Contagion Analysis in the Banking Sector

This paper analyses how an external adverse shock will impact the financial situations of banks and insurance companies and how it will diffuse among these companies. In particular we explain how to disentangle the direct and indirect (contagion) effects of such a shock, how to exhibit the contagion network and how to detect the "superspreaders", i.e. the most important firms involved in the contagion process. This method is applied to a network of 8 large European banks in order to analyze whether the revealed interconnections within these banks differ depending on the underlying measure of banks’ financial positions, namely their market capitalization, the price of the CDS contract written on their default and their book value.


Introduction
The new regulation on financial stability lists global systematically important financial institutions (G-SIFIs), which have to comply with specific regulatory requirements. One of the criteria for a bank to be identified as systematically important is its interconnectedness. In this respect our paper studies how an external adverse shock will impact the financial situations of the banks and insurance companies, and how it will diffuse among these companies. In particular we explain how to disentangle the direct and indirect (contagion) effects of such a shock, how to exhibit the contagion network and how to detect the most important firms involved in the contagion process, that are the "superspreaders", especially the institutions, which are "too interconnected to fail". Such an analysis depends crucially on the way the financial situation (vulnerability) of a bank or an insurance company is measured. In the academic literature as well as in the approaches used to implement Basel regulation, this financial situation (vulnerability) is analyzed in three alternative ways: i) by considering the balance sheets of the firms and typically their accounting equity value defined as the difference between their assets and their liabilities in the firm's book; ii) by analyzing their market values when they are quoted on a stock exchange; iii) by focusing on the information on their potential defaults by means of the prices of their issued bonds, or of the associated Credit Default Swaps (CDS) 4 .
There exist links between these three approaches which have been first mentioned by Merton [Merton (1974)]. Let us denote V t = A t − L t the value of the firm at date t, where A t and L t are the asset and liability components of the balance sheet, respectively. Then, under standard regularity conditions, the firm's market capitalization (or market value) at date t is equal to: 1) and the price of the short-term digital CDS which protects the CDS's holder from a firm's default over the next period of time is: where r t is the short term riskfree rate, Q t the risk-neutral distribution conditional on the information available at date t, V + = max(V, 0), and E Q t the expectation with respect to Q t . Equation (1.1) corresponds to the Merton's interpretation of a stock as a European call written on the asset component with a strike equal to the liability.
The capitalization, the book value of the firm and a CDS price are different notions as are also their rates of change. Typically the stock return Vt − 1, and from CDS t+1 CDSt − 1. To better understand these differences, let us assume a zero riskfree rate r t = 0, a risk-neutral distribution equal to the historical distribution and a latent value of the firm which can be decomposed as: where the shocks u t+1 are independent standard normal variables ∀ t, µ t , σ t the conditional mean and standard error, respectively. Then we get: where ψ(u) = uΦ(u) + ϕ(u), and ϕ, Φ are the probability and cumulative density functions of the standard normal, respectively. This simplified example shows clearly that the three notions capture the first and second-order conditional moments of the underlying risks in the balance sheet in different ways. The CDS price depends on the mean/standard error ratio only after a nonlinear transform, the capitalization on mean and standard error in a more complicated way, while the ex-post observation V + t+1 involves an additional stochastic shock u t+1 .
The aim of our paper is to compare the analysis of contagion based on the rate of changes on the value of the firm, on the capitalization and on the transformed (or standardized) CDS price 5 Φ −1 (CDS t ), respectively. In Section 2 we introduce a linear dynamic model for a joint analysis of rates of changes for several firms. This model allows both for common unobservable exogenous shocks and for contagion phenomena. We study the second-order properties of such models. We develop in Section 3 an approach to estimate the number of underlying factors and the sensitivities of the financial situations of the firms to these factors. The approach is applied to a set of banks and to the three alternative measures of their financial situations. Section 4 describes different estimation methods of the parameters of interest including the contagion matrix. We compare the structure of the estimated contagion matrices according to the selected measures of financial situations. Section 5 concludes. Proofs and additional informations are gathered in Appendices.
2 Dynamic factor model with dynamic frailty and contagion 2.1 The model Let us consider n institutions, and stack the variables of interest [which can be either the rate of change in the book value, in the market capitalization, or in the standardized digital CDS prices of the credit institutions on period (t − 1, t)] in a n-dimensional vector Y t . These variables are preliminary demeaned and satisfying 5 To ensure a transformed CDS price in the same domain of variation as capitalization or the observed value of the firm.
