A Map of the Brazilian Stock Market

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(1) There are many measures of correlation between elements of time series, the most popular being the Pearson correlation coefficent. The drawback of this correlation measure is that it only detects linear relationships between two variables. Two different measures that can detect nonlinear relations are the Spearman and the Kendall tau rank correlations, which measure the extent to which the variation of one variable affects other variable, withouth that relation being necessarily linear. In this work, I chose Spearman’s rank correlation, for it is fairly fast to calculate and it is better at measuring nonlinear relations. The correlation matrix may then be used to create a distance matrix, where distance is a measure of how uncorrelated two log-return series are from one another. There are also many ways to build a distance measure from the correlations between data. In this work, I shall use a different metric from [3], which is a nonlinear mapping of the Pearson correlation coefficients between stock returns. The metric to be considered here differs from the aforementioned metric because it is a linear realization of the Spearman rank correlation coefficient between the indices that are being studied:

dij = 1 − cij , (2) where cij are elements of the correlation matrix calculated using Sperman’s rank correlation. This distance goes from the minimum value 0 (correlation 1) to the maximum value 2 (correlation -1). Using the distance matrix, I shall build two maps (networks) based on the stocks of BM&F-Bovespa. The first map shall be based on a Minimum Spanning Tree (MST), which is a graph where each stock is a node (vertex), connected to one or more nodes. An MST is a planar graph with no intersections in which all nodes are connected and the sum of distances is minimum. The number of connections is the same as the total number of nodes of the network, minus one. This type of representation is very useful for visualizing connections between stocks, but it often oversimplifies the original data. Another representation is called an asset graph, which may be built by establishing a threshold under which correlations are considered, eliminating all other correlations above the said threshold. This also makes the number of original connections drop, simplifying the information given by the original correlation matrix. A three-dimensional map of stocks may be built in such a way that the distances portraied in it are the best approximation to the real distances, and the connections obtained by establishing a threshold are then drawn on such representation. The maps are examples of two networks that can be obtained from the same original data, each with its advantages and drawbacks. Using the two maps, one may then be able to identify which stocks are more dependent on each, and also which ones are more connected to others, what may be useful when building portfolios that minimize risk through diversification [36]. There are measures of how central each node of a network is, establishing its overall importance in the web of nodes, and also ways to visualize the overall distribution of those measures. As we shall see, the maps help verify that stocks that belong to the same types of companies tend to aglommerate in the same clusters, and that stocks with weak correlations often tend to connect at random in those networks. The MST is built in section 2, its centrality measures are shown in section 3, and their cumulative distribution functions are studied in section 4. The assets graphs are built in section 5, and the centrality of one of them are studied in section 6. Section 7 discusses the k-shell decomposition of one of the asset graphs, and section 8 presents a conclusion and general discussion of results.

2 Minimum spanning trees

Minimum spanning trees are networks of nodes that are all connected by at least one edge so that the sum of the edges is minimum, and which present no loops. This kind of tree is particularly useful for representing complex networks, ﬁltering the information about the correlations between all nodes and presenting it in a planar graph. As discussed in the introduction, I shall employ simulations of randomized data in order to establish a distance threshold above which correlations are seen as possibly of random nature. The result of 1000 simulations of randomized data is a lower threshold 0.69 ± 0.02, so distances above this value are represented as dashed lines in the minimum spanning tree diagrams. The minimum spanning tree formed by the stocks of the BM&F-Bovespa which were negotiated every day the stock exchange functioned during 2010 is represented in Figure 1. It has four main hubs: those stocks

2 are VALE3 and VALE5, belonging to Vale, the major mining industry in Brazil and one of the largest in the world, BBDC4, stock from Bradesco, one of the major banks in Brazil, and BRAP4, which is a branch of Bank Bradesco responsible for its participations in other companies. Of local importance are GFSA3, stocks from GAFISA, and PDGR3, stocks from PDG Realty, both construction and materials companies. The stocks are color-coded by main economic sectors, as shown in Figure 2.

CPLE6b CMIG4 b b CMIG3 JBDU4b IGTA3b HBOR3b LUPA3b

b CPFE3 NATU3b CLSC6 b DROG3 TOTS3 TLPP3 b b b b MILK11 UGPA4b COCE5b SULA11 RAPT4b b VLID3b MNPR3b SGPS3b CRUZ3 CNFB4 b b KROT11 b b b BBRK3b EQTL3 MULT3 b ALPA4 b PLAS3 IDNT3b b b TCSA3 ELPL4 VIVO4b BTOW3b b b LAME3 b LREN3 TCSL3 b ENBR3 LIGT3 b BRKM5ODPV3 b b EUCA4b AMBV3b b b MYPK3 BEMA3 CRDE3b b b b TLPP4 INEP3 GSHP3BRSR6 CCRO3 PCAR5 BRML3b b BEEF3 NETC4 b b b MPXE3b b FESA4 b POSI3 b KEPL3b LOGN3b b ELET6b b RENT3b SLED4 OHLB3 b b b CSMG3 TCSL4AMBV4b b b b EZTC3 HYPE3 ELET3 b LAME4 b GRND3 INEP4 BVMF3b GPCP3b

BBAS3 SBSP3 b b AMIL3 b CYRE3b MRVE3b b b ALLL3 BBDC3b DTEX3 BRAP4VALE3 BBDC4 GFSA3 ABCB4b bbb b b b MMXM3b PDGR3 VALE5ITUB3 JHSF3 b bITUB4 b b LLXL3 b RSID3 b ITSA4 PINE4 b b b b CSNA3 BISA3 PSSA3 b CESP6 b GGBR4 b SANB11 b SLCE3 GGBR3 bb b b TEMP3b GOAU4 FIBR3 EVEN3 USIM5 b POMO4b CCIM3 USIM3 b b b b TNLP4b GOLL4b ECOD3b b FRAS4b OGXP3 ETER3b b b b ESTR4 RDCD3 b TNLP3 b IGBR3 b b INPR3b b b b PETR4 b SUZB5b EMBR3SANB3 b JBSS3 CTIP3FLRY3 b MAGG3 GPIV11 BPNM4 UOLL4 PETR3 b b TMAR5 UNIP6 b b b FFTL4b b CZLT11 FJTA4 b BICB4 TAMM4b bFHER3 bTRPL4 PMAM3 SANB4 b b AMAR3b b CIEL3 b WEGE3b b KLBN4b b BRTO4 PFRM3b SFSA4 b b SMTO3b CSAN3 MRFG3 b JFEN3 BRFS3 b CREM3 HAGA4 b b TCNO3 b GETI4 INET3 b MLFT4 MTIG4 LPSB3 b b b CTAX4 b b CARD3 b BTTL4 TEKA4 b b BRTO3 b MNDL4 GETI3HGTX3 b IENG5 b b PRVI3 TOYB4 b b RPMG3 AEDU3 b b TBLE3

b TPIS3 b RNAR3 RPMG4

b b b b TELB3 TELB4 TGMA3 TOYB3 DASA3 b Figure 1. Minimum spanning tree for 2010.

3 b mining, metalurgy, oil, gas, and biofuels b telecommunications and technology b construction, materials, and engineering b banking and ﬁnancial b sugar, ethanol, food, wood, paper, and celulosis b consumer goods and education b health care and sanitation b industry, machines, and equipment b energy b logistics and transportation

Figure 2. Color coding of stocks in Figure 1 according to economic sectors.

The network formed by the stocks of the BM&F-Bovespa which were negotiated every day the stock exchange functioned during 2010 have six main hubs, which are displayed in Figure 3. The stocks are VALE3 and VALE5, belonging to Vale, the major mining industry in Brazil and one of the largest in the world, BBDC4, stock from Bradesco, one of the major banks in Brazil, BRAP4, which is a branch of Bank Bradesco responsible for its participations in other companies, GFSA3, stocks from GAFISA, and PDGR3, stocks from PDG Realty, both construction and materials companies.

