Identifying influential financial stocks using simulation with a two-layer network

Abstract

Risk spillover from one stock to another tends to create a contagion effect in the stock market. Fire sales due to the overlapping portfolios of mutual funds can amplify the contagion risks, leading to a downward spiral of stock prices. In this paper, we simulate this downward spiral phenomenon for the Chinese financial stocks based on a two-layer network structure and aim to identify the influential financial stocks based on their individual induced systemic risks. Our findings show that stock liquidity and the concentration of funds’ holding on stocks play important roles in determining systemically important financial institutions. Our results also confirm the statements of “too-big-to-fail” and “too-interconnected-to-fail” of financial institutions in the Chinese market. Our results show that a more sensitive flow-performance relationship of mutual funds can amplify the contagion risk by 41%. However, the magnitude can be more drastic in a low market liquidity scenario, where the contagion risk is boosted by 160%.

1. Introduction

Systemic risk is the notion of contagion or impact that starts from the failure of one or a group of financial institutions (FIs) and propagates through the financial system and potentially to the real economy [1]. According to the various channels of propagation, contagion can be broadly classified into direct and indirect contagion. The direct contagion arises from counterparty exposures or funding relations [2], while the indirect contagion is usually associated with fire sales due to common asset holdings [3]. When FIs invest in the same assets, their portfolios with common assets are said to overlap. Contagion can occur because of shocks that cause common assets to be devalued, and devaluations can trigger further asset sales [4], known as fire sales. The term fire sale is commonly used to indicate a coerced sale of assets at a depressed price [5]. The phenomenon of depressing prices in the process of fire sale can be referred to as liquidation spirals [6]. When an FI sells illiquid assets, the sale can depress prices due to the lack of unconstrained buyers, which can trigger financial distress for other FIs who hold the same assets. The affected FIs can sell other assets in an attempt to meet their funding needs, which may cause additional sales. Ultimately, downward spirals can occur across seemingly unrelated assets and spread across seemingly unrelated intuitions [3,5]. As a result, fire sales can affect financial market stability by amplifying the effect of an initial shock to FIs that are otherwise unrelated to fundamental asset values, thus increasing volatility [7].

The downward spiral phenomenon exists in the banking and mutual fund sectors. Reference [8] study two contagion channels in the Austrian banking sector by combining empirical data on direct lending with a stylized model on overlapping portfolios. Reference [6] develop a linear model of fire sale spillovers in the banking sector. Their model can be readily estimated using data from bank balance sheets. Their discovery reveals that a shock has a greater overall effect on institutions with higher levels of leverage compared to those with lower leverage. This is because highly leveraged institutions are compelled to sell a larger amount of assets in a rapid liquidation process to sustain their desired capital structure. The banking system is more susceptible to contagion when the most leveraged bank also holds asset classes that are large in dollar terms. Reference [9] extend the model to the mutual fund sector to quantify the vulnerability of asset managers due to systemic asset liquidations. They find that the mutual funds' aggregate vulnerability is generally modest but can become more significant in periods of low market liquidity [9]. A similar study on the fire sale spillovers in the debt markets finds that fire sales induced by investor redemptions have powerful spillover effects among funds holding overlapping assets, hurting peer funds’ performances and flows, and leading to further asset sales with negative bond price impacts [10].

Here we look closely at the stock market's downward spiral phenomenon. When an idiosyncratic shock hits a stock, its price will decrease, potentially causing a series of chained actions. Firstly, the shock will create a risk spillover effect on other stocks in the network, most commonly in the same sector. Secondly, the decreases in the stock prices may result in drops in net values for some mutual funds. Thirdly, investors tend to redeem their mutual fund shares in response to negative performance. Empirical evidence underlines an existence of a positive flow-performance relationship [11,12], which implies that to generate sufficient cash to finance these outflows, mutual funds may have to sell additional assets to the declining market [9]. As a result, the sudden increases in sale orders may create an unexpected illiquidity problem for the sold assets. Stock returns are negatively related to unexpected contemporaneous illiquidity, called the price impact [13]. Therefore, the fire sale spillover may occur, and the depressing stock price may create another round of shock to the stock market, resulting in a downward spiral in stock prices.

Despite China having grown to be one of the major economies in the world, the Chinese stock market has experienced roller coaster dynamics [14]. There were two crashes in 2008 and 2015, respectively. In 2008, the stock market wiped out more than two-thirds of its market value. In 2015, the major index (Shanghai Stock Exchange Composite Index) decreased by more than 70% [14]. The market instability has attracted increasing attention from both practitioners and researchers. Some systemic risk measures have been proposed to identify systemically important financial institutions (SIFIs), such as Delta Conditional Value-at-Risk ($\Delta CoVaR$) [15], systemic risk index (SRISK) [16], marginal expected shortfall [17], and component expected shortfall [18], etc. However, these systemic risk measures ignore the interdependence of the FIs. To capture the connectedness among the FIs in risk modeling, in this paper, we build a dynamic model with a network approach to simulate the downward spiral phenomenon in the stock market to identify the systemic risks of Chinese financial stocks.

Before we move on, it is essential to clarify some notions about risk. The systemic risk in this study comprises two components: the direct risk due to the risk spillover effect through the stock network and the indirect risk associated with the fire sale of mutual funds, measuring how much additional risk will be amplified due to the common asset holding. We focus on financial stocks due to their importance to the economy. The key methods can be summarized as follows. Firstly, our model is based on a two-layer network. More specifically, we build a return spillover stock network to model the direct risk and a fund-stock bipartite network to model the indirect risk due to the fire sale of mutual funds. Secondly, we use a simulation approach to modeling the downward spiral phenomenon. We extend the framework of [19] by incorporating two important factors into the model: the flow-performance relationship of mutual funds and the price impact on the stocks due to the illiquidity problem. Finally, we quantify the systemic risk (including both direct and indirect risk) with the relative systemic loss of market capitalizations induced by each financial stock and rank the stocks based on their contributions to systemic risks. Our analysis consists of three scenarios based on different parameter choices, including the flow-performance relationship and liquidity (see Section 5.4).

Our main finding is that indirect risk due to the fire sale of a mutual fund is moderate in the base scenario according to our proposed risk propagation mechanism. In the base scenario, a 10% shock to a stock value causes an average direct risk of 3.80% in equity loss, and the average indirect risk is 0.46% only, which is 14% of the average direct risk. Our finding is similar to Ref. [9], who find that the indirect risk from fire sales in the US equity market is modest, i.e., a 5% shock on asset values wipes out 1.3% of total equity. However, the value for the indirect loss for the banking sector is much larger. For example [6], find a 50% write-off on all GIIPS1 debt results in a direct loss of 111% of equity, but the indirect loss is much greater as 302% of equity. Reference [4] perform a stress test on the Mexican banking system and find that the average indirect risk to the direct risk is 193%±97% from 2008 to 2013. On the other hand, for a more sensitive flow-performance relationship scenario, we find the indirect risk is increased to 42% of the direct risk. More importantly, for the most illiquid scenario, our results show the indirect risk can be boosted to 160% of the direct risk, which suggests indirect risk can be substantial in a low liquidity environment, and liquidity has a more significant impact on systemic risk than the flow-performance relationship.

Secondly, we explore the influence of each financial sector measured by their average induced systemic risk. The more influential sectors are insurance companies, regional banks, joint-stock banks, and capital servicing companies. In contrast, state-owned banks and diversified financial servicing companies are significantly less influential than other sectors. Thirdly, our results show that a financial stock is more influential if it has greater market liquidity, a greater fund holding ratio, and larger size.

We make two contributions to systemic risk literature. Firstly, unlike many recent studies on identifying influential stocks that only focus on their direct contagion risk [[20], [21], [22], [23]], our model makes an important contribution to the systemic-wide stress tests by considering both the direct and indirect risk. The latter is especially important when market liquidity is under pressure. Our model can serve as the starting point for analyzing the systemic risk of financial stocks and the amplification mechanism of mutual funds. Secondly, different from the existing literature on fire sale spillovers that focus on the FIs, such as banks [6] or asset managers [9,10], we focus on the influences of the assets (stocks). Our study illustrates how an initial shock on a specific financial stock can lead to a more significant systemic loss due to its direct spillover effect and the indirect amplification effect of the overlapping portfolio.

