Cascades in multiplex financial networks with debts of different seniority

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by  Charles D. Brummitt

Cascades in multiplex financial networks with debts of different seniority
Charles D. Brummitt
Center for the Management of Systemic Risk, Columbia University, New York, NY 10027, USA
Teruyoshi Kobayashi∗
Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe 657-8501, Japan
(Dated: May 28, 2015)
Theseniorityofdebt,whichdeterminestheorderinwhichabankruptinstitutionrepaysitsdebts,
isanimportantandsometimescontentiousfeatureoffinancialcrises,yetitsimpactonsystem-wide
stability is not well understood. We capture seniority of debt in a multiplex network, a graph of
nodes connected by multiple types of edges. Here, an edge between banks denotes a debt contract
of a certain level of seniority. Next we study cascading default. There exist multiple kinds of
5
1 bankruptcy, indexed by the highest level of seniority at which a bank cannot repay all its debts.
0 Self-interested banks would prefer that all their loans be made at the most senior level. However,
2 mixingdebts ofdifferentsenioritylevelsmakesthe systemmore stable, inthat itshrinks theset of
network densities for which bankruptcies spread widely. We compute the optimal ratio of senior to
y
junior debts, which we call the optimal seniority ratio, for two uncorrelated Erdo˝s-R´enyi networks.
a
If institutions erode their buffer against insolvency, then this optimal seniority ratio rises; in other
M
words, if default thresholds fall, then more loans should be senior. We generalize the analytical
results to arbitrarily many levels of seniority and to heavy-tailed degree distributions.
7
2 PACS numbers: 89.65.Gh, 89.75.Hc

]
N I. INTRODUCTION vulnerability to large cascades cannot be understood by
G examiningthevulnerabilityofanyonelayertolargecas-
. Theglobalfinancialcrisisof2007–2009spurredaflurry cades [13–20]. In particular, in threshold models, the
n of research on financial contagion using a variety of net- interactions among network layers can make large cas-
i
f work models [1–7]. In most of these models, financial cades easier (if nodes combine influence from multiple
-
q institutions interact in just one way, such as making one types of neighbors in a disjunctive [18] or weighted [19]
[ type of loan to one another. In practice, however, banks way) or more difficult (if nodes combine influence in a

interact in many ways, all of which can contribute to conjunctive way [20]).
3
v a crisis: banks issue loans of different durations and of Inthispaper,wecombinethesetwostrandsofresearch
0 different levels of riskiness; they trade assets with each to study a financial phenomena of interest: how vulner-
0 other;andtheyholdassetsincommon. Modelshaveonly ability to cascading default depends on the seniority of
4 recentlybeguntocapturemorethanoneofthesekindsof debts. Financial institutions (hereafter called “banks”
5 interactions[8–10]. Relativelyunexploredistheseniority for simplicity) hold debt in one another; more generally,
0 of debts, which determines the order in which bankrupt theyhaveexposuretooneanother’sdefaultriskthrough
.
1 institutions repay their debts. Individual creditors pay interbank lending and/or security holding. Each debt
0 close attention to their assets’ seniority levels, but lit- contract has a certain level of seniority. When a bank
5 tle is known about how the system-wide composition of goes into default (i.e., when a bank’s liabilities exceed
1
seniority levels affects the risk of large crises. its assets), the bank sells its remaining assets to pay off
:
v Aconcurrentandmostlyindependentthreadoflitera- whateverliabilitiesitcanindecreasingorderofseniority,
Xi tureinphysicshasstudiedmultiplexnetworks,orgraphs from most senior liability to most junior liability.
with multiple sets of edges (or “network layers”) ; for re- Inourmodel,thereexistmultipletypesofbankruptcy,
r
a views, see [11, 12]. Variants of percolation on multiplex indexed by the highest level of seniority at which a bank
networkshavereceivedmuchattention[13–17]. Closerto cannot repay all its debts. Using a technique called the
financial default contagion are models of threshold cas- cascade condition [18–25], we study whether a small
cade on multiplex networks [18–20]. In these threshold number of initial bankruptcies (of certain types) triggers
models, nodes exist in one of two states (e.g., adopted a a cascade of many bankruptcies.
behaviorornot, gonebankruptornot), andafteranode The model has interesting tradeoffs. Self-interested
adopts the behavior (or goes bankrupt), it influences its banks would prefer to lend at the most senior level in
neighbors in all layers to do so as well. A common order to maximize the assets they would recover in the
conclusion from these models of percolation and thresh- event of its debtors defaulting. However, if all banks
old cascades on multiplex networks is that the system’s lend at the most senior level, then the system is espe-
ciallyvulnerabletolargecascadesofdefault,inthesense
that cascades are likely large for a large interval of net-
work densities. (This extreme case recovers the models
∗ Correspondingauthor. [email protected] in [1, 2].)
2
Bycontrast,thepresenceofdebtsofdifferentseniority
makesthesystemlessvulnerabletobankruptciesspread-
ing widely. For the case of two seniority levels (called
“junior” and “senior”), our main result is to derive the
“optimal” ratio of senior loans to junior loans (for a
simple class of uncorrelated Erd˝os-R´enyi multiplex net-
works). Here,“optimal”meansthatitminimizesthesize
of the interval of network densities for which bankrupt-
cieslikelyspreadwidely. Theoptimalmultiplexnetworks
haveontheorderof50%to100%moreseniorloansthan
juniorones. Asbanks’leverage(i.e.,assetsdividedbynet FIG.1. (Coloronline)Anexampleofabank’sbalancesheet.
worth)increases,this“optimalseniorityratio”increases, Assets (left-hand column) consist of external assets, l = 4
S
meaning that more loans should be senior rather than many senior interbank loans, and lJ = 3 many junior inter-
junior in order to minimize the size of the set of network bank loans. Liabilities (right-hand column) consist of exter-
densities such that cascades are likely large. We also nal liabilities, bS = 2 many senior interbank liabilities, and
b = 3 many junior interbank liabilities. Net worth (a.k.a.
showthattheoptimalseniorityratioisqualitativelysim- J
equity or, for banks, capital) is the difference between assets
ilarforheavy-taileddegreedistributionslikethosefound
and liabilities.
