Diversity among banks may increase systemic risk
Teruyoshi Kobayashi∗
August 2012

Abstract
The problem of how to stabilize the ﬁnancial system has attracted considerable
attention since the global ﬁnancial crisis of 2007-2009. Recently, Beal et al. (2011,“Individual versus systemic risk and the regulator’s dilemma”, Proc Natl Acad Sci
USA 108: 12647-12652) demonstrated that higher portfolio diversity among banks
would reduce systemic risk by decreasing the likelihood of simultaneous defaults. Here,I show that this result is overturned once a ﬁnancial network comes into play. In a
networked ﬁnancial system, the failure of one bank can bring about a contagion of
failure. The optimality of individual risk diversiﬁcation, as opposed to economy-widerisk diversiﬁcation, is thus restored. I also present a new method to quantify how
the diversity of bank size aﬀects the stability of a ﬁnancial system. It is shown thata higher diversity of bank size itself makes the ﬁnancial system more fragile even if
external risk exposure is controlled for. The main reason for this is that larger banks
are more likely to become a “super spreader” of infectious defaults. In this situation
the social cost of letting a bank fail is not uniform and depends on the size of the failingbank. This strongly implies that larger banks are systemically more important thansmaller banks, and preventing large banks from being exposed to high external risks
would therefore be the most eﬀective vaccine against ﬁnancial crisis.
JEL Classiﬁcation: C63, D85, G01.
Keywords: Systemic risk, ﬁnancial crisis, ﬁnancial network.

∗

Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe, Japan. E-mail:kobayashi@econ.kobe-u.ac.jp. Tel, fax: +78-803-6692.

1

Introduction

Over the past decade, simulation of ﬁnancial crisis has been conducted by many researchers
in various ﬁelds (Beal et al., 2011, Upper, 2011, Gai et al., 2011, Gai and Kapadia, 2010,
May and Arinaminpathy, 2010). One reason for this heavy use of simulation is the lackof data on interbank trading. While such data are, at least partially, available in somecountries such as Austria, Italy and the U.S., data accessibility is usually limited to thecentral bankers (Söramaki et al., 2007, Elsinger et al., 2006, Iori et al., 2008). One of themain objectives of simulation-based research is to gain theoretical understanding of how the
topology of a ﬁnancial network aﬀects the likelihood of ﬁnancial crisis. Employing concepts
developed in network theory, such as degree distribution, assortativity, modularity, etc.,
those studies attempt to extract the necessary conditions for a stable ﬁnancial system byconducting simulations under various patterns of network topology (Lenzu and Tedeschi,
2012, Gai and Kapadia, 2010, Nier et al., 2007, Kyriakopoulos et al., 2009).
On the other hand, while the main focus of those simulation studies is on the network
structure, the portfolio structure of individual banks is another key to understanding systemic stability. Interestingly, Beal et al. (2011) recently pointed out that higher portfolio
diversity among banks would reduce systemic risk by decreasing the likelihood of simultaneous defaults. This puts into eﬀect the “regulator’s dilemma”, because it implies thatminimizing the risk of individual failure does not add up to minimizing systemic risk.
In this paper, I show that this result is overturned once a ﬁnancial network comes intoplay. In a networked ﬁnancial system, the failure of one bank may bring about a contagionof failure. The optimality of individual risk diversiﬁcation, as opposed to economy-widerisk diversiﬁcation, is thus restored.
It is also shown that the diversity of the portfolio structure of interbank assets, as well as
the portfolio structure of external assets, can be an important source of systemic fragility.
In this model, the amount of funds a bank lends to bank i relative to bank j depends only
on the relative size of bank i and bank j. This means that the diﬀerence between bankswith respect to interbank risk exposure exclusively reﬂects the heterogeneity of bank size.
The virtue of this property is that the pure eﬀect of bank-size heterogeneity, or of diversityof interbank assets, can be extracted without altering external risk exposure. Although1some of the recent works on systemic risk also explore the role of portfolio diversity in thecontext of a ﬁnancial network (Gai and Kapadia, 2010, Gai et al., 2011), to the best of
my knowledge, this paper is the ﬁrst to provide a methodology that can quantify the pureeﬀects of bank-size heterogeneity or of the diversity of interbank assets. I formally showthat diversity among banks in terms of interbank asset allocation is ineﬃcient from thepoint of view of systemic risk minimization as well as individual risk minimization.
The simulation suggests that an extremely large bank is prone to become a “superspreader” of infectious defaults. I propose two measures to quantify the systemic importance of a bank: infectivity and susceptibility. It is shown that a very large bank is highly
infective in the sense that a failure of the bank will cause contagious defaults with highprobability. By contrast, a highly infective bank is not necessarily susceptible to contagion in the sense that a failure of the bank is in most cases caused by a “fundamental
default”. This strongly implies that larger banks are systemically more important thansmaller banks, and that preventing large banks from being exposed to high external risks
will be the most eﬀective vaccine against contagious defaults.

