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Dynamic topology and allometric scaling behavior on the Vietnamese stock market
Accepted Manuscript Dynamic topology and allometric scaling behavior on the Vietnamese stock market Q. Nguyen, N.K. K. Nguyen, L.H. N. Nguyen
PII: DOI: Reference:
S0378-4371(18)31193-2 https://doi.org/10.1016/j.physa.2018.09.061 PHYSA 20122
To appear in:
Physica A
Received date : 14 May 2018 Revised date : 14 September 2018 Please cite this article as: Q. Nguyen, et al., Dynamic topology and allometric scaling behavior on the Vietnamese stock market, Physica A (2018), https://doi.org/10.1016/j.physa.2018.09.061 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Cover letter and Highlights
Main highlights of manuscript “Dynamic topology and allometric scaling behavior on the Vietnamese stock market” - A star-like MST constructed from the Vietnamese stock price was observed during 2011-2012 - During this period the country was in a financial crisis with interest rate of >30% - It is similar to that of some market during the world-wide financial crisis of 2008 - All temporal MST has an allometric scaling behavior - Their exponent are from 1.2 during the star-like phase, to more than 1.3 otherwise
*Manuscript Click here to view linked References
Dynamic topology and allometric scaling behavior on the Vietnamese stock market Q. Nguyena,∗, N.K.K. Nguyenb , L.H.N. Nguyena a John
Von Neumann Institute, Vietnam National University Ho chi minh City, Ho chi minh City, Vietnam b Van Lang University, Ho chi minh City, Vietnam
Abstract The impact of the Vietnamese financial crisis during 2011-2012 into the stock market was revealed by a structural change of the minimum spanning tree (MST) constructed from the daily stock price. We found that the MST has a star-like structure during this period, similar to that of the German market during the worldwide financial crisis of 2007-2008 [1], and a hierarchical scalefree structure for the rest of time. In addition, we investigate the market from a complex network perspective by analyzing the allometric scaling behavior. We found that all networks have the allometric scaling property, with exponent η ranging from 1.213 ± 0.013 during the financial instability period to about 1.357 ± 0.011 in normal time. These values correspond to a complex “dimension” of the financial market of between 3 and 5, which need to be further investigated in the future. Keywords: financial crisis, complex network, minimum spanning tree, allometric scaling
1. Introduction After the worldwide financial crisis of 2008-2009, it has been shown that traditional financial and economic model needs major revise, especially under ∗ Corresponding
author: Email address: [email protected] (Q. Nguyen)
Preprint submitted to Journal of Physica A
September 14, 2018
rare-but-extreme volatile situations [2, 3, 4, 5, 6, 7]. One of the views is that 5
we consider the financial market as a complex system and use appropriate tools and methods to analyze it. The theory of networks or graphs is one of the methods for studying such complex systems [8, 9, 10, 11, 12, 13, 14]. A network is simply a collection of vertices (or nodes) and edges that link between a pair of nodes. In a financial
10
market, nodes are usually modeled as stock companies and edges between any pair of stocks are created if the co-movement of their prices has some predefined properties. Extensive studies have been done for the network created by using the minimum spanning tree (MST) method. MST was firstly introduced in 1999, when
15
Mantegna [15] found a topological arrangement of stocks traded in a financial market which had associated a meaningful economic taxonomy. Following this work, a lot of subsequent studies constructed on MST have been published, for instance [16, 17, 18, 19, 20, 21, 22]. Another structural network property was found is the power-law distribution
20
of the MST degree. This scale-free behavior (the power-law degree distribution) was found by Vandewalle et al. [23] for the asset tree of US stocks in a limited time window. The scale-free property was also discovered on the dynamic asset trees of many different stock exchanges such as New York (NYSE) [24], Athen (ASE) [25], Warsaw (WSE) [26] and Frankfurt (FSE) [1]. It is found that such
25
a scale-free network is extremely stable under random breakdown but fragile under intentional attacks [27, 28]. The scale-free structure is, therefore, a crucial property for the stability of a financial network under different types of shock [29]. In this work, we analyze the topology and complexity properties and
30
dynamics of the MST of the Vietnamese stock cross-correlation network. We collect data from January 9th, 2008 to December 31st, 2017 and construct a MST graph for each time window of 390 days, using a sliding period of 60 days. We found that most of the time, the MST node degree distribution obeys a power law. Meanwhile, the MST constructed during the period of 2012-2013 2
35
has star-like structure − a large percentage of nodes linked to a super hub (see Figure 2a). The degree of this node is far from the power-law fit (Figure 2b). Before and after this specific period, the MST degree distribution has a normal power-law distribution. We consider that the market has gone through 3 distinct phases, which is similar to the result of [1], in which the second phases
40
happened during the world financial crisis of 2008. Unlike result in [1], the period where the Vietnamese market has the star-like structure is 2012-2013 when the Vietnamese economy was greatly impacted by a financial stressing period between 2011 and mid-2012. During this period the interest rate is particularly high when it went from 10% to as high as 30%/year in a short time
45
of 8 months. In reality, the impact of this period is even more serious than the external impact of the international financial crisis of 2008. To some extends, Lacasa et al. [30] also found a star-like maximum spanning tree with a super hub that directly linked to as much as 50% of all other nodes during the financial instability period.
