Financial Networks and Contagion

TL;DR: In this article, the authors model contagions and cascades of failures among organizations linked through a network of financial interdependencies and identify how the network propagates discontinuous changes in asset values triggered by failures.

Abstract: We model contagions and cascades of failures among organizations linked through a network of financial interdependencies. We identify how the network propagates discontinuous changes in asset values triggered by failures (e.g., bankruptcies, defaults, and other insolvencies) and use that to study the consequences of integration (each organization becoming more dependent on its counterparties) and diversification (each organization interacting with a larger number of counterparties). Integration and diversification have different, nonmonotonic effects on the extent of cascades. Initial increases in diversification connect the network which permits cascades to propagate further, but eventually, more diversification makes contagion between any pair of organizations less likely as they become less dependent on each other. Integration also faces tradeoffs: increased dependence on other organizations versus less sensitivity to own investments. Finally, we illustrate some aspects of the model with data on European debt cross-holdings.

Summary

Introduction

• With low levels of diversification, organizations can be very sensitive to particular others, but the network of interdependencies is disconnected and overall cascades are limited in extent.
• 6 Cabrales, Gottardi, and Vega-Redondo (2013) study the trade-off between the risk-sharing enabled by greater interconnection and the greater exposure to cascades resulting from larger components in the financial network.

I. The Model and Determining Organizations’ Values with Cross-Holdings

• A. Primitive Assets, Organizations, and Cross-Holdings (Analogous notation is used for all matrices.).
• In different settings, Cifuentes, Ferrucci, and Shin (2005) and Gofman (2013) also find that cascades can be nonmonotonic in connectivity.
• In a setting with cross-holdings, there are subtleties in determining the “fair market” value of an organization, and the real economic costs of organizations’ failures.
• The authors briefly review the accounting and the key valuation equations in the absence of failure costs.

C. Discontinuities in Values and Failure Costs

• An important part of their model is that organizations can lose productive value in discontinuous ways if their values fall below certain critical thresholds.
• There are many sources of such discontinuities.
• One detailed and simple microfoundation is laid out in Section IE below.
• These failure costs are subtracted from a failing organization’s cash flow.
• 20 For example, if the failure threshold were based on book values, then two organizations about to fail would be able to avoid failure by exchanging cross-holdings and inflating their book values.

E. A Simple Microfoundation

• To help fix ideas, the authors discuss one simple microfoundation—among many—of the model and the value equations provided above.
• 23 The number b i (v, p) reflects realized failure costs, and is zero when failure does not occur.
• Thus, even though the obligations might initially be in the form of debt, the relevant scenario for their cascades—and the one the model focuses on—is one in which the full promised amounts cannot be met by the organizations.
• While the firm continues to operate, this amount must cover return on capital, wages, benefits, and pension obligations for the owner-operators.
• Liquidation is irreversible and total: a firm cannot partially liquidate its proprietary asset.

F. Equilibrium Existence and Multiplicity

• There always exists a solution—and there can exist multiple solutions—to the valuation equation (multiple vectors v satisfying (5)) in the presence of the discontinuities.
• First, taking other organizations’ values and the values of underlying assets as fixed and given, there can be multiple possible consistent values of organization i that solve equation (5).
• These losses involve time that the asset is left idle, costs of assessing values and holding sales of assets, costs of moving assets to another production venue, and loss of firm-specific capital and knowledge.
• This source of multiple equilibria corresponds to the standard story of self-fulfilling bank runs (see classic models such as Diamond and Dybvig 1983).
• When the authors do discuss multiple equilibria, they will consider only the second novel source of multiplicity—multiplicity due to interdependencies between organizations—rather than the well-known phenomenon of a bank run on a single organization.

G. Measuring Dependencies

• The dependency matrix A takes into account all indirect holdings as well as direct holdings.
• The associated cross-holdings matrix C and the dependency matrix A are as follows.
• One can already see that direct claims—as captured by C and C—can differ quite substantially from the ultimate value dependencies described by A.the authors.

