The 21St Century - Cluster Formation in the S&P 500

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The 21st Century - Cluster Formation in the S&P 500

Maximilian Göbel and Tanya Araújo

REM Working Paper 043-2018

July 2018

REM – Research in Economics and Mathematics Rua Miguel Lúpi 20, 1249-078 Lisboa, Portugal

ISSN 2184-108X

Any opinions expressed are those of the authors and not those of REM. Short, up to two paragraphs can be cited provided that full credit is given to the authors.

The 21st Century - Cluster Formation in the S&P 500

Maximilian Gobel¨ Tanya Vianna de Araujo´ ∗

June 2018

ABSTRACT

Since the beginning of the new millennium, investing in the stock market must have felt like a ride on the roller-coaster. The market went through every state between long-time troughs, trade suspensions and all-time highs. Such turbulent periods give all kinds of portfolio managers a hard time, with correlations of stock returns frequently varying. A deeper understanding of the dynamics within the market and the interlinkages between industry sectors is not only of added value to stock market participants, but also to policy makers, being responsible for implementing measures to counteract possible future macroeconomic imbalances. Therefore, this paper tries to uncover the community structure in the S&P 500 in the years 2000 through 2015 with additional focus on the effect of the Great Recession - especially on the ﬁnancial sector.

∗ISEG/UL - Universidade de Lisboa, Department of Economics; REM - Research in Economics and Mathematics; UECE - Research Unit on Complexity in Economics. 1 Introduction

Since the beginning of the 21st century financial markets have undergone turbulent times. Especially stock markets have experienced several ups and downs, while sucessively breaking through all-time highs. Focusing on the period between 01/03/2000 and 12/31/2015, the S&P 500 Index, which is generally regarded as a proxy for the overall stock market, lost almost 50% within two years after the bursting of the so-called ”dot-com” bubble. Consequently, it reached its sample-period trough on 10/09/2002 at 677.187 points1. Af- terwards, it took the S&P 500 almost exactly four years to fully recapture its peak value of the year 2000. The bullish market persisted even another twelve months, until rumors about falling housing prices, precedingly lax lending standards and a fragil financial system made the S&P 500 turn again on 10/09/2007 - exactly five years after its last trough. Within the following 18 months, the Index lost more than 55% of its turning-point value and - on 03/09/2009 - counted only 1.5 points more than its sample-period record low dated from 10/09/2002. Since then, bearish sentiment seems to have vanished, with not only the S&P 500, but stock indices all around the world ever rising. Between its trough in March 2009 and the end of the assessment period, the S&P 500 score rose by 236% in real terms. The bullish sentiment even persisted until early 2018, when the Index reached its all-time high, ranking at about 4.8 times above its value on 03/09/2009 and adjusted for inflation. A complete assessment of the forces driving these fluctuations during the first decade of this century and the consistently bullish behaviour thereafter, demands the merging of economic, mathematical and psychological sciences. The literature on the behaviour of asset pricing, however, generally assumes random processes to be underlying the dynamics of stock returns (Mantegna, 1999). A vast literature on models, such as Fama & French (1992), tried to identify common factors of single stock returns and to shed further light on the seemingly random behaviour of stock returns. Nevertheless, one commonly agreed perception about stock prices is their procyclical- ity (Cochrane, 1997; Kocherlakota, 1996; Mehra & Prescott, 2003). Thus, the correlation of stock returns plays a major role in portfolio construction and financial modelling. As

1Adjusted for Implicit Price Deﬂator: 31.12.2008 = 100.

1 Marvin (2015) points out, these correlations are, however, not constant over time and may even reverse during times of crisis. The widespread misperception of these dynamics by investors and ﬁnancial product modellers in the beginning of the 2000s is said to have heavily ampliﬁed the severity of the Great Recession (Pozsar et al, 2010). Assessing the co-movements of stock returns and time-varying market structures are, thus, key to understanding and reacting to the effects of a changing macroeconomic environment. Therefore, the purpose of this paper is to shed further light on the community structure of the S &P 500 Index in the years 2000 through 2015, by emphasizing the role of the Great Recession. The remainder of the paper is then structured as follows: section 2 gives a short summary of common clustering techniques, while section 3 presents the method of optimal modularity. Section 4 introduces the method of symbolization in order to facilitate the partition exercise of section 5. The last section concludes.

2 Literature Review

The methods for describing the co-movement of a system’s individual components are summarized in the literature under the notion of cluster formation or community structure. Tan et al (2006) describe clustering as the grouping of items based on empirical data. In another study, Newman & Girvan (2004) define community structure as ”the division of network nodes into groups within which the network connections are dense, but between which they are sparser” (Newman & Girvan, 2004, p.1.). The literature uses the notions of nodes or vertices to describe the single data points of a network, such as the individual stocks of a portfolio. Edges define the connecting links between these data points. Those methods originated initially in the field of physics, but have found entry into the works of economic and social scientists (Newman, 2003). For clustering time-series data, both the definition of an efficient algorithm and an adequate measure of similarity, respectively distance, between nodes are necessary features (Piccardi et al, 2011). To qualify as an appropriate proxy of distance, the respective measure must fulfill three re- quirements, necessary for defining a metric (Mantegna, 1999). Thus, a distance measure d(i, j) between the nodes i and j can only be accounted for as a metric, if the following

2 conditions are jointly fulﬁlled:

d(i, j) = 0 if and only if i = j (1)

d(i, j) = d( j,i) (2)

d(i, j) ≤ d(i,k) + d(k, j) . (3)

Due to condition (1), the Pearson correlation coefﬁcient cannot be regarded as a metric2 and thus, does not qualify as a measure of similarity, respectively distance (Mantegna, 1999). Nevertheless, the literature uses it frequently as a basis for computing metric- compliant distance measures (Mantegna, 1999; Tan et al, 2006). For his analysis of traded stock portfolios, Mantegna (1999) transformed the correlation coefﬁcient of stocks into a measure, equating the Eucledian Distance between two data points: q d(i, j) = 2(1 − ρi j) . (4)

Mantegna (1999) then proceeded by constructing a Minimal Spanning Tree by sequen- tially linking the nodes with the lowest distance measure. This allows to efﬁciently assess the intensity of connections between stocks and different industrial sectors within a given portfolio. His analysis of the S&P 500 between 1989 and 1995 revealed strong intra- industry sector and intra-industry sub-sector connections, suggesting that, statistically, those stocks react to the same economic conditions. In contrast to the correlation of stock returns as a basis for creating a proxy for distance, Marvin (2015) favours company related indicators such as revenues-to-assets ratio or net-income-to-assets ratio, as these are said to show higher persistence across varying economic states. Tan et al (2006) apply a prototype-based algorithm, of which various designs exist (Marvin, 2015). The k-means approach requires each node within a cluster to be closer to the cluster’s prototype than to the prototype of any other cluster. In the case of k-means, the prototype is the mean of the total sum of squared Euclidean distances between all the nodes within a cluster (Tan et al, 2006). Tan et al (2006) prespecify both the number of centroids and the centroids themselves before running the algorithm. The network’s vertices are then connected to that centroid, which minimizes the squared Eucledian distance between the node under consideration and the centroid. Each centroid’s value is 2d(i,i) = 1.

3 updated after each assignment, which can cause inter-cluster movements of already assigned nodes. The process stops as soon as an update does not affect the existing community structure (Tan et al, 2006). The weakness of the k-means approach arises from the necessity to pre-specify the initial centroids and to fix the number of clusters manually. Nevertheless, the two methods described above have a long tradition in the literature. K-means belong to the class of graph partitioning, whereas the Minimal Spanning Tree technique by Mantegna (1999) is a type of hierarchical clustering or community structure detection. While the first assumes the number of clusters to be given exogenously and the number of intra-cluster nodes to be at least approximately fixed, the latter will en- dogenously determine both the number of communities within a network and the number of vertices within each cluster (Newman, 2006; Newman, 2010). Hierarchical clustering can be subdivided into two distinct forms: agglomerative and divisive. The difference between the two approaches results from the direction of filtering the single clusters: the first originates from an initially empty network and successively adds links between items according to a measure of similarity. The divisive approach of hierarchical structuring acts on the assumption of an initially completely specified network, from which the weakest links are removed iteratively (Newman & Girvan, 2004). Newman & Girvan (2004), however, critizise especially the agglomerative partitioning for its incapability to detect already known community structures and the tendency to neglect outliers. For the analysis of the taxonomy of a portfolio of stocks, hierarchical clustering, respectively community structure detection, offers the appropriate features, as the number of clusters - if any - is determined by the network itself (Newman, 2006). Thus, for exam- ining the community structure of the S&P 500 in the various periods between 2000 and 2015, this paper builds up on the community structure detection approach by Newman (2006) - the method of optimal modularity.

3 The Method of Optimal Modularity

The method of optimal modularity was initially introduced by Newman & Girvan (2004) as a measure to quantify the quality of a speciﬁc cluster formation within a network. Isogai (2015) describes it as a ”sort of distance between the actual density of the edges

4 and that of a randomized network” (Isogai, 2015, p.12.), which is most frequently used in modern cluster-detection analyses. The underlying principle of modularity is not - as stated by Newman (2006) - to minimize the number of edges between clusters, but to generate a structure which exhibits fewer edges than usually expected from a random network structure. Thus, Newman (2006) deﬁnes the modularity measure as the difference in the number of edges within a cluster and the number of edges in a random network. A random network is deﬁned as one, in which the number of edges per node is preserved, whereas the connections are spread randomly throughout the network. The crucial assumption is that a random network does not exhibit any community structure (Isogai, 2015), whereas a large modularity suggests a large deviation of the detected community structure from a completely randomized network.

3.1 The Algorithm of Modularity

3.1.1 Division into Two Clusters

As stated in the inital paper on modularity (Newman & Girvan, 2004), the modularity measure Q builds up on Newman’s (2003) assortative mixing, which is based on correlations of any properties of neighbouring nodes within a network. In the case of the S&P 500 Index, assortative mixing would measure the pairwise correlation of stock returns. Thus3, the initial network N consists of n nodes, respectively vertices. The algorithm starts by dividing the network into two communities. If a node i belongs to community

1, si will take the value of 1. If the node belongs to community 2, si will be −1. The number of connections, respectively edges, Ai j, between nodes i and j can either be 1 or

0. Every single Ai j displays an entry in the adjacency matrix of the whole network. Thus, the diagonal elements of the adjacency matrix have a value of 0, as:

Aii = 0 . (5)

What the literature calls the degree of a vertex, ki, is the number of connections between each node i and all the other n − 1 nodes within the network N. Thus, the total number of

3The following description will be mostly based on Newman (2006).

