“ A connectedness analysis of German financial institutions during the financial crisis in 2008

For core financial market activities like risk management and asset pricing, it appears to be crucial to investigate the “connectedness” among financial institutions. In times of economic crises, a suitable measure of connectedness can provide valuable insights of financial markets and helps to understand how institutions influence each other. In particular, depending on contractual obligations between financial institutions, the financial distress at a bank with large systemic impact is likely to cause also distress at other institutions. In the literature, the latter phenomenon is generally tagged by ’contagion’ and can eventually result in severe economic crises. The purpose of this paper is to investigate the connectedness among German financial institutions during the global financial crisis 2007-2009, where the authors focus particularly on 2008 and its height in September 2008 with the bankruptcy of Lehman Brothers. They make use of the definition of connectedness, as it was recently proposed by Diebold and Yilmaz (2014). Their approach relies on analyzing multiple time series of volatilities by a vector autoregressive (VAR) model and a generalized forecast error variance decompositions. It provides several meaningful measures of connectedness and allows for static (average), as well as dynamic (daily time-varying) analyses. The authors show that the connectedness in Germany can be described well by the model.


Introduction 
One central aspect of modern risk management is to analyze the interdependence of certain actors on the financial market. Measuring these interdependences between financial institutions becomes very important during times of economic crises to judge e.g. contagiousness in a market. In particular, since the global financial crisis during 2007-2009, 'connectedness' between financial institutions has been discussed extensively not only in the US, but also in Europe. The Basel Committee comments on the risk of contagion in a market as follows (Basel Commitee on Banking Supervision, 2011, p. 7): "Financial distress at one institution can materially raise the likelihood of distress at other institutions given the network of contractual obligations in which these firms operate. A bank's systemic impact is likely to be positively related to its (inter)connectedness visà-vis other financial institutions".
Analyzing the connectedness in financial markets appears to be central to understanding the inner workings of these markets. It is important for core financial market activities like risk management and asset pricing. For example, connectedness is helpful to analyze key aspects of market risk, credit risk, as well as systemic risk and it is also central to understanding macroeconomic (business cycle) risk. Especially in times of crisis, investigating connectedness can provide valuable insights into questions like how institutions influence each other.
Unfortunately, there exists no natural definition of financial or economic connectedness or a measure for it. In general, connectedness shall be based on contractual obligations between firms. These can be found in the balance sheets of firms, but a high frequency analysis of balance sheets is usually not feasible. A couple of different approaches to conceptualize and to measure connectedness at various levels have been proposed in the literature. Adrian  In this paper, we use the popular concept of connectedness, as it has been recently proposed by Diebold and Yilmaz (2012. Their approach relies on analyzing multiple time series of volatilities by a vector autoregressive (VAR) model and to measure connectedness in several ways based on forecast error variance decompositions. Their concept allows the definition of several natural and insightful measures of connectedness among financial asset returns and volatilities. Furthermore, these variance decompositions define weighted, directed networks which relate their connectedness measures to those used in network literature. The approach of Diebold and Yilmaz has been used in equity return volatility to analyze connectedness among US financial institutions during 1999-2010 in Diebold and Yilmaz (2014), as well among major US and European financial institutions during 2004-2014 in Diebold and Yilmaz (2016). Using policy uncertainty as input data, it has been used by Alter and Beyer (2014), whereas Klößner and Sekkel (2014) consider a modification where the connectedness is defined based on generalized impulse responses instead of variance decompositions.
The contribution of this paper is to measure and analyze the connectedness among financial firms in Germany at various levels based on the approach of Diebold und Yilmaz (2014). Based on raw highfrequency stock price data for 2008, we construct a multiple time series of volatilities for German financial firms listed on the stock exchange and traded on active liquid markets. The data were provided by the Karlsruhe Institute of Technology (KIT). The data set includes all trades on each trading day for each stock on the electronic trading platform XETRA 1 . In particular, we focus on September 2008, the height of the financial crisis from 2007 to 2009, where Lehman Brothers went bankrupt. To analyze the data, we consider both average and daily timevarying connectedness measures.
The paper is organized as follows. Section 1 introduces the concept and describes the methodology of connectedness. In section 2, the results of the connectedness analysis of German financial firms are shown and discussed in detail. The final section concludes.
Note that Σ is equal to the optimal mean squared error (MSE) predictor. As the Φ 's and Σ will be unknown in general, they have to be estimated from data. Having observations ,.., at hand, this can be done by using, e.g., least-squares estimation of the VAR coefficient matrices to get , 1 ,…, and Σ u , by computing the sample variance matrix of the VAR residuals =∑ , 1,…, . As the Φ's are recursively defined by Φ ∑ A Φ with 1,2, …, where A 0 for , we get estimators Φ similarly by using A , 1 ,…, . However, in particular over long time horizons or during crises, the Φ 's will usually not be constant over time, that is, the data generating process (DGP) will not be time-stationary. Hence, it is reasonable to estimate time-varying parameters Φ (t). For this purpose, we consider a uniform one-sided estimation window with width , where for each time point, only the most recent time points enter the estimation with equal uniform weights.
The VMA coefficients contain all contemporaneous and dynamic features of the multivariate time series system. Instead of analyzing these coefficients directly, an alternative way is to use variance decompositions (see, e.g., Lütkepohl, 2007). The variance decomposition indicates the amount of information each variable contributes to the other variables. It determines how much of the -step-ahead forecast error variance of each of the variables in forecasting Y can be explained by shocks. Note that the VAR innovations are generally contemporaneously correlated, but the calculation of variance decompositions does require orthogonal innovations. One common solution to transform the model to get orthogonal innovations is to use the Cholesky factorization. Instead of using this Cholesky approach that depends on the orderings of the variables, Diebold und Yilmaz (2014) propose to use the generalized variance decomposition by Koop et al. (1996) and Pesaran und Shin (1998). This generalized approach uses correlated shocks instead of orthogonal shocks, but factors into the calculation of the distribution of the historically observed errors.

