Financial contagion in interbank networks and
real economy
Asako Chiba∗
December 2016
Abstract
Understanding of financial market as networks is essential in designing
regulations which prevent system-wide breakdown. As a stylized representation of the interbank lending market, this paper construct a model in which
banks’ liabilities form a core-periphery network. The model also includes
endogenously reacting asset prices as contagion proceeds. The main contributions are as follows. First, the simulation shows that when the asset price is
endogenized in the contagious insolvency, the degree of contagion increases
significantly compared with that in exogenous price model. Second, the
analytical study of core-periphery network shows the non-motonotic expansion of insolvency in the strength of the links. Given the exposures between
core banks and peripheral banks, the likelihood of contagious insolvency
is high if the exposures between core banks are in intermediate range. If
the exposures between core banks are very low, the contagion between core
banks are not likely. On the other hand, if they are very high, each bank
has small exposure to the asset price, hence the devaluation of asset does not
have enough influence to damage market value of banks.
∗
Graduate School of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-0033, Japan: [email protected].
1
1
Introduction
Interdependencies among financial institutions generate chain reaction bankruptcy.
Understanding of financial market as networks is essential in designing regulations
which prevent system-wide breakdown. As a stylized representation of the interbank lending market, this paper construct a model in which banks’ liabilities form
a core-periphery network. The model also includes endogenously reacting asset
prices as contagion proceeds. The main contributions are as follows. First, the
simulation shows that when the asset price is endogenized in the contagious insolvency, the degree of contagion increases significantly compared with that in
exogenous price model. Second, the analytical study of core-periphery network
shows the non-motonotic expansion of insolvency in the strength of the links.
Given the exposures between core banks and peripheral banks, the likelihood of
contagious insolvency is high if the exposures between core banks are in intermediate range. If the exposures between core banks are very low, the contagion
between core banks are not likely. On the other hand, if they are very high, each
bank has small exposure to the asset price, hence the devaluation of asset does not
have enough influence to damage market value of banks.
The model borrows much from Elliott, Golub, and Jackson (2014). They study
the expansion of default from one bank to another bank connected through a given
network of liabilities. The cascades of default in their model can be summarized
as follows: If a bank’s value falls below a failure threshold, it discontinuously loses
its full value. This imposes losses on other banks, and these losses then propagate
to others, even those who are not directly connected to the banks that initially
failed. After this contagion, other banks may hit failure thresholds and also lose
their values. An idiosyncratic shock that is relatively small can be amplified in
this way. Focusing mainly on random networks, their main results show how the
probability and the degree of cascades are affected by two feature values of network:
integration and diversification. Integration refers to how much of a bank is held by
final investors, and how much is held by other banks. Diversification refers to how
many others a bank is held by. Each of these two feature values has non-monotonic
effect on the amplification of failure. Financial system is most vulnerable to
widespread cascades when both integration and diversification are intermediate.
If there is no integration, then clearly there is no contagion. As integration
increases, the exposure of banks to each other increases and so contagions become
more likely. Thus, higher level of integration leads to increase in the probability
and impact of contagions. The opposing effect is that a bank is less dependent
on its own primitive assets as it becomes more integrated. Therefore, although
2
integration can empower the amplification, it also decreases the likelihood of the
first failure. Diversification has also trade-offs. Here the exposure of organizations
is fixed but the number of organizations cross-held varies. If the diversification
is low, banks are very sensitive to particular counterparts, but the network of
interconnections is weak. As diversification increases, banks have enough of their
ownerships that are concentrated in particular others so that a cascade take places,
and yet the cross-holdings is strong enough for the contagion to be far-reaching.
Finally, as diversification is extremely high, banks’ portfolios are diversified so that
they become less susceptible to any particular bank’s failure. Putting these results
together, a financial system is most likely to collapse by widespread cascades when
both integration and diversification are intermediate.
There are two differences between their work and this paper. First, this model
includes feedback effect between advance of failure and reduction in asset price.
