by CCP 1, those in the set a

1

+ 1 a

1

+

a

2

} are 2,cleared by CCP and so forth.

The number asset classestotal of novated isto central clearing

A a:=

1

+ · · · + a

M

. with 1The asset classes labels ∈ +{A K} are

not centrally cleared.

21

By samethe reasoning as in Section 3.1, thewe can show that

expectation and variance of exposures net-following centralized

20

This is supported by Table 1 which suggests that, as

increases ain large fat-

tailed network, central bringclearing tends to greater netting benefits. be- That is

cause a fatter tail means greatera number numberof nodes with a substantial of

links, so bilateral netting with individual nodes is relatively less important.

21

We assume here that any given asset atclass is cleared by most one CCP. We

do not theconsider case class onewhere an asset can be cleared by more than CCP.

Our caseresults suggest suchin this section that a is inefficient.

classes. can We redefine K∗

and K

†

as the values of K for which

there is andequality between the left- right-hand sides of (

) and

(†

), respectively.

We can these use expressions to prove the following general

results.

Proposition 6.

1. Merging CCPs will always improve netting efficiency: Given

any constellation

1

= {a

1

a

M

} consider a new constellation

2

= {a

1

+ a

2

a

M

} with one fewer CCP. Then both the expec-

tation and variance of exposures under

2

will be un-lower than

der

1

.

2. It is novate most efficient to asset classes to the largest ex-

isting CCP: Given any constellation

1

= {a

1

a

M

} consider

two alternative constellations

2

= {a

1

+ 1 a

2

a

M

} and

3

=

{a

1

a

2

+ 1 a

M

}. If a

1

a

2

, then both the expectation and vari-

ance of exposures under

2

will be lower than under

3

.

Proof. See the

Appendix.

Proposition

6 shows that merging willCCPs reduce both the

mean and netvariance of exposures, and that it is most efficient

to clear a new asset class through the largest existing CCP. Taken

together, these efficientfindings imply that the most arrangement

would be to have neta single CCP every asset class. easyIt is to

see that this is best, because all exposures netted can be against

one another. However, such an arrangement may optimalnot be


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once a regulator takes into account systemic risk because such a

CCP would be hugely systemically regulator doesimportant. If a

not beingwant too many asset classes cleared by a single CCP,

our expressions (

) (and †

) comparecan help her to different po-

tential clearing arrangements and againstto assess trade-offs some

other factor of suchconcern, as systemic risk.

6. Strategic behavior and asset dynamics

Until now havewe assumed the thethat structure of network is

not affected by the introduction central theof clearing, except for

trades that are novated. Nor does the distribution the exposuresof

change. In reality, dealer respondagents may to the introduction

of CCPa by adjusting their links with other agents, changingand

their this allowbilateral exposures. In section we the introduction

of central theclearing to change network structure and the expo-

sures.

22

We make a between the thedistinction trading network and

clearing network. The trading network reflects the structure of

derivative trading central-relationships prior to the introduction of

ized afternetting. The clearing network reflects the structure cen-

tralized netting. So far, we have assumed the that two networks

are eachidentical. thatSuppose instead network reflects the equi-

librium strategic dealerresult of a network formation process. The

agents trade off various effects, such as:

Bilateral netting benefits: to maximize bilateral netting and

thus on economize margin each agent preferscosts, dealer to

trade more heavily counterparty, and havewith each thus rela-

tively few links.

Search costs: to minimize search costs, each agentdealer

prefers to have andfewer links, consequently to trade more

with each counterparty. entailLinks to core nodes lower search

costs.

Counterparty risk management: to minimize counterparty

credit risk, diversify counterpar-each agentdealer seeks to its

This formulation allows not only the thestructure of network to

change, but the volatility the thealso of in-the-money positions of

contracts, which in may reflect, for example, changes the magni-

tudes of bilateral agents are willingnet positions dealer to take

against one another.

23

In the the netted exposurestrading network, expected total for

a given node i are:

N K

= =E[S f K]

K

2

E[S] (21)

In the clearing network, the quantitycorresponding is:

N K

=

=

=

=

=

=

=

=

= E[

E[

E[

E[

E[

E[

E[

E[

E[

˜

˜

˜

˜

˜

˜

˜

˜

˜

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

˜

˜

˜

˜

˜

˜

˜

˜

Kf

f

f

f

f

f

f

f

f − +1 E[

˜

˜

˜

˜

˜

˜

˜

˜

f

f

f

f

f

f

f

f

f

˜

˜

˜

˜

˜

˜

˜

˜

S

S

S

S

S

S

S

S

S ]

= ˜

K − 12

E[

˜

S] ˜+

12

E[

˜

S] (22)

Define afterto be the exposureschange in expected net the in-

troduction of central ;clearing that we defined in (9) i.e.:

= N K

− N

N

N

N

N

N

N

N

N K

K

K

K

K

K

K

K

K

=

1

1

1

1

1

1

1

1

1

√

√

√

√

√

√

√

√

√2

˜

K − 1E[

˜

S] + E[

˜

S]

−

√

√

√

√

√

√

√

√

√

K SE[ ]

(23)

Central exposuresclearing reduces expected net if and only if

0. When S =

˜

S and = ˜ this is the same condition ( ) as

in andthe case thewhen clearing trading networks were identical.

Proposition 7. Let ˜ E[

˜

˜

˜

˜

˜

˜

˜

˜

˜

S S . increasing] =

=

=

=

=

=

=

=

= E[ ] is with respect to

K if and only if

K

K

K

K

K

K

K

K

K−

−

−

−

−

−

−

−

−1

1

1

1

1

1

1

1

1K

1

1

1

1

1

1

1

1

1

2

.

Proof. Substitution of E[S] into (23) and partially dif entia ngf

f

f

f

f

f

f

f

fe

e

e

e

e

e

e

e

er

r

r

r

r

r

r

r

r ti

ti

ti

ti

ti

ti

ti

ti

ti

with therespect to K yields expression

E[

E[

E[

E[

E[

E[

E[

E[

E[S

S

S

S

S

S

S

S

S]

2

√

√

√

√

√

√

√

√

√2

[ 1K − −

1

1

1

1

1

1

1

1

1

2

− K−

1

1

1

1

1

1

1

1

1

2

]

0.