Generalized variance decomposition.
In comparison to the special case of orthogonal shocks u in (1) that allows an application of a standard variance decomposition, this is not possible if the shocks are correlated. As discussed in Diebold and Yilmaz (2014, p. 5), reduced-form shocks are rarely orthogonal and it is inevitably necessary to make assumptions to identify uncorrelated structural shocks from correlated reduced-form shocks. To address this issue, they suggest to use the generalized variance decomposition (GVD) introduced by Koop et al. (1996) and Pesaran und Shin (1998) instead of using Cholesky factorization. If the latter approach is used, the analysis will unintentionally depend on the ordering of the variables in the VAR system. Nevertheless, the GVD requires normality of the shock distribution, which is the price to pay here. For reaching normality, we shall let our connectedness analysis be based on return volatilities instead of returns, since volatilities tend to be much more serially correlated than returns (compare Diebold und Yilmaz, 2014, p. 14). Further, to make the data more normal-like, we take logarithms.
In general, for variance decompositions, own variance shares are defined to be the fractions of the -step-ahead error variances in forecasting due to shocks to , for 1 ,…, and spillovers to be the fractions of the -step-ahead error in forecasting due to shocks to , for , 1,2,…, , such that . The -step-ahead generalized variance decomposition matrix , , 1, … , is defined to have entries where is a selection vector with -th element unity and zeros elsewhere, Φ is the -th moving-average coefficient matrix, Σ is the covariance matrix of the error terms and is the -th diagonal element of Σ .
Note that the denominator is the forecast error variance of variable and the numerator is the contribution of shocks in variable to the -stepahead forecast error variance of variable . As shocks need not to be orthogonal, forecast error variation contributions do not necessarily sum up to 100, i.e., row sums of are not necessarily equal to 100. Hence, to be able to interpret the entries of a variance decomposition matrix as shares, they have to be scaled. That is, we shall use with ∑ instead of in the following. The entries of can be used to analyze the connectedness between assets and . More precisely, as described in Diebold und Yilmaz (2014), the matrix leads to a so-called Connectedness table, which displays pairwise, as well as system-wide connectedness (see Table 1). The connectedness table is central for understanding the different types of connectedness and their relation that will be defined below.
For a system with variables ,…, , its upper-left -block matrix contains the scaled generalized variance decomposition matrix of thestep-ahead forecast error, i.e.
. Its rightmost column contains row sums, the bottom row contains column sums, and the lower-right element contains the average of the column sums (equal to the average of the row sums), where, in all cases, , i.e. the diagonal elements are excluded. The off-diagonal entries of measure pairwise directional connectedness from j to i and, following the notation in Diebold und Yilmaz (2014), we set ← . ( Note that ← ← in general. The net pairwise directional connectedness from to is defined as The off-diagonal row and column sums, labeled "From Others" and "To Others" in the connectedness table, define total directional connectedness from others to ("from" connectedness) as and the total directional connectedness to others from ("to" connectedness) is defined as Analogously, the net total pairwise directional connectedness ("net" connectedness) is defined as Finally, the grand total of the off-diagonal-entries in divided by (equivalently, the average of the "From Others" column or "To Others" row) measures total connectedness

