Fuzzy Contagion Cascades in Financial Networks

Giuseppe De Marco

1,2

, Chiara Donnini

1

, Federica Gioia

1

and Francesca Perla

1

1

Department of Management and Quantitative Sciences, University of Naples Parthenope,

Via Generale Parisi 13, Napoli 80132, Italy

2

Center for Studies in Economics and Finance, University of Naples Federico II, Italy

Keywords:

Financial Networks, Fuzzy Financial Data, Degree of Default, Fixed Point.

Abstract:

Previous literature shows that ﬁnancial networks are sometimes described by fuzzy data. This paper extends

classical models of ﬁnancial contagion to the framework of fuzzy ﬁnancial networks. The degree of default

of a bank in the network consists in a (real valued) measure of the fuzzy default and it is computed as a ﬁxed

point for the dynamics of a modiﬁed ”ﬁctitious default algorithm”. Finally, the algorithm is implemented in

MATLAB and tested numerically on a real data set.

1 INTRODUCTION

It is well known that the banking system is connected

in a network by the mutual exposures that banks and

other ﬁnancial institutions assume towards each other

and this kind of interbank exposures are recognized

as a source of ﬁnancial crisis known as contagion

cascade. The literature on ﬁnancial stability has in-

creased signiﬁcatively in last years (see for instance

Glassermann and Young (2016) or Hurd (2016) for re-

cent surveys); however, the issue of the lack of precise

information about the overall interbank exposures in

the system has not been exhaustively investigated.

This is an important problem as banks are obliged to

show their exposures within the balance sheet only

few times per year. The present paper studies a ﬁnan-

cial network model under imprecise data in which in-

terconnections are represented by fuzzy numbers and

provide mathematical and computational tools in or-

der to exploit the information arising from this model.

The paper by Eisenberg and Noe (2001) shows

that obligations of all banks within the system are

determined simultaneously by ﬁxed point arguments.

develops an algorithm

1

that converges to a clearing

payment vector but, at the same time, gives informa-

tion about the systemic risk in the systems.

On the other hand, Furﬁne (2003) is the ﬁrst

paper which studies the ﬁnancial contagion arising

from interbank exposures according to more realis-

tic data that are based on daily observations along

a two months period. Furﬁne ﬁnd interbank expo-

1

Known as the ﬁctitious default algorithm.

sures by looking at the transaction data in the Federal

Reserve’s large-value system (Fedwire). More pre-

cisely, he focuses only on federal funds transactions

that are deduced from the Fedwire during February

and March 1998

2

. Furﬁne (2003) found 719 commer-

cial banks trading on the Fedwire and approximately

60000 federal fund transactions in the period taken

into account. In Furﬁne’s approach is that banks are

classiﬁed into four groups according to the volume of

funds traded. The exposure of a bank from one group

in another bank from another group is expressed in

terms of minimum, maximum and average value of

the transactions between the two groups observed in

the sample period. Furﬁne does not fully exploit the

information arising from these data. Our interpreta-

tion of Furﬁne’s data is instead that they can be read

as triangular fuzzy numbers where the minimum and

the maximum are obviously the inf and the sup of the

support and the average is the maximum point for the

membership function. This interpretation is the key

motivation of our paper: on the one hand it shows

that fuzzy data appear naturally in ﬁnancial networks,

on the other hand, we have already a detailed data set

of fuzzy interbank exposures that can be used to run

simulations.

2

Furﬁne identiﬁes federal funds transactions as follows:

ﬁrstly, payments greater than $1 million and ended in ﬁve

zeros were identiﬁed as candidates. For each candidate pay-

ment, another payment between the same two banks in the

opposite direction is searched the following day (plus in-

terest). If such opposite payment is found, then the ﬁrst

payment is considered as federal fund exposure.

Marco, G., Donnini, C., Gioia, F. and Perla, F.

Fuzzy Contagion Cascades in Financial Networks.