Proposition

7 is a generalization of the result in Proposition 1,

which states that the introduction of centralized deliversnetting

fewer benefits as the number of asset classes increases. This result

is in robust to endogenous changes the so trading network long

ties, trading lessforming more links and with each.

As a result of these trade offs, agents form links with several

counterparties, but do not distribute their exposures evenly. This

gives arise to particular trading network structure. In fact, we

may the thesuppose that network structures that were studied in

previous sections optimizing behavior.were the result of Now let

us followingsuppose, that the introduction central theof clearing,

agents are given the theopportunity to re-form network.

To our knowledge, there is no existing academic literature — ei-

ther theoretical the theor empirical — addressing question of how

clearing might differand trading networks following the introduc-

tion of central discussedclearing. As above, it is likely to depend

on agents preferences empirical and so will be an question. Our

approach is to to remain agnostic answerabout the this question,

and instead how consider our model a givencan be adapted to

clearing network.

6.1. Effect on expected net exposures post-CCP

Let degree tradingS denote the distribution thein network, and

˜

S the distribution thedegree in clearing network. Similarly, let

2

and ˜

2

denote each bilateralthe variance of exposure in the trad-

ing and clearing network, respectively. Let f( ) thedenote expected

net exposure between the net-any two linked nodes in trading

work, given that there are asset classes traded between them.

Let

f denote the theexpected net exposure in clearing network.

22

Sections 6 and 7 were added editor to the the thepaper at encouragement of

and referee.

as the the exposuresexpected number of linkages and variance of

in not inclearing network are too much smaller than the trading

network (so that satisfies the condition given thein statement

of Proposition 7).

Interestingly, the the num-possibility that traders will reduce

ber of or oflinkages the variance exposures centralin response to

clearing opens that clearingup the possibility central may be bet-

ter when high. When K is the trading and clearing networks are

the same, then lower makes clearing K central more likely to be

efficient, because the aCCP clears larger proportion of derivatives.

But networkthis longermay no apply if the changes. Suppose that

the clearing network sufficientlyis less tradingconnected thethan

network, so the condition that in Proposition 7 is violated. Then,

even without the the the considering impact of CCP itself, clear-

ing hasnetwork greater tradingbilateral netting thanefficiency the

network because each agent her has trades concentrated among

fewer counterparties. When the introduced,CCP is these bilateral

netting sets are higherdisrupted. The K is, the less disruption is

caused of by the introduction central clearing in one asset class,

meaning a more efficient network.

To illustrate point, supposethis that simplewe have a two-

agent trading degreenetwork with distribution (S∼Bin n, p) and

tha clearint

t

t

t

t

t

t

t

t the

the

the

the

the

the

the

the

the g

g

g

g

g

g

g

g

g ne

ne

ne

ne

ne

ne

ne

ne

netw

tw

tw

tw

tw

tw

tw

tw

twor

or

or

or

or

or

or

or

ork

k

k

k

k

k

k

k

k ash

h

h

h

h

h

h

h

h degree distribution

˜

S n∼ Bin

˜

p . Let

=

√

√

√

√

√

√

√

√

√

2 ˜dan

an

an

an

an

an

an

an

an let =

√

√

√

√

√

√

√

√

√

2 . Then:

hen:

hen:

hen:

hen:

hen:

hen:

hen:

hen:

= n

˜

˜

˜

˜

˜

˜

˜

˜

˜

p

p

p

p

p

p

p

p

p

K − −1 np

p

p

p

p

p

p

p

p

√

√

√

√

√

√

√

√

√

K + E[

˜

S] (24)

23

We continue to assume determined process exposures are by the same as in

Section 3.2. However, here we are allowing agents to re-optimize in a way that

would change the variance of these exposures.


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which is inincreasing K when:

1

K

1 −

p

p

2

(25)

Consider the case

˜

p p. Then there is some K

such that condi-

tion more(25) fails for all K K and isso central clearing efficient

when are there more asset classes in the market. This ei-occurs

ther because the connected theclearing network is less than trad-

ing it hasnetwork, or smaller both).bilateral exposures (or Either

way, it in ismeans that nettingbilateral the clearing network more

efficient tradingthan in the network. Introducing onea CCP in as-

set that. the class disrupts But if one asset class is only smalla

proportion isof the thetotal market, then impact less egregious.

We conclude policymakersthat it is important for considering

the introduction the net-of CCPs to be aware not only of trading

work structure, but also to try to anticipate how dealer agents’

bilateral relationships may the introduction achange following of

CCP. For example, suppose policymakerthat a believes that a likely

consequence aof introducing CCP is that agents increase their bi-

lateral another.risk positions vis-à-vis one Then clearingthe net-

work will be less trading connected thethan network and have

more volatile thatbilateral exposures across the links do exist.

There that nettingis then riska efficiency may a con-be worse as

sequence of ofthe introduction central clearing, not better.

The same cananalysis be applied into our results Section 5 too.

In the model without standard strategic cen-network formation,

trally moreclearing asset classes is always better than clearing

fewer. However, no ifthat may longer be the case each additional

asset class further concentrates the clearing network.

In this section we have focused on ofthe effect strategic net-

work formation on expected net exposures, but a intuitionsimilar

can be applied when indeed anyconsidering the variance, or other

with mean zero and variance

2

S

2

, . uncon-conditional on S The

ditional marketvariance of the risk is — by assumption — the

same tradingin the and clearing desirednetwork, so the equality

is obtained.

Proposition

8 alone does not allow us to pin down the degree

distribution theof clearing network, because it only tells us how

the second moment will change. We can make additional as-an

sumption both networksthat the trading and clearing have a core-

periphery asstructure, described plentyin isSection 4.3. There of

empirical thisand theoretical support for assumption. The empiri-

cal numberliterature discussed . Ahas already been in Section 4.3

of recent theoretical papers have shown that core-periphery struc-

tures can equilibriumarise in as a result of heterogeneity between

agents.

24

In these papers, differ liquidityagents in terms of needs

or ofurgency trade. Those agents who have less need to trade take

on ofthe role a the core.market maker and form The agents who

have greater need periphery.to trade form the

Assumption 1. The tradingrelationship between the and clearing

networks is governed by the following constraints:

1. The trading core-peripherynetwork is a network with parame-

ter set ( a net-N, z, p, ). The clearing network is core-periphery

work with set parameter

N

z

˜

p ˜ . For simplicity, we make

the initial number of core nodes c

0

equal to 0.