H. Avoiding a First Failure

• Before moving on to their main results regarding diversification and integration, the authors provide a result which uses their model to show that there are necessarily trade-offs in preventing the spark that ignites a cascade.
• Before stating the result the authors also introduce the concept of fair trades.30 Fair trades are exchanges of cross-holdings or underlying assets which leave the values of the organizations unchanged at current asset prices.
• It does not incorporate the potential impact of failures of organizations on their values.
• Thus it is a benchmark that abstracts away from the failure costs, which is the right benchmark for the exercise of seeing the impact of trades on first failures.
• It is conceivable that if an organization is at risk of eventual failure but not imminent failure, there could exist some fair trades that would unambiguously make that organization safer: prone to failure at a smaller set of prices.

II. Cascades of Failures: Definitions and Preliminaries

• In order to present their main results, the authors need to first provide some background results and definitions regarding how the model captures cascades, which they present in this section.
• These preliminaries outline how failures cascade and become amplified, a simple algorithm for identifying the waves of failures in a cascade, and their distinction between diversification and integration.

B. Who Fails in a Cascade?

• Again, the authors focus on the best-case equilibrium.
• The authors begin with an example that illustrates these ideas very simply, and then develop the more general analysis.
• The conditional failure frontiers identify a region of multiple equilibria due to interdependencies in the values of the organizations.
• This algorithm provides us with hierarchies of failures.
• 38 The same algorithm can be used to find the set of organizations that fail in the worst-case equilibrium by instead initializing the set 0 to contain all organizations and looking for organizations that will not fail, and so forth.

C. Defining Integration and Diversification

• Before presenting those results (in the next section), the authors define the essential distinction between the two network properties.
• The authors say that a financial system becomes more diversified when the number of cross-holders in each organization i weakly increases and the cross-holdings of all original cross-holders of i weakly decrease.
• A financial system becomes more integrated if the external shareholders of each organization i have lower holdings, so that the total cross-holdings of each organization by other organizations weakly increase.
• Cii for all i, with strict inequality for some i.
• There are changes in cross-holdings that increase diversification but not integration and other changes that increase integration but not diversification.

D. Essential Ingredients of a Cascade

• To best understand the impact of diversification and integration on cascades, it is useful to identify three ingredients that are necessary for a widespread cascade: I. A First Failure: Some organization must be susceptible enough to shocks in some assets that it fails.
• Keeping these different ingredients of cascades in mind will help us disentangle the different effects of changes in cross-holdings.
• As the authors increase integration (without changing each organization’s counterparties), an organization becomes less sensitive to its own investments but more sensitive to other organizations’ values, and so first failures can become less likely while contagion can become more likely conditional on a failure.
• This decreases the circumstances where there can be contagion, making things better with respect to II, while increasing the potential reach of a contagion conditional upon one occurring, making things worse with respect to III.
• Second, both integration and diversification improve matters with respect to at least one of the cascade ingredients above while causing problems along a different dimension.

III. How Do Cascades Depend on the Diversification and Integration of Cross-Holdings?

• The authors begin with some analytic results and then provide additional results via simu- lations for some random network structures.
• Drops in values propagate through the network (as captured by the matrix A), and so the second organization to fail need not be an immediate crossholder, although that would typically be the case.

A. The Consequences of Diversification and Integration: Analytic Results

• —To begin, the authors prove a general result about how integration affects the extent of cascades.
• Given that a first failure occurs, integration only exacerbates the resulting cascade.
• 44 When one allows the number of nodes to become arbitrarily large, then various techniques related to laws of large numbers can be applied to deduce connectedness properties of a random network.
• This is a basic measure of average diversification in the graph that overweights organizations held by many others, and turns out to be the right one for their purposes.
• When integration and diversification are intermediate, so that none of these obstructions to contagion occur, part B of the proposition states that a fraction of organizations fail.

