Modelling systemic risk in financial markets

TESIS DE DOCTORADO Modelling systemic risk in financial markets Andrea Ugolini PROGRAMA DE DOCTORADO EN ECONOMÍA FACULTAD DE CIENCIAS ECONÓMICAS Y EMPRESARIALES Santiago de Compostela 2014 TESIS DE DOCTORADO Modelling systemic risk in financial markets Asdo. Andrea Ugolini PROGRAMA DE DOCTORADO EN ECONOMÍA FACULTAD DE CIENCIAS ECONÓMICAS Y EMPRESARIALES Santiago de Compostela September 2014 AUTORIZACIÓN DO DIRECTOR/TUTOR DA TESI D. Juan Carlos Reboredo Nogueira profesor do Departamento de Fundamentos de Análise Económica, como Director da Tese de Doutoramento titulada «Modelling systemic risk in financial markets» Presentada por D. Andrea Ugolini, alumno do Programa de Doutoramento en Economía: Autorizo a presentación da tese indicada, cosiderando que reúne os requisitos esixidos no artigo 34 do regulamento de Estudos de Doutoramento, e que como Director da mesma non incurre nas causas de abstención establecidas na lei 30/1992 Asdo. Acknowledgements Acknowledgements Giving thanks at the end of a personal and professional journey is an act of humility, to recognize the people that have helped me to achieve my goal. In this space, I have the opportunity to express my gratitude to people dearest to me, and thank them for their support throughout the course of this thesis. First of all, I wish to express my profound gratitude to Professor Dr. D. Juan Carlos Reboredo Nogueira, without his expert guidance and help this thesis would not have been possible. I wish thank you for your openness and encouragement. It was a pleasure and an honour to work with such an inspirational, supportive, and insightful mentor. Your teachings sent me on scientific journeys of discovery and, on a personal level, how to live, love and deal with the world of research. I hope that this collaboration and friendship will endure for a long time. I would not have contemplated studying abroad without the help and support of my dearest parents, Vincenzo and Lucia. Your love, encouragement and guidance throughout my life, has brought me to this stage of my life, culminating in the completion of this thesis. Without you, this thesis would not have be possible. You have made and continue to make sacrifices, so that I can realise my ambitions. For that, I will be eternally grateful. I could not be prouder to be you son. To my Grandparents, Giuseppe and Teresa, thank you for your teachings and blessings that you have been giving me since I was young. You're my source of wisdom. You always refill my heart with joy when I return home. To my dearest aunt Angela, for being like a sister to me, thank you for your support. A very special thanks to two people, who will forever remain close to my heart, aunt Elsa and uncle Marziano, for the love and confidence you gave me. Uncle Marziano, thank you for your phrase, which has been a source of inspiration through difficult times, "Ugo! you can do anything if you want it". To my best friends, Andrea and Daniele, thank you for being awesome friends. Thank you for being there to listen. It's nice to know that even as time passes and the distance grows, interests changes our beautiful friendship will always remain. I cannot forget to say thank you Daniel, Diana, Djamal, Fernando, Jesus, Maria, Óscar and Rafael. You were the perfect traveling companions, good colleagues and great friends. Fate could not have amassed a better group of people. Acknowledgements To Marli, my girlfriend, who has always believed in me, motivated me, inspired me and supported me. She has withstood all the ups and downs with me. Without you, this thesis would not have been possible. You are a wonderful and incredible life partner, to whom I will be grateful forever. Finally, thanks to the whole who of a form of other one has contributed to my enrichment as person. I Contents Introduction ............................................................................................................. 1 1. Literature review ............................................................................................... 5 1.1. Systemic risk measures ............................................................................... 7 1.1.1. Mahalanobis distance .......................................................................... 7 1.1.2. Multivariate density estimators ........................................................... 7 1.1.3. Conditional value at risk ..................................................................... 8 1.1.4. Co-risk ................................................................................................ 8 1.1.5. Systemic and marginal expected shortfalls ........................................... 9 1.1.6. The default intensity model ............................................................... 10 1.1.7. Distress insurance premium ............................................................... 11 1.1.8. Broader hedge fund-based systemic risk measures ............................. 11 1.1.9. Granger causality tests ...................................................................... 12 1.1.10. Simulating a credit scenario ........................................................... 12 1.1.11. GDP stress test .............................................................................. 13 2. Measuring systemic risk in the Spanish financial system: A CoVaR approach . 15 2.1. Introduction ............................................................................................. 15 List of tables VIII Table 4.5: Marginal distribution model. Parameter estimates for MSCI financial index returns by country. ...................................................................... 111 Table 4.6: CoVaR results. ..................................................................................... 123 Table 4.7: Significant test for differences in risk measures. .................................... 131 1 Introduction Three recent crises — the dot-com bubble and the subprime and European sovereign debt crises — have revealed the complex dynamics underpinning the global financial system and how rapidly risk is propagated across markets. Investors, regulators and researchers are thus keen to develop accurate measures of risk transmission between assets and markets. From the investors’ perspective of guaranteeing efficient portfolio diversification, the risk of contagion is essential for their ongoing interest in changes in market linkages. From the regulators’ point of view, risk spillover is important to focalize attention on the maintenance and development of new financial regulatory and institutional rules such as circuit breakers, transaction taxes and short-sale rules. In fact, recognizing important shortcomings in financial supervision, the European Commission and European Central Bank (ECB) created the European Systemic Risk Board (ESRB) at the end of 2010 with the goal of monitoring, at the macro-prudential level, the European financial system and preventing and mitigating any propagation risk within the financial system. The literature contains many definitions of systemic risk. De Bandt and Hartmann (2000) defined it “as the risk of experiencing systemic events in the strong sense” where “strong sense” signifies the spread of news about an institution that has an adverse impact on one or more healthy institutions in a sequential manner. Furfine (2003) distinguished between two kinds of systemic risk: “The first type is the risk that some financial shock causes a set of markets or institutions to simultaneously fail to function efficiently. The second type of systemic is the risk that failure of one or a small number of institutions will be transmitted to others due to explicit financial linkages across institutions”. On the basis of the theoretical model proposed by Diamond and Dybvig (1983), Acharya (2009) defined systemic risk “as the joint failure risk arising from correlation of returns on the asset-side of Introduction 2 bank balance-sheets”. Billio et al. (2010) explained that “systemic risk can be realized as a series of correlated defaults among financial institutions, occurring over a short time span and triggering a withdrawal of liquidity and widespread loss of confidence in the financial system as a whole”. The last two definitions introduce the notion that common investment between banks generates correlation and “herding effects”, thereby generating systemic risk. The International Monetary Fund (IMF), Bank for International Settlements (BIS), Financial Stability Board (FSB) and the ECB focus their attention particularly on the consequences of systemic risk for the real economy. Thus the IMF, BIS, FSB (2009) stated systemic risk to be “the disruption to the flow of financial services that (1) is caused by an impairment of all or parts of the financial system; and (2) has the potential to have serious negative consequence for the real economy”. The ECB (2009) conceptualized systemic risk as a “risk that financial instability becomes so widespread that it impairs the functioning of the financial system to the point where economic growth and welfare suffer materially”. The above brief list of definitions points to the intricacy of the topic and the challenge faced by investors, regulators and researchers in attempting to measure the complexity and dynamics of systemic risk. Using the CoVaR systemic risk measure (Adrian and Brunnermeier, 2011; Girardi and Ergün, 2013), we quantified systemic risk as the impact of the risky situation of a particular financial institution, market or system on the value-at-risk (VaR) of other financial institutions, markets or systems. Our research objectives — potentially of interest to investors, regulators and researchers in equal measure — were as follows: 1. To quantify systemic risk for Spanish financial institution and to account for the quantitative effect on the conditional VaR value with a view to assessing how the fragile position of one particular financial institution could impair the performance of other financial institutions and to determine how much regulatory capital a financial institution would need in order to cover its exposure to this kind of risk. 2. To examine systemic risk in European sovereign debt markets and assess how this risk changed with the onset of the recent European sovereign debt crisis with a view to determining how the deteriorated financial position of a particular sovereign debt market can impair the performance of other sovereign debt markets. Introduction 3 3. To measure the systemic impact of domestic sovereign debt distress on domestic financial systems in Europe and of a potentially distressed Greek debt market on the financial systems of other countries during the recent financial and debt crises with a view to understanding the impact of domestic sovereign distress on a domestic financial system and the impact of Greek sovereign debt distress on other financial systems. The dissertation is laid out as follows: In Chapter 1 we provide a survey of the quantitative measure of systemic risk in the economics and finance literature. In Chapter 2 we examine, using conditional VaR (CoVaR), the systemic risk generated by major Spanish financial institutions in the recent global financial crisis and the European sovereign debt crisis as a systemic risk measure. CoVaR was quantified using quantile regression, multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) and copula approaches. We also describe a novel copula-based approach to computing the CoVaR value, given that copula are flexible modellers of joint distribution and are particularly useful for characterizing the tail behaviour that provides such crucial information for the CoVaR. We found significant increases in systemic risk around the time of the recent global financial crisis and, to a lesser extent, around the time of the European debt crisis. Our evidence also shows that the quantile regression approach was unable to reflect the dynamics of, and sudden changes in, systemic risk. These results have implications for capital regulation in financial institutions and on how systemic risk should be measured. In chapter 3 we study systemic risk in European sovereign debt markets before and after the onset of the Greek debt crisis, taking, as a systemic risk measure, the CoVaR as characterized and computed using copulas. We found sovereign debt markets to be coupled before the debt crisis and systemic risk to be similar for all countries. With the onset of the Greek crisis, debt markets decoupled and the systemic risk of the countries in crisis (excepting Spain) decreased whereas that of the non-crisis countries increased slightly. The systemic risk of the Greek debt market increased for other countries in crisis, especially for Portugal (where systemic risk tripled after the onset of the crisis) and decreased for non-crisis countries. In Chapter 4 we investigated — using the CoVaR measure as characterized and computed using copulas and vine copulas — systemic sovereign debt distress in European domestic financial systems and the systemic risk of a potentially distressed Introduction 4 Greek debt market for other European financial systems countries before and after the onset of the recent financial and debt crises. We found that, before the debt crisis, sovereign debt had a positive systemic risk on European domestic financial systems. However, with the onset of the Greek crisis, the systemic impact of sovereign debt increased for countries in crisis (Greece, Italy and Portugal) whereas it remained stable or reduced for non-crisis countries. Regarding the systemic impact of sovereign Greek debt distress, our evidence indicates that negative impacts were limited to a small set of countries (Belgium, Italy, the Netherlands and Portugal). The dissertation results are described in three research articles with following titles: 1. A CoVaR approach to systemic risk in the Spanish financial system 2. A CoVaR-copula approach to systemic risk in European sovereign debt markets 3. A vine copula-CoVaR approach to systemic sovereign debt risk for the financial sector. The realization of this results was only feasible by accessing to database as Bloomberg or Datastream. Furthermore, with the use of computer programs as: Matlab, R-Project, Eviews and OxMetrics. 5 Chapter 1 1. Literature review Over the last years, researchers have developed a number of systemic risk measures referring to different systemic risk propagation channels. Three main measures can be identified: (1) measures of systemic risk that capture the contagion and exposure effect between institutions; (2) measures of systemic risk that quantify the trigger effect between the financial sector and the real economy; and (3) measures of systemic risk between the financial and public sectors and vice versa. The first category refers to the risk of a failure of one financial institution having a contagion or domino effect on other institutions through the transactions and interconnections linking these institutions. Many researchers have focused their attention on this kind of risk propagation. Segoviano and Goodhart (2009) created a banking stability index that assesses interbank dependence for tail events. Acharya et al. (2010) used systemic and marginal expected shortfall (ES) measures to quantify downside risk and the contribution of individual financial institutions to risk. Allen et al. (2010) proposed a measure of aggregate systemic risk — called CATFIN — to predict declines in aggregate bank lending activity six months in advance. Huang, Zhou and Zhu (2009, 2010, 2012) proposed their distress insurance premium (DIP) measure. Adrian and Brunnermeier (2011) proposed using conditional value-at-risk (CoVaR) to capture possible risk spillovers between financial institutions. Likewise, Brownless and Engle (2012) developed a systemic risk measure, called SRISK, to represent the amount of capital needed to restore minimum capital requirements. Billio et al. (2012) proposed five measures of Chapter 1 6 systemic risk to capture contagion and exposure effects in the relationship between financial institutions. Girardi and Ergün (2013) proposed a new approach to quantifying CoVaR using the joint density for the financial system and financial institution returns. Finally, Halaj et al. (2013) suggested a simple network analysis measure, called the systemic probability index (SPI). Regarding measures of the trigger effect, several authors have developed a systemic risk measure based on interdependence between the financial sector and the real economy. Reinhart and Rogoff (2009a) showed that systemic risk in financial markets increases in crisis periods and has adverse effects that extend to the real economy. Giesecke and Kim (2011) developed the default intensity model (DIM) to capture spillover effects through the complex network of relationships with the real economy. De Nicolò and Lucchetta (2010) proposed a GDP-at-risk model to quantify the impact between the macro-economy, the financial markets and intermediaries. Finally, regarding systemic risk generated between the financial and public sectors, Reinhart and Rogoff (2009b, 2010) documented sovereign distress spread to the financial system when banks held a substantial amount of government debt in their portfolio. Alter and Schuler (2012) examined the relationship between sovereign default risk and domestic banks. Mink and De Haan (2013) analysed the impact of highly volatile Greek bonds on European bank stock prices in 2010 and De Bruyckere et al. (2013) studied contagion between banking and sovereign default risk in Europe through asset, collateral and rating channels. Bhanot et al. (2014) investigated the impact of changes in Greek sovereign yield spreads on stock returns in the financial sector. Battistini et al. (2014) demonstrated that the sovereign debt portfolios of European banks revealed growing home bias during the recent crisis, with domestic sovereign debt holdings growing in line with sovereign solvency risk. Finally, alternative risk measures have been proposed, other than those included in the three categories mentioned above. Engle and Manganelli (2004) developed their conditional autoregressive value at risk (CAViaR) model that uses quantile regression to capture the tail behaviour of returns. De Jonghe (2010) used extreme value theory to measure banks’ systemic risk exposure. Zhou (2010) used multivariate extreme value theory to quantify systemic risk, analysing the relationship between institution size and systemic importance. Finally, Krizman et al. (2011) developed a measure of systemic risk called the absorption ratio that relies on principal component analysis (PCA). Literature review 7 1.1. Systemic risk measures Below we describe the methodologies used to quantify systemic risk measures. In the selection of the systemic risk measures, we would trace briefly all aspects that the literature aims to address. 