Network Structure and Systemic Risk in Banking Systems
Abstract
We present a quantitative methodology for analyzing the potential for contagion and systemic risk in a network of interlinked financial institutions, using a metric for the systemic importance of institutions: the Contagion Index. We apply this methodology to a data set of mutual exposures and capital levels of financial institutions in Brazil in 2007 and 2008, and analyze the role of balance sheet size and network structure in each institution's contribution to systemic risk. Our results emphasize the contribution of heterogeneity in network structure and concentration of counterparty exposures to a given institution in explaining its systemic importance. These observations plead for capital requirements which depend on exposures, rather than aggregate balance sheet size, and which target systemically important institutions.
Network structure and systemic risk in banking systems
Rama Cont
∗
Amal Moussa
†
Edson B. Santos
‡
December 2010. Final revision: April 2012.
Abstract
We present a quantitative methodology for analyzing the potential for contagion and sys-
temic risk in a network of interlinked financial institutions, using a metric for the systemic
importance of institutions: the Contagion Index.
We apply this methodology to a data set of mutual exposures and capital levels of financial
institutions in Brazil in 2007 and 2008, and analyze the role of balance sheet size and network
structure in each institution’s contribution to systemic risk. Our results emphasize the contri-
bution of heterogeneity in network structure and concentration of counterparty exposures to a
given institution in explaining its systemic importance. These observations plead for capital
requirements which depend on exposures, rather than aggregate balance sheet size, and which
target systemically important institutions.
Keywords: default risk, domino effects, balance sheet contagion, scale-free network, default conta-
gion, systemic risk, macro-prudential regulation, random graph.
∗
Center for Financial Engineering, Columbia University, New York. Email: Rama.Cont@columbia.edu
†
Dept of Statistics, Columbia University, New York. Email: am2810@columbia.edu
‡
Banco Central do Brasil, S˜ao Paulo. Email: edson.bastos@bcb.gov.br
1
Contents
1 Introduction 3
1.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 The network structure of banking systems 7
2.1 Counterparty Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 A complex heterogeneous network: the Brazilian banking system . . . . . . . . . . . 8
2.2.1 Distribution of connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Heterogeneity of exposure sizes . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Systemic risk and default contagion 20
3.1 Default mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Loss contagion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Contagion Index of a financial institution . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Is default contagion a significant source of systemic risk? 24
5 What makes an institution systemically important? 30
5.1 Size of interbank liabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Centrality and counterparty susceptibility . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Does one size fit all? The case for targeted capital requirements 36
2
1 Introduction
The recent financial crisis has emphasized the importance of systemic risk, defined as macro-level
risk which can impair the stability of the entire financial system. Bank failures have led in the
recent years to a disruption of the financial system and a significant spillover of financial distress
to the larger economy (Hellwig, 2009). Regulators have had great difficulties anticipating the
impact of defaults partly due to a lack of visibility on the structure of the financial system as well
as a lack of a methodology for monitoring systemic risk. The complexity of the contemporary
financial systems makes it a challenge to define adequate indicators of systemic risk that could help
in an objective assessment of the systemic importance of financial institutions and an objective
framework for assessing the efficiency of macro-prudential policies.
One of the aspects of systemic risk which has been highlighted in the recent crisis has been the
interconnectedness of financial institutions, which increases the probability of contagion of financial
distress. Such externalities resulting from counterparty risk are a major concern for regulators
(Hellwig, 1995; Haldane, 2009) and network models (Allen and Gale, 2000; Boss et al., 2004; Cont
and Moussa, 2010; Nier et al., 2007; Amini et al., 2010; Gai and Kapadia, 2010; Amini et al., 2012)
provide an adequate framework for addressing them. Simulation studies based on network models
have been extensively used by central banks for assessing contagion risk in banking systems; we
refer to the pioneering work of Elsinger et al. (2006a) and the survey of Upper (2011).
Following our earlier work (Cont, 2009; Cont and Moussa, 2010) we introduce and implement a
quantitative methodology for analyzing the potential for contagion and systemic risk in a network
of interlinked financial institutions, using a metric for the systemic importance of institutions –
the Contagion Index– defined as the expected loss to the network triggered by the default of
an institution in a macroeconomic stress scenario (Cont, 2009). The definition of this indicator
takes into account both common market shocks to portfolios and contagion through counterparty
exposures. Contrarily to indicators of systemic risk purely based on market data (Acharya et al.,
2010; Adrian and Brunnermeier, 2008; Zhou et al., 2009), our metric is a forward-looking measure
of systemic importance is based on exposures, which represent potential losses in case of default. We
build on methods proposed in Cont and Moussa (2010) for estimating and analyzing this indicator.
We apply this methodology to a unique and complete data set of interbank exposures and
capital levels provided by the Brazilian Central Bank, and analyze the role of balance sheet size
and network structure in each institution’s contribution to systemic risk. Our results emphasize
the importance of heterogeneity in network structure and the role of concentration of counterparty
exposures in explaining its systemic importance of an institution. These arguments plead for capital
requirements which depend on exposures instead of aggregate balance sheet size and which target
systemically important institutions.
Most of the empirical studies on systemic risk and default contagion in interbank networks
(Sheldon and Maurer, 1998; Furfine, 2003; Upper and Worms, 2004; Wells, 2004; Elsinger et al.,
2006a,b; Mistrulli, 2007) have dismissed the importance of contagion, we find that contagion signif-
icantly contributes to systemic risk in the Brazilian banking system. Our results do not contradict
previous findings but present them in a different light: while most of the aforementioned studies
use indicators averaged across institutions we argue that, given the heterogeneity of the systemic
importance across institutions, the sample average gives a poor representation of the degree of
contagion and conditional measures of risk and should be used. Also, most of these studies are
based on a generous recovery rate assumptions whereby all assets of a defaulting bank are recov-
ered at pre-default value; this is far from reality, especially in the short term –which we focus on
here– where recovery rates are close to zero in practice. Finally, with the exception of Elsinger
et al. (2006a,b), all these studies measure the impact of the idiosyncratic default of a single bank,
3
whereas we use the more realistic setting of stress scenarios where balance sheets are subjected to
common shocks. As in previous studies on other banking systems (Elsinger et al., 2006a,b; Upper,
2011) we find that, while the probability of contagion is small, the loss resulting from contagion
can be very large in some cases.
Our study reveals several interesting features on the structure of the Brazilian financial system
and the nature of systemic risk and default contagion in this system:
∙ Interbank networks exhibit a complex heterogeneous structure, which resembles a directed
scale-free network as defined in Bollob´as et al. (2003): the distributions of number of counter-
parties and exposure sizes are found to be heavy-tailed, with an asymmetry between incoming
and outgoing links. Furthermore, while individual exposures are quite variable in time, these
statistical regularities, which encode the large-scale statistical structure of the network are
shown to be stable across time.
∙ Systemic risk is concentrated on a few nodes in the financial network: while most financial
institutions present only a negligible risk of contagion, a handful of them generate a significant
risk of contagion through their failure.
∙ Ignoring the compounded effect of correlated market shocks and contagion via counterparty
exposures can lead to a serious underestimation of contagion risk. Specifically, market shocks
are found to increase the proportion of contagious exposures in the network, i.e. exposures
that transmit default in all shock scenarios. We are thus led to question the conclusions
of previous studies which dismissed the importance of contagion by looking at pure balance
sheet contagion in absence of market shocks.
∙ Balance sheet size alone is not a good indicator for the systemic importance of financial insti-
tutions: network structure does matter when assessing systemic importance. Network-based
measures of connectivity and concentration of exposures across counterparties –counterparty
susceptibility and local network frailty– are shown to contribute significantly to the systemic
importance of an institution.
∙ Using the Contagion Index as a metric for systemic impact allows a comparative analysis
of various capital allocations across the networks in terms of the resulting cross-sectional
distribution of the Contagion Index. While a floor on the (aggregate) capital ratio is shown
to reduce the systemic impact of defaults of large institutions, imposing more stringent capital
requirements on the most systemic nodes and on the most concentrated exposures is shown
to be a more efficient procedure for immunizing the network against contagion.
1.1 Contribution
Our approach builds on previous theoretical and empirical studies of default contagion in banking
systems (see De Bandt and Hartmann (2000); Upper (2011) for a review of the literature), but also
differs from them both in terms of the methodology used and in terms of the results obtained. In
particular, we are led to revisit some of the conclusions in the previous literature on the magnitude
of contagion risk in interbank networks.
Methodology On the methodological side, most previous studies on contagion in financial net-
works have mostly focused on the stability of the financial system as a whole, either in stylized
equilibrium settings (Allen and Gale, 2000; Freixas et al., 2000; Battiston et al., 2009) or in simula-
tion studies of default cascades (Upper and Worms, 2004; Mistrulli, 2007; Elsinger et al., 2006a,b;
4
Nier et al., 2007). Nier et al. (2007) measure the average number of defaults when the institutions in
the system are subject one at a time to an idiosyncratic shock which wipes out their external assets.
Upper and Worms (2004) and Mistrulli (2007) consider various aggregate measures of contagion:
the number of institutions that default by contagion and the loss as a fraction of the total assets
in the banking system. Elsinger et al. (2006a) also measure contagion by counting the number
of defaults due to counterparty exposure when the system is subject to correlated market shocks.
These studies give insights on the global level of systemic risk in the entire network, but do not
allow to measure the systemic importance of a given financial institution, which is our focus here.
Rather than compute a global measure of systemic risk then allocating it to individual institutions
as in Tarashev et al. (2010); Zhou et al. (2009); Liu and Staum (2011), we use a direct metric of
systemic importance, the Contagion Index (Cont, 2009; Cont and Moussa, 2010), which allows to
rank institutions in terms of the risk they pose to the system by quantifying the expected loss in
capital generated by an institutions default in a macroeconomic stress scenario.