VALE3 BBDC4 GFSA3 BRAP4 b bb b b b PDGR3 VALE5

Figure 3. Main axis of the minimum spanning tree for 2010.

For reasons of clarity, I shall divide the minimum spanning tree into five diagrams, or clusters, each one constructed around one of the five main hubs. The first cluster we shall study in more detail, Figure 4, is the one formed around BRAP4, Bradespar, which is a company created when Bradesco (banking) was dismembered. It manages the participations of Bradesco in other, non-financial companies, particularly CPFL and Vale.

PSSA3 PINE4 b SLCE3 b b LLXL3 b TEMP3 b

MMXM3 b

SULA11 CSMG3 SBSP3 BRAP4 b b b b b VALE3

b b ABCB4 OHLB3 b b ALLL3 HYPE3 b b AMIL3 GPCP3

Figure 4. Cluster around BRAP4.

As can be seen from the excerpt of the minimum spanning tree, it has strong ties with VALE3 (of Vale, a mining industry of which it has about 17% of the stocks). It is immediately surrounded by 13 stocks, three of

4 them of financial background (PINE4 and ABCB4 are both stocks of banks, PSSA3, of an insurance company), health (TEMP3 and AMIL3), sanitation (SBSP3, itself linked with stocks of another sanitation company, CSMG3), mining (VALE3 and MMXM3), petrochemistry (GPCP3), logistics (ALLL3, LLXL2, and OHLB3), and agribusiness (SLCE3). Indirectly and weakly connected (as shown by the dashed line) to this hub are the stocks of another insurance company (SULA11). This is a somewhat mixed cluster, with a variety of stocks orbiting the stocks of a very capitalized bank (Bradesco is the third major bank of Brazil). The second and third clusters are actually the same, and they are represented in different figures in order to make it easier to discern the organization of the stocks arounds VALE3 and VALE5. Beginning with Figure 5, one can see a cluster formed around VALE3, which is strongly correlated with VALE5, as it would be expected. Vale is a mining company, and one can observe that connected to it, on its top, there is a collection of other stocks of mining companies (CSNA3, USIM3, and USIM5), of metalurgy (GGBR3, GGBR4, and GOAU4, all of them from the Gerdau group, and MAGG3), and of petrochemistry (UNIP6). Also connected, indirectly, to VALE3, are the stocks PETR3 and PETR4, of Petrobras, an oil, gas, and biofuel company which is one of the major in the world. Petrobras is responsible for a large amount of the volume traded in the Bovespa, and it is surprising to find it not as a hub itself in the minimum spanning tree representation. A likely explanation is that the main product of Petrobras, petroil, is a comodity negotiated worldwide and so much more dependent on factors like the price of the oil barrel then on internal ones. This is enough to place Petrobras’ stocks apart from the others. At the lower part of Figure 5, one can see a cluster of stocks related with electricity distribution (ELET3 and ELET6, ELPL4, CPFE3, ENBR3, LIGT3, CPLE6, and CMIG3 and CMIG4), most sparsely correlated with one another. To the left, there is a cluster of stocks related with telecommunications (TNPL3 and TNPL4, TMAR5, and BRTO3 and BRTO4). There are also some more stocks, related with engineering and materials, logistics, food, heavy machinery, consumer goods, and health, scattered and not forming any particular cluster. Note that most of these stocks, arranged apparently at random, have weak connections, as shown by the dashed lines.

5 TGMA3 b

TEKA4 b PFRM3 b

CREM3 b

PETR3 MAGG3 b b b IENG5 b PMAM3 b PETR4 UOLL4 b b USIM3 b USIM5 TNLP3 TNLP4 GOAU4 b GGBR4 UNIP6 b b b b b b b INET3 b GGBR3 CSNA3 TMAR5 b BICB4 b TOYB3 VALE3 BRAP4 b b b VALE5 b b RPMG4 b b BEEF3 BRTO4 IGBR3 b RPMG3 CNFB4 INEP4 b b PLAS3 b b FESA4 b ELET3 b INEP3 b ELET6 b BRTO3

b KROT11 b TELB3

ELPL4 b b TELB4

CMIG3 CMIG4 CPLE6 CPFE3 ENBR3 b b b b b

b LIGT3

Figure 5. Cluster around VALE3.

6 Figure 6 shows the cluster around VALE5, which is the same cluster as the one of Figure 5, since VALE3 and VALE5 are intimately connected. At the right of VALE5, one can see a cluster of stocks belonging to companies related to agriculture (FFTL4), food (BRFS3, MRFG3, and RNAR3), paper (the sequence made by FIBR3, SUZB5, and KLBN4), and sugar and ethanol (SMTO3). There are also stocks belonging to other companies, but they apparently do not form clusters.

CARD3 b b PRVI3

RNAR3 CTAX4 FFTL4 b b b b SMTO3 MRFG3 b

NETC4 KEPL3 b b HBOR3 b BEMA3 b SUZB5 KLBN4 FIBR3 b b b

VALE5 VALE3bb b BBDC4 TCSA3 b b BRFS3 b ECOD3 SGPS3 b b LOGN3

b b ESTR4 ETER3

b VLID3 b MNPR3

Figure 6. Cluster around VALE5.

7 The cluster in Figure 7 is built around BBDC4 (closely connected wit BBDC3), which are stocks of Bank Bradesco. It is a denser cluster, comprised in its center by stocks related with banks (ITUB3, ITUB4, and ITSA4, all related with Bank Itau, the second largest in Brazil, SANB3, SANB4, and SANB11, stocks from Bank Santander, BBAS3, stocks of the Bank of Brazil, the largest in Brazil, and BRSR6) or with investment and ﬁnance companies (GPIV11, RDCD3), and BVMF3, wich are stocks of BM&F-Bovespa itself. To the left, there are two stocks of COSAN (CSAN3 and CZLT11), a food, sugar, and ethanol company. To the top and right, there is another cluster, of cyclic consumer goods: beverages (AMBV3 and AMBV4), tobacco (CRUZ3), pharmaceuticals (DROG3), and sandals (ALPA4). Also to be noticed are the sequences RDCD3-CIEL3, both related with companies that operate and sell credit and debit card terminals, and GOLL4-TAMM4, both related with air transport companies. Another small cluster comprises the stocks TCSL3 and TCSL4, and VIVO4, related with mobile telephony companies. Other stocks scatter around the main network without forming any discernible subnetwork.

COCE5 CLSC6 TLPP3 b b b

ALPA4 b

CRUZ3 b b b BRSR6 TLPP4 EUCA4 b

SANB3 b AMBV3 DROG3 b b b SANB4 b AMBV4

TCSL4 CSAN3 CZLT11 SANB11 b b TCSL3 b b b BBDC3 b b BVMF3 b BBAS3 VALE5b b b GFSA3 b BBDC4 b VIVO4 b b b ITUB4 GOLL4 b b POSI3 ITUB3 b ITSA4 b TAMM4 BBRK3 GSHP3 b GPIV11 b FRAS4 b EMBR3 b OGXP3 RDCD3 b b JBSS3

b CIEL3

b b b TCNO3 HAGA4 MLFT4 Figure 7. Cluster around BBDC4.

8 The next cluster, represented in Figure 8, is centered around a construction and real state company, Gaﬁsa (GFSA3), and is composed mostly by the stocks of other construction and real state companies (PDGR3, CYRE3, CRDE3, EVEN3, INPR3, JFEN3, and LPSB3), and by stocks of consumer goods companies (LAME3 and LAME4, LREN3, BTOW3, NATU3, AMAR3, HGTX3, and MTIG4). There is also a small cluster (lower part the ﬁgure) of stocks of electricity distribution companies (GETI3 and GETI4, and TBLE3).

