Quantifying Systemic Risk within the European Union through a simulation of a banking crisis

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Quantifying Systemic Risk within

the European Union through a

simulation of a banking crisis

Derk de Boer

s2714051

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Master thesis: Econometrics, Operations research and Actuarial sciences Supervisor: Prof. Dr. L. Spierdijk

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Quantifying Systemic Risk within the European

Union through a simulation of a banking crisis

Derk de Boer

January 6, 2020

Abstract

The impact of systemic risk became apparent during the global crisis of 2008. Afterward, the interest in developing methods to quantify the level of systemic risk sharply increased. This study aims to quantify the level of systemic risk be-tween the national banking sectors in Europe by evaluating the co-dependence of large equity losses of the national banking with the equity losses of the continent-wide banking system. This leads to the following research question: ”What is the magnitude of an increase in the value-at-risk of one country’s banking sector eq-uity returns on the value-at-risk of its surrounding countries?” The relation be-tween the value-at-risk (VaR) of each nation’s banking sector equity returns and the VaR of the European system is modeled through applying the multivariate multi-quantile conditional autoregressive value-at-risk (MVMQ-CAViaR). The model is evaluated based on its out-of-sample VaR forecasting performance, us-ing the dynamic quantile test (DQ). Finally, the research question is answered using quantile impulse response analysis. This study concludes the GIIPS coun-tries (Greece, Ireland, Italy, Portugal and Spain) are most vulnerable to a sys-temic banking crisis, and the countries with the largest banking sectors com-pared to their equity sector have the largest potential to initiate a systemic banking crisis. The results thereby offer insights that are highly relevant for European central bankers.

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1 Introduction

1.1 Systemic risk in Europe

The global financial crisis of 2008 highlighted the importance of monitoring system-atic risk. According to Karimalis and Nomikos (2018), systemic risk is defined as ”the adverse consequence, for the financial system and the broader economy, of a finan-cial institution being in finanfinan-cial distress”. During the crisis, finanfinan-cial losses spread rapidly across institutions causing devastating effects on the economy as a whole. As Mart´ınez-Jaramillo et al. (2010) argue, systemic risk arises through the spillover effects or contagious effects of large financial losses. As a result, the co-movement of equity prices increases during times of financial distress, as is documented in Longin and Solnik (2001).

Europe is financially a highly integrated region, where a banking crisis in one country profoundly affects the banking sectors of its surrounding countries. So, for national financial authorities, it is crucial to develop capital requirements that incorporate the contagious effects of a crisis in a neighboring country. Otherwise, the sectors find themselves in need of additional capital at the time when it is most difficult to acquire. Therefore, investigating systemic risk is crucial for national central banks, the European central bank, and other financial institutions such as the European stability mechanism (ESM).

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financed by the banking sector itself. However, if the SRF funds are depleted, the ESM has to act as a backstop by providing additional credit. For the bail-in system to function, managing systemic risk is of crucial importance. Otherwise, during crisis periods, the funds are insufficient for saving the financial sector. Consequently, the taxpayer has to step in to save the sector.

Basel III recently introduced the current leading framework for banking regulations. Many European regulatory authorities, such as the European Banking Authority, welcome the new regulations. They provide a high degree of risk sensitivity, and they aim to harmonize credit and operational risk measures. According to Caruana (2010), Basel III also proposes macroprudential measures to mitigate the effects of systemic risk. It aims to decrease the risk by reducing the pro-cyclicality of the capital requirements. However, there exists some skepticism regarding the effectiveness of the new measures. Repullo and Saurina Salas (2011) investigate the measures which aim to prevent procyclical behavior, although they concluded, it fails to achieve its objectives. George (2011) argues, Basel III is a massive step forward, however, it has its shortcomings regarding regulating systemic risk. For instance, the capital requirements are based on risk-weighted assets, which do not represent the underlying risks. As a consequence, different banks tend to hold many similar assets, which enhances the level of systemic risk.

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systemic risk. However, as White et al. (2015) argue, estimating these risk measures is a tedious process since it requires knowledge on the return distribution condition of an institution being in distress.

To summarize, systemic risk has threatened the global economy in the past, and will remain a challenge for the future. This especially holds for Europe, which was profoundly affected by the global financial crisis. To prevent future crises, Europe aims to harmonize banking regulations. However, for this system to be successful, it is of crucial importance to measure and monitor the level of systemic risk. Otherwise, a crisis in one country has devastating effects on the economy of the whole continent.

1.2 Research question

The objective of the study is to quantify the level of systemic risk within the Euro-pean banking sector by analyzing the co-movements of large equity losses across the national banking sectors. The research question is defined as:

What is the magnitude of an increase in the VaR of one country’s banking sector equity returns on the VaR of its surrounding countries?

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Since many European banks hold similar assets, this study expects to find strong linkages across the continent. Consequently, a shock in one country’s VaR substan-tially increases the VaR of the surrounding countries. Reboredo and Ugolini (2015a) investigate systemic risk for a sample of European countries, they found that systemic risk levels are relatively similar during normal times, though higher for the GIIPS countries during the European sovereign debt crisis. Although this study does not investigate a crisis period explicitly, it expects the level of systemic risk to be higher for the GIIPS countries. Furthermore, Langfield and Pagano (2016) found countries with a relatively large banking system, compared to the equity or bond market, have higher levels of systemic risk. In line with their result, this study expects countries with relatively larger banking sectors to have a higher level of systemic risk.

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using the Dynamic Quantile (DQ) test, as described in Engle and Manganelli (2004). The test measures the overall goodness-of-fit of the model by evaluating the bias and the serial dependence of the residuals.

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2 Literature Review

This section describes the previous literature related to systemic risk. First, it dis-cusses the mechanisms through which systemic risk arises, followed by a description of the different measures used for quantifying the level of systemic risk. Finally, it highlights various methods for estimating these risk measures.

2.1 Background systemic risk

According to Acharya et al. (2017), the failing of institutions during periods of finan-cial distress is a negative externality to the economy. During these periods, losses cannot be borne by the financial system itself since there is insufficient credit. The bankruptcy of an institution will further deteriorate the stability of the economy. So, the government has to step in and provide additional funds, as happened during the global financial crisis. Given that financial institutions are not fully liable for the consequences of their default, they do not spend the socially optimal amount of resources on reducing their probability of default. As a solution Acharya et al. (2017), propose regulators to step in and tax to the institutions proportional to their contributions to the level of systemic risk. Such, the incentives of the institutions are aligned with the socially optimal outcome.

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arise as followed: due to a shock in funding rates, the overall market liquidity de-creases, which leads to higher margins, which further depresses the market liquidity. Loss spirals arise when institutions face substantial losses, forcing them to sell their assets, which in turn decreases the asset prices, hence forcing more institutions to sell their assets, causing a further price decrease. The effects are empirically verified in Adams et al. (2014).

2.2 Risk measures

Measuring systemic risk has been proven to be a challenging task. According to Tobias and Brunnermeier (2016), systemic risk builds up in the background and materializes during financial crises. Tobias and Brunnermeier (2016) state the Value-at-Risk (VaR) is the most common risk measure in practice. However, the measure has its shortcomings; it depends on the performance of an institution in isolation, neglecting the relationship between the institution and the financial sector. Hence, it does not incorporate the effects of systemic risk, although, according to Acharya et al. (2017), the presence of systemic risk is a massive motivation for using this risk measure. Furthermore, the VaR is a pro-cyclical risk measure; during times of economic growth, the risk measure is low. As documented in Adrian and Shin (2010), this leads to a simultaneous increase in the leverage among institutions. When the economic growth decreases, the risk measures increase, and hence institutions need to decrease their leverage. This leads to loss spirals; thus, it enhances the level of systemic risk.

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of the financial system’s VaR, given an institution has a return equal to its VaR. Girardi and Erg¨un (2013) modified the ∆CoVaR to give the change of the system’s VaR, given an institution has a return at most at its VaR. This CoVaR approach is extended to the expected shortfall, yielding the Conditional Expected Shortfall (CoES). Acharya et al. (2017) introduced the Systemic Expected Shortfall (SES), which measures the institution’s tendency to be undercapitalized when the system is undercapitalized. Furthermore, Brownlees and Engle (2016) constructed the SRISK measure, which measures the capital shortfall of a firm conditional on a severe mar-ket decline. Afterward, it ranks the institutions based on their contributions to the level of systemic risk. Finally, there exist many more risk measures; Bisias et al. (2012) provide a comprehensive overview.

