Systemic Risk & Financial Contagion in
Eu-rozone Markets
Thomas Veenstra, S2373653 University of Groningen
Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: dr. D. Ronchetti
Systemic Risk & Financial Contagion in Eurozone
Markets
Thomas Veenstra
S2373653
Groningen, March 6, 2018
Abstract: We investigate the financial-contagion mechanism in Eurozone-stock
Contents
1 Introduction . . . 5
1.1 Dependencies and extremes in the Eurozone . . . 5
1.2 Literature Review . . . 7
1.2.1 Modelling approaches . . . 7
1.2.2 Systemic risk in core vs. non-core countries . . . 8
2 Econometric Model . . . 10 2.1 ARMA(m,n)-GARCH(p,q) . . . 10 2.2 Pairwise copulas . . . 11 2.2.1 Sklar’s theorem . . . 11 2.2.2 Example . . . 11 2.2.3 Construction of trees . . . 13 2.2.4 Graphical illustration . . . 14 2.3 Fitting R-vines . . . 16 2.3.1 Bivariate copulas . . . 17
2.3.2 Critique against R-vine models . . . 17
2.4 Systemic risk . . . 18
2.4.1 CoVaR and ∆ CoVaR . . . 18
2.4.2 Other possible measures of systemic risk . . . 19
3 Data . . . 20 3.1 Data preparation . . . 20 4 Results . . . 26 4.1 Univariate models . . . 26 4.1.1 Estimates . . . 26 4.1.2 Correlations of residuals . . . 28 4.2 Multivariate analysis . . . 33 4.2.1 Comparative study . . . 33
4.2.2 Testing the simplifying assumption . . . 34
4.2.3 Network analysis . . . 35
4.3 Systemic risk developments . . . 38
1
Introduction
1.1
Dependencies and extremes in the Eurozone
Continuing increases in the complexity, scale and interconnectedness of financial institutions and financial markets trigger the need for advances in the field of risk management (BIS, 2011; FED, 2011; Randall, 2008). Simultaneously, developments in information technology facilitate the creation and evolution of elaborate risk management programs by providing
the techniques and computing power needed for the calibration of intricate models1.
In order to adequately study the dangers of increasingly interconnected financial markets, we need appropriate dimensions of risk. One of these dimensions is called systemic risk. The
European Central Bank (ECB) defines systemic risk as2:
‘The risk that the inability of one institution to meet its obligations when due will cause other institutions to be unable to meet their obligations when due. Such a failure may cause significant liquidity or credit problems and, as a result, could threaten the stability of or con-fidence in markets.’
Hence, systemic risk is affected by the amount of financial contagion (Leibrock and Gottesman, 2017). The International Monetary Fund (IMF) defines financial contagion as the amount to which a shock in one market cascades to other markets (IMF, 2010). Recent events, such as the political crisis in Catalonia, have shown that geopolitical developments
can result in market risk being transferred from one market to another3. Specifically the
Eurozone is sensitive to such developments, with fragility stemming from the single cur-rency. These recent developments show the importance of accurate methods for modelling the shock-transmission mechanism.
This shock-transmission mechanism can be studied in several ways. An easy and before the crisis very popular method is to model the return distribution of a market or institution by using a Gaussian distribution. However, as documented in many studies (Cont (2001); J.P. Morgan (2009); McNeil et al. (2015)):
1. returns are not independently and identically distributed (iid); 2. the series of absolute or squared returns are serially correlated; 3. volatility of returns is not constant through time; and,
4. the distribution of returns is leptokurtic and skewed.
1Speech by L. H. Meyer, Governor of the Board of Governors of the US Federal Reserve System, before the Bank of Thailand Symposium on Risk Management of Financial Institutions, held in Bangkok on 31 August 2000.
2European Central Bank, 2004, Annual Report: 2004, ECB, Frankfurt, Glossary.
Thus, there is an abundance of research that shows that the Gaussian distribution does not manage to capture return characteristics. In fact, MacKenzie and Spears (2014) refer to Gaussian copulas as ‘the formula that killed Wall Street’. Many alternative solutions have already been proposed in literature - for an overview, see McNeil et al. (2015). The problem with many of these models, however, is that when the number of variables increases, the number of parameters to estimate the variance-covariance matrix increases quadratically. This makes it difficult to extend these models to multivariate settings.
Moreover, many of the models fail to account for the often difficult dependence struc-tures present within markets. This drives the demand for flexible models that allow for asymmetries in losses and models that are easily extendable to portfolios with many compo-nents (McNeil et al., 2015; Oh and Patton, 2017). A solution that is gaining more and more popularity is the use of pairwise copulas (Aas et al., 2009).
The purpose of this paper is twofold. First, we attempt to model the transmission
mechanism of shocks in the Eurozone using state-of-the-art methods from risk management. For this, we use a similar procedure as is done in Oh and Patton (2017). However, instead of using a dynamic factor-copula model, we rely upon the increasingly popular R-vines that allow for studying complex dependence structures (Joe, 1996; Aas et al., 2009; Almeida et al., 2012).
Second, we use techniques from risk management that use the Conditional Value-at-Risk (CoVaR) to compute the level of systemic risk present in financial markets for each Eurozone country (Adrian and Brunnermeier, 2016). The idea behind this method is to determine in what way the returns of one market develop if there is financial turmoil in another market. Brunnermeier and Sannikov (2014) show that a certain amount of procyclicality tends to occur in financial markets. This is due to the fact that risk measures are often low in times of financial prosperity and high in times of crisis. The authors coin this phenomena the ‘volatility paradox’. The CoVaR measure takes this feature into account (Adrian and Brunnermeier, 2016). In this paper, we test whether this expansion and contraction of risk can also be observed in European-stock markets. Hereby, we make a distinction between the markets of countries characterized by strong-economic growth (group-A countries) versus countries which are said to have weak-economic growth (group-B countries) (ECB, 2017b).
The approach of this paper is as follows. First, we apply the modelling approach pre-sented in Aas et al. (2009); Dißmann et al. (2013) and use it to analyze the dependencies between indices in the Eurozone. Second, we will use the calibrated model to test if systemic risk has increased or decreased during the years following the financial crisis of 2007-2009 and the sovereign debt crisis of 2009-2011. Third, we will compute the conditional Value-at-Risks (VaRs) and apply these in a backtesting study.
We adress the following research questions:
1. Did the amount of systemic risk in group-A countries decrease during the years follow-ing the financial crisis?
3. Is the amount of systemic risk lower in group-A- compared to group-B countries? 4. Is there financial contagion between group-A and group-B countries?
5. And related, is the Eurozone-stock market integrated or fragmented?
These questions will be answered using an R-vine model, as presented in Dißmann et al. (2013); Aas et al. (2009). The marginal distributions of index returns are modelled using ARMA-GARCH models. We compute the level of systemic risk by employing similar meth-ods as in Brechmann and Czado (2013) and we will test our hypotheses using simulations in combination with a backtesting study (McNeil et al., 2015). In addition, we will use the dependence structure and systemic-risk measures to analyze the amount of financial conta-gion between markets. Last, we use several benchmark models to check the performance of our model.
1.2
Literature Review
1.2.1 Modelling approaches
The most favoured multivariate dependence model in research is the multivariate Gaussian distribution (MacKenzie and Spears, 2014). However, many fields (i.e. finance, insurance, risk management, etc.) have seen increased needs for other models that are able to capture more complex dependence structures - for instance, models that allow for non-constant dependence structures and asymmetries in tail dependencies.
A widely used group of models is that of the conditional volatility models. A pioneering study is the one by Engle (1982), in which the ARCH model was introduced. The ARCH model is a way of modelling non constant-conditional variance. Later on, these models were extended to GARCH models in Bollerslev (1986). The multivariate variant of these models is described in Bollerslev et al. (1988) - however, a downside of these multivariate models is that the number of parameters needed to be estimated in the variance-covariance matrix grows quadratically with the number of variables, which might pose problems since portfolios are often composed of many stocks.
