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Incorporating Contagion in Portfolio Credit Risk Models Using Network Theory

Anagnostou I Sourabh S Kandhai DDOI10115520186076173Publication date2018Document VersionFinal published versionPublished inComplexityLicenseCC BY

Link to publication

Citation for published version (APA)Anagnostou I Sourabh S amp Kandhai D (2018) Incorporating Contagion in Portfolio CreditRisk Models Using Network Theory Complexity 2018 [6076173]httpsdoiorg10115520186076173

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Download date15 Aug 2021

Research ArticleIncorporating Contagion in Portfolio Credit Risk ModelsUsing Network Theory

Ioannis Anagnostou 12 Sumit Sourabh12 and Drona Kandhai12

1Computational Science Lab University of Amsterdam Science Park 904 1098XH Amsterdam Netherlands2Quantitative Analytics ING Bank Foppingadreef 7 1102BD Amsterdam Netherlands

Correspondence should be addressed to Ioannis Anagnostou ianagnostouuvanl

Received 20 September 2017 Accepted 29 November 2017 Published 8 January 2018

Academic Editor Thiago C Silva

Copyright copy 2018 Ioannis Anagnostou et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Portfolio credit risk models estimate the range of potential losses due to defaults or deteriorations in credit quality Most of thesemodels perceive default correlation as fully captured by the dependence on a set of common underlying risk factors In light ofempirical evidence the ability of such a conditional independence framework to accommodate for the occasional default clusteringhas been questioned repeatedly Thus financial institutions have relied on stressed correlations or alternative copulas with moreextreme tail dependence In this paper we propose a different remedymdashaugmenting systematic risk factors with a contagious defaultmechanism which affects the entire universe of credits We construct credit stress propagation networks and calibrate contagionparameters for infectious defaultsThe resulting framework is implemented on synthetic test portfolios wherein the contagion effectis shown to have a significant impact on the tails of the loss distributions

1 Introduction

One of the main challenges in measuring the risk of abankrsquos portfolio is modelling the dependence between defaultevents Joint defaults of many issuers over a fixed period oftimemay lead to extreme losses therefore understanding thestructure and the impact of default dependence is essentialTo address this problem one has to take into considerationthe existence of two distinct sources of default dependenceOn the one hand performance of different issuers dependson certain common underlying factors such as interest ratesor economic growth These factors drive the evolution of acompanyrsquos financial success which ismeasured in terms of itsrating class or the probability of default On the other handdefault of an issuermay too have a direct impact on the prob-ability of default of a second dependent issuer a phenomenonknown as contagion Through contagion economic distressinitially affecting only one issuer can spread to a significantpart of the portfolio or even the entire system A goodexample of such a transmission of pressure is the Russiancrisis of 1998-1999 which saw the defaults of corporate and

subsovereign issuers heavily clustered following the sovereigndefault [1]

Most portfolio credit risk models used by financialinstitutions neglect contagion and rely on the conditionalindependence assumption according to which conditionalon a set of common underlying factors defaults occur inde-pendently Examples of this approach include the AsymptoticSingle Risk Factor (ASRF) model [2] industry extensionsof the model presented by Merton [3] such as the KMV[4 5] and CreditMetrics [6] models and the two-factormodel proposed recently by Basel Committee on BankingSupervision for the calculation of Default Risk Charge (DRC)to capture the default risk of trading book exposures [7] Aconsiderable amount of literature has been published on theconditional independence framework in standard portfoliomodels see for example [8 9]

Although conditional independence is a statistically andcomputationally convenient property its empirical validityhas been questioned on a number of occasions whereresearchers investigated whether dependence on commonfactors can sufficiently explain the default clustering which

HindawiComplexityVolume 2018 Article ID 6076173 15 pageshttpsdoiorg10115520186076173

2 Complexity

occurs from time to time Schonbucher and Schubert [10]suggest that the default correlations that can be achievedwith this approach are typically too low in comparisonwith empirical default correlations although this problembecomes less severe when dealing with large diversifiedportfolios Das et al [11] use data on US corporations from1979 to 2004 and reject the hypothesis that factor correlationscan sufficiently explain the empirically observed defaultcorrelations in the presence of contagion Since a realisticcredit risk model is required to put the appropriate weight onscenarios where many joint defaults occur one may chooseto use alternative copulas with tail dependence which havethe tendency to generate large losses simultaneously [12] Inthat case however the probability distribution of large lossesis specified a priori by the chosen copula which seems ratherunintuitive [13]

One of the first models to consider contagion in creditportfolios was developed by Davis and Lo [14] They suggesta way of modelling default dependence through infectionin a static framework The main idea is that any defaultingissuer may infect any other issuer in the portfolio Gieseckeand Weber [15] propose a reduced-form model for conta-gion phenomena assuming that they are due to the localinteraction of companies in a business partner network Theauthors provide an explicit Gaussian approximation of thedistribution of portfolio losses and find that typically conta-gion processes have a second-order effect on portfolio lossesLando and Nielsen [16] use a dynamic model in continuoustime based on the notion ofmutually exciting point processesApart from reduced-form models for contagion which aimto capture the influence of infectious defaults to the defaultintensities of other issuers structural models were devel-oped as well Jarrow and Yu [17] generalize existing modelsto include issuer-specific counterparty risks and illustratetheir effect on the pricing of defaultable bonds and creditderivatives Egloff et al [18] use network-like connectionsbetween issuers that allow for a variety of infections betweenfirms However their structural approach requires a detailedmicroeconomic knowledge of debt structure making theapplication of this model in practice more difficult thanthat of Davis and Lorsquos simple model In general sincethe interdependencies between borrowers and lenders arecomplicated structural analysis has mostly been applied toa small number of individual risks only

Network theory can provide us with tools and insightsthat enable us to make sense of the complex interconnectednature of financial systems Hence following the 2008 crisisnetwork-based models have been frequently used to measuresystemic risk in finance Among the first papers to studycontagion using network models was [19] where Allen andGale show that a fully connected and homogeneous financialnetwork results in an increased system stability Contagioneffects using network models have also been investigatedin a number of related articles see for example [20ndash24]The issue of too-central-to-fail was shown to be possiblymore important than too-big-to-fail by Battiston et al in[25] where DebtRank a metric for the systemic impact offinancial institutions was introduced DebtRank was furtherextended in a series of articles see for example [26ndash28]

The need for development of complexity-based tools inorder to complement existing financialmodelling approacheswas emphasized by Battiston et al [29] who called for amore integrated approach among academics from multipledisciplines regulators and practitioners

Despite substantial literature on portfolio credit riskmodels and contagion in finance specifying models whichtake into account both common factors and contagion whiledistinguishing between the two effects clearly still proveschallenging Moreover most of the studies on contagionusing network models focus on systemic risk and theresilience of the financial system to shocks The qualitativenature of this line of research can hardly provide quantitativerisk metrics that can be applied to models for measuring therisk of individual portfolios The aforementioned drawbackis perceived as an opportunity for expanding the currentbody of research by contributing a model that would accountfor common factors and contagion in networks alike Giventhe wide use of factor models for calculating regulatoryand economic capital as well as for rating and analyzingstructured credit products an extended model that can alsoaccommodate for infectious default events seems crucial

Our paper takes up this challenge by introducing aportfolio credit risk model that can account for two channelsof default dependence common underlying factors andfinancial distress propagated from sovereigns to corporatesand subsovereigns We augment systematic factors with acontagion mechanism affecting the entire universe of creditswhere the default probabilities of issuers in the portfolio areimmediately affected by the default of the country wherethey are registered and operating Our model allows forextreme scenarios with realistic numbers of joint defaultswhile ensuring that the portfolio risk characteristics and theaverage loss remain unchanged To estimate the contagioneffect we construct a network using credit default swaps(CDS) time series We then use CountryRank a network-based metric introduced in [30] to quantify the impact ofa sovereign default event on the credit quality of corporateissuers in the portfolio In order to investigate the impact ofourmodel on credit losses we use synthetic test portfolios forwhich we generate loss distributions and study the effect ofcontagion on the associated riskmeasures Finally we analyzethe sensitivity of the contagion impact to rating levels andCountryRank Our analysis shows that credit losses increasesignificantly in the presence of contagion Our contributionsin this paper are thus threefold First we introduce a portfoliocredit risk model which incorporates both common factorsand contagion Second we use a credit stress propagationnetwork constructed from real data to quantify the impact ofdeterioration of credit quality of the sovereigns on corporatesThird we present the impact of accounting for contagionwhich can be useful for banks and regulators to quantifycredit model or concentration risk in their portfolios

The rest of the paper is organized as follows Section 2provides an overview of the general modelling frameworkSection 3 presents the portfolio model with default contagionand illustrates the networkmodel for the estimation of conta-gion effects In Section 4 we present empirical analysis of two

Complexity 3

synthetic portfolios Finally in Section 5 we summarize ourfindings and draw conclusions

2 Merton-Type Models for PortfolioCredit Risk

Most financial institutions use models that are based onsome form of the conditional independence assumptionaccording to which issuers depend on a set of commonunderlying factors Factormodels based on theMertonmodelare particularly popular for portfolio credit risk Our modelextends the multifactor Merton model to allow for creditcontagion In this section we present the basic portfoliomodelling setup outline the model of Merton and explainhow it can be specified as a factor model A more detailedpresentation of themultivariateMertonmodel is provided by[9]

21 Basic Setup andNotations This subsection introduces thebasic notation and terminology that will be used throughoutthis paper In addition we define themain risk characteristicsfor portfolio credit risk

The uncertainty of whether an issuer will fail to meetits financial obligations or not is measured by its probabilityof default For comparison reasons this is usually specifiedwith respect to a fixed time interval most commonly oneyear The probability of default then describes the probabilityof a default occurring in the particular time interval Theexposure at default is a measure of the extent to which oneis exposed to an issuer in the event of and at the time of thatissuerrsquos default The default of an issuer does not necessarilyimply that the creditor receives nothing from the issuer Thepercentage of loss incurred over the overall exposure in theevent of default is given by the loss given default Typicalvalues lie between 45 and 80

Consider a portfolio of119898 issuers indexed by 119894 = 1 119898and a fixed time horizon of 119879 = 1 year Denote by 119890119894 theexposure at default of issuer 119894 and by 119901119894 its probability ofdefault Let 119902119894 be the loss given default of issuer 119894 Denote by119884119894the default indicator in the time period [0 119879] All issuers areassumed to be in a nondefault state at time 119905 = 0 The defaultindicator 119884119894 is then a random variable defined by

119884119894 = 1 if issuer 119894 defaults0 otherwise

(1)

which clearly satisfies P(119884119894 = 1) = 119901119894 The overall portfolioloss is defined as the random variable

119871 fl119898sum119894=1

119902119894119890119894119884119894 (2)

With credit risk in mind it is useful to distinguish poten-tial losses in expected losses which are relatively predictableand thus can easily be managed and unexpected losses whichare more complicated to measure Risk managers are moreconcernedwith unexpected losses and focus on riskmeasuresrelating to the tail of the distribution of 119871

22TheModel ofMerton Credit riskmodels are typically dis-tinguished in structural and reduced-formmodels accordingto their methodology Structural models try to explain themechanism by which default takes place using variables suchas asset and debt values The model presented by Merton in[3] serves as the foundation for all these models Consider anissuer whose asset value follows a stochastic process (119881119905)119905ge0The issuer finances itself with equity and debt No dividendsare paid and no new debt can be issued InMertonrsquosmodel theissuerrsquos debt consists of a single zero-coupon bond with facevalue119861 andmaturity119879The values at time 119905 of equity and debtare denoted by 119878119905 and 119861119905 and the issuerrsquos asset value is simplythe sum of these that is

119881119905 = 119878119905 + 119861119905 119905 isin [0 119879] (3)

Default occurs if the issuer misses a payment to its debthold-ers which can happen only at the bondrsquos maturity 119879 At time119879 there are only two possible scenarios

(i) 119881119879 gt 119861 the value of the issuerrsquos assets is higher than itsdebt In this scenario the debtholders receive 119861119879 = 119861the shareholders receive the remainder 119878119879 = 119881119879 minus 119861and there is no default

(ii) 119881119879 le 119861 the value of the issuerrsquos assets is less thanits debt Hence the issuer cannot meet its financialobligations and defaults In that case shareholdershand over control to the bondholders who liquidatethe assets and receive the liquidation value in lieuof the debt Shareholders pay nothing and receivenothing therefore we obtain 119861119879 = 119881119879 119878119879 = 0

For these simple observations we obtain the below relations

119878119879 = max (119881119879 minus 119861 0) = (119881119879 minus 119861)+ (4)

119861119879 = min (119881119879 119861) = 119861 minus (119861 minus 119881119879)+ (5)

Equation (4) implies that the issuerrsquos equity at maturity 119879 canbe determined as the price of a European call option on theasset value 119881119905 with strike price 119861 and maturity 119879 while (5)implies that the value of debt at 119879 is the sum of a default-freebond that guarantees payment of 119861 plus a short European putoption on the issuerrsquos assets with strike price 119861

It is assumed that under the physical probability measureP the process (119881119905)119905ge0 follows a geometric Brownianmotion ofthe form

119889119881119905 = 120583119881119881119905119889119905 + 120590119881119881119905119889119882119905 119905 isin [0 119879] (6)

where 120583119881 isin R is the mean rate of return on the assets120590119881 gt 0 is the asset volatility and (119882119905)119905ge0 is a Wiener processThe unique solution at time 119879 of the stochastic differentialequation (6) with initial value 1198810 is given by

119881119879 = 1198810 exp((120583119881 minus 12059021198812 )119879 + 120590119881119882119879) (7)

which implies that

ln119881119879 sim N(ln1198810 + (120583119881 minus 12059021198812 )119879 1205902119881119879) (8)

4 Complexity

Hence the real-world probability of default at time 119879measured at time 119905 = 0 is given by

P (119881119879 le 119861) = P (ln119881119879 le ln119861)= Φ( ln (1198611198810) minus (120583119881 minus 12059021198812) 119879

120590119881radic119879 ) (9)

A core assumption of Mertonrsquos model is that asset returns arelognormally distributed as can be seen in (8) It is widelyacknowledged however that empirical distributions of assetreturns tend to have heavier tails thus (9) may not be anaccurate description of empirically observed default rates

23 The Multivariate Merton Model Themodel presented inSection 22 is concerned with the default of a single issuer Inorder to estimate credit risk at a portfolio level a multivariateversion of the model is necessary A multivariate geometricBrownian motion with drift vector 120583119881 = (1205831 120583119898)1015840 vectorof volatilities 120590119881 = (1205901 120590119898) and correlation matrix Σis assumed for the dynamics of the multivariate asset valueprocess (V119905)119905ge0 with V119905 = (1198811199051 119881119905119898)1015840 so that for all 119894

119881119879119894 = 1198810119894 exp((120583119894 minus 121205902119894 )119879 + 120590119894119882119879119894) (10)

where the multivariate random vector W119879 with W119879 =(1198821198791 119882119879119898)1015840 is satisfyingW119879 sim 119873119898(0 119879Σ) Default takesplace if 119881119879119894 le 119861119894 where 119861119894 is the debt of company 119894 Itis clear that the default probability in the model remainsunchanged under simultaneous strictly increasing transfor-mations of 119881119879119894 and 119861119894 Thus one may define

119883119894 fl ln119881119879119894 minus ln1198810119894 minus (120583119894 minus (12) 1205902119894 ) 119879120590119894radic119879

119889119894 fl ln119861119894 minus ln1198810119894 minus (120583119894 minus (12) 1205902119894 ) 119879120590119894radic119879

(11)

and then default equivalently occurs if and only if 119883119894 le119889119894 Notice that 119883119894 is the standardized asset value log-returnln119881119879119894 minus ln1198810119894 It can be easily shown that the transformedvariables satisfy (1198831 119883119898)1015840 sim 119873119898(0 Σ) and their copulais the Gaussian copula Thus the probability of default forissuer 119894 is satisfying 119901119894 = Φ(119889119894) where Φ(sdot) denotes thecumulative distribution function of the standard normaldistribution A graphical representation of Mertonrsquos model isshown in Figure 1 In most practical implementations of themodel portfolio losses are modelled by directly consideringan 119898-dimensional random vector X = (1198831 119883119898)1015840 withX sim 119873119898(0 Σ) containing the standardized asset returnsand a deterministic vector d = (1198891 119889119898) containingthe critical thresholds with 119889119894 = Φminus1(119901119894) for given defaultprobabilities 119901119894 119894 = 1 119898 The default probabilitiesare usually estimated by historical default experience usingexternal ratings by agencies or model-based approaches

TimeT = 1 year

Default threshold di

4

3

2

1

0

minus1

minus2

minus3

minus4

Stan

dard

ised

asse

t ret

urns

A non-default path

Figure 1 In Mertonrsquos model default of issuer 119894 occurs if at time 119879asset value 119881119879119894 falls below debt value 119861119894 or equivalently if 119883119894 fl(ln119881119879119894 minus ln1198810119894 minus (120583119894 minus (12)1205902119894 )119879)120590119894radic119879 falls below the criticalthreshold 119889119894 fl (ln119861119879119894 minus ln1198810119894 minus (120583119894 minus (12)1205902119894 )119879)120590119894radic119879 Since119883119894 sim N(0 1) 119894rsquos default probability represented by the shaded areain the distribution plot is satisfying 119901119894 = Φ(119889119894) Note that defaultcan only take place at time 119879 does not depend on the path of theasset value process

24MertonModel as a FactorModel Thenumber of parame-ters contained in the correlationmatrixΣ grows polynomiallyin119898 and thus for large portfolios it is essential to have amoreparsimonious parametrization which is accomplished usinga factor model Additionally factor models are particularlyattractive due to the fact that they offer an intuitive interpre-tation of credit risk in relation to the performance of industryregion global economy or any other relevant indexes thatmay affect issuers in a systematic way In the following weshow how Mertonrsquos model can be understood as a factormodel In the factormodel approach asset returns are linearlydependent on a vector F of 119901 lt 119898 common underlyingfactors satisfying F sim 119873119901(0 Ω) Issuer 119894rsquos standardizedasset return is assumed to be driven by an issuer-specificcombination 119865119894 = 1205721015840119894F of the systematic factors