2.3 Modelling approaches

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Finally, White et al. (2015) introduced a technique based on the quantile regression, namely the MVMQ-CAViaR. In contrast to the approach of Tobias and Brunner-meier (2016), the MVMQ-CAViaR allows for time-varying volatilities. Furthermore, it is a robust semi-parametric approach that does not require strong distributional assumptions. As Tobias and Brunnermeier (2016) state, the technique can be used to model the CoVaR dynamically. So, it estimates the CoVaR for every time period of the data. Since volatilities, and hence the CoVaR, vary over time, this is a useful feature. However, this approach is not applied in the previous literature.

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3 Research strategy

This section outlines the research strategy. First, it motivates the research method, followed by a short description of the method. Afterward, a description of how to apply the method is given by providing a step-by-step plan.

3.1 Motivation modelling strategy

The main objective of this study is to investigate the effect of an increase in the VaR of one European country’s banking sector on the VaR of the surrounding coun-tries. To answer the research question, this study simulates a shock to the banking sector of each country. Then, it observes its effects on the VaR of the surrounding countries. Afterward, the simulations are used to determine which shock has the most substantial effect on the VaR of the countries, and which country is the most vulnerable to the shocks.

The study applies the MVMQ-CAViaR model of White et al. (2015). As White et al. (2015) argue, it has three advantages over traditional methods. First, it is based on the quantile regression technique, which is more robust to outliers than least squares regressions. This especially holds considering fat-tailed distributions, as is docu-mented in Engle and Manganelli (2004). Second, it is a semi-parametric technique, and hence, it imposes minimal assumptions on the data generating process. To finish, it models the object of interest, namely the VaR, directly in a multivariate framework. Similar to a vector autoregression (VAR), the model has its associated impulse response function (IRF), which allows for the shock simulations.

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there are some downsides of using the VaR. According to Acerbi and Tasche (2002), the VaR does not satisfy all criteria that a risk measure must satisfy. It violates the subadditivity criterion, which states that the risk of two separate assets is greater or equal as the risk of both assets in one portfolio. Implying the VaR of two assets combined could be lower than the VaR of the associated portfolio. As an alternative for the VaR, Acerbi and Tasche (2002) suggest using the expected shortfall. Since the MVMQ-CAViaR methodology is not yet extended to the expected shortfall, this study uses the VaR.

Since this study evaluates a large sample of European countries, it is impossible to model the VaR of each country’s banking sector simultaneously. This requires a tremendous amount of parameters. Hence the model would suffer from the curse of dimensionality. Therefore, it is necessary to generate an aggregate banking quarter and observe the effects of each country on the aggregate sector. For simplicity, the aggregate banking sector is referred to as the European banking sector, although it does not contain the data of all the European countries.

The European sector is derived using the weighted returns of the national banking sectors. This process is described in the data analysis section. Then, the shocks are simulated for each country’s banking sector and the European banking sector. After collecting the results, the countries are ranked based on their contribution to systemic risk and their vulnerability to systemic risk. So, the contribution to systemic risk is derived by observing the effect of the country’s VaR on the European banking sector, while the susceptibility derived by observing the effect of a shock to the VaR of the European banking sector.

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than the VaR is equal to the confidence level. Since this study is interested in the relationship among large losses, it selects low values as confidence levels. To mimick a significant crisis, the confidence level needs to approach zero. However, there is a trade-off between choosing a low confidence level and model accuracy. For confidence levels close to zero, the model requires additional observations to obtain consistent estimates. This study uses the percent and 1-percent confidence level. The 5-percent is selected since it commonly used in practice. The 1-5-percent is selected to capture the more extreme shock while respecting the problem of needing sufficient data.

The models are evaluated based on their out-of-sample VaR forecasting performance. There are multiple criteria for a ’good’ forecasting model; the VaR forecasts should be unbiased and serial independent. They are serial dependent if the exceedances of the model’s predicted VaR values appear in clusters. To test for these criteria, this study applies the out-of-sample dynamic quantile (DQ) test, as introduced in Engle and Manganelli (2004). As a benchmark, the same tests are applied to the VaR forecasts of the univariate CAViaR models. Furthermore, the models have different functional specifications. As a backtest, the functional specifications are also evaluated by the out-of-sample DQ test. The methodology section provides a technical description of the MVMQ-CAViaR, the DQ-tests, and the impulse response functions. Furthermore, the section illustrates how the technique is related to the currently used risk measures.

3.2 Step-by-step plan

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Figure 1: Modeling strategy

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and select the confidence levels of the VaR. The following paragraphs describe Figure 1.

The first step consists of generating the aggregate banking sector. As stated above, the primary motivation for using an aggregated banking sector is to reduce the number of parameters within the model. The sector is constructed using the weighted returns of the individual countries, Section 5.1 describes the exact derivation. The second step consists of estimating the CAViaR model on the univariate series. The univariate models serve multiple purposes. First, the parameter estimates of the univariate model are used as initial values of the MVMQ-CAViaR. Then, the VaR forecasts of the CAViaR serve as a benchmark for the multivariate model, as is described later in the process. Finally, it is used to determine the optimal functional specification of the model. The different options are described in Section 4.1.2. The out-of-sample DQ test evaluates the performance of each functional specifications. The process continues by estimating the MVMQ-CAViaR model. Section 4.2 de-scribes the technical details of the model. The DQ tests evaluate its performance by testing its out-of-sample VaR forecasts on biasedness and serial dependence. The out-of-sample DQ test results of the CAViaR function as a benchmark for the mul-tivariate test results. If the mulmul-tivariate model’s performance is worse than the performance of the univariate model, then there exists a probability that the model is overfitting the data.

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the country’s VaR are measured. Then, the results are collected, and the process is repeated for all countries. The simulation is done using the quantile impulse response function, which is described in Section 4.2.2. To compactly present the effects, the total impact is summarized into one measure. Finally, the countries are ranked based on these measures.

The research question is answered by evaluating the effects of the country on the Eu-ropean banking sector since the EuEu-ropean banking sector represents the surrounding countries. The effects of a shock to the European banking sector on the national sec-tors, hint which countries are most vulnerable to shocks. However, in this framework, it is impossible to observe how each country affects each other directly.

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4 Methodology

This section provides a technical description of the models used in this study. First, it provides a short overview of the quantile regression methodology, followed by the introduction of both univariate CAViaR and the MVMQ-CAViaR. Then, it explains the DQ-tests and the QIRF. Finally, there is a discussion on the relation between the CAViaR methodology and current systemic risk measures.

4.1 Background

4.1.1 Quantile Regression

Before introducing the main model, it is useful to consider the background of the quantile regression. Koenker and Bassett (1978) introduced the quantile regression in 1978, it is a flexible method that allows for estimating the full conditional dis-tribution of the dependent variable. In contrast to the classical regression, which minimizes the squared error function, the quantile regression minimizes the absolute error function. A particular case of the quantile regression is the least absolute devi-ations regression (LAD), which is a regression technique that evaluates the median instead of the mean. According to Engle and Manganelli (2004), the LAD regression outperforms the classical regression in terms of robustness, especially if the data generation process has fat tails. Currently, the quantile regression has numerous applications; an overview is provided by Yu et al. (2003).

Before introducing the quantile regression, consider the basic statistical notions. Let the cumulative distribution function (CDF), of a random variable Y be defined as

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The τ ’the quantile of Y , denoted by qτ is defined as

qτ = inf{y : F (y) ≥ τ }, (2)

for τ ∈ (0, 1).

Now, consider a n-sized sample y1, · · · ynof Y . Let xi be a k-vector of regressors and

let ετ,i be the error term for i = 1, · · · , n. Finally, let βτ be a k-vector of unknown

parameters. The model is given by

yi = x0iβτ + ετ,i, (3)

q_τ(ετ,i|xi) = 0, (4)

here the distribution of the error term can be left unspecified. The parameters are estimated as followed ˆ β_τ = arg min β ( 1 n n X i=1 τ − I(yi < x0iβ)(y − x 0 iβ) ) , (5)

where I(·) denotes the standard indicator function. Generally, equation (10) is solved using a Nelder-Mead algorithm. Using the parameter estimates, the conditional quantiles of Y are estimated as

ˆ

qτ,i = x0iβˆτ.