Recent studies such as Acharya et al. (2010); Zhou et al. (2009) have also proposed measures of
systemic importance based on market data such as CDS spreads or equity volatility. By contrast
to these methods which are based on historical market data, our approach is a forward-looking,
simulation-based approach based on interbank exposures (Cont, 2009). Exposure data, which repre-
sent potential future losses, are available to regulators, should be used as an ingredient in evaluating
systemic importance and interconnectedness. As argued in Cont (2009), since exposures are not
publicly available, even if market variables correctly reflect public information they need not reflect
the information contained in exposures, so exposures-based indicators are a useful complement to
market-based indicators.
With the exception of Elsinger et al. (2006a,b), most simulation studies of contagion in banking
networks examine the sole knock-on effects of the sudden failure of a single bank by considering
an idiosyncratic shock that targets a single institution in the system. Upper and Worms (2004)
estimate the scope of contagion by letting banks go bankrupt one at a time and measuring the
number of banks that fail due their exposure to the failing bank. Sheldon and Maurer (1998) and
Mistrulli (2007) also study the consequences of a single idiosyncratic shock affecting individual
banks in the network. Furfine (2003) measures the risk that an exogenous failure of one or a small
number of institutions will cause contagion. These studies fail to quantify the compounded effect
of correlated defaults and contagion through network externalities. Our study, on the contrary,
shows that common market shocks to balance sheets may exacerbate contagion during a crisis
and ignoring them can lead to an underestimation of the extent of contagion in the network. We
argue that, to measure adequately the systemic impact of the failure of a financial institution, one
needs to account for the combined effect of correlation of market shocks to balance sheets and
balance sheet contagion effects, the former increasing the impact of the latter. Our simulation-
based framework takes into account common and independent market shocks to balance sheets, as
well as counterparty risk through mutual exposures.
The loss contagion mechanism we consider differs from most network-based simulations, which
consider the framework of Eisenberg and Noe (2001) where a market clearing equilibrium is defined
through a clearing payment vector with proportional sharing of losses among counterparties in
case of default (Eisenberg and Noe, 2001; Elsinger et al., 2006a,b; M¨uller, 2006). This leads to an
endogenous recovery rate which corresponds to a hypothetical situation where all bank portfolios are
simultaneously liquidated. This may be an appropriate assumption model for interbank payment
systems, where clearing takes place at the end of each business day, but is not a reasonable model
for the liquidation of defaulted bank portfolios. Our approach is, by contrast, a stress-testing
approach where, starting from the currently observed network structure, capital levels are stressed
5
by macroeconomic shocks and a risk measure computed from the distribution of aggregate loss.
We argue that, since bankruptcy procedures are usually slow and settlements may take up several
months to be effective, creditors cannot recover the residual value of the defaulting institution
according to such a hypothetical clearing mechanism, and write down their entire exposure in
the short-run, leading to a short term recovery rate close to zero. In absence of a global default
resolution mechanism, this seems a more reasonable approach.
Studies on simulated network structures have examined the variables that affect the global level
of systemic risk in the network (Nier et al., 2007; Battiston et al., 2009) such as the connectivity,
concentration, capital levels, but the main results (such as the level of contagion and the role of
interconnectedness) strongly depend on the details of the model and the structure of the network,
which have left open whether these conclusions hold in actual banking networks. On the other
hand, most of the empirical studies have only partial information on the bilateral exposures in
the network, and estimate missing exposures with a maximum entropy method (Sheldon and
Maurer, 1998; Upper and Worms, 2004; Wells, 2004; Elsinger et al., 2006a,b; Degryse and Nguyen,
2007). However, the maximum entropy method is found to underestimate the possibility of default
contagion (Mistrulli, 2007; van Lelyveld and Liedorp, 2006; Cont and Moussa, 2010). Our study,
by making use of empirical data on all bilateral exposures, avoids this caveat.
Results Our empirical findings on the network structure of the Brazilian financial system are
–qualitatively and quantitatively– similar to statistical features observed in the Austrian financial
system (Boss et al., 2004). This suggests that these features could be a general characteristic of
interbank networks, and it would interesting to check whether similar properties are also observed
in other interbank networks.
While most of the empirical studies on systemic risk and default contagion in interbank networks
have dismissed the importance of contagion, our study reveals that the risk of default contagion is
significant in the Brazilian financial system. We show examples in which the expected loss resulting
from the default of an institution can exceed by several multiples the size of its interbank liabilities.
In contrast with Elsinger et al. (2006a), we find that scenarios with contagion are more frequent
than those without contagion when grouped by number of fundamental defaults. This difference
in results is due to two reasons. First, our metric, the Contagion Index, measures the magnitude
of loss conditional to the default of a given institution, instead of averaging across all defaults as
in Elsinger et al. (2006a). We argue that these conditional measures provide a better assessment
of risk in a heterogeneous system where the sample average may be a poor statistic. Second, we
use a heavy-tailed model for generating the common shocks to balance sheets: we argue that this
heavy-tailed model is more realistic than Gaussian factor models used in many simulation studies.
We find that macroeconomic shocks play an essential role in amplifying contagion. Specifically,
we observe that the proportion of contagious exposures increases considerably when the system is
subject to a market shock scenario, thus creating additional channels of contagion in the system.
The Contagion Index, by compounding the effects of both market events and counterparty exposure,
accounts for this phenomenon.
Our study also complements the existing literature by studying the contribution of network-
based local measures of connectivity and concentration to systemic risk. Previous studies on
simulated network structures have examined the contribution of aggregate measures of connectivity
and concentration such as increasing the probability that two nodes are connected in an Erd¨os-
Renyi graph, or increasing the number of nodes in the system (Battiston et al., 2009; Nier et al.,
2007). We introduce two measures of local connectivity: counterparty susceptibility, which measures
the susceptibility of the creditors of an institution to a potential default of the latter, and local
6
network frailty which measures rhow network fragility increases when a given node defaults, and
argue that these indicators provide good clues for localizing sources of contation in the network.
The impact of capital requirements in limiting the extent of systemic risk and default contagion
has not been explored systematically in a network context. Analogies with epidemiology and peer-
to-peer networks (Cohen et al., 2003; Madar et al., 2004; Huang et al., 2007) suggest that, given
the heterogeneity of nodes in terms of systemic impact, targeted capital requirements may be more
effective than uniform capital ratios. We argue that
∙ targeting the most contagious institutions is more effective in reducing systemic risk than
increasing capital ratios uniformly across all institutions, and
∙ capital requirements should not simply focus on the aggregate size of the balance sheet
but depend on their concentration/distribution across counterparties: a minimal capital-to-
exposure ratio allows to reduce channels of contagion in the network by reducing the number
of ’contagious links’.
1.2 Outline
Section 2 introduces a network model for a banking system and describes their structure and
statistical properties using empirical data from the Brazilian banking system. Section 3 introduces
a quantitative approach for measuring contagion and systemic risk, following Cont (2009). Section
4 applies this methodology to the Brazilian financial system. Section 5 investigates the role of
different institutional and network characteristics which contribute to the systemic importance of
Brazilian financial institutions. Section 6 analyzes the impact of capital requirements on these
indicators of systemic risk and uses the insights obtained from the network model to examine
the impact of targeted capital requirements which focus on the most systemic institutions and
concentrated exposures.
2 The network structure of banking systems
2.1 Counterparty Networks
Counterparty relations in financial system may be represented as a weighted directed graph, or a
network , defined as a triplet 𝐼 = (𝑉, 𝐸, 𝑐), consisting of
∙ a set 𝑉 of financial institutions, whose number we denote by 𝑛,
∙ a matrix 𝐸 of bilateral exposures: 𝐸
𝑖𝑗
represents the exposure of node 𝑖 to node 𝑗 defined
as the (mark-to-)market value of all liabilities of institution 𝑗 to institution 𝑖 at the date of
computation. It is thus the maximal short term loss of 𝑖 in case of an immediate default of 𝑗.
∙ 𝑐 = (𝑐(𝑖), 𝑖 ∈ 𝑉 ) where 𝑐(𝑖) is the capital of the institution 𝑖, representing its capacity for
absorbing losses.
Such a network may be represented as a graph in which nodes represent institutions and links
represent exposures.
We define the in-degree 𝑘
𝑖𝑛
(𝑖) of a node 𝑖 ∈ 𝑉 as the number of its debtors and out-degree
𝑘
𝑜𝑢𝑡
(𝑖) the number of its creditors:
𝑘
𝑖𝑛
(𝑖) =
∑
𝑗∈𝑉
1
{𝐸
𝑖𝑗
>0}
, 𝑘
𝑜𝑢𝑡
(𝑖) =
∑
𝑗∈𝑉
1
{𝐸
𝑗𝑖
>0}
, (1)
7
The degree 𝑘(𝑖) of a node 𝑖 is defined as 𝑘(𝑖) = 𝑘
𝑖𝑛
(𝑖) + 𝑘
𝑜𝑢𝑡
(𝑖) and measures its connectivity.
Although all institutions in the network are not banks, we will refer to the exposures as “inter-
bank” exposures for simplicity. We denote 𝐴(𝑖) the interbank assets of financial institution 𝑖, and
𝐿(𝑖) its interbank liabilities:
𝐴(𝑖) =
∑
𝑗∈𝑉
𝐸
𝑖𝑗
, 𝐿(𝑖) =
∑
𝑗∈𝑉
𝐸
𝑗𝑖
, (2)
We now give an example of such a network and describe its structure and topology.
2.2 A complex heterogeneous network: the Brazilian banking system
The Brazilian financial system encompasses 2400 financial institutions chartered by the Brazilian
Central Bank and grouped into three types of operation: Type I are banking institutions that have
commercial portfolios, Type III are institutions that are subject to particular regulations, such as
credit unions, and Type II represent all other banking institutions. Despite their reduced number
(see table 1), financial institutions of Type I and II account for the majority (about 98%) of total
assets in the Brazilian financial system (see table 2). We therefore consider in the Brazilian data
set only Type I and Type II financial institutions which is a very good proxy for the Brazilian
financial system. Most of the financial institutions belong to a conglomerate (75% of all financial
institutions of Type I and II). Consequently, it is quite meaningful to analyze the financial system
from a consolidated perspective where financial institutions are classified in groups that are held by
the same shareholders. Only banking activities controlled by the holding company are considered
in the consolidation procedure. The accounting standards for consolidation of financial statements
were established by Resolutions 2,723 and 2,743, BCB (2000a,b), and they are very similar to IASB
and FASB directives. If we regard financial institutions as conglomerates, the dimension of the
exposures matrices reduces substantially, see table 1 for the number of financial conglomerates in
the Brazilian financial system after the consolidation procedure.
