Does a CCP reduce counterparty risk in a
heterogeneous network?
Rod Garratt (FRBNY) and Peter Zimmerman (BoE)
CEMLA, Seminar on Network Analysis and Financial Stability Issues, 2014
The views expressed in this paper are those of the authors and do not necessarily reflect those of the
Federal Reserve Bank of New York, the Federal Reserve System or the Bank of England.
Netting Efficiency
 One of the benefits of central clearing is that it reduces the
aggregate level of exposures in the system by netting offsetting
claims.
 This is true in a world where all asset classes are
simultaneously novated to a CCP.
 Duffie and Zhu (2011) shows that when a single asset class (or
a subset of the asset classes) is switched to central clearing,
bilateral exposures may increase, as a result of reduced netting
opportunities across pairs of banks, resulting in an overall loss
in netting efficiency.
2
Netting Efficiency
1
1
A
-1
1
A
B
-1
B
-1
-1
1
CCP
(i) central clearing in a single asset is bad
A
-1
B
1
A
1
1
-1
-1
C
1
B
1
-1
1
-1
CCP
C
-1
(ii) central clearing in a single asset is good
3
Netting Efficiency
 In both cases introducing a CCP in all assets would minimize
net exposures.
 It is the fact that some assets are less conducive to central
clearing than others that makes this (ideal) solution impractical.
4
Duffie and Zhu (2011)
 Duffie and Zhu’s results are not based on actual exposure
networks.
 Rather they express the total exposure of dealer i to dealer j of
all positions in asset class k by a random variable
 They specify a distribution that is assumed to generate these
exposures and their results are based on expected exposures
with respect to this distribution.
 Because they assume symmetry in the distributions across all
dealers, they can compute the expected total net exposure with
or without a CCP by looking at a single dealer.
5
Duffie and Zhu (2011)
 Taking as given the distribution used to generate exposures,
they are able to express conditions for whether or not a CCP is
beneficial solely in terms of the number of assets classes (K)
and the number of dealers (N).
𝑁2
𝐾<
4(𝑁 − 1)
 Advantage: it relates the question of whether or not the
introduction of a CCP in a single asset class is beneficial or not
to easily observable parameters.
 Disadvantage: the implied networks are too homogeneous and
lack the scale-free property typically associated with financial
networks (Soramaki et al. 2007, and Ianaoka et al., BoJ
working paper, 2004).
6
Barabási and Albert (1999)
 Barabási and Albert (1999) argue that two features are
essential to capture real world networks: growth and
preferential treatment.
 To incorporate the growing character of the network, they
propose that the network construction begin with a small
number (m0) of nodes.
 At every time step a new node is added, which is linked to m
existing nodes.
 To incorporate preferential attachment, they assume that the
probability that a new node will be connected to an existing
node i is given by 𝑘𝑖 / 𝑗 𝑘𝑗 where ki is the connectivity of node i.
 This network evolves into a scale-free network (the number of
links k originating from a given node exhibits a power law
distribution P(k)=k-α) with power α=3.
7
Dorogovtsev, Mendes and Samukin (2001)
 Barabási and Albert obtain results that hold when the number of
nodes becomes very large
 Assumption of a large network is not necessarily realistic for our
purposes
 Duffie and Zhu consider a network of size 12, which is the number
of entities that, at the time of writing their paper, had partnered
with ICE Trust to create a CCP for clearing credit default swaps.
 Table 1 in Galbiati and Soramaki (2012) covers a wider range of
CCP networks; none of them have more than 60 members.
 We use a simpler version that has an exact solution for a
network of any size
 DMS Network
8
DMS Network
1. Begin time t = 2 with 3 nodes. Each has two links connecting
to one another.
2. At each step a new node is added with two links. Choose an
existing link at random: each is chosen with equal probability.
Connect the new node to the two nodes which share that link.
This process generates an
undirected network of size N
=t+1, which tends towards a
scale-free network with
exponent a = 3 as t becomes
large.
9
t=100
t=100
t=500
t=1000
Degree Densities for t=100, 500 and 1000 (blue, pink,
green). Power function parameter a=2,3.
Distribution of S
 for a given t
𝑃𝑡 𝑠
𝑡
𝑠−1
𝑠
=
𝑃𝑡−1 𝑠 − 1 + 1 −
𝑃𝑡−1 𝑠
𝑡 + 1 2𝑡 − 3
2𝑡 − 3
1
+
1[𝑠=2]
𝑡+1
for 𝑡 ≥ 3 with the initial condition 𝑃2 𝑠 = 1[𝑠=2]
 Exact solution for any t is messy, but obtainable (see DMS)
 In the limit
12
lim 𝑃𝑡 𝑠 =
𝑡→∞
𝑠(𝑠 + 1)(𝑠 + 2)
 Point 1: we can obtain finite and limit predictions for the
expected number of links each node has: 𝐸𝑡 (𝑆) = 𝑡−1
𝑗=2 𝑗𝑃𝑡 (𝑗)
15
Bilateral Exposures
We now need to determine the value (or weight) of each link.