This paper is structured in the following. Section 2 provides the literature review. Section 3 discusses the data, followed by the methodology in Section 4. Section 5 presents the research results and findings, while Section 6 concludes the paper.

2. Literature

There are two main strands of literature for systemic risk. A first strand of literature aims to derive global measures of systemic risk, such as $\Delta CoVaR$, SRISK, MES, etc. These measures generally use market data to estimate systemic risk. One of the advantages of using market data is that it allows these methods to quickly capture the drifts of the risks. However, these methods tend to ignore the sources of risk and the mechanism of risk propagation. The other strand of literature looks at specific sources of systemic risk, such as contagion and amplification. A network model is generally involved for analysis in this line of research. Our study is related to the later strand, so we limit our literature review to the network approach for risk contagion and amplification. Specifically, we focus on two aspects: (1) application and development of spillover network, and (2) multilayer network analysis. More comprehensive review can refer to some recent surveys, see Refs. [24,25].

The first aspect of literature is the application and development of spillover network. The correlation network is one of the early methods to model the stock network among companies. This method is based on specific correlation indicators using market data. The method is proposed by Ref. [26] to analyze the similarity of stock behaviors among the stocks in the Dow Jones Industrial Average. It has become a frequently used method in literature [[27], [28], [29]]. The major limitation of this method is that it can only measure the correlated information but cannot describe the direction of contagion. To overcome this limitation, several network methods are developed to uncover contagion sources: the return spillover network, the volatility spillover network, and the extreme risk spillover network. These methods are based on different information spillovers, i.e., return, volatility, or extreme risk. Their common characteristic is that they are all based on market data that are available in real-time and at high frequency.

Return spillover networks, also called the Granger-causality networks, measure the connectedness among financial entities based on returns (c.f. [[30], [31], [32], [33], [34]]). For example, Ref. [35] develop a return spillover network using Granger causality in vector auto-regressive (VAR) models. They find that financial institutions with a larger tail risk, higher return on equity, lower turnover rate, and lower assets growth rate are associated with greater systemic importance. A recent work by Ref. [36] uses Granger causality networks to investigate the interconnectedness and systemic risk of the Chinese FIs, and their method is able to identify the financial crises and adverse financial events.

Volatility spillover networks, such as the variance decomposition frame-based network [37], and the GARCH model-based network [[38], [39], [40]], assume that a large shock to a stock will not only increase the volatility of its asset return but also increase the volatility of others. Thus, these networks model the connectedness among stocks based on price volatility. With the volatility spillover method, Ref. [41] find that, when considering the risk contagion level, the securities sector plays a risk-leading role, followed by the banking and insurance sectors. Reference [39] find that FIs with larger sizes and higher asset growth rates tend to be associated with greater systemic importance.

Extreme risk spillover networks, such as the tail event driven network (TENET) [42], quantify the connectedness among stocks based on the higher moment of asset returns and reflect the tail risk spillover in the extreme market movements. For example, Ref. [43] use an extreme risk spillover network to study the risk of the FIs in the US market. The TENET method is applied to study the extreme risk spillover effects of the Chinese FIs [44,45]. Reference [44] find large banks and insurers usually exhibit systemic importance, but some small firms are systemically important due to their high level of incoming connectedness. Similar results are nuanced by Ref. [45], who find that SIFIs are concentrated in the banking and insurance industries. A recent study by Ref. [46] adopts the TENET method to explore the systemic risk of the banking industry along the Belt and Road, and find the intra-regional tail risk spillovers are remarkably stronger than the inter-regional tail risk spillovers.

The limitation of a single layer network is that it may not be able to reflect the comprehensive risk characteristics of FIs. Thus, the other aspect of literature is related to multilayer network analysis for systemic risk. Some recent works construct multilayer networks by combining different layers to capture different types of risks. For instance Ref. [47], construct multilayer information spillover networks, including return spillover layer, volatility spillover layer and extreme risk spillover layer, to investigate the risk spillovers among 30 Chinese FIs. Reference [48] construct three single-layer networks for stock market based on Spearman correlation coefficient, grey relational analysis and maximum information coefficient. Reference [49] construct the multilayer network combining the Pearson correlation network, Granger causality network and nonlinear relation network. Reference [50] construct the multilayer network by considering liability and cross-holding of shares between FIs. Reference [51] construct the multilayer network using Granger causality by considering the stocks layer and their respective convertible bonds layer. Reference [52] use stock return layer and sentiment layer to study the differences between investor sentiment connectedness and stock return connectedness.

However, these studies, on one hand, tend to focus on the direct spillover risks and ignore the indirect risks, i.e., fire sale. On the other hand, they do not consider the potential effects from the interactions between the layers.

The former limitation is addressed in some recent papers. For example, Ref. [53] study two contagion channels in the Australian banking sector by combining empirical data on direct lending with a stylized model on overlapping portfolios. Reference [4] study the Mexican banking system by assessing its systemic risk consisting of direct and indirect risks measured by the DebtRank method. Reference [6] analyze the networks of common asset exposures in the EU and the US with aggregate data on asset classes. Reference [9] extend the [6] model to the US equity market to quantify the indirect contagion risk due to the fire sale of overlapping portfolios. A similar study is made by Ref. [10] on the fire sale spillovers in the debt markets.

The later limitation is addressed in Ref. [19], which develop a risk transfer framework between a two-layer network to model both the direct and indirect risk of the Chinese stock market. The merit of [19] is building a framework that simulates the interactions between the stock market and mutual funds market despite its model lacking some important factors, such as the follow-performance relationship of mutual funds and the price impact.

Inspired by these studies, this paper proposes a simulation approach to quantify the systemic risks of the Chinese financial stocks considering both the direct and indirect risks. We choose the mean spillover approach to construct the risk spillover network since existing works (c.f. [30,35]) find it is able to build the lead-lag relationships among stocks that are important to measure systemic risk. In addition, we build a two-layer network to allow interactions between layers for risk transfers between the stock market and fund market. Our model also overcomes some shortcomings of [19], i.e., flow-performance relationship and price impact.

3. Data

Our dataset is retrieved from Wind Economics Database. Firstly, we use the portfolio holding of mutual funds at the end of 2020 to construct a fund-stock bipartite network. Secondly, we use 1-min high-frequency trading data for the fourth quarter of 2020 to build the return spillover stock networks. The advantage of using high-frequency data is that it can capture the highly dynamic nature of the stock market. Each stock network is built based on the data length of 5 trading days (1200 trading minutes), and we employ a rolling window technique to build 56 stock networks for the quarter (60 trading days). Thirdly, we collect the daily trading data, including the daily return and trading volume for the year's second half, for modeling the stock pair-wise correlation and illiquidity ratio. The illiquidity ratio is calculated based on five trading days data, corresponding to the stock network. The correlation for each pair of stocks is calculated based on their daily return for the previous 60 trading days (approximately a quarter).

There are 116 financial stocks as of 2020 based on the sectoral classification of Wind. Four stocks were newly listed during the second half of 2020, i.e., Xiamen Bank, Guolian Securities and China International Capital Corporation and Zhongtai Securities. They are excluded given they are at the price discovery stage and lack some trading data. As a result, there are 112 financial stocks in this study, which can be classified into six sectors, according to Wind. The state-owned banking sector includes six banks, the joint-stock sector includes nine banks, and the regional banking sector includes 21 banks. The capital market servicing sector (hereafter, Capitals) includes 59 companies, mainly securities companies, trust companies, or futures trading companies. The insurance sector (hereafter, Insurance) includes six companies, while the diversified financial servicing sector (hereafter, Diversified) includes the remaining 11 companies. Table 1 shows the descriptive statistics of our sample.

Table 1.