inrealinterbanklendingnetworks[26–28]. Theseresults
indicate what types of “seniority–multiplex” interbank
networks are most insusceptible to large cascades of de-
gant theory and indicative suggestions [29]. Although fi-
fault.
nancialnetworkschangeovertime,thereisevidencethat
Themostrecentschemeforfinancialregulation,called
their statistical properties (such as degree distributions)
Basel III, emphasizes the liability structure of individual
can be somewhat stable [28].
banks, but it does not take into account how seniority
of debts affect systemic risk. Our results suggest that
seniorityofdebtcouldbeanimportantfocusoffinancial
regulators and of macro-prudential policies. B. Model
Furthermore, this work contributes to the rapidly
growing area of research on contagion in multiplex net- 1. Multiplex network of loans of different seniority
works by introducing a model in which nodes can “ac-
tivate” in more than one way, which we call “multilevel
WeconsideramultiplexnetworkconsistingofN nodes
contagion”. An interpretation of this model for the
and M layers. Each node represents a financial institu-
contagious spread of fads, products, etc. in social net-
tion (or a “bank” for simplicity). Each layer is a set
works [21] is that the kinds of neighbors a person influ-
of directed edges (i.e., ordered pairs of nodes) that de-
ences depends on how excited that person is. Thus, the
noteloansand,moregenerally,creditexposuresbetween
model resembles the multistage model of complex conta-
banks. The layer to which an edge belongs is the loan’s
gion [25] but on multiplex networks.
seniority, and the vector of layers is sorted in increasing
order of seniority. When a bank goes bankrupt, it re-
pays its liabilities in decreasing order of seniority; that
A. Related literature is, it repays its most senior liabilities first, its second-
most senior liabilities second, etc. For the moment, we
Theoretical studies of contagion in financial networks consider a two-layer (“duplex”) network, and the M =2
generally fall into two classes [29]. Some studies con- many levels of seniority are called “junior” and “senior”
sideraparticularinstanceofafinancialnetwork, usually (labeled J and S), respectively.
a complicated and in some cases empirically measured The number of banks to which a bank lends funds of
object. Other studies consider probability distributions seniority α ∈ {J,S} (i.e., the bank’s out-degree in layer
over financial networks. α) is denoted by l . Likewise, the number of banks from
α
Our study belongs to the latter class, but two pa- whichabankborrowsfundsofseniorityα(i.e.,thebank’s
pers closely related to our analysis belong to the for- in-degree in layer α) is denoted by bα. For simplicity,
mer [30, 31]. Elsinger [30] and Gourieroux et al. [31] we assume that the sizes of loans (i.e., the notional val-
also assume a senior–junior structure of interbank as- ues) are the same, so we can consider the networks as
sets. Whereas they obtain a clearing payment vector for unweighted. Thus, the out-degree and the in-degree cor-
agivennetwork(asinEisenbergandNoe[32]),westudy respond to the bank’s total volume of lending and bor-
the cascade condition for ensembles of random networks rowing, respectively.
in the same spirit as Watts [21] and followup work in Banksmayalsoholdassetsandissueliabilitiesoutside
physics [18–20, 22–25, 33–35]. The former approach can the banking system. (For instance, external creditors
be useful to regulators faced with a real system on the maylendtobanksintheformofdemanddeposits.) The
verge of a crisis, whereas the latter approach offers ele- balance sheet of a typical bank is illustrated in Fig. 1.
3
2. Multiple types of default Consider some point in time during a cascade. For a
certain bank, let m denote the number of its junior-
J
borrowers that have junior-defaulted, and let m de-
In a model with two levels of seniority of debt, there S
notethenumberofitssenior-borrowersthathavesenior-
are two types of defaults (bankruptcies): junior-defaults
defaulted. This bank is in junior-default if and only if
and senior-defaults. A bank is in junior-default if and
its losses m +m exceeds the bank’s equity w, and this
only if its capital (a.k.a. equity or net worth; see Fig. 1) J S
bankisinsenior-defaultifandonlyifitslossesm +m
is negative, meaning that it cannot pay all of its junior J S
exceed the sum of its equity w and its junior liabilities
debts. Abankisinsenior-default ifandonlyifitscapital
b .
is so far below zero that it cannot pay its all of its senior J
To exploit the solution technique for threshold cas-
debts nor any of its junior debts. Note that banks in
cades in networks [18–20, 22, 23], we express these in-
senior-default are necessarily in junior-default as well.
equalities in terms of losses relative to total out-degree
We assume that when a bank goes into junior-default,
l +l : the bank is in default at layer α∈{J,S} if and
itcannotrepayanyofitsjunior-levelliabilities,andwhen J S
only if
a bank goes into senior-default, it cannot pay any of its
junior liabilities nor any of its senior liabilities. Said dif- (cid:40)
fαeriesnt1l0y0,%wefoarssinutmerebathnaktlilaobssiligtiievsenofdseefnauiolrtitoyf osfenαioarintdy mlJJ ++lmSS >Rα ≡ llwJJ+++wbllJSS iiff αα==JS (1)
lower. This assumption departs from previous cascade
models of financial contagion, such as that of Gai and [In Appendix A, we generalize the thresholds R in
α
Kapadia [1], in which loss given default is 100% for all Eq. (1) to the general case of M ≥ 1 layers.] Figure 2
interbank liabilities, no matter their seniority. Table I illustrates a bank in junior-default and a bank in senior-
makes this assumption concrete: the value of a bank’s default in two ways: as losses on a balance sheet (top
debt is equal to some positive value (say, 1 unit of cur- row) and as contagion in a multiplex network (bottom
rency) if the bank is solvent (i.e., if its equity is nonneg- row).
ative); the value of the bank’s junior debt falls to zero The dynamics of multilevel default contagion can
when the bank becomes insolvent, and the value of its be described as a multiplex version of the threshold
senior debt falls to zero when the bank’s equity passes a model [21, 24, 33, 36]. (Other multiplex generalizations
nonpositive threshold value (derived later to be −bJ). were studied in [18–20].) The “response functions” for
each layer are
TthAeBvLalEueI.oRfietlsajtuionnioshridpebbetstwaenednstehneioerqdueitbytsohfealdbobryroitwsecrreadnid- FJ((cid:126)l,(cid:126)b,m(cid:126))≡(cid:40)1 if mlJJ++mlSS >RJ (2a)
tors. Here,weassumethatjuniorandseniordebtsareequally 0 otherwise
valuable when the borrower is solvent, so values of debts are (cid:40)
eitherzeroor$1. Iftheborrower’sequityisnegativebutsuf- FS((cid:126)l,(cid:126)b,m(cid:126))≡ 1 if mJl+Jm+SlS−bJ >RJ , (2b)
ficiently large (≥ −b ), then it can pay its senior debts but 0 otherwise
J
none of its junior debts; the borrower is in “junior-default”.