2

Model 1: Uniform bank size

There are N interconnected banks whose balance-sheet sizes are identical. The asset sideof bank i’s balance sheet consists of risky external assets, ai , interbank assets, li , andthe riskless asset, bi . In general each bank holds K kinds of risky external assets and NNinterbank assets, where ai = K
k=1 ai,k and li =j=i li,j . The liability side of bank i’sbalance sheet consists of interbank liability, p̄i , deposits, di and networth, wi . The balance
sheet condition implies that ai + bi + li = p̄i + di + wi , ∀ i ∈ [1, N ]. The relative proportionsof ai , bi and li are common to all banks, while the portfolio structure of external risk assets,(ai,1 , . . . , ai,K ), and the portfolio structure of interbank assets, (li,1 . . . . , li,N ) are allowed
to vary from bank to bank.
The amount of bank i’s borrowings from bank j is expressed as πi,j p̄i , where πi,j denotesthe relative weight of bank i’s borrowings from j, and thereby j=i πi,j = 1, ∀ i ∈ [1, N ].
Let the N -by-N matrix Π have πi,j as its (i, j)-th elements. The diagonal elements2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


li,j =j=iN
πj,i p̄j ,j=i

In vector form,l = Π p̄.
All the oﬀ-diagonal elements of Π are assumed to be 1/(N − 1) in Model 1.
Bank i tries to pay back the full amount of interbank liability p̄i after the external asset
values are realized, but the funds available at that time are expressed as

πj,i pj + ãi + bi − di ,

(1)

j=i

where ã denotes the ex-post external assets. It should be pointed out that deposits di arereserved because deposits are senior to interbank assets. I also assume the following three
conditions: (a) limited liability; (b) the priority of debt claims; and (c) the proportionalityof the amount of repayment after default (Eisenberg and Noe, 2001). The actual amountof repayment, pj , is not necessarily equal to its face value, p̄j , since bank j’s loss of assets
might be greater than its networth, wj . This also means that the solvency of a bankdepends on the debtors’ solvency. Speciﬁcally, the market clearing vector, p∗ , is expressed
as a ﬁxed point of the map, Φ : [0, p̄] → [0, p̄] deﬁned as
Φ(p) = p̄ ∧ (Π p + ã + b − d),(2)where ∧ denotes the meet operator, such that x∧y = (min(x1 , y1 ), . . . , min(xn , yn )). Underthe reasonable conditions (a), (b) and (c), Eisenberg and Noe prove that a ﬁxed point of (2)exists and is unique if the ﬁnancial network is a complete graph and i (ãi + bi − di ) > 0.
They also show that the unique ﬁxed point is the solution of the following problem:
max f (p) s.t. p ≤ Π p + ã + b − dp∈[0,p̄]for any strictly increasing function f (·) : RN → R.
The simulation procedure is as follows. First, the size and the composition of thebalance sheet are chosen. Given the networth and the asset-capital ratio, the size of the3balance sheet is automatically determined. In model 1, it is assumed that the networth
is identical across banks. Interbank assets and liabilities are determined simultaneously
as one assigns Π and p̄. The ratio of the total amount of external risk asset to networth
is ﬁxed and common to all banks. Deposits and safe assets are determined residually sothat the LHS and the RHS of the balance sheet are equal. Second, the random returnsof K external risk assets are drawn. Let nnc be the number of defaults at this stage (i.e.,“fundamental defaults”). Third, given the funds available for the repayment of interbank
loans, ã + b − d, the unique market-clearing vector, p∗ , is calculated. Let nc be the numberof defaults after contagion takes place. Note that nc −nnc represents the number of defaults
caused purely by contagion.
Following Beal et al., I assume that social costs, C, depend on the number of bankdefaults, n: C(n) = ns , s > 0. The expected cost is thus
E[C] =

Nq(n)ns ,n=1

where q(·) is the probability function. The case of s > 1 means that simultaneous multiple
bank defaults will be more costly than the sum of single bank defaults.

