Document Type : Original Article
Authors
 Masoome Ramezani ^{1}
 Mohammad Osoolian ^{} ^{} ^{2}
 Mostafa Zandieh ^{3}
 Seyed Ali Hosseiny Esfidvajani ^{4}
^{1} Ph.D. Candidate, Department of Finance, Science and Research Branch, Islamic Azad University, Tehran, Iran.
^{2} Assistant Prof., Department of Finance, Shahid Beheshti University, Tehran, Iran.
^{3} Prof. of Industrial Engineering, Department of Industrial Management and Information Technology, Faculty of Management and Accounting, Shahid Beheshti University, Tehran, Iran.
^{4} Associate Prof., Department of Physics, Shahid Beheshti University, Tehran, Iran.
Abstract
Examining the importance and influence of financial market companies is one of the main issues in the field of financial management because sometimes the collapse of a stock exchange company can affect an entire financial market. One systematic way to analyze the significance and impacts of companies is to use complex networks based on Interaction Graphs (IGs). There are different methods for quantifying the edge weight in an IG. In this method, the graph vertices represent the stock exchange companies that are connected by weighted edges (corresponding to the extent to which they relate to each other). In this paper, using the GARCH model (1,1) and the Clayton copula, we obtained the lower tail dependence interaction network of the first 52 companies of the Tehran Stock Exchange in terms of average market value, between June 2017 and October 2020. Then, based on the minimum spanning tree of the interaction network, we divided the companies into different communities. Using this classification, it was observed that the companies of the first group (Food Industry) and the second group (Oil Refinery) have the greatest impact on other companies. We also calculated the central indexes of the minimum spanning tree for each company. According to the results, the companies of the third group (Steel) have the highest average in the central indicators.
Keywords
Introduction
Examining the importance and influence of financial market companies is one of the main issues in the field of financial management. Because sometimes the collapse of a stock exchange company can affect an entire financial market. An example is the collapse of Lehman Brothers Holding, which eventually led to the 2008 global financial crisis. One systematic way to analyze the significance and impacts of companies is to use complex networks based on Interaction Graphs (IGs) (Yang et al., 2020). In this method, the graph vertices represent the stock exchange companies that are connected by weighted edges (corresponding to the extent to which they relate to each other). Converting an IG to a tree in which loops have been removed reduces the complexity of the problem and simplifies system analysis. In this study, we used the Minimum Spanning Tree (MST) to analyze the results.
There are different methods for quantifying the edge weight in an IG. Among these methods, we can mention the Pearson Correlation (PerC) coefficient (Patro et al., 2013; Wiliński et al., 2013) and, Transfer Entropy (TE) (Ardalankia et al., 2020; Kwon & Yang, 2008; Osoolian & Koushki, 2020). Here we use Lower Tail Dependence (LTD) to measure the edge weight in the IG, which is a measure of the coordinated behavior of companies in negative returns (Wang & Xie, 2016). This quantity can be used in portfolio risk management. To calculate LTD, we need models through which we can obtain the marginal and joint Probability Distribution Functions (PDFs) of the returns of companies. Due to the fattailed behavior of the return PDF in financial markets (Chakraborti et al., 2011), we have used the GARCH(1,1) model to calculate the marginal PDFs (Lee & Hansen, 1994). To obtain joint PFDs we used the Clayton copula (Wen et al., 2019).
The structure of the present article is as follows: First, we briefly review using IGs in financial market analysis. Next, we have introduced the quantities and models used. Then, the results of the research are presented. And finally, we discuss and conclude the results.
Literature Review
The idea of using IGs in financial market analysis goes back to Mantegna. Using PerCbased IG, he was able to obtain a hierarchical structure for the companies under study. In other words, he was able to use the IG to divide companies into groups that had the same field of activity (Mantegna, 1999).
Bonanno et al. investigated the topological structure of the MST in financial markets. They have shown that for real markets, MST has a structure that cannot be reconstructed from random data, even in the first approximation (Bonanno et al., 2003).
Wiliński et al. have shown that the MST of the Frankfurt Stock Exchange has a unique topological structure (superstar) during the financial crisis between 2007 and 2008 (Wiliński et al., 2013). Therefore, it can be used to predict the occurrence of financial crises. A similar study on the impact of the global financial crisis on the South Korean financial markets IG has been conducted by Nobi et al. (Nobi et al., 2014).
Kwon and Yang have used the TEbased IG to determine the direction of information flow in financial markets. They have shown that information flows from US financial markets to Asian financial markets. They also use the MST to show that the S&P500 market is the primary source of information among financial markets (Kwon & Yang, 2008).
Wang et al. investigated the differences between the PerCbased and Partial Correlation (ParC)based MST. Based on central indexes, they have shown that the ParCbased MST provides more reliable results than the PerCbased MST (Wang et al., 2018).
EngUthaiwat has shown that the topology of the IG can be used to predict the return of the portfolio (EngUthaiwat, 2018). Peralta and Zareei have also presented a method for the optimal selection of the portfolio using the IG (Peralta & Zareei, 2016).
Research Methodology
The statistical population of this study is the 100 large companies of the Tehran Stock Exchange based on the average market value (https://www.fipiran.ir/). Among them, companies that had more than 3 months of interruption in daily stock exchange transactions were eliminated. Finally, the remaining 52 companies were selected as statistical sample companies. For these companies, the LTDbased IG was drawn. We used the Clayton copula and the GARCH(1,1) model to calculate the LTD. To obtain the parameters of the Clayton copula and the GARCH(1,1) model, we used the Maximum Likelihood Estimation (MLE) method. From the IG, we obtained the MST for simplicity in analyzing the results. Using MST, we divided the studied companies into different groups in terms of dependency. Finally, we calculated the central indexes associated with the MST.
 Interaction Network Construction
 Lower Tail Dependence
In probability theory, the LTD of a pair of random variables is a measure of their coordinated motion in the tail of the probability distribution. The LTD ( ) of a pair of random variables is defined as follows:
(1) 
, 
where is the inverse of the Cumulative Distribution Function (CDF) .
Similarly, the Upper Tail Dependence (UTL) is defined as follows:
(2) 