Specifically, this model describes the price of each proprietary asset decreases
as the number of failed banks increases. This extension reflects the frequently
observed fact that asset prices fall together rapidly when the bubble is bursting.
This paper provides results on the systemic behaviour when each bank’s individual
asset price reacts against the contagion of failure. Second, this model mainly focus
on core-periphery network. As several empirical studies show, interbank networks
consists of two types of banks: local banks and global banks, or, retail banks and
wholesale banks. The former depends relatively high on real sectors rather than
financial sectors, whereas the latter is engaged heavily on interbank borrowing and
lending. In other words, global banks are linked to each other and play roles as
hub nodes, with local banks linked to some small number of global banks. As
a well description of the reality, this paper distinguishes global banks and local
banks and models as core-periphery networks.
As Brunnermeier and Oehmke (2012) review, the recent financial crisis had a
wide-spread effect triggered by a single event for the study of financial economics.
The view that financial interlinkages amplify the shock has been gaining traction
among many economists. A large body of work has focused on the role of
interconnections among banks as amplifiers of shocks. Most of this literature
takes interbank structure as exogenously given and analyzes the likelihood of
contagion by comparative statistics. This strand of literature dates back to the
works of Allen and Gale (2013), Freixas, Parigi, and Rochet (2000) and Eisenberg
and Noe (2001). Using a network with four banks, Allen and Gale (2013) illustrate
that the network structure among banks determines whether contagion occurs or
not. When all banks are completely connected to each other, the impact of a shock
is so small that there is no contagion, since the shock is shared among banks.
3
However, if each bank has links to only a fraction of others, those neighbors of
the bank initially hit incur substantial losses to liquidate long-term assets, thus
bring other banks into the contagion. Freixas et al. (2000) show that networks
may also be fragile with money-center banks and the peripheral banks only linked
to the center. Eisenberg and Noe (2001) show that the set of payments of banks
that satisfies their obligations uniquely exists under mild regularity conditions,
and provide an algorithm which corresponds to a process of sequential default.
Various algorithms to cauculate steady state are organized and presented by Upper
(2011). Cifuentes, Ferrucci, and Shin (2005) incorporate fire sales into the setting
of Eisenberg and Noe (2001); their approach is further extended by Gai and
Kapadia (2010) and Feinstein (2015). Gai and Kapadia (2010), using a standard
network model of epidemics characterized by its degree distribution, suggest that
financial systems are robust-yet-fragile tendency: the effects can be extremely
widespread depending on the structure of the network and on the location of the
target node. Feinstein (2015) provides a framework for modeling the financial
system with multiple illiquid assets, where the model of Cifuentes et al. (2005)
incorporate a single asset. Some of the more recent studies, including Elliott et
al. (2014), provide a comparative analysis of the extent of financial contagion as
a function of the structure of the linkages among financial institutions. Awiszus
and Weber (2015) incorporate direct liabilities, cross-holdings and fire sales and
characterize the equilibrium as the vector of clearing payments and the price
of the common illiquid asset exposed to price effects. In contrast, some of the
recent literature focus on the endogenous formation of linkages and study the
implications. Zawadowski (2013), Babus (2014), Farboodi (2014) and Acemgolu
and Tahbaz-Salehi (2015) explicitly model interbank liabilities to investigate the
relationship between counterparty risk and the equilibrium interest rates. Farboodi
(2014) studies a model of endogenous intermediation among debt financed banks.
She shows that if some banks have access to investment, equilibrium networks have
core-periphery structures and may not be efficient. Acemgolu and Tahbaz-Salehi
(2015) also consider the endogenous formation of financial linkages, but interest
rate is endogenously determined in their model, where Farboodi’s model takes
the face value of the contracts and the allocation rule of intermediation rents as
exogenously given. This strand of literature points out that banks come into contract
voluntarily, and that the structures of the networks should be equilibrium objects
endogenously determined. In other words, regulations based on the network
analysis that do not regard the network structure as the equilibrium are subject to
the critique suggested by Lucas (1976). Although these criticisms are reasonable,
it is still meaningful to examine the resulting cascades with the network fixed. The
4
concept of network structure exogenously given is valid under the assumption of
short-term story. If one is to model the resulting contagion just after the triggering
event of crisis, it suffices to fix network structure, because there is no room for
the agents to optimize linkages. In this sense, my work is to study of shortterm behavior of banks on networks, especially when asset prices are treated as
endogenous factor.