119883119894 = radic120573119894119865119894 + radic1 minus 120573119894120598119894 (12)

where 119865119894 and 1205981 120598119898 are independent standard normalvariables and 120598119894 represents the idiosyncratic risk Conse-quently 120573119894 can be seen as a measure of sensitivity of 119883119894to systematic risk as it represents the proportion of the 119883119894variation that is explained by the systematic factors Thecorrelations between asset returns are given by

120588 (119883119894 119883119895) = cov (119883119894 119883119895) = radic120573119894120573119895cov (119865119894 119865119895)= radic1205731198941205731198951205721015840119894Ω120572119895

(13)

since 119865119894 and 1205981 120598119898 are independent and standard normaland var(119883119894) = 1

Complexity 5

TimeT = 1 year

TimeT = 1 year

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

dCdsdC

dnsdC

Default thresholds for issuer Ci Default thresholds for issuer Ci

Figure 2 Under the standard Merton model the default threshold 119889119862119894 for corporate issuer 119862119894 is set to be equal to Φminus1(119901119862119894 ) Under theproposed model the threshold increases in the event of sovereign default making 119862119894rsquos default more likely as the contagion effect suggests

3 A Model for Credit Contagion

In the multifactor Merton model specified in Section 24 thestandardized asset returns 119883119894 119894 = 1 119898 are assumed tobe driven by a set of common underlying systematic factorsand the critical thresholds 119889119894 119894 = 1 119898 are satisfying119889119894 = Φminus1(119901119894) for all 119894 The only source of default dependencein such a framework is the dependence on the systematicfactors In themodel we propose we assume that in the eventof a sovereign default contagion will spread to the corporateissuers in the portfolio that are registered and operating inthat country causing default probability to be equal to theirCountryRank In Section 31 we demonstrate how to calibratethe critical thresholds so that each corporatersquos probabilityof default conditional on the default of the correspondingsovereign equals its CountryRank while its unconditionaldefault probability remains unchanged In Section 32 weshow how to construct a credit stress propagation networkand estimate the CountryRank parameter

31 Incorporating Contagion in Factor Models Consider acorporate issuer 119862119894 and its country of operation 119878 Denoteby 119901119862119894 the probability of default of 119862119894 Under the standardMertonmodel default occurs if119862119894rsquos standardized asset return119883119862119894 falls below its default threshold 119889119862119894 The critical threshold119889119862119894 is assumed to be equal to Φminus1(119901119862119894) and is independentof the state of the country of operation 119878 In the proposedmodel a corporate is subject to shocks from its country ofoperation its corresponding state is described by a binarystate variable The state is considered to be stressed in theevent of sovereign default In this case the issuerrsquos defaultthreshold increases causing it more likely to default asthe contagion effect suggests In case the corresponding

sovereign does not default the corporates liquidity state isconsidered stable We replace the default threshold 119889119862119894 with119889lowast119862119894 where119889lowast119862119894=

119889sd119862119894

if the corresponding sovereign defaults

119889nsd119862119894

otherwise

(14)

or equivalently

119889lowast119862119894 = 1119884119878=1119889sd119862119894

+ 1119884119878=0119889nsd119862119894

(15)

We denote by 119901119878 the probability of default of the countryof operation and by 120574119862119894 the CountryRank parameter whichindicates the increased probability of default of 119862119894 given thedefault of 119878 An example of the new default thresholds isshown in Figure 2 Our objective is to calibrate 119889sd

119862119894and 119889nsd

119862119894in such way that the overall default rate remains unchangedand P(119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 Denote by

1206012 (119909 119910 120588) fl 12120587radic1 minus 1205882 exp(minus1199092 + 1199102 minus 2120588119909119910

2 (1 minus 1205882) )

Φ2 (ℎ 119896 120588) fl intℎminusinfin

int119896minusinfin

1206012 (119909 119910 120588) 119889119910 119889119909(16)

the density and distribution function of the bivariate standardnormal distribution with correlation parameter 120588 isin (minus1 1)

6 Complexity

Note that 119889lowast119862119894(120596) = 119889sd119862119894for 120596 isin 119884119862119894 = 1 119884119878 = 1 sub 119884119878 = 1

and 119889lowast119862119894(120596) = 119889nsd119862119894

for 120596 isin 119884119862119894 = 1 119884119878 = 0 sub 119884119878 = 0 Werewrite P(119884119862119894 = 1 | 119884119878 = 1) in the following way

P (119884119862119894 = 1 | 119884119878 = 1)= 1P (119884119878 = 1)P (119884119862119894 = 1 119884119878 = 1)

= 1119901119878P [119883119862119894 lt 119889sd119862119894 119883119878 lt 119889119878]

= 1119901119878Φ2 (119889sd119862119894 119889119878 120588119878119862119894)

(17)

Using the above representation and given 119889119878 = Φminus1(119901119878) and120588119878119862119894 one can solve the equation

P (119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 (18)

over 119889sd119862119894 We proceed to the derivation of 119889nsd

119862119894in such way

that the overall default probability remains equal to 119901119862119894 Thisconstraint is important since contagion is assumed to haveno impact on the average loss Clearly

119901119862119894 = P (119884119862119894 = 1)= P (119884119862119894 = 1 119884119878 = 1) + P (119884119862119894 = 1 119884119878 = 0)= P (119884119862119894 = 1 | 119884119878 = 1)P (119884119878 = 1)

+ P (119884119862119894 = 1 119884119878 = 0)(19)

and thus

P (119884119862119894 = 1 119884119878 = 0) = 119901119862119894 minus 120574119862119894 sdot 119901119878 (20)

The left-hand side of the above equation can be representedas follows

P (corpdef cap nosovdef) = P [119883119862119894 lt 119889nsd119862119894

119883119878 gt 119889119878]= P [119883119862119894 lt 119889nsd

119862119894] minus P [119883119862119894 lt 119889nsd

119862119894 119883119878 lt 119889119878]

= Φ (119889nsd119862119894

) minus Φ2 (119889nsd119862119894

119889119878 120588119878119862119894) (21)

By use of the above and given 119889119878 = Φminus1(119901119878) and 120588119878119862119894 one cansolve the previous equation over 119889nsd

119862119894

32 Estimation of CountryRank In this section we elaborateon the estimation of the CountryRank parameter [30] whichserves as the probability of default of the corporate condi-tional on the default of the sovereign In addition we providedetails on the construction of the credit stress propagationnetwork

321 CountryRank In order to estimate contagion effectsin a network of issuers an algorithm such as DebtRank

[31] is necessary In the DebtRank calculation process stresspropagates even in the absence of defaults and each nodecan propagate stress only once before becoming inactiveThelevel of distress for a previously undistressed node is givenby the sum of incoming stress from its neighbors with amaximum value of 1 Summing up the incoming stress fromneighboring nodes seems reasonable when trying to estimatethe impact of one node or a set of nodes to a network ofinterconnected balance sheets where links represent lendingrelationships However when trying to quantify the prob-ability of default of a corporate node given the infectiousdefault of a sovereign node one has to consider that there issignificant overlap in terms of common stress and thus bysummingwemay be accounting for the same effectmore thanonce This effect is amplified in dense networks constructedfrom CDS data Therefore we introduce CountryRank as analternative measure which is suited for our contagion model

We assume that we have a hypothetical credit stresspropagation network where the nodes correspond to theissuers including the sovereign and the edges correspondto the impact of credit quality of one issuer on the otherThe details of the network construction will be presented inSection 322 Given such a network the CountryRank of thenodes can be defined recursively as follows

(i) First we stress the sovereign node and as a result itsCountryRank is 1

(ii) Let 120574119878 be the CountryRank of the sovereign and let119890(119895119896) denote the edge weight between nodes 119895 and 119896Given a node 119862119894 let 119901 = 11987811986211198622 sdot sdot sdot 119862119894minus1119862119894 be a pathwithout cycles from the sovereign node 119878 to the node119862119894 The weight of the path 119901 is defined as

119908 (119901) = 120574119878119890(12) sdot sdot sdot 119890(119894minus1119894) (22)

where 119890(119895119896) are the respective edge weights betweennodes 119895 and 119896 for 119895 isin 1 119894 minus 1 and 119896 isin 2 119894Let 1199011 119901119898 be the set of all acyclic paths fromthe sovereign node to the corporate node 119862119894 and let119908(1199011) 119908(119901119898) be the correspondingweightsThenthe CountryRank of node 119862119894 is defined as

120574119862119894 = max1le119895le119898

119908(119901119895) (23)

In order to compute the conditional probability of defaultof a corporate given the sovereign default analytically wewould need the joint distribution of probabilities of defaultof the nodes which has an exponential computational com-plexity and it is therefore intractable Thus we approximatethe conditional probability by choosing the path with themaximum weight in the above definition for CountryRank

The example in Figure 3 illustrates calculation of Coun-tryRank for a hypothetical network The network consistsof a sovereign node 119878 and corporate nodes 1198621 1198622 1198623 1198624The edge labels indicate weights in network between twonodes We initially stress the sovereign node which results ina CountryRank of 1 for node 119878 In the next step the stresspropagates to node 1198621 and as a result its CountryRank is 09

Complexity 7

09 08

02

03 06

03

S = 1

C1 C2

C3C4

(a) Sovereign node in stress

09 08

02

03 06

03

S = 1

C2

C3C4

C1= 09

(b) Stress propagates to node 1198621

09 08

02

03 06

03

S = 1

C3C4

C1= 09 C2

= 08

(c) Stress propagates to node 1198622

09 08

02

03 06

03

S = 1

C4

C1= 09 C2

= 08

C3= 048

(d) Stress propagates to node 1198623

09 08

02

03 06

03

S = 1

C1= 09 C2

= 08

C3= 048C4

= 027

(e) Stress propagates to node 1198624

Figure 3 Illustration of the CountryRank parameter using a hypothetical network The subfigures (a)ndash(e) show the propagation of stress inthe network starting from the sovereign node to corporate nodes At each step the stress spreads to a node using the path with the maximumweight from the sovereign node

Then node1198622 gets stressed giving it aCountryRank value 08For node 1198623 there are two paths from node 119878 so we pick thepath through node1198622 having a higher weight of 048 Finallythere are three paths from node 119878 to node 1198624 and the pathwith maximum weight is 027

322 Network Construction Credit default swap spreads aremarket-implied indicators of probability of default of anentity A credit default swap is a financial contract in whicha protection seller A insures a protection buyer B against thedefault of a third party C More precisely regular couponpayments with respect to a contractual notional 119873 and afixed rate 119904 the CDS spread are swapped with a payment of119873(1 minus RR) in the case of the default of C where RR the so-called recovery rate is a contract parameter which representsthe fraction of investment which is assumed be recovered inthe case of default of C

Modified 120598-Draw-Up We would like to measure to whatextent changes in CDS spreads of different issuers occursimultaneously For this we use the notion of a modified120598-draw-up to quantify the impact of deterioration of creditquality of one issuer on the other Modified 120598-draw-up is analteration of the 120598-draw-ups notion which is introduced in[32] In that article the authors use the notion of 120598-draw-ups to construct a network which models the conditionalprobabilities of spike-like comovements among pairs of CDSspreads A modified 120598-draw-up is defined as an upwardmovement in the time series in which the amplitude of themovement that is the difference between the subsequentlocal maxima and current local minima is greater than athreshold 120598 We record such local minima as the modified 120598-draw-ups The 120598 parameter for a local minima at time 119905 is setto be the standard deviation in the time series between days119905 minus 119899 and 119905 where 119899 is chosen to be 10 days Figure 5 shows

8 Complexity

the time series of Russian Federation CDS with the calibratedmodified 120598-draw-ups using a history of 10 days for calibra-tion

Filtering Market Impact Since we would like to measure thecomovement of the time series 119894 and 119895 we exclude the effectof the external market on these nodes as followsWe calibratethe 120598-draw-ups for the CDS time series of an index that doesnot represent the region in question for instance for Russianissuers we choose the iTraxx index which is the compositeCDS index of 125 CDS referencing European investmentgrade credit Then we filter out those 120598-draw-ups of node119894 which are the same as the 120598-draw-ups of the iTraxx indexincluding a time lag 120591That is if iTraxx has amodified 120598-draw-up on day 119905 thenwe remove themodified 120598-draw-ups of node119894 on days 119905 119905 + 1 119905 + 120591 We choose a time lag of 3 days forour calibration based on the input data which is consistentwith the choice in [32]

Edges After identifying the 120598-draw-ups for all the issuers andfiltering out the market impact the edges in our network areconstructed as follows The weight of an edge in the creditstress propagation network from node 119894 to node 119895 is theconditional probability that if node 119894 has an epsilon draw-upon day 119905 then node 119895 also has an epsilon draw-up on days119905 119905 + 1 119905 + 120591 where 120591 is the time lag More preciselylet 119873119894 be the number of 120598-draw-ups of node 119894 after filteringusing iTraxx index and119873119894119895 epsilon draw-ups of node 119894 whichare also epsilon draw-ups for node 119895 with the time lag 120591Then the edge weight119908119894119895 between nodes 119894 and 119895 is defined as119908119894119895 = 119873119894119873119894119895 Figure 6 shows the minimum spanning tree ofthe credit stress propagation network constructed using theCDS spread time series data of Russian issuers

Uncertainty in CountryRank We test the robustness of ourCountryRank calibration by varying the number of daysused for 120598-parameter The figure in Appendix B shows thatthe 120598-parameter for Russian Federation CDS time seriesremains stable when we vary the number of days We initiallyobtain time series of 120598-parameters by calculating standarddeviation in the last 119899 = 10 15 and 20 days on all localminima indices of Russian Federation CDS Subsequently wecalculate the mean of the absolute differences between theepsilon time series calculated and express this in units of themean of Russian Federation CDS time series The percentagedifference is 138 between the 10-day 120598-parameter and 15-day 120598-parameter and 222 between the 10-day and 20-day120598-parameters

Further we quantify the uncertainty in CountryRankparameter as follows For an corporate node we calculate theabsolute difference in CountryRank calculated using 119899 = 15and 20 days with CountryRank using 119899 = 10 days for the 120598-parameter We then calculate this difference as a percentageof the CountryRank calculated using 10 days for 120598-parameterfor all corporates and then compute their mean The meandifference between CountryRank calibrated using 119899 = 15days and 119899 = 10 days is 684 and 119899 = 20 days and 119899 = 10days is 973 for the Russian CDS data set

Table 1 Systematic factor index mapping

Factor IndexEurope MSCI EUROPEAsia MSCI AC ASIANorth America MSCI NORTH AMERICALatin America MSCI EM LATIN AMERICAMiddle East and Africa MSCI FM AFRICAPacific MSCI PACIFICMaterials MSCI WRLDMATERIALSConsumer products MSCI WRLDCONSUMER DISCRServices MSCI WRLDCONSUMER SVCFinancial MSCI WRLDFINANCIALSIndustrial MSCI WRLDINDUSTRIALS

Government ITRAXX SOVX GLOBAL LIQUIDINVESTMENT GRADE

4 Numerical Experiments

We implement the framework presented in Section 3 tosynthetic test portfolios and discuss the corresponding riskmetrics Further we perform a set of sensitivity studies andexplore the results

41 FactorModel Wefirst set up amultifactorMertonmodelas it was described in Section 2 We define a set of systematicfactors thatwill represent region and sector effectsWe choose6 region and 6 sector factors for which we select appropriateindexes as shown in Table 1 We then use 10 years of indextime series to derive the region and sector returns 119865119877(119895)119895 = 1 6 and 119865119878(119896) 119896 = 1 6 respectively and obtainan estimate of the correlation matrix Ω shown in Figure 7Subsequently we map all issuers to one region and one sectorfactor 119865119877(119894) and 119865119878(119894) respectively For instance a Dutch bankwill be associated with Europe and financial factors As aproxy of individual asset returns we use 10 years of equityor CDS time series depending on the data availability foreach issuer Finally we standardize the individual returnstime series (119883119894119905) and perform the following Ordinary LeastSquares regression against the systematic factor returns

119883119894119905 = 120572119877(119894)119865119877(119894)119905 + 120572119878(119896)119865119878(119896)119905 + 120598119894119905 (24)

to obtain 119877(119894) 119878(119894) and 120573119894 = 1198772 where1198772 is the coefficient ofdetermination and it is higher for issuers whose returns arelargely affected by the performance of the systematic factors

42 Synthetic Test Portfolios To investigate the properties ofthe contagion model we set up 2 test portfolios For theseportfolios the resulting risk measures are compared to thoseof the standard latent variable model with no contagionPortfolio A consists of 1 Russian government bond and 17bonds issued by corporations registered and operating in theRussian Federation As it is illustrated in Table 2 the issuersare of medium and low credit quality Portfolio B representsa similar but more diversified setup with 4 sovereign bonds

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

Complexity 3

synthetic portfolios Finally in Section 5 we summarize ourfindings and draw conclusions

2 Merton-Type Models for PortfolioCredit Risk

Most financial institutions use models that are based onsome form of the conditional independence assumptionaccording to which issuers depend on a set of commonunderlying factors Factormodels based on theMertonmodelare particularly popular for portfolio credit risk Our modelextends the multifactor Merton model to allow for creditcontagion In this section we present the basic portfoliomodelling setup outline the model of Merton and explainhow it can be specified as a factor model A more detailedpresentation of themultivariateMertonmodel is provided by[9]

21 Basic Setup andNotations This subsection introduces thebasic notation and terminology that will be used throughoutthis paper In addition we define themain risk characteristicsfor portfolio credit risk

The uncertainty of whether an issuer will fail to meetits financial obligations or not is measured by its probabilityof default For comparison reasons this is usually specifiedwith respect to a fixed time interval most commonly oneyear The probability of default then describes the probabilityof a default occurring in the particular time interval Theexposure at default is a measure of the extent to which oneis exposed to an issuer in the event of and at the time of thatissuerrsquos default The default of an issuer does not necessarilyimply that the creditor receives nothing from the issuer Thepercentage of loss incurred over the overall exposure in theevent of default is given by the loss given default Typicalvalues lie between 45 and 80

