The Risk Spillover Within a Financial System: Evidence from China

Abstract

In this paper, by proposing a new risk measure called CoRVaR, we explore the risk spillover effect within a financial system consisting of the banking, security and insurance industries. To be precise, we examine the risk spillover effect from each industry to the other. Moreover, we also examine the risk spillover effect from each industry to the system. The dependence structure between industries is modeled using the vine copula. By employing the Monte Carlo simulation technique, an empirical study via data from Chinese financial markets is implemented. Finally, backtesting and comparisons with existing results are also performed. It turns out that the proposed risk measure can well quantify the risk spillover effect. In addition, the copula employed can also well reveal the interdependence structure of the financial system.

1. Introduction

Quantifying financial risks plays an important role in financial risk management. A systemic risk measure quantifies the risk of a financial system consisting of finitely many financial institutions. Systemic risk measures can stem from univariate and multivariate risk measures. For univariate risk measures, Artzner et al. [1] axiomatically initiated coherent risk measures for financial positions. Föllmer and Schied [2] and Frittelli and Rosazza Gianin [3] introduced convex risk measures, generalizing coherent risk measures. For a comprehensive review, we refer to Föllmer and Schied [4]. For multivariate risk measures, Burgert and Rüschendorf [5] first introduced multivariate coherent and convex risk measures via an axiomatic approach. For more related studies about multivariate risk measures, we refer to Rüschendorf [6,7], Ekeland and Schachermayer [8], Ekeland et al. [9], Wei and Hu [10], Guillen et al. [11], Gong et al. [12], and the references therein.

There are two main streams within the study of systemic risk measures. One is the axiomatic approach. Chen et al. [13] first studied systemic risk measures via an axiomatic approach on finite probability space. Kromer et al. [14] studied systemic risk measures on general probability space, generalizing the framework of Chen et al. [13].

The other is the constructive approach. Adrian and Brunnermeier [15] established a conditional value at risk (CoVaR) systemic risk measure based on VaR that can evaluate the contributions of individual financial institutions to the systemic risk. Cui et al. [16] proposed the Conditional Expected Shortfall (CoES) systemic risk measure based on ES, and used quantile regression to estimate CoES. From a time-series perspective, Adrian and Bunnermeier [15] used a bivariate GARCH to predict financial risks by assuming a bivariate Gaussian distribution.

In order to effectively control the systemic risk of a financial system, it is important to well understand the risk spillover effects within the financial system. Once we evaluate various risk spillover effects, then we can quantitatively monitor the cross-institution and cross-industrial-sector risk contagion in the financial system. Finally, we can reach the aim of risk control by exercising relevant risk management strategies. Therefore, the objective of this paper is to design effective systemic risk measures to quantify risk spillover effects.

In this paper, we establish a new systemic risk measure, which we refer to as conditional range value at risk (CoRVaR). Precisely, by proposing the CoRVaR systemic risk measure, we explore the risk spillover effects within a financial system consisting of the banking, security and insurance industries. We examine risk spillover effects from each industry to the other. Moreover, we also examine risk spillover effects from each industry to the system. The dependence structure between the industries is modeled using the vine copula. By employing the Monte Carlo simulation technique, an empirical study via data from the Chinese financial market is implemented. Finally, backtesting and comparisons with existing results are also performed. It turns out that the proposed systemic risk measure can well quantify the risk spillover effect. In addition, the copula employed can also well reveal the interdependence structure of the financial system.

We would also like to mention that in the present paper, the dependence structure between individual institutions and industries is modeled by general copulas, with the help of the Akaike Information Criterion (AIC) to fit the copulas. On the other hand, in the majority of the reference studies, the dependence structure was assumed to follow a Gaussian copula; for instance, see [17,18,19,20].

The rest of this paper is organized as follows. Section 2 introduces the preliminaries, including the modeling of marginal distributions and the interdependence structure, as well as the introduction of systemic risk measures and their estimation. Section 3 is devoted to the description of the data. In Section 4, an empirical study is carried out. Finally, the conclusion is summarized in Section 5.

2. Model Set-Up

In this section, we present the model setup and preparations, as well as the introduction of the CoRVaR systemic risk measure and its estimation.

2.1. Marginal Distribution Modeling

It is well known that a financial time series usually has characteristics such as non-normality, a fat tail, a skewed distribution, a leverage effect and volatility clustering. In order to capture these characteristics, many studies adopt the GARCH model. Although the GARCH model can deal with conditional heteroscedasticity, it cannot well reflect the financial leverage effect. To reflect the leverage effect, Glosten et al. [21] suggested the GJR-GARCH model, generalizing the GARCH model. In this paper, we adopt the AR-GJR-GARCH model to describe the marginal distribution.

Next, we briefly recall the AR-GJR-GARCH model. Arbitrarily fix a time period $T>0$. For each $t=2,\cdots ,T$, the AR-GJR-GARCH model is described as

$$r_t=\mu +c{r}_{t-1}+{ϵ}_t,$$

(1)

$${ϵ}_t={\sigma }_t{Z}_t,$$

(2)

$${\sigma }_t^2=w+\alpha {ϵ}_{t-1}^2+\beta {\sigma }_{t-1}^2+\gamma {ϵ}_{t-1}^{2}I({ϵ}_{t-1}<0),$$

(3)

where $r_t$ is the logarithmic return at time t; ${ϵ}_t$ means the residual of $r_t$ at time t; ${\sigma }_t^2$ is the conditional variance of ${ϵ}_t$; and $Z_t$ is the standardized residual.

In addition, $\mu$ is a constant term; w stands for the long-term average volatility; $\gamma$ is a constant presenting the leverage effect; and positive and negative $\gamma$ reflect the positive and negative leverage effect, respectively. $I\left(A\right)$ stands for the indicator function of set A.

In the empirical studies below, based on the Akaike Information Criterion (AIC), the distribution of the standardized residual $Z_t$ will be fitted within three distributions: the skewed Student distribution (sstd), the generalized error distribution (ged) and the skewed generalized error distribution (sged). The observed standardized residual ${\widehat{Z}}_t$ can be defined as follows:

$${\widehat{Z}}_t={\displaystyle \frac{r_t-\widehat{\mu }-\widehat{c}{r}_{t-1}}{{\widehat{\sigma }}_t}},$$

(4)

where ${\widehat{\sigma }}_t$ is the estimated standard deviation in Equation (3), and $\widehat{\mu }$ and $\widehat{c}$ are estimated parameters in Equation (1).

2.2. Dependence Structure Modeling

In order to evaluate the systemic risk of a financial system as accurately as possible, an important issue is the description of the possible interdependence structure between the individual institutions of the financial system. Hoyland and Wallace [22] proposed scenario tree generation to describe the interdependence structure between finitely many financial positions. Kaut [23] further suggested copula-based scenario tree generation. It is well known that copulas are a powerful tool to describe the interdependence structure between components (i.e., marginals) of a random vector. Nguyen and Liu [24] suggested the vine copula via a series of binary (pair) copulas. There are three types of vine copulas, C-vine, D-vine and R-vine, among which C-vine and D-vine can be considered as special cases of R-vine; for instance, see [25]. Taking into account the fact that R-vine is more flexible than C- and D-vine, in this paper, we will use an R-vine copula to describe the interdependence structure between our financial institutions and industries.

