Journal of the Korean Physical Society, Vol. 66, No. 8, April 2015, pp. L1153∼L1159
Letters
Structural Changes in the Minimal Spanning Tree and the Hierarchical Network in the Korean Stock Market around the Global Financial Crisis Ashadun Nobi Department of Physics, Inha University, Incheon 402-751, Korea and Department of Computer Science and Telecommunication Engineering, Noakhali Science and Technology University, Sonapur Noakhali-3802, Bangladesh
Seong Eun Maeng, Gyeong Gyun Ha and Jae Woo Lee∗ Department of Physics, Inha University, Incheon 402-751, Korea (Received 29 December 2014, in final form 26 February 2015) This paper considers stock prices in the Korean stock market during the 2008 global financial crisis by focusing on three time periods: before, during, and after the crisis. Complex networks are extracted from cross-correlation coefficients between the normalized logarithmic return of the stock price time series of firms. The minimal spanning trees (MSTs) and the hierarchical network (HN) are generated from cross-correlation coefficients. Before and after the crisis, securities firms are located at the center of the MST. During the crisis, however, the center of the MST changes to a firm in heavy industry and construction. During the crisis, the MST shrinks in comparison to that before and that after the crisis. This topological change in the MST during the crisis reflects a distinct effect of the global financial crisis. The cophenetic correlation coefficient increases during the crisis, indicating an increase in the hierarchical structure during in this period. When crisis hits the market, firms behave synchronously, and their correlations are higher than those during a normal period. PACS numbers: 05.45.Tp, 89.75.-k, 89.65.Gh Keywords: Stock market, Complex network, Minimal spanning tree, Hierarchical network, Power law DOI: 10.3938/jkps.66.1153
I. INTRODUCTION The 2008 global financial crisis has had considerable influence on the global stock market. The crash initiated from the U.S. housing crisis in 2007 [1–7], and the collapse of Lehman Brothers was one of the worst bankruptcies with global effects. The bankruptcy of Lehman Brothers triggered a crisis in the U.S. financial market and induced a chain reaction across the world. Effects of local stock markets depend on each country’s economic situation. Many methods of complex systems have been applied to financial systems to describe these dynamic features [8–17]. In particular, the idea of networks has been applied to many complex systems [18–21], and the tools of complex networks have been applied to the stock market to characterize the complex dynamics of markets [22–31]. Cross-correlation coefficients are calculated from return time series between a pair of stock indices [9–12]. Then, sparse networks are extracted using some specific filtering methods such as the threshold method, the min∗ E-mail:
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imal spanning tree, the hierarchical network, and the planar maximally filtered graph [24, 25, 29–32]. Crosscorrelation and network analysis methods have been successfully applied to many stock markets [8, 10–12, 33]. The minimal spanning tree (MST) has been applied to financial markets in the U.S. stock market [8]. Bonnano et al. [22] investigated the MST and the hierarchical network (HN) in the U.S. equity market. Vandewalle et al. [34] observe the scale-free degree distribution of the MST in the U.S. stock market. Onnela et al. [24] report a scale-free behavior of the MST in the New York Stock Exchange. The effects of financial crises on stock markets have been examined by using many methods of complex networks [24, 29–36]. Onnela et al. [24] observed different power-law exponents of asset trees between usual and crash periods. Kumar and Deo [29] observed a chain-like MST during the global financial market crisis. Kantar et al. [30] showed that Turkish firms were not influenced by the global financial crisis in the Turkish stock market [30]. Recently, the network topology of the German stock market around the time of the global financial crisis has been studied [35]. During the global financial crisis,
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Journal of the Korean Physical Society, Vol. 66, No. 8, April 2015
Table 1. Statistical values of the cross-correlation coefficients obtained from three time periods: before, during, and after the crisis. The average cross-correlation coefficient and ¯ and its standard deviation are represented by the symbols C σ, respectively. Time period Before crisis During crisis After crisis
¯ C 0.23 0.35 0.15
σ 0.10 0.11 0.10
Skewness 2.80 1.04 3.80
Kurtosis 27.2 14.0 35.4
network structures of financial markets showed considerable changes because of large fluctuations in the market dynamics [25, 35]. This paper focuses on the effects of a global financial crisis by considering the MST and the HN in the Korean stock market. The rest of this paper is organized as follows: Section II introduces the data set and statistical properties of the local stock market. Section II investigates the MST across three periods, namely before, during, and after the crisis, and reports changes in the MST during the global financial crisis. Section IV considers the HN of prices of local stocks, and Section V concludes.
