An AI approach to measuring financial risk
Lining Yu, Wolfgang Karl Härdle, Lukas Borke, Thijs Benschop
AAn AI approach to measuring financial risk ∗ Lining Yu † Wolfgang Karl Härdle ‡ Lukas Borke § Thijs Benschop ¶ Abstract
AI artificial intelligence brings about new quantitative techniques to assess the stateof an economy. Here we describe a new measure for systemic risk: the FinancialRisk Meter (FRM). This measure is based on the penalization parameter ( λ R package RiskAnalytics is another tool with the purpose of integrating and fa-cilitating the research, calculation and analysis methods around the FRM project.The visualization and the up-to-date FRM can be found on hu.berlin/frm. ∗ Financial support from the Deutsche Forschungsgemeinschaft (DFG) via SFB 649 “ÖkonomischesRisiko”, IRTG 1792 “High-Dimensional Non-Stationary Times Series” and Sim Kee Boon Institute forFinancial Economics, Singapore Management University,as well as the Czech Science Foundation undergrant no. 19-28231X, the Yushan Scholar Program and the European Union’s Horizon 2020 researchand innovation program ”FIN-TECH: A Financial supervision and Technology compliance training pro-gramme” under the grant agreement No 825215 (Topic: ICT-35-2018, Type of action: CSA), Humboldt-Universität zu Berlin, is gratefully acknowledged. † Research associate at Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. - Center for AppliedStatistics and Econometrics, IRTG 1792, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099Berlin, Germany. Email: [email protected]. ‡ Ladislaus von Bortkiewicz Professor of Statistics, C.A.S.E. - Center for Applied Statistics and Econo-metrics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany; Wang YananInstitute for Studies in Economics, Xiamen University, 422 Siming Road, Xiamen 361005, China; SimKee Boon Institute for Financial Economics, Singapore Management University, 90 Stamford Road,Singapore 178903, Singapore; Department of Mathematics and Physics, Charles University Prague, KeKarlovu 2027/3, 12116 Praha 2, Czech. Email: [email protected]. § Research associate at Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. - Center for AppliedStatistics and Econometrics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Ger-many. Email: [email protected]. ¶ Research associate at Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. - Center for AppliedStatistics and Econometrics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Ger-many. Email: [email protected]. a r X i v : . [ q -f i n . R M ] S e p eywords : Systemic Risk, Quantile Regression, Value at Risk, Lasso, Parallel Comput-ing, Financial Risk Meter JEL : C21, C51, G01, G18, G32, G38.
This is a post-peer-review, pre-copyedit version of an article published in the SingaporeEconomic Review. The final authenticated version is available online at: http://dx.doi.org/10.1142/S0217590819500668
Systemic risk is hazardous for the stability of financial markets, since the failure of onefirm may impact the stability of the whole system. There are various definitions ofsystemic risk. One of the most popular definitions is introduced in Schwarcz (2008).He defined systemic risk as a trigger event, such as an economic shock or institutionalfailure, causing a chain of bad economic consequences, sometimes referred to as dominoeffect. This definition indicates that interlinkages and interdependencies in a system ormarket are very crucial for controlling systemic risk. The financial crisis in 2008 is anexample. After the bankruptcy of Lehman Brothers, several more financial cooperationsbankrupted as a result of their interlinkages with Lehman Brothers. Consequently, therehas been a surge in the interest in measuring and controlling systemic risk since the 2008crisis, which has led to an increase in the research on this topic.Several methodologies for measuring systemic risk have been proposed. Adrian and Brun-nermeier (2016) proposed CoVaR, the value at risk of financial institutions conditionalon the other institutions being under distress, which uses two linear quantile regressions.Hautsch et al. (2015) refined this algorithm by introducing linear quantile lasso regres-sion with a fixed penalization parameter λ for each company to select the relevant riskdrivers. Fan et al. (2016) and