Various alternative solutions have been proposed to solve this issue. For example, one may use a Constant-Conditional Correlation (CCC) model (Bollerslev, 1990). However, many researchers have shown that the dependence structure of returns of stocks is not constant over time, see e.g. Longin (2001). Moreover, regulatory institutions indicate that ‘time varying correlations should be taken into account’ (BIS, 2011). One solution for this may be a Dynamic-Conditional Correlation (DCC) model (Engle, 2000), or extensions of this model where asymmetries in volatilities are allowed. Another solution is the use of copula models.
than with big gains (McNeil et al., 2015). Archimedean copulas do allow for asymmetries, but often only allow for a single parameter that controls the dynamics of the dependence structure, which might be too restrictive in the presence of a complex dependence structure. A solution that is gaining popularity in literature is the use of blocks of pairwise copulas (Aas et al., 2009), where the authors conceptualize the idea to use bivariate copulas for several combinations of variables and connect these in an efficient way. Which pairs of variables are used, and their corresponding conditioning variables, is determined using a sequence of trees that when combined form a network. These trees are also called vines, where the most general group of vines is referred to as regular vines (R-vines). Within this group of vines, two other important groups can be distinguished: the group of canonical vines (C-vines) and the group of drawable vines (D-vines), where each group has a different way of composing the density of joint returns. Consider, for instance, Dißmann et al. (2013) for an extensive review of these models.
1.2.2 Systemic risk in core vs. non-core countries
In most research on systemic risk and financial contagion, the focus is on the interconnec-tivity of banks. That is, most researchers consider transaction data between banks within the money market. This network of interbank transactions gives an indication of the expo-sures that banks have with counterparties. Subsequently, financial contagion is analyzed by considering the effect of a default of one bank on the banking system as a whole (through the contagion that takes place within the network). Examples of this approach can be found in, for instance, Lehar (2005), who analyzed the interconnections between North-American banks, Markose et al. (2010), where the ‘too interconnected to fail’ principle is analyzed and Mistrulli (2011) who analyzed the bilateral exposures in the Italian money market.
These approaches focus on one component of financial contagion which is the result of a combination of liquidity risk - a bank defaults because it cannot meet its obligations - and counterparty risk - the risk that a counterparty fails to meet its future obligations. This approach is certainly an important component of financial contagion, albeit not the only source, because banks also take positions in the stock- and bond market and failures in these markets may spread through the system to other banks.
In Poledna et al. (2015) the authors explain that financial contagion can be seen as a network consisting of several layers (more commonly referred to as multiplex network ). These layers each model some components of financial contagion. Several of these components are: • Interbank lending and borrowing - this component is modelled using interbank
trans-actions;
• Positions in the mortgage market; • Investments in equity;
In this paper we focus on modelling the latter two components; that is, we consider the source of contagion resulting from participation in the stock market. Market-risk contagion is also known as ‘comovement’, ‘volatility spillover’ and ‘interdependence’. King and Wadhwani (1990), Barberis et al. (2005) and Aloui and Hkiri (2014) analyze the comovement of stocks. They show that there are contagion effects in the markets and that this contagion effect has grown during the last several years. Volatility spillover has been analyzed in Mensi et al. (2013) where the spillover effect between commodities and the S&P500 is analyzed. He et al. (2015) study the interdependence between the Chinese market and the rest of the world and show that reforms in the Chinese financial markets have ‘significantly enhanced market interdependence’. In Upper (2007), several of these models are studied and the author concludes that there is a need for models that are capable of assigning probabilities to certain stress events.
Our approach is similar to the one presented in Brechmann and Czado (2013). We extend their network analysis by including more market indices and by exploring the dependence structure in substantially more detail. Moreover, we consider a different class of models that allow for time-varying dependence structures and apply the models to analyze the spillover effect between group-A and group-B Eurozone countries.
2
Econometric Model
2.1
ARMA(m,n)-GARCH(p,q)
Consider the random vector of losses Yt= (Y1,t, . . . , Yd,t) for t = 1, . . . , T , and integer d the
number of equity indices. Before modelling the dependence structure, we first focus on the
marginal distribution of each component of Yt. In specific, assume that each variable Yi,t is
distributed according to the following specification:
φi(B)yi,t = ψi(B)εi,t (1)
εi,t =phi,tηi,t (2)
hi,t = αi,0+ p X j=1 αi,jε2i,t−j + q X k=1 γi,khi,t−k, (3)
with polynomials φi(B) = 1 − φi,1B − · · · − φi,mBm and ψi(B) = 1 + ψi,1B + · · · + ψi,nBn
without common factors; B is defined as the backward-shift operator - that is, Bnyt= yt−n;
The orders of the ARMA-GARCH model are denoted by the integer variables m, n, p and
q; ηi,t is a white-noise sequence with mean zero and unit variance. The coefficients of the
process are constrained so that αi,0 > 0, αi,j ≥ 0 and γi,j ≥ 0. The GARCH part of the model
has finite unconditional variance if Pp
j=1αi,j+
Pq
k=1γi,k < 1 (Bollerslev, 1986). Moreover,
the process yi,t is strictly stationary and ergodic if all the roots of the equations φi(B) = 0
and ψi(B) = 0 lie outside the unit circle (Ling and Li, 1997). Last, we model the innovations
using the student-t distribution.
The idea of our modelling approach is that we try to capture the marginal behaviour of the processes with the marginal distribution, as specified above, and that a copula captures all the remaining dependence among the residuals. Since we study nineteen different market
indices, it is difficult to find one model that fits all series. For this reason, we use an
algorithmic procedure to find the best fitting ARMA-GARCH model. We choose among all models with the orders as decribed in Table 1.
Tab. 1: Orders of the mean- and variance model Mean model Component Order Moving average 0,1,2 Autoregressive 0,1,2 Variance model Component Order Moving average 0,1,2 Autoregressive 0,1,2
This leaves 34 = 81 possible models. Among all models without unit root, we select the
Observe that we allow for parameterizations with positive autoregressive order in the mean model. Several empirical studies have shown that return series are poorly autocorre-lated (McNeil et al., 2015; Cont, 2001) - however, Spierdijk et al. (2012), Fama and French (2012) and Balvers and Wu (2006) find evidence of momentum and mean reversion in eq-uity returns. Due to this lack of consensus in literature, we look for an extremely flexible parameterization and allow for m 6= 0. For robustness, we also calibrate the models where we set the autoregressive order in the mean model to zero.
2.2
Pairwise copulas
2.2.1 Sklar’s theorem
Consider a random vector of losses X = (X1, . . . , Xd)
0
∼ F with d the amount of market
indices, where each component of X has marginal distribution Fi, with i ∈ {1, . . . , d}. Sklar’s
theorem (Sklar, 1959) states that there exists a copula C such that
F (x1, . . . , xd) = C1,...,d(F1(x1), . . . , Fd(xd)) , (4)
where C denotes a copula of d dimensions. Instead of calibrating the joint-distribution func-tion and then computing the marginal-distribufunc-tion funcfunc-tions (which is tradifunc-tionally done), we model the marginal distributions first and connect them in some way to get the joint-distribution function.
Sklar also shows that if absolute continuity and differentiability of F is satisfied and if
F1, . . . , Fd are strictly increasing, we get that
f (x1, . . . , xd) = " d Y k=1 fk(xk) # c1,...,d(F1(x1), . . . , Fd(xd)) , (5)
where capital letters denote distribution functions and small letters denote densities.
The marginal distribution functions (F1, . . . , Fd) are modelled using the
ARMA(m,n)-GARCH(p,q) specification presented in (3). The residuals from the fitted model constitute a new time series in which, in theory, all autocorrelation is removed. The next subsection explains how to proceed to model the residual time-series by using an R-vine model.
2.2.2 Example
The following example illustrates the concept of simplified pairwise-copula modelling.
Con-sider the situation where we are dealing with three market indices (i.e. d = 3). Let
X = (X1, X2, X3) ∼ F with F differentiable and absolutely continuous. Moreover, let F1(x1),
F2(x2) and F3(x3) denote the corresponding strictly increasing marginal distribution
func-tions with density funcfunc-tions f1(x1), f2(x2) and f3(x3). We decompose the joint-probability
density function f123(x1, x2, x3) as follows:
To get a pair-copula construction (PCC), we rewrite each component of the right hand side of (6) in terms of copula densities.