2.2.1. Vine Copula

A copula is a multivariate function that connects marginal distribution functions to a multivariate joint distribution function. Precisely, let F be the joint distribution function of a random vector $X=(X_1,\cdots ,X_n)$ with marginal distribution functions $F_1\left(x_1\right),\cdots ,F_n\left(x_n\right)$. Then, according to Sklar’s Theorem, there exists a copula C such that for all $x=(x_1,\cdots ,x_n)\in {\mathbb{R}}^n$,

$$F(x_1,x_2,\cdots ,x_n)=C(F_1\left(x_1\right),\cdots ,F_n\left(x_n\right)).$$

(5)

Moreover, if the random vector $X=(X_1,\cdots ,X_n)$ has a continuous distribution, then the copula C is unique. In the case where the joint distribution function F has a joint density function f, it holds that

$$f(x_1,\cdots ,x_n)=c(F_1\left(x_1\right),\cdots ,F_n\left(x_n\right))f_1\left(x_1\right)\cdots f_n\left(x_n\right),$$

(6)

where c is the copula density of C, and $f_i$ is the density function of $X_i,1\le i\le n$.

Vine copulas are based on a series of binary copulas. There have been candidates rich enough for the binary copula, such as the class of elliptical copulas including Gaussian and Student’s copulas, the class of Archimedean copulas including Frank and Joe copulas, and so on. Next, we introduce the R-vine copula. A vine on n elements is defined by a set of a sequence of trees $T_i=(N_i,E_i)$ for $i=1,\cdots ,n-1$, with $N_i$ and $E_i$ denoting the sets of nodes and edges of tree $T_i$, respectively. The definition of regular vine (R-vine) copulas can be found in Dißmann et al. [25]. We briefly recall the definitions of R-vine and R-vine copula.

Definition 1.

$V=(T_1,\cdots ,T_n)$ is an R-vine on n elements if the following applies:

1. 

$T_1$ is a tree with nodes $N_1=1,\cdots ,n$ and a set of edges denoted by $E_1$.

2. 

For $i=2,\cdots ,n-1$, $T_i$ is a tree with nodes $N_i=E_{i-1}$ and an edge set $E_i$.

3. 

For $i=2,\cdots ,n-1$ and $a,b\in E_i$ with $a=a_1{a}_2$ and $b=b_1{b}_2$, it must hold that #$\{a\cap b\}=1$ (proximity condition), where # denotes the cardinality of a set.

To better understand the definition of R-vine, an illustrative figure is presented in Figure 1 for C-vine and D-vine, respectively, which are two special cases of R-vine.

We now, in turn, introduce the R-vine copula. According to [26], the complete union $U_{e_i}$ of an edge $e_i=a,b\in E_i$ is the set of all indices that this edge contains. For two nodes a and b that are connected by an edge $e_i$, the conditioned sets $c_{e_i,a}$ and $c_{e_i,b}$ and the conditioning set $D_{e_i}$ are defined as the symmetric difference and the intersection of the complete unions of a and b, respectively, that is, $D_{e_i}=U_a\cap U_b$, $C_{e_i,a}=U_a\setminus D_{e_i},C_{e_i,b}=U_b\setminus D_{e_i}$. Then, the multivariate joint density function $f_{R-vine}$ of $X=(X_1,\cdots ,X_n)$ can be written as

$$f_{R-vine}={\displaystyle \prod _{k=1}^n}{f}_k\left(x_k\right){\displaystyle \prod _{k=1}^{n-1}}\prod _{e\in E_k}{c}_{j\left(e\right),i\left(e\right)\left|D\right(e)}(F\left(x_{j\left(e\right)}\right|x_{D\left(e\right)}),F\left(x_{i\left(e\right)}\right|x_{D\left(e\right)})),$$

(7)

where $x_{D\left(e\right)}$ stands for variables in $D_e$, that is, $x_{D\left(e\right)}=\left\{x_k\right|k\in D_e\}$, $x_{j\left(e\right)}=\left\{x_j\right|x_{j\left(e\right)}\in C_{e,j}\}$ and $x_{i\left(e\right)}=\left\{x_i\right|i\in C_{e,i}\}$. $f_i$ denotes the density function corresponding to $F_i$ for $i=1,\cdots ,n$, $F(·|·)$ is the conditional distribution, $f_k\left(x_k\right)$ is the probability density of variable $x_k$, $E_k$ is the set of edges in the kth tree, $j\left(e\right)$ and $i\left(e\right)$ are two nodes with edge e, and $c_{j\left(e\right),i\left(e\right)\left|D\right(e)}$ denotes the copula of two variables with edge e.

Meanwhile, from the above formula, it is obvious that the decomposition is not unique. Two special cases of vine copulas are canonical vine (C-vine) and drawable vine (D-vine) copulas. The corresponding decomposition of the joint density function $f_{R-vine}$ given in (7) can be expressed by

$$f_{C-vine}={\displaystyle \prod _{k=1}^n}{f}_k(x_k){\displaystyle \prod _{j=1}^{n-1}}{\displaystyle \prod _{i=1}^{n-j}}{c}_{i,i+j|1,\cdots ,j-1}(F(x_j\mid x_1,\cdots ,x_{j-1}),F(x_{i+j|x_1,\cdots ,x_{j-1}})),$$

(8)

$$f_{D-vine}={\displaystyle \prod _{k=1}^n}{f}_k\left(x_k\right){\displaystyle \prod _{j=1}^{n-1}}{\displaystyle \prod _{i=1}^{n-j}}{c}_{i,i+j\mid i+1,\cdots ,i+j-1}(F\left(x_i\right|x_{i+1},\cdots ,x_{i+j-1}),F(x_{i+j}\mid x_{i+1},\cdots ,x_{i+j-1})).$$

(9)

Here, we give a brief illustrative graph for the four-dimensional C-vine copula tree in Figure 1a and D-vine copula tree in Figure 1b. We find that the relation between trees ${\mathrm{T}}_1$, ${\mathrm{T}}_2$ and ${\mathrm{T}}_3$ is that they go forward one by one.

2.2.2. Vine Copula Grouped Model

In a study by Chen and Hao [27], they considered the financial institutions of different industrial sectors, and used a grouped vine copula model to deal with the dependence among financial assets. In this paper, we adopt the grouped vine copula method, which can accurately describe the dependence structure for systemic risk measurement. First, we classify the financial institutions into different financial industrial sectors according to their business features. Second, we describe the dependence structure of the financial institutions within the same industrial sector using vine copulas, and then describe the dependence structure between the different industrial sectors. Consequently, the (total) random gain of the entire financial system can be computed by the weighted sum of random gains of all financial assets held by all financial institutions within the financial system. For a better understanding, we describe the vine copula under the categorical structure presented in (10) below.  

$$S\left(C_X\right)\left\{\begin{array}{cc}{X}_1\left(C_{X_1}\right)\hfill & \left\{\begin{array}{c}{X}_{11}\left(F_{11}\right)\hfill \\ ⋮\hfill \\ X_{1n_1}\left(F_{1n_1}\right)\hfill \end{array}\right.\hfill \\ \hfill & ⋮\hfill \\ X_N\left(C_{X_N}\right)\hfill & \left\{\begin{array}{c}{X}_{N1}\left(F_{N1}\right)\hfill \\ ⋮\hfill \\ X_{N{n}_N}\left(F_{N{n}_N}\right)\hfill \end{array}\right.\hfill \end{array}\right.$$

(10)

In (10), $X_{i1},\cdots ,X_{i{n}_i}$ denote the asset returns of financial institutions. $F_{i1},\cdots ,F_{i{n}_i}$ are the marginal distribution functions of $X_{ij},1\le j\le n_i$, belonging to sector $i,1\le i\le N$. $X_i$ denotes the total random gain of financial institutions within industrial sector $i,1\le i\le N$. S stands for the total random gain of the entire financial system, that is, $S={\displaystyle \sum _{i=1}^N}{\displaystyle \sum _{j=1}^{n_i}}{w}_{ij}{X}_{ij}$, where $w_{ij}$ is the contribution weight of institution j in the industrial sector i to the financial system. $C_{X_i},i=1,\cdots ,N$ are the vine copulas within industrial sectors, and $C_X$ denotes the vine copula among different industrial sectors.

Compared with the ordinary copula from (5), the advantage of the vine copula grouped method lies in the fact that it allows different dependence structures for different industrial sectors.