1.1.1. Mahalanobis distance Kritzman and Li (2010) defined “financial turbulence” as the statistical unusualness of a set of returns given historical behaviour patterns, including extreme price movements, decoupling of correlated assets and convergence of uncorrelated assets. To quantify turbulence they used the Mahalonabis distance (Mahalanobis, 1927). Given the returns for a particular period of n assets, the turbulence index was formally defined as:     1 ' t t t d y y      , (1.1) where: t d = turbulence at time t t y = vector of asset returns at time t (n  1) = sample average vector of historical returns (n  1)  = sample covariance matrix of historical returns (n  n) When applying the turbulence index to two assets, we consider the difference between the return for the first asset at time t and the mean return and the difference between the return for the second asset at time t and the mean return and calculate the covariance between the returns for these two assets. We then take the absolute value of the final calculation so that the turbulence index is always positive. The information provided by the financial turbulence index is helpful because assets that may be negatively correlated during normal economic conditions may become positively correlated during times of high turbulence. This systemic risk measure, which can be used for stress tests of asset portfolios, provides a realistic estimate of possible losses arising from a systemic event. 1.1.2. Multivariate density estimators Segoviano and Goodhart (2009) developed a systemic risk measure based on the banking system’s multivariate density (BSMD). Considering the banking system as Chapter 1 8 a portfolio of banks, with each bank as part of the portfolio, probabilities of distress can be obtained by estimating the BSMD using a multivariate density methodology (Segoviano, 2006). It is also possible to estimate banking stability measures from the BSMD. Segoviano and Goodhart (2012) used this methodology to examine and quantify relative changes in stability over time in the following cases: (1) general distress in the banking system; (2) distress between specific banks; and (3) distress in the system associated with a specific bank. Given the BSMD, the authors proposed a set of measures of systemic risk, namely, joint probability of default, the banking stability index and the distress dependence matrix. 1.1.3. Conditional value at risk Adrian and Brunnermeier (2011) proposed the CoVaR measure for systemic risk, which captures possible risk spillovers between financial institutions by providing information on the VaR of the financial system conditional on the fact that a financial institution is in distress. These authors also calculated the systemic risk contribution of an institution as the ∆CoVaR, which measures the difference between the CoVaR under financial distress and the CoVaR in the benchmark state. Formally, the CoVaR can be defined as the q-quantile of a conditional distribution:   j|i i qq Pr X CoVaR |X =VaR =q, ji (1.2) with the systemic risk contribution defined as: j|i j|i j|i q q 50% CoVaR CoVaR CoVaR   . (1.3) The authors proposed using quantile regression to compute the CoVaR: i i i i i t t tM system| system| system| 1 CoVaR VaR      , (1.4) where tM1 denotes a set of explanatory variables. 1.1.4. Co-risk An IMF Global Financial Stability Report (IMF, 2009a) proposed the co-risk methodology to estimate co-movements between the credit default swap (CDS) spreads for several financial institutions. This methodology assesses direct and indirect financial linkages that may arise from exposure to common risks (similar business models, common accounting practices, etc). Like the CoVaR, co-risk Literature review 9 employs quantile regression to estimate co-movement between risk factors for financial institutions in distress. The co-risk measure is formally defined as: k i ,i i ,i j i CDS R CDS           , (1.5) where i CDS is the credit default swap spread of the institution i, j CDS is the credit default swap spread of the institution j, i R denotes common aggregated risk factors,  denotes the quantile (usually the 95th) corresponding to a distress period and where ,i  are the parameter estimates that quantify the input of firms as the credit risk of firm i at quantile  . Hence, the conditional co-risk measure is given by: k 95 95,i i 95, j j i i, j i R CDS (95) CoRisk 100 1 CDS (95)                 , (1.6) where i CDS (95) and j CDS (95) are the CDS spreads of institution i and j corresponding to the 95th percentile and where 95  , 95,i  and 95, j  are the quantile regression parameters at the 95th level. 1.1.5. Systemic and marginal expected shortfalls Acharya et al. (2010) showed how a financial institution’s contribution to systemic risk could be measured and priced as the systemic expected shortfall (SES). The SES represents a propensity to be undercapitalized when the system as a whole is undercapitalized. The SES can be quantified using three measures: (1) the outcome of stress tests performed by regulatory bodies; (2) the decline in equity valuations of large financial firms during a crisis; and (3) the widening of the credit default swap spread of large financial institutions. The same authors also developed leading indicators denominated the marginal expected shortfall (MES) and leverage (LVG) defined as the ratio of the quasi-market value of assets market to the value of equity. They define the MES of a financial institution as its short-run expected equity loss conditional on the market taking a loss greater than its VaR at  %. Formally it is expressed as:     i,t i,t 1 m,t 1 ,t t 1 MES r | r q r C ,=E      (1.7) or Chapter 2 16 against systemic financial distress. Using CDS data, Segoviano and Goodhart (2009) constructed a banking stability index with which to assess interbank dependence for tail events. Moreno and Peña (2012) provided evidence regarding the suitability of using CDS data to estimate systemic risk. Acharya et al. (2010) introduced systemic expected shortfall and marginal expected shortfall as indicators to quantify downside risk and the contributions of financial institutions to risk. Brownless and Engle (2012) developed a systemic risk measure called SRISK, representing the amount of capital needed to restore a minimal capital requirement. Allen et al. (2010) proposed a measure of aggregate systemic risk called CATFIN that can predict declines in aggregate bank lending activity six months in advance. Billio et al. (2012) proposed five measures of systemic risk that capture contagion and exposure effects in relationships between financial institutions. Engle and Manganelli (2004) developed a conditional autoregressive value at risk (CAViaR) model that uses quantile regression to capture the tail behaviour of returns. Recently, Adrian and Brunnermeier (2011) proposed conditional VaR (CoVaR) as a new measure of systemic risk. CoVaR captures possible risk spillovers between financial institutions by providing information on the VaR of the financial system conditional on the fact that a financial institution is in distress, 1 with the systemic risk contribution of an institution measured as the difference between the CoVaR under financial distress and the CoVaR in its benchmark state. More recently, Girardi and Ergün (2013) generalized the CoVaR measure by assuming that the conditioning financial distress event should refer to the return of the financial institution being less than or equal to its VaR, rather than merely being equal to its VaR, as proposed in Adrian and Brunnermeier (2011). Girardi and Ergün (2013) also proposed a new approach to quantifying systemic risk that differs from the quantile regression approach proposed by Adrian and Brunnermeier (2011); it consists of using a multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model to characterize joint density between the financial system and financial institution returns and to obtain the CoVaR value by numerically solving a double integral. Below we quantify systemic risk for Spanish financial institutions using the CoVaR measure and also account for the quantitative effect on the CoVaR value of 1 López-Espinosa et al. (2012) identified the main determinants of systemic risk for a set of large international banks using the CoVaR measure proposed by Adrian and Brunnermeier (2011). Measuring systemic risk in the Spanish financial system: A CoVaR approach 17 using quantile regression and MGARCH. We also propose a novel copula-based approach to computing the CoVaR value, given that copulas are more flexible in modelling joint distributions and are particularly useful for characterizing tail behaviour, which provides such crucial information for the CoVaR. Estimating the CoVaR through copulas is also computationally less cumbersome than using the MGARCH approach. The procedure involves two steps: first, given the confidence level of the VaR and the CoVaR, we obtain the cumulative probability of the CoVaR from the copula; and second, we invert the marginal distribution function for this cumulative probability and so obtain the value of the CoVaR. We studied systemic risk for financial institutions listed on the Spanish stock exchange using weekly data for the period January 2003 to February 2013. Our evidence shows that systemic risk displays dynamic behaviour that is well captured by the copula and MGARCH approach to computing the CoVaR; quantile regression, however, is unable to capture the dynamics and abrupt changes in the value of systemic risk. More specifically, we found significant increases in systemic risk during the recent global financial crisis and the European debt crisis that quantile regression was unable to capture or underestimated. The fact that the copula approach, on average, indicated greater systemic risk than the MGARCH approach is consistent with the time-varying evidence of tail dependence reported by the copula. The remainder of the chapter is laid out as follows. Section 2.2 characterizes systemic risk, quantile regression, MGARCH and copula approaches to the CoVaR, Section 2.3 presents our data, Section 2.4 reports the results and Section 2.5 concludes the chapter. 2.2. Methodology Several systemic risk measures have been proposed in the literature to quantify the impact of a potentially risky financial institution on the financial system as a whole. For our research, we chose to use VaR — as arguably the measure most widely employed by financial institutions — to quantify systemic risk as the effect of the risky situation of a particular financial institution on the VaR of the financial system overall, specifically, the CoVaR (Adrian and Brunnermeier, 2011; Girardi and Ergün, 2013). Chapter 2 18 2.2.1. CoVaR The CoVaR of the financial system is the VaR of the financial system conditional on the fact that a given financial institution is in financial distress. Let be the returns of the financial system and let be the returns of the financial institution i. The CoVaR for a confidence level and time t can be formally defined as the -quantile of the conditional distribution of : , (2.1) where the financial distress situation of the financial institution i is represented by the fact that , where is the VaR for the financial institution i, measuring the maximum loss that financial institution i may experience for a confidence level and a specific time horizon t, that is, the -quantile of the return distribution for the financial institution i: . Thus, from a statistical point of view, computing the CoVaR value consists of determining the quantile of a conditional distribution. In addition, the systemic risk contribution of a particular financial institution i can be defined as the difference between the CoVaR for a confidence level and the VaR of the financial system conditional on the fact that financial institution i is in a benchmark state, measured as the median of the return distribution of institution i (the VaR value for ). This measure, called delta CoVaR ( CoVaR), is formally defined as: . (2.2) Below we describe different approaches to computing the value of the CoVaR. 2.2.2. Quantile regression Adrian and Brunnermeier (2011) proposed using quantile regression to compute the CoVaR in such a way that the VaR of the financial institution i and a set of explanatory variables determine the quantile of the conditional distribution of . We can thus obtain information on the VaR of the financial system conditional on the fact that the returns of financial institution i are in its VaR, , that is, . We do this by characterizing the conditional quantile function of as: s t X i t X 1  s t X    s s|i i i t , t t, Pr X CoVaR |X VaR = t  ii t t, X VaR i t, VaR 1  ii tt X VaR    , Pr( ) 1 0.5    s|i s|i s|i, 0.5 s|i, 0.5 t ,t ,t ,t CoVaR CoVaR CoVaR CoVaR         s t X ii t t, X =VaR   s s|i i i t , t t, Pr X CoVaR |X =VaRt th  s t X Measuring systemic risk in the Spanish financial system: A CoVaR approach 19 , (2.3) where is the conditional distribution function of given the set x of explanatory variables, includes a set of explanatory variables and the quantile regression coefficient determines the dependence relationship between the and the VaR of the financial system. The CoVaR value at each time t is computed as the estimated value of the quantile regression given by Eq. (2.3) for the corresponding confidence levels of and . 2 Quantile regression is computationally simple but has the disadvantage that it provides information only on the CoVaR conditional on the fact that and not on the fact that ; this fact has repercussions for the CoVaR values, as was discussed in Girardi and Ergün (2013). In addition, using quantile regression requires computation of the . Adrian and Brunnermeier (2011) proposed computing the in Eq. (2.3) by means of quantile regression for the distribution of , where only the explanatory variables in were included in order to allow the value of to change over time. With the aim of accounting for the effect of heteroskedasticity and fat tails of the return distribution on the value of the , we followed a different approach to computing the . Specifically, we modelled the returns of each financial institution through an autoregressive (AR) moving average (MA) model, specifically, ARMA(p,q), specified as: , (2.4) where p and q are non-negative integers and and are the AR and MA parameters, respectively. The stochastic process , where — the conditional variance of whose dynamic is reflected in a threshold generalized autoregressive conditional heteroskedasticity (TGARCH) specification (Zakoian, 1994; Glosten et al., 1993) — is given by: 2 Adrian and Brunnermeier (2011) proposed a specification of the quantile regression function that is slightly different from Eq. (2.3). As the explanatory variable they include i t X instead of i t, VaR  and use the estimated parameter values to predict the CoVaR by substituting i t X by i t, VaR  . In this way, they assume that estimated parameter values are equal across different quantile regression functions.   ss s|i i t, t Xx Q ( |x) inf b|F (b|x) VaR M 1         s x F (b| x) s t X t M1 s|i  i t, VaR 1 1 ii t t, X =VaR ii t t, X VaR i t, VaR i t, VaR i t X t M1 i t, VaR i t, VaR i t, VaR pq i i i i t 0 j t j h t h t j 1 h 1 XX             j  h  i t i,t i,t z   2 i,t  i t  Chapter 2 20 , (2.5) where is a constant, is the variance prediction error for the previous period (the generalized autoregressive conditional heteroskedasticity (GARCH) component), represents the volatility shock for the previous period (the autoregressive conditional heteroskedasticity (ARCH) component) and captures leverage effects. When takes values greater than zero the future conditional variance will increase proportionally more after a negative shock that after a positive shock of the same magnitude. is an i.i.d. random variable with mean zero and unit variance that follows a Hansen (1994) skewed-t density distribution given by: , (2.6) where and are the degrees of freedom parameter ( ) and the symmetric parameter ( ), respectively. The constants a, b and c are given by , and . This distribution converges to the standard Gaussian as and and to the symmetric Student-t distribution as and is finite. From Eqs. (2.4)-(2.6) we can compute the at each time t as: , (2.7) where is the quantile of the standardized skewed-t distribution for probability . This is the value for the VaR that we considered in Eq. (2.3) in order to estimate the CoVaR through quantile regression. 2.2.3. MGARCH In order to obtain the value of the VaR conditional on the event , Girardi and Ergün (2013) proposed an alternative procedure based on the joint distribution of and . From Eq. (2.1), the CoVaR can be expressed as: d m m 2 2 i 2 i i,t h i,t h k t k r t r h 1 k 1 r 1 ()                      2 i,t l  i tr  r  r  i,t z it it bz a it it bz a it bc z a b fz bc z a b                                   , , ( 1) 2 2 1 , 21 ,( 1) 2 2 1 , 21 1 ( ; , ) 1   2    11      ac 2 1 4   ba 2 2 2 13        c1 22 ( 2)         0    0  i t, VaR pp i i i 2 1 t, 0 j t j h t h i,t i,t j 1 h 1 VaR X z ( )                 1 i,t z ( )   ii t t, X VaR s t X i t X Measuring systemic risk in the Spanish financial system: A CoVaR approach 21 . (2.8) Given that , Eq. (2.8) can be written as: . (2.9) Hence, the value of the CoVaR can be obtained by numerically solving the following double integral: , (2.10) for , where denotes the joint probability density. Thus, computing the CoVaR involves knowledge of the joint distribution of and . According to Girardi and Ergün (2013), the joint distribution of and can be obtained using MGARCH and time-varying dynamic conditional correlation (DCC). This MGARCH-DCC model, initially proposed by Engle (2002), considers the mean and volatility dynamics of the returns as given by Eqs. (2.4) and (2.5). The bivariate stochastic terms are given by , where , has a standardized bivariate Student-t distribution and where is a variance-covariance matrix where the variance of each stochastic component is given by Eq. (2.5) and the covariance between the two stochastic components is given by , where is the correlation coefficient between the returns of the system and the financial institution i. The correlation matrix is given by , where is a 2x2 diagonal matrix, with the conditional variance of each variable located along the main diagonal. Engle (2002) proposed characterizing the dynamics of the conditional correlations as follows: , , (2.11) where is the unconditional covariance matrix for the standardized residuals, is the diagonal matrix and and are parameters. Once we estimated the MGARCH-DCC model, we had all the necessary information on the elliptical , so, for the given value of the we could numerically solve Eq. (10) to obtain the CoVaR for each time period.       