Networks of Companies from Stock Price Correlations J. Kertész 1,2, L. Kullmann 1, J.-P. Onnela 2, A. Chakraborti 2, K. Kaski 2, A. Kanto 3 1 Department of Theoretical Physics Budapest University of Technology and Economics, Hungary 2 Laboratory of Computational Engineering Helsinki University of Technology, Finland 3 Dept of Quantitative Methods in Economics and Management Science Helsinki School of Economics, Finland

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 time dependencies 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.

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

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: P GE (t) Daily logarithmic price: lnP GE (t) Daily logarithmic return: r GE (t)=lnP GE (t) – lnP GE (t-1) Data: price and return

Time series of asset returns Return matrix: Data is divided into M time-windows of width T displaced by  T, thus getting M matrices window width T step length  T time t

For each window a correlation matrix is defined with elements being the equal time correlation coefficients: where r i, r j  R t, ..  denotes time average. Transformation to distance-matrix with elements: Minimum spanning tree (MST), which is a graph linking N vertices (stocks) with N-1 edges such that the sum of distances is minimum. Efficient algorithms. Correlations and distances

Central vertex To characterise positions of companies in the tree the concept of central vertex is 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 v i giving minimum value for mean occupation layer ( l(t,v i ) ); Global.

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%

Asset tree and clusters Business sectors (Forbes) Yahoo data

Potts superparamagnetic clustering Kullmann, JK, Mantegna Antiferromagnetic bonds

Mismatch between tree clusters and business sectors? 1.Random price fluctuations introduce noise to the system 2.Business sector definitions vary by institutions (Forbes…) 3.Historical data should be matched with a contemporary business sector definition 4.Classifications are ambiguous and less informative for highly diversified companies 5.MST classification mechanism imposes constraint 6.Uniformity and strength of correlations vary by business sector (c.f. Energy sector vs. Technology) Asset tree clustering

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

Asset tree: topology change Normal market topology crash topology Yahoo data

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

Tree evolution: multi-step survival Within the first region decay is exponential After this there is cross-over to power law behaviour:  (t,k) ~ t - - z T (y) t 1/2 (y) t 1/2 =0.12 T Half life vs. window width Connections survived vs. time Power law decay: z ≈1.2

Evolution of graphs and trees Overlap of edges in asset graph G t and asset tree T t as a function of time Overlap of edges in asset graph G t and asset tree T t as a function of normalized number of edges, averaged over time

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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

Distribution of vertex degrees L: normal R: crash

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 w i so that the overall portfolio risk is minimized for a given portfolio return (minimum risk portfolio is uniquely defined)

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

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

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 )

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 strongest and 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) C i = # of  -s / [k(k-1) / 2] where k is the degree of node i

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Elementary graph concepts Graph size: number of edges in the graph (variable) Graph order: number of vertices in the graph (constant) Spanned graph order: number of vertices in the subgraph spanned by the edges, thus excluding the isolated vertices (variable) These definitions can be applied also to clusters (two types) (1) edge cluster (2) vertex cluster Edge clusters are more meaningful in the asset graph context

Cluster growth The growth patterns of clusters can be divided into four topologically different types: (I) Create a new cluster (two nodes and the incident edge) when neither of the two end nodes are part of an existing edge cluster (spanned cluster order +2, size +1) (II) Add a node and the incident edge to an already existing edge cluster (spanned cluster order +1, size +1) (III) Merge two edge clusters by adding an edge between them (combined spanned cluster size +1) (IV) Add an edge to an already existing edge cluster, thus creating a cycle in it (spanned cluster size +1)

Cluster growth empiricalrandom N=477

Spanned graph order empiricalrandom N=477

Number of vertex clusters empiricalrandom N=477

Cluster size for edge clusters empiricalrandom N=477

Vertex degree distribution empiricalrandom p=0.01 N=477

Vertex degree distribution empiricalrandom p=0.25 N=477

Clustering coefficient empiricalrandom N=116

Mean clustering coefficient empiricalrandom N=116

NO TIME REVERSAL SYM. ON THE MARKETS Physics close to equilibrium: Time reversal symmetry (TRS)  Detailed balance  Symmetric correlation functions, Fluctuation Dissipation Th. (FDT) No fundamental principle forcing TRS on the market. In contrast: The elementary process, a transaction is irreversible: Though the price is set by equilibrating supply and demand, both parties (or at least one of them ) feel that the transaction is for their advantage and would not agree to revert it. Possibility of Asymmetry in the cross correlation functions Differences between the decay of spontaneous fluctuations and of response to external perturbations

Time dependent cross correlations log return of stock A between t and t  t Correlation fn between returns of company A and B It depends on  t and . Is it symmetric? Difficulties: trade not syncronized, frequencies are very different bad signal/noise ratio Approptiate averaging

Toy model to test the method: Persistent 1d random walk (increment x   1): We take two such walks, which are correlated, with increments x and y The correlation function can be calculated: We corrupt the data to have similar quality to real ones Only 1% of the data are kept. (  o =200,  =1000,  =0.99)

The measured correlations on a finite set of data depends on the averaging procedure (moving average) The appropriate choice is  t min   t   o DATA set: Trade And Quote, companies tick by tick 54 days: 195 companies traded more than times  t = 100s but results checked for s.

Results We measure  max, C(  max ), and R = C(  max )/noise Consider I  max I > 100, C(  max ) > 0.04, and R > 6 as ‘effect’ Not all pairs of comp’s show the effect Peak not only shifted but also asymmetric Large, frequently traded companies ‘pull’ the smaller ones Weak effect and short characteristic time (minutes) XON: Exxon (oil) ESV: Ensco (oil wells)

No chains Many leaders for a follower Many followers for a leader Disconnected graph Directed network of influence

Conclusions Networks constructed from cross correlations of stock price time series (MST, PAC) Though C ij noisy, much information content, useful for portfolio optimization MST robust, reasonable classification, interesting dyn. at crash-time Clusters (branches) not equally correlated, PAC reveals differences, separation of noise from info Asymmetric time dependent cross correlations lead to directed network of influence