A model of financial contagion with variable asset returns
may be replaced with a simple threshold model of cascades
Teruyoshi Kobayashi1∗
1Graduate School of Economics, Kobe University,
2-1 Rokkodai, Nada, Kobe 657-8501, Japan.
E-mail: kobayashi@econ.kobe-u.ac.jp
Tel, fax: +81-78-803-6692

Abstract
I show the equivalence between a model of financial contagion and the widely-usedthreshold model of global cascades proposed by Watts (2002). The model financial
network comprises banks that hold risky external assets as well as interbank assets.
It turns out that there is no need to construct the balance sheets of banks if theshadow threshold of default is appropriately defined in accordance with the stochastic
fluctuations in external assets.
Keywords: financial network, cascades, financial contagion, systemic risk.
JEL codes: G01, G18.

1Introduction

Bank default is contagious. A failure of one bank can be spread through the financial
network, generating default cascades. Over the past few years, many researchers in various
fields of natural and social sciences, such as physicists, ecologists and economists, havebeen tackling the question of how to prevent financial contagion (e.g., Nier et al., 2007,Soramaki et al., 2007, Gai and Kapadia, 2010, Gai et al., 2011, Lenzu and Tedeschi, 2012,
Kobayashi, 2013, Kobayashi and Hasui, 2013).
However, there is no wide agreement among researchers about how to construct a modelof financial contagion. Different research groups use different models, which would makeit difficult to establish a consensus about policy implications.1
In this letter, I show the equivalence between a model of financial contagion and thewidely-used threshold model of global cascades proposed by Watts (2002).2 Basically, thefinancial network models require researchers to construct the banks’ balance sheets. Then,the influence of a bank failure is examined by sequentially undermining the interbank assetsof the lenders. Second-round defaults would occur if the fraction of defaulted borrowers
among total borrowers exceeds a certain threshold.
These mechanisms are closely related to those of the Watts model of cascades. I findthat there is no need to construct the balance sheets of banks as long as the threshold ofdefault is appropriately defined in accordance with the fluctuations in asset values.

2The models

2.1

A model of financial contagion

The model of financial contagion used in this paper is an extended version of the model
used by Gai and Kapadia (2010). The only essential difference is that I take into accountstochastic fluctuations in the value of external assets. A typical bank’s balance sheet isillustrated in Figure 1.
1
2

See Lorenz et al. (2009) and Upper (2011) for a survey of the literature.
See, for example, Dodds and Watts (2004), Gleeson and Cahalane (2007), Watts and Dodds (2007) andGleeson (2013).

1safe assets:!

deposits: "!
interbank assets:'! ( )+ *+! #$+interbank liabilities:#$!

external risk assets:&!

net worth: %!

Fig. 1. A typical bank’s balance sheet.

There are N banks in the financial market. Bank i holds risky external assets, ai ,interbank assets, li , and risk-less assets, bi . In the liability side of the balance sheet, there
are deposits, di , interbank liabilities, p̄i , and net worth, wi . The balance-sheet condition
implies that ai + li + bi = di + p̄i + wi .
Banks are connected to each other by lending and borrowing. The existence of alending-borrowing relationship is expressed as a link or an edge. In network theory, thenumber of outgoing links is called out-degree, while the number of incoming links is calledin-degree. The direction of links represents the flow of funds at the time of initial lending.
The amount of bank i’s borrowings from bank j is expressed as πij p̄i , where πij denotesPthe relative weight of bank i’s borrowings from j, and thereby j6=i πij = 1, i = 1, . . . , N .
PoutThe amount of bank i’s total interbank assets is given by li = N
j6=i πji p̄j = ki lu , where
kiout , or out-degree, is the number of banks to which the bank lends, and lu is the unit sizeof interbank lending.
Bank i will default ifp̄i >Xπji pj + ãi + bi − di ,

