This contribution presents a network of interdependent firms in which the spatial diffusion of the business relations is described by an entropy spatial interaction model. This network is used in a credit risk model in order to take into account the counterparty risk and describe the resulting contagion effects.credit risk contagion, networks, counterparty risk, entropy spatial models

Antonella Basso

Diana Barro

English

Research Papers in Economics

Department of Applied Mathematics,  University of Venice  
 
 
 
WORKING PAPER SERIES 
 
 
 
 
 
 
 
 
Barro Diana  and  Basso Antonella 
 
 
A network of business relations to model 
counterparty risk 
 
 
 
 
 
 
 
 
 
 
Working Paper n. 171/2008 
November 2008 
 
ISSN: 1828-6887 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
This Working Paper is published under the auspices of the Department of Applied 
Mathematics of the Ca’ Foscari University of Venice. Opinions expressed herein are 
those of the authors and not those of the Department. The Working Paper series is 
designed to divulge preliminary or incomplete work, circulated to favour discussion 
and comments. Citation of this paper should consider its provisional nature. A network of business relations to model
counterparty risk¤
Diana Barro Antonella Basso
<d.barro@unive.it> <basso@unive.it>
Dept. of Applied Mathematics Dept. of Applied Mathematics
University of Venice and SSAV University of Venice and SSAV
(November 2008)
Abstract. This contribution presents a network of interdependent ¯rms in which the spatial
di®usion of the business relations is described by an entropy spatial interaction model. This
network is used in a credit risk model in order to take into account the counterparty risk
and describe the resulting contagion e®ects.
Keywords: credit risk contagion, networks, counterparty risk, entropy spatial models.
JEL Classi¯cation Numbers: D61, C63.
MathSci Classi¯cation Numbers: 65C05, 90B99.
Correspondence to:
Antonella Basso Dept. of Applied Mathematics, University of Venice
Dorsoduro 3825/e
30123 Venezia, Italy
Phone: [++39] (041)-234-6914
Fax: [++39] (041)-522-1756
E-mail: basso@unive.it
¤ The authors were supported in part by a PRIN grant of MIUR.1 Introduction
The main aim of this contribution is to present a model for the description of the dependence
among defaults of ¯rms in an interrelated economy. The dependence we are interested in is
the one implied by the so called counterparty risk, de¯ned by [6] as the risk that the default
of a ¯rm's counterparty might a®ect its own default probability.
The problem is widely studied and it is of interest for the analysis of the credit risk of
portfolios of dependent positions, such as, for example, portfolios of bank loans to enter-
prises. Di®erent approaches have been proposed to tackle this issue.
A ¯rst class of models introduces dependence through the presence of common factors,
which determine the behavior of the economy and are linked to the business cycle. In
this way it is possible to reduce the dimension of the correlation matrix which has to be
considered, turning the attention from the correlation among positions to the correlation
among factors, which are usually considerably fewer than the number of positions. An
advantage of this widely accepted and widely applied approach is that conditionally on the
factors the positions are mutually independent.
Nevertheless, empirical evidence gathered in a number of studies shows that the depen-
dence among defaults cannot be fully explained using only factor models (see, for example,
[6], [4], [2]). In particular, default events exhibit clustering behaviors, especially in recession
periods, and ¯rms seem to show correlation in their defaults due not only to a dependence
on some common risk factors, but also to some ¯rm-speci¯c risks. The counterparty risk
introduces an additional source of dependence and a mechanism of contagion through which
the ¯nancial distress can spread in the economy.
Recent empirical studies support the presence of contagion e®ects in the di®usion of
defaults (see, for example, [4] and the references therein). Taking into consideration this
evidence, we propose to extend a factor model that describes the value of a ¯rm in order to
include an idiosyncratic term which takes into account the business connections with other
¯rms in the economy.