the following dynamic factor model: where the error terms u t , v t are zero-mean, serially independent, with second-order moments: (2.2) F t gives the values at date t of K unobservable factors, called dynamic frailties in the credit risk literature [see Duffie, Eckner, Horel, Saita (2009)]. The second subsystem of (2.1) and the noncorrelation condition (2.2) means that these factors (frailties) have a (strong) exogenous dynamics. This type of models is known under the acronym of FAVAR for factor augmented vector autoregressive [see Bernanke, Boivin, Eliasz (2005)], but includes in our case parameter restrictions for factor exogeneity.
The first subsystem shows that the variables of interest (either the rate of changes on the value of the firm, or on the capitalization, or on the transformed CDS price) are dynamically dependent through the effects of the common exogenous factors, measured by the (n, K) matrix of beta coefficients, and through the effects of their lagged value, measured by the (n, n) contagion matrix C. For expository purpose we have not introduced an intercept in the return (resp. changes in standardized price) equation. Indeed, whenever the process (Y t ) is stationary, this intercept can be set to zero by considering the demeaned return (or demeaned change in price) Y t −Ȳ and after this transformation, we can assume E(F t ) = 0. The joint process (Y t , F t ) is asymptotically stationary if both the eigenvalues of C and Φ have a modulus strictly smaller than one (see Appendix 2, Lemma 1).
We can find in the literature on systemic risk special cases of dynamic model (2.1), with either common factor only, or contagion only. We review in Table A.1 in Appendix 1 this literature distinguishing the models with factor only, either observed, or unobserved, and the models with contagion only. We also mention if they are applied to balance sheet data, stock data, bond data, or CDS data.
In this paper, we focus on the contagion matrix C for different indicators of institutions' financial vulnerability (i.e. either its book value, its market value, or its CDS standardized price). We prefer to let unobservable the factors F t to prevent a bad selection of observable factors from contaminating the estimation of matrix C. Moreover, model (2.1) considers jointly the return and exogenous factor dynamics. This approach is particularly useful for the purpose of predictions, risk measures, or stress-tests. Indeed, it takes into account the uncertainty on the future values of the factors as well as the implied dependence between future returns (resp. changes in standardized prices) due to the factors 6 .
As usual the latent factors are defined up to an invertible linear transformation. Thus we introduce identification restrictions on the parameters.
Proposition 1 (Identification restrictions): Without loss of generality, we can assume either IR1: Ω = Id, or IR2: B B = Id, if B is with full column rank K. Moreover the autoregressive matrix Φ can be chosen triangular, and a sign convention on the columns of B can be fixed.
Proof: The first identification restriction is obtained by the change of factor F → QF with Q = Ω −1/2 , where Ω −1/2 denotes the inverse of the square root of the symmetric positive definite matrix Ω. The second identification restriction with Q = (B B) −1/2 . Finally, we can also apply an orthogonal matrix Q, that is a change of orthonormal basis to get a triangular Φ matrix by the Gram-Schmidt process.
QED 6 The literature on factor models considers frequently observable factors, such as for instance a market return, the inflation rate ... (see Table A.1 in Appendix 1). Observable factors are often treated in a misleading way, when we are interested in prediction, risk measures, or stress-tests. Theoretically, the prediction of these future values requires a dynamic model (as the second subsystem of (2.1)), but very often in practice they are predicted by deterministic scenarios. Moreover it is not checked in general that these observable factors are exogenous, while the exogeneity condition is required for the right interpretation of matrix C as capturing all the contagion.
The second-identification restriction IR 2 shows that what matters is not the matrix B itself, but more the vector space spanned by BF t when F t varies. The condition B B = Id means that the columns of B can be chosen as an orthonormal basis of this vector space with dimension K.
Finally, a significant part of the literature on contagion considers that the returns are uncorrelated with their own past and analyzes the structure of conditional heteroscedasticity. By model (2.1), we follow the opposite approach focusing on the expression of expected returns and assuming conditional homoscedasticity. In particular, we expect B and C to be significant. The main reason for this significance is the specificity of financial institutions. The asset components of their balance sheets can be seen as portfolios of basic assets. Loosely speaking we have: where j is the index of the basic asset, p jt its price, and a jt its quantity. If the portfolio allocation is crystallized a jt = a j , ∀t, and if the returns on the basic asset are i.i.d, the return on J j=1 a j p j,t will also be close to i.i.d. But the role of a financial institution is to update frequently its portfolio with observed prices, that is, the allocations a j,t are functions of current and past prices. This allocation adjustment will destroy the i.i.d. property of the changes in asset value, in particular their serial independence, even if this property is satisfied on the basic assets.