LUPA3 NATU3 b b BTOW3 b LAME3 BTTL4 b b b b IDNT3 LREN3 HGTX3 FHER3 b b b LAME4 CCRO3 b TPIS3 b EVEN3 b CYRE3 CRDE3 b b

BBDC4b b b PDGR3 GFSA3 b AMAR3

BPNM4BISA3 POMO4 LPSB3 b b b b b SFSA4

INPR3 b b b FJTA4 JFEN3

GETI4 b b AEDU3 b MTIG4

b GETI3 b TBLE3

b DASA3 Figure8. Cluster around GFSA3.

9 The last cluster, Figure 9, is also centered around the stock of a construction and real state company, PDG Realty (PDGR3), and is composed mostly of stocks of other construction and real state companies (GFSA3, EZTC3, RSID3, MRVE3, JHSF3, and CCIM3), real state management of shopping centers (BRML3, MULT3, and IGTA3), and of building materials (DTEX3). Other stocks belong to electricity, logistics and transportation, consumer goods and stocks of other types of companies. MNDL4 GRND3 b b

SLED4 FLRY3 b b

CESP6 CTIP3 b b CCIM3 b b b TRPL4 JHSF3 JBDU4 b RENT3 b RSID3 b

PDGR3 EZTC3 GFSA3 b b b b EQTL3 WEGE3 b

MRVE3 b b ODPV3 b MPXE3 b MILK11

MYPK3 b b b b DTEX3 RAPT4 PCAR5 b BRKM5 b TOYB4

MULT3 IGTA3 b b b BRML3 b TOTS3 b UGPA4

Figure 9. Cluster around PDGR3.

3 Centrality measures

In network theory, the centrality of a node, or how influential the vertex is in the network, is an important measurement which is handled in a number of diferent ways. In what follows, I perform an analysis of the centrality of vertices in the network depicted by the MST of the last section according to five different definitions of centrality. I also do some analysis of the frequency distribution of each centrality measure in the network and which are the stocks that are more central according to each definition. Node degree Next, I analize the node degree of the stocks in the network. The node degree of a node (stock in our case) is the number of connections it has in the network. Most of the stocks have low node degree, and some of them have a large number associated with them. The latter are called hubs, and are generally nodes that are

10 more important in events that can change the network. Table 1 and Figure 10 show the node degree frequency distribution of the network of the BM&F-Bovespa 2010.

Node degree Frequency 1 105 frequency 2 45 3 18 100 4 7 5 8 80 6 3 7 0 60 8 1 9 0 40 10 1 11 0 20 12 1 13 1 node degree 1 2 3 4 5 6 7 8 9 10 11 12 13 Table 1. Node degree distribution. Figure 10. Node degree frequency distribution.

The stocks with highest node degree are BRAP4 (node degree 13), BBDC4 (node degree 12), VALE5 (node degree 10), and VALE3 (node degree 8). Node strength The strength of a node is the sum of the correlations of the node with all other nodes to which it is connected. If C is the matrix that stores the correlations between nodes that are linked in the minimum spanning tree, then the node strength is given by n n k Ns = X Cik + X Ckj , (3) i=1 j=1 where Cij is an element of matrix C. In our network, the vertices with highest node strength are BBDC4 (Ns = 6.98), BRAP4 (Ns = 5.37), VALE5 (Ns = 4.72), and VALE3 (Ns = 4.50). The node strenght frequency distribution is shown in Figure 11.

frequency

node strength 1 2 3 4 5 6 7 Figure 11. Node strength frequency distribution.

Eigenvector centrality Node degree may be seen as a bad representation of how important a node is, for it doesn’t take into account how important the neighbors of a node may be. As an example, one node may have a low dregree, but it may be connected with other nodes with very high degree, so it is, in some way, inﬂuent. A measure that takes

11 into account the degree of neighbouring nodes when calculating the importance of a node is called eigenvector centrality. In order to define it properly, one must first define an adjency matrix, A, whose elements aij are 1 if there is a connection between nodes i and j and zero otherwise. If one now considers the eigenvectors of the adjency matrix, and choosing its largest value, one then may define the eigenvector with largest eigenvalue by the equation AX = λX , (4) where X is the eigenvector with the largest eigenvector λ. The eigenvector centrality of a node i is then defined as the ith element of eigenvalue X: Ec = xi , (5) where xi is the element of X in row i. In the present network, the vertices with highest eigenvector centralities are BBDC4 (Ec = 0.502), VALE5 (Ec = 0.373), VALE3 (Ec = 0.294), and BRAP4 (Ec = 0.257). The eigenvector centrality frequency distribution is shown in Figure 12.

frequency

100

eigenvector centrality 0.1 0.2 0.3 0.4 0.5 Figure 12. Eigenvector centrality frequency distribution.

Betwenness The betweennes centrality measures how much a node lies on the paths between other vertices. It is an important measure of how much a node is important as an intermediate between other nodes. It may be deﬁned as n k nij Bk = , (6) c X m i,j=1 ij where nij is the number of shortest paths (geodesic paths) between nodes i and j that pass through node k and mij is the total number of shortest paths between nodes i and j. Our network is fully connected, so we need not worry about mij being zero. The betweenness centrality frequency distribution is shown in Figure 13.

frequency 160

140

120

100

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 betweenness Figure 13. Betweenness centrality frequency distribution.

12 The vertices with highest betweenness centrality are BBDC4 (Bc = 12010), VALE5 (Bc = 10490), VALE3 (Bc = 9197), and GFSA3 (Bc = 9010). There are 106 vertices with zero betweenness centrality, what means that no shortest path between any two vertices in the network pass through those vertices. Closeness Another measure of centrality is the closeness centrality, which measures the average distance between one node and all the others. It is deﬁned as the measure of the mean geodesic distance for a given node i, which is given by n 1 ℓ = d , (7) i n X ij j=1 where n is the number of vertices and dij is a geodesic (minimum path) distance from node i to node j. This measure is small for highly connected vertices and large for distant or poorly connected ones. In order to obtain a measure that is large for highly connected nodes and small for poorly connected ones, one then deﬁnes the inverse closeness centrality of node i as i 1 Cc = . (8) ℓi

The vertices with highest inverse closeness centrality are BBDC4 (Cc = 0.00107), VALE5 (Cc = 0.00106), GFSA3 (Cc = 0.00100), and VALE3 (Cc = 0.00099). The inverse closeness centrality frequency distribution is shown in Figure 14. frequency

inverse closeness (×10−3) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Figure 14. Inverse closeness centrality frequency distribution.

4 Cumulative distribution function

Note that all frequency distributions, with the exception of the one for inverse closeness centrality (the same happens for the frequency distribution of the closeness centrality), are exponentialy decreasing. One say that those frequency distributions follow a power law of the type

−α pk = ck , (9) where pk is the frequency distribution for the value k, and c and α are constants. This is a characteristic of a diversity of complex systems, and it happens in the study of earthquakes, the world wide web, networks of scientific citations, of film actors, of social interactions, protein interactions, and many other topics [37]. Networks whose centrality measures follow this type of distribution are often called scalle-free networks, and this behavior can best be visualized if one plots a graph of the cumulative frequency distribution of a centrality in terms of the centrality values, both in logarithmic representation. Figure 15 shows the logarithm of the cumulative frequency distributions as functions of the logarithm of the five types of centrality measures we used in the last section.

13 100 100 101 10−1 101

10−1 10−1

10−2

10−3 a) Node degree x cumulative distribution b) Node strength x cumulative distribution

10−3 10−2 10−1 100 100 100 101 102

10−1 10−1

10−2 10−2

10−3 10−3 c) Eigenvector centrality x cumulative distribution d) betweenness centrality x cumulative distribution

10−3.4 10−3.2 10−3 100

10−1

Figure 15. Log-log plots of the cumu- 10−2 lative frequency distributions as functions of centrality measures. e) Inverse closeness x cumulative distribution

Were a network a pure scalle-free one, then the graphic of the logarithm of the cumulative frequency distribution as a function of the logarithm of a centrality measure would be a straight line. One may notice that this behavior is aproximately followed by the intermediate levels of all centrality measures, except for inverse closeness centrality. The exactly same behavior is seen in, as examples, the world wide web or networks of citations.