Connectedness analysis of German financial institutions
In this section, we make use of the connectedness tools described in Section 1 to monitor and characterize the evolution of connectedness among German financial institutions during 2008. We proceed in four steps. First, in section 2.1, we describe the data set that we use as the basis for our connectedness analysis. Next, we conduct a static and a dynamic connectedness analysis in sections 2.2 and 2.3, respectively. In section 2.4, we take a closer look at the impact of Lehman Brothers' bankruptcy. Finally, in section 2.5, we comment on the robustness of our results with respect to the choice of model parameters.
where describes the stock price at interval 1, … ,104 and trading day 1, … ,253. Figure 1 shows the daily realized volatility for each considered firm. Instead of balance sheet information, we use return volatilities, which depend (thus, not only) on the forward-thinking assessment of brokers. Hence, we consider volatility connectedness, as suggested by Diebold und Yilmaz (2014), because volatility tracks investors' fear (e.g., "VDAX" or "VDAX New" 3 ) and it is crisis sensitive, whereas crises will be of much interest to us.

Static connectedness analysis.
Based on the volatility data pre-processed, as described in section 2.1 above, a static connectedness analysis is conducted. The analysis is static in the sense that we set → ∞ in our analysis in this section leading to an "in average" connectedness analysis. Furthermore, we use a VAR(3) approximating model and a forecast horizon of 1 2. The latter choice was used also in Diebold und Yilmaz (2014) and was motivated, e.g., by the 10day Value at Risk (VaR) required under the Basel accord. In section 2.5, we discuss the robustness to the choice of VAR order and forecast horizon. Table 3 shows the connectedness among the considered institutions. For each firm, the diagonal entries of the upper-left matrix ("own connectedness") are the largest ones in each column, especially for IKB and comdirect bank. However, in most of the cases, the total directional connectedness ("From Others" or "To Others") is larger than the "own connectedness". The total connectedness, the mean of the total directional connectedness ("from" and "to" are equal by definition), is of medium size 46.51. The total connectedness describes the average impact of connectedness. Total dire e series of t m" and "net" . We observe s (systematica "from" conn lained by Di , in particula smitted to oth ) institution i se also larger quite distincti smitted instit ness has a wid already obse ure 2, we s nectedness d most firms i explained by ual stocks d cks had been ure 4 shows m" direction ms of their m % quantiles, r means of "fro r distribution discussed alr nectedness is nectedness a m" connecte ge of the "to ve that "from ereas "to" co ng the crisis. el in Figure 3 l connectedn 8 is mainly c he "to" conne urs in one me large exten In Figure 3, edness ("to", ancial instituo" connectedver time than enomenon is ) by the fact pected to be a larger (cenexpected to since shocks ity and as the "to" connecnnectedness. ectedness in f directional ably as well ber. This can hat hit indid and these cks.
for "to" and eight firms in m, 25%-and by definition, ess are equal, y differently. on of "from" iation of "to" stribution of ontrast to the over, we ob-  We begin with the connectedness analysis on September 11. As the liquidity crisis started on September 9, the banks did not trust each other anymore and were not willing to lend money to other banks. The EURIBOR 1 increased for September and October. The total connectedness on September 11 reaches its second lowest value for the whole year and the "own" connectedness is rather large. Only a few net pairwise connections between the firms are present. The firms want to leave the system and to be independent from the other firms.