These exposures, reported at six dates (June 2007, December 2007, March 2008, June 2008,
September 2008 and November 2008) cover various sources of risk:
1. fixed-income instruments (certificate of deposits and debentures);
2. borrowing and lending (credit risk);
3. derivatives (including OTC instruments such as swaps);
4. foreign exchange and,
5. instruments linked to exchange-traded equity risk.
Derivatives positions were taken into account at their market prices when available, or at fair value
when a model-based valuation was required.
The data set also gives the Tier I and Tier 2 capital of each institution, computed according to
guidelines provided in Resolution 3,444 BCB (2007a) of the Brazilian Central Bank, in accordance
with the Basel I and II Accords. Tier 1 capital is composed of shareholder equity plus net income
(loss), from which the value of redeemed preferred stocks, capital and revaluation of fixed assets
reserves, deferred taxes, and non-realized gains (losses), such as mark-to-market adjustments from
securities registered as available-for-sale and hedge accounting are deducted. Tier 2 capital is equal
to the sum of redeemed preferred stocks, capital, revaluation of fixed assets reserves, non-realized
8
Type Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08 Dec-08
Multiple Bank 135 135 135 136 139 139 140
Commercial Bank 20 20 21 20 20 18 18
Development Bank 4 4 4 4 4 4 4
Savings Bank 1 1 1 1 1 1 1
Investment Bank 17 17 17 18 18 18 17
Consumer Finance Company 51 52 51 56 55 55 55
Security Brokerage Company 113 107 114 107 107 107 107
Exchange Brokerage Company 48 46 48 46 46 45 45
Security Distribution Company 132 135 133 133 136 136 135
Leasing Company 40 38 41 37 36 36 36
Real Estate Credit Company and Savings and Loan Association 18 18 18 18 18 17 16
Mortgage Company 6 6 6 6 6 6 6
Development Agency 12 12 12 12 12 12 12
Total Banking Institutions of Type I and II 597 591 601 594 598 594 592
Credit Union 1.461 1.465 1.460 1.466 1.460 1.457 1.453
Micro-financing Institution 54 52 54 48 46 45 47
Total Banking Institutions Type III 2.112 2.108 2.115 2.108 2.104 2.096 2.092
Non-Banking Institutions 332 329 333 324 317 318 317
Total Banking and Non-Banking Institutions 2444 2.437 2.448 2.432 2.421 2.414 2.409
Table 1: Number of financial institutions by type of operation for the Brazilian financial system. Source: Sisbacen.
9
Assets in Billions of USD Jun-07 % Dec-07 % Mar-08 % Jun-08 % Sep-08 % Dec-08 %
Banking - Type I 1,064.8 87.1 1,267.7 87.8 1,366.9 87.9 1,576.0 87.7 1,433.2 88.0 1,233.6 87.5
Banking - Type II 129.6 10.6 142.7 9.9 152.7 9.8 179.4 10.0 160.1 9.8 148.3 10.5
Banking - Type I and II 1,194.5 97.7 1,410.4 97.7 1,519.6 97.7 1,755.4 97.7 1,593.2 97.8 1,382.0 98.0
Banking - Type III 17.7 1.5 21.5 1.5 23.7 1.5 28.3 1.6 24.1 1.5 19.1 1.4
Non-Banking 10.4 0.9 12.8 0.9 12.5 0.8 14.4 0.8 11.4 0.7 9.3 0.7
Total Financial System 1,222.6 100.0 1,444.8 100.0 1,555.8 100.0 1,798.1 100.0 1,628.8 100.0 1,410.4 100.0
Number of Conglomerates Jun-07 % Dec-07 % Mar-08 % Jun-08 % Sep-08 % Dec-08 %
Banking - Type I 102 5.4 101 5.4 101 5.4 101 5.4 103 5.5 101 5.4
Banking - Type II 32 1.7 32 1.7 32 1.7 33 1.8 34 1.8 35 1.9
Banking - Type I and II 134 7.1 133 7.1 133 7.1 134 7.2 137 7.3 136 7.3
Banking - Type III 1,440 76.8 1,440 77.0 1,436 77.0 1,441 77.0 1,442 76.9 1,438 77.0
Non-Banking 302 16.1 298 15.9 297 15.9 296 15.8 296 15.8 294 15.7
Total Financial System 1,876 100.0 1,871 100.0 1,866 100.0 1,871 100.0 1,875 100.0 1,868 100.0
Table 2: Representativeness of Brazilian financial institutions in terms of total Assets and number. The total assets were converted
from BRL (Brazilian Reals) to USD (American Dollars) with the following foreign exchange rates (BRL/USD): 1.9262 (Jun-07), 1.7713
(Dec-07), 1.7491 (Mar-08), 1.5919 (Jun-08), 1.9143 (Sep-08), and 2.3370 (Dec-08). Source: Sisbacen.
10
gains (losses), and complex or hybrid capital instruments and subordinated debt. We shall focus
on Tier 1 capital as a measure of a bank’s capacity to absorb losses in the short term.
Financial conglomerates in Brazil are subject to minimum capital requirements. The required
capital is a function of the associated risks regarding each financial institution’s operations, whether
registered in their balance sheets (assets and liabilities) or not (off-balance sheet transactions), as
defined in Resolution 3,490, BCB (2007b). The required capital is computed as 𝑐
𝑟
= 𝛿 × Risk Base
where the 𝛿 = 11% and the risk base is the sum of credit exposures weighted by their respective
risk weights, foreign currency and gold exposures, interest rate exposures, commodity exposures,
equity market exposures, and operational risk exposures. It is important to highlight that the
exposures considered in the computation of the risk base include not only interbank exposures but
also exposures to all counterparties.
Table 3 presents some descriptive statistics of these variables.
In-Degree Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
Mean 8.56 8.58 8.75 8.98 8.99 7.88
Standard Deviation 10.84 10.86 10.61 11.15 11.32 11.02
5% quantile 0 0 0 0 0 0
95% quantile 30.50 29.30 30.45 31 32 30.60
Maximum 54 54 51 57 60 62
Out-Degree Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
Mean 8.56 8.58 8.75 8.98 8.99 7.88
Standard Deviation 8.71 8.82 9.02 9.43 9.36 8.76
5% quantile 0 0 0 0 0 0
95% quantile 26 26 27.90 29.25 30.20 27.40
Maximum 36 37 39 41 39 44
Exposures (in billions of BRL) Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
Mean 0.07 0.05 0.05 0.05 0.05 0.08
Standard Deviation 0.77 0.32 0.32 0.30 0.38 0.54
5% quantile 0.00 0.00 0.00 0.00 0.00 0.00
95% quantile 0.20 0.17 0.17 0.18 0.19 0.35
Maximum 23.22 9.89 9.90 9.36 12.50 15.90
Relative Exposures (𝐸
𝑖𝑗
/𝑐(𝑖)) Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
Mean 0.23 0.20 0.04 0.04 0.03 0.05
Standard Deviation 1.81 1.62 0.16 0.17 0.06 0.21
5% quantile 0.00 0.00 0.00 0.00 0.00 0.00
95% quantile 0.70 0.59 0.20 0.21 0.16 0.18
Maximum 49.16 46.25 4.57 5.17 0.69 6.02
Distance Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
Mean 2.42 2.42 2.38 2.38 2.33 2.35
Standard Deviation 0.84 0.85 0.84 0.82 0.77 0.78
5% quantile 1 1 1 1 1 1
95% quantile 4 4 4 4 3 4
Maximum (Diameter) 5 6 6 6 5 6
Table 3: Descriptive statistics of the number of debtors (in-degree), number of creditors (out-
degree), exposures, relative exposures (ratio of the exposure of institution 𝑖 to institution 𝑗 to the
capital of 𝑖), and distance between two institutions (nodes) in the network.
Figure 1 illustrates the Brazilian interbank network in December 2007. It is observed to have a
heterogeneous and complex structure, some highly connected institutions playing the role of “hubs”
11
Figure 1: Brazilian interbank network, December 2007. The number of financial conglomerates is
𝑛 = 125 and the number of links in this representation at any date does not exceed 1200.
while others are at the periphery.
2.2.1 Distribution of connectivity
Casual inspection of the graph in figure 1 reveals the existence of nodes with widely differing
connectivity. This observation is confirmed by further analyzing the data on in-degrees and out-
degrees of nodes. Figures 2 and 3 show, respectively, the double logarithmic plot of the empirical
complementary cumulative distribution for the in-degree
ˆ
ℙ(𝐾
𝑖𝑛
⩾ 𝑘) and out-degree
ˆ
ℙ(𝐾
𝑜𝑢𝑡
⩾ 𝑘)
for 𝑘 ⩾ 1. We notice that the tails of the distributions exhibit a linear decay in log-scale, suggesting
a heavy Pareto tail.
This observation is confirmed through semiparametric tail estimates. Maximum likelihood
estimates for the tail exponent 𝛼 and tail threshold 𝑘
𝑚𝑖𝑛
(Clauset et al., 2009) are shown in
Table 4 for the in-degree, out-degree and degree distributions. Maximum likelihood estimates for
ˆ𝛼 range from 2 to 3. The results are similar to the findings of Boss et al. (2004) for the Austrian
network.
We test the goodness-of-fit of the power law tails for in-degree, out-degree and degree via the
12
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
In Degree
Pr(K ≥ k)
α = 2.1997
k
min
= 6
p−value = 0.0847
Network in June 2007
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
In Degree
Pr(K ≥ k)
α = 2.7068
k
min
= 13
p−value = 0.2354
Network in December 2007
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
In Degree
Pr(K ≥ k)
α = 2.2059
k
min
= 7
p−value = 0.0858
Network in March 2008
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
In Degree
Pr(K ≥ k)
α = 3.3611
k
min
= 21
p−value = 0.7911
Network in June 2008
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
In Degree
Pr(K ≥ k)
α = 2.161
k
min
= 6
p−value = 0.0134
Network in September 2008
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
In Degree
Pr(K ≥ k)
α = 2.132
k
min
= 5
p−value = 0.0582
Network in November 2008
Figure 2: Brazilian interbank network: distribution of in-degree.
