The illiquidity network of stocks in China's market crash
TThe illiquidity network of stocks in China’s market crash
Xiaoling Tan a , Jichang Zhao a,b a School of Economics and Management, Beihang University, Beijing, China b Beijing Advanced Innovation Center for Big Data and Brain Computing, Beijing, China
Abstract
The stock market of China experienced an abrupt crash in 2015 and evapo-rated over one third of the market value. Given its associations with fear andfine-resolutions in frequency, the illiquidity of stocks may offer a promising per-spective of understanding and even signaling the market crash. In this study,by connecting stocks that mutually explain illiquidity fluctuations, a illiquiditynetwork is established to model the market. It is found that as compared tonon-crash days, the market is more densely connected on crash days due toheavier but more homogeneous illiquidity dependencies that facilitate abruptcollapses. Critical socks in the illiquidity network, in particular the ones in sec-tor of finance are targeted for inspection because of their crucial roles in takingover and passing on the losing of illiquidity. The cascading failures of stocksin market crash is profiled as disseminating from small degrees to high degreesthat usually locate in the core of the illiquidity network and then back to theperiphery. And by counting the days with random failures in previous five days,an early single is implemented to successfully warn more than half crash days,especially those consecutive ones at early phase. Our results would help marketpractitioners like regulators detect and prevent risk of crash in advance.
Keywords: illiquidity, complex network, market crash, cascading failures,warning signals ∗ Corresponding author: [email protected]
Preprint submitted to Elsevier April 7, 2020 a r X i v : . [ q -f i n . C P ] A p r . Introduction The stock market occupies the most profound role in the financial systemsof modern economies like China. An abrupt stock market crash, like the oneof 2015 that evaporated around 15 trillion yuan in wealth, therefore could be acartographic shock to the economics and bring about huge losses to the wholesociety. In fact, how to understand the market crash and implement early warn-ings has been an important issue and trending topic not only in finance but alsointerdisciplinary fields after the crisis. While it is conventionally thought thatmarket crash might be a typical black-swan event, which is hardly predicteddue to sophisticated factors beyond and unexpected entanglements with ex-ternal systems. Nevertheless, the associations between investor behaviors, likeexpectations, emotions and imitations and the market performance, especiallytheir power in return predictions [1, 2], imply that trading behaviors may pro-vide a new but promising perspective of probing and warning the market crash.In particular, details of every trading decision in high-frequency records fur-ther offer a manner of big-data proxy to investigate the collective behavior ofinvestors, either before, during or after the market crash.Liquidity, referring to the spread between bid price and ask price, inherentlyreflects expectations of investors towards the future performance of stocks intheir elementary trading decisions. And illiquidity, which inversely originatesfrom the pessimism of investors, would thus increase the crash risk since it dis-solves the effective price information and disseminates panic across the market.Given the significant impact from investor emotions, especially the negativeones [3, 4], illiquidity can also be contagious, e.g., scared investors on stocks ofilliquidity incline to sell out other stocks on hands to keep their own liquidityand reluctantly result in more stocks of illiquidity. Hence, in order to modelthe market crash from a system view, it would be natural to connect stocks ofsimilar illiquidity fluctuations and build a network to represent the market. Inthe accordingly established illiquidity network, links among stocks stands forthe possibilities of cascading crash across the market, suggesting a new angle2f profiling the market crash dynamics. Though it is indeed not a new idea totransform a market into a network, linking stocks in terms of illiquidity is rarelyvisited. More importantly, different from previous networking models of mutualfund sharing [5] or price co-movements, illiquidity can be captured dynamicallyin a fine-resolution, i.e., in the most minimum decision granularity of bid andask. It means that in terms of elementary decisions in trading and their con-tagions, the illiquidity network provides a very micro-perspective of the marketcrash.Although there are lots of literatures on stock market crash, results on