B. The Different Roles of Diversification and Integration: Simulations on Random Networks

• The authors now show that the analytic results of the previous section hold in other classes of simulated random networks.
• —To illustrate how increased diversification and increased integration affect the number of organizations that fail in a cascade following the failure of a single organization’s assets, the authors specialize the model.
• When d is sufficiently low (1.5 or below), then the authors see the percentage of organizations that fail is less than 20.
• In summary, there is constantly a trade-off between II and III, but initially III dominates as diversification leads to dramatic changes in the connectedness of the network.
• For very high levels of integration, each organization begins to carry something close to the market portfolio, and so any first failure caused by the devaluation of a single proprietary asset becomes less likely.

IV. Alternative Network Structures

• Additional insights emerge from examining some other random graph models of financial interdependencies.
• The x-axis corresponds to the diversification level (the expected degree in the random network of cross-holdings).
• The two figures work with different failure thresholds and depict how the size of cascades varies with the level of integration c ranging from 0.1 to 0.5.

A. A Core-Periphery Model

• As a stylized representation of the interbank lending market, the authors examine a core-periphery model where 10 large organizations are completely connected among themselves, and each of 90 smaller organizations has one connection to a random core organization.
• They identify a clique of 25 completely connected banks (including the very largest ones), and thousands of less connected peripheral regional and local banks.
• 57 Note that in this model the diversification structure is essentially fixed given the structure of ten completely interconnected organizations and the peripheral ones each having one connection; the only randomness comes from the random attachment of each peripheral organization to a single core organization.
• Once the core organizations become sufficiently integrated among themselves, starting around C CC = 0.29, the core organization’s failure begins to cascade to other core organizations, and then wider contagion occurs.
• How far this ultimately spreads is governed by the combination of integration levels.

B. A Model with Segregation among Sectors

• Second, the authors consider a model which admits segregation among different segments of an economy—for instance among different countries, industries, or sectors.
• There are ten different groups of ten nodes each.
• This captures the difference between integration across industries and integration within industries.
• Varying this difference leads to the results captured in Figure 7.
• So at high levels of homophily, lower-degree networks are actually more robust.

C. Power Law Distributions

• The authors also examined networks with more extreme degree distributions, such as a power-law distribution.
• More extreme exponents in the power law actually lead to smaller contagions on average, but also lead to larger contagions conditional on some high-degree organization’s failure.

D. Correlated and Common Assets

• An important concern that emerged from the recent financial crisis is that many organizations may have investments with correlated payoffs, which could potentially Notes:.
• The more interesting part is that the increase occurs abruptly at a particular level of correlation.
• Organizations that are short the common asset might escape a cascade triggered by a shock to that asset.
• The authors close the paper with an illustration of the model with data on the crossholdings of debt among six European countries (France, Germany, Greece, Italy, Portugal, and Spain).
• Such losses may arise for various reasons: discontinuous changes in government policies of how to make use of fiscal streams; government decisions not to honor obligations (at which point it makes sense to do so discontinuously); discontinuities in the fiscal streams themselves (due to strikes, discontinuous changes in foreign investments, bank runs, and so forth).

A. The Data

• Data on the cross-holdings are for the end of December 2011 from the BIS (Bank for International Settlements) Quarterly Review (Table 9B).
• The data looks at the immediate borrower rather than the 58 See Upper (2011) for a nice review of the empirical literature simulating the effects of shocks to financial systems.
• To convert the above matrix into their fractional cross-holdings matrix, C, the authors then estimate the total amount of debt issued by each country.
• The arrows show the way in which decreases in value flow from country to country.

B. Cascades

• The authors examine the best equilibrium values for various levels of θ.
• Greece’s value has already fallen by well more than 10 percent, and so it has hit its failure point for all of the values of θ that the authors look at.
• Table 1 records the results of these simulations.
• Once Portugal fails, then Spain fails due to its poor initial value and its exposure to Portugal.
• The authors reemphasize that the cascades are off the equilibrium path, but that understanding the dependency matrix and the hierarchical structure of potential cascades can improve policy interventions.

VI. Concluding Remarks

• Based on a simple model of cross-holdings among organizations that allows discontinuities in values, the authors have examined cascades in financial networks.
• First, diversification and integration are usefully distinguished as they have different effects on financial contagions.
• The trade-offs can also be related to important realistic aspects of a network, such as its core-periphery and segregation structure.
• As underlying asset prices change, the differences between organizations’ values and their failure thresholds change.
• This completes the proof of the proposition.