13
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
Out Degree
Pr(K ≥ k)
α = 1.9855
k
min
= 5
p−value = 0
Network in June 2007
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
Out Degree
Pr(K ≥ k)
α = 3.4167
k
min
= 15
p−value = 0.166
Network in December 2007
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
Out Degree
Pr(K ≥ k)
α = 3.4
k
min
= 16
p−value = 0.1937
Network in March 2008
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
Out Degree
Pr(K ≥ k)
α = 2.911
k
min
= 12
p−value = 0.0874
Network in June 2008
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
Out Degree
Pr(K ≥ k)
α = 2.4302
k
min
= 9
p−value = 0.0006
Network in September 2008
10
0
10
1
10
2
10
−3
10
−2
10
−1
10
0
Out Degree
Pr(K ≥ k)
α = 2.8861
k
min
= 11
p−value = 0.0893
Network in November 2008
Figure 3: Brazilian interbank network: distribution of out-degree.
14
In-Degree Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08 Mean
ˆ𝛼 2.19 2.70 2.20 3.36 2.16 2.13 2.46
ˆ𝜎 (ˆ𝛼) 0.48 0.46 0.47 0.53 0.47 0.44 0.48
ˆ
𝑘
𝑖𝑛,𝑚𝑖𝑛
6 13 7 21 6 5 9.7
Out-Degree Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08 Mean
ˆ𝛼 1.98 3.41 3.40 2.91 2.43 2.88 2.83
ˆ𝜎 (ˆ𝛼) 0.63 0.59 0.48 0.43 0.41 0.49 0.51
ˆ
𝑘
𝑜𝑢𝑡,𝑚𝑖𝑛
5 15 16 12 9 11 11.3
Degree Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08 Mean
ˆ𝛼 2.61 3.37 2.29 2.48 2.27 2.23 2.54
ˆ𝜎 (ˆ𝛼) 0.52 0.47 0.48 0.41 0.43 0.35 0.44
ˆ
𝑘
𝑚𝑖𝑛
17 34 12 15 12 10 16.7
Exposures* Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08 Mean
ˆ𝛼 1.97 2.22 2.23 2.37 2.27 2.52 2.27
ˆ𝜎 (ˆ𝛼) 0.02 0.60 0.21 0.69 0.38 0.98 0.48
ˆ
𝐸
𝑚𝑖𝑛
39.5 74.0 80.0 101.7 93.4 336.7 120.9
*values in millions of BRL (Brazilian Reals)
Table 4: Statistics and maximum likelihood estimates for the distribution of in/out degree: tail
exponent 𝛼, tail threshold for in-degree 𝑘
𝑖𝑛,𝑚𝑖𝑛
, out-degree 𝑘
𝑜𝑢𝑡,𝑚𝑖𝑛
, degree 𝑘
𝑚𝑖𝑛
, and exposures
𝐸
𝑚𝑖𝑛
.
one-sample Kolmogorov-Smirnov test with respect to a reference power law distribution. The
results in figures 2 and 3 provide evidence for the Pareto tail hypothesis at the 1% significance
level.
The precise pattern of exposure across institutions may vary a priori in time: it is therefore
of interest to examine whether the large scale structure of the graph, as characterized by the
cross-sectional distributions of in- and out-degrees, is stationary, that is, may be considered as
time-independent. Comparing quantiles of the degree distributions at different dates ( figure 4)
shows that the empirical distribution of the degree, in-degree and out-degree are in fact stable
over time, even though the observations span the turbulent period of 2007-2008. A two-sample
Kolmogorov-Smirnov test for consecutive dates produces p-values greater than 0.6, suggesting that
the null hypothesis of stationarity of the degree distribution cannot be rejected. These results
show that, while individual links continuously appear and disappear in the network, statistical
regularities such as degree distributions which encode the large-scale statistical topology of the
network are stable in time.
2.2.2 Heterogeneity of exposure sizes
The distribution of interbank exposures is also found to be heavy-tailed, with Pareto tails. Figure 5
shows the existence of a linear decay in the tail of the double logarithmic plot for the empirical
distribution of exposure sizes. Maximum likelihood estimates for the tail exponent 𝛼 and the tail
cutoff 𝑘
𝑚𝑖𝑛
for the distribution of exposures are shown in Table 4. Note that an interbank asset
for an institution is an interbank liability for its counterparty, thus, the distribution of interbank
liability sizes is the same. The only difference is how these exposures are allocated among the
financial institutions in the network. Figure 5 shows evidence for Pareto tails in the exposure
15
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p−value = 0.99998
p−value = 0.99919
p−value = 0.99998
p−value = 0.9182
p−value = 0.9182
Q−Q Plot of In Degree
Pr(K(i)≤ k)
Pr(K(j)≤ k)
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p−value = 0.99234
p−value = 0.99998
p−value = 0.99998
p−value = 0.99234
p−value = 0.84221
Q−Q Plot of Out Degree
Pr(K(i)≤ k)
Pr(K(j)≤ k)
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p−value = 0.99998
p−value = 0.9683
p−value = 0.9683
p−value = 0.99234
p−value = 0.64508
Q−Q Plot of Degree
Pr(K(i)≤ k)
Pr(K(j)≤ k)
Jun/07 vs. Dec/07
Dec/07 vs. Mar/08
Mar/08 vs. Jun/08
Jun/08 vs. Sep/08
Sep/08 vs. Nov/08
45
o
line − (i) vs. (j)
Jun/07 vs. Dec/07
Dec/07 vs. Mar/08
Mar/08 vs. Jun/08
Jun/08 vs. Sep/08
Sep/08 vs. Nov/08
45
o
line − (i) vs. (j)
Jun/07 vs. Dec/07
Dec/07 vs. Mar/08
Mar/08 vs. Jun/08
Jun/08 vs. Sep/08
Sep/08 vs. Nov/08
45
o
line − (i) vs. (j)
Figure 4: Scatterplot of the the empirical cumulative distributions at consecutive dates for the
degree, in-degree and out-degree in the Brazilian interbank network.
distributions at all dates.
Most financial institutions in Brazil have sufficient Tier 1 capital to cover their interbank
exposures. However, some institutions have interbank exposures which total much higher than
their Tier 1 capital: these nodes can be very sensitive to counterparty defaults and, as we will see
in section 5.2 they may play a crucial role in the contagion of losses across the network.
Another interesting observation is that financial institutions which are highly connected tend
to have larger exposures. We investigate the relationship between the in-degree 𝑘
𝑖𝑛
(𝑖) of a node
𝑖 and its average exposure size 𝐴(𝑖)/𝑘
𝑖𝑛
(𝑖) and also examine the relation between the out-degree
𝑘
𝑜𝑢𝑡
(𝑖) and the average liability size 𝐿(𝑖)/𝑘
𝑜𝑢𝑡
(𝑖) and between 𝑘(𝑖) and 𝐴(𝑖)/𝑘(𝑖) by computing
the Kendall tau for each of these pairs. Table 5 displays the Kendall tau 𝜏
𝐾𝑒𝑛𝑑𝑎𝑙𝑙
coefficients that
measure the statistical dependence between the variables, and their respective p-values. The results
show that the in-degree and the average interbank asset size, as well as the out-degree and the
average interbank liability size, show positive dependence.
2.2.3 Clustering
The clustering coefficient of a node is defined as the ratio of the number of its links between its
neighbors to the total number of possible links among its neighbors (Watts and Strogatz, 1998):
this ratio, between 0 and 1, tells how connected among themselves the neighbors of a given node
are. In complete graphs, all nodes have a clustering coefficient of 1 while in regular lattices the
clustering coefficient shrinks to zero with the degree.
A property often discussed in various networks is the small world property (Watts and Strogatz,
1998) which refers to networks where, although the network size is large and each node has a small
number of direct neighbors, the distance between any two nodes is very small compared to the
network size. Boss et al. (2004) report that in the Austrian interbank network any two nodes
are on average 2 links apart, and suggest that the Austrian interbank network is a small-world.
However, a small graph diameter is not sufficient to characterize the small world property: indeed,
complete networks have diameter one but are not ”small worlds”. The signature of a small world
16
10
−9
10
−7
10
−5
10
−3
10
−1
10
1
10
−4
10
−3
10
−2
10
−1
10
0
Exposures × 10
−10
in BRL
Pr(X ≥ x)
α = 1.9792
x
min
= 0.0039544
p−value = 0.026
Network in June 2007
10
−9
10
−7
10
−5
10
−3
10
−1
10
−4
10
−3
10
−2
10
−1
10
0
Exposures × 10
−10
in BRL
Pr(X ≥ x)
α = 2.2297
x
min
= 0.0074042
p−value = 0.6
Network in December 2007
10
−9
10
−7
10
−5
10
−3
10
−1
10
−4
10
−3
10
−2
10
−1
10
0
Exposures × 10
−10
in BRL
Pr(X ≥ x)
α = 2.2383
x
min
= 0.008
p−value = 0.214
Network in March 2008
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10
−4
10
−3
10
−2
10
−1
10
0
Exposures × 10
−10
in BRL
Pr(X ≥ x)
α = 2.3778
x
min
= 0.010173
p−value = 0.692
Network in June 2008
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10
−4
10
−3
10
−2
10
−1
10
0
Exposures × 10
−10
in BRL
Pr(X ≥ x)
α = 2.2766
x
min
= 0.0093382
p−value = 0.384
Network in September 2008
10
−9
10
−7
10
−5
10
−3
10
−1
10
1
10
−3
10
−2
10
−1
10
0
Exposures × 10
−10
in BRL
Pr(X ≥ x)
α = 2.5277
x
min
= 0.033675
p−value = 0.982
Network in November 2008
Figure 5: Brazilian interbank network: distribution of exposures in BRL.