3

Results of Model 1

Figure 1 illustrates how the expected costs depend on the distance between banks, D, andthe distance from the optimal portfolio, G, which are deﬁned as
D ≡
G ≡


1
|ai,k − aj,k | ,2N (N − 1)
i
j
k



1  

(a
−
1/K)

.
i,k


N
k

(3)

(4)

i

Beal et al. demonstrate that 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 ofsimultaneous defaults. This is reconﬁrmed in the simulation without contagion (Figures
1A, 1C). On the other hand, it turns out that this outcome does not hold true once thepossibility of contagion is taken into account (Figures 1B, 1D). In the latter case a bank’s

4default undermines the lenders’ balance sheets, which would give rise to further defaults.
In the presence of such a negative spillover eﬀect, the optimal combination of (D, G) moves
toward the origin compared to the case without contagion. It follows that the stability of
an interconnected ﬁnancial system will be achieved by ensuring the stability of individual
banks.

4

Model 2: Variable bank size

Previous studies on ﬁnancial stability usually assume that a ﬁnancial system consists ofidentical banks in terms of the size of balance sheets. This assumption is ubiquitous
because introducing diﬀerence in bank size generally entails heterogeneity on the asset-sideof the balance sheets. Note that an increase of one bank’s interbank borrowing is always
accompanied by a rise in the lenders’ interbank assets. The relative amounts of ai , bi andli thus generally vary with bank-size distribution, which generates heterogeneity of risk
exposure. This heterogeneity prohibits the researcher from extracting the pure eﬀect ofbank-size diversity on ﬁnancial stability.
Here, I show a way to examining the relation between bank-size diversity and ﬁnancial
stability, keeping the relative size of ai , bi and li ﬁxed for all i ∈ [1, N ]. The idea is that theelements of Π and p̄, which were exogenous in the previous section, are adjusted to ensurethat bank-size diﬀerence is reﬂected only in the portfolio of interbank assets. Speciﬁcally,
I assume the following:j=i+1

πj,i+1 p̄jj=i πj,i p̄j

πj+1,i p̄j+1
πj,i p̄j


πi,j

ωi+1
, ∀ i ∈ [1, N − 1],ωiωj+1 α=,ωj∀ i ∈ [1, N ], j ∈ [1, N − 1], i = j,(6)

= 1, ∀ i ∈ [1, N ],

(7)

=

(5)

j=i

where ωi denotes the size of bank i’s balance sheet. Eq.(5) requires that the ratio of li+1 toli be equal to the relative size of bank i + 1 to bank i. This also ensures that the fractionof interbank assets in the balance sheet is identical between banks. Eq.(6) states that theamount of bank i’s interbank lending to j +1 relative to j, li,j+1 /li,j , is equal to the relative
5