Figures

Figure 3. Example Random Networks (Plotted here with undirected edges) for Different Levels of Diversification

Figure 4. How Integration (the Fraction c of a Typical Portfolio Held by Other Organizations) Affects the Percentage of Organizations Failing (Averaged over 1,000 simulations)

Figure 1. With Positive Cross-Holdings the Discontinuities in Values Generated by the Failure Costs Can Result in Multiple Equilibria and Cascades of Failure

Figure 7. Ten Groups of Ten Organizations Each

Table 1—Hierarchies of Cascades in the Best-Case Equilibrium Algorithm, as a Function of the Failure Threshold θ

Figure 6. The Consequences of Failure in the Core-Periphery Model

Figure 8. Interdependencies in Europe

Figure 5. How Integration Affects the Percentage of “First Failures”

Figure 2. How Diversification (the Average Number of Other Organizations that an Organization Cross-Holds) Affects the Percentage of Organizations Failing (Averaged over 1,000 simulations)

Full text

American Economic Review 2014, 104(10): 3115–3153
http://dx.doi.org/10.1257/aer.104.10.3115
3115
Financial Networks and Contagion
†
By M E, B G,  M O. J *
We study cascades of failures in a network of interdependent nan-
cial organizations: how discontinuous changes in asset values
(e.g., defaults and shutdowns) trigger further failures, and how this
depends on network structure. Integration (greater dependence on
counterparties) and diversication (more counterparties per orga-
nization) have different, nonmonotonic effects on the extent of cas-
cades. Diversication connects the network initially, permitting
cascades to travel; but as it increases further, organizations are
better insured against one another’s failures. Integration also faces
trade-offs: increased dependence on other organizations versus less
sensitivity to own investments. Finally, we illustrate the model with
data on European debt cross-holdings. (JEL D85, F15, F34, F36,
F65, G15, G32, G33, G38)
Globalization brings with it increased nancial interdependencies among many
kinds of organizations—governments, central banks, investment banks, rms, etc.—
that hold each other’s shares, debts, and other obligations. Such interdependencies
can lead to cascading defaults and failures, which are often avoided through massive
bailouts of institutions deemed “too big to fail.” Recent examples include the US
government’s interventions in AIG, Fannie Mae, Freddie Mac, and General Motors;
and the European Commission’s interventions in Greece and Spain. Although such
bailouts circumvent the widespread failures that were more prevalent in the nine-
teenth and early twentieth centuries, they emphasize the need to study the risks
created by a network of interdependencies. Understanding these risks is crucial to
designing incentives and regulatory responses which defuse cascades before they
are imminent.
In this paper we develop a general model that produces new insights regard-
ing nancial contagions and cascades of failures among organizations linked
through a network of nancial interdependencies. Organizations’ values depend on
each other—e.g., through cross-holdings of shares, debt, or other liabilities. If an
* Elliott: Division of Humanities and Social Sciences, California Institute of Technology, 1200 E. California
Blvd., Pasadena, CA 91125 (e-mail: melliott@caltech.edu); Golub: Department of Economics, Harvard University,
Littauer Center, 1805 Cambridge St., Cambridge, MA 02138 (e-mail: ben.golub@gmail.com); Jackson: Department
of Economics, Stanford University, 579 Serra Mall, Stanford, CA 94305, the Santa Fe Institute, and CIFAR (e-mail:
jacksonm@stanford.edu). Jackson gratefully acknowledges nancial support from NSF grant SES-0961481 and
grant FA9550-12-01-0411 from AFOSR and DARPA, and ARO MURI award No. W911NF-12-1-0509. All authors
thank Microsoft Research New England Lab for research support. We thank Jean-Cyprien Héam, Scott Page,
Gustavo Peralta, Ployplearn Ravivanpong, Alp Simsek, Alireza Tahbaz-Salehi, and Yves Zenou, as well as three
referees and many seminar participants for helpful comments. The authors declare that they have no relevant or
material nancial interests that relate to the research described in this paper.
†
Go to http://dx.doi.org/10.1257/aer.104.10.3115 to visit the article page for additional materials and author
disclosure statement(s).