2.3. Risk Measures

In this section, we briefly introduce the risk measures that we will use in the following sections. In finance, the random gain of a financial institution (or a financial industrial sector or financial position) can be described by a random variable X defined on some probability space. Let $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$ be a fixed probability space, and all the random variables involved in the sequel are supposed to be defined on it. The value at risk (VaR) risk measure is one of the most widely used risk measures in practice. In this paper, we will incorporate one more temporal parameter into the definition of VaR besides the confidence level. Denote by ${\mathsf{\Omega }}_{t-1}$ the information set available up to time $t-1.$ Let $c\in (0,1).$ Then, the VaR of a random gain X at the confidence level $1-c$ and at the tth period is given by

$$\begin{array}{cc}\hfill {\mathrm{VaR}}_{c,t}\left(X\right)& :=\inf \{r\in \mathbb{R}:\mathbb{P}(-X\le r|{\mathsf{\Omega }}_{t-1})\ge 1-c\}\hfill \\ \hfill & =-\sup \{r\in \mathbb{R}:\mathbb{P}(X<r|{\mathsf{\Omega }}_{t-1})\le c\}\hfill \\ \hfill & =\inf \{m\in \mathbb{R}:\mathbb{P}(X+m<0|{\mathsf{\Omega }}_{t-1})\le c\}.\hfill \end{array}$$

(11)

In the financial context, for a random gain X, the quantity ${\mathrm{VaR}}_{c,t}\left(X\right)$ is the smallest amount of capital at time $t-1$ that, if added to X and invested in the risk-free asset, keeps the probability of a negative outcome below the level c. Alternatively, the quantity ${\mathrm{VaR}}_{c,t}\left(X\right)$ is understood as the (minimal) capital requirement for X in order to satisfy the regulatory requirement. Thus, the larger the quantity of VaR is, the higher the riskiness is. For more details, we refer to Föllmer and Schied [4].

Although VaR has been widely used in practice, it does not satisfy the property of subadditivity. Subadditivity is one form that reflects the diversification advantage in finance. Subadditivity is one of the four properties requested by the coherent risk measures proposed by Artzner et al. [1]. On the other hand, VaR focuses on the amount of loss at a single confidence level $1-c$, and does not take into account the distributional aspects within the tail. Hence, $\mathrm{VaR}$ is not sensitive to extreme events (i.e., events with potentially huge losses but very small probabilities). Thus, $\mathrm{VaR}$ might provide an underestimated capital requirement, and therefore might give a less robust estimate. Acerbi and Tasche [28] proposed the Expected Shortfall (ES), which is a valid alternative to VaR and belongs to the class of coherent risk measures.

Throughout this paper, we assume that all random gains involved X are continuous with probability density function $f_X.$ Then, the ES of a random gain X at the confidence level $1-c$ and at the tth period is given by

$$\begin{array}{cc}\hfill {\mathrm{ES}}_{c,t}\left(X\right)& :={\displaystyle \frac{1}{c}}{\int }_0^c{\mathrm{VaR}}_{\theta ,t}\left(X\right)d\theta \hfill \\ \hfill & =\mathrm{E}[-X|-X\ge {\mathrm{VaR}}_{c,t}\left(X\right)]\hfill \\ \hfill & =-{\displaystyle \frac{1}{c}}{\int }_{-\infty }^{-{\mathrm{VaR}}_{c,t}\left(X\right)}x{f}_X\left(x\right)dx.\hfill \end{array}$$

(12)

Similar to $\mathrm{VaR}$, the quantity of ${\mathrm{ES}}_{c,t}\left(X\right)$ is also understood as the (minimal) capital requirement at time t. Similar to VaR, the larger the quantity of ES is, the higher the riskiness is. $\mathrm{ES}$ conquers the insensitivity of $\mathrm{VaR}$ to extreme events, because ${\mathrm{ES}}_{c,t}\left(X\right)$ is an average of $\mathrm{VaR}$ from a given confidence level $1-c$ to confidence level 1. Nevertheless, $\mathrm{ES}$ might provide an overestimated capital requirement because it integrates $\mathrm{VaR}$ from confidence level $1-c$ to confidence level 1. Thus, $\mathrm{ES}$ might provide an overly robust estimate.

From the point of view of the statistical robustness of the risk measure estimators, Cont et al. [29] proposed the range value at risk (RVaR) risk measure. Statistical robustness is important for risk measure estimators via historical data since it is responsible for the stable behavior of the risk measure estimators.

The RVaR of a random gain X across a confidence level interval $[1-\beta ,1-\alpha ]$ is defined by

$${\mathrm{RVaR}}_{\alpha ,\beta }\left(X\right):=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\beta -\alpha }}{\int }_{\alpha }^{\beta }{\mathrm{VaR}}_{\theta }\left(X\right)d\theta ,\hfill & \mathrm{if}\alpha <\beta ,\hfill \\ & \\ {\mathrm{VaR}}_{\beta }\left(X\right),\hfill & \mathrm{if}\alpha =\beta .\hfill \end{array}\right.$$

(13)

When $\alpha <\beta ,$ RVaR can be rewritten as

$${\mathrm{RVaR}}_{\alpha ,\beta }\left(X\right)={\displaystyle \frac{\beta {\mathrm{ES}}_{\beta }\left(X\right)-\alpha {\mathrm{ES}}_{\alpha }\left(X\right)}{\beta -\alpha }}.$$

(14)

Clearly, ${\mathrm{RVaR}}_{\alpha ,\beta }$ includes ${\mathrm{ES}}_{\beta }$ (i.e., $\alpha :=0$) and ${\mathrm{VaR}}_{\beta }$ (i.e., $\alpha :=\beta$) as special cases.

In the definition of ${\mathrm{RVaR}}_{\alpha ,\beta }$, the lower confidence level $1-\beta$ can represent a $normal$ confidence level (say, a regulatory confidence level). The higher confidence level $1-\alpha$ can represent a $tolerance$ level up to which one can tolerate catastrophic losses. Hence, compared with $\mathrm{VaR}$ and $\mathrm{ES}$, $\mathrm{RVaR}$ can well balance the robustness of capital requirement estimates and sensitivity to extreme events. For a more theoretical analysis about the robustness and sensitivity of $\mathrm{RVaR}$, we refer to Cont et al. [29]. We would also like to mention that the specification of the higher confidence level $1-\alpha$ is usually subjective.

Systemic risk measures are supposed to quantify the systemic risk of a financial system, which consists of individual financial institutions. It is usually a little difficult to measure systemic risk accurately and reasonably. Compared with traditional risk measures focusing on the risk of individual institutions, Adrian and Brunnermeier [15] proposed the CoVaR risk measure to evaluate the contribution of each individual institution to the risk of the financial system. They also indicated that the CoVaR is general enough to analyze the risk spillover from one institution to another throughout the financial system.

In the remainder of this paper, since we arbitrarily fix a time period, we will drop the time parameter t in all subscripts for notational simplicity.