4.1.2 CAViaR

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was to establish a method which forecasts the VaR in a dynamic context, without requiring strong distributional assumptions on the error term. Since volatility of returns is clustered, large losses tend to be clustered as well. Consequently, the CAViaR generally follows a GARCH specification, which incorporates the effects of the clustering. Section 3.2 discusses the purpose of the CAViaR for this study. To illustrate the technique, consider a vector of equity returns, denoted by y, with elements yt for t = 1, · · · , T . Again let qτ be the quantile, which is given by the

function ft(βτ), where βτ is a k-vector of unknown parameters, such

qτ,t = ft(βτ). (6)

Following Engle and Manganelli (2004), the function is specified in different man-ners, depending on the assumption of the underlying data generating process. If the standard errors follow a GARCH process, the function follows the Symmetric absolute value specification, which is defined as

ft(βτ) = β1+ β2ft−1(βτ) + β3|yt−1|. (7)

Similar as in volatility models, it is assumed that the quantile depends symmetrically on the positive and negative returns. So, the function depends on the absolute value of the dependent variable. If this assumption does not hold, the function is modified to incorporate the effects of the asymmetry. Hence the function follows the asymmetric slope specification,

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where (x)+ _{= max(x, 0) and (x)}− _{= max(−x, 0). Furthermore, if the variance,}

instead of the standard error, follows a GARCH process, the function follows the indirect GARCH specification, which is given by

ft(βτ) = β1+ β2ft−12 + β3yt−12

1₂

. (9)

The estimation of the CAViaR is similar to the minimization problem in Equation 10, hence ˆ β_τ = arg min β ( 1 T T X i=1 τ − I(yi < ft(βτ))(y − ft(βτ)) ) . (10) Consequently, ˆ qτ,t = ft( ˆβτ). (11)

Under the assumptions mentioned in Engle and Manganelli (2004) the estimator is consistent.

4.2 MVMQ-CAViaR

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4.2.1 Model specification

The model is the MVMQ extension of the CAViaR. So, consider p time series instead of one. Furthermore, consider the symmetric absolute value function specification, given by Equation 7. The multivariate model for a specific quantile τ is given by

qτ,t = c + A|yt−1| + Bqτ,t−1, (12)

where qτ,t and yt are p-vectors, c is a p-vector of parameters and A, B are p × p

matrices of parameters.

Consider the MVMQ-CAViaR model with for p series and l different quantiles. The parameters are estimated as

ˆ β_τ = arg min β ( 1 T T X t=1 p X i=1 l X j=1

τ − I(yt,i < ft,i,j(βτ))(yt,i− ft,i,j(βτ))

!)

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Under the assumptions mentioned in White et al. (2015), the estimates are consistent, and the estimator follows an asymptotically normal distribution. Let ∇fi,j,t denote

the gradient of fi,j,t and let Fi,j,t denote the density of the error term conditioned on

the information available at period t − 1. The asymptotic distribution is given by

T12( ˆβ

τ − βτ)

d

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where Q = n X i=1 p X j=1

E Fi,j,t(0)∇fi,j,t(β)∇fi,j,t(β)0, (15)

V = E(ηtηt0), (16) η = n X i=1 p X j=1

∇fi,j,t(β) τ − I(yi,t < qi,j,t). (17)

4.2.2 Quantile impulse response function

The MVMQ-CAViaR allows for the estimation of quantile impulse response functions (QIRF). The resulting impulse response functions (IRF) differs from the traditional IRF associated with a VAR. Where the traditional IRFs simulate shocks to the error term of a VAR, the QIRF simulates a shock in the dependent variable. For the QIRF, it is impossible to find out the development of the dependent variable, since only the quantile is of interest. So, the QIRF is more restrictive, since it only covers the change of the quantile, qi,t keeping the dependent variable, yi,t, constant. Therefore,

White et al. (2015) named it the pseudo quantile impulse response function. This study refers to this pseudo quantile impulse response function as the QIRF. The part below illustrates how the QIRF operates. In this study, the QIRF is used for the shock simulation, as mentioned in Figure 1.

Consider the series y = (· · · , ys−1, ys, ys+1, · · · ), suppose there is a one time shock of

size δ at period s. Now, the new series is given by ˜y = (· · · , ys−1, ys+ δ, ys+1, · · · ).

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which is given by QIRFi_h(δ, y) = ˜qt+h− qt+h, (18) QIRFh(δ, y) =      QIRF1_h(δ, y) .. . QIRFn_h(δ, y)      , (19)

where ˜qt+hand qt+hare the quantiles associated with ˜y and y respectively. Consider

For illustrative purposes, consider a shock, (δ1,1, δ1,2), which is applied to the y1

series. The QIRF is given by

QIRF_h(δ1, y) =      A(δ1,1, δ1,2), for h = 1, Bh−1_A(δ 1,1, δ1,2), for h > 1. (21)

Consequently, when a shock, (δ2,1, δ2,2) is applied to the y1 series. The QIRF is given

by QIRF_h(δ2, y) =      A(δ2,1, δ2,2), for h = 1, Bh−1_A(δ 2,1, δ2,2), for h > 1. (22)

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ac-count a shock is simulated based on a Cholesky identification. So based on the Choleksy structure, either δ1,2 or δ2,1 is equal to zero.

The asymptotic distribution of the QIRF is derived using the Delta method, recall Equation 18, which can be reformulated as

QIRFi_h(δ, y) = ˜qt+h− qt+h (23)

= ˜ft(βτ) − ft(βτ), (24)

where ˜ft(βτ) and ft(βτ) correspond to the series ˜y and y respectively. So, the QIRF

is a function of β, and hence it is possible to apply the delta method. Consider the asymptotic distribution of β, as defined in Equation 14. Applying the delta method yields T12 QIRF[ h(δ, y) −QIRFh(δ, y) d −→ N 0, GhQ−1VQ−1G0h, (25) where Gh = ∂QIRF_h( ˜Yt; β) ∂β0 . (26)

The confidence interval of the quantile impulse response functions are derived from the asymptotic distribution.

4.2.3 Dynamic quantile tests

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statistic and the out-of-sample DQ statistic. The HIT statistic is generally used in quantile regression models and the DQ statistic is introduced by Engle and Man-ganelli (2004). The HIT statistic evaluates the biasedness of the quantile estimates, while the DQ statistic evaluates both the biasedness and the serial dependence of the quantile estimates. This section first provides these properties in mathematical terms. Afterward, it introduces the test statistics.

A quantile model, in the context of time series, has unbiased estimates if the series of quantile forecasts satisfy the following condition

Pr(yt < ft(βτ)) = τ, ∀ t ∈ 1, · · · , T, (27)

As a second condition, the series is unautocorrelated if

I yt< ft(βτ) ∼ i.i.d., ∀ t ∈ 1, · · · , T. (28)

As Engle and Manganelli (2004) argue, even if the underlying generating data process is serial dependent, the quantile estimates should provide a series satisfying Equation 28.

The Hit for the τ ’th quantile at time t is defined as

Hitτ,t = I yt< ft(βτ) − τ. (29)

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statistic which is defined as HITτ,T = 1 T T X t=1 Hitτ,t. (30)

Using Equation 29, the conditions of Equations 27 and 28 are reformulated as E(Hitt) = 0 and E(Hitt|Hitt−l) = 0, for l = 1, · · · , t − 1. These statements are

evaluated using the out-of-sample DQ statistic.

The out-of-sample DQ statistic is based on the one step out-of-sample quantile fore-casts. Let Ti and T0 denote the number of in-sample observations and out-of-sample

observations respectively. Let ˆβ_i denote the in-sample parameter estimates and let HitT0( ˆβi) be a T0-vector containing the out-of-sample forecasted Hits. Finally, let

X( ˆβ) be a matrix containing lagged values of Hitτ,t and other instrumental variables,

such as a constant. The out-of-sample DQ statistic is defined as

DQos=

Hit0_t( ˆβ_i)X( ˆβ_i) X0( ˆβ_i)X( ˆβ_i)−1X0( ˆβ_i)Hitt( ˆβi)

τ (1 − τ ) . (31)

Under the null hypothesis DQos d

−→ χ2 q.

