Comput Econ (2015) 46:519–537DOI 10.1007/s10614-014-9475-2

Spatial Interaction Model of Credit Risk Contagionin the CRT Market

Tingqiang Chen · Xindan Li · Jining Wang

Accepted: 27 October 2014 / Published online: 1 November 2014© Springer Science+Business Media New York 2014

Abstract In this paper, we introduce an entropy spatial model of credit risk contagionin the credit risk transfer (CRT) market that considers the effects of spatial, industry-specific, regional financial and individual factors of the CRT market on credit riskcontagion. We use numerical simulation to describe the influence and active mecha-nism of the spatial distance and transmission capacity between banks and investors inthe CRT market. We also assess bank asset quality and credit risk transfer capability,as well as investor asset scale and risk preference level, bank financial developmentlevel, investors in the area and the weight of investors in the area relative to creditrisk contagion. This model contributes to the explicit formalization of the connec-tion between probability and spatial factors and provides new ideas and a theoreticalframework for considering credit risk contagion in a spatial context, which has greatrelevance for credit risk management.

Keywords Credit risk contagion · Entropy spatial model · CRT market ·Spatial distance · Risk preference

1 Introduction

This study mainly aims to investigate the effects of the behaviour andheterogeneity of

T. Chen (B) · J. WangSchool of Economics and Management, Nanjing Tech University,Nanjing 211816, People’s Republic of Chinae-mail: tingqiang88888888@163.com

T. Chen · X. LiSchool of Management and Engineering, Nanjing University,Nanjing 210093, People’s Republic of China

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banks and investors, spatial distance factors, and geographical heterogeneity on creditrisk contagion in the credit risk transfer (CRT) market.

The recent U.S. subprime mortgage crisis and the European sovereign debt crisismotivated the creation of models in which credit risk contagion has repercussions onother investors in the financial market, especially in the CRT market. In particular,this influence can be modelled through the introduction of credit risk transfer and thecredit risk contagion effect it induces.

Over the last few years, credit risk contagion induced by the credit default of debtorshas become a concern for all financial institutions, including banks and financialadministrations. Different approaches have been proposed in the literature to analysecredit risk contagion.Davis andLo (2000, 2001) proposed a purely probabilisticmodelof credit risk contagion in a portfolio of bonds. Jarrow and Yu (2001) proposed a creditrisk contagionmodel based on existing reduced-formmodels to include default intensi-ties dependent on the default of a counterparty. This model included jump processes inthe set of state variables, thereby capturing the interdependence among several defaultprocesses. Frey and Backhaus (2003) extended the approach of Jarrow and Yu (2001)and assumed that the default intensity of a firm was directly affected by the defaultof other firms in the portfolio. Their model considered an intensity-based dynamicmodel for dependent defaults. Kodres and Pritsker (2002) developed a multiple asset,rational expectation model of asset prices to explain financial market contagion. Neuand Kuehn (2004) generalised existing structural models for credit risk to capture theeffect of counterparty defaults on economic capital allocated to the loan portfolios ofbanks. Focardi and Fabozzi (2004) proposed a credit risk contagion model that usespercolation and random graphs. Giesecke (2004) introduced a credit risk contagionmodel based on interacting particle systems that showed the relative strength of cycli-cal correlations and contagion. Moreover, Giesecke (2006) was the first to attemptintegrating macroeconomic effects in which cyclical default correlations and conta-gion phenomena were created. Such effects were associated with the local interactionof debtors with their business partners. Cossin and Schellhorn (2007) developed astructural model of credit risk in a network economy, in which any firm can lend toany other firm, such that each firm is subject to counterparty risk either from directborrowers or from remote firms in the network. Egloff et al. (2007) proposed a simplemodel of credit contagion, which included macroeconomic and microstructural inter-dependencies among debtors within a credit portfolio. Martin and Marrison (2007)presented an approach to modelling credit risk contagion that involved the spread ofcredit events among related companies. Kchia and Larsson (2011) provided a generalframework capable of handling many non-ordered default times arbitrarily under thebasic assumption that conditional joint densities exist.

A number of recent studies have considered credit risk contagion in theCRTmarket.Haworth and Reisinger (2006) proposed a numerical approach to modelling firm valuedynamics and default events. This approach enables the valuation of basket creditdefault swap spreads in a first passage framework with both asset correlation anddefault contagion. Theirmodel could be used as a powerful tool for analysing the effectof the spread of different dependence assumptions and parameter values. Allen andCarletti (2006) proposed amodelwith banking and insurance sectors. Themodel showsthat credit risk transfer can be beneficial because it improves risk sharing. However,

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credit risk transfer can also induce contagion and lead to a Pareto reduction in welfare.Haworth et al. (2008) developed a two-dimensional structural framework for valuingcredit default swaps and corporate bonds in the presence of default contagion. Jorionand Zhang (2009) provided the first empirical analysis of credit contagion throughdirect counterparty effects. They found that bankruptcy announcements cause negativeabnormal equity returns and increase CDS spreads for creditors. In addition, creditorswith large exposure are more likely to suffer from financial distress later. Chen andHe (2012a, b) proposed network models of credit risk contagion in the CRT market.These models consider the effect of investor risk preference and the network structurethat consists of investors on credit risk contagion. Chen et al. (2013a, b) proposednonlinear dynamic models of credit risk contagion in the CRT market. These modelsconsidered the effects of some nonlinear factors on credit risk contagion. They alsodiscussed the dynamic evolution behaviour of credit risk contagion in the CRTmarket.