17
𝑘
𝑖𝑛
vs. 𝐴/𝑘
𝑖𝑛
Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
𝜏
𝐾𝑒𝑛𝑑𝑎𝑙𝑙
0.28 0.25 0.22 0.26 0.24 0.21
(p-value) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
𝑘
𝑜𝑢𝑡
vs. 𝐿/𝑘
𝑜𝑢𝑡
Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
𝜏
𝐾𝑒𝑛𝑑𝑎𝑙𝑙
0.27 0.28 0.31 0.32 0.34 0.30
(p-value) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
𝑘 vs. 𝐴/𝑘 Jun-07 Dec-07 Mar-08 Jun-08 Sep-08 Nov-08
𝜏
𝐾𝑒𝑛𝑑𝑎𝑙𝑙
0.24 0.24 0.21 0.23 0.23 0.23
(p-value) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
Table 5: Brazilian interbank network: Kendall 𝜏
𝐾𝑒𝑛𝑑𝑎𝑙𝑙
coefficients for in-degree 𝑘
𝑖𝑛
vs. interbank
assets 𝐴, out-degree 𝑘
𝑜𝑢𝑡
vs. interbank liabilities 𝐿, and degree 𝑘 vs. exposures 𝑤.
network is that, while the diameter is bounded or slowly increasing with the number of nodes, the
degree remains small (or bounded) and the clustering coefficient of nodes remain bounded away
from zero (Cont and Tanimura, 2008). In the Brazilian financial system, we observe nodes with an
arbitrary small clustering coefficient across all time periods (Figure 6). This absence of uniform
clustering shows that the Brazilian financial system is not a small world network.
Figure 6 shows the relationship between the local clustering coefficient and number of degrees
for the Brazilian interbank network. The negative slope of the plots shows that financial institutions
with few connections (small degree) have counterparties that are very connected to each other (large
clustering) while financial institutions with many connections (large degree) have counterparties
with sparsely connected neighbors.
18
0 20 40 60 80 100
0
0.1
0.2
0.3
0.4
0.5
June 2007
Degree
Local Clustering Coefficient
0 20 40 60 80 100
0
0.1
0.2
0.3
0.4
0.5
December 2007
Degree
Local Clustering Coefficient
0 20 40 60 80
0
0.1
0.2
0.3
0.4
0.5
March 2008
Degree
Local Clustering Coefficient
0 20 40 60 80 100
0
0.1
0.2
0.3
0.4
0.5
June 2008
Degree
Local Clustering Coefficient
0 20 40 60 80 100
0
0.1
0.2
0.3
0.4
0.5
September 2008
Degree
Local Clustering Coefficient
0 20 40 60 80 100 120
0
0.1
0.2
0.3
0.4
0.5
November 2008
Degree
Local Clustering Coefficient
Figure 6: Degree vs. clustering coefficient for the Brazilian interbank network. The grey line is the average clustering coefficient.
19
3 Systemic risk and default contagion
Once the network structure linking balance sheets has been identified, one is interested in examining
the consequences of the failure of a given node or set of nodes in the network on the stability of
the network and locate the node or nodes whose failure would pose the highest threat to network
stability. We now define two indicators of default contagion and systemic impact for a financial
institution – the Default Impact and the Contagion Index– following Cont (2009). These indicators
aim at quantifying the impact of the default of a given institution in terms of the (expected) loss it
incurs for other institutions in the network, taking into account both balance sheet contagion and
common shocks affecting balance sheets.
3.1 Default mechanism
Default occurs when an institution fails to fulfill a legal obligation such as a scheduled debt pay-
ment of interest or principal, or the inability to service a loan. Typically, this happens when the
institution does not hold enough liquid assets to meet its contractual obligations i.e. due to a
shortage of liquidity.
Insolvency happens when the net worth of an institution is reduced to zero, i.e. losses exceed
capital, while illiquidity occurs when reserves in liquid assets, such as cash and cash equivalents,
are insufficient to cover short term liabilities. While illiquidity leads to default, in principle in-
solvency may not necessarily entail default as long as the institution is able to obtain financing
to meet payment obligations. Nevertheless, in the current structure of the financial sector where
financial institutions are primarily funded through short-term debt, which must be constantly re-
newed, insolvent institutions would have great difficulties in raising liquidity as their assets lose
in value. Indeed, renewal of short term funding is subject to the solvency and creditworthiness of
the institution. In practice, insolvency leads to illiquidity which in turn leads to default unless, of
course, a lender of last resort such as the central bank intervenes.
Thus, in line with various previous studies, we consider default as generated by insolvency. In
practice, this may be defined as a scenario where losses in asset value exceed Tier 1 capital. If
Tier 1 capital is wiped out, the institution becomes insolvent which is very likely to generate a loss
of short term funding leading to default. One must bear in mind, however, that other scenarios to
default may exist which may amplify the contagion phenomena described below, so our assessments
should be viewed as a lower bound on the magnitude of contagion.
We recognize that institutions may default due to lack of liquidity even when just a portion of
their Tier 1 capital is wiped out: the example of Bear Stearns is illustrative in this sense (Cox,
2008). However, given the current funding structure of financial institutions through short term
debt, absent a government bailout, insolvency due to market losses which exceed the level of capital
will most probably lead to a loss of funding opportunities and credit lines and entail default. Also,
it is difficult argue that illiquidity in absence of insolvency will systematically lead to default: as
argued by Lo (2011), uncertainty about bank solvency was more central than illiquidity in the
recent financial crisis.
Thus, our estimates for the extent of default contagion will, if anything, lead to lower bounds
for its actual extent in absence of government intervention.
3.2 Loss contagion
When a financial institution (say, 𝑖) defaults, it leads to an immediate writedown in value of all its
liabilities to its creditors. These losses are imputed to the capital of the creditors, leading to a loss
20
of 𝐸
𝑗𝑖
for each creditor 𝑗. If this loss exceeds the creditor’s capital i.e. 𝐸
𝑗𝑖
> 𝑐
𝑗
this leads to the
insolvency of the institution 𝑗, which in turn may generate a new round of losses to the creditors of
𝑗. This domino effect may be modeled by defining a loss cascade, updating at each step the losses
to balance sheets resulting from previously defaulted counterparties:
Definition 1 (Loss cascade). Consider an initial configuration of capital reserves (𝑐(𝑗), 𝑗 ∈ 𝑉 ).
We define the sequence (𝑐
𝑘
(𝑗), 𝑗 ∈ 𝑉 )
𝑘≥0
as
𝑐
0
(𝑗) = 𝑐(𝑗) and 𝑐
𝑘+1
(𝑗) = max(𝑐
0
(𝑗) −
∑
{𝑖,𝑐
𝑘
(𝑖)=0}
(1 − 𝑅
𝑖
)𝐸
𝑗𝑖
, 0), (3)
where 𝑅
𝑖
is the recovery rate at the default of institution 𝑖. (𝑐
𝑛−1
(𝑗), 𝑗 ∈ 𝑉 ), where 𝑛 = ∣𝑉 ∣ is the
number of nodes in the network, then represents the remaining capital once all counterparty losses
have been accounted for. The set of insolvent institutions is then given by
𝔻(𝑐, 𝐸) = {𝑗 ∈ 𝑉 : 𝑐
𝑛−1
(𝑗) = 0} (4)
Remark 1 (Fundamental defaults vs defaults by contagion). The set 𝔻(𝑐, 𝐸) of defaulted institu-
tions may be partitioned into two subsets
𝔻(𝑐, 𝐸) = {𝑗 ∈ 𝑉 : 𝑐
0
(𝑗) = 0}
| {z }
Fundamental defaults
∪
{𝑗 ∈ 𝑉 : 𝑐
0
(𝑗) > 0, 𝑐
𝑛−1
(𝑗) = 0}
| {z }
Defaults by contagion
where the first set represents the initial defaults which trigger the cascade –we will refer to them as
fundamental defaults– and the second set represents the defaults due to contagion.
The default of an institution can therefore propagate to other participants in the network
through the contagion mechanism described above. We measure the impact of the default event
triggering the loss cascade by the loss incurred across the network during the default cascade:
Definition 2 (Default Impact). The Default Impact 𝐷𝐼(𝑖, 𝑐, 𝐸) of a financial institution 𝑖 ∈ 𝑉 is
defined as the total loss in capital in the cascade triggered by the default of 𝑖:
𝐷𝐼(𝑖, 𝑐, 𝐸) =
∑
𝑗∈𝑉
𝑐
0
(𝑗) − 𝑐
𝑛−1
(𝑗), (5)
where (𝑐
𝑘
(𝑗), 𝑗 ∈ 𝑉 )
𝑘≥0
is defined by the recurrence relation (3), with initial condition is given by
𝑐
0
(𝑗) = 𝑐(𝑗) for 𝑗 ∕= 𝑖 and 𝑐
0
(𝑖) = 0.
It is important to note that the Default Impact does not include the loss of the institution
triggering the cascade, but focuses on the loss this initial default inflicts to the rest of the network:
it thus measures the loss due to contagion.
Here we have chosen to measure the impact of a default in terms of loss in capital. If one adopts
the point of view of deposit insurance, then one can use an alternative measure, which is the sum
of deposits across defaulted institutions:
𝐷𝐼(𝑖, 𝑐, 𝐸) =
∑
𝑗∈𝔻(𝑐,𝐸)
𝐷𝑒𝑝𝑜𝑠𝑖𝑡𝑠(𝑗).
The contagion mechanism described above is similar to the one presented in Furfine (2003);
Upper and Worms (2004); Mistrulli (2007). Since liquidation procedures are usually slow and
21
settlements may take up several months to be effective, creditors cannot recover the residual value
of the defaulting institution according to such a hypothetical clearing mechanism, and write down
their entire exposure in the short-run, leading to a short term recovery rate of zero. In absence of a
clearing mechanism, this approach seems more reasonable than the one proposed by Eisenberg and
Noe (2001) which corresponds to a hypothetical situations where all portfolios are simultaneously
liquidated. Eisenberg and Noe (2001) focused on payment systems, where clearing takes place at
the end of each business day, but is not a reasonable model for the liquidation of defaulted bank
portfolios. Finally, we note that this model does not capture medium- or long-term contagion:
maintaining exposures constant over longer term horizons, as in (Elsinger et al., 2006a) is unrealistic
since exposures and capital levels fluctuate significantly over such horizons.