5 edges, m, within the network is:

1 1 m = ∑ki = ∑Ai j . (6) 2 i 2 i The expected number of edges between vertices i and j, which can also be thought of as the probability of a random edge connecting vertices i and j, is given by:

k k i j . 2m

Remembering the goal of a clustering algorithm being to generate fewer than expected edges between communities of a network, the modularity measure Q is hence a function of the difference between the number of connections linking vertices i and j, Ai j, which

kik j are detected by the algorithm, and the expected number of edges 2m , while preserving the degree of each vertex i and j: 1 kik j Q = ∑ Ai j − sis j . (7) 4m i j 2m

If nodes i and j belong to the same cluster, the last term ensures that Q will take on a positive value and vice versa. Equation (7) can alternatively be written as:

1 Q = sTBs , (8) 4m where s is the column vector with items si. B is a symmetric matrix consisting of the following elements:

k k B = A − i j , (9) i j i j 2m which Newman (2006) calls the modularity matrix. As the elements within each row and each column of the modularity matrix sum to 0,

Bi j always displays an eigenvector composed only of ones and an eigenvalue equal to 0. This feature accounts for the possibility that any division of the network N into several communities is suboptimal.

Next, all eigenvectors ei of the modularity matrix, Bi j, will be normalized to ui, such T that each entry of ei = (a,b,c) will be divided by the length of the vector:

T p 2 2 2 p 2 2 2 p 2 2 2 ui = a/ a + b + c ,b/ a + b + c ,/ a + b + c . (10)

6 The length of the normalized eigenvectors ui will thus be equal to unity and two distinct normalized eigenvectors ui and u j are in addition orthogonal:

ui ⊥ u j , such that each eigenvector ui is orthonormal. Making use of: n s = ∑ aiui , (11) i=1 and the following dot-product:

T ai = ui • s , (12) the modularity measure Q of (7) can be transformed into: n 1 T 1 T 2 Q = ∑aiui B∑a ju j = ∑ ui • s βi , (13) 4m i j 4m i=1 where βi is the eigenvalue corresponding to the normalized eigenvector textb f ui of the modularity matrix B. As the network N still exists as a whole and hasn’t been divided into any distinct communities, yet, the goal is now the partition into two clusters, such that the modularity measure Q is maximizied. Therefore, the eigenvalues βi with i = 1,...,n, corresponding to the orthonormal eigenvectors ui, will be ordered with decreasing value. Because of: n 1 T 2 Q = ∑ ui • s βi , (14) 4m i=1 the maximization of Q would require s to be parallel to the eigenvector ui corresponding to the largest eigenvalue βi, which is i = 1 by convention. The reason is, that if s was parallel to u1, then - due to orthogonality of two distinct eigenvectors ui and u j - the following condition would hold:

s • u j = 0 with s k ui for i 6= j . (15)

As the elements of s are restricted to ±1, a parallelization of s and ui is not possible. The closest approximation is a maximization of:

T max s • u1 .

7 This is done by assigning a value of +1 to si, if the corresponding entry in u1 displays a positive preﬁx and vice versa. Thus, the vertices i of the network N are assigned to each community according to the preﬁx of their corresponding value in u1. The value of an element i in eigenvector u1 gives an indication of the magnitude of change in modularity Q, if the corresponding node i was assigned to the other community. Thus, assigning vertices with a high value in u1, will have a larger impact on the maximization of Q than a vertex with lower value would have. This is especially important for continuing the clustering process for a further separation of the network N.

3.1.2 Division into N Clusters

After the ﬁrst round of clustering, a further partitioning of the two clusters might be possible. The established community structure with all its vertices and both inter- and intra-cluster edges will be preserved and the algorithm will keep dividing each community g into two subgroups. The variable to be maximized is now the change in the modularity measure, ∆Q: " # 1 ∆Q = ∑ Bi jsis j − ∑ Bi j (16) 4m i, j∈g i, j∈g " # 1 = ∑ Bi j − δi j ∑ Bik sis j (17) 4m i, j∈g k∈g 1 = sT B(g)s , (18) 4m where δi j is the Kronecker symbol, which takes on a value of 1 if i and j are located in (g) the same community and 0 otherwise. B is a ngxng matrix, with ng being the number of vertices within cluster g, which was generated in the previous round. The modularity matrix of cluster g, B(g), consists of the following entries, describing the connection between vertices i and j:

(g) Bi j = Bi j − δi j ∑ Bik . (19) k∈g

As the rows and columns of B from (8) sum to 0, so do the rows and columns of B(g). Thus, the same approach as mentioned above can be applied to maximize ∆Q.

8 The algorithm stops and the optimal number of clusters within network N is detected, if ∆Q ≤ 0, respectively any additional inter-cluster movement of vertices within the given community structure will not result in a positive ∆Q. This is equivalent to the leading (g) eigenvalue β1 of B taking the value of 0.

3.1.3 Transformation of Modularity for Stock Market Applications

When applying the modularity measure to time-series portfolio analysis, the algorithm has to be slightly adjusted. Following Piccardi et al (2011), equation (7): 1 kik j Q = ∑ Ai j − sis j , (20) 4m i j 2m will be transformed into a weighted network analysis, where the nodes i and j can be thought of as two distinct time-series of lenght T. The degree of node i, ki, will be substituted by a measure of strength , wi, of the time-series i. A measure for the strength of the link, respectively similarity, between the time-series i and j, wi j > 0, will replace the adjacency matrix Ai j as follows:

1 wiw j Q = ∑ wi j − sis j , (21) 4w i j 2w where the total number of edges within network N is replaced by the total weight of edges,w, in the network:

w = ∑wi . (22) i For measuring the similarity between time-series i and j the literature does not refer to a unique approach. Some researchers use distance measures, which are based on various techniques to calculate correlation coefﬁcients, such as given in equation (4) (Eleuterio´ et al, 2012). Others, such as Piccardi et al (2011), model the similarity between two distinct time series, wi j, as a non-linear function of the Euclidean distance edi j: ∂ f di j wi j = f edi j , < 0 . (23) ∂edi j By transforming the Euclidean distance measure, the authors assign a higher importance to highly similar time-series. However, all the various distance measures, di j, fulﬁll jointly conditions (1) through (3).

9 3.2 Criticism on Modularity

By refering to Fortunato & Barthelemy´ (2007), Isogai (2015) mentions the problem of resolution limit, which describes the weakness of modularity-based algorithms to detect relatively small communities within a network. The modularity measure struggles with distinguishing a single cluster from a combination of weakly interdependent small clusters. Isogai (2015) tries to circumvent this issue by first identifying the optimal community structure by global modularity maximization and then proceeding by applying the same approach for each detected cluster individually on a localized level. Isogai (2015) also gives reason for caution concerning the Pearson correlation coefficient: as the distribution of stock returns might not be normal but exhibit fat tails - especially during times of crises as volatility increases - the linear correlation coefficient may display upwardly distorted values. Thus, Isogai (2015) first applies a GARCH to separate volatilities and returns. The generated residuals are then used to compute the correlation matrix for clustering. However, Isogai (2015) remarks that the level of modularity, whether based on the linear correlation matrix or on the GARCH-filtered one, does not reveal large differences. Only the statistical significance of the modularity Q is reduced, when based on the linear correlation coefficient.

4 Methodology

4.1 The Method of Symbolization

After discussing modularity as the benchmark according to which our network will be divided into distinct communities, this section will shed further light on how to prepare the raw data for community-structure analysis, before section 5 examines the actual clustering dynamics of the portfolio of 296 S&P 500 stocks. It is important to keep in mind that the method of symbolization is only a technique to prepare the underlying raw data for the priorly described algorithm of modularity, which is the main criterion of for this paper’s community structure detection procedure. As already mentioned in Section 2, a community structure analysis of a portfolio of stocks requires an appropriate distance measure between individual stocks. The following

10 assessment will thus be based on the methodology of symbolization as applied by Brida & Risso (2007), Brida & Risso (2010) and Piccardi et al (2011). In particular, the two- step symbolization methodology of the latter paper will serve as the benchmark for the following data transformation:

Given the one-day opgged¨ differences of stock prices, rit, of n stocks of time-series- length T, calculated as: pi,t rit = log , (24) pi,t−1

- pi,t being the stock price of company i at time t - the first step is to generate a cumulative normal probability distribution for each of the n stocks over time T. The second step then calls for symbolization: according to pre-defined thresholds, the normalized logged stock returns, rit, will be assigned a specific symbol as described in Piccardi et al (2011):  1 if P(r ) ≤ 1  it 3  sit = 1 2 (25) 2 if 3 < P(rit) ≤ 3   3 otherwise. This three-fold classification allows for a better differentiation between the different states of the economy and capturing the resulting volatility (Brida & Risso, 2007; Brida & Risso, 2010). Intervalls of equal length are furthermore said to optimally account for noisy time-series data (Molgedey & Ebeling, 2000). As proposed by Piccardi et al (2011), we proceed by computing the Euclidean distance between the time-series of any two stocks in our portfolio, based on the symbolization-algorithm explained above: 1 2 2 di j = ∑ sit − s jt (26) t As modularity, which determines the final community structure of our portfolio of S&P 500 companies, measures the similarity between individual stocks, the Euclidean distances, di j, have to be further transformed, such that high values of di j symbolize a low degree of similarity and vice versa. Piccardi et al (2011) remarks that unlike other networks, in which not every single node exhibits a theoretical relationship with any other node of the system,4 a portfolio of stocks displays edges between any two of its con- stituents. This makes the detection of a community structure much more difficult and

4For example social network analyses.

11 requires the elimination of the weakest links between any two stocks to allow for a meaningful partition. In order to differentiate between weaker and stronger relationships, the

Euclidean distances, di j, will be transformed by a non-linear and downward-sloping function f (•) resulting in the second round of symbolization.

The non-linear transformation of the (n(n−1))/2 Euclidean distances, di j, is achieved by computing a cumulative distribution function, such that:

number of di j ≤ d Π(d ) = Pd ≤ d = . (27) i j i j n(n − 1) Symbolization now adopts the downward-sloping feature of f (•):  1 if (d ) ≤ 0.025  Π i j   0.1 if 0.025 < Π(di j) ≤ 0.05 wi j = f (di j) = (28)  0.01 if 0.05 < Π(di j) ≤ 0.1   0.001 otherwise.