 As in Duffie and Zhu, we assume that if two dealers have a link
in one asset class, then they have a link in all K asset classes.
 For each pair of connected nodes i and j, we generate the net
𝑘
exposures 𝑋𝑖𝑗
, k=1,…,K, as K iid normal random variables with
mean 0 and standard deviation s.
𝑌𝑖𝑗𝐾
= max
𝐾
𝑘=1
𝑘
𝑋𝑖𝑗
,0
denotes node i’s netted exposure to node j
𝑓 𝐾 = 𝐸 𝑌𝑖𝑗𝐾
denotes expected net exposure between any two nodes
16
f(K)
𝐾
𝑓 𝐾 = 𝐸 𝑌𝑖𝑗𝐾 = 𝐸 max
𝑘=1
∞
=
0
𝑘
𝑋𝑖𝑗
,0
∞
=
max 𝑥, 0 𝑝𝐾 𝑥 𝑑𝑥
−∞
𝑥𝑝𝐾 𝑥 𝑑𝑥
where 𝑝𝐾 𝑥 is the pdf of
𝑘
𝐾
𝑘=1 𝑋𝑖𝑗
𝑘
Since 𝑋𝑖𝑗
~𝑁 0, 𝜎 2 , we know that
.
𝑘
𝐾
𝑋
𝑘=1 𝑖𝑗 ~ 𝑁
0, 𝐾𝜎 2 .
Thus
∞
𝑓 𝐾 =
0
𝑥2
x
𝑒𝑥𝑝 −
dx =
2
2𝐾𝜎
2𝜋𝐾𝜎
1
𝐾
𝜎
2𝜋
 Point 2: we can obtain an expression for the expected net
exposure associated with each link
17
I. Expected netting efficiency
𝑌𝑖𝑗𝐾 = 𝐸 𝐸
𝜑𝑁,𝐾 = 𝐸
𝑗∈𝐽𝑖
𝑌𝑖𝑗𝐾 |𝑆
= 𝐸 𝑆𝑓(𝐾)
𝑗∈𝐽𝑖
= 𝐸 𝑆 𝑓(𝐾)
 With a CCP net exposure of a given node i becomes
𝑆
𝑌𝑖𝑗𝐾−1 + 𝑌𝑖,𝐶𝐶𝑃
|𝑆
𝜑𝑁,𝐾 = 𝐸 𝐸
= 𝐸 𝑆𝑓 𝐾 − 1 + 𝑓(𝑆)
𝑗∈𝐽𝑖
= 𝐸 𝑆 𝑓 𝐾 − 1 + 𝐸 𝑓(𝑆)
 Change in expected net exposure
𝜑𝑁,𝐾 − 𝜑𝑁,𝐾 = 𝐸 𝑆 𝑓 𝐾 − 1 + 𝐸 𝑓(𝑆) − 𝐸 𝑆 𝑓(𝐾)
18
I. Expected netting efficiency
 Change in expected net exposure
𝜑𝑁,𝐾 − 𝜑𝑁,𝐾 = 𝐸 𝑆 𝑓 𝐾 − 1 + 𝐸 𝑓(𝑆) − 𝐸 𝑆 𝑓 𝐾 < 0
iff
𝐾+ 𝐾−1<
DMS network: 𝐸𝑡 𝑆 =
𝐸𝑆
~
𝐸 𝑆
∞
𝐸𝑆
𝐸
𝑆
4𝑡−2
𝑡+1
𝑠=2
→ 4 and hence for large t,
4
= 1.73 < 2 + 1
12
𝑠(𝑠 + 1)(𝑠 + 2)
Result 1. CCP reduces netting efficiency for all K>2.
(Strengthens D&Z)
19
II. Variance of net exposures
Can show that
𝑣𝑁,𝐾 − 𝑣𝑁,𝐾 < 0
2 𝐾−1<
iff
𝑉𝑎𝑟 𝑆 − 𝑉𝑎𝑟
𝐸 𝑆 3/2 − 𝐸[𝑆]𝐸
𝑆
𝑆
DMS network: R.H.S increases without bound as t gets large
Why? Asymptotically, 𝐸 𝑆 2 is infinite, but 𝐸 𝑆 𝑚 will be finite for
any 𝑚 < 2.