Consider a portfolio of119898 issuers indexed by 119894 = 1 119898and a fixed time horizon of 119879 = 1 year Denote by 119890119894 theexposure at default of issuer 119894 and by 119901119894 its probability ofdefault Let 119902119894 be the loss given default of issuer 119894 Denote by119884119894the default indicator in the time period [0 119879] All issuers areassumed to be in a nondefault state at time 119905 = 0 The defaultindicator 119884119894 is then a random variable defined by

119884119894 = 1 if issuer 119894 defaults0 otherwise

(1)

which clearly satisfies P(119884119894 = 1) = 119901119894 The overall portfolioloss is defined as the random variable

119871 fl119898sum119894=1

119902119894119890119894119884119894 (2)

With credit risk in mind it is useful to distinguish poten-tial losses in expected losses which are relatively predictableand thus can easily be managed and unexpected losses whichare more complicated to measure Risk managers are moreconcernedwith unexpected losses and focus on riskmeasuresrelating to the tail of the distribution of 119871

22TheModel ofMerton Credit riskmodels are typically dis-tinguished in structural and reduced-formmodels accordingto their methodology Structural models try to explain themechanism by which default takes place using variables suchas asset and debt values The model presented by Merton in[3] serves as the foundation for all these models Consider anissuer whose asset value follows a stochastic process (119881119905)119905ge0The issuer finances itself with equity and debt No dividendsare paid and no new debt can be issued InMertonrsquosmodel theissuerrsquos debt consists of a single zero-coupon bond with facevalue119861 andmaturity119879The values at time 119905 of equity and debtare denoted by 119878119905 and 119861119905 and the issuerrsquos asset value is simplythe sum of these that is

119881119905 = 119878119905 + 119861119905 119905 isin [0 119879] (3)

Default occurs if the issuer misses a payment to its debthold-ers which can happen only at the bondrsquos maturity 119879 At time119879 there are only two possible scenarios

(i) 119881119879 gt 119861 the value of the issuerrsquos assets is higher than itsdebt In this scenario the debtholders receive 119861119879 = 119861the shareholders receive the remainder 119878119879 = 119881119879 minus 119861and there is no default

(ii) 119881119879 le 119861 the value of the issuerrsquos assets is less thanits debt Hence the issuer cannot meet its financialobligations and defaults In that case shareholdershand over control to the bondholders who liquidatethe assets and receive the liquidation value in lieuof the debt Shareholders pay nothing and receivenothing therefore we obtain 119861119879 = 119881119879 119878119879 = 0

For these simple observations we obtain the below relations

119878119879 = max (119881119879 minus 119861 0) = (119881119879 minus 119861)+ (4)

119861119879 = min (119881119879 119861) = 119861 minus (119861 minus 119881119879)+ (5)

Equation (4) implies that the issuerrsquos equity at maturity 119879 canbe determined as the price of a European call option on theasset value 119881119905 with strike price 119861 and maturity 119879 while (5)implies that the value of debt at 119879 is the sum of a default-freebond that guarantees payment of 119861 plus a short European putoption on the issuerrsquos assets with strike price 119861

It is assumed that under the physical probability measureP the process (119881119905)119905ge0 follows a geometric Brownianmotion ofthe form

119889119881119905 = 120583119881119881119905119889119905 + 120590119881119881119905119889119882119905 119905 isin [0 119879] (6)

where 120583119881 isin R is the mean rate of return on the assets120590119881 gt 0 is the asset volatility and (119882119905)119905ge0 is a Wiener processThe unique solution at time 119879 of the stochastic differentialequation (6) with initial value 1198810 is given by

119881119879 = 1198810 exp((120583119881 minus 12059021198812 )119879 + 120590119881119882119879) (7)

which implies that

ln119881119879 sim N(ln1198810 + (120583119881 minus 12059021198812 )119879 1205902119881119879) (8)

4 Complexity

Hence the real-world probability of default at time 119879measured at time 119905 = 0 is given by

P (119881119879 le 119861) = P (ln119881119879 le ln119861)= Φ( ln (1198611198810) minus (120583119881 minus 12059021198812) 119879

120590119881radic119879 ) (9)

A core assumption of Mertonrsquos model is that asset returns arelognormally distributed as can be seen in (8) It is widelyacknowledged however that empirical distributions of assetreturns tend to have heavier tails thus (9) may not be anaccurate description of empirically observed default rates

23 The Multivariate Merton Model Themodel presented inSection 22 is concerned with the default of a single issuer Inorder to estimate credit risk at a portfolio level a multivariateversion of the model is necessary A multivariate geometricBrownian motion with drift vector 120583119881 = (1205831 120583119898)1015840 vectorof volatilities 120590119881 = (1205901 120590119898) and correlation matrix Σis assumed for the dynamics of the multivariate asset valueprocess (V119905)119905ge0 with V119905 = (1198811199051 119881119905119898)1015840 so that for all 119894

119881119879119894 = 1198810119894 exp((120583119894 minus 121205902119894 )119879 + 120590119894119882119879119894) (10)

where the multivariate random vector W119879 with W119879 =(1198821198791 119882119879119898)1015840 is satisfyingW119879 sim 119873119898(0 119879Σ) Default takesplace if 119881119879119894 le 119861119894 where 119861119894 is the debt of company 119894 Itis clear that the default probability in the model remainsunchanged under simultaneous strictly increasing transfor-mations of 119881119879119894 and 119861119894 Thus one may define

119883119894 fl ln119881119879119894 minus ln1198810119894 minus (120583119894 minus (12) 1205902119894 ) 119879120590119894radic119879

119889119894 fl ln119861119894 minus ln1198810119894 minus (120583119894 minus (12) 1205902119894 ) 119879120590119894radic119879

(11)

and then default equivalently occurs if and only if 119883119894 le119889119894 Notice that 119883119894 is the standardized asset value log-returnln119881119879119894 minus ln1198810119894 It can be easily shown that the transformedvariables satisfy (1198831 119883119898)1015840 sim 119873119898(0 Σ) and their copulais the Gaussian copula Thus the probability of default forissuer 119894 is satisfying 119901119894 = Φ(119889119894) where Φ(sdot) denotes thecumulative distribution function of the standard normaldistribution A graphical representation of Mertonrsquos model isshown in Figure 1 In most practical implementations of themodel portfolio losses are modelled by directly consideringan 119898-dimensional random vector X = (1198831 119883119898)1015840 withX sim 119873119898(0 Σ) containing the standardized asset returnsand a deterministic vector d = (1198891 119889119898) containingthe critical thresholds with 119889119894 = Φminus1(119901119894) for given defaultprobabilities 119901119894 119894 = 1 119898 The default probabilitiesare usually estimated by historical default experience usingexternal ratings by agencies or model-based approaches

TimeT = 1 year

Default threshold di

4

3

2

1

0

minus1

minus2

minus3

minus4

Stan

dard

ised

asse

t ret

urns

A non-default path

Figure 1 In Mertonrsquos model default of issuer 119894 occurs if at time 119879asset value 119881119879119894 falls below debt value 119861119894 or equivalently if 119883119894 fl(ln119881119879119894 minus ln1198810119894 minus (120583119894 minus (12)1205902119894 )119879)120590119894radic119879 falls below the criticalthreshold 119889119894 fl (ln119861119879119894 minus ln1198810119894 minus (120583119894 minus (12)1205902119894 )119879)120590119894radic119879 Since119883119894 sim N(0 1) 119894rsquos default probability represented by the shaded areain the distribution plot is satisfying 119901119894 = Φ(119889119894) Note that defaultcan only take place at time 119879 does not depend on the path of theasset value process

24MertonModel as a FactorModel Thenumber of parame-ters contained in the correlationmatrixΣ grows polynomiallyin119898 and thus for large portfolios it is essential to have amoreparsimonious parametrization which is accomplished usinga factor model Additionally factor models are particularlyattractive due to the fact that they offer an intuitive interpre-tation of credit risk in relation to the performance of industryregion global economy or any other relevant indexes thatmay affect issuers in a systematic way In the following weshow how Mertonrsquos model can be understood as a factormodel In the factormodel approach asset returns are linearlydependent on a vector F of 119901 lt 119898 common underlyingfactors satisfying F sim 119873119901(0 Ω) Issuer 119894rsquos standardizedasset return is assumed to be driven by an issuer-specificcombination 119865119894 = 1205721015840119894F of the systematic factors

119883119894 = radic120573119894119865119894 + radic1 minus 120573119894120598119894 (12)

where 119865119894 and 1205981 120598119898 are independent standard normalvariables and 120598119894 represents the idiosyncratic risk Conse-quently 120573119894 can be seen as a measure of sensitivity of 119883119894to systematic risk as it represents the proportion of the 119883119894variation that is explained by the systematic factors Thecorrelations between asset returns are given by

120588 (119883119894 119883119895) = cov (119883119894 119883119895) = radic120573119894120573119895cov (119865119894 119865119895)= radic1205731198941205731198951205721015840119894Ω120572119895

(13)

since 119865119894 and 1205981 120598119898 are independent and standard normaland var(119883119894) = 1

Complexity 5

TimeT = 1 year

TimeT = 1 year

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

dCdsdC

dnsdC

Default thresholds for issuer Ci Default thresholds for issuer Ci

Figure 2 Under the standard Merton model the default threshold 119889119862119894 for corporate issuer 119862119894 is set to be equal to Φminus1(119901119862119894 ) Under theproposed model the threshold increases in the event of sovereign default making 119862119894rsquos default more likely as the contagion effect suggests

3 A Model for Credit Contagion

In the multifactor Merton model specified in Section 24 thestandardized asset returns 119883119894 119894 = 1 119898 are assumed tobe driven by a set of common underlying systematic factorsand the critical thresholds 119889119894 119894 = 1 119898 are satisfying119889119894 = Φminus1(119901119894) for all 119894 The only source of default dependencein such a framework is the dependence on the systematicfactors In themodel we propose we assume that in the eventof a sovereign default contagion will spread to the corporateissuers in the portfolio that are registered and operating inthat country causing default probability to be equal to theirCountryRank In Section 31 we demonstrate how to calibratethe critical thresholds so that each corporatersquos probabilityof default conditional on the default of the correspondingsovereign equals its CountryRank while its unconditionaldefault probability remains unchanged In Section 32 weshow how to construct a credit stress propagation networkand estimate the CountryRank parameter

31 Incorporating Contagion in Factor Models Consider acorporate issuer 119862119894 and its country of operation 119878 Denoteby 119901119862119894 the probability of default of 119862119894 Under the standardMertonmodel default occurs if119862119894rsquos standardized asset return119883119862119894 falls below its default threshold 119889119862119894 The critical threshold119889119862119894 is assumed to be equal to Φminus1(119901119862119894) and is independentof the state of the country of operation 119878 In the proposedmodel a corporate is subject to shocks from its country ofoperation its corresponding state is described by a binarystate variable The state is considered to be stressed in theevent of sovereign default In this case the issuerrsquos defaultthreshold increases causing it more likely to default asthe contagion effect suggests In case the corresponding

sovereign does not default the corporates liquidity state isconsidered stable We replace the default threshold 119889119862119894 with119889lowast119862119894 where119889lowast119862119894=

119889sd119862119894

if the corresponding sovereign defaults

119889nsd119862119894

otherwise

(14)

or equivalently

119889lowast119862119894 = 1119884119878=1119889sd119862119894

+ 1119884119878=0119889nsd119862119894

(15)

We denote by 119901119878 the probability of default of the countryof operation and by 120574119862119894 the CountryRank parameter whichindicates the increased probability of default of 119862119894 given thedefault of 119878 An example of the new default thresholds isshown in Figure 2 Our objective is to calibrate 119889sd

119862119894and 119889nsd

119862119894in such way that the overall default rate remains unchangedand P(119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 Denote by

1206012 (119909 119910 120588) fl 12120587radic1 minus 1205882 exp(minus1199092 + 1199102 minus 2120588119909119910

2 (1 minus 1205882) )

Φ2 (ℎ 119896 120588) fl intℎminusinfin

int119896minusinfin

1206012 (119909 119910 120588) 119889119910 119889119909(16)

the density and distribution function of the bivariate standardnormal distribution with correlation parameter 120588 isin (minus1 1)

6 Complexity

Note that 119889lowast119862119894(120596) = 119889sd119862119894for 120596 isin 119884119862119894 = 1 119884119878 = 1 sub 119884119878 = 1

and 119889lowast119862119894(120596) = 119889nsd119862119894

for 120596 isin 119884119862119894 = 1 119884119878 = 0 sub 119884119878 = 0 Werewrite P(119884119862119894 = 1 | 119884119878 = 1) in the following way

P (119884119862119894 = 1 | 119884119878 = 1)= 1P (119884119878 = 1)P (119884119862119894 = 1 119884119878 = 1)

= 1119901119878P [119883119862119894 lt 119889sd119862119894 119883119878 lt 119889119878]

= 1119901119878Φ2 (119889sd119862119894 119889119878 120588119878119862119894)

(17)

Using the above representation and given 119889119878 = Φminus1(119901119878) and120588119878119862119894 one can solve the equation

P (119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 (18)

over 119889sd119862119894 We proceed to the derivation of 119889nsd

119862119894in such way

that the overall default probability remains equal to 119901119862119894 Thisconstraint is important since contagion is assumed to haveno impact on the average loss Clearly

119901119862119894 = P (119884119862119894 = 1)= P (119884119862119894 = 1 119884119878 = 1) + P (119884119862119894 = 1 119884119878 = 0)= P (119884119862119894 = 1 | 119884119878 = 1)P (119884119878 = 1)

+ P (119884119862119894 = 1 119884119878 = 0)(19)

and thus

P (119884119862119894 = 1 119884119878 = 0) = 119901119862119894 minus 120574119862119894 sdot 119901119878 (20)

The left-hand side of the above equation can be representedas follows

P (corpdef cap nosovdef) = P [119883119862119894 lt 119889nsd119862119894

119883119878 gt 119889119878]= P [119883119862119894 lt 119889nsd

119862119894] minus P [119883119862119894 lt 119889nsd

119862119894 119883119878 lt 119889119878]

= Φ (119889nsd119862119894

) minus Φ2 (119889nsd119862119894

119889119878 120588119878119862119894) (21)

By use of the above and given 119889119878 = Φminus1(119901119878) and 120588119878119862119894 one cansolve the previous equation over 119889nsd

119862119894

32 Estimation of CountryRank In this section we elaborateon the estimation of the CountryRank parameter [30] whichserves as the probability of default of the corporate condi-tional on the default of the sovereign In addition we providedetails on the construction of the credit stress propagationnetwork

321 CountryRank In order to estimate contagion effectsin a network of issuers an algorithm such as DebtRank

[31] is necessary In the DebtRank calculation process stresspropagates even in the absence of defaults and each nodecan propagate stress only once before becoming inactiveThelevel of distress for a previously undistressed node is givenby the sum of incoming stress from its neighbors with amaximum value of 1 Summing up the incoming stress fromneighboring nodes seems reasonable when trying to estimatethe impact of one node or a set of nodes to a network ofinterconnected balance sheets where links represent lendingrelationships However when trying to quantify the prob-ability of default of a corporate node given the infectiousdefault of a sovereign node one has to consider that there issignificant overlap in terms of common stress and thus bysummingwemay be accounting for the same effectmore thanonce This effect is amplified in dense networks constructedfrom CDS data Therefore we introduce CountryRank as analternative measure which is suited for our contagion model

We assume that we have a hypothetical credit stresspropagation network where the nodes correspond to theissuers including the sovereign and the edges correspondto the impact of credit quality of one issuer on the otherThe details of the network construction will be presented inSection 322 Given such a network the CountryRank of thenodes can be defined recursively as follows

(i) First we stress the sovereign node and as a result itsCountryRank is 1

(ii) Let 120574119878 be the CountryRank of the sovereign and let119890(119895119896) denote the edge weight between nodes 119895 and 119896Given a node 119862119894 let 119901 = 11987811986211198622 sdot sdot sdot 119862119894minus1119862119894 be a pathwithout cycles from the sovereign node 119878 to the node119862119894 The weight of the path 119901 is defined as

119908 (119901) = 120574119878119890(12) sdot sdot sdot 119890(119894minus1119894) (22)

where 119890(119895119896) are the respective edge weights betweennodes 119895 and 119896 for 119895 isin 1 119894 minus 1 and 119896 isin 2 119894Let 1199011 119901119898 be the set of all acyclic paths fromthe sovereign node to the corporate node 119862119894 and let119908(1199011) 119908(119901119898) be the correspondingweightsThenthe CountryRank of node 119862119894 is defined as

120574119862119894 = max1le119895le119898

119908(119901119895) (23)

In order to compute the conditional probability of defaultof a corporate given the sovereign default analytically wewould need the joint distribution of probabilities of defaultof the nodes which has an exponential computational com-plexity and it is therefore intractable Thus we approximatethe conditional probability by choosing the path with themaximum weight in the above definition for CountryRank

The example in Figure 3 illustrates calculation of Coun-tryRank for a hypothetical network The network consistsof a sovereign node 119878 and corporate nodes 1198621 1198622 1198623 1198624The edge labels indicate weights in network between twonodes We initially stress the sovereign node which results ina CountryRank of 1 for node 119878 In the next step the stresspropagates to node 1198621 and as a result its CountryRank is 09

Complexity 7

09 08

02

03 06

03

S = 1

C1 C2

C3C4

(a) Sovereign node in stress

09 08

02

03 06

03

S = 1

C2

C3C4

C1= 09

(b) Stress propagates to node 1198621

09 08

02

03 06

03

S = 1

C3C4

C1= 09 C2

= 08

(c) Stress propagates to node 1198622

09 08

02

03 06

03

S = 1

C4

C1= 09 C2

= 08

C3= 048

(d) Stress propagates to node 1198623

09 08

02

03 06

03

S = 1

C1= 09 C2

= 08

C3= 048C4

= 027

(e) Stress propagates to node 1198624

Figure 3 Illustration of the CountryRank parameter using a hypothetical network The subfigures (a)ndash(e) show the propagation of stress inthe network starting from the sovereign node to corporate nodes At each step the stress spreads to a node using the path with the maximumweight from the sovereign node