50
Parallely, we compute a number of specific measures of a complex network and analyze their temporal dynamics. We particularly focus on the allometric coefficient measure, which characterizes the complexity structure of a graph. This allometric exponent η lies between 1 and 2 for two extreme network structures: 1+ for a star network where all nodes (except the center one) connect
55
to a single center-node; 2− for a chain-like network where all nodes are linked one after another [31]. We found a temporal agreement of η together with the topology phase change mentioned above: η varies between 1.22 and 1.25 during the 2nd phase and distinguishes higher (from 1.26 to 1.36) during the 1st and 3rd phase (Figure 5). Using allometric exponent to characterize the financial
60
network complexity is a novel idea and according to our knowledge, only a few works have been done, for instance, [32, 33]. Using a single market index time series, Qian et al. [32] constructed a visibility graph where each data point is a node and an edge is drawn to connect two nodes according to the rule that the two corresponding data points can see each other in the diagram of the time
65
series. MST graph is extracted from this visibility graph using the minimum 3
weight and no-loop rule. The authors examined 30 worldwide stock indices and found a universal value of the allometric exponent, η = 1.264 ± 0.002. While our MST is constructed from multiple time series of stock prices, its allometric coefficients are found to be in agreement with this study. 70
In another work, Lautier et al. [33] analyzed the MST constructed from future contracts in 14 derivatives markets, which is a subset of a larger graph of 250 future contracts with different maturities [34]. The authors computed the corresponding allometric coefficients and found a value of 1.493. Analysis of allometric coefficient in financial MST suggests that the shape of
75
the financial network is rather complex and stands in between the two asymptotic topologies, the star and chain- structure, with more bias to the starstructure. This allometric exponent η is, in fact, the fitting power of the relation between the 2 variables defined for each node A and C in an iterative manner as follow: Ai =
X
Aj + 1 and Ci =
j
80
X
Cj + Ai
(1)
j
where j stands for all nodes connected from node i [31]. Then the allometric scaling relation is picked out by the power-law relation between Ci and Ai : C ∼ Aη
(2)
where the leaf nodes with A = C = 1 have to be ejected from fitting the scaling exponent η [35]. In the river network example in [31], A stands for the total water flow coming 85
from the sub-basin area around each node and C stands for the total water flow that goes through this node through the drainage direction. From the perfect power-law relation between A and C found here (Figure 4 below), naturally, we suggest that there must be some meanings for A and C in the financial network. We propose that A could be considered as the total transactional volume. For
90
C, we need a measure that quantifies the impact of a stock throughout the market. Further detail analysis is therefore expected in the future. The paper is organized as follows. In Section 2, we describe the data and
4
methods used in this work. Section 3 presents the empirical findings: the topology analysis in 3.1 and the complexity analysis in 3.2. Finally, Section 4 sum95
marizes the results and concludes.
2. Methods and data In this work, we consider companies listed in the Hochiminh Stock Exchange (HSX). The other stock exchange, the Hanoi Stock Exchange (HNX) is quite small in term of market capital (about 10 times smaller) and therefore is ne100
glected in this study. We analyze the daily stock price in HSX from January 9th, 2008 to December 31st, 2017, consisting of 2496 trading days. In order to guaranty a statistical stability and a temporal sensitivity, we select a window length of T = 390 trading days and roll forward at τ = 60 days interval. As a result, we obtain 36 temporal windows.
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In each window, we select stocks whose number of trading days are more than 80% of the total window length and have a minimum trading volume of 1000 shares/day. Because the Vietnamese market is emerging, the market size is growing fast, newly listed companies appeared regularly during our studying period. In consequence, the number of stocks in each window increases from
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119 to 266. Cross-correlation matrix. In each time window, let N is the number of stocks satisfying our filtering conditions. The daily return ri (t) of stock i i = 1, N at day t t = 1, T is calculated by: ri (t) = ln (Si (t)) − ln (Si (t − 1))
(3)
where Si (t) is the closing price of stock i at day t. 115
Then the correlation coefficient cij between stock i and stock j is computed using all complete pairs of observations on their daily returns, i.e. cij =
hri (t).rj (t)i − hri (t)i . hrj (t)i σi .σj
5
(4)
where σi , σj are the standard deviations of daily return series of stock i and stock j respectively and h·i denotes the average over time. The N × N matrix C = (cij ) is called the cross-correlation matrix between 120
the N stocks. Stock cross-correlation network. : From cij , a metric dij is defined by [15]: dij =
q
2 (1 − cij ) .
(5)
dij is non-negative and get value of 0 if and only if cij equals 1. The N × N matrix D = (dij ) is considered as the distance matrix between the N nodes representing stocks. The stock cross-correlation network is fully 125
connected with N (N − 1) /2 edges whose weights are the distances between the nodes. This weighted complete graph contains all the possible connections and their strengths in the stock network. Minimum spanning tree of stock cross-correlation network. In order to reduce the high number of edge N (N − 1) /2 of the complete network, MST can be
130
used. The MST is unique and corresponds to the shortest path covering all the nodes of the initial network without loops. This MST has only N − 1 edges and it seems to extract the most important information contained in the full network. For the study of systematic risk, MST is considered to be the most probable path for the transmission of a price shock.