Forsufficientlynegativeequityoftheborrower,theborrower
where (cid:126)l ≡ (l ,l ),(cid:126)b ≡ (b ,b ) and m(cid:126) ≡ (m ,m ). If
cannot pay neither its junior debts nor its senior debts; the J S J S J S
the outputs of F and F are both 0, then the bank is
borrower is in “senior-default”, and all its debts have value J S
zero. solvent; if FJ gives 1, then the bank is in junior-default;
if F gives 1, then the bank is in senior-default (and in
S
equity status junior debt value senior debt value junior-default).
≥0 solvent $1 $1
We note in passing that a special case of this model
<0 junior-default 0 $1
coincides with a special case of the “multistage complex
<−b senior-default 0 0
J contagion” model of Melnik et al. [25]; for details, see
Appendix B.
II. MULTILEVEL CASCADES ON 2-LAYER
3. Thresholds for junior- and senior-default MULTIPLEX NETWORKS OF JUNIOR AND
SENIOR LOANS
Initially, all banks are solvent, and a small fraction
of banks are chosen to be in junior-default or in senior- This multiplex model adds a new dimension of com-
default. These initial defaults could be caused by the plexity to previous threshold models [1–3, 21, 22, 24] by
loss of external assets. These bank failures cause losses considering multiple levels of seniority of debt, so to un-
for their creditors, who may default as a result, which derstand the model we start by simplifying in two other
may cause one or more of their creditors to default, and dimensions, the distribution of networks and the distri-
so on, resulting in a cascade of default. bution of thresholds.
4
FIG. 2. (Color online) An example of the balance sheet (and the corresponding multiplex network visualizations) of a bank
thatis(a)injunior-default: m +m >R (l +l )≡w and(b)insenior-default: m +m >R (l +l )≡w+b . Circular
J S J J S J S S J S J
and spiky-shaped nodes are solvent and insolvent (a.k.a. bankrupt, in default), respectively, in that layer. Edges point from
lender to borrower. Senior loans are dashed green arrows; junior loans are solid red arrows. In panel (a), the bank labeled A
suffers losses of m +m = 2 on its assets side (assets are listed on the left-hand side of the bank’s balance sheet) because
J S
m =1ofits(l =3many)juniordebtorsareinjunior-defaultandbecausem =1ofits(l =4many)seniordebtorsarein
J J S S
senior-default. TheselossesexceedbankA’sequity(w),sobankAcannotpayoffallits(b many)juniorliabilities. Becauseof
J
theassumptionthatlossgivendefaultatlayerαis100%forallcreditorsinthatlayer,allofA’sjuniorcreditorsdonotreceive
repayments on their loan to A, and we say that A goes into junior-default. However, bank A does not go into senior-default
because it can still pay its senior liabilities (i.e., because m +m ≤ w+b ). In panel (b), bank A suffers so many losses
J S J
(m +m =5) that it cannot repay its junior creditors nor its senior creditors, so bank A goes into senior-default.
J S
First, each layer is modeled as an independent, lo- 1. Recursion equations
cally treelike, random directed network generated by the
configuration model. The sparsity and independence Consider a bank chosen uniformly at random, and let
of the layers implies that the number of “overlapping the network hang down like a tree from this root node
edges” (i.e., junior and senior loans between the same bydoingabreadth-firstsearchalongthedirectionsofthe
pairofbanks)isnegligiblysmallforsufficientlylargenet- edges. Wedefineφα astheprobabilitythatanodet+1
t+1
works [37]. hops from the bottom of the tree is in α-default because
Second, following previous studies [1–3], we assume sufficiently many of its debtors (i.e., its out-neighbors)
that every bank has the same junior-default threshold are in default. The nodes at the bottom of the tree are
RJ = w/(lJ + lS). This assumption implies that all initially in α-default with probability φα0. To compute
banks have the same ratio of capital to total interbank φα fromφα,weconditiononwhetheranodelocatedt+
t+1 t
assets. In the real world, banks’ capital-to-asset ra- 1hopsabovetheleavesofthetreeisinitiallyinα-default,
tios are not identical, of course, but they are somewhat which occurs independently with probability φα. If this
0
similar [38]. It is straightforward to consider hetero- node is not initially in α-default, then we condition on
geneous capital-to-asset ratios by considering heteroge- the number of debtors of this node that have defaulted.
neous thresholds [3, 22]. For a sufficiently large graph, the fractions of nodes
in junior- and senior-default at the end of the cascade
are well approximated by a fixed point (φJ ,φS ) of the
∞ ∞
recursion equations
A. Analytical approximation
φα =g(α)(φJ,φS)
t+1 t t
Let φJ and φS denote the probabilities that a node is (cid:88) (cid:88)
0 0 ≡φα+(1−φα) pJ,loanpS,loanpJ,borrow
initially in junior-default or in senior-default, where 0< 0 0 lJ lS bJ
φJ (cid:28)1 and φS ∈[0,φJ]. Next we adapt to this model a lJ+lS≥1 bJ
0 0 0
tweicdhenlyiq,utehefocrasccaalcduelacotinndgitwiohne,tihnetrrocdausccaeddeisnli[k2e1l,y2s2p]raenadd × (cid:88)lJ (cid:88)lS BmlJJ(φJt)BmlSS(φSt)Fα((cid:126)l,(cid:126)b,m(cid:126)), (3)
generalizedtoothermultiplexnetworkmodelsin[18–20]. mJ=0mS=0
5
for α ∈ {J,S}, where pα,borrow and pα,loan denote the In Appendix C, we simplify the Jacobian matrix and
bα lα
in- and out-degree distributions, respectively, on layer show that, for this model, its largest eigenvalue equals
α ∈ {J,S}. In Eq. (3), we approximated the numbers its trace.