Then node1198622 gets stressed giving it aCountryRank value 08For node 1198623 there are two paths from node 119878 so we pick thepath through node1198622 having a higher weight of 048 Finallythere are three paths from node 119878 to node 1198624 and the pathwith maximum weight is 027

322 Network Construction Credit default swap spreads aremarket-implied indicators of probability of default of anentity A credit default swap is a financial contract in whicha protection seller A insures a protection buyer B against thedefault of a third party C More precisely regular couponpayments with respect to a contractual notional 119873 and afixed rate 119904 the CDS spread are swapped with a payment of119873(1 minus RR) in the case of the default of C where RR the so-called recovery rate is a contract parameter which representsthe fraction of investment which is assumed be recovered inthe case of default of C

Modified 120598-Draw-Up We would like to measure to whatextent changes in CDS spreads of different issuers occursimultaneously For this we use the notion of a modified120598-draw-up to quantify the impact of deterioration of creditquality of one issuer on the other Modified 120598-draw-up is analteration of the 120598-draw-ups notion which is introduced in[32] In that article the authors use the notion of 120598-draw-ups to construct a network which models the conditionalprobabilities of spike-like comovements among pairs of CDSspreads A modified 120598-draw-up is defined as an upwardmovement in the time series in which the amplitude of themovement that is the difference between the subsequentlocal maxima and current local minima is greater than athreshold 120598 We record such local minima as the modified 120598-draw-ups The 120598 parameter for a local minima at time 119905 is setto be the standard deviation in the time series between days119905 minus 119899 and 119905 where 119899 is chosen to be 10 days Figure 5 shows

8 Complexity

the time series of Russian Federation CDS with the calibratedmodified 120598-draw-ups using a history of 10 days for calibra-tion

Filtering Market Impact Since we would like to measure thecomovement of the time series 119894 and 119895 we exclude the effectof the external market on these nodes as followsWe calibratethe 120598-draw-ups for the CDS time series of an index that doesnot represent the region in question for instance for Russianissuers we choose the iTraxx index which is the compositeCDS index of 125 CDS referencing European investmentgrade credit Then we filter out those 120598-draw-ups of node119894 which are the same as the 120598-draw-ups of the iTraxx indexincluding a time lag 120591That is if iTraxx has amodified 120598-draw-up on day 119905 thenwe remove themodified 120598-draw-ups of node119894 on days 119905 119905 + 1 119905 + 120591 We choose a time lag of 3 days forour calibration based on the input data which is consistentwith the choice in [32]

Edges After identifying the 120598-draw-ups for all the issuers andfiltering out the market impact the edges in our network areconstructed as follows The weight of an edge in the creditstress propagation network from node 119894 to node 119895 is theconditional probability that if node 119894 has an epsilon draw-upon day 119905 then node 119895 also has an epsilon draw-up on days119905 119905 + 1 119905 + 120591 where 120591 is the time lag More preciselylet 119873119894 be the number of 120598-draw-ups of node 119894 after filteringusing iTraxx index and119873119894119895 epsilon draw-ups of node 119894 whichare also epsilon draw-ups for node 119895 with the time lag 120591Then the edge weight119908119894119895 between nodes 119894 and 119895 is defined as119908119894119895 = 119873119894119873119894119895 Figure 6 shows the minimum spanning tree ofthe credit stress propagation network constructed using theCDS spread time series data of Russian issuers

Uncertainty in CountryRank We test the robustness of ourCountryRank calibration by varying the number of daysused for 120598-parameter The figure in Appendix B shows thatthe 120598-parameter for Russian Federation CDS time seriesremains stable when we vary the number of days We initiallyobtain time series of 120598-parameters by calculating standarddeviation in the last 119899 = 10 15 and 20 days on all localminima indices of Russian Federation CDS Subsequently wecalculate the mean of the absolute differences between theepsilon time series calculated and express this in units of themean of Russian Federation CDS time series The percentagedifference is 138 between the 10-day 120598-parameter and 15-day 120598-parameter and 222 between the 10-day and 20-day120598-parameters

Further we quantify the uncertainty in CountryRankparameter as follows For an corporate node we calculate theabsolute difference in CountryRank calculated using 119899 = 15and 20 days with CountryRank using 119899 = 10 days for the 120598-parameter We then calculate this difference as a percentageof the CountryRank calculated using 10 days for 120598-parameterfor all corporates and then compute their mean The meandifference between CountryRank calibrated using 119899 = 15days and 119899 = 10 days is 684 and 119899 = 20 days and 119899 = 10days is 973 for the Russian CDS data set

Table 1 Systematic factor index mapping

Factor IndexEurope MSCI EUROPEAsia MSCI AC ASIANorth America MSCI NORTH AMERICALatin America MSCI EM LATIN AMERICAMiddle East and Africa MSCI FM AFRICAPacific MSCI PACIFICMaterials MSCI WRLDMATERIALSConsumer products MSCI WRLDCONSUMER DISCRServices MSCI WRLDCONSUMER SVCFinancial MSCI WRLDFINANCIALSIndustrial MSCI WRLDINDUSTRIALS

Government ITRAXX SOVX GLOBAL LIQUIDINVESTMENT GRADE

4 Numerical Experiments

We implement the framework presented in Section 3 tosynthetic test portfolios and discuss the corresponding riskmetrics Further we perform a set of sensitivity studies andexplore the results

41 FactorModel Wefirst set up amultifactorMertonmodelas it was described in Section 2 We define a set of systematicfactors thatwill represent region and sector effectsWe choose6 region and 6 sector factors for which we select appropriateindexes as shown in Table 1 We then use 10 years of indextime series to derive the region and sector returns 119865119877(119895)119895 = 1 6 and 119865119878(119896) 119896 = 1 6 respectively and obtainan estimate of the correlation matrix Ω shown in Figure 7Subsequently we map all issuers to one region and one sectorfactor 119865119877(119894) and 119865119878(119894) respectively For instance a Dutch bankwill be associated with Europe and financial factors As aproxy of individual asset returns we use 10 years of equityor CDS time series depending on the data availability foreach issuer Finally we standardize the individual returnstime series (119883119894119905) and perform the following Ordinary LeastSquares regression against the systematic factor returns

119883119894119905 = 120572119877(119894)119865119877(119894)119905 + 120572119878(119896)119865119878(119896)119905 + 120598119894119905 (24)

to obtain 119877(119894) 119878(119894) and 120573119894 = 1198772 where1198772 is the coefficient ofdetermination and it is higher for issuers whose returns arelargely affected by the performance of the systematic factors

42 Synthetic Test Portfolios To investigate the properties ofthe contagion model we set up 2 test portfolios For theseportfolios the resulting risk measures are compared to thoseof the standard latent variable model with no contagionPortfolio A consists of 1 Russian government bond and 17bonds issued by corporations registered and operating in theRussian Federation As it is illustrated in Table 2 the issuersare of medium and low credit quality Portfolio B representsa similar but more diversified setup with 4 sovereign bonds

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

4 Complexity

Hence the real-world probability of default at time 119879measured at time 119905 = 0 is given by

P (119881119879 le 119861) = P (ln119881119879 le ln119861)= Φ( ln (1198611198810) minus (120583119881 minus 12059021198812) 119879

120590119881radic119879 ) (9)

A core assumption of Mertonrsquos model is that asset returns arelognormally distributed as can be seen in (8) It is widelyacknowledged however that empirical distributions of assetreturns tend to have heavier tails thus (9) may not be anaccurate description of empirically observed default rates

23 The Multivariate Merton Model Themodel presented inSection 22 is concerned with the default of a single issuer Inorder to estimate credit risk at a portfolio level a multivariateversion of the model is necessary A multivariate geometricBrownian motion with drift vector 120583119881 = (1205831 120583119898)1015840 vectorof volatilities 120590119881 = (1205901 120590119898) and correlation matrix Σis assumed for the dynamics of the multivariate asset valueprocess (V119905)119905ge0 with V119905 = (1198811199051 119881119905119898)1015840 so that for all 119894

119881119879119894 = 1198810119894 exp((120583119894 minus 121205902119894 )119879 + 120590119894119882119879119894) (10)

where the multivariate random vector W119879 with W119879 =(1198821198791 119882119879119898)1015840 is satisfyingW119879 sim 119873119898(0 119879Σ) Default takesplace if 119881119879119894 le 119861119894 where 119861119894 is the debt of company 119894 Itis clear that the default probability in the model remainsunchanged under simultaneous strictly increasing transfor-mations of 119881119879119894 and 119861119894 Thus one may define

119883119894 fl ln119881119879119894 minus ln1198810119894 minus (120583119894 minus (12) 1205902119894 ) 119879120590119894radic119879

119889119894 fl ln119861119894 minus ln1198810119894 minus (120583119894 minus (12) 1205902119894 ) 119879120590119894radic119879

(11)

and then default equivalently occurs if and only if 119883119894 le119889119894 Notice that 119883119894 is the standardized asset value log-returnln119881119879119894 minus ln1198810119894 It can be easily shown that the transformedvariables satisfy (1198831 119883119898)1015840 sim 119873119898(0 Σ) and their copulais the Gaussian copula Thus the probability of default forissuer 119894 is satisfying 119901119894 = Φ(119889119894) where Φ(sdot) denotes thecumulative distribution function of the standard normaldistribution A graphical representation of Mertonrsquos model isshown in Figure 1 In most practical implementations of themodel portfolio losses are modelled by directly consideringan 119898-dimensional random vector X = (1198831 119883119898)1015840 withX sim 119873119898(0 Σ) containing the standardized asset returnsand a deterministic vector d = (1198891 119889119898) containingthe critical thresholds with 119889119894 = Φminus1(119901119894) for given defaultprobabilities 119901119894 119894 = 1 119898 The default probabilitiesare usually estimated by historical default experience usingexternal ratings by agencies or model-based approaches

TimeT = 1 year

Default threshold di

4

3

2

1

0

minus1

minus2

minus3

minus4

Stan

dard

ised

asse

t ret

urns

A non-default path

Figure 1 In Mertonrsquos model default of issuer 119894 occurs if at time 119879asset value 119881119879119894 falls below debt value 119861119894 or equivalently if 119883119894 fl(ln119881119879119894 minus ln1198810119894 minus (120583119894 minus (12)1205902119894 )119879)120590119894radic119879 falls below the criticalthreshold 119889119894 fl (ln119861119879119894 minus ln1198810119894 minus (120583119894 minus (12)1205902119894 )119879)120590119894radic119879 Since119883119894 sim N(0 1) 119894rsquos default probability represented by the shaded areain the distribution plot is satisfying 119901119894 = Φ(119889119894) Note that defaultcan only take place at time 119879 does not depend on the path of theasset value process

24MertonModel as a FactorModel Thenumber of parame-ters contained in the correlationmatrixΣ grows polynomiallyin119898 and thus for large portfolios it is essential to have amoreparsimonious parametrization which is accomplished usinga factor model Additionally factor models are particularlyattractive due to the fact that they offer an intuitive interpre-tation of credit risk in relation to the performance of industryregion global economy or any other relevant indexes thatmay affect issuers in a systematic way In the following weshow how Mertonrsquos model can be understood as a factormodel In the factormodel approach asset returns are linearlydependent on a vector F of 119901 lt 119898 common underlyingfactors satisfying F sim 119873119901(0 Ω) Issuer 119894rsquos standardizedasset return is assumed to be driven by an issuer-specificcombination 119865119894 = 1205721015840119894F of the systematic factors

119883119894 = radic120573119894119865119894 + radic1 minus 120573119894120598119894 (12)

where 119865119894 and 1205981 120598119898 are independent standard normalvariables and 120598119894 represents the idiosyncratic risk Conse-quently 120573119894 can be seen as a measure of sensitivity of 119883119894to systematic risk as it represents the proportion of the 119883119894variation that is explained by the systematic factors Thecorrelations between asset returns are given by

120588 (119883119894 119883119895) = cov (119883119894 119883119895) = radic120573119894120573119895cov (119865119894 119865119895)= radic1205731198941205731198951205721015840119894Ω120572119895

(13)

since 119865119894 and 1205981 120598119898 are independent and standard normaland var(119883119894) = 1

Complexity 5

TimeT = 1 year

TimeT = 1 year

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

dCdsdC

dnsdC

Default thresholds for issuer Ci Default thresholds for issuer Ci

Figure 2 Under the standard Merton model the default threshold 119889119862119894 for corporate issuer 119862119894 is set to be equal to Φminus1(119901119862119894 ) Under theproposed model the threshold increases in the event of sovereign default making 119862119894rsquos default more likely as the contagion effect suggests

3 A Model for Credit Contagion

In the multifactor Merton model specified in Section 24 thestandardized asset returns 119883119894 119894 = 1 119898 are assumed tobe driven by a set of common underlying systematic factorsand the critical thresholds 119889119894 119894 = 1 119898 are satisfying119889119894 = Φminus1(119901119894) for all 119894 The only source of default dependencein such a framework is the dependence on the systematicfactors In themodel we propose we assume that in the eventof a sovereign default contagion will spread to the corporateissuers in the portfolio that are registered and operating inthat country causing default probability to be equal to theirCountryRank In Section 31 we demonstrate how to calibratethe critical thresholds so that each corporatersquos probabilityof default conditional on the default of the correspondingsovereign equals its CountryRank while its unconditionaldefault probability remains unchanged In Section 32 weshow how to construct a credit stress propagation networkand estimate the CountryRank parameter

31 Incorporating Contagion in Factor Models Consider acorporate issuer 119862119894 and its country of operation 119878 Denoteby 119901119862119894 the probability of default of 119862119894 Under the standardMertonmodel default occurs if119862119894rsquos standardized asset return119883119862119894 falls below its default threshold 119889119862119894 The critical threshold119889119862119894 is assumed to be equal to Φminus1(119901119862119894) and is independentof the state of the country of operation 119878 In the proposedmodel a corporate is subject to shocks from its country ofoperation its corresponding state is described by a binarystate variable The state is considered to be stressed in theevent of sovereign default In this case the issuerrsquos defaultthreshold increases causing it more likely to default asthe contagion effect suggests In case the corresponding

sovereign does not default the corporates liquidity state isconsidered stable We replace the default threshold 119889119862119894 with119889lowast119862119894 where119889lowast119862119894=

119889sd119862119894

if the corresponding sovereign defaults

119889nsd119862119894

otherwise

(14)

or equivalently

119889lowast119862119894 = 1119884119878=1119889sd119862119894

+ 1119884119878=0119889nsd119862119894

(15)

We denote by 119901119878 the probability of default of the countryof operation and by 120574119862119894 the CountryRank parameter whichindicates the increased probability of default of 119862119894 given thedefault of 119878 An example of the new default thresholds isshown in Figure 2 Our objective is to calibrate 119889sd

119862119894and 119889nsd

119862119894in such way that the overall default rate remains unchangedand P(119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 Denote by

1206012 (119909 119910 120588) fl 12120587radic1 minus 1205882 exp(minus1199092 + 1199102 minus 2120588119909119910

2 (1 minus 1205882) )

Φ2 (ℎ 119896 120588) fl intℎminusinfin

int119896minusinfin

1206012 (119909 119910 120588) 119889119910 119889119909(16)

the density and distribution function of the bivariate standardnormal distribution with correlation parameter 120588 isin (minus1 1)

6 Complexity

Note that 119889lowast119862119894(120596) = 119889sd119862119894for 120596 isin 119884119862119894 = 1 119884119878 = 1 sub 119884119878 = 1

and 119889lowast119862119894(120596) = 119889nsd119862119894

for 120596 isin 119884119862119894 = 1 119884119878 = 0 sub 119884119878 = 0 Werewrite P(119884119862119894 = 1 | 119884119878 = 1) in the following way

P (119884119862119894 = 1 | 119884119878 = 1)= 1P (119884119878 = 1)P (119884119862119894 = 1 119884119878 = 1)

= 1119901119878P [119883119862119894 lt 119889sd119862119894 119883119878 lt 119889119878]

= 1119901119878Φ2 (119889sd119862119894 119889119878 120588119878119862119894)

(17)

Using the above representation and given 119889119878 = Φminus1(119901119878) and120588119878119862119894 one can solve the equation

P (119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 (18)

over 119889sd119862119894 We proceed to the derivation of 119889nsd

119862119894in such way

that the overall default probability remains equal to 119901119862119894 Thisconstraint is important since contagion is assumed to haveno impact on the average loss Clearly

119901119862119894 = P (119884119862119894 = 1)= P (119884119862119894 = 1 119884119878 = 1) + P (119884119862119894 = 1 119884119878 = 0)= P (119884119862119894 = 1 | 119884119878 = 1)P (119884119878 = 1)

+ P (119884119862119894 = 1 119884119878 = 0)(19)

and thus

P (119884119862119894 = 1 119884119878 = 0) = 119901119862119894 minus 120574119862119894 sdot 119901119878 (20)

The left-hand side of the above equation can be representedas follows

P (corpdef cap nosovdef) = P [119883119862119894 lt 119889nsd119862119894

119883119878 gt 119889119878]= P [119883119862119894 lt 119889nsd

119862119894] minus P [119883119862119894 lt 119889nsd

119862119894 119883119878 lt 119889119878]

= Φ (119889nsd119862119894

) minus Φ2 (119889nsd119862119894

119889119878 120588119878119862119894) (21)

By use of the above and given 119889119878 = Φminus1(119901119878) and 120588119878119862119894 one cansolve the previous equation over 119889nsd

119862119894

32 Estimation of CountryRank In this section we elaborateon the estimation of the CountryRank parameter [30] whichserves as the probability of default of the corporate condi-tional on the default of the sovereign In addition we providedetails on the construction of the credit stress propagationnetwork

