A Network Analysis of the Volatility of High

A Network Analysis of the Volatility of
High-Dimensional Financial Series
Matteo Barigozzi
London School of Economics
Marc Hallin
Université libre de Bruxelles
Statistical Network Science and its Applications
Isaac Newton Institute Cambridge, UK
25–26 August 2016
In a nutshell - aim and methods
1
We study the network of S&P100 stocks’ volatilities;
2
for financial data no network structure pre-exists the observations
⇒ we consider Long Run Variance Decomposition Networks;
Diebold and Yılmaz, 2014
3
large dimensional system of time series poses difficulties in estimation
⇒ we use factor models and lasso-type regressions as solutions;
4
financial shocks have an economic meaning
⇒ we identify shocks by means of recursive identification scheme;
5
high connectedness means high uncertainty
⇒ we extract volatilities as measures of fear or lack of confidence.
Introduction
1/43
In a nutshell - results
JPM
APC
OXY
PFE
KO
CAT
BAC
USB
APA
C
WFC
DVN
RTN
AMZN
TXN
COP
BMY
INTC
HAL
XOM
NOV
TGT
T
AIG
FDX
DD
LOW
HD
CVS
ALL
BRK.B
LMT
HPQ
SBUX
MS
NKE
EMC
DIS
MMM
QCOM
CL
WAG
DOW
WMB
PG
UPS
F
AMGN
BK
CMCSA
COF
PEP
SPG
MRK
2000-2013
Introduction
GILD
BA
GD
RTN
MDT
UTX
GS
INTC
C
MS ALL
TWX
TGT
HD
CVS
EXC
BRK.B
AXP
DD
NOV
UPS
DELL
CATMRK
PFE
WAG HAL
QCOM
JNJ
EMR CMCSA
MMM
BMY
LLY
NKE
SO
VZ
COST
WMT
CL
HPQ
DOW
ABT
EMR
BAX
LOW
AEP
AAPL
WFC COF
MCD
F
AAPL
AMZN
BAX
TWX
FDX
WMT ORCL
MCD
UNH
IBM
USB
HON
SPG
WMB
BAC
IBM
FCX
ABT
NSC
GS
DIS
HON
LLY
SO
AEP
UNP
JNJ
CSCO
MDT
VZ
NSC
MSFT
ORCL
KO
JPM
DELL
UNP
GE COST
BA AMGN
AIG
MSFT
MO
APC
SBUX
BK
SLB
CVX
CVX
EBAY EXC
GE
UTX
LMT
MO
GD
OXY
DVN APA
COP SLB
XOM
EMC
TXN
FCXGILD
PEP
CSCO
PG
AXP
T
UNH
EBAY
2007-2008
Great Financial Crisis
2/43
Data
A panel of stock daily returns
r = rit |i = 1, . . . , n, t = 1, . . . , T
from Standard & Poor’s 100 index;
n = 90 assets from 10 sectors: Consumer Discretionary, Consumer Staples,
Energy, Financial, Health Care, Industrials, Information Technology, Materials,
Telecommunication Services, Utilities;
T = 3457 days from 3rd January 2000 to 30th September 2013;
from returns we extract volatilities which are unobserved;
we are in a large n, T setting.