2. agentsThe number of dealer is inthe same both networks: N =

˜

N.

3. The nodes sameproportion of core is the in both networks: z =

˜

z.

4. Market risk sameis the in both networks; that is,

Proposition 8 holds.

The number canof dealer agents be beassumed to exoge-

feasible summary statistic.

7. A thecomputational approach to assessing benefits of

central clearing

In this showsection, we how our model acan be used by prac-

titioner — for example, wishes a policy maker who to determine

whether or not the introduction central of clearing will improve

netting efficiency. aWe present computational approach which al-

lows for the estimation the withof benefits knowledge of only a

few parameters, rather than entirethe network structure. We al-

low clearingthe trading and networks to be describeddifferent, as

in inSection 6. theWe focus here on measuring change expected

netting efficiency, N K

− N K

but it is straightforward to extend

the framework the nettedto measuring changes in variance of ex-

posures summarytoo, or indeed any other feasible statistic.

We make that willingnessan assumption the overall of each

agent to bear market risk does not change following the introduc-

tion of central clearing. Market risk is defined as the volatility of

an each This anagent’s total exposures in asset class. measures

agent’s asset class.need eachor desire to trade This is based on a

premise that marketparticipation in the is not affected by the in-

troduction of CCP, ofthe although the choices counterparties may

be.

This each agentassumption means that bears the same ex-

pected risk system tradingmarket in the in both the and clearing

network. This is in whichcontrast ,to counterparty credit risk a CCP

can doesand generally affect.

Proposition 8. If agents bear the same expected market risk before

and clearing,after of centralthe introduction then

2

E[S

2

] ˜=

2

E[

˜

S

2

].

Proof. See the theAppendix. The idea of proof is as follows. For

each agent eachand asset class, risk normally distributedmarket is

nously the introduction centraldetermined and not affected by of

clearing. Similarly, are the core nodes assumed to have intrinsic

differences from the peripheral nodes, and so they themake up

same nodesproportion of coreboth networks. For example, the

may represent major broker-dealers who have relationships with

a large financial number of smaller institutions and non-financial

customers in a range of different markets. These existing relation-

ships counterparties,mean lower search theircosts for and so they

tend to form links not iswith each other whether or a CCP intro-

duced.

Given the parameter clearingset the theof trading network,

network thehas andfour parameters three constraints. This makes

set aof feasible clearing networks one-dimensional set. We use

˜

p

to parameterize this Clearing set. networks with higher values of

˜

p have more connections. By Proposition 8, that implies that each

bilateral willlink have a lower variance.

Conditional number nodes tradingon a of core C∼ Bin(N, z), the

network has the following distribution:

S C| ∼C

N

C − +1 Bin

N C p−

+

1 −C

N

Bin

C p

(26)

and similarly clearingfor the network. It is straightforward to com-

pute unconditionalthe mean of S:

E[S] = N − 1 z

z p z+ 2 1 −

(27)

but the other unconditional moments are more complicated. To as-

sess conditionsthe for a exposuresreduction of net we need half-

moments, which not in do exist closed form. Therefore it will be

necessary to take a computational approach.

24

See, for example, Aldasoro et al., 2016; and 2015; al.,Chang Zhang, Hollifield et

2017 Van (2016)and der Leij et al. .


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Table 3

Robustness checks using core-periphery a network with parameters N = 38 K = 5 z = 0 10 p =

10 c

0

= 0. The table shows the mean, variance and 99% VaR of netted exposures before and after

the introduction of a CCP, under resultsvarious assumptions. These are obtained computationally

by simulating the network one million times eachin case.

Base model Correlated Different variance Student’s t

Mean No CCP 0.62 0.74 2.29 0.56

With CCP 0.75 0.84 0.65 0.94

Relative change 0.20 0.14 0.17 −0.59

Variance No CCP 2.58 4.71 35.11 2.40

With CCP 2.91 4.37 2.64 4.96

Relative change 0.13 0.10 0.05 −0.88

99% VaR No CCP 7.79 7.4010.84 28.65

With CCP 8.17 11.11

9.92 7.70

Relative change 0.05 0.02 −0.65 0.04

To compute post-CCP,the efficiencynetting pre- and a planner

needs as asestimates parametersof the (N, z, p, ), well the num-

ber beof asset classes these canK. suggestWe that all estimated

using summary statistics, and thus do not require knowledge of

the full network. Post-crisis, regulators are obtaining better infor-

mation about so estimationtrading networks, and of these param-

eters should be possible. For further details, see empirical papers

such as Craig and von Peter (2014). theThe results of previous

section suggest may the that outcomes be non-monotonic in pa-

rameters, so the having precise estimates for parameters is very

important.

As

˜

p is only observable ex post, a planner’s best approach is to

form a the prior over likely connectivity theof clearing network

99% all eachVaR to report of the previous results. For of these, we

generate reportthe network one million times and the statistics in

Table 3.

As shown meanby point C ,in Fig. 3 both and variance increase

upon introduction aof CCP. Using leads similar VaR to decisions

as using assumingthe metrics.other two For these parameters,

positively exposures thecorrelated or using Student’s t makes CCP

mildly more effective (i.e. smaller increasea in the metric), but the

direction the theof change is same.

The asset classmost striking case exposuresis when in K has a

higher parame-variance than the other asset classes. Under these

ters, introducing a aCCP leads to substantive improvement in net-

ting thatunder all meaningmetrics, a different decision would be

and use that priorto compute a distribution netted exposures.for

Future empirical research may help to refine priors over

˜

p as regu-

lators can introducinglook at the effect of central clearing in other

networks.

7.1. Robustness checks

We use our computational approach whether any to check of

our modelling assumptions drive the results.

25

In particular, we

follow (Cont and Kokholm, 2014) the mayin looking at how results

change under circumstances:the following

1. exposures positivelyare correlated;

2. exposures in different areasset classes not identically dis-

tributed;

3. a distribution theother than normal is used to generate expo-

sures;

4. the effectiveness assessed comparingof central clearing is by

Value-at-Risk netted exposures, the(VaR) of rather than mean

or variance.

