Network versus portfolio structure in financial systems
Teruyoshi Kobayashi∗
April 19, 2013

Abstract
The question of how to stabilize financial systems has attracted considerable attention since the global financial crisis of 2007-2009. Recently, Beal et al. (“Individual
versus systemic risk and the regulator’s dilemma”, Proc Natl Acad Sci USA 108: 1264712652, 2011) demonstrated that higher portfolio diversity among banks would reduce
systemic risk by decreasing the risk of simultaneous defaults at the expense of a higher
likelihood of individual defaults. In practice, however, a bank default has an externality in that it undermines other banks’ balance sheets. This paper explores how each ofthese different sources of risk, simultaneity risk and externality, contributes to systemicrisk. The results show that the allocation of external assets that minimizes systemicrisk varies with the topology of the financial network as long as asset returns have negative correlations. In the model, a well-known centrality measure, PageRank, reflects anappropriately defined “infectiveness” of a bank. An important result is that the mostinfective bank need not always be the safest bank. Under certain circumstances, themost infective node should act as a firewall to prevent large collective defaults. The
introduction of a counteractive portfolio structure will significantly reduce systemic
risk.

∗

Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan.

E-mail: kobayashi@econ.kobe-u.ac.jp. Tel, fax: +81-78-803-6692.

1

Introduction

In the past 5 years, essential differences between micro- and macro-prudential policies havebeen emphasized by many academic researchers, central bankers and regulators (Beale etal., [1], Upper, [2], May and Arinaminpathy, [3]). Since the failures of Bear Stearns andLehman Brothers in 2008, there seems to have been wide agreement that micro-prudential
policies do not necessarily add up to a macro-prudential policy. An ongoing challenge forregulators is to find the best macro-prudential policy that stabilizes the whole financial
system, minimizing the expense of individual bank failures.
One of the main objectives of recent studies on systemic risk has been to gain a theoretical understanding of how the topology of the financial network affects the likelihoodof a financial crisis. By employing basic concepts developed in network theory, such as
percolation, assortativity and clustering, those studies attempt to extract the necessary
conditions for a stable financial system by conducting simulations under various patternsof network topology (Lenzu and Tedeschi, [4], Gai and Kapadia, [5], Gai et al., [6], Nier etal., [7], Kyriakopoulos et al., [8]). The basic idea behind these studies is that a financial
network must be robust enough to withstand default cascades. That is, the extent of an
externality to which a bank failure would give must be minimized.
While those studies focus mainly on the topological aspect of financial networks, theportfolio structure of individual banks is another key to understanding systemic stability.1
Beale et al. [1] recently pointed out that higher portfolio diversity among banks would
reduce systemic risk by decreasing the likelihood of simultaneous defaults, which I call“simultaneity risk” throughout this paper. This puts into effect the “regulator’s dilemma”
or the “diversity-diversification trade-off”, because minimizing the risk of individual failure(i.e., portfolio diversification) does not add up to minimizing systemic risk. Ibragimov etal. [9] and Wagner [10], [11] made similar arguments.
The main purpose of this paper is to show how the systemically optimal allocation of
external risk assets would vary with network structure. To do so, both the externality andthe simultaneity risk need to be taken into account. In a financial market with simultaneity
risk only, where all banks are completely isolated from each other, each bank is necessarily
1

In this paper, any type of financial institution is called a “bank’’ for brevity.

1

anonymous. From a macro-prudential viewpoint, it does not matter which banks should
diversify their portfolios. The only concern is how many banks should do so. However,once a network structure is introduced, the interconnected banks are not generally anonymous because the possibility of generating default cascades could differ across banks. The
systemic importance of each bank can vary in accordance with its topological location in
the network.2 The problem of how many banks should be diversified is replaced with the
problem of which banks should be diversified.
In the model of this paper, an appropriately defined “infectiveness” of an individualbank is closely correlated with a well-known centrality measure, PageRank. Other centrality measures, such as the degree centrality, the eigenvector centrality, and the betweenness
centrality, do not provide correct information about the infectivity of a bank. It is also
shown that the topological fragility of a whole network can be well captured either by
network entropy or by the Herfindahl-Hirschman Index (HHI). On average, a more concentrated financial network is topologically more fragile than a decentralized network.
An important result of the analysis is that the most infective bank need not always
be the safest bank. This might be counterintuitive because, by definition, the extent of
externality is the strongest for the most infective, and at the same time the biggest, bank.
A basic reason for this result is that a single default of the biggest bank could be lesscontributive to systemic risk than a simultaneous failure of multiple banks. If collective
defaults are more costly than a single default of the biggest bank, then the biggest bankshould act as a firewall to prevent large default cascades. More specifically, the biggest
bank should hold external assets that are negatively correlated with the assets held byother banks. It is shown that the introduction of such a counteractive portfolio structure
will significantly reduce systemic risk.
This paper also reveals that any topological characteristics of the financial network are
not necessarily a correct measure of systemic risk. If each bank has independent external
assets, then there is a negative (positive) relationship between network entropy (HHI) andsystemic risk. However, this correlation disappears once the asset allocation is optimized.
Assigning appropriate assets to appropriate banks will eliminate the differences in systemic2

For classical studies on the information content of network, see Rashevsky [12] and Trucco [13].

2risk stemming from the topological dissimilarity and thereby reduce the level of systemic
risk.