Entropy has been widely applied in the study of spatial interaction theory as a mea-sure of uncertainty in the system. Such application is called entropy spatial interactiontheory. Entropy spatial interaction models disperse agents from an origin among alldestinations, instead of assigning all of them to the nearest one. Moreover, entropymodels can be obtained as an optimal solution for mathematical programming prob-lems in which maximizing the entropy of the system maximizes the dispersion ofthe origin destination flows (see Wilson 1970). Thus, these models have become aneffective tool for analysing the interaction flows between a set of origins and a setof destinations (Clarke et al. 1998; Celik and Guldmann 2007; O’Kelly 2010). Theseflows generally represent sets of agents that move from an origin to a destination(Barro and Basso 2010). Entropy spatial interaction theory has recently been appliedto economic and financial systems to analyse the effect of the spatial locations of eco-nomic subjects and information flows in the economic system. Wilson (2010) intro-duced a new framework for constructing spatial interactions and associated locationmodels. The models contribute to multiregional demographic and input-output mod-els, transport models and location models, which have applications in retail. Gordon(2010) proposed an entropy spatial interaction model based on economic principlesand random utility maximization. Barro and Basso (2010) introduced an entropy spa-tial interaction model that considers counterparty risk in a network of interdependentfirms. The model describes the presence of business relations among different firmsand show the effects of credit contagion on the credit risk of a portfolio of bank loans.Chen and He (2013) introduced an entropy spatial model of credit risk contagionbased on the study by Barro and Basso (2010). The latter study considered the effectof spatial distance and nonlinear coupling between banks and investors, as well asthe risk preference of investors, on credit risk contagion in the CRT network. In aneconomic system, this research produces an entropy spatial model that mainly useseffective information regarding the spatial distance matrix of the economic subjects ineconomic networks and the weight distribution of network nodes tomaximize entropy.However, existing research on economic and financial systems have not consideredthe risk preference and heterogeneity of economic subjects and the heterogeneity ofgeographical locations.

In this paper, we make several contributions to the study of credit risk contagionin the CRT market. First, we introduce an entropy spatial model of credit risk transfer

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that considers the effect of the behaviour and heterogeneity of banks and investors,spatial distance factors and geographical heterogeneity on credit risk transfer in theCRT market. This model mainly analyses the liquidity ratios of credit risk throughcredit risk transfer from banks to investors. Second, we build a value contagion modelusing the discrete time of multi-name credit derivatives based on the study by Bassoand Barro (2005); Barro and Basso (2010). Subsequently, an entropy spatial model ofcredit risk contagion that consists of the above two models is established. Finally, weanalyse the proposed model through a series of numerical simulations.

This paper is structured as follows: Section 2 presents some assumptions and definesthe notation used in this paper. Section 3 presents the entropy spatial model of creditrisk transfer in the CRTmarket. Section 4 discusses the entropy spatial model of creditrisk contagion in the CRTmarket. Section 5 highlights the effects of the behaviour andheterogeneity of banks and investors, spatial distance factors and geographical het-erogeneity on credit risk contagion in the CRT market through numerical simulations.The final section contains concluding remarks.

2 Notations and Assumptions

To model risk contagion in the process of credit risk transfer in the CRT market, someassumptions must be made, and the involved notation must be defined. First, we onlyconsider banks and institutional investors among CRT participants. We assume thatbanks and institutional investors constitute a tiered structure networkwithmulti-origindestinations. More specifically, a bank can transfer credit risk to multiple investors,and the credit risk of an investor originates from multiple banks. Second, we assumethat the transferors and transferees of credit risk are banks and institutional investors,respectively. These players are the origins and destinations of the CRT network. Third,we assume that credit risk transfer behaviour is unidirectional in the hierarchicalnetwork with multi-origin destinations. Bank credit risk is transferred to institutionalinvestors. In otherwords, we do not consider the internalmutual transfer in the bankingsystem.

The notation used in this paper can be summarized as follows:

• Im = {1, . . . ,m} is the set of banks in the CRT market and is also the set of originsof the CRT network.

• Jn = {1, . . . , n} is the set of institutional investors in the CRT market, i.e., the setof destinations of the CRT network.

• Ti j , for i ∈ Im and j ∈ Jn is the flow of credit derivatives from bank i to investorj , which are generated by the model.

• Oi = ∑nj=1 Ti j , for i ∈ Im is the total flow of credit derivatives going out from

bank i , observed in the CRT network.• Hj = ∑n

i=1 Ti j , for j ∈ Jn is the total flow of credit derivatives that investor jreceives as observed in the CRT network.

• T = ∑mi=1

∑nj=1 Ti j = ∑m

i=1 Oi = ∑nj=1 Hj is the total flow of credit derivatives

as observed in the CRT network.• di j , for i ∈ Im and j ∈ Jn is the physical distance between bank i and investor

j . In modern society, although physical space distance is not a key factor of credit

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risk contagion, the adverse selection problems, moral hazard and transaction costcaused by the asymmetric information as a result of spatial distance are some keyfactors that affect credit risk contagion. If the complexity of internal mutual transfercredit risk in the banking system is not considered, we assume that di j ≥ 1.