The DQ test assumes, under the null hypothesis, the HIT series are unbiased and serial independent. So, if the null hypothesis is rejected, the test suggests these conditions do not hold. Therefore, this study states the series passed the DQ test if the null hypothesis is not rejected.

4.3 Related risk measures

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Let {yt} denote a equity return series. The VaR of the series is defined as

VaRτ = inf{y : F (y) ≥ τ } = qτ (32)

So, the VaR represents the τ ’th quantile. Therefore, the model can be rewritten in terms of VaRs instead of the quantiles.

In its most general form, the CoVaR is implicitly defined as

Prnyi ≤ CoVaRi|ϕ(yτ j) | yj = ϕ(yj)

o

= τ, (33)

Where ϕ(yj) denotes some event of yj. In most cases the event is set equal to

the VaR of yj, hence ϕ(yj) = V aRjτ, and yi denotes the return series of the sector

while yj denotes the return series of a single institution. Furthermore, Tobias and

Brunnermeier (2016) introduced the ∆CoVaR, which is defined as

∆CoVaRi|j_τ = CoVaRi|VaR

j τ

τ − CoVaR

i|VaRj0.5

τ . (34)

To illustrate how the MVMQ-CAViaR is related to these risk measures consider the MVMQ-CAViaR specification given by equation (12) for two return series. Let

VaRτ,t = (VaR1τ,t,VaR2τ,t)0. The model can be rewritten as

VaRτ,t = c + A|yt−1| + BVaRτ,t−1. (35)

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Let δ = (0,VaRj_τ,t/a2,1)0, where ai,j are elements of A. Then, ∆ \CoVaR1|2_τ = h X s=1 [ QIRF_s(δ, y). (36)

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5 Data analysis

This section introduces the dataset and its main characteristics. First, it describes how the data is obtained. Afterward, the section presents the summary statistics. Finally, it investigates the distribution of the return series using histograms and QQ-plots.

5.1 Data preparation

The study investigates the banking sectors of a large number of European countries within the European Union, Appendix A presents the list of all included countries. The countries are selected based on their data availability, where the data is ob-tained from Thomson Reuters Datastream. It contains the daily data of the national banking sectors share price indices from January 2012 to September 2019. The time frame is chosen due to its homogeneity. It consists of 2026 observations per country, which results in approximately 270 observations per country per year.

Due to different national holidays in the national stock markets, the dataset contains missing values. The number of missing values per country is reported in Appendix B. Instead of removing all dates with missing values, this study interpolates the data using a simulation technique, as is described in Appendix C. The variable of interest is the equity return of the national banking sectors, while the dataset contains the share prices. To transform the data from share prices to returns, the study opts for using log returns. The transformation is given by Equation 37,

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to the curse of dimensionality. To reduce the number of parameters, a European banking sector is generated based on the weighted returns of the countries. The weights are based on each banking sector’s relative size. Hence,

˜ r_tsec = n X i=1 wi_tr˜_ti, (38)

here ˜rtdenotes the normal return instead of the log return. Appendix D provides the

exact derivation of the European banking sector’s equity returns. The weights are based on the size of the banking sector for each country. Since the data on the size is not available, it is approximated using yearly data on national GDP, combined with the asset to GDP ratio data. These datasets are obtained from the World Bank. Appendix E presents an oversight of the size of each banking sector.

Finally, the countries split into different groups to investigate which common traits drive the differences in sensitivity to the shocks. The first criterion is based on the region of the countries. This study defined three regions, Northwestern Europe, GIIPS and Eastern Europe. Second, the countries are ranked based on the relative size of their banking sector compared to the equity markets. Then, three groups are created, the first contains the countries with the largest relative sector, followed by the medium sector and the smallest sector. Appendix E presents the groups for each countries, and discusses how the data is obtained.

5.2 Summary statistics

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signifi-cantly, ranging from −0.08 to −0.63. The standard deviations have a similar range. Interestingly, the deviations of Ireland and Greece are substantially higher, while the returns of those countries are lower. This contradicts the financial theory, which argues; higher standard deviations should be compensated by higher returns.

According to Cont (2001), the series should have heavy tails, implying large devi-ations of the mean occur more often, as is predicted by the normal distribution. The kurtosis of most series is higher than 3, indicating the tails are fatter than the tails associated with the normal distribution. The skewness of most series is approx-imately equal to zero; solely Bulgaria has a high negative skewness. The median returns are centered around zero. Finally, the VaR given in the table refers to the empirical VaR over the whole length of the time series. The 5-percentile VaR ranges from −0.076 to −0.018, and the 1-percentile VaR ranges from −0.137 to −0.034. Hence, the difference between the series is substantial. In both cases, the VaR of the European sector is close to the lowest VaR of the sample. This is expected since the European banking sector is more diversified than the national sectors. The results for the VaR are comparable to the standard deviations, in the sense that countries with the highest standard deviation have the highest VaR. Note, in this case, the highest VaR refers to the highest negative VaR value. The countries Greece and Ireland both have the highest standard deviations and the highest VaRs.

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Table 1: Summary Statistics

Mean Min Max Stdev Skew Kurt Median VaR5 VaR1

AUT 0.0002 -0.1044 0.0869 0.0163 -0.2089 2.9874 0.0006 -0.0253 -0.0462 BEL 0.0006 -0.1377 0.0822 0.0187 -0.2079 3.8256 0.0004 -0.0281 -0.0508 BGR -0.0003 -0.6346 0.0961 0.0199 -15.9124 508.4572 -0.0006 -0.0217 -0.0383 HRV 0.0003 -0.0601 0.0627 0.0122 0.0730 2.2476 0.0001 -0.0196 -0.0315 CYP -0.0016 -0.1698 0.1522 0.0252 0.3035 6.4881 -0.0031 -0.0379 -0.0768 CZE 0.0001 -0.0738 0.0552 0.0131 -0.1829 2.2092 0.0002 -0.0215 -0.0338 DNK 0.0001 -0.0769 0.0721 0.0140 -0.1962 3.1927 0.0002 -0.0220 -0.0411 FRA 0.0002 -0.1815 0.0873 0.0175 -0.4556 7.1344 0.0005 -0.0274 -0.0456 DEU -0.0006 -0.1385 0.1179 0.0186 -0.0502 4.1007 -0.0008 -0.0306 -0.0498 GRC -0.0029 -0.3541 0.2307 0.0482 -0.7160 7.9360 -0.0030 -0.0762 -0.1370 IRL 0.0000 -0.2307 0.1430 0.0260 -0.4781 7.5040 0.0006 -0.0386 -0.0626 ITA -0.0002 -0.2407 0.1092 0.0220 -0.5700 8.6458 -0.0000 -0.0355 -0.0586 NLD -0.0002 -0.1446 0.1177 0.0170 -0.0794 6.0323 -0.0006 -0.0262 -0.0443 NOR 0.0005 -0.1025 0.0718 0.0144 -0.2226 3.6741 0.0008 -0.0238 -0.0388 POL 0.0000 -0.0593 0.0573 0.0119 -0.0979 1.8474 -0.0002 -0.0184 -0.0309 PRT -0.0007 -0.1261 0.1664 0.0250 0.1025 3.6318 -0.0013 -0.0413 -0.0703 ROU 0.0004 -0.2030 0.1036 0.0136 -1.5282 32.1615 0.0005 -0.0179 -0.0366 ESP -0.0002 -0.2000 0.1775 0.0188 -0.3327 12.7589 -0.0002 -0.0285 -0.0464 SWE 0.0001 -0.0965 0.0507 0.0128 -0.4800 3.8490 0.0004 -0.0206 -0.0343 SEC -0.0002 -0.1517 0.0644 0.0143 -0.5645 7.5208 -0.0001 -0.0240 -0.0371