s s|i i i t , t t, ii t t, Pr X CoVaR , X VaR = Pr X VaR t    ii t t, Pr X VaR    s s|i i i t , t t, Pr X CoVaR , X VaR = t   s|i i t, t, CoVaR VaR s i s i t t t t t pdf X , X dX dX     s|i , CoVaRt   si t t t pdf X , X s t X i t X s t X i t X 12 t t t z ss t t t ()     t s,t i,t z (z z ) t  22 si,t si,t s,t i,t  si,t 11 22 t t t t R D D   t D 11 22   t t t t R diag(Q ) Q diag(Q ) 1 1 1 1 t t t t Q ( )Q ( ) Q                Q t diag(Q ) t Q     si t t t pdf X , X i t, VaR Chapter 2 22 2.2.4. Copulas We can employ copulas 3 to compute the CoVaR. From Eq. (2.9), we can express the CoVaR in terms of the joint distribution function of and , , as: . (2.12) Furthermore, according to Sklar’s (1959) theorem, the joint distribution function of two continuous random variables can be expressed in terms of a copula function. Hence, Eq. (2.12) can be written as: , (2.13) where is a copula function and and denote the marginal distributions of and , respectively, such that and . Thus, from Eq. (2.13) we can compute the CoVaR value following a two-step procedure. First, given that and given the copula specification, for a confidence level we compute the cumulative probability for the CoVaR, , by solving from Eq. (2.13). Next, from , we invert the marginal distribution function of to obtain the CoVaR, hence, . Computing the CoVaR using copulas offers three main advantages. First, copulas offer more flexibility in separate modelling the marginals and dependence, which is especially important when the marginals have different characteristics or when dependence is not linear — in particular, when the joint distribution displays different forms of tail dependence (crucial for the values of the CoVaR). Second, computation of the CoVaR through copulas is less computationally burdensome than using the MGARCH approach involving the numerical resolution of a double integral. Third, computation of the is not necessary, as we only need information on the confidence level for , which is exogenously determined. Obtaining the CoVaR through copulas requires specification of the marginals and the copula function. The marginals we used are given by Eqs. (2.4)-(2.6); in order to characterize different patterns of dependence, different copula specifications, 3 For further analysis of copulas, see Joe (1997) and Nelsen (2006). An overview of copula applications to finance can be found in Cherubini et al. (2004). Mainik and Schaanning (2012) provided the first representation of the CoVaR in terms of copulas. s t X i t X si tt XX F, si tt s i i tt XX F CoVaR VaR | ,, ,( , )      si tt s i i tt XX C F CoVaR F VaR | ,, ( ), ( )    C(, ) s t X F i t X F s t X i t X s t si t X u F CoVaR | , ()   i t i t X v F VaR , ()   v 1 u u s t X s t s|i 1 t, X CoVaR F (u)   i t, VaR  i t, VaR  Measuring systemic risk in the Spanish financial system: A CoVaR approach 23 as reported in Table 2.1, were used. We also captured time-varying dependence by assuming that copula parameters change over time. For the Gaussian and the Student-t copulas, we adopted an ARMA(1,q)-type process (Patton, 2006) for the linear dependence parameter : , (2.14) where is the modified logistic transformation that keeps the value of in (-1,1). For the Student-t copula, is replaced by . We also considered time-varying dependence for the Gumbel copula and for its rotated version by assuming that the parameters follow the dynamics given by the following equation: . (2.15) Overall, the family of eleven copulas considered here can be classified as follows: (1) symmetric copulas, with either tail dependence (Student-t and time-varying Student-t copulas) or tail independence (Gaussian, time-varying Gaussian and Plackett copulas) and (2) asymmetric tail dependence copulas (Gumbel, rotated Gumbel, BB7, BB1 and time-varying Gumbel and rotated Gumbel copulas). The parameters of the marginal and copula models were estimated using a twostep procedure called inference functions for margins (Joe and Xu, 1996). We first estimated the marginal models by maximum likelihood (ML) and then transformed each filtered standardized return series (standardized residual) into its uniform marginal via the probability integral transform, thus obtaining and . Using this information, we then estimated the copula function parameters by ML. The number of lags in the mean and variance equations for each series was selected according to the Bayesian information criterion (BIC) and Akaike information criterion (AIC). The performance of the different copula models was evaluated using the AIC adjusted for small-sample bias, as in Breymann et al. (2003) and Reboredo (2011; 2013). t    q11 1 t 0 1 t 1 2 t i t i j1 q(u ) (v )                      xx (x) e e       1 11 t  1(x)   1 t (x)   q t t t i t i qjuv           1 11 t ˆ u t ˆ v Chapter 2 24 Table 2.1: Copula specifications Name Copula Parameter Structure dependence Gaussian   11 N C (u,v; ) (u), (v)       No tail dependence. Student-t      11 ST C (u,v; , ) T(t (u),t (v)) Symmetric tail dependence. Gumbel                    1 G C (u,v; ) exp logu log v 1 Upper tail dependence and lower tail independence. When 1 two variables are independent. Rotated Gumbel         RG G C (u,v; ) u v 1 C (1 u,1 v; ) 1 Upper tail independence and lower tail dependence. BB7     1 1 BB7 C (u,v; , ) 1 1 1 1 u 1 1 v 1                                      1 , 0 Differing degrees of upper and lower tail dependence. Plackett                               2 P 1 C (u,v; ) 1 1 u v 1 1 u v 4 1 uv 21 0 , 1 Symmetric tail independence. BB1     1 1 CG C (u,v; , ) u 1 v 1 1                      0 1 Asymmetric tail dependence. Measuring systemic risk in the Spanish financial system: A CoVaR approach 25 2.3. Data We empirically examined systemic risk for nine listed Spanish institutions using daily prices for the period 16 January 2003 to 28 February 2013. The set included seven banks (BBVA, Banco Santander, Banco Sabadell, Banco Popular, Bankinter, Banesto and Banco de Valencia) and two insurance companies (Catalana Occidente and Mapfre). To capture the behaviour of the whole financial system, we used the MSCI Spain Financials Index. Data were obtained from Bloomberg and returns for price data were computed on a continuous compounding basis. We also considered daily information for a set of variables that were included in in Eq. (2.3), as follows (all these data were obtained from Bloomberg): (a) Daily IBEX-35 volatility, computed as the standard deviation of the daily returns in a backward window of three months. (b) The first difference of the 12-month Treasury bill rate. (c) The slope of the yield curve measured as the difference between the 10-year and the 12-month Treasury bill rates. (d) The credit spread determined as the difference between interest rates for corporate and government 10-year maturity bonds. (e) Daily market returns obtained from the IBEX-35 general market index. Figure 2.1 shows the time-series plot for the returns of the studied institutions ( ) and the MSCI ( ). All series display the usual characteristics of financial returns — including volatility clustering and fat tails — and also show abrupt changes around the onset of the global financial crisis (mid-2008) and European sovereign debt crisis (at the end of 2009). Figure 2.2 depicts the temporal dynamics of the set of variables considered as explanatory variables in the quantile regression. 1t M i t X s t X Chapter 2 32 Figure 2.3: Time series plot for value-at-risk at the 95% confidence level. 2.4.2. Quantile regression results With the value of the VaR for each financial institution and the set of explanatory variables described in the data section, we estimated the CoVaR for the financial system using the quantile regression reflected in Eq. (2.3). Table 2.6 reports estimates for a 95% confidence level ( and ). Given that the quantil regression parameter was significant and had the expected sign for all the financial institutions, the VaR for each institution had a significant impact on the VaR of the financial system. Regarding the explanatory variables, we found that market volatility and yield curve slope were significant and had the expected sign in all cases; other explanatory variables were only significant in some cases; and the IBEX-35 returns were non-significant in all the cases. For each financial institution the CoVaR value at each time t was computed as the estimated value of the corresponding quantile regression. Table 2.9 reports descriptive statistics for the CoVaR, showing values that were greater for the banks than for the insurance companies and, in general, with values reflecting the size of the bank. Likewise, larger banks generated more instability in CoVaR values than small banks or insurance companies. -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Valencia VaR 95% Banesto Bankinter BBVA Popular Sabadell Santander Catalana Mapfre Valencia 1 95% 1 95%    Measuring systemic risk in the Spanish financial system: A CoVaR approach 33 Table 2.6: Quantile regression estimates at the 95% confidence level. Note. The quantile regression coefficients, are as follows: = coefficient of the quantile regression for the returns of each institution ( ), = coefficient of the quantile regression for market volatility (IBEX-35), = coefficient of the quantile regression for the 12-month Treasury bill variation rate, = coefficient of the quantile regression for the yield curve slope, = coefficient of the quantile regression for the credit spread, = coefficient of the quantile regression for the market returns (IBEX-35), Numbers in brackets indicate standard error: * Significance at 10%. ** Significance at 5%. *** Significance at 1%. 2.4.3. MGARCH results We estimated the DCC model for returns composed of the financial system paired with each financial institution. Table 2.7 reports the results for the parameters and degrees of freedom, as reflected in Eq. (2.11), for the bivarate Student-t distribution. All the parameters were significant, providing consistent evidence in favour of timevarying dependence and fat tails according to the estimated degrees of freedom. Figure 2.4 depicts correlation dynamics for each financial institution, showing that institutional interdependence with the general financial system varies according to the institution’s size, with no significant change with the onset of the global | | | | | | , 1, 1, 1, 1, 1, ( ) 12 . , si i si i si i si i si i si i t p t p t p t p t p t p Q x VaR MkVol TrBill Slope cr spread Mkreturns                      0.01,   Banesto 0.834*** -0.091*** -0.114 -0.156*** -0.072 -0.008 (0.058) (0.006) (0.219) (0.035) (0.110) (0.023) Bankinter 0.637*** -0.111*** 0.179 -0.188*** 0.591*** -0.013 (0.049) (0.008) (0.316) (0.034) (0.101) (0.022) BBVA 0.790*** -0.026*** 0.051 -0.100*** -0.202*** -0.014 (0.021) (0.003) (0.096) (0.014) (0.058) (0.009) Popular 0.766*** -0.061*** 0.059 0.004 -0.013 -0.038 (0.085) (0.012) (0.375) (0.050) (0.213) (0.027) Sabadell 0.719*** -0.134*** 0.049 -0.147*** 0.379** -0.022 (0.048) (0.006) (0.299) (0.034) (0.175) (0.023) Santander 0.634*** -0.054*** -0.288*** -0.131*** -0.447*** 0.001 (0.026) (0.004) (0.109) (0.020) (0.044) (0.011) Valencia 0.145*** -0.183*** -0.011 -0.177*** 0.669*** -0.005 (0.025) (0.007) (0.413) (0.033) (0.143) (0.025) Catalana 0.272*** -0.161*** 0.084 -0.306*** 0.485*** -0.007 (0.042) (0.012) (0.615) (0.038) (0.178) (0.028) Mapfre 1.017*** -0.043*** -0.219 -0.291*** 0.017 -0.008 (0.087) (0.010) (0.368) (0.034) (0.110) (0.021) Chapter 2 34 financial crisis or of the European sovereign debt crisis. With this correlation coefficient and marginal model information we have all the information on the joint distribution function of the financial system and the financial institution at each time t. Hence, taking the VaR of a given financial institution computed through its marginal return model, we can compute the CoVaR value by numerically solving Eq. (2.10) for the CoVaR and and . Table 2.7: Dynamic conditional correlation (DCC) model estimates. Banesto Bankinter BBVA Popular Sabadell Santander Valencia Catalana Mapfre DoF 5.220* 6.110* 6.270* 5.682* 4.850* 5.139* 4.900* 6.294* 7.185 (15.65) (14.61) (13.91) (15.21) (18.27) (17.21) (19.49) (13.59) (11.42) α 0.056* 0.056* 0.044* 0.038* 0.030* 0.035* 0.017* 0.010* 0.025* (3.71) (5.50) (3.73) (5.23) (3.11) (3.74) (3.20) (3.75) (3.11) β 0.917* 0.696* 0.659* 0.696* 0.638* 0.692* 0.686* 0.689* 0.673* (33.10) (51.49) (42.56) (90.82) (71.76) (77.19) (175.60) (314.30) (59.91) Note. In accordance with the DCC model, α and β are the parameters estimated with the Student-t distributed errors. An asterisk (*) indicates rejection of the null hypothesis at 5%. Figure 2.4: Time series plot for dynamic conditional correlation (DCC). Descriptive statistics for the CoVaR values are reported in Table 2.9. The evidence on systemic risk is qualitatively similar to that reported for the quantile regression approach, even though the size of the CoVaR was quite different. The CoVaR values obtained using the MGARCH approach were considerably reduced with respect to those obtained using the quantile regression approach. This is consistent with the idea that conditioning on the fact that and not on the fact that has a quantitative impact on the value of the CoVaR (an effect that was also discussed by Girardi and Ergün, 2013). 1 95% 1 95%   -0.2 0 0.2 0.4 0.6 0.8 1Rho DCC Banesto Bankinter BBVA Popular Sabadell Santander Valencia Catalana Mapfre  ii t t, X VaR ii t t, X VaR Measuring systemic risk in the Spanish financial system: A CoVaR approach 35 2.4.4. Copula results Table 2.8 reports parameter estimates for the nine pairs given by the MSCI Spain Financials Index returns matched with each institution’s returns. Both static and dynamic copula specifications indicate that all the financial institutions co-moved with the general financial index. Parameter estimates for all the static copulas were significant, as also was the case for most of the time-varying copula specifications. Considering the AIC values corrected for small-sample bias, we found evidence of time-varying dependence and lower tail dependence, with the time-varying rotated Gumbel copula offering the best fit for all institutions, except for BBVA and Catalana Occidente, for which the time-varying Student-t performed better, thereby providing evidence of symmetric tail dependence. From the best copula specification and following the same two-step procedure as described above, we obtained the CoVaR value at the 95% confidence level for the financial system conditional on the VaR of each institution at the 95% confidence level. We also obtained the CoVaR value using Eq. (2.2). Figure 2.5 depicts the dynamic behaviour of the estimated CoVaR and CoVaR (in percentage) and reports — to enable assessment and comparison of the impact of different methodological approaches — the CoVaR values estimated using the quantile regression and MGARCH approaches. The graphical evidence for the CoVar points to two conclusions. First, trends in the CoVaR values were consistent across different financial institutions, with systemic risk increasing around the onset of the global financial and the European sovereign debt crises. Significant changes also occurred in the CoVaR values in April 2011, when the systemic risk of Spanish financial institutions increased due to Portugal requesting a financial bailout. Second, there were significant differences in CoVaR values obtained using quantile regression as compared to using the MGARCH and copula approaches. Specifically, quantile regression CoVaR values were higher and also relatively stable through the sampling period, with little variation once the crises broke, indicating that the real systemic risk was underestimated. In contrast, MGARCH and copula CoVaR values were lower and reflected dynamics that adjusted much better to crisis events; thus, values were reduced when the crises broke and increased in more stable periods, thereby indicating a better adjustment to the changing economic environment in the latter years of the sampling period.   Chapter 2 36 Table 2.8: Copula model estimates for institutions vs the MSCI Index. Panel A: Parameter estimates for time-invariant copulas Copula Banesto Bankinter Bbva Popular Sabadell Santander Valencia Catalana Mapfre Gaussian 0.661 0.680 0.947 0.742 0.666 0.967 0.438 0.404 0.586 AIC -1375.837 -1485.457 -5458.490 -1918.773 -1404.743 -6575.424 -508.015 -424.749 -1005.770 Student-t 0.667* 0.692* 0.951* 0.755* 0.684* 0.970* 0.459* 0.411* 0.589* (0.01) (0.01) (0.00) (0.01) (0.01) (0.00) (0.02) (0.01) (0.01) 6.175* 5.300* 4.055* 4.907* 4.065* 3.389* 5.426* 12.943* 8.622* (1.95) (1.47) (0.48) (0.46) (0.41) (0.36) (0.74) (1.81) (0.87) AIC -1434.531 -1581.141 -5714.405 -2037.900 -1568.800 -6861.588 -588.128 -437.843 -1043.008 Gumbel 1.749* 1.813* 4.629* 2.057* 1.802* 5.860* 1.385* 1.314* 1.578* (0.03) (0.03) (0.08) (0.03) (0.03) (0.10) (0.02) (0.02) (0.03) AIC -1205.439 -1324.791 -5320.482 -1797.433 -1298.704 -6361.999 -469.609 -356.186 -883.238 Rotated Gumbel 1.838* 1.915* 4.794* 2.125* 1.903* 6.197* 1.411* 1.336* 1.623* (0.03) (0.03) (0.08) (0.04) (0.03) (0.11) (0.02) (0.02) (0.03) AIC -1460.203 -1594.086 -5528.936 -1983.757 -1565.309 -6685.885 -568.721 -422.484 -1019.961 BB7 1.428* 1.454* 4.034* 1.738* 1.448* 4.103* 1.233* 1.185* 1.369* (0.04) (0.05) (0.00) (0.08) (0.09) (0.04) (0.03) (0.03) (0.04) 1.126* 1.217* 4.263* 1.327* 1.218* 6.311* 0.547* 0.453* 0.798* (0.05) (0.05) (0.00) (0.06) (0.04) (0.21) (0.04) (0.04) (0.04) AIC -1431.329 -1552.662 -5355.069 -1931.303 -1530.783 -6214.699 -551.679 -415.580 -1018.554 Plackett 9.732* 11.279* 99.679* 15.385* 11.115* 186.395* 4.516* 3.581* 6.682* (0.52) (0.57) (4.53) (0.75) (0.57) (29.81) (0.26) (0.20) (0.35) AIC -1387.328 -1567.059 -5534.123 -2016.034 -1525.218 -6787.020 -568.881 -428.177 -981.821 BB1 0.721* 0.753* 0.906* 0.659* 0.763* 1.036* 0.383* 0.322* 0.515* (0.05) (0.06) (0.04) (0.05) (0.06) (0.07) (0.04) (0.04) (0.05) 1.348* 1.384* 3.352* 1.612* 1.372* 4.090* 1.195* 1.161* 1.300* (0.03) (0.03) (0.05) (0.04) (0.03) (0.12) (0.03) (0.02) (0.03) AIC -1459.027 -1590.462 -5646.838 -2004.665 -1562.220 -6754.743 -563.207 -426.512 -1039.673               Measuring systemic risk in the Spanish financial system: A CoVaR approach 37 Panel B: Parameter estimates for time-varying copulas. Banesto Bankinter Bbva Popular Sabadell Santander Valencia Catalana Mapfre TVP-Gaussian -0.269 0.076 -1.294 -0.517* -0.039* 5.000 0.030 -0.041 -0.144* (1.67) (0.13) (9.37) (0.10) (0.00) (9.58) (0.02) (0.92) (0.07) 0.176 0.300* 0.201 0.184* 0.262* 0.116 0.241* 0.045 0.083* (5.81) (0.06) (0.18) (0.04) (0.01) (2.35) (0.05) (2.46) (0.03) 2.695 2.070* 5.000 3.121* 2.286* -1.022 1.909* 2.200 2.469* (3.73) (0.24) (10.05) (0.17) (0.02) (6.16) (0.08) (1.49) (0.15) AIC -1433.80 -1552.50 -5471.60 -1983.83 -1554.62 -6575.19 -636.04 -473.50 -1028.27 TVP-Student-t -0.198* -0.072 -14.21* -0.657* -0.037* -12.743* 0.059 0.003 2.183 (0.02) (0.11) (0.00) (0.08) (0.00) (5.05) (0.05) (0.02) (2.03) 0.131* 0.156* 0.03* 0.103* 0.189* -0.014 0.205* 0.097* 0.249 (0.02) (0.03) (0.00) (0.03) (0.02) (0.01) (0.04) (0.03) (0.14) 2.543* 2.388* 18.80* 3.354* 2.297* 17.521* 1.843* 2.034* -1.707 (0.04) (0.19) (0.00) (0.14) (0.01) (5.20) (0.15) (0.08) (3.55) 6.221* 6.677* 5.00* 5.770* 6.666* 3.171* 7.646* 17.118* 9.488* (0.46) (1.14) (0.00) (0.35) (0.68) (0.33) (2.84) (2.18) (1.93) AIC -1480.20 -1618.48 -5736.39 -2085.60 -1642.09 -6861.18 -684.12 -481.04 -1046.55 TVP-Gumbel 0.772* 0.595 1.768* 0.946* 1.121* 1.417* 0.675* 0.389 1.505* (0.32) (0.38) (0.17) (0.43) (0.24) (0.04) (0.12) (0.36) (0.20) 0.222 0.300* 0.106* 0.198 0.147 0.167* 0.279* 0.329 -0.194 (0.12) (0.13) (0.03) (0.14) (0.08) (0.00) (0.05) (0.19) (0.11) -1.626* -1.390 -5.000* -2.095* -2.779* -3.801* -1.977* -1.084* -2.060* (0.58) (0.77) (0.57) (0.95) (0.57) (0.39) (0.25) (0.50) (0.29) AIC -1306.20 -1456.58 -5464.95 -1957.46 -1538.34 -6626.42 -660.92 -389.85 -925.14 TVP-Rotated Gumbel 0.368* 0.477* 1.162* 0.473* 0.932* 1.480* 0.448* -0.167 1.381 (0.04) (0.04) (0.05) (0.04) (0.08) (0.07) (0.07) (0.10) (2.72) 0.370* 0.340* 0.195* 0.336* 0.197* 0.160* 0.358* 0.621* -0.111 (0.01) (0.01) (0.01) (0.01) (0.03) (0.01) (0.02) (0.05) (1.37) -0.761* -1.027* -2.230* -0.868* -2.050* -3.810* -1.425* -0.351* -1.948 (0.12) (0.10) (0.29) (0.13) (0.14) (0.63) (0.18) (0.13) (2.29) AIC -1553.51 -1731.97 -5679.02 -2122.33 -1756.68 -6913.55 -739.91 -447.49 -1066.97 Notes. The table reports the maximum likelihood (ML) estimates for the different copula models for MSCI and the series indicated in each column. Standard error values (in brackets) and the AIC values adjusted for small-sample bias are provided for the different copula models. The minimum Akaike information criterion (AIC) value indicates the best copula fit. For the TVP-Gaussian and TVP-Student-t copulas, q in Eq. (2.14) was set to 10. An asterisk (*) indicates significance at the 5% level. 