Introduction
3/43
Methods
Long Run Variance Decomposition Network (LVDN)
weighted and directed graph;
the weight associated with edge (i, j) represents the proportion of
h-step ahead forecast error variance of variable i which is accounted
for by the innovations in variable j;
completely characterised by the infinite vector moving average (VMA)
representation given by Wold’s classical representation theorem;
for a generic process Y the model reads
Y t = D(L)e t ,
e t ∼ w .n.(0, I)
where
e t are orthonormal shocks with an economic meaning
D(L) are impulse response functions (IRF) which give the network
Introduction
4/43
Methods
Goal 1 - estimate a VMA:
estimate and invert VAR (classical approach);
in a large dimensional setting ⇒ curse of dimensionality;
two main solutions in time series:
1
factor models;
Forni, Hallin, Lippi, Reichlin, 2000, Bai and Ng, 2002, Stock and Watson, 2002,
Forni, Hallin, Lippi, Zaffaroni, 2015, Barigozzi and Hallin, 2016
2
lasso-type penalised regressions;
Tibshirani, 1996, Zou and Hastie, 2005, Yuan and Lin, 2006, Zou, 2006,
Peng, Wang, Zhou, Zhu, 2009
used to analyse two complementary features of financial markets:
1
effect of global shocks ⇒ pervasive risk, non-diversifiable;
Ross, 1976, Chamberlain and Rothschild, 1983, Fama and French, 1993
2
effect of idiosyncratic shocks ⇒ systemic risk, limited diversifiability;
Gabaix, 2011, Acemoglu, Carvalho, Ozdaglar, Tahbaz-Salehi, 2012
Introduction
5/43
Methods
Goal 2 - identify the shocks:
given a VMA any invertible linear transformation of the shocks is a
statistically valid representation;
to attach an economic meaning to the shocks ⇒ to identify;
assume orthonormality or even independence;
recursive identification schemes, i.e. choose shocks’ ordering.
Introduction
6/43
Generalised Dynamic Factor Model
(GDFM)
Consider a generic n × T panel of time series Yn such that
A1 Yn is strongly stationary;
A2 its spectral density Σn (θ) exists with eigenvalues λj,n (θ);
A3 there exists a q < n not depending on n such that:
a λq,n (θ) → ∞ as n → ∞;
b λq+1,n (θ) < M < ∞ for any n ∈ N.
Factors and Networks
7/43
Generalised Dynamic Factor Model
(GDFM)
Under A1-A3
Ynt = Xnt + Znt = Bn (L)ut + Znt
(1)
i u is q-dimensional and ut ∼ w .n.(0, I)
⇒ global shocks;
ii Bn (L) is n × q polynomials with squared summable coefficients
⇒ IRFs to global shocks;
iii q spectral eigenvalues of Xn diverge as n → ∞;
⇒ strong (auto)correlation among components of Xn ;
iv n spectral eigenvalues of Zn are bounded for any n ∈ N;
⇒ weak, but not zero, (auto)correlation among components of Zn ;
v Xn and Zn are mutually orthogonal at every lead and lag.
Notice that
model and assumptions are defined in the limit n → ∞;
under A1-A2, model (1) and A3 are equivalent.
Factors and Networks
8/43
Idiosyncratic component - VMA
The idiosyncratic component always admits the Wold decomposition
Znt = Dn (L)ent
vi en is n-dimensional and ent ∼ w .n.(0, I)
⇒ idiosyncratic shocks;
vii Dn (L) is n × n polynomials with squared summable coefficients
⇒ IRFs to idiosyncratic shocks.
Factors and Networks
9/43
Shocks
Two sources of variation:
1
few (q) global shocks, u, with pervasive effect
due to condition (iii) of diverging eigenvalues;
2
many (n) idiosyncratic shocks, en , with limited, but not null, effect
due to condition (iv) of bounded eigenvalues
⇒ no sparsity assumption is made.
We first control for the global effects and then we focus on the effect of
idiosyncratic shocks measured through the LVDN.
Factors and Networks
10/43
Long-Run Variance Decomposition Network
(LVDN)
Vertices set V = {1 . . . n};
edges set
ELVDN =
(i, j) ∈ V × V| lim
h→∞
wijh
6= 0
edges weights
Ph−1
wijh = 100
2
k=0 dk,ij
Pn Ph−1 2
`=1
k=0 dk,i`
where dk,ij is entry of Dnk such that Dn (L) =
!
P∞
k=0 Dnk L
k;
wijh is the proportion of h-step ahead forecast error variance of Zi
which is accounted for by the innovations in Zj .