(1)

j6=i

where ãi and pj stand for the ex-post values of external assets and interbank liabilities,respectively. It should be pointed out that deposits, di , are reserved because deposits are2

senior to interbank assets.
Provided that there is no loss in the interbank assets, banks will default with probability
δ due to a loss of external assets. δ is called the probability of fundamental defaults, whichis assumed to be common across banks. If bank j failed, then bank j’s creditors lose all of
their credits extended to bank j. Some of these creditors may fail due to the loss of theirinterbank assets. Accordingly, the creditors of the creditors of bank j may fail as well,because they lose their credits extended to the failed banks.
Given the unit size of interbank lending, lu , the sizes of interbank assets and liabilities
are given by the structure of the interbank network. To ensure that the probability offundamental defaults is the same across banks, the size of external assets relative to thenet worth, θaw , is fixed. For those banks that have a positive amount of interbank assets,
the ratio of total interbank assets to net worth, θlw , is also fixed. If bank i has so many
incoming links that its liability side would be bigger than the sum of interbank assets andexternal assets, then the risk-less asset, bi , is imposed to adjust the asset side. Otherwise,
deposits, di , is imposed to adjust the liability side.

2.2

The Watts model of global cascades

Here, let us summarize the Watts’ (2002) threshold model of cascades. In this model, eachnode in the network can take one of the two states, “flipped” or “not flipped”. The network
is undirected.
Let k and m denote the degree and the number of flipped neighbors, respectively. The
algorithm of cascading behavior in the Watts model is thatnode i flips if mi > φki ,node i does not flip otherwise,
where φ ∈ [0, 1] is the threshold of flipping. This means that the threshold number offlipped neighbor nodes above which node i will flip is bφki c, where bxc denotes the floor
function that returns the maximum integer smaller than x.

33Equivalence between a financial network model and a threshold model

Here I show the equivalence between the financial network model and a modified versionof the threshold model. Let us consider the financial network model. Recall that the valueof external assets, ai , may vary stochastically from bank to bank. Let ∆ai ≡ ãi − ai be thereturn of external assets. If ∆ai = 0, bank i will fail if the fraction of defaulted neighbors
exceeds bwi /li c. Notice that there is no possibility of contagious default if wi > li , as longas there is no loss in the external assets.
If ∆ai 6= 0, on the other hand, the threshold of the fraction of defaulted neighbors
depends on the realization of asset returns. If a bank gains positive (negative) returns fromthe external assets, then the bank becomes more resilient against (susceptible to) the failureof neighbor banks. Let φ̃i be the “shadow” threshold of default for bank i ∈ {i | kiout > 0}.
It follows thatφ̃i ===

wi + ∆aili
kiout lu /θlw + ∆ai
kiout lu
1∆ai+ out
θlw
ki lu

(2)

for i ∈ {i | kiout > 0}. Notice that the values of θlw , lu and kiout are already given by theresearcher. It turns out that if asset returns, ∆ai , follow a distribution of mean zero and1variance σ 2 , then the shadow threshold φ̃i follows a distribution of mean θlwand variance2σ. More generally, the p.d.f. of φ̃i , defined as fi (·), is given bykout luifi (r) = kiout lu · gi (r · kiout lu − ki lu /θlw ), i ∈ {i | kiout > 0}.

(3)where gi (·) is the p.d.f. of ∆ai .
Suppose, for instance, that asset returns follow a normal distribution with mean zero.
Let z̃ ≡ F −1 (δ), where F −1 is the inverse CDF of the standard normal distribution. The
standard deviation of asset returns, σi , such that the probability of fundamental default

4

(a)

(b)

1

1

financial contagion modelthreshold model

0.9

0.9

0.8

0.8

frequency of crisis

average fraction of failed banks

financial contagion modelthreshold model

0.7

0.6

0.5

0.4

0.3

0.7

0.6

0.5

0.4

0.3

0.2

0.2

0.1

0.1

001

2

3

4

567890010average degree

12345678910

average degree

Fig. 2. Simulation results under a model of financial contagion and a threshold model.
Note: “Financial crisis” is defined as a situation in which at least 5% of banks go bankrupt.