DOI: 10.5220/0006530603010305

In Proceedings of the 10th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2018) - Volume 2, pages 301-305

ISBN: 978-989-758-275-2

Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

301

In this paper, we follow the Eisenberg and Noe

(2001) approach: we use ﬁxed point arguments to

show existence of clearing vectors and construct a

suitable adjustment of the ﬁctitious default algorithm

to our fuzzy model. In our approach, the balance sheet

of each bank is a triangular fuzzy number, called fuzzy

net worth; then, we construct index of fuzzy default

functions that assign to every fuzzy net worth a real

(or crisp) degree of fuzzy default. For each degree of

fuzzy default, clearing payment vectors are then con-

structed. We focus on two speciﬁc models: the op-

timistic δ model and the pessimistic σ model. They

depend on the way the fuzzy net worth is greater than

0; namely, the optimistic model measures the set of

all the alpha-cuts having not empty intersection with

R

+

, while, the pessimistic model measures the set of

all the alpha-cuts which are subsets of R

+

. The vector

of degrees of default is characterized as a ﬁxed point

of the dynamics of a modiﬁed ﬁctitious default algo-

rithm. Finally, the algorithm is implemented in MAT-

LAB using Furﬁne’s (2003) data; simulations show

that contagion spreads only within smaller banks as

it was also shown in Furﬁne 2003.

2 FUZZY NUMBERS

In this section we recall some key notions and re-

sults from the theory of fuzzy numbers that are re-

quired in our model (see, for example, Buckley and

Eslami (2002), Klir and Yuan (1995) and Zimmer-

mann (2001) for extensive surveys and references).

Given a universal set X, a fuzzy subset A of X is a

function which associates with each point in X a real

number in the interval [0, 1]. That function is called

membership function.

A fuzzy number n is a particular fuzzy subset of R,

with membership function denoted by µ

n

, such that

1. the core of n, i.e. the set co(n) = {x ∈ X | µ

n

(x) =

1}, is non-empty;

2. the α-cuts of n, i.e. the sets {x ∈ X |µ

n

(x) > α},

are all closed, bounded, intervals, for every α ∈

]0, 1]

3

;

3. the support of n, i.e. the set supp(n) = {x ∈

X |µ

n

(x) > 0}, is bounded

4

.

3

We remark that for each α ∈]0, 1] the α-cut of n is

always [n(α), n(α)]. Moreover if 0 < α

1

< α

2

6 1, then

n[α

2

] ⊆ n[α

1

].

4

We remind that the core of n is n[1], while the support

of n is not n[0], since the 0-cut is always the whole universal

set. Moreover, since the support of n has to be bounded,

there exists a positive real number l so that the support of n

is a subset of [−l, l].

A fuzzy number n is said to be positive if

infsupp(n) > 0, or, equivalently, if supp(w

i

) ⊆ R

+

;

n is said to be negative if sup supp(n) < 0, or, equiv-

alently, if supp(w

i

) ⊆ R

−

. We can trivially observe

that there are fuzzy numbers that are not positive nei-

ther negative. With abuse of notation we will indicate

that n is positive (negative) with n > 0 (n < 0).

A triangular fuzzy number n is a continuous fuzzy

number

5

such that the core is a singleton, i.e.

co(n) = {ˆn}. Denote with n = inf supp(n) and n =

supsupp(n), then the triangular fuzzy number n is de-

noted by n = (n, ˆn, n), while its membership function

is deﬁned as follows

µ

n

(x) =

x − n

ˆn − n

, if n 6 x 6 ˆn;

x − n

ˆn − n

, if ˆn < x 6 n;

0, otherwise.

We denote by N the set of triangular fuzzy numbers.

For the computation of the sum of triangular fuzzy

numbers and the product of a triangular fuzzy number

by a real number, we can use the following rule:

Given three triangular fuzzy numbers n = (n, ˆn, n),

m = (m, ˆm, m), l = (l,

ˆ

l, l) and a real number a,

i) n + m − l = (m + n − l, ˆn + ˆm −

ˆ

l, n + m − l)

ii) an =

(an, a ˆn, an), if a > 0;

(an, a ˆn, an), if a < 0;

0, if a = 0.

(1)

3 NETWORKS OF BANKS

Banks, Balance Sheets and Fuzzy Default

The market consists in a set of banks I = {1, 2, . . . , n}.

Each bank i is characterized by its balance sheet

which, in turn, consists in assets and liabilities.