Descriptive data by sectors.

	Total assets (CNY)	Total equities (CNY)	MC (CNY)	FHR (%)
Regional bank
N	21	21	21	21
Mean	8.68E+11	6.57E+10	4.95E+10	1.13
Std	8.39E+11	6.41E+10	4.91E+10	2.88
Min	1.39E+11	1.13E+10	8.94E+09	0.00
Max	2.90E+12	2.21E+11	2.12E+11	12.77
Joint-stock bank
N	9	9	9	9
Mean	5.99E+12	4.82E+11	3.42E+11	2.05
Std	2.27E+12	1.92E+11	3.09E+11	2.47
Min	2.05E+12	1.33E+11	8.68E+10	0.00
Max	8.36E+12	7.30E+11	1.11E+12	6.28
State-owned bank
N	6	6	6	6
Mean	2.25E+13	1.87E+12	1.02E+12	0.39
Std	9.37E+12	8.91E+11	5.88E+11	0.37
Min	1.07E+13	6.73E+11	3.33E+11	0.02
Max	3.33E+13	2.91E+12	1.78E+12	1.09
Capitals
N	59	59	59	59
Mean	1.77E+11	3.99E+10	6.65E+10	0.88
Std	2.39E+11	4.46E+10	7.59E+10	2.33
Min	3.36E+09	1.15E+08	2.45E+09	0.00
Max	1.05E+12	1.86E+11	3.80E+11	14.33
Insurance
N	6	6	6	6
Mean	3.01E+12	3.46E+11	5.90E+11	2.15
Std	3.47E+12	3.47E+11	6.12E+11	2.71
Min	2.39E+11	3.46E+10	2.41E+10	0.01
Max	9.53E+12	9.88E+11	1.59E+12	6.72
Diversified
N	11	11	11	11
Mean	7.02E+10	1.62E+10	1.87E+10	0.02
Std	8.50E+10	1.60E+10	1.97E+10	0.08
Min	7.87E+08	6.25E+08	7.79E+08	0.00
Max	2.50E+11	4.16E+10	6.67E+10	0.27

Note: This table reports the descriptive statistics for six financial sectors in China, including total assets, total equities, market capitalization, and the fund holding ratio. The sample includes 21 regional banks, nine joint-stock banks, six state-owned banks, 59 capital servicing companies (Capitals), six insurances companies (Insurance), and 11 diversified capital servicing companies (Diversified). CNY is short for Chinese yuan. FHR: fund holding ratio. $FH{R}_i\left(t\right)={\sum }_{f}h{v}_{fi}\left(t\right)/M{C}_i\left(t\right)$ where $h{v}_{fi}\left(t\right)$ is the holding value of fund $f$ on stock $i$ at time t. $M{C}_i$: Market capitalization of stock $i$ at the end of 2020. The number of stocks, the mean, standard deviation, minimum, and maximum are reported.

4. Methodology

In this part, we present our methodology. Firstly, we construct a return spillover network among the financial stocks using Granger causality in a VAR form, described in Section 4.1. Then, we use the mutual funds’ portfolio holding data to build a fund-stock bipartite network in Section 4.2. Section 4.3 describes the main framework of the simulation mechanism through the two-layer network. Fourthly, the risk contagion mechanism through the VAR network is described in Section 4.4. Lastly, we define the measure for systemic risk in Section 4.5.

4.1. The return spillover network (“VAR network”)

The continuous return of stock $i$ at time $t$ is represented as $r_i\left(t\right)$, defined in Eq. (1) as:

$$r_i\left(t\right)=\ln \left(\frac{p_i\left(t\right)}{p_i\left(t-1\right)}\right)\text{,}$$ (1)

where $p_i\left(t\right)$ represents the closing price of stock $i$ at time $t\text{.}$ We follow [35,54] to express the VAR model in the following form as shown in Eq. (2):

$$r\left(t\right)=\mathrm{\Phi }\left(0\right)+\mathrm{\Phi }\left(1\right)r\left(1\right)+\cdots +\mathrm{\Phi }\left(k\right)r\left(t-k\right)+\epsilon \left(t\right)\text{,}$$ (2)

where $r\left(t\right)=\left(r_i\left(t\right),r_j\left(t\right)\right)$ is a two-dimensional column vector of return series under consideration, $\mathrm{\Phi }\left(0\right)$ is a two-dimensional vector, $\mathrm{\Phi }\left(i\right)\left(i=\mathrm{1,2},\cdots ,k\right)$ is a $2\times 2$ matrix that measures the dynamic dependence of $r\left(t\right)$, and $\epsilon \left(t\right)$ is a sequence of serially uncorrelated random vector with mean zero in a normal distribution. Since the stock market is one of the most dynamic trading markets where prices are adjusted quickly, we use the 1-min-high frequent trading data to examine causal relationships among stocks. The following VAR system in Eqs. (3), (4) is utilized to conduct the Granger-causality test:

$$r_i\left(t\right)={\phi }_i+\sum _{l=1}^k{\phi }_{ii}\left(l\right)r_i\left(t-l\right)+\sum _{l=1}^k{\phi }_{ij}\left(l\right)r_j\left(t-l\right)+{\epsilon }_i\left(t\right)\text{,}$$ (3)

and

$$r_j\left(t\right)={\phi }_j+\sum _{l=1}^k{\phi }_{jj}\left(l\right)r_j\left(t-l\right)+\sum _{l=1}^k{\phi }_{ji}\left(l\right)r_i\left(t-l\right)+{\epsilon }_j\left(t\right)\text{,}$$ (4)

where ${\phi }_{ij}\left(l\right)$ is the $\left(i,j\right)$ th element of $\mathrm{\Phi }\left(l\right)\left(l=\mathrm{1,2},\cdots ,k\right)$. The coefficients ${\phi }_{ii}\left(l\right)$ and ${\phi }_{jj}\left(l\right)$ describe the lead-lag relationship between the respective stocks' own returns, while the coefficients ${\phi }_{ij}\left(l\right)$ and ${\phi }_{ji}\left(l\right)$ quantify the Granger-causality between the respective stocks $i$ and $j$. Since we are modeling highly frequent dynamic activities (i.e., 1-min trading activity), so we only consider the significant (at 1% level) lead-lag relationships in 1 lag. As the stock market is highly efficient with prices quickly adjusting, using long lags may not accurately reflect this characteristic. If stock $i$ and $j$ reinforce each other, for simplicity, we only keep the higher coefficient, which implies a stronger impact from one stock to the other.

Denote $E$ as the set of edges of a stock network $G$, and $V$ as the set of nodes (i.e., stocks). The stock network can be described as $G=\left(V,E\right)$. A stock network containing $n$ nodes can be represented as a $n\times n$ matrix of $c_{i,j}$, where $c_{i,j}=1$ indicates node $i$ can cause node $j$, and $c_{i,j}=0$ indicates no causal relation from node $i$ to node $j$. Therefore, the matrix of the stock network is a binary matrix, and the diagonal of the matrix is 0 since we do not consider autocorrelations. The network can be represented in the following form as shown in Eq. (5):

$$G=\left(V,E\right)=\left[\begin{array}{cccc}0& c_{12}& \cdots & c_{1n}\\ c_{21}& 0& \cdots & c_{2n}\\ ⋮& ⋮& 0& ⋮\\ c_{n1}& \cdots & \cdots & 0\end{array}\right]\text{,}$$ (5)

where

$$c_{i,j}=\{\begin{array}{cc}1& \mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}i\mathrm{t}\mathrm{o}j\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\text{.}$$

We use a rolling window of 5 trading days (1200 trading minutes) to build the VAR networks for our analysis. Since our sample contains 60 trading days in the fourth quarter of 2020, we end up with 56 VAR networks. An illustration of VAR network can be referred to Panel (A) of Fig. 1.

Fig. 1.

Illustration of networks, a VAR network in Panel (A), a fund-stock bipartite network in Panel (B), and a two-layer network in Panel (C) combining a mutual fund layer and a stock layer.