Härdle et al. (2016) use a nonlinear Single Index Model(SIM) combined with a variable selection technique to select risk factors. In their applica-tion, they use data on 200 financial companies and 7 macro variables to estimate CoVaR.During the estimation procedure, a time-varying penalization parameter λ is generated.This series has a striking pattern: higher values correspond to financial crisis times andlower values correspond to stable periods. This observation has led to the idea to use thepenalization parameter λ itself as a measure for systemic risk. The time-varying featureis not observed in Hautsch et al. (2015), who applied a fixed λ for each firm, and nottime varying.Fan et al. (2016) provide the λ series for single companies. In contrast, we would like tosee the overall behavior of λ . Härdle et al. (2016) compare the linear quantile lasso modeland SIM, and conclude that SIM is more suited than the linear model, but that the linearquantile lasso model is also valid in terms of backtesting. Indeed if one generates λ seriesfor 100 firms with more than 300 observations each, then the application of SIM is notrealistic. Since linear quantile lasso is easier to apply and time saving, we decided to use2t to compute the FRM.We use log return data from the 100 largest US publicly traded financial institutionsas well as 6 macro variables. Our model is based on daily log returns of these financialinstitutions. The time period under consideration runs from April 5, 2007 until September23, 2016 and covers several documented financial crises (2008, 2011). We observe thatthe pattern of this risk measure is more informative on financial risk. The shape andvolatility of the series correspond to the market volatility and financial events with alarge impact on systemic risk are clearly visible. Therefore, this series of averaged λ maybe called a Financial Risk Meter (FRM). Zbonakova et al. (2017) apply linear quantilelasso regression to analyze the behavior of the λ series. They find that λ is sensitiveto the changes of volatility, which provide the theoretical evidence for the FRM to be asystemic risk measure, as high volatility indicates high risk.We introduce the methodology of the FRM, describe the risk levels, the computationalimplementation as well as possible visualizations. We compare the FRM with othersystemic risk measures, such as VIX (Hallett, 2009), SRISK (see Brownlees and Engle,2016) as well as the Google trends of key words related to financial crises (see Preis et al.,2013). We find that the FRM and these risk measures mutually Granger cause, whichindicates the validity of the FRM as a systemic risk measure.The remainder of this paper is organized as follows. In Section 2 the methodology used toconstruct our FRM, which is quantile lasso modeling, is presented. Section 3 presents thedata, computational challenge and the visualization of the results. Section 4 shows thevalidity of our FRM as a measure for financial risk by comparing with other financial riskmeasures. Section 5 concludes, the financial institutions applied in this paper is listed inSection 6 Appendix. All the R In this section we describe the methodology and algorithm used to compute FRM. Sincethe penalization parameters are computed based on an L -norm (LASSO) quantile linearregression, this regression framework is introduced first. Within this framework, thepenalization parameter λ is exogenous. Since the FRM is distilled from the selectedpenalization parameter, we subsequently discuss methods to select λ . Following Härdle et al. (2016), we introduce the quantile lasso regression model. Let m be the number of macro variables describing the state of the economy, k the number offirms under consideration, j ∈ { , . . . , k } . Then p = k + m − t ∈ { , . . . , T } is the time point with T the total number of observations(days). s is the index of moving window, s ∈ { , . . . , ( T − ( n − } , where n is the length3f window size. Then the quantile lasso regression is defined as: X sj,t = α sj + A s, > j,t β sj + ε sj,t , (1)where A sj,t def = " M st − X s − j,t , M st − the m dimensional vector of macro variables, X s − j,t is the p − m dimensional vector of log returns of all other firms except firm j at time t andin moving window s , α sj is a constant term and β