II. THE LOCAL STOCK MARKET AND STATISTICAL PROPERTIES
Cij =< ri (t)rj (t) > − < ri (t) >< rj (t) > .
(2)
Table 1 summarizes the statistical properties of crosscorrelation coefficients for the three time periods. The average cross-correlation coefficient during the crisis increases in comparison to those before and after the crisis. During the crisis, firms showed stronger interactions than in other periods. In that period, the return time series of each firm fluctuated heavily and showed a synchronous dynamic behavior. The average volatilities were calculated as v¯ = 0.019, 0.025, 0.017 for before, during, and after the crisis, respectively. The average volatility during the crisis was higher than those in the other periods. The average volatility after the crisis recovered to the level before the crisis. Skewness and kurtosis values during the crisis were lower than those before and after the crisis. The distribution of cross-correlation coefficients was asymmetric to the positive side based on a positive skewness value. All distributions are sharper than the normal distribution in all three periods. The deviation of the cross-correlation coefficients before the crisis from the normal period provided a red flag for the coming stock market crash. The lower value of the average cross-correlation coefficient after the crisis indicated that the market was more stable than in any other period.
1. THE MINIMAL SPANNING TREE
Daily closing stock prices of each firm in the Korea stock market are analyzed from June 2, 2006, to December 30, 2010. There were 185 surviving stocks in the Korean Composite Stock Price Index (KOSPI) 200 during the analysis period. The time span was divided into three parts based on volatility. Around the global financial crisis from 3 December 2007 to 30 June 2009, higher volatilities were observed. In that period the market was in unstable. The period before the crisis covered the time from June 2, 2006, to November 30, 2007. During this period, stress from the U.S. mortgage crisis partly influenced the Korean stock market. The period from December 3, 2007, to June 30, 2009, is considered as the crisis period. In that period, Lehman Brothers declared bankruptcy on September 15, 2008. The crash immediately had considerable influence on the global stock market. Finally, the period from July 1, 2009, to December 30, 2010, is the period after the crisis. In that period, some countries recovered to their normal states. However, many countries remained affected by the crisis. Let the closing stock price of a firm be Pi (t) at time t. For each window, calculate normalized returns on the stock price of each firm as ri (t) = [ln Pi (t) − ln Pi (t − 1)]/σi ,
12]. Then, we define cross-correlation coefficients between firms as
(1)
where σi is the standard deviation of the time series [10,
A MST is constructed by calculating the distance matrix of indices [23,24,34]. The distance matrix is defined as (3) dij = 2(1 − Cij ), where dij = 0 if the price time series of indices i and j are perfectly correlated and dij = 2 if they are perfectly anticorrelated. The MST is constructed following Kruskal’s algorithm to find the N − 1 most important correlated pairs of indices among N (N − 1)/2 possible pairs. Asset trees are constructed to visualize a complex network before, during, and after the global financial crisis. The numbers on the MST correspond to the top 10 vertices of a high degree from Figs. 1 to 3. The top 10 hubs in each MST list are shown in Table 2. Before the crisis, the top hub is the Ssangyong Cement Industry (number 168) in Fig. 1. Four financial firms (vertex numbers are 34, 38, 149, and 181) are located around the center of the MST. There is a topological transition in the asset tree during the crisis such that Hanjin Heavy Industries and Construction (20th largest firm in Korea; vertex 62) comes out as a center hub, like a dragon king, in Fig. 2. This reflects a dynamic topological transition from an asset tree
Structural Changes in the Minimal Spanning Tree and the Hierarchical Network · · · – Ashadun Nobi et al.