of neighbors in junior and senior default as independent
binomial random variables, the probability mass func-
tions of which are written using the shorthand notation B. Cascade regions
Bl (φ) ≡ (cid:0)l(cid:1)φm(1−φ)l−m]. This approximation works
m m
well when the network layers are independent and hence
If many banks are in junior-default, the fraction of
the number of overlapping edges is negligibly small [37].
banks in senior-default could be large or small. Thus,
(If overlap is significant, then one could adapt the tech-
there are two other cascade conditions: one for junior-
niques introduced in a study of mutual percolation [16]
defaults and another for senior-defaults. Next we derive
to this threshold model.) Variants of Eq. 3 appear in
theseconditionsinsomegenerality(forensemblesofmul-
several studies of this threshold model; see, for exam-
tiplex networks with independent layers that are speci-
ple, [22, Eqs. (1)–(3)], [23, Eqs. (1) and (4)], [18, Eqs.
fied by their degree distributions). We also illustrate all
(1)–(3)], [19, Eqs. (11)–(13)], [25, Eqs. (A9)–(A11)], [20,
three cascade conditions (junior, senior, and multiplex)
Eqs. (1)–(3)] and, for a rigorous treatment, [7, Theorem
for a particular case of interest: a multiplex network of
1]; the closest variant is the study of directed (but not
two uncorrelated Erd˝os-R´enyi networks of mean degree
multiplex) networks [24, Eqs. (14) and (15)].
(cid:104)l (cid:105) and (cid:104)l (cid:105), respectively.
J S
Inshort,weapproximatethemultiplexgraphasatree
according to outgoing edges, and φα is the probability
t+1
thatabank(locatedt+1hopsfromtheleavesofthetree)
1. Cascades of junior-default (ignoring senior-default)
goesintoα-default becauseitwasinitiallyinα-defaultor
because of the defaults of its debtors. Note from Eq. (2)
First consider contagion only on junior loans. The
that going into senior-default due to defaulted debtors
“junior-only” cascade condition is that a junior-default
implies having gone into junior-default due to defaulted
debtors [i.e., F ((cid:126)l,(cid:126)b,m(cid:126)) = 1 implies F ((cid:126)l,(cid:126)b,m(cid:126)) = 1], so causes on average at least one more junior-default, or
S J equivalently that the top-left entry of the Jacobian ma-
the fraction in senior-default never exceeds the fraction
trix, J , exceeds 1. In Appendix C, we show that this
in junior-default [i.e., φS ≤φJ for all t≥0]. JJ
t t junior-only cascade condition is the inequality
For convenience, we rewrite the recursions (3) as
(cid:18)φJt+1(cid:19)=(cid:18)g(J)(φJt,φSt)(cid:19). (4) JJJ =E[lJ(11>RJ(lJ+lS))]>1. (7)
φS g(S)(φJ,φS)
t+1 t t The set of parameters satisfying this junior-only cas-
Iterating Eq. (4) to a fixed point (φJ ,φS ) gives the ex- cade condition (7) is shown in orange in Fig. 3(a) for
∞ ∞
pected fractions of banks in junior-default and senior- uncorrelated Erd˝os-R´enyi networks of mean degree (cid:104)lJ(cid:105)
default, respectively, at the end of the cascade. and (cid:104)lS(cid:105). If the senior-layer is the empty graph (i.e., if
(cid:104)l (cid:105)=0), then we recover standard cascade models on a
S
single-layer network [1–3, 21, 22, 24]. Creating more se-
2. Multiplex cascade condition niorloans(i.e.,increasing(cid:104)lS(cid:105))shrinksthe“junior-only”
cascade region. This shrinking occurs because a bank’s
equityw =R (l +l )increaseswithitsnumberofsenior
Nowweobtainasimpleexpressionthatapproximately J J S
loans l [recall Eq. (1)]. Thus, as the expected number
captures,inthelimitofnetworksizeapproachinginfinity, S
(cid:104)l (cid:105) of senior loans per bank increases, a typical bank’s
for which parameters a vanishingly small seed triggers a S
buffer against junior-default (w) increases, which makes
cascadethatresultsinafinitefractionofbanksinjunior-
a junior-default less likely to cause other junior-defaults.
orsenior-default. Thefirst-ordercascadecondition isthe
linear instability of Eq. (4) at the origin (0,0), which
provides a sufficient condition for the numbers of junior-
andsenior-defaultedbankstogrow. TheJacobianmatrix 2. Cascades of senior-default (ignoring junior-default)
of the right-hand side of Eq. (4) is denoted by
In a similar way, consider contagion along senior loans
(cid:18) (cid:19)
J J
J ≡ JJ JS . (5) only. The “senior-only” cascade condition is that a
J J
SJ SS senior-defaultcausesonaverageatleastonemoresenior-
default, or equivalently that the bottom-right entry of
Thefirst-ordercascadeconditionisthatthelargesteigen-
theJacobianmatrix,J ,exceeds1. InAppendixC,we
value of the Jacobian matrix, λ (J), exceeds 1: SS
max show that this senior-only cascade condition is
(cid:112)
J +J + (J −J )2+4J J
λmax(J)≡ JJ SS JJ2 SS JS SJ JSS =E[lS(11−bJ>RJ(lJ+lS))]
>1. (6) =pJ,borrowE[l 1 ]>1. (8)
0 S 1>RJ(lJ+lS)
6
The first-order cascade condition considers defaults trig- (�)
gfaeureltds bdyuethteo dmeJfau+ltmoSf j=ust1)o;ntehuosu,t-intecigahpbtuorre(si.see.,nidoer-- =〈〉�� 7 ������������-����������������������������������λ�����(�>)>� �
defaults of banks that lack junior debts (bJ = 0), which �� 6 ������-��������������������>�
explains the factor pJ,borrow in Eq. (8). We found that ��
0
capturingdefaultscausedbytwobankruptcreditors(us- ��
�
ing the second-order cascade condition) was not neces- �� 5
sary to approximate simulations well, as shown in Ap- ��� 〈��〉�������≈����
pendix D and in Fig. 6. �� 4 〈��〉
��
Theparameterssatisfyingthissenior-onlycascadecon- �� 〈��〉=〈��〉
dition(8)areshowningreeninFig.3(a) foruncorrelated �� 3
Erd˝os-R´enyinetworksofmeandegree(cid:104)lJ(cid:105)and(cid:104)lS(cid:105). This ����
regionshrinkswiththedensityofjuniorloans(cid:104)lJ(cid:105)forthe �� 2
same reason that the junior-only cascade region shrinks �
�
with (cid:104)lS(cid:105). ���� 1
�
�
0
3. Multiplex cascades 0 1 2 3 4 5 6 7
����������������������������������=〈� 〉
�
TheblueregioninFig.3(a)isthemultiplexcascadere-
(�)
gion, the set of average out-degrees ((cid:104)l (cid:105),(cid:104)l (cid:105)) such that
J S
the first-order cascade condition [inequality (6)] is satis-
fied. Note that, as pointed out in [18, page 3], we have 6
�
λmax(J)≥max{JJJ,JSS}. Thus, a sufficient condition ��
forthemultiplexcascadeconditiontoholdisthatatleast ��� 4
one kind of default α ∈ {J,S} is supercritical (meaning �
�
that Jαα > 1) even in the absence of the other kind of ��� ����������� 〈��〉 ≈����
default. Thus, the blue region in Fig. 3(a) contains both �� 2 〈��〉
the orange and green regions.