Let c23(F2(x2), F3(x3)) denote the density of the copula C23(F2(x2), F3(x3)) corresponding
to the distribution F23(x2, x3) of the pair X2, X3. We use the following identity:
f23(x2, x3) = c23(F2(x2), F3(x3))f2(x2)f3(x3). (7)
Using this expression, we can decompose f2|3(x2|x3) as
f2|3(x2|x3) =
f23(x2, x3)
f3(x3)
= c23(F2(x2), F3(x3))f2(x2). (8)
We can factorize f1|23(x1|x2, x3) using copula density c12|3, which is the copula density
corre-sponding to F12|3(x1, x2|x3), and copula density c13, which is the copula density corresponding
to distribution F13 of pair X1, X3. That is,
c12|3(F1|3(x1|x3), F2|3(x2|x3); x3) = f12|3(x1, x2|x3) f1|3(x1|x3)f2|3(x2|x3) , (9) so we have f1|23(x1|x2, x3) = f12|3(x1, x2|x3) f2|3(x2|x3) = c12|3(F1|3(x1|x3), F2|3(x2|x3); x3)f1|3(x1|x3) = c12|3(F1|3(x1|x3), F2|3(x2|x3); x3)c13(F1(x1), F3(x3))f1(x1). (10)
Using these expressions we may rewrite (6) as follows f123(x1, x2, x3) =f1(x1)f2(x2)f3(x3)
· c13(F1(x1), F3(x3))c23(F2(x2), F3(x3))
· c12|3(F1|3(x1|x3), F2|3(x2|x3); x3). (11)
This expression is also known as the full PCC (Aas et al., 2009).
By assuming that the conditioned variables X1 and X2 are independent given the
con-ditioning variable X3, we reduce the number of levels in the pair-copula construction. Not
making this assumption complicates matters in higher dimensions - in general it is quite dif-ficult to estimate a copula that depends additionally on several conditioning variables. For
this reason, it is common practice to make the simplifying assumption that X1 and X2 are
independent given the conditioning variable X3, so that c12|3(F1|3(x1|x3), F2|3(x2|x3); x3) = 1.
This leaves us with the factorization:
f123(x1, x2, x3) =f1(x1)f2(x2)f3(x3) · c13(F1(x1), F3(x3))c23(F2(x2), F3(x3)), (12)
which is a probability density function as long as the simplifying assumption is satisfied.
The interesting thing about this approach is that the copulas c13 and c23 can be chosen
2.2.3 Construction of trees
The previous example shows the case in which we consider only three dimensions. Suppose now that we extend our analysis to d dimensions. Bedford and Cooke (2001) introduce a general graphical framework that is more commonly known as the regular vine (R-vine) structure - with graphical in the sense that it allows the user to visualize dependencies in a network. R-vines are based on a sequence of trees decreasing in size in each step. The formal definition of R-vines is as follows:
1. Tree 1:
• d nodes given by the random variables X1, . . . , Xd;
• d − 1 edges which are pair-copula densities between X1, . . . , Xd.
2. Tree j
• d + 1 − j nodes which are the edges of tree j − 1; • d − j edges which are conditional pair-copula densities.
Important here is the proximity condition: one can only join two nodes in tree j + 1 if the corresponding edges in tree j share a node.
In order to study in more detail the properties and characteristics of vines we define the following:
• Define N := (N1, . . . , Nd−1) to be the set of nodes for each tree, and define E :=
(E1, . . . , Ed−1) to be the set of corresponding edges;
• We link each edge e = j(e), k(e)|D(e) ∈ Ei with a pair-copula density cj(e),k(e)|D(e),
where j(e) and k(e) are conditioned nodes and D(e) the conditioning set;
• X is the vector of returns and XD(e)is the subvector of X where we select those indices
that are contained in D(e).
Under the assumptions presented above, the joint R-vine density is given by
f (x1, . . . , xd) = d−1 Y i=1 Y e∈Ei cj(e),k(e)|D(e) Pair-copula densities · d Y k=1 fk(xk) Marginal densities , (13) where
cj(e),k(e)|D(e) := cj(e),k(e)|D(e) F (xj(e)|XD(e)), F (xk(e)|XD(e)) . (14)
impose some kind of structure to organize the composition of the joint density.
Both C- and D-vines are subcases of the more general R-vines. For C-vines, each tree that is created has a unique node that is connected to all other nodes (i.e. the all roads lead to Rome principle). C-vines are appropriate whenever there is a pivoting variable. D-vines use the same reasoning as R-vines; however, each tree that is formed constitutes a path (i.e. no node has degree greater than two). The formal definition of these graphs is as follows:
• D-vines are defined based on their path-tree structure, resulting in the joint-probability density: f (x1, . . . , xd) = "d−1 Y j=1 d−j Y i=1 ci,i+j|i+1,...,i+j−1 # · " d Y k=1 fk(xk) # . (15)
• For C-vines we use the fact that each tree contains one node that functions as pivot, i.e. f (x1, . . . , xd) = "d−1 Y j=1 d−j Y i=1 cj,j+1|1,...,j−1 # · " d Y k=1 fk(xk) # . (16)
In this paper, the main focus is on R-vines, but we use C- and D-vines as benchmarks for the performance of the R-vines. A downside of the use of R-vines is that the number of possible tree structures grows exponentially with the number of variables, which makes fitting models computationally a lot more cumbersome. This problem was adressed in Brechmann et al. (2012) - the authors recognize that there is a need to find a balance between computational speed and accuracy when modelling R-vines. Their solution is to capture the most important dependencies in the first number of trees, and to capture the dependence in the remaining number of trees by using computationally less demanding copulas. The idea results in two procedures known as truncated R-vines and pairwisely-simplified R-vines.
We call an R-vine level K pairwisely truncated if all pairwise-copulas with conditioning set larger or equal to K are estimated using the independence copula. Similarly, we call an R-vine level K pairwisely simplified if all pair copulas with conditioning set larger or equal to K are estimated using bivariate-Gaussian distributions. Hence, in both cases, we try to find the best fit for the variables with conditioning set smaller than K. The first K variables can be seen as capturing the most important dependencies in the data.
2.2.4 Graphical illustration
In order to make the above procedure more intuitive, we use the example given in Figure 1.
T1 displays a possible tree in the first level of the R-vine. For each edge {i, k} with nodes
i, k ∈ {1, 2, . . . , 6} and i 6= k we select a copula and estimate its parameters. Next, we use
these to transform loss observations xi and xk to get pseudo observations ui,k = ˆFi|k(xi|xk)
and uk,i = ˆFk|i(xk|xi).
The successive tree, denoted by T2, is fitted on these pseudo observations. Note that we
can only connect one node to another node in tree j if the corresponding edges in tree j − 1 share a node (the proximity condition). The next step consists of estimating the pair copula corresponding to each pair of edges, denoted by {i, k|D} with nodes i, k ∈ {1, 2, . . . , 6}, i 6= k and set of conditioning nodes D ⊂ {1, 2 . . . , 6} \ {i, k}, and computing the new pseudo
observations using the corresponding conditional distribution functions ˆFi|k,Dand ˆFk|i,D. The
remaining trees are obtained in a similar manner. For a more comprehensive description of this algorithm, see Dißmann et al. (2013).
1 2 3 4 5 6 1,2 2,4 2,3 4,5 5,6 T1 1,2 2,4 2,3 4,5 5,6 1, 4|2 3, 4|2 2, 5|4 4, 6|5 T2 1, 4|2 3, 4|2 2, 5|4 4, 6|5 1, 5|2, 4 3, 5|2, 4 2, 6|4, 5 T3 1, 5|2, 4 3, 5|2, 4 2, 6|4, 5 1, 3|2, 4, 5 3, 6|2, 4, 5 T4 1, 3|2, 4, 5 1, 6|2, 3, 4, 5 3, 6|2, 4, 5 T5
2.3
Fitting R-vines
The calibration of R-vines can be divided into three separate parts. The first part consists of finding the right tree structure. Second, for each tree structure, we should find appropriate pair copulas, and third, for each pair copula, we should find the relevant parameters. The following section explains step-by-step how this is done.