2

Simultaneity risk

Systemic risk in the financial market consists of at least two independent sorts of risks:simultaneity risk and externality. Simultaneous defaults could occur regardless of whether
or not banks have common assets or borrow from each other. In this sense, simultaneity risk
always exists if there are multiple banks in the financial market. If banks are interconnected
through an interbank market, then there is a possibility of default cascades, which amplify
the impact of the initial bank failure. It can be said that a failure of an interconnectedbank has an externality in that it necessarily increases the other banks’ default probability.
In this section, I focus on simultaneity risk by considering a situation in which eachbank is fully isolated and there is no interbank trading. Beale et al. [1] show that portfolio
diversity among banks can be desirable for the financial system as a whole because it
reduces the likelihood of simultaneous defaults. I revisit their result in the following.
Following Beale et al. [1], I assume that social costs, C, depend on the number of bankdefaults, n: C(n) = ns , s ∈ [1, ∞). The expected cost is thus given as
E[C] =NX

q(n)ns ,n=1

where N is the number of banks and q(·) is the probability function for the number ofdefault banks. The case of s > 1 means that simultaneous defaults of multiple banks will
be more costly than a sequence of single bank failures. A rationale for the convexity of thecost function is that the cost in the real economic sector will increase in superlinearly as
the number of bank defaults increases. For example, the extent of liquidity hoarding would
become greater as the number of bank defaults increases, making liquidity shortages forfirms and households more severe.
For simplicity, suppose that there are only two kinds of external assets: bank-specificassets and fully diversified assets. In the case of full diversity, each bank has a bankspecific asset that is independent of any other bank-specific assets. On the other hand,3full diversification requires every bank to hold a common fully diversified asset. As abenchmark case, it is instructive to consider a situation in which each asset return follows
an i.i.d. Student t-distribution with degree of freedom v. At one extreme, there is nobenefit of portfolio diversification if v = 1. Diversity is thus always desirable in this case.
For v > 1, we need to compare the cost and the benefit of diversity with those under
diversification. Let K denote the variety of external assets. If we restrict our attention to
the case of N = K, in which case full diversity is made possible by the minimum numberof assets, then the expected costs under full diversity are given as
E[C]diversity =NXn
N −n sn ,N Cn p (1 − p)

(1)

n=0

where the threshold of default is determined such that default occurs with probability p if
the bank has a single asset. Notice that the variance of asset returns does not matter inthe case of full diversity because each bank has only one asset. Since eq.(1) is equivalent
to the s-th moment of binomial distribution B(N, p), this converges to the s-th moment of
normal distribution N(N p, N p(1 − p)) as N goes to infinity.
At the other extreme of v = ∞, or if asset returns follow a normal distribution, diversification is more likely to be a good option. If asset returns obey i.i.d. normal, N(0, σ 2 ),the expected costs under full diversification are given by
E[C]diversif ication∞

=

N
X

∞
HN
(n)ns

n=0
∞
= HN
(N )N s= FN (x)N s ,(2)∞ (·) is the probability function for the number of defaults under full diversification.
where HN

The second equality follows from the fact that all banks share an identical portfolio. FN (·)
is the CDF of N(0, σ 2 /N ), and x is the threshold value of the diversified asset below which
the banks go bankrupt. It follows that

√


Z x
N
y2
FN (x)N = N · √dy.
exp − 22σ /N
σ 2π −∞ss(3)

No matter how big s is, the presence of N in the exponential term suggests that the
expected costs will converge to zero as N goes to infinity. However, when N is sufficiently
small, an increase in N will lead to higher expected costs.
4

Figure 1 illustrates how the expected costs change with the size of N (= K). As noted
above, the benefit of diversification increases as the degree of freedom, v, goes up. According to the figure, if v = 3, then diversity is more desirable than diversification as long as
N < 30. If v = ∞, on the other hand, then full diversification will be the optimal choice
for N > 4. This reconfirms the point noted by Ibragimov et al. [9] that a fatter-tailed
distribution makes diversification more dangerous and thus diversity more attractive. The
extent of simultaneity risk depends heavily on the thickness of the tail of distribution.
Figure 2 depicts expected costs as a function of the number of diversified banks. It is clearthat the optimal fraction of diversified banks increases with N (= K).

3

Externality

3.1

Basic framework

Thus far, it has been assumed that a financial market consists of isolated banks only. In thissection, I introduce interbank networks and see how the previous results will be altered.
Introducing a network structure not only correlates the states of individual banks, butalso differentiates each bank’s systemic importance. In the presence of interbank trading,a default of one bank could generate further defaults. A default cascade, or contagion,
is more likely to occur when a more “infective bank” fails. Therefore, regulators shouldinternalize the potential externality effects, the extent of which could differ from bank tobank.
Let us describe a model of the interconnected financial market here. A typical bank’sbalance sheet is illustrated by Figure 3. It is assumed that the unit volume of interbank
lending is fixed for all banks and that the composition of the balance sheet is common to allbanks, which means that more interconnected banks are bigger in size. This commonalityof balance sheet structure ensures that the probability of “fundamental default” is prima
facie identical across banks regardless of bank size. By fundamental default, I mean the
failure of a bank caused purely by the loss of external assets. In this model, the differencesin the topological location in the network are reflected solely by the portfolio of interbank
assets.