3.3 Contagion Index of a financial institution
The Default Impact of an institution is conditional on the level of capital buffers held by different
institutions and these may in fact decrease in an unfavorable stress scenario such as an economic
downturn which adversely affects bank portfolios. Losses in asset value resulting from macroe-
conomic shocks, in addition to generating correlation in market risk across bank portfolios, also
contribute to amplifying the magnitude of contagion: by depleting the capital buffer of banks, they
increase the impact of a given default event and make the network less resilient to defaults. This
points to the need of integrating both common macroeconomic shocks and contagion effects when
measuring systemic risk in interbank networks, a point that was already recognized by Elsinger
et al. (2006a).
To take into account the impact of macroeconomic shocks, we introduce a (negative) random
variable 𝑍 which represents the magnitude of such a common shock. This variable 𝑍 is then scaled
to generate a loss of 𝜖
𝑖
(%) in the (Tier 1) capital of institution 𝑖 with a severity that depends on
the capital and the creditworthiness of each institution: those with higher default probabilities are
more affected by a macroeconomic shock.
Macroeconomic shocks affect bank portfolios in a highly correlated way, due to common expo-
sures of these portfolios. This correlation has been found to be significantly positive in banking
systems across different countries (Lehar, 2005). Moreover, in market stress scenarios fire sales may
actually exacerbate such correlations (Cont and Wagalath, 2011). In many stress-testing exercises
conducted by regulators, the shocks applied to various portfolios are actually scaled version of the
same random variable i.e. perfectly correlated across portfolios. To generalize this specification
while conserving the idea that macroeconomic shocks should affect portfolios in the same direction,
we considering a co-monotonic model for macroeconomic shocks (Cont, 2009):
𝜖(𝑖, 𝑍) = 𝑐(𝑖)𝑓
𝑖
(𝑍) (6)
where the 𝑓
𝑖
are strictly increasing functions with values in (−1, 0].
In each loss scenario, defined by a vector of capital losses 𝜖, one can compute, as in Definition
2, the Default Impact 𝐷𝐼(𝑖, 𝑐 + 𝜖(𝑍), 𝐸) of a financial institution, computed as above but in the
network with stressed capital buffers 𝑐 + 𝜖(𝑍). A macroeconomic stress scenario corresponds to a
scenario where 𝑍 takes very negative values. A plausible set of stress scenarios for stress testing
purposes may be defined by using a low quantile 𝛼 of 𝑍:
ℙ(𝑍 < 𝛼) = 𝑞
where 𝑞 = 5% or 1% for example. Similar definitions based on quantiles of macroeconomic losses
were proposed by Zhou et al. (2009).
22
We define the Contagion Index 𝐶𝐼(𝑖, 𝑐, 𝐸) of an institution 𝑖 as its expected Default Impact
when the network is subject to such a macroeconomic stress scenario (Cont, 2009):
Definition 3 (Contagion Index). The Contagion Index 𝐶𝐼(𝑖, 𝑐, 𝐸) (at confidence level 𝑞) of insti-
tution 𝑖 ∈ 𝑉 is defined as its expected Default Impact in a market stress scenario:
𝐶𝐼(𝑖, 𝑐, 𝐸) = 𝔼 [𝐷𝐼(𝑖, 𝑐 + 𝜖(𝑍), 𝐸)∣𝑍 < 𝛼] (7)
where the vector 𝜖(𝑍) of capital losses is defined by (6) and 𝛼 is the 𝑞-quantile of the systematic
risk factor 𝑍: ℙ(𝑍 < 𝛼) = 𝑞.
In the examples given below, we choose for 𝛼 the 5% quantile of the common factor 𝑍, which
corresponds therefore to a (mild) market stress scenario whose probability is 5%, but obviously
other choices of quantile levels are perfectly feasible.
The Contagion Index 𝐶𝐼(𝑖, 𝑐, 𝐸) measures the systemic impact of the failure of an institution
by the expected loss –measured in terms of capital– inflicted to the network in the default cascade
triggered by the initial default of 𝑖. In this way, it jointly accounts for network contagion effects
and correlations in portfolio losses through common shocks.
The idea of jointly examining macroeconomic shocks to balance sheet and contagion has also
been examined in Elsinger et al. (2006a). However, unlike the metrics used by Elsinger et al.
(2006a), the definition of the Contagion Index involves conditioning on stress scenarios. This
conditioning is essential to its interpretation: in this way, the Contagion Index focuses on the
stability of banking system in a stress scenario rather than exploring the average outcome of a
macroeconomic shock.
The computation of this index involves the specification (6) for the joint distirbution of shocks
affecting balance sheets. Other specifications –static or dynamic, factor-based or copula-based– are
possible, but the co-monotonic shocks leads to desirable monotonicity properties for the Contagion
Index, viewed as a risk measure (Cont, 2009). Note that, given the specification (6) with 𝑓
𝑖
(𝑍) >
−1, we have 𝑐(𝑖) + 𝜖(𝑖, 𝑍) > 0 so defaults are not caused by the market shocks alone. However,
since 𝜖(𝑖, 𝑍) ≤ 0, capital buffers are lowered in stress scenarios so we have
𝐷𝐼(𝑖, 𝑐 + 𝜖, 𝐸) ≥ 𝐷𝐼(𝑖, 𝑐, 𝐸) and thus 𝐶𝐼(𝑖, 𝑐, 𝐸) ≥ 𝐷𝐼(𝑖, 𝑐, 𝐸).
In the examples below, we model 𝑍 as a negative random variable with a heavy-tailed distribution
𝐹 and an exponential function for 𝑓
𝑖
:
𝜖(𝑖, 𝑍) = 𝑐(𝑖) (exp(𝜎
𝑖
𝑍) − 1) (8)
where 𝜎
𝑖
is a scale factor which depends on the creditworthiness, or probability of default 𝑝
𝑖
, of
institution 𝑖. For example, a possible specification is to choose 𝜎
𝑖
such that 𝑝
𝑖
corresponds to the
probability of losing 90% of the Tier 1 capital in a market stress scenario:
𝜎
𝑖
= −
log(10)
𝐹
−1
(𝑝
𝑖
)
. (9)
Default probabilities are obtained from historical default rates given by credit ratings for the firms
at the date corresponding to the simulation.
23
4 Is default contagion a significant source of systemic risk?
Most empirical studies of interbank networks have pointed to the limited extent of default con-
tagion (Sheldon and Maurer, 1998; Furfine, 2003; Upper and Worms, 2004; Wells, 2004; Elsinger
et al., 2006a,b; Mistrulli, 2007). However, almost all these studies (with the exception Elsinger
et al. (2006a,b)) examine the sole knock-on effects of the sudden failure of a single bank by an
idiosyncratic shock, thus ignoring the compounded effect of both correlated market events and
default contagion. A market shock affecting the capital of all institutions in the network can con-
siderably reduce capital buffers in the network, which makes it more vulnerable to potential losses
and increases the likelihood of large default cascades.
The data set of exposures in the Brazilian financial system allows to compute, the Default
Impact and the Contagion Index for each financial institution in the Brazilian network. The
Contagion Index is computed by Monte Carlo simulation using the model specified in Section 3.3,
using the procedures described in Cont and Moussa (2010).
Figure 7 shows the cross-sectional distribution of the size of the average default cascade gener-
ated by the initial default of a single node. We observe that while for most institutions this number
is close to zero (which indicates no contagion), for a very small number of institutions this number
can be as high as 3 or 4, meaning that the initial default of some nodes can trigger the default of
up to 3 or 4 other nodes. This is a signature of contagion.
Another indicator of default contagion is the ratio of the Contagion Index of a bank to its
interbank liabilities. As shown in Figure 8, the Contagion Index can significantly exceed the
interbank liabilities for the most systemic nodes.
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Size of default cascade
Frequency
Network in June 2007
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Size of default cascade
Frequency
Network in December 2007
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Size of default cascade
Frequency
Network in March 2008
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
Size of default cascade
Frequency
Network in June 2008
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Size of default cascade
Frequency
Network in September 2008
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
Size of default cascade
Frequency
Network in November 2008
Figure 7: Distribution of the size of default cascades (number of defaults).
24
Figure 8: Ratio of the Contagion Index to interbank liabilities: the Contagion Index can exceed
the size of interbank liabilities by a large factor during periods of economic stress.
Both the Default Impact and the Contagion Index exhibit heavy tailed cross-sectional distribu-
tions (Figure 9), indicating the existence of a few institutions that present a high contagion risk to
the financial system (up to 10% of the total capital of the network) while most institutions exhibit
a small risk. As shown in figure 9, that the proportion of nodes with large Default Impact and large
Contagion Index is the highest during June 2007 and December 2007. These periods correspond
to the onset of the subprime mortgage crisis in the United States.
Figure 10 displays the cross-sectional distribution of the ratio of the Contagion Index to the
Default Impact at different periods. We observe that the Contagion Index may, for some nodes,
significantly exceed the Default Impact, showing that common shocks to balance sheets seem to
amplify contagion, by reducing the capital buffer available to financial institutions and rendering
them more susceptible to default.
Exposures that are not covered by an adequate amount of capital to sustain their loss in the
event of default constitute channels of contagion across the system. We will call such exposures
contagious exposures:
Definition 4 (Contagious Exposure). An exposure 𝐸
𝑖𝑗
is called contagious if it exceeds the capital
available to 𝑖: 𝐸
𝑖𝑗
> 𝑐(𝑖).