14 5 Asset Graphs

A two dimensional representation of non-overlapping connections between vertices, such as the minimum spanning tree, has some misleading features. Two nodes that are actualy close to each other may appear far or unconnected, and one node that is just very weakly connected may establish a connection in the MST at random. A three dimensional representation is often a better representation, and one can be built using distance as a guide. As an example, one may use an algorithm that minimizes the sum of the squares of the diﬀerences between the real distances and the ones portraied in a three dimensional graph. Other, more advanced techniques, can also be used, such as principal coordinates analysis. By using the latter, one obtains the following three dimensional representation of the stocks of BM&F-Bovespa (Figure 16).

IGBR3 BTTL4 b TEKA4 INET3b b b IENG5 KLBN4 TNLP4 b VALE5b b SANB4 VALE3b PETR3PETR4 b USIM3b bb LIGT3 CMIG4 RPMG3SLCE3ESTR4 CSNA3USIM5SUZB5b b CMIG3 b UOLL4b b KEPL3b b b b GGBR4GOAU4 b b SGPS3b GGBR3b b RPMG4GRND3b BEEF3 b ELET6 TELB4 b UNIP6BEMA3b b BRAP4 b b MILK11PRVI3b PMAM3b JBDU4CSMG3b FIBR3 CPFE3 PFRM3b b b b b NETC4b b GPCP3TPIS3WEGE3b b TLPP4TNLP3 RNAR3b b MNDL4b PLAS3TCNO3 b b CPLE6 MNPR3b bINEP4b b VIVO4SBSP3 b TELB3 TOYB4b b ECOD3ALLL3EMBR3TBLE3GETI4ELPL4ELET3b b b PINE4b MTIG4 b b b b b TCSL3TCSL4b b TMAR5 OHLB3b MLFT4b SANB3 BTOW3 bb b LUPA3b b GETI3b b SLED4 INEP3b BRTO3b UGPA4BRTO4 RDCD3 VLID3 b CNFB4b FFTL4b BRFS3NATU3b b b b b MAGG3b GPIV11CRUZ3ENBR3b TRPL4b TGMA3LOGN3HAGA4CTAX4b b b SANB11b b b b BICB4b b LLXL3MRFG3MMXM3 b BBDC4 COCE5b FRAS4b b b b TEMP3b ALPA4 CZLT11b TLPP3 FESA4 bSMTO3FJTA4b CCRO3b b CIEL3 FLRY3b CREM3CLSC6IDNT3b b b ITUB4CESP6ITSA4b TOYB3 b bb b JBSS3TAMM4BVMF3b b b b KROT11JFEN3DROG3 bBBAS3b ITUB3b b DASA3b b BRKM5BBDC3b b GSHP3FHER3ETER3SULA11TOTS3SFSA4b HYPE3POSI3b b CTIP3b b CARD3MYPK3b b b b AMIL3INPR3CRDE3CSAN3b b b PSSA3b b ABCB4b b b b b b OGXP3 EQTL3 LAME3 b b TCSA3b AMBV3 LPSB3 b LAME4AMBV4b AMAR3 b IGTA3 b b GOLL4 EUCA4b MPXE3BBRK3BRSR6b RENT3BISA3b b b b b LREN3b b EZTC3PCAR5 b CCIM3b BPNM4b AEDU3 b b MULT3 JHSF3ODPV3b HBOR3 b b b b BRML3 DTEX3 b GFSA3b CYRE3b HGTX3POMO4 b b b RAPT4 b RSID3 EVEN3 b b PDGR3MRVE3 b b Figure 16. Three dimensional representation of stocks of the BM&F-Bovespa.

The picture is not very enlightening when printed on paper, but oﬀers a good three-dimensional view when viewed on a computer. In particular, it may be used in order to study the distributions of stocks of the same type of companies, what is done in Appendix B. Here, we shall use the three dimensional representation in order to study asset graphs, which are networks built on the distance matrix by establishing thresholds under which connections are considered. As an example, one may build a network of those nodes whose distances with the others are below or equal to T = 0.5. This will probably exclude many of the connections and some of the nodes of the original network. In what follows, I perform an analysis of some asset graphs obtained by selecting thresholds that go from 0.1 to 0.7, which is the limit at which random noise starts to become absolute. At T = 0.1 (Figure 17), the only connections are those between BBDC3 and BBDC4, both stocks from Bradesco (banking), GGBR3, GGBR4, and GOAU4 from Gerdau (metalurgy), ITUB4 and ITSA4 from Itau (banking), PETR3 and PETR4 from Petrobras (petroleum and gas), and between VALE3 and VALE5, both stocks from Vale (mining). PETR3 VALE3b PETR4 b bb GGBR4 b b VALE5 b GGBR3 GOAU4

BBDC4 b ITUB4 b b b ITSA4 BBDC3 Figure 17. Connections bellow threshold T = 0.1.

15 At T = 0.2 (Figure 18), the earlier connections are joined by a small cluster of stocks RPMG3 and RPMG4, of the Refinaria de Petróleo Manguinhos (petroleum refinement), and by another small cluster formed by USIM3 and USIM4, of Usiminas (mining). Connections are established between BRAP4, Bradespar, which is an investiment branch of Bank Bradesco which has 17% of the control of Vale, and VALE3 and VALE5. A cluster is formed with stocks from Bank Bradesco and Bank Itau, now with the joining of ITUB3.

PETR3 VALE3b PETR4 RPMG3 b bb b USIM3 b b GGBR4 b b VALE5 b b USIM5 b GGBR3 GOAU4 RPMG4 BRAP4

BBDC4 b ITUB4 ITUB3 b b b b ITSA4 BBDC3

Figure 18. Connections bellow threshold T = 0.2.

At T = 0.3 (Figure 19), the network is joined by the pairs TELB3-TELB4 of Telebras (telecommunications), INEP3-INEP4 of Inepar (construction), TNLP3-TNLP4 of Telemar (telecommunications), CMIG3-CMIG4 of CEMIG (electricity), and AMBV3-AMBV4 of Ambev (beverages). The network formed by the stocks of Gerdau now joins with the stocks of Usiminas, and with the newcommer CSNA3 of Companhia Siderúrgica Nacional (metalurgy). The new network represents a cluster of stocks of companies working in mining and meatlurgy, close to but still not connected with Vale. A new newtork is formed by the stocks GFSA3 of Gafisa, CYRE3 of Cyrela, RSID3 of Rossi Residencial, PDGR3 of PDG Realty, and MRVE3 of MRV Engenharia e Participa¸cões, all of them in the construction business.

PETR3 TNLP4 b VALE3b PETR4 b bb CMIG4 RPMG3 CSNA3 CMIG3 b b USIM3b b b GGBR4 b b b USIM5VALE5 b TELB4 b b b GGBR3 GOAU4 RPMG4 INEP4 BRAP4 b b b TNLP3 TELB3 b BBDC4 INEP3 b ITUB4 ITUB3 b b b b ITSA4 AMBV3BBDC3 b b AMBV4

GFSA3 b b CYRE3 RSID3 b b b MRVE3 PDGR3

Figure 19. Connections bellow threshold T = 0.3.

For T = 0.4 (Figure 20, with the new stocks in the network in red, for better visualization), the mining and metalurgy network joins with the ﬁnancial network, and the construction network becomes denser, with more connections established between its nodes. We also have the newcommer isolated pairs TCSL3-TCSL4 of TIM (telecommunications), CIEL3 of Cielo and RDCD3 of Redecard, both operating in the business of electronic cards, SUZB5 of Suzano Papel e Celulose and FIBR3 of Fibria Celulose, both operating in the paper production market, and GOLL4 of Gol and TAMM4 of TAM, both pertaining to airlines. The stocks of two banks, SANB11 of Bank Santander and BBAS3 of the Bank of Brazil connect with the ﬁnancial network via the stocks of Bradesco, and the stocks CZLT11 of Cosan (sugar and ethanol) connect with CSNA3 (metalurgy).