0  1  2  0  1  2         Chapter 2 38 Figure 2.5 also provides information on CoVaR dynamics, with the value remaining relatively stable over the sampling period, but registering sudden reductions when dependence between a financial institution and the overall system reduced, thereby increasing the value of the CoVaR. In particular, there was a significant reduction in CoVaR and  CoVaR for Banco de Valencia towards the end of the sampling period when this bank went bankrupt. However, this bankruptcy had little impact on the system as a whole, given the decoupling that occurred. Table 2.9 summarizes descriptive CoVaR and CoVaR statistics for the three approaches. The average CoVaR values obtained with the MGARCH model and bivariate Student-t density and copula models were similar and much lower than for the quantile regression approach. Moreover, fluctuations in the quantile regression CoVaR values were much less than for the other two procedures, as confirmed by the differences between maximum and minimum values. We can therefore conclude that quantile regression underestimates systemic risk. Descriptive statistics for CoVaR allow us to conclude that the systemically important financial institutions in the Spanish financial sector were BBVA and Santander and that systemic risk increases with the size of the financial institution. Figure 2.5: Estimates of CoVaR (left axis) and ∆CoVaR (right axis) for the MSCI Index with respect to each institution.    0.5 0.55 0.6 0.65 0.7 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Banesto CoVaR 95% Copula CoVaR 95% Regression CoVaR 95% multivariate Garch ∆CoVaR 95% Copula Measuring systemic risk in the Spanish financial system: A CoVaR approach 39 Figure 2.5: (Continued) 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Bankinter CoVaR 95% Copula CoVaR 95% Regression CoVaR 95% multivariate Garch ∆CoVaR 95% Copula 0.65 0.66 0.67 0.68 0.69 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 BBVA CoVaR 95% Copula CoVaR 95% Regression CoVaR 95% multivariate Garch ∆CoVaR 95% Copula Chapter 2 40 Figure 2.5: (Continued) 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Popular CoVaR 95% Copula CoVaR 95% Regression CoVaR 95% multivariate Garch ∆CoVaR 95% Copula -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Sabadell CoVaR 95% Copula CoVaR 95% Regression CoVaR 95% multivariate Garch ∆CoVaR 95% Copula Measuring systemic risk in the Spanish financial system: A CoVaR approach 41 Figure 2.5: (Continued) 0.65 0.66 0.67 0.68 0.69 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Santander CoVaR 95% Copula CoVaR 95% Regression CoVaR 95% multivariate Garch ∆CoVaR 95% Copula -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Valencia CoVaR 95% Copula CoVaR 95% Regression CoVaR 95% multivariate Garch ∆CoVaR 95% Copula Chapter 3 48 sector. De Bruyckere et al. (2013) studied contagion between bank and sovereign default risk in Europe through asset, collateral and rating channels. Bhanot et al. (2014) investigated the impact of changes in Greek sovereign yield spreads on stock returns in the financial sector. Similarly, Mink and De Haan (2013) analysed the impact of highly volatile Greek bonds on European bank stock prices in 2010. Alter and Schuler (2012) examined the relationship between sovereign default risk and domestic banks. In addition, using credit default swaps, other studies examined sovereign risk contagion among eurozone countries (see, e.g., Missio and Watzka, 2011; Arezki et al., 2011; Alter and Beyer, 2012; Caporin et al., 2013). However, even though systemic risk is an important dimension of contagion that enables to quantify the impact of extreme downward movement in one market on other markets, no study has yet examined systemic risk in European sovereign debt markets and how this risk has changed with the onset of the recent European sovereign debt crisis. This chapter attempts to fill this gap, contributing, in particular, to the existing literature in two ways. First, we characterize the CoVaR systemic risk measure—as proposed by Adrian and Brunnermeier (2011) and generalized by Girardi and Ergün (2013)—in terms of copulas. CoVaR captures possible risk spillovers between markets by providing information on the value-at-risk (VaR) of a market, conditional on the fact that another market is in financial distress. Using copulas, the value of the CoVaR can be obtained in a two-step procedure. Given the cumulative probability of the VaR of the market in financial distress and the confidence level for the CoVaR, we can compute the cumulative probability for the CoVaR from a copula function. We then can invert the marginal distribution function for this cumulative probability in order to obtain the value of the CoVaR. From a computational point of view, this approach is more tractable than other parametric approaches; it is also more flexible, given that copula functions, by providing a measure of both average dependence and upper and lower tail dependence (joint extreme movements), enable the dependence structure of stochastic variables to be fully described. This information is crucial to determining the VaR of one variable, which is conditional on the fact that another variable takes values below or equal to its own VaR. In fact, (lower) tail dependence of a copula function naturally provides this information, but at the limit. Second, for a sample of sovereign bond benchmark price indices for France, Germany and the Netherlands, for GIIPS economies (Greece, Ireland, Italy, Portugal and Spain) and for an overall sovereign bond price index for the European Economic Systemic risk in European sovereign debt markets: A CoVaR-copula approach 49 and Monetary Union (EMU), for the period January 2000 to October 2012, we provide evidence of strong co-movement between European debt markets and the EMU index before the onset of the European debt crisis. All sovereign debt markets shared a similar trend in systemic risk, which was very similar in size across markets. However, with the onset of the European sovereign debt crisis, we find evidence of the decoupling of debt markets in such a way that GIIPS markets negatively correlate, on average, with the EMU index returns, displaying, in general, market independence at the tails. As a result, systemic risk for the GIIPS markets, with the exception of Spain, was reduced, whereas systemic risk for the non-crisis countries experienced a significant upsurge as a result of a high degree of comovement with the EMU index. Finally, we examined the systemic risk impact of the Greek debt market on other European debt markets, finding that before the Greek debt crisis, Greek systemic risk was relatively low and different across countries. However, after crisis onset, Greek systemic risk increased for the countries in crisis, especially for Portugal, where systemic risk tripled overall. For countries not in crisis the systemic impact of the Greek debt crisis was less, given that the debt markets of these countries decoupled from the Greek market. The remainder of the chapter is laid out as follows: in Section 3.2 we outline the copula approach to the CoVaR, in Section 3.3 we present data and in Section 3.4 we discuss the results. Finally, Section 3.5 concludes the chapter. 3.2. Methodology With the aim of quantifying systemic risk between assets, institutions or markets, different systemic risk measures have been proposed in the literature (see, e.g., Huang et al., 2009; Segoviano and Goodhart, 2009; Acharya et al., 2010; Allen et al., 2010; Zhou, 2010, Adrian and Brunnermeier, 2011; Brownlees and Engle, 2011; Billio et al., 2012; Girardi and Ergün, 2013; Gravelle and Li, 2013). Given that VaR is arguably the most widely employed risk measure by investors, financial institutions and regulators, in this study we evaluate systemic risk between European debt markets using the CoVaR measure proposed by Adrian and Brunnermeier (2011) and generalized by Girardi and Ergün (2013). Chapter 3 50 3.2.1. CoVaR and copulas The CoVaR for the European debt market is the VaR for the European debt market as a whole, conditional on the fact that a given debt market is in financial distress. Let d t R be the returns for the debt market as a whole and let j t R be the returns for debt market j. The CoVaR, formally defined as the  -quantile of the conditional distribution of d t R , is as follows: d d j j j t t t t R CoVaR R VaR | ,, Pr( | )      , (3.1) where jt VaR , is the VaR for debt market j, measuring the maximum loss that debt market j may experience for a confidence level 1 and a specific time horizon, that is, the  -quantile of the return distribution for the debt market j: jj tt R VaR , Pr( )     . Therefore, computing the CoVaR consists of determining the quantile of a conditional distribution, or, alternatively, of an unconditional bivariate distribution if we express Eq. (3.1) as: d d j j j t t t t jj tt R CoVaR R VaR R VaR | ,, , Pr( , ) Pr( )      . (3.2) Given that jj tt R VaR , Pr( )     , the CoVaR in Eq. (3.2) can be expressed as: d d j j j t t t t R CoVaR R VaR | ,, Pr( , )      . (3.3) Girardi and Ergün (2013) proposed to compute for the CoVaR in Eq. (3.3) by numerically solving a double integral: d j j tt CoVaR VaR d j d j t t t t t R R R Rf d d | ,, ( , )       (3.4) for given levels of  ,  and jt VaR , ; and where dj t t t RRf( , ) is the bivariate density of d t R and j t R . In this chapter we propose to compute the CoVaR through copulas. 5 Note that Eq. (3.3) can be expressed in terms of the joint distribution function of d t R and j t R , dj tt RR F, , as: 5 For further analysis on copulas, see Joe (1997) and Nelsen (2006). An overview of copula applications to finance can be found in Cherubini et al. (2004). Mainik and Schaanning (2012) provide the first representation of the CoVaR in terms of copulas. Systemic risk in European sovereign debt markets: A CoVaR-copula approach 51 dj tt d j j tt RR F CoVaR VaR | ,, ,( , )    , (3.5) and that, according to Sklar’s (1959) theorem, the joint distribution function of two continuous variables can be expressed in terms of a copula function. Hence, Eq. (3.5) can be written as: C u v( , )   , (3.6) where C(·,·) is a copula function, d t dj t R u F CoVaR | , ()   and j t jt R v F VaR , ()   and where d t R F and j t R F are the marginal distribution functions of d t R and j t R , respectively. Given its copula representation in Eq. (3.6), the CoVaR can be computed from that equation through copulas in a two-step procedure: 1) We obtain the value of d t dj t R u F CoVaR | , ()   . Since C u v( , )   , where  ,  and v are given (note that v ), from the copula function specification we can solve to determine the value of u . 2) Taking u , we can obtain the CoVaR value as the quantile of the distribution of d t R , with a cumulative probability equal to u , by inverting the marginal distribution function of d t R : d t dj tR CoVaR F u |1 ,()   . Computing the CoVaR through copulas has two main advantages. First, since copulas allow separate modelling of the marginals and dependence structures, they offer great flexibility in modelling marginals. This flexibility is crucial for the computation of VaR and the modelling of dependence structures with different tail dependence characteristics, such as the tail independence and symmetric or asymmetric tail dependence that is especially relevant for the computation of the CoVaR measure. Furthermore, copulas are useful when the joint distribution function is not elliptical, when the traditional dependence measure given by the linear correlation coefficient is insufficient to describe the dependence structure (see Embrechts et al. 2003). This is especially relevant when the bivariate Gaussian or Student-t distributions (both widely used in multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models) do not adequately represent the joint distribution function of the data. Second, computation of the CoVaR using copulas is computationally more tractable than obtaining the value of CoVaR using Eq. (3.4), as the equation requires numerical resolution of a double integral and VaR computation for the financially distressed market. Note that, using a copula Chapter 3 52 characterization of the CoVaR, we only need information on the cumulative probability for the VaR and not the value of the VaR itself. Adrian and Brunnermeier (2011) and Girardi and Ergün (2013) define the systemic risk contribution of a particular market j as the delta CoVaR (  CoVaR), which is the difference between the VaR of the European debt market as a whole conditional on the distressed state of market j ( jj tt R VaR ,  ) and the VaR of the European debt market as a whole conditional on the benchmark state of market j, considering it as the median of the return distribution of market j, or, alternatively, the VaR for an 0.5 . The systemic risk contribution of market j is thus defined as: d j d j d j d j t t t t CoVaR CoVaR CoVaR CoVaR | | | , 0.5 | , 0.5 , , ,         . (3.7) 3.2.2. Marginal distribution and copula models The marginal models and copula specifications we used to compute the CoVaR measure for the European sovereign debt markets are described as follows. For the marginal models, following Bhanot et al. (2014), we consider that the conditional mean of the market returns for a European debt market j are given as a function of common and specific factors. Common factors are given by interbank interest rate changes, measured by changes in the Euribor rate (E), with a dummy variable (CRISIS) denoting periods before and after the onset of the European sovereign debt crisis as 0 and 1, respectively. Included as specific factors are the stock market index returns ( jt r, ) and market volatility ( jt vol , ) for each country and lagged values for the debt market returns. Thus, the marginal model for market j is specified as: p jj t j j h t h j j t j j t j t j t j t h R R r vol E Crisis ,0 , , , , 1                  , (3.8) with j t j t j t z , , ,    , where jt 2 ,  is the conditional variance, given by a threshold generalized autoregressive conditional heteroskedasticity (TGARCH) specification (Zakoian, 1994; Glosten et al., 1993): r m m j,t j j,k j,t k j,h j,t h j,h j,t h j t k h h b a d Crisis 2 2 2 1 1 1                     , (3.9) Systemic risk in European sovereign debt markets: A CoVaR-copula approach 53 where j  is a constant, j,t k 2  is the GARCH component and j,t h  is the autoregressive conditional heteroskedasticity (ARCH) component and where  captures leverage effects. If 0 , then the future conditional variance will proportionally increase more following a negative shock than following a positive shock of the same magnitude. The crisis dummy variable is included in the volatility specification to take into account the potential effect of the sovereign debt crisis on volatility. jt z, is a i.i.d. random variable with zero mean and unit variance that follows a Hansen’s (1994) skewed-t density distribution. It is given by: jt jt bz a jt jt bz a jt bc z a b fz bc z a b , , ( 1) 2 2 1 , 21 ,( 1) 2 2 1 , 21 1 ( ; , ) 1                                   , (3.10) where  and  are the degrees of freedom parameter ( 2    ) and the symmetric parameter ( 11    ), respectively. The constants a, b and c are given by   ac 2 1 4   , ba 2 2 2 13    ,     c1 22 ( 2)         . This distribution converges to the standard Gaussian as 0 and    , and to the symmetric Student-t distribution as 0 and  is finite. We used seven different copula specifications to capture different characteristics of dependence, as follows: 1) The bivariate Gaussian is the most commonly employed distribution in the finance literature. It is defined by   11 N C (u,v; ) (u), (v)       , where  is the bivariate standard normal cumulative distribution function with correlation  between X and Y and where 1(u)   and 1(v)   are standard normal quantile functions. It has no tail dependence. 2) The Student-t is useful for capturing symmetric tail dependence. It is given by      11 ST C (u,v; , ) T(t (u),t (v)) , where T is the bivariate Student-t cumulative distribution function with degree-of-freedom parameter  and correlation  and where 1 t (u)   and 1 t (v)   are the quantile functions of the univariate Student-t distribution with degree-of-freedom parameter  . Chapter 3 54 3) The Gumbel copula is asymmetric and displays upper tail dependence and lower tail independence. It is given by                    1 G C (u,v; ) exp logu log v . Note that the two variables are independent when 1 . 4) The rotated Gumbel copula displays upper tail independence and lower tail dependence. It is given by         RG G C (u,v; ) u v 1 C (1 u,1 v; ) . 5) The BB7 copula allows for different degrees of upper and lower tail dependence. It is defined as:     1 1 BB7 C (u,v; , ) 1 1 1 1 u 1 1 v 1                                      , (3.11) with, 1 0 . 