321 CountryRank In order to estimate contagion effectsin a network of issuers an algorithm such as DebtRank

[31] is necessary In the DebtRank calculation process stresspropagates even in the absence of defaults and each nodecan propagate stress only once before becoming inactiveThelevel of distress for a previously undistressed node is givenby the sum of incoming stress from its neighbors with amaximum value of 1 Summing up the incoming stress fromneighboring nodes seems reasonable when trying to estimatethe impact of one node or a set of nodes to a network ofinterconnected balance sheets where links represent lendingrelationships However when trying to quantify the prob-ability of default of a corporate node given the infectiousdefault of a sovereign node one has to consider that there issignificant overlap in terms of common stress and thus bysummingwemay be accounting for the same effectmore thanonce This effect is amplified in dense networks constructedfrom CDS data Therefore we introduce CountryRank as analternative measure which is suited for our contagion model

We assume that we have a hypothetical credit stresspropagation network where the nodes correspond to theissuers including the sovereign and the edges correspondto the impact of credit quality of one issuer on the otherThe details of the network construction will be presented inSection 322 Given such a network the CountryRank of thenodes can be defined recursively as follows

(i) First we stress the sovereign node and as a result itsCountryRank is 1

(ii) Let 120574119878 be the CountryRank of the sovereign and let119890(119895119896) denote the edge weight between nodes 119895 and 119896Given a node 119862119894 let 119901 = 11987811986211198622 sdot sdot sdot 119862119894minus1119862119894 be a pathwithout cycles from the sovereign node 119878 to the node119862119894 The weight of the path 119901 is defined as

119908 (119901) = 120574119878119890(12) sdot sdot sdot 119890(119894minus1119894) (22)

where 119890(119895119896) are the respective edge weights betweennodes 119895 and 119896 for 119895 isin 1 119894 minus 1 and 119896 isin 2 119894Let 1199011 119901119898 be the set of all acyclic paths fromthe sovereign node to the corporate node 119862119894 and let119908(1199011) 119908(119901119898) be the correspondingweightsThenthe CountryRank of node 119862119894 is defined as

120574119862119894 = max1le119895le119898

119908(119901119895) (23)

In order to compute the conditional probability of defaultof a corporate given the sovereign default analytically wewould need the joint distribution of probabilities of defaultof the nodes which has an exponential computational com-plexity and it is therefore intractable Thus we approximatethe conditional probability by choosing the path with themaximum weight in the above definition for CountryRank

The example in Figure 3 illustrates calculation of Coun-tryRank for a hypothetical network The network consistsof a sovereign node 119878 and corporate nodes 1198621 1198622 1198623 1198624The edge labels indicate weights in network between twonodes We initially stress the sovereign node which results ina CountryRank of 1 for node 119878 In the next step the stresspropagates to node 1198621 and as a result its CountryRank is 09

Complexity 7

09 08

02

03 06

03

S = 1

C1 C2

C3C4

(a) Sovereign node in stress

09 08

02

03 06

03

S = 1

C2

C3C4

C1= 09

(b) Stress propagates to node 1198621

09 08

02

03 06

03

S = 1

C3C4

C1= 09 C2

= 08

(c) Stress propagates to node 1198622

09 08

02

03 06

03

S = 1

C4

C1= 09 C2

= 08

C3= 048

(d) Stress propagates to node 1198623

09 08

02

03 06

03

S = 1

C1= 09 C2

= 08

C3= 048C4

= 027

(e) Stress propagates to node 1198624

Figure 3 Illustration of the CountryRank parameter using a hypothetical network The subfigures (a)ndash(e) show the propagation of stress inthe network starting from the sovereign node to corporate nodes At each step the stress spreads to a node using the path with the maximumweight from the sovereign node

Then node1198622 gets stressed giving it aCountryRank value 08For node 1198623 there are two paths from node 119878 so we pick thepath through node1198622 having a higher weight of 048 Finallythere are three paths from node 119878 to node 1198624 and the pathwith maximum weight is 027

322 Network Construction Credit default swap spreads aremarket-implied indicators of probability of default of anentity A credit default swap is a financial contract in whicha protection seller A insures a protection buyer B against thedefault of a third party C More precisely regular couponpayments with respect to a contractual notional 119873 and afixed rate 119904 the CDS spread are swapped with a payment of119873(1 minus RR) in the case of the default of C where RR the so-called recovery rate is a contract parameter which representsthe fraction of investment which is assumed be recovered inthe case of default of C

Modified 120598-Draw-Up We would like to measure to whatextent changes in CDS spreads of different issuers occursimultaneously For this we use the notion of a modified120598-draw-up to quantify the impact of deterioration of creditquality of one issuer on the other Modified 120598-draw-up is analteration of the 120598-draw-ups notion which is introduced in[32] In that article the authors use the notion of 120598-draw-ups to construct a network which models the conditionalprobabilities of spike-like comovements among pairs of CDSspreads A modified 120598-draw-up is defined as an upwardmovement in the time series in which the amplitude of themovement that is the difference between the subsequentlocal maxima and current local minima is greater than athreshold 120598 We record such local minima as the modified 120598-draw-ups The 120598 parameter for a local minima at time 119905 is setto be the standard deviation in the time series between days119905 minus 119899 and 119905 where 119899 is chosen to be 10 days Figure 5 shows

8 Complexity

the time series of Russian Federation CDS with the calibratedmodified 120598-draw-ups using a history of 10 days for calibra-tion

Filtering Market Impact Since we would like to measure thecomovement of the time series 119894 and 119895 we exclude the effectof the external market on these nodes as followsWe calibratethe 120598-draw-ups for the CDS time series of an index that doesnot represent the region in question for instance for Russianissuers we choose the iTraxx index which is the compositeCDS index of 125 CDS referencing European investmentgrade credit Then we filter out those 120598-draw-ups of node119894 which are the same as the 120598-draw-ups of the iTraxx indexincluding a time lag 120591That is if iTraxx has amodified 120598-draw-up on day 119905 thenwe remove themodified 120598-draw-ups of node119894 on days 119905 119905 + 1 119905 + 120591 We choose a time lag of 3 days forour calibration based on the input data which is consistentwith the choice in [32]

Edges After identifying the 120598-draw-ups for all the issuers andfiltering out the market impact the edges in our network areconstructed as follows The weight of an edge in the creditstress propagation network from node 119894 to node 119895 is theconditional probability that if node 119894 has an epsilon draw-upon day 119905 then node 119895 also has an epsilon draw-up on days119905 119905 + 1 119905 + 120591 where 120591 is the time lag More preciselylet 119873119894 be the number of 120598-draw-ups of node 119894 after filteringusing iTraxx index and119873119894119895 epsilon draw-ups of node 119894 whichare also epsilon draw-ups for node 119895 with the time lag 120591Then the edge weight119908119894119895 between nodes 119894 and 119895 is defined as119908119894119895 = 119873119894119873119894119895 Figure 6 shows the minimum spanning tree ofthe credit stress propagation network constructed using theCDS spread time series data of Russian issuers

Uncertainty in CountryRank We test the robustness of ourCountryRank calibration by varying the number of daysused for 120598-parameter The figure in Appendix B shows thatthe 120598-parameter for Russian Federation CDS time seriesremains stable when we vary the number of days We initiallyobtain time series of 120598-parameters by calculating standarddeviation in the last 119899 = 10 15 and 20 days on all localminima indices of Russian Federation CDS Subsequently wecalculate the mean of the absolute differences between theepsilon time series calculated and express this in units of themean of Russian Federation CDS time series The percentagedifference is 138 between the 10-day 120598-parameter and 15-day 120598-parameter and 222 between the 10-day and 20-day120598-parameters

Further we quantify the uncertainty in CountryRankparameter as follows For an corporate node we calculate theabsolute difference in CountryRank calculated using 119899 = 15and 20 days with CountryRank using 119899 = 10 days for the 120598-parameter We then calculate this difference as a percentageof the CountryRank calculated using 10 days for 120598-parameterfor all corporates and then compute their mean The meandifference between CountryRank calibrated using 119899 = 15days and 119899 = 10 days is 684 and 119899 = 20 days and 119899 = 10days is 973 for the Russian CDS data set

Table 1 Systematic factor index mapping

Factor IndexEurope MSCI EUROPEAsia MSCI AC ASIANorth America MSCI NORTH AMERICALatin America MSCI EM LATIN AMERICAMiddle East and Africa MSCI FM AFRICAPacific MSCI PACIFICMaterials MSCI WRLDMATERIALSConsumer products MSCI WRLDCONSUMER DISCRServices MSCI WRLDCONSUMER SVCFinancial MSCI WRLDFINANCIALSIndustrial MSCI WRLDINDUSTRIALS

Government ITRAXX SOVX GLOBAL LIQUIDINVESTMENT GRADE

4 Numerical Experiments

We implement the framework presented in Section 3 tosynthetic test portfolios and discuss the corresponding riskmetrics Further we perform a set of sensitivity studies andexplore the results

41 FactorModel Wefirst set up amultifactorMertonmodelas it was described in Section 2 We define a set of systematicfactors thatwill represent region and sector effectsWe choose6 region and 6 sector factors for which we select appropriateindexes as shown in Table 1 We then use 10 years of indextime series to derive the region and sector returns 119865119877(119895)119895 = 1 6 and 119865119878(119896) 119896 = 1 6 respectively and obtainan estimate of the correlation matrix Ω shown in Figure 7Subsequently we map all issuers to one region and one sectorfactor 119865119877(119894) and 119865119878(119894) respectively For instance a Dutch bankwill be associated with Europe and financial factors As aproxy of individual asset returns we use 10 years of equityor CDS time series depending on the data availability foreach issuer Finally we standardize the individual returnstime series (119883119894119905) and perform the following Ordinary LeastSquares regression against the systematic factor returns

119883119894119905 = 120572119877(119894)119865119877(119894)119905 + 120572119878(119896)119865119878(119896)119905 + 120598119894119905 (24)

to obtain 119877(119894) 119878(119894) and 120573119894 = 1198772 where1198772 is the coefficient ofdetermination and it is higher for issuers whose returns arelargely affected by the performance of the systematic factors

42 Synthetic Test Portfolios To investigate the properties ofthe contagion model we set up 2 test portfolios For theseportfolios the resulting risk measures are compared to thoseof the standard latent variable model with no contagionPortfolio A consists of 1 Russian government bond and 17bonds issued by corporations registered and operating in theRussian Federation As it is illustrated in Table 2 the issuersare of medium and low credit quality Portfolio B representsa similar but more diversified setup with 4 sovereign bonds

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

Complexity 5

TimeT = 1 year

TimeT = 1 year

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

3

2

1

0

minus1

minus2

minus3

Stan

dard

ised

asse

t ret

urns

dCdsdC

dnsdC

Default thresholds for issuer Ci Default thresholds for issuer Ci

Figure 2 Under the standard Merton model the default threshold 119889119862119894 for corporate issuer 119862119894 is set to be equal to Φminus1(119901119862119894 ) Under theproposed model the threshold increases in the event of sovereign default making 119862119894rsquos default more likely as the contagion effect suggests

3 A Model for Credit Contagion

In the multifactor Merton model specified in Section 24 thestandardized asset returns 119883119894 119894 = 1 119898 are assumed tobe driven by a set of common underlying systematic factorsand the critical thresholds 119889119894 119894 = 1 119898 are satisfying119889119894 = Φminus1(119901119894) for all 119894 The only source of default dependencein such a framework is the dependence on the systematicfactors In themodel we propose we assume that in the eventof a sovereign default contagion will spread to the corporateissuers in the portfolio that are registered and operating inthat country causing default probability to be equal to theirCountryRank In Section 31 we demonstrate how to calibratethe critical thresholds so that each corporatersquos probabilityof default conditional on the default of the correspondingsovereign equals its CountryRank while its unconditionaldefault probability remains unchanged In Section 32 weshow how to construct a credit stress propagation networkand estimate the CountryRank parameter

31 Incorporating Contagion in Factor Models Consider acorporate issuer 119862119894 and its country of operation 119878 Denoteby 119901119862119894 the probability of default of 119862119894 Under the standardMertonmodel default occurs if119862119894rsquos standardized asset return119883119862119894 falls below its default threshold 119889119862119894 The critical threshold119889119862119894 is assumed to be equal to Φminus1(119901119862119894) and is independentof the state of the country of operation 119878 In the proposedmodel a corporate is subject to shocks from its country ofoperation its corresponding state is described by a binarystate variable The state is considered to be stressed in theevent of sovereign default In this case the issuerrsquos defaultthreshold increases causing it more likely to default asthe contagion effect suggests In case the corresponding

sovereign does not default the corporates liquidity state isconsidered stable We replace the default threshold 119889119862119894 with119889lowast119862119894 where119889lowast119862119894=

119889sd119862119894

if the corresponding sovereign defaults

119889nsd119862119894

otherwise

(14)

or equivalently

119889lowast119862119894 = 1119884119878=1119889sd119862119894

+ 1119884119878=0119889nsd119862119894

(15)

We denote by 119901119878 the probability of default of the countryof operation and by 120574119862119894 the CountryRank parameter whichindicates the increased probability of default of 119862119894 given thedefault of 119878 An example of the new default thresholds isshown in Figure 2 Our objective is to calibrate 119889sd

119862119894and 119889nsd

119862119894in such way that the overall default rate remains unchangedand P(119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 Denote by

1206012 (119909 119910 120588) fl 12120587radic1 minus 1205882 exp(minus1199092 + 1199102 minus 2120588119909119910

2 (1 minus 1205882) )

Φ2 (ℎ 119896 120588) fl intℎminusinfin

int119896minusinfin

1206012 (119909 119910 120588) 119889119910 119889119909(16)

the density and distribution function of the bivariate standardnormal distribution with correlation parameter 120588 isin (minus1 1)

6 Complexity

Note that 119889lowast119862119894(120596) = 119889sd119862119894for 120596 isin 119884119862119894 = 1 119884119878 = 1 sub 119884119878 = 1

and 119889lowast119862119894(120596) = 119889nsd119862119894

for 120596 isin 119884119862119894 = 1 119884119878 = 0 sub 119884119878 = 0 Werewrite P(119884119862119894 = 1 | 119884119878 = 1) in the following way

P (119884119862119894 = 1 | 119884119878 = 1)= 1P (119884119878 = 1)P (119884119862119894 = 1 119884119878 = 1)

= 1119901119878P [119883119862119894 lt 119889sd119862119894 119883119878 lt 119889119878]

= 1119901119878Φ2 (119889sd119862119894 119889119878 120588119878119862119894)

(17)

Using the above representation and given 119889119878 = Φminus1(119901119878) and120588119878119862119894 one can solve the equation

P (119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 (18)

over 119889sd119862119894 We proceed to the derivation of 119889nsd

119862119894in such way

that the overall default probability remains equal to 119901119862119894 Thisconstraint is important since contagion is assumed to haveno impact on the average loss Clearly

119901119862119894 = P (119884119862119894 = 1)= P (119884119862119894 = 1 119884119878 = 1) + P (119884119862119894 = 1 119884119878 = 0)= P (119884119862119894 = 1 | 119884119878 = 1)P (119884119878 = 1)

+ P (119884119862119894 = 1 119884119878 = 0)(19)

and thus

P (119884119862119894 = 1 119884119878 = 0) = 119901119862119894 minus 120574119862119894 sdot 119901119878 (20)

The left-hand side of the above equation can be representedas follows

P (corpdef cap nosovdef) = P [119883119862119894 lt 119889nsd119862119894

119883119878 gt 119889119878]= P [119883119862119894 lt 119889nsd

119862119894] minus P [119883119862119894 lt 119889nsd

119862119894 119883119878 lt 119889119878]

= Φ (119889nsd119862119894

) minus Φ2 (119889nsd119862119894

119889119878 120588119878119862119894) (21)

By use of the above and given 119889119878 = Φminus1(119901119878) and 120588119878119862119894 one cansolve the previous equation over 119889nsd

119862119894

32 Estimation of CountryRank In this section we elaborateon the estimation of the CountryRank parameter [30] whichserves as the probability of default of the corporate condi-tional on the default of the sovereign In addition we providedetails on the construction of the credit stress propagationnetwork

321 CountryRank In order to estimate contagion effectsin a network of issuers an algorithm such as DebtRank

[31] is necessary In the DebtRank calculation process stresspropagates even in the absence of defaults and each nodecan propagate stress only once before becoming inactiveThelevel of distress for a previously undistressed node is givenby the sum of incoming stress from its neighbors with amaximum value of 1 Summing up the incoming stress fromneighboring nodes seems reasonable when trying to estimatethe impact of one node or a set of nodes to a network ofinterconnected balance sheets where links represent lendingrelationships However when trying to quantify the prob-ability of default of a corporate node given the infectiousdefault of a sovereign node one has to consider that there issignificant overlap in terms of common stress and thus bysummingwemay be accounting for the same effectmore thanonce This effect is amplified in dense networks constructedfrom CDS data Therefore we introduce CountryRank as analternative measure which is suited for our contagion model

We assume that we have a hypothetical credit stresspropagation network where the nodes correspond to theissuers including the sovereign and the edges correspondto the impact of credit quality of one issuer on the otherThe details of the network construction will be presented inSection 322 Given such a network the CountryRank of thenodes can be defined recursively as follows

(i) First we stress the sovereign node and as a result itsCountryRank is 1

(ii) Let 120574119878 be the CountryRank of the sovereign and let119890(119895119896) denote the edge weight between nodes 119895 and 119896Given a node 119862119894 let 119901 = 11987811986211198622 sdot sdot sdot 119862119894minus1119862119894 be a pathwithout cycles from the sovereign node 119878 to the node119862119894 The weight of the path 119901 is defined as

119908 (119901) = 120574119878119890(12) sdot sdot sdot 119890(119894minus1119894) (22)

where 119890(119895119896) are the respective edge weights betweennodes 119895 and 119896 for 119895 isin 1 119894 minus 1 and 119896 isin 2 119894Let 1199011 119901119898 be the set of all acyclic paths fromthe sovereign node to the corporate node 119862119894 and let119908(1199011) 119908(119901119898) be the correspondingweightsThenthe CountryRank of node 119862119894 is defined as