Marginal dynamics of returns
Let us consider the identification restriction IR2: B B = Id. By premultiplying both sides of the first subsystem (2.1) by B , we get: and then by substitution in the second subsystem: Since the combination u t + Bv t − BΦB Cu t−1 can be written as a vector moving average w t − Θw t−1 , say, we deduce that the dynamic of (Y t ) alone can be written as a VARMA(2,1) model, with restriction on the two matrix autoregressive coefficients.

State space representation
Model (2.1)-(2.2) is a Vector AutoRegressive (VAR) model with partial observability. This model admits a state space representation. More precisely let us introduce the state variable Z t = (Y t , F t ) . We have: Thus the linear Kalman filter can be used to compute recursively the linear predictions of future values of Y , the filtered values of the unobservable factors as well as the values of a Gaussian pseudo-likelihood function [see e.g. Reinsel (1993), Section , Gourieroux, Monfort (1997), Section ].

Second-order properties
By considering the VAR dynamics of state process (Z t ), we can deduce its first and second-order moments of the joint process (Y t , F t ) (see Appendix 2). In particular, the autocovariance of the observable process depends on the lag in the way given in Proposition 2.
Proposition 2: The second-order properties of (Y t , F t ) are the following: i) The autocovariance function Γ Y (h) of the observable process Y is equal to: ii) The unconditional covariance between the observable process Y and the frailty Cov(Y, F ) is solution of: By Proposition 2.i), we get the standard component C h Γ Y (0) for a VAR dynamic for the observable process (Y t ) with autoregressive matrix C plus the term D h Cov(F, Y ) due to the unobservable factor. D h is a rather complicated function of h. Indeed, the unobservability of F implies a VAR(∞) dynamic with an infinite autoregressive lag, when process (Y t ) is considered alone.
The unconditional covariance between Y and F is solution of a system of Riccati equations, which in general has to be solved numerically. As seen below, the different formulas are greatly simplified for a single factor model.

The single factor model
Let us consider the case K = 1 and the model: We get: We deduce: Then the expressions of the unconditional variances and covariance become (see Appendix 2):

Comparing datasets with different frequencies
The indicators of banks' financial vulnerabilities are not necessarily be available at the same frequency. In particular, indicators based on accounting data are accessible at a lower frequency than the market based ones. To facilitate the comparison with the analysis on market data, we can consider the daily model (2.1) for the returns on book values. Due to the state space representation, the model's form remain unchanged at lower frequency: say, with h = 60 opening days at quarterly frequency.

The impulse response
Let us finally consider the consequence of a temporary shock on the exogenous systematic factor set at date t: δ ≡ dv t . By Lemma 1 in Appendix 2, the values Y t+h , F t+h at horizon h will be modified as: In particular: dY t+h = D h δ. The matrix multiplier D h can be decomposed as: The first component on the right hand side is the direct effect of the exogenous systemic shock, whereas the second component gives the indirect effect due to contagion.