If the link 𝑖 → 𝑗 represents a contagious exposure, the default of 𝑗 leads to the default of 𝑖
in all stress scenarios. Thus, the subgraph constituted of contagious exposures will be a primary
support for the propagation of default cascades: the larger this subgraph, the larger the extent of
contagion. In a stress scenario in which balance sheets are subjected to negative market shocks,
25
new contagious exposures may appear, leading to a higher degree of contagion. Figure 11 shows
the graph of contagious exposures (black) in the Brazilian network in June 2007, with, in red, the
exposures that become contagious once a (particular) set of correlated market shocks is applied
to balance sheets. As shown in Amini et al. (2010) using large-network asymptotics, the path of
contagion in large networks is concentrated on the subgraph of contagious exposures, so keeping
track of such exposures is a natural idea in the monitoring of contagion.
Figure 12 presents the proportion of contagious exposures in the Brazilian system, their ex-
pected proportion under stress test scenarios, and their expected proportion in scenarios where the
level of common downward shocks to balance sheets exceeds its 5% quantile. We find that corre-
lated market shocks may increase the proportion of contagious exposures considerably, so ignoring
market risk when assessing contagion effects can lead to a serious underestimation of the extent of
default contagion.
26
10
−4
10
−3
10
−2
10
−1
10
−3
10
−2
10
−1
10
0
k
P(DI>k)
Network in June 2007
Network in December 2007
Network in March 2008
Network in June 2008
Network in September 2008
Network in November 2008
10
−4
10
−3
10
−2
10
−1
10
−3
10
−2
10
−1
10
0
k
P(CI>k)
Network in June 2007
Network in December 2007
Network in March 2008
Network in June 2008
Network in September 2008
Network in November 2008
Figure 9: Brazilian interbank network: distribution of the default impact and the Contagion Index
on the logarithmic scale. The highest values of Default Impact and Contagion Index are observed
in June 2007 and December 2007.
27
Figure 10: Default impact vs Contagion Index: the Contagion Index can be up to fifteen times
larger than the Default Impact for some nodes.
28
Figure 11: Network of contagious exposures before (dashed lines) and after (dashed and red lines)
market shocks.
29
Jun 2007 Dec 2007 Mar 2008 Jun 2008 Sep 2008 Nov 2008
0
0.005
0.01
0.015
0.02
0.025
0.03
Proportion of contagious exposures
a
b
c
Figure 12: Proportion of contagious exposures (a) in the initial network, (b) averaged across market
shock scenarios, (c) averaged across scenarios where common factor falls below 5% quantile level.
5 What makes an institution systemically important?
Previous studies on contagion in financial networks (Allen and Gale, 2000; Battiston et al., 2009;
Elsinger et al., 2006a; Nier et al., 2007) have examined how the network structure may affect the
global level of systemic risk but do not provide metrics or indicators for localizing the source of
systemic risk within the network. The ability to compute a Contagion Index for measuring the
systemic impact of each institution in the network, enables us to locate the institutions which have
the largest systemic impact and investigate their characteristics.
We first investigate (section 5.1) the effect of the size, measured in terms of interbank liabilities
or assets on the Contagion Index. Then we examine (section 5.2) the effect of network structure on
the Contagion Index and define, following Cont and Moussa (2010), network-based indicators of
connectivity counterparty susceptibility and local network frailty, which are shown to be significant
factors for contagion.
5.1 Size of interbank liabilities
Size is generally considered a factor of systemic importance. In our modeling approach, where
losses flow in through the asset side and flow out through the liability side of the balance sheet, it
is intuitive that, at least at the first iteration of the loss cascade, firms with large liabilities to other
nodes will be a large source of losses for their creditors in case of default. Accordingly, interbank
liabilities are highly correlated with any measure of systemic importance. A simple plot on the
logarithmic scale of the Contagion Index against the interbank liability size reveals a strong positive
relationship between the interbank liabilities of an institution in the Brazilian financial system and
its Contagion Index (see figure 13). A linear regression of the logarithm of the Contagion Index on
the logarithm of the interbank liability size supports this observation: interbank liabilities explains
27% of the cross-sectional variability of the Contagion Index.
30
Therefore, balance sheet size does matter, not surprisingly. However, the size of interbank
liabilities does not entirely explain the variations in the Contagion Index across institutions: the
interbank liability size does exhibit a strong positive relationship with the Contagion Index, but
the ranking of institutions according to liability size does not correspond to their ranking in terms
of systemic impact (see figure 13).
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
Interbank Liabilities
Contagion Index
Figure 13: Contagion Index versus total interbank liabilities (logarithmic scale), June 2007.
Model: log(𝐶𝐼) = 𝛽
0
+ 𝛽
1
log(𝐿) + 𝜖
Coefficients Standard error t-statistic 𝑅
2
𝑏
0
= 4.53 2.64 1.71 27%
𝑏
1
= 0.69** 0.13 5.01
* significant at 5% confidence level
** significant at 1% confidence level
Table 6: Log-log cross-sectional regression of the Contagion Index (expressed in percentage of the
total network capital) on the interbank liability in June 2007.
Table 7, where nodes are labeled according to their decreasing ranking in terms of the Contagion
Index, shows that they all have interbank liabilities less than the 90% quantile of the cross sectional
interbank liability sizes. This suggests that factors other than size contribute to their systemic
importance.
Rank Contagion Index (Billions BRL) Number of creditors Interbank liability (Billions BRL)
1 3.48 25 1.64
2 3.40 21 0.97
3 2.09 20 1.10
4 1.78 20 0.60
5 1.45 34 1.59
Network median 0.0007 20 0.52
90%-quantile 0.53 28 2.07
Table 7: Analysis of the five most contagious nodes in June 2007.
31
5.2 Centrality and counterparty susceptibility
Table 7 shows that, while the large size of liabilities of the node with the highest Contagion Index
can explain its ranking as the most systemic node, this is not the case, for instance, for the fourth
most systemic node whose interbank liabilities and number of counterparties are in line with the
network average. This shows that balance sheet size alone or simple measures of connectenedness
such as number of counterparties are not good proxies for systemic importance. This points to a
more subtle role of network structure in explaining the cross-sectional variability in the Contagion
Index. As shown in figure 14 the five most systemic nodes are not very connected and just have
few contagious exposures (in red) but, as shown in figure 15, their creditors are heavily connected
and many of their cross-exposures are contagious exposures (in the sense of Definition 4). This
motivates to define indicators which go beyond simple measures of connectivity and size: following
Cont and Moussa (2010), we define indicators which attempt to quantify the local impact of a
default:
Figure 14: Subgraph of the five institutions with highest Contagion Index and their creditors in
the network in June 2007. Non contagious exposures are indicated by dotted links. Contagious
exposures are bold links.
32
Figure 15: Subgraph of the five institutions with highest Contagion Index and their first and
second-order neighbors in the network in June 2007.Non contagious exposures are indicated by
dotted links. Contagious exposures are bold links.
Definition 5. Susceptibility coefficient
The susceptibility coefficient of a node is the maximal fraction of capital wiped out by the default
of a single counterparty.
𝜒(𝑖) = max
𝑗∕=𝑖
𝐸
𝑖𝑗
𝑐(𝑖)
A node with 𝜒(𝑖) > 100% may become insolvent due to the default of a single counterparty.
Counterparty risk management in financial institutions typically imposes an upper limit on this
quantity.
Definition 6. Counterparty susceptibility
The counterparty susceptibility 𝐶𝑆(𝑖) of a node 𝑖 is the maximal (relative) exposure to node 𝑖 of its
counterparties:
𝐶𝑆(𝑖) =
max
𝑗,𝐸
𝑗𝑖
>0
𝐸
𝑗𝑖
𝑐(𝑗)
33
𝐶𝑆(𝑖) is thus a measure of the maximal vulnerability of creditors of 𝑖 to the default of 𝑖.
Definition 7. Local network frailty
The local network frailty 𝑓(𝑖) at node 𝑖 is defined as the maximum, taken over counterparties
exposed to 𝑖, of their exposure to 𝑖 (in % of capital), weighted by the size of their interbank liability:
𝑓(𝑖) = max
𝑗,𝐸
𝑗𝑖
>0
𝐸
𝑗𝑖
𝑐(𝑗)
𝐿(𝑗)
Local network frailty combines the risk that the counterparty incurs due to its exposure to node
𝑖, and the risk that the (rest of the) network incurs if this counterparty fails. A large value 𝑓(𝑖)
indicates that 𝑖 is a node whose counterparties have large liabilities and are highly exposed to 𝑖.
The analysis of the creditors of the five most systemic institutions in the network (see table 8)
indicates that, whereas the size of interbank liabilities fails to explain their high Contagion Index,
this is better understood by looking at the number of creditors and the size of interbank liabilities of
the counterparties, as well as the counterparty susceptibility and local network frailty. We observe
that the five most systemic nodes have each at least one highly connected counterparty with a large
interbank liability size and exhibit in general a high counterparty susceptibility and local network
frailty.
Ranking max
𝑗,𝐸
𝑗𝑖
>0
𝑘
𝑜𝑢𝑡
(𝑗) max
𝑗,𝐸
𝑗𝑖
>0
𝐿(𝑗) (Billions BRL) 𝐶𝑆(𝑖) 𝑓 (𝑖) (Billions BRL)
1 36 11.23 1.23 8.78
2 34 23.27 1.65 3.15
3 36 23.27 3.68 46.92
4 36 2.91 2.28 0.97
5 36 11.23 5.88 1.52
Network median 36 2.91 0.19 0.07
90%-quantile 36 23.27 3.04 6.89
Table 8: Analysis of the counterparties of the five most contagious nodes in June 2007.
Figure 16 shows that institutions with a high Contagion Index tend to have a large interbank
liability, local network frailty and counterparty susceptibility. To investigate the relevance of these
measures of connectivity and centrality, we perform a logistic regression of the indicator of the
Contagion Index being higher than 1% of the total network capital, using as instrumental variables
interbank liabilities, counterparty susceptibility and local network frailty.
The outputs of the logistic regression are summarized in table 9. We observe that counterparty
susceptibility and local network frailty contribute significantly to the probability of observing a
large Contagion Index
1
: positive coefficients at the 1% significance level and a very high pseudo-
𝑅
2
.