crashforecasts are still inadequate and more efforts are desperate. Unlike many emerg-ing financial markets, however, the China stock market is unique since it is dom-inated by individual investors [6]. Contrary to their institutional counterparts,individual investors are more emotional and susceptible, meaning they are morelikely to be scared, spread panic and overly react to external disturbs. Theyeven imitate trading strategies and help forge the herding in market. Thesecharacteristics might undermine the challenges that make crash hard to predictand suggest the possibility of detecting the crash of China market at early days.In terms of illiquidity, the trading behaviors in extreme market situations canbe finely examined from the micro perspective, helping identify the sources ofmarket volatility and extreme stock price movements. In addition, the anomalyin the evolution of illiquidity networks can also be probed from the differencesbetween crash days to non-crash days, which paves the way to develop thewarning signals of market crash.Inspired by above motivations, this study aims to profile, explain and warnthe China market crash through the illiquidity network. The illiquidity of stocksis defined and derived from 2.3 billion trades in 2015, from which profoundassociations between illiquidity and negative emotions of investors like fear aredisclosed. The illiquidity dependency between stocks, measured by the mutualinformation, can surprisingly distinguish crash days from those non-crash ones.And it is also inspiring that the market is more connected and homogeneousdue to heavier and lower-deviated illiquidity dependencies on crash days. While3n the illiquidity network, influential stocks in crash are found to be the oneswith large capital values or belonging to the sector of finance. The dynamicsof the crash is also profiled in the illiquidity network as cascading failures oflosing illiquidity from stocks of smaller degrees to the ones of higher degreesthat usually locate in the core and then out to the fringe. More importantly,an early signal, which simply counts the days without systemic failures in awindow of previous five days is presented to accurately warn more than halfcrash days in 2015. Our results decently demonstrate the power of illiquiditynetwork in understanding market crash of China and would help practitionersin particularly the regulators inspect risky stocks and prevent possible crash inadvance.The rest of the paper is organized as follows. Section 2 reviews literatures.Section 3 introduces our datasets and the methodology of measuring illiquidity.Section 4 presents the results from illiquidity networks. Section 5 concludes thepaper with a brief summary and suggestions for future research.
2. Literature review
Due to the late development of China’s stock market and the obvious gapwith developed foreign markets, there have been some unique features of theChinese stock market discussed among the academic scholars and practitioners.On the one hand, Yao et al. indicated that Chinese investors exhibit differentlevels of herding behavior [7]. On the other hand, Xing and Yang found thatthe increased correlation among the stocks could ignite market crash [8]. Fur-ther, Tian et al. found that institutional investors (primarily pension funds)provide stabilizing effect during extreme market-down days [9], unlike Dennisand Strickland who revealed that institutional investors magnify extreme mar-ket movements by buying (selling) more on return-up (return-down) days inthe U.S. markets [10]. Although there are many related studies in either Chinamarket or foreign ones, no detailed explanations and early warning signals ofstock market crash have been given to prevent risks. In the meantime, the4ominant occupation of individual investors in China market also implies thepossible abnormality in trading behaviors that can be sensed and detected aswarnings before the crash.In fact, previous efforts have already suggested that the stock market crashis closely related to illiquidity. Amihud et al. presented evidence linking thedecline in stock prices to increased illiquidity using the method of bid-ask spreadduring the market crash [11]. As return is more comparable to price, relatedresearch on associations between return and illiquidity has increased rapidly.Amihud and Bekaert et al. stated that there is a positive correlation betweenstock returns and illiquidity in terms of the daily ratio of absolute stock returnto its dollar volume and the proportion of zero return days, respectively [12,13]. Furthermore, Nagel indicated that the main reason of the evaporation ofliquidity during crash is the increasing expected returns of liquidity [14]. Evenmore inspiring, measuring