Estimation of the number of factors and of their effects
In a static factor model without contagion (C = 0) and no dynamic (ϕ = 0), the number of factors, the factors and the beta coefficients are usually obtained by applying a principal component analysis, based on the spectral decomposition of the historical variance-covariance matrix of the observable variables. We use a similar approach valid for model (2.1) with both contagion and factor dynamics. The new approach allows for the estimation of the number of factors and of the betas, but does not provide approximations of the factor themselves. This approach is based on the notion of directions immunized to shocks on the factors.
As mentioned in Section 2.1, the possible effects of the factors on the vector of returns (resp. changes in standardized price) belong to the space E(B) generated by the columns of matrix B.
Let us now consider an element γ of the space orthogonal to E(B). We get: These directions define portfolio allocations, which are immunized against the latent common factors. By considering these directions, we also eliminate the effect on Y t of lagged values of Y with a lag larger or equal to 2.
Let us now consider the linear regression of γ Y t on both Y t−1 , Y t−2 : The theoretical regression coefficients are: We deduce that: by inverting by blocks.
Additional Restriction AR: Rank C 2 = K.
Then we get the following property.
Proposition 3: Under the additional restriction AR the immunizing vectors γ ∈ E(B) ⊥ are the vectors of the kernel of C 2 .
Corollary 1: For any positive definite (n, n) matrix S, the matrix C 2 SC 2 admits exactly n − K eigenvalues equal to zero. A basis of orthonormal eigenvectors of this matrix is such that the eigenvectors associated with the non zero eigenvalues are the column vectors of a B matrix , which satisfies BB = Id, and the eigenvectors associated with the zero eigenvalues are immunizing vectors.
We deduce from Corollary 1 consistent estimation methods for the number of factors K, a matrix B of beta coefficients and a basis of immunizing vectors. They follow the steps below.
step 1 : Compute the estimated multivariate partial autocovariance of order 2 by substituting in the expression of C 2 the autocovariances by their sample coun-terpartsĈ 2 .
step 2 : Select a metric S and perform the spectral decomposition ofĈ 2 SĈ 2 , with the eigenvalues written in a decreasing order.
step 3 : Estimate K by the first-orderK for which the eigenvalues ofĈ 2 SĈ 2 are non significant.
step 4 : Estimate B by considering the firstK orthonormal eigenvectors as columns ofB.
step 5 : Estimate a basis of immunizing vectors by considering the next n −K orthonormal eigenvectors.
Such an approach is the analogue for Vector Autoregressive process of the analysis of codependence directions introduced for Vector Moving Average processes by Gourieroux, Peaucelle (1983), (1993) 7 .
There exist several possible choices of the metric S.
we get: with a simple interpretation of the test statistics as a multivariate estimated partial autocorrelation.
ii) It is also possible to choose S in order to get some optimality properties of some test statistics based onĈ 2 SĈ 2 , such as the sum of its eigenvalues ξ 1 = T r(Ĉ 2 SĈ 2 ), or ξ 3 = largest eigenvalue ofĈ 2 SĈ 2 [see Gourieroux, Monfort, Renault (1999), for a discussion].
In fact the additional restriction Rank C 2 = K could be tested and be the basis of a pretest procedure. Typically, when the additional restriction is rejected, the approach above can be extended by increasing the lag in autoregression (3.1) defining the immunizing directions. For a given lag p, we will estimate the partial autocovariance: and look for its kernel.

Estimation of the parameters for a given number of factors
The approach of Section 3 provides neither estimates of the contagion matrix C, nor the factor dynamics characterized by Φ and Σ, nor filtered factors. We provide below estimation methods of all parameters, once the number of factors is known.

Pseudo-maximum likelihood
As already noted in Section 2.2 the Gaussian pseudo-likelihood function is easy to compute numerically by applying the linear Kalman filter to the state space representation (2.3)-(2.4). It is also easy to compute numerically the pseudo-maximum likelihood estimates by maximizing this function. To avoid identification problems, this optimization has to be done under the identification restriction: which is easier to take into account in the optimization problem than the second identification restriction: BB = Id, Φ triangular (IR2). Note that the Gaussian pseudo likelihood function under IR2 is exactly the Gaussian likelihood function of the constrained VARMA(2,1) model derived in Section 2.2.