2
2.1
Model
Banks and interbank networks
There are n banks and M assets. The price of asset k is denoted pk . Banks hold
assets and liabilities to other banks. Asset holdings are described as follows. Let
Dik be the share of the value of asset k held by bank i and let "asset-holdings matrix"
D ∈ RN ×M denote the matrix whose (i, k)th entry is equal to Dik . Holdings of
liabilities are described as follows. For any bank i and j, let Cij denote the fraction
of bank j’s value of assets held by bank i. Cii = 0 for each i. Let "cross-holdings
matrix" C ∈ RN ×N denote a network where there is a directed link from bank i
to bank j if i holds a positive fraction of bank j’s assets, so that Cij > 0. The
remainder of the fractions after all these interbank cross-holding areP
accounted for
is capital, or shares held by other investors than banks. Ĉii = 1 − j Cji denote
this capital of bank i, which is assumed to be positive.
2.2
Value of banks
The accounting and the key valuation equations are set as follows. Let Vi denote
the total value of bank i’s asset. Vi is the sum of the value of bank i’s individual
assets and the value of its liabilities to other banks:
X
X
Vi =
Dik pk +
Cij Vj
(1)
j
k
In the matrix form, equation (1) is written as
and solved to yield
V = Dp + CV,
(2)
V = (I − C)−1 Dp
(3)
5
, where V ∈ RN ×1 denotes the vector of banks’ asset values. Let vi denote the
market value of bank i, which is defined as the value of assets which belong to
final investors of bank i.
vi = Ĉii Vi
(4)
Therefore,
v = ĈV = Ĉ(I − C)−1 Dp = ADp
(5)
, where v ∈ RN ×1 denotes the vector of banks’ market values. Equation (5) shows
that the value of a bank equals the sum of the values of its final claims on individual
assets, with bank i having a fraction Aij of j’s direct holdings of individual assets.
The dependency matrix A is substantially different from C and Ĉ, which capture
the direct claims, because A takes into account all indirect holdings as well as
direct holdings.
2.3
Insolvency
A bank lose its value discontinuously if its market value falls below the threshold. If
vi , the value of bank i, falls below some threshold level vi , bank i becomes insolvent
and incurs insolvency cost βi (p). This discontinuity brings nonlinearities into the
financial system. When this insolvency cost is taken into account, each bank’s
balance sheet is directly affected. The equity value of bank i becomes:
X
X
(6)
Dik pk − βi Ivi <vi
Vi =
Cij Vj +
j
k
where Ivi <vi is an indicator variable taking value 1 if vi < vi and 0 otherwise. This
leads to a new version of (3)
V = (I − C)−1 (Dp − b(v, p))
(7)
where bi (v, p) = βi (p)Ivi <vi . Correspondingly, (5) is re-written as
v = Ĉ(I − C)−1 (Dp − b(v)) = A(Dp − b(v, p))
(8)
An entry Aij of the dependency matrix denotes the proportion of j’s failure costs
that i incurs when j fails as well as i’s claims on the individual assets that j directly
holds. If bank j fails and incurs insolvency costs of βj , then i’s value decreases
by Aij βj .