One possibility for fitting an R-vine model would be to select each possible R-vine struc-ture and then, for each strucstruc-ture, find the best fitting pair copulas and their corresponding parameter specifications. However, the number of possible R-vine structures on d variables
grows very rapidly, with as many as d!
2 × 2
d−2
d different structures (Morales-Napoles, 2016).
Therefore, in Dißmann et al. (2013), it is suggested to use a sequential-heuristic method for selecting the tree structure.
This sequential-heuristic approach is more commonly referred to as the sequential method. The idea behind this method is to capture the strongest dependencies present in the data in the first number of trees. For each edge between two variables, we can use several measures of dependency. For instance,
• Kendall’s τ ; • Spearman’s ρ;
• Akaike- or Bayesian Information Criteria (respectively AIC and BIC); • Likelihood ratio tests like the one in Vuong (1989).
However, performing a likelihood-ratio test or computing the information criteria requires us to fit all possible pair-copula models. This is a computationally demanding task which becomes even more difficult when the number of variables increase. Therefore, one possibility is to use correlations as edge weights, where we use Kendall’s τ as correlation measure. The last step involves using an algorithm to compute a maximum-spanning tree for each tree, which can be done by using the algorithm presented in Prim (1957). The resulting specification is used as R-vine tree structure. The whole procedure above is also referred to as Dißmann’s algorithm (Dißmann et al., 2013).
A downside of the procedure above is that we might not find a global optimum in terms of model fit of the whole R-vine, since we consider each tree structure individually and do not take the latter tree structures into account. Despite of this, Dißmann et al. (2013) advocate the use of this approach because
1. we model the most important dependencies in the first model;
2. the copula used in the first tree have the greatest influence on the model fit;
2.3.1 Bivariate copulas
For each of the pairs of variables above we need to select a copula model. For this, we have the choice between a wide range of different copulas. For an extensive overview, see e.g. Nelsen (2006). Our selection of copulas is made on the basis of tail symmetry, tail dependence and positive or negative dependence arguments.
In this paper we consider, among others, the one-parameter copulas presented in Table 2. The table shows the different (a)symmetry- and tail-dependence properties of each copula. This wide array of characteristics is important, because it allows us to calibrate an R-vine model that is capable of dealing with complex dependency structures. Table 2 displays a subset of the copulas that we are using. The full list of copulas is given in the Appendix in Table 11.
Tab. 2: Pair-copula families used for R-vine specification
Copula Tail assymetry Tail dependence
Gaussian No No
Student-t No Yes
Gumbel (survival Gumbel) Yes Upper (lower)
Rotated Gumbel (90- & 270 degrees) Yes No
Frank No No
We use the previously mentioned AIC criteria to select the relevant copula families. More-over, if the correlation values in higher trees tend to be low, we replace the copula by the independence copula (Nelsen, 2006). This way, we end up with a pairwisely-truncated R-vine. Lastly, we estimate the copula parameters by using maximum-likelihood
estima-tion. Let u ∈ [0, 1]n denote a sequence of standardized residuals and let l(·) denote the
likelihood function. Czado (2010) shows that in three dimensions maximizing the log-likelihood function amounts to maximizing:
l(θ|u) = T X t=1 [log c12(u1t, u2t|θ12) + log c23(x2t, x3t|θ23) (17) + log c13|2(F (x1t|x2t, θ12), F ((x2t|x3t, θ23)|θ13|2)], (18)
where θ denotes the parameter vector, θ12, θ23and θ13|2 denote the parameter values of the
corresponding copula densities. Here, the independence assumption between the conditioning variables and the parameters of the copula corresponding to the conditioned variables is crucial.
2.3.2 Critique against R-vine models
and the conditioning variables. Accordingly, this simplifying assumption needs to be tested first. Ingrid et al. (2010) show that even when the simplifying assumption does not hold, the R-vine models may still provide a good fit to the data.
An example of a test for the simplifying assumption is the one presented in Kurz and Spanhel (2017). The test is innovative because it does not suffer from the curse of di-mensionality, unlike many other proposed tests (Derumigny and Fermanian, 2016), and is computationally feasible due to the use of decision trees. In the same paper, it is shown that the simplifying assumption holds for most Eurozone index-related data.
2.4
Systemic risk
2.4.1 CoVaR and ∆ CoVaR
Measuring systemic risk has been discussed in many different papers - for an overview, see for instance ECB (2010). One approach by Adrian and Brunnermeier (2016) is the use of Conditional Value-at-Risk (more commonly known as CoVaR). McNeil et al. (2015); Jorion (2006) define Value-at-Risk as
V aRα(X) = qα(FL) = FL←(α),
where FL(.) denotes the loss distribution function, FL←(α) denotes the inverse distribution
function evaluated at α%, and qα denotes the quantile function. Then, in Adrian and
Brunnermeier (2016) the conditional VaR is defined as
P rLj|C(Li) ≤ CoV aRj|C(Lα i)= α%,
with C(Li) the conditioning event of institution i. We may distinguish different contributions
to CoVaR by certain institutions by considering the ∆CoV aRj|iα :
∆CoV aRj|iα = CoV aRj|Li=V aRiα
α − CoV aR
j|Li=V aRi 50
α .
Adrian and Brunnermeier (2016) propagate this measure as both forward looking and coun-tercyclical. In other words, systemic risk accumulates in good times and releases in times of depression. In the same paper, it is also stressed that the CoVaR measure is directional.
In this paper, the focal point will be on two different effects. First, the effect that index i has on the system as a whole; i.e,
∆CoV aRsystem|iα = CoV aRsystem|Li=V aRiα
α − CoV aRsystem|L
i=V aRi 50
α .
And secondly, the quantiles of the loss distribution given by the Value-at-Risk.
It is because of the aforementioned properties that the ∆CoV aR is an interesting and in-formative complement alongside the more commonly used VaR. In specific, the VaR measures the risk development of stocks on an an sich basis - independent of market developments. The ∆CoV aR complements this measure by including the comovement of indices with their counterparts and allows for asymmetries in this process.
2.4.2 Other possible measures of systemic risk
3
Data
3.1
Data preparation
We consider data from country indices, where we only examine those countries that are in the Eurozone. A special distinction is made between countries belonging to group A and countries belonging to group B (ECB, 2017b). Countries that neither belong to group A nor group B are collected within the ‘Rest of Europe’ group, denoted by RoE. In previous years, group A and B were also denoted as, respectfully, ‘non-vulnerable’ and ‘vulnerable’, ‘non-distressed’ and ‘distressed’ and ‘non-core’ and ‘core’. The labelling of the countries
reflects the economic growth vis-`a-vis financial distress that occurs or has occured in each of
the countries. Table 3 shows an overview of each of the groups.
Economic growth moves conjointly with returns and losses observed in the equity markets (Levine and Zervos, 1998) - and therefore moves side-by-side with the expansion and con-traction of risk. Grouping the equity indices beforehand can give us an interesting tool for analyzing the development of risk through time, for equity markets in countries experiencing low economic growth and for those with better economical prospects. Aside from the country grouping, we also list the credit ratings for each country, with credit ratings from the major rating agencies (Moody’s, S&P and Fitch). These credit ratings are collected for reference date December 25, 2017. Be mindful of the fact that the ratings reflect merely a snapshot of the risk present in each country. The ratings prove to be interesting as they show that the economic-growth indicators are to a certain extent correlated with the country-risk profile. Observe, for instance, that countries in group B have substantially lower credit ratings than those in group A.
Tab. 3: Indices and identifiers per country.