illiquidity, e.g., through bid-ask spread, is deeplyrooted in the minimum decision granularity of daily trading and thus can beinherently derived from highly frequent trading records of investors. And also,illiquidity contains future economic information which can be employed for stockmarket forecasting [15, 16]. Therefore, it is feasible to explore stock marketcrash from the perspective of illiquidity, but existing examinations still lackexplanations, cascading dynamics, and warning signals of the crash.Illiquidity may also be influenced by both internal and external factors in-cluding stock attributes, policies and industry, which should be considered inunderstanding the market crash. Stoll et al. suggested that stock attributessuch as market value, volume and volatility can significantly reshape the stockilliquidity [15, 16, 17]. On the other hand, An et al. found that macro economicfactors such as media independence, policy uncertainty, default risk and fund-ing conditions have a remarkable impact on illiquidity [18, 19, 20, 21]. Theseevidences imply that stocks can be well profiled in terms of illiquidity and moreimportantly, external shocks to the market can also be absorbed and thus sensedthrough illiquidity. In addition, the illiquidity of individual stocks co-varies witheach other [22, 23, 24, 25, 26], suggesting in essence that illiquidity can be con-5agious across the market.Modeling market as a network of stocks to examine the crash is a new andpromising approach in recent efforts. Stocks can be connected due to pricecorrelations or common investors [5]. By removing failed stocks, e.g., reachingthe down-limit and transactions being suspended, the market crash can thenbe reflected through the falling apart of the network. The topology evolutionbefore and after the 2008 financial crisis of South African, Korean and Chinasstock markets were investigated [27, 28, 29], respectively, in which the minimumspanning trees (MST) are carefully examined. Li and Pi proposed a complexnetwork based method to understand the effects of the 2008 global financialcrisis on global main stock index [30]. besides, Bosma et al. use network cen-trality to identify the position of the financial industry in the network, whichcan be a significant predictors of bailouts [31]. In particular, the turbulence in20152016 were probed by transforming China stock market into a complex net-work, showing that there exist influential stocks and sectors within the marketcrash [5, 32]. Nevertheless, connecting stocks because of illiquidity associationsis rarely considered in constructing the market network. The absence of estab-lishing illiquidity networks in existing studies on market crash will spark up newperspectives in this paper.To sum up, although extensive efforts have been devoted on the associationbetween stock illiquidity and market crash, few insights are available on illiq-uidity networks based on high-frequency transaction data. Given the closenessbetween stock illiquidity and both internal and external factors of the market,probing the crash from the perspective of illiquidity networks could offer moreinsightful observations and explanations. Moreover, the dominance of individ-ual investors in China stock market also indicates that the trading abnormality,which can be grasped by illiquidity and its contagion in a fine resolution couldproduce novel signals to warn risks before the crash. From a interdisciplinaryview, a big-data proxy based on tremendous trading records before, during andafter the 2015 crash of Chinas stock market will be employed to measure illiq-uidity, establish networks, examine crash dynamics and detect warning signals.6 . Dataset and methods
The data sample employed in this study consists of stocks selected from theShenzhen Stock Exchange and the Shanghai Stock Exchange in 2015, i.e., morethan 2500 stocks and a total of 244 trading days. In particular, transactionrecords of the minimum trading decision granularity include ask price, ask vol-ume, bid price, and bid volume for every second of every stock. The datasetis provided by the Wind Information (Wind Info), a leading integrated serviceprovider of financial data in China.
June 12nd:
SHCOMP index hit new high of 5178.19
June 27th:
The People's Bank of China cut interest rate by 0.25% to release the liquidity
July 2nd:
Special verification of suspected market manipulation by CSRC
July 4th, July 5th:
Premier Li returned to rescue the market
July 8th:
The China Insurance Regulatory Commission had relaxed the regulatory ratio of blue-chip stocks invested in insurance funds
Aug.24th:
Aug.25th:
June 19th:
The first market crash, 1067 stocks down to lower limit
Figure 1: Review of key events of the market crash in 2015.