Asymptotic Least Squares
An alternative consistent estimation method can be based on the moment restrictions given in Proposition 2 i). Let us for instance consider the two first restrictions written for h = 1, 2. We get: (4.1) If Cov(F, Y ) is let free, we get 2n 2 restrictions to find the parameters C, B, Φ, Cov(F, Y ), that include n 2 + nK + K(K + 1)/2 + nK = n 2 + 2nK + K(K + 1)/2 independent parameters. The order condition for identification is satisfied if: 2n 2 > n 2 + 2nK + K(K + 1)/2 ⇐⇒ n 2 − 2nK > K(K + 1)/2 Under this order restriction we can apply an asymptotic least-squares approach to estimate C, B, Φ, and Θ = Cov(F, Y ). The ALS estimates are solutions of the optimization problem: where T r denotes the trace operator. The criterion above is written without weighting for expository purpose and Θ plays the role of a nuisance parameter in the optimization above. Indeed the parameters of interest are the autoregressive parameters C, B, Φ. The last parameter Σ can then be deduced from the historical estimate of V (Y ) by using the expression of V (Y ) given in Lemma 2 in Appendix 2.i). Note finally that the first subsystem in (4.1) can also be written as: C * is the standard estimator of the contagion matrix for a model without exogenous factor, that is with B = 0. Thus the omission of the exogenous factor will generally induce a bias on the estimated contagion matrix. For instance in a model with a single factor with positive effects on the returns: there is an overestimation of the contagion matrix whenever ϕ > 0.

Application to the financial system
In our application, we focus on 8 of the biggest financial institutions in the euro area, namely: Banco Santander SA, BNP Paribas SA, Commerzbank AG, Crédit Agricole SA, Deutsche Bank AG, Intesa San Paolo SPA, Société Générale SA, and Unicredit SPA. Thus, Germany, France, Italy and Spain are represented in our sample, which allows us to investigate both intra-and international contagion among banks in the euro area.
We convert stock prices and equity values in order to make them comparable with CDS prices, which are quoted in US Dollar. More precisely, we consider the variable Y i,t either as: • the rate of change in the market capitalization of institution i at time t, i.e.
, where Cap i,t stands for the market capitalization of institution i in USD at time t; • the variation in the standardized CDS price for institution i at time t, i.e.
• the rate of change in the equity value of institution i at time t, i .e.
, where Equ i,t stands for the book value of institution i in USD at time t. For this application, we define banks' equity value as their Tier 1 capital 8 .
For each approach, the dependent variable Y t gathers the 8 individual variables Y i,t . Data on institutions' market capitalization and CDS come from Bloomberg, while data on the banks' equity value are obtained from Bankscope. Our sample starts in January 2, 2007, and ends on December 20, 2012. In this section, the application builds on quarterly data. An extension to daily data (which are available for CDS prices and market capitalization) is presented in Appendix 3.

Estimation of the number of factors
For all datasets, the decrease in the series of eigenvalues shows a clear break after the first eigenvalue, which suggests a number of factor equal to one for all types of data, even if the second eigenvalue is almost significant for the changes in book values.