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Other measures. • The normalized average of shortest path length is defined by: l=
hlij i N
(6)
where lij is the length of shortest path from node i to node j, h·i is the average over all pairs of nodes. This normalized procedure helps to compare the shortest path lengths, in average, of a price shock’s propagation 140
in our 36 temporal windows.
6
• Same sector ratio: the fraction of the number of MST’s edges which connect two stocks belonging to a same business sector. The sector classification is obtained from www.cophieu68.vn. • Survival ratio: is defined as the ratio of number of common edges found 145
in the two consecutive trees to the average number of edges of the two consecutive trees. It measures the stability of the MST topology over time.
3. Results and discussion 3.1. Topology analysis 150
A scaled-free MST, constructed from the window between Mars 31st, 2009 and October 19th, 2010 with N = 140 stocks, is shown in Figure 1a. In Figure 1b, the corresponding degree of distribution f (k) vs. node degree k is shown in a log-log plot. This distribution fits well with a power-law equation, using the least square method, which results in a slope equals to −1.755 ± 0.14. This
155
value is significantly lower than the ones found in the German market [1] but very closes to the result found in the Warsaw Stock Exchange (WSE) [26], which equals −1.97 ± 0.13, and result from S&P 500 companies mentioned in [36], which approximates to −1.8. The figure 1a represents the business sector of each node from which the same sector ratio will be calculated, which is relatively
160
small. The temporal evolution of the same sector ratio will be represented in the Figure 4c. The small value of the same sector ratio which results in a weak sector segmentation in the MST graph, however, is not the main focus of this paper and will be addressed in a future work. As we move forward, we found that a giant node emerges with increasing
165
degree, which corresponds to the stock SSI (the Saigon Securities Incorporation) - a stock brokerage company. The network starts to form a hub around SSI during the period of 2011 - 2012. The most remarkable star-like MST is found by the time window between May 16th, 2012 and December 02nd, 2013 with N = 232 stocks, as presented in the Figure 2a. This is also the reason why the 7
(a)
(b) Figure 1: (a) The MST of stock cross-correlation network constructed from the window
8
03/31/2009 - 10/19/2010. Nodes’ color represent the business sectors. As there is a few telecommunication companies, they are grouped with IT sector. Both energy and utilities sector in Vietnam are dominated by Oil and Gas companies, these two sectors are also grouped by many analysts. (b) its degree distribution.
170
resulting length of this specific window of 390 trading, was used as our window length throughout the whole time frame. In this MST, the SSI’s degree is 67, in another word, it links directly to 28.9% of the network’s nodes. (This connecting proportion is extremely high when compared to ones of other nodes in this MST which just range from 0.4%
175
to 5.6%). Consequently, this MST has a star-like structure like the one found in [1]. Figure 2b shows its degree distribution and we can clearly see that the point corresponding to SSI’s degree is located far from the rest. However, by neglecting this point, the remained distribution fits well a power law whose slope equals to −2.23 ± 0.22.
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In Figure 3a, we show the topology of the MST obtained from the time window between January 14th, 2014 and August 18th, 2015. There are N = 249 stocks selected in this time window. We found that the network returns to a hierarchical scale-free MST. Its corresponding degree distribution, shown in Figure 3b, follows a good power law whose slope equals to −1.93 ± 0.16. As we
185
can see, the most connected nodes of the MST in Figure 1 and Figure 3 only own a connecting proportion of 11.5% and 8.9%, respectively, in comparison to the SSI in the MST in Figure 2 which links to 28.9% of the network’s nodes. The transition between the 1st, 2nd and the 3rd MST is somewhat gradual if only look at the graphs. Next, we compute other quantitative network
190
measures in order to find a proper dynamical analytical description of the network structure. We suppose that the normalized average shortest path length described in Section 2 would be a good candidate measure to identify a star-like network, as suggested by [37]. In a star-like network, the super hub serves as an intermediate center. For all nodes linked to this point, the shortest path
195
between any pair of them goes through the center node and consists of 2 edges only. In consequence, the network normalized average shortest path length is expected to low in comparison to that of other network structures. Figure 4a represents the temporal evolution of the normalized average shortest path length. We found that the value for the second MST is clearly lower
200
than the rests. This dynamic helps us to define the 3 distinguished phases as 9
(a)
(b) Figure 2: (a) The MST of stock cross-correlation network constructed from the window 10 05/16/2012 - 12/02/2013 and (b) its degree distribution.
(a)
(b) Figure 3: (a) The MST of stock cross-correlation network constructed from the window 11 01/14/2014 - 08/18/2015 and (b) its degree distribution.
(a)
(b)
(c) Figure 4: Temporal evolution of (a) the normalized average shortest path (left axis) and CPI (right axis); (b) the survival ratio and (c) the same sector ratio. The time index reports the final date of the corresponding window.