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THE AMERICAN ECONOMIC REVIEW
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organization’s value becomes sufciently low, it hits a failure threshold at which
it discontinuously loses further value; this imposes losses on its counterparties,
and these losses then propagate to others, even those who did not interact directly
with the organization initially failing. At each stage, other organizations may hit
failure thresholds and also lose value discontinuously. Relatively small, and even
organization-specic, shocks can be greatly amplied in this way.
1
In our model, organizations hold primitive assets (any factors of production or
other investments) as well as shares in each other.
2
The basic network we start with
describes which organizations directly hold which others. Cross-holdings lead to a
well-known problem of inating book values,
3
and so we begin our analysis by deriv-
ing a formula for a noninated “market value” that any organization delivers to nal
investors outside the system of cross-holdings. This formula shows how each organi-
zation’s market value depends on the values of the primitive assets and on any failure
costs that have hit the economy. We can therefore track how asset values and failure
costs propagate through the network of interdependencies. An implication of failures
being complementary is that cascades occur in “waves” of dependencies. Although in
practice these might occur all at once, it can be useful to distinguish the sequence of
dependencies in order to gure out how they might be avoided. Some initial failures
are enough to cause a second wave of organizations to fail. Once these organizations
fail, a third wave of failures may occur, and so on. A variation on a standard algo-
rithm
4
then allows us to compute the extent of these cascades by using the formula
discussed above to propagate the failure costs at each stage and determine which
organizations fail in the next wave. Policymakers can use this algorithm in conjunc-
tion with the market value formula to run counterfactual scenarios and identify which
organizations might be involved in a cascade under various initial scenarios.
With this methodology in hand, our main results show how the probability of
cascades and their extent depend on two key aspects of cross-holdings: integration
and diversication. Integration refers to the level of exposure of organizations to
each other: how much of an organization is privately held by nal investors, and how
much is cross-held by other organizations. Diversication refers to how spread out
cross-holdings are: is a typical organization held by many others, or by just a few?
Integration and diversication have different, nonmonotonic effects on the extent of
cascades.
If there is no integration, then clearly there cannot be any contagion. As integration
increases, the exposure of organizations to each other increases and so contagions
become possible. Thus, on a basic level, increasing integration leads to increased
exposure, which tends to increase the probability and extent of contagions. The
countervailing effect here is that an organization’s dependence on its own primitive
1
The discontinuities incurred when an organization fails can include the cost of liquidating assets, the (tempo-
rary) misallocation of productive resources, as well as direct legal and administrative costs. Given that efcient
investment or production can involve a variety of synergies and complementarities, any interruption in the ability to
invest or pay for and acquire some factors of production can lead to discontinuously inefcient uses of other factors,
or of investments. See Section IC for more details.
2
We model cross-holdings as direct (linear) claims on values of organizations for simplicity, but the model
extends to all sorts of debt and other contracts as discussed in Section 2 in the online Appendix.
3
See Brioschi, Buzzacchi, and Colombo (1989) and Fedenia, Hodder, and Triantis (1994).
4
This sort of algorithm is the obvious one for nding extreme points of a lattice, and so is standard in a variety
of equilibrium settings. Ours is a variation on one from Eisenberg and Noe (2001).