• vi j , for i ∈ Im and j ∈ Jn is the spatial transmission capacity between banki and investor j . It is the concentrated expression of developed degree of trafficand network information technology between two different geographical locations.It can reduce the information asymmetry of the spatial distance between bank iand investor j significantly, reduce the transaction costs of credit risk transfer, andincrease the probability of credit risk transfer.

• ri j = Ti jT , for i ∈ Im and j ∈ Jn is the share of the total flow of credit derivatives

that is transferred from bank i to investor j .• oi = Oi

T , for i ∈ Im is the share of total flow of credit derivatives from bank i .

• h j = HjT , for j ∈ Jn is the share of total flow of credit derivatives investor j . It

reflects the credit risk concentration of investor j .• ηi j , for i ∈ Im and j ∈ Jn is the ability of bank i transfer credit derivatives toinvestor j , whose level of credit risk exceeds the average risk level. It is a scale thatmeasures creditor transfer credit risk. It is also an important element that depictsthe credit risk contagion effect.

• β j , for j ∈ Jn is the risk preference level of investor j .• Qi , for i ∈ Im is the asset quality of bank i .• Q j , for j ∈ Jn is the asset scale of investor j .• c j , for j ∈ Jn is the financial development level of investor j in the area, and0 ≤ c j ≤ 1.

• φi , for i ∈ Im is the financial development level of bank j in the area, and 0 ≤ φi ≤1.

• ψ j , for j ∈ Jn is the weight of investors j in the area.

The geographic location factors of banks and investors can increase the complexityof credit risk transfer in the CRT market. Thus, in our study we use the financialdevelopment level of investors in the area, the weight of investors in the area andthe spatial transmission capacity between banks and investors to depict the effect ofgeographical heterogeneity on credit risk transfer in the CRT market. Then, the effectof the geographical heterogeneity on credit risk contagion is depicted.

3 Entropy Spatial Model of Credit Risk Transfer in the CRT Market

3.1 Analysis of Spatial Attraction of Credit Risk Transfer

Research on thematerial flow at different geographical spaces indicated that the attrac-tion ofmaterial flows in different geographic spaces exhibits the attenuation lawof spa-tial distance (O’Kelly 2010). In spatial structure theory, some research results indicatethat regional economic behaviour also exhibits the attenuation law of spatial distance(Krugman 1997; Fujita et al. 1999). Recent economic research also confirmed thateconomic behaviour at different geographical locations also exhibits the attenuation

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law of spatial distance (Gordon 2010; Barro and Basso 2010). The attraction functionof the economic behaviour of different spatial distance was discussed in studies byO’Kelly (2010), Gordon (2010) and Barro and Basso (2010), who provide a wide andcommon definition as follows:

f (di j ) = e−γ di j (1)

where γ ∈ R+ is a real parameter. This definition can intuitively and clearly depict the

attenuation law of the spatial distance of economic behaviour in different geographicallocations. However, if Eq. (1) is used to define the attraction function between banki and investor j in the process of credit risk transfer, which cannot reflect the effectsof the different financial development levels in different geographical locations, thedifference in spatial transmission capability among different geographical locations,banks asset quality, investors assets scale and the weight of investors in the area oncredit risk transfer. Thus, we consider we consider the limitation of the aforementioneddefinition. In reference to “Newton’s gravitational law”, we propose the attractionmodel of credit risk transfer between banks and investors as the following:

f (c j , ψ j , Qi , Q j , di j , vi j ) = (1 + c j )ψ j kQiaQ j

b

(di j/vi j )ξ

(2)

where k, a, b and ξ are empirical coefficients. In general, a = b = 1 (Krugman 1997;Fujita et al. 1999). Fujita et al. (1999) determined that the value of xi might be closeto 1. This definition can satisfy the spatial distance attenuation law of the attractionof credit risk transfer and sufficiently reflect the effects of the financial developmentlevel of investors in the area, the spatial transmission capability between differentgeographical locations, bank asset quality, investor assets scale and the weight ofinvestors in the area on credit risk transfer.

3.2 Analysis of Banks Credit Risk Transfer Ability

Credit risk transfer is a kind an effective financial tool for dispersing the credit risk ofbanks. In this tool, banks are transferors that serve a vital function in the process ofcredit risk transfer. Credit risk transfer is also critical part of credit risk contagion inthe CRT market. According to the aforementioned assumption, ηi j is the capability ofbank i to transfer credit risk when the level of credit risk exceeds the average risk levelof investor j . This element is also important for depicting the credit risk contagion.In the CRT market, a higher complexity of CRT instruments results in stronger risktransfer concealment and stronger transfer credit risk. At the same time, the strongerinfectivity of credit risk in the CRT market. In fact, the capability of banks to transfercredit risk is affected mainly by two factors.

(1) Financial development level. Financial development level refers to the coordi-nated development of regional finance and economy and is also a concentrated man-ifestation of regional financial innovation and globalization. Financial developmentnot only promotes the change in regional financial architecture, but also facilitates

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regional financial innovation and provides an endless supply of intrinsic power andinnovative tools for bank transfer credit risk. Thus, the financial development levelof banks in the area is a key factor that affects bank transfer credit risk. A higherfinancial development level of banks in an area result in a higher complexity and moretypes of CRT instruments. In other words, banks can more easily transfer credit risk toinvestors. In addition, the financial development level of investors in the area is also akey factor that affects the risk preference of investors. A higher financial developmentlevel cause a higher level of risk preference for investors.