4.2. The fund-stock bipartite network (“the F–S network”)

We use a bipartite network to represent holding relationships from mutual funds to stocks. Let $f$ denote a mutual fund and $F$ as the set of mutual funds, where $f\in F$. In the fund-stock network, we use $h$ to represent the holding relation from a fund to a stock. If fund $f$ holds stock $i$ (i.e., $h_{f,i}=1$), it implies a directional link from node $f$ (in the mutual fund layer) to node $i$ (in the stock layer). If $h_{f,i}=0$, there is no holding relation. Let $W$ represent a set of the holding values for the edges, i.e., $w_{f,i}$ represents the holding value of fund $f$ on stock $i$, where $w_{f,i}\in W$. Therefore, we can use a holding matrix with $m$ funds and $n$ stocks to represent the bipartite network $B$, as shown in Eq. (6):

$$B=\left[\begin{array}{ccc}{h}_{11}& \cdots & h_{1n}\\ ⋮& \ddots & ⋮\\ h_{m1}& \cdots & h_{mn}\end{array}\right]\text{,}$$ (6)

where

$$h_{f,i}=\{\begin{array}{cc}1& \mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}f\mathrm{t}\mathrm{o}i\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\text{,}$$

and $W$ can be represented as in Eq. (7):

$$W=\left[\begin{array}{ccc}{w}_{11}& \cdots & w_{1n}\\ ⋮& \ddots & ⋮\\ w_{m1}& \cdots & w_{mn}\end{array}\right]\text{.}$$ (7)

An illustration of network structure is depicted in Fig. 1, including a VAR network in Panel (A), a F–S network in Panel (B) and a two-layer network in Panel (C).

4.3. Simulation mechanism through the two-layer network

The simulation mechanism is the key to this study. Our model is based on the framework of [19] that follows a recursive, iterative flow to simulate the risk contagion between the two layers. In their model, the authors assume that an entire fund will be liquidated in case of redemption. Nevertheless, empirical evidence shows that investors tend to redeem partial, not entire, funds in response to negative market performance ([11,55]). Therefore [19], may overestimate the price impacts of stock redemption.

For the Chinese financial market, we start by calibrating the model by allowing the funds to sell a fraction ($\tau$) of its portfolio when facing redemption pressures due to a decrease in the portfolio value ($\vartheta$). Then, we incorporate the flow-performance relation ($\gamma$) of mutual fund to determine the amount of portfolios to be sold to meet the redemption needs. In the third step, we consider the liquidity factor of each stock to determine its price impact on the market. In summary, the key of the two-layer network structure is that it allows interactions between the stock layer and the fund layer. On the one hand, the decreases of stock prices cause funds' values to drop. On the other hand, funds’ actions on fire sales trigger the stocks prices to decrease further. Hence, it creates a phenomenon of downward spiral of prices. The flowchart is depicted in Fig. 2, described in the following steps.

• (1)The simulation begins with an initial shock to stock $i$ that causes its price to change by ${\kappa }_i$. $\kappa$ represents an initial shock to the stock price in percentage, with $\kappa \in \left[-1,0\right]\text{.}$ For instance, ${\kappa }_i=-0.1$ indicates an initial price drop of 10% of stock $i$.
• (2)The initial shock to stock $i$ will spill over to other stocks through the VAR network. The contagion mechanism comprises two key components, the return spillover mechanism and the spreading route. The former determines the magnitude of the impact from one stock to another, whereas the latter describes how the contagion propagates through the VAR network. The detailed mechanism is described in Section 4.4.
• (3)After the contagion process in the VAR network is completed, we calculate ($\vartheta$) of the portfolio based on the F–S network. Here $\vartheta$ denotes the change of portfolio value of the mutual fund in percentage, calculated as ${\vartheta }_f\left(t\right)=\frac{v_f\left(t\right)-v_f\left(t-1\right)}{v_f\left(t-1\right)}$, where ${\vartheta }_f\left(t\right)$ is the change of portfolio value for fund $f$ from time $t-1$ to $t$, and $v_f\left(t\right)$ is the portfolio value for fund $f$ at time $t$. Let $\epsilon$ be a threshold controlling the termination of the recursive iteration algorithm. Here we choose a small number, $\epsilon ={10}^{-5}$ for the minimal change in portfolio value, that is, when the change in the portfolio value ($\vartheta$) is smaller than this threshold ($\epsilon$), i.e., $\left|\vartheta \right|<\epsilon$, the risk contagion process is terminated. If $\left|\vartheta \right|>\epsilon$, the mutual funds are forced to sell a fraction ($\tau$) of their portfolio holdings to meet the redemption needs.
• (4)At this step, we discuss how the fraction of portfolio ($\tau$) is determined. The existence of a flow-performance relationship in the mutual fund literature has become a “stylized fact” - there is a positive relationship between fund's past performance and their future net flow (see Refs. [12,[55], [56], [57], [58], [59], [60], [61]]). The estimation equation for the relationship is in Eq. (8):

$$Flow{s}_f\left(t\right)=a+\gamma \cdot r_f\left(t-1\right)+{\epsilon }_f\left(t\right)\text{,}$$ (8)

where $Flow{s}_f\left(t\right)$ is the net inflows of fund $f$ at period $t$, and $r_f\left(t-1\right)$ is the return of fund $f$ in the last period $t-1$, while $\gamma$ is the fund's specific flow-performance sensitivity parameter. Therefore, the fraction of portfolio to be sold is deemed as the outflow of the fund, and its value can be calculated based on the product of the flow-performance parameter ($\gamma$) and the return (decrease) of the fund's value ($\vartheta$), e.g., ${\tau }_f\left(t\right)=\gamma \cdot {\vartheta }_f\left(t-1\right)$.

Figure 2.

Flowchart of risk contagion simulation. This flowchart illustrates the mechanism of risk contagion through a two-layer network. $\kappa$: an initial shock to a stock price in percentage $\kappa \in \left[-1,0\right]$. $\gamma$: the parameter of flow-performance relation of a mutual fund. $\tau$: a fraction of portfolio to be sold where $\tau \in \left[0,1\right]$. $\vartheta$: the percentage change of portfolio value of mutual fund where $\vartheta \in \left[-1,0\right]$. $\epsilon$: a threshold to exit the iteration algorithm. The simulation of the risk contagion processes starts with some variable settings, then passes an initial shock $\kappa$ on a stock's price (step 1). The shocked stock would spread its contagion risk to other stocks through the VAR network, and we calculate the new prices of the affected stocks in the network (step 2). The decreases in stock prices cause the values of relevant mutual funds to decrease by $\vartheta$ (step 3). In step 4, the mutual funds are forced to sell a fraction ($\tau$) of their portfolios to meet the redemption pressures, where $\tau =\gamma \bullet \vartheta$. In step 5, we calculate the price impact on stocks in response to the selling of stocks based on the Amihud $ILLIQ\text{.}$ The price impacts of the affected stocks would become the second round of stocks in the system, so the simulation continues to iterate. The simulation would be terminated when the decreases in the portfolio values are very minimal, e.g., $\left|\vartheta \right|<\left|\epsilon \right|$.

Reference [55] study the Chinese equity market and find that, during a stable period,2 the average value for $\gamma$ is 0.1404 and 0.4280 for the growth period. The value for $\gamma$ in other markets does not show large variations. For instance Ref. [9], find the value of $\gamma$ to be 0.2748 in the US market. Reference [12] analyze the flow-performance relationships across 28 countries. They find the average value of $\gamma$ for low-performing funds to be 0.086 and the high-performing funds to be 0.362 [12]. Therefore, we follow the finding of [55] and set $\gamma =0.1404$ as the base case and $\gamma =0.4280$ for a more sensitive scenario of the flow-performance relationship. In addition, in line with many simulation studies in the literature (c.f. [3,6,9,62,63]), we assume the mutual funds sell all the composites in their portfolio on a pro-rata basis.