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Vol. 104 no. 10
assets decreases as it becomes integrated. Thus, although integration can increase
the likelihood of a cascade once an initial failure occurs, it can also decrease the
likelihood of that rst failure.
With regard to diversication, there are also trade-offs, but on different dimen-
sions. Here the overall exposure of organizations is held xed but the number of
organizations cross-held is varied. With low levels of diversication, organizations
can be very sensitive to particular others, but the network of interdependencies is
disconnected and overall cascades are limited in extent. As diversication increases,
a “sweet spot” is hit where organizations have enough of their cross-holdings
concentrated in particular other organizations so that a cascade can occur, and
yet the network of cross-holdings is connected enough for the contagion to be
far-reaching. Finally, as diversication is further increased, organizations’ port-
folios are sufciently diversied so that they become insensitive to any particular
organization’s failure.
Putting these results together, an economy is most susceptible to widespread nan-
cial cascades when two conditions hold. The rst is that integration is intermediate:
each organization holds enough of its own assets that the idiosyncratic devaluation
of those assets can spark a rst failure, and holds enough of other organizations for
failures to propagate. The second condition is that organizations are partly diversi-
ed: the network is connected enough for cascades to spread widely, but nodes don’t
have so many connections that they are well-insured against the failure of any coun-
terparty. Our analysis of these trade-offs includes both analytical results on a class
of networks for which the dynamics of cascades are tractable, as well as simulation
results on other random cross-holding networks.
In the simulations, we examine several important specic network structures. One
is a network with a clique of large “core” organizations surrounded by many smaller
“peripheral” organizations, each of which is linked to a core organization. This emu-
lates the network of interbank loans. There we see a further nonmonotonicity in
integration: if core organizations have low levels of integration, then the failure of
some peripheral organization is contained, with only one core organization failing;
if core organizations have middle levels of integration, then widespread contagions
occur; if core organizations are highly integrated, then they become less exposed to
any particular peripheral organization and more resistant to peripheral failures. A
second model is one with concentrations of cross-holdings within sectors or other
groups. As cross-holdings become more sector-specic, particular sectors become
more susceptible to cascades, but widespread cascades become less likely. The level
of segregation at which this change happens depends on diversication. With lower
diversication, cascades disappear at lower rates of segregation—it takes less segre-
gation to fragment the network and prevent cascades.
We also consider what a regulator or government might do to mitigate the possi-
bility of cascades of failures. Preventing a rst failure prevents the potential ensuing
cascade of failures, and it might be hoped that a clever reallocation of cross-holdings
could achieve this. Unfortunately, we show that any fair exchange of cross-holdings
or assets involving the organization most at risk of failing makes that organization
more likely to fail at some asset prices close to the current asset prices. Making the
system unambiguously less susceptible to a rst failure necessitates bailing out the
organization most at risk of failing.