Fig. 1. (Color online) The MST plots the time period before the crisis. Firms are marked by small circles and numbers, and edges represent the distance between firms. The square symbol represents the hub vertex, and triangles correspond to firms belonging to the financial sector. The star is Samsung Electronics (144), the most influential firm in the Korean stock market. Numbers on the vertex are used to prevent a crowded figure from labeling names of firms.
Fig. 2. (Color online) The MST plots the time period during the crisis. During the crisis, Hanjin Heavy Industries and Construction (HHIC; 62) comes out as a central hub. There are several local hubs before and after the crisis. HHIC, a legend in heavy industry, is one of the world’s best specialists in heavy industry equipped with digital technologies. During the crisis, this may play an important role in resisting the shock from the crisis. As a result, it becomes a central hub on the MST during the crisis. The large numbers on the MST correspond to the top 10 vertices of a high degree.
with several local hubs before the crisis to a super hublike asset tree during the crisis. Firms in the financial sector form only a minor branch in the MST.
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Fig. 3. (Color online) The MST plots the time period after the crisis. Hyundai Motor (90) comes out as a hub. There are several local hubs after the crisis. Daewoo Securities (34) become the center in the MST. After the crisis, financial firms recover their roles in the MST.
After the crisis, this super-hub disappears, as shown in Fig. 3. This type of transition is also found in the German stock exchange (FSE) [35]. The superstar-like hub indicates an unstable market state and thus a largegroup and compact structure of indices to resist a crisis shock. After the crisis, two firms (34 and 38) in the financial sector become the center of a star-like MST. In this period, the main hub is Hyundai Motor (90). In every period, Samsung Electronics is located in the minor branch of the MST. On the other hand, before and after the crisis, a tree containing several hubs indicates a stable or meta-stable market state and (by extension) several groups and a large structure of indices. The crisis is like a natural disaster. During the last few years, much attention has been paid to the degree distribution of graphs called scale-free trees, in which the degree distribution follows a power law. Scale-free networks have been extensively employed in economic systems [23, 31, 34, 37, 38]. Figure 4 shows the degree distribution of the MST. The degree distribution PM ST (k) of the MST follows a power law such that PM ST (k)∼k−γm , where k is the degree of the vertex. The exponent γm implies that the mean size or the second moment of the degree diverges. The MST in Fig. 9 follows a scaling law with the exponent γm = 1.98(36) (r2 = 0.96), which is lower than in normal periods. However, during the crisis period, γm = 1.84(45) (r2 = 0.92), which is lower than that in the usual period. Similar behaviors of the degree exponents are observed for the threshold network. The lower exponents during the crisis are due to a shrinking tree, which means that nodes
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Journal of the Korean Physical Society, Vol. 66, No. 8, April 2015
Table 2. The top 10 hubs in the MST during the three periods are summarized. Before the crisis, several firms in the financial sector belong to the top 10 hubs. However, during the crisis, firms in heavy industry occupy the top 10 hubs. After the crisis, there are no dominant sectors occupying the top 10 hubs.
Ranking
Vertex number
1
168
2
181
3
Before the crisis Company name Ssangyong cement industry
Degree
Vertex number
20
62
Woori investment securities
11
44
173
Taihan electric wire
8
163
4
149
Samsung securities
8
31
5
34
Daewoo securities
8
168
6
38
7
170
7
87
7
177
8
135
Posco coated colour steel
7
11
9
112
Kumho industrial Co. Ltd.
6
86
10
115
Kwang dong pharmaceutical Co. Ltd.
6
52
Daishin securities Hyundai hisco Co. Ltd.
with high degrees increase. The size of the tree is characterized by the average tree length of the MST. The average tree length is defined as [18–21] L(t) =
1 M ST d , N ij
(4)
where N is the total number of vertices in the tree and ST is the shortest path length between two vertices dM ij i and j. The average tree lengths are L = 1.05, 0.92, and 1.1 for before, during, and after the crisis, respectively. During the crisis, the MST is smaller than it is in other periods, which indicates that market behaviors are tightly correlated. In addition, the lower exponent before the crisis than in the usual period is a precursor of the market crash. After the crisis, the exponent γm
During the crisis Company name Hanjin heavy industries and construction Dongbu steel Co. Ltd.
