## Measuring systemic risk in the Euro area

Introduction
We analyze the time evolution of systemic risk in Europe using the WorldScope lists of the financial industry for the european countries. The analysis is based on the cross-sectional distribution of systemic risk measures such as Marginal Expected Shortfall, Delta CoVaR, SRISK, SES and network connectedness measures. Moreover, we consider Shannon entropy as in Billio et al. (2015) on the estimated cross-sectional systemic risk measures. These measures are conceived at a single institution level for the financial industry in the Euro area and capture different features of the financial market during the period of stress. To estimate systemic risk measures, we use a rolling window approach with a window size of 252 daily observations, which corresponds approximately to one year of daily observations.

 Marginal Expected Shortfall (MES)

$MES_t^i$ is defined as the expected value of $X_t^i$ when the market $X_t^{sys}$ is below a given quantile $q$ (Acharya et al., 2010). \begin{equation*} MES_t^i = \mathbb{E}\left(X_t^i |X_t^{sys} \leq q_{5\%}\right), \end{equation*} where $X_t^i$, $t=1,\ldots,T$ denotes the series of asset returns for the asset $i$ and $X_t^{sys}$ is the market return.

Systemic Expected Shortfall (SES)
SES is the financial institution's contribution to systemic risk , its propensity to be undercapitalized when the system as a whole is undercapitalized. \begin{equation*} SES_i = a + bMES_i + cLVG_i + \varepsilon_i \end{equation*} where $LVG_i$ is the leverage ratio:
$(BookAsset-BookEquity+MarketValue)/MarketValue$.

 SRISK

Brownlees and Engle (2015) define the capital shortfall of firm $i$ on day $t$ as \begin{equation*} CS_{it} = kA_{it}-W_{it} = k(D_{it}+W_{it})-W_{it}, \end{equation*} where $W_{it}$ is the market value of equity, $D_{it}$ is the book value of debt, $A_{it}$ is the value of assets and $k$ is the prudential capital fraction ($\approx 8\%$).

• The CS can be viewed as the negative working capital of the firm (when positive the firm experiences distress).
They define a systemic risk event as a market decline below a threshold $C=10\%$ over a time horizon $h=22$. Thus,
\begin{eqnarray*} SRISK_{it} &=& E_t(CS_{it+h}|R_{mt+1:t+h} < C),\\ &=& kE_t(D_{it+h}|R_{mt+1:t+h} < C)+\\ &-&(1-k)E_t(W_{it+h}|R_{mt+1:t+h} < C), \end{eqnarray*} where $R_{mt+1:t+h}$ is the arithmetic multi-period market return. By assuming that in case of a systemic event debt cannot be renegotiated $\Rightarrow E_t(D_{it+h}|R_{mt+1:t+h} < C)=D_{it}$.
Therefore, it follows that
\begin{eqnarray*} SRISK_{it} &=& kD_{it}-(1-k)W_{it}(1+LRMES_{it}),\\ &=& W_{it}[kLVG_it-(1-k)LRMES_{it}-1], \end{eqnarray*} where $LVG_{it}$ denotes the leverage ratio $(D_{it}+W_{it})/W_{it}$ and $LRMES_{it}=E_t(R_{it+1:t+h}|R_{mt+1:t+h} < C)$.
• SRISK is a function of the size of the firm, its degree of leverage, and its expected equity devaluation conditional on a market distress.
• LRMES is obtained using a GARCH-DCC model.

 $\Delta$CoVaR

$\Delta$CoVaR represents the value at risk (VaR) of the financial system conditional on institutions being under distress. \begin{eqnarray*} && \mathbb{P}(X_t^i\leq VaR_{it,q}) = q,\\ && \mathbb{P}(X_t^{sys}\leq CoVaR_{sys,i,t,q}|X_t^i = VaR_{i,t,q}) = q \end{eqnarray*} $\Delta$CoVaR (Adrian and Brunnermeier, 2011) is defined as the difference between the CoVaR conditional on an institution being under distress and the CoVaR in the median of the institution, that is \begin{equation*} \Delta CoVaR_q^{sys|i} = CoVaR_q^{j|i} - CoVaR_{50\%}^{sys|i}, \end{equation*} where $X_t^i$ is the asset return value of the institution $i$ and $X_t^{sys}$ represents the system.