Factors and Networks
11/43
Long-Run Variance Decomposition Network
(LVDN)
The operative definition of LVDN requires to fix h;
the weights are normalised
n
1 X h
wij = 1,
100
j=1
n
1 X h
wij = n;
100
i,j=1
we define the FROM and TO degrees as
δiFROM =
n
X
wijh ,
δjTO =
n
X
j=1
j6=i
wijh ;
i=1
i6=j
a measure of total connectedness is given by
δ TOT =
Factors and Networks
n
n
i=1
j=1
1 X FROM
1 X TO
δi
=
δj .
n
n
12/43
Idiosyncratic component - VAR
To estimate a VMA we assume
A4 Zn has the sparse-VAR(p) representation
Fn (L)Znt = vnt ,
vnt ∼ w .n.(0, C−1
n )
P
where Fn (L) = pk=0 Fnk Lk with Fn0 = I and det(Fn (z)) 6= 0 for any
z ∈ C such that |z| ≤ 1, and Cn has full-rank. Moreover, Fnk and
Cn−1 are sparse matrices.
Notice that
we assume sparsity of VAR and not of VMA for estimation purposes but the argument
of condition (iv) of bounded eigenvalues still holds;
for convenience in the identification step, we parametrise the covariance matrix of the
VAR innovations by means of its inverse Cn
Factors and Networks
13/43
Idiosyncratic component - VAR
As a by-product, we have a Long-Run Granger Causality Network (LGCN)
Edges set
(
)
p
X
ELGCN = (i, j) ∈ V × V|
fkij 6= 0
k=0
it captures the leading/lagging conditional linear dependencies;
Dahlhaus and Eichler, 2003, Eichler, 2007
under A4 the LGCN is likely to be sparse but the LVDN is not
necessarily sparse;
the economic interpretation of the LGCN is not as straightforward as
that of the LVDN, and the LGCN therefore is of lesser interest for the
analysis of financial systems and we consider it just as a tool to derive
the LVDN;
cfr. with traditional macroeconomic analysis where IRF, i.e. VMA
coefficients, rather than VAR ones, are the object of interest for policy
makers.
Factors and Networks
14/43
Identification
From the VAR and VMA of Zn we have
Dn (L) = (Fn (L))−1 Rn
where Rn is such that it makes the shocks R−1
n vn = en orthonormal.
choosing Rn is equivalent to identifying the shocks;
choose Rn as the lower triangular matrix such that
0
Cov(vn ) = C−1
n = Rn Rn
then
0
0
0 −1
−1
= R−1
=I
Cov(en ) = R−1
n Rn Rn Rn
n Cov(vn )Rn
but this choice depends on the ordering of the shocks, a given order of
shocks defines which component we choose to hit first.
Factors and Networks
15/43
Identification
We use vn ’s partial correlation structure
the partial correlation between vi and vj is
−[Cn ]ij
ρij = p
[Cn ]ii [Cn ]jj
associated is the Partial Correlation Network (PCN), with edges set
EPCN = (i, j) ∈ V × V| ρij 6= 0
by A4, the PCN is a sparse network;
Peng, Wang, Zhou, Zhu, 2009
order shocks by decreasing eigenvector centrality in the PCN;
we are considering the case in which the most contemporaneously
interconnected node is firstly affected by an unexpected shock, and
then, by means of the subsequent impulse response analysis, we study
the propagation of such shock through the whole system.
Factors and Networks
16/43
Returns vs. Volatilities
returns are observed but volatilities are unobserved
⇒ factor model for returns to extract volatilities in a multivariate way
Barigozzi and Hallin, 2016
in a univariate setting a generic model for volatility reads as
a(L)rt = ηt ,
ηt ∼ w .n.(0, σ 2 ),
ηt2 = f (ηt−1 . . . η0 )
for example stochastic volatility models where log η 2 is an AR(1)
2
log ηt2 = c + a1 log ηt−1
+ νt .