16 PETR3 TNLP4 b VALE3b PETR4 b bb CMIG4 RPMG3 CSNA3SUZB5 CMIG3 b b USIM3b b b b GGBR4 b b b USIM5VALE5 b TELB4 b b FIBR3 b GGBR3b GOAU4 RPMG4 INEP4 BRAP4 b b b TCSL3 TNLP3bb TELB3 b RDCD3 SANB11 TCSL4 b b BBDC4 INEP3 b CZLT11 b TAMM4 ITUB4b b ITUB3 b b BBAS3b b b CIEL3ITSA4 AMBV3BBDC3 b b b AMBV4 GOLL4

GFSA3 b b CYRE3 RSID3 b b b MRVE3 PDGR3 Figure 20. Connections bellow threshold T = 0.4.

For T = 0.5 (Figure 21, with the new stocks in the network highlighted in red), all three major networks, mining and metalurgy, financial, and construction, are now joined, with a central role played by the financial network. BISA3 of Brookfield Incorpora¸cões, EVEN3 of Even Construtora e Incorporadora, EZTC3 of EZETEC, all of the building industry, join the building companies network. BRML3 of BR Malls and MULT3 of Multiplan, both of the real state business (shopping centers), form a pair close to but not connected with the building industry network.

PETR3 TNLP4 b VALE3b PETR4 b bb CMIG4 RPMG3 CSNA3SUZB5 CMIG3 b b USIM3b b b b GGBR4 b b b USIM5VALE5 b TELB4 b b FIBR3 b GGBR3b GOAU4 b CPLE6 RPMG4 INEP4 BRAP4 b b GETI4 b b TCSL3TNLP3TMAR5 GETI3 bb b TELB3 b b BRTO4 RDCD3 SANB11 TCSL4b b MMXM3 b BBDC4 INEP3 b b b ITUB4b CZLT11 TAMM4BVMF3b b b ITUB3b BBAS3b b b CIEL3ITSA4 LAME3 BBDC3 b AMBV3 b LAME4b b BISA3b LREN3b EZTC3 AMBV4b b MULT3 GOLL4 b BRML3 b GFSA3 b b CYRE3 RSID3 EVEN3 b b b b MRVE3 PDGR3 Figure 21. Connections bellow threshold T = 0.5.

BRTO4 of Brasil Telecom and TMAR5 of Telemar Norte Leste, both of telecommunications companies, join TNLP4 of Telemar in order to form a telecommunications network. GETI3 and GETI4, both of AES Tietê (electricity) form an isolated pair. LAME3 and LAME4 of Lojas Americanas and LREN3 of Lojas Renner, both of the retailing business, form another separate cluster. MMXM3 of MMX Minera¸cão e Metálicos join Vale and Bradespar in the mining and metalurgy network. The pair of stocks of Inepar (construction) connects with Vale (mining), the pair of Telemar connects with Telemar Norte Leste, the stocks of Gol (airliner) connect with the ones of Bradesco (banking) and Gerdau (metalurgy). BVMF3, which are stocks of the BM&F-Bovespa (financial), connect with the ones of Bradesco, Bradespar, Itau (all in the banking and financial sectors), Cyrella (building), and Gerdau (metalurgy). Petrobras (petroleum and gas) connects with Bradesco (banking) and Gerdau (metalurgy).

17 For T = 0.6 (Figure 22, with the new stocks in the network highlighted in red), and T = 0.7, connections between sectors become more frequent, and already existing clusters become denser. More stocks take part of the complete network now, even those that have weaker correlations with other stocks. From T = 0.8 onwards, noise starts to takes over, and new connections are not reliable.

KLBN4 PETR3 TNLP4 b b VALE3b PETR4 b bb CMIG4 RPMG3 CSNA3SUZB5 CMIG3 b b USIM3b b b b GGBR4 b b b b VALE5 b ELET6 KEPL3USIM5 b TELB4 b PMAM3 b FIBR3 b b GGBR3b GOAU4 b b CPLE6CPFE3 RPMG4 INEP4 BRAP4 VIVO4 b b EMBR3GETI4ELPL4ELET3b b b b b TCSL3b b TMAR5 SBSP3TNLP3b b SANB3 b b b GETI3BRTO3b BRTO4 RDCD3 TELB3 LUPA3b FFTL4b b b b b SANB11 TCSL4 b MMXM3 b b INEP3MAGG3 b BBDC4 LOGN3BICB4b LLXL3b b b b b b CZLT11b TAMM4b ITUB4 SMTO3 CCRO3b ITUB3BVMF3b b b IDNT3 b b b CESP6 BBAS3b b CIEL3 FESA4b b ITSA4 LAME3INPR3b JBSS3b POSI3BBDC3 FHER3 b AMBV3 b b b LAME4CRDE3b b b BISA3OGXP3b b b LREN3b b EZTC3TCSA3BBRK3AMBV4b b IGTA3b GOLL4 b b MULT3 BRSR6 b RENT3 b PCAR5 CCIM3BPNM4BRML3 b GFSA3b JHSF3 DTEX3b b CYRE3 RSID3 EVEN3 b b b b MRVE3 PDGR3 Figure 22. Connections bellow threshold T = 0.5.

6 Centrality measures for the asset graph

I shall now analize some of the centrality measures seen in sections 3 and 4 but now applying them to the network obtained by ﬁxing a threshold T = 0.7 (just bellow the noisy region) and considering only those distances that are bellow it. I begin by analyzing the node degree. Table 2 and Figure 23 show the node degree distribution.

Node degree Frequency 0 22 10 47 frequency 20 22 30 10 40 14 40 50 13 60 16 70 8 30 80 9 90 6 100 4 20 110 3 120 6 130 3 10 140 2 150 3 160 2 node degree 10 20 30 40 50 60 70 80 90 100110120130140150160 Table 2. Node degree distri- Figure 23. Node degree frequency distribution. bution for the asset graph.

The highest node degrees belong to BBDC4 and ITUB4 (node degree 103), BRAP4 (node degree 101), ITSA4 (node degree 95), BBDC3, and VALE5 (node degree 95). Figure 24 represents the acumulated frequency distribution of the node degree plotted against the node degree.

18 100 100 101 102

10−1

10−2 Figure 24. Node degree x cumulative distribution

Next, calculating the node strenght, one can see that the nodes with highest values are BBDC4 (Ns = 86.87), ITUB4 (Ns = 85.05), BRAP4 (Ns = 81.22), and ITSA4 (Ns = 79.93). The probability distribution function is shown in Figure 25, together with the log-log plot of the cumulative distribution as a function of the node strength (Figure 26).

frequency 50

40 10−2 10−1 30 10−1 20

10 10−2 node strength 10 20 30 40 50 60 70 80 90 Figure 26. Node strength x cumulative distribution. Figure 25. Node strength frequency distribution. The eigenvector centrality, given by equation (4), measures the centrality of a node by giving weights to the nodes that are immediately connected with it. The highest values for eigenvector centrality are those of BBDC4 and ITUB4 (Ec = 0.172), ITSA4 (Ec = 0.167), BBDC3 and BRAP4 (Ec = 0.163), and VALE5 (Ec = 0.160). The probability distribution function is shown in ﬁgure 25, together with the log-log plot of the cumulative distribution as a function of the eigenvector centrality (ﬁgure 26).

frequency 10−3 10−2 10−1 100 100 40

30 10−1 20

10 10−2

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 eigenvector centrality Figure 28. Eigenvector centrality x Figure 27. Eigenvector centrality cumulative distribution frequency distribution. If we now consider betweenness as a measure of centrality, then the highest values occur for BRAP4 (Bc = 950), BBDC4 (Bc = 722), ITUB4 (Bc = 701), and VALE5 (Bc = 629). The probability distribution function is shown in Figure 29, together with the log-log plot of the cumulative distribution as a function of the betweenness centrality (Figure 30).