sj is a p × s .The regression is performed using L -norm quantile regression proposed by Li and Zhu(2008): min α sj ,β sj n − s +( n − X t = s ρ τ (cid:16) X sj,t − α sj − A s, > j,t β sj (cid:17) + λ sj k β sj k , (2)where λ sj is the penalization parameter, and the check function ρ τ ( u ) is: ρ τ ( u ) = | u | c | I ( u ≤ − τ | , where c = 1 corresponds to quantile regression. The L -norm quantile linear regressioncan be used to select relevant covariates (other firms and macro state variables) for eachfirm. λ Since Equation (2) has an L loss function and an L -norm penalty term, the optimiza-tion problem is an L -norm quantile regression estimation problem. The choice of thepenalization parameter λ sj is crucial. There are several options to select λ sj , e.g. withthe Bayesian Information Criterion (BIC) or using the Generalized Approximate Cross-Validation criterion (GACV). Yuan (2006) conducted simulations and concluded thatGACV outperforms BIC in terms of statistical efficiency. Therefore, we determine λ sj with the GACV criterion in the FRM model and set λ sj as the solution of the followingminimization problem:min GACV ( λ sj ) = min P s +( n − t = s ρ τ (cid:16) X sj,t − α sj − A s > j,t β sj (cid:17) n − df , where df is a measure of the effective dimensionality of the fitted model. The advantageof GACV is that it also works for p > n , which can be important for the FRM if themoving window size is small.To compute the FRM, we perform the regression analysis as described above and selectthe λ s, ∗ j for each firm j using GACV. The Financial Risk Meter is defined as the averagelambdas over the set of k firms for all windows:4 RM def = 1 k k X j =1 λ ∗ j To compute the FRM, we use data from 100 US publicly traded financial institutions aswell as six macro variables. The selection of financial companies is based on the NASDAQcompany list and based on the market capitalization. The selected companies are the100 US publicly traded financial institutions with the largest market capitalization, seeTable 13 in Appendix. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . . . Figure 1:
The x -axis represents the number of firms ordered by market capitalization and the y -axisthe percentage of total market capitalization. FRM_per_capInitially, we used data on the 200 US publicly traded financial institutions with thelargest market capitalization to compute the FRM. However, the smaller companies inthis set change regularly over the time period under consideration (2007-2016) due to,for instance, bankruptcies. This leads to obvious data download issues and therefore weuse only 100 firms. Figure 1 shows the cumulative market capitalization of US financialfirms. The x -axis represents the firms ordered by market capitalization and the y -axis thecumulative market capitalization. We observe that the largest 100 firms cover more than
55% of the total market capitalization of all companies in the US financial market andwe therefore can restrict our analysis to 100 firms. Furthermore, the results of estimatingthe FRM based on 100 or 200 firms are very similar if the moving window size is thesame. Figure 2 plots both FRM series with the window size n = 126, the shape and thetrends of them are similar. . . . . . . Figure 2:
FRM with 100 firms (black) and FRM with 200 firms (grey), moving window size n = 126. FRM_compare_nf . . . . . . Figure 3:
FRM with different moving window size, n = 63 (black) and n = 126 (grey), both series arescaled into the interval [0,1], from July 6, 2007 until September 23, 2016. FRM_compare_wsWe select six macro state variables to represent the general state of the economy: 1) theimplied volatility index, VIX from Yahoo Finance; (2) the changes in the three-month6reasury bill rate from the Federal Reserve Bank of St. Louis; (3) the changes in the slopeof the yield curve corresponding to the yield spread between the ten-year Treasury rateand the three-month bill rate from the Federal Reserve Bank of St. Louis; (4) the changesin the credit spread between BAA-rated bonds and the Treasury rate from the FederalReserve Bank of St. Louis; (5) the daily S&P500 index returns from Yahoo Finance, and(6) the daily Dow Jones US Real Estate index returns from Yahoo Finance.To compute the FRM we employ the tail parameter τ = 0 .