1
The Adjusted Pseudo-𝑅
2
in a logistic regression is defined as 1 − log 𝐿(𝑀)/ log 𝐿(0)((𝑛 − 1)/(𝑛 − 𝑘 − 1)) where
log 𝐿(𝑀) and log 𝐿(0) are the maximized log likelihood for the fitted model and the null model, 𝑛 is the sample size
and 𝑘 is the number of regressors.
34
Figure 16: Counterparty susceptibility (upper figure) and local network frailty (lower figure) of the
most systemic nodes (with a Contagion Index higher than 1% of the network capital) and the less
systemic nodes (with a Contagion Index smaller than 1% of the network capital).
35
Model: 𝑙𝑜𝑔𝑖𝑡(𝑝(𝐶𝐼 > 1%)) = 𝛽
0
+ 𝛽
1
log(𝐿) + 𝛽
2
log(𝐶𝑆) + 𝜖
Coefficients Std error Adjusted Pseudo-𝑅
2
ˆ
𝛽
0
= -20.85** 7.96 93.46%
ˆ
𝛽
1
= 0.96* 0.39
ˆ
𝛽
2
= 0.98* 0.40
Model: 𝑙𝑜𝑔𝑖𝑡(𝑝(𝐶𝐼 > 1%)) = 𝛽
0
+ 𝛽
1
log(𝐿) + 𝜖
Coefficients Std error Adjusted Pseudo-𝑅
2
ˆ
𝛽
0
= -29.24** 7.11 94.54%
ˆ
𝛽
1
= 1.39** 0.34
Model: 𝑙𝑜𝑔𝑖𝑡(𝑝(𝐶𝐼 > 1%)) = 𝛽
0
+ 𝛽
1
log(𝐶𝑆) + 𝜖
Coefficients Std error Adjusted Pseudo-𝑅
2
ˆ
𝛽
0
= -1.46** 0.37 43.36%
ˆ
𝛽
1
= 1.31** 0.33
Model: 𝑙𝑜𝑔𝑖𝑡(𝑝(𝐶𝐼 > 1%)) = 𝛽
0
+ 𝛽
1
log(𝐿) + 𝛽
2
log(𝑓) + 𝜖
Coefficients Std error Adjusted Pseudo-𝑅
2
ˆ
𝛽
0
= -43.20** 11.06 97.76%
ˆ
𝛽
1
= 1.05** 0.39
ˆ
𝛽
2
= 0.97** 0.29
Model: 𝑙𝑜𝑔𝑖𝑡(𝑝(𝐶𝐼 > 1%)) = 𝛽
0
+ 𝛽
1
log(𝑓) + 𝜖
Coefficients Std error Adjusted Pseudo-𝑅
2
ˆ
𝛽
0
= -21.32** 4.75 93.79%
ˆ
𝛽
1
= 0.95** 0.22
* significant at 5% confidence level ** significant at 1% confidence level
Table 9: Marginal contribution of the interbank liabilities, counterparty susceptibility and local
network frailty to the Contagion Index.
We also test for the differences in median between the counterparty susceptibility of the insti-
tutions with a Contagion Index higher than 1% of the total network capital and the counterparty
susceptibility of those with a Contagion Index smaller than 1% of the total network capital. The
Wilcoxon signed-rank test rejects the hypothesis of equal medians at the 1% level of significance.
The median of the counterparty susceptibility of the institutions with a high Contagion Index (2.29)
is significantly higher than the median of the counterparty susceptibility of the institutions with a
small Contagion Index (0.06). Similarly, the median of the local network frailty of the institutions
with a high Contagion Index (18.79 billion BRL) is significantly higher than the median of the
local network frailty of the institutions with a small Contagion Index (0.02 billion BRL).
6 Does one size fit all? The case for targeted capital requirements
Capital requirements are a key ingredient of bank regulation: in the Basel Accords, a lower bound
is imposed on the ratio of capital to (risk-weighted) assets. It is clear that globally increasing
the capital cushion of banks will decrease the risk of contagion in the network, but given the
heterogeneity of systemic importance, as measured for instance by the Contagion index, it is not
clear whether a uniform capital ratio for all institutions is the most efficient way of reducing
systemic risk. Indeed, using a uniform capital ratio penalizes neither the systemic importance
36
of the institutions not the concentration of risk on a few counterparties, two features which our
analysis points to as being important.
Indeed, recent debates on regulatory reform have considered the option of more stringent capital
requirements on systemically important institutions. The analysis described above points to two
types of ’targeted’ capital requirements.
A first, natural, idea consists in imposing (more stringent) capital requirements on the most
systemic institutions in the network. This may be done by first computing the Contagion Index of
all institutions and imposing a higher capital ratio on, say, the 5 most systemic nodes.
A second method is to target the ’weak links’ in the network which correspond to exposures
which constitute a high fraction of capital. Such exposures have a higher probability of becoming
”contagious exposures” (in the sense of Definition 4) when capital buffers are reduced in a stress
scenario. Such ’weak links’ may be strengthened by imposing a minimal capital-to-exposure ra-
tio which, unlike the aggregate capital ratios currently implemented in Basel II, would penalize
concentration of exposures across few counterparties.
Studies in epidemiology or the spread of viruses in peer-to-peer networks (Cohen et al., 2003;
Madar et al., 2004; Huang et al., 2007) have explored similar problems in the context of immu-
nization of heterogeneous networks to contagion. Madar et al. (2004) study various immunization
strategies in the context of epidemic modeling and show that a targeted immunization strategy that
consists in vaccinating first the nodes with largest degrees is more cost-effective than a random or
exhaustive immunization scheme.
These ideas may be implemented using capital ratios:
∙ Minimum capital ratio: institutions are required to hold a capital equal to or higher than
cover a portion 𝜃 of their aggregae interbank exposure:
𝑐(𝑖) = max(𝑐(𝑖), 𝜃𝐴(𝑖)) (10)
∙ Minimum capital-to-exposure ratio: institutions are required to hold a level of capital which
covers a portion 𝛾 of their largest interbank exposure:
𝑐(𝑖) = max(𝑐(𝑖),
max
𝑗∕=𝑖
(𝐸
𝑖𝑗
)
𝛾
) (11)
This penalizes nodes with large counterparty susceptibility (Definition 6) and local network
frailty (Definition 7), which is a desirable feature.
To assess the impact on systemic risk of these different schemes for the allocation of capital
across institutions, we compare the cross-sectional distribution of the Contagion Index in the
Brazilian network when
(a) a minimum capital ratio is applied to all financial institutions in the network (non-targeted
capital requirements),
(b) a minimum capital ratio applied only to the 5% most systemic institutions (targeted capital
requirements),
(c) a minimum capital -to-exposure ratio is applied to the 5% most systemic institutions (disag-
gregated and targeted capital requirements),
by computing in each case the average of 5% largest Contagion Indexes (i.e. the 5% tail conditional
expectation of the cross sectional distribution of Contagion Index).
37
Figure 17 displays the result of our simulation for the Brazilian network: for each allocation
of capital across nodes, we represent the average Contagion Indices for the 5% most systemic
institutions versus the total capital allocated to the network. Assets and liabilities are identical for
all networks considered. Comparing cases a. and b. shows that targeted capital requirements can
achieve the same reduction in the size of default cascades while requiring less capital.
Another observation is that setting a minimum capital-to-exposure ratio (cass c.) seems to
be slightly more effective than simply setting a minium ratio of capital to aggregate interbank
assets. The reason is intuitive: a minimum capital-to-exposure penalizes concentrated exposures
and reduces the number of contagious exposures in the network. In this case the gain is not very
large, but the exact magnitude of the resulting decrease in systemic risk depends on the network
configuration.
Figure 17: Comparison of various capital requirement policies: (a) imposing a minimum capital
ratio for all institutions in the network, (b) imposing a minimum capital ratio only for the 5% most
systemic institutions, (c) imposing a minimum capital-to-exposure ratio for the 5% most systemic
institutions.
We conclude that, given the heterogeneity of banks in terms of size, connectivity and systemic
importance,
∙ targeting the most contagious institutions is more effective in reducing systemic risk than
increasing capital ratios uniformly across all institutions, and
∙ capital requirements should not simply focus on the aggregate size of the balance sheet
but depend on their concentration/distribution across counterparties: a minimal capital-to-
exposure ratio can be a more effective way of controling contagious exposures.
38
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- "This subclass of multivariate conditional risk measures corresponds to the idea that we first aggregate the risk factors X and then evaluate the risk of the aggregated values. In fact many prominent examples of multivariate conditional risk measures are of type (1.4), for instance the Contagion Index of Cont et al. (2013) or the SystRisk of Brunnermeier and Cheridito (2014) from the systemic risk literature. Chen et al. (2013) were the first to axiomatically describe this intuitive type of multivariate risk measures on a finite state space, and in Kromer et al. (2016) this has been extended to general L p -spaces, whereas the conditional framework was studied in Hoffmann et al. (2016). "
[Show abstract] [Hide abstract] ABSTRACT: We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced Hoffmann et al. (2016). Further, in analogy to the univariate case in F\"ollmer (2014), we prove that under law-invariance strong consistency implies that multivariate conditional risk measures are necessarily multivariate conditional certainty equivalents.- "2407). We also refer to Battiston et al. (2012) and Cont et al. (2013) for a similar reasoning and Memmel et al. (2012) for empirical evidence for the frequent occurrence of a recovery rate close to zero. Standard balance sheet considerations lead us to the following definition. "
[Show abstract] [Hide abstract] ABSTRACT: We develop a structural default model for interconnected financial institutions in a prob-abilistic framework. For all possible network structures we characterize the joint default distribution of the system using Bayesian network methodologies. Particular emphasis is given to the treatment and consequences of cyclic financial linkages. We further demonstrate how Bayesian network theory can be applied to detect contagion channels within the financial network, to measure the systemic importance of selected entities on others, and to compute conditional or unconditional probabilities of default for single or multiple institutions.- "Early studies on systemic risk networks focused on linkages arising from actual exposures based on balance sheet information. However, due to data accessibility issues, little empirical work has been done in this area (Cont et al., 2013; Georg, 2013 ). Meanwhile, several statistical and econometric methods have been advanced to study interdependencies, contagion and spillover effects from observed market data. "
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