19 frequency

50 2 40 10

30 1 20 10

100 200 300 400 500 600 700 800 900 1000 10−4 10−3 10−2 10−1 betweenness centrality Figure 29. Betweenness centrality Figure 30. Betweenness centrality x frequency distribution. cumulative distribution The last centrality measure I shall analize is inverse closeness centrality, for which the highest values occur for −4 −4 −4 BBDC4 and ITUB4 (Cc = 1.866×10 ), BRAP4 and ITSA4 (Cc = 1.864×10 ), BBDC3 (Cc = 1.863×10 ), −4 and VALE5 (Cc = 1.862 × 10 ). The probability distribution function is shown in Figure 31, together with the log-log plot of the cumulative distribution as a function of the inverse closeness centrality (Figure 32).

frequency

40 −1 30 10 −1 20 10

10 10−2

1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 Figure 32. Inverse closeness centrality x − inverse closeness centrality (×10 4) cumulative distribution Figure 31. Inverse closeness centrality frequency distribution.

One thing to be noticed is that the frequency distributions of the centrality measures of the asset graph are more distant from what would be expected from a scale-free network. This may be due to the great amonut of noise that comes with the choice of threshold. For higher threshold values, this diﬀerence increases, and for lower threshold values, it decreases.

7 K-shell decomposition

Another important centrality measure which can be applied only to the asset graph (and not to the MST) is k-shell decomposition, which consists on classifying a vertice according to the connections it makes, and also considers if it is in a region of the network that is also highly connected. This decomposition is frequently used in the study of the propagation of diseases and of information, and also shed some light on ﬁnancial networks. It consists of considering all nodes with degree 1, assigning to them k = 1, and striping the network from them. Then, one looks at the remaining vertices with degree 2 or less, and assigns to them k = 2. Repeating the procedure, all vertices with k = 2 are removed, and one then looks for vertices with degree 3 or less. The process goes on until all nodes are removed. What one then obtains are shells of vertices that increase in importance as k goes larger. For our asset graph network for T = 0.7, there are 30 shells, and the stocks that belong to it are represented in Figure 33.

20 bc bc bc bc bc VALE5bc bc VALE3bcb PETR3PETR4 bc USIM3bcb bcbbcb CSNA3USIM5SUZB5bcb bc bc bc bc bc bcb bcb bcb GGBR4GOAU4 bc bc bc GGBR3bcb bcb bc bcb bc bc bc BRAP4 bc bc bc bc bcb FIBR3 bc bc bc bc bc bcb bc bc bc bc bc bc bc bc bc bcINEP4bc bc bc bc bcb bc bc bc bc bc bc bc bc bc TCSL4bc bc bc bc BTOW3 bcbcb bc bc bc bc bcb bc bc BRTO4 bc bc FFTL4bc bc bcb bc bc bc MAGG3bcb bc bc bcb bc bc SANB11bc bc bc bc bc bc MMXM3 bcb BBDC4 bc bc bc bcb bc bc CZLT11bc bc bc bcb bc bc bc bc bc ITUB4ITSA4bc bc bcbc bc JBSS3TAMM4BVMF3bcb bc bcb bc bcbBBAS3bcb ITUB3bcb bc bc bc BBDC3bc bcb FHER3bc bc bc bc bcb bc bc bc bc INPR3 bc bc bc bc bc bc bcb bc bc bc bc OGXP3 bcb bc bc bc LAME4bc bc bcb bc GOLL4 bc bc RENT3BISA3bcb bc bc bc bc bcb bcb bc bc bc bc bc bc bc bc bc bc bc GFSA3bc CYRE3bcb bcb bc bc bc RSID3 bcb bc PDGR3MRVE3 bcb bcb Figure 33. Core (k = 30) of the asset graph.

There is a strong dependence of k-shell position and node degree, as can be seen in Figure 34. With the exception of the innermost shell, there is an almost linear relation between the two measures.

Node degree

bb b 100 b b b

b b 80 b b bb b bbb b b bb 60 b b bb bb b b b b b b b b b b b 40 b b b b b b b b b b bbb b b b b b b b bbb b b b b bb b b b b b 20 b bb b b b b b bb b b bb bbb b b b bb b bb b b bbb bbbbbbb bbbbbbbbb bbbbbbbbbbb k-shell 10 20 30 Figure 34. Node degree as a function of k-shell number.

8 Conclusion

Using the concepts of minimum spanning trees and asset graphs, both based on the correlation matrix of log-returns of the BM&F-Bovespa for the year 2010, maps of that stock exchange were built, and the network structure so obtained was examined. One could see that that the BM&F-Bovespa has a clustered structure roughly based on economic activity sectors, and gravitates around some key stocks. The results may be used in order to obtain a better knowledge of the structure of this particular stock exchange, and maybe devise diversification strategies for portfolios of stocks belonging to it. There was also the opportunity to compare two different representations of the same complex system, each one with its benefits and maladies.

Acknowledgements

I thank for the support of this work by a grant from Insper, Instituto de Ensino e Pesquisa. I am also grateful to Leonidas Sandoval Neto (my father and Economist), who helped me clarify some details, and to Gustavo Curi Amarante, who collected the data. This article was written using LATEX, all ﬁgures were made

21 using PSTricks, and the calculations were made using Matlab, Ucinet and Excel. All data are freely available upon request on [email protected].

A Stocks, codes, and sectors

Here I display, in alphabetical order, the stocks that are being used in the present work, together with the companies they represent and the sectors those belong to. They are not all the stocks that are negotiated in the BM&F-Bovespa, for only the ones that were negotiated every day the stock market opened are being considered. So, all the stocks have high liquidity and there is no missing data in the time series being used.

Code Company Sector ABCB4 Banco ABC Banking AEDU3 Anhanguera Educacional Education ALLL3 América Latina Log´ıstica Logistics and transportation - railways ALP A4 Alpargatas Consumer goods - clothing and footwear AMAR3 Lojas Marisa Consumer goods - clothing and footwear AMBV 3 AMBEV Consumer goods - beverages AMBV 4 AMBEV Consumer goods - beverages AMIL3 Amil Participa¸cões Health Care BBAS3 Banco do Brasil Banking BBDC3 Bradesco Banking BBDC4 Bradesco Banking BBRK3 Brasil Brokers Construction, materials, and engineering BEEF 3 Minerva Consumer goods - food BEMA3 Bematech Technology, internet, and call-centers BICB4 Bicbanco Banking BISA3 Brookfield Incorpora¸cões Construction, materials, and engineering BVMF 3 BM&F-BOVESPA Financial BPNM4 Banco Panamericano Banking BRAP 4 Bradespar Investments BRFS3 Brasil Foods Consumer goods - food BRKM5 Braskem Petrochemistry BRML3 BR Malls Retail - real states BRSR6 BANRISUL Banking BRTO3 Brasil Telecom Fixed line telecommunications BRTO4 Brasil Telecom Fixed line telecommunications BTOW 3 B2W - Companhia Global do Varejo Consumer goods - electronics BTTL4 Battistella Adm Investments BVMF 3 BM&F-BOVESPA Financial CARD3 CSU Cardsystem Services - diverse CCIM3 Camargo Correa Construction, materials, and engineering CCRO3 CCR Logistics and transportation CESP 6 CESP Electric energy CIEL3 Cielo Financial CLSC6 Celesc Electric energy CMIG3 CEMIG Electric energy CMIG4 CEMIG Electric energy CNFB4 Confab Metalurgy COCE5 Coelce Electric energy CPFE3 CPFL Electric energy CPLE6 COPEL Electric energy CRDE3 CR2 Empreendimentos Imobiliários Construction, materials, and engineering CREM3 Cremer Health Care CRUZ3 Souza Cruz Consumer goods - Tobacco CSAN3 COSAN Sugar an etanol CSMG3 COPASA Sanitation CSNA3 Cia Siderúrgica Nacional Metalurgy