12
shown by the inserted vertical lines. We also highlight the point corresponding to the three MSTs illustrated in Figure 1, 2 and 3. Other measures also confirm this phase’s separation, meanwhile less clearly. In Figure 4b, the survival ratio is calculated in order to analyze the stability of 205
the MST. Using the sliding period of 60 days, the survival ratio remains high with a value greater than 0.95 most of the time. It is lowest at the end of phase I and at the beginning of phase II when the organization of the star-like structure happens. It remains high and stable throughout the rest of phase II. It decreases when phase III begins since the super hub has dissolved, but much
210
less important than when it was created. We found that there is a part of the super hub’s connection unbroken and SSI is still the most important hub during phase III. In Figure 4c, we calculate the ratio of edges that connect stocks from a same sector. It was known that MST can help to identify stock groups or sectors thus we would like to verify how the change of the network structure affects the
215
sector segmentation. We found that the same sector ratio is in average lowest during the 2nd phase. It is understandable because sector groups are usually located in local branches of the MST, so when the super hub is created, many of these branches are dissolved. The MSTs’ topology with sectors as node colors has confirmed this observation (they are not shown in this paper but available
220
upon request). An important question is that what make the MST of stock cross-correlation network change through the three mentioned phases. As mention in the introduction, in the year 2011, Vietnam was in a financial stressing situation where interest rate jumped as high as 30% (see Figure 4a (right axis) where the inter-
225
est rate as a function of time is shown). This fact happened when the country was in a recession that caused VNIndex, the index represents all stocks listed on HSX, to decrease 27.5% in that year (Source: Bloomberg). From 2012 to 2013, the interest rate decreased gradually and the stock market also retrieved from the bottom.
230
In this situation, the economy or, in particular, the financial market can be considered as in a systemic risk period, which results in the topological 13
structure’s change of the stock cross-correlation network’s MST. Inversely, if there is any early topological structure change in the same direction, it could be an indication of the market crisis. 235
3.2. Allometric analysis In this section, we study the allometric scaling property of our network. For each MST, we carry out an allometric analysis using the method described in Section 2: two sequences of A and C are calculated and the allometric exponent is estimated using linear regression of log A on log C. We found that all the trees
240
exhibit nice allometric scaling behaviors. Figure 5 shows the allometric scaling behaviors of 3 MSTs selected from 3 phases that are presented previously. For the MST from the 2nd phase which has a star-like structure, η = 1.213 ± 0.013, while η is significantly higher in the other two phases: 1.289 ± 0.011 in the 1st phase and 1.301 ± 0.011 in the 3rd phase. Those values behave as predicted by
245
[31]: a star-like network will have a value close to 1+ (the MST in phase II) and a chain-like network will have a value close to 2− (two MSTs in the 1st and 3rd phase). Figure 6 shows the temporal dynamic of the allometric exponent together with that of the normalized average shortest path length. We found a good synchronization between these two measures. A shorter normalized
250
average shortest path results in a low allometric exponent, which corresponds to a more star-like structure, and vice versa. Finally, we provide some interpretations of the variables A and C in our experiments. In the river network example in [31], A stands for the total water flow coming from the sub-basin area around each node and C stands for the
255
total water flow that goes through this node through the drainage direction. We suppose that in our financial context, the analogy of A is the average transactional volume of each stock, while C represents the total impact of the stock toward the network. A is obviously measurable and available, however, C is less quantifiable. Therefore, we suggest using the scaling relationship in order
260
to estimate C, given A, the average daily volume. In case of a financial crisis, the value of C could represent how important the stock influences other stocks 14
Figure 5: The allometric scaling behaviors of three MSTs constructed on the three phases.
in the market. For illustration, in Figure 7 we plot the MSTs taken from phase II and phase III with log(C) as the nodes’ size. We found that in phase 2, there is only one 265
stock that has high impact (C). If there is a shock, it is very probable that it starts from this stock and transferred almost instantly to the whole network. In contrast, with the network in phase III, there are several important stocks that lie along the main path. A shock, if it happened among one of those stocks, will go to other important stock before spreading throughout the network. This
270
dynamic may have a considerable importance in studying the systemic risk.
4. Conclusion In summary, basing on the network constructed from stocks trading in Hochiminh Stock Exchange (Vietnam), we have investigated the temporal dynamics of its minimum spanning tree. We found that the market went through 275
3 phases: a hierarchical scale-free MST structure, then a star-like MST structure during the period of 2011-2012, then again a hierarchical scale-free MST structure. This star-like structure of MST was found in German market during the world financial crisis in 2008, therefore, we suppose that the Vietnamese 15
Figure 6: The temporal dynamic of the allometric exponent and the normalized average shortest path length.
financial stressing period of 2010-2011 is the root of the structural change in the 280
observed MSTs. The dynamic of other measures such as the normalized average shortest path length and the survival ratio also confirmed the phase changes. In addition, we analyzed the allometric scaling behavior of the network’s MST and found that the allometric scaling property is applied to the MST all the investigating time. Furthermore, we found the scaling exponent varies with the
285
structure change of the network. It is as low as 1.213 when the MST is star-like and is in average 1.300 when the MST is scale-free. The value of η may help us to estimate the impact of the stock during a crisis time as well as to estimate the “dimension” of the MST’s structure. Further investigation of the allometric scaling behavior is expected in the future.
290
5. Acknowledgements This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2017-42-01. We are grateful to the anonymous referee for showing us a wrong comment (that has been removed), suggesting us the reference [34] and other helpful suggestions. 16
(a)
(b)
Figure 7: The MSTs taken from (a) phase II and (b) phase III with log(C) as the nodes’ size.