Then, for identifying the stock market crash, the crash days are definedas days whose number of stocks being sell-off to the down limit (the allowedmaximum one-day drop of a stock, i.e., ten percent of its closing price last day)is more than 800 . Specifically, as seen in Figure 1, in 2015, there are 17 tradingdays on which the stock market was crashed, including June 19th, June 26th,June 29th, July 1st, July 2nd, July 3rd, July 6th, July 7th, July 8th, July 15th,July 27th, Aug.18th, Aug.24th, Aug.25th, Sept.1st, Sept.14th, Oct.21st. And7ther days before or after these crash ones will be defined as non-crash days andconsist the counterparts for further comparison.
The transaction data is full of noise due to the too much frequent occurrencesof quote. In order to filter out noise and smooth the data, a fixed time window ofone minute is selected to average the spread. Note that as compared to previousstudy, one minute is short enough to reflect the investment behavior of investorsat the smallest decision granularity. Besides, it is necessary to convert the lengthof data sequence into 237 minutes for every stock in a day for the reason that theShenzhen Stock Exchange adopts collective bid for the last three minutes. Withrespect to the illiquidity, various methods have been presented to calculate itfor different occasions and purposes. The methods on low-frequency data workgreat when high-frequency data is not available [12, 33, 34, 35, 36], but it isstill undeniable that approaches based on high-frequency data perform bettersince richer information and higher accuracy [19, 37]. Here the illiquidity isexpected to sense the minimum decisions in trading behavior, hence the bid-askspread based on high-frequency records, which is always considered to be thebest method, is selected to measure illiquidity [38, 39]. Moreover, it is knownthat the size of the transaction has a great impact on illiquidity, we furtherupdate the measure by adding the quoted amount as the weight of the spread.The illiquidity can be noted as I t = ( (cid:80) i =1 A it V it − (cid:80) j =1 B jt V jt ) P mid,t · , (1)where A it is the ask price of investor i at time t, V it is the ask volume of investor i at time t , B jt is the bid price of investor j at time t , V jt is the ask volume ofinvestor j at time t , P mid,t is the mean of ask price and bid price at time t . Itcan be learned from the definition that the lower the weighted spread, the lowerthe transaction coast and the lower the illiquidity.The potential capability of the illiquidity in understanding the market crashcan be simply illustrated in Figure 2, in which the market index is negatively8 a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) I t illiquiditymarket index m a r k e t i n d e x Figure 2: Illiquidity with stock index. I t is the illiquidity we measured and market indexrepresents the CSI 300 Index. The correlation between illiquidity and market index is -0.64with p -value 0.00. The red dots indicate the crash days. associated with the fluctuation of illiquidity we measured. In fact, China’s stockmarket had experienced a period of ups and downs in the year of 2015, in whichperiod more than ten days of crash erupted in succession. Specifically, theilliquidity continued a decreasing trend before June and at this stage investorseasily completed transactions due to lowering cost and the market index keptsoaring. In contrast, the illiquidity demonstrated an abrupt increase in Juneand August, in which months the crash densely occurred and resulted hightransaction cost, inactive investors and falling market index. These observationsconfirm the previously disclosed association between illiquidity and crash inChina’s stock market and inspire the following investigations from the novelperspective of illiquidity network. 9 : : : : : : : : : time v o l u m e ask volumebid volume : : : : : : : : : time v o l u m e (a) ask volumebid volume : : : : : : : : : time v o l u m e ask volumebid volume J u n e t h J u n e t h J u n e t h J u l y s t J u l y n d J u l y r d J u l y t h J u l y t h J u l y t h J u l y t h J u l y t h A u g . t h A u g . t h A u g . t h S e p t . s t S e p t . t h O c t . s t date(day) F r e q u e n c y (b) no askno bidno quotation M a y t h M a y t h M a y t h M a y t h M a y t h M a y t h M a y t h M a y t h M a y t h M a y t h M a y t h M a y t h M a y s t M a y n d date(day) F r e q u e n c y (c) no askno bidno quotation Figure 3: Trading behaviors in crash and non-crash days. (a) shows the ask and bid volumein market crash day of June 26th, the stocks are randomly selected from the sample. Thefirst sub-graph shows the stock that not losing