Di®erent contributions in the literature rely on the introduction of network-like struc-
tures to model the dependence among positions in a portfolio of risks, see for example [3],
[4], [7]. To model the connections among ¯rms in the economy we propose to introduce
a weighted network based on a spatial interaction model, with directed arcs which can
accounts both for the presence of business relations and for their intensity.
The structure of the paper is as follows. In Section 2, we introduce the use of net-
works to model business relations. Section 3 describes our network model including spatial
interaction, while in Section 4 we describe the credit contagion model. Section 5 concludes.
2 Networks for business relations
Many phenomena where interaction plays a crucial role are modeled using networks. There
are di®erent types of networks which allow to describe di®erent behavior of a system of in-
teracting or connected elements. The structure of a network can be analyzed by considering
some important properties such as the connectivity of each vertex and the strength of each
1interaction.
The graphical representation of networks enables us to easily grasp the essence of the
interactions. The elements of the system are usually represented using nodes while (di-
rected) edges describe the (directed) relationships which occur among nodes. Each arc can
be assigned a weight, which is usually interpreted as the intensity of the relation.
Many recent contributions in the credit risk literature introduce network structures to
explicitly model the interactions among the positions in a portfolio of ¯rms, in order to
describe the microeconomic foundations of dependence and thus capture the e®ect of a
contagion mechanism.
Some of these approaches try to adapt some models from physics in order to describe the
economic interactions; see, for example, [7], in which the authors apply a lattice gas model
to describe couplings among counterparties. Other contributions in the literature propose to
use weighted networks to give a stylized description of the interaction among counterparties
and introduce some simplifying hypotheses on the structure of the dependence to make the
model analytically tractable; see, for example, [4], [3]. In [10] the authors apply queueing
theory to analyze the behavior of networks with looping lending relationships. Other recent
applications of network theory to the analysis of credit risk problems include the description
of interbank payment °ows; see e.g. [8]. Furthermore, in [5] some business connection
variables such as the intensity of the relationships and the distance are considered as relevant
explanatory variables to describe the recovery rates of bank loans in Germany.
3 A business relation network with spatial interaction
In order to study the propagation of the defaults in a system of ¯rms and its e®ects on the
assessment of credit risk, we build a model in which the business connections are modeled
using a directed weighted network where the nodes represent the ¯rms, directed edges
between ¯rms represent the business connections and the weight associated to each edge
gives a measure of the intensity of the connection.
The existence of the directed arc (k;i), connecting the origin node k to the destination
node i, means that the (possible) default of ¯rm k will cause some ¯nancial distress to ¯rm
i. The precise way in which we model this distress will be discussed in next session, but it
relies on the existence of a client-supplier relation.
More precisely, we choose as an indicator of the presence of a connection with k which
may cause distress to i the fact that ¯rm k is a client of ¯rm i. The basic idea is that
if a client which represents a signi¯cant percentage of the turnover of ¯rm i, let us say a
percentage above a given threshold, defaults, this causes di±culties to ¯rm i and may result
in a serious distress for the ¯rm.
A natural choice for the measure of the intensity of the connection is given by the the
percentage of sales on the turnover of the ¯rm; therefore, the weight associated to the
edge (k;i), wki, is given by the percentage of sales to client k on the turnover of ¯rm i.
An example of a network of business connections is shown in ¯gure 1, which depicts the
incoming and outgoing arcs for two generic nodes k and i which are connected; notice that
the presence of the arc (k;i) indicates that ¯rm k is a client of ¯rm i and the weight wki