6) The Plackett copula is a symmetric copula which, like the Gaussian copula, exhibits tail independence, even though the dependence for large joint realizations is less than for the Gaussian copula. It is given by:                               2 P 1 C (u,v; ) 1 1 u v 1 1 u v 4 1 uv 21 . (3.12) 7) The Clayton-Gumbel or BB1 copula allows for asymmetric tail dependence. It is specified as     1 1 CG C (u,v; , ) u 1 v 1 1                      , (3.13) with 0 , 1 . In addition, we considered possible time-varying dependence by allowing the parameters of some copulas to vary according to a specific evolution equation. For the Gaussian and the Student-t copulas, we specified the linear dependence parameter t  so that it evolves according to a model with 1 autoregressive term and q moving-average terms, that is, an ARMA(1,q)-type process (Patton, 2006):  q t t t j t j qj(u ) (v )               11 1 0 1 1 2 1 , (3.14) where    xx (x) e e       1 11 is the modified logistic transformation that keeps the value of t  in (-1,1). The dependence parameter is explained by a constant, 0 , by an autoregressive term, 1 , and by the average product over the last q observations of the transformed variables, 2 . For the Student-t copula, 1(x)   is substituted by Systemic risk in European sovereign debt markets: A CoVaR-copula approach 55  1 t (x) . Similarly, we consider time-varying dependence for the Gumbel and its rotated version by assuming that their parameters follow the dynamics represented by the following equation: q t t t j t j qjuv 1 11           . (3.15) Overall, the copula family considered here can be classified as either symmetric copulas with tail dependence (Student-t and time-varying Student-t copulas) or tail independence (Gaussian, time-varying Gaussian and Plackett) or as asymmetric copulas with tail dependence (Gumbel, rotated Gumbel, BB7, BB1, time-varying Gumbel and rotated Gumbel). The parameters of the marginal and copula models are estimated using a twostep procedure called inference for the margins (Joe and Xu, 1996). The likelihood function is given by: dj tt d j d j t t t t t RR R R c u v f R f R( , ) ( , ) ( ) ( )f , (3.16) where c(u,v) is the copula density and d t d t R fR() and j t j t R fR() are the marginal densities of d t R and j t R , respectively. The log likelihood function can thus be decomposed as the sum of the log likelihood function of the marginals plus the log likelihood of the copula density. Thus, in a first step, we estimate the parameters of the marginal distributions separately by maximum likelihood and, in a second step, we estimate the parameters of the copula by solving the following problem:      T tt t1 ˆˆ arg max lnc(u ,v ; ) , (3.17) where  are the copula parameters and d t d t R t d ˆ ˆ u F (R ; ) and j t j t t j Rˆ ˆ v F (R ; ) are pseudo-sample observations for the copula. Under standard regularity conditions, this two-step estimation is consistent and the parameter estimates are asymptotically efficient and normal (see Joe, 1997). The number of lags in the mean and variance equations for each series is selected according to the Akaike information criteria (AIC) and the performance of the different copula models is evaluated using the AIC adjusted for small-sample bias, as in Breymann et al. (2003) and Reboredo (2011; 2013). Chapter 3 56 3.3. Data We empirically evaluated systemic risk in European sovereign debt markets by considering weekly data for sovereign bond benchmark price indices for France, Germany, the Netherlands, GIIPS markets and the overall sovereign bond price index for the EMU. Benchmark bond price indices were sourced from Datastream for 10-year maturities covering the period 7 January 2000 to 26 October 2012. Thus, we evaluated the VaR of the European sovereign debt market, represented by the EMU, conditional on the fact that a specific European debt market was in financial distress. Figure 3.1 displays the benchmark bond price return dynamics (computed on a continuous compounding basis) for all the debt markets considered, and also for the EMU index returns, showing differences in the size and timing of price movements (especially relevant after the onset of the debt crisis at the end of 2009). Price volatility significantly changed for the GIIPS markets with the onset of the debt crisis, whereas volatility dynamics for the non-crisis countries (Germany, France and the Netherlands) remained relatively stable. Table 3.1 reports descriptive statistics for bond price returns. The average returns were similar across different debt markets and the corresponding standard deviations were larger for GIIPS markets than for the non-crisis markets. Also, differences between the maximum and minimum price returns show that price ranges were greater for GIIPS. Negative values for skewness were more pronounced for Greece than for the other debt markets (suggesting a greater probability of large decreases), with Ireland, Italy and Spain showing positive skewness. All return series showed high values for the kurtosis statistic, consistent with fat tails in the returns distributions; in fact, the Jarque-Bera test strongly rejected the normality of the unconditional distribution for all the series. The Ljung-Box statistic suggested the presence of serial correlation only in GIIPS returns. The autoregressive conditional heteroskedasticity-Lagrange multiplier (ARCH-LM) statistic indicated that ARCH effects could be found in all the returns series. Finally, the results of the Dickey and Fuller (1979) and Phillips and Perron (1988) non-stationarity tests and the Kwiatkowski et al. (1992) stationarity test confirm that all debt return series were stationary. For the explanatory variables, we obtained weekly data on stock market indices for each country from Datastream, and also on the Eurostoxx 50 index for the EMU. Stock market volatility for each index at any time was computed as the standard deviation of the daily returns in a backward window of three months. Data Systemic risk in European sovereign debt markets: A CoVaR-copula approach 57 for the Euribor at one year were also sourced from Datastream. Figure 3.2 depicts the stock price return dynamics, showing an abrupt change around the onset of the debt crisis that was common to all stock markets. This fact is corroborated by the behaviour of the market volatility data depicted in Figure 3.3. Chapter 3 64 Table 3.3: Descriptive statistics for market volatility and Euribor at 1 year. France Germany Greece Ireland Italy Netherlands Portugal Spain EMU Euribor 1y Mean 0.103 0.106 0.116 0.094 0.100 0.100 0.076 0.102 0.261 0.000 Std. Dev. 0.045 0.048 0.048 0.047 0.048 0.052 0.035 0.044 0.106 0.001 Maximum 0.263 0.247 0.234 0.302 0.250 0.281 0.205 0.248 0.813 0.003 Minimum 0.044 0.045 0.047 0.034 0.035 0.040 0.024 0.041 0.116 -0.003 Skewness 1.280 1.178 0.426 2.066 0.915 1.497 1.272 0.855 1.540 -0.468 Kurtosis 4.712 3.740 2.130 8.599 3.374 4.855 5.467 3.790 6.238 6.835 J-B 264.06* 169.69* 41.27* 1347.99* 97.15* 345.17* 349.50* 98.70* 556.05* 433.73* Q(20) 7600.1 8054.8 8983.6 9428.9 8098.5 7964.8 7634.3 7518.3 5613.7 442.53 [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] ARCH 2382.7 1974.5 1047.3 3307.9 1831.2 3532.8 2140.5 2212.1 69.9 9.6 [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] Notes. See Table 3.1 notes. Systemic risk in European sovereign debt markets: A CoVaR-copula approach 65 Table 3.4: Pearson correlation coefficients. France Germany Greece Ireland Italy Netherlands Portugal Spain EMU France 0.69 -0.09 0.09 0.22 0.80 -0.02 0.23 0.69 Germany 0.95 -0.36 -0.10 -0.19 0.93 -0.16 -0.09 1.00 Greece 0.81 0.78 0.27 0.26 -0.28 0.29 0.22 -0.36 Ireland 0.82 0.80 0.85 0.40 0.00 0.51 0.35 -0.10 Italy 0.89 0.87 0.91 0.86 -0.08 0.29 0.75 -0.19 Netherlands 0.95 0.96 0.86 0.86 0.92 -0.12 0.00 0.93 Portugal 0.90 0.89 0.91 0.88 0.94 0.94 0.25 -0.16 Spain 0.94 0.94 0.86 0.87 0.93 0.97 0.95 -0.09 EMU 0.95 1.00 0.78 0.80 0.87 0.96 0.89 0.94 Notes. EMU indicates European Economic and Monetary Union. The lower triangular matrix reports the Pearson’s correlation coefficient in the pre-onset period and the upper triangular matrix shows the Pearson’s correlation coefficient in the post-onset period. 3.4. Empirical results 3.4.1. Marginal model results Table 3.5 displays estimation results for the marginal models specified in Eqs. (3.8)- (3.10) for sovereign debt returns. Marginal models were estimated by considering different combinations of the parameters p, r and m for values ranging from zero to a maximum lag of two; the most suitable model was selected according to AIC values. The evidence reported in Table 3.5 indicates that stock market returns had a negative impact on sovereign debt returns for all debt markets except Greece, Ireland, Italy and Portugal; stock market volatility, meanwhile, had no significant impact on average returns except for French debt returns and for the EMU index. Interest rate dynamics had a significant negative impact on debt returns in all series. The results for the crisis dummy variable indicate that the sovereign debt crisis negatively impacted average returns only in Greece and Portugal. Regarding debt return volatility, the empirical results confirm that volatility was persistent across different debt markets and leverage effects were hardly observed (Ireland, Italy and Spain were the exceptions). Consistent with the descriptive evidence on nonnormality and fat tails reported in Table 3.1, the estimated values for the degrees of freedom and the symmetry parameter of the skewed Student-t distribution confirm that the error terms are not normal and in some cases are asymmetric. We also checked the goodness-of-fit of our marginal models. The last rows of Table 3.5 indicate that neither autocorrelation nor ARCH effects remain in the residuals of the marginal models. Furthermore, we tested for the adequacy of the Chapter 3 66 skewed-t distribution model, testing the null hypothesis that the standardized model residuals were uniform (0,1) by comparing the empirical distribution and the theoretical distribution function using the well-known Kolmogorov-Smirnov, Cramér-von Mises and Anderson-Darling tests. The p-values for these tests, reported in last three rows of Table 3.5, indicate that, for either of the marginal models, the null of the correct specification of the distribution function could not be rejected at the 5% significance level. Overall, our goodness-of-fit tests indicate that the marginal distribution models were not mis-specified, so the copula model could correctly capture dependence between debt markets. 3.4.2. Copula model results We considered the potential effects of the European sovereign debt crisis on dependence by estimating copula models for the preand post-onset periods, delimitating both periods according to the information reported by the crisis dummy variable. We also considered the systemic risk of each country for the European debt market as a whole and the systemic risk of Greece—the main country affected by the debt crisis—for other European debt markets. For the pre-onset and post-onset periods we estimated eight copula pairs for the EMU index returns and each country’s debt index returns; we also estimated seven copula pairs for the Greek debt index returns and the debt market returns for each of the other countries. Tables 3.6 and 3.7 report the copula model results for the EMU paired with each European country in the preand post-onset periods, respectively. The evidence from both static and time-varying copulas indicate that all the debt markets strongly co-moved with the EMU index, providing consistent evidence of tail dependence and time-varying (TVP) dependence. In fact, according to the AIC values, TVP-rotated Gumbel copula and the Student-t offered the best fit for all markets, meaning that there was lower tail dependence and, in one case, symmetric tail dependence. Table 3.7 reports the results for the post-onset period, showing how the shape of dependence changed completely. With the onset of the crisis, European debt markets decoupled: GIIPS debt markets moved in the opposite direction to the general debt index, whereas the non-crisis countries (France, Germany and the Netherlands) continued to strongly co-move (mainly Germany) with the EMU index as in the pre-onset period. Tail dependence results also indicated that Greece and Portugal decoupled from the EMU index under extreme market movements, since, according to the AIC, the best copula fit displayed tail independence (the Gaussian copula for Greece and the Plackett copula for Portugal). The other countries in Systemic risk in European sovereign debt markets: A CoVaR-copula approach 67 crisis displayed lower symmetric tail dependence — as represented by the TVPStudent-t copula — in the post-onset period compared to the pre-onset period. For the non-crisis countries, tail dependence also changed, with France and the Netherlands showing lower tail independence and with Germany showing asymmetric tail dependence. Obviously, such changes in the dependence structure, and in particular in tail dependence, have implications for systemic risk that will be considered below. Tables 3.8 and 3.9 report results for the copula models for Greece paired with each European country in the preand post-onset periods. The evidence provided by the static and TVP copulas shows that the Greek debt market strongly co-moved with all the European debt markets, indicating lower tail dependence and upper tail independence. According to the AIC, the TVP-rotated Gumbel copula was the model that best fit all countries, with the exception of Italy, where the TVP-Gumbel copula offered the best fit, displaying upper tail dependence and lower tail independence with Greece. Table 3.9 reports dependence results for the post-onset period, showing that Greek debt market dependence on other European markets completely changed with the onset of the debt crisis. The Greek market decoupled from the non-crisis countries (France, Germany and the Netherlands), moving in the opposite direction on average and showing independence at the tails. The Greek debt market continued to co-move with the other countries in crisis in the post-onset period, although the intensity of co-movement decreased considerably. According to the AIC, the best copula fit revealed that, in times of extreme downturns, the Greek debt market did not co-move with the Irish, Italian, Portuguese or Spanish debt markets. There was also upper tail independence with the Greek market for all these markets except Portugal. It is important to point out that, according to our copula results, the Greek crisis at its depth did not have any spillover effect on the general EMU market index, given that this index is mainly based on non-crisis countries. This evidence is consistent with the results reported in Bhanot et al. (2014) for a multivariate GARCH model. Chapter 3 68 Table 3.5: Parameter estimates for the marginal distribution models. France Germany Greece Ireland Italy Netherlands Portugal Spain EMU Mean 0  0.001* 0.001* 0.001 0.000 0.001 0.001 0.001* 0.001* 0.001* (2.18) (1.99) (0.85) (0.37) (1.77) (1.58) (2.10) (1.96) (2.04) AR(1) -0.112* -0.091* -0.042 -0.125* -0.121* -0.094* (-2.64) (-2.18) (-1.00) (-3.07) (-2.99) (-2.27) Euribor -4.726* -4.388* -4.754* -4.957* -4.673* -4.429* -5.017* -4.824* -4.574* (-8.97) (-9.03) (-11.24) (-9.78) (-11.60) (-9.70) (-10.47) (-11.28) (-9.07) EMU Index -0.068* -0.069* -0.003 -0.013 -0.012 -0.068* -0.014 -0.026* -0.091* (-5.74) (-6.30) (-0.39) (-1.06) (-0.91) (-5.91) (-1.05) (-2.33) (-6.21) Market Vol. -0.011* -0.009 -0.003 0.000 -0.008 -0.007 -0.015 -0.008 -0.005* (-1.92) (-1.93) (-0.45) (0.05) (-1.68) (-1.29) (-1.82) (-1.49) (-2.03) Dummy 0.001 0.001 -0.012* -0.001 0.000 0.001 -0.003* -0.001 0.001 (1.197) (1.50) (-4.32) (-0.90) (-0.23) (1.49) (-2.16) (-1.33) (1.72) Variance  1.968* 1.531* 0.035* 0.019* 1.291* 1.870* 0.014* 0.015* 1.567* (2.35) (2.35) (3.74) (2.16) (2.54) (2.073) (3.05) (2.64) (2.27) 1  0.087* 0.085* 0.034 -0.003 0.000 0.087* 0.038 0.010 0.094* (3.04) (2.68) (0.50) (-0.08) (0.01) (2.543) (1.47) (0.38) (2.72) 1  0.867* 0.885* 0.851* 0.884* 0.890* 0.863* 0.886* 0.882* 0.879* (29.61) (37.38) (21.51) (20.26) (40.50) (24.11) (48.30) (33.18) (34.66)  0.002 -0.020 0.158 0.162* 0.166* 0.005 0.104 0.147* -0.028 (0.04) (-0.49) (1.29) (3.45) (3.37) (0.094) (1.840) (2.226) (-0.65) Dummy 0.000* 0.000* 0.000* 0.000* 0.000* 0.000* 0.000* 0.000* 0.000* (32.81) (99.43) (87.63) (151.6) (42.44) (75.75) (227.2) (186.3) (93.75) Asymmetry -0.034 -0.116* -0.165* -0.115* -0.117 -0.073 -0.127* -0.144* -0.091 (-0.61) (-2.31) (-2.68) (-2.12) (-1.85) (-1.26) (-1.93) (-2.48) (-1.69) Tail 6.827* 8.538* 3.348* 5.257* 6.424* 8.390* 5.614* 7.252* 8.489* (72.08) (39.20) (4.02) (14.36) (105.6) (117.2) (39.93) (83.69) (128.8) LogLik 2404.47 2412.65 2120.94 2259.65 2390.82 2428.22 2203.12 2332.55 2414.79 LJ 18.129 16.061 24.227 16.418 29.691 19.782 18.374 19.710 17.921 [0.514] [0.653] [0.233] [0.629] [0.075] [0.472] [0.498] [0.412] [0.528] LJ 2 22.954 23.810 27.523 27.278 15.443 19.416 21.691 12.057 26.933 [0.192] [0.161] [0.051] [0.074] [0.631] [0.367] [0.246] [0.844] [0.080] ARCH 1.065 1.403 1.456 1.365 0.826 0.969 0.939 0.603 1.549 [0.383] [0.113] [0.090] [0.133] [0.682] [0.498] [0.537] [0.912] [0.060] K-S [0.986] [0.946] [0.967] [0.892] [0.882] [0.903] [0.940] [0.948] [0.531] C-vM [0.990] [0.864] [0.938] [0.812] [0.989] [0.926] [0.989] [0.996] [0.740] A-D [0.992] [0.911] [0.963] [0.896] [0.998] [0.907] [0.992] [0.999] [0.806] Note. The table presents the maximum likelihood estimates and the z statistics (in parentheses) for the parameters of the marginal distribution models given by Eqs. (8)-(10). LogLik is the log-likelihood value. LJ denotes the Ljung-Box statistic for serial correlation in the residual model calculated with 20 lags. LJ2 denotes the Ljung-Box statistic for serial correlation in the squared residual model calculated with 20 lags. ARCH is Engle’s LM test for the ARCH effect in the residuals up to 20th order. K-S, C-vM and A-D denote the Kolmogorov-Smirnov, Cramér-von Mises and Anderson-Darling tests for adequacy of the skewed-t distribution model. P values (in square brackets) below 0.05 indicate rejection of the null hypothesis. An asterisk (*) indicates significance at 5%. EMU indicates European Economic and Monetary Union. Systemic risk in European sovereign debt markets: A CoVaR-copula approach 69 Table 3.6: Estimates for copula models in the period before crisis onset. European Economic and Monetary Union (EMU) vs. selected countries. Panel A: Parameter estimates for time-invariant copulas. Copula France Germany Greece Ireland Italy Netherlands Portugal Spain Gaussian  0.936* 0.981* 0.834* 0.835* 0.868* 0.943* 0.880* 0.909* (0.00) (0.00) (0.01) (0.10) (0.01) (0.00) (0.01) (0.01) AIC -1067.493 -1680.860 -603.613 -608.745 -713.692 -1123.293 -756.375 -890.382 Student-t  0.952* 0.983* 0.858* 0.868* 0.881* 0.948* 0.897* 0.916* (0.01) (0.00) (0.01) (0.01) (0.01) (0.00) (0.01) (0.01)  1.828* 3.510* 2.841* 2.322* 4.214* 2.495* 3.247* 3.646* (0.26) (0.80) (0.53) (0.36) (0.99) (0.36) (0.59) (0.82) AIC -1266.306 -1760.424 -688.291 -724.814 -759.001 -1214.324 -824.799 -936.630 Gumbel  4.674* 7.955* 2.764* 2.840* 3.001* 4.593* 3.174* 3.528* (0.18) (0.30) (0.10) (0.11) (0.11) (0.17) (0.12) (0.13) AIC -1117.217 -1687.119 -619.432 -643.766 -715.031 -1115.740 -748.857 -865.385 Rotated Gumbel  4.971* 7.969* 2.809* 2.862* 2.981* 4.759* 3.270* 3.662* (0.19) (0.30) (0.10) (0.11) (0.11) (0.18) (0.12) (0.14) AIC -1186.789 -1688.961 -644.116 -657.189 -704.506 -1166.714 -784.269 -901.423 BB7  4.162* 4.929* 2.421* 2.563* 2.801* 4.116* 2.711 3.073* (0.15) (0.00) (0.25) (0.18) (0.29) (0.22) (2.14) (0.49)  4.770* 8.747* 2.051* 1.930* 1.877* 4.461* 2.469* 2.937* (0.27) (0.03) (0.39) (0.18) (0.22) (0.31) (1.22) (0.85) AIC -1136.301 -1604.134 -633.823 -645.491 -696.002 -1131.798 -748.137 -870.589 Plackett  134.543* 294.951* 31.904* 40.619* 37.994* 110.522* 46.854* 56.622* (13.34) (32.99) (3.49) (4.60) (3.25) (15.88) (4.46) (5.43) AIC -1256.681 -1691.174 -672.451 -733.420 -755.312 -1189.036 -827.383 -921.424 BB1  1.051* 0.827* 0.679* 0.599* 0.479* 0.947* 0.730* 0.783* (0.15) (0.13) (0.12) (0.12) (0.11) (0.14) (0.12) (0.13)  3.264* 5.876* 2.150* 2.268* 2.496* 3.282* 2.433* 2.668* (0.21) (0.34) (0.12) (0.13) (0.14) (0.21) (0.14) (0.16) AIC -1198.671 -1746.093 -660.372 -677.892 -739.670 -1187.867 -796.358 -918.379     Chapter 3 70 Panel B: Parameter estimates for time-varying copulas. Copula France Germany Greece Ireland Italy Netherlands Portugal Spain TVP-Gaussian 0  0.697 -0.611 -1.198* -0.870 -1.749 -1.050 -1.836 -1.484 (4.02) (17.72) (0.03) (70.06) (66.30) (22.01) (10.09) (452.12) 1  0.111 0.422 0.309* 0.443 0.078 -0.132 0.249 -0.007 (0.06) (1.64) (0.02) (5.03) (2.76) (0.33) (0.27) (2.44) 2  2.796 5.000 4.087* 3.658 5.000 5.000 5.000 5.000 (4.29) (16.58) (0.05) (106.19) (73.45) (23.11) (11.74) (501.79) AIC -1064.763 -1691.338 -628.580 -656.319 -710.828 -1121.542 -762.466 -886.352 TVP-Student 0  7.669* 44.715 5.629* 7.100* 9.028* -0.050 6.440 7.761 (0.93) (115.39) (1.03) (1.09) (1.94) (0.11) (6.81) (20.72) 1  -0.017 0.001 -0.082 -0.051 -0.091 0.337 -0.289 -0.031 (0.02) (0.00) (0.04) (0.03) (0.09) (0.28) (0.29) (0.11) 2  -3.975* -40.634 -3.406* -5.000* -6.974* 0.934 -3.260 -5.000 (1.07) (117.53) (1.20) (1.21) (2.22) (1.16) (7.34) (22.96)  1.951* 3.508* 2.344* 2.084* 3.822* 5.032* 11.020* 3.518 (0.29) (1.18) (0.37) (0.29) (0.84) (2.14) (5.62) (2.04) AIC -1269.201 -1756.383 -688.087 -723.210 -755.618 -1214.939 -275.838 -932.676 