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PII: DOI: Reference:
S0378-4371(18)31193-2 https://doi.org/10.1016/j.physa.2018.09.061 PHYSA 20122
To appear in:
Physica A
Received date : 14 May 2018 Revised date : 14 September 2018 Please cite this article as: Q. Nguyen, et al., Dynamic topology and allometric scaling behavior on the Vietnamese stock market, Physica A (2018), https://doi.org/10.1016/j.physa.2018.09.061 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Cover letter and Highlights
Main highlights of manuscript “Dynamic topology and allometric scaling behavior on the Vietnamese stock market” - A star-like MST constructed from the Vietnamese stock price was observed during 2011-2012 - During this period the country was in a financial crisis with interest rate of >30% - It is similar to that of some market during the world-wide financial crisis of 2008 - All temporal MST has an allometric scaling behavior - Their exponent are from 1.2 during the star-like phase, to more than 1.3 otherwise
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Dynamic topology and allometric scaling behavior on the Vietnamese stock market Q. Nguyena,∗, N.K.K. Nguyenb , L.H.N. Nguyena a John
Von Neumann Institute, Vietnam National University Ho chi minh City, Ho chi minh City, Vietnam b Van Lang University, Ho chi minh City, Vietnam
Abstract The impact of the Vietnamese financial crisis during 2011-2012 into the stock market was revealed by a structural change of the minimum spanning tree (MST) constructed from the daily stock price. We found that the MST has a star-like structure during this period, similar to that of the German market during the worldwide financial crisis of 2007-2008 [1], and a hierarchical scalefree structure for the rest of time. In addition, we investigate the market from a complex network perspective by analyzing the allometric scaling behavior. We found that all networks have the allometric scaling property, with exponent η ranging from 1.213 ± 0.013 during the financial instability period to about 1.357 ± 0.011 in normal time. These values correspond to a complex “dimension” of the financial market of between 3 and 5, which need to be further investigated in the future. Keywords: financial crisis, complex network, minimum spanning tree, allometric scaling
1. Introduction After the worldwide financial crisis of 2008-2009, it has been shown that traditional financial and economic model needs major revise, especially under ∗ Corresponding
author: Email address: [email protected] (Q. Nguyen)
Preprint submitted to Journal of Physica A
September 14, 2018
rare-but-extreme volatile situations [2, 3, 4, 5, 6, 7]. One of the views is that 5
we consider the financial market as a complex system and use appropriate tools and methods to analyze it. The theory of networks or graphs is one of the methods for studying such complex systems [8, 9, 10, 11, 12, 13, 14]. A network is simply a collection of vertices (or nodes) and edges that link between a pair of nodes. In a financial
10
market, nodes are usually modeled as stock companies and edges between any pair of stocks are created if the co-movement of their prices has some predefined properties. Extensive studies have been done for the network created by using the minimum spanning tree (MST) method. MST was firstly introduced in 1999, when
15
Mantegna [15] found a topological arrangement of stocks traded in a financial market which had associated a meaningful economic taxonomy. Following this work, a lot of subsequent studies constructed on MST have been published, for instance [16, 17, 18, 19, 20, 21, 22]. Another structural network property was found is the power-law distribution
20
of the MST degree. This scale-free behavior (the power-law degree distribution) was found by Vandewalle et al. [23] for the asset tree of US stocks in a limited time window. The scale-free property was also discovered on the dynamic asset trees of many different stock exchanges such as New York (NYSE) [24], Athen (ASE) [25], Warsaw (WSE) [26] and Frankfurt (FSE) [1]. It is found that such
25
a scale-free network is extremely stable under random breakdown but fragile under intentional attacks [27, 28]. The scale-free structure is, therefore, a crucial property for the stability of a financial network under different types of shock [29]. In this work, we analyze the topology and complexity properties and
30
dynamics of the MST of the Vietnamese stock cross-correlation network. We collect data from January 9th, 2008 to December 31st, 2017 and construct a MST graph for each time window of 390 days, using a sliding period of 60 days. We found that most of the time, the MST node degree distribution obeys a power law. Meanwhile, the MST constructed during the period of 2012-2013 2
35
has star-like structure − a large percentage of nodes linked to a super hub (see Figure 2a). The degree of this node is far from the power-law fit (Figure 2b). Before and after this specific period, the MST degree distribution has a normal power-law distribution. We consider that the market has gone through 3 distinct phases, which is similar to the result of [1], in which the second phases
40
happened during the world financial crisis of 2008. Unlike result in [1], the period where the Vietnamese market has the star-like structure is 2012-2013 when the Vietnamese economy was greatly impacted by a financial stressing period between 2011 and mid-2012. During this period the interest rate is particularly high when it went from 10% to as high as 30%/year in a short time
45
of 8 months. In reality, the impact of this period is even more serious than the external impact of the international financial crisis of 2008. To some extends, Lacasa et al. [30] also found a star-like maximum spanning tree with a super hub that directly linked to as much as 50% of all other nodes during the financial instability period.