The bank’s assets are:

i) Outside assets c

i

: aggregate claims of bank i on

nonﬁnancial entities;

ii) In-network assets p

ki

, for each k 6= i. Each p

ki

is

the claim of bank i on bank k, that is, a payment

obligation of bank k to bank i and is the aggregate

exposure of bank i in the bank k.

The bank’s liabilities include:

5

A continuous fuzzy number is a fuzzy number having a

continuous membership function.

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

302

i) Obligations b

i

to nonﬁnancial entities;

ii) Obligations p

ik

, for each k 6= i, to the bank k.

In the literature, the matrix (p

ik

)

n

i,k=1

is the adja-

cency matrix of an directed network, called ﬁnancial

network. Each node is a bank, and a directed edge

runs from node i to node k if bank i has a payment

obligation to node k. In this case we say that bank i is

connected to bank k. All entities outside the network

can be represented through a single node representing

the ”outside”.

The difference between the bank i’s assets and li-

abilities is the net worth w

i

. Following the previous

literature, we assume that all debt obligations have

equal priority and the assets are distributed to cred-

itors from each bank k in proportion η

k

, where η

k

= 1

if the bank k is able to honor all its debts with cer-

tainty, η

k

∈]0, 1[ if it is possible that k is not able to

honor all its debts

6

.

Therefore the asset side of node i’s balance sheet

is given by

c

i

+

∑

k6=i

η

k

p

ki

and the liability side by

b

i

+

∑

k6=i

p

ik

.

The node’s net worth is

w

i

= c

i

+

∑

k6=i

η

k

p

ki

− b

i

−

∑

k6=i

p

ik

. (2)

The previous formula of the net worth is standard

in the literature on contagion (see Glasserman and

Young (2016) or Hurd (2016)). Aim of this paper

is to extend the previous in case of triangular fuzzy

numbers. It is well known that in the crisp case, the

default of a bank corresponds to a negative net worth.

In the framework of the present paper, in which the

net worth is a fuzzy number, the classical ”binary”

concept of default is inadequate. Therefore a suitable

deﬁnition of fuzzy default must be given.

Deﬁnition 3.1. We say that:

i) A bank i defaults with certainty if its net worth

w

i

< 0, (i.e. supp(w

i

) ⊆ R

−

).

ii) A bank i does not default if w

i

> 0, (i.e.

supp(w

i

) ⊆ R

+

).

ii) A bank i incurs in a fuzzy default if supp(w

i

) ∩

R

−

6=

/

0

7

.

6

The term ’possible’ refers to the situation of fuzzy de-

fault as it will be explained below.

7

We remark that a certainty default is a particular case

of a fuzzy default.

Degree of Default as a Fixed Point

In this subsection we construct a model which gives

for every bank and every fuzzy net worth a reason-

able vector of proportions η and a proper measure

of default. The model will be assigned by a pair of

functions (Λ, g) which speciﬁes a supposed degree of

default Λ(ω) given a fuzzy default ω (that is, Λ is

defuzziﬁed ”degree of default”), and the proportions

η

k

= g(λ

k

) for every bank k and every degree of de-

fault λ

k

. The pair (Λ, g) will be asked to satisfy spe-

ciﬁc properties.

Recall that for every triangular fuzzy number n,

n = sup supp(n), n = inf supp(n) and ˆn is the element

of the core. Then,

Deﬁnition 3.2. Let %

L

be the binary relation on N

deﬁned by

n %

L

m ⇐⇒

i) n > m,

ii) n > m,

iii) ˆn > ˆm.

We say that n is L-related to m, if n %

L

m.

Moreover

Deﬁnition 3.3. We say that a function Λ : N → R is

L-decreasing if and only if

n %

L

m =⇒ Λ(n) > Λ(m). (3)

Then we introduce the model as follows

Deﬁnition 3.4. A defuzziﬁed measure of default is a

pair of functions (Λ, g) where

i) Λ : N → [0, 1] is a L-decreasing function such that

Λ(w) =

0 if w > 0 (supp(w) ⊆ R

+

)

1 if w < 0 (supp(w) ⊆ R

−

)

Λ is called index of fuzzy default and λ

i

= Λ(w

i

)

represents the degree of default of bank i when its

net worth is w

i

.

ii) g : [0, 1] → [0, 1] is a decreasing function such that

g(0) = 1. The term η

i

= g(λ

i

) gives the proportion

of debts that bank i is ”supposed” to distribute to

the other banks if i incurs in a degree of default

equal to λ

i

.