Estimation
17/43
Returns vs. Volatilities
in the large n case we use an autoregressive representation of the
GDFM
Forni, Hallin, Lippi, Zaffaroni, 2015
An (L)r nt = ηnt + ξnt
ηnt = Hn ut with Hn full-rank and n × q and ut ∼ w .n.(0, I)
⇒ global shocks to returns, that is market shocks;
An (L) is block-diagonal with blocks of size q + 1;
ξn has bounded spectral eigenvalues, is idiosyncratic such that it has
the sparse VAR representation
Fn (L)ξnt = v nt
Estimation
18/43
Returns vs. Volatilities
centred log-volatilities are defined as
σnt = log(ηnt + v nt )2 − E[log(ηnt + v nt )2 ]
we assume a GDFM also for log-volatilities
σnt = χσ,nt + ξσ,nt
Aσ,n (L)χσ,nt = Hn,σ εt
ε is Q-dimensional and εt ∼ w .n.(0, I)
⇒ global shocks to volatilities, that is risk market shocks;
Hn,σ is n × Q and H0n,σ Hn,σ = nI
⇒ global shocks are pervasive;
An (L) is block-diagonal with blocks of size Q + 1;
ξσ,n has bounded spectral eigenvalues, is idiosyncratic such that it has
the sparse VAR representation
Fn (L)ξσ,nt = νnt
Estimation
19/43
Estimation in one slide
GDFM estimation is based on the following “tools” (details omitted):
Spectral density matrix, dynamic PCA, autocovariances by inverse
Fourier transform, Yule Walker equations;
Forni, Hallin, Lippi, Zaffaroni, 2015
run GDFM estimation twice:
Barigozzi and Hallin, 2016
1
2
on returns to obtain shocks u and vn for computing log-volatilities;
on volatilities to obtain the idiosyncratic component ξσ,n ;
estimate a sparse VAR on ξσ,n , some options are
1
2
3
4
Estimation
elastic net Zou and Hastie, 2005
group lasso Yuan and Lin, 2006, Nicholson, Bien, Matteson, 2014, Gelper, Wilms, Croux, 2016
adaptive lasso Zou, 2006, Kock and Callot, 2015, Barigozzi and Brownlees, 2016
other penalties are also possible Hsu, Hung, Chang, 2008, Abegaz and Wit, 2013
20/43
LVDN
The VMA for idiosyncratic volatilities
ξσ,nt = (Fn (L))−1 νnt
= (Fn (L))−1 Rn R−1
n νnt
= Dn (L)ent
ent ∼ w .n.(0, I) by construction
⇒ idiosyncratic shocks to volatilities;
Rn identified using centrality in the PCN of νnt (estimated via lasso);
truncate (Fn (L))−1 Rn at lag h = 20 (one month);
LVDN weights are given by the entries of Dn (L)
⇒ weighted directed non-sparse network;
LVDN can be made sparse by some other form of thresholding.
Estimation
21/43
The effect of global shocks
The VMA for common volatilities
χσ,nt = (An (L))−1 Hσ,n Kεt
= Bn (L)εt
εt ∼ w .n.(0, I) by construction
⇒ global shocks to volatilities;
K for identification (easy since there are few shocks);
truncate (An (L))−1 Hσ,n K at lag h = 20 (one month);
from the entries of Bn (L) we can compute percentages of h-step
ahead forecast error variances due to the global shocks.
Estimation
22/43
Data
Adjusted daily closing prices pit ;
90 daily stock returns from S&P 100 index rit = 100∆ log pit ;
10 sectors: Consumer Discretionary, Consumer Staples, Energy, Financial, Health
Care, Industrials, Information Technology, Materials, Telecommunication Services,
Utilities;
two periods 2000-2013 and 2007-2008;
from returns we extract log-volatilities;
data are not standardised.
Results
23/43
Number of factors
look at the behaviour of the spectral eigenvalues;
Hallin and Liška, 2007
one global shock in returns q = 1;
one global shock in volatilities Q = 1;
in both cases the global shock explains about 40% of total variance
Rπ
λ1,n (θ)dθ
EV = Pn −πR π
' 0.4
i=1 −π λi,n (θ)dθ
the idiosyncratic shocks account for about 60% of total variance.
Results
24/43
Effects of global volatility shocks
Sector
Cons. Disc.
Cons. Stap.
Energy
Financial
Health Care
Industrials
Inf. Tech.
Materials
Telecom. Serv.