size of bank j + 1 to the power α. With this condition bank-size diﬀerences are directly
reﬂected in the portfolio of interbank assets. The case of α > 1 means that the eﬀect ofbank-size heterogeneity is magniﬁed.
Note that there are (N 2 − 1) restrictions in total. Since the number of oﬀ-diagonal
elements of Π is N (N − 1) and the vector p̂ = (p̄2 , . . . p̄N ) has N − 1 elements, all the(N 2 − 1) conditions may be satisﬁed by appropriately choosing the elements of Π andp̂. p̄1 is used as the numéraire, which standardizes the amounts of interbank loans andinterbank assets. I ﬁnd that the solution for Π and p̂ respectively satisfy 0 < πi,j < 1,
∀ (i, j) ∈ {(i, j) : i ∈ [1, N ], j ∈ [1, N ], i = j} and p̄i > 0, ∀ i ∈ [2, N ]. Details of
the derivation are shown in the Appendix. Note that the Eisenberg and Noe’s suﬃcientcondition can be applied since the ﬁnancial network is a complete graph.
As noted above, a virtue of this method is that heterogeneity of bank size is reﬂected
exclusively in the portfolio of interbank assets, so that ai /ωi , bi /ωi and li /ωi are common
to all i ∈ [1, N ] regardless of the size of the bank.
This property allows us to evaluate the inﬂuence of bank-size distribution on systemicrisk without aﬀecting the probability of fundamental default. It should be pointed out thatsuch irrelevancy of bank-size distribution may not hold true if some of the elements of p∗
are zero, in which case the liability structure could aﬀect the likelihood of defaults. To avoid
this situation, p̄1 should be set at a suﬃciently large value. The parameter conﬁgurationis shown in the appendix.

5

Results of Model 2

Figures 2A, 2B and 2C illustrate the total number of defaults under uniform bank size
and under diverse bank size. The index of the bank is in ascending order with respect
to bank size. The distribution of bank size is illustrated in Figure 4A. The most notablefeature is that a failure of the largest bank is less frequent under diverse bank size thanunder uniform size, while the opposite is true for smaller banks. When one of the banks
in a ﬁnancial system is very large, the largest bank is unlikely to fail unless it fails by
fundamental default. In contrast, higher diversity of bank size increases the likelihood of a6

failure of smaller banks because smaller banks are more susceptible to contagious defaults.
Figure 3 shows the extent of “infectivity” and “susceptibility”. By the infectivity ofbank i, I mean the total probability of generating contagious defaults conditional on asingle fundamental default of bank i. By the susceptibility of bank i, I mean the total
probability of suﬀering from infectious defaults conditional on a single fundamental default
of bank j = i. As the ﬁgure shows, a failure of the largest bank is highly infective while its
susceptibility is rather low. This is because the smaller banks have a large risk exposure to
the largest bank while the largest bank has a relatively more uniform risk exposure. The
portfolio of interbank assets, as well as the portfolio of external assets, is quite importantin evaluating systemic stability. Other things being equal, more heterogeneous exposure in
the interbank market leads to a more unstable ﬁnancial system.
This point is also conﬁrmed in Figure 4 in terms of expected costs. This ﬁgure illustrates
the relation between the expected social costs and the Herﬁndahl-Hirschman Index (HHI),deﬁned as
HHI =

N


Si2 ,

i=1

where Si ∈ [0, 1] is the market share of bank i. The higher the HHI, the more concentrated
the market is. Figure 4 implies that the probability of a severe contagion becomes higher as
the ﬁnancial market becomes more concentrated. Although the network structure always
takes a form of a complete graph, the introduction of bank-size diversity will make it easierfor infectious defaults to be transmitted.
The question is how we could prevent a large bank from being a “super spreader’’ of infectious defaults. I here suggest that the large bank’s portfolio of external risk assets should
be diversiﬁed preferentially. One measure useful for this purpose is a correlation betweenbank size and the l1 distance from the optimal portfolio, GI = (GI(1), . . . , GI(N )), whereGI(i) ≡



|ai,k − 1/K| , ∀ i ∈ [1, N ].

(8)kFigure 5 indicates that the expected social cost is the lowest when there is a strong
negative correlation between bank size and GI. This corresponds to a situation in whichlarger banks have less risky portfolios of external risk assets, i.e., (ai,1 , . . . ai,K ) is more
uniformly distributed around 1/K than smaller banks. The systemic risk of an interbank
7market can be eﬀectively lowered by having the “too-big-too-fail” banks diversify theirexternal risks.