05. To find the a stable windowsize, n , we have to make a trade-off. We find that the lasso selection technique performsworse if the window size is too small. Since we use daily data, the moving window sizeshould be larger than 50, so that the estimation for each window is more precise. Theresults of using different window sizes (we have considered window sizes n = 63 (onequarter) and n = 126 (half a year)) are shown in Figure 3. The larger the window size,the more lagged, but also the smoother the plot is. Cross correlation can be used todetermine the time delay of a time series, which we apply here for the estimate of theFRM with n = 63 and the FRM with n = 126. In Figure 4 and Table 1, the largestautocorrelation between FRM with n = 63 and the lagged FRM with n = 126 is 0 . −
29 to lag −
22. We conclude that the FRM with n = 63 leads the FRM n = 126by at least 22 periods. From all the preceding we set the moving window size to n = 63. −30 −20 −10 0 10 20 30 . . . . . . Lag c r o ss − c o rr e l a t i on FRM100_63 & FRM100_126
Figure 4:
Cross correlation between FRM with n = 63 and FRM with n = 126, where the number offirms is 100. FRM_compare_wsFor each firm we have 2 ,
386 daily observations and 105 covariates (99 firms and 6 macro-state variables). The FRM is the average of the λ ’s computed from the 100 individualfirms. The λ ’s for the individual firms are more volatile and less smooth than the averageover 100 firms and therefore more robust to reflect the impact from financial events onsystemic risk. Figure 5 illustrates this by plotting the λ of firm Wells Fargo (the largestfirm by market capitalization) and the FRM.7 ag -30 -29 -28 -27 -26 -25 -24 -23 -22 -21Cross correlation 0.963 0.964 0.964 0.964 0.964 0.964 0.964 0.964 0.964 0.963 Table 1:
Cross correlation between the estimates of the FRM with n = 63 and FRM with n = 126. . . . . . . Figure 5:
FRM (black) and λ of Wells Fargo (grey), both series are scaled into interval [0,1], from April5, 2007 until September 23, 2016. FRM_compare_of
We wrote a script to automatically download the data from Yahoo Finance and FederalReserve Bank of St. Louis. The R package quantmod is used. More details and the scriptare available from Quantnet ( FRM_download_data).The L -norm quantile regression used to generate the λ series is computationally intensiveand therefore time-consuming, if applied sequentially for a large number of firms, seefor instance the code from Quantnet ( FRM_lambda_series). Therefore, we considerparallel computing in R to reduce the computation time. R offers several algorithms forperformance computing, such as lapply , mclapply , parLapply , for and foreach . For ourpurposes the foreach loops is the fastest solution, which we use for implementation.We use the doParallel and foreach packages in R as developed and proposed by Calaway,Weston, Tenenbaum and Analytics (2015) and Calaway, Weston and Analytics (2015),see also Kane et al. (2013). Since we have 100 financial firms we use the foreach loopstwice: the first loop is for the 100 financial firms with the second loop nested in the firstloop to perform the moving window estimation. The speed of computation is increasedconsiderably, the script is available from Quantnet: FRM_parallel_compute. To implement the visualization of the FRM, we use the JavaScript framework D3.js (orjust D3 for Data-Driven Documents), which is a JavaScript library for producing dynamic,interactive data visualizations in web browsers. The QuantNetXploRer is a good exampleof D3 in power. More information about the D3 architecture, its various designs and theD3-based QuantNetXploRer can be found in Bostock et al. (2011) and Borke and Härdle(2017b).
Figure 6:
The graph of Financial Risk Meter (FRM).
FRM_parallel_computeFigure 6 illustrates the D3-based FRM visualization, and more examples e.g. for Asia,Europe are available on Althof et al. (2019).
Figure 6 shows the FRM series from April 5, 2007 through September 23, 2016. The FRMhas no theoretical upper bound. In the time frame under consideration, the maximum9alue is 0 . . .
8% in the US, which was the highest sinceJune 1992. Another peak around the fourth quarter of 2011 coincides with the declinein stock markets in August 2011, which was due to fears of contagion of the Europeansovereign debt crisis to Spain and Italy.Therefore one may state that the peaks of FRM series identify financial events and theirimpact on financial and systemic risk. The minimum of the FRM series in the time periodunder consideration is observed in August 26, 2014, with a value of 0 . For convenience and following the color scheme of US homeland security office we dividerisk into five levels with different classifications and colors. The levels of risk are definedas different intervals of ratios for the FRM. These ratios are computed based on the pastvalues of the FRM. As shown in Figure 7, we have five levels of risk with five color codes.The current risk level is determined by the ratio based on all past FRM observations intowhich the current λ falls. Table 2 presents the risk levels as well as the colors, descriptionsand ratios of the risk levels.Color Risk level description FRM ratioGreen Low risk of crisis in the financial market. < >
80A financial crisis is imminent or happening right now.
Table 2:
Risk levels, color codes and ratios for FRM
As an example, on September 23, 2016 the value of FRM was 0 . . . igure 7: Risk levels of FRM as low risk of crisis in the financial market with color green. On the website the currentrisk level is marked with a cross as shown in Figure 7 for this example.