We examine separately,these using the global OTC derivatives

network (identified as point C ) ain Figs. 2 and 3 and assuming

core-periphery structure with 5 0N = 38 K = z = 10 p = 10 c

0

=

0. (We follow Cont and Kokholm, 2014) in implementing each of

the the assume exposuresrobustness checks. For first, we that are

normally distributed and identically with but withunit variance

correlation coefficient second, there= 1. theFor we assume that

is correlationzero but exposures thein centrally cleared asset class

K has double the exposures thestandard deviation of in other as-

set the the withclasses. For third, we use Student’s t-distribution

3 degrees of freedom and unit variance.

26

For the fourth, we use

25

We thank an anonymous this suggestion. referee for

26

See Cont and (2014) Kokholm for an explanation this assumption.of Using a

distribution mean make results dependentwith infinite variance or will the heavily

on a few rare extreme observations.

made. thatThe reason is exposures in asset class K are larger than

those in against other asset classes, themso netting each other

(in is againsta CCP) more efficient themthan netting smaller ex-

posures with (as bilateral If, instead, netting). exposures in asset

class introducing lessK had a lower variance, a CCP would be ef-

fective. All of ofthese results are findingsin line with the Cont and

Kokholm (2014).

8. Implications for policy

In 2009, clearingG-20 centralLeaders called for in a variety of

derivatives markets (in particular high-volume creditstandardized

default and interest rate swaps). beenThis has now introduced

into in in legislation various member countries; for example the

United States through Title VII 2010of ofthe Dodd-Frank Act and

in the the InfrastructureEuropean Union through European Market

Regulation (EMIR) of 2012. What can our usanalysis tell about

the given need for regulatory intervention? In other words, that

the dealer agents have notchosen to set aup CCP themselves,

what market mandatingfailure does regulator addressa by central

clearing?

Margin requirements in a derivatives network are related to

netted exposures. assume theLet us that dealers are less averse to

volatility theand associated extreme outcomes than is socially op-

timal. case,This may thebe for example, high marginif or volatile

requirements impose externalities markets as-on for the collateral

sets (Murphy et al., 2014), problems mean that dealersor if agency

take excessive therisks. In such cases, dealers would not wish to

introduce maycentralized netting, even though it be socially op-

timal do socialto so. A planner may determine that the optimal

policy measure central although theis to mandate clearing, deal-

ers themselves may thedisagree. This explains need to introduce

regulation which mandates that dealers use CCP.a


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Following 2016the JuneUnited Kingdom’s vote in to leave the

European EuropeanUnion, the Commission is considering changes

to the theoversight of clearing of euro-denominated swaps, most

of which is indone London.

27

One mandate thatlikely option is a

clearing swaps doneof such must be within Union.the European

This existingwould mean disruption to netting sets among CCPs in

the UK and EU (see O’Malia (2017)). Or model can help to assess

the efficiency possible impact of this change on netting in these

CCPs. outcome ofProposition 6 predicts that the this will be to

make beneficiaries thisLondon-based less efficientCCPs and the of

in However, isthe efficient.EU more as London currently the larger

financial willcentre, the effectoverall be worse aggregate netting

efficiency.

9. Concluding remarks

We show nettinghow the introduction of centralized in a sin-

gle asset class affects both the nettedmean and variance of dealer

exposures, the thedepending on underlying structure of network.

Centralized decreasenetting is more likely to both mean and vari-

ance if the network is larger, smaller or if is there a number of

asset classes traded network. in the However, centralized netting

brings netting reliesfewer benefits if the network on a num-small

ber of ofkey nodes for most its links.

This has welfare implications because net dealer exposures re-

late to aggregate counterparty inrisk the network and to liquidity

where

1

1

1

1

1

1

1

1

1

r

+

1

1

1

1

1

1

1

1

1

s

= =1. arEquality holds if and only if bs except in the

trivial case when T ais constant.

Let thereus assume that Proposition 2 is false, and that is some

non-trivial network for which K ∗

≥K

†

. We can write this as:

1

4

T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

T

1

+

+

+

+

+

+

+

+

+

T

T

T

T

T

T

T

T

T

1

1

1

1

1

1

1

1

1

T

2

2

≥1

4

T

4

− T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

− T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

+

+

+

+

+

+

+

+

+ T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

1

1

1

1

1

1

1

1

1

T

3

− T

2

T

1

2

+ 1

⇐⇒

T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

T

1−

T

T

T

T

T

T

T

T

T

1

1

1

1

1

1

1

1

1

T

2

2

T

3

− T

2

T

1

2

≥ T

4

− T

2

2

− T

2

+ T

2

1

2

We can take square roots of both sides of this expression with-

out changing the direction theof inequality, longso we are sure

that sidethe square root of each is non-negative. The following

statements thatensure it is indeed the case.

T

4

− T

2

2

− T

2

+ T

2

1

= −Var[ ]S Var

Var

Var

Var

Var

Var

Var

Var

Var[

[

[

[

[

[

[

[

[

√

√

√

√

√

√

√

√

√

S] 0≥ ;

T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

T

1−

T

T

T

T

T

T

T

T

T

1

1

1

1

1

1

1

1

1

T

2

=

E[

E[

E[

E[

E[

E[

E[

E[

E[S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]−

E[

√

√

√

√

√

√

√

√

√

S]

E[S]

≥ ;0 and

T

3

− T

2

T

1

= E[T

3

] − E[T

2

]E[T ] 0≥

The first two are obviously true, because S≥ 1 with probability 1.

T e canh

h

h

h

h

h

h

h

h third be proved using Proposition 9 with 2 0a = b = r =

3

3

3

3

3

3

3

3

3

2

s = 3.

That means that sideswe can take square roots of both of the

late to aggregate counterparty inrisk the network and to liquidity

needs. For example, if centralized netting reduces both the expec-

tation and net variance of exposures, it is likely to be beneficial.

If centralized netting increases both the expectation and variance

of net it isexposures, then likely to reduce welfare. show thatWe

there cases decision-makerswill be where the must trade off lower

variance against nota buthigher mean, vice versa. This means that

some aversion to tovolatility may be required favor the introduc-

tion of such CCP. sucha centralized netting system, as a In cases,

whether centralized netting improves welfare or not will depend

upon controls,details of risk as aswell the thepreferences of deal-

ers, policymakers and any other relevant agents.

