Conditional graphical models for systemic risk estimation

ARTICLE IN PRESS

JID: ESWA

[m5G;September 14, 2015;17:18]

Expert Systems With Applications xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Expert Systems With Applications journal homepage: www.elsevier.com/locate/eswa

Conditional graphical models for systemic risk estimation Paola Cerchiello1, Paolo Giudici∗

Q1

University of Pavia, Department of Economics and Management, Italy

a r t i c l e

i n f o

a b s t r a c t

MSC: 00-01 99-00

Financial network models are a useful tool to model interconnectedness and systemic risks in banking and finance. Recently, graphical Gaussian models have been shown to improve the estimation of network models and, consequently, the interpretation of systemic risks.

Keywords: Conditional independence Network models Financial risk management

This paper provides a novel graphical Gaussian model to estimate systemic risks. The model is characterised by two main innovations, with respect to the recent literature: it estimates risks considering jointly market data and balance sheet data, in an integrated perspective; it decomposes the conditional dependencies between financial institutions into correlations between countries and correlations between institutions, within countries. The model has been applied to study systemic risks among the largest European banks, with the aim of identifying central institutions, more subject to contagion or, conversely, whose failure could result in further distress or breakdowns in the whole system. The results show that, in the transmission of systemic risk, there is a strong country effect, that reflects the weakness or the strength of the underlying economies. Besides the country effect, the most central banks are those larger in size. © 2015 Elsevier Ltd. All rights reserved.

1

Q2 Q3

Q4

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

1. Introduction During the latest financial crisis, started in 2007, impairment losses arising from loan books and security portfolios have substantially increased the probability of default of banks. Interconnectedness between financial institutions, through interbank lending and common sovereign or corporate exposures, have further amplified such probability, giving rise to the so-called “systemic risk” effect. To enhance the future resilience of the banking sector, a new regulatory framework, the so-called Basel III package, has been proposed, implying more stringent capital requirements for financial institutions (Basel committee, 2011). The effectiveness of the new regulatory framework to prevent banking default and financial crisis is an open problem, particularly as regulations themselves are still in progress, and may thus benefit from the results of research findings in the field. Research studies on bank failures can be classified in two main streams: scoring models and market models. Scoring models are typically based on financial fundamentals, taken from publicly available balance sheets. Their diffusion has followed the seminal paper by Altman, 1978 and has induced the ∗

Corresponding author. Tel.: +39 0382984351. E-mail addresses: [email protected] (P. Cerchiello), [email protected] (P. Giudici). 1 Tel.: +39 0382984348.

production of scoring models for banks themselves: noticeable examples are Sinkey (1975), Tam and Kiang (1992) and Cole and Gunther (1998). The development of the Basel regulation and the recent financial crisis have further boosted the literature on scoring models for banking failure predictions, especially under the so-called CAMELS framework (see e.g. Arena, 2005; Davis and Dilruba, 2008 and Klomp and Haan, 2012). Scoring models have been extended in different ways: interesting developments include the incorporation of macroeconomic components (see e.g. Kanno, 2013; Koopman, Lucas, and Schwaab, 2011; Mare, 2012 and Kenny, Kostka, and Masera, 2013) and the explicit consideration of the credit portfolio, as in the Symbol model of De Lisa, Zedda, Vallascas, Campolongo, and Marchesi (2011), that allows stress tests of banking asset quality and capital, as emphasized by Halaj (2013). Recent extensions are aimed at overcoming the rare nature of bank defaults: noticeable examples include Betz, Oprica, Peltonen, and Sarlin (2014) and Calabrese and Giudici (2015). The problem with scoring models is that they are mostly based on balance sheet data, which have, differently from the market, a low frequency of update (annual or, at best, quarterly) and do depend on subjective management choices. They may thus be good to predict defaults (especially in the medium term) but not in the assessment of systemic risks, which occur very dynamically and with short notice. Market models originate from the seminal paper of Merton (Merton, 1974), in which the market value of a bank’s assets, typically modelled as a diffusion process, is insufficient to meet its

http://dx.doi.org/10.1016/j.eswa.2015.08.047 0957-4174/© 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

JID: ESWA 2

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112

ARTICLE IN PRESS

[m5G;September 14, 2015;17:18]

P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx

liabilities. Due to its practical limitations, Merton’s model has been evolved into a reduced form (Vasicek, 1984), leading to a widespread diffusion of the resulting model, and the related implementation in Basel II credit portfolio models. In order to implement market models, diffusion process parameters and, therefore, bank default probabilities, can be obtained on the basis of share price data that can be collected almost in real time from financial markets. Market data are relatively easy to collect, are public, and are quite objective. On the other hand, they may not reflect the true fundamentals of the underlying financial institutions, and may lead to a biased estimation of the probability of failure, especially in the long-term (see e.g. Idier, Lamé, & Mésonnier, 2013). The high frequency of market data makes them suitable to study systemic risks, which describe the additional probability of default that derives from the interconnectedness network of a bank. A comprehensive review on systemic risk models is provided in Benoit, Colliard, Hurlin, and Perignon (2015). Specific measures of systemic risk have been proposed by Acharya, Pedersen, Philippon, and Richardson (2010) Adrian and Brunnermeier (2010) Brownlees, Hans, and Nualart (2014) Huang, Hao, and Haibin (2011) Segoviano and Goodhart (2009). All of these approaches are built on financial market price information, on the basis of which they lead to the estimation of appropriate quantiles of the estimated loss probability distribution of a financial institution, conditional on a crash event in the financial market. They however do not address the issue of how risks are transmitted between different institutions. Trying to address this aspect of systemic risks, researchers have recently introduced network models. In particular, Billio, Getmansky, Lo, and Pelizzon (2012) propose several econometric measures of connectedness based on principal component analysis and Grangercausality networks; Diebold, Demirer, and Yilmaz (2014) propose Vector Autoregressive models, augmented with a LASSO type estimation procedure, aimed at selecting the significant links in a network model; Hautsch, Schaumburg, and Schienle (2013) and Peltonen, Piloiou, and Sarlin (2015) propose tail dependence network models aimed at overcoming the bivariate nature of the available systemic risk measures. Network models, albeit elegant and visually attractive, are based on the assumption of full connectedness among all institutions, which make their estimation and interpretation quite difficult, especially when a large number of them is being considered. To tackle the previous limitation, Ahelegbey, Billio, and Casarin (2015) and Giudici and Spelta (2015) have recently introduced graphical vector autoregressive models, which can account for partial connectedness, expressed in terms of conditional independence constraints. A similar line of research has been followed by Barigozzi and Brownlees (2013) and Brownlees et al. (2014) who have introduced multivariate Brownian processes with a correlation structure that is determined by a conditional independence graph. Our contribution follows the latter perspective, and apply graphical network models to systemic risk. Besides the applied contribution, we add two main innovations. First, we consider a novel approach, that uses both financial market and balance sheet data, joining the financial market and the scoring perspective, in a single model. To our knowledge, this is the first paper that has such an integrated perspective. Second, we allow correlations between financial institutions to be decomposed into a systemic country effect plus an idiosyncratic bank-specific effect. This, following what assumed for the asset returns in CAPM models (Sharpe, 1964), improves substantially both model estimation and interpretation of the results. The rest of the paper is organized as follows. In Section 2, we introduce our proposed model. In Section 2.1 we describe the empirical results obtained with the application of our model to data that concern the largest European banks. Finally, Section 2.2 contains some concluding remarks.