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Table 2: Correlation Table

AUT BEL BGR HRV CYP CZE DNK FRA DEU GRC IRL ITA NLD NOR POL PRT ROU ESP SWE SEC AUT 1.00 0.63 0.03 0.03 0.12 0.28 0.49 0.70 0.58 0.27 0.46 0.65 0.44 0.47 0.35 0.45 0.21 0.60 0.52 0.74 BEL 0.63 1.00 0.04 0.07 0.08 0.24 0.52 0.78 0.61 0.26 0.54 0.70 0.48 0.49 0.33 0.45 0.16 0.64 0.56 0.79 BGR 0.03 0.04 1.00 0.03 0.05 0.03 0.05 0.04 0.01 0.05 0.04 0.05 0.02 0.05 0.03 0.05 0.02 0.04 0.05 0.05 HRV 0.03 0.07 0.03 1.00 0.01 0.08 0.05 0.06 0.05 0.01 0.04 0.03 0.00 0.05 0.02 0.04 0.05 0.04 0.03 0.06 CYP 0.12 0.08 0.05 0.01 1.00 0.09 0.08 0.10 0.08 0.23 0.06 0.10 0.06 0.07 0.11 0.13 0.04 0.08 0.07 0.13 CZE 0.28 0.24 0.03 0.08 0.09 1.00 0.21 0.27 0.22 0.18 0.18 0.24 0.18 0.23 0.27 0.18 0.15 0.24 0.23 0.31 DNK 0.49 0.52 0.05 0.05 0.08 0.21 1.00 0.59 0.49 0.23 0.42 0.53 0.44 0.52 0.32 0.36 0.19 0.49 0.59 0.63 FRA 0.70 0.78 0.04 0.06 0.10 0.27 0.59 1.00 0.77 0.28 0.59 0.83 0.58 0.57 0.39 0.54 0.16 0.79 0.63 0.93 DEU 0.58 0.61 0.01 0.05 0.08 0.22 0.49 0.77 1.00 0.26 0.51 0.71 0.56 0.47 0.33 0.44 0.15 0.68 0.55 0.88 GRC 0.27 0.26 0.05 0.01 0.23 0.18 0.23 0.28 0.26 1.00 0.19 0.29 0.25 0.21 0.20 0.21 0.10 0.28 0.21 0.38 IRL 0.46 0.54 0.04 0.04 0.06 0.18 0.42 0.59 0.51 0.19 1.00 0.54 0.40 0.38 0.25 0.38 0.12 0.51 0.43 0.64 ITA 0.65 0.70 0.05 0.03 0.10 0.24 0.53 0.83 0.71 0.29 0.54 1.00 0.53 0.47 0.33 0.53 0.15 0.77 0.56 0.89 NLD 0.44 0.48 0.02 0.00 0.06 0.18 0.44 0.58 0.56 0.25 0.40 0.53 1.00 0.42 0.27 0.34 0.13 0.50 0.47 0.65 NOR 0.47 0.49 0.05 0.05 0.07 0.23 0.52 0.57 0.47 0.21 0.38 0.47 0.42 1.00 0.34 0.34 0.19 0.49 0.56 0.61 POL 0.35 0.33 0.03 0.02 0.11 0.27 0.32 0.39 0.33 0.20 0.25 0.33 0.27 0.34 1.00 0.26 0.14 0.32 0.34 0.44 PRT 0.45 0.45 0.05 0.04 0.13 0.18 0.36 0.54 0.44 0.21 0.38 0.53 0.34 0.34 0.26 1.00 0.09 0.51 0.38 0.58 ROU 0.21 0.16 0.02 0.05 0.04 0.15 0.19 0.16 0.15 0.10 0.12 0.15 0.13 0.19 0.14 0.09 1.00 0.14 0.16 0.20 ESP 0.60 0.64 0.04 0.04 0.08 0.24 0.49 0.79 0.68 0.28 0.51 0.77 0.50 0.49 0.32 0.51 0.14 1.00 0.53 0.85 SWE 0.52 0.56 0.05 0.03 0.07 0.23 0.59 0.63 0.55 0.21 0.43 0.56 0.47 0.56 0.34 0.38 0.16 0.53 1.00 0.68 SEC 0.74 0.79 0.05 0.06 0.13 0.31 0.63 0.93 0.88 0.38 0.64 0.89 0.65 0.61 0.44 0.58 0.20 0.85 0.68 1.00

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5.3 Data visualization

Having presented the summary statistics, this subsection provides supplementary visualizations of the data. These visualizations aim to investigate the tail behavior of the series. A large motivation for applying the MVMQ-CAViaR is the stylized fact that return series present heavy tails. So, this fact should be present in the data. The histograms of the returns are presented in Figure 2. According to the histograms, most return series have a leptokurtic distribution. Hence, most returns are centered around the mean, while there are some significant outliers. According to Cont (2001), it is a stylized fact that returns display many small gains and occasionally have large losses. However, the histograms indicate the return distribution is generally central. Figure 3 presents the QQ-plots of the series against the normal distribution. All series violate normality, both the left and right tails of the distribution display more extreme behavior as is predicted by the normal distribution.

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Figure 2: Histograms banking sector equity return

(a) AUT (b) BEL (c) BGR (d) HRV

(e) CYP (f) CZE (g) DNK (h) FRA

(i) DUE (j) GRC (k) IRL (l) ITA

(m) NLD (n) NOR (o) POL (p) PRT

(q) ROU (r) ESP (s) SWE (t) SEC

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Figure 3: QQ-plots equity return series.

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6 Empirical Results

This section performs the empirical analysis of this study. First, it discusses the model specification. Then it estimates all the models, followed by the evaluation of its out-of-sample VaR forecasting performance. Then, the subsequent subsection performs the shock analysis. Finally, the results are averaged for different groups, to investigate whether countries with specific characteristics are more vulnerable to systemic risk.

6.1 Model specification

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Table 3: Functional specification evaluation

Symmetric Absolute Error Unsymmetric Absolute Error Indirect GARCH

5% 1% 5% 1% 5% 1%

HIT DQ HIT DQ HIT DQ HIT DQ HIT DQ HIT DQ

AUT -0.015 0.791 -0.005 0.994 -0.010 0.671 -0.008 0.940 -0.020 0.691 -0.005 0.994 BEL 0.009 0.284 -0.000 0.946 0.004 0.169 -0.008 0.870 0.017 0.059 -0.003 0.951 BGR 0.009 0.899 -0.003 0.992 -0.015 0.611 -0.003 0.999 0.019 0.004 -0.003 0.966 HRV -0.025 0.392 0.002 0.000 -0.038 0.027 -0.008 0.747 -0.023 0.504 0.005 0.002 CYP -0.001 0.179 -0.008 0.941 -0.023 0.259 -0.008 0.943 0.004 0.089 -0.005 0.983 CZE -0.033 0.240 0.012 0.038 -0.033 0.241 -0.005 0.994 0.019 0.018 0.015 0.044 DNK 0.029 0.001 0.005 0.089 0.027 0.036 0.002 0.111 0.031 0.028 0.005 0.142 FRA 0.017 0.618 0.002 0.002 0.007 0.840 -0.010 1.000 0.012 0.770 -0.000 0.912 DEU 0.014 0.033 0.005 0.021 0.004 0.071 0.002 0.998 0.012 0.029 0.005 0.972 GRC -0.023 0.386 -0.010 1.000 -0.018 0.433 -0.010 1.000 -0.018 0.630 -0.005 0.954 IRL -0.013 0.778 -0.005 0.990 -0.028 0.079 -0.008 0.943 -0.018 0.658 -0.000 1.000 ITA 0.004 0.147 0.002 0.982 -0.020 0.467 -0.005 0.992 0.002 0.118 -0.003 1.000 NLD -0.003 0.404 -0.000 1.000 -0.003 0.154 -0.000 0.999 -0.003 0.230 0.002 0.974 NOR -0.008 0.743 -0.005 0.986 0.002 0.554 0.002 0.637 0.009 0.651 0.002 0.984 POL 0.007 0.820 -0.000 0.809 0.002 0.144 -0.000 0.991 0.012 0.598 0.002 0.955 PRT 0.036 0.037 -0.000 0.001 -0.018 0.544 -0.000 0.001 0.041 0.000 -0.000 0.001 ROU 0.004 0.773 0.005 0.939 -0.001 0.370 0.005 0.809 -0.001 0.639 0.007 0.014 ESP -0.006 0.198 -0.003 0.999 -0.006 0.144 -0.010 1.000 0.002 0.070 -0.003 0.998 SWE 0.002 0.504 0.007 0.000 -0.008 0.773 0.005 0.000 -0.003 0.366 0.010 0.000 Mean 0.014 0.433 0.004 0.617 0.014 0.347 0.005 0.788 0.014 0.324 0.004 0.676

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Table 6.1 suggests that the differences between the forecasting performance of under the different specifications is small. The bias is approximately equal for all model specifications since the HIT statistics give 0.014 and 0.004 for the 5% VaR and the 1% models, respectively. Solely the DQ statistics differ. Under the GARCH specification, the model obtains the best performance for the 5% VaR model, while the absolute symmetric error has the best performance for the 1% VaR model. However, the differences are not substantial. They could easily differ if another forecasting period is used. This study opts for the absolute symmetric error specification. Since the performance of the 1% VaR model is more relevant. Besides, the absolute symmetric error specification is more commonly used in the previous literature.