A criticism of work that analyzes changes in financial network

structures that thatresult from shocks or policy changes is nodes

in notthe network (financial institutions) are assumed to opti-

mally theiradjust behavior following the event studied. This prac-

tice arises in part because of difficulties withassociated accurately

modeling market scenar-optimal network formation in particular

ios. there canWe show that are trade-offs that lead to ambiguous

conclusions. about theOur approach is to remain agnostic outcome

of behavioral effects, but to identify criteria under which such ef-

fects are likelymore or less to affect conclusions.

Appendix

Proof of 2Proposition

As notational shorthand, define T

j

= E[T

j

] = E[S

j 2

] where S is

the distribution the degree of network. Note that, because S can

only non-negative take values, T

j

is real-valued non-negativeand

for all j≥0. In this shall make proof we repeated use of the fol-

lowing anresult, which is immediate consequence Hölder’sof in-

equality.

Proposition 9. For any r a b1, ≥0, ≥ 0,

T

a b+

≤ T

1

1

1

1

1

1

1

1

1

r

ar

T

1

1

1

1

1

1

1

1

1

s

bs

27

See, for example, https://www.ft.com/content/d3754014-30da-11e7-9555-23ef563ecf9a.

s = 3.

inequality above obtain:to

T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

T

1−

T

T

T

T

T

T

T

T

T

1

1

1

1

1

1

1

1

1

T

2

T

3

− T

2

T

1

≥ T

4

− T

2

2

− T

2

+ T

2

1

⇐⇒

T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

T

1−

T

T

T

T

T

T

T

T

T

1

1

1

1

1

1

1

1

1

T

2

T

3

− T

2

2

+ T

2

1

≥ T

4

− T

2

2

− T

2

+ T

2

1

⇐⇒

T

T

T

T

T

T

T

T

T

2

2

2

2

2

2

2

2

2

T

1−

T

T

T

T

T

T

T

T

T

1

1

1

1

1

1

1

1

1

T

2

T

3

≥ T

4

− T

2

⇐⇒ T

2

2

T

3

+ T

1

T

2

2

≥ T

1

T

2

T

4

+ T

2

1

T

3

Now we shall show that thisuse Proposition 9 to is not true for

any non-trivial network, and thus obtain a contradicti .on

on

on

on

on

on

on

on

on

Setting a =

4

4

4

4

4

4

4

4

4

3

b =

2

2

2

2

2

2

2

2

2

3

r = 3 s =

3

3

3

3

3

3

3

3

3

2

we have T

2

T

1

1

1

1

1

1

1

1

1

3

4

T

2

2

2

2

2

2

2

2

2

3

1

. Setting

a =

8

8

8

8

8

8

8

8

8

3

b =

1

1

1

1

1

1

1

1

1

3

r =

3

3

3

3

3

3

3

3

3

2

s = 3 we have T

3

T

2

2

2

2

2

2

2

2

2

3

4

T

1

1

1

1

1

1

1

1

1

3

1

. Multiply these to-

gether gives T

2

T

3

3

3

3

3

3

3

3

3

T

1

T

4

4

4

4

4

4

4

4

4

, soand T

2

2

T

3

T

1

T

2

T

4

.

Setting a =

3

3

3

3

3

3

3

3

3

2

b =

1

1

1

1

1

1

1

1

1

2

r = 2 s = 2 in Proposition 9, we can show

T

2

2

T

1

T

3

. This means T

1

T

2

2

T

2

1

T

3

. gives Adding these up T

2

2

T

3

+

T

1

T

2

2

T

1

T

2

T

4

+ T

2

1

T

3

and we have our contradiction.

We have shown that K ∗

≥K

†

can only trivialbe true for a net-

work. Thus, the result is proved.


Bounds for K∗

For K∗

, the upper bound N

N

N

N

N

N

N

N

N

2

4 1N−

is achieved if P 1 1S N= − =

that is, lowerif have and andwe a Duffie Zhu network, the bound

of 1 is achieved if S can ly valueson

on

on

on

on

on

on

on

on tak

k

k

k

k

k

k

k

ke

e

e

e

e

e

e

e

e on 0

0

0

0

0

0

0

0

0 or

or

or

or

or

or

or

or

or 1.

1.

1.

1.

1.

1.

1.

1.

1.

Let us write K ∗

x =

1

1

1

1

1

1

1

1

1

4

x +

1

1

1

1

1

1

1

1

1

x

2

where x =

E[

E[

E[

E[

E[

E[

E[

E[

E[S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]

. In the feasible

region x≥1, K∗

has and isa 1minimum at x = increasing for x 1.

Thus the

he

he

he

he

he

he

he

he werlo

lo

lo

lo

lo

lo

lo

lo

lo bound for K∗

is is1. This attained i.e.when x = 1;

E[ E[S] =

√

√

√

√

√

√

√

√

√

S S] which implies that only takes values on 0 and 1.

Now we consider bound. Suppose therethe upper is some S for

whic

c

c

c

c

c

c

c

ch

h

h

h

h

h

h

h

h K ∗ x ≥ N

N

N

N

N

N

N

N

N

2

4 1N−

. Then, since K∗

∗

∗

∗

∗

∗

∗

∗

∗

) (

(

(

(

(

(

(

(

(x

x

x

x

x

x

x

x

x is a n

n

n

n

n

n

n

n

n creasingin

in

in

in

in

in

in

in

in function,

x ≥√

√

√

√

√

√

√

√

√

N S− 1. That means that E[ ] ≥√

√

√

√

√

√

√

√

√

N − 1E[

√

√

√

√

√

√

√

√

√

S] which can only

be true Duffie if have P 1 1; S N= − = that is, we the and Zhu

network.


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Bounds for K

†

For K

†

, the lower bound 1 is achieved when Var Va[ ]S = r

r

r

r

r

r

r

r

r[

√

√

√

√

√

√

√

√

√

S];

i.e. when S only values easy seetakes on or0 1. It is to from the

definition of K

†

that attain lower thanit cannot a value 1.

The proof bound complicated,for the upper is more so we

first explain heuristically how it findproceeds. We are trying to

a candidate probability vector p∈ [0, 1)

N

which maximises K

†

( )p

subject to

i

p

i

= −1 (where i takes values in {0 1 N 1}).