Let us recall the definitions of CoVaR and related concepts introduced by Adrian and Brunnermeier [15]. Denote by $X^i$ the random gain of the financial institution i (or financial industrial sector i), by $X^{\mathrm{syst}}$ the random gain of the financial system, by ${\mathrm{VaR}}_c^i$ the VaR of $X^i$ at the confidence level $1-c$, and by ${\mathrm{Median}}^i$ the median of $X^i$, that is,

$$\begin{array}{c}\hfill \mathbb{P}(X^i\le {\mathrm{Median}}^i)=\mathbb{P}(X^i\ge {\mathrm{Median}}^i)={\displaystyle \frac{1}{2}}.\end{array}$$

Note that ${\mathrm{Median}}^i=-{\mathrm{VaR}}_{50\%}^i$. Denote by $\mathbb{C}\left(X^i\right)$ a certain event determined by $X^i$. Let $c\in (0,1)$. Denote by ${\mathrm{CoVaR}}_c^{j\left|\mathbb{C}\right(X^i)}$ the VaR of $X^j$ conditional on an event $\mathbb{C}\left(X^i\right)$ at the confidence level $1-c$, that is, ${\mathrm{CoVaR}}_c^{j\left|\mathbb{C}\right(X^i)}$ is given by

$$\begin{array}{cc}\hfill {\mathrm{CoVaR}}_c^{j\left|\mathbb{C}\right(X^i)}& :=\inf \{x\in \mathbb{R}:\mathbb{P}(-X^j\le x|\mathbb{C}\left(X^i\right))\ge 1-c\}\hfill \\ \hfill & =-\sup \{x\in \mathbb{R}:\mathbb{P}(X^j<x|\mathbb{C}\left(X^i\right))\le c\}\hfill \\ \hfill & =\inf \{m\in \mathbb{R}:\mathbb{P}(X^j+m<0|\mathbb{C}\left(X^i\right))\le c\}.\hfill \end{array}$$

(15)

By the continuity of the conditional distribution function, we know that

$$\mathbb{P}(X^j\le -{\mathrm{CoVaR}}_c^{j|\mathbb{C}(X^i)}|\mathbb{C}(X^i))=c,$$

(16)

since $X^j$ is continuous.

We write ${\mathrm{CoVaR}}_c^{j|-X^i={\mathrm{VaR}}_c^i}$ and ${\mathrm{CoVaR}}_c^{j|-X^i={\mathrm{VaR}}_{50\%}^i}$ for ${\mathrm{CoVaR}}_c^{j\left|\mathbb{C}\right(X^i)}$ when $\mathbb{C}\left(X^i\right)$ is defined as $\{-X^i={\mathrm{VaR}}_c^i\}$ and $\{-X^i={\mathrm{VaR}}_{50\%}^i\}$, respectively. When $X^j$ is set as $X^{\mathrm{syst}}$, we also write ${\mathrm{CoVaR}}_c^{\mathrm{syst}|-X^i={\mathrm{VaR}}_c^i}$ for ${\mathrm{CoVaR}}_c^{j|-X^i={\mathrm{VaR}}_c^i}$. Then, according to Adrian and Brunnermeier [15], the risk contribution $\Delta {\mathrm{CoVaR}}_c^{j|i}$ of institution i to institution j at confidence level $1-c$ is defined as

$$\Delta {\mathrm{CoVaR}}_c^{j|i}:={\mathrm{CoVaR}}_c^{j|-X^i={\mathrm{VaR}}_c^i}-{\mathrm{CoVaR}}_c^{j|-X^i={\mathrm{VaR}}_{50\%}^i}.$$

(17)

Denote by ${\mathrm{ES}}_c^i$ the ES of $X^i$ at confidence level $1-c$, that is,

$$\begin{array}{c}\hfill {\mathrm{ES}}_c^i:={\displaystyle \frac{1}{c}}{\int }_0^c{\mathrm{VaR}}_{\theta }^{i}d\theta =\mathrm{E}[-X^i\mid -X^i\ge {\mathrm{VaR}}_c^i].\end{array}$$

Similarly, we can respectively define ${\mathrm{CoES}}_c^{j\mid -X^i={\mathrm{VaR}}_c^i}$, ${\mathrm{CoES}}_c^{\mathrm{syst}\mid -X^i={\mathrm{VaR}}_c^i}$ and $\Delta {\mathrm{CoES}}_c^{j|i}$ as follows:

$${\mathrm{CoES}}_c^{j\mid -X^i={\mathrm{VaR}}_c^i}:=\mathrm{E}[-X^j\mid -X^j\ge {\mathrm{CoVaR}}_c^{j\mid -X^i={\mathrm{VaR}}_c^i}],$$

(18)

$$\begin{array}{c}\hfill {\mathrm{CoES}}_c^{\mathrm{syst}|-X^i={\mathrm{VaR}}_c^i}:={\mathrm{CoES}}_c^{j|-X^i={\mathrm{VaR}}_c^i},\mathrm{when}{X}^j\mathrm{is}\mathrm{set}\mathrm{as}{X}^{\mathrm{syst}}\end{array}$$

and

$$\begin{array}{c}\hfill \Delta {\mathrm{CoES}}_c^{j|i}:={\mathrm{CoES}}_c^{j|-X^i={\mathrm{VaR}}_c^i}-{\mathrm{CoES}}_c^{j|-X^i={\mathrm{VaR}}_{50\%}^i}.\end{array}$$

For more details, we refer to Adrian and Brunnermeier [15].

Inspired by Adrian and Brunnermeier [15] and Cont et al. [29], we introduce a new risk measure, CoRVaR, which can be used to evaluate systemic risk.

Definition 2.

Denote by ${\mathrm{CoRVaR}}_{\alpha ,\beta }^{j\left|\mathbb{C}\right(X^i)}$ the $\mathrm{RVaR}$ of $X^j$ across a confidence level interval $[1-\beta ,1-\alpha ]$ conditional on an event $\mathbb{C}\left(X^i\right)$ determined by $X^i$, that is, ${\mathrm{CoRVaR}}_{\alpha ,\beta }^{j\left|\mathbb{C}\right(X^i)}$ is defined by

$${\mathrm{CoRVaR}}_{\alpha ,\beta }^{j\left|\mathbb{C}\right(X^i)}:=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\beta -\alpha }}{\int }_{\alpha }^{\beta }{\mathrm{CoVaR}}_{\theta }^{j\left|\mathbb{C}\right(X^i)}d\theta ,\hfill & \mathit{if}\alpha <\beta ,\hfill \\ & \\ {\mathrm{CoVaR}}_{\alpha }^{j\left|\mathbb{C}\right(X^i)},\hfill & \mathit{if}\alpha =\beta .\hfill \end{array}\right.$$

(19)

An alternative definition by means of conditional expectation is given by

$$\begin{array}{c}\hfill {\mathrm{CoRVaR}}_{\alpha ,\beta }^{j\left|\mathbb{C}\right(X^i)}:=\mathbb{E}[-X^j|{\mathrm{CoVaR}}_{\alpha }^{j|-{\mathrm{X}}^i={\mathrm{VaR}}_{\alpha }^i}\le -{\mathrm{X}}^j\le {\mathrm{CoVaR}}_{\beta }^{j|-{\mathrm{X}}^i={\mathrm{VaR}}_{\beta }^i}]\end{array}$$

(20)

$$\begin{array}{c}\hfill {\mathrm{CoRVaR}}_{\alpha ,\beta }^{\mathrm{syst}\left|\mathbb{C}\right({\mathrm{X}}^i)}:={\mathrm{CoRVaR}}_{\alpha ,\beta }^{j\left|\mathbb{C}\right({\mathrm{X}}^i)},\mathrm{when}{\mathrm{X}}^j\mathrm{is}\mathrm{set}\mathrm{as}{\mathrm{X}}^{syst}\end{array}$$

and

$$\begin{array}{c}\hfill \Delta {\mathrm{CoRVaR}}_{c_1,c_2}^{j|i}:={\mathrm{CoRVaR}}_{c_1,c_2}^{j|-X^i={\mathrm{RVaR}}_{c_1,c_2}^i}-{\mathrm{CoRVaR}}_{c_1,c_2}^{j|-X^i=-Media{n}^i}.\end{array}$$

Note that $\mathrm{RVaR}$ is coherent, that is, it admits the properties of monotonicity, positive homogeneity, translation invariance and subadditivity. Similar to the relation of $\mathrm{RVaR}$ with $\mathrm{ES}$ and $\mathrm{VaR}$, ${\mathrm{CoRVaR}}_{\alpha ,\beta }$ includes ${\mathrm{CoES}}_{\beta }$ (i.e., $\alpha :=0$) and ${\mathrm{CoVaR}}_{\beta }$ (i.e., $\alpha :=\beta$) as special cases.