5

The asset side of bank i’s balance sheet consists of risky external assets, ai , interbank
assets, li , and a riskless asset, bi . At the maximum, each bank may hold K kinds of risky
PN
Pexternal assets and N − 1 interbank assets, where ai = K
j6=i li,j . The
k=1 ai,k and li =liability side of bank i’s balance sheet consists of interbank liability, p̄i , deposits, di , and net
worth, wi . The balance sheet condition implies that ai + bi + li = p̄i + di + wi , i = 1, . . . , N .
The size of the balance sheet is determined such that the relative proportions of ai , bi , andli are common to all interconnected banks, given the unit size of interbank assets. The
portfolio of external risk assets, (ai,1 , . . . , ai,K ), can vary from bank to bank.
The amount of bank i’s borrowings from bank j is expressed as πi,j p̄i , where πi,j denotesPthe relative weight of bank i’s borrowings from j, and thereby j6=i πi,j = 1, i = 1, . . . , N .
Let the N -by-N matrix Π have πi,j as its (i, j)-th element. The diagonal elements are all
zero because no banks lend to themselves. The amount of bank i’s total interbank assets,li , can be given as
li =

N
X

li,j =

j6=i

NX

πj,i p̄j ,j6=i

In vector form,l = Π0 p̄.
Bank i tries to pay back the full amount of interbank liability p̄i after the values ofexternal assets are realized, but the funds available at that time are expressed asX

πj,i pj + ãi + bi − di ,

(4)

j6=i

where ã denotes the ex-post value of external assets. It should be pointed out that deposits,di , are reserved because deposits are senior to interbank assets. I also impose three natural
conditions: (a) limited liability; (b) the priority of debt claims; and (c) the proportionalityof the amount of repayment after default (c.f., Eisenberg and Noe [14]). The actual amountof repayment, pj , is not necessarily equal to its face value, p̄j , since the sum of bank j’s
loss of interbank and external assets might be greater than its net worth, wj .
More precisely, the market-clearing vector, p∗ , is expressed as a fixed point of the map,
Φ : [0, p̄] → [0, p̄] defined as
Φ(p) = p̄ ∧ (Π0 p + ã + b − d),6(5)Table 1. Baseline parameters

cost parameter

capital/asset

external asset/capital

I.B. asset/capital

I.B. liability/capitalswi
ai +bi +li

ai /wili /wip¯i /wi

4

.1

5

4

4

where ∧ denotes the meet operator, such that x ∧ y = (min(x1 , y1 ), . . . , min(xn , yn )).
Under the reasonable conditions (a), (b), and (c), Eisenberg and Noe [14] prove that afixed point of (5) exists and is unique if the financial network is strongly connected andPi (ãi + bi − di ) > 0. They also show that the unique fixed point is the solution to thefollowing linear program:max f (p) s.t. p ≤ Π0 p + ã + b − dp∈[0,p̄]for any strictly increasing function f (·) : RN → R.
The simulation procedure is as follows. First, the topology of the network structure
is given. Given the unit size of interbank assets, the balance sheet size and thereby thevolume of each component are automatically determined. In order to apply the EisenbergNoe theorem, it is assumed that any two banks that are connected by an edge lend to eachother, i.e., edges are bidirectional. This is the simplest assumption that ensures that any
interconnected networks are also strongly interconnected. Second, random returns of K
external risk assets are drawn and fundamental defaults are observed. Third, given thefunds available for the repayment of interbank loans, ã + b − d, the unique market-clearing
vector, p∗ , is calculated using the Eisenberg-Noe algorithm. Because a fundamental default
may cause contagion, the number of default banks would increase at this stage, dependingon the structure of the network. Such additional defaults are called contagious defaults. It
P
is assumed that if i (ãi + bi − di ) ≤ 0, then all banks go bankrupt.
The baseline parameters are summarized in Table 1. The return of each external asset
is assumed to follow an i.i.d. normal distribution with mean zero. The variance of each
asset return is determined such that the probability of fundamental default is equal to p if
the bank holds a single external asset.

7

3.2

Independent external assets

Suppose that N = 5 and K = 3. Figure 4 illustrates how the expected costs depend on
the distance between banks, D, and the distance from the optimal portfolio, G, which aredefined as
D ≡

XXX
1
|ai,k − aj,k | ,2N (N − 1)

(6)

1 X X
(ai,k − 1/K) .
N

(7)

i

G ≡

k

j

k

i

D = G = 0 corresponds to the case of full diversification. Beale et al. [1] demonstratethat when s > 1, the social costs tend to be reduced by increasing D. This is because
the greater the distance between banks, the lower the probability of simultaneous defaults.
This is reconfirmed by the simulation without an interbank network (Figure 4a). However,it turns out that this outcome does not hold true once the possibility of contagion istaken into account (Figures 4b, 4c, 4d). The introduction of externality moves the optimal
combination of (D, G) toward the origin. It is interesting to note that the optimal value ofD reaches zero as all banks become interconnected. I found that this phenomenon remains
unchanged even when the relative amount of interbank assets, li /wi , is reduced to 1. A
simple rationale is that as banks become interconnected, the financial system takes theform of one “big bank’’. Thus, full diversification becomes desirable because it attains theminimum probability of default for the “big bank” by analogy with a single individual bank.
This implies that the stability of an interconnected financial system will be best achievedby micro-prudential policies. One may think that if the probability of fundamental default
is very small, then diversification might not be desirable. It is true that a smaller possibilityof individual failure will make a default cascade less likely, but it also reduces simultaneityrisk. The latter effect is likely to dominate the former, suggesting that the advantage ofdiversification is greater than that of diversity.
To examine in greater detail how the topology of the network structure affects financial
fragility, Figure 5 illustrates eight patterns of five-bank network topologies. Although a fivebank network is unrealistically small, a single “bank’’ in this network could also be viewed
as a group of financial institutions that have an identical portfolio. Each network structure8

is named after its degree distribution. For example, structure (b) 2-1 contains nodes thathave degree 1 or 2. Figure 6 is a visualization of the likelihood of contagious defaults, where
each bank is assumed to hold one bank-specific external asset. The number of dots in the(i, j)-th element indicates the expected number of defaults of bank j per 1000-time single
defaults of bank i. Based on this information, Figure 7 shows the extents of “infectivity”
and “susceptibility”. By the infectivity of bank i, I mean the average number of contagious
defaults generated by a single fundamental default of bank i. The susceptibility of bank irepresents the ratio of the number of times that bank i suffers from infectious defaults tothe number of bank i’s fundamental defaults. In order to see the topological importanceof each node in the network, PageRank (Brin and Page [15]) yi is also shown in the figure,where
yi = α