(5) The sales of stocks may create an illiquidity problem and depress the stock prices in the short run. According to Ref. [13], illiquidity reflects the impact of order flow on price, which is the discount that a seller concedes or a premium a buyer pays when executing a market order. The relation between the order flow or transaction volume and price change is known as the price impact. Here we follow the Amihud illiquidity ratio to quantify the price impact (see Refs. [9,13]). The Amihud illiquidity ratio is the average across stocks of the daily ratio of absolute stock return to dollar volume [13]. It can be interpreted as the daily price response associated with one dollar of the trading volume. The illiquidity ratio for stock $i$, $ILLI{Q}_i$, is defined as in Eq. (9):

$$ILLI{Q}_i=\frac{1}{N}\sum _t^N\frac{\left|r_i\left(t\right)\right|}{vo{l}_i\left(t\right)}\text{,}$$ (9)

where $\left|r_i\left(t\right)\right|$ is the absolute daily return of stock $i$ on day $t$, $vo{l}_i\left(t\right)$ represents the trading volume in dollar amount for stock $i$ on day $t$, and $N$ is the number of trading days in the period. To be consistent with the data length in constructing a VAR network, here we set $N=5$ to measure $ILLIQ$ for each stock and use the rolling window technique to calculate different $ILLIQ$ s for the corresponding VAR networks. Once calculated, the price impacts on the affected stocks become the sources of shocks for the next round of risk contagion. So that simulation process goes to Step 2. The simulation keeps iterating until the impacts are below the threshold $\epsilon$ for ${\forall }_f\in F$. After the simulation process is terminated, we quantify its induced systemic risk, as defined in Section 4.5.

4.4. Mechanism for risk contagion in the VAR network

The directional VAR network makes it possible to determine the risk spillover and to trace the risk contagion route alone with the directional edges. We follow [19] to use the Pearson correlation ($\rho$) of returns between two stocks to determine the magnitude of return spillover. Given the return of stock $j$ at time t, denoted as $r_j\left(t\right)$, its estimated return of stock $i$, ${\hat{r}}_i\left(t\right)$, can be determined with the correlation between two stocks, i.e., ${\hat{r}}_i\left(t\right)={\rho }_{i,j}\left(t\right)\cdot r_j\left(t\right)$. We then follow [21] for the mechanism of the contagion route. Firstly, we use the breadth-first-search method to obtain the contagion route and model daily returns. This implies that we assume the spillover can only pass once to each stock for every round of contagion within a VAR network. However, if the market encounters another round of contagion, e.g., due to the price impact, then spillover will be transmitted for another round of shock. Secondly, if a stock is concurrently affected by several stocks, we take the maximum magnitude of the spillover effects, i.e., $r_i\left(t\right)=\max \left({\rho }_{i,j}\left(t\right)\times r_j\left(t\right),j\in \mathbb{M}\right)$, where $\mathbb{M}$ is a set of stocks directed to stock $i$.

Fig. 3 illustrates the mechanism of the risk contagion route. For instance, as shown in Step 1, if stock $i$ undergoes an initial shock, its risk simultaneously spreads to stock $j$ and $u$. In Step 2, stock $g$ receives the spillovers by stock $j$ and $u$ concurrently, then the magnitude of the return spillover on stock $g$ would be based on the average spillover of stock $j$ and $u$. In Step 3, stock $g$ spreads its risk simultaneously to stock $m$ and $n$. In Step 4, stock $m$ spreads its risk to stock $v$. It is worth noticing here that although there is a direct route from stock $m$ to $n$, stock $n$ is not affected by stock $m$ at this step because $n$ is already affected by stock $g$ at the previous step, since we assume each stock can be only influenced once at the daily level.

Fig. 3.

Risk contagion spreading mechanism of a VAR network. The risk contagion starts with an initial shock on stock $i$, then its risk spreads to stock $j$ and $u$ simultaneously, shown in Step 1. Both stock $j$ and $u$ spread their influences simultaneously to stock $g$, and the influence received by stock $g$ is calculated based on the maximum influences it received, shown in Step 2. In Step 3, stock $g$ spreads its influence on stock $m$ and $n$ simultaneously. At Step 4, stock $m$ spread its influence on stock $v$ only, but not to stock $n$ based on the assumption that all stocks only receive influence once for each round of propagation.

4.5. Measure of systemic risk

Based on the DebtRank method of measuring risk [2,64], we measure the systemic risk with the relative systemic loss induced by an initial shock on a stock. Specifically, systemic risk is defined as in Eq. (10):

$$S{R}_i=\frac{{\sum }_{j,j\ne i}^{S}M{C}_j\left(0\right)-{\sum }_{j,j\ne i}^{S}M{C}_j\left(t\right)}{{\sum }_{j,j\ne i}^{S}M{C}_j\left(0\right)}\text{,}$$ (10)

where ${SR}_i$ measures the relative loss of market capitalizations of the entire system as induced by an initial shock to stock $i$, excluding the initial shortfall of the causing stock itself (e.g., stock $i$). ${\sum }_{j,j\ne i}^{S}M{C}_j\left(t\right)$ is denoted as the aggregate market capitalizations for stocks in the system ($j\in S$) excluding the initial shocked stock $i$. The larger the $SR$ is, the greater systemic loss of the stock induced to the system so the greater influence of the stock. In addition, we break down the $SR$ into two components, the direct risk component, which measures the spillover risk due through the VAR network, and the indirect risk component, which measures the risk associated with the fire sales, showing the amplification effect due to the mutual funds' overlapping portfolios. Thus the ${SR}_i$ can be rewritten into Eq. (11):

$$S{R}_i=\underset{\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}}{\underbrace{\frac{{\sum }_{j,j\ne i}^{S}M{C}_j\left(0\right)-{\sum }_{j,j\ne i}^{S}M{C}_j\left(1\right)}{{\sum }_{j,j\ne i}^{S}M{C}_j\left(0\right)}}}-\underset{\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}}{\underbrace{\frac{{\sum }_{j,j\ne i}^{S}M{C}_j\left(1\right)-{\sum }_{j,j\ne i}^{S}M{C}_j\left(t\right)}{{\sum }_{j,j\ne i}^{S}M{C}_j\left(0\right)}}}\text{.}$$ (11)

So systemic risk is the sum of direct risk and indirect risk. The former is measured by the relative systemic loss between the initial values (time = 0) and their value after the spillover in the VAR network (time = 1). The latter is measured by the relative systemic loss between the values before the fire sales (time = 1) and the final values after the contagion terminated (time = $t$).

5. Results

5.1. The VAR networks

Before identifying the influential financial stocks, it is important to test the validity of the network. Empirical evidence shows that stocks may have a higher correlation when the stock market suffers from fluctuations [40]. To this end, we calculate the densities of the VAR networks and compare them with the mean value of the Financials Index (Wind tick. 399240) in the same sliding windows used to build the VAR networks. As shown in Fig. 4, the density of the networks has the same trend in their rise and fall as the mean value of the Financials Index, consistent with [40,65]. The Pearson correlation test finds the change of densities of the VAR networks is significantly correlated with the return of the Financial Index ($\rho =0.348,p=0.009$), which suggests that when the stock market is more turbulent, the financial stocks will have more spillovers from each other.

Fig. 4.

Movement of the financial index and the network density. This figure shows the movement of the mean value of the Financial Index (Wind ticker 399240, the red dashed line with the left y-axis) in the same sliding windows and the density of the VAR networks (the solid blue line with the right y-axis). It demonstrates a similar trend of movement in their rises and falls. The correlation for their changes is 0.348 at 1% level of significance. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

5.2. The F–S network

Based on the mutual funds' portfolio holding data as of 2020, the F–S network consists of 112 stocks, 3233 funds, and 8312 edges in total. The stocks' degrees, measured by how many funds hold a position in the stock, follow a power law distribution as shown in Fig. 5, which means few nodes have large number of degrees while most of nodes have fewer degrees. It indicates that some stocks are in many funds' portfolios, yet some stocks are held by none of the funds. For instance, Ping An Insurance Group has the largest number of degrees as it appears in 1817 funds’ portfolios.