(2) The risk preference level of investors. According to behavioural finance, the riskpreference of investors is an external factor that affects risk transfer and contagion.Risk-averse investors would prefer to curb the spread and contagion of financial risk,whereas risk-seeking investors would prefer to promote the spread and contagion offinancial risk. In the CRT market, the risk preference level of credit risk transfereeshas an important effect on the capability of banks to transfer credit risk to investors(Allen and Carletti 2006; Santos 2006).

According to the above analysis, we attempt to combine financial developmentlevel and investor risk preference level in modelling the capability of banks to transfercredit risk to investors as follow:

(1 + φi )ηi j(1−β j

1−c j ) (3)

3.3 Entropy Space Model of Credit Risk Transfer

In this contribution, we introduce an entropy spatial model of credit risk transfer thatconsiders the spatial distance and transmission capacity between banks and investorsin the CRT market, bank asset quality and credit risk transfer capability, investor assetscale and risk preference level, the financial development level of banks, investorsin the area and the weight of investors in the area. In this model, an entropy spatialinteraction model describes the spatial transfer of credit risk. According to earlierstudies (O’Kelly 2010; Gordon 2010; Barro and Basso 2010; Chen and He 2013),we consider the heterogeneity of geographical locations, as well as the behaviouralcharacteristics and heterogeneity of banks and investors.We propose an entropy spatialinteraction model that describes the credit risk transfer of bank i to investor j in theCRT market as follows:

Ri j = Ai Oi B j Hj f (c j , ψ j , Qi , Q j , di j , vi j )(1 + φi )ηi j(1−β j

1−c j ) (4)

where Ai denotes the effect of bank i in the financial system, Ai ∈ [0, 1]; Bj denotesthe effect of investor j on the CRT market, Bj ∈ [0, 1]; i ∈ Im ; j ∈ Jn ;

∑

iRi j = Hj ;

∑

jRi j = Oi .

According to the aforementioned assumption and Eq. (4), we can obtain the entropyspatial interaction model for the share of the total transferred-in credit risk of investor

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j when bank i transfers credit risk to investor j as follows:

ri j = Ai Oi B j Hj f (c j , ψ j , Qi , Q j , di j , vi j )

T(1 + φi )ηi j

(1−β j1−c j ) (5)

That is,

ri j = aioi b j h j f (c j , ψ j , Qi , Q j , di j , vi j )(1 + φi )ηi j(1−β j

1−c j ) (6)

where ai ∈ [0, 1]; b j ∈ [0, 1]; h j =∑m

i=1 Ti j∑mi=1

∑nj=1 Ti j

; i ∈ Im ; j ∈ Jn ;∑

iri j = h j ;

∑

jri j = oi .

The model defined using Eq. (6) may be useful in that ri j is the probability that acredit derivative in the CRTmarket will belong to the flow of credit derivatives movingfrom bank i to investor j . Thus, the information on credit risk transfer from bank ito investor j can be fully extracted and described in the form of a probability. Theentropy spatial interaction model in Eq. (6) is a doubly constrained model. However,we mainly investigate the effects of credit risk transfer from banks to investors oncredit risk contagion in the CRTmarket. Thus, we can obtain an entropy spatial modelconstrained only by investors as follows:

ri j = aib j h j f (c j , ψ j , Qi , Q j , di j , vi j )(1 + φi )ηi j(1−β j

1−c j ) (7)

where ai ∈ [0, 1]; b j ∈ [0, 1]; h j =∑m

i=1 Ti j∑mi=1

∑nj=1 Ti j

; i ∈ Im ; j ∈ Jn ;∑

iri j = h j ;

∑

jri j = 1.

We combined Eq. (2) with Eq. (7) to obtain to the following:

ri j = aib j h j(1 + c j )ψ j kQi

aQ jb

(di j/vi j )ξ

(1 + φi )ηi j(1−β j

1−c j ) (8)

where ai ∈ [0, 1]; b j ∈ [0, 1]; h j =∑m

i=1 Ti j∑mi=1

∑nj=1 Ti j

; i ∈ Im ; j ∈ Jn ;∑

iri j = h j ;

∑

jri j = 1; k, a, b and ξ are empirical coefficients.

According to Eq. (8), the entropy of credit risk transfer is not only closely relatedto the spatial distance di j between bank i and investor j but also to the transmissioncapacity between banks and investors in the CRTmarket, banks asset quality and creditrisk transfer capability, investors asset scale and risk preference level, bank financialdevelopment level, investors in the area and the weight of investors in the area. Thus,the model can sufficiently reflect the effects of the heterogeneity of economic agents,the heterogeneity of the geographic locations of economic agents, and the spatialdistance on credit risk transfer in the CRT market.

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4 Entropy Spatial Model of Credit Risk Contagion in the CRT Market

In the CRTmarket, banks optimize and reorganize the credit risk from debtors and thentransfer them to investors through credit derivatives to reduce the financing cost andrisk concentration, satisfy the regulatory capital requirements, andmaximize their owninterests. Therefore, if the credit status of debtors changes or a credit default occurs,the value of credit derivatives will fluctuate and affect the banks and investors that areassociated with the debtors. Moreover, when the loss rate caused by the fluctuation ofthe value of credit derivatives is greater than the given threshold value θ , banks andinvestors will exhibit similar credit default behaviour, and then the contagion effect ofcredit risk will occur (Chen and He 2013).