X

Aij

j

yj
+ β.
kjout

Aij is the (i, j)-th element of the adjacency matrix, which takes a value of one if bank jlends to bank i and zero otherwise. Notice that A is symmetric in this model. kjout is theout-degree of node j, α = .85 and β = 1.
It turns out that a bank’s infectivity is very closely correlated with its PageRank. In
fact, this property can be obtained only if PageRank is used as the centrality measure.
Other centrality measures, such as the betweenness centrality, the Katz centrality, thecloseness centrality, and the eigenvector centrality, are not necessarily correlated positively
with infectivity. As for network (b), for instance, only PageRank can correctly report that
nodes 2 and 3 are equally the most influential. Other centrality measures predict that node
1 is the most important. The reason for this difference is that PageRank assigns a highcentrality to a node that is pointed to by a node that has a small number of outgoing edges.
Bank 1 in network (b) is located at the center of the network, but its counterparties, banks
2 and 3, each have two edges. On the other hand, banks 2 and 3 are each linked to a bankthat has only one edge, banks 4 and 5, respectively. In the latter case, the failure of bank
2 (3) will cause the failure of bank 4 (5) with high probability. Therefore, a bank withhigher PageRank becomes more infective because a higher-ranking bank collects a larger
fraction of credit from each of its counterparties.
While PageRank is a good indicator of the topological importance of individual banks,9

it is also useful to construct a measure of the topological fragility of a financial system as
a whole. Figure 8a illustrates the expected costs under full diversity and network entropy,
SN , defined as3
SN = −

NX

pj log pj , where pj =j=1

# of bank j’s links.
# of total links in the network(8)

Figure 8b shows the relationship between the expected costs and the Herfindahl-HirschmanIndex (HHI), where
HHIN =

N
X

p2j .

(9)

j=1

It turns out that SN and HHIN have clear negative and positive relationships with the
expected costs under full diversity, respectively. This implies that the topological riskinessintrinsic to a financial network is well captured by the degree of market concentration. If
the number of links is quite similar among banks, which means that the size of the banks
is similar, then the financial system is relatively stable. In contrast, if a small number ofbanks own a large fraction of the total links in the network, i.e., if there are a small numberof giant banks, then the financial system becomes fragile. Network (d) is the most notable
example. This property is unchanged if PageRank is used to define pj instead of degree.
Notice that a simple measure of connectivity, such as the average degree, does not provide
accurate information on the level of systemic risk.
Figure 9 illustrates the optimal allocation of external assets when N = 5. If there is nointerbank trading, then the optimal situation for society is that any one bank diversifies its
portfolio and the other four have independent bank-specific assets. In contrast, if there isinterbank trading and all banks become interconnected, then the socially optimal state isthat all banks diversify their portfolios. This outcome holds true regardless of the topologyof the financial network. This implies that the previous argument for the case of N = 5
and K = 3 continues to hold for a more general structure of a financial network. It could
be said that micro-prudential policies will add up to a macro-prudential policy as long as
the returns of original assets are independent.
3

Wilhelm and Hollunder [16] propose various information-theoretic measures for network characteriza-tion. Sato [17] employs entropy measures to describe the state of foreign exchange markets.

10

3.3

Correlated external assets

Although, as PageRank suggested, the topological importance of each bank differs substantially, the above results indicate that the most desirable state for an interconnectedfinancial market is that every bank fully diversifies its portfolio. Recall that a presumption
for this argument is that the returns of external assets are independent of each other. In
the following, I consider a more general case in which external assets may be correlated.
Consider again the case of N = 5. The set of external assets, {ai }, available for banks
is now given as
5{a1 , a2 , a3 , a4 , a5 , a6 } = {a1 , −ρa1 +(1−ρ)â2 , a3 , ρa3 +(1−ρ)â4 , a5 ,

1X
ai },
5
i=1

where ρ ∈ [0, 1], and the return of each element of (a1 , â2 , a3 , â4 , a5 ) follows an independent
normal distribution. Notice that if ρ > 0, assets 1 and 2 have a negative correlation whileassets 3 and 4 are positively correlated. Asset 5 is independent of assets 1-4. For the1+ρ 2
σ , where σvariance of each asset not to be dependent on ρ, it is assumed that σ̂ 2 = 1−ρ