liquidity in crash day, and the ohter two showstocks that losing liquidity in crash day. When one of the ask or the bid does not exist, orneither of them exists, the stock loses liquidity. (b) shows the quotation of buyers and sellersin crash days, in which frequency is defined as how often each action occurred every minuteof the crash day, no ask means no buyers quote and no bid means no buyers quote. (c) showsthe quotation of buyers and sellers in non-crash days. . Results It is supposed that trading behaviors, especially the elementary actions likeask and bid of high frequency, would be essentially influenced by shocks likemarket crash. As can be seen in Figure 3(a), when stocks approached downlimit on crash days, the volume of bid experienced an abrupt decline and thenvanished, contrarily the ask volume soared, implying that many investors wereforced to sell off shares owing to panic selling and risk prevention. However,approaching down limit might also happen on non-crash days. To further tes-tify the impact from market crash on trading behaviors, we randomly selectten crash days and non-crash days to compose two different groups and com-pare the occurrence occupations of no-ask, no-bid, and no-quote when stocksexperienced down limit. It is unexpected that crash days can be surprisinglydistinguished from non-crash days. Specifically, as can be seen in Figure 3(b),no quotations, which would result in liquidity losing, mainly comes from no-bidon crash days instead of no-ask on non-crash days. This disguising impact frommarket crash to trading behaviors further suggest that in terms of illiquidity,whose calculations are based on both ask and bid, would inherently sense thefootprints of market crash from the novel angle of trading decisions.Zero volume of bid but soaring amount of ask suggests that on crash daysinvestors are anxious and their anxiety are accumulating. As can be seen inFigure 4(a), it is interesting that the maximum volume of ask in fact logarith-mically grows with the duration of losing illiquidity, i.e., no-bid. At this stage,investors can be easily affected by others, especially the spread of pessimism.This logarithmic-like relationship also indicates that the longer the no-bid lasts,the more anxious the investors are and the soar of ask eventually slows down.The saturation of ask volume can be explained that investors will become lesspanic when more information is obtained. From this perspective, trading actionslike ask can be directly connected to investor emotions and intuitively, illiquiditythat based on spread of ask and bid should be coupled with emotions, especially11
30 60 90 120 150 180 210 max duration of illiquidity (minute) m a x a s k v o l u m e max duration of illiquidity (minute) m a x a s k v o l u m e J a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) illi q u i d i t y illiquidityfear f e a r (b)(a) Figure 4: Max duration of illiqudity due to no bid. (a) shows the correlation between themax ask volume and the max duration of illiqudity, which indicates the longest duration oflosing liquidity. Note that there may be several periods of losing liquidity per stock in a day.The stock is randomly selected from the sample, and other stocks have similar relationships.(b) shows the correlation between investors’ fear and the illiquidity, whose value is 0.44 with p -value 0.00. In the stock market of China, the actual interactions coupled within stocksare extraordinarily important because of susceptible investors. While most ex-13sting models forge links between stocks mainly based on the similarity of time-series, e.g., of price and measures of Pearson and Partial correlations are ex-tensively employed [41, 42, 43, 44]. However, the relationship between stocks istoo complicated and should not be too much simplified to neglect trading be-haviors, investor emotions and their possible contagions. Taking the limitationsof linear correlations into account, here we use mutual information to measurethe nonlinear dependency between illiquidity of stock pairs. In fact ,the powerin reflecting nonlinear dependency of mutual information in networking markethave been previously demonstrated and emphasized [32, 45, 46].
Figure 5: The normalized mutual information (NMI) of illiquidity. (a) shows the distributionsof NMI of illiquidity on both crash(June 26th, June 29th) and non-crash(June 24th, June 25th)days. It is clear that the globally averaged NMI is getting larger while the standard deviationis getting smaller when the stock market is approaching a turmoil. (b) shows the mean andstandard deviation of NMI of illiquidity with all the transaction days over the year of 2015.