This study mainly investigates the effect of the credit status of debtors and thebehaviour of banks relative to credit risk transfer on investors in the CRT market. Todescribe the contagion effect of credit risk, a discrete time model of value contagionis introduced based on the discrete time model proposed in Basso and Barro (2005).This model focuses on the asset value of debtors in a multi-name credit derivative,similar to a structural approach. Moreover, when debtor s experiences a serious creditoccurrence, such as credit default or bankruptcy, the loss rate can be greater thanthe given threshold value θ . According to Jarrow and Yu (2001), a multi-name creditderivative is assumed to be composed of credit assets of the core debtor and themicro debtor. The effect of the credit default of debtor s on the value of a multi-namecredit derivative is described by the sum of two terms: a macroeconomic componentHs(i)(t), modelled using a factor model that describes the influence of the industryenvironment, and amicroeconomic component Ps(i)(t), which describes the contagioneffect caused by the past distresses and mainly measures the financial distress ofbanks by observing the past default of debtors included in the credit derivatives. Thiscomponent is a random disturbance term of the external environment. Thus, a valuecontagion discrete time model caused by the credit default of debtor s is as follows:

VCs(i)(t) = Hs(i)(t) + Ps(i)(t) + εs(i)(t) (9)

where s = 1, 2, . . . , S; t = 0, 1, 2, . . .; and i ∈ Im . εs(i)(t) is a random disturbanceterm that describes the effect of the external environment, including the macroeco-nomic, financial, industry, and legal environment, on the credit default of debtors.εs(i)(t) is assumed to be normal distribution with zero mean and standard deviationσεs(i) .

To prevent the underestimation of the correlation, give that theDuffiemodel ignoresthe credit risk contagion effect, the Duffie model is expanded to include industry-specific factors contained in the state variables to depict credit risk contagion effects inthe industry. Thus, a factor model that describes the effect of the industry environmentHs(i)(t) on credit risk contagion of multi-name credit derivatives is proposed:

Hs(i)(t) =L∑

l=1

[αs(i)l Yl(t)] (10)

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where l = 1, 2, . . . , L denotes different industry types; and Yl(t) denotes the specificfactor of l industry that describes the effect of defaulted debtors on the credit defaultof other debtors, such as the share of the totalthat have defaulted in the industry, theindustry total liabilities, etc. αs

l denotes the degree of influence of the specific factorof l industry to the credit default of debtors,

∣∣αs

l

∣∣ ≤ 1.

The contagion effect Ps(i)(t) of the past default of debtor s mainly describes theeffect of the past default of observed debtor s on financial distress of banks. Thus, thefinancial distress measure Ds(i)(t) of bank i at time t is assumed to originate from thedebtors included in the credit derivatives. This measure is compared with the averagedefault rate p(t) observed in the economy at time t . Thus the financial distressmeasureDi (t) can be described by the following equation:

Ds(i)(t) = p(t) −⎡

⎣∑

v∈ψs (t)

Iv(t)wsv(t) + p(t)ms(t)

⎤

⎦ (11)

where ψs(t) denotes the set of the major debtors included in the credit derivatives,wsv(t) denotes the importance of the debtor s included in the credit derivatives, andwsv(t) = (the value of credit assets of core debtor v)/(the value of the credit deriva-tives), wsv > θ . This formula implies that the loss rate caused by the change in thecredit status of core debtors may be greater than the given threshold value θ . Moreover,the loss rate can be greater than that of the core debtors v that experience a serious creditevent, such as credit default or bankruptcy, and it can trigger credit risk contagion effect.ms(t) = 1− ∑

v∈ψs (t)wsv(t) denotes the important degree of all minor debtors included

in the credit derivatives.Moreover, the credit default of eachminor debtor results in thefluctuation of the value of credit derivatives, which is far less than the given thresholdvalue θ . Thus, the credit defaults of a single minor debtor cannot cause the financialdistress of banks. Iv(t) is an indicator function that the core debtors v included in thecredit derivatives experience credit defaults at time t , and Iv(t) is defined as

Iv(t) =⎧⎨

⎩

1 if the core debtors v included in the creditderivatives experience credit defaults at time t

0 otherwise(12)

Equation (11) only considers the credit defaults of the major debtors. Accordingto Basso and Barro (2005), a case in which the percent amount of minor debtorsequal to the average default rate p(t) of the economy is assumed to be credit default.Furthermore, the financial distress measure Ds(i)(t) is a real number that is negativeif at time t , the default rate of debtors included in the credit derivatives is higher thanthe average default rate of the economy p(t). The case is null if the value is equal top(t) and has a positive value if the value is lower than p(t).