and σ̂ denote the standard deviations of ai and âi , respectively. If σ̂ is so defined, thenP
the variance of the diversified asset a6 , var[ 15 5i=1 ai ], is also independent of the value of
ρ. Thus, the set of external assets used here differs from the previous one only in terms of
the presence of correlations. It is assumed that banks are allowed to hold one of the six
external assets. The number of asset portfolio patterns to be examined is 6N = 7776 foreach network structure. Random returns of each asset are drawn 5000 times.
Figure 10 shows the optimal allocation of external assets. There are two notable features: first, neither of the positively correlated assets, a3 and a4 , nor the independent asset,
a5 , is used in interconnected networks, while both of the negatively correlated assets, a1
and a2 , are used in every network structure. This reflects the fact that the two negatively
correlated assets are advantageous in that they generate little possibility of simultaneous
defaults.
Second, the topologically most influential node, in terms of both infectivity and PageRank, is not necessarily assigned the diversified asset, a6 . In networks (e), (f) and (g), forexample, the infectivity measure and PageRank both indicate that node 1 is the mostimportant, but asset 1 is assigned to node 1 in all cases. This might be somewhat counter11

intuitive because the diversified asset ensures the lowest probability of fundamental default.
Note that if p = .2, then the probability of fundamental default by a bank that holds thediversified asset, a6 , is approximately .03. Thus, the likelihood of a fundamental default
by a bank with an asset other than a6 is about 6.7 times higher than that by a diversified
bank. This implies that in certain network structures, something is more important thaninfectivity from the standpoint of systemic stability.
The reason why the most infective bank need not be the safest is twofold: first, collective
fundamental defaults would be more influential than single defaults. To see this, Figure 11
shows the contributions of various combinations of collective defaults to the total expected
costs. It turns out that in all networks but (d), a single default of node 1 is not the main
contributor to systemic risk. Rather, double and triple defaults are systemically more
important than a single default of node 1. This suggests that double and triple defaults
be avoided to reduce systemic risk, which makes it desirable to have the banks’ balance
sheets move in counteractive ways. As for network (d), however, node 1 is so infectivethat it should be protected with the highest priority. In this sense, node 1 of network (d)could be viewed as a “super spreader’’. Second, a highly connected node is intrinsicallysafer than a poorly connected one. This can be confirmed by Figure 7, which indicatesthat a node with higher infectivity tends to exhibit lower susceptibility. This is because,
given the external asset portfolio, interbank assets become more diversified as the numberof links increases. Such a higher extent of diversification of interbank assets creates roomfor external assets to be less diversified.
Figure 12 shows how the optimal allocation of assets varies with the degree of convexityof the cost function. Only networks (b) and (e) are shown, but essential results hold truealso for other network structures. There are two notable phenomena. First, the number ofbanks that hold asset 6 decreases with s. When s is equal to or larger than 8, it is optimal
that only one bank holds the diversified asset. This is because simultaneity risk becomes
more severe as s increases. At this point, one may wonder why holding asset 1 or 2 would
be systemically less costly than holding asset 6. To simplify the situation, suppose that
there are three isolated banks and the regulator considers two types of asset allocation:(a) (a1 , a2 , a2 ) and (b) (a1 , a6 , a6 ). It is assumed that the probabilities of fundamental

12

default are p for holding a1 or a2 and q (< p) for holding a6 . Since a1 and a2 are negatively
correlated, suppose that there is no risk of double default for any combination of two banks
such that one has a1 and the other has a2 . a1 and a2 are assumed to be uncorrelated witha6 . For patterns (a) and (b), the expected costs can be respectively calculated as
Ca = p + p · 2 s ,(10)

Cb = p(1 − q) + q(1 − p) · 2s + pq · 3s .

(11)

Note that there is no risk of triple defaults in the case of (a). It follows that Cb ≥ Ca if
and only ifp−q
3 ≥1+ 1+
2s
pq
s

(12)

Since p > q, (12) does not hold if s = 1. However, (12) surely holds for values of s greater ∗∗than or equal to a certain threshold s∗ such that 3s = 1 + 1 + p−q
2s . This simple
pq
example clearly show that the benefit of a counteractive portfolio structure is increasing
in s. Although diversified assets increase the soundness of individual balance sheets, thisdoes not necessarily lead to a reduction of systemic risk if the cost of simultaneous defaults
is sufficiently large (i.e., s > s∗ ). For sufficiently large values of s, the risk of triple
defaults becomes significant even though its likelihood is quite low. Thus, introducing
a counteractive portfolio structure becomes more desirable as a larger size of collective
defaults begins to contribute to systemic risk.
The second point to be noted is that the safest node would differ as the value of svaries. As shown in the figure, node 1 is assigned asset 1 when s = 4 but asset 6 whens = 15. Recall that node 1 is not the most infective bank in network (b), so that it
is not correct to conjecture that the most infective node should be the safest when s is
sufficiently large. A possible explanation is that node 1 should act as a firewall, because
it is located at the center of the network. For example, if s = 8, there is a possibility of
triple fundamental defaults by banks that each have (a1 , a1 , a6 ) or (a2 , a2 , a6 ). Although
there is little possibility of quadruple fundamental default, it is highly likely that node 1
would suffer from infectious default if both of its counterparties fail. In contrast, in the
optimal portfolio structure under s = 15, there is little possibility that both of bank 1’s
counter parties fail by fundamental default. Therefore, if a large size of collective defaults
13

is highly contributive to the expected costs, then it is systemically beneficial to protect thenode that would effectively act as a firewall. In the terminology of network theory, node 1
is called a “cut point’’ in the sense that removing node 1 will separate the original network
into two isolated components. From this viewpoint, protecting the cut point will preventa collective default of one component from transmitting to the other.
Figure 13 illustrates how the introduction of counteractive portfolio structures reduces
expected costs. It shows that the relative benefit of introducing negatively correlated
assets varies with the network structure. An important fact is that the relative costs under
optimal asset allocation are not necessarily correlated with those under full diversity. Recall
that the case of full diversity exhibits pure effects of network topology on systemic risk.
Although the relative extent of systemic risk under full diversity has a negative correlation
with network entropy, such a correlation disappears once a system-wide portfolio structureis optimized. This implies that the fragility of a financial system is totally unknown until
asset allocation is taken into account. In other words, optimization of asset allocation could
eliminate the differences in systemic risk stemming from differences in network topology.