This transformation now allows to clearly differentiate between the degrees of similarity between any two stocks of our portfolio. Thus, the following analysis will omit the weakest interlinkages, depicted by wi j = 0.001, in order to display a meaningful partition of our network. The drawback of this approach is the loss of information incorporated in 90% of the initial edges. Nevertheless, the strongest 10% of relationships in this portfolio now form the weights for the modularity algorithm to reveal the portfolio’s community structure.

4.2 Gephi’s Modularity Measure

Having applied symbolization as a filtering technique to allow only the strongest relationships among companies to enter the community structure detection algorithm, this section briefly explains the modularity algorithm implemented in the software-package Gephi. It is based on Blondel et al (2008) and resembles a modification of the initial modularity measure as presented in section 3.1. Modularity is a widely used measure in the research literature, but its practical application is computationally extensive, especially for network structures with more than a million nodes (Blondel et al, 2008). For this reason, Blondel et al (2008) suggest an approximation procedure, which bypasses this obstacle.

12 Blondel’s et al (2008) modification of Newman’s (2006) modularity is a two-stage approach: at first, each single node i in the network N is assigned to a different community. Thus, initially the number of communities will equal the total number of nodes n in the network N. The algorithm proceeds by comparing the increase in modularity, if node i was attached to each of its neighbouring communities. Optimizing modularity thus requires to merge node i with the adjacent community, which generates the largest positive increase in modularity Q.5 The first stage terminates as soon as no positive increase in Q can be detected. As Blondel et al (2008) give to consider, researchers being sensitive to computation time, shall be cautious about the order of node evaluation. As this paper’s analysis only comprises a couple hundred of nodes, time issues are not a concern. The second states builds up on the already established community structure throughout stage one. However, the nodes i are now represented by the communities detected in the first stage. Therefore, the weight of a link between two of the new nodes i is the sum of the weights of all the edges connecting nodes of the two first-stage-communities. Working with these new weights, the algorithm of stage one is now applied again, where the nodes of the second stage are defined as the communities of the first stage. In short, the second stage of the algorithm can be described as a merging of communities which max- imizes the increase in modularity Q until no further improvement can be detected. Then once again, a new network is constructed, with the new nodes being the communities detected in the previous round and weights correspond again to the sum of the weighted links between two communities. Adjacent communities are merged so as to maximize the increase in modularity. Blondel et al (2008) emphasize in particular the advantage of the short running time needed for appropriate results and the quick reduction of the number of communities. Furthermore, the resolution limit problem, which describes the struggle of modularity to detect small clusters, is mitigated by the algorithm’s first stage (Blondel et al, 2008). Thus, this paper’s methodological approach is two-fold: at first the method of symbolization is applied to filter out the roughly 10% strongest relationships among companies based on the Euclidean Distance between the one-day logged differences of their stock prices. Contrary to Granovetter’s (1973) theory, which holds rather the weak links

5see section 3.1 for deﬁnition of variables.

13 between nodes accountable for stronger and more widespread spill-over effects of information, this paper’s analysis follows Piccardi et al (2011) by focusing on the strong relationships only. The second stage of the assessement comprises the help of Gephi’s modularity algorithm, based on Blondel et al (2008), to identify cluster formation in the underlying network.

5 Clustering in the Standard and Poor’s 500 Index

5.1 Period: 01/03/2000 - 12/31/2015

5.1.1 Descriptive Statistics

The underlying data set comprises 296 companies of the Standard&Poor’s 500 Index, covering the period from 01/03/2000 to 12/31/2015. As shown in Table 1, these companies sort themselves into 11 distinct industrial sectors, with Consumer Discretionary representing the largest number of companies, such as Amazon, Ford, Nike or Macy’s. The Telecommunication Services sector, in contrast, only covers AT&T, Centurylink and Verizon Communications. The strongest time-series correlation of stock returnsbetween two single entities is 0.88 and was detected in the Real Estate sector between Avalon Communities and Equity Residential. In terms of average market-value6, Exxon Mobile was with $352,797.54 billion the largest company, followed by General Electric with $298,148.52 billion. The smallest company according to the market-value indicator was the Information Technology company Flir Systems averaging $2,776.80 billion over the fifteen-year period. Applying the filtering techniques described in section 4.1, the data set for the final clustering analysis reduces to 263 companies showing at least one interconnection with any of the other entities. As Table 2, the number of industry sectors was preserved, with Consumer Discretionary still forming the largest group of companies followed by Financials as displayed in Table 2. In terms of interlinkages, Table 3 proves Financials to be the most interconnected sector, forming relationships with 36.53% of all companies

6Deﬁned as stated by Bloomberg: total current market value of all of a company’s outstanding shares; capitalization is a measure of corporate size.

14 within the network. Financial companies are therefore twice as highly interconnected as the second largest industry sector, Industrials, representing 17.73% of all degrees. On the single entity level, the company with the most interlinkages is the insurance and investment company Lincoln National Corporation with 165 degrees, followed by American International Group and The Hartford Financial Services Group, Inc. with 164 interlinkages remaining after having deleted the 90% weakest links from the initial network. Ranking as the 211th largest company in terms of market-capitalization7, Lincoln National Corporation counts 76 degrees more than the second largest company General Electric (89) and 144 more than the leading Exxon Mobile (21). This already suggests, that a clear-cut relationship between a company’s size and its interconnectedness within the network does not exist. Indeed, the 5% largest companies only represent 4.18% of the network’s edges. On an aggregate level, the average degree of an entity within the system is slightly higher than 33, whereas 13 companies are left with being connected to only one of the other 262 firms within the system. A comparison of the average degree with its median value of 21 suggests that the portfolio consists of a relatively small number of companies being highly interconnected with the other companies within the system: the five most interconnected companies - which all correspond to the financial sector - comprise almost 9% of the network’s interlinkages. With only one of them - CitiGroup - ranking among the top-ten largest companies, this is further evidence against a direct relationship between size and interconnectedness.

5.1.2 Community Structure

As already stated in the introductory paragraph, a closer look at the community structure of a portfolio of S&P 500 companies shall not only shed further light on portfolio composition strategies, but also provide further insights into industrial networking. Figure 1 shows the clustering result of all 263 companies for the period 01/03/2000 to 12/31/2015. The colours represent the 21 different clusters within the overall network resulting in a modularity of 0.595. This measure is in line with previous research as already

7Deﬁned as stated by Bloomberg: total current market value of all of a company’s outstanding shares; capitalization is a measure of corporate size.

15 Figure 1: Community Structure of 263 Companies of the S&P 500 Index

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MICROCHIP TECH. FIRSTENERGY MICROSOFT STRYKER NETAPP IRON MOUNTAIN CITRIX SYS. AMEREN LAM RESEARCH TIFFANY & CO V F PAYCHEX INTERPUBLIC GROUP PINNACLE WEST CAP.

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NORTHERN TRUST COMERICA CORNING FISERV NEXTERA ENERGY US BANCORP DUKE ENERGY CENTERPOINT EN. MORGAN STANLEY BOSTON PROPERTIESAVALONBAY COMMNS. HARTFORD FINL.SVS.GP. KIMCO REALTY MOTOROLA SOLUTIONS APPLE AMPHENOL 'A' AMERICAN INTL.GP. SEMPRA EN. CMS ENERGY HARRIS LINCOLN NATIONAL SUNTRUST BANKS HARLEY-DAVIDSON CINTAS FIFTH THIRD BANCORP T ROWE PRICE GROUP NISOURCE CITIGROUP M&T BANK PNC FINL.SVS.GP. PERKINELMER SIMON PROPERTYEQUITY GROUP RESD.TST.PROPS. SHBI JP MORGAN CHASEHUNTINGTON & CO. BCSH. CAMPBELL SOUP XEROX CAPITAL ONE FINL. AES STATE STREET BOEING ROBERT HALF INTL. FLIR SYSTEMS WELLS FARGO & COPROLOGIS MARSH & MCLENNAN BANK OF AMERICA KIMBERLY-CLARK UNITED TECHNOLOGIES EXPEDITOR INTL.OF WASH. FRANKLIN RESOURCES CHUBB KELLOGG

HONEYWELL INTL. TORCHMARK ZIONS BANCORP. BB&T FEDEX GENERAL ELECTRIC GENERAL MILLS PACCAR KEYCORP LENNAR 'A' PROCTER & GAMBLE DAVITA HEALTHCARE PTNS. ROPER TECHNOLOGIES ALLSTATE PROGRESSIVE OHIO TEXTRON AFLAC PEPSICO NORFOLK SOUTHERN UNUM GROUP CINCINNATI FINL. ONEOK SEALED AIR SNAP-ON JOHNSON CONTROLS AON CLASS A FASTENAL DANAHER PULTEGROUP COLGATE-PALM. DOVER ILLINOIS TOOL WORKS CIGNA LOEWS LEGGETT&PLATT COCA COLA MASCO EMERSON ELECTRICAVERY DENNISON 3M D R HORTON FORD MOTOR EATON VERIZON COMMUNICATIONS PARKER-HANNIFIN LEUCADIA NATIONAL

FLOWSERVE HUMANA ROCKWELL AUTOMATION PFIZER STANLEY BLACK & DECKER WILLIAMS WHIRLPOOL DEERE PPG INDUSTRIES BALL UNITEDHEALTH GROUP DOW CHEMICAL CUMMINS WEYERHAEUSER NEWELL RUBBERMAID AT&T AETNA CATERPILLAR ECOLAB EASTMAN CHEMICAL INGERSOLL-RANDGENUINE PARTS VULCAN MATERIALS EQT NOBLE ENERGY AIR PRDS.& CHEMS. PRAXAIR INTERNATIONAL PAPER BRISTOL MYERS SQUIBB CENTURYLINK OCCIDENTAL PTL. BORGWARNER FMC INTL.FLAVORS & FRAG. HESS HELMERICH & PAYNE JACOBS ENGR. CHEVRON RANGE RES. UNION PACIFIC GOODYEAR TIRE & RUB. DEVON ENERGY

PENTAIR NUCOR EXXON MOBIL PIONEER NTRL.RES.NEWFIELD EXPLORATION SHERWIN-WILLIAMS APACHE

ANADARKO PETROLEUM

SCHLUMBERGER ALEXION PHARMS. BAKER HUGHES MARATHON OILNATIONAL OILWELL VARCO EOG RES. AMGEN FREEPORT-MCMORAN VALERO ENERGY MERCK & COMPANY CABOT OIL & GAS 'A' CELGENE HALLIBURTON CONOCOPHILLIPS

GILEAD SCIENCES JOHNSON & JOHNSON

Note: Colours display different clusters. Colours do not have any interpretational purpose, but are just randomly applied for a more convenient visualization.