Group Country ID Country-Credit Ratings
Moody’s S&P Fitch
A Austria ATX Index Aa1 AA+ AA+
A Belgium BEL20 Index Aa3 AAu
AA-A Finland HEX Index Aa1 AA+ AA+
A France CAC Index Aa2 AAu AA
A Germany DAX Index Aaa AAAu AAA
A The Netherlands AEX Index Aaa AAAu AAA
B Greece ASE Index Caa2 B-
B-B Ireland ISEQ Index A2 A+ A+
B Italy FTSEMIB Index Baa2 BBBu BBB
B Portugal PSI20 Index Ba1 BBB-u BBB
B Spain IBEX Index Baa2 BBB+ BBB+
RoE Cyprus CYSMFTSE Index (P)Ba3 BB+ BB
RoE Estonia TALSE Index WR AA- A+
RoE Latvia RIGSE Index A3 A-
A-RoE Lithuania VILSE Index A3 A-
A-RoE Luxemburg LUXXX Index Aaa AAA AAA
RoE Malta MALTEX Index N.A. A- A+
RoE Slovakia SKSM Index A2 A+ A+
RoE Slovenia SVSM Index Baa1 A+
A-Source: Bloomberg. Moody’s sovereign rating for malta is unavailable.
Observe that the time frame that we study here includes periods in which several of the countries mentioned in 3 were not yet present in the Eurozone. This is done because it allows us to study the development of financial contagion through time, with more-and-more countries entering the Eurozone. Aside from this, ECB (2017b) employs the same approach. For a complete description of when all countries entered the Eurozone, see ECB (2017a).
Table 4 describes how we differentiate between the various time frames - where we make a separation based on corresponding market developments. This division of time into several stages gives us the opportunity to monitor the development of risk on a macro scale - taking into account major developments in Eurozone-equity markets that had a substantial effect on risk expansion and contraction. The division also allows us to get an impression of the development of the contagion mechanism through time, and thereby gives an indication of the fragmentation within market responses.
In the case of the Slovenian index, there are two indices that might be considered: the Ljubljana Stock Exchange Composite Index, which was in operation until October 15, 2010. The other is the Slovenian Blue-Chip Index, which started in 2006 and is still in operation today. Taking into account that our dataset overlaps both periods of time, we need to find a proper way to include both indices in the analysis.
Tab. 4: Division of data based on time periods
Time period Description
2003-2006 Pre-crisis period
2007-2009 Banking crisis
2010-2013 Sovereign-debt crisis
2014-2017 Post-crisis period
moment where the new index becomes operational we switch to the new index. This is done by indexing the new index to the value of the old index. Each subsequent index value is then calculated as change of the new index times the previous value of the indexed series. The upside of this approach is that it leaves the return values intact and does not result in a major spike in the Slovenian index when switching from one index to another.
Another important component of our data is that among the constituents of each in-dex, there should be a wide array of industries. Specifically, the financial sector should be respresented, since financial intermediaries have been shown to be an important source of contagion in the past (Billio et al., 2010). In the Appendix, we show that for snapshot date January 1, 2018, each of the indices is composed of at least one financial-services firm. The indices with the least financial-services firms as constituents are Portugal and Slovakia.
Let Vt,i denote the value of index i at day t. We compute the compound return Rt,i of
index i at time t by using the following transformation: Rt,i = log(Vt,i) − log(Vt−1,i),
and then losses are computed as the negative value of the returns.
Due to public holidays in several countries, we do not have data available for each weekday in each country. We need to ‘clean’ our data so that we take this feature into account. To do so, we replace those days in which no return is available by linearly interpolating between the returns that are available. The downside of this approach is that we artificially create returns
that were not present beforehand. However, the alternative of deleting the observation
alltogether results in significant loss of information for the other indices. Therefore, we choose to interpolate in the case when there is a public holiday. One can find a similar procedure in, for instance, Satchell and Scowcroft (2003). We test the robustness of this approach by running our models for the case where we delete all data and for the case where we choose different points to interpolate on.
Tab. 5: Summary statistics of loss data - fitted on data from 2003 to 2017.
Statistic Mean St. Dev. Min Median Max
aex −0.000126 0.013281 −0.100283 −0.000577 0.095903 dax −0.000370 0.013609 −0.107975 −0.000941 0.074335 bel20 −0.000181 0.012104 −0.093340 −0.000488 0.083193 cac −0.000137 0.013677 −0.105946 −0.000465 0.094715 ase 0.000223 0.018671 −0.134311 −0.000447 0.144131 hex −0.000130 0.013640 −0.088500 −0.000615 0.092318 atx −0.000279 0.014769 −0.120210 −0.001143 0.102526 ibex −0.000123 0.014238 −0.134836 −0.000784 0.131853 psi20 0.000024 0.011875 −0.101959 −0.000377 0.103792 iseq −0.000126 0.013940 −0.097331 −0.000609 0.139636 ftsemib 0.000024 0.015040 −0.108742 −0.000724 0.133314 luxxx −0.000207 0.012735 −0.091043 −0.000488 0.111586 talse −0.000445 0.010523 −0.120945 −0.000563 0.070459 rigse −0.000429 0.011772 −0.115946 −0.000255 0.078586 vilse −0.000521 0.010157 −0.110015 −0.000536 0.119369 maltex −0.000238 0.006619 −0.047385 −0.000056 0.047367 sksm −0.000215 0.011041 −0.118803 0.000000 0.095775 svsm −0.000096 0.010144 −0.083584 −0.000257 0.084311 cysmftse 0.000549 0.022377 −0.164664 0.000211 0.142304
Based on 3857 observations for each index.
Observe that, consistent with the stylized facts, there is little persistency in the losses, but a high level of persistency in the squared- and absolute time series. In addition to this,
the Jarque-Bera-4 and a Shapiro-Wilkinson5 test rejects the null hypothesis of normality for
each serie of losses at the 1% significance level.
Figure 2 shows the empirical 95%VaR estimates as well as the empirical ∆CoVaR esti-mates through time. Specifically, the first figure illustrates the development of the VaR per index per year from 2003 to 2017, where we differentiate between the three groups of coun-tries. It can be observed here that the VaR goes from values around 0.02 and 0.03 in 2003 for, respectively, group-A countries and group-B countries to values close to 0.08 in 2008. From here, the values decrease gradually until they are back at their normal level (which happens around 2016). Notice also that the VaR of Cyprus remains high for a prolonged amount of time. This is due to an abnormal amount of market stress in the Cyprese markets. Figure 2b shows the development of the ∆CoVaR through time. A positive value here
4Jarque, Carlos M.; Bera, Anil K. (1980). ”Efficient tests for normality, homoscedasticity and serial independence of regression residuals”. Economics Letters. 6 (3): pages 255–259.
A B RoE 2004 2008 2012 2016 2004 2008 2012 2016 2004 2008 2012 2016 0.02 0.04 0.06 year V aR Group A B RoE Label aex ase atx bel20 cac cysmftse dax ftsemib hex ibex iseq luxxx maltex psi20 rigse sksm svsm talse vilse
(a) Empirical 95%VaR estimates per year
A B RoE 2004 2008 2012 2016 2004 2008 2012 2016 2004 2008 2012 2016 0.00 0.01 0.02 0.03 0.04 year dCoV aR Group A B RoE Label aex ase atx bel20 cac cysmftse dax ftsemib hex ibex iseq luxxx maltex psi20 rigse sksm svsm talse vilse
(b) Empirical delta CoVaR - calculated for each year
means that there is a positive comovement effect between the index and the market, i.e. stress in the index goes hand in hand with stress in the market return. A negative value is an indication of a negative comovement effect, i.e. the market is stressed more in periods where the index is at its median return compared to when the index attains its VaR.
4
Results
In this section we present the results of our modelling strategy. The first part focuses on the fitting of the R-vine model. Second, we discuss the topic of model fit, specifically related to the residuals of the univariate models as well as the simplifying assumption of independence between the conditioned variables and the conditioning variables. Third, we take a closer look at the residuals of the model and analyze the corresponding network. The R-vine model is accompanied by several benchmark models that highlight the performance of the former. In the fourth section we analyse the risk measures. Last, we end with a backtesting study in which we test our model based on the risk-management criteria presented in McNeil et al. (2015).