4

Discussion and conclusion

The recent literature addresses systemic risk mainly from two different and independent
viewpoints: asset allocation and network structure. A recent study by Beale et al. [1] isan example of the former. It is important to note, however, that their argument misses
a crucial aspect of actual financial systems: interconnectivity. On the other hand, manystudies, such as Gai and Kapadia [5], Gai et al. [6], and Hurd and Gleeson [18], focus on
investigating the relationship between network structure and systemic risk, employing amethodology developed in network theory. In these studies, the role of asset commonality
is missing. The model of this paper incorporates both viewpoints, considering a situationin which the extent of systemic risk depends not only on how banks choose their asset
portfolio, but also on how banks are interconnected with each other.
The optimal allocation of external assets that minimizes systemic risk naturally depends
on the extent of both simultaneity risk and externality. The analysis reveals that the use

14

of negatively correlated assets will be quite effective in reducing systemic risk by lowering
the probability of collective fundamental defaults. An important result is that the mostinfective bank, or the topologically most influential bank indicated by PageRank, need notalways be the safest. Under certain circumstances, the most infective node is required toact as a firewall to prevent large collective defaults. From this point of view, it appears
that the widely used term “systemically important financial institution (SIFI)’’ is harderto define than is usually thought. Recently, Caccioli et al. [19] also argue that increasing
the capital buffer of most highly connected banks may not affect systemic risk.
The optimization of system-wide asset allocation also reveals that the relative fragilityof a financial network structure is unknown until asset allocation is taken into account.
This is quite important because systemic risk tends to be considered directly related to
network topology. However, differences in the amount of systemic risk due to differencesin network topology could be eliminated to a large extent by appropriately regulating theasset holdings of banks.
In recent years, there has been wide agreement that countercyclical capital buffers
are needed to reduce economic fluctuations. This is based on the idea that the previously
employed simple capital requirement could amplify business cycles due to the procyclicalityof capital caused by the regulation itself. Roughly speaking, this paper shows that thesearguments can also be applied in a crossectional dimension. In order to reduce the riskthat the balance sheets of a large number of banks fluctuate in sync, a certain fraction ofbanks should have balance sheets that fluctuate counteractively. This will lower the levelof simultaneity risk, ensuring bank diversity.
Some issues need to be addressed in future research. First, the validity of certain
characteristic measures of network structures, such as modularity and assortativity, in
determining asset allocation should be explored in more detail. To do this, a larger-scalefinancial network needs to be considered, but computation will be highly time-consuming.
Second, the interconnectivity of banks needs to be treated in a more general form. In thismodel, it is assumed that banks are strongly connected in order to apply the Eisenberg-Noe
algorithm in calculating market-clearing vectors. However, the actual interbank market is
not necessarily strongly connected, and there are many one-directional edges and weakly

15

connected components (Söramaki et al. [20], Iori et al. [21], Imakubo and Soejima [22]).
Another algorithm needs to be employed to explore such a complex financial network.

Acknowledgments
I would like to thank Takashi Kamihigashi, Wataru Takahashi, Takayuki Tsuruga, seminar
participants at Kobe University and two anonymous referees for their helpful comments.
I also thank Kohei Hasui for the excellent research assistance. Financial support fromthe Japan Securities Scholarship Foundation and KAKENHI 24243044 and 13277309 are
gratefully acknowledged.

References[1] N. Beal, D.G. Rand, H. Battey, K. Croxson, R.M. May, and M.A. Nowak, P. Natl.
Acad. Sci. USA 108, 12647 (2011)[2] C. Upper, J. Financ. Stability. 7, 111 (2011)[3] R.M. May and N. Arinaminpathy, J. R. Soc. Interface. 7, 823 (2010)[4] S. Lenzu and G. Tedeschi, Physica. A 391, 4331 (2012)[5] P. Gai and S. Kapadia, P. Roy. Soc. A 466, 2401 (2010)[6] P. Gai, A. Haldane, and S. Kapadia, J. Monetary Econ. 58, 453 (2011)[7] E. Nier, J. Yang, T. Yorulmazer, and A. Alentorn, J. Econ. Dyn. Cont. 31, 2033
(2007)[8] F. Kyriakopoulos, S. Thurner, C. Puhr, and S.W. Schmitz, Eur. Phys. J. B 71, 523(2009)[9] R. Ibragimov, D. Jaffee, and J. Walden, J. Financ. Econ. 99, 333 (2011)[10] W. Wagner, J. Financ. Intermed. 19, 373 (2010)[11] W. Wagner, J. Financ. 66, 1141 (2011)16