Importantly,however,theblueregioncontainspairsof
0
intermediateedgedensities((cid:104)l (cid:105),(cid:104)l (cid:105))outsidethejunior-
J S 0 2 4 6 8
only and senior-only cascade regions. In this part of pa- ��������������〈� 〉/〈� 〉
� �
rameterspace,neitherjunior-defaultsnorsenior-defaults
alone suffice to cause global cascades (i.e., J < 1 and
JJ FIG. 3. (Color online) (a) The multiplex cascade region
J < 1), but a large cascade of defaults can neverthe-
SS [the edge-density parameters ((cid:104)lJ(cid:105),(cid:104)lS(cid:105)) satisfying Eq. (6),
less occur. These cascades, akin to the “multiplexity- drawn as a blue region with a thick, solid boundary] con-
facilitated cascades” in [18, 20], result from the interac- tainsthejunior-onlyandsenior-onlycascadeconditionsgiven
tion of junior- and senior-defaults. by Eqs. (7) and (8) and drawn as an orange region (with a
dashedboundary)andgreenregion(withadottedboundary),
respectively. Moreover,themultiplexcascaderegionisasym-
metric in the densities of junior and senior loans, ((cid:104)l (cid:105),(cid:104)l (cid:105)).
C. Optimal seniority ratio J S
The optimal seniority ratio, defined in Eq. (9), is drawn as a
black line. This ratio (cid:104)l (cid:105)/(cid:104)l (cid:105)≈1.79 minimizes the cascade
S J
Note that the multiplex cascade region, show in blue
window,thesetofmeandegreessatisfyingthefirst-ordercas-
inFig.3(a),isnotsymmetricwithrespecttotheaverage cadecondition,Eq.(6). Thecaseofequallymanyjuniorand
out-degrees in the two layers, (cid:104)lJ(cid:105) and (cid:104)lS(cid:105). This asym- senior loans (on average) is shown with a dashed line, which
metry of the cascade region results from the asymmetric has a slightly larger cascade window. (b) The cascade win-
role that junior and senior loans play in the response dow as a function of the ratio of senior loans to junior loans,
function [Eq. (1)]. Given this asymmetry, what ratio of (cid:104)lS(cid:105)/(cid:104)lJ(cid:105), achieves a minimum size at ≈1.79.
junior and senior loans is optimal in some sense?
Suppose that the average total degree (cid:104)l (cid:105) + (cid:104)l (cid:105) is
J S
kept constant and that we tune the fraction of loans
that are junior. For some loan densities (cid:104)lJ(cid:105) + (cid:104)lS(cid:105), satisfying 7=(cid:104)lJ(cid:105)+(cid:104)lS(cid:105) in Fig. 3(a), the cascade condi-
the multiplex cascade condition (6) holds no matter the tion (6) holds if and only if the fraction of junior loans
fraction of junior loans. For example, the line segment is very small or somewhat large. In cases like this one,
{((cid:104)l (cid:105),(cid:104)l (cid:105)) : 4 = (cid:104)l (cid:105) + (cid:104)l (cid:105)} is contained within the having an intermediate fraction of junior debts prevents
J S J S
multiplex cascade region in Fig. 3(a), so defaults spread defaults from spreading widely.
widely no matter the composition of seniority levels. In Now suppose that we can set the ratio (cid:104)l (cid:105)/(cid:104)l (cid:105) of
S J
other cases, such as the line segment of mean degrees theaveragenumbersofseniorandjuniorloansperbank,
7
but the total density of loans (cid:104)l (cid:105)+(cid:104)l (cid:105) can vary. For In practice, the number of defaults of any type is of pri-
S S
instance, the number of loans may increase during an mary concern, so our definition of the optimal seniority
economic boom and decrease during a recession, yet the ratio in Eq. (9) ignores types of defaults.
fraction of loans that are senior does not change. Thus,
we are considering rays through the origin in Fig. 3(a).
For “seniority ratios” σ ≥ 0, define the cascade win- D. Effect of varying the thresholds (ratio of equity
dow tobethesetofaveragedegrees((cid:104)l (cid:105),(cid:104)l (cid:105))suchthat to interbank assets)
S J
(cid:104)l (cid:105)/(cid:104)l (cid:105)=σandsuchthatthefirst-ordercascadecondi-
S J
tion(6)holds. Figure3(b)showsthiscascadewindowas How does the cascade region (and its optimal senior-
a function of the seniority ratio (cid:104)lS(cid:105)/(cid:104)lJ(cid:105). Note that the ity ratio) change as the banks’ threshold RJ changes?
sizeofthecascadewindowisminimizedatauniquevalue Here we keep the assumption that banks all have the
of (cid:104)lS(cid:105)/(cid:104)lJ(cid:105), which is approximately 1.79 in Fig. 3. We samejunior-defaultthresholdRJ, butwevaryRJ. The
define the optimal seniority ratio to be this minimizer, multiplex network is still a pair of uncorrelated Erd˝os-
or more precisely, R´enyi random graphs with mean degrees (cid:104)l (cid:105) and (cid:104)l (cid:105).