As pointed out previously, compared with VaR and ES, RVaR can well balance robustness and sensitivity. Similarly, compared with CoVaR and CoES, CoRVaR can also well balance robustness and sensitivity. From the representation of coherent risk measures, a robust estimate of a risk measure can be represented by a maximal expected loss over a set of plausible probability measures. Hence, in the empirical study that follows, we use the mean of losses as a robust estimate of the capital requirement, and standard deviation (Std) to reflect the sensitivity for extreme losses; see the table in Section 4.3.2.

2.4. Algorithm of Calculation of Systemic Risk Measures

In this subsection, we introduce the calculation algorithms for systemic risk measures including CoVaR, CoES and CoRVaR. In order to generate samples from the standardized residual $Z_t$ as in (2), we make use of Monte Carlo simulation. For static and dynamic settings, we introduce the corresponding algorithms below.

2.4.1. Algorithm in Static Setting

Since a vine copula grouped model is employed to describe the dependence structure of the financial system, we need to distinguish the intra-group copula from the inter-group copula. The detailed implementation is described as follows.

Step 1: Generate samples from the standard residual $Z_t$ for each $2\le t\le T$. First, we generate random numbers from the uniform distribution $U(0,1)$. Then, using the probability integral transform, we obtain samples from $Z_t$.

Step 2: Calculate the risk measures CoRVaR, CoES and CoVaR. First, by substituting the samples from $Z_t$ to (2), we obtain the return $r_t$. By quantile regression, we then calculate CoVaR, CoES and CoRVaR, as shown in (23) below.

2.4.2. Algorithm in Dynamic Setting

In order to better capture the dynamics of the sequence of returns $r_t$, we employ a rolling Monte Carlo simulation to simulate the marginal distributions required for the vine copula grouped model. By using rolling Monte Carlo simulation, we can obtain multiple sets of simulated data. The use of rolling Monte Carlo simulation makes clear the changes in the risk measures. Meanwhile, such high-frequency predictions of returns can be consistent with actual market returns in the real market.

An algorithm for calculating the dynamic risk measures CoVaR, CoES and CoRVaR is provided in Algorithm 1 below.

Algorithm 1 Rolling Monte Carlo Simulation

n samples, window size $n_1$. Number of institutions belonging to banking industry $n_b$, number of institutions belonging to security industry $n_s$, number of institutions belonging to insurance industry $n_i$. Then, return $r_t$ at time t.   1: for  $i\in [0,n-n_1]$  do   2:        for $b\in [1,n_b$] do   3:              Select samples $i,\cdots ,n_1+i-1$ from bth banking.   4:              Construct the marginal distribution $F_b$.   5:              Calculate $\widehat{\mu }$, $\widehat{\omega }$, $\widehat{\alpha }$, $\widehat{\beta }$, $\widehat{\gamma }$ ⟵ samples $i,\cdots ,n_i+i-1$.   6:              Calculate ${ϵ}_{i+1,b}$, with $\widehat{r_{i+1,b}}=\widehat{{\mu }_i}+\widehat{c}{r}_i$ and $r_{i+1}.$   7:        end for   8:        Simulate the Copula function $F_b(F_1^{-1}({ϵ}_{i+1,1}),\cdots ,F_{n_b}^{-1}({ϵ}_{i+1,n_b})).$   9:        for $s\in [1,n_s$] do 10:              Select samples $i,\cdots ,n_1+i-1$ from $s_{th}$ banking. 11:              Construct the marginal distribution $F_s$. 12:              Calculate $\widehat{\mu }$, $\widehat{\omega }$, $\widehat{\alpha }$, $\widehat{\beta }$, $\widehat{\gamma }$ ⟵ samples $i,\cdots ,n_1+i-1$. 13:              Calculate ${ϵ}_{i+1,s}$, with $\widehat{r_{i+1,s}}=\widehat{{\mu }_i}+\widehat{c}{r}_i$ and $r_{i+1}.$ 14:        end for 15:        Simulate the Copula function $F_s(F_1^{-1}({ϵ}_{i+1,1}),\cdots ,F_{n_s}^{-1}({ϵ}_{i+1,n_s})).$ 16:        for $i\in [1,n_i$] do 17:              Select the samples $i,\cdots ,n_1+i$ from $i_{th}$ banking. 18:              Construct the marginal distribution $F_i$. 19:              Calculate $\widehat{\mu }$, $\widehat{\omega }$, $\widehat{\alpha }$, $\widehat{\beta }$, $\widehat{\gamma }$ ⟵ samples $i,\cdots ,n_1+i-1$. 20:              Calculate ${ϵ}_{i+1,i}$, with $\widehat{r_{i+1,i}}=\widehat{{\mu }_i}+\widehat{c}{r}_i$ and $r_{i+1}.$ 21:        end for 22:        Simulate the Copula $F_i(F_1^{-1}\left({ϵ}_{i+1,1}\right),\cdots ,F_{n_i}^{-1}\left({ϵ}_{i+1,n_i}\right)).$ 23:        Simulate the Copula $F_{sys}^{-1}({ϵ}_{i+1,b},{ϵ}_{i+1,i},{ϵ}_{i+1,s})$. 24:        Set samples $t_1\cdots t_{1000}$ from $U(0,1)$. Calculate $r_b$,$r_s$ and $r_i$. 25:        Calculate the risk measures CoVaR, CoES and CoRVaR as above-mentioned. 26: end for    Output: The risk measures at different running epoch $n_1+i,\cdots ,n$.

3. Data Description

In this section, we describe the data used in the following empirical analysis. Since the banking, security and insurance industries are three important ingredients of a national financial system, we consider a financial system consisting of the banking, security and insurance industries. Precisely, we choose a total of 20 financial institutions from the Chinese A-share market. The financial system consists of ten banks, seven security agents and three insurance companies, as is displayed in

Table 1. Selected financial institutions.

Industry	Financial Institutions
Banking	Industrial and Commercial Bank of China (601398), China Construction Bank (601939),
	Agricultrual Bank of China (601288), Bank of China (601988),
	Bank of Communications (601328), China Merchants Bank (600036),
	Industrial Bank (601166), China Citic Bank (601998),
	China Minsheng Bank (600016), China Everbright Bank (601818)
Security	Citic Securities (600030), Huatai Securities (601688), Guotai Junan (601211),
	China Merchants Securities (600999), Haitong Securities (600837), GF Securities (000776)
	Guosen Securities (002736)
Insurance	China Life (601628), Ping An Insurance (601318), China Pacific Insurance Company (601601)

Notes: The number in parentheses is the stock code of the financial institution.

below. We use the daily closing price from the China Stock Market and Accounting Research (CSMAR) database from 13 October 2017 to 13 October 2023. After data preprocessing, there are a total of 1434 valid observations. Following the traditional approach for calculating asset returns, we choose log-return as our variable. The calculation of log-return is given by

$$X_{i,t}=\ln P_{i,t}-\ln P_{i,t-1},$$

(21)

where $P_{i,t}$ stands for the daily closing price of the stock of financial institution i at time t.

In

Table 2. Statistics of different financial industries.

	Banking	Security	Insurance
Mean	−0.000104106	−0.0001941232	−0.00003399299
Std	0.01109873	0.01801919	0.01919151
Max	0.08126908	0.09529105	0.09212155
Min	−0.0933514	−0.1056447	−0.08780622
Kurtosis	8.132977	5.576644	2.132066
Skewness	0.05364904	0.2845737	0.3627719
J-B	3968 ***	1885.5 ***	304.95 ***
Q(15)	34.429 ***	29.919 **	25.388 **
LM(5)	153.51 ***	74.986 ***	46.21 ***
ADF	−11.616 ***	−10.944 ***	−11.96 ***

*** denotes significance at 1% level. ** denotes significance at 5% level.

, we provide some basic statistics about the financial industries studied. From Table 2, we can observe the following:

(1)

The skewness of logarithmic returns across different financial industries is generally non-zero, and the kurtosis is relatively large. These two statistics indicate that the logarithmic returns exhibit the typical “peaked and fat-tailed” distribution, a common characteristic of financial data that should be taken into account in empirical analysis.