Proposition 3 stipulates that maximising we are over the set for

which K

†

( ) so withp is defined, we can exclude any vectors p any

element with the value 1. We call these excluded points ‘vertices’.

We will show that K

†

( ) thep is maximised on boundary of this

feasible set. That means space we reduce our focus from a with

N Ndimensions to one with − 1 the dimensions. By same argu-

ment K

†

( ) the set p must be maximised on boundary of that too

which has andN − 2 dimensions, so on inductively. Eventually we

are reduced to one-dimensional that is, line segments; the sub-

spaces non-zero elements. thiswhere atp has most two At point,

we cannot employ anythe inductive argument more, because the

boundaries theseof line which segments are atthe vertices are

not incontained the set . feasible for p Therefore we know K

†

( )p

is line andmaximised theseon segments, we can then finish the

of points 1)in [0,

N

with at least two zero elements, one of which

is in position i

.

Repeating infer this argument N − 2 times canwe that is

maximised on ofthe set points 1)in [0,

N

which have exactly two

non-zero elements. We cannot anygo further, because the bound-

ary pointsof of this subspace is the set with only one non-zero

element; that is, the thevertices, which we know do not lie in

feasible set .of p

Consider subspace,a general point p in this with its non-zero

elements at wherei and j, i

i

i

i

i

i

i

i

i j. Suppose p

i

= and p

j

= −1 for

some 0 1. Then

n

n

n

n

n

n

n

n E

E

E

E

E

E

E

E

E

m

m

m

m

m

m

m

m

m

= i

i

i

i

i

i

i

i

i

m

+ −1 j

m

for each m, and so:

p =

j i−

2

−

j

j

j

j

j

j

j

j

j −

−

−

−

−

−

−

−

−√

√

√

√

√

√

√

√

√ i

2

j i−

j −√

√

√

√

√

√

√

√

√ i

(33)

with the theterms in cancelling out in numerator and denomi-

nator is line inof (33). This means that constant along any the

subspace, maximiseand andwe simpl

l

l

l

l

l

l

l

ly

y

y

y

y

y

y

y

y en

n

n

n

n

n

n

n

n e

e

e

e

e

e

e

e

ed

d

d

d

d

d

d

d

d to

to

to

to

to

to

to

to

to c

c

c

c

c

c

c

c

ch

h

h

h

h

h

h

h

hoose

oose

oose

oose

oose

oose

oose

oose

oose i

i

i

i

i

i

i

i

i j to (33).

Noting that j − i =

j −√

√

√

√

√

√

√

√

√ i

j +

√

√

√

√

√

√

√

√

√

i we can cancel terms to

obtain:

p =

=

=

=

=

=

=

=

=

j

j

j

j

j

j

j

j

j +

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

i −

1

j

j

j

j

j

j

j

j

j +

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

i

i

i

i

i

i

i

i

i

(34)

proof thes maximal value.by examining e

e

e

e

e

e

e

e

e to

to

to

to

to

to

to

to

to ndfi

fi

fi

fi

fi

fi

fi

fi

fi the

Write p =

Var

Var

Var

Var

Var

Var

Var

Var

Var[ ]S −

−

−

−

−

−

−

−

−Var

Var

Var

Var

Var

Var

Var

Var

Var[

√

√

√

√

√

√

√

√

√

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[S

1 5

]−E[S]E[

√

√

√

√

√

√

√

√

√

S]

. ( ) 0,Since p ≥ maximising K

†

=

1

1

1

1

1

1

1

1

1

4

2

+ 1 is equivalent canto maximising ).(p We write down the

following Kuhn–Tucker conditions:

p

i

+

i

+ = 0 ∀i (28)

i

p

i

= 0 ∀i (29)

i

p

i

− 1

= 0 (30)

where each

i

is the theLagrange multiplier for constraint p

i

≥0,

and is the theLagrange multiplier for constraint

i

p

i

= 1. As we

are aretrying the maximum , theto find of Lagrange multipliers

all non-negative.

Computing the partial derivative of , becomes:Eq. ( )28

28

28

28

28

28

28

28

28

−

i

+

2

= E

1 5

− E

1

E

0 5

i

2

− 2iE

1

− +i 2

√

√

√

√

√

√

√

√

√

iE

0

0

0

0

0

0

0

0

0 5

5

5

5

5

5

5

5

5

− E

2

− E

2

1

− E

1

+ E

2

0 5

i

1 5

− iE

0 5 −√

√

√

√

√

√

√

√

√

iE

1

(31)

where v is andthe ,denominator of E

m

is shorthand for E[S

m

].

Multiplying (31) by p

i

and summing over i gives:

−N

2

= E

1 5

− E

1

E

0 5

E

2

− 2E

2

1

− E

1

+ 2E

2

0 5

− E

2

− E

2

1

− E

1

+ E

2

0 5

E

1 5

− 2E

1

E

0 5

= E

1

E

0 5

E

2

− E

1

E

1 5

+ E

0 5

E

0 5

E

1 5

− E

2

1

(32)

But all arethe theterms on right-hand side of (32) greater than or

equal to zero,

28

while arethose on the left-hand side less than or

equal to zero. Therefore each term must be exactly equal to zero,

and is is notso S a constant. At such points, defined, so we can

surmise ( ) the the that p is maximised on boundary of feasible

set .for p

This boundary is the set of points 1)in [0,

N

with at least one

zero thatelement. Suppose ( ) a thep is maximised on point on

boundary where the positionelement in i

is zero. Then we can try

to maximise subspace,( ) thep on corresponding which has N − 1

dimensions. thatRepeating above,the analysis we can see must

be maximised boundary subspace,on the of this which is the set

28

We showed this in the proof of Proposition 2, where we used Proposition 9 to

show that T

1T

4

≥T

2T

3

and T

1

T

3

≥ T

2

2

with equality if and only if T is a constant.

p =

=

=

=

=

=

=

=

=

j

j

j

j

j

j

j

j

j +

+

+

+

+

+

+

+

+ i −

j

j

j

j

j

j

j

j

j +

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

which is maximised when

j +

√

√

√

√

√

√

√

√

√

i is as large as p ble.ossi

ossi

ossi

ossi

ossi

ossi

ossi

ossi

ossi This

means that j = N − 1 and and i = N − 2 so 2p =

√

√

√

√

√

√

√

√

√

N − 2 and

K

†

= −N 1.