[12] N. Rashevsky, Bull. Math. Biophys., 17, 229 (1955)[13] E. Trucco, Bull. Math. Biophys., 18, 129 (1956)[14] L. Eisenberg and T. H. Noe, Manage. Sci. 47, 236 (2001)[15] S. Brin, L. Page, Comput. Netw.. 30, 107 (1998)[16] T. Wilhelm and J. Hollunder, Physica A, 385, 385 (2007)[17] A.-H. Sato, in Agent-Based Approaches in Economic and Social Complex Systems VI:
Post-Proceedings of The AESCS International Workshop 2009 (Agent-Based Social
Systems), edited by S.-H. Chen et al. (Springer, Tokyo, 2010), p.3[18] T.R. Hurd and J.P. Gleeson, arXiv: 1110.4312, (2011)[19] F. Caccioli, M. Shrestha, C. Moore, J. D. Farmer, arXiv:1210.5987 (2012)[20] K. Söramaki, M. Bech, J. Arnold, R. Glass, and W. Beyeler, Physica. A 379, 317(2007)[21] G. Iori, G. De Masi, O.V. Precup, G. Gabbi, and G. Caldarelli, J. Econ. Dyn. Cont.
32, 259 (2008)[22] K. Imakubo and Y. Soejima, Monetary and Economic Studies 28, 107 (2010)

17

8000

diversity
diversification: v=3
diversification: v=5
diversification: v=7
diversification: normal

7000

expected costs

6000
10

5000

8

6

4000
4

2

3000

0

2000
2

4

6

8

10

12

1000

0
0

20

40

60

80

# assets
Fig. 1. Expected costs under full diversity and full diversification: asset returns obey Student
t-distribution with degree of freedom v. There is no interbank network. The number of assets is
assumed to be equal to the number of banks. Random returns are drawn 200,000 times for each
asset. p = .1 and s = 4.

18

N =2

N =6

N =4

2

12

p=.1
p=.2

N =8

25

N =10

40

70
60

10

20
30

1.5

50

8
15

40

6

20

1

30

10
4

20

10
0.5

5

2

10

0
0

1

2

0
0

1

N =12

2

3

4

0
0

N =14

140

2

4

6

0
0

2

N =16

200

4

6

8

4

6

8

10

16

20

N =20
700

400

600

250
150

100

2

N =18

300

120

0

500
200

300

80

400
100

150
200

60

300

100
40

200

50

100
50

20
0

0
0

3

6

9

12

100

0
0

3

6

9

12 14

0
0

4

8

12

16

0
0

4

8

12

16 18

0

4

8

12

Fig. 2. Expected costs with varying numbers of diversified banks. Horizontal axis: numberof diversified banks. Vertical axis: expected costs. Banks other than diversified ones have bankspecific assets. Each asset return is assumed to follow an independent standard normal distribution.
There is no interbank network. Random returns are generated 20,000 times for each asset. s = 4.

Safe assets (୧ )Deposits (୧ )Interbank assets(σ ߨ௝௜ ‫݌‬ഥఫ )Interbank liabilities(‫݌‬ഥప )External risk assets(୧ )Networth (୧ )Fig. 3. A typical bank’s balance sheet.

19

(b) # links = 2(a) # links = 0

20
16

0.6

0.6

14
15

G

10

G

12 0.4

0.4

8

0.2

10

0.2

6
0
0

0.2

0
0

0.4

5
0.2

0.4

D
(d) complete graph

D(c) # links = 4
20

0.4

G

15

20

0.6

0.4

15

G

0.6

0.2

0.2

10

10
0
0

0.2

0
0

0.4

0.2

0.4

D

D

Fig. 4. Expected costs with various combinations of D and G (defined by eqs.(6) and (7), respectively). Combinations of (D, G) that attain relatively low (high) expected costs are indicated bycold (warm) colors. N =5, K=3 and p=.1. (a) There is no interbank network. (b) and (c) The
numbers of links are 2 and 4, respectively. (d) Complete graph. Random returns are drawn 20,000
times for each of 5,000 portfolio patterns. The combination of (D, G) that minimizes the expected
cost is indicated by a gray sphere.

20

(a) 0-0(b) 2-1(c) 2-2(d) 4-1(f) 3-2-1(g) 4-3-1(h) 4-4

1
3

2
4

5
(e) 4-2

Fig. 5. Eight patterns of interbank networks. Each network structure is named after its degree
distribution.

21

(a) 0 - 0
1

2

3

4

(b) 2 - 1
5

1

2

3

4

(c) 2 - 2
5

1

2

3

4

(d) 4 - 1
5

1

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

5

5

(e) 4 - 2
1

2

3

4

(f) 3 - 2 - 1
5

1

2

3

4

1

2

3

4

3

4

5

(h) 4 - 4(g) 4 - 3 - 1
5

2

5

1

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

5

5

2

3

4

5

Fig. 6. Visualization of the likelihood of contagious defaults under the eight patterns of network
structure shown in Fig.5. The number of dots in the (i, j)-th element indicates the expected numberof defaults of bank j per 1000-time single defaults of bank i. Full diversity is assumed. p = .2.

22

(b) 2 - 1(a) 0 - 0
1(c) 2 - 2

infectivity
susceptibility

PageRank(right scale).8.5(d) 4 - 1

.25 1

1

.251.5

.5

.8

.4

.2

.2

.6

.6

.4

.4

1
.3

0
.15
-.5

.2

-1

.1

0
1

2

3

4

5

1

(e) 4 - 2

2

3

4

.25 1

.8

.4

.1
1

2

3

4

5

(h) 4 - 4
.2

1

.25

.8
.2

.6

.6

.4

.1 .4

.2

.2

.15
.2

4

0

5

.15

.4
.15

3

4

.8

.6

2

3

.2

.6

.1

2

(g) 4 - 3 - 1

.8

0

.1
1

.25 1

.2

.2

.2

0

5

(f) 3 - 2 - 1

1

1.15 .5
.2

0

5

.15

.1
1

2

3

4

.05

0

5

0
1

2

3

4

5

.1
1

2

3

4

5

Fig. 7. Infectivity, susceptibility and PageRank under the eight patterns of network structure shownin Fig.5. Horizontal axis: bank index. Vertical axis (left): infectivity (blue) and susceptibility(green). Vertical axis (right): PageRank (red solid line). The infectivity of bank i is defined as theaverage number of contagious defaults generated by a single fundamental default of bank i. The
susceptibility of bank i represents the ratio of the number of times that bank i suffers from infectious
defaults to the number of bank i’s fundamental defaults. Full diversity is assumed. p = .2.