Estimation of the parameters
We provide in Table 1 the results of the estimation of the contagion matrix C by Gaussian pseudo-maximum likelihood for the 3 different datasets, setting the number of factors to K = 1. We observe a total number of significant connections equal to 42 for the capitalization, 54 for the CDS prices and 50 for the accounting data, to be compared with 64 possible connections. Thus this system of banks is highly interconnected. The rather large number of connections on returns show that both the efficient market hypothesis and the standard two funds theorem are not satisfied. As announced before we have first to focus on the conditional mean before considering conditional variance for financial institutions. However the different banks do not play the same role. For instance, the return on Commerzbank's stocks depends neither on its lagged return, nor on the return of the other banks once the effect of the common factor has been taken into account. On the other hand, the Commerzbank has an effect on all the other banks. Another extreme example is Société Générale, which is affected by all the other banks for the three different series, and affects all the other banks, except Commerzbank, for the changes in standardized CDS prices and book values. The estimated latent factor, or dynamic frailty, is presented in Figure 2 for the three datasets. The three dynamic frailties feature negative autocorrelation (see Table 2) and share a certain degree of commonality, in particular between the market-based frailties. This is formally emphasized in Table 3 and 4, which present the correlations between the three frailties, as well as the factor loadings obtained from a principal component analysis on the 3 factors. Table 4 in particular highlights the existence of a common factor, which loads uniformly on the three frailties and explains about 80% of their overall variation. The second factor distinguishes market-based indicators from accounting-based one, whereas the last factor mainly discriminates the market capitalization frailty from the CDS one. However, we see on Figure     As mentioned before, the factor are defined up to some multiplicative scalar. The factors in Figure 2 have been constructed to be comparable in magnitude (same mean and variance), but also for their interpretation in terms of risk. This is easily seen on Table 6 which provides the factor loading: an increase in the factor value implies a decrease in the stock returns, in CDS prices or book values, respectively. As expected from financial theory the common factor affects all stock returns with the largest influence on the Commerzbank. This compensates the absence of additional effect of the lagged return already discussed. However the interpretation of the frailty as a common factor is no longer valid when we consider the analysis based on either CDS price or book values, since in both cases only one beta is significant, for BNPP and Commerzbank, respectively. The interpretation of the significant beta is now different. If all betas were equal to 0, the dynamic model would be a VAR(1) model, with short memory features. The presence of significant beta introduces longer memory, and this longer memory is channeled through a single bank (either Commerzbank or BNPP).   Table 6: Matrix of loadings to the frailty factor obtained from banks' market capitalization, CDS prices or accounting data ( * significant at 10%, * * significant at 5%).

Concluding remarks
The aim of this paper is to explain how to disentangle the direct and indirect (contagion) effects of the exogenous shocks on the financial situations of the banks or insurance companies. For this purpose we have considered a linear dynamic model with both common frailty and autoregressive feature, and introduce an appropriate methodology to estimate the contagion matrix, the sensitivity of institutions to the factors and to reconstitute the underlying factor paths. Our method has been applied to a set of eight banks and different measures of their financial position, measured by the change in their market capitalization, CDS prices or book values, respectively. Even if we might expect similar results for the three different measures, the analysis shows that the revealed interconnections are significantly different, in particular between market data and accounting data. In practice, the European banking system contains much more than 8 banks and the extension of the methodology will require methods for large scale factor models based on sparse estimator or Lasso [see e.g. Barigozzi, Brownlees (2013) for such an approach in a model without frailty].

Second-order properties
By considering the VaR dynamics of process (Z t ) given in (2.3), we get: These expressions can be used to derive the second-order properties of the observable process (Y t ).
Lemma 1: We have: In particular system (2.1) is stable iff the eigenvalues of Φ and C have a modulus strictly smaller than one.

QED
Lemma 2: We have: where: Proof: We have: By identification we deduce the recursive system:

QED
These lemmas are used to deduce equations satisfied by the autocovariance function of the observable process (Y t ). Indeed we have: By considering the first block diagonal element of this product, we get the recursion in Proposition 2. Table 8: Contagion matrix obtained from CDS prices ( * significant at 10%, * * significant at 5%), at daily frequency.
As expected, we get a smaller number of interconnections, that is a total of 5 for market capitalization and 3 for the CDS prices. These small numbers of connections can provide much larger numbers when the frequency of the observations diminishes, due to the effect of time aggregation on the contagion matrix discussed in Section 2.5, that is the transformation C → C h = C 60 .        Table 12: Loadings to the frailty factor ( * significant at 10%, * * significant at 5%) As in the case of the quarterly analysis the first factors show a significantly different dynamics. In particular the factor for the capitalization features more important periods of high volatility both for the magnitude of the volatility and the length of the period, especially during the 2008 financial crisis.