(2)

The mean of the logarithmic returns of the three industries is close to zero. Notably, the standard deviation of the banking industry is smaller than that of the security and insurance industries, suggesting that the banking sector is more stable—a finding consistent with the reality of China’s financial market.

(3)

The skewness of all three financial industries is greater than zero, indicating that their returns follow a right-skewed distribution. This indicates that positive returns tend to be more frequently observed across all three financial industries over the chosen time period.

In Table 2, we also give the test statistics for normality, autocorrelation, ARCH effects, and stationarity. The Jarque–Bera test shows that the null hypothesis of normality is strongly rejected for all industry returns. The Ljung-Box Q test shows the significant autocorrelation among all three industries, and the ARCH-LM test indicates volatility clustering in all industry returns. We also perform the ADF test, which confirms that all logarithmic return series are stationary. These tests ensure the robustness of our model.

4. Empirical Study

In this section, we illustrate the proposed model by means of implementation with the real data described in Section 3. To simplify and ensure the efficiency of the implemented data, we adopt the AR-GJR-GARCH (1.1) model, as in (1)–(3). We also adopt maximum likelihood estimation (MLE) to estimate the model parameters. In both static and dynamic settings, we calculate the risk measures VaR, ES, CoVaR, CoES and CoRVaR, respectively.

4.1. Distribution Functions of Institutions and Industries

In this subsection, we make goodness-of-fit of distribution functions of 20 institutions and three industries based on the AIC, respectively. The candidates for goodness-of-fit are the skewed Student’s t distribution (sstd), the generalized error distribution (ged) and the skewed generalized error distribution (sged). The results of the goodness-of-fit of the distributions are displayed in

Table 3. Distribution of standardized residuals by financial institution.

	sstd	ged	sged
Banking	601398, 601939, 601328,	601288	601988, 600036
	601166, 601998, 600016,
	601818
Security	600030, 601688, 601211,	-	-
	600999, 600837, 000776,
	002736
Insurance	601628	-	601318, 601601

Notes: The numbers in this table are the stock codes of the 20 financial institutions studied.

and

Table 4. Parameter estimation results of the marginal distribution models.

	Banking	Security	Insurance
$\mu$	0.00007 (0.00025)	−0.00018 (0.00040)	0.00001 (0.00044)
c	−0.02404 (0.02661)	−0.06437 (0.02308)	−0.03607 (0.02748)
w	0.00000 *** (0.00000)	0.00001 *** (0.00000)	0.00003 *** (0.00001)
$\alpha$	0.12513 *** (0.02126)	0.06217 *** (0.01113)	0.04990 *** (0.01790)
$\beta$	0.83551 *** (0.01802)	0.92621 *** (0.00968)	0.85321 *** (0.04306)
$\gamma$	−0.05150 (0.03430)	−0.02943 (0.01938)	0.01430 (0.02902)
skewness	1.09572 *** (0.03855)	1.11222 *** (0.03976)	1.13621 *** (0.04132)
shape	4.67036 *** (0.56362)	3.38530 *** (0.29336)	1.25515 *** (0.06313)
LL	4593.067	3919.389	3710.5
${\mathrm{Z}}_t$	sstd	sstd	sged

*** denotes significance at 1% level.

, respectively.

From Table 3, we observe that except for three institutions, all the remaining institutions follow the skewed Student’s t distribution. From Table 4, we observe that except for the insurance industry, the other two industries follow the skewed Student’s t distribution. Meanwhile, We observe that although the leverage coefficients $\gamma$ are non-zero, they are not statistically significant (p-value > 0.1). Therefore, the leverage effects in this observed period of time can be neglected.

We would also like to mention that if we use negative log-return instead of log-return to fit the AR-GJR-GARCH model, then Table 2 and Table 4 become

Table 5. Statistics of different financial industries.

	Banking	Security	Insurance
Mean	0.000104106	0.0001941232	0.00003399299
Std	0.01109873	0.01801919	0.01919151
Max	0.0933514	0.1056447	0.08780622
Min	−0.08126908	−0.09529105	−0.09212155
Kurtosis	8.132977	5.576644	2.132066
Skewness	−0.05364904	−0.2845737	−0.3627719
J-B	3968 ***	1885.5 ***	304.95 ***
Q(15)	34.429 ***	29.919 **	25.388 **
LM(5)	153.51 ***	74.986 ***	46.21 ***
ADF	−11.616 ***	−10.944 ***	−11.96 ***

*** denotes significance at 1% level. ** denotes significance at 5% level.

and

Table 6. Parameter estimation results of the marginal distribution models.

	Banking	Security	Insurance
$\mu$	−0.000075 (0.000251)	0.000183 (0.000406)	−0.000013 (0.000394)
c	−0.024041 (0.026613)	−0.064374 (0.023082)	−0.036008 (0.027675)
w	0.000008 *** (0.000001)	0.000011 *** (0.000001)	0.000033 *** (0.000012)
$\alpha$	0.073642 *** (0.016911)	0.032739 *** (0.008954)	0.064065 *** (0.025828)
$\beta$	0.835494 *** (0.018069)	0.926216 *** (0.009698)	0.852997 *** (0.039188)
$\gamma$	0.051495 (0.034210)	0.029444 (0.019342)	−0.014094 (0.026760)
skewness	0.912654 *** (0.032116)	0.899106 *** (0.032124)	0.880075 *** (0.031360)
shape	4.671370 *** (0.563955)	3.385191 *** (0.293529)	1.255074 *** (0.062956)
LL	4593.067	3919.389	3710.5
${\mathrm{Z}}_t$	sstd	sstd	sged

*** denotes significance at 1% level.

below, respectively.

We find that there are no significant differences appearing. Hence, in the following, we use log-return to implement the empirical study.

4.2. Results in Static Setting

In this subsection, we present the specific pairwise copulas for institutions within the same industry and for the three industries together, respectively. We also calculate the risk measures VaR, ES, CoVaR, CoES and CoRVaR. The relevant results are displayed in Figure 2 and

Table 7. Risk spillover effect among the three industries.

Industry	${\mathit{c}}_1$	${\mathit{c}}_2$	VaR	ES	CoVaR	CoES	CoRVaR
Banking	2.5%		0.01926	0.02840
	1%		0.02630	0.03783
Banking to Security	2.5%				0.04763	0.07200
	1%				0.06286	0.09537
	0.5%	1%					0.06745
	0.5%	2.5%					0.05688
Banking to Insurance	2.5%				0.04454	0.05641
	1%				0.05969	0.07315
	0.5%	1%					0.06441
	0.5%	2.5%					0.05289
Security	2.5%		0.03144	0.04836
	1%		0.04377	0.06636
Security to Banking	2.5%				0.02760	0.03966
	1%				0.03919	0.05521
	0.5%	1%					0.04502
	0.5%	2.5%					0.03531
Security to Insurance	2.5%				0.04630	0.05648
	1%				0.05810	0.07125
	0.5%	1%					0.06154
	0.5%	2.5%					0.05203
Insurance	2.5%		0.03637	0.04755
	1%		0.04616	0.05811
Insurance to Banking	2.5%				0.02818	0.04041
	1%				0.03902	0.05505
	0.5%	1%					0.04349
	0.5%	2.5%					0.03506
Insurance to Security	2.5%				0.04979	0.07545
	1%				0.07127	0.10844
	0.5%	1%					0.07656
	0.5%	2.5%					0.06131

and

Table 8. Risk spillover from industries to the system.