Newman & Girvan (2004) suggested that typical modularity measures range between 0.3 and 0.7. In his analysis of 30 time-series of the Dow Jones Industrial Average between 2001 and 2006, Piccardi et al (2011) detected 7 clusters with a modularity of 0.679. The clustering depicted in Figure 1 is characterized by an agglomeration of several clusters which seem to exhibit not only strong intra- but also inter-cluster linkages, whereas the two clusters in black and orange are quite isolated. The network is comple- mented by a large number of smaller clusters with apparently weaker links to the bigger communities. As Figure 1 only differentiates between different communities, a closer look at the

16 distribution of the several sectors among clusters might provide a depper understanding of intra-industry co-movements of companies. Figure 2 depicts again a partition in patterns of clusters. Whereas the sectors portrayed by Figures 2c, 2d, 2e and 2i are quite concentrated on forming a cluster on their own, the stock returns of sectors such as Consumer Discretionary or Health Care, Figures 2a and 2f, are spread out among several clusters. Thus, the latter two sectors do not seem to be affected by intra-industry developements but are rather influenced by extra-industry- sector shocks. After an overview of the community structure formation and the distribution of industry sectors among clusters, a look at the specific composition of the single clusters might allow for a more thorough insight into industry sector dynamics: therefore, the colours in Figure 3 now represent the different sectors and the partition of Figure 4 shows the composition of the largest communities within the network. Worth noting is again the two-fold picture in Figure 3: on the one hand, strong inter-industry sector linkages of a subset of sectors including Financials and Industrials, and on the other hand two seemingly isolated sector-specific clusters of the Energy and Utilities sector. Although only hardly visible, the Real Estate seems to be highly interconnected with the financial sector. However, only an extension of the assessment period permits an analysis of possible reasons for this formation, such as the contribution of increased securitization of mortgages. Nevertheless, its central positioning already indicates that a crisis hitting the financial system may infiltrate other sectors or even the whole real economy. Figure 4 presents the largest nine clusters and the remaining ones as an aggregate in Figure 4j. Preserving the sector colours, Figure 4a confirms the interconnectedness of financial companies not only with own-sector entities and proves its role as a catalysator of sector-specific risk. Figures 4d and 4e emphasize the impression of Figure 3 as largely isolated communities, whereas also at least some Real Estate companies form a unique community as referenced by Figure 4g.

5.2 Period: 12/01/2007 - 06/30/2009

Having examined the cluster formation of a subset of the S&P 500 Index for the whole ﬁrst 15 years of the 21st century, this section provides an assessment of the community

17 Figure 2: Composition of Industry Sectors by Clusters

O REILLY AUTOMOTIVE FORD MOTOR

AUTOZONE OMNICOM GROUP

BORGWARNER GOODYEAR TIRE & RUB. STARBUCKS NEWELL RUBBERMAID CAMPBELL SOUP

GENUINE PARTS INTERPUBLIC GROUP

STANLEY BLACK & DECKER GENERAL MILLS WHIRLPOOL VIACOM 'B' COLGATE-PALM.

CBS 'B' TIME WARNER SNAP-ON

BEST BUY

YUM! BRANDS AMAZON.COM KELLOGG HARLEY-DAVIDSON

COMCAST 'A' PEPSICO PULTEGROUP V F KIMBERLY-CLARK TARGET

GAP DARDEN RESTAURANTS COCA COLA TIFFANY & CO LEGGETT&PLATT RALPH LAUREN CL.A

TJX PROCTER & GAMBLE LOWE'S COMPANIES D R HORTON CARNIVAL

NORDSTROM MACY'S COSTCO WHOLESALE WALGREENS BOOTS ALLIANCE

ROSS STORES LENNAR 'A'

HOME DEPOT KOHL'S NIKE 'B' CVS HEALTH WAL MART STORES

(a) Consumer Discretionary (1.76%) (b) Consumer Staples (0.3%)

CONOCOPHILLIPS XCEL ENERGY LEUCADIA NATIONAL EXXON MOBIL CENTERPOINT EN. BAKER HUGHES FRANKLIN RESOURCES EDISON INTL. T ROWE PRICE GROUP

AFLAC AES E*TRADE FINANCIAL WILLIAMS

VALERO ENERGY CHEVRON STATE STREET PG&E WELLS FARGO & CO UNUM GROUP HARTFORD FINL.SVS.GP. PPL NEWFIELD EXPLORATION NEXTERA ENERGY CMS ENERGY AMERICAN INTL.GP. BERKSHIRE HATHAWAY 'B' MARATHON OIL NOBLE ENERGY KEYCORP TORCHMARK SEMPRA EN. LOEWS BANK OF AMERICA

HESS OCCIDENTAL PTL. LINCOLN NATIONAL HALLIBURTON PINNACLE WEST CAP. MORGAN STANLEY ENTERGY EXELON ZIONS BANCORP. PNC FINL.SVS.GP. JP MORGAN CHASE & CO. AMEREN EOG RES. DUKE ENERGY CINCINNATI FINL. ANADARKO PETROLEUM HELMERICH & PAYNE HUNTINGTON BCSH. CHARLES SCHWAB CITIGROUP

NISOURCE BB&T US BANCORP SCHLUMBERGER APACHE FIFTH THIRD BANCORP DOMINION RESOURCES CONSOLIDATED EDISON FIRSTENERGY PROGRESSIVE OHIO NATIONAL OILWELL VARCO SUNTRUST BANKS NORTHERN TRUST

RANGE RES. WEC ENERGY GROUP DEVON ENERGY MARSH & MCLENNAN COMERICA DTE ENERGY CAPITAL ONE FINL. PIONEER NTRL.RES. AON CLASS A ALLSTATE

PUB.SER.ENTER.GP.

ONEOK SCANA CABOT OIL & GAS 'A' M&T BANK PEOPLES UNITED FINANCIAL SOUTHERN CHUBB EQT

UNION PACIFIC AETNA QUEST DIAGNOSTICS CENTURYLINK NORFOLK SOUTHERN LOCKHEED MARTIN

HUMANA FASTENAL

MEDTRONIC BOEING JACOBS ENGR. DAVITA HEALTHCARE PTNS. EXPEDITOR INTL.OF WASH.

PENTAIR JOHNSON CONTROLS

UNITEDHEALTH GROUP CELGENE NORTHROP GRUMMAN

MCKESSON LABORATORY CORP.OF AM. HDG. PARKER-HANNIFIN FEDEX WASTE MANAGEMENT CIGNA EATON CATERPILLAR

DOVER

GENERAL DYNAMICS ALEXION PHARMS. GILEAD SCIENCES HONEYWELL INTL.

STRYKER WATERS ROPER TECHNOLOGIES EQUIFAX THERMO FISHER SCIENTIFIC CUMMINS DANAHER DEERE TEXTRON

RAYTHEON 'B' ILLINOIS TOOL WORKS

ROBERT HALF INTL. PACCAR AMGEN PFIZER ROCKWELL AUTOMATION CARDINAL HEALTH PERKINELMER VERIZON COMMUNICATIONS MERCK & COMPANY INGERSOLL-RAND EMERSON ELECTRIC FLOWSERVE CINTAS

JOHNSON & JOHNSON UNITED TECHNOLOGIES AMERISOURCEBERGEN GENERAL ELECTRIC

3M BRISTOL MYERS SQUIBB DENTSPLY SIRONA AT&T MASCO

(f) Health Care (0.48%) (g) Telecommunications (0.05%) (h) Industrials (5.98%)

HARRIS EASTMAN CHEMICAL AUTODESK NUCOR FLIR SYSTEMS PROLOGIS

WEYERHAEUSER

APPLE PRAXAIR

MICROSOFT INTERNATIONAL BUS.MCHS. ADOBE SYSTEMS FREEPORT-MCMORAN APARTMENT INV.& MAN.'A' SYMANTEC

AIR PRDS.& CHEMS. QUALCOMM TOTAL SYSTEM SERVICES FMC BOSTON PROPERTIES SIMON PROPERTY GROUP AMPHENOL 'A' PPG INDUSTRIES CISCO SYSTEMS ANALOG DEVICES KLA TENCOR

MICRON TECHNOLOGY ECOLAB FISERV DOW CHEMICAL XEROX PUBLIC STORAGE CA

NETAPP XILINX AVALONBAY COMMNS. INTERNATIONAL PAPER LAM RESEARCH SEALED AIR EQUITY RESD.TST.PROPS. SHBI KIMCO REALTY

INTUIT INTL.FLAVORS & FRAG. AUTOMATIC DATA PROC. ORACLE ADVANCED MICRO DEVC. BALL INTEL TEXAS INSTRUMENTS

HCP AVERY DENNISON

CORNING CITRIX SYS. PAYCHEX APPLIED MATS. SHERWIN-WILLIAMS IRON MOUNTAIN MICROCHIP TECH. VENTAS VULCAN MATERIALS

MOTOROLA SOLUTIONS HP

(i) Information Technology (6.07%) (j) Real Estate (1.1%) (k) Materials (1.67%)

Note: Numbers in brackets indicate the ratio of intra-industry sector edges to total number of edges within the network.

18 Figure 3: Industry Sector distribution of 263 Companies of the S&P 500 Index

NEWFIELD EXPLORATIONCONOCOPHILLIPS MARATHON OIL OCCIDENTAL PTL. GENERAL MILLS AETNA BAKER HUGHES NOBLE ENERGY VALERO ENERGY HESS HELMERICH & PAYNE EOG RES. KELLOGG

UNITEDHEALTH GROUP RANGE RES. FREEPORT-MCMORAN HALLIBURTON SCHLUMBERGER APACHE DEVON ENERGY AUTOZONE LOCKHEED MARTIN ANADARKO PETROLEUM CAMPBELL SOUP PIONEER NTRL.RES. HUMANA

NORTHROP GRUMMAN KIMBERLY-CLARK EQT COLGATE-PALM. MEDTRONIC CHEVRON CABOT OIL NATIONAL& GAS 'A' OILWELL VARCO CELGENE ALEXION PHARMS. EXXON MOBIL

QUEST DIAGNOSTICS O REILLY AUTOMOTIVE APPLE UNION PACIFIC FLIR SYSTEMS YUM! BRANDS FORD MOTOR EXELON RAYTHEON 'B' NORFOLK SOUTHERN FASTENAL FIRSTENERGY AES WASTE MANAGEMENT

LABORATORY CORP.OF AM. HDG. NUCOR COMCAST 'A' NEXTERA ENERGY DUKE ENERGY DAVITA HEALTHCARE PTNS.STRYKER ENTERGY GENERAL DYNAMICS BORGWARNER GOODYEAR TIRE & RUB. WILLIAMSEXPEDITOR INTL.OF WASH.