50
Parallely, we compute a number of specific measures of a complex network and analyze their temporal dynamics. We particularly focus on the allometric coefficient measure, which characterizes the complexity structure of a graph. This allometric exponent η lies between 1 and 2 for two extreme network structures: 1+ for a star network where all nodes (except the center one) connect
55
to a single center-node; 2− for a chain-like network where all nodes are linked one after another [31]. We found a temporal agreement of η together with the topology phase change mentioned above: η varies between 1.22 and 1.25 during the 2nd phase and distinguishes higher (from 1.26 to 1.36) during the 1st and 3rd phase (Figure 5). Using allometric exponent to characterize the financial
60
network complexity is a novel idea and according to our knowledge, only a few works have been done, for instance, [32, 33]. Using a single market index time series, Qian et al. [32] constructed a visibility graph where each data point is a node and an edge is drawn to connect two nodes according to the rule that the two corresponding data points can see each other in the diagram of the time
65
series. MST graph is extracted from this visibility graph using the minimum 3
weight and no-loop rule. The authors examined 30 worldwide stock indices and found a universal value of the allometric exponent, η = 1.264 ± 0.002. While our MST is constructed from multiple time series of stock prices, its allometric coefficients are found to be in agreement with this study. 70
In another work, Lautier et al. [33] analyzed the MST constructed from future contracts in 14 derivatives markets, which is a subset of a larger graph of 250 future contracts with different maturities [34]. The authors computed the corresponding allometric coefficients and found a value of 1.493. Analysis of allometric coefficient in financial MST suggests that the shape of
75
the financial network is rather complex and stands in between the two asymptotic topologies, the star and chain- structure, with more bias to the starstructure. This allometric exponent η is, in fact, the fitting power of the relation between the 2 variables defined for each node A and C in an iterative manner as follow: Ai =
X
Aj + 1 and Ci =
j
80
X
Cj + Ai
(1)
j
where j stands for all nodes connected from node i [31]. Then the allometric scaling relation is picked out by the power-law relation between Ci and Ai : C ∼ Aη
(2)
where the leaf nodes with A = C = 1 have to be ejected from fitting the scaling exponent η [35]. In the river network example in [31], A stands for the total water flow coming 85
from the sub-basin area around each node and C stands for the total water flow that goes through this node through the drainage direction. From the perfect power-law relation between A and C found here (Figure 4 below), naturally, we suggest that there must be some meanings for A and C in the financial network. We propose that A could be considered as the total transactional volume. For
90
C, we need a measure that quantifies the impact of a stock throughout the market. Further detail analysis is therefore expected in the future. The paper is organized as follows. In Section 2, we describe the data and
4
methods used in this work. Section 3 presents the empirical findings: the topology analysis in 3.1 and the complexity analysis in 3.2. Finally, Section 4 sum95
marizes the results and concludes.
2. Methods and data In this work, we consider companies listed in the Hochiminh Stock Exchange (HSX). The other stock exchange, the Hanoi Stock Exchange (HNX) is quite small in term of market capital (about 10 times smaller) and therefore is ne100
glected in this study. We analyze the daily stock price in HSX from January 9th, 2008 to December 31st, 2017, consisting of 2496 trading days. In order to guaranty a statistical stability and a temporal sensitivity, we select a window length of T = 390 trading days and roll forward at τ = 60 days interval. As a result, we obtain 36 temporal windows.
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In each window, we select stocks whose number of trading days are more than 80% of the total window length and have a minimum trading volume of 1000 shares/day. Because the Vietnamese market is emerging, the market size is growing fast, newly listed companies appeared regularly during our studying period. In consequence, the number of stocks in each window increases from
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119 to 266. Cross-correlation matrix. In each time window, let N is the number of stocks satisfying our filtering conditions. The daily return ri (t) of stock i i = 1, N at day t t = 1, T is calculated by: ri (t) = ln (Si (t)) − ln (Si (t − 1))
(3)
where Si (t) is the closing price of stock i at day t. 115
Then the correlation coefficient cij between stock i and stock j is computed using all complete pairs of observations on their daily returns, i.e. cij =
hri (t).rj (t)i − hri (t)i . hrj (t)i σi .σj
5
(4)
where σi , σj are the standard deviations of daily return series of stock i and stock j respectively and h·i denotes the average over time. The N × N matrix C = (cij ) is called the cross-correlation matrix between 120
the N stocks. Stock cross-correlation network. : From cij , a metric dij is defined by [15]: dij =
q
2 (1 − cij ) .
(5)
dij is non-negative and get value of 0 if and only if cij equals 1. The N × N matrix D = (dij ) is considered as the distance matrix between the N nodes representing stocks. The stock cross-correlation network is fully 125
connected with N (N − 1) /2 edges whose weights are the distances between the nodes. This weighted complete graph contains all the possible connections and their strengths in the stock network. Minimum spanning tree of stock cross-correlation network. In order to reduce the high number of edge N (N − 1) /2 of the complete network, MST can be
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used. The MST is unique and corresponds to the shortest path covering all the nodes of the initial network without loops. This MST has only N − 1 edges and it seems to extract the most important information contained in the full network. For the study of systematic risk, MST is considered to be the most probable path for the transmission of a price shock.