Degree
Vertex number
20
90
9
34
SK holdings
9
30
Daesang
8
85-
8
140
8
21
8
38
7
106
6
153
6
133
Ssangyoung cement industries STX offshore and shipbuilding Unio steel Co. Ltd. LG electronics Hyundai heavy industries Co. Ltd. Doosan construction And Engineering
After the crisis Company Degree name Hyundai Motor 12 Daewoo securities Daelim industrial Co. Ltd. Hyundai glovis Co. Ltd. S&T Holding Co. Ltd. Binggrae Co. Ltd. Daishin securities Korea industrial Co. Ltd. Seah bestell corp. Poongsan holding corp.
10
7
6
6
5 5
5
5
4
= 2.2(9) (r2 = 0.88), which is close to the usual value for the Korean stock market, indicates a more uniform degree. Therefore, the Korean market can be divided into three states: metastable (before the crisis), unstable (during the crisis), and completely stable (after the crisis). The exponent γm varies from 2.1 to 2.2 for the usual period based on data from 2002 to 2005. The slight deviation of the exponent γm from the normal period is a red flag for the market crash. The exponent found for the MST from the Korean stock market for the normal and the crash periods are analogous to the exponents for the New York Stock Exchange (NYSE) during the financial crisis of 1986 [24]. Then, the Korean stock market can be assumed to complement the NYSE.
Structural Changes in the Minimal Spanning Tree and the Hierarchical Network · · · – Ashadun Nobi et al.
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Fig. 4. (Color online) Log-log plots of the degree distribution function represent a function of the degree in the MST (a) before the crisis, (b) during the crisis, and (c) after the crisis. Degree exponents are γm ≈ 1.98(36) before the crisis, γm ≈ 1.84(45) for during the crisis, and γm ≈ 2.2(9) after the crisis.
III. THE HIERARCHICAL NETWORK A dendrogram gives an alternative representation of the network and shows the full hierarchical structure [29, 30]. It is used to strengthen hierarchical structures. At the first level of the dendrogram, there are N-singleton clusters. As the vertical scale of the dendrogram is moved, clusters combine until all vertices are contained in a single community at the top of the dendrogram. The hierarchical structure of financial markets has been analyzed for the Dow Jones Industrial Average (DJIA) and
Fig. 5. (Color online) Dendrograms of financial indices represent 30 firms. HNs are obtained (a) before, (b) during, and (c) after the crisis. Numbers representing individual firms are given in a supplementary Excel file. The height of the dendrogram decreases during the crisis, but increases before and after the crisis. Firms in construction (2, 30), communication (8, 17), and medical supplies (13, 25) make a tight bond in every period.
the S&P 500 (10). The tree-like dendrogram provides a meaningful economic taxonomy. The average linkage hierarchical clustering algorithm
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is applied to the distance matrix to produce the dendrogram. A tree-like dendrogram of the Korean stock market is constructed during, before, and after the crisis. Here, 30 stocks are taken because if all stocks has been used, the figure would be crowded, clearly explaining the figure, as shown in Fig. 5, might have been difficult. Here, the focus is on the height of the nearest clusters in the HN in all periods. If the range of the distance dij ≤ 1 is restricted, then the numbers of vertex pairs in that range are 31 (before the crisis), 82 (during the crisis), and 27 (after the crisis) out of 184 pairs (not shown in Fig. 5). The numbers of pairs between 1.0 ≤ dij ≤ 1.2 are 102 (before the crisis), 80 (during the crisis), and 66 (after the crisis), and for the distance dij > 1.2, the numbers are 51 (before the crisis), 22 (during the crisis), and 91 (after the crisis). The height between the nearest clusters decreases in the HN during the crisis. Of course, these decreasing trends are not true for all pairs of firms. Throughout all examined periods, some firms form tight bonds with one another. The cluster before the crisis is reorganized with new firms during and after the crisis (Fig. 10). This suggests that when a crisis shocks the stock market, firms can protect themselves from the shock by making loose or strong bonds with other firms or by finding some other factors to relax the shock. The cophenetic correlation coefficient CCC is defined as [13,39] ¯ × (cij − c¯)] [ i