20
II. Variance of net exposures
Fat-tailed networks: 𝑃(𝑆 = 𝑠) ~𝑠 −𝛼
𝐸 𝑆𝑚 ~
𝑁−1
𝑠 𝑚−𝛼 𝑑𝑠
𝑍
where Z is some constant. Thus, as 𝑁 → ∞
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 + 𝑂( 𝑁 − 1 𝑚−𝛼+1 ) 𝑖𝑓 𝑚 ≠ 𝛼 − 1
𝑚
𝐸𝑆 ~
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 + log 𝑁 − 1
𝑖𝑓 𝑚 = 𝛼 − 1
21
II. Variance of net exposures
Can show that
𝑣𝑁,𝐾 − 𝑣𝑁,𝐾 < 0
2 𝐾−1<
iff
𝑉𝑎𝑟 𝑆 − 𝑉𝑎𝑟
𝐸 𝑆 3/2 − 𝐸[𝑆]𝐸
𝑆
𝑆
DMS network: R.H.S increases without bound as t gets large
Means when you write the variance in terms of expectations
operators, everything is finite except for an 𝐸 𝑆 2 term in the
numerator. Therefore RHS increases without bound as the
network becomes larger.
Result 2. If agent’s preferences put any weight on volatility,
then for large enough network a CCP would be viewed as
beneficial.
22
Finite Case: Tradeoff between mean and variance
23
Other results (different network structures)
 Homogeneous networks
 In D&Z, 𝑃 𝑆 = 𝑠 = 1 when 𝑠 = 𝑁 − 1 and 0 otherwise. So
𝐸 𝑆 = 𝑁 − 1 and Var 𝑆 = 0.
 CCP beneficial iff 𝐾 + 𝐾 − 1 < 𝑁 − 1
 Core-periphery networks
 Nodes in completely connected core attached to periphery nodes
with probability p
 Can express any number of links for any randomly selected node
as a mix of binomial distributions plus a constant
 R.H.S. of iff conditions increase without bound as network gets
large
▫ CCP unequivocally good
 Trade off exists for smaller networks
24
Literature Review
 Anderson and Perez Saiz (2013) and Cox, Garvin and Kelly
(2013) apply the Duffie and Zhu framework to explore the issue
of interoperability between CCPs
 they retain the assumption of a homogeneous network
 Renault (2010) and Borovkova and Lalaoui El Mouttalibi (2013)
examine the effect of different network configurations and CCP
arrangements on default risk – with the latter finding that more
homogeneous networks may be more resilient – while Song,
Sowers and Jones (2014) study the effect of network structure
on the maximum exposure risk of the CCP itself.
 not interested in the effect of a CCP on netted exposures
 look at how the default contagion dynamics of the system
change as a result of the introduction of a CCP.
25
Literature Review
 Cont and Kokholm (2014) extends D&Z in a different way to
ours: they relax the assumption of normality of link weights, but
they do not vary the pattern of network links.
 In particular, they allow for different distributions and
heterogeneity of risk across asset classes.
 They look not only at expected exposures but also tails
(99% quantile of total exposure and 99% quantile of
conditional expectation).
 They don’t look at variance.
 use a simulation approach with two asset classes; don’t
adopt an analytical approach.
26
Literature Review
 Jackson and Manning (2007) and Galbiati and Soramaki (2012)
use different approaches to examine the desirability of tiering—
restricting direct access to the CCP to a limited set of
counterparties.
 adequacy of margin requirements
27
Justification for theoretical approach/
Concluding remarks
 In general, the data is being collected from now, but it is not
back-dated, and in particular we don’t have data over the 2008
crisis period
 Our approach is applicable to growing networks or in cases of
financial innovation.
 Suppose a regulator hosts a market which is expected to
grow (in terms of numbers of participants) over the next few
years.
 Our paper provides insight as to whether a CCP is beneficial
or not.
▫ If the network is expected to grow to a very large size, the CCP
is likely to be beneficial.
▫ But if the number of potential participants is bounded at a fairly
low number, then the CCP may provide less efficient netting
than bilateral trading does.
28
Justification for theoretical approach/
Concluding remarks
 Having more data is only a panacea if you know how it all fits
together.
 Suppose that a regulator can collect the network data for
each individual asset class.
 It would then need to fit these together to form the overall
exposure network across all K asset classes.
 In reality it will be difficult to identify the nodes in each
network with the same corporate entity.
▫ eg, the part of MegaBank Corp. which trades CDS might be a
different corporate entity to that which trades T-bills.
 Our approach allows the regulator not to be concerned
about the identity of each node, but to use the pattern of the
network as a whole.
29
Thank you!
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