Can financial crisis be detected? Laplacian energy measure

ABSTRACT

How to rapidly and accurately detect the financial crisis is one of the fundamental and challenging problems in the field of financial risk management. This paper aims to develop a novel network characteristic indicator to deal with this issue. Specifically, we select the daily closing price of stocks spanning from 2006 to 2020 in China’s A-share market to establish a series of complex networks, and extract Laplacian energy measure as a new network indicator. By employing the method of seasonal-trend decomposition procedure based on loess, the proposed indicator successfully detects the global financial crisis, the Eurozone debt crisis, the Chinese stock market crash, the Sino-US trade friction and the COVID-19 pandemic. Furthermore, compared with the traditional topological indicators (e.g. global efficiency, average clustering coefficient, characteristic path length and network density), the proposed indicator demonstrates the outstanding characteristics of higher identification accuracy, wider application range and faster response speed. Lastly, the robustness of the Laplacian energy measure in the financial crisis detection is further confirmed in the US, UK, German, French and Spanish stock markets.

KEYWORDS:

• Financial crisis
• complex network
• Laplacian energy
• network structure
• seasonal-trend decomposition procedure based on loess (STL)

Acknowledgements

The authors are grateful to the editor and anonymous reviewers for their constructive comments, which led to a significant improvement of our original manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The 2008 global financial crisis is always interpreted as the ‘gale of creative destruction’ by economists and economic historians (Brem, Nylund, and Viardot 2020).

2 Closeness is a measure of the degree to which a node is near all other nodes in a network, which can be calculated by the inverse of the sum of the shortest distances between each node and every other node in the network (Wang, Li, and Chen 2006).

3 LE-oriented indicator: inspired by the dominant view of Laplacian energy, this paper aims to construct a leading indicator to rapidly and accurately detect the financial crisis.

4 Let $L$ be a $n\times n$ square matrix. If there is a vector $X\in {\mathbb{R}}^n\ne 0$ such that: (6) $L\text{X}=\mu X,$(6) for some scalar $\mu$, then $\mu$ is called the eigenvalue of $L$ with corresponding eigenvector $X$ (Marcus and Minc 1988).

5 China’s A-share market is a comprehensive representation of the Chinese stock market since it is the aggregation of the Shanghai (SSE) and Shenzhen (SZSE) stock markets, and it comprises the companies most traded by domestic investors compared with B-share and H-share (Xing and Yang 2019).

6 The S&P 500 index has a good overview of the US stock market since it is a capitalization-weighted measurement of the largest 500 companies listed on stock exchanges (Yousfi et al. 2021).

7 The FTSE 100 index comprises of the top 100 qualified companies with the highest market capitalization, a group which occupies almost 85% of the London Stock Exchange’s total capitalization (Davies and Studnicka 2018).

8 Traded on the Frankfurt Stock Exchange, the DAX 30 index comprises the 30 largest industrial and financial listed firms (Georg-Schaffner and Prinz 2021).

9 Except for the French and Spanish stock markets, the time period of the above-mentioned samples is from January 2006 to December 2020. Due to the lack of data on the French and Spanish stock markets in 2006, we have to choose the daily closing price from January 2007 to December 2020 as an alternative.

10 Wind database is available at https://www.wind.com.cn.

11 Due to the heterogeneity of different stock markets and the time lag of risk propagation, the divisions of crisis periods are typically country-specific.

12 The subdivision length ($h$) represents the number of divisions for the $\left[0,1\right]$ interval, $cp=\frac{i}{h}\left(1=1,2,\cdot \cdot \cdot ,h\right)$.

13 More details (i.e. Supplementary-REJF-2021-0319) are available at the author’s homepage: https://www.csust.edu.cn/stxy/info/1028/2492.htm.

14 In fact, the λ-LE graphs in monthly, quarterly and semi-annual networks are similar.

15 Similar to the method adopted in (Wen et al. 2020), the logistic distribution will be confirmed if $p>0.1$. In this paper, $60$ p-values of the fitted curves for the quarterly networks in Figure 4 remain above $0.1$, and the results of K-S test in the monthly and semi-annual networks (all p-values remain above $0.1$) are also obtained.