J S
IfbankshaveasmallercapitalratioR ,thendefaultbe-
(cid:104)l (cid:105) J
S :=argmin|{r ≥0:λ (J)≥1}| (9a) comesmorelikely,sothemultiplexcascaderegiongrows,
(cid:104)lJ(cid:105)optimal σ≥0 max as shown in Fig. 4(a). Conversely, as RJ increases, the
cascade region shrinks, and it can even splinter into two
where r is the distance from the origin, | · | denotes
regions[seethepurpleregionsforR =0.25inFig.4(a)].
Lebesgue measure on R, and λ (J) is evaluated at J
max In cases like this one with two disjoint cascade regions,
theoptimalratiodefinedinEq.(9)doesnotexistbecause
r rσ
(cid:104)lJ(cid:105)= √1+σ2, (cid:104)lS(cid:105)= √1+σ2, (9b) the set {r ≥ 0 : λmax(J) ≥ 1} is empty for some σ. [A
sensible definition for the optimal seniority ratio could
be the slope that maximizes the distance from the ray
where we have changed from polar to Cartesian coor-
through the origin (with that slope) to the two cascade
dinates via (r,arctanσ) (cid:55)→ ((cid:104)l (cid:105),(cid:104)l (cid:105)). If this optimal
J S
regions.]
seniority ratio ((cid:104)l (cid:105)/(cid:104)l (cid:105)) is greater than one (as
S J optimal
in Fig. 3), then senior debts should be issued more than The asymmetry between junior and senior loans also
junior debts (on average) for the sake of minimizing the changeswiththejunior-defaultthresholdRJ. Inparticu-
size of the set of network densities such that bankrupt- lar,theoptimalseniorityratioincreasesasRJ decreases,
cies spread widely. That is, if a financial network keeps as shown in Fig. 4(b).
approximately the same ratio of senior to junior loans In some financial crises, many banks find themselves
((cid:104)l (cid:105)/(cid:104)l (cid:105)) but changes the total number of loans (say, in financial distress, because of losses in external assets,
S J
overthecourseofabusinesscycle), thenthereshouldbe for instance. The model studied here approximates this
more senior loans than junior loans in order to minimize scenario as a small junior-default threshold RJ for every
the risk of a large cascade of defaults. bank (and a small fraction of banks initially in default).
Intuitively,seniorloansarestabilizingbecausetheyare DecreasingRJ enlargesthecascaderegion, asillustrated
lesssignificantearlyinacascade: anodeinjunior-default in Fig. 4(a). However, the banks can reduce systemic
causes losses (of assets) for its lenders (in-neighbors) in risk if they increase, on average, the fraction of debts
thejuniorlayeronly. Thus,changingsomeloansfromju- that are senior. Figure 4(b) shows the optimal amount
nior to senior makes large cascades less likely, and hence of that increase.
it shrinks the cascade window. However, not all loans
should be senior, because that case is identical to all
loans being junior, which has a dangerously large cas- III. MORE THAN TWO SENIORITY LEVELS
cade window [look at the horizontal and vertical axes of
Fig. 3(a)]. In between those extremes lies an optimal ra- Now we show that mixing among several seniority
tio of senior to junior loans. That optimal ratio exceeds levels can further reduce systemic risk (in the sense of
one[thedashedblack45-degreelineinFig.3(a)]because shrinking the cascade region).
senior loans are less contagious, in the sense that nodes We generalize the model to M ≥ 1 seniority levels.
in junior-default can still repay their senior debts. Let (1,2,...,M) denote the indices of the seniority lev-
In Fig. 7 of Appendix D, we show that the three els in increasing order of seniority, and let R denote
1
types of cascade regions shown in Fig. 3(a) (junior-only, the threshold for default at the most junior layer. The
senior-only, and multiplex) closely agree with the fixed response function for default at level i, derived in Ap-
point of the recursion equations (4) and with numerical pendix A, is
simulations of the model for uncorrelated Erd˝os-R´enyi
(cid:40)
multiplex networks. These comparisons also show that 1 if (cid:80)M m −(cid:80)i−1 b >R (cid:80)M l
within the multiplex cascade region many nodes end up Fi((cid:126)l,(cid:126)b,m(cid:126))= s=1 s k=1 k 1 s=1 s .
0 otherwise
in junior-default, but many nodes are in senior-default
(10)
only when senior loans greatly outnumber junior loans.
8
(�) 14 ceeding one. The (i,j) entry of JM is
〉=〈���� 12 ����-�����������==������������������ JiMj =EM(cid:104)lj11−(cid:80)ik−=11bk>R1((cid:80)Mα=1lα)(cid:105) (11a)
����� 10 ����==������� = i(cid:89)−1pk0,borrowEM(cid:104)lj11>R1((cid:80)Mα=1lα)(cid:105) (11b)
�� k=1
�
� 8 i−1 M
� (cid:89) (cid:88) (cid:89)
���� = pk0,borrow psls,loanlj. (11c)
���� 6 k=1 (cid:126)l:(cid:80)Mα=1lα<1/R1s=1
��� Here, products of the form (cid:81)0 are defined to be 1,
� 4 k=1
��� andtheexpectationsEM areoverp1l1,loan,...,pMlM,loan and
���� 2 over pb11,borrow,...,pMbM−−11,borrow. The expression in (11b)
��� equals that in (11a) because the threshold R1 > 0, so
the indicator in (11a) is 1 only if the bank has no liabil-
2 4 6 8 10 12 14 ities of seniority k (i.e., b = 0) for all k = 1,2,...,i−
k
����������������������������������=〈��〉 1; furthermore, because a bank’s in- and out-degrees
are independent, the indicator in Eq. (11a) factors
(�)22..23 to 1bk=0∀k∈{1,2,...,i−1}11>R1((cid:80)Mα=1lα). Equation (11c)
writesouttheexpectationinEq.(11b)bysummingover
〉
〈�� 2.1 (cid:126)l∈{0,1,2,...}M such that (cid:80)M l <1/R .