Industry to System	${\mathit{c}}_1$	${\mathit{c}}_2$	CoVaR	CoES	CoRVaR
Banking to System	1%		0.05127	0.06721
	2.5%		0.04097	0.05410
	0.5%	1%			0.05670
	0.5%	2.5%			0.04862
Security to System	1%		0.04951	0.05964
	2.5%		0.03557	0.04407
	0.5%	1%			0.05426
	0.5%	2.5%			0.04251
Insurance to System	1%		0.04727	0.06540
	2.5%		0.03506	0.04886
	0.5%	1%			0.05212
	0.5%	2.5%			0.04278

, respectively. Finally, backtesting is done.

In Figure 2, t, N, F, C, SG, G and J respectively represent the t-copula, Gaussian copula (normal distribution function), Frank-copula, Clayton-copula, ${180}^{\circ }$-rotated Gumbel-copula (SG), Gumbel-copula and Joe-copula. In Figure 2a, the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 respectively represent the institutions with codes 601398, 601939, 601288, 601988, 601328, 600036, 601166, 601998, 600016 and 601818 in the banking industry. In Figure 2b, the numbers 1, 2, 3, 4, 5, 6 and 7 respectively represent the institutions with codes 600030, 601688, 601211, 600999, 600837, 000776 and 002736 in the security industry. In Figure 2c, the numbers 1, 2 and 3 respectively denote the institutions with codes 601628, 601318 and 601601 in the insurance industry. In Figure 2d, the numbers 1, 2 and 3 denote the banking industry, the security industry and the insurance industry, respectively.

Consistent with Figure 2d,

Table 9. Vine copulas for the industries.

Tree	Edge	Copula	Par	Par2	tau	utd	ltd
$T_1$	(3, 1)	t	0.73	4.54	0.52	0.39	0.39
	(3, 2)	t	0.63	4.66	0.44	0.31	0.31
$T_2$	(2, 1; 3)	t	0.26	10.59	0.17	0.02	0.02

gives the copulas between the three industries. Precisely, from Table 9, we observe the following:

(1)

The dependence structure between either the insurance and banking industries or insurance and security industries obeys the t-copula. Meanwhile, from the coefficient $\tau$, we can see that there is a small but strong positive correlation relationship between the insurance industry and both the banking industry and security industry. From the coefficients utd and ltd, we also find that there exists tail dependence. All the above observations are in accordance with intuition.

(2)

Conditional on the insurance industry, the dependence structure between the security and banking industries also follows the t-copula. There is also a weak positive correlation and weak tail dependence.

In Table 7, $1-c_1$ and $1-c_2$ represent two confidence levels. From the CoRVaR in Table 7, we can observe the following:

(1)

The risk spillover effects between the three industries are all positive, which is consistent with those of the risk measures CoVaR and CoES.

(2)

The extent of the risk spillover effect from one industry to another may vary, which is also consistent with what can be seen with the risk measures CoVaR and CoES. For instance, the risk spillover effect from the banking industry to the security industry is stronger than that from the banking industry to the insurance industry. Moreover, the risk spillover between two industries may be asymmetric.

(3)

The quantities of CoRVaR are between those of CoVaR and CoES.

(4)

The banking industry is the least risky among the three industries because its capital requirement (i.e., the values of VaR and ES) is the lowest.

Table 8 displays the risk spillover effects from the three industries to the entire system, respectively. From CoRVaR in Table 8, we can see that the risk spillover effect from the banking industry to the system is the strongest among the three industries, which is in accordance with intuition.

We end this subsection with backtesting. The backtesting is designed to gauge accuracy and effectiveness via historical data. Motivated by Zhang and Nadarajah [30], we adopt a binomial distribution test to compile backtesting CoRVaR, as well as CoVaR and CoES. The binomial distribution test says that if $I_t$ is independent and identically distributed and $\mathbb{P}[I_{t+1}\left(\alpha \right)=1]=\alpha$, then the total number of violations H has a binomial distribution $B(n,\alpha )$ with mean $\mathbb{E}\left(H\right)=n\alpha$ and variance $VaR\left(H\right)=n\alpha (1-\alpha )$.

According to Jorion [31], the test statistic T is given by

$$T={\displaystyle \frac{H-n\alpha }{\sqrt{n\alpha (1-\alpha )}}}$$

(22)

Its asymptotic null distribution is the standard normal distribution. According to Adrian and Brunnermeier [15], the risk spillover from institution j to institution i for the q-quantile can be estimated by

$$CoVa{R}_q^i:=Va{R}_q^{i|-X^i=Va{R}_q^i}={\widehat{\alpha }}_q^j+{\widehat{\beta }}_q^{j}Va{R}_q^j.$$

(23)

Based on (23), we can obtain the results of backtesting CoVaR, CoES and CoRVaR, which are displayed in

Table 10. The backtesting for the different systemic risk measures.

Industry to System	${\mathit{c}}_1$	${\mathit{c}}_2$	CoVaR	p-Value	CoES	p-Value	CoRVaR	p-Value
Banking to System	1%		0.3624	0.6415	1.8317	0.9665
	2.5%		0.0708	0.5282	0.9215	0.8216
	0.5%	1%					1.4235	0.9227
	0.5%	2.5%					0.6486	0.7417
Security to System	1%		1.4071	0.9203	1.5813	0.9431
	2.5%		0.0906	0.50361	1.1017	0.8647
	0.5%	1%					1.6595	0.9514
	0.5%	2.5%					1.0771	0.8592
Insurance to System	1%		1.5687	0.9416	1.0605	0.8555
	2.5%		1.2715	0.8982	1.2835	0.9003
	0.5%	1%					1.8800	0.9699
	0.5%	2.5%					0.6444	0.7403

.

From the p-values in Table 10, we can see that CoVaR, CoES and CoRVaR successfully pass the backtesting.

4.3. Results in Dynamic Setting

In the preceding Section 4.2, we calculated the static risk measures VaR, ES, CoVaR, CoES and CoRVaR, as well as risk spillover quantities, by making use of the total 1434 data points. In this subsection, we will calculate these risk measures and their risk spillovers in a dynamic setting. To complete this purpose, we will use rolling Monte Carlo simulation and adopt Algorithm 1 to calculate risk measures. The models for constructing the vine copulas and calculating the risk measures are basically the same as those in the static setting, except for the setting of a window size. In the rolling procedure, we set a fixed window size of 1300. Hence, we have 134 quantities for each risk measure. Moreover, we also make comparisons between them.

4.3.1. Results in the Case of Rolling Once

In this subsection, we display the results for the case of rolling once, that is, the results of using the first 1300 data points. We begin with the display of the copula trees among the three industries by rolling once in Figure 3.

From tree one ${\mathrm{T}}_1$ in Figure 3, we can see that the inter-industry dependence structure is the same as that in tree ${\mathrm{T}}_1$ from Figure 2d. Nevertheless, from tree two ${\mathrm{T}}_2$ in Figure 3, we can see the dependence structure between banking and security conditioning on insurance obeys the rotated survival Gumbel-copula (SG), which is different from what was observed in Figure 2d.

For more details about the inter-industry dependence structure, please see

Table 11. The estimation of vine copulas after rolling once.

Tree	Edge	Copula	Par	Par2	tau	utd	ltd
$T_1$	(3, 1)	t	0.75	4.18	0.54	0.42	0.42
	(3, 2)	t	0.64	4.50	0.44	0.31	0.31
$T_2$	(2, 1; 3)	SG	1.19	-	0.16	-	0.21

.

Compared with the coefficients in Table 9, Table 11 shows that the dependence relation appears stronger.

In

Table 12. Risk spillover effects after rolling once.