Zbonakova et al. (2017) analyze the factors affecting the value of λ and summarize that λ depends on three major factors: the variance of the error term, the correlation structure ofthe covariates and the number of non-zero coefficients of the model. Since high volatilityindicates high risk in finance and the number of non-zero coefficients is related to theconnectedness of the financial firms, they provide more theoretical evidence for the FRMas a risk measure. In their application, they find a co-integration relationship betweenˆ λ and other systemic risk measures. We extend their idea and use Granger causalityanalysis to validate FRM. We select three measures: VIX (see Hallett, 2009), SRISK (seeBrownlees and Engle, 2016) as well as the Google trends of the key word "financial crisis"(see Preis et al., 2013). 11or the causality analysis we first need to introduce the Vector Autoregression (VAR)model briefly. Lütkepohl (2005) proposes the VAR(P) model as follows: y t = α + A y t − + A y t − + · · · + A P y t − P + u t , (3)where y t def = ( y t , . . . , y Kt ) > , A i are fixed ( K × K ) coefficient matrices, u t is a K dimen-sional process. The coefficients could be estimated by applying multivariate least squaresestimation. In order to perform the Granger causality test, the vector of endogenousvariables y t is split into two subvectors y t and y t with dimensions ( K ×
1) and ( K × K = K + K . Then the VAR(P) model can be rewritten as follows: y t = y t y t = α α + A , A , A , A , y ,t − y ,t − + · · · + A ,P A ,P A ,P A ,P y ,t − P y ,t − P + u t u t (4)The null hypothesis of the Granger causality test is that the subvector y t does notGranger-cause y t , which is defined as A ,i = 0 for i = 1 , , . . . , P . The alternativehypothesis states that the subvector y t Granger-causes y t and is defined as: ∃ A ,i =0 for i = 1 , , . . . , P . The test statistic follows an F distributions with P K K and KJ − n ∗ degrees of freedom, where J is the sample size and n ∗ equals the total numberof parameters in the above VAR(P) model. The VIX series is often addressed as a “fear index” and can be interpreted as a measurefor systemic risk (Hallett, 2009). For reasons of comparability, we standardize these twoseries by setting the lowest value in the sample to zero and the highest to one. Figure 8plots the standardized FRM series (thick black line) and the VIX series (thin red line).The plot shows that both indicators move in the same direction, with the VIX series alittle more volatile. We also get some evidence of some financial events by observing thecorresponding volatility levels of the FRM and VIX. For example, in the end of 2008there is a sharp upward trend of FRM, whereas the upward trends dominates VIX aswell, which corresponds to the bankruptcy of Lehman Brothers on September 15, 2008.Both FRM and VIX have higher values between 2008 and 2010, which corresponds to thetime period of the financial crises. After 2013 the values of FRM are relative stable at alow level, while there is similar pattern of VIX, which shows signs of the slow recovery ofthe global economy from the recession.Before we perform the Granger causality test, we test for stationarity of both time serieswith the Augmented Dickey-Fuller (ADF) test. The results of the test are shown inTable 3. For the FRM series, the p value is larger than 0 .
05, so we cannot reject thenull hypothesis, i.e. the FRM series may have a unit root and may be non-stationary.We reject the null hypothesis for the VIX series with a p value smaller than 0 .