becomes δ is given asσi ==−wi, ∀ i,z̃
−kiout lu, for i ∈ {i | kiout > 0}.
θlw z̃

(4)

It follows that the shadow threshold follows a normal distribution with mean 1/θlw andthe standard deviation 1/(θlw z̃).
In this way, the model of financial contagion shown above can be expressed as a simple
threshold model by introducing the concept of shadow threshold. Intuitively, the shadow
threshold will become smaller as the returns of external assets take a lower value, meaningthat the bank becomes more susceptible to default contagion. Those banks that have anegative value of φi will fail at the beginning, including the banks with no interbank assets.3
This corresponds to the case of fundamental defaults in the model of financial contagion.
Figure 2 shows the simulation results under both models. The parameter values are as
31xNotice that Prob(φ̃i < 0) = Prob( θlw− θlw< 0) = F (x < z̃) = δ, where x is a random variable fromz̃

the standard normal distribution.

5

follows: N = 1000, θaw = 7, θlw = 3, lu = 1 and δ = .01. These parameters suggest thatthe ratio of capital to total risky assets is .1 for those banks that have a positive amountof interbank assets and 1/7 ≈ .143 for the other banks. Given the average degree, ErdosRényi (directed) random networks are generated 20 times, and asset returns, or shadow
thresholds, are generated 1000 times for each network structure.

4Conclusion and discussion

In this letter, I showed that the introduction of shadow thresholds allows us to treat amodel of financial contagion as a simple threshold model of cascades. There is no need toconstruct the balance sheets of banks. However, it is not clear whether this method canbe applied to the other versions of the financial network models. For example, θlw and lu
are treated as constants in this model, but they generally vary across banks in practice.
Another methodology will be needed in such more general cases.

Acknowledgments
I thank Kohei Hasui for his excellent research assistance. Financial support from KAKENHI 25780203 and 24243044 is gratefully acknowledged.

References
Dodds, P.S., Watts, D.J., 2004. Universal behavior in a generalized model of contagion.
Phys. Rev. Lett. 92, 218701.
Gai, P., Kapadia, S., 2010. Contagion in financial networks. Proc. Roy. Soc. A. 466,2401-2423.
Gai, P., Haldane, A., Kapadia, S., 2011. Complexity, concentration and contagion. J.
Monetary. Econ. 58(5), 453-470.
Gleeson, J.P., 2013. Binary-state dynamics on complex networks: pair approximation andbeyond. Phys. Rev. X. 3, 021004.
6

Gleeson, J.P., Cahalane, D.J., 2007. Seed size strongly affects cascades on random networks. Phys. Rev. E. 75, 056103.
Kobayashi, T., 2013. Network versus portfolio structure in financial systems. Eur. Phys.
J. B. 86:434.
Kobayashi, T., Hasui, K., 2013. Efficient immunization strategies to prevent financialcontagion. arXiv: 1308.0652.
Lenzu, S., Tedeschi, G., 2012. Systemic risk on different interbank network topologies.
Physica A. 391(18), 4331-4341.
Lorenz, J., Battiston, S., Shweitzer, F., 2009. Systemic risk in a unifying framework for
cascading processes on networks. Eur. Phys. J. B. 71(4), 441-460.
Nier, E., Yang, J., Yorulmazer, T., Alentorn, A., 2007. Network models and financial
stability. J. Econ. Dyn. Cont. 31(6), 2033-2060.
Soramäki, K., Bech, M.L., Arnold, J., Glass, R.J., Beyeler, W.E., 2007. The topology ofinterbank payment flows. Physica A. 379(1), 317-333.
Upper, C., 2011. Simulation methods to assess the danger of contagion in interbank
markets. J. Financ. Stability. 7(3), 111-125.
Watts, D.J., 2002. A simple model of global cascades on random networks. P. Natl. Acad.
Sci. USA. 99(9), 5766-5771.
Watts, D.J., Dodds, P.S., 2007. Influentials, networks, and public opinion formation. J.
Consumer. Res. 34(4), 441-458.

7