Fig. 5.

Distribution of stocks' degrees in the F–S bipartite network. The power law probability distributions follow the form $p\left(x\right)\propto x^{-\alpha }$, where $x$ is the observed number of the holding funds in this case, and $\alpha$ is the scaling parameter. The figure shows a power law distribution with $\alpha =1.71$ , which indicates some stocks are in many funds' portfolios, but some stocks are none.

The degree of overlapping is important for the indirect risk (see Refs. [3,4,9]). The degree of stock in the F–S network shows the popularity of the stock in the funds' portfolios. Nevertheless, it does not consider the size of the holdings. As an alternative, we calculate fund holding ratios to measure each stock's concentration of shares by mutual funds to evaluate the impact of fund holding on stocks [4]. The fund holding ratio is a value-weighted measure. Similar to Ref. [4], we define stock $i$’s fund holding ratio at time t, $FH{R}_i\left(t\right)$, as a ratio of all the funds' holding values in the stock $i$ at time t, $h{v}_{fi}\left(t\right)$, to the stock $i$’s total market capitalization at the same time t, $M{C}_{i,t}$, as shown in Eq. (12) below:

$$FH{R}_i\left(t\right)=\frac{{\sum }_{f}h{v}_{fi}\left(t\right)}{M{C}_i\left(t\right)}\text{.}$$ (12)

Table 2 shows the ranking orders for the fund holding ratio and stock's degree. The ranking orders by FHR are Insurance, Joint-stock, Regional, Capitals, State-owned, and Diversified. However, regarding stock's degree, the ranking order is Insurance, joint-stock bank, a state-owned bank, regional bank, Capitals and Diversified. The effect of FHR on indirect risk is discussed in Section 5.4.2 regarding the result of indirect risk.

Table 2.

Sectoral fund holding data.

Sector	Fund holding ratio		Fund holding degree
	Ranking	%	Ranking	Degree
Insurance	1	2.15	1	393
Joint-stock bank	2	2.05	2	320
Regional bank	3	1.13	4	37
Capitals	4	0.88	5	28
State-owned bank	5	0.39	3	104
Diversified	6	0.02	6	1

Note: This table reports the average fund holding ratio and fund holding degree by sector. The ranking in terms of ratio and degree is different, particularly for the state-owned bank. The comparison shows although the state-owned bank sector has an average of 104° and ranks No. 3, in terms of ratio, it has only 0.39% on average and ranks No. 5.

5.3. Liquidity

Illiquidity can amplify contagion risk in crises [7]. When the market's demand for illiquid assets is less than perfectly elastic, sales depress the market prices of such assets. Marking to market the asset book can induce a further round of endogenously generated sales of assets, depressing prices further and inducing further sales [66]. Therefore, we conduct a scenario analysis for illiquidity on systemic risk for robustness. We calculate the average daily $ILLIQs$ for 2020 to assess its variation, as depicted in Fig. 6. An outlier on Feb 3, 2020, in Panel A, was the first trading day after the outbreak of Covid-19. The pandemic caused the market index to drop by 9.47% that day. 79 out of 112 financial stocks reached the daily price bottom limit, and many selling orders could not be executed due to the lack of buying orders. For this most illiquid trading day, its average $ILLIQ$ was approximately 97 times higher than that for the most liquid day in the year. However, this is an extreme event. If we exclude such an event, the $ILLIQs$ between the most illiquid day and the most liquid day for the year still differ by approximately 12 times, as shown in Panel B. Therefore, we use 12 as a multiplier for the liquidity scenario analysis (Scenario 3 in Section 5.4).

Fig. 6.

Daily illiquidity ratio for 2020. Panel A shows an outlier, the most illiquid day dated Feb 3, 2020. It was the first trading day after the outbreak of Covid-19. The $ILLIQ$ for that day is 97.57 times higher than that for the most liquid day in the year. Panel B shows the $ILLIQs$ excluding the outlier, the $ILLIQs$ for the most illiquid day and the most liquid day differ by 12.08 times.

5.4. Simulation results

In the following, we assume an initial shock of −10% to each stock at a time and perform the simulations to obtain the $SR$ s induced by each financial stock. The direct risk is relevant to the VAR network structure. Based on the rolling window technique, we perform the simulation with 56 different VAR networks and draw a conclusion based on the average of 56 results. As mentioned, we differentiate three scenarios with regards to the choices of parameters of the flow-performance relationship and liquidity:

• -Scenario 1 (base case): the flow-performance relationship parameter is based on the finding of [55] for the stable period of the Chinses stock market, i.e., $\gamma =0.1404$.
• -Scenario 2 (flow-performance sensitive case): setting $\gamma =0.4280$, based on the average value for the flow-performance relationship of the growth period [55].
• -Scenario 3 (liquidity sensitive case): based on the base case, $\gamma =0.1404$, we increase the illiquidity parameter by 12 times, i.e., the price impact parameter × 12.

We discuss the results for the direct risk, indirect risk, and $SR$ separately under these three scenarios.

5.4.1. Direct risk

The direct risk in our model, as defined in Section 4.5, is the contagion risk in the VAR network that is not involved with the fire sale. As mentioned above, the parameter settings for the three scenarios are only relevant to the fire sale, which means the indirect risks vary in different scenarios, but the direct risk remains the same. According to the results of direct risks shown in Fig. 7, the Top 5 financial stocks are CITIC Securities (4.87%), Guotai Junan Securities (4.76%), Ping An Insurance (4.71%), East Money Information (4.53%), and Shanghai Pudong Development Bank (4.52%). The number for the direct risk indicates how much systemic loss is induced by a shock to each FI. For instance, a 10% decrease in the value of CITIC Securities will result in a 4.87% decrease in the aggregate market values of the entire financial sector (the system in our case). CITIC Securities and Guotai Junan Securities are the top-tier stockbrokers in China and are ranked first and fifth in market capitalization as of the end of 2020. Ping An Insurance is the largest insurance company in China, one of the Global Systematically Important Insurances (G-SIIs). East Money Information is a leading online based stockbroker in China. Shanghai Pudong Development Bank is one of the leading joint-stock banks among the top influential banks in the Chinese banking sector [44,67].

Fig. 7.

Ranking of financial stocks by direct risk. This figure ranks the stocks in terms of direct contagion risk based on the average result of 56 VAR networks. Their range is between 0.16% (Anxin Trust) and 4.87% (CITIC Securities), with a mean of 3.45%.

The average direct risks by sectors, as shown in Table 3, are as follows: Insurance (3.95%), joint-stock bank (3.84%), regional bank (3.72%), Capitals (3.53%), state-owned bank (2.76%) and Diversified (2.26%). The pair-wise t-tests in Table 4 do not show any significant difference in direct risk among the Insurance, joint-stock bank, regional bank, and Capitals. However, state-owned banks and Diversified sectors induce significantly lower direct risk than other sectors. Our finding is consistent with [47] in the senses that regional banks and joint-stock banks contribute more spillover shocks and are the risk emitters, while the state-owned banks undertake spillover shocks and are the risk receivers, despite the sizes of regional banks and joint-stock banks are smaller than state-owned banks [47]. Insurance and Capitals also have high direct risks due to the high sensitivity to the market [68].

Table 3.

Sectoral analysis of systemic risk.