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Other measures. • The normalized average of shortest path length is defined by: l=
hlij i N
(6)
where lij is the length of shortest path from node i to node j, h·i is the average over all pairs of nodes. This normalized procedure helps to compare the shortest path lengths, in average, of a price shock’s propagation 140
in our 36 temporal windows.
6
• Same sector ratio: the fraction of the number of MST’s edges which connect two stocks belonging to a same business sector. The sector classification is obtained from www.cophieu68.vn. • Survival ratio: is defined as the ratio of number of common edges found 145
in the two consecutive trees to the average number of edges of the two consecutive trees. It measures the stability of the MST topology over time.
3. Results and discussion 3.1. Topology analysis 150
A scaled-free MST, constructed from the window between Mars 31st, 2009 and October 19th, 2010 with N = 140 stocks, is shown in Figure 1a. In Figure 1b, the corresponding degree of distribution f (k) vs. node degree k is shown in a log-log plot. This distribution fits well with a power-law equation, using the least square method, which results in a slope equals to −1.755 ± 0.14. This
155
value is significantly lower than the ones found in the German market [1] but very closes to the result found in the Warsaw Stock Exchange (WSE) [26], which equals −1.97 ± 0.13, and result from S&P 500 companies mentioned in [36], which approximates to −1.8. The figure 1a represents the business sector of each node from which the same sector ratio will be calculated, which is relatively
160
small. The temporal evolution of the same sector ratio will be represented in the Figure 4c. The small value of the same sector ratio which results in a weak sector segmentation in the MST graph, however, is not the main focus of this paper and will be addressed in a future work. As we move forward, we found that a giant node emerges with increasing
165
degree, which corresponds to the stock SSI (the Saigon Securities Incorporation) - a stock brokerage company. The network starts to form a hub around SSI during the period of 2011 - 2012. The most remarkable star-like MST is found by the time window between May 16th, 2012 and December 02nd, 2013 with N = 232 stocks, as presented in the Figure 2a. This is also the reason why the 7
(a)
(b) Figure 1: (a) The MST of stock cross-correlation network constructed from the window
8
03/31/2009 - 10/19/2010. Nodes’ color represent the business sectors. As there is a few telecommunication companies, they are grouped with IT sector. Both energy and utilities sector in Vietnam are dominated by Oil and Gas companies, these two sectors are also grouped by many analysts. (b) its degree distribution.
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resulting length of this specific window of 390 trading, was used as our window length throughout the whole time frame. In this MST, the SSI’s degree is 67, in another word, it links directly to 28.9% of the network’s nodes. (This connecting proportion is extremely high when compared to ones of other nodes in this MST which just range from 0.4%
175
to 5.6%). Consequently, this MST has a star-like structure like the one found in [1]. Figure 2b shows its degree distribution and we can clearly see that the point corresponding to SSI’s degree is located far from the rest. However, by neglecting this point, the remained distribution fits well a power law whose slope equals to −2.23 ± 0.22.
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In Figure 3a, we show the topology of the MST obtained from the time window between January 14th, 2014 and August 18th, 2015. There are N = 249 stocks selected in this time window. We found that the network returns to a hierarchical scale-free MST. Its corresponding degree distribution, shown in Figure 3b, follows a good power law whose slope equals to −1.93 ± 0.16. As we
185
can see, the most connected nodes of the MST in Figure 1 and Figure 3 only own a connecting proportion of 11.5% and 8.9%, respectively, in comparison to the SSI in the MST in Figure 2 which links to 28.9% of the network’s nodes. The transition between the 1st, 2nd and the 3rd MST is somewhat gradual if only look at the graphs. Next, we compute other quantitative network
190
measures in order to find a proper dynamical analytical description of the network structure. We suppose that the normalized average shortest path length described in Section 2 would be a good candidate measure to identify a star-like network, as suggested by [37]. In a star-like network, the super hub serves as an intermediate center. For all nodes linked to this point, the shortest path
195
between any pair of them goes through the center node and consists of 2 edges only. In consequence, the network normalized average shortest path length is expected to low in comparison to that of other network structures. Figure 4a represents the temporal evolution of the normalized average shortest path length. We found that the value for the second MST is clearly lower
200
than the rests. This dynamic helps us to define the 3 distinguished phases as 9
(a)
(b) Figure 2: (a) The MST of stock cross-correlation network constructed from the window 10 05/16/2012 - 12/02/2013 and (b) its degree distribution.
(a)
(b) Figure 3: (a) The MST of stock cross-correlation network constructed from the window 11 01/14/2014 - 08/18/2015 and (b) its degree distribution.
(a)
(b)
(c) Figure 4: Temporal evolution of (a) the normalized average shortest path (left axis) and CPI (right axis); (b) the survival ratio and (c) the same sector ratio. The time index reports the final date of the corresponding window.