2. Methodology In this section we first review graphical Gaussian models and, then, present our methodological proposal. Graphical models are based on the idea that interactions among random variables in a system can be represented in the form of graphs, whose nodes represent the variables and whose edges show their interactions. For an introduction to graphical models see, for example, Lauritzen and Wermuth (1989) Pearl (1988) Wermuth and Lauritzen (1990) Whittaker (1990) and Edwards (1990). 2.1. Graphical Gaussian models Let g = (V, E ) be an undirected graph, with vertex set V = {1, . . . , n}, and edge set E = V × V, a binary matrix, with elements eij , that describes whether pairs of vertices are (symmetrically) linked between each other (ei j = 1), or not (ei j = 0). If the vertices V of the graph g are put in correspondence with a vector of random variables X = X1 , ..., Xn , the edge set E induces conditional independence on X via the so-called Markov properties, see Wermuth and Lauritzen (1990). More precisely, the pairwise Markov property determined by the graph g states that, for all 1 ≤ i < j ≤ n,

114 115 116 117 118 119 120 121

122 123 124 125 126 127 128 129 130 131

ei j = 0 ⇐⇒ Xi ⊥ X j |XV \{i, j} ; that is, the absence of an edge between vertices i and j is equivalent to independence between the random variables Xi and Xj , conditionally on all other variables XV\{i, j} . Here we are concerned with quantitative random variables and, therefore, the graphical model we assume is a graphical Gaussian model, specified as follows. Let X = (X1 , . . . , Xn ) ∈ Rn be a random vector distributed according to a multivariate normal distribution N (μ,  ). In this paper, without loss of generality, we will assume that the data are generated by a stationary process, and, therefore, μ = 0. In addition, we will assume throughout that the covariance matrix  is non singular. Let K =  −1 be the inverse covariance matrix, or precision matrix, with elements kij . Whittaker (1990) proved that the pairwise Markov property implies that the following equivalence holds, for graphical Gaussian models:

132 133 134 135 136 137 138 139 140 141 142 143 144 145 146

Xi ⊥ X j |XV \{i, j} ⇐⇒ ρi jV = 0, where

147

−ki j

ρi jV = 

kii k j j

denotes the ijth partial correlation, that is, the correlation between Xi and Xj conditionally on the remaining variables XV\{i, j} . Therefore, given an undirected graph g = (V, E ), a graphical Gaussian model can be defined as the family of all N-variate normal distributions N (0, g ) that satisfy the constraints induced by a graph g on the variance-covariance matrix, in terms of zero partial correlations. Statistical inference for graphical models can be of two kinds: quantitative learning, which means that, given a graphical structure, with associated Markov properties, data are employed to estimate the unknown parameters of the model; and structural learning, which means that the graphical structure itself is estimated on the basis of the data. Here we focus on structural learning, as our aim is to infer from the data the network model that best describes the interrelationships between the financial institutions we consider. To achieve this aim, we now recall the expression of the likelihood of a graphical Gaussian model, on which structural learning will be based. For a given graph g, consider a sample X of size n from P = N (0, g ), and let S be the corresponding observed variancecovariance matrix. For a subset of vertices A ⊂ N, let  A denote the

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

113

148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167

ARTICLE IN PRESS

JID: ESWA

[m5G;September 14, 2015;17:18]

P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx

3

R

Y1

Y2

Y3

Y4

Y5

Y6

Y7

Y8

Y9

Y10

Y11

Y12

Y13

Y14

Y15

Y16

Y17

Y18

Y19

Fig. 1. Representationof plain Gaussian graphical model.

168 169 170 171 172

173 174

variance–covariance matrix of the variables in XA , and define with SA the corresponding observed submatrix. When the graph g is decomposable the likelihood of a graphical Gausssian model specified by P nicely decomposes as follows (see e.g. Dawid & Lauritzen, 1993):

 p(xC |C ) p(x| , g) = C∈C , p S∈S (xS |S )

where C and S, respectively denote the set of cliques and separators of the graph G, and:

P (xC |C ) = (2π )−

n|C | 2



|C |−n/2 exp[−1/2 tr SC (C )

 −1

]

186

and similarly for P(xS | S ). Note that the likelihood depends on the parameter  and on g, which indicates the set of cliques and separators that factorise the likelihood, and determine which submatrices of  to consider. Structural learning can be achieved replacing  with its maximum likelihood estimator, the observed sample variance covariance matrix S (constrained by g) and comparing graphical models in terms of their resulting maximised likelihood. In practice, a stepwise model selection algorithm is implemented, starting from a reference model (such as the independence one) and checking whether, at each step, an additional edge can be added or removed, according to a pairwise statistical testing procedure.