On September 12, Deutsche Bank became the greatest single shareholder of Postbank. The total connectedness went up. Noticeable is the substantially increased impact of Postbank on the other firms. More precisely, the from directional connectedness of Postbank is very high with ∘← 522.93 and the net pairwise connectedness from Postbank to the single institutions ranges between 47.45 and 86.02. The large connectedness from Postbank to the others remains till September 17. September 15 is also called "Black Monday". Lehman Brothers was bankrupt. The total connectedness falls. The pairwise connectedness between the firms decreases too, except the impact of Postbank to the others. The stock prices of the firms lose about 22% to 33% of their value in the following days.
On September 18, Central Banks of America, Europe and Japan offer more than 180 billion US-Dollar to reduce tensions on the financial market. Banks can lend up to 40 billion Euro for one day, as well as Euro quick tenders from the European Central Bank. The total connectedness rises and the net pairwise connections become very strong.
On September 19, the US administration worked on a 700 billion US Dollar plan to rescue banks. The fund was to protect several banks. Some stock prices begin to rise up again. In Germany, the Ba-1 EURIBOR is the interest rate at which banks lend to each other in Europe. Till September 22, the total connectedness remained high. In these days, several shocks occurred that affected the whole system such that we can hardly connect each single shock to its trigger firm. But for the single days, we can divide the institutions into groups. On September 18, the net-transmitters are Commerzbank, comdirect bank and Allianz and the net-receivers are Aareal Bank, IKB and Münchner Rück. On the September 19, the nettransmitters are Commerzbank, comdirect bank and Allianz and the net-receivers are Deutsche Bank, Postbank, Aareal Bank, IKB and Münchner Rück. On September 22, the net-transmitters are comdirect bank and Allianz and the net-receivers are Münchner Rück and Deutsche Bank. On September 23, the total connectedness began to fall and the strong pairwise connections began to vanish from the system. Mitigating the impacts of the financial crisis and stabilizing the system, the Federal government of Germany lent money to crisis-affected banks and gave guarantees.

Sep 18
Banks can lend up to 40 billion Euro for one day to overcome their liquidity difficulties.

Sep 19
The US administration decides to stabilize the banking system with 700 billion USD. Hence, a lot of stocks rise, e.g., Commerzbank. Prohibition of short sales by BaFin. Several stocks of banks and insurance companies were affected. Fig. 7. We plotted all net pairwise directional connectedness more than 5. The black edges describe connectedness between 5 and 8, the blue edges between 8 and 10 and the red ones more than 10. The vertices of Allianz and Münchner Rück are marked green as they are insurance companies. The node size is an indicator for the stock market capitalization 2.5. Robustness check. In conclusion of this section, we discuss the robustness of connectedness analysis with respect to the choice of model parameters. These are the VAR order p, the rolling window size and the forecast horizon H. Furthermore, we consider the Cholesky factorization in comparison to the generalized variance decomposition. The corresponding analyses with different parameters are not reported in this paper, but can be requested from the authors. 100 and varying , we nearly see no differences in total connectedness. If differrences occur, then, they are small. With decreased window size 7 5 and varying , we observe differences between the variation of the models and the preferred model with 100 and 1 2 rises with the size of . In non-crisis times, the total connectedness is overestimated, but in crisis times, there are hardly any deviations between the models. Increasing the window size to 125 and varying leads to larger differences between this model and the chosen model for smaller values of . Especially in crisis times, there are almost no differences. In non-crisis times, the total connectedness is underestimated. Qualitatively, the evolution over the year turns out to be very similar for 75 and 125. We conclude that the results are pretty robust to the choice of rolling window size and forecast horizon.

