Networks of Companies from StockPrice Correlations

J. Kertész1,2, L. Kullmann1, J.-P. Onnela2, A. Chakraborti2, K. Kaski2,

A. Kanto3

1Department of Theoretical PhysicsBudapest University of Technology and Economics, Hungary

2Laboratory of Computational EngineeringHelsinki University of Technology, Finland

3Dept of Quantitative Methods in Economics and Management ScienceHelsinki 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 understandingand 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:

PGE(t)

Daily logarithmic price:

lnPGE(t)

Daily logarithmic return:

rGE(t)=lnPGE(t) – lnPGE(t-1)

Data: price and return

For each window a correlation matrix is defined with

elements being the equal time correlation coefficients:

where ri ,rj Rt, .. 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

11 where,2222

t

ij

jjii

jijitij

rrrr

rrrr

02 where,)1(2

tij

NN

ttij

tij dd D

NN

t

C

TN

t

R

Central vertex

To characterise positions of companies in the tree theconcept 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 vi giving minimum value for mean occupation layer (l(t,vi)); 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 clustersBusiness sectors (Forbes)

Yahoo data

Potts superparamagnetic clustering

Kullmann, JK, Mantegna

Antiferromagneticbonds

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 onthe asset tree, concept of mean occupation layeris 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 beused: exhibit similar behaviour Robustness

N

i

tic v

Nvtl

1

)lev(1

),(

Asset tree: topology change

Normal market topology crash topology

Yahoodata

Robustness of dynamic asset tree topology measured asthe ratio of surviving connections when moving by one step:

Single-step survival ratio:

Robustness: single-step survival

1

1

1

ttt EE

N

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) t1/2 (y)

2 0.22 4 0.46 6 0.75

t1/2=0.12T

Half life vs. window width

Connections survived vs. time

Power lawdecay: z ≈1.2

ktttktt EEE

N

1, 1

1

k=0

k=1

k=2

k=4

k=6

k=12

k=24

k=36

k=48

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 data1.01.21.08.1

2.2

Distribution of vertex degrees

• L: normal• R: crash

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

1

min

1

1

1,

N

ii

N

iii

jij

N

jii

w

rwr

rrww

and

given

),Cov(2

1

0 iw :selling-short No

Weighted portfolio layerHow 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).

Portfolio layer

No short-selling Short-selling

portfolio layer mean occupation layer

Static c.v. Static c.v.

Dynamic 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 FCGTree 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 thedata)

Ci = # of -s / [k(k-1) / 2] where k is the degree of node i

size= 0

size= 10

size= 20

size= 30

size= 40

size= 50

size= 60

size= 70

size= 80

size= 90

size= 100

size= 120

size= 140

size= 160

size= 180

size= 200

size= 300

size= 400

size= 500

size= 600

size= 700

size= 800

size= 900

size=1000

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

empirical random

N=477

Spanned graph order

empirical random

N=477

Number of vertex clusters

empirical random

N=477

Cluster size for edge clusters

empirical random

N=477

Vertex degree distribution

empirical random

p=0.01N=477

Vertex degree distribution

empirical random

p=0.25N=477

Clustering coefficient

empirical random

N=116

Mean clustering coefficient

empirical random

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 tt

Correlation fn between returns of company A and BIt 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 onesOnly 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, 10000 companies tick by tick54 days: 195 companies traded more than 15000 timest = 100s but results checked for 50-500s.

Results

• We measure max, C(max), and R = C(max)/noise• Consider Imax 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 Cij noisy, much information content, useful forportfolio 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