187

2.2. Proposal

175 176 177 178 179 180 181 182 183 184 185

188 189 190 191 192 193 194 195 196 197 198 199 200 201 202

In high dimensional settings, such as those occurring in systemic risk modelling, there is a high chance that model selection algorithms used for structural learning get trapped, spending much time to learn local optimum structures which might not be optimal at a global level. In addition, it may be difficult to extract interpretable information from a learned structure that is large and shows many interrelationships. A possible solution to the above problems is to add more structure to graphical models, and this is what we propose. To ease understanding, and without loss of generality, we will from now on refer the notation to the specific systemic risk problem we face. Assume that Ri,j,t is a random variable representing the return for the ith bank in the jth country at time t and that R¯ jt is the mean return of all banks present in country j, at time t. The joint distribution of all bank returns can then be factorised, using the country mean variables, as follows:

P (R11t , . . . , RIJt ) = P (R11t , . . . , RIJt |R¯ 1t . . . , R¯ Jt ) ∗ p(R¯ 1t , . . . , R¯ Jt ). 203 204 Q5 205

Our main modelling assumption concerns the conditional distribution of bank returns, P (R11t , . . . , RIJt |R¯ 1t , . . . , R¯ Jt ). For any given time point we assume that:

P {( R11t , . . . , RI1t ), (R12t , . . . , RI2t ), (R1Jt , . . . , RIJt ) |R¯ 1t , . . . , R¯ Jt }





Country1

206





Country2







CountryJ

= P (R11t , . . . , RI1t |R¯ 1t ) ∗ P (R12t , . . . , RI2t |R¯ 2t ) ∗ . . . ∗ × P (R1Jt , . . . , RIJt |R¯ Jt ).

that is, returns of banks from different countries are assumed to be independent, conditionally on the knowledge of their country means. From an economic viewpoint, this assumption relies on the fact that, at least in Europe, countries and banks are strongly interconnected and, as the recent history has shown, the main source of market volatility of bank share prices is the state of the economy of the country where they operate. According to the previous assumption, we can factorise the joint distribution of bank returns (and, correspondingly, the likelihood), into J + 1 terms: J bank return distributions for the J considered countries plus the joint distribution of the country mean returns. This clearly simplifies computations and, in particular, the search for the best graphical model, that can be carried out locally, in each of the J + 1 likelihood terms. The best model will be the product of J + 1 best (sub)graphical models, each of which is the mode of the distribution of all possible graphs for the corresponding subset of variables. Fig. 1 below graphically represents our assumed model, in terms of conditional independence assumptions. For illustrative purposes, J = 19, as in the application Section. Fig. 1 emphasises once more that our proposed model is made up of J + 1 distributions: one for the vector of country means, and J for the vectors of bank returns, within each country. Each of the J + 1 joint distributions can be described by a graphical Gaussian model, as follows. First, the vector of all country mean returns, R¯ jt , is a graphical Gaussian model, with mean μt , the overall mean. The overall mean return can be subtracted out, leading to the extra returns Z j,t = R¯ jt − R¯t distributed as follows:

Z j,t = ∼ N(0, c ), t = 1, . . . , n,

209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234

(1)

where  c indicate the (non singular) variance–covariance matrix between country mean extra returns. According to the pairwise Markov property the following then holds, for any pair of countries j, j :

235 236 237

e j j = 0 ⇐⇒ ρ j j V = 0. a missing edge between two countries is equivalent to a zero partial correlation between the corresponding extra-returns. The zero partial correlation constraints on the matrix  c are to be estimated from the data, in a (local) graphical Gaussian model selection procedure based only on the country mean returns. Second, we assume that, for any given country j, the vector of all bank returns Ri,j,t , conditionally on the country means, is distributed as a graphical Gaussian model with mean μj,t , the country mean. Indeed the (observed) country mean return can be subtracted out, leading to the extra returns Yi, j,t = Ri, j,t − R¯ j,t be distributed as follows:

Yi, j,t ∼ N(0,  j ), t = 1, . . . , n,

238 239 240 241 242 243 244 245 246 247

(2)

where  j indicates the (non singular) variance–covariance matrix between bank extra returns of a country j. According to the pairwise Markov property, the following then holds, for any pair of banks (i, i ):

eii = 0 ⇐⇒ ρii V = 0 :

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

207 208

248 249 250 251

JID: ESWA 4

252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278

ARTICLE IN PRESS P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx

a missing edge between two banks is equivalent to a zero partial correlation between the corresponding extra-returns. The zero partial correlation constraints on the matrix  j are to be estimated from the data, in a (local) graphical Gaussian model selection procedure, based exclusively on the extra returns for that country. The proposed model can be extended to take into account covariates that may explain the returns and their correlations. We will consider two types of covariates: macroeconomic ones, that may affect returns at the country level, and microeconomic ones, that may affect returns at the bank level. This will allow us “to mix” market data with bank specific balance sheet data, and with macroeconomic data as well. We remark that, to our knowledge, this is the first paper that models systemic risk using more than one data source. Covariates can be introduced in our model as further conditioning variables. For example, we may condition country mean returns on macroeconomic variables, and check whether the correlations between countries are due to common macroeconomic cycles. Or we may condition bank returns on microeconomic balance sheet ratios and check whether the correlations between banks in a country are due to common bank management policies. Mathematically, the introduction of covariates can be done maintaining the hierarchical structure seen before, specifying a linear regression model of bank returns on the covariates, both at the country and at the bank level and, then, model the regression residuals with a graphical Gaussian model. More formally, we first assume that, for each country j:

Yt ∼ N(X β ,  j ), t = 1, . . . , n, 279

where  j is such that, for any pair of banks (i, i ):

eii = 0 ⇐⇒ ρii V = 0, 280 281 282 283

and X is a data matrix containing balance sheet micro economic explanatory variables with β the corresponding vector of regression coefficients. Second, we assume that:

Zt ∼ N(W  , c ), t = 1, . . . , n, 284

where c is such that, for any pair of countries (j, j ):

e j j = 0 ⇐⇒ ρ j j V = 0, 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308

[m5G;September 14, 2015;17:18]

and W is a data matrix containing macro-economic explanatory variables with  the corresponding vector of regression coefficients. The described introduction of covariates in the model may change the conditional independence structure between the extra returns, consistently with the effect of introducing missing variables in a graphical model, that may induce paradoxes such as Berkson’s or Simpson’s (see e.g. Whittaker, 1990; or Didelez and Pigeot, 1998). From an applied viewpoint, it is better to focus model interpretation on the more complete model, that includes covariates, as it does represent a better state of knowledge. For example, a strong correlation of the macroeconomic variables of two countries may create a spurious correlation in c that is “explained away” in c when macroeconomic variables are taken into account. Or, at the bank-specific level, a common management strategy of two banks, reflected in similar balance sheet variables, may induce a correlation that is “explained away” when conditioning on such variables. On the other hand, a non significant dependence between the macroeconomic variables of two countries can make them conditionally independent in c but, when conditioning on such variables in c , the two countries become dependent, reflecting the fact that the two countries are perceived by the market as dependent, even though, macroeconomically, they are not. The same paradox can occur when a non significant dependence occurs between the balance sheet variables.

3. Application

309

In this section, we apply our proposed model to the estimation of the systemic risk of European banks. Europe is an interesting test case as banking systems, that differ among many different countries, are progressing towards the integration in a single Banking Union. We consider only large banks, whose total assets are greater than 30 billion euros, and are therefore included in the European Central Bank comprehensive assessment review, for the Eurozone countries. To obtain a more complete representation of European risks, we also consider countries in the European Union, not belonging to the Eurozone, as well as Switzerland and Norway. We consider only publicly listed banks, for which market data are available. In the case of a banking group with more entities that satisfy the above criteria, we consider only the controlling entity. The complete list of the 61 considered financial institutions is in Table 1, with the corresponding ticker code acronyms. Table 1 contains, besides bank names and their codes, their prevalent country, and their Total Assets from the last available balance sheet (in thousands of euro), at 4Q 2013. For each bank, we have collected data on their reported (quarterly) balance sheet, as well as on their (daily) market performance, from Bankscope, for the period 2009–2013. Concerning balance sheets, as they are published in the quarter that follows their reference period, the corresponding data has been shifted by one quarter, to make it consistent with the period in which they are published. In more detail, from the balance sheet of each bank we have extracted the ratio indicators suggested in the C.A.M.E.L. approach extensively employed in scoring models (see for example Arena, 2005), for a total of 9 candidate explanatory variables: Leverage (equity/total assets) and Tier1 (capital/risk weighted assets), that measure Capital adequacy; Coverage (loan reserves/gross loans) and Impairments (loan losses/gross losses), that measure Asset quality; Assets (total assets), that measure Management; NIM (net interest margin) and ROAA (return/average assets), that measure Earnings; Liquid (liquid assets/total assets) and Loans (loans/assets) that measure Liquidity. We found some missing data, especially for ratios which are less common; we have decided to impute them using a regression based approach, when the missing variable is strongly correlated with another (as it is the case for Coverage and Impairments), or a moving average approach, in other cases. Market performance has instead been measured, consistently with balance sheet data, taking the average quarterly share prices for each bank. Average prices have then been converted into returns. Let Pt be the average price of a bank in a quarter t. Its return can then be defined as: Rt = log(Pt /Pt−1 ), where t is a quarter and t − 1 the quarter that precedes it. Besides the above bank-specific data, we have extracted from Eurostat the main macroeconomic information from the countries to which the considered banks belong to, for the same period under consideration. We remark that the described data acquisition process and, in particular, the choice of the balance sheet indicators, is consistent with what done by researchers in the field of banking and finance: see, for example, the recent paper by Beck et al. (2014). We now present the main results from the application of our conditional graphical Gaussian models. According to our assumptions, the likelihood factorises into J + 1 terms: one for country means, and J for the considered countries, whose bank extra returns are assumed independent, given the country means. We can search the best graphical model by maximising separately each of the J + 1 terms. To ease the illustration of the results, in the following we present the obtained modal graph for the country means and for a selection of representative countries, among the J considered. Fig. 2 shows the graphical network model between countries that is selected by the stepwise graphical model selection procedure implemented in the software R, package gRim. Within gRim we have

310

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

311 312 313 314 315 316

Q6

317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373

Q7

ARTICLE IN PRESS

JID: ESWA

[m5G;September 14, 2015;17:18]

P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx Table 1 The considered banks.

374 375 376 377 378

Bank name

Code

Country

T.assets

HSBC Holdings Deutsche Bank BNP Paribas Credit Agricole Barclays Royal Bank of Scotland Banco Santander Societe Generale ING Groep NV Lloyds Banking Group UBS UniCredit Credit Suisse Group Nordea Bank AB Intesa Sanpaolo Banco Bilbao Vizcaya Argentaria Commerzbank Natixis-BPCE Group Standard Chartered Danske Bank Dexia DnB ASA Skandinaviska Enskilda Banken Svenska Handelsbanken KBC Groep NV CIC Credit Mutuel Group Banca Monte dei Paschi di Siena Swedbank Erste Group Bank Banco de Sabadell Banco Popular Espanol Raiffeisen Bank International Unione di Banche Italiane Banco Popolare Allied Irish Banks National Bank of Greece Banco Comercial Portuguese Banco Espirito Santo Wustenrot and Wurttembergische Mediobanca Piraeus Bank Eurobank Ergasias Banca popolare Emilia Romagna Alpha Bank Bankinter Banca Popolare di Milano Banca Carige PKO Bank Polski Aareal Bank Pohjola Bank Oyj Banco BPI Bank Pekao OTP Bank Jyske Bank Banca Popolare di Sondrio IKB Deutsche Industriebank Komercni Banka Credito Emiliano Credito Valtellinese