The past credit defaults of debtors included in the credit derivatives is assumed tohave a long-term effect on the current credit risk of banks, and the effect of long-termmemory has an exponential decay in time relative to the influence of the past creditdefault (see, for example, Basso and Barro 2005; Barro and Basso 2010). The time of

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the past credit defaults of debtor s of bank i is assumed to be τ , with τ > 1. Thus, theeffect factor of the past credit defaults of the debtor s of bank i on its current financialdistress is λs(i), and 0 ≤ λs(i) < 1. Therefore, at time t , the overall financial distressPi (t) of bank i is measured as the sum of the effects of all the past credit default of itsdebtor s as follow:

Pi (t) =∞∑

τ=1

λτs(i)Ds(i)(t − τ) (13)

Namely,

Ps(i)(t) =∞∑

τ=1

λτs(i)

[

p(t − τ)

−[ ∑

v∈ψs (t−τ)

Iv(t − τ)wsv(t − τ) + p(t − τ)ms(t − τ)

]]

(14)

Equation (14) depicts the contagion effect that the past credit default of the debtorsincluded in the credit derivatives owned by the bank i on its current financial distress.Therefore, the value contagion discrete time model of the credit derivatives owned bybank i can be obtained by integrating Eqs. (10) and (14) into Eq. (9):

VCs(i)(t) =L∑

l=1

[αs(i)l Yl(t)] +

∞∑

τ=1

λτs(i)

[

p(t − τ)

−[ ∑

v∈ψs (t−τ)

Iv(t − τ)wsv(t − τ) + p(t − τ)ms(t − τ)

]]

+ εs(i)(t)

(15)

According to Basso and Barro (2005); Barro and Basso (2010), if the value of λs(i)is sufficiently small, only the term

∑∞τ=1 λτ

s(i) p(t−τ) is non negligible in the practicalapplication.

To achieve capital safety and improve operation efficiency, bank i can moderatelytransfer credit risks brought by the debtors to investors in the CRT market. Thus,the change in the credit status of the debtors included in the credit derivatives has acontagion effect on investors because of the role of credit risk transfer in the CRTmarket. The entropy spatial model of credit risk contagion in the CRT market isobtained by employing (8) and (15):

πi j = ai b j h j

(1 + c j )ψ j kQa

i Qbj

(di j/vi j )

ξ(1 + φi )η

(1−β1−c jj )

i j

⎡

⎣L∑

l=1

[αs(i)l Yl (t)] +

∞∑

τ=1

λτs(i)

×⎡

⎣p(t − τ) −⎡

⎣∑

v∈ψs (t−τ)

δv(t − τ)wsv(t − τ) + p(t − τ)ms(t − τ)

⎤

⎦

⎤

⎦ + εs(i)(t)

⎤

⎦

(16)

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The constraint conditions of model (16) can be obtained as follows:

∑

i

ai b j h j(1 + c j )ψ j kQi

aQ jb

(di j/vi j )ξ

(1 + φi )ηi j(1−β j

1−c j ) = h j (17)

∑

i

ai b j h j(1 + c j )ψ j kQi

aQ jb

(di j/vi j )ξ

(1 + φi )ηi j(1−β j

1−c j ) = 1 (18)

where ai ∈ [0, 1]; bi ∈ [0, 1]; i ∈ Im ; j ∈ Jn ; k, a, b, ξ are empirical coefficients.πi j denotes that investor j will suffer from credit risk because of the transfer of creditderivatives form bank i to investor j .

According to Eqs. (16)–(18), the entropy spatial model of credit risk contagion canmaximize the entropy of credit risk contagion in the CRT market by satisfying thegiven constraints and can depict the effects of the spatial distance and transmissioncapacity between banks and investors in the CRT market, the asset quality and creditrisk transfer capability of banks, the asset scale and risk preference level of investors,the financial development level of banks and investors in the area and the weight ofinvestors in the area on credit risk contagion in the CRT market.

5 Simulation Analysis of Credit Risk Contagion in the CRT Market

Given the absence of a large amount of time series data for empirical tests, numericalsimulation analysis is the most effective testing method. Such analysis is conductedby considering the different values of the parameters in the entropy spatial model.Moreover, the credit derivatives that are mainly constituted by the credit loans ofmedium-sized and small enterprises are particularly considered. The following areassumed: the number of banks m = 500 and the number of investors n = 200 in theCRTmarket. The average credit default rate p = 0.06 in the economy, the impact factorλs = 0.15 of the past credit defaults of debtor s and the disturbance level σ = 0.4 ofexternal environment. According to Krugman (1997) and Fujita et al. (1999), assumea = b = 1, ξ = 0.9, and k = 0.08. The number of the credit derivatives of the flow inthe CRT market is also assumed to be equal to 10,000, and the number of bank i thatowns credit derivatives is assumed to be a normal distribution in which the mean isequal to 20 and the standard deviation is 4. The ratio of the number of major debtorsand minor debtors is also assumed equal to 3

7 . For the influence of industry-specificfactors, this study adopts a factor simulation of mean reversion process with a driftparameter that is equal to 0.5 and volatility of 0.08. The long-term average is equal to 1.

5.1 Analysis of the Effect of the Influence, Asset Quality, and Capability of CreditRisk Transfer of Banks on Credit Risk Contagion in the CRT Market

For investors in the CRT market, on the one hand, the influence and asset quality ofbanks have a strong “certification effect” that can attract investors to buy risk assets ofbanks. On the other hand, the “certification effect” of banks can reduce the conscious-ness of risk identify of investors, potentially increase the risk level of investors and

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Spatial Interaction Model 531

enhance the contagion effects of credit risk. Figure 1 shows that the contagion effectof credit risk is positively related to the influence and asset quality of banks becauseof the positive feedback of the “certification effect” of banks on investors.