4.1
Univariate models
4.1.1 Estimates
Table 6 shows the results of the fitted univariate ARMA-GARCH models for the DAX and the ASE on pre-crisis data and on crisis data. We test a variety of ARMA-GARCH models with different orders, ranging between 0 and 2 for the mean model and between 0 and 1 for the variance model. The table displays the estimated coefficients and the corresponding standard errors. Observe that the algorithm fitted an ARMA(2,2)-GARCH(1,0) model on the pre-crisis DAX data, whereas it fitted an ARMA(2,2) model on pre-crisis-period data. Note that we obtain a positive order for the auroregressive component of the ARMA part - however, note that the coefficient AR1 is positive and that AR2 is negative. It is difficult to interpret these outcomes as supportive of a momentum effect given the negative autocorrelation coefficient of the second lag.
Tab. 6: Estimation results of the univariate models for the DAX and the ASE - fitted on pre-crisis and banking-crisis data.
DAX
Pre-crisis Banking crisis
Est. S.E. t value Pr(> |t|) Est. S.E. t value Pr(> |t|)
mu 0.0009 0.0001 12.1065 0.0000 0.0001 0.0005 0.2067 0.8362 ar1 1.8686 0.0014 1334.9146 0.0000 0.0845 0.0042 19.8870 0.0000 ar2 -0.8744 0.0013 -671.6663 0.0000 -0.9825 0.0037 -264.2732 0.0000 ma1 -1.9336 0.0001 -29876.9957 0.0000 -0.0990 0.0003 -322.8196 0.0000 ma2 0.9375 0.0001 41795.3755 0.0000 0.9872 0.0001 13785.1833 0.0000 omega 0.0000 0.0001 0.9613 0.3364 0.0004 0.0001 4.9527 0.0000 beta1 0.9980 0.0001 12950.4940 0.0000 - - - -shape 5.8155 0.6555 8.8717 0.0000 2.9703 0.3612 8.2224 0.0000 ASE
Pre-crisis Banking crisis
Est. S.E. t value Pr(> |t|) Est. S.E. t value Pr(> |t|)
mu 0.0011 0.0003 3.6704 0.0002 -0.0001 0.0006 -0.2344 0.8147 ar1 0.3797 0.0076 49.7895 0.0000 -0.6751 0.0693 -9.7491 0.0000 ar2 -0.9813 0.0056 -176.7295 0.0000 -0.8504 0.1187 -7.1650 0.0000 ma1 -0.3635 0.0029 -123.5758 0.0000 0.7126 0.0758 9.3969 0.0000 ma2 0.9884 0.0001 14194.9797 0.0000 0.8182 0.1274 6.4230 0.0000 omega 0.0001 0.0000 14.6153 0.0000 0.0004 0.0001 5.5358 0.0000 shape 5.5403 0.8995 6.1595 0.0000 3.1578 0.4286 7.3679 0.0000
4.1.2 Correlations of residuals
Figures 3 to 6 show the remaining correlations present in the residuals. The correlations are computed using Kendall’s tau. In each of the figures, the upper-left corner shows a frequency distribution of the residual correlations. The heatmaps show that there is quite a lot of correlation present between residuals of western countries, whereas the correlation levels between Baltic and eastern countries are considerably lower. It can also be noted that there tends to be higher correlations between countries that are geographically close, for instance, Greece and Cyprus or Portugal and Spain. Notice that the correlations between the remaining country-indices in group RoE are generally quite low.
4.2
Multivariate analysis
4.2.1 Comparative study
In this subsection, we provide a comparative analysis of the models described above. Recall that we fit a mixed-copula R-vine model (where we consider both one- and two-dimensional copulas), a Gaussian R-vine copula, a Student-t R-vine copula, a C-vine copula and a mixed-copula model with only one-parameter mixed-copulas. For the results, see table 7. Notice that the R-vine model gives the best fit, as illustrated by the AIC (where the lowest value is indicative of the best fit). We perform a Vuong test (Vuong, 1989) to disentangle the performance of the R-vine model relative to the other popular alternatives. Notice that the mixed R-vine model outperforms several popular competing models, such as the Student-t R-vine copula model and the C-vine copula model.
Tab. 7: Comparative study of different R-vine models. Notice that we compare R-vine mod-els with (i) all copulas mentioned in Czado (2010), (ii) only normal copulas and (iii) only Student-t copulas. Moreover, we fit (v) a C-vine model with the set of copulas as in Czado (2010) and (vi) a mixed R-vine mode with only those copulas that are listed below.
Mixed (all) Normal Student-t C-vine Mixed (1-6)
Num. of par. 157 98 160 164 142 Independence 267 263 269 265 268 Gaussian 6 98 24 3 6 Student-t 45 0 68 48 49 Clayton 3 0 0 5 6 Gumbel 7 0 0 5 11 Frank 15 0 0 13 13 Joe 0 0 0 0 1 AIC -53, 783 -51, 220 -53, 578 -53, 432 -53, 669 Log likelihood 27, 048 25, 708 26, 949 26, 880 26, 976
P-value Vuong test − 0 0.0001 0 0.0001
The first column of Table 7 shows that relatively few edges are fitted with the Gaussian copula. This is in line with our expectation. The Student-t copula, on the other hand, is fitted 45 times, indicative of the amount of kurtosis present in the tails of the bivariate distributions. Also note that the first column shows 157 fitted parameters, of which 6 + 45 + 3 + 7 + 15 = 76 are fitted with the most famous one-parameter copulas. The remaining edges are fitted with either less famous one-parameter copulas or two-parameter copulas. The fact that there is a large number of edges fitted with two-parameter copulas might be taken as indicative of a complex dependence structure between the variables.
fitted models. This is in line with the research done in MacKenzie and Spears (2014). In addition, notice that the mixed R-vine model with all copulas outperforms the other models both in terms of AIC and in terms of log likelihood. The Vuong test in the last row confirms that this difference in the log likelihoods is statistically significant at the 1% significance level.
Regarding the robustness of our interpolation method, we calibrate our models on a dataset with (i) all public holidays removed, and (ii) an interpolation technique where we interpolate taking an average of all previous two days and next two days. The latter technique does not impact our results - i.e. we find the same trees and similar estimates. The removal of public holidays however yields the estimation of some of the univariate models unstable in the sense that some contain a unit root - that is, the series tend to diverge in the long run. Thus, the modelling approach is robust to the kind of interpolation technique considered, but not to removal of public holidays altogether. This fragility is likely the result of too few data available for robust estimation.
In addition to this robustness test, we fit our model using a pairwisely-truncated model for truncation levels between two and nineteen. For each of these models, we take the log likelihood and perform a Vuong test to asses whether there is one level at which the model fit is statistically equivalent to the full model (without truncation). We find that for the model fitted on the full dataset, from truncation level 15 until 19 the model does not significantly perform worse compared to the model where we do not truncate at all. For the models fitted on the smaller datasets, the results are similar with truncation levels between tree 13 and 16.
4.2.2 Testing the simplifying assumption
Tab. 8: Results of the simplifying-assumption test - reported is the equivalent-correlation test (Ingrid et al., 2010) with a significance level of 5%.
Name Description
Data Pre-crisis
Period 2003-2006
Dimension 19
n 1039
ECORR test Fail to
results reject H0
Name Description
Data Banking crisis
Period 2007-2009
Dimension 19
n 784
ECORR test Fail to
results reject H0 Name Description Data SD-crisis Period 2010-2013 Dimension 19 n 1043
ECORR test Fail to
results reject H0
Name Description
Data Post crisis
Period 2007-2009
Dimension 19
n 992
ECORR test Fail to
results reject H0
4.2.3 Network analysis
aex dax bel20 cac ase hex atx ibex psi20 iseq ftsemib luxxx talse rigse vilse maltex sksm svsm cysmftse
Fig. 7: R-vine network - pre-crisis, 2003-2006
aex dax bel20 cac ase hex atx ibex psi20 iseq ftsemib luxxx talse rigse vilse maltex sksm svsm cysmftse
aex dax bel20 cac ase hex atx ibex psi20 iseq ftsemib luxxx talse rigse vilse maltex sksm svsm cysmftse
Fig. 9: R-vine network - sovereign-debt crisis, 2010-2013
aex dax bel20 cac ase hex atx ibex psi20 iseq ftsemib luxxx talse rigse vilse maltex sksm svsm cysmftse
Another finding is that there is no clear clustering within the B group. That is, it seems as if during the 2003-2006 period there was not much fragmentation within the correlation between stock markets. This result is surprising, as at this stage many of the countries in the analysis are not yet part of the Eurozone and the corresponding European Monetary Union (EMU), which might lead to fragmentation (especially in, for instance, the Baltic countries). Strikingly, this result changes when one considers the other figures. In both crisis periods, one may observe that group-B countries start to cluster together - with in the last figure almost all group-B countries connected by edges. Even though each group-B country is still connected to every group-A country, the amount of edges between both groups has increased. This might be taken as an indication of markets and their correlation structure having become more fragmented in the aftermath of the crisis.