12
shown by the inserted vertical lines. We also highlight the point corresponding to the three MSTs illustrated in Figure 1, 2 and 3. Other measures also confirm this phase’s separation, meanwhile less clearly. In Figure 4b, the survival ratio is calculated in order to analyze the stability of 205
the MST. Using the sliding period of 60 days, the survival ratio remains high with a value greater than 0.95 most of the time. It is lowest at the end of phase I and at the beginning of phase II when the organization of the star-like structure happens. It remains high and stable throughout the rest of phase II. It decreases when phase III begins since the super hub has dissolved, but much
210
less important than when it was created. We found that there is a part of the super hub’s connection unbroken and SSI is still the most important hub during phase III. In Figure 4c, we calculate the ratio of edges that connect stocks from a same sector. It was known that MST can help to identify stock groups or sectors thus we would like to verify how the change of the network structure affects the
215
sector segmentation. We found that the same sector ratio is in average lowest during the 2nd phase. It is understandable because sector groups are usually located in local branches of the MST, so when the super hub is created, many of these branches are dissolved. The MSTs’ topology with sectors as node colors has confirmed this observation (they are not shown in this paper but available
220
upon request). An important question is that what make the MST of stock cross-correlation network change through the three mentioned phases. As mention in the introduction, in the year 2011, Vietnam was in a financial stressing situation where interest rate jumped as high as 30% (see Figure 4a (right axis) where the inter-
225
est rate as a function of time is shown). This fact happened when the country was in a recession that caused VNIndex, the index represents all stocks listed on HSX, to decrease 27.5% in that year (Source: Bloomberg). From 2012 to 2013, the interest rate decreased gradually and the stock market also retrieved from the bottom.
230
In this situation, the economy or, in particular, the financial market can be considered as in a systemic risk period, which results in the topological 13
structure’s change of the stock cross-correlation network’s MST. Inversely, if there is any early topological structure change in the same direction, it could be an indication of the market crisis. 235
3.2. Allometric analysis In this section, we study the allometric scaling property of our network. For each MST, we carry out an allometric analysis using the method described in Section 2: two sequences of A and C are calculated and the allometric exponent is estimated using linear regression of log A on log C. We found that all the trees
240
exhibit nice allometric scaling behaviors. Figure 5 shows the allometric scaling behaviors of 3 MSTs selected from 3 phases that are presented previously. For the MST from the 2nd phase which has a star-like structure, η = 1.213 ± 0.013, while η is significantly higher in the other two phases: 1.289 ± 0.011 in the 1st phase and 1.301 ± 0.011 in the 3rd phase. Those values behave as predicted by
245
[31]: a star-like network will have a value close to 1+ (the MST in phase II) and a chain-like network will have a value close to 2− (two MSTs in the 1st and 3rd phase). Figure 6 shows the temporal dynamic of the allometric exponent together with that of the normalized average shortest path length. We found a good synchronization between these two measures. A shorter normalized
250
average shortest path results in a low allometric exponent, which corresponds to a more star-like structure, and vice versa. Finally, we provide some interpretations of the variables A and C in our experiments. In the river network example in [31], A stands for the total water flow coming from the sub-basin area around each node and C stands for the
255
total water flow that goes through this node through the drainage direction. We suppose that in our financial context, the analogy of A is the average transactional volume of each stock, while C represents the total impact of the stock toward the network. A is obviously measurable and available, however, C is less quantifiable. Therefore, we suggest using the scaling relationship in order
260
to estimate C, given A, the average daily volume. In case of a financial crisis, the value of C could represent how important the stock influences other stocks 14
Figure 5: The allometric scaling behaviors of three MSTs constructed on the three phases.
in the market. For illustration, in Figure 7 we plot the MSTs taken from phase II and phase III with log(C) as the nodes’ size. We found that in phase 2, there is only one 265
stock that has high impact (C). If there is a shock, it is very probable that it starts from this stock and transferred almost instantly to the whole network. In contrast, with the network in phase III, there are several important stocks that lie along the main path. A shock, if it happened among one of those stocks, will go to other important stock before spreading throughout the network. This
270
dynamic may have a considerable importance in studying the systemic risk.
4. Conclusion In summary, basing on the network constructed from stocks trading in Hochiminh Stock Exchange (Vietnam), we have investigated the temporal dynamics of its minimum spanning tree. We found that the market went through 275
3 phases: a hierarchical scale-free MST structure, then a star-like MST structure during the period of 2011-2012, then again a hierarchical scale-free MST structure. This star-like structure of MST was found in German market during the world financial crisis in 2008, therefore, we suppose that the Vietnamese 15
Figure 6: The temporal dynamic of the allometric exponent and the normalized average shortest path length.
financial stressing period of 2010-2011 is the root of the structural change in the 280
observed MSTs. The dynamic of other measures such as the normalized average shortest path length and the survival ratio also confirmed the phase changes. In addition, we analyzed the allometric scaling behavior of the network’s MST and found that the allometric scaling property is applied to the MST all the investigating time. Furthermore, we found the scaling exponent varies with the
285
structure change of the network. It is as low as 1.213 when the MST is star-like and is in average 1.300 when the MST is scale-free. The value of η may help us to estimate the impact of the stock during a crisis time as well as to estimate the “dimension” of the MST’s structure. Further investigation of the allometric scaling behavior is expected in the future.
290
5. Acknowledgements This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2017-42-01. We are grateful to the anonymous referee for showing us a wrong comment (that has been removed), suggesting us the reference [34] and other helpful suggestions. 16
(a)
(b)
Figure 7: The MSTs taken from (a) phase II and (b) phase III with log(C) as the nodes’ size.
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