By calculating the normalized mutual information (NMI) of illiquidity seriesin minute between all pairs of stocks, we first try to profile the distributionsof illiquidity dependency of the market on both crash and non-crash days. Ascan be found in randomly selected samples in Figure 5(a), the globally aver-aged NMI is getting larger while the standard deviation (e.g., the broadnessof the distribution) is getting smaller when the stock market is approachinga turmoil. Drawing a mean-standard deviation graph with all the transaction14ays over the year of 2015 for ease of observation, see as Figure 5(b), it is clearthat the average mutual information will increase and the standard deviationwill decrease while in the crash days, indicating that the illiquidity networkwill become more closely connected and more homogeneously coupled when themarket is in a bad situation. Because of pessimism, investors become cautiousand unwilling to participate in the transaction, which abruptly increases andspreads illiquidity across the market and results in a crash. Besides, we also findthat the market crash demonstrates a lasting effect because the days after thecrash show the same characteristics as the day in the crash. However, regardingto the days before the crash, as seen in Figure 5(b), they overlap with those ofnon-crash and hardly demonstrate any distinct features, suggesting that fromthe global and static view there is no warning signal can be detected. It in-spires us to investigate the illiquidity network from more in-depth and dynamicperspectives further.In building an illiquidity network, links are weighted as NMI between theirends illiquidity, while not all links are necessarily kept and those with lessweights, which might relatively represent random dependency among stocksinstead of plausible paths for illiquidity contagion, would be removed. Specif-ically, the size of the giant connected component (GCC) is taken into accountfor locating the critical threshold of link weight [47, 5], i.e., the value beyondwhich the size of GCC starts to decline rapidly will be set as the threshold foreach trading day (see Appendix Figure A1(a)). And links with weights belowthe threshold will then be omitted since their removals trivially influence theconnectivity of the market structure. The ratio of GCC in illiquidity networksfluctuates and significantly increase on crash days, suggesting consistently thatthe market will be more connected and coupled in crashes (see Appendix FigureA1(b)). High illiquidity dependency could facilitate spread of illiquidity acrossthe market and low deviation of illiquidity dependency would further lead to anabrupt collapse of the network. The positive associations between GCC ratiosof illiquidity networks and market crash indicate that the refined structures bythresholds of link weights can be proper models of networking market.15 a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) s e c t o r r a t i o ( R ij ) (a) financeinformation technologymanufacturingreal estateservice J a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) s t y l e r a t i o ( R ij ) (b) large-cap-valuemid-cap-valuesmall-cap-value Figure 6: The degree-weighted occurrence proportions on different sections and capital stylesin illiquidity networks.(a) shows the proportions on different sections. Note that the pro-portions in other sections are very similar except for the financial. Therefore, only a fewrepresentative industries are selected to simplify the picture. (b) shows the proportions ondifferent capital styles of stock values, the large-cap-value is the most critical group in marketcrash. As for growth stocks and balanced stocks, the results are the same, they are not shownhere in the figure. R ij , is defined to identify key group i of stocks on trading day j . Specifically, R ij = n ij /n j N ij /N j , (2)where n ij is the occurrence of stocks belonging to group (sector or style) i andit is summed over all links in the network of j day, n j is the occurrence of allstocks and it is summed over all links in the network of j day, N ij is the numberof group i in the network of j day, N j is the number of unique stocks in thenetwork of j day. Accordingly, the group of stocks with higher R ij will occupymore links in the market, meaning heavier dependency on other stocks illiquidityand greater odds of taking over or passing on crash risk. It is unexpected that,as can be seen Figure 6(a), the sector of finance constantly occupies the highestproportion in Chinas stock market, especially on crash days. As for the capitalstyle, the style of large capitalization, i.e., the large-cap-value is the most criticalgroup in market crash (see Figure 6(b)). Both observations suggest that stocksin finance, especially those of large capital values, should be targets of inspectionfor market regulators.The falling-apart of Chinas market in crash was consisted by waves of stockscompletely losing illiquidity, i.e., declining to the down limit [48]. These failurewaves produced peaks in number of newly failed stocks (see Appendix FigureA3). Assuming that each wave of failure can be identified by a peak, thenstocks that failed before the peak could be seeding failures that lead to thecorresponding wave of losing illiquidity. Then sectors with more stocks failedbefore peaks might be causes of the following collapse and thus could be targets17 an.