6

Discussion

Over the past 5 years the diﬀerence between micro- and macro-prudential policies has beenemphasized by many academic economists, central bankers and other regulators. Since thefailure of Bear Stearns and Lehman 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 for regulators is to search for the best macro-prudential policy
to stabilize the ﬁnancial system as a whole.
One extreme of the arguments is that the portfolio of risk assets should be diversiﬁed
from bank to bank so that simultaneous multiple failures would not occur. The negativeside of this proposal is that systemic stability requires the self-sacriﬁce of individual banks.
It is important to note that such an argument misses a crucial aspect of actual ﬁnancial
systems: interconnectivity. Almost by construction, a ﬁnancial system is a networked
system in which individual banks are connected with each other via the interbank market.
It is widely recognized that a crash in the interbank market was one of the most importantfactors leading Bear Stearns and Lehman to fail.
In this paper I show that diversity among banks should be minimized in order toenjoy maximum ﬁnancial stability. The main implications can be summarized as follows:First, every individual bank should be encouraged to minimize its own external risk. In aninterconnected ﬁnancial system, independently minimizing the risk of individual default will
also minimize systemic risk by reducing the probability of contagious defaults. Therefore,high diversity among banks in terms of their portfolios of external assets will not be optimal.
Second, diversity of bank size itself tends to make the ﬁnancial system more fragile. This isbecause an extremely large bank is prone to become a “super spreader’’. While the balance
sheets of large banks are not so sensitive to the failure of other banks, smaller banks arehighly susceptible to the failure of large banks. In such a “skewed’’ ﬁnancial system,
preventing large banks from being exposed to high external risks is the most eﬀective

8

vaccine against systemic turmoil. At the same time, this also implies that the regulators
should supervise the network structure thoroughly. Systemic risk increases as the extent ofheterogeneity in the interbank asset allocation increases, given the risk exposure of external
assets. Heterogeneity of interbank trading itself may be an important source of ﬁnancial
crisis. A more concrete investigation into the relationship between network structure andﬁnancial stability is left for future research.

7

Appendix

7.1

Baseline parameters

Throughout the paper, the number of banks, N , and assets, K, are 5 and 3, respectively.
The mean bank size is 100 for both Model 1 and Model 2. p̄i /wi = 4 ∀ i ∈ [1, N ] in Model
1, and p̄1 /w1 = 4 in Model 2. ai /wi = 2 and wi /ωi = 0.1, ∀ i ∈ [1, N ] in both models. The
parameter α in Eq.(6) is 1.2. The systemic cost parameter, s, is set at 1 or 4.

7.2

Generating the loss of external risk assets

The portfolio of external risk assets is chosen as follows: ﬁrst, the fraction of asset 1 heldby bank i, afi,1 ≡ ai,1 /ai , is set at 1/K for all i. Second, bank i s relative holding of asset
2, afi,2 , is taken from uniform distribution on [0, 1 − afi,1 ]. The fraction of asset 3 is thentaken from uniform distribution on [0, 1 − afi,1 − afi,2 ], and so forth. Finally, the relativeK−1 famount of asset K must satisfy afi,K = 1 − k=1
ai,k .
I assume that the loss of each asset follows a normal distribution and the probability
that the asset loss takes a value greater than x0 is 1 − p0 if the bank invests in a single
asset. Let p0 ≡ F (x0 | μ, σ0 ) denote the CDF of a normal distribution with mean μ andstandard deviation σ0 . If μ = 0, the CDF can be written as

 x√0
σ0 2
2
1
2
1+ √
F (x0 | 0, σ0 ) =e−t dt .
2
π 0
Let pw ≡ F (w| 0, σw ) be the probability that the asset loss is lower than the networth, w.
To ensure that the probability of fundamental default takes an arbitrary value 1 − p0 , weneed to have pw = p0 , or w/σw = x0 /σ0 . It follows that σw = wσ0 /x0 . The ex post asset
9value thus leads toθσ0ãi,k = max 0, ai,k − ai,kxk , ∀ i ∈ [1, N ], k ∈ [1, K].
x0

where the capital-asset ratio, θ = wi /ai , is common to all i and xk ∼ i.i.n.(0, 1), ∀ k ∈0[1, K]. This implies that the rate of return of asset k is max(−1, θσx0 xk ). Note that if

ai,k = ai , ∀ k ∈ [1, K], then the probability of bank i’s default is 1 − p0 . p0 is set at 0.9 inthe simulation.