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Finally, we illustrate the model in the context of cross-holdings of European debt.
While there is a growing literature on networks of interdependencies in nancial
markets,
5
our methodology and results are different from any that we are aware
of, especially the results on nonmonotonicities in cascades due to integration and
diversication.
An independent study by Acemoglu, Ozdaglar, and Tahbaz-Salehi (2012)—as
well as related earlier studies of Gouriéroux, Héam, and Monfort (2012) and Gai
and Kapadia (2010)—are the closest to ours.
6
They each examine how shocks prop-
agate through a network based on debt holdings or interbank lending, where shocks
lead an organization to pay only a portion of its debts. They are also interested in
how shocks propagate as a function of network architecture. However, beyond the
basic motivation and focus on the network propagation of shocks, the studies are
quite different and complementary. The main results of Acemoglu, Ozdaglar, and
Tahbaz-Salehi (2012) characterize the best and worst networks from a social plan-
ner’s perspective. For moderate shocks a perfectly diversied pattern of holdings
is optimal, while for very large shocks perfectly diversied holdings become the
worst possible.
7
Our focus is on the complementary question of what happens for
intermediate shocks and for a variety of networks. To this end, we consider a class of
random networks and ask how the consequences of a given moderate shock depend
on diversication and integration. The results highlight that intermediate levels of
diversication and integration can be the most problematic.
Gai and Kapadia (2010) made two observations. First: rare, large shocks may have
extreme consequences when they occur—a point elaborated upon in the subsequent
literature discussed above. Second, a shock of a given magnitude may have very
different consequences depending on where in the network it hits and on the aver-
age connectivity of the network. Gai and Kapadia develop these points in a standard
model of epidemics in which the network is characterized by its degree distribution.
An innovation of our model is to go beyond the degree distribution of a network and
calculate equilibrium (xed-point) values and interdependencies for organizations.
Doing so allows us to distinguish an important dimension of nancial networks:
integration, which can be varied independently of diversication. Building on that,
we show how diversication and integration each affect the ingredients of nancial
cascades—and the nal outcomes—in different and nonmonotonic ways. In doing
so, we recover, as a special case, Gai and Kapadia’s observation that cascades can
5
For example, see Rochet and Tirole (1996); Kiyotaki and Moore (1997); Allen and Gale (2000); Eisenberg
and Noe (2001); Upper and Worms (2004); Cifuentes, Ferrucci, and Shin (2005); Leitner (2005); Allen and Babus
(2009); Lorenz, Battiston, and Schweitzer (2009); Gai and Kapadia (2010); Wagner (2010); Billio et al. (2012);
Demange (2012); Diebold and Yilmaz (2011); Dette, Pauls, and Rockmore (2011); Gai, Haldane, and Kapadia
(2011); Greenwood, Landier, and Thesmar (2012); Ibragimov, Jaffee, and Walden (2011); Upper (2011); Acemoglu
et al. (2012); Allen, Babus, and Carletti (2012); Cohen-Cole, Patacchini, and Zenou (2012); Gouriéroux, Héam, and
Monfort (2012); Alvarez and Barlevy (2013); Glasserman and Young (2013); and Gofman (2013).
6
Cabrales, Gottardi, and Vega-Redondo (2013) study the trade-off between the risk-sharing enabled by greater
interconnection and the greater exposure to cascades resulting from larger components in the nancial network.
Their focus is also on some benchmark networks (minimally connected and complete ones) and they examine
which ones are best for different distributions of shocks. Again, our work is complementary not only in terms of
distinguishing diversication and integration but also analyzing comparative statics for intermediate network struc-
tures and nding nonmonotonicities there.
7
Shaffer (1994) also identies a trade-off between risk sharing and systemic failures. While diversied portfo-
lios reduce risk, they also result in organizations holding similar portfolios and a system susceptible to simultaneous
failures. See also Ibragimov, Jaffee, and Walden (2011) and Allen, Babus, and Carletti (2012).

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Vol. 104 no. 10
be nonmonotonic in connectivity.
8
But we also gain key new results on when and
how the “danger zone” of intermediate diversication can be blunted by changing
the level of integration in the system. Finally, we study how the integration of a
nancial network interacts with a core-periphery structure and with segregation, and
other correlation structures.
I. The Model and Determining Organizations’ Values with Cross-Holdings
A. Primitive Assets, Organizations, and Cross-Holdings
There are n organizations (e.g., countries, banks, or rms) making up a set
N = {1, … , n}.
The values of organizations are ultimately based on the values of primitive assets
or factors of production—from now on simply assets—M = {1, … , m}. For con-
creteness, a primitive asset may be thought of as a project that generates a net ow
of cash over time.
9
The present value (or market price) of asset k is denoted p
k
. Let
D
ik
≥ 0 be the share of the value of asset k held by (i.e., owing directly into) orga-
nization i and let D denote the matrix whose (i, k)th entry is equal to D
ik
. (Analogous
notation is used for all matrices.)
An organization can also hold shares of other organizations. For any i, j ∈ N the
number C
ij
≥ 0 is the fraction of organization j owned by organization i, where
C
ii
= 0 for each i.
10
The matrix C can be thought of as a network in which there is a
directed link from i to j if i owns a positive share of j, so that C
ij
> 0.
11
Paths in this
network are called ownership paths. We also sometimes work with a graphical rep-
resentation of C where directed links point in the opposite direction, the direction in
which value (and loss of value) ows. We call the paths in that network cascade paths.
After all these cross-holding shares are accounted for, there remains a
share

C
ii
:= 1 −
∑
j∈N
C
ji
of organization i not owned by any organization in the
system—a share assumed to be positive.
12
This is the part that is owned by outside
shareholders of i, external to the system of cross-holdings. The off-diagonal entries
of the matrix