TVP-Gumbel  1.412* 1.744* 1.661* 1.449* 1.238* 1.353* 1.603* 1.892 (0.11) (0.24) (0.41) (0.29) (0.61) (0.10) (0.40) (1.13)  0.172* 0.138* 0.095 0.148* 0.176 0.179* 0.112 0.049 (0.01) (0.02) (0.09) (0.05) (0.12) (0.01) (0.08) (0.21)  -4.429* -5.000* -4.880* -4.422* -3.295 -4.174* -4.757* -5.000 (0.88) (1.82) (1.48) (1.17) (2.08) (0.84) (1.56) (4.11) AIC -1275.875 -1730.546 -698.930 -768.086 -766.341 -1198.528 -799.470 -903.540 TVP-Rotated Gumbel  1.320* 1.702* 1.316* 1.457* 0.773* 1.264* 1.461* 1.696* (0.06) (0.23) (0.25) (0.21) (0.05) (0.09) (0.21) (0.36)  0.180* 0.144* 0.167* 0.149* 0.266* 0.186* 0.147* 0.111 (0.01) (0.01) (0.05) (0.03) (0.01) (0.01) (0.04) (0.06)  -3.451* -5.000* -3.716* -4.494* -1.664* -3.172* -4.309* -5.000* (0.53) (2.52) (0.99) (0.91) (0.32) (0.67) (0.91) (1.47) AIC -1335.555 -1757.076 -725.442 -782.999 -766.521 -1237.901 -842.849 -963.021 Notes. The table reports the ML estimates for the different copula models for the EMU index returns and debt index returns for the European countries indicated. Standard error values (in brackets) and Akaike information criterion (AIC) values adjusted for small-sample bias are provided for the different copula models. The minimum AIC value (in bold) indicates the best copula fit. For the TVP-Gaussian and TVP-Student-t copulas, q in Eq. (3.13) was set to 10. An asterisk (*) indicates significance at the 5% level. Systemic risk in European sovereign debt markets: A CoVaR-copula approach 71 Table 3.7: Estimates for copula models in the period after crisis onset. European Economic and Monetary Union (EMU) vs. selected countries. Panel A: Parameter estimates for time-invariant copulas. Copula France Germany Greece Ireland Italy Netherlands Portugal Spain Gaussian  0.679* 0.994* -0.370* -0.156* -0.182* 0.914* -0.194* -0.103 (0.03) (0.00) (0.06) (0.07) (0.07) (0.01) (0.07) (0.07) AIC -94.074 -686.627 -20.824 -1.812 -3.188 -279.631 -3.928 0.364 Student-t  0.684* 0.994* -0.362* -0.175* -0.154 0.916* -0.192* -0.108 (0.04) (0.00) (0.06) (0.09) (0.09) (0.01) (0.08) (0.08)  10.311 17.672 59.756 5.179* 5.393 11.644 7.905 3.810* (7.53) (13.18) (643.5 2) (2.43) (2.86) (20.36) (4.69) (1.57) AIC -93.371 -685.348 -18.805 -5.746 -5.473 -278.957 -4.300 -4.192 Gumbel  1.821* 13.132* 1.000* 1.000* 1.000* 3.475* 1.000* 1.000* (0.12) (0.88) (0.10) (0.08) (0.08) (0.23) (0.08) (0.08) AIC -89.144 -666.026 2.026 2.026 2.026 -263.297 2.026 2.026 Rotated Gumbel  1.838* 13.029* 1.000* 1.005* 1.000* 3.558* 1.000* 1.000* (0.12) (0.87) (0.11) (0.02) (0.09) (0.24) (0.09) (0.09) AIC -88.574 -666.476 2.026 1.973 2.026 -267.907 2.026 2.026 BB7  1.666* 5.733* 1.001* 1.001* 1.001* 3.113* 1.001* 1.001* (0.19) (0.30) (0.21) (0.38) (0.35) (0.17) (0.38) (0.43)  0.913* 10.474* 0.001 0.001 0.001 2.635* 0.001 0.001 (0.21) (0.95) (0.99) (0.95) (0.95) (0.17) (0.95) (0.94) AIC -88.226 -588.400 4.261 4.109 4.159 -258.116 4.178 4.119 Plackett  9.939* 665.125* 0.366* 0.507* 0.606* 48.488* 0.540* 0.700* (2.00) (130.30) (0.08) (0.13) (0.15) (10.71) (0.13) (0.18) AIC -94.752 -648.064 -15.609 -4.192 -2.079 -270.526 -4.663 0.164 BB1  0.403* 0.924* 0.001 0.001 0.001 0.675* 0.001 0.001 (0.18) (0.25) (0.84) (0.87) (0.87) (0.22) (0.85) (0.86)  1.568* 9.389* 1.001 1.001 1.001 2.720* 1.001 1.001 (0.15) (1.04) (0.66) (0.57) (0.58) (0.28) (0.64) (0.59) AIC -92.345 -686.060 4.327 4.135 4.188 -274.670 4.215 4.132     Chapter 3 72 Panel B: Parameter estimates for time-varying copulas. Copula France Germany Greece Ireland Italy Netherlands Portugal Spain TVP-Gaussian 0  2.916 5.000 -0.845 -0.185 -0.290 5.000 -0.388 -0.003 (41.03) (83.18) (0.49) (0.18) (2.50) (8.76) (1.36) (0.27) 1  0.116 0.015 -0.372 -0.248 1.130 -0.284 -0.148 1.201* (5.85) (25.41) (0.48) (0.22) (0.86) (0.38) (0.47) (0.33) 2  -2.000 1.000 0.126 1.160 -2.000 -1.761 0.116 -2.000* (63.09) (56.93) (1.02) (0.77) (5.33) (9.33) (6.93) (0.05) AIC -90.021 -682.559 -17.566 0.373 -9.741 -276.408 -0.009 -4.988 TVP-Student 0  1.595 106.808 -0.838* -0.021 -0.044 6.440 -0.583 -0.070 (2.07) (161.24) (0.26) (0.03) (0.04) (6.81) (1.24) (0.12) 1 -0.195 0.155 -0.362 -0.110* 0.181 -0.289 -0.099 0.299 (0.27) (0.32) (0.29) (0.06) (0.14) (0.29) (0.37) (0.23) 2 0.363 -101.562 0.132 2.159* 1.600* -3.260 -0.817 0.807 (2.71) (161.13) (0.66) (0.10) (0.48) (7.34) (6.21) (0.93)  6.862 17.439 100.000* 5.309* 4.833* 11.020* 7.982 4.420* (5.16) (19.73) (0.00) (1.48) (1.93) (5.62) (5.82) (1.88) AIC -89.577 -681.335 -15.464 -8.163 -15.034 -275.838 -0.194 -6.759 TVP-Gumbel  2.167* 1.984* 0.000 0.000 0.000 0.771* 0.000 0.000 (0.62) (0.23) (1.00) (1.00) (1.00) (0.21) (1.00) (1.00)  -0.154 0.123* 0.000 0.000 0.000 0.258* 0.000 0.000 (0.21) (0.02) (1.00) (1.00) (1.00) (0.04) (1.00) (1.00)  -5.000* -4.619 0.000 0.000 0.000 -1.107 0.000 0.000 (1.25) (2.46) (1.00) (1.00) (1.00) (0.76) (1.00) (1.00) AIC -112.068 -663.314 6.175 6.163 6.167 -263.027 6.169 6.163 TVP-Rotated Gumbel  0.847 2.920 0.000 -1.399* 0.000 0.633* 0.000 0.000 (0.48) (1.82) (1.00) (0.53) (1.00) (0.10) (1.00) (1.00)  0.252 0.052 0.000 1.001 0.000 0.286* 0.000 0.000 (0.13) (0.13) (1.00) (0.62) (1.00) (0.02) (1.00) (1.00)  -2.317 -5.000 0.000 1.248 0.000 -0.655 0.000 0.000 (1.24) (12.52) (1.00) (0.95) (1.00) (0.39) (1.00) (1.00) AIC 1.000 -662.873 6.173 4.807 6.161 -267.848 6.161 6.160 Notes. See Table 3.6 notes. Systemic risk in European sovereign debt markets: A CoVaR-copula approach 73 Table 3.8: Estimates for the copula models in the period before crisis onset. Greece vs. country. Panel A: Parameter estimates for time-invariant copulas. Copula France Germany Ireland Italy Netherlands Portugal Spain Gaussian  0.856* 0.834* 0.916* 0.932* 0.867* 0.944* 0.915* (0.01) (0.01) (0.01) (0.00) (0.01) (0.00) (0.01) AIC -671.901 -602.133 -932.624 -1035.026 -707.130 -1128.174 -925.587 Student-t  0.878* 0.857* 0.940* 0.949* 0.880* 0.952* 0.931* (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01)  2.318* 3.061* 1.343* 1.521* 2.287* 2.489* 2.094* (0.43) (0.57) (0.16) (0.19) (0.34) (1.01) (0.29) AIC -772.265 -682.688 -1147.700 -1200.938 -784.636 -1213.367 -1044.899 Gumbel  3.060* 2.743* 4.266* 4.647* 3.065* 4.812* 3.960* (0.12) (0.10) (0.17) (0.18) (0.12) (0.18) (0.15) AIC -704.236 -616.276 -982.597 -1073.074 -710.077 -1125.291 -932.318 Rotated Gumbel  3.035* 2.798* 4.496* 4.766* 3.148* 4.971* 4.200* (0.11) (0.10) (0.17) (0.18) (0.12) (0.18) (0.16) AIC -710.350 -640.877 -1049.016 -1108.588 -745.980 -1165.662 -1002.412 BB7  2.836 2.399* 3.639* 4.126* 2.692* 5.275* 3.353* (3.81) (0.23) (0.25) (0.01) (0.22) (0.04) (0.17)  2.143* 2.029* 4.483* 4.269* 2.553* 5.074* 4.036* (0.29) (0.14) (0.41) (0.03) (0.17) (0.00) (0.23) AIC -712.089 -630.231 -1022.834 -1068.123 -740.602 -1126.389 -969.510 Plackett  43.107* 31.535* 125.294* 143.040* 44.410* 117.270* 82.852* (6.44) (3.10) (6.78) (14.70) (3.60) (11.82) (5.60) AIC -763.799 -666.266 -1137.193 -1209.029 -768.939 -1190.713 -1030.318 BB1  0.609* 0.674* 1.093* 0.861* 0.806* 0.925* 1.068* (0.12) (0.13) (0.16) (0.12) (0.14) (0.15) (0.23)  2.425* 2.140* 2.942* 3.409* 2.290* 3.474* 2.750* (0.14) (0.13) (0.19) (0.20) (0.14) (0.22) (0.22) AIC -740.509 -656.994 -1061.755 -1130.593 -763.160 -1190.198 -1011.073     Chapter 3 80 Figure 3.4: (Continued) -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 Italy CoVaR Copula CoVaR Student-t ∆CoVaR -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 Netherlands CoVaR Copula CoVaR Student-t ∆CoVaR Systemic risk in European sovereign debt markets: A CoVaR-copula approach 81 Figure 3.4: (Continued) -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 Portugal CoVaR Copula CoVaR Student-t ∆CoVaR -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 Spain CoVaR Copula CoVaR Student-t ∆CoVaR Chapter 3 82 Student-t distribution or using a copula approach. Thus, regarding the pre-onset period, the copula approach to the CoVaR only had computational, not modelling, advantages, given that co-movement between debt markets and the EMU index was very high, with tail dependence also high for all copula specifications. In fact, there is no significant difference between the tail dependence arising from the TVProtated Gumbel copula (the best copula fit for almost all the markets) and the lower tail dependence arising from the Student-t copula. As a result, the CoVaR values for these two copula specifications did not differ too much, as Table 3.10 reports. Regarding the post-onset period, the dynamics of the estimated CoVaR—as displayed in Figure 3.4 to the right of the vertical line and according to the descriptive statistics in Table 3.10—indicate that values increased for the GIIPS, with the exception of Spain. This is consistent with the fact that the debt markets of countries in crisis decoupled from other debt markets after the onset of the European sovereign debt market crisis. In fact, the dependence structure drastically changed in that the tail dependence of the pre-onset period dissipated after the onset of the debt crisis. The rise in the CoVaR values indicate that the systemic risk of the crisis countries was reduced, a finding corroborated by the behaviour of the  CoVaR values, which, in some cases, dropped to negative values as a result of opposite movement between the country in question and the EMU index returns. In the case of Spain, lower tail dependence with the EMU index persisted, so CoVaR values dropped, thus furnishing evidence of an increase in systemic risk in the Spanish sovereign debt market. This result is consistent with the concerns of government authorities and financial media; it is also consistent with the fact that the Spanish debt market could be the drive-belt for the debt crisis in peripheral countries, bringing its repercussions to the hard core of Europe and generalizing the crisis to all European debt markets. In contrast, for the non-crisis countries, the CoVaR values dropped significantly, indicating that French, German and Dutch debt market risk became more systemic than in the pre-onset period. This result was a consequence of their decoupling from debt markets after the onset of the debt crisis; it was also due to the fact that the markets continued to strongly co-move with the EMU index. Systemic risk in European sovereign debt markets: A CoVaR-copula approach 83 Table 3.10: Summary CoVaR and  CoVaR statistics for the European Economic and Monetary Union (EMU) and selected countries. Before crisis onset After crisis onset Country Method Mean Std. Max Min Mean Std. Max Min France Student-t -3.566 0.979 -1.325 -7.914 -4.914 0.748 -3.314 -7.290 Copula -3.568 0.979 -1.325 -7.915 -4.086 0.650 -2.112 -5.560 ∆CoVaR 100.70 40.42 487.64 43.26 61.32 23.03 119.87 6.90 Germany Student-t -3.569 0.979 -1.325 -7.915 -5.131 0.806 -3.396 -7.856 Copula -3.569 0.979 -1.325 -7.914 -5.131 0.806 -3.396 -7.856 ∆CoVaR 100.71 40.42 487.68 43.26 96.77 10.39 137.42 75.64 Greece Student-t -3.536 0.958 -1.313 -7.825 -2.810 0.495 -1.585 -4.556 Copula -3.560 0.966 -1.325 -7.877 -0.897 0.302 -0.092 -1.916 ∆CoVaR 100.56 40.46 487.67 43.25 -48.43 8.56 -33.43 -88.59 Ireland Student-t -3.511 0.924 -1.313 -7.576 -2.154 1.713 0.000 -6.862 Copula -3.534 0.936 -1.325 -7.803 -3.001 1.149 -0.896 -6.704 ∆CoVaR 99.98 40.86 487.72 2.56 54.28 18.96 120.16 5.07 Italy Student-t -3.566 0.979 -1.324 -7.913 -5.129 0.806 -3.394 -7.853 Copula -3.568 0.978 -1.325 -7.904 -3.490 1.063 -0.231 -6.337 ∆CoVaR 100.69 40.42 487.66 43.25 66.13 26.43 118.17 -47.64 Netherlan ds Student-t -3.566 0.979 -1.324 -7.913 -5.129 0.806 -3.394 -7.853 Copula -3.569 0.979 -1.325 -7.915 -5.130 0.806 -3.395 -7.855 ∆CoVaR 100.71 40.42 487.64 43.26 96.73 10.39 137.36 75.60 Portugal Student-t -3.556 0.973 -1.313 -7.893 -2.691 0.480 -1.491 -4.386 Copula -3.568 0.978 -1.325 -7.912 -1.916 0.391 -0.887 -3.285 ∆CoVaR 100.69 40.42 487.65 43.25 -6.75 0.95 -4.95 -10.82 Spain Student-t -3.562 0.977 -1.320 -7.893 -3.253 0.942 -0.622 -6.540 Copula -3.568 0.977 -1.325 -7.911 -3.793 0.896 -1.760 -6.890 ∆CoVaR 100.70 40.42 487.65 43.26 78.65 13.56 125.20 43.60 Notes. The table reports descriptive CoVaR and  CoVaR statistics (percentages) at the 99% confidence level for the EMU and the VaR at the 99% level for selected countries in the preand postonset periods using the best copula fit and the multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model with bivariate Gaussian and Student-t distributions. Std., Max and Min denote standard deviation, maximum and minimum, respectively. Chapter 3 84 Empirical evidence for the post-onset period also revealed differences for CoVaR measures computed using different methods. In general, for the non-crisis countries, dependence continued to be strong after the onset of the debt crisis, with lower tail dependence changing in some cases. Tail dependence for France was lower, explaining the differences between the copula approach and the bivariate Student-t approach that can be observed in Figure 3.4 and in Table 3.10. However, tail dependence for Germany remained high through different copula specifications, so there was no almost difference between the CoVaR values obtained through the BB1 copula or the Student-t approach. This evidence was similar for the Netherlands, with no significant differences. Results for the countries in crisis in the post-onset period (Figure 3.4) provide striking evidence of differences in CoVaR values as computed using different approaches. These differences can be explained in terms of (lower) tail dependence: for the Greek and Portuguese markets there was tail independence, so the CoVaR values were reduced with respect to the values obtained using a bivariate Student-t distribution. Also, when there was (lower) tail dependence, as for the Italian, Portuguese and Spanish debt markets, CoVaR values computed using the copula approach differed with respect to the values for the bivariate Student-t distribution. Hence, one of the advantages of using copulas to compute the CoVaR lies in the fact that copulas offers more flexibility in terms of fitting dependence (and particularly tail dependence) than parametric bivariate distributions; consequently, they yield a more accurate CoVaR measure. Overall, our results can be summarized as follows. Before the onset of the debt crisis, the dynamics and extent of systemic risk in European sovereign debt markets were similar, evidence consistent with high co-movement or coupling between debt markets. However, debt markets decoupled with the onset of the debt crisis, with crisis countries even displaying negative dependence. As a result, systemic risk decreased for the GIIPS debt markets, although not for Spain, and systemic risk increased for the non-crisis countries. Our results have implications for investors in debt markets. First, evidence regarding decoupling after the onset of the crisis indicates that investors could find hedging opportunities using sovereign debt instruments; this would not have been possible before the crisis given the coupling between debt markets. Furthermore, our results regarding tail independence and reductions in systemic risk indicate that, after the onset of the debt crisis, investors could achieve downside risk reductions with a portfolio that included sovereign debt from different countries, mainly for Systemic risk in European sovereign debt markets: A CoVaR-copula approach 85 sovereign debt markets that moved independently or in the opposite direction to the EMU index. 3.4.3.1. Systemic risk in the Greek debt market Given that Greece was the leading protagonist of the European sovereign debt crisis, we estimated the CoVaR and  CoVaR values for each European debt market conditional on the VaR of the Greek debt market in order to evaluate the systemic impact of the Greek crisis and its differential effects across debt markets before and after the onset of the crisis. The dynamics of the CoVaR and  CoVaR for all debt markets in the preand post-onset periods (delimited by a vertical line) are depicted in Figure 3.5. As with Figure 3.4, for each country we included information on the CoVaR computed using the methodology proposed by Girardi and Ergün (2013). The graphical evidence indicates that Greek systemic risk was low and relatively stable in the pre-onset period. The impact of the global financial crisis is reflected in an abrupt fall in the CoVaR value. Summary statistics for the CoVaR and  CoVaR (in percentages) for the pre-onset period, as reported in Table 11, confirm that the systemic risk of the Greek debt market for other European debt markets was, on average, lower for the non-crisis countries than for the crisis countries. However, Greek systemic risk drastically changed from the onset of the Greek debt crisis, with the CoVaR value associated with the crisis countries experiencing a huge reduction. In contrast, non-crisis countries experienced an increase in the CoVaR value given that they decoupled from the Greek market. Moreover, CoVaR volatility increased substantially for the countries in crisis as a result of the uncertainty of the debt markets and the implementation of stabilization policies as formulated by the European Central Bank and the International Monetary Fund, provoking sudden changes in investor expectations. This evidence on systemic risk dynamics is consistent with the idea that the crisis had spillover effects on countries with weak economic fundamentals, and also had contagion effects, given that countries like France, Germany and the Netherlands—with no great economic difficulties—reduced their conditional VaR. As for the crisis countries, Portugal experienced the greatest impact, with a fall in its CoVaR values of up to -15% on average, followed by Ireland (-13%), Spain (-7.6%) and Italy (-5.9%). This evidence suggests that the Greek debt crisis particularly affected Portugal. This is consistent with the concerns of the financial media regarding fears that the Greek crisis could rapidly extend to Portugal. Chapter 3 86 Table 3.11: Summary CoVaR statistics for selected European debt markets and Greece. Before crisis onset After crisis onset Country Method Mean Std. Max Min Mean Std. Max Min France Student-t -3.980 1.080 -1.680 -10.249 -3.658 1.312 -1.573 -8.207 Copula -4.023 1.119 -1.688 -10.274 -1.876 0.716 -0.244 -4.275 ∆CoVaR 114.24 43.10 550.10 4.81 -7.96 2.38 -5.54 -32.22 Germany Student-t -3.621 0.945 -1.271 -8.041 -2.776 0.451 -1.699 -4.437 Copula -3.643 0.950 -1.284 -8.031 -0.871 0.289 0.060 -1.784 ∆CoVaR 101.31 43.90 597.29 44.24 -49.84 9.18 -33.79 -106.74 Ireland Student-t -5.560 2.175 -2.777 -15.875 -9.927 8.789 0.000 -36.085 Copula -5.581 2.188 -2.830 -15.958 -13.335 5.624 -4.798 -29.465 ∆CoVaR 146.57 43.82 702.18 89.91 79.62 1.86 84.25 72.99 Italy Student-t -4.402 1.786 -1.641 -14.113 -7.399 3.086 -3.624 -15.844 Copula -4.252 1.724 -0.563 -13.585 -5.962 2.502 -3.027 -12.845 ∆CoVaR 111.88 28.45 321.86 22.50 59.53 2.30 66.84 47.97 Netherlands Student-t -3.520 1.052 -1.282 -10.208 -3.083 0.754 -1.689 -5.313 Copula -3.547 1.051 -1.341 -10.226 -1.157 0.357 -0.048 -2.382 ∆CoVaR 100.28 39.38 513.80 44.45 -33.56 6.99 -22.27 -91.58 Portugal Student-t -5.330 2.321 -2.399 -18.431 -23.683 8.277 -6.366 -42.892 Copula -5.335 2.324 -2.405 -18.432 -15.167 4.916 -3.932 -28.369 ∆CoVaR 135.06 27.30 355.42 82.11 44.68 21.55 95.68 1.14 Spain Student-t -4.203 1.625 -1.692 -14.474 -9.511 2.915 -4.271 -16.290 Copula -4.214 1.630 -1.696 -14.484 -7.612 2.402 -3.424 -13.192 ∆CoVaR 111.00 32.44 457.41 62.43 52.23 3.07 61.98 45.01 Notes. The table reports descriptive CoVaR and  CoVaR statistics (percentages) at the 