So, the models consist of two series, the first series represents a national banking sector and the second series represents the European banking sector. the series are given by Equations 39 and 40.

qτ,t,1 = c1+ a11|yt−1,1| + a12|yt−1,2| + b11qτ,t−1,1+ b12qτ,t−1,2, (39)

qτ,t,2 = c2+ a21|yt−1,1| + a22|yt−1,2| + b21qτ,t−1,1+ b22qτ,t−1,2. (40)

The model’s variables represent the following; qτ,t,i represents the VaRτ and yt,i

represents the equity returns, at time t for i = 1, 2 . The model’s parameters consist of the constants, denoted by ci, parameters associated with the absolute returns,

denoted by ai,j, and the parameters associated with the lagged VaR values, denoted

by bi,j, for i = 1, 2 and j = 1, 2.

If the volatilities of the series are highly clustered, the parameter estimates ˆb1,1, and

ˆ_b

2,2 are expected to be close to one and significant. The estimates of the most interest

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spillover effects between the series. If the parameter estimate, ˆa1,2, is significantly

different from zero, it implies the returns of the European banking sector significantly affects the VaR of the national sector. If the estimate, ˆa2,1, is significantly different

from zero, the returns of the national sector significantly affect the European banking sector.

6.2 Model estimation

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Figure 4: VaR forecasts

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Due to the large number of models, the parameter estimates are compactly sum-marized by comparing the average of all estimates. Table 6.2 presents the averages, the standard deviations, the minimum and maximum of all coefficients. Appendix K provides the complete overview. Following the discussion of the previous paragraphs, the parameter estimates of interest are the contagion estimates, ˆa12 and ˆa21.

Fur-thermore, the other parameters estimates of interest are, ˆb11 and ˆb22, the clustering

parameters.

For the 5% MVMQ-CAViaR, the averages for, ˆa12 and ˆa21, are −0.034 and −0.028

respectively, while for the 1% models the average parameter estimates are −0.070 and −0.030 respectively. So, the 1% models indicate the VaR of the national banking sectors depends stronger on the absolute returns of the European banking sector than the VaR of the European sector depends on the national banking sectors. For the 5% models, this difference is minimal. Furthermore, the parameter estimates ˆa12

ranges from −0.166 to 0.017 and from −0.386 to 0.176 for the 1% and 5% models, respectively. So, for the lower quantile, the estimates become more volatile. A similar trend is observed for the estimates ˆa2,1, where the range increases from −0.087 to

0.023 for the 5% models, to −0.206 to 0.141. A possible explanation for these changes is that the model becomes less accurate as the quantile approaches zero.

The clustering parameter, ˆb11, for the 5% models ranges from 0.048 and 0.996, with

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Table 4: Mean of the estimates 5% MVMQ-CAViaR 1% MVMQ-CAViaR ˆ c1 â11 â12 ˆb11 ˆb12 ˆc1 â11 â12 ˆb11 ˆb12 Mean -0.002 -0.141 -0.034 0.825 0.04 -0.006 -0.283 -0.07 0.578 0.232 Stdev 0.002 0.128 0.056 0.242 0.102 0.012 0.332 0.16 0.531 0.481 Min -0.01 -0.474 -0.166 0.048 -0.056 -0.039 -1.424 -0.386 -0.838 -0.091 Max 0.000 -0.015 0.017 0.996 0.304 0.003 0.047 0.176 0.985 1.813 ˆ c2 â21 â22 ˆb21 ˆb22 ˆc2 â21 â22 ˆb21 ˆb22 Mean -0.002 -0.028 -0.105 0.08 0.74 -0.003 -0.03 -0.152 0.13 0.717 Stdev 0.006 0.03 0.05 0.216 0.5 0.007 0.088 0.091 0.351 0.512 Min -0.023 -0.087 -0.201 -0.065 -0.671 -0.032 -0.206 -0.302 -0.173 -0.876 Max 0.002 0.023 -0.013 0.869 0.965 0.002 0.141 0.021 1.248 0.953

Table presenting the mean (Mean), standard deviation (Stdev), minimum (Min), maximum (Max) of the parameter estimates of all different models. The mean is obtained by averaging all parameters given in Appendix K, the other statistics are obtained similarly. The 5% and 1% MVMQ-CAViaR in the first row indicate which models are used to obtain the parameters estimates. The parameters are defined by Equations 39 and 40.

To more thoroughly investigate the contagion parameter estimates, Table 6.2 pro-vides the number of cases where the estimates are significantly different from zero. Again the parameters ˆa21 are more often significant than the parameters ˆa12,

al-though this diminishes under the 1% significance level. There exist more significant estimates for ˆa21, while this is not the case for the estimates of ˆa12. So, a shock to

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Table 5: Significance contagion parameters 10% 5% 1% ˆ a12 â21 aˆ12 â21 aˆ12 â21 5% MVMQ-CAViaR 1 5 1 3 1 1 1% MVMQ-CAViaR 2 4 2 4 2 2

Table presenting the number of significant contagion pa-rameters of the MVMQ-CAViaR models. The papa-rameters are defined by Equations 39 and 40. The percentages, 10%, 5% and 1%, in the fist row, denote different significance levels at which the estimates are significant. Again the 5% and 1% MVMQ-CAViaR indicate which model is used.

6.3 Model performance

As discussed in Section 3.1, the performance of the models is evaluated on its biased-ness and the serial dependence of the out-of-sample VaR forecasts. To perform the forecasts, the parameters are re-estimated based on 80% of the sample, the leftover 20% of the sample is used for forecasting. Since the model requires the information of the previous period, all forecasts are one period ahead. Due to computational issues, there is no rolling window. Hence, the parameters are estimated once and used for the entire forecasting period. The biasedness of the forecasts is evaluated by comparing the out-of-sample HIT statistics with the in-sample HIT estimates of the univariate CAViaR on the out-of-sample data. The out-of-sample DQ statistics assess the serial dependence. The estimation process of the statistic is presented in Appendix H.

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5% models, and the last three columns present the results for the 1% models. In both cases, the first two entries give the HIT for the multivariate and univariate estimates, respectively, while the last entry provides the p-values of the DQ statistic. The last two rows of the table represent the mean of the results and the number of series, which passed the DQ test at the 5% significance level, respectively. Note the mean of the HIT series is obtained by taking the average of the absolute HIT series, such it more clearly describes the deviations from the true quantile.