HSBA DBK BNP ACA BARC RBS SAN GLE INGA LLOY UBSN UCG CSGN NDA ISP BBVA

UK Germany France France UK UK Spain France Netherlands UK Switzerland Italy Switzerland Sweden Italy Spain

2040674.62 2012329.00 1907290.00 1842361.00 1782411.78 1569494.14 1269628.00 1250696.00 1168632.00 1105756.67 1041208.74 926827.50 764250.28 677309.00 673472.00 637785.00

CBK KN STAN DANSKE DEXB DNB SEB.A

Germany France UK Denmark Belgium Norway Sweden

635878.00 528370.00 482417.01 466755.96 357210.00 308173.44 285875.07

SHB.A KBC CC BMPS

Sweden Belgium France Italy

277776.39 256886.00 235732.00 218882.20

SWED.A EBS SAB POP RBI

Sweden Austria Spain Spain Austria

215194.90 213824.00 161547.10 157618.10 136116.00

UBI BP AIB ETE BCP BES WUW

Italy Italy Ireland Greek Portugal Portugal Germany

132433.70 131921.40 122516.00 104798.80 89744.00 83690.80 77192.90

MB TPEIR EUROB BPE ALPHA BKT PMI CRG PKO ARL POH1S BPI PEO OTP JYSK BPSO IKB BAAKOMB CE CVAL

Italy Greek Greek Italy Greek Spain Italy Italy Poland Germany Finland Portugal Poland Hungary Denmark Italy Germany Czech Republic Italy Italy

72841.30 70406.20 67653.00 61637.80 58357.40 58165.90 52475.00 49325.80 47308.75 45734.00 44623.00 44564.60 36909.55 34694.23 34585.96 32349.10 31593.70 31295.88 30748.70 29896.10

chosen to follow a forward approach, that moves from the full independence model (with no edges) by incrementally adding the edge that maximally increases the log-likelihood, if significant, and otherwise stop. The same package and approach has been followed to search the best (local) model for each country.

Fig. 2 shows that the largest and strongest economies of Europe (DE, FR, GB) are related, and form an isolated clique. Southern and/or weaker economies (ES, PT, IE, AT, HU) are also related, with IT acting as a central agent of contagion. Other economies, typically smaller, are (weakly) connected to the latter, with a position that depends on the performance of their banks. In terms of systemic risk, Fig. 2 shows that, besides IT, the most contagious/subject to contagion countries are NL (with its multinational bank, ING) and GR (with its troubled economy). The links in Fig. 2 may indeed be due to correlations in the macroeconomic cycles. This can be checked looking at the selected graphical models on the residuals from the regression of country mean returns on macroeconomic variables, reported in Fig. 3. The structure of Fig. 3 is similar to that of Fig. 2: this indicates that the relationships among country mean returns are not due only to macroeconomic factors. We however underline two important differences. On one hand, CH gets linked to the strongest economies, making them connected with the others, and more central: this, presumably, on the basis of market portfolio strategies. On the other hand, FI is removed from the group of weaker economies, revealing that the link with IT that appears in Fig. 2 may be due to a similar economic cycle, but it is not supported by market allocation strategies. According to Fig. 3 the most contagious/subject to contagion economies are: IT, CH, GB, PL and IE: weaker economies and financial hubs, as one would expect. For the sake of completeness, we report the results from the regression of country mean returns on macroeconomic variables, on which the graph of residuals in Fig. 3 is based, in Table 2. Table 2 shows that, while the GDP affects positively the mean returns, the Unemployment rate and the Inflation rate are negatively related, at the overall European level. The result for Unemployment and GDP indicates that downturn periods are associated to lower returns. The result for the inflation rate indicates that in the considered deflationary period, the rate is negatively related with the returns. These results are consistent with the economic literature (see e.g. Beck et al., 2014). Having estimated the country-to-country connections, we now look in detail at the interdependencies between banks, within countries. Figs. 4–7 present the selected graphical models for four Eurozone countries, two from northern Europe (France and Germany) and two from southern Europe (Spain and Italy). Fig. 4 shows that, in the case of Germany, the selected graph contains three independent cliques: one that contains the large commercial banks (CBK and DBK), and one that contains the smaller ones (ARL and WUW), with the specialized bank IKB on its own, with the lowest systemic risk. Fig. 5 shows that, in France, all banks are pairwise connected, apart from KN, which has three connections, and therefore, appears the most risk systemic. Fig. 6 shows that in Spain there is one central clique, composed of the largest two banks, BBVA and SAN, connected with the medium sized banks BKT and POP. SAB and CABK are instead disconnected. Therefore, in Spain, the largest banks are also the most risk systemic. Fig. 7 shows the more complex case of Italy. The most connected bank appears to be PMI, whereas the large bank UCG and the investment bank MB, together with CVAL, are disconnected. Besides PMI, the banks that are more risk systemic are the large sized ISP and the medium size BPSO. We now investigate whether the previously found connections can be explained by a similar management strategy, as described by balance sheet ratios or are, instead, due to other latent variables, such as market portfolio strategies. To achieve this aim, we consider the graphical model selected on the basis of the residuals from the regression model of bank extra returns on the balance sheet ratios. Table 3 presents, in a combined form, the results of the regressions for the four considered countries.

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

5

379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444

JID: ESWA 6

ARTICLE IN PRESS

[m5G;September 14, 2015;17:18]

P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx

Fig. 2. Links among countries based on Gaussian graphical model.