7.3

Generating bank-size distribution

In order to capture the inﬂuences purely caused by bank-size heterogeneity, a sequence ofbank-size distribution needs to be generated. Here, I consider a growth process that allows
bank size to deviate gradually from the uniform distribution and converge to a power-law
distribution. The basic process of the bank-size growth is given byi
ωi,t+1 = ζt+1
ωi,t ,(9)i , i =∈ [1, N ] are independently and identically distributed growth rates. It iswhere ζt+1i
important to note that ζt+1
is independent of the bank size. I assume that banks of size

100 are uniformly distributed at the beginning, and then randomly chosen banks grow by
the rate of ζ ∈ {.95, 1.1} at each step. As Gabaix pointed out, this simple process produces
a log-normal distribution without a steady state. Some frictions need to be added for a
power-law distribution to arise in a steady state (Gabaix, 1999). Here, I let the lower boundof wi be 5. Throughout the growth process the average size of banks is kept unchanged. I
repeat the growth process 1000 times.

7.4

Derivation of Π and p̂

From Eqs.(5), (6) and (7), the following conditions can be obtained:
π13 = π12
π14 = π12

ω3
ω2
ω4
ω2

1+

ω3
ω1

1+

ω2
ω1



1+
1+

ω2
ω1

α
α

ω4
ω1

+
+

ω3
ω1
α

ω4
ω1
α

+
10

+

ω3
ω1

α
α

+ ... +

ωN
ω1

+ ... +

ωN
ω1

ω4
ω1
α

+

α

+ ... +

ω5
ω1

α

α
α,
ωN
ω1

+ ... +

(10)
α
ωN
ω1

α,

(11)

..
.
Inserting these equations into Eq.(7) yields⎡
  1 + ω3 α +ω1ω3π12 ⎣1 +ω2 1 + ω2 α +

+

ω4
ω2



1+
1+

ω2
ω1

ω3
ω1
α

ω1
α

+

ω3
ω1

+

α

ω4
ω1

ω4
ω1
α

ω4
ω1
α

+

α

ωN
ω1
α

+ ... +
α

α

⎤

ωN
ω1

+ ... +

ω5
ω1

α

ωN
ω1

+ ... +

ωN
ω1

+ ... +

α + ...

⎦=1(12)

Thus π12 , π13 ,. . . , π1N are obtained from Eqs.(10), (11) and (12). From Eq.(7), we have
π2,1 p̄2 + π2,3 p̄2 + π2,4 p̄2 + . . . + π2,N p̄2 = p̄2 .

(13)

It follows that
π2,1 p̄2 +

ω2
ω1

α


π1,3 p̄1 +

ω2
ω1

α


π1,4 p̄1 + . . . +

ω2
ω1

α
π1,N p̄1 = p̄2 .

(14)

Using the relation
π2,1 p̄2 = π1,2 p̄1

ω1
ω2

1+

ω3
ω1

1+

ω3
ω2

α
α

+

ω4
ω1

+

ω4
ω2

α
α

+ ... +

ωN
ω1

+ ... +

ωN
ω2

α
α,

(15)

we obtain π2,1 ,. . . , π2,N and p̄2 . In the same manner, π3,1 ,. . . π3,N and p̄3 can be derived
by noting that
π3,1 p̄3 + π3,2 p̄3 + π3,4 p̄3 + . . . + π3,N p̄3 = p̄3

(16)

and


ω3
ω2

α


π2,1 p̄2 +

ω3
ω1

α


π1,2 p̄1 +

ω3
ω1

α


π1,4 p̄1 + . . . +

ω3
ω1

α
π1,N p̄1 = p̄3 .

(17)

In this way, the other elements of Π and p̂ can also be computed recursively. A Matlab
program for solving these equations is available from the author upon request.

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11

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[13] Upper, C. (2011) Simulation methods to assess the danger of contagion in interbank
markets. Journal of Financial Stability 7, 111-125.

