Equity Importance Modeling With Financial Network and Betweenness Centrality Zhao Zhao 1 Guanhong Pei 1 Fei Huang 1 Xiaomo Liu 2 1 NDSSL,Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA, U.S.A. 2 Computer Science Department, Virginia Tech, Blacksburg, VA, U.S.A. 1 {zhaozhao, somehi, huangf}@vbi.vt.edu 2 xiaomliu@vt.edu Abstract Financial market has been investigated from many perspectives. The recent emerging financial network methods model the market as a network. The edge between two vertices, or two equities are modeled as the correlation coefficient of the return on the prices of two equities in a period. A common question one can pose is that how can we determine the importance of an equity in the market. From a financial perspective, market-cap is an indicator of importance. But what is its correspondence from a financial network point-of-view? Degree distribution is an intuitive answer. However, after investigating on the betweenness centrality, which is another importance measurement of the graph, we found a high correlation between the betweenness centrality and market-cap. Degree of a vertex, in our case, denotes the number of equities that are highly correlated to it, which shows its local importance and lacks the capability to reflect its importance from a market-wide view. However, betweenness centrality of a vertex encodes global information. It measures the level to which a vertex is need by others along shortest path. In the experiment, we build a financial network using 473 stocks out of the SP-500 pool, during a one year period from June, 2008 to June, 2009. Betweenness centrality is calculated based on the known fasted Brandes algorithm. It is found that the average market-cap of the 20 stocks with largest degree is 29.0 billion dollars, slightly larger than that of the SP-500 stocks, which is 23.5 billion dollars. However, the average market-cap of the stocks with 20 largest betweenness is 50.5 billion dollars, more than twice of the SP- 500 s average market-cap. The market-cap-betweenness plots also shows an upward tendency, meaning that betweenness values are positive correlated with the market-caps, while the market-cap-degree plots doesn t show such phenomenon. Keywords: financial network, betweenness centrality JEL codes: G10 1
1 Introduction Financial market has been investigated from many perspectives. The recent emerging financial network methods model the market as a network, or a graph G(V, E), in which each vertex v V represents an equity and each edge e E represents the relation between two equities [8][4][5]. The edge between two vertices, or two equities are modeled as the correlation coefficient of the return on the prices of two equities in a given period. Given a predefined threshold, two vertices that has the correlation coefficient larger than that are assigned an edge between them. Financial network model provides an approach for people to investigate an equity in the background of the whole market. Several applications involved with data mining are carried out on such financial network model. In [7], the degree distribution of the financial network is investigated which was proven to satisfy the power-law. Clusters, which represents the sectors in the market[5], or independent set, which denotes the possibility of the portfolio diversification[7], are also introduced. An concern involving financial network is that, if an equity has some characteristics in the finance point-of-view, such as major influence on the market, lower Beta, higher Sharpe Ratio, etc., does it have some correspondingly special structure in the financial network? The current financial network researches have rarely touched this concern. One major challenge is that it is hard to define what kind of the structure information can be explored in terms of financial perspective. Also most graph measurement calculation involves vast computation, which is not easy to implement. Despite of the challenges, there are still some market characteristics which can be explored from financial network. In this paper, we explore what equity importance means in the language of financial network. In [6], the author considers degree (the number of vertices connected to a vertex) of a vertex in the financial network to be a quantity that can measure an equity s importance or influence over the market, for the reason that large-degree vertex v has more links in the market graph, which shows that a large number of vertices are highly correlated with it. However, degree only measures the direct influence of a vertex v to its neighbors. It does not take into account that how v influences those vertices that are not connected to it, which is a majority in the market. In this paper, we investigate on another measurement, centrality [1][3], to show that it is a better indicator of equity importance than degree. In the work, we use market-cap of an equity as the benchmark of the importance of a stock in the market, from a finance perspective. The paper is organized as follows: Section 2 defines the model of our financial network. Section 3 introduce the betweenness centrality on a financial network. In section 4, through experiments, we show that the market-cap has higher correlation with betweenness centrality than degree. Section 5 concludes this paper. 2