We have isalready shown the second part true, because = 3

for the soDMS network and K

†

grows without bound according to

Table 1. (In fact this will be true for any scale-free network.) Now

we just ofneed to show that the asymptotic value K ∗

— which we

know tends to a finite limit

it

it

it

it

it

it

it

it is—

—

—

—

—

—

—

—

— smaller than 2.

Observe Ethat

t

S = 4

4

4

4

4

4

4

4

4t

t

t

t

t

t

t

t

t−

−

−

−

−

−

−

−

−2

2

2

2

2

2

2

2

2

t+1

→ 4 thein DMS network, since at

each step of the network construction process four additional links

are created (each new has andnode an in-link an out-link with

two existing nodes). (Dorogovtsev et al., 2001) show that:

lim

t→∞

P

t

s

=12

s

s + 1

s + 2

and so the term

E

E

E

E

E

E

E

E

E

t

t

t

t

t

t

t

t

t

[

[

[

[

[

[

[

[

[S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E

t

[

√

√

√

√

√

√

√

√

√

S]

is asymptotically equal to:

4 12

∞

∞

∞

∞

∞

∞

∞

∞

∞

s=2

1

√

√

√

√

√

√

√

√

√

s

s + 1

s + 2

= 2 17

This gives us an asymptotic value for K ∗

= 1 2. 73 Therefore in

the the netted exposures,infinite limit CCP never reduces expected

except in the case 1; the thetrivial where K = i.e. when CCP clears

only asset class.


Let us use and and V to denote E[S] Var[ ]S respectively, for

a given . , theN Then, as N→∞ we can make use of following ap-

proximations:

E[

√

√

√

√

√

√

√

√

√

S] =

0 5

E

1 +

S

− 1

0 5

≈

0 5

1 −V

8

2

(35)

and

E[S

3

3

3

3

3

3

3

3

3

2

] =

1 5

E

1 +

S

− 1

1 5

≈

1 5

1 +

+

+

+

+

+

+

+

+

3V8

2

(36)

In both cases we have andexpanded the 1binomial series around

neglected cubic higher-order Substituting theseand terms. into

our expressions for K ∗

and K

†

, we find that both K∗

∼O( ) and

K

†

∼O( ).


,

,

,

,

,

,

,

,



ARTICLE

ARTICLE

ARTICLE

ARTICLE

ARTICLE

ARTICLE

ARTICLEARTICLE IN

IN

IN

IN

IN

IN

ININ PRESS

PRESS

PRESS

PRESS

PRESS

PRESS



1. Let K∗1

and K∗2

represent the values of K ∗

for constellations

1

and

2

respectively, as given equation (by

). showWe need to

that K ∗1

K ∗2

. the distribution theAs A and S are same under both

constellations, canwe show that:

K ∗1

K ∗2

⇐⇒

1

A

K

K

K

K

K

K

K

K

K∗

∗

∗

∗

∗

∗

∗

∗

∗1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+

K∗1

− A

1

A

K

K

K

K

K

K

K

K

K∗

∗

∗

∗

∗

∗

∗

∗

∗2

2

2

2

2

2

2

2

2

+

+

+

+

+

+

+

+

+

K∗2

− A

⇐⇒

M

M

M

M

M

M

M

M

M

m

m

m

m

m

m

m

m

m=1

√

√

√

√

√

√

√

√

√

a

m

−1

·E[

E[

E[

E[

E[

E[

E[

E[

E[

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+ a

a

a

a

a

a

a

a

a

2

2

2

2

2

2

2

2

2

+

+

+

+

+

+

+

+

+

M

M

M

M

M

M

M

M

M

m=3

√

√

√

√

√

√

√

√

√

a

m

−1

·E[

E[

E[

E[

E[

E[

E[

E[

E[

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]

⇐⇒

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

2

2

2

2

2

2

2

2

2

√

√

√

√

√

√

√

√

√

a

1

+ a

2

which is true for any a

1

, a

2

0.

Now let us define K

†

1

and K

†

2

similarly. We have:

K

†

K

†

⇐⇒

A +

+

+

+

+

+

+

+

+ 1 Var[S] −

a

1

+ +1

M

m=2

2

2

2

2

2

2

2

2

√

√

√

√

√

√

√

√

√

a

m

2

Var[

√

√

√

√

√

√

√

√

√

S]

a

1

+ +1

M

m=2

2

2

2

2

2

2

2

2

√

√

√

√

√

√

√

√

√

a

m

E[S

3

3

3

3

3

3

3

3

3

2

] − E[S]E[

√

√

√

√

√

√

√

√

√

S]

A +

+

+

+

+

+

+

+

+ 1 Var[S] −

a

2

+ +1

m=2

2

2

2

2

2

2

2

2

√

√

√

√

√

√

√

√

√

a

m

2

Var[

√

√

√

√

√

√

√

√

√

S]

a

2

+ +1

m=2

2

2

2

2

2

2

2

2

√

√

√

√

√

√

√

√

√

a

m

E[S

3

3

3

3

3

3

3

3

3

2

] − E[S]E[

√

√

√

√

√

√

√

√

√

S]

⇐⇒ A + 1 [ ]Var S

1

a

1

+ 1 +

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

a

2

+

M

m=3

3

3

3

3

3

3

3

3

√

√

√

√

√

√

√

√

√

a

m

−1

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+

a

2

+ 1 +

+

+

+

+

+

+

+

+

M

m=3

3

3

3

3

3

3

3

3

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

m

m

m

m

m

m

m

m

m

+Var[

√

√

√

√

√

√

√

√

√

S]

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+

a

2

+ −1

a

1

+ −1√

√

√

√

√

√

√

√

√

a

2

0 (39)

which is always true since a

1

a

2

. theThus all parts of proposition

are proved.