(a)

(b)
60
PageRank-based
degree-based

55

expected costs

expected costs

60

network (d)
50
45
40
35
1.4

1.5

1.6

1.7

entropy55

degree-based
PageRank-based

50
45
40
35
0.2

0.25

0.3

HHI

Fig. 8. Expected costs under full diversity. All the possible patterns of 5-bank interconnected
network topologies are considered (21 patterns in total). p = .2. (a) Negative relationship between
entropy and expected costs. Green square: entropy is based on PageRank. Blue circle: entropy
is based on node degree. (b) Positive relationship between the Herfindahl-Hirschman Index (HHI)and expected costs.

23

(a) 0-0(b) 2-1(c) 2-2(d) 4-1(f) 3-2-1(g)(g)4-34-3-1(h) 4-4

1
3

2
4

5
(e) 4-2

assets 1:

2:

3:

4:

5:

6:

Fig. 9. The optimal allocation of external assets for each topology. Assets 1-5 are bank-specificassets whose returns are independent (ρ = 0). Asset 6 (dark purple circle) is the diversified asset,
defined as the average value of assets 1-5. p = .2.

24

(a) 0-0(b) 2-1(c) 2-2(d) 4-1(f) 3-2-1(g) 4-3-1(h) 4-4

1
3

2
4

5
(e) 4-2

assets 1:

2:

3:

4:

5:

6:

Fig. 10. The optimal allocation of external assets for each topology. Assets 1 and 2 (black andlight-blue triangles) have a negative correlation, while assets 3 and 4 (brown and blue squares) havea positive correlation. Asset 5 (white circle) is independent of assets 1-4. Asset 6 (purple circle) isthe diversified asset, defined as the average value of assets 1-5. ρ = .8 and p = .2.

25

(a) 0 - 0(b) 2 - 1

.1

double triple.2(1)(1,2,3)

(2,3)

.08

quadruple(d) 4 - 1

.1

(1,3)

quintuple.08 single.06(c) 2 - 2

.1

.08

.06

.06

(1,2,3,4)(3,4,5)

(1,2)

.16
.12

(1,2)

(1,4,5)

(1)
.04

.04

.04

.08

.02

.02

.02

.04

0
0

10

20

30

0
0

(e) 4 - 2

.08

10

20

0
0

30

(f) 3 - 2 - 1

.1

(1,3)

20

30

(1,2)

(1,2)

.08

.06

.06

.04

.04

.04

.04

.02

.02

.02

.02

20

30

0
0

10

20

30

(1,2,4)(1,2,4)
.06

10

20

(1,4)

.06

0
0

10

.1

.08

(1,5)

0
0

(1,2,3)(h) 4 - 4

.1

(1,4)
.08

10

(g) 4 - 3 - 1

.1

(1,2)

(1,5)

0
0

30

10

20

30

0
0

(1,2)

10

(3,4,5)20

30

Fig. 11. Contributions of various size of collective defaults to the total expected costs. Horizontal
axis: index of collective defaults: {1,2, ..., 31} = {(1), (2), (3), (4), (5), (1,2), (1,3), (1,4), (1,5),(2,3), (2,4), (2,5), (3,4), (3,5), (4,5), (1,2,3), (1,2,4), (1,2,5), (1,3,4), (1,3,5), (1,4,5), (2,3,4), (2,3,5),(2,4,5), (3,4,5), (1,2,3,4), (1,2,3,5), (1,2,4,5), (1,3,4,5), (2,3,4,5), (1,2,3,4,5)}. Numbers in parentheses are indexes of banks that default simultaneously. Vertical axis: percentage of the cost of thecorresponding collective default to the total expected costs. Full diversity is assumed. ρ = 0 and
p = .2.

26‫ܛ‬ൌ૛ൌ ૡૡ‫ ܛܛ‬ൌ‫ܛ‬ൌ૝‫ ܛ‬ൌ ૚૞(b) 2-1(e) 4-2

assets 1:

2:

3:

4:

5:

6:

Fig. 12. The optimal allocation of external assets with different values of s (i.e., concavity of the costfunction). Assets 1 and 2 (black and light-blue triangles) have a negative correlation, while assets
3 and 4 (brown and blue squares) have a positive correlation. Asset 5 (white circle) is independentof assets 1-4. Asset 6 (purple circle) is the diversified asset, defined as the average value of assets
1-5. ρ = .8 and p = .2.

27

expected costs

60
optimalfull diversification
full diversity

50
40
30
20
10
0
1.4

1.45

1.5

1.55

1.6

1.65

entropyFig. 13. The relationship between the expected cost and the entropy based on PageRank. All thepossible patterns of 5-bank interconnected network topologies are considered. Blue solid circles:
expected costs under optimal allocation. Green stars: expected costs under full diversification. Red
squares: expected costs under full diversity. p = .2. In deriving the optimal asset allocations, ρ
is set at .8. The figure shows how the introduction of negatively correlated assets would reduce
systemic risk.

28