Networks of Companies from Stock Price Correlations

1. Networks of Companies from StockPrice Correlations J. Kertész1,2,L. Kullmann1, J.-P. Onnela2, A. Chakraborti2,K. Kaski2, A. Kanto3 1Department of Theoretical Physics Budapest University of Technology and Economics, Hungary 2Laboratory of Computational Engineering Helsinki University of Technology, Finland 3Dept of Quantitative Methods in Economics and Management Science Helsinki School of Economics, Finland

2. Motivation • Financial market is a self-adaptive complex system; many interacting units, obvious networking. Networks: • Cooperation Most important and most difficult • Activity, ownership • Similarity • Temporal aspects • Networks generated by timedependencies • Time dependent networks • Revealing NW structure is crucial for understanding and also for pragmatic reasons (e.g., portfolio opt.) Many groups active: Palermo, Rome, Seoul etc.

3. Outline • Classification by Minimum Spanning Trees (MST) (Mantegna) • Temporal evolution • Relation to portfolio optimization • Correlations vs. noise: Parametric aggregational classification • Temporal correlations: Directed NW of influence

4. Data: price and return • Daily price data for N=477 of NYSE stocks (CRSP of U. of Chicago), such as GE, MOT, and KO • Time span S=5056 trading days: Jan 1980 – Dec 1999 • Daily closure price of GE: • PGE(t) • Daily logarithmic price: • lnPGE(t) • Daily logarithmic return: • rGE(t)=lnPGE(t)– lnPGE(t-1)

5. Correlations and distances For each window a correlation matrix is definedwith elements being the equal time correlation coefficients: where ri ,rjRt, ..denotes time average. Transformationto distance-matrix with elements: Minimum spanning tree(MST), which is agraph linkingN vertices (stocks) with N-1 edges such that the sum of distances is minimum. Efficient algorithms.

6. Central vertex • To characterise positions of companies in the tree the • concept of central vertexis introduced: • Reference vertex to measure locations of other vertices, needed to extract further information from asset trees • Central vertex should be a company whose price changes strongly affect the market;three possible criteria: • (1) Vertex degree criterion: vertex with the highest vertex degree, i.e., the number of incident edges; Local. • (2) Weighted vertex degree criterion: vertex with the highest correlation coefficient weighted vertex degree; Local. • (3) Center of mass criterion: vertex vigiving minimum value for mean occupation layer (l(t,vi));Global.

7. Central vertex: comparison (1) Vertex degree criterion (local): GE: 67.2% (2) Weighted vertex degree criterion (local): GE: 65.6% (3) Center of mass criterion (global): GE: 52.8%

8. Asset tree and clusters Business sectors (Forbes) Yahoo data

9. Potts superparamagnetic clustering Kullmann, JK, Mantegna Antiferromagnetic bonds

10. Asset tree clustering • Mismatch between tree clusters and business sectors? • Random price fluctuations introduce noise to the system • Business sector definitions vary by institutions (Forbes…) • Historical data should be matched with a contemporary business sector definition • Classifications are ambiguous and less informative for highly diversified companies • MST classification mechanism imposes constraint • Uniformity and strength of correlations vary by business sector (c.f. Energy sector vs. Technology)

11. Mean occupation layer In order to characterise the spread of vertices on the asset tree, concept of mean occupation layer is introduced: where vc is the central vertex, lev(vi) denotes the level of vertex vi, such that lev(vc) = 0. Both static and dynamic central vertex may be used: exhibit similar behaviour  Robustness

12. Asset tree: topology change Normal market topologycrash topology Yahoo data

13. Robustness: single-step survival Robustness of dynamic asset tree topology measured as the ratio of surviving connections when moving by one step: Single-step survival ratio: • T = 4 years, T = 1 month

14. Tree evolution: multi-step survival Connections survived vs. time • Within the first region decay • is exponential • After this there is cross-over • to power law behaviour: •  (t,k) ~ t--z Power law decay:z ≈1.2 Half life vs. window width T (y)t1/2(y) 2 0.22 4 0.46 6 0.75 t1/2=0.12T

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24. Distribution of vertex degrees The topological nature of the network is studied by analysing the distribution of vertex degrees: • Power law distribution would indicate scale-free topology, a feature unexpected by random network models • Vandewalle et al. find for one year data while we found • Power law fit ambiguous due to limited range of data

25. Distribution of vertex degrees • L:normal • R: crash

26. Portfolio optimisation In the Markowitz portfolio optimisation theory risks of financial assets are characterised by standard deviations of average returns of assets: The aim is to optimise the asset weights wiso that the overall portfolio risk is minimized for a given portfolio return (minimum risk portfolio is uniquely defined)

27. Weighted portfolio layer • How are minimum risk portfolio assets located • on graph? • Weighted portfolio layer is defined • by imposing no short-selling, i.e. wi  0, and it is compared with the mean occupation layer l(t).

28. Portfolio layer No short-selling Short-selling Static c.v. Static c.v. Dynamic c.v. Dynamic c.v. portfolio layer mean occupation layer

29. Correlations vs.noise • Correlation matrix contains systematics and noise. • MST: Non-parametric, unique classification scheme, but! • Even for uncorrelated random matrix MST would lead to • classification… • Meaningful clustering and robustness already signalize significance. • Different methods to separate noise from information: • Eigenvalue spectra (Boston, Paris) • Independent/principal component analysis(economists)

30. Here: Building up the FCG • Tree condition may ignore important correlations. • (General classification problem) • Visualization through Parametrized Aggregated Classification(PAC): Add links one by one to the graph, according to their rank, started by the strongestand ended with a Fully Connected Graph (FCG). Strongly correlated parts get early interconnected, clustering coefficient becomes high. • Price time series data for a set of 477 companies. • Window width T=1000 business days (4 years), • located at the beginning of the 1980’s • Comparison with random graph (obtained by shuffling the • data) Ci= # of -s / [k(k-1) / 2] where k is the degree of node i

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