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij agriculture Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij transportation Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij finance Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij service Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij comprehensive Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij retailing Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij construction Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij extractive Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij real estate Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij manufacturing Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij electricity,gas,water Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij communication and cultural Jan.5th Apr.7th July 3rd Sept.29thDec.29th date(day) S ij information technology Figure 7: The significance of failing before peak. The red dots indicate crash days, and theblue dots indicate non-crash days. S ij = R bpji − R bprji , so S ij may be positive or negative. If S ij is positive, which means stocks within i tend to fail before peaks. It is obvious that the sectorof finance failed most before peaks on crash days. In contrast, the sectors like manufacturingand information technology perform similarly both on crash and non-crash days. Note that S ij can not be calculated for all stocks since some of them might not appear in the illiquiditynetwork due to good liquidity, especially on non-crash days. R bpij is thus defined to target critical sectors, which can be calculated as R bpij = n bpij /N bpj N ij /N j , where n bpij is the number of stocks failed before peaks in group i on day j , N bpj is the number of stocks failed before peaks on day j . To testify the significanceof failing before peaks, the timings of fail for all the stocks of one trading dayare also randomly shuffled to get a random value of R bpij , which is denoted as R bprij for comparison to test significance. Then for group i , its significance ofbeing seeds that probably lead to a wave of failures on day j can be definedsimply as S ij = R bpij − R bprij . Intuitively, S ij will be much greater than 0 if stockswithin i tend to fail before peaks. Consistent with our above observation, thesector of finance, as can be seen in Figure 7, failed most before peaks, especiallyon crash days. In the contrary, the significance of sectors like manufacturingand information technology just fluctuates around zero with trivial deviations.It again suggests that stocks of finance in Chinas market might be sinks oreven triggers that produce illiquidity and spread it across the market. In termsof inspecting these stocks of finance, market practitioners, in particular theregulators, could sense warnings from their abnormal variations on illiquidity.The illiquidity network can also track the dynamics of market crash. Con-sidering peaks of newly failed stocks can be interfaces to split failure cascades,the timing distance between the timing of losing illiquidity to the peak timinginherently measures at which stage the stock join the crash cascade. Specif-ically, for negative distances, smaller ones stand for the early collapse, whilefor positive distances, greater ones represent the later failures in the crash (seeFigure 8(a)). We then examine the function between stock degree and the ab-solute value of time distance, as can be seen in Figure 8(b), it is found that thedegree, in particular the maximum degree in each bin, is negatively correlatedwith the distance. This negative association indicates that stocks fails nearlythe peak timing are those with high degrees, while these fail at the early stateor at the ending of the crash possess small degrees. Thats to say, the crash19 the distance to the peak d e g r ee before peakafter peak 20 10 0 10 20 the distance to the peak d e g r ee before peakafter peak10 absolute distance from peak d e g r ee Fitted line for max degreesmax degrees (b)(a)
Figure 8: The correlation between stock degree and the timing distance of losing illiquidity tothe peak. (a) shows the degrees of stocks that decline to the down limit before and after thepeak. How to find and determine the peak of stocks decline to the down limit is illustratedin Appendix Figure A3. (b) shows that greater the absolute distance, smaller the degrees ofstocks (y-axis is logarithmic). The correlation between the maximum degree and the absolutedistance is -0.66 with p -value 0.00. Above illustrations solidly suggest the associations between illiquidity net-work and market crash. Assuming market crash being systemic failure ratherthan random error, stocks failed together in a short interval, e.g., ten min-utes, should be inherently entangled with each other due the contagion of los-ing illiquidity and therefore connected in our built illiquidity network. Thenthe non-randomness of failures within a short interval i can be defined as w i = e nf n f ( n f − / , where n f stocks got to the down limit in i simultaneously, e n f is the number of links among them that captured in the illiquidity networkbuilt on the corresponding day and n f ( n f − / w i represents more like-lihoods of systemic failures instead of random errors, i.e., signs of crash. And wd j =
This research was supported by National Natural Science Foundation ofChina (Grant No. 71871006). The authors also appreciate valuable commentsand constructive suggestions from Dr. Shan Lu.