05 and12
008 2010 2012 2014 2016 . . . . . . Figure 8:
Scaled FRM (thick black line) and VIX (thin red line)
FRM_VIXconclude that the VIX series is stationary. We do not need to consider the co-integrationproblem, since only if both series are non-stationary, we should take into account theco-integration. There is a trade-off between using the original data and the transformed(differenced) data to find the causality relationship. Sims (1980) prefers to use the originaldata. He argues that VAR with non-stationary variables may provide important insights,if one is interested in the nature of relationships between variables. Brooks (2014) alsostates that differencing will destroy information on any long-run relationships betweenthe series. However, other people argue that the original non-stationary data might leadto untrusted estimation, see Yule (1926) and Granger and Newbold (1974). In our case,we consider both the original data and transformed data.Series p valuesFRM 0 . . Table 3: p values of ADF test for stationarity FRM_VIXFirstly, we consider the original data. We choose the VAR order according to four crite-ria: the Akaike information criterion (AIC), the Hannan-Quinn information criterion(HQ), the Schwarz criterion (SC) and the Prediction Error Criterion (FPE), see Ta-ble 4. While HQ and SC suggest an order 3 VAR process, AIC and FPE suggest an order20 process. We fit both VAR models with order 3 and order 20. Next, we check the13odel AIC HQ SC FPEFRM and VIX 20 3 3 20DFRM and VIX 19 8 5 19
Table 4:
Suggested order for VAR process by different criteria
Model Order VAR PT (asymptotic) PT (adjusted) BG ESFRM and VIX 3 < . × − < . × − . × − . × −
11 2 . × − . × − . × − . × − < . × − < . × − . × − . × − DFRM and VIX 5 2 . × − . × − . × − . × − . × − . × − . × − . × −
11 2 . × − . × − . × − . × −
19 1 . × − . × − . × − . × − Table 5: p values of model selection tests Cause Effect p valuesFRM VIX 4 . × − VIX FRM 6 . × − DFRM VIX 6 . × − VIX DFRM 8 . × − Table 6: p values of Granger causality test FRM_VIXautocorrelation of the residuals to decide the optimal order. Four tests are carried out:the asymptotic Portmanteau Test, the adjusted Portmanteau Test, the Breusch-GodfreyLM test and the Edgerton-Shukur F test. The null hypothesis of these tests is that thereis no first order autocorrelation among residuals. Choosing order 3 and 20 leads to therejection of all these tests (cf. Table 5). Subsequently we try the other orders and findthat with order 11 both the Breusch-Godfrey LM test and the Edgerton-Shukur F testsare passed. Therefore, we select order 11. The autocorrelation function of the residuals isplotted in Figure 9. Table 6 shows the results of the Granger causality test. All p valuesare smaller than 0 .
05 which indicates that the null hypothesis is rejected. Therefore,FRM Granger causes VIX, and also VIX Granger causes FRM.Next, we consider the transformed series. Since FRM is non-stationary, we take the firstdifference. The transformed series is called as DFRM. In Table 3 we see that DFRM is14 . . . FRM . . . FRM & VIX −30 −20 −10 0 . . . VIX & FRM . . . VIX
Figure 9:
Autoregression functions of FRM and VIX
FRM_VIX . . . DFRM . . . DFRM & VIX −30 −20 −10 0 . . . VIX & DFRM . . . VIX
Figure 10:
Autoregression functions of DFRM and VIX
FRM_VIX15tationary. Then the same procedure as before is performed. While HQ suggests an order8 process, SC suggest an order 5, and AIC and FPE both suggest an order 19 (cf. Table4). After checking the four tests for autocorrelation of the residuals, we conclude thatthe optimal order is 19. Although it does not pass the autocorrelation test, the p value isclose to the critical value 0 .
05, and the autocorrelation function confirms this result (cf.Table 5 and Figure 10). The result of the Granger causality test is summarized in Table6. We find that all p values are significantly smaller than 0 .
05, which indicates that thenull hypothesis is rejected. Therefore we conclude that DFRM Granger causes VIX, andalso VIX Granger causes DFRM.
SRISK is a macro-finance measure of systemic risk (Acharya et al., 2012; Brownlees andEngle, 2016). Our data on SRISK for the US are obtained from V-Lab . We alsostandardize SRISK, so that both series are comparable on the same scale. Figure 11plots the standardized FRM series (thick black line) and the SRISK series (thin blueline). We see that there is a peak in the first quarter of 2008 for SRISK, but afterwardsFRM and SRISK have similar patterns. Especially during the beginning of 2010 and thebeginning of 2012, the two series have a similar shape. . . . . . . Figure 11:
Scaled FRM (thick black line) and SRISK (thin blue line)
FRM_SRISKWe perform the same procedure as in section 4.1. The results of the ADF test for theSRISK series in Table 7 show that the series is non-stationary. Since the FRM seriesis neither stationary, we consider the co-integration of them. From Granger (1988) weknow that if both series are co-integrated, then there must be Granger causality betweenthem in at least one way. We perform the Engle Granger 2-step test for co-integration,which is suitable for bivariate time series. In the first step, the linear regression of FRM See the Systemic Risk Analysis Welcome Page: https://vlab.stern.nyu.edu/welcome/risk/ . Table 7: p values of ADF test for stationarity for FRM and SRISK Explanatory (Cause) Response (Effect) Value of test-statistic Critical value at 5%FRM SRISK -3.1 -1.95SRISK FRM -2.7 -1.95
Table 8:
Results of Engle Granger 2-step co-integration test
FRM_SRISKon SRISK is carried out, i.e. FRM is the explanatory variable and SRISK the responsevariable. In the second step, we test the residuals of the aforementioned linear regression.If these residuals are stationary, then there is co-integration of FRM and SRISK. Thenull hypothesis of this test is that the residuals are non-stationary. The result of thistest are summarized in Table 8. We conclude that FRM and SRISK are co-integrated, inother words, FRM Granger causes SRISK. If we regress SRISK on FRM, i.e. SRISK isthe explanatory variable and FRM the response variable, we also conclude that SRISKand FRM are co-integrated, which indicates that SRISK Granger causes FRM. We thusconclude that there is mutual causality between FRM and SRISK.17 .3 FRM versus Google Trends
Finally, we analyze the relationship between FRM and Google Trends (GT) for the key-word "financial crisis". Google Trends provides data on the search volume of particularwords and phrases relative to the total search volume. This can be disaggregated bycountries. If a keyword is more frequently searched for, this might indicate a particularinterest. Preis et al. (2013) analyzed the data related to finance from Google Trends, andfind that Google Trends data did not only reflect the current state of the stock markets,but may have also been able to forecast certain future trends. We use Google Trendsfor the keyword "financial crisis", assuming that more people will search for this term ifthey feel the risk for a financial crisis is high. The Google Trends data are weekly data.To allow for comparison with the FRM we apply cubic interpolation to estimate dailydata from the weekly Google Trends series. This series is compared with the daily FRMseries. Figure 12 plots both the daily FRM series as well as the cubic interpolated GoogleTrends daily series. Both series are standardized to the interval zero-one for comparison.We observe some co-movement between both series, but no continuous superiority of theGoogle Trends above FRM. . . . . . . Figure 12:
Scaled FRM (thick black line) and Google Trends (thin green line)
FRM_GTThe ADF test shows that the GT series is stationary (cf. Table 9). We perform two testsfor the relationship between the two series. Firstly, we consider the original data of FRM,then we consider the transformed data. We perform four criteria to find the optimal orderof VAR model. As the results in Table 10 show, all the criteria suggest an order 20 VARprocess. Therefore, we apply an order 20 VAR model. Next, the autocorrelation of theresiduals is tested. Although none of the tests can be passed (cf. Table 11), we have nobetter choice for the order than 20. The autocorrelation function of residuals are plottedin Figure 13. Table 12 shows the results of the Granger causality test. All p valuesare significantly smaller than 0 .
05, which indicates that the null hypothesis is rejected.Therefore, FRM Granger causes GT, and GT Granger causes FRM.For the first differenced FRM, i.e. DFRM, the same procedure is used. In Table 10 all18he criteria suggest an order 20 VAR process. The result of the autocorrelation testsare presented in Table 10. Although none of the tests is passed, we still use order 20.The autocorrelation function of the residuals is shown in Figure 14. Table 12 shows theresults of the Granger causality test. All p values are significantly smaller than 0 . . . Table 9: p values of ADF test for stationarity for FRM and GT Model AIC HQ SC FPEFRM and GT 20 20 20 20DFRM and GT 20 20 20 20