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Table 6: Model performance

5% MVMQ-CAViaR 1% MVMQ-CAViaR

HITmv HITuv DQ HITmv HITuv DQ

AUT 1.17 0.93 0.13 -0.26 -0.01 0.98 BEL 3.40 2.65 0.01 -0.26 -0.26 0.99 BGR -2.04 -2.78 0.59 -0.26 0.23 1.00 HRV -0.06 -0.06 0.20 -0.75 -0.26 0.94 CYP 2.41 -3.02 0.00 -0.51 -0.75 0.97 CZE 3.64 2.65 0.00 0.73 0.73 0.00 DNK 0.43 1.17 0.98 0.23 0.23 0.00 FRA 1.42 1.42 0.03 -0.26 0.23 1.00 DEU -2.78 -2.28 0.13 -1.00 -1.00 1.00 GRC -0.06 -1.30 0.29 -0.26 -0.51 1.00 IRL 0.68 0.43 0.01 0.23 -0.75 0.98 ITA -0.31 -0.80 0.39 -0.01 -0.01 1.00 NLD 0.93 0.68 0.98 -0.26 -0.26 0.75 NOR 1.42 0.19 0.13 -0.01 -0.01 0.85 POL 4.14 3.64 0.00 -0.51 -0.01 0.99 PRT 0.68 0.43 0.67 0.73 0.48 0.02 ROU -0.56 -0.56 0.19 -0.51 -0.26 0.99 ESP 0.68 0.19 0.56 0.48 0.23 0.00 Mean 1.45 1.33 0.32 0.47 0.39 0.76 passed DQ 13 15

Table evaluating the performance of 5% and 1% MVMQ-CAViaR models, using the out-of-sample HIT statistics of the MVMQ-CAViaR (HITmv) and of the CAViaR (HITuv), the statistic is defined by

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6.4 Shock analysis

Following Figure 1, the following steps consist of simulating the shocks to the returns of the VaR of the national banking sectors and to the VaR of the European banking sectors. In order to perform the simulations, this study applies the QIRF. The technical details of the QIRF are discussed in Section 4.2.2, and the estimation procedure is outlined in Appendix I.

To estimate the ∆CoVaR, a shock of two standard deviations is applied to the equity returns of each series. Since the returns are contemporaneously correlated, this study applies a Choleksy identification scheme. Hence, it is assumed the European sector directly affects its own returns and the returns of the national banking sector, while the national banking sector returns solely affects the returns of the European sector in future periods. The shocks are first applied to the national banking sectors, and second to the European banking sector. The development of the ∆CoVaR is followed for 50 periods.

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VaR increases substantially.

Following Figure 1 and Section 3.2, the final step consist of sorting the results based on their ∆CoVaR estimates. According to White et al. (2015), one downside of the comparison is; since all models are estimated separately, the results are based on assuming a different VaR series for the European sector’s returns. Hence, the results for each country are specific, and comparisons across pairs can lead to misleading results. Since the current research has yet to solve this problem, the rankings remain based on the different information sets. Table 7 presents the final results. The first, third, fifth and seventh column presents the ∆CoVaR estimates associated with Figures 5, 6, 7 and 8 respectively.

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Figure 5: 5% ∆CoVaR, European banking sectors

(q) ROU (r) ESP (s) SWE

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Figure 6: 5% ∆CoVaR, national banking sectors

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Figure 7: 1% ∆CoVaR, European banking sector

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Figure 8: 1% ∆CoVaR, national banking sectors

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Table 7: ∆CoVaR estimates

5% ∆CoVaR 1% ∆CoVaR

C2S S2C C2S S2C

FRA* -4.388** NOR* -2.586* PRT* -7.515** GRC -10.297 ITA* -3.949** IRL* -2.544* SWE* -6.939** NOR -5.529**

DEU* -3.942** GRC -1.173 ITA* -6.492* IRL -3.234*

HRV* -3.784** NLD* -1.063 POL* -5.889* DNK -2.909*

POL -3.661** AUT* -0.752* NLD* -5.563* CZE* -2.572

NOR* -3.618** ROU* -0.504 ESP -5.496* NLD* -2.179

NLD* -3.61** BEL* -0.404 FRA* -5.488 ROU -1.905

SWE* -3.555** POL* -0.394* HRV* -5.428* AUT -1.839

ESP* -3.358** ITA* -0.394 AUT* -5.382* HRV* -1.551

BEL -3.123** CZE* -0.265 DNK* -5.084* SWE* -1.354*

PRT* -3.006** FRA* -0.168 GRC* -4.176* BEL* -1.352

AUT* -2.661** BGR* -0.122 DEU* -3.788 PRT* -1.304*

IRL* -2.336* CYP* -0.106 BEL* -3.503* POL* -1.052

GRC* -2.211* DNK* 0.007 IRL* -3.1 CYP* -0.051

ROU* -1.933* HRV* 0.508 ROU -2.502 BGR* -0.019

DNK* -1.809* ESP* 0.572 CZE* -1.823* ESP 0.195

BGR* -1.522 PRT* 0.673 NOR* -1.504 FRA 0.41

CZE* -0.6 DEU* 0.741 BGR -0.717 ITA 0.835

CYP* 0.879 SWE* 1.983 CYP* 2.022 DEU* 0.97

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6.5 Explanatory analysis

Figure 9: ∆CoVaR, geographic groups

(a) 5%, country to sector (b) 5%, sector to country

(c) 1%, country to sector (d) 1%, sector to country

Figure presenting the average ∆CoVaR estimates for the Northwestern European countries (NW), GIIPS countries (GIIPS) and Eastern Eu-ropean countries (EAST). 5% refers to 5% ∆CoVaR estimates and 1% refer to 1% ∆CoVaR estimates. Country to sector indicates a shock is applied to national banking sectors, while sector to country indicates a shock is applied to the European banking sector.

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Figure 10: ∆CoVaR, relative banking size groups

(a) 5%, country to sector (b) 5%, sector to country

(c) 1%, country to sector (d) 1%, sector to country

Figure presenting the average ∆CoVaR estimates for the Northwestern European countries (NW), GIIPS countries (GIIPS) and Eastern Euro-pean countries (EAST). 5% refers to 5%-MVMQ-CAViaR and 1% refers to 1%-MVMQ-CAViaR. Country to sector indicates a shock is applied to national banking sectors, while sector to country indicates a shock is applied to the European banking sector.

Appendix E presents the exact overview.

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for the 1% models. The graphs indicate that Eastern European countries contribute less to the level of systemic risk compared to the other groups. Furthermore, the graphs indicate the GIIPS countries are most vulnerable to a shock to the European banking sector. This is as expected since these countries experienced the most sub-stantial economic turmoil in the past years. Furthermore, Appendix O presents the in-sample VaR forecasts for the different groups. Clearly, over the whole period, the VaR of the GIIPS countries exceeds the VaR of the other groups.

For the second comparison, the countries are grouped based on the relative size of their banking sector compared to the equity markets. The results are presented in Figure 10, here the blue lines represent the countries with the largest relative sectors, the red line represents the middle countries and the green lines represent the countries with the smallest sector. Similar to the previous comparison, for the 5% models, there do not exist substantial differences across the groups. However, for the 1% models, there exist pronounced differences. The countries with the largest relative sectors have the most substantial contributions to systemic risk, while the countries with the smallest sector have the lowest contributions.

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7 Conclusion

The global financial crisis of 2008 stressed the importance of measuring systemic risk. Neglecting systemic risk gives rise to a situation where the default of a single institution has devastating effects on the economy of the whole continent. As a consequence of the global financial crisis, Europe is aiming to centralize its financial risk management through harmonizing banking regulations and constructing a bail-in system. For these developments to function, it is of importance to bail-incorporate the effects of systemic risk. Therefore, this study aims to quantify the level of systemic risk within the national banking sectors of a large number of European countries. This led to the following research question:

”What is the magnitude of an increase in the VaR of one country’s banking sector equity returns on the VaR of its surrounding countries?”

This study applied the MVMQ-CAViaR model, as introduced by White et al. (2015). It is a semi-parameter technique, and hence it imposes minimal assumptions on the data generation process. Furthermore, the method is based on the quantile regression methodology, which allows for to modeling the object of interest, the VaR, directly. The research question is answered using the QIRF. A shock is applied to the VaR of each country’s banking sector returns, and the effects on the European banking sector are observed. Then, a shock is applied to the VaR of the European banking sector returns, and its effects on the VaR of the national banking sectors are observed. This process is performed for both the 5% and 1% MVMQ-CAViaR. Afterward, the study investigated if the GIIPS countries or countries with a relatively large banking sector have higher levels of systemic risk.