6
2.4
Asset price
This paper introduces the price of each bank’s individual asset endogenously
reacting to the expansion of insolvency. The basic economic idea is that, if the
contagious default expands on the financial networks and more banks fail, asset
price is pushed down. Assuming the individual asset of each bank is tradable in
some market, it faces decreasing demand just after the realization of triggering
event of crisis. Thus, asset price declines. Awiszus and Weber (2015) assume
all the banks hold the same asset and express the price of this single asset as an
inverse demand function of the number of unit of the asset which has been sold
as of then. Although each bank in this model is assumed to hold different asset
from one another, same argument holds and thus asset prices can be modeled
in similar manner. To integrate this idea into the baseline network model of
contagious defaults, I assume that price of each asset is expressed an exogenously
given positive continuous function. In particular, using parametric exponential
function, the price of individual asset held by bank i is written as follows.
pi = exp(−γx)
(9)
, where x denotes the total number of assets sold-off at that time. This setting
implicitly incorporates the mutual effect between financial system and real economy, abstracting away from describing details of the whole mechanism, which is
complicated in reality.
2.5
Equilibrium
The equilibrium consists of {p, v} which satisfies equation (8) and (9). The time
horizon in this model is as follows. In the first period, cross-holdings is determined,
given feature values. Each bank’s asset price is initially 1. Then one of the
peripheral bank’s asset price falls to 0. In the second period, each bank’s market
value is recalculated. A bank with market value below the threshold becomes
insolvent and its asset price falls to 0. Each bank’s asset price reacts negatively
to the number of insolvent banks. In the next period, each bank’s market value is
recalculated and some banks newly become insolvent. This process is continued
until the set of insolvent banks does not change before and after the period.
7
2.6
Description of core-periphery networks
This paper focus on a specific type of network, consisting of global banks playing
as hub nodes and local banks connected to limited number of global banks. I
call it "core-periphery" network and describe the details below. There are two
types of banks: Nc of them, called core banks, have mutually directed liabilities
to each other, and the rest N − Nc of them, called peripheral banks, have mutually
directed liabilities with one of the core banks. The cross-holdings matrix C is
characterized by the strength of links to each other: how much of bank’s liability
is held by other banks, and how much by outside investors. Let ccc denote the
fraction of a core bank’s asset value held by other core banks and ccp denote the
fraction of a core bank’s asset value held by its neighboring peripheral banks. In
other words, a core bank ’s exposure to other core banks is ccc and the exposure
to the peripheral banks is ccp . In addition, let the banks have different assets with
each other. Then, M = N and D = I. This setting gives a simple approximation
of the interbank market in the real world, where regional banks borrow from and
lend to a particular large bank and those large banks are connected to each other
in the global financial market. Figure 1 shows some examples with N = 12 and
Nc = 3.
Figure 1: Examples of core-periphery network
8
3
3.1
Simulation and analysis
simulation
This section shows the result of simulation and detailed analysis. Figure 2 illustrates how the number of insolvent banks changes as the level of core exposures
ccc , peripheral exposures ccp and the price sensitivity γ change. Here, Nc = 10,
N = 100 and θ = 0.98. The left figure shows the number of insolvent banks with
γ = 0, which corresponds to the case where the asset price is fixed through the
expansion of insolvency. The middle and the right figures show the result with
γ = 5 × 10−6 and γ = 5 × 10−5 . In each figure, the vertical axis is ccc , integration
between core banks, and the horisontal axis is cpc , integration between a core bank
and a peripheral bank. These result exhibit two noticeable things. First, in the right
figure, the number of insolvent banks are significantly high compared with that in
the left figure. This shows that the effect of endogenous asset price has substantial
effect on the expansion of insolvency, even if its sensitivity is very small. Second,
given the peripheral exposure ccp , the number is nonlinear in the core exposure
ccc . Starting from ccc = 0, the degree of contagion increases as ccc increases, but
from a certain point, it turns into decrease. The degree of contagion is low when
the strength of links between core banks are either very low or very high, which is
consistent with the experiment in Elliott et al. (2014). The next section gives the
formal explanation to this phenomena.
9
Figure 2: The number of failed banks on core-periphery networks
3.2
Analysis
The phenomenon observed in the previous section can be analytically verified.