Comparison to Cholesky factorization.
Concluding our robustness assessment, we compare our results using the generalized variance decomposition to corresponding results using the Cholesky factorizations. As the latter results crucially depend on the ordering of the variables, it is not suitable for assessing pairwise and total directional connectedness, but it should be robust for total connectedness.
Given the dependence of the Cholesky approach on the ordering, we considered the maximal, the minimal and the average value of total connectedness. The latter are computed considering all possible permutations of the variables 1 . This approach is preferred by Klößner and Wagner (2012) to the method propagated by Diebold und Yilmaz (2014) based on GVDs. Diebold and Yilmaz (2014) used 100 random orderings of the realized stock return volatilities to get the total connectedness by Cholesky factorization and averaged over these 100 orderings. The differences between the GVD approach and the approach of Klößner und Wagner (2012) for the total connectedness are very small for the average, whereas the deviations are huge for the maximum and minimum values. Qualitatively, the evolution of the total connectedness by the GVD and Cholesky factorization are quite similar. Furthermore, for the average Cholesky approach, we varied also the window size ∈ 75, 100, 125 and the forecast horizon ∈ 6,12,18 . Irrespective of the choice of the rolling window size and the predictive horizon, we observed only small deviations of total connectedness during 2008 between GVD and Cholesky factorization. Especially in crisis time (September and October), the deviations vanish. Overall, we get that the results are pretty robust to the Cholesky factorization.

Conclusion
We have conducted a connectedness analysis, as proposed by Diebold und Yilmaz (2014), for eight German financial institutions during the financial crisis in 2008. Using high-frequency intraday stock trading data, we calculate the daily realized return volatility, which serves as the basis of our analysis. We provide several versions of volatility connectedness that help to understand the interplay between financial institutions. In particular, these measures allow to study the evolution of connectedness during crises. In general, connectedness in Germany can be depicted well by using the approach of Diebold and Yilmaz (2014) based on VAR models and GVDs. Our empirical results are nicely interpretable and lead to helpful insight. For example, we see that with the exception of IKB, all firms affected by the financial crisis show large static connectedness. However, a closer look at dynamic connectedness measures reveals that IKB is dominated by one huge peak in "to" connectedness which results in very large static "own" connectedness. Also, as expected, we find pronounced connectedness between firms that, indeed, have contractual obligations, as, e.g., Postbank and Deutsche Bank or Münchner Rück and Allianz. The dynamic analysis shows that connectedness measures can react quickly on shocks occurring at the market. For example, the shocks triggered by IKB, Lehman Brothers or insurance companies immediately lead to an increase of total connectedness. Not expected was the rather low connectedness between Deutsche Bank and the other banks. Nevertheless, this result turns out to plausible indeed, as Deutsche Bank has by far the largest market capitalization among them and it is the only German bank that can be looked upon as a global player. Hence, Deutsche Bank plays in a different league leading to low connectedness to the other German banks. Finally, robustness checks indicate that our empirical results are quite robust to different parameter choices. In summary, the values resulting from this method should not be taken as absolutes, but rather as an indication of connectedness among financial firms. Nevertheless, as it gives insights in the interdependencies between financial institutions it can be helpful tool, e.g., for risk management and asset pricing.