Table 10:
Suggested order for VAR process by different criteria
Model Order PT (asymptotic) PT (adjusted) BG ESFRM and GT 20 < . × − < . × − < . × − < . × − DFRM and GT 20 < . × − < . × − < . × − < . × − Table 11: p values of model selection tests Cause Effect p-valuesFRM GT 1 . × − GT FRM 2 . × − DFRM GT 6 . × − GT DFRM 4 . × − Table 12: p values of Granger causality test FRM_GT19 − . . . . FRM − . . . . FRM & GT −30 −20 −10 0 − . . . . GT & FRM − . . . . GT Figure 13:
Autoregression functions of FRM and GT
FRM_GT − . . . . DFRM − . . . . DFRM & GT −30 −20 −10 0 − . . . . GT & DFRM − . . . . GT Figure 14:
Autoregression functions of DFRM and GT
FRM_GT20
Conclusion
In this paper we propose and develop an AI based measure for systemic risk in financialmarkets: the Financial Risk Meter (FRM). The FRM is a measure for systemic risk basedon the penalty term λ of the linear quantile lasso regression, which is defined as theaverage of the λ series over the 100 largest US publicly traded financial institutions. Theimplementation is carried out by using parallel computing. The risk levels are classifiedby five levels. The empirical result shows that our Financial Risk Meter can be a goodindicator for trends in systemic risk. Compared with other systemic risk measures, suchas VIX, SRISK, Google Trends with the keyword “financial crisis”, we find that the FRMand VIX, FRM and SRISK, FRM and GT mutually granger cause one another, whichmeans that our FRM is a good measure of systemic risk for the US financial market.All the codes of FRM are published on with keyword FRM. The R package RiskAnalytics (Borke, 2017b) is another tool with the purpose of integratingand facilitating the research, calculation and analysis methods around the FRM project(Borke, 2017a). The up-to-date FRM can be found on hu.berlin/frm.21 A pp e nd i x : F i n a n c i a l I n s t i t u t i o n s W F C W e ll s F a r go & C o m p a n y A O NA o np l c J P M J P M o r ga n C h a s e & C o A LL A ll s t a t e C o r p o r a t i o n B A C B a n k o f A m e r i c a C o r p o r a t i o n B E N F r a n k li n R e s o u r ce s , I n c . CC i t i g r o up I n c . S T I Sun T r u s t B a n k s , I n c . A I G A m e r i c a n I n t e r n a t i o n a l G r o up , I n c . M C O M oo d y ’ s C o r p o r a t i o n G S G o l d m a nS a c h s G r o up , I n c . P G R P r og r e ss i v e C o r p o r a t i o n U S B U . S . B a n c o r p A M P A m e r i p r i s e F i n a n c i a l S e r v i ce s , I n c . AX P A m e r i c a n E x p r e ss C o m p a n y A M T D T D A m e r i t r a d e H o l d i n g C o r p o r a t i o n M S M o r ga nS t a n l e y H I G H a r t f o r d F i n a n c i a l S e r v i ce s G r o up , I n c . B L K B l a c k R o c k , I n c . T R O W T . R o w e P r i ce G r o up , I n c . M E T M e t L i f e , I n c . N T R S N o r t h e r n T r u s t C o r p o r a t i o n P N C P N C F i n a n c i a l S e r v i ce s G r o up , I n c . ( T h e ) M T B M & T B a n k C o r p o r a t i o n B K B a n k O f N e w Y o r k M e ll o n C o r p o r a t i o n ( T h e ) F I T B F i f t h T h i r d B a n c o r p S C H W T h e C h a r l e s S c h w a b C o r p o r a t i o n I V Z I n v e s c o P l c C O F C a p i t a l O n e F i n a n c i a l C o r p o r a t i o n LL o e w s C o r p o r a t i o n P R U P r ud e n t i a l F i n a n c i a l, I n c . E F X E q u i f a x , I n c . T R V T h e T r a v e l e r s C o m p a n i e s , I n c . P F G P r i n c i p a l F i n a n c i a l G r o up I n c C M E C M E G r o up I n c . R F R e g i o n s F i n a n c i a l C o r p o r a t i o n C B C hubb C o r p o r a t i o n M K L M a r k e l C o r p o r a t i o n MM C M a r s h & M c L e nn a n C o m p a n i e s , I n c . F N FF i d e li t y N a t i o n a l F i n a n c i a l, I n c . BB T BB & TC o r p o r a t i o n L N C L i n c o l n N a t i o n a l C o r p o r a t i o n I C E I n t e r c o n t i n e n t a l E x c h a n g e I n c . C B G C B R E G r o up , I n c . S TT S t a t e S t r ee t C o r p o r a t i o n K E Y K e y C o r p A F L A fl a c I n c o r p o r a t e d N D A Q T h e NA S D A QO M X G r o up , I n c . C I N F C i n c i nn a t i F i n a n c i a l C o r p o r a t i o n C A CCC r e d i t A cce p t a n ce C o r p o r a t i o n C NA C NA F i n a n c i a l C o r p o r a t i o n B R O B r o w n & B r o w n , I n c . 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