120574119862119894 = max1le119895le119898

119908(119901119895) (23)

In order to compute the conditional probability of defaultof a corporate given the sovereign default analytically wewould need the joint distribution of probabilities of defaultof the nodes which has an exponential computational com-plexity and it is therefore intractable Thus we approximatethe conditional probability by choosing the path with themaximum weight in the above definition for CountryRank

The example in Figure 3 illustrates calculation of Coun-tryRank for a hypothetical network The network consistsof a sovereign node 119878 and corporate nodes 1198621 1198622 1198623 1198624The edge labels indicate weights in network between twonodes We initially stress the sovereign node which results ina CountryRank of 1 for node 119878 In the next step the stresspropagates to node 1198621 and as a result its CountryRank is 09

Complexity 7

09 08

02

03 06

03

S = 1

C1 C2

C3C4

(a) Sovereign node in stress

09 08

02

03 06

03

S = 1

C2

C3C4

C1= 09

(b) Stress propagates to node 1198621

09 08

02

03 06

03

S = 1

C3C4

C1= 09 C2

= 08

(c) Stress propagates to node 1198622

09 08

02

03 06

03

S = 1

C4

C1= 09 C2

= 08

C3= 048

(d) Stress propagates to node 1198623

09 08

02

03 06

03

S = 1

C1= 09 C2

= 08

C3= 048C4

= 027

(e) Stress propagates to node 1198624

Figure 3 Illustration of the CountryRank parameter using a hypothetical network The subfigures (a)ndash(e) show the propagation of stress inthe network starting from the sovereign node to corporate nodes At each step the stress spreads to a node using the path with the maximumweight from the sovereign node

Then node1198622 gets stressed giving it aCountryRank value 08For node 1198623 there are two paths from node 119878 so we pick thepath through node1198622 having a higher weight of 048 Finallythere are three paths from node 119878 to node 1198624 and the pathwith maximum weight is 027

322 Network Construction Credit default swap spreads aremarket-implied indicators of probability of default of anentity A credit default swap is a financial contract in whicha protection seller A insures a protection buyer B against thedefault of a third party C More precisely regular couponpayments with respect to a contractual notional 119873 and afixed rate 119904 the CDS spread are swapped with a payment of119873(1 minus RR) in the case of the default of C where RR the so-called recovery rate is a contract parameter which representsthe fraction of investment which is assumed be recovered inthe case of default of C

Modified 120598-Draw-Up We would like to measure to whatextent changes in CDS spreads of different issuers occursimultaneously For this we use the notion of a modified120598-draw-up to quantify the impact of deterioration of creditquality of one issuer on the other Modified 120598-draw-up is analteration of the 120598-draw-ups notion which is introduced in[32] In that article the authors use the notion of 120598-draw-ups to construct a network which models the conditionalprobabilities of spike-like comovements among pairs of CDSspreads A modified 120598-draw-up is defined as an upwardmovement in the time series in which the amplitude of themovement that is the difference between the subsequentlocal maxima and current local minima is greater than athreshold 120598 We record such local minima as the modified 120598-draw-ups The 120598 parameter for a local minima at time 119905 is setto be the standard deviation in the time series between days119905 minus 119899 and 119905 where 119899 is chosen to be 10 days Figure 5 shows

8 Complexity

the time series of Russian Federation CDS with the calibratedmodified 120598-draw-ups using a history of 10 days for calibra-tion

Filtering Market Impact Since we would like to measure thecomovement of the time series 119894 and 119895 we exclude the effectof the external market on these nodes as followsWe calibratethe 120598-draw-ups for the CDS time series of an index that doesnot represent the region in question for instance for Russianissuers we choose the iTraxx index which is the compositeCDS index of 125 CDS referencing European investmentgrade credit Then we filter out those 120598-draw-ups of node119894 which are the same as the 120598-draw-ups of the iTraxx indexincluding a time lag 120591That is if iTraxx has amodified 120598-draw-up on day 119905 thenwe remove themodified 120598-draw-ups of node119894 on days 119905 119905 + 1 119905 + 120591 We choose a time lag of 3 days forour calibration based on the input data which is consistentwith the choice in [32]

Edges After identifying the 120598-draw-ups for all the issuers andfiltering out the market impact the edges in our network areconstructed as follows The weight of an edge in the creditstress propagation network from node 119894 to node 119895 is theconditional probability that if node 119894 has an epsilon draw-upon day 119905 then node 119895 also has an epsilon draw-up on days119905 119905 + 1 119905 + 120591 where 120591 is the time lag More preciselylet 119873119894 be the number of 120598-draw-ups of node 119894 after filteringusing iTraxx index and119873119894119895 epsilon draw-ups of node 119894 whichare also epsilon draw-ups for node 119895 with the time lag 120591Then the edge weight119908119894119895 between nodes 119894 and 119895 is defined as119908119894119895 = 119873119894119873119894119895 Figure 6 shows the minimum spanning tree ofthe credit stress propagation network constructed using theCDS spread time series data of Russian issuers

Uncertainty in CountryRank We test the robustness of ourCountryRank calibration by varying the number of daysused for 120598-parameter The figure in Appendix B shows thatthe 120598-parameter for Russian Federation CDS time seriesremains stable when we vary the number of days We initiallyobtain time series of 120598-parameters by calculating standarddeviation in the last 119899 = 10 15 and 20 days on all localminima indices of Russian Federation CDS Subsequently wecalculate the mean of the absolute differences between theepsilon time series calculated and express this in units of themean of Russian Federation CDS time series The percentagedifference is 138 between the 10-day 120598-parameter and 15-day 120598-parameter and 222 between the 10-day and 20-day120598-parameters

Further we quantify the uncertainty in CountryRankparameter as follows For an corporate node we calculate theabsolute difference in CountryRank calculated using 119899 = 15and 20 days with CountryRank using 119899 = 10 days for the 120598-parameter We then calculate this difference as a percentageof the CountryRank calculated using 10 days for 120598-parameterfor all corporates and then compute their mean The meandifference between CountryRank calibrated using 119899 = 15days and 119899 = 10 days is 684 and 119899 = 20 days and 119899 = 10days is 973 for the Russian CDS data set

Table 1 Systematic factor index mapping

Factor IndexEurope MSCI EUROPEAsia MSCI AC ASIANorth America MSCI NORTH AMERICALatin America MSCI EM LATIN AMERICAMiddle East and Africa MSCI FM AFRICAPacific MSCI PACIFICMaterials MSCI WRLDMATERIALSConsumer products MSCI WRLDCONSUMER DISCRServices MSCI WRLDCONSUMER SVCFinancial MSCI WRLDFINANCIALSIndustrial MSCI WRLDINDUSTRIALS

Government ITRAXX SOVX GLOBAL LIQUIDINVESTMENT GRADE

4 Numerical Experiments

We implement the framework presented in Section 3 tosynthetic test portfolios and discuss the corresponding riskmetrics Further we perform a set of sensitivity studies andexplore the results

41 FactorModel Wefirst set up amultifactorMertonmodelas it was described in Section 2 We define a set of systematicfactors thatwill represent region and sector effectsWe choose6 region and 6 sector factors for which we select appropriateindexes as shown in Table 1 We then use 10 years of indextime series to derive the region and sector returns 119865119877(119895)119895 = 1 6 and 119865119878(119896) 119896 = 1 6 respectively and obtainan estimate of the correlation matrix Ω shown in Figure 7Subsequently we map all issuers to one region and one sectorfactor 119865119877(119894) and 119865119878(119894) respectively For instance a Dutch bankwill be associated with Europe and financial factors As aproxy of individual asset returns we use 10 years of equityor CDS time series depending on the data availability foreach issuer Finally we standardize the individual returnstime series (119883119894119905) and perform the following Ordinary LeastSquares regression against the systematic factor returns

119883119894119905 = 120572119877(119894)119865119877(119894)119905 + 120572119878(119896)119865119878(119896)119905 + 120598119894119905 (24)

to obtain 119877(119894) 119878(119894) and 120573119894 = 1198772 where1198772 is the coefficient ofdetermination and it is higher for issuers whose returns arelargely affected by the performance of the systematic factors

42 Synthetic Test Portfolios To investigate the properties ofthe contagion model we set up 2 test portfolios For theseportfolios the resulting risk measures are compared to thoseof the standard latent variable model with no contagionPortfolio A consists of 1 Russian government bond and 17bonds issued by corporations registered and operating in theRussian Federation As it is illustrated in Table 2 the issuersare of medium and low credit quality Portfolio B representsa similar but more diversified setup with 4 sovereign bonds

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

6 Complexity

Note that 119889lowast119862119894(120596) = 119889sd119862119894for 120596 isin 119884119862119894 = 1 119884119878 = 1 sub 119884119878 = 1

and 119889lowast119862119894(120596) = 119889nsd119862119894

for 120596 isin 119884119862119894 = 1 119884119878 = 0 sub 119884119878 = 0 Werewrite P(119884119862119894 = 1 | 119884119878 = 1) in the following way

P (119884119862119894 = 1 | 119884119878 = 1)= 1P (119884119878 = 1)P (119884119862119894 = 1 119884119878 = 1)

= 1119901119878P [119883119862119894 lt 119889sd119862119894 119883119878 lt 119889119878]

= 1119901119878Φ2 (119889sd119862119894 119889119878 120588119878119862119894)

(17)

Using the above representation and given 119889119878 = Φminus1(119901119878) and120588119878119862119894 one can solve the equation

P (119884119862119894 = 1 | 119884119878 = 1) = 120574119862119894 (18)

over 119889sd119862119894 We proceed to the derivation of 119889nsd

119862119894in such way

that the overall default probability remains equal to 119901119862119894 Thisconstraint is important since contagion is assumed to haveno impact on the average loss Clearly

119901119862119894 = P (119884119862119894 = 1)= P (119884119862119894 = 1 119884119878 = 1) + P (119884119862119894 = 1 119884119878 = 0)= P (119884119862119894 = 1 | 119884119878 = 1)P (119884119878 = 1)

+ P (119884119862119894 = 1 119884119878 = 0)(19)

and thus

P (119884119862119894 = 1 119884119878 = 0) = 119901119862119894 minus 120574119862119894 sdot 119901119878 (20)

The left-hand side of the above equation can be representedas follows

P (corpdef cap nosovdef) = P [119883119862119894 lt 119889nsd119862119894

119883119878 gt 119889119878]= P [119883119862119894 lt 119889nsd

119862119894] minus P [119883119862119894 lt 119889nsd

119862119894 119883119878 lt 119889119878]

= Φ (119889nsd119862119894

) minus Φ2 (119889nsd119862119894

119889119878 120588119878119862119894) (21)

By use of the above and given 119889119878 = Φminus1(119901119878) and 120588119878119862119894 one cansolve the previous equation over 119889nsd

119862119894

32 Estimation of CountryRank In this section we elaborateon the estimation of the CountryRank parameter [30] whichserves as the probability of default of the corporate condi-tional on the default of the sovereign In addition we providedetails on the construction of the credit stress propagationnetwork

321 CountryRank In order to estimate contagion effectsin a network of issuers an algorithm such as DebtRank

[31] is necessary In the DebtRank calculation process stresspropagates even in the absence of defaults and each nodecan propagate stress only once before becoming inactiveThelevel of distress for a previously undistressed node is givenby the sum of incoming stress from its neighbors with amaximum value of 1 Summing up the incoming stress fromneighboring nodes seems reasonable when trying to estimatethe impact of one node or a set of nodes to a network ofinterconnected balance sheets where links represent lendingrelationships However when trying to quantify the prob-ability of default of a corporate node given the infectiousdefault of a sovereign node one has to consider that there issignificant overlap in terms of common stress and thus bysummingwemay be accounting for the same effectmore thanonce This effect is amplified in dense networks constructedfrom CDS data Therefore we introduce CountryRank as analternative measure which is suited for our contagion model

We assume that we have a hypothetical credit stresspropagation network where the nodes correspond to theissuers including the sovereign and the edges correspondto the impact of credit quality of one issuer on the otherThe details of the network construction will be presented inSection 322 Given such a network the CountryRank of thenodes can be defined recursively as follows

(i) First we stress the sovereign node and as a result itsCountryRank is 1

(ii) Let 120574119878 be the CountryRank of the sovereign and let119890(119895119896) denote the edge weight between nodes 119895 and 119896Given a node 119862119894 let 119901 = 11987811986211198622 sdot sdot sdot 119862119894minus1119862119894 be a pathwithout cycles from the sovereign node 119878 to the node119862119894 The weight of the path 119901 is defined as

119908 (119901) = 120574119878119890(12) sdot sdot sdot 119890(119894minus1119894) (22)

where 119890(119895119896) are the respective edge weights betweennodes 119895 and 119896 for 119895 isin 1 119894 minus 1 and 119896 isin 2 119894Let 1199011 119901119898 be the set of all acyclic paths fromthe sovereign node to the corporate node 119862119894 and let119908(1199011) 119908(119901119898) be the correspondingweightsThenthe CountryRank of node 119862119894 is defined as

120574119862119894 = max1le119895le119898

119908(119901119895) (23)

In order to compute the conditional probability of defaultof a corporate given the sovereign default analytically wewould need the joint distribution of probabilities of defaultof the nodes which has an exponential computational com-plexity and it is therefore intractable Thus we approximatethe conditional probability by choosing the path with themaximum weight in the above definition for CountryRank

The example in Figure 3 illustrates calculation of Coun-tryRank for a hypothetical network The network consistsof a sovereign node 119878 and corporate nodes 1198621 1198622 1198623 1198624The edge labels indicate weights in network between twonodes We initially stress the sovereign node which results ina CountryRank of 1 for node 119878 In the next step the stresspropagates to node 1198621 and as a result its CountryRank is 09

Complexity 7

09 08

02

03 06

03

S = 1

C1 C2

C3C4

(a) Sovereign node in stress

09 08

02

03 06

03

S = 1

C2

C3C4

C1= 09

(b) Stress propagates to node 1198621

09 08

02

03 06

03

S = 1

C3C4

C1= 09 C2

= 08

(c) Stress propagates to node 1198622

09 08

02

03 06

03

S = 1

C4

C1= 09 C2

= 08

C3= 048

(d) Stress propagates to node 1198623

09 08

02

03 06

03

S = 1

C1= 09 C2

= 08

C3= 048C4

= 027

(e) Stress propagates to node 1198624

Figure 3 Illustration of the CountryRank parameter using a hypothetical network The subfigures (a)ndash(e) show the propagation of stress inthe network starting from the sovereign node to corporate nodes At each step the stress spreads to a node using the path with the maximumweight from the sovereign node

Then node1198622 gets stressed giving it aCountryRank value 08For node 1198623 there are two paths from node 119878 so we pick thepath through node1198622 having a higher weight of 048 Finallythere are three paths from node 119878 to node 1198624 and the pathwith maximum weight is 027

322 Network Construction Credit default swap spreads aremarket-implied indicators of probability of default of anentity A credit default swap is a financial contract in whicha protection seller A insures a protection buyer B against thedefault of a third party C More precisely regular couponpayments with respect to a contractual notional 119873 and afixed rate 119904 the CDS spread are swapped with a payment of119873(1 minus RR) in the case of the default of C where RR the so-called recovery rate is a contract parameter which representsthe fraction of investment which is assumed be recovered inthe case of default of C

Modified 120598-Draw-Up We would like to measure to whatextent changes in CDS spreads of different issuers occursimultaneously For this we use the notion of a modified120598-draw-up to quantify the impact of deterioration of creditquality of one issuer on the other Modified 120598-draw-up is analteration of the 120598-draw-ups notion which is introduced in[32] In that article the authors use the notion of 120598-draw-ups to construct a network which models the conditionalprobabilities of spike-like comovements among pairs of CDSspreads A modified 120598-draw-up is defined as an upwardmovement in the time series in which the amplitude of themovement that is the difference between the subsequentlocal maxima and current local minima is greater than athreshold 120598 We record such local minima as the modified 120598-draw-ups The 120598 parameter for a local minima at time 119905 is setto be the standard deviation in the time series between days119905 minus 119899 and 119905 where 119899 is chosen to be 10 days Figure 5 shows

8 Complexity

the time series of Russian Federation CDS with the calibratedmodified 120598-draw-ups using a history of 10 days for calibra-tion

Filtering Market Impact Since we would like to measure thecomovement of the time series 119894 and 119895 we exclude the effectof the external market on these nodes as followsWe calibratethe 120598-draw-ups for the CDS time series of an index that doesnot represent the region in question for instance for Russianissuers we choose the iTraxx index which is the compositeCDS index of 125 CDS referencing European investmentgrade credit Then we filter out those 120598-draw-ups of node119894 which are the same as the 120598-draw-ups of the iTraxx indexincluding a time lag 120591That is if iTraxx has amodified 120598-draw-up on day 119905 thenwe remove themodified 120598-draw-ups of node119894 on days 119905 119905 + 1 119905 + 120591 We choose a time lag of 3 days forour calibration based on the input data which is consistentwith the choice in [32]

Edges After identifying the 120598-draw-ups for all the issuers andfiltering out the market impact the edges in our network areconstructed as follows The weight of an edge in the creditstress propagation network from node 119894 to node 119895 is theconditional probability that if node 119894 has an epsilon draw-upon day 119905 then node 119895 also has an epsilon draw-up on days119905 119905 + 1 119905 + 120591 where 120591 is the time lag More preciselylet 119873119894 be the number of 120598-draw-ups of node 119894 after filteringusing iTraxx index and119873119894119895 epsilon draw-ups of node 119894 whichare also epsilon draw-ups for node 119895 with the time lag 120591Then the edge weight119908119894119895 between nodes 119894 and 119895 is defined as119908119894119895 = 119873119894119873119894119895 Figure 6 shows the minimum spanning tree ofthe credit stress propagation network constructed using theCDS spread time series data of Russian issuers