Rank	Sector	SR (%)	Direct risk (%)	Indirect risk (%)	Indirect/Direct (%)
Panel A. Scenario 1
1	Insurance	4.48	3.95	0.53	13.51
2	Joint-stock bank	4.36	3.84	0.52	13.68
3	Regional bank	4.21	3.72	0.50	13.33
4	Capitals	4.03	3.53	0.50	14.11
5	State-owned bank	3.14	2.76	0.38	13.65
6	Diversified	2.59	2.26	0.32	14.17
	Average	3.80	3.34	0.46	13.74
Panel B. Scenario 2
1	Insurance	5.57	3.95	1.62	40.98
2	Joint-stock bank	5.43	3.84	1.59	41.47
3	Regional bank	5.22	3.72	1.50	40.45
4	Capitals	5.04	3.53	1.51	42.82
5	State-owned bank	3.91	2.76	1.15	41.45
6	Diversified	3.24	2.26	0.97	43.06
	Average	4.74	3.34	1.39	41.71
Panel C. Scenario 3
1	Insurance	10.15	3.95	6.21	157.16
2	Joint-stock bank	9.94	3.84	6.10	158.83
3	Regional bank	9.50	3.72	5.78	155.35
4	Capitals	9.34	3.53	5.81	164.46
5	State-owned bank	7.18	2.76	4.41	159.69
6	Diversified	6.03	2.26	3.77	166.40
	Average	8.69	3.34	5.34	160.32

Note: This table reports the systemic risk and its breakdown of direct and indirect risk by sectors in three scenarios. The average direct risk under the three scenarios is the same at 3.34%. The indirect risk for scenario 1 (in Panel A) is 0.46%, amplified by 13.74%, for scenario 2 (in Panel B) is 1.39%, amplified by 41.71%, for scenario 3 (in Panel C) is 5.34%, amplified by 160.32%. In terms of $SR$ by sectors, the ranking order remains consistent in three scenarios, whereas the Insurance ranks No. 1, followed by the joint-stock bank, the regional bank, Capitals, state-owned bank and Diversified.

Table 4.

Test for mean difference of direct risk by sector.

	Regional bank	Joint-stock bank	Capitals	Insurance	State-owned bank	Diversified
Regional bank	N/A
Joint-stock bank	0.464	N/A
Capitals	−1.294	−1.144	N/A
Insurance	0.963	0.334	1.674	N/A
State-owned bank	−3.938***	−3.240***	−3.038**	−3.747***	N/A
Diversified	−4.114***	−3.750***	−3.514***	−4.132***	−1.219	N/A

***: p < 1%, **: p < 5%, *: p < 10%.

Note: This table reports the statistics and corresponding p-values for the mean difference t-test of the direct risks among sectors. The results show that there is no significant difference between a regional bank, joint-stock bank, Capital and Insurance. However, the state-owned bank and Diversified are significantly lower than other sectors.

Furthermore, we calculate the time-varying direct risk based on the VAR networks and find the direct risk is correlated with the network density ($\rho =0.387,p=0.003$), which reveals more links among the stocks lead to a higher density of the network so that the greater spillover effects.

5.4.2. Indirect risk

The contagion risk can be amplified due to the overlapping portfolio, which varies with different choices of parameters for the flow-performance relation and liquidity. Fig. 8 shows the sensitivity of indirect risks under three different scenarios. The direct risk, shown in the grey-shaded areas, is an average of 3.34%. The average for the indirect risk is 0.48% for scenario 1 (the red-shaded areas) and 14% for the direct risk. In the scenario of a more sensitive flow-performance relation (scenario 2), the average indirect risk is increased to 1.45% (the green and the red shaded areas), amplified by 42% based on the direct risk. In the highly illiquid scenario (scenario 3), the average indirect risk is 5.57% (the blue, green, and red shaded areas), and the contagion risk is amplified much greater by 160%.

Fig. 8.

Breakdown of systemic risk in three scenarios based on 56 VAR networks. The grey shaded area is the direct risk, the indirect risk for scenario 1 is in the red shaded areas, scenario 2 is in the combined green and red shaded areas, and scenario 3 is in the combined blue, green, and red shaded areas. The risk is calculated based on the average of all the stocks on each VAR network. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

The correlation test in Panel A of Table 5 shows that the indirect risks in three scenarios are highly correlated, which shows the ranking order of financial stocks by the indirect risks remains consistent in the three scenarios. On the other hand, the indirect risks in the three scenarios are all positively correlated with the FHR ($\bar{\rho }=0.422$)3 and size ($\bar{\rho }=0.433$), but negatively correlated with ILLIQ ($\bar{\rho }=-0.511$), which suggests a stock with a greater proportion of shares held by funds and with higher liquidity tends to have greater indirect risk.

Table 5.

Risk variable correlation test.

	Scenario 1	Scenario 2	Scenario 3	FHR	ILLIQ	MC
Panel A. Indirect risk
Scenario 1	N/A
Scenario 2	1.000***	N/A
Scenario 3	1.000***	1.000***	N/A
FHR	0.420***	0.419***	0.416***	N/A
ILLIQ	−0.510***	−0.511***	−0.512***	−0.168*	N/A
MC	0.434***	0.433***	0.432***	0.422***	−0.531***	N/A
Panel B. Systemic risk
Scenario 1	N/A
Scenario 2	1.000***	N/A
Scenario 3	0.997***	0.999***	N/A
FHR	0.380***	0.388***	0.401***	N/A
ILLIQ	−0.530***	−0.527***	−0.521***	−0.168*	N/A
MC	0.430***	0.432***	0.432***	0.422***	−0.531***	N/A

***: p < 1%, **: p < 5%, *: p < 10%.

Note: This table reports the correlations among the indirect risks in Panel A, and Systemic Risk, in Panel B, for the financial stocks in three scenarios and their correlations with the fund holding ratio (FHR), illiquidity ratio ($ILLIQ$), and market capitalization (MC). The table in Panel A shows the values of indirect risk under three scenarios are significantly correlated with a coefficient of 1.00, which indicates the ranking orders of stocks by indirect risk remain consistent, not affected by the parameter of flow-performance relationship or liquidity. FHR is significantly correlated with indirect risk, suggesting a stock that has a greater proportion of shares held by mutual funds tends to have greater indirect risk. The $ILLIQ$ is significantly correlated with the indirect risks, which indicates more liquid stocks tend to have greater indirect risks. MC positively correlates with indirect risks, suggesting large stocks have greater indirect risks. Similar results for systemic risk are shown in Panel B. Systemic risks in different scenarios are highly correlated, and systemic risk is also positively correlated with FHR and MC, and negatively correlated with $ILLIQ$.

5.4.3. Systemic risk

The $SR$ is the sum of direct and indirect risk. Given the drastic differences of indirect risks among three different scenarios, the $SR$ s vary accordingly. The ranking of individual financial stock is shown in Fig. 9. The average $SR$ for scenario 1 (in blue) is 3.92%, for scenario 2 (the combined of blue and green) is 4.90%, and for scenario 3 (the aggregate of blue, green, and red) is 9.02%. According to the base scenario (scenario 1), the top 10 financial stocks by $SR$ are CITIC Securities (5.58%), Guotai Junan Securities (5.43%), Ping An Insurance (5.34%), East Money Information (5.20%), China Merchants Bank (5.14%), Shanghai Pudong Development Bank (5.12%), Ping An Bank (5.11%), Huaxi Securities (5.11%), China Merchants Securities (5.10%) and Industrial Securities (4.88%). Notably, among the top 10 influential stocks, 6 belong to the Capitals, another 3 are joint-stock banks, and the remaining one is an insurance company. The $SRs$ among different scenarios are highly correlated (${\rho }_{\mathrm{1,2}}=1.000,{\rho }_{\mathrm{1,3}}=0.997,{\rho }_{\mathrm{2,3}}=0.999$), as shown in Panel B of Table 5. In addition, the $SR$ is significantly negatively correlated with $ILLIQ$ ($\bar{\rho }=-0.526$), positively correlated with the FHR ($\bar{\rho }=0.390$) and positively correlated with the market capitalization ($\bar{\rho }=0.431$). It suggests that a financial stock is more influential if it has greater market liquidity, greater FHR, and larger size.

Fig. 9.