PENTAIR JACOBS ENGR. FEDEXTOTAL SYSTEM SERVICES BERKSHIRE HATHAWAY 'B' SEMPRA EN. AMERENDOMINION RESOURCES BOEING CATERPILLAR PROCTER & GAMBLE CIGNA AUTODESK FMC GILEAD SCIENCES PINNACLE WEST CAP. EATONJOHNSON CONTROLS VIACOM 'B' CMS ENERGY PPL SOUTHERN WHIRLPOOL STARBUCKS ROCKWELL AUTOMATION NEWELL RUBBERMAID ECOLAB AIR PRDS.& CHEMS. OMNICOM GROUP HARRIS ROPER TECHNOLOGIES XCEL ENERGY PUB.SER.ENTER.GP. PARKER-HANNIFIN DARDEN RESTAURANTS FLOWSERVE WEC ENERGY GROUPSCANA DANAHER CUMMINS DOVER INTERPUBLIC GROUP EDISON INTL. DEERE EASTMAN CHEMICAL CENTERPOINT EN. CBS 'B' MICROSOFT PACCAR PG&E ILLINOIS TOOL WORKS DTE ENERGY INGERSOLL-RAND TIME WARNER CONSOLIDATED EDISON HONEYWELL INTL. PRAXAIR HARLEY-DAVIDSON ADOBE SYSTEMS CENTURYLINK EMERSON ELECTRIC CINTAS PPG INDUSTRIES LEUCADIA NATIONAL TEXTRON INTERNATIONAL BUS.MCHS. XILINX NISOURCE AMPHENOL 'A' ONEOK STANLEY BLACK & DECKER DOW CHEMICAL STATE STREET TEXAS INSTRUMENTS GENUINE PARTS FRANKLIN RESOURCES LINCOLN NATIONAL INTEL BALL SNAP-ONSEALED AIR COCA COLA HARTFORD FINL.SVS.GP. CISCO SYSTEMS UNITED TECHNOLOGIES T ROWE PRICE GROUP BANK OF AMERICA FISERV CITRIX SYS. SYMANTEC AVERY DENNISON WELLS FARGO & CO GENERAL ELECTRIC AMERICAN INTL.GP. INTL.FLAVORS & FRAG. CITIGROUP ANALOG DEVICESKLA TENCOR MICRON TECHNOLOGY INTERNATIONAL PAPER PEPSICO 3M LOEWS HUNTINGTON BCSH. CA AMGEN TORCHMARK AFLAC QUALCOMM FIFTH THIRD BANCORP INTUIT APPLIED MATS. KEYCORP ROBERT HALF INTL. NETAPP VULCAN MATERIALS ORACLE PROLOGIS MORGAN STANLEY ADVANCED MICRO DEVC. WEYERHAEUSERLEGGETT&PLATT UNUM GROUP JP MORGAN CHASE & CO. MCKESSON MICROCHIP TECH. ZIONS BANCORP. LAM RESEARCH PNC FINL.SVS.GP. CORNING NORTHERN TRUST E*TRADE FINANCIAL PULTEGROUP PAYCHEX MASCO CINCINNATI FINL. MOTOROLA SOLUTIONS AMAZON.COM BB&T SHERWIN-WILLIAMS SUNTRUST BANKS WALGREENS BOOTS ALLIANCE HP ALLSTATE CHARLES SCHWAB CAPITAL ONEUS FINL. BANCORP XEROX COMERICA TIFFANY & CO WATERS CARDINAL HEALTH AUTOMATIC DATA PROC. EQUIFAX PERKINELMER GAP AMERISOURCEBERGEN CARNIVAL APARTMENT INV.& MAN.'A' MARSH & MCLENNANM&T BANK AT&T RALPH LAUREN CL.A LENNAR 'A' KIMCO REALTY BOSTON PROPERTIES V F PUBLIC STORAGE BEST BUY ROSS STORES PROGRESSIVE OHIOSIMON PROPERTY GROUP AON CLASS A CHUBB THERMO FISHER SCIENTIFIC NORDSTROM HOME DEPOT MACY'S CVS HEALTH EQUITY RESD.TST.PROPS. SHBI PEOPLES UNITED FINANCIAL D R HORTON AVALONBAY COMMNS. LOWE'S COMPANIES TARGET TJX HCP NIKE 'B' IRON MOUNTAIN VENTAS

JOHNSON & JOHNSON KOHL'S

PFIZER COSTCO WHOLESALE DENTSPLY SIRONA

BRISTOL MYERS SQUIBB VERIZON COMMUNICATIONSWAL MART STORESMERCK & COMPANY

19 Figure 4: Composition of Clusters by Industry Sectors

FREEPORT-MCMORAN COMCAST 'A' NUCOR TOTAL SYSTEM SERVICES FORD MOTOR WASTE MANAGEMENT BORGWARNER

PENTAIR NEWELL RUBBERMAID FLIR SYSTEMS FASTENAL ROCKWELL AUTOMATION VULCAN MATERIALS HARLEY-DAVIDSON FMC ROBERT HALF INTL. BOEING

BERKSHIRE HATHAWAY 'B' EATON XEROX CATERPILLAR BANK OF AMERICA JACOBS ENGR. TORCHMARK DAVITA HEALTHCARE PTNS. E*TRADE FINANCIAL PARKER-HANNIFIN STATE STREET T ROWE PRICE GROUP CUMMINS

PROLOGIS ECOLAB AIR PRDS.& CHEMS. FLOWSERVE DOVER EQUIFAX GOODYEAR TIRE & RUB. WELLS FARGO & CO HUNTINGTON BCSH. MEDTRONIC LEUCADIA NATIONAL LINCOLN NATIONAL CITIGROUP TEXTRON EASTMAN CHEMICAL JOHNSON CONTROLS GENERAL ELECTRIC DANAHER FRANKLIN RESOURCES AFLAC STARBUCKS DEERE INGERSOLL-RAND HARTFORD FINL.SVS.GP. PACCAR FIFTH THIRD BANCORP AMERICAN INTL.GP. STRYKER HONEYWELL INTL. LOEWS FEDEX PRAXAIR UNUM GROUP KEYCORP CHARLES SCHWAB ILLINOIS TOOL WORKS ZIONS BANCORP. DOW CHEMICAL ROPER TECHNOLOGIES JP MORGAN CHASE & CO.MORGAN STANLEY WATERS BALL SEALED AIR CARNIVAL BB&T EMERSON ELECTRIC NORTHERN TRUST PERKINELMER CAPITAL ONE FINL. PPG INDUSTRIES WHIRLPOOL SNAP-ON SUNTRUST BANKS PNC FINL.SVS.GP. AVERY DENNISON GENUINE PARTS CINCINNATI FINL.

US BANCORP COMERICA STANLEY BLACK & DECKER IRON MOUNTAIN UNITED TECHNOLOGIES M&T BANK THERMO FISHER SCIENTIFIC 3M EXPEDITOR INTL.OF WASH. PROGRESSIVE OHIO CHUBB INTL.FLAVORS & FRAG. AON CLASS A LEGGETT&PLATT MARSH & MCLENNAN INTERNATIONAL PAPER ALLSTATE DENTSPLY SIRONA WEYERHAEUSER PEOPLES UNITED FINANCIAL SHERWIN-WILLIAMS

(a) Largest Cluster: 61 Nodes; (21.77%) (b) 2nd largest Cluster: 46 Nodes; (13.04%)

APPLE CONOCOPHILLIPS HARRIS NEWFIELD EXPLORATION

AUTODESK

OCCIDENTAL PTL. INTERNATIONAL BUS.MCHS. MARATHON OIL ADOBE SYSTEMS

NOBLE ENERGY MICRON TECHNOLOGY XILINX MICROSOFT HELMERICH & PAYNE

AMPHENOL 'A' EOG RES. TEXAS INSTRUMENTS SCHLUMBERGER INTEL BAKER HUGHES HESS FISERV ADVANCED MICRO DEVC. APACHE CITRIX SYS. CINTAS

CA PIONEER NTRL.RES. CISCO SYSTEMS SYMANTEC HALLIBURTON RANGE RES. ANADARKO PETROLEUM INTUIT APPLIED MATS. VALERO ENERGY KLA TENCOR DEVON ENERGY AUTOMATIC DATA PROC.

ANALOG DEVICES

QUALCOMM EXXON MOBIL

ORACLE EQT NATIONAL OILWELL VARCO MICROCHIP TECH. CHEVRON PAYCHEX AMAZON.COM CABOT OIL & GAS 'A' NETAPP

LAM RESEARCH MOTOROLA SOLUTIONS WILLIAMS

AES HP CORNING ONEOK

EXELON TIFFANY & CO RALPH LAUREN CL.A FIRSTENERGY NEXTERA ENERGY

V F

DUKE ENERGY GAP SEMPRA EN. ENTERGY DOMINION RESOURCES BEST BUY

NORDSTROM HOME DEPOT PPL CMS ENERGY SOUTHERN

XCEL ENERGY ROSS STORES AMEREN MACY'S

WEC ENERGY GROUP PINNACLE WEST CAP. TARGET

SCANA NIKE 'B'

CENTERPOINT EN. LOWE'S COMPANIES

PUB.SER.ENTER.GP. TJX DTE ENERGY

CONSOLIDATED EDISON EDISON INTL. WAL MART STORES

KOHL'S

PG&E NISOURCE COSTCO WHOLESALE

(e) 5th largest Cluster: 22 Nodes; (5.23%) (f) 6th largest Cluster: 16 Nodes; (1.1%)

KELLOGG BOSTON PROPERTIES

APARTMENT INV.& MAN.'A'

GENERAL MILLS COLGATE-PALM. KIMCO REALTY

PUBLIC STORAGE

AVALONBAY COMMNS. CAMPBELL SOUP

SIMON PROPERTY GROUP

KIMBERLY-CLARK COCA COLA

HCP

EQUITY RESD.TST.PROPS. SHBI

VENTAS PROCTER & GAMBLE PEPSICO

(g) 7th largest Cluster: 9 Nodes; (0.83%) (h) 8th largest Cluster: 8 Nodes; (0.25%)

LOCKHEED MARTIN OMNICOM GROUP UNITEDHEALTH GROUP CELGENE

AETNA VIACOM 'B' RAYTHEON 'B' NORTHROP GRUMMAN

ALEXION PHARMS. HUMANA AUTOZONE

YUM! BRANDS GILEAD SCIENCES NORFOLK SOUTHERN

O REILLY AUTOMOTIVE

UNION PACIFIC LABORATORY CORP.OF AM. HDG. MCKESSON

GENERAL DYNAMICS DARDEN RESTAURANTS CIGNA CBS 'B' QUEST DIAGNOSTICS

MASCO CARDINAL HEALTH AMERISOURCEBERGEN

CENTURYLINK AMGEN LENNAR 'A' WALGREENS BOOTS ALLIANCE TIME WARNER PULTEGROUP PFIZER CVS HEALTH

AT&T JOHNSON & JOHNSON

MERCK & COMPANY D R HORTON

INTERPUBLIC GROUP VERIZON COMMUNICATIONS BRISTOL MYERS SQUIBB

(i) 9th largest Cluster: 5 Nodes; (0.25%) (j) 12 smallest Clusters combined: 35 Nodes; (0.83%)

Note: Numbers in brackets indicate the ratio of intra-cluster edges to total number of edges within the network.