Uncertainty in CountryRank We test the robustness of ourCountryRank calibration by varying the number of daysused for 120598-parameter The figure in Appendix B shows thatthe 120598-parameter for Russian Federation CDS time seriesremains stable when we vary the number of days We initiallyobtain time series of 120598-parameters by calculating standarddeviation in the last 119899 = 10 15 and 20 days on all localminima indices of Russian Federation CDS Subsequently wecalculate the mean of the absolute differences between theepsilon time series calculated and express this in units of themean of Russian Federation CDS time series The percentagedifference is 138 between the 10-day 120598-parameter and 15-day 120598-parameter and 222 between the 10-day and 20-day120598-parameters

Further we quantify the uncertainty in CountryRankparameter as follows For an corporate node we calculate theabsolute difference in CountryRank calculated using 119899 = 15and 20 days with CountryRank using 119899 = 10 days for the 120598-parameter We then calculate this difference as a percentageof the CountryRank calculated using 10 days for 120598-parameterfor all corporates and then compute their mean The meandifference between CountryRank calibrated using 119899 = 15days and 119899 = 10 days is 684 and 119899 = 20 days and 119899 = 10days is 973 for the Russian CDS data set

Table 1 Systematic factor index mapping

Factor IndexEurope MSCI EUROPEAsia MSCI AC ASIANorth America MSCI NORTH AMERICALatin America MSCI EM LATIN AMERICAMiddle East and Africa MSCI FM AFRICAPacific MSCI PACIFICMaterials MSCI WRLDMATERIALSConsumer products MSCI WRLDCONSUMER DISCRServices MSCI WRLDCONSUMER SVCFinancial MSCI WRLDFINANCIALSIndustrial MSCI WRLDINDUSTRIALS

Government ITRAXX SOVX GLOBAL LIQUIDINVESTMENT GRADE

4 Numerical Experiments

We implement the framework presented in Section 3 tosynthetic test portfolios and discuss the corresponding riskmetrics Further we perform a set of sensitivity studies andexplore the results

41 FactorModel Wefirst set up amultifactorMertonmodelas it was described in Section 2 We define a set of systematicfactors thatwill represent region and sector effectsWe choose6 region and 6 sector factors for which we select appropriateindexes as shown in Table 1 We then use 10 years of indextime series to derive the region and sector returns 119865119877(119895)119895 = 1 6 and 119865119878(119896) 119896 = 1 6 respectively and obtainan estimate of the correlation matrix Ω shown in Figure 7Subsequently we map all issuers to one region and one sectorfactor 119865119877(119894) and 119865119878(119894) respectively For instance a Dutch bankwill be associated with Europe and financial factors As aproxy of individual asset returns we use 10 years of equityor CDS time series depending on the data availability foreach issuer Finally we standardize the individual returnstime series (119883119894119905) and perform the following Ordinary LeastSquares regression against the systematic factor returns

119883119894119905 = 120572119877(119894)119865119877(119894)119905 + 120572119878(119896)119865119878(119896)119905 + 120598119894119905 (24)

to obtain 119877(119894) 119878(119894) and 120573119894 = 1198772 where1198772 is the coefficient ofdetermination and it is higher for issuers whose returns arelargely affected by the performance of the systematic factors

42 Synthetic Test Portfolios To investigate the properties ofthe contagion model we set up 2 test portfolios For theseportfolios the resulting risk measures are compared to thoseof the standard latent variable model with no contagionPortfolio A consists of 1 Russian government bond and 17bonds issued by corporations registered and operating in theRussian Federation As it is illustrated in Table 2 the issuersare of medium and low credit quality Portfolio B representsa similar but more diversified setup with 4 sovereign bonds

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

Complexity 7

09 08

02

03 06

03

S = 1

C1 C2

C3C4

(a) Sovereign node in stress

09 08

02

03 06

03

S = 1

C2

C3C4

C1= 09

(b) Stress propagates to node 1198621

09 08

02

03 06

03

S = 1

C3C4

C1= 09 C2

= 08

(c) Stress propagates to node 1198622

09 08

02

03 06

03

S = 1

C4

C1= 09 C2

= 08

C3= 048

(d) Stress propagates to node 1198623

09 08

02

03 06

03

S = 1

C1= 09 C2

= 08

C3= 048C4

= 027

(e) Stress propagates to node 1198624

Figure 3 Illustration of the CountryRank parameter using a hypothetical network The subfigures (a)ndash(e) show the propagation of stress inthe network starting from the sovereign node to corporate nodes At each step the stress spreads to a node using the path with the maximumweight from the sovereign node

Then node1198622 gets stressed giving it aCountryRank value 08For node 1198623 there are two paths from node 119878 so we pick thepath through node1198622 having a higher weight of 048 Finallythere are three paths from node 119878 to node 1198624 and the pathwith maximum weight is 027

322 Network Construction Credit default swap spreads aremarket-implied indicators of probability of default of anentity A credit default swap is a financial contract in whicha protection seller A insures a protection buyer B against thedefault of a third party C More precisely regular couponpayments with respect to a contractual notional 119873 and afixed rate 119904 the CDS spread are swapped with a payment of119873(1 minus RR) in the case of the default of C where RR the so-called recovery rate is a contract parameter which representsthe fraction of investment which is assumed be recovered inthe case of default of C

Modified 120598-Draw-Up We would like to measure to whatextent changes in CDS spreads of different issuers occursimultaneously For this we use the notion of a modified120598-draw-up to quantify the impact of deterioration of creditquality of one issuer on the other Modified 120598-draw-up is analteration of the 120598-draw-ups notion which is introduced in[32] In that article the authors use the notion of 120598-draw-ups to construct a network which models the conditionalprobabilities of spike-like comovements among pairs of CDSspreads A modified 120598-draw-up is defined as an upwardmovement in the time series in which the amplitude of themovement that is the difference between the subsequentlocal maxima and current local minima is greater than athreshold 120598 We record such local minima as the modified 120598-draw-ups The 120598 parameter for a local minima at time 119905 is setto be the standard deviation in the time series between days119905 minus 119899 and 119905 where 119899 is chosen to be 10 days Figure 5 shows

8 Complexity

the time series of Russian Federation CDS with the calibratedmodified 120598-draw-ups using a history of 10 days for calibra-tion

Filtering Market Impact Since we would like to measure thecomovement of the time series 119894 and 119895 we exclude the effectof the external market on these nodes as followsWe calibratethe 120598-draw-ups for the CDS time series of an index that doesnot represent the region in question for instance for Russianissuers we choose the iTraxx index which is the compositeCDS index of 125 CDS referencing European investmentgrade credit Then we filter out those 120598-draw-ups of node119894 which are the same as the 120598-draw-ups of the iTraxx indexincluding a time lag 120591That is if iTraxx has amodified 120598-draw-up on day 119905 thenwe remove themodified 120598-draw-ups of node119894 on days 119905 119905 + 1 119905 + 120591 We choose a time lag of 3 days forour calibration based on the input data which is consistentwith the choice in [32]

Edges After identifying the 120598-draw-ups for all the issuers andfiltering out the market impact the edges in our network areconstructed as follows The weight of an edge in the creditstress propagation network from node 119894 to node 119895 is theconditional probability that if node 119894 has an epsilon draw-upon day 119905 then node 119895 also has an epsilon draw-up on days119905 119905 + 1 119905 + 120591 where 120591 is the time lag More preciselylet 119873119894 be the number of 120598-draw-ups of node 119894 after filteringusing iTraxx index and119873119894119895 epsilon draw-ups of node 119894 whichare also epsilon draw-ups for node 119895 with the time lag 120591Then the edge weight119908119894119895 between nodes 119894 and 119895 is defined as119908119894119895 = 119873119894119873119894119895 Figure 6 shows the minimum spanning tree ofthe credit stress propagation network constructed using theCDS spread time series data of Russian issuers

Uncertainty in CountryRank We test the robustness of ourCountryRank calibration by varying the number of daysused for 120598-parameter The figure in Appendix B shows thatthe 120598-parameter for Russian Federation CDS time seriesremains stable when we vary the number of days We initiallyobtain time series of 120598-parameters by calculating standarddeviation in the last 119899 = 10 15 and 20 days on all localminima indices of Russian Federation CDS Subsequently wecalculate the mean of the absolute differences between theepsilon time series calculated and express this in units of themean of Russian Federation CDS time series The percentagedifference is 138 between the 10-day 120598-parameter and 15-day 120598-parameter and 222 between the 10-day and 20-day120598-parameters

Further we quantify the uncertainty in CountryRankparameter as follows For an corporate node we calculate theabsolute difference in CountryRank calculated using 119899 = 15and 20 days with CountryRank using 119899 = 10 days for the 120598-parameter We then calculate this difference as a percentageof the CountryRank calculated using 10 days for 120598-parameterfor all corporates and then compute their mean The meandifference between CountryRank calibrated using 119899 = 15days and 119899 = 10 days is 684 and 119899 = 20 days and 119899 = 10days is 973 for the Russian CDS data set

Table 1 Systematic factor index mapping

Factor IndexEurope MSCI EUROPEAsia MSCI AC ASIANorth America MSCI NORTH AMERICALatin America MSCI EM LATIN AMERICAMiddle East and Africa MSCI FM AFRICAPacific MSCI PACIFICMaterials MSCI WRLDMATERIALSConsumer products MSCI WRLDCONSUMER DISCRServices MSCI WRLDCONSUMER SVCFinancial MSCI WRLDFINANCIALSIndustrial MSCI WRLDINDUSTRIALS

Government ITRAXX SOVX GLOBAL LIQUIDINVESTMENT GRADE

4 Numerical Experiments

We implement the framework presented in Section 3 tosynthetic test portfolios and discuss the corresponding riskmetrics Further we perform a set of sensitivity studies andexplore the results

41 FactorModel Wefirst set up amultifactorMertonmodelas it was described in Section 2 We define a set of systematicfactors thatwill represent region and sector effectsWe choose6 region and 6 sector factors for which we select appropriateindexes as shown in Table 1 We then use 10 years of indextime series to derive the region and sector returns 119865119877(119895)119895 = 1 6 and 119865119878(119896) 119896 = 1 6 respectively and obtainan estimate of the correlation matrix Ω shown in Figure 7Subsequently we map all issuers to one region and one sectorfactor 119865119877(119894) and 119865119878(119894) respectively For instance a Dutch bankwill be associated with Europe and financial factors As aproxy of individual asset returns we use 10 years of equityor CDS time series depending on the data availability foreach issuer Finally we standardize the individual returnstime series (119883119894119905) and perform the following Ordinary LeastSquares regression against the systematic factor returns

119883119894119905 = 120572119877(119894)119865119877(119894)119905 + 120572119878(119896)119865119878(119896)119905 + 120598119894119905 (24)

to obtain 119877(119894) 119878(119894) and 120573119894 = 1198772 where1198772 is the coefficient ofdetermination and it is higher for issuers whose returns arelargely affected by the performance of the systematic factors

42 Synthetic Test Portfolios To investigate the properties ofthe contagion model we set up 2 test portfolios For theseportfolios the resulting risk measures are compared to thoseof the standard latent variable model with no contagionPortfolio A consists of 1 Russian government bond and 17bonds issued by corporations registered and operating in theRussian Federation As it is illustrated in Table 2 the issuersare of medium and low credit quality Portfolio B representsa similar but more diversified setup with 4 sovereign bonds

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

8 Complexity

the time series of Russian Federation CDS with the calibratedmodified 120598-draw-ups using a history of 10 days for calibra-tion

Filtering Market Impact Since we would like to measure thecomovement of the time series 119894 and 119895 we exclude the effectof the external market on these nodes as followsWe calibratethe 120598-draw-ups for the CDS time series of an index that doesnot represent the region in question for instance for Russianissuers we choose the iTraxx index which is the compositeCDS index of 125 CDS referencing European investmentgrade credit Then we filter out those 120598-draw-ups of node119894 which are the same as the 120598-draw-ups of the iTraxx indexincluding a time lag 120591That is if iTraxx has amodified 120598-draw-up on day 119905 thenwe remove themodified 120598-draw-ups of node119894 on days 119905 119905 + 1 119905 + 120591 We choose a time lag of 3 days forour calibration based on the input data which is consistentwith the choice in [32]

Edges After identifying the 120598-draw-ups for all the issuers andfiltering out the market impact the edges in our network areconstructed as follows The weight of an edge in the creditstress propagation network from node 119894 to node 119895 is theconditional probability that if node 119894 has an epsilon draw-upon day 119905 then node 119895 also has an epsilon draw-up on days119905 119905 + 1 119905 + 120591 where 120591 is the time lag More preciselylet 119873119894 be the number of 120598-draw-ups of node 119894 after filteringusing iTraxx index and119873119894119895 epsilon draw-ups of node 119894 whichare also epsilon draw-ups for node 119895 with the time lag 120591Then the edge weight119908119894119895 between nodes 119894 and 119895 is defined as119908119894119895 = 119873119894119873119894119895 Figure 6 shows the minimum spanning tree ofthe credit stress propagation network constructed using theCDS spread time series data of Russian issuers

Uncertainty in CountryRank We test the robustness of ourCountryRank calibration by varying the number of daysused for 120598-parameter The figure in Appendix B shows thatthe 120598-parameter for Russian Federation CDS time seriesremains stable when we vary the number of days We initiallyobtain time series of 120598-parameters by calculating standarddeviation in the last 119899 = 10 15 and 20 days on all localminima indices of Russian Federation CDS Subsequently wecalculate the mean of the absolute differences between theepsilon time series calculated and express this in units of themean of Russian Federation CDS time series The percentagedifference is 138 between the 10-day 120598-parameter and 15-day 120598-parameter and 222 between the 10-day and 20-day120598-parameters

Further we quantify the uncertainty in CountryRankparameter as follows For an corporate node we calculate theabsolute difference in CountryRank calculated using 119899 = 15and 20 days with CountryRank using 119899 = 10 days for the 120598-parameter We then calculate this difference as a percentageof the CountryRank calculated using 10 days for 120598-parameterfor all corporates and then compute their mean The meandifference between CountryRank calibrated using 119899 = 15days and 119899 = 10 days is 684 and 119899 = 20 days and 119899 = 10days is 973 for the Russian CDS data set

Table 1 Systematic factor index mapping

Factor IndexEurope MSCI EUROPEAsia MSCI AC ASIANorth America MSCI NORTH AMERICALatin America MSCI EM LATIN AMERICAMiddle East and Africa MSCI FM AFRICAPacific MSCI PACIFICMaterials MSCI WRLDMATERIALSConsumer products MSCI WRLDCONSUMER DISCRServices MSCI WRLDCONSUMER SVCFinancial MSCI WRLDFINANCIALSIndustrial MSCI WRLDINDUSTRIALS

Government ITRAXX SOVX GLOBAL LIQUIDINVESTMENT GRADE

4 Numerical Experiments

We implement the framework presented in Section 3 tosynthetic test portfolios and discuss the corresponding riskmetrics Further we perform a set of sensitivity studies andexplore the results

41 FactorModel Wefirst set up amultifactorMertonmodelas it was described in Section 2 We define a set of systematicfactors thatwill represent region and sector effectsWe choose6 region and 6 sector factors for which we select appropriateindexes as shown in Table 1 We then use 10 years of indextime series to derive the region and sector returns 119865119877(119895)119895 = 1 6 and 119865119878(119896) 119896 = 1 6 respectively and obtainan estimate of the correlation matrix Ω shown in Figure 7Subsequently we map all issuers to one region and one sectorfactor 119865119877(119894) and 119865119878(119894) respectively For instance a Dutch bankwill be associated with Europe and financial factors As aproxy of individual asset returns we use 10 years of equityor CDS time series depending on the data availability foreach issuer Finally we standardize the individual returnstime series (119883119894119905) and perform the following Ordinary LeastSquares regression against the systematic factor returns

119883119894119905 = 120572119877(119894)119865119877(119894)119905 + 120572119878(119896)119865119878(119896)119905 + 120598119894119905 (24)

to obtain 119877(119894) 119878(119894) and 120573119894 = 1198772 where1198772 is the coefficient ofdetermination and it is higher for issuers whose returns arelargely affected by the performance of the systematic factors

42 Synthetic Test Portfolios To investigate the properties ofthe contagion model we set up 2 test portfolios For theseportfolios the resulting risk measures are compared to thoseof the standard latent variable model with no contagionPortfolio A consists of 1 Russian government bond and 17bonds issued by corporations registered and operating in theRussian Federation As it is illustrated in Table 2 the issuersare of medium and low credit quality Portfolio B representsa similar but more diversified setup with 4 sovereign bonds

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

Complexity 9

Table 2 Rating classification for the test portfolios

Rating Portfolio A Portfolio BIssuers Issuers

AAA - 000 3 375AA - 000 3 375A - 000 22 2750BBB 1 556 39 4875BB 15 8333 9 1125B 2 1111 3 375CCCC - 000 1 125

Table 3 Sector classification for the test portfolios

Sector Portfolio A Portfolio BIssuers Issuers

Materials 5 2778 12 1500Consumer products - 000 12 1500Services 3 1667 19 2375Financial 7 3889 25 3125Industrial - 000 6 750Government 3 1667 6 750

issued by Germany Italy Netherlands and Spain and 76 cor-porate bonds by issuers from the aforementioned countriesThe sectors represented in Portfolios A and B are shown inTable 3 Both portfolios are assumed to be equally weightedwith a total notional of euro10 million

43 Credit Stress Propagation Network We use credit defaultswap data to construct the stress propagation network TheCDS raw data set consists of daily CDS liquid spreads fordifferent maturities from 1 May 2014 to 31 March 2015for Portfolio A and 1 July 2014 to 31 December 2015 forPortfolio B These are averaged quotes from contributorsrather than exercisable quotes In addition the data set alsoprovides information on the names of the underlying refer-ence entities recovery rates number of quote contributorsregion sector average of the ratings from Standard amp PoorrsquosMoodyrsquos and Fitch Group of each entity and currency of thequote We use the normalized CDS spreads of entities for the5-year tenor for our analysis The CDS spreads time series ofRussian issuers are illustrated in Figure 4