16 Simple calculation shows that $\alpha <{10}^{-6}$ and $\text{β}>2.40$.

17 The slope indicates the rate of change and the steepness of a regression line.

18 In this paper, we apply Sturges’ rule to determine the number of groups in equidistant grouping (Scott 2009). In addition, $9$ groups of the $180$ monthly networks and $4$ groups of the $30$ semi-annual networks are obtained.

19 The right-skewed distribution will be figured out if $SK>0$. The mode of the data set that follows the right-skewed distribution is the highest point of the histogram, yet the mean fall to the right of the peak, visually.

20 The financial crisis is the externalization form of extreme swings in the stock market, and the structural change is the internal cause of it.

21 We also introduce the method of $STL$ to statistically capture the outliers of time series figures of the above indicators in monthly, quarterly and semi-annual networks.

22 The supplementary material (i.e. Supplementary-REJF-2021-0319) is available at the author’s homepage: https://www.csust.edu.cn/stxy/info/1028/2492.htm.

23 Since the negative impact of the COVID-19 pandemic for the Chinese stock market is mainly concentrated in the first quarter of 2020, the identification effect of the LE-m in the semi-annual networks is not significant.

24 Part of the reason is that the ACC is highly dependent on the local structure information rather than the global, the CPL is not capable for networks with isolated points, and the ND is susceptible to the influence of the network size.

25 For more local and global views of monthly and semi-annually figures, please see the supplementary material (i.e. Supplementary-REJF-2021-0319) at the author’s homepage: https://www.csust.edu.cn/stxy/info/1028/2492.htm.

26 Figure 21 is inaccessible to detect the crisis of the Catalonia independence movement.

Additional information

Funding

Research for this paper was supported by the National Natural Science Foundation of China [Nos. 72192800, 72101035, 71471020] and the Excellent Youth Foundation of Educational Committee of Hunan Provincial [No.21B0339].

Notes on contributors

Chuangxia Huang

Chuangxia Huang received the BS degree in Mathematics in 1999 from National University of Defense Technology, Changsha, China. From September 2002, he began to pursue his MS degree in Applied Mathematics at Hunan University, Changsha, China, and from April 2004, he pursued his PhD degree in Applied Mathematics in advance at Hunan University. He received the PhD degree in June 2006. He is currently a Professor of Changsha University of Science and Technology, Changsha, China. He is the author of more than 80 journal papers. His research interests are in the areas of complex network and financial risk management.

Yunke Deng

Yunke Deng was born in Guizhou, China, in 1996. She received the BS degree in financial mathematics in 2019 from Capital University of Economics and Business, Beijing, China, and the MA degree in applied statistics in 2022 from Changsha University of Science and Technology, Changsha, China. She is a PhD student at University of Chinese Academy of Sciences. Her research interests include complex network, financial risk management and economic analysis and forecasting.

Xin Yang

Xin Yang received the BS degree in Finance in 2011 from Central South University of Forestry and Technology, Changsha, China. From September 2011, he began to pursue his MS degree in School of Economics & Management at Changsha University of Science & Technology, Changsha, China. From September 2014, he pursued his PhD degree in Business School at Central South University and received the PhD degree in December 2017. He is currently a lecturer of Changsha University of Science and Technology, Changsha, China. His research interests are in the areas of complex network and financial risk management.

Xiaoguang Yang

Xiaoguang Yang received his BS degree in Applied Mathematics and his PhD degree in Computational Mathematics from Tsinghua University in 1986 and 1993 respectively. He is a Professor at Academy of Mathematics and Systems Science, Chinese Academy of Sciences and University of Chinese Academy of Sciences. He currently serves as the President of System Engineering Society of China. He has published more than 300 journal papers. His research interests include risk management, financial market, and game theory.

Jinde Cao

Jinde Cao received the BS degree from Anhui Normal University, Wuhu, China, the MS degree from Yunnan University, Kunming, China, and the PhD degree from Sichuan University, Chengdu, China, all in mathematics/applied mathematics, in 1986, 1989, and 1998, respectively. He is an Endowed Chair Professor, the Dean of the School of Mathematics at Southeast University. Prof. Cao was a recipient of the National Innovation Award of China, Gold medal of Russian Academy of Natural Sciences, Obada Prize. He is elected as a member of the Academy of Europe, a foreign member of Russian Academy of Engineering, a member of the European Academy of Sciences and Arts, a foreign member of Russian Academy of Natural Sciences.