K

1

K

2

⇐⇒

A SVar[ ] −

M

m=1

1

1

1

1

1

1

1

1

√

√

√

√

√

√

√

√

√

a

m

2

Var[

√

√

√

√

√

√

√

√

√

S]

M

m=1

1

1

1

1

1

1

1

1

√

√

√

√

√

√

√

√

√

a

m

E[S

3

3

3

3

3

3

3

3

3

2

] − E[S]E[

√

√

√

√

√

√

√

√

√

S]

A SVar[ ] −

√

√

√

√

√

√

√

√

√

a

1

+ a

2

+

M

m=3

3

3

3

3

3

3

3

3

√

√

√

√

√

√

√

√

√

a

m

2

Var[

√

√

√

√

√

√

√

√

√

S]

√

√

√

√

√

√

√

√

√

a

1

+ a

2

+

M

m=3

3

3

3

3

3

3

3

3

√

√

√

√

√

√

√

√

√

a

m

E[S

3

3

3

3

3

3

3

3

3

2

] − E[S]E[

√

√

√

√

√

√

√

√

√

S]

⇐⇒

AVar

Var

Var

Var

Var

Var

Var

Var

Var[

[

[

[

[

[

[

[

[S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

M

m=1

1

1

1

1

1

1

1

1

√

√

√

√

√

√

√

√

√

a

m

−

M

M

M

M

M

M

M

M

M

m=1

√

√

√

√

√

√

√

√

√

a

m

Var[

√

√

√

√

√

√

√

√

√

S]

A SVar[ ]

√

√

√

√

√

√

√

√

√

a

1

+ a

2

+

M

m=3

3

3

3

3

3

3

3

3

√

√

√

√

√

√

√

√

√

a

m

−

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+ a

a

a

a

a

a

a

a

a

2

2

2

2

2

2

2

2

2

+

+

+

+

+

+

+

+

+

M

M

M

M

M

M

M

M

M

m=3

√

√

√

√

√

√

√

√

√

a

m

Var[

√

√

√

√

√

√

√

√

√S]

⇐⇒ 0 AVar[ ]S

1

√

√

√

√

√

√

√

√

√

a

1

+ a

2

+

M

M

M

M

M

M

M

M

M

m

m

m

m

m

m

m

m

m=

=

=

=

=

=

=

=

=3

3

3

3

3

3

3

3

3

√

√

√

√

√

√

√

√

√

a

m

−1

M

m=1

1

1

1

1

1

1

1

1

√

√

√

√

√

√

√

√

√

a

m

+

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

a

2 −√

√

√

√

√

√

√

√

√

a

1

+ a

2

Var[

√

√

√

√

√

√

√

√

√

S] (37)

which is always true since

e

e

e

e

e

e

e

e

√

√

√

√

√

√

√

√

√

a

1

+

√

√

√

√

√

√

√

√

√

a

2

√

√

√

√

√

√

√

√

√

a

1

+ a

2

for any a

1

, a

2

0.

Thus the thefirst part of proposition is proved.

2. Let K ∗2

and K∗3

represent the values of K ∗

for constellations

2

and

3

respectively. We need to show that K ∗2

K ∗3

. We have:

K ∗2

K ∗3

⇐⇒

1

A +

+

+

+

+

+

+

+

+ 1

1

1

1

1

1

1

1

1

K

K

K

K

K

K

K

K

K∗

∗

∗

∗

∗

∗

∗

∗

∗2

2

2

2

2

2

2

2

2

+

+

+

+

+

+

+

+

+

K∗2

− +A 1

1

A + 1

K

K

K

K

K

K

K

K

K∗

∗

∗

∗

∗

∗

∗

∗

∗3

3

3

3

3

3

3

3

3

+

+

+

+

+

+

+

+

+

K∗3

− +A 1

⇐⇒

a

1

+ 1 +

+

+

+

+

+

+

+

+

M

M

M

M

M

M

M

M

M

m=2

√

√

√

√

√

√

√

√

√

a

m

−1

·E[

E[

E[

E[

E[

E[

E[

E[

E[

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

√

√

√

√

√

√

√

√

√

S]

a

2

+ 1 +

+

+

+

+

+

+

+

+

m=

=

=

=

=

=

=

=

=2

√

√

√

√

√

√

√

√

√

a

m

−1

·E[

E[

E[

E[

E[

E[

E[

E[

E[

S

S

S

S

S

S

S

S

S]

]

]

]

]

]

]

]

]

E[

E[

E[

E[

E[

E[

E[

E[

E[

√

√

√

√

√

√

√

√

√

S]

⇐⇒

a

a

a

a

a

a

a

a

a

1

+ 1 +

+

+

+

+

+

+

+

+

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

2

2

2

2

2

2

2

2

2

√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

+

+

+

+

+

+

+

+

+

a

a

a

a

a

a

a

a

a

2

2

2

2

2

2

2

2

2

+

+

+

+

+

+

+

+

+ 1

⇐⇒

a

1

+ −1√

√

√

√

√

√

√

√

√

a

a

a

a

a

a

a

a

a

1

1

1

1

1

1

1

1

1

a

2

+ −1√

√

√

√

√

√

√

√

√

a

2

(38)

which is true since a

1

a

2

and

d

d

d

d

d

d

d

d

√

√

√

√

√

√

√

√

√

X + −1√

√

√

√

√

√

√

√

√

X is a func-decreasing

tion.

Now let us define K

†

2

and K

†

3

similarly. We have:

K

†

2

K

†

3


Consider asset classa asingle agent i and single k. theIn

trading network, the agent has S

i

links, (0,each distributed N

2

).

As these are all exposures the they to same asset class, are per-

fectly correlated and so the total market risk conditional on S

i

is

M

ik

|S

i

∼ N 0

2

S

2

i

. Then market risk is:

Var[M

ik

] E=

Var[M

ik

|S

i

]

+ Var

E[M

ik

|S

i

]

= E

2

S

2

i

] + Var[0] =

2

E[S

2]

(40)

For the the procedure givesclearing network, exactly same

market risk of ˜

2

E[

S

2

] thefor each agent in each asset class. As

number asset classes doesof agents and not change, conservation

of mustmarket means thatrisk the two terms be equal.

The perfect assumption of correlation assumes thethat agent

has no offsetting derivative positions. can this,We allow for so

long as hedgingthe extent of such is inthe same the trading

and clearing networks. Let h

ik

be the proportion of positions in as-

set class k which do not represent hedging positions. Then overall

market risk tradingin the network is h

ik

2

E[S

2

] and market risk in

the clearing network is h

ik ˜

2

E[

S

2

] so Proposition 8 still holds.

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