445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463

From Table 3 note that the significant variables differ considerably among countries, with a degree of similarity present between the two southern countries, ES and IT and, on the other hand, between the two northern countries, DE and FR. In more detail, for both ES and IT, Asset quality has an important negative sign on the returns (as expected), with an important difference between the two countries: while in Italy what matters is the Coverage of credit losses, in Spain both Impairments and Coverage are significant: higher Impairments are related with lower returns, and higher coverages partly correct this effect, increasing returns. Besides asset quality, the only other variable that significantly affect the returns, in Italy, is Tier1, with a positive effect, as expected. In Spain, instead, further significant variables are Liquid and NIM. While the former has the expected positive sign, the sign of the net interest margin is instead counter-intuitive. One possible way to interpret it is in “reversing” the causal chain: banks with more difficulties, reflected by a lower return, have been prompted to improve, increasing the interest margin. None of the remaining variables are significant in the two southern countries.

The structure of the estimated linear regression model for France and Germany is different from the previous one. For both countries, the significant ratios concern Asset quality, Management and Liquidity, with neither Capital adequacy nor Earnings being significant. Asset quality acts through the Impairments variable, significant and negative for both countries (as expected); Management through the Assets variable, significant and positive for both countries (as expected); Liquidity through the Loans variable (for Germany) and the Liquid variable for France. None of the remaining variables are significant in the two northern countries. To summarise, the findings in Table 3 emphasize the importance of the country effect, as similar countries have a similar structure of significant explanatory variables. As done for the country mean returns we now evaluate whether the introduction of balance sheet variables changes the dependence structure between the banks described in Figs. 3–6. Figs. 8–11 present the selected graphical model for the residuals of the same four countries: Germany, France, Spain and Italy.

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482

JID: ESWA

ARTICLE IN PRESS

[m5G;September 14, 2015;17:18]

P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx

7

Fig. 3. Links among countries based on Gaussian graphical model, conditional on macroeconomic variables.

483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499

Comparing Figs. 8–11 with Figs. 4–7 note that some new links are added, and others are deleted. In more detail: the graph of Germany and that of France are unchanged, revealing that the relationship between bank returns are not determined univocally by balance sheet ratios but, rather, by other factors, such as portfolio diversification strategies, based on the economical strength of the underlying country. In the case of Spain, the only change concerns the position of POP which is now connected to SAN rather than BBVA. This means that SAN becomes the most central bank, from a systemic risk viewpoint. In Italy, the differences concern the positions of BPE which is now connected to ISP, rather than BMPS. This means that, when balance sheet data are taking into account, BMPS becomes less contagious, and ISP more contagious. This is good news, given the difficult situation of BMPS at the moment and conversely the good state of ISP. Note, once, more, the difference between northern and southern countries: the estimated systemic risk network is affected by microe-

Table 2 Linear regression of country means on macroeconmic variables. Dependent variable: Country mean return GDP Un Inf. Constant Observations R2 Adjusted R2 Residual std. error F statistic

0.470∗∗∗ (0.066) −0.056∗∗∗ (0.012) −0.872∗∗∗ (0.305) 0.701∗∗∗ (0.148) 1,220 0.060 0.058 2.286 (df = 1216) ∗∗∗ 25.957 (df = 3; 1216)

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

ARTICLE IN PRESS

JID: ESWA 8

[m5G;September 14, 2015;17:18]

P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx Table 3 Linear regressions of bank returns on balance sheet ratios. Dependent variable: Share IT

DE

FR

ES

Leverage

0.340 (0.223) 0.218∗∗ (0.094) −0.170∗ (0.099)

Tier1 Coverage

−0.384∗∗∗ (0.116) 1.480∗∗ (0.671)

Impairments Assets_std

0.547∗∗ (0.236) −0.394∗∗∗ (0.148)

0.833 (0.511) −0.497∗ (0.293) 0.574∗ (0.328)

−0.914∗∗ (0.455)

NIM ROAA Liquid

Constant

−1.249 (0.823)

0.121∗∗ (0.047) −3.733∗ (1.887)

Observations R2 Adjusted R2 Residual std. error F statistic

240 0.028 0.019 2.500 (df = 237) 3.376∗∗ (df = 2; 237)

100 0.116 0.088 2.357 (df = 96) 4.199∗∗∗ (df = 3; 96)

Loans

0.040∗∗∗ (0.015)

0.059∗ (0.035)

−0.648 (0.600)

−0.692 (1.329)

100 0.079 0.040 1.699 (df = 95) 2.042∗ (df = 4; 95)

120 0.106 0.066 1.966 (df = 114) 2.694∗∗ (df = 5; 114)

SAB

IKB CBK DBK

SAN

BKT

ARL WUW BBVA Fig. 4. Selected graph of Germany bank returns. (ARL=Aareal Bank) (CBK=Commerzbank) (DBK=Deutsche Bank) (IKB=IKB Deutsche Industriebank) (WUW=Wustenrot and Wurttembergische).

CABK POP

GLE Fig. 6. Selected graph of Spain bank returns. (BKT=Bankinter) (BBVA=Banco Bilbao Vizcaya Argentaria) (CABK=Caixabank) (POP=Banco Popular Espanol) (SAB=Banco de Sabadell) (SAN=Banco Santander).

KN

BNP

ACA CC Fig. 5. Selected graph of France bank returns. (ACA=Credit Agricole) (BNP=BNP Paribas) (CC=Credit Industriel et Commercial) (GLE=Societe’ Generale) (KN=Natixis).

conomic factors only for the latter, weaker economically and, therefore, more fragile from a banking point of view. In general, the most central banks are those larger in size, consistently with the economic literature (as shown, in particular, in the recent paper by Laeven, Ratnovski, and Tong (2014).

500

4. Concluding remarks

505

This paper suggest a novel model to estimate systemic risks, along the network based research perspective suggested by Billio et al. (2012). Graphical Gaussian models can aid estimation and interpretation of network models, as discussed, for example, in Ahelegbey et al. (2015) Brownlees et al. (2014) and Giudici and Spelta (2015). Here we propose a novel graphical Gaussian model, that can usefully decompose dependencies between financial institutions into correlations between countries and correlations between institutions, within countries. It is the first model that, in the estimation

506

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

501 502 503 504

507 508 509 510 511 512 513 514

ARTICLE IN PRESS

JID: ESWA

[m5G;September 14, 2015;17:18]

P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx

9

SAB UCG CVAL CRG

BP MB

BKT

BBVA

BPSO

SAN CE

PMI BMPS

CABK POP ISP

BPE

UBI

Fig. 10. Selected graph of Spain bank residuals. (BKT=Bankinter) (BBVA=Banco Bilbao Vizcaya Argentaria) (CABK=Caixabank) (POP=Banco Popular Espanol) (SAB=Banco de Sabadell) (SAN=Banco Santander).