In addition, the capability of credit risk transfer of banks is also a key factor thataffects credit risk contagion in the CRTmarket. The capability of credit risk transfer ofbanks can generally cause banks to transfer and disperse their credit risk reasonably,improve the utilization efficiency and liquidity of their owned capital, obtain higherprofit and bigger development space, enhance the contagion effects of credit risk andincrease the probability of credit risk contagion. Figure 1 also shows that the contagioneffect of credit risk is a diminishing marginal and is positively related to the capabilityof credit risk transfer of banks. In fact, the capability of credit risk transfer of banksmainly depends on financial innovation. Banksmainly use scientific and effective CRTtool innovation to transfer and disperse their own credit risk, but financial innovationcan potentially enhance the depth and breadth of credit risk contagion and can bringnew credit risk (Allen and Carletti 2006). However, financial innovation has a slowdevelopment process and is a limited and diminishing marginal that gives rise to creditrisk contagion.

Figure 1 also shows that the effect of the capability of credit risk transfer of bankson credit risk contagion is significantly enhanced along with the increase in the “certi-fication effect” of the influence and asset quality of banks. Moreover, the “certificationeffect” of the asset quality of banks on credit risk contagion is more significant thanthe capability of credit risk transfer and the influence of banks, and the effect of theinfluence of banks on credit risk contagion is more significant than the capability ofcredit risk transfer of banks.

5.2 Analysis of the Effect of the Influence, Asset Scale, Risk Preference Level, andConcentration Degree of Credit Risk of Investors on Credit Risk Contagion inthe CRT Market

In the CRT market, investors are the terminals of credit risk transfer of banks. Theinfluence, asset scale and risk preference level of investors will affect the concentrationof credit risk in themselves, the behaviour of credit risk transfer of banks and theeffectiveness and influence degree of credit risk contagion. Figure 2 shows that thecontagion effect of credit risk is positively related to the level of risk concentrationon investors and the influence, asset scale and risk preference level of investors in theCRTmarket. Moreover, the effect of the risk preference level of investors on credit riskcontagion is a diminishing marginal. The effect of the level of risk concentration oninvestors and the risk preference level of investors on credit risk contagion is enhancedalong with the increase in the influence and asset scale of investors.

Figure 2 shows that the effect of the level of risk concentration on investors on creditrisk contagion is more significant than the influence, asset scale and risk preferencelevel of investors. The effect of the asset scale of investors on credit risk contagion ismore significant than the influence and risk preference level of investors, whereas theeffect of the influence of investors on credit risk contagion is more significant than therisk preference level of investors in the CRT market.

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00.2

0.40.6

0.81

0

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0.4

0.6

0.8

10

0.5

1

1.5

2

2.5

3

3.5

4

x 10−3

ai

bj=0.6; c

j=0.75; h

j=0.15; ψ

j=0.2; φ

i=0.6; Q

j=20; d

ij=50; v

ij=0.75; η

ij=0.7; β

j=0.7

Qi

πij

(a)

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

1

2

3

4

5

6

7

8

x 10−4

ai

bj=0.6; c

j=0.75; h

j=0.15; ψ

j=0.2; φ

i=0.6; Q

i=0.2; Q

j=20; d

ij=50; v

ij=0.75; β

j=0.7

ηij

πij

(b)0

0.20.4

0.60.8

1

0

0.2

0.4

0.6

0.8

10

0.5

1

1.5

2

2.5

3

x 10−3

Qi

ai=0.7; b

j=0.6; c

j=0.75; h

j=0.15; ψ

j=0.2; φ

i=0.6; Q

j=20; d

ij=50; v

ij=0.75; β

j=0.7

ηij

πij

(c)

Fig. 1 Effect of the influence, asset quality and capability of credit risk transfer of banks on credit riskcontagion in the CRT market. a Plot of the function πi j as function in ai and Qi . b Plot of the function πi jas function in ai and ηi j . c Plot of the function πi j as function in Qi and ηi j

5.3 Analysis of the Effect of the Spatial Distance and Transmission CapacityBetween Banks and Investors on Credit Risk Contagion in the CRT Market

In the financial market, the existence of geographic spatial distance inevitably resultsin information asymmetry between banks and investors (Carling and Lundberg 2005),but the transmission capacity among regions can reduce the information asymmetryand information transmission cost (Krugman 1997; Fujita et al. 1999). Thus, the trans-mission capacity between the regions can also reduce the cost of credit risk transfer andpotentially contribute to the credit risk contagion in the CRT market. Figure 3 showsthat the contagion effect of credit risk is positively related to the transmission capacitybetween banks and investors and is negatively related to the spatial distance betweenbanks and investors. Moreover, the effect of the spatial distance between banks andinvestors on credit risk contagion has significant “local effect”, but the transmissioncapacity between banks and investors can change this effect. When the probability ofcredit risk transfer is increased, the possibility of a CRT market credit risk contagionis also increased.