Ranking of financial stocks by $SR$ in three scenarios. This figure shows the ranking of $SR$ based on the average simulation results with 56 VAR networks. The $SR$ for scenario 1 is shown in blue, scenario two is shown as the combination of blue and green, and scenario three is shown as the combination of blue, green, and red. For example, the $SR$ for CITIC Securities is 5.58% for scenario 1, 7.00% for scenario 2, and 13.00% for scenario 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

On the other hand, concerning the result of $SR$ by sectors, two findings can be summarized. Firstly, although the choice of parameters would amplify the indirect risk, the sectoral rankings of influences measured by SR in three scenarios remain unchanged, as shown in Table 3. The insurance sector has the highest $SR$, followed by the joint-stock bank, regional bank, Capitals, state-owned banks, and Diversified. Secondly, similar to the finding of the direct risk, the pair-wise comparisons of the mean difference of sectoral $SR$ in three scenarios, in Table 6, show that there is no significant difference of $SRs$ among the regional bank, joint-stock bank, Capitals, and Insurance sectors. However, the state-owned bank and the Diversified sectors are significantly lower than other sectors. Therefore, the financial stocks can be broadly classified into the highly influential sectors, including the regional bank, joint-stock bank, Capitals, and Insurance, and the low influential sectors, the state-owned bank and the Diversified.

Table 6.

Statistics for mean difference of SR by sectors.

	Regional bank	Joint-stock bank	Capitals	Insurance	State-owned bank	Diversified
Scenario 1
Regional bank	N/A
Joint-stock bank	0.505	N/A
Capitals	−1.125	−1.086	N/A
Insurance	0.984	0.314	1.591	N/A
State-owned bank	−3.926***	−3.240***	−3.110**	−3.738***	N/A
Diversified	−4.065***	−3.723***	−3.529***	−4.095***	−1.198	N/A
Scenario 2
Regional bank	N/A
Joint-stock bank	0.559	N/A
Capitals	−0.880	−1.001	N/A
Insurance	1.014	0.288	1.472	N/A
State-owned bank	−3.904***	−3.234***	−3.213**	−3.724***	N/A
Diversified	−3.993***	−3.679***	−3.551***	−4.042***	−1.167	N/A
Scenario 3
Regional bank	N/A
Joint-stock bank	0.637	N/A
Capitals	−0.423	−0.828	N/A
Insurance	1.067	0.250	1.254	N/A
State-owned bank	−3.856***	−3.198***	−3.404***	−3.701***	N/A
Diversified	−3.845***	−3.573***	−3.580***	−3.936***	−1.104	N/A

***: p < 1%, **: p < 5%, *: p < 10%.

Note: This table reports the t-statistics for the pair-wise comparisons of the mean difference for the $SR$ s among different sectors. It shows that, in all scenarios, the regional bank, joint-stock bank, Capitals, and Insurance sectors have no significant difference between each other at 5% level. However, the state-owned bank and Diversified sectors are significantly lower than other sectors at 1%, except for scenario 2, where the state-owned bank is significantly lower than the Insurance at 5% level.

5.5. General discussions

5.5.1. Interactions of the two-layer network

Our model is designed to capture the spillover risk through the VAR network and fire sale contagion through the bipartite network. However, our result for systemic risk is not simply the sum of risks from two individual networks. This is because our model allows risk transfer between the stock market and the mutual fund market, so it captures the interactions between the layers that creates the downward spiral phenomenon of stock price. Similar finding is nuance by Ref. [4], which study the systemic risk in the Mexican banking sector considering direct risk from the interbank liability and indirect risk from bank's portfolio overlap. They find the average DebtRank by summing up two individual networks is 0.08. However, by combining two networks and allowing interactions between layers, they find the average DebtRank is increased to 0.25. Therefore, focusing only on direct spillover effect underestimates the total SR, particularly in a low liquidity environment. For instance, without considering the fire sale effect, the average contagion risk is 3.34% (see Table 3), however, with incorporating fire sale and allowing interactions between the layers, the risk is elevated to 8.69% for the most illiquid case (scenario 3), amplified by 160%.

5.5.2. Policy implication

Firstly, our study relates to the literature on the “too-large-to-fail” or “too-interconnected-to-fail” for SIFIs. Our results confirm both statements. We find $SR$ is significantly correlated the market capitalization $\left({\bar{\rho }}_{SR,MC}=0.431\right)$ even though our model does not explicitly consider the size factor. For the connectedness, we find $SR$ is not only significantly correlated with the out-degree in the VAR network ($\bar{\rho }=0.673$), but also the degree in the F–S network ($\bar{\rho }=0.297$), see Table 7. Furthermore, our results also show the importance of liquidity (${\bar{\rho }}_{SR,ILLIQ}=-0.526$) and FHR (${\bar{\rho }}_{SR,FHR}=0.390$) in identifying the SIFIs. To summarize, our suggestion to policymakers on classifying a SIFI is not only to focus on its size and connectedness but also on its liquidity in the market and the amplification effect from the mutual funds.

Table 7.

Correlation test between SR with network degree.

	SR
	Scenario 1	Scenario 2	Scenario 3	Average
Average out degree: VAR network	0.656***	0.670***	0.694***	0.673
Degree: F–S network	0.294***	0.297***	0.299***	0.297

***: p < 1%.

Note: This table reports the correlation test for SRs in three scenarios with the stock's degree in the VAR and F–S networks. The results show that the correlations are significant at 1% for all three scenarios.

Secondly, the banking sector is the most influential sector in the Chinese financial system (c.f. [23,44,69]), mainly because of its large size [35,68,70] or their tight connections to economics [71]. However, the banking sector should be broken down into sub-sectors and analyzed separately since their sizes vary drastically. The average total assets for the state-owned banks are about 3.8 times that for the joint-stock banks and 28 times that for the regional banks. However, despite the much larger state-owned banks, our findings show their influences are significantly lower than the smaller banks, the regional banks, and the joint-stock banks, which suggests analyzing the banking sector in a mix may not be able to unveil the insights of the banking risks.

Thirdly, based on the scenario analysis results, the liquidity factor has a greater impact on amplifying the contagion risk than the flow-performance relationship of mutual funds. The more sensitive flow-performance relationship results in a 42% amplification of contagion risk. However, the extreme illiquid scenario could result in a 160% amplification of its original contagion risk. It suggests that in some cases with abnormal market phenomena, e.g., the Covid19 pandemic, the regulator could act as a liquidity provider to the market to keep the systemic risk under control.

6. Conclusion

This paper uses a simulation approach to identify the influential financial stocks in China based on a two-layer network driven by the market trading data and the portfolio holding data of mutual funds. The two-layer structure contains a return spillover network that models the contagion risks among the stocks and a fund-stock bipartite network that models the amplification effect of contagion due to the fire sale of mutual funds due to common asset holdings. The systemic risk is measured by the relative systemic loss of market capitalizations induced by the initial shock on a stock.

Based on 56 time-varying VAR networks and the funds’ portfolio holdings as of 2020, we find a financial stock is more influential if it is more liquid in the market, held by more mutual funds, and larger in size. The sectoral analysis shows that regional banks, joint-stock banks, Capitals, and Insurance are among the highly influential sectors that are the risk emitters in the system. In contrast, the state-owned bank and Diversified sectors are the low influential sectors and the risk receivers. The scenario analysis shows that the flow-performance relation of mutual funds and the liquidity factor greatly impact amplifying the contagion risk. By contrast, a more illiquidity scenario tends to amplify greater contagion risks (by 160%) than a more sensitive flow-performance relation scenario (by 42%). Our results make suggestions to the policymakers. On the one hand, when identifying SIFIs, ones should focus on not only their size and connections but also their market liquidity and the fund holding concentration. On the other hand, in some extreme cases with low market liquidity, the regulator could play a role as the liquidity provider to the market to keep the systemic risk manageable.

In terms of limitations, several aspects can be improved for future study. Firstly, our choice for the parameter for the flow-performance relation of a mutual fund is based on literature findings. If data are available, ones can perform an analysis to obtain a more dynamic parameter for the flow-performance relation. Secondly, our stock network is based on the return spillover. One can build the network using other techniques, e.g., volatility spillover or extreme events-driven network, to make a comparison of the consistency of the results. All these areas are believed to be important research opportunities for future study.

Author contribution statement

Shiqiang Lin: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Hairui Zhang: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Data availability statement

Data will be made available on request.

Declaration of interest's statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.