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Table S1: Sector and cap style of stocks sector Agriculture, Communication and cultural,Comprehensive,Construction,Electricity, gas, water, Extractive, Financial, Information technology,Manufacturing, Real estate, Retailing, Service, Transportationstyle Small-cap-growth, Small-cap-balance, Small-cap-valueMid-cap-growth, Mid-cap-balance, Mid- cap-valueLarge-cap- growth, Large- cap-balance, Large- cap-value30 .7 0.8 0.9 1.0 link weight(NMI) t h e s i z e o f G CC maxsecond 02468101214 0.7 0.8 0.9 1.0 link weight(NMI) t h e s i z e o f G CC maxsecond 02468101214 0.7 0.8 0.9 1.0 link weight(NMI) t h e s i z e o f G CC maxsecond 02468101214 J a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) T h r e s h o l d (b)(a) Figure A1: The threshold of link weights. (a) shows the sizes of the giant connected component(GCC) and the second largest connected component as the threshold of link weights increase.The value beyond which the size of GCC starts to decline rapidly will be set as the thresholdfor each trading day, it is found that the value can be well captured when the decline of size ismore than 1%. Considering that the size of the second largest connected component is small,the GCC can well represent the entire network. (b) shows that the threshold fluctuates withtime, but increases greatly during the crash days (the red dots indicate crash days), suggestingconsistently that the market will be more connected and coupled in crash. a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) t h e s i z e o f li n k s start of market crash (June 19th) (a) new linksreduced linksunchanged links J a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) r a t i o (b) the ratio of unchanged links Figure A2: The evolution of links in illiquidity networks. (a) shows the size of new links,reduced links and unchanged links for consecutive two trading days. It is found that theChinas stock market evolves in a high frequency, especially on crash days. (b) shows the ratioof unchanged links which indicates that only 10% links kept on average for consecutive twotrading days. : : : : : : : : : time nu m b e r o f f a il e d s t o c k s : : : : : : : : : time nu m b e r o f f a il e d s t o c k s : : : : : : : : : time nu m b e r o f f a il e d s t o c k s : : : : : : : : : time nu m b e r o f f a il e d s t o c k s Figure A3: The peaks of stocks down to limit. The principle of determining the peak is thatthe number of stocks mentioned above is the largest relative to the previous period and thesubsequent period. As can be seen, there may be multiple peaks within a day. a n . t h F e b . n d M a y . t h A p r . t h M a y t h J u n e r d J u l y r d J u l y s t A u g . t h S e p t . t h N o v . r d D e c . s t D e c . t h date(day) r a t i o (a) J a n . t h F e b . t h M a y . r d A p r . r d M a y n d J u n e t h J u l y t h A u g . t h S e p t . t h O c t . n d N o v . r d D e c . t h date(day) F r e q u e n c y (b) Figure A4: The warning signal. (a) shows the likelihoods of systemic failures instead ofrandom errors. The red bar indicates crash days, and the blue bar indicates non-crash days.(b) shows the occurrence of w d = 0, i.e., the daily non-randomness is zero within five days,which is denoted as N w d =0 to construct a warning signal. Specifically, smaller N w d =0 suggestsmore systemic failures and greater odds of leading to market crash. As can be seen, an abruptdecline of N w d =0 can be detected one day earlier. It indicates that if N w d =0 = 0 in theprevious five days, a warning signal can be implemented to warn a market crash in the nextday.= 0 in theprevious five days, a warning signal can be implemented to warn a market crash in the nextday.