22 Code Company Sector CTAX4 Contax Technology, internet, and call-centers CTIP 3 CETIP Financial CY RE3 Cyrela Brazil Construction, materials, and engineering CZLT 11 COSAN Consumer goods - food DASA3 DASA - Diagnósticos da América Health Care DROG3 Drogasil Consumer goods - pharmaceuticals DTEX3 Duratex Construction, materials, and engineering ECOD3 Brasil Ecodiesel Oil, gas, and biofuels ELET 3 Eletrobras Electric energy ELET 6 Eletrobras Electric energy ELPL4 Eletropaulo Electric energy EMBR3 Embraer Air space company ENBR3 EDP - Energias do Brasil Electric energy EQTL3 Equatorial Electric energy ESTR4 Estrela Consumer goods - toys ETER3 Eternit Construction, materials, and engineering EUCA4 Eucatex Wood and paper EVEN3 Even Construtora e Incorporadora Construction, materials, and engineering EZTC3 EZTEC Empreendiments e Participa¸cões Construction, materials, and engineering FESA4 FERBASA Metalurgy FFTL4 Valefert Agribusiness FHER3 Fertilizantes Heringer Agribusiness FIBR3 Fibria Celulose Paper and celulosis FJTA4 Forjas Taurus Machinery and equipament FLRY 3 Fleury Health Care FRAS4 Fras-Le Transport material GET I3 AES Tietê Electric energy GET I4 AES Tietê Electric energy GF SA3 Gafisa Construction, materials, and engineering GGBR3 Gerdau Metalurgy GGBR4 Gerdau Metalurgy GOAU4 Metalúrgica Gerdau Metalurgy - metalurgia GOLL4 GOL Logistics and transportation - airline GP CP 3 GPC Participa¸cões Petrochemistry GP IV 11 GP Investments Investments GRND3 Grendene Consumer goods - clothing and footwear GSHP 3 General Shopping Brasil Retail HAGA4 Haga Materiais de constru¸cão HBOR3 Helbor Construction, materials, and engineering HGTX3 Hering Consumer goods - clothing and footwear HYPE3 Hypermarcas Consumer goods IDNT 3 Ideiasnet Technology, internet, and call-centers IENG5 INEPAR Construction, materials, and engineering IGBR3 IGB Electronics Electronics IGT A3 Iguatemi Retail INEP 3 Inepar Indústria e Constru¸cões Machinery and equipament INEP 4 Inepar Indústria e Constru¸cões Machinery and equipament INET 3 INEPAR Telecomunica¸cões Telecomunica¸cões INPR3 Inpar Construction, materials, and engineering ITSA4 Itausa Banking ITUB3 Itau Unibanco Banking ITUB4 Itau Unibanco Banking JBDU4 JB Duarte Investments JBSS3 JBS Consumer goods - food JFEN3 João Fortes Engenharia Construction, materials, and engineering JHSF 3 JHSP Participa¸cões Construction, materials, and engineering KEPL3 Kepler Weber Machinery and equipament

23 Code Company Sector KLBN4 Klabin Paper and celulosis KROT 11 Kroton Educacional Education LAME3 Lojas Americanas Consumer goods - retail LAME4 Lojas Americanas Consumer goods - retail LIGT 3 Light Electric energy LLXL3 LLX Log´ıstica Logistics and transportation LOGN3 Log-in Log´ıstica Intermodal Logistics and transportation LPSB3 Lopes Brasil Construction, materials, and engineering LREN3 Lojas Renner Consumer goods - clothing LUPA3 Lupatech Machinery and equipament MAGG3 Magnesita Refratários Metalurgy MILK11 LAEP Investments Consumer goods - food MLFT 4 Jereissati Participa¸cões Fixed line telecommunications MMXM3 MMX Minera¸cão e Metálicos Minera¸cão MNDL4 Mundial Consumo - eletrodomésticos MNPR3 Minupar Consumer goods - food MPXE3 MPX Energia Electric energy MRFG3 MARFRIG Consumer goods - food MRVE3 MRV Engenharia e Participa¸cões Construction, materials, and engineering MTIG4 Metalgráfica Igua¸cu Consumer goods - wraping materials MULT 3 Multiplan Retail MYPK3 IOCHPE Maxion Transportation material NATU3 Natura Consumer goods - cosmetics NETC4 NET Telecommunications ODPV 3 Odontoprev Health Care OGXP 3 OGX Petróleo e Gás Oil, gas, and biofuels OHLB3 Obrascon Huarte Lain Brasil Logistics and transportation P CAR5 Pão de A¸cucar Consumer goods - retail P DGR3 PDG Realty Construction, materials, and engineering PETR3 Petrobras Oil, gas, and biofuels PETR4 Petrobras Oil, gas, and biofuels PFRM3 Proframa Consumer goods - pharmaceuticals PINE4 PINE Banking PLAS3 Plascar Transport material PMAM3 Paranapanema Transport material POMO4 Marcopolo Transport material POSI3 Positivo Informática Technology, internet, and call-centers PRVI3 Providência Materials - diverse PSSA3 Porto Seguro Insurance RAPT 4 Randon Transportation material RDCD3 Redecard Financial RENT 3 Localiza Rent a Car Logistics and transportation - rental RNAR3 Renar Macas Consumer goods - food RPMG3 Refinaria de Petróleos Manguinhos Oil, gas, and biofuels RPMG4 Refinaria de Petróleos Manguinhos Oil, gas, and biofuels RSID3 Rossi Residencial Construction, materials, and engineering SANB11 Banco Santander Banking SANB3 Santander Banking SANB4 Santander Banking SBSP 3 Sabesp Sanitation SFSA4 Sofisa Banking SGPS3 Spring Global Consumer goods - clothing and footwear SLCE3 SLC Agr´ıcola Agribusiness SLED4 Saraiva Consumer goods - books SMTO3 São Martinho Sugar and etanol SULA11 Sul América Insurance SUZB5 Suzano Paper and celulosis Paper and celulosis

24 Code Company Sector TAMM4 TAM Logistics and transportation - airline TBLE3 Tractebel Electric energy TCNO3 Tecnosolo Engenharia Construction, materials, and engineering TCSA3 Tecnisa Construction, materials, and engineering TCSL3 TIM Mobile telecommunications TCSL4 TIM Mobile telecommunications TEKA4 TEKA Tecelagem TELB3 Telebras Fixed line telecommunications TELB4 Telebras Fixed line telecommunications TEMP 3 Tempo Health Care T GMA3 Tegma Gestão Log´ıstica Logistics and transportation TLPP 3 Telesp Fixed line telecommunications TLPP 4 Telesp Fixed line telecommunications T MAR5 Telemar Norte Leste Fixed line telecommunications TNLP 3 TELEMAR Fixed line telecommunications TNLP 4 TELEMAR Fixed line telecommunications TOTS3 Totvs Technology, internet, and call-centers TOYB3 TEC TOY Consumer goods - toys TOYB4 TEC TOY Consumer goods - toys TPIS3 Triunfo Logistics and transportation TRPL4 CTEEP Electric energy UGP A4 Ultrapar Investments UNIP 6 Unipar Petrochemistry UOLL4 UOL Technology, internet, and call-centers USIM3 Usiminas Metalurgy USIM5 Usiminas Metalurgy V ALE3 Vale Minera¸cão V ALE5 Vale Minera¸cão VIVO4 VIVO Mobile telecommunications VLID3 Valid Services - diverse W EGE3 WEG Machinery and equipament

Table 3. Stocks, companies, and sectors to which they belong.

B Sector distribution

In the following ﬁgures, I present the distribution of stocks according to sectors in three dimensional maps. One may see that stocks from the same sectors tend to aglommerate.