12

A

C

expected costs without contagion: s = 1
0.7

expected costs without contagion: s = 4
0.7

-1.8
0.6

0.6

2.6

0.5

2.4

-2

0.5

0.4

-2.2

2.2

G

G

0.4

2

0.3

0.3

-2.4

1.8
0.2

0.2
-2.6
1.6

0.1

0.1
-2.8

1.4

0
0

B

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

D

D

expected costs after contagion: s = 1

expected costs after contagion: s = 4

D 0.7

0.7

0.4

-1.6

3
0.6

0.6

2.9

-1.8
2.8
0.5

0.5

2.7

-2

2.6

0.4

G

0.4

G

-2.2

2.5
0.3

0.3

2.4

-2.4

2.3

0.2

0.2

2.2
-2.6
0.1

0.1

0

0
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2.1
2

-2.8

0

0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

D

D

Fig. 1: The logged expected costs as a function of D and G. The optimal combination of(D, G) is shown by the gray sphere. Random draws are conducted 20000 times for 5000
patterns of external asset portfolio. The size of banks is identical. (A) Expected social costs
without a ﬁnancial network: s = 1. (B) Expected social costs with a ﬁnancial network:s = 1. (C) Expected social costs without a ﬁnancial network: s = 4. (D) Expected social
costs with a ﬁnancial network: s = 4. The introduction of an interbank network moves
the optimal distance between banks, D, toward the origin. This is because the failure of
one bank may generate contagious defaults. In the presence of a ﬁnancial network, saving
individual banks will save the ﬁnancial system as a whole.

13

A

x 10

# of fundamental defaults

6

B

3.5

x 10

# of defaults

6

C

3.5

x 10

# of contagious defaults

6

3.5

most diversified
uniform
3

3

3

2.5

2.5

2.5

2

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0

0
1

2

3

4

5

bank index

0
1

2

3

bank index

4

5

1

2

3

4

5

bank index

Fig. 2: The eﬀects of bank-size distribution on the frequency of defaults. (A) Total numberof fundamental defaults. (B) Total number of defaults after contagion takes place. (C)Total number of contagious defaults. The failure of an extremely large bank is less frequent
than that of smaller banks.

14contagion likelihood matrix: uniform

A

1

2

3

4

5

C

uniform
0.35

1
0.3

0.25

2

0.2
3
0.15

4

0.1

infectivity
susceptibility

0.05
5
0
1

contagion likelihood matrix: most diversified
1

B

2

3

4

5

D

1

2

3

4

5

4

5

most diversified
0.9
0.8
0.7

2
0.6
0.5
3
0.4
0.3
4
0.2
0.1

5

0
1

2

3

Fig. 3: Susceptibility and infectivity. (A) The contagion likelihood matrix under uniformbank size. The number of plots in the (i, j)-th element indicates the expected number ofdefaults of bank j per 1000-time single defaults of bank i. (B) The contagion likelihood
matrix under heterogeneous bank size. (C) The degree of infectivity and susceptibility under uniform bank size. The infectivity of bank i is calculated as the sum of the oﬀ-diagonal
elements of the i-th row of the contagion likelihood matrix. The susceptibility of bank i
is calculated as the sum of the oﬀ-diagonal elements of the i-th column of the contagion
likelihood matrix. (D) The degree of infectivity and susceptibility under heterogeneousbank size. Note that an extremely large bank is highly infective but not very susceptible.

15

A

distribution of bank size

average expected costs: s = 4

B

450

14

total
fundamental defaults

400

350

12

300

10

250

8

200

150
6

100

4
50

0
1

2

3

4

5

bank index

2
1

1.5

2

2.5

3

3.5

HHI

Fig. 4: The eﬀect of bank-size diversity on expected costs. (A) The sequence of generated bank-size distributions. (B) The relation between expected costs and the HerﬁndahlHirschman Index (HHI). Systemic risk will increase as the interbank market becomes moreconcentrated.

16

4

7

relative expected cost: s = 4

6

uniform
most diversified
y = .999x + 2.536, R2 = .266, p = .000

5

4

3

2

1
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

corr(GI, bank size)

Fig. 5: The expected relative costs and the correlation between GI and bank size. GI is the
distance from the optimal portfolio of external risk assets. The relative costs are deﬁned
as the ratio of total costs to the costs of fundamental defaults. The upward-sloping lineis the best ﬁt under the OLS using the simulated data under diversiﬁed bank size. This
indicates that expected cost of contagion will be reduced as larger banks’ external risks are
decreased.

17