20 structure during the Great Recession - the biggest economic turmoil since the Great De- pression - with a particular emphasis on the financial sector. However, the accompanying question adresses the definition of the crisis period: was the trigger of the crisis the fall in housing prices, being represented by the turning point in the widely used Case-Shiller Index? Was it the onset of tightening refinancing conditions in the interbank market in the summer of 2007, when money market mutual funds struggled to not break the buck, or was it the run on the investment bank Bear Stearns in March 2008? Another, more quantitative criterion to identify abnormal periods, especially in the stock market, is the Volatility Index (VIX) of the Chicago Board Options Exchange. The VIX represents the 30-day expected volatility of the U.S. stock market, using prices of call- and put-options on the S&P 500 Index as a basis for calculation (CBOE, 2018). Cerutti et al (2017) define a period to be characterized by financial stress, if the close-of-quarter value of the VIX surpasses the value of 30. Evaluating this proxy for risk-aversion and uncertainty of market participants (Rey, 2015) on a monthly basis accordingly, identifys the crisis period to be starting in September 2008 and finishing in April 2009. Although the stock market is said to be strongly correlated with the business cycle, using the VIX as a criterion for crisis identification might cause biased results, as the calculation of the VIX uses data derived from the same variables serving as raw data for this paper’s assessment process. Therefore, referring to the recession-dating indicator of the National Bureau of Eco- nomic Research, this paper assumes the Great Recession to be described by the 19 con- secutive months between December 2007 and the end of June 2009.

5.2.1 Descriptive Statistics

Before investigating the community structure of the same initial 296 stocks within the above-defined period of economic turmoil, a look at some stylized facts may already give an idea about the exposure of certain industry sectors to the recent financial crisis. After applying the symbolization and filtering techniques of section 4.1, again 263 companies form the input for the clustering based on the modularity measure. Neverthe- less, Figure 7 demonstrates the effect of a period characterized by financial distress on the dynamics of one-day logged differences of stock returns in the S&P 500. Both, the

21 shift in correlations to higher levels and the overall reduction in the pairwise Euclidean distances, proxying the similarity of the portfolio’s companies, are an indication of increased co-movement of stocks during theGreat Recession. This also caused the set of companies under investigation to slightly differ from the previous analysis, as only 252 companies entered both algorithms. These results are also apparent from Tables 2 and 3: once again all 11 industry sectors were covered in the analysis with Consumer Discretionary still covering the largest number of companies (42), but now followed by both Financials and Industrials with 37 entities. In terms of interconnectedness, the sector of Financials suffered a drop of almost 15 percentage points to 22.96%, which underscores the aforementioned increased insta- bility of financial institutions during the recent financial crisis. Furthermore, the sector’s representation in the top-five interconnected companies dropped to one, while Consumer Discretionary covered 60% and one company of the Industrials sector complements that group. Despite being the industry sector to represent most companies in the sample, Con- sumer Discretionary was only weakly interconnected within the system relative to other sectors. However, it was that sector, which experienced the largest rise of interlinkages with more than 70%. The Real Estate sector could experience an equal rise in strong relationships. Zooming in on the individual company level, it was Loews Corporation which became the most interlinked company during the crisis. Being classified as part of the Financials by Standard & Poor’s, Loews Corporation describes itself as ”one of the largest diversified companies in the United States, with businesses in the insurance, energy, hospitality and packaging industries” (Loews, 2018). Nevertheless, it only ranked number 126 in terms of overall period’s market value. A look at the five most interconnected companies during the crises and their ranking according to the full period’s average market value tempts to infer a negative correlation of a companies size, proxied by its market capitalization, and similarity, proxied by the Euclidean Distance, during times of crisis. The Pearson correlation coefficient of 0.13, however, invalidates such a conclusion. With the average degree of a company having slightly fallen to 32 and the median degree having risen to 23, the distribution of the degrees seemed to have become more center-driven. Even though, the number of firms keeping only a single interrelation with

22 any of the remaining 262 firms, rose from 13 to 19. However, also the share of total degrees covered by the five most interconnected companies dropped from almost 9% to 6.5%. Even though these figures might suggest that the Great Recession might have countributed to a higher level of isolation on a single company basis, Figure 7b clearly shows the overall tendency of company-dynamics to converge to similarity during times of economic turmoil.

5.2.2 Community Structure

This subsection is intended to shed further light on the effect of the recent financial crisis on the community structure of a sub-portfolio of the S&P 500 Index by means of modularity. In particular, the following analysis focuses on whether a period of increased financial stress tends to make companies stick together, resulting in less clusters, or whether companies try to seperate in order to isolate themselves from possible negative spill-over effects. An increased variety of clusters would as well indicate a growing risk-sensitivity and willingness for in-depth company analysis by stock market investors during times of bearish economic outlooks. Compared to the whole assessment horizon, the modularity increased during the crisis period from 0.595 to 0.621, indicating a slightly more clear-cut cluster formation. Remarkable, however, is the decrease in the number of identified clusters from 21 to 12. This leaves plenty of room for causal interpretation: on the one hand, the reduction in clusters is evidence for the impact of the Great Recession not being limited to an individual market segment, as the the crisis is said to have emerged from the housing sector, spilling over to financial instiutions and to the whole real economy, . On the other hand, the shift in clusters as well as the stock return dynamics depicted in Figure 7 reflect the effects of fire-sales triggering the shift to safe havens, due to increased risk-aversion and the fear of large losses. Figure 5 depicts the community structure of the 263 companies with each industry sector being coloured differently. Compared to Figure 3, the emergence of the Consumer Discretionary sector is apparent. A second look suggests a slight shift and a tendency towards isolation by the Financials. The Energy sector seems to have moved from a quite isolated position in Figure 3 to a more interlinked community, especially with Industrials.

23 The increased modularity, indicating a more clear-cut differentiation between clusters, is hardly visible and can only be confirmed by a closer look. Nevertheless, both figures confirm the non-uniform community structure of the stock market across varying states of the economy. Last but not least, Figure 6 allows a comparison of the financial sector’s interconnectedness with the system as a whole. A first look suggests the sector’s loss of its central role as doubts about its stability became widespread among market participants. Even though a loss of aggregate interconnectedness, symbolized by the amount of degrees as displayed in Table 3, might not be apparent from the visualizations, the size of the nodes being connected to the financial sector has changed. Whereas the nodes were rather small in Figure 6a, the company-specific amount of connections of entities being linked to the financial sector has risen in the months representing the crisis period. Thus, the financial sector seemed to have either shifted its interlinkages to companies, which are highly interconnected themselves, or the companies, to which the financial sector had already established relationships with, became highly interconnected throughout the period of increased financial stress themselves. Understanding the reasons and drivers of the generally changed landscape of the network structure is crucial for understanding the dynamics of not only stock markets, but also of investors’ decision making across varying economic states. A more in-depth assessment is open to future research.

6 Conclusion

The ﬁrst 15 years of the 21st century have been marked by several ups and downs in the stock market. As the stock market is said to mimic the course of the business cycle and thus the behaviour of the overall economy, a more thorough understanding of the market’s dynamics may allow policy makers to base their decisions on improved economic fore- casts. Undoubtably, knowledge about the differences in aggregate stock market behaviour and the interplay of various industry sectors during times of crises and normal times is of great importance to portfolio managers and stock market participants. This paper, hence, tried to shed further light on the market dynamics, proxied by 296

24 Figure 5: Industry Sector distribution of 263 Companies of the S&P 500 Index (12/01/2007 - 06/30/2009)

CELGENE

MCDONALDS STARBUCKS AMAZON.COM AON CLASS A SYSCO WALGREENS BOOTS ALLIANCE GILEAD SCIENCES

DARDEN RESTAURANTS MATTEL INTERNATIONAL BUS.MCHS. HASBRO NIKE 'B' YUM! BRANDS NETAPP TJX CITRIX SYS. MICROSOFT GENUINE PARTS