BTTL4 bc b bc bc bc bc bc SANB4 bc b bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc BRAP4 bc bc bc bc JBDU4b bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc PINE4bc bc bc bc bc bc bc bc b bc SANB3 bcbc bc bc bc b bc bc bc UGPA4 RDCD3 bc bc bc b bc b bc bc bc GPIV11bc bc bc bc b SANB11bc bc bc bc BICB4bc bc b BBDC4 b bc bc bc b bc bc bc bc bc bc CIEL3 bc bc bc bc ITUB4ITSA4b bc bcbc bc BVMF3b bc b bc bcBBAS3bc ITUB3b bc bc bc BBDC3b b SFSA4bc bc b CTIP3bc bc bc bc bc b bc bc b bc bc ABCB4bc bc bc bc bc b bc bc bc bc bc bc bc bc bc BRSR6bc bc bc bc bc b bc bc bc bc BPNM4bc bc b bc bc bc bc bc bc bc bc bc bc bc bc

bc bc bc bc

Figure B1. Banking (red), ﬁnancial (blue), and holdings (black) sectors.

25 bc bc bc bc bc VALE5bc bc VALE3b PETR3PETR4 bc USIM3b bb RPMG3 CSNA3USIM5b bc bc b bc bc bc b bc GGBR4GOAU4 bc bc bc GGBR3b b RPMG4 bc b b UNIP6bc bc bc bc b bc bc bc bc bc bc bc bc bc GPCP3 bc bc b bc bc bc bc bc bc bc bc bc bc bc ECOD3 bc bc bc bc b bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc CNFB4bc bc bc bc bc bc b MAGG3bc bc bc b bc bc bc bc bc bc bc bc MMXM3 bc bc bc bc b bc bc bc FESA4 bc bc bc bc b bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc BRKM5bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc b bc bc OGXP3 b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc

bc bc bc bc

Figure B2. Mining (red), ironworks (blue), and petroleum (black) sectors.

bc bc bc bc bc bc bc bc bc bc bcbc LIGT3 CMIG4 bc b CMIG3 b bc bc bc bc bc bc b bc bc bc bc bc bc ELET6 bc bc bc b bc bc bc CSMG3bc CPFE3 bc bc bc bc b bc b bc bc bc bc bc bc bc CPLE6 bc bc bc bc SBSP3 b bc bc TBLE3GETI4ELPL4ELET3bc b bc bc bc bc bc b b b b bc bc bcbc bc bc bc GETI3bc bc bc b bc bc bc bc bc bc bc bc bc ENBR3bc TRPL4bc bc bc bc b b bc bc bc bc bc COCE5bc bc bc bc bc b bc bc bc bc bc bc CLSC6bc bc bc CESP6bc bc bbc bc bc b bc bc KROT11 bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc EQTL3 bc b bc bc bc bc bc bc bc MPXE3bc bc bc b bc bc bc bc bc bc bc AEDU3 bc bc b bc bc bc bc bc bc bc bc bc bc bc

bc bc bc bc

Figure B3. Electricity (red), sanitation (blue), and education (black) sectors.

bc INET3bc bc b IENG5 TNLP4 bc bc b bc bc bc bcbc bc bc bc UOLL4bc bc bc bc bc bc bc b bc bc bc bc bc TELB4 bc BEMA3bc bc bc b bc b bc bc bc bc bc bc NETC4bc bc bc b TLPP4TNLP3 bc bc bc b b bc bc bc bc VIVO4 bc TELB3 bc bc b bc b bc bc bc bc bc bc TCSL3TCSL4bc bc TMAR5 bc MLFT4bc bb b bc b bc bc bc BRTO3bc BRTO4 bc bc b bc b bc bc bc bc bc bc CTAX4bc bc bc bc bc bc bc bc b bc bc bc bc bc bc bc bc TLPP3 bc bc bc b bc IDNT3bc bc bc bc bc bcbc b bc bc bc bc bc bc bc bc bc bc bc bc TOTS3bc POSI3bc bc bc bc bc b bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc

bc bc bc bc

Figure B4. Telecommunications (red), cable TV (blue), and technology, internet, and call-centers (black) sectors.

26 bc bc bc bc bc bc bc bc bc bc bcbc SLCE3 bc bc bc bc b bc bc bc bc bc bc bc bc bc bc BEEF3 bc MILK11bc bc b bc bc bc bc bc b bc bc bc bc bc bc bc bc RNAR3bc bc bc bc bc MNPR3b bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc FFTL4bc BRFS3bc bc bc bc bc b b bc bc bc bc bc bc bc bc bc bc MRFG3 bc bc bc b bc bc bc CZLT11bc bcSMTO3bc b bc bc b bc bc bc bc bcbc bc JBSS3 bc bc bc bc b bc bc bc bc bc bc bc FHER3bc bc bc bc b bc bc bc bc CSAN3bc bc bc bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc

bc bc bc bc

Figure B5. Food (red), sugar and ethanol (blue), and agribusiness (black) sectors.

bc bc bc bc IENG5 b bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc TCNO3 bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc HAGA4 bc bc bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc JFEN3 bc bc bc bc b bc bc bc GSHP3ETER3SULA11bc bc bc b bc b bc b bc INPR3CRDE3bc bc bc PSSA3bc bc bc b b bc b bc bc bc TCSA3bc LPSB3 b bc b IGTA3 bc bc bc BBRK3b BISA3bc bc bc b bc bc b EZTC3 bc CCIM3b bc b bc MULT3 JHSF3bc HBOR3 b b bc b BRML3 DTEX3 b GFSA3b CYRE3b b bc bc bc RSID3 EVEN3 b b PDGR3MRVE3 b b

Figure B6. Construction, materials, and engineering (red), real state (blue), and insurance (black).

IGBR3 b TEKA4 bc b bc KLBN4 bc b bc bc bc bc bcbc ESTR4 SUZB5bc bc bc bc bc b bc bc b bc bc SGPS3bc bc bc b bc bc b bc bc bc bc bc bc FIBR3 PFRM3bc bc bc bc bc b bc b bc bc bc MNDL4bc bc bc bc b bc bc bc TOYB4bc bc bc bc bc b MTIG4 bc bc bc bc bc bc bc bc b BTOW3 bcbc bc bc bc bc b SLED4 bc bc b bc bc NATU3bc bc bc bc bc bc CRUZ3bc b bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc ALPA4 bc bc b bc bc TOYB3 bc bc bc bc bc bc bcbc bc bc bc bc b bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc LAME3 bc bc b AMBV3 bc LAME4AMBV4b AMAR3 bc b b EUCA4b bc bc b bc bc bc LREN3bc bc PCAR5 b bc b bc bc bc bc bc bc bc bc bc bc bc b bc bc

bc bc bc bc

Figure B7. Consumer goods (red), electronics (bue), and wood and paper (black).

27 bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc PMAM3bc bc bc PRVI3bc bc bc bc b b bc bc bc bc TPIS3bc bc bc b bc PLAS3 bc bc bc bc b bc bc bc bc ALLL3EMBR3 bc bc bc bc bc b b bc bc bc bc OHLB3bc bc bcbc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc TGMA3 bc bc bc bc bc b b bc bc LLXL3 bc bc FRAS4b bc bc bc bc b bc bc CCRO3bc bc bc bc bc TAMM4b bc bc bcbc bc bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc MYPK3bc bc bc bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc GOLL4 bc bc RENT3 b bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc POMO4 bc bc b RAPT4 b

bc bc bc bc

Figure B8. Logistics and transportation (red), materials for transport and general (blue), airplane industry (black).

bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc KEPL3bc bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc WEGE3bc bc bc bc b bc bc bc bcINEP4bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc LUPA3bc bc bc bc INEP3b bc VLID3 bc b bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc FJTA4bc bc bc bc bc b bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc CARD3bc bc bc bc bc bc bc b bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc

bc bc bc bc

Figure B9. Equipments and machinery (red), general services for payment (blue).

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