〈〉/�� 2.0 ObservefromEq.(11b)thatαJ=1Mαconsists1ofrowsthat
�� 1.9 arelinearlydependent. Thus,JM hasrankoneandonly
�
��� 1.8 one non-zero eigenvalue. Because the trace of a matrix
�
1.7 equals the sum of its eigenvalues, the largest eigenvalue
of JM is its trace, which, from Eq. (11c), is given by
1.6
0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24
������-������������������ M i−1 M
(cid:88)(cid:89) (cid:88) (cid:89)
trJM = pk,borrow ps,loanl .(12)
0 ls i
FreIlGat.iv4e.t(oCtohleoirroinntleinrbe)anIkf baassnektsslho+ldlles(si.ec.a,piiftathle(etqhureitsyh)olwd i=1k=1 (cid:126)l:(cid:80)Mα=1lα<1/R1s=1
J S
for junior-default, RJ, decreases), then (a) the cascade re- Therefore, we can analytically express λ (JM) by us-
gion (the set of parameters satisfying the first-order cascade max
ing the degree distributions and the threshold value R
condition(6)]enlarges,and(b)theoptimalratioofseniorto 1
on the most junior layer. This expression immediately
junior loans increases. The first-order cascade condition (6)
gives the first-order cascade condition, trJM >1.
considers defaults that occur due to one defaulted neighbor
(fordetails, seeAppendixC).Thus, whenR fallsbelowthe
J
inverseofaninteger,1/k,nodeswithtotaldegreek andwith
one defaulted neighbor will junior-default, so the cascade re- B. Distinguishing more seniority levels reduces risk
gionchangesatRJ =1/k,andhencetheoptimalratiojumps of large cascades of default
up at R =1/k.
J
If we realize that not all loans have the same se-
niority level, how does vulnerability to large cascades
change? Forsimplicity,weconsider“splitting”anErd˝os-
R´enyirandomgraphintoM independent,identicallydis-
By convention, sums of the form (cid:80)0 are zero.
k=1 tributedlayers(asin[18]). Thatis,wecompareasingle-
layer Erd˝os-R´enyi random graph with mean out-degree
z to a multiplex Erd˝os-R´enyi random graph with mean
out-degree z/M in each of the M layers.
The Jacobian matrix JM-ER for an M-layer directed
Erd˝os-R´enyirandomgraphwithedgedensity(cid:104)l (cid:105)ineach
i
A. First-order cascade condition layer i∈{1,2,...,M} is, from Eq. (11),
The cascade condition, a straightforward generaliza- JiMj -ER = (cid:104)li(cid:105)Γ((cid:100)Γ1/((cid:100)R11/(cid:101)R−(cid:101)1−,(cid:80)1)Mi=1(cid:104)li(cid:105))exp(cid:34)−(cid:88)j−1(cid:104)lk(cid:105)(cid:35),
1
tionoftheoneforM =2layersinSec.IIA2, isthatthe k=1
JacobianmatrixJM ∈RM×M haslargesteigenvalueex- (13)
9
where the incomplete gamma function Γ(x,y) ≡ numberofsenioritylevels(M)grows. Itispossiblethatif
(cid:82)∞ux−1e−udu and the gamma function Γ(x)≡Γ(x,0). there were more seniority levels, then banks would erode
y
Figure 5 shows the resulting cascade region (i.e., the their buffers against insolvency, R1, thereby pushing the
parameters satisfying trJM-ER > 1) for M ∈ {1,2,3,4} systembacktowardthecascaderegion, wherebankrupt-
manyErd˝os-R´enyilayersthateachhavemeanout-degree cies likely spread widely.
(cid:104)l (cid:105) = z/M. Here, the parameters are the threshold R
i 1
for default at the most junior level (horizontal axis) and
the mean total out-degree z (vertical axis). IV. HEAVY-TAILED DEGREE
DISTRIBUTIONS
30
Real networks of loans among banks show evidence
� ����-������������������ of somewhat heavy-tailed degree distributions [26–28].
= 25 �=��������������� For instance, the network of Brazilian banks in 2007–
�
� �=���������������� 2008 [28] has in- and out-degree distributions that have
�
� �=���������������� tails approximated by a power law ∝k−γ with exponent
�� 20
� �=���������������� γ ≈ 2.8, and the mean in- and out-degree are both ap-
�
�� proximately 8.5.
�
��� 15 To capture such heavy-tailed degree distributions, we
�� usethestaticmodelofscale-freenetworks[39,40],which
�
� allowstuningthemeandegreeandtheexponentγ ofthe
� 10
� degree distribution [41]. Because we lack access to data
�
��� onsenioritylevelsofloans,wefirstcreatearandomgraph
��� 5 with a certain mean out-degree, and then we assign a
� fraction of directed edges (chosen uniformly at random)
to be junior, and the remaining are senior.
0 Figure 6 shows that, in numerical simulations and in
0.05 0.10 0.15 0.20 0.25 0.30 0.35
the analytical approximation (6), the fraction of loans
���������� ��������������������
� that are junior can determine whether large cascades of
defaults tend to occur. Notice the “valley” in Fig. 6:
FIG. 5. Distinguishing more types of seniority reduces the
there is a narrow range of “optimal” values of the frac-
sizeofthecascaderegion. Inotherwords,withmoretypesof
tionofjuniorloans(horizontalaxisofFig.6)thatshrinks
seniority, the expected size of a default cascade is negligibly
the set of mean-degrees such that cascade size is large.
small for more pairs of thresholds R and expected number
1
of loans per bank z =(cid:80)M (cid:104)l (cid:105). Here we consider splitting a This optimal fraction of junior loans is approximately
i=1 i
directed Erdo˝s-R´enyi network into M ∈ {1,2,3,4} many in- 0.30, which translates to an optimal seniority ratio of
dependent, identically distributed layers, each with the same (cid:104)l (cid:105)/(cid:104)l (cid:105) ≈ (1−0.30)/0.30 ≈ 2.3, which is qualitatively
S J
mean degree (cid:104)li(cid:105)=z/M. similar to the optimal seniority ratio 1.79 found for mul-
tiplex Erd˝os-R´enyi networks in Fig. 3.