2k i
wki
Figure 1: Example of network of business connections.
gives the per cent sales to k on the turnover of i. Of course, di®erent and more general
choices could be made, but this de¯nition of the business connections is straightforward and
fairly natural.
In our model the edges in the network are determined by considering a spatial dimension
and resorting to an entropy spatial interaction model constrained to origins which takes
into account both the economic weight of the di®erent ¯rms and their distance; on spatial
interaction models see, for example, [12].
As it is known, entropy spatial interaction models can be obtained by maximizing the
entropy of the system under the available information on the distance matrix between the
origin and destination nodes and on the weights assigned to these origins and destinations,
which account for the size of the outgoing (for origins) or incoming (for destinations) °ows.
Let us indicate with n the number of ¯rms in the system, and let us consider a
generic ¯rm i 2 f1;2;:::;ng. Let us denote by ¼ki the probability that ¯rm k, with
k 2 f1;2;:::;ng, is connected to ¯rm i, i.e. that it is a client of ¯rm i, and by dki the
distance between ¯rms k and i. Moreover, let s(k) denote the economic sector of ¯rm k
and let Wk be a weight representing the relative attractiveness of ¯rm k; in the spatial
interaction model the weight Wk can naturally be de¯ned as the overall turnover of ¯rm k.
Let us observe that the probability ¼ki that ¯rm k is a client of ¯rm i coincides with
the probability that the directed arc (k;i) exists in the network and can be decomposed
into the product of the probability Ps(i)s(k) that a client of i pertains to sector s(k) and the
probability Qki that ¯rm k is a client of ¯rm i conditional to the fact that the client of i
pertains to sector s(k)
¼ki = Ps(i)s(k) Qki: (1)
In order to compute these probabilities we assume that the probability Ps(i)s(k) that a
client of i pertains to sector s(k) (i.e. to the same sector of ¯rm k) is the same for all ¯rms
in the same sector, and that it coincides with the percentage of the output of sector s(i)
sold to sector s(k), so that it can be computed as follows
Ps(i)s(k) = Probfclient of i 2 s(k)g =
As(i)s(k)
PS
j=1 As(i)j
; (2)
where S is the number of sectors and the S £ S matrix A =
¡
As(i)s(k)
¢
is the input-output
table of the economy, in which the element As(i)s(k) represents the output of sector s(i) sold
3to sector s(k).
Moreover, in order to compute the conditional probability Qki we apply an entropy
spatial interaction model constrained to origins which makes this probability depend on the
relative attractiveness of the di®erent ¯rms and their distance as follows
Qki = Probfk is a client of i j client of i 2 s(k)g = Wk
e¡®s(i)dki
P
j2Ss(k) Wje¡®s(i)dji (3)
where Ss(k) denotes the set of ¯rms in sector s(k) and ®s(i) is a positive real parameter,
dependent on the sector s(i), which determines the relevance of the e®ect of distance on the
business relations.
It can be seen that the sum of the probabilities ¼ki on all k 2 f1;2;:::;ng is equal to
1 for all nodes i 2 f1;2;:::;ng.
Once determined which arcs exist in the network, we need to compute their weights,
where the weight wki of arc (k;i) is given by the per cent sales to client k on the turnover
of ¯rm i. Formally, let us indicate with I(k;i) the indicator function of arc (k;i), which
takes value 1 if the arc (k;i) exists and value 0 otherwise. We make the assumption that
the weight wki on the arc (k;i) is proportional to the output of sector s(i) sold to sector
s(k) and to the weight Wk that determines the relative attractiveness of ¯rm k, i.e. to the
turnover of ¯rm k, as follows
wki =
As(i)s(k)
PS
j=1 As(i)j
Wk I(k;i) P
j2Ss(k) Wj I(j;i)
: (4)
The business relation network de¯ned in such a way will be used in next section to
model the contagion of defaults in the system.
4 The credit contagion model
The network of ¯rms described in the previous section can be applied to model the coun-
terparty risk, which gives rise to a contagion e®ect and may provoke clustering of defaults
during crisis periods.
To this aim we use the network of business relations to model the connections in the
system of ¯rms applying a discrete time model analogous to that proposed in [1]; this model