Fig. 7. Selected graph of Italy bank returns. (BMPS=Banca Monte dei Paschi di Siena) (BP=Banco Popolare) (BPE=Banca popolare dell’Emilia Romagna) (BPSO=Banca Popolare di Sondrio) (CE=Credito Emiliano) (CRG=Banca Carige) (CVAL=Credito Valtellinese) (ISP=Intesa Sanpaolo) (MB=Mediobanca) (PMI=Banca Popolare di Milano) (UBI=Unione di Banche Italiane) (UCG=UniCredit).

MB CRG BMPS

IKB

BP

BPSO

CBK DBK

PMI CE

UBI ISP

ARL WUW

UCG

CVAL BPE

Fig. 8. Selected graph of Germany bank residuals. (ARL=Aareal Bank) (CBK=Commerzbank) (DBK=Deutsche Bank) (IKB=IKB Deutsche Industriebank) (WUW=Wstenrot and Wrttembergische).

Fig. 11. Selected graph of Italy bank residuals. (BMPS=Banca Monte dei Paschi di Siena) (BP=Banco Popolare) (BPE=Banca popolare dell’Emilia Romagna) (BPSO=Banca Popolare di Sondrio) (CE=Credito Emiliano) (CRG=Banca Carige) (CVAL=Credito Valtellinese) (ISP=Intesa Sanpaolo) (MB=Mediobanca) (PMI=Banca Popolare di Milano) (UBI=Unione di Banche Italiane) (UCG=UniCredit).

GLE

KN

BNP

ACA CC Fig. 9. Selected graph of France bank residuals. (ACA=Crdit Agricole) (BNP=BNP Paribas) (CC=Credit Industriel et Commercial) (GLE=Societe Generale) (KN=Natixis).

of risks, mixes balance sheet with market data, in an integrated perspective. We have applied our proposed model to the study of systemic risks among the largest European banks, with the aim of identifying central institutions, more subject to contagion or, conversely, whose failure could result in further distress or breakdowns in the whole system. Our results shows that, in the transmission of systemic risks, there is a strong country effect, that reflects the weakness and the strength of the underlying economies. Besides the country effect, the most central banks are those larger in size. The advantage of graphical network models, with respect to fully connected models, such as those discussed by Billio et al. (2012) Diebold et al. (2014) Hautsch et al. (2013) and Peltonen et al. (2015), is mainly a simplified method for model selection and parameter estimation, based on conditional independence restrictions, which allows computations to localise on subsets of the graph. This, in turn, simplifies model interpretation. The correlated disadvantage is that the assumed conditional independence assumptions may be too restrictive, in practice. Conditional independence of bank returns in different countries may be one of those assumptions.

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534

JID: ESWA 10

ARTICLE IN PRESS P. Cerchiello, P. Giudici / Expert Systems With Applications xxx (2015) xxx–xxx

560

Another potential imitation of our proposed models is that they are based on correlation values and, therefore, may not be suited to describe dependencies among the tails of the distributions, which may be relevant in a financial crisis context. This limitation may be overcome combining the models of Diebold et al. (2014) Hautsch et al. (2013) and Peltonen et al. (2015) with a multivariate graphical modelling perspective. From an applied viewpoint, the advantage of our model is a joint modelling of balance sheet data with market data. Joint modelling allows to take advantage of the high frequency of market data, correcting them with low frequency balance sheet data, which better reflect the characteristics of the examined financial institutions. The disadvantage of what proposed is that, often, balance sheet data are not reliable or missing and, therefore, especially, for small banks, its added value may be limited. Another limitation of the proposed database is the lack of real expert opinions, apart from market data, which are obviously biased by portfolio allocation aspects. This limitation could be overcome inserting in the model further sources of data: for example, analysts opinions, ratings and other “big data” sources, once their quality is being assessed (as in Cerchiello & Giudici, 2015). Finally, from a wider expert and intelligence systems perspective, it may be of interest to compare the proposed model against others on the basis of observed (or partially observed) default data. In this sense it may be interesting to apply what proposed to the database described by Holopainen and Sarlin (2015).

561

Uncited references

562 564

Altman, 1968; Billio, Caporin, Panzica, Pelizzon, 2015; Brownlees, EngleC, Volatility, 2011; Brunnermeier, Oehmke, 2012; Lauritzen, 1996.

565

Acnowledgements

566

569

The authors acknowledge financial support from the PRIN project MISURA: multivariate models for risk assessment. They also acknowledge useful comments and suggestions from the referees, that have considerably improved the paper.

570

References

535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 Q8 550 551 552 553 554 555 556 557 558 559

563

567 568

571 572 573 574 575 Q9576 577 578 579 580 581 582 583 584 585 586 587 588 589 Q10 590 591

[m5G;September 14, 2015;17:18]

Acharya, V. V., Pedersen, L. H., Philippon, T., & Richardson, M. (2010). Measuring systemic risk (p. 1002). Federal Reserve Bank of Cleveland. Working paper. Adrian, T., & Brunnermeier, M. K. (2010). CoVaR (p. 348). Federal Reserve Bank of New York. Staff reports. Ahelegbey, D., Billio, M., & Casarin, R. (2015). Bayesian graphical models for structural vector autoregressive processes. Journal of Applied Econometrics. Altman, E. I. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The Journal of Finance, 23(4), 589–609. Arena, M. (2005). Bank failures and bank fundamentals: A comparative analysis of Latin America and East Asia during the nineties using bank-level data. Journal of Banking and Finance, 32, 299–310. Barigozzi, M. Brownlees, C. (2013). Nets: Network estimation for time series. Working paper. Benoit, S., Colliard, J., Hurlin, C. Perignon, C. (2015). Where the risks lie: A survey on systemic risk. Working paper. Basel committee on banking supervision, A global regulatory framework for more resilient banks and banking systems, Bank for International Settlements. (2011). Betz, F., Oprica, S., Peltonen, T. A., & Sarlin, P. (2014). Predicting distress in European banks. Journal of Banking and Finance, 45(C), 225–241. Billio, M., Caporin, M., Panzica, R. Pelizzon, L. (2015). Networkconnectivity and systemic risk. Technical report.