O REILLY AUTOMOTIVE NORDSTROM XILINX ADOBE SYSTEMS WASTE MANAGEMENT FASTENAL CVS HEALTH MACY'S UNION PACIFIC VULCAN MATERIALS ANALOG DEVICES AUTOMATIC DATA PROC. FISERV ROSS STORES KLA TENCOR LAM RESEARCH LOWE'S COMPANIES INTUIT EXPEDITOR INTL.OF WASH. AUTODESK BEST BUY TIFFANY & CO ORACLE QUALCOMM FLIR SYSTEMS PAYCHEX KOHL'S NORFOLK SOUTHERN BALL GAP FEDEX TEXAS INSTRUMENTS DANAHER RALPH LAUREN CL.A V F HP COCA COLA CINTAS SYMANTEC APPLIED MATS. HOME DEPOT MASCO TARGET ECOLAB MICROCHIP TECH. SOUTHWEST AIRLINES WEYERHAEUSER ROBERT HALF INTL. LEGGETT&PLATT AUTOZONE BAXTER INTL. MARSH & MCLENNAN PARKER-HANNIFIN GOODYEAR TIRE & RUB. EQUIFAX STANLEY BLACK & DECKER AMPHENOL 'A' JOHNSON CONTROLS CISCO SYSTEMS INTEL SHERWIN-WILLIAMS NORTHROP GRUMMAN SEALED AIR HARRIS ELECTRONIC ARTS PACCAR CARNIVAL ILLINOIS TOOL WORKS PERKINELMER ROCKWELL AUTOMATION NEWELL RUBBERMAID HARLEY-DAVIDSON PPG INDUSTRIES EATON PEPSICO D R HORTON EMERSON ELECTRIC CHARLES SCHWAB PENTAIR WATERS WHIRLPOOL SNAP-ON COSTCO WHOLESALE FRANKLIN RESOURCES DOVER TOTAL SYSTEM SERVICES DEERE PUBLIC STORAGE ROPER TECHNOLOGIES ESTEE LAUDER COS.'A' T ROWE PRICE GROUP NUCOR KIMCO REALTY DOW CHEMICAL LABORATORY CORP.OF AM. HDG. AMERISOURCEBERGEN HONEYWELL INTL. UNITED TECHNOLOGIES AVALONBAY COMMNS. AVERY DENNISON HCP INGERSOLL-RAND VALERO ENERGY SIMON PROPERTY GROUP 3M VENTAS CBS 'B' CATERPILLAR APARTMENT INV.& MAN.'A' FLOWSERVE BOSTON PROPERTIES PULTEGROUP LENNAR 'A' JACOBS ENGR. COLGATE-PALM. CINCINNATI FINL. OMNICOM GROUP AIR PRDS.& CHEMS. PROLOGIS XEROX FREEPORT-MCMORAN NORTHERN JPTRUST MORGAN CHASE & CO. LOEWS CUMMINS MCKESSON LINCOLN NATIONAL NATIONAL OILWELL VARCO EQUITY RESD.TST.PROPS. SHBI EXXON MOBIL UNUM GROUP PRAXAIR MORGAN STANLEY LEUCADIA NATIONAL BAKER HUGHES BB&T INTL.FLAVORS & FRAG. EASTMAN CHEMICAL M&T BANK ALLSTATE RAYTHEON 'B' BANK OF AMERICA HESS BORGWARNER CONOCOPHILLIPS ANADARKO PETROLEUM ABBOTT LABORATORIES FMC TORCHMARK AFLAC COMERICA WELLS FARGO & CO US BANCORP WILLIAMS GENERAL ELECTRIC COMCAST 'A' OCCIDENTAL PTL. APACHE PNC FINL.SVS.GP. CITIGROUP TIME WARNER LOCKHEED MARTIN KEYCORP NOBLE ENERGY EOG RES. INTERNATIONAL PAPER CA CHEVRON STATE STREET TEXTRON MARATHON OIL HUNTINGTON BCSH. NEWFIELD EXPLORATION CAPITAL ONE FINL. SCHLUMBERGER VIACOM 'B' SUNTRUST BANKSZIONS BANCORP. EQT HELMERICH & PAYNE IRON MOUNTAIN RANGE RES. CENTURYLINK HARTFORD FINL.SVS.GP. E*TRADE FINANCIAL CHUBB INTERPUBLIC GROUP HALLIBURTON PROGRESSIVE OHIO GENERAL DYNAMICSBOEING CABOT OIL & GAS 'A' FIFTH THIRD BANCORP ONEOK PIONEER NTRL.RES. CIGNA DEVON ENERGY AT&T MERCK & COMPANY AMERICAN INTL.GP. PEOPLES UNITED FINANCIAL MOTOROLA SOLUTIONS AMEREN WESTERN DIGITAL PROCTER & GAMBLETHERMO FISHER SCIENTIFIC

CORNING CARDINAL HEALTH GENERAL MILLS JOHNSON & JOHNSON QUEST DIAGNOSTICS VERIZON COMMUNICATIONS SEMPRA EN. MEDTRONIC AES CAMPBELL SOUP DUKE ENERGY AETNA KELLOGG DOMINION RESOURCES EXELON XCEL ENERGY UNITEDHEALTH GROUP MCCORMICK & COMPANY NV. DTE ENERGY PG&E CMS ENERGY HUMANA EDISON INTL. BRISTOL MYERS SQUIBB CONSOLIDATED EDISON PINNACLE WEST CAP. PPL PFIZER KIMBERLY-CLARK NEXTERA ENERGY PUB.SER.ENTER.GP. CENTERPOINT EN.

WEC ENERGY GROUP SOUTHERN ENTERGY FIRSTENERGY SCANA NISOURCE

25 Figure 6: Interconnectedness of the Financial Sector

VIACOM 'B' NIKE 'B' BEST BUY

V F O REILLY AUTOMOTIVE STARBUCKS WASTE MANAGEMENT LABORATORY CORP.OF AM.BERKSHIRE HDG. HATHAWAY 'B' RALPH LAUREN CL.A

MICROSOFT NORDSTROM QUEST DIAGNOSTICSGAP TOTAL SYSTEM SERVICES

INTERNATIONAL BUS.MCHS. SYMANTEC NETAPP OMNICOM GROUP DENTSPLY SIRONA TARGET PAYCHEX

TIFFANY & CO MACY'S CITRIX SYS. COSTCO WHOLESALE STRYKER

HP LOWE'S COMPANIES RAYTHEON 'B' CVS HEALTH YUM! BRANDSCOMCAST 'A' INTERPUBLIC GROUP LOCKHEED MARTIN LAM RESEARCH EQUIFAX IRON MOUNTAIN TIME WARNER DARDEN RESTAURANTS VENTAS INTEL NORTHROP GRUMMAN AUTOZONE WALGREENSWATERS BOOTS ALLIANCE MICROCHIP TECH. AMAZON.COM KOHL'S THERMO FISHER SCIENTIFIC TJX HOME DEPOT XILINX GENERAL DYNAMICS CARDINAL HEALTH PUBLIC STORAGE ROSS STORES AUTOMATIC DATA PROC. ORACLE ADVANCED MICRO DEVC. CBS 'B' CENTERPOINT EN. KLA TENCOR XEROX HCP MCKESSON EDISON INTL. INTUIT AES CARNIVAL WAL MART STORES TEXAS INSTRUMENTS AMERISOURCEBERGEN APPLIED MATS. AVALONBAY COMMNS. MEDTRONIC XCEL ENERGY QUALCOMM CISCO SYSTEMS AMPHENOL 'A' EXELON HARLEY-DAVIDSON FIRSTENERGY CA PINNACLE WEST CAP. ADOBE SYSTEMS ROBERT HALF INTL. CONSOLIDATED EDISON HARTFORD FINL.SVS.GP. AMEREN CINTAS CHARLES SCHWAB PERKINELMER BOSTON PROPERTIES PPL MICRON TECHNOLOGY WEC ENERGY GROUP PACCAR PG&E WILLIAMS SCANA LENNAR 'A' CORNING ROPER TECHNOLOGIES AUTODESK NORTHERN TRUST PUB.SER.ENTER.GP. FISERV AMERICAN INTL.GP. APARTMENT INV.& MAN.'A' E*TRADE FINANCIAL PEOPLES UNITED FINANCIAL SOUTHERN HARRIS DOMINION RESOURCES CMS ENERGY ANALOG DEVICES SIMON PROPERTY GROUP JOHNSON CONTROLS LINCOLN NATIONAL COMERICA DTE ENERGY DOVER HONEYWELL INTL. US BANCORP ENTERGY FLIR SYSTEMS EQUITY RESD.TST.PROPS. SHBI MOTOROLA SOLUTIONS MORGAN STANLEY KIMCO REALTY SUNTRUST BANKS NEXTERA ENERGY UNITED TECHNOLOGIES DUKE ENERGY HUNTINGTON BCSH.FIFTH THIRD BANCORP FORD MOTOR SEMPRA EN. APPLE M&T BANK DAVITA HEALTHCARE PTNS. EMERSON ELECTRIC SNAP-ON MARSH & MCLENNAN DANAHER BOEING CITIGROUP NISOURCE PNC FINL.SVS.GP. CHUBB MASCO EXPEDITOR INTL.OF WASH. CAPITAL ONE FINL. CAMPBELL SOUP BORGWARNER JP MORGAN CHASE & CO. WELLS FARGO & CO KELLOGG KIMBERLY-CLARK PROCTER & GAMBLE PARKER-HANNIFIN ILLINOIS TOOL WORKS BB&T PROLOGIS 3M ZIONS BANCORP. GENERAL MILLS T ROWE PRICE GROUP FRANKLIN RESOURCES PROGRESSIVECIGNA OHIO PEPSICO VERIZON COMMUNICATIONS BANK OF AMERICA FLOWSERVE TORCHMARK AON CLASS A ONEOK NORFOLK SOUTHERNFASTENAL KEYCORP ALLSTATE FEDEX COLGATE-PALM. COCA COLA PULTEGROUP GENERAL ELECTRIC EATON STATE STREET D R HORTON AFLAC UNUM GROUP CINCINNATI FINL. CUMMINS DOW CHEMICAL HUMANA UNITEDHEALTH GROUP PFIZER AIR PRDS.& CHEMS. AETNA TEXTRON LOEWSLEUCADIA NATIONAL EQT AT&T LEGGETT&PLATT BRISTOL MYERS SQUIBB NEWELL RUBBERMAID DEERE SEALED AIR HELMERICH & PAYNE NOBLE ENERGYWHIRLPOOL CATERPILLAR EASTMAN CHEMICAL

HESS INGERSOLL-RAND BALL OCCIDENTAL PTL. RANGE RES.

FMC AVERY DENNISON VULCAN MATERIALS NEWFIELD EXPLORATION JACOBS ENGR. ROCKWELL AUTOMATION UNION PACIFIC PIONEER NTRL.RES.