The key parameter is the dependency matrix A. A bank is judged to be insolvent
based on its market value, which reflects the set of insolvent banks before that
period. Since D = I in (5), (i, j)the entry in A measures the change in bank iś
market value when bank jś asset fall by 1, which exactly means bank j has become
insolvent in the case of exogenous asset price. In other words, if Aij is large, bank
i is likely to become insolvent following bank j. For this reason, this section gives
a closer to look at the matrix A. The cross-holdings matrix C can be decomposed
in four blocks.
Cc Cp
C≡
(10)
Cp0 O
where Cc ∈ RNc ×Nc denotes the links among core banks and Cp ∈ RNc ×(N −Nc )
denotes those between core banks and peripheral banks. Since peripheral banks
do not have links with each other, the bottom-right block which represents the
links among peripheral banks is O ∈ R(N −Nc )×(N −Nc ) . These blocks in C are
characterized by the strength of links to each other: how much of bank’s liability
10
is held by other banks, and how much by outside investors. Let ccc denote the
fraction of a core bank’s asset value held by other core banks and ccp denote the
fraction of a core bank’s asset value held by its neighboring peripheral banks. In
other words, a core bank ’s exposure to other core banks is ccc and the exposure to
the peripheral banks is ccp .
Each block of cross-holdings matrix is then,
ccc
(110 − I) ∈ RNc ×Nc
Nc − 1
Nc
Cp = ccp
L ∈ RNc ×(N −Nc )
N − Nc
Cc =
, where L is a Nc × (N − Nc ) matrix such that Lij equals 1 if bank i and bank
j are linked, and 0 otherwise. 110 is the Nc × Nc matrix of ones.
With ccc and cpc , fraction of each bank’s value held by outside investors is
denoted as follows.
ccc I O
Ĉ =
O ccp I
Hereafter, this paper focus on the specific core-periphery network where
(i) each core bank has links with exactly the same number of peripheral banks,
and
(ii) each peripheral bank has links with only one of the core banks.
, without loss of generality.
The network in this model contains seven types of connection. Each entry in
A represents the type of connection and they are interpreted as follows:
1. Core bank’s exposure against its own proprietary asset: Acc_ii , diagonal
elements of Acc
2. Core bank’s exposure against another core bank’s proprietary asset: Acc_ij ,
off-diagonal elements of Acc
3. Peripheral bank’s exposure against its own proprietary asset: App_ii , diagonal
elements of App
4. Peripheral bank’s exposure against another peripheral bank which shares the
same core bank: App_ij_linked
11
5. Peripheral bank’s exposure against another peripheral bank which does not
share the same core bank: App_ij_nonlinked
6. Core bank’s exposure against a linked peripheral bank: Acp_linked
7. Core bank’s exposure against a non-linked peripheral bank: Acp_nonlinked
Figure 3 and 4 show the example with N = 12, Nc = 3, ccc = 0.5 and
ccp = 0.3.
Figure 3: Example of core-peripheryl network
12
Figure 4: Elements in Matrix A
In this setting, each entries can be explicitly derived as a function of ccc and
ccp . Given the set of N and Nc in a certain range, Acc_ij and Acp_linked are
significantly high compared to others, apart from Acc_ii and App_ii . In other words,
each core bank’s market value is highly dependent on the price of asset held by its
directly linking banks. By focusing on these two entries, following propositions
are obtained.
Proposition 1 Let Aij_cc > 0 denote the change in core bank i’s market value per
unit change in core bank j ’s asset price. Fixing ccp ,
limccc →0
∂
Aij_cc > 0
∂ccc
,
limccc →1−ccp
, and
∂
Aij_cc < 0
∂ccc
∂2
Aij_cc < 0
∂c2cc
13
Proposition 2 Let Aij−cp_linked > 0 denote the change in core bank i’s market
value per unit change in its linking peripheral bank j’s asset price. Fixing ccp ,
∂
Aij_cp_linked < 0
∂ccc
Proposition 1 is interpreted as follows. Given the exposures between a core
bank and a peripheral bank which cross-hold liabilities to each other, a core bank
is exposed to relatively high risk of insolvency after another core bank if the
exposures between core banks are in the intermediate range. The intuition is
obtained by assuming the opposite case, where the exposures between two core
banks are extremely low or high. If these core exposures are very low, links
between core banks have relatively low importance in this financial network, so
any core bank is less likely to be affected by another core bank. On the other hand,
if they are very high, the links of cross-holdings between core banks is strong. The
shock generated by one core bank’s insolvency is more likely to be shared across
solvent core banks. Therefore, every single core bank’s market value is not so
damaged as to fall below the threshold. Figure 5 and figure 6 illustrate this.