44 Simulation Study In order to generate portfolio lossdistributions and derive the associated risk measures weperform Monte Carlo simulations This process entails gen-erating joint realizations of the systematic and idiosyncraticrisk factors and comparing the resulting critical variableswiththe corresponding default thresholds By this comparisonwe obtain the default indicator 119884119894 for each issuer and thisenables us to calculate the overall portfolio loss for thistrial The only difference between the standard and thecontagion model is that in the contagion model we firstobtain the default indicators for the sovereigns and theirvalues determine which default thresholds are going to be

1000

800

600

400

CDS

spre

ad (b

ps)

200

CDS spreads of Russian entities20

14-0

6

2014

-08

2014

-10

2014

-12

2015

-02

2015

-04

Oil Transporting Jt Stk Co TransneftVnesheconombankBk of MoscowCity MoscowJSC GAZPROMJSC Gazprom NeftLukoil CoMobile TelesystemsMDM Bk open Jt Stk CoOpen Jt Stk Co ALROSAOJSC Oil Co RosneftJt Stk Co Russian Standard BkRussian Agric BkJSC Russian RailwaysRussian FednSBERBANKOPEN Jt Stk Co VIMPEL CommsJSC VTB Bk

Figure 4 Time series of CDS spreads of Russian issuers

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

10 Complexity

Table 4 Portfolio losses for the test portfolios and additional risk due to contagion

(a) Panel 1 Portfolio A

Quantile Loss standard model Loss contagion model Contagion impact

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Average loss 71807 71691

(b) Panel 2 Portfolio B

Quantile Loss standard model Loss contagion model Contagion impact

99 373013 379929 6915 29950 471497 520467 48971 109990 775773 1009426 233653 309999 1350279 1847795 497516 37

Average loss 44850 44872

May

201

4

Jun

2014

Jul 2

014

Aug

2014

Sep

2014

Oct

201

4

Nov

201

4

Jan

2015

Feb

2015

CDS spreadLocal minima

Local maximaModified epsilon draw-up

Modified epsilon draw-up for Russian Fedn

Dec

201

4

Mar

201

5

100150200250300350400450500550

CDS

spre

ad (b

ps)

Figure 5 Time series for Russian Federation with local minimalocal maxima and modified 120598-draw-ups

used for the corporate issuers The quantiles of the generatedloss distributions as well as the percentage increase due tocontagion are illustrated in Table 4 A liquidity horizon of 1year is assumed throughout and the figures are based on asimulation with 106 samples

For Portfolio A the 9990 quantile of the loss distri-bution under the standard factor model is euro2258857 whichcorresponds to approximately 23 of the total notional Thisfigure jumps to euro4968393 (almost 50 of the notional)under the model with contagion As shown in Panel 1contagion has a minimal effect on the 99 quantile whileat 995 9990 and 9999 it results in an increase of108 120 and 61 respectively This is to be expected

as the probability of default for Russian Federation is lessthan 1 and thus in more than 999 of our trials defaultwill not take place and contagion will not be triggered ForPortfolio B the 9990 quantile is considerably lower underboth the standard and the contagion model at euro775773 or8 of the total notional and euro1009426 or 10 of the totalnotional respectively reflecting lower default risk One canobserve that the model with contagion yields low additionallosses at 99 and 995 quantiles with a more significantimpact at 9990 and 9999 (30 and 37 respectively)An illustration of the additional losses due to contagion isgiven by Figure 8

45 Sensitivity Analysis In the following we present a seriesof sensitivity studies and discuss the results To achieve acandid comparison we choose to perform this analysis on thesingle-sovereign Portfolio AWe vary the ratings of sovereignand corporates as well as the CountryRank parameter todraw conclusions about their impact on the loss distributionand verify the model properties

451 Sovereign Rating We start by exploring the impact ofthe credit quality of the sovereign Table 5 shows the quantilesof the generated loss distributions under the standard latentvariable model and the contagion model when the rating ofthe Russian Federation is 1 and 2 notches higher than theoriginal rating (BB) It can be seen that the contagion effectappears less strong when the sovereign rating is higher Atthe 999 quantile the contagion impact drops from 120to 62 for an upgraded sovereign rating of BBB The dropis even higher when upgrading the sovereign rating to Awith only 11 additional losses due to contagion Apart fromhaving a less significant impact at the 999 quantile it isclear that with a sovereign rating of A the contagion impactis zero at the 99 and 995 levels where the results of the

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

Complexity 11

Oil Transporting Jt Stk Co Transneft

Vnesheconombank

Bk of Moscow

City Moscow

JSC GAZPROM

JSC Gazprom Neft

Lukoil Co

Mobile Telesystems

MDM Bk open Jt Stk Co

Open Jt Stk Co ALROSA

OJSC Oil Co RosneftJt Stk Co Russian Standard Bk

Russian Agric Bk

JSC Russian Railways

Russian Fedn

SBERBANK

OPEN Jt Stk Co VIMPEL Comms

JSC VTB Bk

Figure 6 Minimum spanning tree for Russian issuers

Europe

Asia

North America

Latin America

Middle East and Africa

Pacific

Materials

Consumer products

Services

Financial

Industrial

Government

Euro

pe

Asia

Nor

th A

mer

ica

Latin

Am

eric

a

Mid

dle E

ast a

nd A

fric

a

Paci

fic

Mat

eria

ls

Con

sum

er p

rodu

cts

Serv

ices

Fina

ncia

l

Indu

stria

l

Gov

ernm

ent

090

075

060

045

030

Figure 7 Estimated systematic factor correlation matrix Ω

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

12 Complexity

6

5

4

3

2

1

0

LossContagion add-on

Loss

(mill

ions

)

99

Portfolio A

995 999 9999Quantile ()

Portfolio B

LossContagion add-on

99 995 999 9999Quantile ()

0

025

05

075

1

125

15

175

Loss

(mill

ions

)Figure 8 Additional losses due to contagion for Portfolios A and B

Table 5 Varying the sovereign rating

Sovereign rating Quantile Lossstandard model

Losscontagion model

Contagion impact

BB

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

BBB

99 1115153 1115153 - 09950 1443009 1490755 47746 39990 2229742 3613625 1383883 629999 3496264 5432236 1935972 55

A

99 1114583 1114583 - 09950 1443009 1443009 - 09990 2229737 2469922 240185 119999 3455199 5056639 1601439 46

contagion model match those of the standard model This isto be expected since a rating of A corresponds to a probabilityof default less than 001 and as explained in Section 44when sovereign default occurs seldom the contagion effectcan hardly be observed

452 Corporate Default Probabilities In the next test theimpact of corporate credit quality is investigated As Table 6illustrates contagion has smaller impact when the corporatedefault probabilities are increased by 5 which is in linewith intuition since the autonomous (not sovereign induced)default probabilities are quite high meaning that they arelikely to default whether the corresponding sovereign defaultsor not For the same reason the impact is even less significantwhen the corporate default probabilities are stressed by 10

453 CountryRank In the last test the sensitivity of the con-tagion impact to changes in the CountryRank is investigatedIn Table 7 we test the contagion impact when CountryRankis stressed by 15 and 10 respectively The results arein line with intuition with a milder contagion effect forlower CountryRank values and a stronger effect in case theparameter is increased

5 Conclusions

In this paper we present an extended factor model forportfolio credit risk which offers a breadth of possibleapplications to regulatory and economic capital calculationsas well as to the analysis of structured credit products In theproposed framework systematic risk factors are augmented

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

Complexity 13

Table 6 Varying corporate default probabilities

Corporate default probabilities Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1250570 135417 129950 1443579 3003949 1560370 1089990 2333935 4968393 2634458 1139999 3584506 5713486 2128979 59

Stressed by 10

99 1162329 1260422 98093 89950 1503348 3003949 1500602 1009990 2375570 4968393 2592823 1099999 3642099 5713486 2071387 57

Table 7 Varying CountryRank

CountryRank Quantile Lossstandard model

Losscontagion model

Contagion impact

Unstressed

99 1115153 1162329 47176 49950 1443579 3003949 1560370 1089990 2258857 4968393 2709536 1209999 3543441 5713486 2170045 61

Stressed by 5

99 1115153 1162329 47176 49950 1443579 3196958 1753379 1219990 2258857 5056634 2797777 1249999 3543441 5713495 2170054 61

Stressed by 10

99 1115153 1162329 47176 49950 1443579 3389398 1945818 1359990 2258857 5296249 3037392 1349999 3543441 5801727 2258286 64

with an infectious default mechanism which affects theentire portfolio Unlike models based on copulas with moreextreme tail behavior where the dependence structure ofdefaults is specified in advance our model provides anintuitive approach by first specifying the way sovereigndefaults may affect the default probabilities of corporateissuers and then deriving the joint default distribution Theimpact of sovereign defaults is quantified using a credit stresspropagation network constructed from real data Under thisframework we generate loss distributions for synthetic testportfolios and show that the contagion effect may have aprofound impact on the upper tails

Our model provides a first step towards incorporatingnetwork effects in portfolio credit riskmodelsThemodel can

be extended in a number of ways such as accounting for stresspropagation from a sovereign to corporates even withoutsovereign default or taking into consideration contagionbetween sovereigns Another interesting topic for futureresearch is characterizing the joint default distribution ofissuers in credit stress propagation networks using Bayesiannetwork methodologies which may facilitate an improvedapproximation of the conditional default probabilities incomparison to the maximum weight path in the currentdefinition of CountryRank Finally a conjecture worthy offurther investigation is that a more connected structure forthe credit stress propagation network leads to increasedvalues for the CountryRank parameter and as a result tohigher additional losses due to contagion

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

14 Complexity

EnBW Energie BadenWuerttemberg AG

Allianz SE

BASF SEBertelsmann SE Co KGaABay Motoren Werke AGBayer AGBay Landbk GirozCommerzbank AGContl AGDaimler AGDeutsche Bk AGDeutsche Bahn AGFed Rep GermanyDeutsche Post AGDeutsche Telekom AGEON SEFresenius SE amp Co KGaAGrohe Hldg Gmbh

0

100

200

300

400

CDS

spre

ad (b

ps)

2015

-08

2014

-10

2015

-10

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

Hannover Rueck SEHeidelbergCement AGHenkel AG amp Co KGaALinde AGDeutsche Lufthansa AGMETRO AGMunich ReTUI AG

Robert Bosch GmBHRheinmetall AGRWE AGSiemens AGUniCredit Bk AGVolkswagen AGAegon NVKoninklijke Ahold N VAkzo Nobel NVEneco Hldg N VEssent N VING Groep NVING Bk N VKoninklijke DSM NVKoninklijke Philips NVKoninklijke KPN N VUPC Hldg BVKdom NethNielsen CoNN Group NVPostNL NVRabobank NederlandRoyal Bk of Scotland NVSNS Bk NVStmicroelectronics N VUnilever N VWolters Kluwer N V

Figure 9 Time series of CDS spreads ofDutch andGerman entities

Appendix

A CDS Spread Data

The data used to calibrate the credit stress propagationnetwork for European issuers is the CDS spread data ofDutch German Italian and Spanish issuers as shown inFigures 9 and 10

B Stability of 120598-Parameter

The plot in Figure 11 shows the time series of the epsilonparameter for different number days used for 120598-draw-upcalibration

2015

-08

2014

-10

2015

-10

ALTADIS SABco de Sabadell S ABco Bilbao Vizcaya Argentaria S ABco Pop EspanolEndesa S AGas Nat SDG SAIberdrola SAInstituto de Cred OficialREPSOL SABco SANTANDER SAKdom SpainTelefonica S AAssicurazioni Generali S p AATLANTIA SPAMediobanca SpABco Pop S CCIR SpA CIE Industriali RiuniteENEL S p AENI SpAEdison S p AFinmeccanica S p ARep ItalyBca Naz del Lavoro S p ABca Pop di Milano Soc Coop a r lBca Monte dei Paschi di Siena S p AIntesa Sanpaolo SpATelecom Italia SpAUniCredit SpA

2015

-12

2014

-08

2014

-12

2015

-02

2015

-04

2015

-06

50

100

150

200

250

300

CDS

spre

ad (b

ps)

Figure 10 Time series of CDS spreads of Spanish and Italianentities

Epsilon parameter for varying number of calibrationdays for Russian Fedn CDS

10 days15 days20 days

6020100 5030 40

Index of local minima

0

10

20

30

40

50

60

70

Epsil

on p

aram

eter

(bps

)

Figure 11 Stability of 120598 parameter for Russian Federation CDS

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013

Complexity 15

Disclosure

The opinions expressed in this work are solely those of theauthors and do not represent in anyway those of their currentand past employers

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors are thankful to Erik Vynckier Grigorios Papa-manousakis Sotirios Sabanis Markus Hofer and Shashi Jainfor their valuable feedback on early results of this workThis project has received funding from the European UnionrsquosHorizon 2020 research and innovation programme underthe Marie Skłodowska-Curie Grant Agreement no 675044(httpbigdatafinanceeu) Training for BigData in FinancialResearch and Risk Management

References

[1] Moodyrsquos Global Credit Policy ldquoEmergingmarket corporate andsub-sovereign defaults and sovereign crises Perspectives oncountry riskrdquo Moodyrsquos Investor Services 2009

[2] M B Gordy ldquoA risk-factor model foundation for ratings-basedbank capital rulesrdquo Journal of Financial Intermediation vol 12no 3 pp 199ndash232 2003

[3] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo The Journal of Finance vol 29 no2 pp 449ndash470 1974

[4] J R Bohn and S Kealhofer ldquoPortfolio management of defaultriskrdquo KMV working paper 2001

[5] P J Crosbie and J R Bohn ldquoModeling default riskrdquo KMVworking paper 2002

[6] J P Morgan CreditmetricsmdashTechnical Document JP MorganNew York NY USA 1997

[7] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standardsrdquo Bankof International Settlements 2004

[8] D Duffie L H Pedersen and K J Singleton ldquoModelingSovereign Yield Spreads ACase Study of RussianDebtrdquo Journalof Finance vol 58 no 1 pp 119ndash159 2003

[9] A J McNeil R Frey and P Embrechts Quantitative riskmanagement Princeton Series in Finance Princeton UniversityPress Princeton NJ USA Revised edition 2015

[10] P J Schonbucher and D Schubert ldquoCopula-DependentDefaults in Intensity Modelsrdquo SSRN Electronic Journal

[11] S R Das DDuffieN Kapadia and L Saita ldquoCommon failingsHow corporate defaults are correlatedrdquo Journal of Finance vol62 no 1 pp 93ndash117 2007

[12] R Frey A J McNeil and M Nyfeler ldquoCopulas and creditmodelsrdquoThe Journal of Risk vol 10 Article ID 11111410 2001

[13] E Lutkebohmert Concentration Risk in Credit PortfoliosSpringer Science amp Business Media New York NY USA 2009

[14] M Davis and V Lo ldquoInfectious defaultsrdquo Quantitative Financevol 1 pp 382ndash387 2001

[15] K Giesecke and S Weber ldquoCredit contagion and aggregatelossesrdquo Journal of Economic Dynamics amp Control vol 30 no5 pp 741ndash767 2006

[16] D Lando and M S Nielsen ldquoCorrelation in corporate defaultsContagion or conditional independencerdquo Journal of FinancialIntermediation vol 19 no 3 pp 355ndash372 2010

[17] R A Jarrow and F Yu ldquoCounterparty risk and the pricing ofdefaultable securitiesrdquo Journal of Finance vol 56 no 5 pp1765ndash1799 2001

[18] D Egloff M Leippold and P Vanini ldquoA simple model of creditcontagionrdquo Journal of Banking amp Finance vol 31 no 8 pp2475ndash2492 2007

[19] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[20] P Gai and S Kapadia ldquoContagion in financial networksrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 466 no 2120 pp 2401ndash2423 2010

[21] P Glasserman and H P Young ldquoHow likely is contagion infinancial networksrdquo Journal of Banking amp Finance vol 50 pp383ndash399 2015

[22] M Elliott B Golub and M O Jackson ldquoFinancial networksand contagionrdquo American Economic Review vol 104 no 10 pp3115ndash3153 2014

[23] D Acemoglu A Ozdaglar andA Tahbaz-Salehi ldquoSystemic riskand stability in financial networksrdquoAmerican Economic Reviewvol 105 no 2 pp 564ndash608 2015

[24] R Cont A Moussa and E B Santos ldquoNetwork structure andsystemic risk in banking systemsrdquo in Handbook on SystemicRisk J-P Fouque and J A Langsam Eds vol 005 CambridgeUniversity Press Cambridge UK 2013

[25] S Battiston M Puliga R Kaushik P Tasca and G CaldarellildquoDebtRank Too central to fail Financial networks the FEDand systemic riskrdquo Scientific Reports vol 2 article 541 2012

[26] M Bardoscia S Battiston F Caccioli and G CaldarellildquoDebtRank A microscopic foundation for shock propagationrdquoPLoS ONE vol 10 no 6 Article ID e0130406 2015

[27] S Battiston G Caldarelli M DrsquoErrico and S GurciulloldquoLeveraging the network a stress-test framework based ondebtrankrdquo Statistics amp Risk Modeling vol 33 no 3-4 pp 117ndash138 2016

[28] M Bardoscia F Caccioli J I Perotti G Vivaldo and GCaldarelli ldquoDistress propagation in complex networksThe caseof non-linear DebtRankrdquo PLoS ONE vol 11 no 10 Article IDe0163825 2016

[29] S Battiston J D Farmer A Flache et al ldquoComplexity theoryand financial regulation Economic policy needs interdisci-plinary network analysis and behavioralmodelingrdquo Science vol351 no 6275 pp 818-819 2016

[30] S Sourabh M Hofer and D Kandhai ldquoCredit valuationadjustments by a network-based methodologyrdquo in Proceedingsof the BigDataFinance Winter School on Complex FinancialNetworks 2017

[31] S Battiston D Delli Gatti M Gallegati B Greenwald and JE Stiglitz ldquoLiaisons dangereuses increasing connectivity risksharing and systemic riskrdquo Journal of Economic Dynamics ampControl vol 36 no 8 pp 1121ndash1141 2012

[32] R Kaushik and S Battiston ldquoCredit Default Swaps DrawupNetworks Too Interconnected to Be Stablerdquo PLoS ONE vol8 no 7 Article ID e61815 2013