4.3
Systemic risk developments
Figures 11 to 14 plot both the VaR and the ∆CoVaR for all the indices. These values are based on simulated data from the R-vine model. We differentiate based on group - the left panel represents group A, the middle panel represents group B and the right panel the remaining Eurozone indices - and on time - respectively pre-crisis, banking crisis, sovereign-debt crisis and post-crisis.
The figures are interesting because of several reasons. First and foremost, one may
observe that there seems to be no clear correlation between Value-at-Risk and ∆CoVaR. In fact, it seems that at many times when there is no spread in the Value-at-Risk, there is a relatively large spread between the ∆CoVaR estimates. This is one of the reasons why Adrian and Brunnermeier (2016) advocate the use of ∆CoVaR - the VaR is a static measure that does not take the tail behaviour of losses into account relative to other losses.
The figures show that in the pre-crisis period (Figure 11), the VaR was rather similar among indices. The ∆CoVaR, however, shows that there are many differences among the indices. Recall that the ∆CoVaR measures the quantile of the market return given that some specific index is stressed (losses are above its 95%VaR) versus the quantile of market return given that the index is at its median. Thus, a high ∆CoVaR implies that the market is stressed more when some specific index is under stress compared to when the index is performing regularly. This relationship is not causal, since there might be other indices that are stressed at the same time. Nonetheless, the measure does give an interesting insight into the correlation structure behind market indices.
aex dax hex ase ibex psi20 iseq ftsemib luxxx talse rigse vilse maltex sksm svsm cysmftse A B RoE 0.01 0.02 0.03 0.04 0.05 0.06 0.01 0.02 0.03 0.04 0.05 0.06 0.01 0.02 0.03 0.04 0.05 0.06 0.000 0.002 0.004 0.006 0.008 VaR dCoV aR
aex dax bel20 cac hex ase ibex psi20 iseq ftsemib luxxx talse rigse vilse maltex sksm svsm cysmftse A B RoE 0.04 0.05 0.06 0.07 0.04 0.05 0.06 0.07 0.04 0.05 0.06 0.07 0.00 0.01 0.02 0.03 0.04 0.05 VaR dCoV aR
aex dax hex ase ibex psi20 iseq luxxx talse rigse vilse maltex sksm svsm cysmftse A B RoE 0.02 0.04 0.06 0.08 0.02 0.04 0.06 0.08 0.02 0.04 0.06 0.08 0.000 0.005 0.010 0.015 0.020 VaR dCoV aR
aex hexatx ase ibex psi20 luxxx talse rigse vilse maltex sksm svsm cysmftse A B RoE 0.02 0.04 0.06 0.02 0.04 0.06 0.02 0.04 0.06 0.000 0.005 0.010 0.015 VaR dCoV aR
4.4
Backtesting study
Table 9 shows the results of the backtesting study of the R-vine models, performed both on crisis- and non-crisis data. The backtesting study consists of estimating the VaR for various rolling windows of three-year loss data. For each of these windows, the forecasted VaR is compared to the actual loss for the next day. The amount of exceedances of the estimated VaR forecast is documented and the rows in Table 9 show the fraction per index. Note that
this number should be close to 0.01 or 0.05, for V aR.99 and V aR.95, respectively.
We use the binomial test and the runs test (McNeil et al., 2015) to test whether the exceedances of the VaR are, respectively:
1. Binomially distributed with p = .01 or p = 0.05; and 2. Independently distributed.
The results of the test are displayed in Table 9 and 10. The table shows - per index - the fraction of times the estimated VaR is exceeded by the losses during the next day. The additional two columns give the p-value of the binomial test and the p-value of the runs test for independence of these exceedances. Note that the fractions of exceedances per indices are similar for some indices - for instance, the DAX and the AEX. This is a result of the small backtesting sample of 50 rolls.
The results of the backtesting study are interesting because of several reasons. First, no-tice that we fail to reject the null hypothesis that the exceedances are binomially distributed with p = 0.05 across both crisis- and non-crisis periods, with p-values around 0.5. This can be taken as indication that the model manages to forecast the 95%VaR quite accurately. However, the results of the runs test indicate that exceedances tend not to occur on an in-dependent basis. It seems as if the model is not always able to take the volatility clustering into account - that is, exceedances of the VaR tend to cluster together.
Tab. 9: Exceedances of backtesting study at 95%VaR - respectfully on crisis and non-crisis data - for each dataset fitted on a rolling window of 750 observations with 50 rolls.
P-value
Fraction Binomial Runs
aex 0.020 0.520 0.000 dax 0.020 0.520 0.000 bel20 0.020 0.520 0.000 cac 0.020 0.520 0.000 ase 0.020 0.520 0.000 hex 0.020 0.520 0.000 atx 0.039 1.000 0.071 ibex 0.020 0.520 0.000 psi20 0.059 0.742 0.000 iseq 0.020 0.520 0.000 ftsemib 0.020 0.520 0.000 luxxx 0.098 0.110 0.397 talse 0.078 0.324 0.014 rigse 0.039 1.000 0.000 vilse 0.020 0.520 0.000 maltex 0.020 0.520 0.000 sksm 0.039 1.000 0.071 svsm 0.039 1.000 0.071 cysmftse 0.078 0.324 0.014 P-value
Fraction Binomial Runs
Tab. 10: Exceedances of backtesting study at 99%VaR - respectfully on crisis and non-crisis data - for each dataset fitted on a rolling window of 750 observations with 50 rolls.
P-value
Fraction Binomial Runs
aex 0.216 0.000 0.000 dax 0.196 0.000 0.071 bel20 0.196 0.000 0.000 cac 0.098 0.000 0.000 ase 0.098 0.000 0.000 hex 0.196 0.000 0.000 atx 0.176 0.000 0.000 ibex 0.157 0.000 0.071 psi20 0.176 0.000 0.071 iseq 0.216 0.000 0.071 ftsemib 0.157 0.000 0.000 luxxx 0.235 0.000 0.701 talse 0.216 0.000 0.071 rigse 0.216 0.000 0.000 vilse 0.216 0.000 0.071 maltex 0.118 0.000 0.000 sksm 0.098 0.000 0.000 svsm 0.157 0.000 0.071 cysmftse 0.039 0.015 0.071 P-value
Fraction Binomial Runs
5
Discussion
In this paper, we use a step-wise procedure to build up a model that is capable of analyzing complex dependency structures. This step-wise procedure allows for an extensive amount of flexibility. However, care should also be taken at each step, since errors in each step influence the model as a whole.
Step one of the approach consists of finding suitable models for the marginal distribu-tions. Due to computational factors, we have decided to select ARMA-GARCH models with Student-t distributed residuals. Even though this model has been proven to be successful in previous applications in literature, it should be noted that the fact that we obtain a positive autoregressive order for the mean model is more difficult to interpret economically - aside from providing a good model fit. One might also interpret this finding as a sign of overfitting. It might be that we get different results by applying a different kind of marginal model. In future studies, it might be interesting to consider different (i) ARMA-GARCH specifications, (ii) a different distribution for the error term, or (iii) a different marginal model altogether. In the network analysis section, we have shown that the correlation mechanism between the residuals of the indices has become more fragmented during the years following the crisis. This fragmentation might have been caused due to the fact that all group-B countries have received bail-out funds in the past due to a large amount of non-performing loans (NPLs) on their balance sheet. These excessive amounts of NPLs resulted in credit-rating downgrades both on a country and a firm level, limiting market access. In fact, Greece has seen a long period with very limited access to the money markets due to the fact that their bonds had junk bond status (Eijffinger, 2012; De Pascalis, 2017). This also implies that in order for companies in these countries to get funding, they might need to turn towards countries with lower credit ratings, such as their group-B counterparts.