Billio, M., Getmansky, M., Lo, A. W., & Pelizzon, L. (2012). Econometric measures of connectedness and systemic risk in the finance and insurance sectors. Journal of Financial Economics, 104(3), 535–559. Brownlees, C., EngleC, R. F., & Volatility, T. (2011). Correlation and tails for systemic risk measurement. New York University. Technical report. Brownlees, C., Hans, C. Nualart, E. (2014). Bank credit risk networks: Evidence from the eurozone crisis. Working paper. Brunnermeier, M. K., & Oehmke, M. (2012). Bubbles, financial crises, and systemic risk (p. 18398). National Bureau of Economic Research, INC. NBER. Working papers Calabrese, R., & Giudici, P. (2015). Estimating bank default with generalised extreme value regression models. Journalof the Operational Research society, 1–10. Cerchiello, P., & Giudici, P. (2015). How to measure the quality of financial tweets. Qualityand Quantity. Cole, R. A., & Gunther, J. W. (1998). Predicting bank failures: A comparison of on-and off-site monitoring systems. Journal of Financial Services Research, 13(2), 103–117. Davis, E. P., & Dilruba, K. (2008). Comparing early warning systems for banking crises. Journal of Financial stability, 4(2), 89–120. Dawid, A. P., & Lauritzen, S. L. (1993). Hyper Markov laws in the statistical analysis of decomposable graphical models. Annals of Statistics, 21, 317–1272. De Lisa, R., Zedda, S., Vallascas, F., Campolongo, F., & Marchesi, M. (2011). Modelling deposit insurance scheme losses in a basel 2 framework. Journal of Financial Services Research, 40(3), 123–141. Didelez, V., & Pigeot, I. (1998). Maximum likelihood estimation in graphical models with missing data. Biometrika, 85(4), 960–966. Diebold, F. X., Demirer, M., & Yilmaz, K. (2014). Estimatingglobal bank interconnectedness. Technical report. Edwards, D. (1990). Hierarchical interaction models (with discussion). Journal of the Royal Statistical Society Series B, 52, pp. 3–20 and 51–72. Giudici, P., & Spelta, A. (2015). Graphical network models for international financial flows. Journal of Business and Economic Statistics, in press. Halaj, G. (2013). Optimal asset structure of a bank – bank reactions to stressful market conditions. In Working paper series: Vol. 1533. European Central Bank. Hautsch, N., Schaumburg, J., & Schienle, N. (2013). Forecasting systemic impact in financial networks. International Journal of Forecasting, 30, 781–794. Holopainen, M. Sarlin, P. (2015). Toward robust early-warning models: A horse-race, ensembles andmodel uncertainty. Technical report. Huang, X., Hao, Z., & Haibin, Z. (2011). Systemic risk contributions, macroprudential regulation and policy. BIS Papers Chapters, 60, 36–43. Idier, J., Lamé, G., & Mésonnier, J. S. (2013). How useful is the marginal expected shortfall for the measurement of systemic exposure? a practical assessment: 348. Banque de France. Working papers. Kanno, M. (2013). Credit migration forecasting and correlation between business and credit cycles. Kanagawa University. Technical report. Kenny, G., Kostka, T., & Masera, F. (2013). Can macroeconomists forecast risk? event-based evidence from the euro area spf: 1540. European Central Bank. Working paper series. Klomp, J., & Haan, J. (2012). Banking risk and regulation: Does one size fit all? Journal of Banking and Finance, 36(12), 3197–3212. Koopman, S. J., Lucas, A., & Schwaab, B. (2011). Modeling frailty-correlated defaults using many macroeconomic covariates. Journal of Econometrics, 162(2), 312–325. Laeven, L., Ratnovski, L. Tong, H. (2014). Banksize and systemic risk. IMF technical report. Lauritzen, S. L. (1996). Graphical models. Oxford University Press. Lauritzen, S. L., & Wermuth, N. (1989). Graphical models for associations between variables, some of which are quantitative and some qualitative. Annals of Statistics, 17, 31–57. Mare, D. S. (2012). Contribution of macroeconomic factors to the prediction of small bank failures. Available at SSRN 2050029. Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 2, 449–471. Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Mateo, CA, USA: Morgan Kaufmann publishers. Peltonen, T., Piloiou, A. Sarlin, P. (2015). Network linkagesto predict bank distress. Technical report. Segoviano, M. A., & Goodhart, C. A. (2009). Banking stability measures. IMF Working Papers, 09(04). Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3), 425–442. Sinkey, J. F. (1975). A multivariate statistical analysis of the characteristics of problem banks. The Journal of Finance, 30(1), 21–36. Tam, K. Y., & Kiang, M. Y. (1992). Managerial applications of neural networks: The case of bank failure predictions. Management Science, 38(7), 926–947. Vasicek, O. A. (1984). Credit valuation. KMV corporation. Wermuth, N., & Lauritzen, S. L. (1990). On substantive research hypotheses, conditional independence graphs and graphical chain models (with discussion). Journal of the Royal Statistical Society, Series B, 52(1), 21–50. Whittaker, J. (1990). Graphical models in applied multivariate statistics. Chichester: John Wiley.

Please cite this article as: P. Cerchiello, P. Giudici, Conditional graphical models for systemic risk estimation, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.08.047

592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668

Q11 Q12

Q13

Q14

Q15

Q16

Q17