 IN network degree
• A network is defined as a set of nodes $V_t=\{1,2,\ldots,n\}$ and directed arcs (edges) between nodes.
• The network can be represented through an $n_t$-dimensional adjacency matrix $A_t$, with the element $a_{ijt}=1$ if there is an edge from $i$ directed to $j$ with $i,j\in V_t$ and 0 otherwise.
• The matrix $A_t$ is estimated by using a pairwise Granger causality approach to detect the direction and propagation of the relationships between the institutions (Billio et al., 2012).
The $IN_{it}$ network degree is defined as
\begin{equation*} IN_{it}=\sum_{j=1}^{n_t} a_{ijt}. \end{equation*}
 OUT network degree
• A network is defined as a set of nodes $V_t=\{1,2,\ldots,n\}$ and directed arcs (edges) between nodes.
• The network can be represented through an $n_t$-dimensional adjacency matrix $A_t$, with the element $a_{ijt}=1$ if there is an edge from $i$ directed to $j$ with $i,j\in V_t$ and 0 otherwise.
• The matrix $A_t$ is estimated by using a pairwise Granger causality approach to detect the direction and propagation of the relationships between the institutions (Billio et al., 2012).
The $OUT_{it}$ network degree is defined as \begin{equation*} OUT_{it}=\sum_{j=1}^{n_t} a_{jit}. \end{equation*}
 In-Out network degree
• A network is defined as a set of nodes $V_t=\{1,2,\ldots,n\}$ and directed arcs (edges) between nodes.
• The network can be represented through an $n_t$-dimensional adjacency matrix $A_t$, with the element $a_{ijt}=1$ if there is an edge from $i$ directed to $j$ with $i,j\in V_t$ and 0 otherwise.
• The matrix $A_t$ is estimated by using a pairwise Granger causality approach to detect the direction and propagation of the relationships between the institutions (Billio et al., 2012).
The $IO_{it}$ network degree is defined as \begin{equation*} IO_{it}=\sum_{j=1}^{n_t} a_{ijt}+\sum_{j=1}^{n_t} a_{jit}. \end{equation*}
 Dynamic Causality Index (DCI)
• A network is defined as a set of nodes $V_t=\{1,2,\ldots,n\}$ and directed arcs (edges) between nodes.
• The network can be represented through an $n_t$-dimensional adjacency matrix $A_t$, with the element $a_{ijt}=1$ if there is an edge from $i$ directed to $j$ with $i,j\in V_t$ and 0 otherwise.
• The matrix $A_t$ is estimated by using a pairwise Granger causality approach to detect the direction and propagation of the relationships between the institutions (Billio et al., 2012).
The $DCI_t$ is defined as \begin{equation*} DCI_t = {n_t\choose 2}^{-1} \sum_{i=1}^{n_t}\sum_{j=1}^{n_t}a_{ijt}. \end{equation*} If $(DCI_t-DCI_{t-1})>0$, there is an increase of system interconnectedness.

 Entropy Measures

The analysis is based on the cross-sectional distribution of systemic risk measures by considering two classes of these measures:

1. Tails of the financial returns that captures the co-dependence between financial institutions and the market ($\Delta$CoVaR and MES).
$\Rightarrow$ Theoretical models showing that shocks to volatility or to tail risk provoke common fluctuations across firms (Acemoglu et al., 2013).
2. Network linkages among financial institutions (IO network degree).
$\Rightarrow$ Skewness and fat tails suggest the presence of heterogeneity in the linkages among institutions: a large majority of financial institutions have low degree, but a small number (HUBS-SIFI) have a high number of linkages (Acemoglu, 2012).
• Structural changes in the proximity of a systemic event: the relevant or frail financial institutions are probably be the first to react and thus to provoke a structural change in the cross-sectional
• Billio et al.(2015) aims to exploit the ability of the entropy indicator to detect this heterogeneity and time variations in the financial system
• They do not impose any assumption on the cross sectional distribution of the risk measures (non parametric approach).

Entropy is used in a variety of fields to characterize the complexity of a system and to summarize the information content of a distribution.

• Let $\boldsymbol{\pi}_{t}=(\pi_{1t},\ldots,\pi_{mt})$, $t=1,\ldots,T$, with $\pi_{jt}\geq 0$, $\Sigma_j\pi_{jt}=1$, be a sequence of probability vectors.
• In the considered case, these vectors represents a sequence of cross-sectional distributions of a given systemic risk measure (InOut degree, MES and $\Delta$CoVaR) for a set of financial assets available at time $t$ in the market.
• We apply to $\boldsymbol{\pi}_{t}$ the Shannon Entropy, \begin{equation*} H_S(\boldsymbol{\pi}_t)=-\sum_{j=1}^m\pi_{jt}\log\pi_{jt} \textrm{ where }m<\infty, \end{equation*}
• Other types of entropies can be considered (i.e., Tsallis, 1988 and Rényi, 1961).

References