 Copula
The joint CDF of several continuously uniform random variables on the set [0,1] is called a Copula. In other words, if is a random vector with , then the copula
is defined as follows:
(3) 

See Appendix 1 for details on Copula.
 GARCH(p,q) Model
In economics, the Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) is a statistical model for time series data. In this model, the time series of the return of a financial institution is given as follows:
(4) 

(5) 

where (random variable with normal distribution with mean zero and variance one), is the timedependent variance, and are the free parameters of the model. These parameters can be estimated using the real value of in a time series based on the MLE method.
As can be seen, in this model the variance of the random variable , itself is also a random variable. It can be shown that in the GARCH(1,1) model the mean value of variance is equal to . In obtaining the time series of variance, we use this value as the starting point of variance. In other words, we put .
 Maximum Likelihood Estimation
Suppose is a random variable with PDF that depends on the parameter . If are the results of experiments on the random variable , we want to get the best estimate for the value of the parameter in terms of these results. In the Maximum Likelihood Estimation method, the best estimate for the parameter is the value that maximizes the following likelihood function:
(6) 

 Assortativity
For a graph whose vertices are divided into groups, the assortativity coefficient is a measure that quantifies the quality of the grouping. The value of this coefficient is in the set [0,1]. The closer the value to one indicates more edges in groups relative to edges that connect the groups. The assortativity coefficient of a connected graph is defined as follows:
(7) 

where is the total number of edges, is the weight of the edge connecting vertices and , is the degree of the vertice , is the Kronecker delta, and, is a binary function that is equal to one when vertices and belong to the same group otherwise, it is equal to zero.
 Central Indexes
 Node Degree
The degree of each vertice or node is equal to the number of edges connected to that node. The degree of node is defined as follows:
(8) 