C are dened to be 0.
Cross-holdings are modeled as linear dependencies in this paper, and we now
briey discuss the interpretation of this. We view the functional form as an approxi-
mation of debt contracts around and below organizations’ failure thresholds—the
8
In different settings, Cifuentes, Ferrucci, and Shin (2005) and Gofman (2013) also nd that cascades can be
nonmonotonic in connectivity.
9
The primitive assets could be more general factors: prices of inputs, values of outputs, the quality of organi-
zational know-how, investments in human capital, etc. To keep the exposition simple, we model these as abstract
investments and assume that net positions are nonnegative in all assets.
10
It is possible to instead allow C
ii
> 0, which leads to some straightforward adjustments in the derivations
that follow; but one needs to be careful in interpreting what it means for an organization to have cross-holdings in
itself—which effectively translates into a form of private ownership.
11
Some denitions: a path from i
1
to i
ℓ
in a matrix M is a sequence of distinct nodes i
1
, i
1
, … , i
ℓ
such that
M
i
r+1
i
r
> 0 for each r ∈ {1, 2, … , ℓ − 1}. A cycle is a sequence of (not necessarily distinct) nodes i
1
, i
1
, … , i
ℓ
such that M
i
r+1
i
r
> 0 for each r ∈ {1, 2, … , ℓ − 1} and M
i
1
i
r
> 0.
12
This assumption ensures that organizations’ market values (discussed below) are well dened. It is slightly
stronger than necessary. It would sufce to assume that, for every organization i, there is some j such that

C
jj
> 0
and there is an ownership path from j to i. An organization with

C
ii
= 0 would essentially be a holding company,
and the important aspect is to have an economy where there are at least some organizations that are not holding
companies and some outside shareholders that no organizations have claims on.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Financial networks and contagion" ?

The authors study cascades of failures in a network of interdependent financial organizations: how discontinuous changes in asset values ( e. g., defaults and shutdowns ) trigger further failures, and how this depends on network structure. Diversification connects the network initially, permitting cascades to travel ; but as it increases further, organizations are better insured against one another ’ s failures. 

Q2. What future works have the authors mentioned in the paper "Financial networks and contagion" ?

This presents interesting challenges for future research. For example, counterfactual scenarios can be run using the algorithm. To determine the marginal effect of saving a set of organizations, the failure costs of those organizations can be set to zero and the algorithm run with and without their failure costs. This set of organizations can be compared to the set of organizations that fail under other interventions, including doing nothing. 

Q3. How can the authors determine the marginal effect of saving a set of organizations?

To determine the marginal effect of saving a set of organizations, the failure costs of those organizations can be set to zero and the algorithm run with and without their failure costs. 

Q4. What is the main limiting force of the network?

Then II dominates: once the network is connected, the main limiting force is the extent to which the failure of one organization sparks failures in others, which is decreasing with diversification. 

Q5. Why do the authors see more organizations fail in a cascade?

Since network components are larger, the failure of any one organization infects more other organizations, and more organizations are drawn into the cascade. 

Q6. What is the definition of a solution for organization values in equation (5)?

A solution for organization values in equation (5) is an equilibrium set of values, and encapsulates the network of cross-holdings in a clean and powerful form, building on the dependency matrix A. 

Q7. What is the cumulative failure cost to the economy?

for example, the first K organizations end up failing in the cascade, the cumulative failure costs to the economy are β 1 + ⋯ + β K , which can greatly exceed the drop in asset value that precipitated the cascade. 

Q8. What is the value left to the owner-operators/shareholders?

If the value left to the owner-operators/shareholders is sufficiently low (below some outside option value of their time or effort), they may choose to cease operations. 

Q9. What is the meaning of the term ownership paths?

The matrix C can be thought of as a network in which there is a directed link from i to j if i owns a positive share of j, so that C ij > 0.11 Paths in this network are called ownership paths. 

Q10. How does it illustrate the simplicity of the approach?

it illustrates the simplicity of the approach and makes it clear that much more accurate simulations could be run with access to precise cross-holdings data, default costs, and thresholds. 

Q11. How do you think failure costs will be reduced?

It might be hoped that organizations will reduce the scope for cascades of failures by minimizing their failure costs and reducing the threshold values at which they fail.