describes the asset value of ¯rms by taking into account the counterparty risk of the ¯rms
connected by arcs in the network.
The asset value of ¯rm i at time t, Vi(t), is modeled as the sum of three components: a
macroeconomic component F, modeled using a factor model which takes into account the
in°uence of the business cycle (on factor models see, for example, [11], [9]), a microeconomic
component M which introduces a contagion e®ect due to the business connections with other
¯rms and a residual idiosyncratic term ":
Vi(t) = Fi(t) + Mi(t) + "i(t) t = 0;1;::: (5)
4The value of the macroeconomic component Fi(t) is described by the following factor
model:
Fi(t) =
J X
j=1
¯
s(i)
j Yj(t) t = 0;1;:::; (6)
where Y (t) = (Y1(t);Y2(t);:::;YJ(t)) is the vector of the values at time t of the driving
factors, s(i) represents the economic sector of ¯rm i and ¯s
j is the weight of factor j for the
¯rms of sector s. The driving factors Yj, with j 2 f1;2;:::;J g, follow a vector stochastic
process modeled according to the nature of the macroeconomic factors considered.
The microeconomic component, Mi(t), takes into account the e®ects of past distresses
on the health of ¯rm i and is de¯ned as follows
Mi(t) = ¹s(i)
1 X
¿=1
¸¿
s(i)
"
p(t ¡ ¿) ¡
X
k2Ci(t¡¿)
±k(t ¡ ¿)wki(t ¡ ¿)
#
; (7)
where ¹s 2 R+ is a real parameter dependent on the economic sector of the ¯rm, ¸s, with
0 · ¸s < 1, is a dampening factor which determines the distress memory of ¯rms in sector
s, p(t) is the average default rate of the economy at time t, Ci(t) is the set of clients of ¯rm
i, i.e. the subset of nodes in the network which are origins of arcs entering node i, ±k(t) is
the indicator function of default of client k which takes value 1 if client k defaults at time
t and wki is the weight of arc (k;i), given by the percentage of the sales to client k on the
turnover of ¯rm i.
The quantity within brackets represents a measure of the distress undergone by ¯rm i
at time (t ¡ ¿) due to the defaults observed among its clients. This distress measure is the
di®erence between the average default rate of the economy, p(t ¡ ¿), and the percentage of
turnover of ¯rm i sold to clients which defaulted in period (t¡¿); therefore it has a negative
value if the ¯rm su®ered a rate of defaults of clients higher than the average rate of the
economy, and a positive value in the opposite case. Notice that this distress measure a®ects
the health of the ¯rm with a one-period delay and its e®ects are dampened according to an
exponential decay in time.
Furthermore, we have a residual idiosyncratic term "i(t), which is assumed to be nor-
mally distributed with zero mean and standard deviation ¾"i. As is usual in factor models,
we assume that the residual idiosyncratic terms "1(t);"2(t);:::;"N(t) are both mutually
independent and independent of the factors Y1;Y2;:::;YJ.
As in a structural approach, in our model a ¯rm defaults when the value of its asset
falls below a given threshold.
From equations (7) and (5) we can see that the default of a client lowers the value of the
microeconomic component and thus the asset value of the ¯rm; as a result, the probability
of crossing the default threshold for the ¯rm increases.
This is the mechanism through which the contagion of defaults spreads in the system.
In the network of ¯rms, the contagion of defaults starts with the default of a node and
spreads along the directed edges which connect the ¯rms, proceeding in the direction of the
edges.
55 Concluding remarks
In this contribution we tackle the issue of modeling the microeconomic relations among the
¯rms in a network economy in order to describe the contagion mechanism which spreads
the ¯nancial distress in the system when a ¯rm defaults.
To this aim we de¯ne a weighted network based on a spatial interaction model to de-
scribe the business relations that take place among the ¯rms and we include the e®ect of
the microeconomic interactions as an additional idiosyncratic term in a factor model that
describes the value of the ¯rms in the network. A default occurs when the value of a ¯rm
cross a given default threshold and has repercussions on the ¯nancial health of the con-
nected ¯rms in the subsequent periods, thus representing a possible source of contagion of
defaults.
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7

A network of business relations to model counterparty risk

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