 Node Strength
The strength of node is defined as follows:
(9) 

where is the LTD of nodes and .
 Betweenness Centrality
Betweenness centrality is the way to determine the extent to which a node affects the flow of information in a graph. It is often used to find groups that act as bridges from one part of a graph to another. The betweenness centrality of node is defined as follows:
(10) 

where is the number of the shortest path from node to the node and, is the number of the shortest path from node to the node that crosses the node .
 Closeness Centrality
Closeness centrality shows the average distance of that vertex from other vertices. The larger the closeness centrality of a node, the closer that node is to the other nodes. The closeness centrality of node is defined as follows:
(11) 

where is the distance of the shortest path from node to node .
Data Analyses
 Interaction Network Construction
The data used in this study is the time series of the closed stock prices of 52 companies on the Tehran Stock Exchange in the period 2017/6/22 to 2020/10/21. Using this data, we obtained the logarithmic time series of stock prices of companies as follows:
(12) 

Then, using the GARCH(1,1) model (Eq.9 and Eq.10), we calculated the time series of variance through the following recursive relation:
(13) 

(14) 

In this model, is a random variable with a normal distribution function with a mean of zero and a variance of , in other words
(15) 

We used the MLE method to estimate the parameters }. The best estimates for the value of these parameters using the time series are values that maximize the following likelihood function:
(16) 


The values obtained for these parameters for each company are given in See Appendix 2 for company names. Error! Reference source not found.
In this study, we used the Clayton copula to calculate the LTD of two companies. We used the MLE method to obtain the parameter of the Clayton copula between the two companies. The likelihood function used to estimate the parameter is as follows:
(17) 


where and are the normal CDF for the return of companies and , respectively, and are obtained as follows:
where erf is the error function.
After obtaining the parameter for each pair of companies, we obtained their LTD using Eq.8. The mean, variance, maximum, and, minimum of LTDs are given in Table 1.
Table 1. LTD statistics of companies
Mean 
Variance 
Min 
Max 
0.199 
0.164 
0.000 
0.878 
To draw the MST, we first converted the LTD matrix to the distance matrix as follows:
(18) 

Figure 1 shows this matrix. The corresponding MST is depicted in Figure 2.
Figure 1. LTDDistance Matrix
Figure 2. LDTBased MST
Using the FindGraphCommunities command in Mathematica software, the present companies were divided into different groups on MST. These groups are listed in Table 2 along with the mean value of the LTD of each group.
Table 2. Dividing companies into different groups on MST
Group Number 
Group Name 
Companies 
Mean LTD 
1 
Food Industry 
Glucosan (21), Pars Minoo Industrial Co. (32), Iran China Clay Ind. (36), Iran Khodro Investment Development Co. (41), Shahid Ghandi Production Factories Co. (46) 

2 
Oil Refinery 
Isfahan Oil Refinery (3), Tehran Oil Refinery (4), Bandar Abbas Oil Refinery (5), Parsian Oil and Gas Development Group (8), Tabriz Oil Refinery (14), Civil Pension Fund Investment (22), Khark Petrochemical Co. (45) 

3 
Steel 
Mobarakeh Steel Co. (1), Khouzestan Steel Co. (11), Behran Oil Co. (28), Iran Alloy Steel Co. (37), Sepahan Oil Co. (42), South Kaveh Steel Co. (47), Iranian Investment Petrochemical Group (50) 

4 
Communication 
Mobile Telecommunication Company of Iran (6), Informatics Services Co. (15), Golrang Industrial Group (20), Iran Chemical Industries Investment Co. (23), National Investment Co. (29), Asan Pardakht Persian (34), Telecommunication Company of Iran (44), MAPNA Group (48) 

5 
Petrochemical 
Omid Investment Co. (10), Jam company (13), Pardis Petrochemical Co. (16), Mobin Petrochemical Co. (18), Shazand Petrochemical Co. (26), Middle East Bank (27), Kermanshah Petrochemical Ind. (38), Iran Khodro (40), Persian Gulf Fajr Energy Co. (51) 

6 
Pharmacy 
Dr. Abidi Pharmacy (19), Carton Iran (30), Pars Oil (33), Bahman Group (35), Zahravi Pharmaceutical Co. (39), Tehran Stock Exchange (43), Iranian Aluminium Co. (52) 