99% confidence level for the EMU and the VaR at the 99% level for Greece in the preand post-onset periods using the best copula fit and the multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model with bivariate Gaussian and Student-t distributions. Std., Max and Min denote standard deviation, maximum and minimum, respectively. Systemic risk in European sovereign debt markets: A CoVaR-copula approach 87 Figure 3.5: CoVaR (left axis) and ∆CoVaR (right axis) estimates for Greece with respect to selected countries. -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 Germany CoVaR Copula CoVaR Student-t ∆CoVaR Chapter 3 88 Figure: 3.5: (Continued) -8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Ireland CoVaR Copula CoVaR Student-t ∆CoVaR -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Italy CoVaR Copula CoVaR Student-t ∆CoVaR Systemic risk in European sovereign debt markets: A CoVaR-copula approach 89 Figure: 3.5: (Continued) -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 Netherlands CoVaR Copula CoVaR Student-t ∆CoVaR -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 -0.45 -0.35 -0.25 -0.15 -0.05 0.05 0.15 0.25 0.35 0.45 Portugal CoVaR Copula CoVaR Student-t ∆CoVaR Chapter 4 96 European financial systems were affected, they were affected to different degrees. For core countries Greek sovereign debt continued to play a diversification role (even though the intensity was less and more varied); whereas for peripheral countries like Belgium and the Netherlands, Greek sovereign systemic effects exacerbated the risk associated with their financial systems. This study adds to the burgeoning literature on the European sovereign debt crisis regarding links between sovereign debt markets and domestic and general financial sectors. De Bruyckere et al. (2013), for instance, examined contagion between bank and sovereign default risk in Europe through asset, collateral and rating channels. Bhanot et al. (2014) investigated how stock returns in the financial sector in crisis and non-crisis European countries were affected by the yield spreads for Greek sovereign debt. Mink and De Haan (2013) studied the impact of highly volatile Greek bonds on European bank stock prices in 2010. Alter and Schuler (2012) examined the relationship between sovereign default risk and domestic banking risk. For credit default swaps, other studies examined sovereign risk contagion among eurozone countries (e.g., Missio and Watzka, 2011; Arezki et al., 2011; Alter and Beyer, 2012; Caporin et al., 2013) and the effects of sovereign debt default risk on the financial stability of the eurozone (Radev, 2012). However, underrepresented in this literature is examination of the systemic impact of domestic sovereign distress on domestic financial systems or the impact of Greek sovereign debt distress on the financial systems of other countries. This chapter fills this gap by attempting to quantify CoVaR for financial and sovereign debt crises using procedures based on copulas and vine copulas. The remainder of the chapter is laid out as follows: in Section 4.2 we outline the copula and vine-copula approaches to CoVaR, in Section 4.3 we present our data and in Section 4.4 we discuss the results. Finally, Section 4.5 concludes the chapter. 4.2. Methodology We quantified the systemic impact of the sovereign debt market on the financial system using the CoVaR measure as introduced by Adrian and Brunnermeier (2011) and generalized by Girardi and Ergün (2013). CoVaR for the financial system of a country is VaR for the financial system conditional on the fact that the sovereign debt market is in financial distress. Let f t x be the returns for the financial system at time t and let d t x be the returns for the sovereign debt market at time t. Hence, A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 97 CoVaR for a confidence level of (1 ) can be formally characterized as the  - quantile of the conditional distribution of f t x as follows: f f d d d t t t t x CoVaR x VaR | ,, Pr( | )      , (4.1) where dt VaR , denotes the VaR of the debt market that measures the maximum loss that may be experienced by the sovereign debt market for a confidence level 1 at time t. Formally, it is the  -quantile of the return distribution for the debt market: dd tt x VaR , Pr( )     . We can compute CoVaR by determining the quantile of a conditional distribution or by using the quantile of an unconditional bivariate distribution, given that Eq. (4.1) can be written as: f f d d d t t t t dd tt x CoVaR x VaR x VaR | ,, , Pr( , ) Pr( )      , (4.2) or alternatively as: f f d d d t t t t x CoVaR x VaR | ,, Pr( , )      . (4.3) 4.2.1. CoVaR with copulas To obtain CoVaR from Eq. (4.3), we used copulas to characterize the joint distribution function. 10 Eq. (4.3) can be expressed in terms of the joint distribution function of f t x and d t x , fd F, , as f d d f d t t F CoVaR VaR | , , , ( , )    ; furthermore, Sklar’s (1959) theorem relates the joint distribution function and the copula as follows: fd f d t t f d F x x C u u ,( , ) ( , ) , (4.4) where C(·,·) is a copula function, f f f t u F x() and d d d t u F x() and where f F and d F are the marginal distribution functions of f t x and d t x , respectively. Consequently, we can express Eq. (4.3) in terms of copulas as:   f d d f t d t C F CoVaR F VaR | ,, ( ), ( )    . (4.5) Hence, CoVaR can be computed from Eq. (5) using a simple two-step procedure: 10 For further analysis on copulas, see Joe (1997) and Nelsen (2006). An overview of copula applications to finance can be found in Cherubini et al. (2004). Mainik and Schaanning (2012) provide the first representation of CoVaR in terms of bivariate copulas. Chapter 4 98 (1) We obtain the value of fd ft F CoVaR | , ()  from Eq. (4.5). Given that fd C u u( , )   , where  ,  and d u are given (note that d u ), from the copula function specification we can solve to determine the value of fd f f t u F CoVaR | , ()   . (2) From u we can obtain CoVaR as the quantile of the distribution of f t x , with a cumulative probability equal to u , by inverting the marginal distribution function of f t x : fd t f f CoVaR F u |1 ,()   . The use of copulas to obtain CoVaR is appealing because of their flexibility, compared to parametric bivariate functions, in allowing separate modelling of the marginals and the dependence structure. This is crucial because marginals and dependence functions may have different tail dependence characteristics that may affect CoVaR. Furthermore, computing CoVaR using the above two-step procedure is simple and only requires information on the confidence levels. In fact, tail dependence information from copulas naturally provides a measure of CoVaR, even though it does so at the limits. 4.2.2. CoVaR with vine copulas Copula CoVaR provides useful information in a bivariate setup. However, since we wanted to consider systemic risk affecting several markets—i.e., the impact of Greek debt distress on the financial system and on other debt markets—we needed to consider dependence in more than two dimensions. We thus considered vine copulas, 11 since these account for a multivariate distribution that combines three or more marginal distributions in a joint distribution. Vine copulas are multivariate copulas that are generated through a hierarchical construction that is decomposed into a cascade of bivariate copulas called pair-copulas, where each bivariate paircopula captures conditional dependence between two variables. Thus, a vine construction requires pairs of original variables and pairs of conditional distributions of recomputed variables to be modelled. Since we wished to analyse the systemic impact of a distressed sovereign debt market (Greece) and foreign debt markets on national financial systems, we 11 In the statistical literature vine copulas were introduced by Joe (1997) and were extended by Aas et al. (2009) for risk management purposes. Some applications of vine copulas in finance include, among others, Chollete et al. (2009), Aas and Beng (2009), Low et al. (2013) and Weiß and Supper (2013). A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 99 considered a vine copula with three variables. Let d t x* be the returns for the foreign debt market at time t with distribution function d d t d F x u * ** () . According to Bedford and Cooke (2001), the joint density of financial, national and foreign debt returns can be expressed as the product of the marginal densities and a set of conditional bivariate copulas as:       f d d f d d d f d f d d f d d d f d f d d d f d d d d d d f d d f x x x c F x x F x x c F x F x c F x F x f x f x f x * * * * , | * | * | * , * * ** , * * * ( , , ) ( | ), ( | ) ( ), ( ) ( ), ( ) ( ) ( ) ( )  , (4.6) where c(·,·) denotes the copula density and f(·) the marginal densities and where f d d c, | * is referred to as the pair-copula. The conditional distribution functions in Eq. (4.6) for any two random variables x and y can be obtained as (Joe, 1997): xy C F x F y F x y Fy ,( ( ), ( )) ( | ) ()   . (4.7) The decomposition in Eq. (4.6) is a canonical or C-vine copula model where the initial node of the vine copula hierarchical structure is given by the returns of the foreign debt market; alternatively, the decomposition in Eq. (4.6) is given by a Dvine copula model like that represented in Figure 4.1, where each edge corresponds to a bivariate copula density and the first three T1 (upper) nodes correspond to the marginals. We adopted this hierarchical structure as we were interested in the systemic impact of the Greek national debt market on the financial system, given specific foreign debt market circumstances, or the systemic effect of sovereign Greek debt distress on the financial systems of other countries. Figure 4.1: A D-Vine copula hierarchical structure Now, in this multivariate conditional setup, CoVaR is given by:   f f d d d d d d t t t t t x CoVaR x VaR x | , * | * * ,, Pr | ,      , (4.8) which can be expressed in terms of the conditional joint distribution function as   f d d d d f d d t t F CoVaR VaR | , * | * , | * , , ,    . Hence, to obtain CoVaR from the vine-copula f d* d f,d* d*,d (f, d*) (d*, d) (f, d|d*) T1 T2 Chapter 4 100 specification, we have to take into account information provided by the conditional joint distribution function of f t x and d t x , given, in terms of the copula, as:     f d d d f d d f d d f d d d F x x x x C u u u u ** , | * , | * * * ( | ),( | ) ( | ),( | ) . (4.9) Thus, CoVaR can be obtained from the vine copula using a three-steep procedure: (1) For given values of  , d u and d u* and for the copula specification in Eq. (4.9) we can solve to determine the value of fd uu * ( | ) . (2) From the value of fd uu * ( | ) we obtain the value of f u by solving from the conditional distribution of f t x given by Eq. (4.7). (3) From f u we obtain CoVaR as the quantile of the distribution of f t x , with a cumulative probability equal to u , by inverting the marginal distribution function of f t x : f d d t f f CoVaR F u | , * 1 ,()   . Following this three-step procedure we obtain information on CoVaR of the financial system in a given country in a situation of debt market distress, taking into account the foreign debt market situation. Furthermore, we can also consider the CoVaR of the financial system under two other market scenarios: (1) both national and foreign debt markets are distressed ( dd tt x VaR ,  and dd tt x VaR ** ,  ); (2) only the foreign debt market is distressed ( dd tt x VaR ** ,  ). In the first case, the returns in both debt markets are below or equal to their VaR figures, so d u and d u* , where 1 is the confidence level for the VaR of the foreign debt market. The estimation procedure is identical to the three-steep procedure described above except regarding the value of d u* . In the second case, we only have information on d u* and for that information we have to obtain the value of d u from the bivariate copula for both debt markets. Once we have this information we follow the threestep procedure described above. 4.2.3. Marginal distribution and copula models The marginal models and copula specifications used to compute the CoVaR measures are described as follows. To account for the usual characteristics of financial return distributions, such as leverage, fat tails and asymmetries, we considered that the conditional mean and A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 101 variance of returns ( t r ) are given by an autoregressive moving average (ARMA) model with p and q lags specified as: pq t j t j j t h t jh rr 0 11             , (4.10) where t t t z   and where t 2  is the conditional variance, given by a threshold generalized autoregressive conditional heteroskedasticity (TGARCH) specification (Zakoian, 1994; Glosten et al., 1993): r m m t k t k h t h h t h k h h ba 2 2 2 1 1 1                   , (4.11) where  is a constant, tk 2  is the GARCH component and th  is the ARCH component. The parameter  captures asymmetric effects in such a way that when 0 , the future conditional variance will proportionally increase more following a negative shock than following a positive shock of the same magnitude. t z is a zero mean and unit variance i.i.d. random variable that follows a Hansen’s (1994) skewed-t density distribution given by:     t t bz a t t bz a t bc z a b fz bc z a b ( 1) 2 2 1 21 ( 1) 2 2 1 21 1 ( ; , ) 1                              , (4.12) where  and  are the degrees of freedom parameter ( 2    ) and the symmetric parameter ( 11    ), respectively. The constants a, b and c are given by   ac 2 1 4   , ba 2 2 2 13    and     c1 22 ( 2)         . This distribution converges to the standard Gaussian as 0 and as    , and to the symmetric Student-t distribution as 0 and  is finite. We used seven different copula specifications to capture different characteristics of dependence: tail independence (Gaussian and Plackett), symmetric tail dependence (Student-t) and asymmetric tail dependence (Gumbel, Rotated Gumbel, BB7 and Symmetric Joe-Clayton (SJC)). Table 4.1 summarizes the main features of all the static and dynamic copula functions that were employed in the empirical analysis. We estimated the parameters of the marginal and bivariate copula models following the inference function for margins procedure (Joe and Xu, 1996), which consists of first estimating the parameters of the marginal distributions separately Chapter 4 102 using maximum likelihood and then estimating the parameters of the copula using the pseudo-sample observations for the copula given by the probability integral transformation of the standardized residuals for the marginals. For the vine copula (second tree of Figure 4.1), we recomputed the pseudo-sample observations through the copulas estimated for the first tree. This sequential estimation procedure was introduced by Aas et al. (2009) and later examined in Hobæk Haff (2013). The number of lags in the mean and variance equations for each series was selected according to the Akaike information criteria (AIC) and the different copula models were evaluated using the AIC adjusted for small-sample bias, as in Breymann et al. (2003) and Reboredo (2011; 2013). 4.3. Data We empirically examined the systemic risk effect of sovereign debt distress on the financial sector by considering six eurozone core countries (Austria (AT), Belgium (BE), Finland (FI), France (FR), Germany (DE) and Netherlands (NL)) and four peripheral countries (Italy (IT), Greece (GR), Portugal (PT) and Spain (ES)). For each country we considered weekly data for benchmark bond price indices for 10year maturities and for the MSCI financial price index. Data were sourced from Datastream and Bloomberg and cover the period 23 December 1999 to 25 May 2012. With this data we evaluated the following: (1) the impact of a distress event in one country’s debt market on its financial sector as represented by the MSCI financial index; (2) the impact of a distress event in the Greek debt market on the banking sector of other European countries; and (3) the impact of a simultaneous distress event in the Greek and domestic debt markets on the domestic financial system. A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 103 Table 4.1: Copula models. Name Copula Parameter Structure dependence Gaussian   11 N C (u,v; ) (u), (v)        No tail dependence. UL 0    Student-t 11 ST C (u,v; , ) T(t (u),t (v))      , Symmetric tail dependence:  U L 1 2t 1 1 / 1             Gumbel                    1 G C (u,v; ) exp logu log v 1 Asymmetric tail dependence: L0 , 1 U22     Rotated Gumbel         RG G C (u,v; ) u v 1 C (1 u,1 v; ) 1 Asymmetric tail dependence: L0 , 1 U22     BB7     1 1 BB7 C (u,v; , ) 1 1 1 1 u 1 1 v 1                                      1 , 0 Asymmetric tail dependence: 1 L2  , 1 U22     Plackett               2 P 1 C (u,v; ) 1 1 u v 1 1 u v 4 1 uv 21                 0 , 1 No tail dependence. UL 0    SJC   U L U L U L SJC JC JC C (u,v; , ) 0.5 C (u,v; , ) C (1 u,1 v; , ) u v 1              where      U 2 1/ log 2 ,      L 2 1/ log , JC C ( ) is similar to BB7 C ( ) U(0,1) L(0,1) Upper and lower tail independence: U0 and L0 Notes. We also captured time-varying dependence by assuming that copula parameters change over time. For the Gaussian and Student-t copulas, we assumed an ARMA(1,q)-type process (Patton, 2006) for the linear dependence parameter t  : q11 1(u ) (v ) j1 t1 0 1 t 1 2 t i t i q                    , where  xx (x) e e 1 11 1       is the modified logistic transformation that keeps the value of t  in (-1,1). For the Student-t copula, 1(x)   is replaced by 1 t (x)   . We considered time-varying dependence for the Gumbel and Rotated Gumbel copulas by assuming that the parameters reflect the dynamics given by the following equation: quv tt jt i t i q 1 11           . Finally, we considered time-varying dependence for the symmetrized Joe-Clayton (SCJ) copula by assuming that Uq U1uv j1 t2 0,U 1,U t 1 2,U t i t i q                  and Lq L1uv j1 t2 0,L 1, L t 1 2,L t i t i q                  , where  x (x) e 1 11 2      is the logistic transformation used to keep U  and L  in (0,1) Chapter 4 104 Figures 4.2 and 4.3, which display the benchmark bond and MSCI price dynamics for all ten countries considered in our analysis, show differences in the size and timing of price movements in debt markets and the financial sector that become especially relevant after the onset of the debt crisis at the end of 2009. Price volatility changed significantly around the period of the recent global financial crisis for the MSCI index and around the period of the European debt crisis for debt price benchmarks, to degrees that differed significantly across core and peripheral countries. A superficial inspection of the data shows that co-movement between debt and financial sectors was different across countries and also changed with the onset of the debt crisis. Table 4.2 reports descriptive statistics for bond and financial price returns computed on a continuous compounding basis. Average returns were similarly close to zero in all the debt and financial markets and standard deviations were larger for the financial markets than for the debt markets. Also, differences in maximum and minimum values show that price ranges were greater for financial markets than for debt markets and greater for peripheral countries than for core countries. Negative values for skewness were common across markets and countries, with the exception of the Belgian, Italian and Spanish debt markets. All return series showed fat tails; the kurtosis statistic took high values and the Jarque-Bera test strongly rejected the normality of the unconditional distribution for all the series. The Ljung-Box statistic indicated that some return series displayed temporal correlation, whereas the ARCH-Lagrange multiplier (ARCH-LM) statistic indicated that ARCH effects could be found in all the return series. Finally, in order to take into account the effects of the European sovereign debt crisis on expected returns and on volatility, we considered a crisis dummy variable in the mean and the variance of the marginal models that identified sample periods for before (value set to 0) and after (value set to 1) the onset of the European sovereign debt crisis. The crucial point here was to determine when the European sovereign debt crisis started. Following Bhanot et al. (2014), we took this date to be November 2009, as this was when investors became concerned regarding the quality of Greek debt; this concern developed in response to the Greek government’s revelation that its deficit amounted to 12.7% of gross domestic product and not the previously announced 6.7%. The fact that the impact of the crisis was different across debt markets is likely to have affected the dependence relationships between markets. A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 105 Figure 4.2: Time series plot of weekly sovereign bond price indices. 