Remark 3.5. Every decreasing function g is suitable

from a theoretical point of view even if, in examples

and simulations, we will consider the simple func-

tional form

g(λ

i

) = 1 − λ

i

,

which represents a good approximation of the relation

between the likelihood of default and expected rate of

debt repayment.

Fuzzy Contagion Cascades in Financial Networks

303

Note also that the assumption of a decreasing re-

lation between degree of default λ

i

and the propor-

tion η

i

is natural as the greater is the likelihood of

default the lower is the perception of solvability of i

and, therefore, the lower is the expected rate of debt

repayment.

Let (Λ, g) be a defuzziﬁed measure of default. De-

note with

H

i

((η

k

)

k6=i

) := c

i

+

∑

k6=i

η

k

p

ki

− b

i

−

∑

k6=i

p

ik

,

and with F

i

: [0, 1]

n−1

→ [0, 1] the function deﬁned by

F

i

((λ

k

)

k6=i

) = Λ

H

i

(g(λ

k

))

k6=i

.

Let F : [0,1]

n

→ [0, 1]

n

be the function deﬁned by

F(λ

1

, ...,λ

n

) =

(F

1

((λ

k

)

k6=1

), ...,F

n

((λ

k

)

k6=n

)) = (F

i

((λ

k

)

k6=i

))

i=1,...,n

.

(4)

Then, it follows that

Proposition 3.6. Let (Λ, g) be a defuzziﬁed mea-

sure of default and λ = (λ

1

, ...,λ

n

) ∈ [0, 1]

n

where

λ

i

= Λ(w

i

) is the degree of default associated to a net

worth w

i

, for every bank i = 1, . . . ,n. Then λ is a ﬁxed

point for F, i.e. λ = F(λ).

We have

Theorem 3.7. Let (Λ, g) be a defuzziﬁed measure of

default, then the function F, deﬁned as in (4), admits

a ﬁxed point.

4 OPTIMISTIC AND

PESSIMISTIC INDEXES OF

FUZZY DEFAULT

This section focuses on two particular examples of in-

dexes of fuzzy default which have an interesting inter-

pretation and allow for simple computations.

Deﬁnition 4.1. The function Λ

σ

: N → [0, 1] deﬁned

by

Λ

σ

(w) =

µ

w

(0), if ˆw > 0;

1, if ˆw < 0.

∀w ∈ N (5)

is said to be pessimistic index of default.

and

Deﬁnition 4.2. The function Λ

δ

: N → [0, 1] deﬁned

by

Λ

δ

(w) =

1 − µ

w

(0), if ˆw 6 0;

0, if ˆw > 0;

∀w ∈ N (6)

is said to be optimistic index of default.

Interpretation

1: If there is no default with certainty (w > 0), then

Λ

δ

(w) = Λ

σ

(w) = 0.

2: If there is default with certainty (w < 0), then

Λ

δ

(w) = Λ

σ

(w) = 1.

3: Λ

δ

(w) 6 Λ

σ

(w) ∀w ∈ N .

4: Suppose that there is fuzzy default with ˆw > 0.

That is

w < 0 < ˆw < w

This fuzzy net worth represents the situation in

which the values that are the most likely to occur

are positive, but there is yet the possibility that

negative values occur, even if with a small mem-

bership.

The optimistic index gives a degree Λ

δ

(w) = 0.

The pessimistic index gives a degree Λ

σ

(w) =

µ

w

(0) which is the measure of the range interval

[0, µ

w

(0)] of all the values α whose α-cuts include

at least a negative value. Intuitively, the larger is

the interval then more likely is the possibility of

default.

5: Suppose that there is fuzzy default with ˆw < 0.

That is

w < ˆw < 0 < w

This fuzzy net worth represents the situation in

which the values that are the most likely to oc-

cur are negative but there is yet the possibility that

positive values occur, even if with a small mem-

bership.