Notice in Fig. 5 that the cascade region shrinks with Notice in Fig. 6 that the empirical mean out-degree in
the number M of seniority levels, meaning that the sys- the Brazilian network (approximately 8.5) [28] intersects
tem is less vulnerable to large cascades of default. (By the black regions (where the mean cascade size is large)
contrast, in the model in [18], the cascade region grows wheneverthefractionofjuniorloansislargeorsmall. We
with M.) For some capital-to-interbank-asset ratios R , emphasizethatwedonotknowthecapitallevelsofthese
1
the cascade region is even eliminated as M grows (e.g., Brazilian banks, which affects the threshold for junior-
R1 =0.3 for M =1). default,RJ. However,Fig.6suggeststhatsenioritylevels
The reason the cascade region shrinks with M is that of loans in real-world networks could potentially affect
making a loan more senior makes it less significant early whether default might spread widely or not.
in a cascade. For instance, if M = 1, then a defaulted
nodereducestheassetsofallofitslenders(in-neighbors),
whereas if M > 1, then only the in-neighbors in layer 1 V. CONCLUSION AND DISCUSSION
lose an asset, so less contagion occurs. The decline in
importance of more senior layers is captured by the mul- We model seniority of debt using a multiplex network,
tiplicative factor (cid:81)i−1 pk,borrow in Eq. (11c) and by the and we study the likelihood of bankruptcy spreading
k=1 0
exponential factor in Eq. (13). In short, distinguishing widelyusingageneralizedcascadecondition. Themodel
more types of seniority makes more interbank debts in- has interesting tradeoffs. Whereas self-interested banks
significant (early in a crisis); hence a systemic crisis is wouldprefertomaketheirloansatthemostseniorlevel,
less likely. a mixture of junior and senior debts optimally reduces
This result hinges on the assumption that banks’ systemic risk, in that it minimizes the size of the set
buffers against insolvency R remain constant as the of network densities such that cascades are likely large.
1
10
away, such as interest rates for different seniority levels,
���������ϕ�
10 ��� which affect the incentives of banks.
Traditionally, cascades of losses in financial networks
���
8 havebeenstudiedusingtheEisenberg-Noealgorithm[32]
���
for computing a market-clearing payment vector (with
6 ���
�� ��� losses shared proportionally), which was recently ex-
�� 4 tended to multiple seniority levels [31]. The Eisenberg-
� �
����- 0.0 0.2 0.4 0.6 0.8 1.0 Ntimoeesa,lbguotritahsmCoinst eectoanlo.m[2i8c]apllyoinmteoaunti,ntghfeulalignorintohrmmaisl
�� 10 ���������ϕ� not so reasonable for modeling an extreme event such as
��
� ��� thedefaultofalargebank. Weinsteadtakeanapproach
��� 8 ��� typical in physics of studying ensembles of financial net-
�
��� works [1, 21]. We show that cascades on networks with
6
��� differentsenioritymaybeanalyzedwithavariantofmul-
4 ��� tiplex threshold cascade models studied in physics [18–
0.0 0.2 0.4 0.6 0.8 1.0 � 20].
���������������������������� The seniority of debt is just one dimension of multi-
plexity in financial networks. Other dimensions of debt
include the maturity of loans and whether loans are col-
FIG. 6. (Color online) For heavy-tailed degree dis-
lateralized [8, 9]. Furthermore, other important interac-
tributions like those in the Brazilian interbank lending
network [28], having ≈ 30% of loans be junior (and the rest tions include the trade of financial assets [43] and com-
senior) reduces systemic risk, in that it minimizes the size mon exposures (i.e., multiple institutions holding the
of the set of mean total out-degrees such that bankruptcy same financial assets) [9, 44].
spreads widely. The gray-scale background shows the We call our model “multilevel contagion” because a
averagefractionofbanksinjunior-default(toppanel)andin node can become “contagious” in one layer without nec-
senior-default (bottom panel) in simulations on 2400 banks, essarily becoming contagious on another layer (e.g., a
begun from 10 banks initially in senior-default, averaged
bank can be in junior-default but not in senior-default).
over 30 simulations. The green lines mark the boundary of
By contrast, in other models of threshold contagion in
the cascade region [λ (J) = 1 with J computed from
max multiplex networks [18–20] and in variants of percola-
Eq. (C4)]. The junior-default threshold of every bank is
tiononmultiplexnetworks[13–17],whenanodechanges
R = 0.18, as in Fig. 3. Directed graphs are generated with
J
thestaticmodel[39,40]without-degreedistributionshaving state in one layer, it necessarily changes state in other
tail behavior ∝ k−γ with γ = 2.83, which approximates the layers. Multilevelcontagionmightgiveinsightintoother
value found in the Brazilian interbank lending network [28]. kinds of social contagion. For example, one might use it
A fraction of loans (horizontal axis) chosen uniformly at to model the adoption of a new product if “fans” of the
random are junior, and the rest are senior. The “optimal” product spread influence to neighbors of one type but
fraction of junior loans (that minimizes the height of the “super-fans” spread influence to neighbors of two types
green line) is ≈ 0.30. For the mean out-degree (vertical
(amultiplexversionofthemultistagecomplexcontagion
axis)of≈8.5observedforBrazilianbanks[28],theexpected
model [25]). We hope our model will stimulate further
cascadesizeissmallifandonlyifanintermediatefractionof
researchonthesemanytypesofcascadesinsocial,finan-
loans are junior.
cial, and other multiplex networks.
ACKNOWLEDGMENTS
Such disparity between individual interests and system-
wide stability is the “regulator’s dilemma” [42]. For two C.D.B. is supported by the James S. McDonnell Post-
typesofseniority, andforedgesplaceduniformlyatran- doctoral Fellowship in Studying Complex Systems. T.K.
dom(i.e.,uncorrelatedErd˝os-R´enyinetworks)oraccord- issupportedbyKAKENHI25780203and24243044. The
ing to a heavy-tailed degree distribution, we show that authorsthanktheIsaacNewtonInstituteforMathemat-
theoptimalmultiplexnetworkshaveontheorderof50% ical Sciences (Cambridge, UK) for support and hospital-
to 100% more senior debts than junior debts. To give ity during the programme Systemic Risk: Mathematical
theoretical support for policy suggestions, however, will ModellingandInterdisciplinaryApproaches, wherework
requireconsideringotherfactorsthatwehaveabstracted on this paper was undertaken.
Appendix A: Thresholds of default for M levels of seniority
Here we consider the general case of M ≥ 1 levels of seniority of interbank debt (rather than just two, junior and
senior), and we label the levels of seniority by α∈{1,2,...,M}. We choose the convention that debts of seniority α