2 Financial Network Model The financial network is built based on correlations among stocks returns. Here we are going to build a financial network G(V, E) on N stocks. The price time series of stock i is denoted as P(i) =< P(i, 1), P(i, 2),..., P(i, T) >. Here T is the selected time window of the price time series. In modern equity analysis framework, return, rather than absolute price, are often used to analyze equity or portfolio historic performance. Therefore, we use returns time series to calculate the correlations. The return of a stock i at moment t is denoted as R(i, t) = P(i,t) P(i,t 1) P(i,t 1), and the time series is R(i) =< R(i, 1), R(i, 2),..., R(i, T) >. The correlation coefficient of stock i and j is written as: ρ ij = R(i, t)r(j, t) R(i, t) R(j, t) ((R(i, t) 2 R(i, t) 2 )(R(j, t) 2 R(j, t) 2 )) 1/2 (1) A higher correlation ρ between two stocks means that the return of the two stocks has larger tendency to move in the same direction. It is very costly if we build the financial network on all the pairs of correlations, since for that we will have N(N 1)/2 edges, in which each links a pair of stocks. To simplify the financial network, we define a threshold γ. Only the edge of the pair of stocks whose correlation coefficient ρ ij > γ will be remained. Also, in the generated financial network, the values of correlation coefficient among stocks are no longer considered, or, the network is unweighted. All the edges are equally weighted 1. Figure 1 shows a 5-node financial network before and after edge removing, with predefined threshold γ = 0.6. a a b 0.75 0.36 0.65 c b c 0.4 0.33 0.81 0.84 0.61 0.5 e 0.72 d e d Figure 1: Example of financial network after removing the edges with the correlation coefficient less than threshold γ = 0.6 3
3 Degree and Betweenness Centrality in a Network We defined the neighbors of vertex v as N(v). The Degree of a vertex v in a financial network G(V, E) is defined as the number of its neighbors, or N(v). For example, in the right graph of Fig. 1, vertex a has two vertices b and c connected to it, so the degree of a is 2. To reach the calculation of betweenness centrality, we need first introduce the concept of shortest path. A shortest path between a pair of vertices s and t is the path that the sum of its constituent edges is minimized. For example, in the right graph of Fig. 1, (b, d), (b, e, d) and (b, a, c, e, d) are all paths from vertex b to d. However, since the length of (b, d) is 1, which is the minimum among all three paths, it is the shortest path. The betweenness centrality C B (v) of vertex v, is defined as following: Calculate σ(s, t v), the number of shortest path connecting any pairs of vertices s and t, which passes v, Divide σ(s, t v) by σ(s, t), the number of shortest path between s and t. Sum up the quantities obtained in step 2 for all pairs of vertices in the graph to acquire betweenness of v, C B (v). The step is also shown in Eq. 2. Betweenness centrality measures the level to which a vertex is needed by others along shortest path, and can be considered as an indicator of the importance of a vertex. C B (v) = s =v t σ(s, t v) σ(s, t) We use the known fastest Brandes algorithm to calculate the betweenness centrality of each vertex[2]. 4 Experiments In the experiment, based on correlation coefficient of the return on the price among stocks, we implemented a financial network generator to produce the network from public date-sorted data[9]. The financial network is generated using 473 stocks out of the SP-500 pool, during a one year period from June, 2008 to June, 2009. Threshold is set to be 0.8. Fig. 2 and Fig. 3 shows the distribution of the degree and betweenness centrality of the financial network, respectively. It shows that most of the vertices have relative low degree and betweenness, and both distributions present power-law. It is found that the average market-cap of the 20 stocks with largest degree is 29.0 billion dollars, slightly larger than that of the SP-500 stocks, which is 23.5 billion dollars. However, the average market-cap of the stocks with 20 largest (2) 4
1000 # of nodes 100 10 1 degree Figure 2: Degree distribution of the financial network. 1000 # of nodes 100 10 1 betweenness Figure 3: Betweenness distribution of the financial network. 5
betweenness is 50.5 billion dollars, more than twice of the SP-500 s average market-cap. Fig. 5 is the plot for betweenness VS. market cap, which shows an upward tendency, meaning that betweenness values are positive correlated with the market-caps of the stocks. Fig. 4 is the plot for degree VS. market cap, which doesn t show such phenomenon. 140000 120000 100000 market cap 80000 60000 40000 20000 0 degree Figure 4: Degree VS. market cap market cap 200000 180000 160000 140000 120000 100000 80000 60000 40000 20000 0 betweenness Figure 5: Betweenness VS. market cap 6
5 Conclusion This paper proposed an equity importance measurement based on the betweenness centrality of the financial network. Comparing with traditional degree measurement, betweenness centrality, due to its encoding of the global information of the network, can better represent the equity importance. Experiments validate the positive relation between betweenness and market cap of a stock, and compare with vertex s degree. References [1] J.M. Anthonisse. The rush in a directed graph. Technical Report BN 9/71, Stichting Mathematisch Centrum, 1971. [2] U Brandes. A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 2001. [3] L.C. Freeman. A set of measures of centrality based upon betweeness. Sociometry, 1977. [4] K. Kaski J. Kertesz J.-P. Onnela, A. Chakraborti and A. Kanto. Asset trees and asset graphs in financial markets. Physica Scripta, 2003. [5] F. Brisbois N. Vandewalle and X. Tordoir. Non-random topology of stock markets. Quantitative Finance, 2001. [6] Sergiy Butenko Vladimir Boginski and Panos M. Pardalos. On structural properties of the market graph. Innovations in financial and economic networks, 2001. [7] Sergiy Butenko Vladimir Boginski and Panos M. Pardalos. Statistical analysis of financial networks. Computational Statistics and Data Analysis, 2005. [8] Sergiy Butenko Vladimir Boginski and Panos M. Pardalos. Mining market data: A network approach. Computers and Operations Research, 2006. [9] Zhao Zhao. Financial network generator (open source project). http://code.google.com/p/financial-network-generator/. 7