The pessimistic index gives a degree Λ

δ

(w) = 1.

The optimistic index gives a degree Λ

δ

(w) = 1 −

µ

w

(0) which is the measure of the range interval

[µ

w

(0), 1] of all the values α whose α-cuts include

all negative values. Intuitively, even in this case,

the larger is the interval then more likely is the

possibility of default.

6: The case w < ˆw = 0 < w obviously gives Λ

δ

(w) =

0 and Λ

σ

(w) = 1, which is the largest possible dif-

ferences between the two degrees. This sounds

reasonable as ˆw = 0 is the case in which uncer-

tainty is maximal, since there are no reasons to

believe that negative values occur more likely than

positive ones and viceversa.

5 FUZZY CONTAGION

In this section, we propose a model of default cascade

in the our framework in which net worths are triangu-

lar fuzzy numbers. In particular, we extend the classi-

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

304

cal ”ﬁctitious default algorithm” introduced in Eisen-

berg and Noe (2001) (see also Glasserman and Young

(2016)) to our fuzzy framework. The dynamic pro-

cess is constructed in general for arbitrary defuzziﬁed

measures of fuzzy contagion, but we will look also at

the particular cases of the Λ

δ

and Λ

σ

functions deﬁned

in the previous section.

The Contagion Dynamics

The exogenous shock is parametrized as a vector

x = (x

1

, . . . ,x

n

)

where each x

i

is a triangular fuzzy number which rep-

resents the exogenous shock which affects the (ante-

shock) capital of bank i which, in turn, is character-

ized by the difference b

i

− c

i

8

. In particular, we study

the fuzzy default cascade step by step, computing in

each step the net worth and the associated degree of

default and show the convergence of the dynamics to

a ﬁxed point.

In particular the cascade is constructed as follows:

- at step h = 1 the exogenous shock x occurs. The

net worth of each bank i is given by:

w

1

i

= H

i

((η

0

k

)

k6=i

) = c

i

+

∑

k6=i

η

0

k

p

ki

−b

i

−

∑

k6=i

p

ik

−x

i

,

where each η

0

k

= 1 meaning that every bank is

solvable before the exogenous shock.

For each i, we then compute

λ

1

i

= Λ(w

1

i

), η

1

i

= g(λ

1

i

).

- at each step h, given the the vectors

w

h−1

= (w

h−1

1

, w

h−1

2

, . . . ,w

h−1

n

)

λ

h−1

= (λ

h−1

1

, λ

h−1

2

, . . . ,λ

h−1

n

)

η

h−1

= (η

h−1

1

, η

h−1

2

, . . . ,η

h−1

n

),

we compute, for each i,

w

h

i

= H

i

((g(λ

h−1

k

))

k6=i

)

= c

i

+

∑

k6=i

g(λ

h−1

k

)p

ki

− b

i

−

∑

k6=i

p

ik

− x

i

λ

h

i

= Λ(w

h

i

)

η

h

i

= g(λ

h

i

),

- therefore we get a sequence (λ

h

)

h∈N

⊂ [0, 1]

n

as

follows: By construction,

λ

h

i

= Λ

H

i

((g(λ

h−1

k

))

k6=i

)

=

8

Indeed the exogenous shock could also have been im-

plicitly included in the capital b

i

−c

i

without any exogenous

parameter x. Therefore it could be possible to study the ﬁ-

nancial contagion for every choice of capital b

i

− c

i

.

F

i

(λ

h−1

k

)

k6=i

∀i = 1, . . . , n ;∀h ∈ N;

being F = (F

1

, . . . ,F

n

), it follows that

λ

h

= F

λ

h−1

∀h ∈ N.

It immediately follows that the stationary points

for this sequence are ﬁxed points for the function

F, called degree of fuzzy default. We will show

below the convergence of the sequences in the op-

timistic and in the pessimistic models.

Remark 5.1 (Simulations). The algorithm, proposed

in the present paper, has been implemented in MAT-

LAB and tested numerically on a real ﬁnancial data

set in order to analyze the contagion dynamics for the

case of fuzzy input data. The analysis of the result of

our simulation conﬁrms Furﬁne’s prediction that only

small banks may be affected by contagion.

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831.

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305