7 
Investment 
National Iranian Copper Industries Co. (2), Tamin Petroleum & Petrochemical Investment Co. (7), Ghadir Investment Co. (9), Islamic Republic of Iran Shipping Line Group (12), National Development Investment Group (17), Tamin Pharmaceutical Investment Co. (24), Shiraz Petrochemical Co. (25), Iran Transfo (31), Mobarakeh Steel Co. (49) 

Considering that only the food companies in the list of 52 companies i.e. Glucosan and Pars Minoo Industrial Co. were in the first group, we named this group Food Industry. A similar argument is made about the Pharmacy group. The nomination of other groups is selected according to the number of dominant companies in that group that have the same field of activity.
 Assortativity
The assortativity coefficient of grouping was 0.897. This value indicates that the grouping performed has a highresolution quality.
 Central Indexes
Table 3 shows the average of the central indexes for the classified groups in Table 2. The highest node strength is related to the first group (Food Industry) and the highest node degree, betweenness centrality, and, closeness centrality is related to the third group (Steel). Table 4 also shows the top five companies in each central index. As can be seen, Shazand Petrochemical Co. is among the top 5 companies in three of the four central indexes. Bandar Abbas Oil Refinery and Iran Khodro Investment Development Co. are also among the top five companies in two of the four central indexes.
Table 3. Average central indexes of groups
Group 
Node Degree 
Node Strength 
Betweenness Centrality 
Closeness Centrality 
1 
2.000 
1.521 
242 
0.217 
2 
1.857 
1.050 
41 
0.191 
3 
2.286 
1.353 
395 
0.249 
4 
1.875 
0.978 
59 
0.200 
5 
2.000 
1.169 
161 
0.185 
6 
1.857 
1.029 
68 
0.165 
7 
1.889 
0.851 
96 
0.167 
Table 4: The first 5 companies in the central indexes
Rank 
Node Degree 
Node Strength 
Betweenness Centrality 
Closeness Centrality 
1 
Bandar Abbas Oil Refinery 
Shazand Petrochemical Co. 
Iran Khodro Investment Development Co. 
South Kaveh Steel Co. 
2 
Mobarakeh Steel Co. 
Khouzestan Steel Co. 
Iranian Investment Petrochemical Group 
Mobin Petrochemical Co. 
3 
Mobile Telecommunication Company of Iran 
Bandar Abbas Oil Refinery 
Shazand Petrochemical Co. 
Persian Gulf Fajr Energy Co. 
4 
National Development Investment Group 
Parsian Oil and Gas Development Group 
Carton Iran 
Telecommunication Company of Iran 
5 
Shazand Petrochemical Co. 
Iran Transfo 
Pars Minoo Industrial Co. 
Iran Khodro Investment Development Co. 
Discussion and Conclusion
In the study, the impact of the selected 52 companies from the Tehran Stock Exchange (based on market value) on each other in the period from June 2017 to October 2020 was examined. The mathematical tool used in this study was the lower tail dependence of companies. This quantity was calculated using the GARCH(1,1) model and the Clayton copula. By obtaining the lower tail dependence of the companies, their interaction graph was obtained. By drawing the minimum spanning tree from the interaction graph, the existing companies were divided into different groups based on their impacts on each other. The assortativity coefficient of the minimum spanning tree in this division was 0.897, which indicates the appropriate resolution of the groups. According to the results, it can be seen that the companies in the first (Food Industry) and second (Oil Refinery) groups have the greatest impact on other companies in the period under review. Also, companies in the sixth (Pharmacy) and seventh (Investment) groups have the least impact on other groups. By calculating the central indexes, it was observed that the third group (Steel) had the highest average in three indexes (node degree, betweenness centrality, and, closeness centrality) of the four studied indexes. Also, Shazand Petrochemical Co., Bandar Abbas Oil Refinery, and, Iran Khodro Investment Development Co. were among the first companies in the central indexes.
Based on the above results, the policy implications of this article are as follows:
 By systematically categorizing companies based on lower tail dependencies, regulators can apply different protection policies to different groups in times of financial crisis.
 By identifying key nodes with respect to central indexes, financial regulators can effectively monitor to reduce the spread
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest concerning the research, authorship and, or publication of this article.
Funding
The authors received no financial support for the research, authorship and, or publication of this article.