100 110 120 130 140 150 160 Austria 100 110 120 130 140 150 160 170 Belgium 100 110 120 130 140 150 160 170 180 190 200 Finland 100 110 120 130 140 150 160 170 France 100 110 120 130 140 150 160 Germany 0 20 40 60 80 100 120 140 Greece 130 140 150 160 170 180 190 200 Italy 100 110 120 130 140 150 160 Netherlands 60 80 100 120 140 160 180 Portugal 150 160 170 180 190 200 210 220 230 240 250 Spain Chapter 4 112 Figure 4.4: Times series plots for parameter estimates of the best copula fits between domestic financial systems and sovereign debt returns. 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 Austria: Plackett -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 Belgium: TVP-Gaussian 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Finland: Plackett 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 France: Plackett 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Germany: Plackett -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 Greece: TVP-Student -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 Italy: TVP-Student -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 Netherlands: TVP-Gaussian -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 Portugal: TVP-Student -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Spain: TVP-Student A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 113 Figure 4.6 displays the dynamics of the parameter estimates for the best copula model fit between the domestic sovereign debt and sovereign Greek debt return pairs. The evidence was conclusive in the period before the onset of the global financial crisis: all the debt markets strongly co-moved with the sovereign Greek debt market. However, thereafter—and mainly in the aftermath of the Greek debt crisis—European debt markets decoupled from the Greek market (markets in the core countries more so than in the peripheral countries), with dependence continuing to be positive, although less intense, in the peripheral countries. Of the core countries, Belgium was a particular case in that it exhibited high tail dependence at a specific times after the Greek crisis; a similar pattern was also observed for the Netherlands. Figure 4.7 displays the dynamics of the parameter estimates for the best paircopula model for domestic financial and sovereign debt returns conditional on sovereign Greek debt returns:   f d d f d f d d d d d c F x x F x x , | * | * * | * * ( | ), ( | ) . The empirical evidence is consistent with the evidence reported for the copula linking domestic financial and sovereign debt returns. Evidence on tail independence was found for all the core countries except Austria and Belgium. Dependence was static for Finland, France and the Netherlands but was time-varying for Austria, Belgium and Germany. Moreover, for Austria and Germany, dependence did not experience significant changes with the onset of the European debt crisis; in Belgium, dependence turned positive after 2011. Regarding peripheral countries, evidence of time-varying dependence was found for Italy and Spain and evidence of static dependence was found for Portugal. Dependence increased in the aftermath of the European debt crisis in Italy and, to a lesser extent, in Spain. 4.4.3. Systemic risk results Using the best copula and vine-copula fits, we obtained the CoVaR for each time period following the twoand three-step procedures described above. We obtained CoVaR at the 95% confidence level ( 0.05 ), conditional on the VaR for sovereign debt returns at the 95% confidence level ( 0.05 or/and 0.05 ). 13 Below we present the results for CoVaR using bivariate copulas and then using vine copulas. 13 Results at the 99% confidence level, which were consistent with the results reported here, are available on request. Chapter 4 114 Figure 4.5: Times series plots for parameter estimates of the best copula fits between domestic financial systems and sovereign Greek debt returns. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Austria: TVP-Gaussian -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Belgium: TVP-Student -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Finland: TVP-Gaussian -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1France: TVP-Student -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Germany: TVP-Student -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Italy: TVP-Gaussian -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Netherlands: TVP-Gaussian -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Portugal: TVP-Student -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Spain: TVP-Student A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 115 Figure 4.6: Times series plots for parameter estimates of the best copula fits between domestic sovereign and Greek sovereign debt returns. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Austria: TVP-SCJ Lower Tail Upper Tail 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Belgium: TVP-SCJ Lower Tail Upper Tail 0 2 4 6 8 10 12 Finland: TVP-Gumbel 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1France: TVP-SCJ Lower Tail Upper Tail 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Germany: TVP-SCJ Lower Tail Upper Tail 0 2 4 6 8 10 12 14 16 Italy: TVP-Rotated Gumbel 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Netherlands: TVP-SCJ Lower Tail Upper Tail 0 1 2 3 4 5 6 7 8 9 10 Portugal: TVP-Rotated Gumbel 0 2 4 6 8 10 12 14 Spain: TVP-Gumbel Chapter 4 116 Figure 4.7: Times series plots for parameter estimates of the best copula fits between domestic financial systems and domestic sovereign debt returns conditional on sovereign Greek debt returns. -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Austria: TVP-Student -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Belgium: TVP-Student 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Finland: Plackett 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 France: Plackett -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0Germany: TVP-Gaussian -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Italy: Gaussian 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Netherlands: Plackett 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 Portugal: BB7 Theta delta -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Spain: TVP-Gaussian A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 117 Figure 8 depicts the results for CoVaR dynamics throughout the sample period, with the post-onset financial and debt crisis periods indicated as shaded areas. For each figure representing the systemic risk of domestic sovereign debt for the financial systems for each country, we also included information on the financial system VaR so as to allow comparisons between VaR and CoVaR data. For each country in Table 6, the first two rows provide information on average VaR and CoVaR for the entire sample and for the preand post-onset European sovereign debt crisis periods (dated according to the dummy variable). Our evidence shows that, in the pre-crisis period, domestic sovereign debt played a diversification role for domestic financial systems in the eurozone, as indicated by CoVaR figures that were greater than VaR figures. This was particularly relevant for the core countries where systemic risk reductions were greater than for the peripheral countries. This evidence confirms the diversification role played by sovereign debt across European countries in the pre-crisis period, with the intensity of this role varying across countries depending on the degree of dependence between sovereign and financial sector returns and the weight of sovereign debt in bank portfolios. As our copula results show, dependence between domestic financial systems and sovereign debt markets changed in a different way across countries on the outbreak of the sovereign Greek debt crisis that had an impact on systemic risk. Figure 4.8 shows that sovereign debt for the core countries continued to play a diversification role for the financial system, given that CoVaR figures were, in general, greater than VaR figures; the only exception was Belgium, where, from the end of 2010, sovereign debt distress increased VaR. Table 4.6 shows, in fact, that average CoVaR figures were below the average VaR figures for all the core countries (except Belgium). However, for the peripheral countries, the opposite effect was observed: the systemic impact of sovereign debt increased considerably for Greece, Italy and Portugal, while remaining relatively stable for Spain. This evidence is consistent with the change in dependence observed for peripheral countries after the onset of the debt crisis; an increase in (positive) dependence swept away the diversification effects of domestic sovereign debt on domestic financial systems. Chapter 4 118 Figure 4.8: VaR(f) and CoVaR(f|d) -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Austria VaR (f) CoVaR (f|d) -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Belgium VaR (f) CoVaR (f|d) A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 119 Figure 4.8: (Continued) -0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Finland VaR (f) CoVaR (f|d) -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 France VaR (f) CoVaR (f|d) Chapter 4 120 Figure 4.8: (Continued) -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Germany VaR (f) CoVaR (f|d) -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Greece VaR (f) CoVaR (f|d) A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 121 Figure 4.8: (Continued) -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Italy VaR (f) CoVaR (f|d) -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Netherlands VaR (f) CoVaR (f|d) Chapter 4 128 Figure 4.9: (Continued) -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Italy CoVaR (f|d=0.05,d*) CoVaR (f|d,d*=0.05) CoVaR (f|d=0.05,d*=0.05) -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Netherlands CoVaR (f|d=0.05,d*) CoVaR (f|d,d*=0.05) CoVaR (f|d=0.05,d*=0.05) A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 129 Figure 4.9: (Continued) -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Portugal CoVaR (f|d=0.05,d*) CoVaR (f|d,d*=0.05) CoVaR (f|d=0.05,d*=0.05) -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-08 Dec-09 Dec-10 Dec-11 Spain CoVaR (f|d=0.05,d*) CoVaR (f|d,d*=0.05) CoVaR (f|d=0.05,d*=0.05) Chapter 4 130 Overall, our CoVaR results indicate that: (1) domestic sovereign debt had a systemic impact on domestic financial systems across European countries that was positive in the sense of increasing financial system VaR; (2) this systemic effect changed with the onset of the European debt crisis in peripheral countries like Greece, Italy and Portugal, where the systemic impact of sovereign debt increased, thereby reducing financial system CoVaR; (3) the systemic impact of a potential Greek debt default was mainly limited to Belgium, Italy, the Netherlands and Portugal; and finally, (4) for the remaining countries this event does not add much value with respect to the CoVaR figures obtained without considering the impact of the Greek sovereign debt. 4.4.4. Statistical test Empirical evidence reported in Section 4.3 describes systemic risk behaviour before and after crisis onset with no testing of statistical significance. Below we use statistical significance testing to draw robust conclusions on sovereign debt systemic risk for financial systems. We compared cumulative distribution for VaR and/or CoVaR data using the Kolmogorov-Smirnov (KS) bootstrapping test as proposed by Abadie (2002) and applied by Bernal et al. (2014) to compare CoVaR figures. This test measures the difference between two cumulative quantile functions relying on the empirical distribution function and without considering any underlying distribution function. It is defined as:     mn x m n mn KS sup F x G x mn 1 2      , (4.13) where   m Fx and   n Gx are the cumulative VaR or CoVaR distribution functions, respectively, and n and m are the size of the two samples. With this statistic we tested four hypotheses, both before and after crisis onset:  Hypothesis 1: H0: CoVaR(f|d) > VaR(f)  Hypothesis 2: H0: CoVaR(f|d) > CoVaR(f|d=0.05,d*)  Hypothesis 3: H0: CoVaR(f|d) > CoVaR(f|d,d*=0.05)  Hypothesis 4: H0: CoVaR(f|d) > CoVaR(f|d=0.05,d*=0.05) The first hypothesis examines whether sovereign debt contributes to downside risk in the financial system, whereas the remaining hypotheses examine whether systemic risk in one country is affected by the Greek debt market under normal or exceptional circumstances (denominated scenarios (1)-(3) above). Table 4.7 reports results for the KS statistic and the associated bootstrap p-values under the null A Vine-copula CoVaR approach to systemic sovereign debt risk for the financial… 131 hypothesis, yielding statistical evidence that is fully consistent with the empirical evidence described in Section 4.3. Table 4.7: Significant test for differences in risk measures. Hypothesis 1 Hypothesis 2 Hypothesis 3 Hypothesis 4 Stat p-value Stat p-value Stat p-value Stat p-value Austria Before crisis onset 0.000 0.999 0.033 0.567 0.024 0.724 0.033 0.544 After crisis onset 0.000 0.999 0.098 0.271 0.008 0.988 0.008 0.990 Belgium Before crisis onset 0.000 0.999 0.010 0.950 0.285 0.000 0.031 0.601 After crisis onset 0.256 0.000 0.158 0.032 0.000 0.999 0.023 0.925 Finland Before crisis onset 0.000 0.999 0.056 0.194 0.109 0.002 0.021 0.782 After crisis onset 0.000 0.999 0.105 0.219 0.030 0.873 0.030 0.874 France Before crisis onset 0.000 0.999 0.017 0.842 0.085 0.022 0.027 0.652 After crisis onset 0.000 0.999 0.030 0.878 0.023 0.923 0.015 0.964 Germany Before crisis onset 0.000 0.999 0.308 0.000 0.816 0.000 0.101 0.004 After crisis onset 0.000 0.999 0.752 0.000 0.722 0.000 0.526 0.000 Greece Before crisis onset 0.081 0.031 After crisis onset 0.895 0.000 Italy Before crisis onset 0.010 0.949 0.041 0.413 0.213 0.000 0.291 0.000 After crisis onset 0.617 0.000 0.466 0.000 0.150 0.046 0.248 0.000 Netherlands Before crisis onset 0.000 0.999 0.000 0.999 0.083 0.029 0.000 0.999 After crisis onset 0.008 0.988 0.000 0.999 0.000 0.999 0.000 0.999 Portugal Before crisis onset 0.029 0.636 0.000 0.999 0.000 0.999 0.002 0.997 After crisis onset 0.504 0.000 0.406 0.000 0.000 0.999 0.113 0.179 Spain Before crisis onset 0.010 0.950 0.025 0.718 0.106 0.003 0.015 0.878 After crisis onset 0.128 0.110 0.113 0.173 0.045 0.751 0.023 0.924 Note. The bootstrap Kolmogorov-Smirnov tests whether the values of different risk measures follow (or not) the same cumulative distribution function (CDFs) in the preand post-onset crisis periods. The null hypotheses are considered as follows: Hypothesis 1 = H0: CoVaR(f|d) > VaR(f) Hypothesis 2 = H0: CoVaR(f|d) > CoVaR(f|d=0.05,d*) Hypothesis 3 = H0: CoVaR(f|d) > CoVaR(f|d,d*=0.05) Hypothesis 4 = H0: CoVaR(f|d) > CoVaR(f|d=0.05,d*=0.05). 4.5. Conclusions We have provided empirical evidence, for the periods before and after the onset of the recent financial and debt crises, of (1) the systemic impact of domestic sovereign debt distress on domestic financial systems in European countries and (2) the potential systemic impact of a distressed Greek debt market on the financial systems of other European countries. We measured systemic risk using the CoVaR measure, Chapter 4 132 as proposed by Adrian and Brunnermeier (2011) and generalized by Girardi and Ergün (2013). CoVaR measures VaR for a financial system conditional on the fact that the debt market is in distress. We computed CoVaR data using both bivariate and vine-copula models, given that copulas can flexibly account for dependence, most especially for tail dependence, which is crucial to determining CoVaR data. To estimate CoVaR, we adopted (1) a two-step procedure that accounted for the impact of domestic debt distress on domestic financial systems, and (2) a three-step procedure that—taking into account the link between domestic financial and sovereign debt markets—accounted for the systemic impact of a potential Greek default on the financial systems of other European countries. Using a sample of MSCI financial and sovereign bond benchmark indices for six eurozone core countries (Austria, Belgium, Finland, France, Germany and the Netherlands) and four peripheral countries (Italy, Greece, Portugal and Spain) for the period 2000 to 2012, we estimated copula and vine-copula models—in order to characterize the dependence structure between financial and sovereign debt markets—and then computed CoVaR figures. Our evidence indicates that there were substantial differences in the systemic impact of sovereign debt in the periods before and after the onset of the European debt crisis. In the pre-onset period sovereign debt was observed to have a positive systemic risk effect in reducing financial system VaR. This impact can be explained in terms of the diversification effect of sovereign debt on bank portfolios, with even Greek sovereign debt playing a diversification role across financial systems in the eurozone, except in Portugal. However, in the post-onset crisis period the picture was quite different, with domestic sovereign debt having a negative systemic impact on domestic financial systems as CoVaR fell. This evidence was found for all the peripheral countries; as for the core countries, sovereign debt continued to play a diversification role, having a positive impact on CoVaR. This positive impact can be explained by the fact that the negative impact of the sovereign debt crisis was not fully transmitted to the core countries. Regarding the systemic impact of Greek sovereign debt distress on the financial systems of other countries, we found all financial systems in Europe to be affected, but to differing degrees. For core countries, after the onset of the debt crisis, Greek sovereign debt continued to play a diversification effect, although this effect was less intense and more varied. 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