Figure 5: Contagion from a core bank to another core bank.
14
Figure 6: Decrease in core bank i’s market value[%] caused by 1-unit fall in core
bank j’s asset price. ccp = 0.25, Nc = 10 and N = 100.
Proposition 2 means that given the exposures between a core bank and a
peripheral bank which cross-hold liabilities to each other, a core bank is exposed
to relatively high risk of insolvency after its neighboring peripheral bank if the
exposures between these two are very low. The intuition is similar to proposition
1. Figure 7 and figure 8 illustrate this.
15
Figure 7: Contagion from a peripheral bank to its neighboring core bank.
Figure 8: Decrease in core bank i’s market value[%] caused by 1-unit fall in its
linking peripheral bank j’s asset price. ccp = 0.25, Nc = 10 and N = 100.
Taking these two effects together into account, one can see that if the core
16
exposures are very low, global contagions are unlikely, and if core-exposures are
very high, both global contagions and local contagions are unlikely (figure 9).
Otherwise, i.e. if the core exposures are in intermediate range, neither global
prevention nor local prevention functions, hence the system-wide contagions are
likely.
Figure 9: Decrease in a core bank’s market value[%] caused by 1-unit fall in its
linking banks’ asset price. ccp = 0.25, Nc = 10 and N = 100.
17
4
Robustness check
4.1
Number of banks
4.2
Elasticity
4.3
Asset demand function
5
Conclusion
This paper construct a model in which banks’ liabilities form a core-periphery
network with endogenously reacting asset prices as contagion proceeds, as a
stylized representation of the interbank lending market. There are two main
findings. First, when the asset price is endogenized in the contagious insolvency,
the degree of contagion increases significantly compared with that in exogenous
price model. Second, given the exposures between core banks and peripheral
banks, the likelihood of contagious insolvency is high if the exposures between
core banks are in intermediate range. If the exposures between core banks are very
low, the contagion between core banks are not likely. On the other hand, if they
are very high, each bank has small exposure to the asset price, so the devaluation
of asset does not have enough influence to damage market value of banks.
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A
Proof of propositions
I − C can be decomposed as follows:
I − Cc −Cp
I −C =
−Cp0
I
Its inverse matrix is then,
(I − C)
−1
=
Z
ZCp
0
Cp Z I + Cp0 ZCp
where
Z = (I − Cc − Cp Cp0 )−1 = (αI − β110 )−1
α=1+
Nc
ccc
−
c2
Nc − 1 (N − Nc ) cp
19
(11)
β=
Using Z −1 = α−1 I +
(I − C)−1 =
ccc
Nc − 1
β
110 ,
α(α−Nc β)
β
β
α−1 I + α(α−N
110
(α−1 I + α(α−N
110 )Cp
c β)
c β)
β
β
Cp0 (α−1 I + α(α−N
110 ) I + Cp0 (α−1 I + α(α−N
110 )Cp
c β)
c β)
Ĉ =
(1 − ccc − ccp )I
0
0
(1 − ccp )I
!
(12)
By (10), (11) and (12),
A=
(1 − ccc − ccp )Z
(1 − ccc − ccp )ZCp
Acc Acp
≡
(1 − ccp )Cp0 Z
(1 − ccp )(I + Cp0 ZCp )
A0cp App
Two propositions are derived by checking the partial derivatives of ln Acc_ij
and ln Acp_linked with respect to ccc .
20