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• The magnitude of an increase in the 5% VaR of the national banking sectors on the VaR of the surrounding countries ranges form −4.342% to 1.131%. The countries with the highest ∆CoVaR estimates are France, Italy and Germany. These countries have the largest banking sectors of the sample. For 12 out of the 19 countries the ∆CoVaR estimates are significant, and for 15 countries, there is a significant increase in the VaR during one period. Furthermore, this study found no substantial difference between the different groups. Even the average ∆CoVaR estimates of the GIIPS countries is approximately equal to the estimates of the Eastern and Northwestern European countries. For the relative size of the banking sectors, there do not exist profound differences as well.

• The magnitude of an increase in the 5% VaR of the European banking sector on the VaR of the national sectors ranges from −2.586% to 1.983%. The countries which are most vulnerable to a banking crisis are Norway, Ireland and Greece. Out of these three countries, two are GIIPS countries. In this case, all three countries do not have a substantial sized banking sector. None of the countries have ∆CoVaR estimates that are significant for the whole period. For 7 out of the 19 countries, the effects are positive, implying the level of systemic risk decreases. This contradicts the theoretical expectations, although the results are insignificant for every period. So, it might be explained by the presence of overfitting of the data. Similar to the 5% ∆CoVaR estimates, of the previous part, there do not exist notable differences among the different regional or relative bank size groups.

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on the VaR of the European banking sector ranges from −7.515% to 2.022%. The countries which have the largest magnitude are Portugal, Sweden and Italy. In contrast to the results for the 5% models, the absolute size of the banking sector is less relevant since the countries, Portugal and Sweden, have small sectors. However, the size of the relative banking sector has a profound impact. The ∆CoVaR increase is the most substantial for countries with a large relative sector. For the GIIPS countries the ∆CoVaR estimates differ among the group. Countries as Portugal, Italy and Spain have a high estimate, while it is low for Greece and Portugal. For 15 out of the 19 countries the ∆CoVaR estimates are significant at one period. This study found a similar magnitude of an increase in the VaR of the Northwestern European countries and the GIIPS countries. The magnitude of the Eastern European countries is lower than the other groups, indicating the Eastern European countries remain mostly unaffected by a shock to the European banking sector.

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have a higher magnitude as for the other groups. Furthermore, solely the estimates for the countries with a small banking sector are lower.

This study has some limitations regarding its analysis. First, a large number of QIRFs remained insignificant. Furthermore, the study investigated the magnitude of an increase in the VaR of one country affects the VaR of its surrounding countries. However, it assumes the VaR of a banking sector increases when applying a shock to the absolute returns of the sector. Theoretically, this should hold, though not all models agree. In some cases, the VaR decreases as the absolute returns increase. Furthermore, following the discussion of Section 4.2.2, the QIRFs for the different nations depend on different information sets. Consequently, the VaR estimates for the European banking sector differ for each model. Hence naive comparison among the different models is misleading. Finally, the study

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8 Discussion & Recommendations

This section first discusses the contributions of the study with regards to the aca-demic literature. It highlights the shortcomings and suggestions for further research. Afterward, this section gives recommendations for risk managers, especially for Eu-ropean central bankers.

8.1 Academic discussion

This study contributes to the academic literature by applying the MVMQ-CAViaR methodology to the field of systemic risk. This approach is previously applied to investigate systemic risk by White et al. (2015). Although this study mainly followed the approach of White et al. (2015), there exist some differences. First, White et al. (2015) focused on the effects of leverage on the contributions to systemic risk, while this study is mainly concerned with the quantification of systemic risk. Then, this study investigated if the levels of systemic risk differ per region and if it differs for countries with different bank sizes relative to their equity market size.

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systemic shock to the country, neglecting the effects of a shock to the country on the system. This study found the latter effect more present.

Besides evaluating if countries with relatively large banking sectors have higher levels of systemic risk, this study evaluates how the different geographic regions relate the systemic risk. The analysis found the GIIPS countries to be most vulnerable to a shock in the European banking sector, which is in line with Reboredo and Ugolini (2015a). Furthermore, this study found the Eastern European countries to remain mostly unaffected by a systemic crisis. Since these countries have a small weight to the European banking sector, it is possible there exist significant risk spillovers between them. Moreover, a similar argument is possible for Northwestern European countries. Since their weight to the European banking sector is large, the systemic risk may be mainly driven by spillovers from within the region. Therefore, for future research, it is interesting to investigate if there exist distance decay effects. This entails investigating if the co-dependence of large losses is greater with a system consisting of closely located countries than with a greater system also including further located countries.

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Another limitation is the number of insignificant contagion parameters, and con-sequently, the number of insignificant ∆CoVaR estimates. A possible explanation for these results is the fact that the study evaluates the tail of the distribution. As a result applying the extremal quantile regression, as described in Chernozhukov et al. (2005) and Chernozhukov and Fern´andez-Val (2011) is more appropriate. How-ever, as White et al. (2015) argued, applying the extremal quantile regression to the MVMQ-CAViaR methodology has not yet been established. Moreover, it is not straightforward since the MVMQ-CAViaR relies on nonlinear quantile regression, and the extremal quantile methodology is not yet extended to nonlinear regressions. Another possible explanation for the number of insignificant estimates is the tran-quility of the dataset. Although there are some economic shocks present in the time series, the data does not include a substantial economic crisis. Whereas the data of White et al. (2015) covers the global financial crisis of 2008.

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8.2 Practical recommendations

The most practical information is derived from Section 6.4 and 6.5. This study found the GIIPS countries are most vulnerable to a continental banking crisis. As discussed in Section 1.1, Europe aims to harmonize its banking regulations. These banking regulations need to be structured to decrease the dependence of these countries on the continental banking system. Moreover, the capital requirements should incorporate the higher vulnerability of the GIIPS countries. The countries themselves should, as argued in Acharya et al. (2010), enforce banks to insure their capital against large losses.

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References

Acerbi, Carlo and Dirk Tasche (2002). Expected shortfall: a natural coherent alter-native to value at risk. Economic notes 31 (2), 379–388.

Acharya, Viral, Lasse Pedersen, Thomas Philippon, and Matthew Richardson (2010). A tax on systemic risk. Regulating Wall Street: The Dodd-Frank Act and the New Architecture of Global Finance. John Wiley and Sons, New York , 121–142.

Acharya, Viral, Lasse H Pedersen, Thomas Philippon, and Matthew Richardson (2017). Measuring systemic risk. The Review of Financial Studies 30 (1), 2–47. Adams, Zeno, Roland F¨uss, and Reint Gropp (2014). Spillover effects among

finan-cial institutions: A state-dependent sensitivity value-at-risk approach. Journal of Financial and Quantitative Analysis 49 (3), 575–598.

Adrian, Tobias and Hyun Song Shin (2010). Liquidity and leverage. Journal of financial intermediation 19 (3), 418–437.

Bisias, Dimitrios, Mark Flood, Andrew W Lo, and Stavros Valavanis (2012). A survey of systemic risk analytics. Annu. Rev. Financ. Econ. 4 (1), 255–296. Brownlees, Christian and Robert F Engle (2016). Srisk: A conditional capital

short-fall measure of systemic risk. The Review of Financial Studies 30 (1), 48–79. Brunnermeier, Markus K and Lasse Heje Pedersen (2008). Market liquidity and

funding liquidity. The review of financial studies 22 (6), 2201–2238. Caruana, Jaime (2010). Basel iii: towards a safer financial system.

Quantifying Systemic Risk within the European Union through a simulation of a banking crisis

Quantifying Systemic Risk within

the European Union through a

simulation of a banking crisis

Derk de Boer

s2714051

Quantifying Systemic Risk within the European

Union through a simulation of a banking crisis

Derk de Boer

January 6, 2020

Contents

1

Introduction

1.1

Systemic risk in Europe

1.2

Research question

2

Literature Review

2.1

Background systemic risk

2.2

Risk measures

2.3

Modelling approaches

3

Research strategy

3.1

Motivation modelling strategy

3.2

Step-by-step plan

4

Methodology

4.1

Background

4.2

MVMQ-CAViaR

4.3

Related risk measures

5

Data analysis

5.1

Data preparation

5.2

Summary statistics

5.3

Data visualization

6

Empirical Results

6.1

Model specification

6.2

Model estimation

6.3

Model performance

6.4

Shock analysis

6.5

Explanatory analysis

7

Conclusion

8

Discussion & Recommendations

8.1

Academic discussion

8.2

Practical recommendations

References