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020

4060

80100

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ai=0.7; h

j=0.15; c

j=0.75; ψ

j=0.2; φ

i=0.6; Q

i=0.2; d

ij=50; v

ij=0.75; η

ij=0.7; β

j=0.7

Qj

πij

(a)

00.2

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hj

ai=0.7; b

j=0.6; c

j=0.75; ψ

j=0.2; φ

i=0.6; Q

i=0.2; Q

j=20; d

ij=50; v

ij=0.75; η

ij=0.7

βj

πij

(b)

00.2

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ai=0.7; h

j=0.15; c

j=0.75; ψ

j=0.2; φ

i=0.6; Q

i=0.2; Q

j=20; d

ij=50; v

ij=0.75; η

ij=0.7

βj

πij

(c)

020

4060

80100

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ai=0.7; b

j=0.6; h

j=0.15; c

j=0.75; ψ

j=0.2; φ

i=0.6; Q

i=0.2; d

ij=50; v

ij=0.75; η

ij=0.7

βj

πij

(d)

Fig. 2 Effect of investors’ influence, asset scale, risk preference level, and concentration degree of creditrisk on credit risk contagion in the CRT market. a Plot of the function πi j as function in b j and Q j . b Plotof the function πi j as function in b j and β j . c Plot of the function πi j as function in b j and β j . d Plot ofthe function πi j as function in Q j and β j

Figure 3 shows that the transmission capacity between banks and investors canenhance the “local effect” of credit risk contagion in the CRT market, thus enhancingmaking the effect of the spatial distance between banks and investors on credit riskcontagion. The transmission capacity between banks and investors can also increasethe depth and scope of credit risk contagion in the CRT market.

5.4 Analysis of the Effect of the Financial Development Level of Banks andInvestors in the Area and the Weight of Investors in the Area on Credit RiskContagion in the CRT Market

In the whole economic system, financial development is consistent with economicdevelopment, which is the fundamental driving force of financial development. Finan-cial development also facilitates financial innovation and development of financial ser-vices to sustain economic development. Moreover, the regional financial developmentis conducive to realize the aggregation effect of capital and improve the efficiencyof resource use, social investment levels and liquidity of financial capital. The risk

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Fig. 3 Plot of the function πi j as function in the spatial distance di j and transmission capacity vi j betweenbanks and investors in the CRT market

preference level of investors and the concentration degree of credit risk on investorsalso increase.

Figure 4 shows that the contagion effect of credit risk is positively related to thefinancial development level of banks and investors in the area and is a positivelyincreasing marginal related to the weight of investors in the area. The effect of thefinancial development level of banks in the area on credit risk contagion is also moresignificant than that of the financial development level of investors in the area. Thefigure also shows that when the financial development level of investors in the area isvery low, the effect of the financial development level of banks on credit risk contagionis more significant than that of the financial development level and the weight ofinvestors in the area. The effect of the weight of investors in the area on credit riskcontagion is also more significant than that of the financial development level ofinvestors in the area. However, when the financial development level of investors inthe area is very high, the effect of the weight of investors in the area on credit riskcontagion is more significant than that of the financial development level of banks inthe area.

6 Conclusion

In this study, an entropy spatialmodel of credit risk contagion is introduced. Thismodelconsiders the effect of the behaviour and heterogeneity of banks and investors, spatialdistance factors and geographical heterogeneity on credit risk contagion in the CRTmarket. The effects of the spatial distance and transmission capacity between banksand investors in the CRT market, the asset quality and credit risk transfer capability ofbanks, the asset scale and risk preference level of investors, the financial developmentlevel of banks and investors in the area and the weight of investors in the area oncredit risk contagion in the CRT market are determined through numerical simulation

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Spatial Interaction Model 535

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x 10−4

cj

ai=0.7; bj=0.6; h

j=0.15; φ

i=0.6; Q

i=0.2; Q

j=20; d

ij=50; v

ij=0.75;η

ij=0.7; β

j=0.7

ψj

πij

(a)

00.2

0.40.6

0.81

00.2

0.40.6

0.812

4

6

8

10

12

x 10−4

ψj

ai=0.7; bj=0.6; h

j=0.15; c

j=0.75; Q

i=0.2; Q

j=20; d

ij=50; v

ij=0.75;η

ij=0.7; β

j=0.7

φi

πij

(b)0

0.20.4

0.60.8

1

00.2

0.40.6

0.812

3

4

5

6

7

8

x 10−4

cj

ai=0.7; bj=0.6; h

j=0.15; ψ

j=0.2; Q

i=0.2; Q

j=20; d

ij=50; v

ij=0.75;η

ij=0.7; β

j=0.7

φi

πij

(c)

Fig. 4 Effect of the financial development level of banks and investors in the area, and the weight ofinvestors in the area on credit risk contagion in the CRT market. a Plot of the function πi j as function inc j and ψ j . b Plot of the function πi j as function in ψ j and φi . c Plot of the function πi j as function in c jand φi

analysis. This study will provide a new analysis framework for the research of creditrisk contagion in the CRTmarket and is also an important supplement to the structuralmodel of credit risk contagion.

However, this study can be further expanded to cover other topics. For example, thestudy considers the internal relations among banks, investors, and debtors; the locationfactor of debtors; and the internal interaction transfer of credit risk in a banking system,among others. Therefore, future study will focus on these issues.

Acknowledgments We wish to express our gratitude to the referees for their invaluable comments. Thisstudy was supported by the National Natural Science Foundation of China (Nos. 71173098, 71271109, and71301078), China Postdoctoral Science Foundation (2014M561626), the Philosophy and Social SciencesResearch Funded Projects in Colleges and Universities of Jiangsu (2014SJB081).

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