PPG INDUSTRIES INTL.FLAVORS & FRAG. ANADARKODEVON PETROLEUMSCHLUMBERGER ENERGY STANLEY BLACK & DECKER PRAXAIR APACHE CENTURYLINK EOG RES. INTERNATIONAL PAPER NATIONALCABOT OILWELL OIL VARCO& GAS 'A' MARATHON OIL BAKER HUGHES MERCK & COMPANY CONOCOPHILLIPS ALEXION PHARMS. VALERO ENERGY FREEPORT-MCMORAN HALLIBURTON AMGEN CELGENEGILEAD SCIENCESJOHNSON & JOHNSON

GOODYEAR TIRE & RUB. CHEVRON ECOLAB PENTAIR EXXON MOBIL NUCOR

WEYERHAEUSER

GENUINE PARTS

SHERWIN-WILLIAMS

(a) 03/01/2000 - 12/31/2015: 36.53% of all edges

RALPH LAUREN CL.A

TIFFANY & CO BALL FASTENALVULCAN MATERIALS BEST BUY INTUIT

YUM! BRANDS D R HORTON ELECTRONIC ARTS V F SHERWIN-WILLIAMS FEDEX PAYCHEX LOWE'S COMPANIES KOHL'S TARGETHOME DEPOT NIKE 'B' O REILLY AUTOMOTIVE ROSS STORES GENUINE PARTS AON CLASS A DANAHER MASCO

AMPHENOL 'A' MACY'S ADOBE SYSTEMS LEGGETT&PLATT HARLEY-DAVIDSON FISERV IRON MOUNTAIN

COSTCO WHOLESALE EXPEDITOR INTL.OF WASH. WHIRLPOOL CARNIVAL TJX DARDEN RESTAURANTS AUTOMATIC DATA PROC. AUTOZONE STARBUCKS MARSH & MCLENNAN LOCKHEED MARTIN CISCO SYSTEMS NORDSTROM ROBERT HALF INTL. INTEL UNITED TECHNOLOGIES RAYTHEON 'B' SYSCO GAP AVERY DENNISON MCDONALDS HP SYMANTEC SEALED AIR MATTEL CINTAS WALGREENS BOOTS ALLIANCE AMAZON.COMHASBRO CELGENE STANLEY BLACK & DECKER ILLINOIS TOOLINTERNATIONAL WORKS NETAPP BUS.MCHS. EQUIFAX GENERALNORTHROP DYNAMICS GRUMMANORACLE MICROSOFTGILEADBOEING SCIENCES CITRIX SYS. XILINX FRANKLIN RESOURCES UNION PACIFIC NEWELL RUBBERMAID ESTEE LAUDER COS.'A' FLIR SYSTEMS JOHNSON CONTROLS MCCORMICK & COMPANY NV. CVS HEALTH PACCAR AUTODESK PPG INDUSTRIESTEXAS INSTRUMENTSPERKINELMER HCP LABORATORYQUALCOMM CORP.OF AM. HDG. LENNAR 'A' SNAP-ON ANALOG DEVICES NORFOLK SOUTHERN PUBLIC STORAGE EQUITY RESD.TST.PROPS. SHBI APPLIED MATS. KLA TENCOR WEYERHAEUSER CATERPILLAR T ROWE PRICE GROUP LAM RESEARCH PARKER-HANNIFIN DEERE VENTAS GOODYEAR TIREROCKWELL & RUB. AUTOMATION PENTAIR PULTEGROUP AVALONBAY COMMNS.CHARLES SCHWAB THERMO FISHER SCIENTIFIC EMERSON WASTEELECTRIC MANAGEMENTPEPSICO MICROCHIP TECH. PEOPLES UNITED FINANCIAL HONEYWELL INTL. SOUTHWESTWATERS AIRLINES COLGATE-PALM.MCKESSON OMNICOM GROUP ABBOTTCOCA COLA LABORATORIESHARRIS BAXTER INTL. APARTMENT INV.& MAN.'A' VALERO ENERGYNUCOR BB&T KIMCO REALTYNORTHERN TRUST AMERISOURCEBERGEN MORGAN STANLEY BORGWARNER FREEPORT-MCMORANANADARKO PETROLEUMEATON 3M CINCINNATI FINL. ROPER TECHNOLOGIES LOEWS JP MORGAN CHASE & CO. INGERSOLL-RAND INTL.FLAVORS & FRAG. TOTAL SYSTEM SERVICES SIMON PROPERTY GROUP LINCOLN NATIONAL PROCTER & GAMBLE HUMANA CUMMINS COMERICA ONEOK US BANCORP UNUM GROUP HELMERICH & PAYNE FMC DOVER NATIONAL OILWELL VARCO WILLIAMS AETNAHESS WELLS FARGO & CO AIR PRDS.& CHEMS. EOG RES. BAKER HUGHES BOSTON PROPERTIES SCHLUMBERGERQUEST DIAGNOSTICSRANGE RES. ALLSTATE CONOCOPHILLIPS PNC FINL.SVS.GP. EXXON MOBIL CA CITIGROUP LEUCADIA NATIONAL M&T BANK HALLIBURTONNOBLE OCCIDENTALENERGY PTL. TORCHMARK PRAXAIRDEVON ENERGY ECOLAB HUNTINGTON BCSH. WESTERNPIONEER DIGITALCABOT NTRL.RES. OIL & GAS 'A' KEYCORP BANK OF AMERICA DOW CHEMICAL NEWFIELD EXPLORATION FIRSTENERGYUNITEDHEALTH GROUP COMCAST 'A' CAPITAL ONE FINL. STATE STREET CARDINAL HEALTH SUNTRUST BANKS APACHEFLOWSERVE JACOBS ENGR. EXELON XEROX EDISON INTL. ZIONS BANCORP. CHUBB AFLAC FIFTH THIRD BANCORP PROLOGIS PUB.SER.ENTER.GP. AMERICAN INTL.GP. KIMBERLY-CLARK MOTOROLAINTERNATIONAL SOLUTIONS PAPER AES PROGRESSIVE OHIO CMS ENERGY CHEVRONNEXTERA ENERGY CIGNA CORNING EASTMAN CHEMICALGENERAL ELECTRIC EQT ENTERGY GENERAL MILLS DTE ENERGY HARTFORD FINL.SVS.GP. NISOURCE TIME WARNER AMEREN CBS 'B' PPL AT&T SCANA INTERPUBLIC GROUP

TEXTRON SEMPRA EN. DOMINION RESOURCES DUKE ENERGY MERCK & COMPANY KELLOGGPINNACLE WEST CAP. MARATHON OIL CENTERPOINT EN. PFIZER XCEL ENERGY JOHNSON & JOHNSON CENTURYLINK WEC ENERGYSOUTHERN GROUP BRISTOL MYERS SQUIBBCAMPBELL SOUPPG&ECONSOLIDATED EDISON

E*TRADE FINANCIAL

VERIZON COMMUNICATIONSMEDTRONIC

VIACOM 'B'

(b) 12/01/2007 - 06/30/2009: 22.92% of all edges

26 companies of the S&P 500 between 2000 and end of 2015. Using the measure of modularity combined with symbolization allowed a detailed assessment of cluster formation in the stock market. The Euclidean distance of one-day logged differences of stock prices served as a measure to describe the similarity between two single stocks. The data revealed strong interlinkages of the ﬁnancial sector with the overall community, whereas the Great Recession lead to a sharp decline in the sector’s interconnectedness. Moreover, a direct relationship between a company’s size and its connectivity to the overall system could not be backed up by the empirical analysis. The value of the modularity measure conforms with earlier studies of network analyses and allows to claim that a certain community structure is indeed prevailing in the stock market as well. Even though the visualizations show a quite dense picture with no trivial cluster differentiation, they clearly demonstrate the non-stationarity of partition, varying with the state of the economy. Although this paper’s analysis is limited to a time frame of 15 years, it may serve as an encouragement for further research extending the assessment period and analyz- ing patterns during other times of economic stress. Furthermore, the determinants driving the speciﬁc partitioning and the reasons for their strong non-stationarity are still unknown.

27 Appendix

Table 1: Number of Companies per Industry Sector - before Symbolization -

Absolute Percent

Industrials 38 12.84% Health Care 35 11.82% Information Technology 36 12.16% Utilities 23 7.77% Financials 38 12.84% Materials 17 5.74% Consumer Discretionary 45 15.20% Energy 24 8.11% Real Estate 12 4.05% Consumer Staples 25 8.45% Telecommunication Services 3 1.01%

Sum 296 100.00%

28 Table 2: Number of Companies per Industry Sector - after Symbolization -

Industry Sector Period

01/03/2000 - 12/31/2015 12/01/2007 - 06/30/2009

Absolute Percent Absolute Percent Industrials 36 13.69% 37 14.07% Health Care 25 9.51% 22 8.37% Information Technology 34 12.93% 33 12.55% Utilities 23 8.75% 23 8.75% Financials 38 14.45% 37 14.07% Materials 16 6.08% 16 6.08% Consumer Discretionary 40 15.21% 42 15.97% Energy 24 9.13% 24 9.13% Real Estate 12 4.56% 12 4.56% Consumer Staples 12 4.56% 14 5.32% Telecommunication Services 3 1.14% 3 1.14%

Sum 263 100.00% 263 100.00%

29 Table 3: Number of Degrees per Industry Sector - after Symbolization -

Industry Sector Period

01/03/2000 - 12/31/2015 12/01/2007 - 06/30/2009

Absolute Percent Absolute Percent Industrials 1547 17.73% 1652 19.60% Health Care 142 1.63% 103 1.22% Information Technology 957 10.97% 816 9.68% Utilities 489 5.60% 425 5.04% Financials 3188 36.53% 1932 22.92% Materials 662 7.59% 774 9.18% Consumer Discretionary 774 8.87% 1281 15.20% Energy 476 5.45% 665 7.89% Real Estate 445 5.10% 734 8.71% Consumer Staples 39 0.45% 39 0.46% Telecommunication Services 7 0.08% 9 0.11%

Sum 8726 100.00% 8430 100.00%

Table 4: Number of Degrees - Summary Statistics - after Symbolization -

Period

01/03/2000 - 12/31/2015 12/01/2007 - 06/30/2009

Total Degrees 8726 8430 Maximum Degrees 165 117 Minimum Degree 1 1 Frequency of Minimum 13 19 Average Degree 33.18 32.05 Median Degree 21 23

30 Figure 7: Dynamics of Stock Returns

(a) Correlation of Stock Returns

Correlation of Stock Returns 296 Stocks of S&P 500 4000 3000 2000 Frequency 1000 0 0 .2 .4 .6 .8 1 Correlation

12/01/2007 - 30/06/2009 01/03/2000 - 12/31/2015

(b) Euclidean Distance of Stock Returns

Full-Period Standardized Euclidean Distance of Stock Returns 296 Stocks of S&P 500 4000 3000 2000 Frequency 1000 0 30 40 50 60 70 Standardized Eucledian Distance

12/01/2007 - 30/06/2009 01/03/2000 - 12/31/2015

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