Another explanation might be that the constituents of the indices of group-B indices have converged, with convergence in the sense that the same industries are booming in all indices, resulting in similar movements in the indices. However, notice that it is difficult to impossible to test these theories without having (i) bank-transaction data, or (ii) more information about the constituents of the indices.
Figures 7 to 10 also show that several group-A countries have a high amount of degree centrality. This degree centrality might be due to the fact that each of these indices consist of several global systemically important banks - more commonly referred to as G-SIBs (FSB,
2017). Several examples of such banks are Soci´et´e G´en´erale, HSBC and Credit Agricole. Our
findings here are in line with those in Acharya et al. (2012) and Engle et al. (2015), where the authors show that several banks with headquarters in group-A countries are within the top-10 of banks with the highest amount of systemic risk.
more research is required to provide conclusive evidence.
The results presented in Figures 11 to 14 are illustrative of the anti-cyclical manner in which risk tends to develop. This development might be indicative of the volatility paradox (Adrian and Brunnermeier, 2016). The speed at which this development occurs is likely different between the groups of countries, as shown in Figure 2b. Moreover, the expansion and contraction of risk tends to evolve faster for countries in group A.
The backtesting study provides an interesting insight into the quality of VaR estimates, both on crisis and non-crisis data. Observe here that the VaR-forecast quality seems to deteriorate in times of crisis. It should be noted that it is not the goal to choose the model with the best backtesting properties since this would likely result in overfitting - however, poor backtesting properties can be indicative of a larger problem of model fit. It would be interesting to see if we can find a specification of the R-vine model, or a different model, in which shocks have a larger effect on the estimated risk measures. One such model might, for instance, be the one used in Oh and Patton (2017).
6
Conclusion
Market indices tend to be subject to complex mechanisms that control their dependency structure. These mechanisms are shaped by, among others, supply and demand, geopolitical developments, trade deals as well as industry-specific evolutions. This motivates the use of models that are capable of modeling such dependencies, such as the R-vine copula model.
In risk management, the current golden standard is the Value-at-Risk measure. However, by applying an R-vine model, we have demonstrated that using only Value-at-Risk may not suffice as proxy for risk. This is due to the fact that VaR is calculated on an ‘an sich’ basis, without taking into account the spillover effects that stocks might have on other stocks. A possible complement for the VaR is the ∆CoVaR. This measure does take into account the interaction that stocks might have with the system as a whole.
In this paper, we have used the ∆CoVaR to show that the volatility paradox is a prominent factor for group-A indices as well as for group-B indices. That is, systemic risk seems to expand in times of crisis and contract in times without turmoil. The speed at which this process happens is likely to be different between indices, as shown in Figure 2b.
Moreover, by considering the networks produced for each of the four time periods, we have shown that market movements have seen an evolution from an integrated-market response before the crisis to a slightly fragmented-market response in the aftermath of the crises. This move seems to go hand-in-hand with the division into group-A and group-B countries, and might be a result of the limited market access of group-B countries during and after the banking- and the sovereign-debt crisis.
A suggestion for future research would be to include several additional layers to our network, such as the money markets, the bond markets, the Foreign-Exchange (FX) market and the Overnight-Indexed Swaps (OIS) market. This can be done by adding, among others, the EURIBOR rate, bond rates for each country, the EURUSD swap rates as well as the OIS swap rates. The inclusion of these markets gives a comprehensive overview of the spillover effects between countries and markets through time.
References
Aas, K., C. Czado, A. Frigessi, and H. Bakken (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics 44 (2), 182–198.
Acar, E. F., C. Genest, and J. Neˇslehov´a (2012). Beyond simplified pair-copula constructions.
Journal of Multivariate Analysis 110, 74 – 90.
Acharya, V., R. Engle, and M. Richardson (2012, May). Capital shortfall: A new approach to ranking and regulating systemic risks. American Economic Review 102 (3), 59–64. Adrian, T. and M. Brunnermeier (2016). Covar. American Economic Review 106 (7), 1705–
1741. Predicting and measuring a financial institution’s contribution to systemic risk that internalizes externalities and avoids procyclicality.
Almeida, C., C.A Czado, and H. Manner (2012, 02). Modeling high dimensional time-varying dependence using d-vine scar models. Applied Stochastic Models in Business and Industry. Aloui, C. and B. Hkiri (2014). Co-movements of gcc emerging stock markets: New evidence
from wavelet coherence analysis. Economic Modelling 36 (C), 421–431.
Baig, T. and I. Goldfajn (1999, Jun). Financial market contagion in the asian crisis. IMF Staff Papers 46 (2), 167–195.
Balvers, Ronald J. and Yangru Wu (2006). Momentum and mean reversion across national equity markets. Journal of Empirical Finance 13 (1), 24 – 48.
Barberis, N., A. Shleifer, and J. Wurgler (2005). Comovement. Journal of Financial Eco-nomics 75 (2), 283–317.
Bedford, T. and R. M. Cooke (2001, Aug). Probability density decomposition for condition-ally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence 32 (1), 245–268.
Billio, M., M. Getmansky, A. Lo, and L. Pelizzon (2010, July). Econometric measures of systemic risk in the finance and insurance sectors. Working Paper 16223, National Bureau of Economic Research.
BIS (2011). Basel III: A Global Regulatory Framework for more resilient Banks and Banking Systems. Bank for International Settlements.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. EERI Re-search Paper Series EERI RP 1986/01, Economics and Econometrics ReRe-search Institute (EERI), Brussels.
Bollerslev, T., R. Engle, and J. Wooldridge (1988). A capital asset pricing model with time-varying covariances. Journal of Political Economy 96 (1), 116–31.
Brechmann, C. and C. Czado (2013). Risk management with high-dimensional vine copulas: An analysis of the euro stoxx 50. Statistics Risk Modeling 30 (4), 307–342.
Brechmann, E. C., C. Czado, and K. Aas (2012). Truncated regular vines in high dimensions with application to financial data. Canadian Journal of Statistics 40 (1), 68–85.
Brunnermeier, M. K. and Y. Sannikov (2014, February). A macroeconomic model with a financial sector. American Economic Review 104 (2), 379–421.
Claessens, S., M. Ayhan, L. Laeven, and F. Valencia (2013, January). Understanding finan-cial crises: Causes, consequences, and policy responses. Technical report.
Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1, 223–236.
Czado, C. (2010). Pair-Copula Constructions of Multivariate Copulas, pp. 93–109. Berlin, Heidelberg: Springer Berlin Heidelberg.
Dale, G., R. Merton, and B. Zvi (2007, November). New framework for measuring and man-aging macrofinancial risk and financial stability. Working Paper 13607, National Bureau of Economic Research.
De Pascalis, F. (2017). Credit Ratings and Market Over-reliance: An International Legal Analysis. Brill-Nijhoff International Trade Law Series.
Derumigny, A. and J.D. Fermanian (2016, December). About Tests of the ‘Simplifying’ Assumption for Conditional Copulas. ArXiv e-prints.
Dißmann, J., E.C. Brechmann, C. Czado, and D. Kurowicka (2013). Selecting and estimating regular vine copulae and application to financial returns. Computational Statistics Data Analysis 59 (C), 52–69.
ECB (2010). Analytical models and tools for the identification and assessment of systemic risks - financial integration in Europe. ECB working paper .
ECB (2017a). Euro area 1999-2015. from https://www.ecb.europa.eu/euro/intro/html/map.en.html . ECB (2017b). Financial integration in Europe. ECB working paper .
Eijffinger, S. (2012, 11). Rating agencies: Role and influence of their sovereign credit risk assessment in the eurozone. JCMS Journal of Common Market Studies 50.