Industry	${\mathit{c}}_1$	${\mathit{c}}_2$	VaR	ES	CoVaR	CoES	CoRVaR
Banking	2.5%		0.01882	0.02743
	1%		0.02575	0.03607
Banking to Security	2.5%				0.04719	0.06870
	1%				0.06223	0.08882
	0.5%	1%					0.06687
	0.5%	2.5%					0.05596
Banking to Insurance	2.5%				0.04410	0.05526
	1%				0.05912	0.07206
	0.5%	1%					0.06359
	0.5%	2.5%					0.05210
Security	2.5%		0.02374	0.03603
	1%		0.03287	0.04906
Security to Banking	2.5%				0.02468	0.03474
	1%				0.03472	0.04791
	0.5%	1%					0.03986
	0.5%	2.5%					0.03143
Security to Insurance	2.5%				0.03990	0.05059
	1%				0.05283	0.06499
	0.5%	1%					0.05631
	0.5%	2.5%					0.04675
Insurance	2.5%		0.03422	0.04439
	1%		0.04310	0.05406
Insurance to Banking	2.5%				0.02735	0.03813
	1%				0.03058	0.04237
	0.5%	1%					0.03661
	0.5%	2.5%					0.03379
Insurance to Security	2.5%				0.04849	0.07048
	1%				0.06900	0.09811
	0.5%	1%					0.07402
	0.5%	2.5%					0.05925

, we display all values of risk measures VaR and ES and risk spillovers CoVaR, CoES and CoRVaR. The trends in the risk spillovers are similar to those in Table 7.

In

Table 13. Risk spillovers after rolling once.

Industry to System	${\mathit{c}}_1$	${\mathit{c}}_2$	CoVaR	CoES	CoRVaR
Banking to System	1%		0.05085	0.06239
	2.5%		0.04056	0.05191
	0.5%	1%			0.05605
	0.5%	2.5%			0.04846
Security to System	1%		0.04367	0.05143
	2.5%		0.03139	0.03855
	0.5%	1%			0.04783
	0.5%	2.5%			0.03738
Insurance to System	1%		0.04574	0.05885
	2.5%		0.03405	0.04547
	0.5%	1%			0.05050
	0.5%	2.5%			0.04148

, we display risk spillovers CoVaR, CoES and CoRVaR from each industry to the system in the case of rolling once.

From the values of CoRVaR in Table 13, we find that the severity of the risk spillover effect from most to least severe is ranked in the order of the banking industry, insurance industry and security industry. This reveal is in accordance with intuition, because both the banking and insurance industries should be the two most important and first ingredients of a financial system. Meanwhile, such a rank of severity of risk spillover is also consistent with that ranked by CoVaR and CoES, respectively. These findings are also consistent with those of Zhang et al. [30] and Cui [16].

4.3.2. Risk Measures and Risk Spillovers in Dynamic Setting

In this subsection, we display the values of risk measures VaR and ES and risk spillovers CoVaR, CoES and CoRVaR in a dynamic setting. Since we have a total of 1434 data points and a window size of 1300, we can roll 134 times to get 134 quantities for each risk measure and risk spillover, respectively.

We begin with a discussion about the risk spillovers from one industry to another.

In Figure 4, we display the risk measures VaR and ES and risk spillovers CoRVaR, CoVaR and CoES in the banking and security industries.

In Figure 5, we display the risk measures VaR and ES and risk spillovers CoRVaR, CoVaR and CoES in the banking and insurance industries, respectively.

In Figure 6, we display the risk measures VaR and ES and risk spillovers CoRVaR, CoVaR and CoES in the security and insurance industries, respectively.

From Figure 4, Figure 5 and Figure 6, we can observe the following:

(1)

The banking industry is the least risky among the three industries, which is also consistent with the static risk measures VaR and ES in Table 7 and risk measures VaR and ES after rolling once in Table 12.

(2)

The risk spillover effect from the banking industry to each of the other two industries is greater than that from each of the other two industries to the banking industry. This phenomenon is mainly due to the fact that the banking industry plays a dominant role in the Chinese financial system. Hence, controlling the risk spillover from the banking industry to other industries could be helpful to keep the entire financial system running stably. Thus, banking regulation should be continuously concerned.

(3)

The risk spillover effect from the insurance industry to the security industry is greater than that from the security industry to the insurance industry. This phenomenon reveals that the insurance industry plays a more important role in the Chinese financial system than the security industry.

Next, we turn to discuss the risk spillovers from each industry to the system.

In Figure 7 and Figure 8, we display the values of CoVaR, CoES and CoRVaR of the system under each industry.

From Figure 7 and Figure 8, we find the following:

(1)

The values of CoVaR, CoES and CoRVaR are positive, which means that there does exist risk spillover from each industry to the system.

(2)

The curve of CoRVaR is between those of CoES and CoVaR. This empirically illustrates CoRVaR’s balancing between robustness and sensitivity.

To compare the robustness and sensitivity of CoRVaR with those of CoVaR and CoES, we calculate the means and variances of these risk measures over the entire observation period, and construct corresponding box plots. These box plots are intended to provide a visual comparison of CoRVaR’s performance with those of CoVaR and CoES.

In

Table 14. Statistical summary of quantitative performance of CoVaR, CoES and CoRVaR.

Industry and System	CoVaR		CoES		CoRVaR
	Mean	Std	Mean	Std	Mean	Std
Banking to Security	0.049306	0.005776	0.072678	0.008741	0.058961	0.007193
Banking to Insurance	0.046231	0.005807	0.058272	0.007344	0.054812	0.006929
Security to Banking	0.026009	0.001495	0.037765	0.002390	0.033321	0.001982
Security to Insurance	0.042047	0.002421	0.053766	0.003505	0.049221	0.002785
Insurance to Banking	0.028487	0.002092	0.041102	0.003469	0.035438	0.002652
Insurance to Security	0.050282	0.003303	0.074045	0.005233	0.061755	0.004195
Banking to System	0.042666	0.005748	0.055536	0.007085	0.050579	0.006702
Security to System	0.033247	0.002090	0.041880	0.003060	0.040059	0.002628
Insurance to System	0.035498	0.002659	0.048811	0.003740	0.043182	0.003295

, the confidence level $(1-c)$ for CoVaR and CoES is 97.5%, while the confidence level interval $[1-c_2,1-c_1]$ for CoRVaR is $[97.5\%,99.5\%]$. As shown in Table 14, the means of CoRVaR over the entire observation period consistently lie between those of CoVaR and CoES. This intermediate position indicates that CoRVaR provides an intermediately robust estimate for the risk spillover. Moreover, the standard deviations (Std) of CoRVaR are notably lower than those of CoES, implying that CoRVaR exhibits lower sensitivity to risk fluctuations than CoES, but higher sensitivity than CoVaR.

Finally, we draw pictures of the box plots of CoVaR, CoES and CoRVaR, respectively. In Figure 9, the two confidence levels $c_1$ and $c_2$ are respectively set to be 2.5% and 0.5%. From Figure 9, we find that for the risk spillover either in the three industries or from each industry to the system, CoRVaR can well balance robustness and sensitivity, compared with CoVaR and CoES.

5. Conclusions

In this paper, we suggest a new risk measure called CoRVaR to explore the risk spillover effect in a financial system. In both a static setting and dynamic setting, empirical analysis illustrates the effectiveness of CoRVaR from the perspective of robustness and sensitivity. Over the course of this empirical study, rolling Monte Carlo simulation is employed. In addition, comparisons with CoVaR and CoES are also made.

We would like to mention that the interaction term in Equation (3) is defined for negative residuals, i.e., $\gamma {ϵ}_{t-1}^{2}I({ϵ}_{t-1}<0)$. An interesting topic for further study could consider an interaction term defined for positive residuals, i.e., $\gamma {ϵ}_{t-1}^{2}I({ϵ}_{t-1}>0)$, instead of negative residuals.