Utilities
Total
2000-2013
8.87
10.54
11.61
11.89
9.38
8.50
10.00
8.35
10.07
10.82
100
2007-2008
8.82
10.14
18.44
14.40
8.01
7.97
6.94
9.79
8.04
7.47
100
Percentages of 20-step ahead forecast error variances due to the global shock.
Results
25/43
Sparse VAR for idiosyncratic component
VAR order selected: p = 5 - Elastic net
1
1
45
45
90
1
45
2000-2013
90
90
1
45
90
2007-2008
density = 0.53
density = 0.86
negative weights in blue, positive weights in red.
Results
26/43
Sparse VAR for idiosyncratic component
VAR order selected: p = 5 - Group lasso
1
1
45
45
90
1
45
2000-2013
90
90
1
45
90
2007-2008
density = 0.14
density = 0.32
negative weights in blue, positive weights in red.
Results
27/43
PCN for idiosyncratic innovations
1
1
45
45
90
1
45
2000-2013
90
90
1
45
90
2007-2008
density = 0.06
density = 0.24
negative weights in blue, positive weights in red.
Results
28/43
PCN for idiosyncratic innovations
2000-2013
JPM JP Morgan Chase & Co.
C Citigroup Inc.
BAC Bank of America Corp.
APA Apache Corp.
WFC Wells Fargo
COP Conoco Phillips
OXY Occidental Petroleum Corp.
DVN Devon Energy
SLB Schlumberger
MS Morgan Stanley
2007-2008
BAC Bank of America Corp.
USB US Bancorp
JPM JP Morgan Chase & Co.
MS Morgan Stanley
WFC Wells Fargo
DVN Devon Energy
GS Goldman Sachs
AXP American Express Inc.
COF Capital One Financial Corp.
UNH United Health Group Inc.
Eigenvector centrality in the PCN.
Results
29/43
LVDN
1
1
45
45
90
1
45
90
90
1
45
90
2000-2013
2007-2008
weights below the 95th percentile in grey, between the 95th and 97.5th percentiles in blue,
between the 97.5th and 99th percentiles in yellow, and above the 99th percentile in red.
Results
30/43
LVDN
percentiles
2000-2013
2007-2008
50th
0.02
0.17
90th
0.13
0.71
95th
0.20
1.00
97.5th
0.29
1.28
99th
0.48
1.76
max
4.29
4.53
Selected percentiles of LVDN weights.
Results
31/43
LVDN - sparse
1
1
45
45
90
1
45
90
90
1
45
90
2000-2013
2007-2008
Thresholded LVDN, non-zero weights in red.
Results
32/43
LVDN - sparse
JPM
APC
OXY
PFE
KO
CAT
BAC
USB
APA
C
WFC
DVN
AMZN
TXN
GE
UTX
BMY
INTC
HAL
XOM
NOV
CSCO
MDT
AIG
FDX
DD
LOW
HD
CVS
ALL
BRK.B
LMT
HPQ
SBUX
NKE
EMC
DIS
MMM
QCOM
CL
MS
WAG
IBM
DOW
WMB
PG
UPS
F
AAPL
BK
LOW
AMGN
GILD
GD
COF
PEP
MS ALL
TWX
TGT
HD
BRK.B
DD
VZ
COST
NOV
UPS
DELL
CATMRK
AXP
PFE
WAG HAL
QCOM
DOW
TXN
FCXGILD
MRK
PEP
CSCO
PG
AXP
T
UNH
2000-2013
WMT
CL
HPQ
ABT
JNJ
EMR CMCSA
MMM
BMY
LLY
NKE
SO
SPG
BAX
UTX
GS
INTC
C
CMCSA
EMR
RTN
MDT
CVS
EXC
AEP
BA
AAPL
AMZN
BAX
WFC COF
MCD
F
SPG
WMB
TWX
FDX
WMT ORCL
MCD
UNH
USB
HON
MSFT
ORCL
KO
BAC
IBM
FCX
ABT
NSC
GS
DIS
HON
LLY
SO
AEP
UNP
JNJ
UNP
NSC
BK
MSFT
MO
APC
SBUX
GE COST
BA AMGN
AIG
JPM
DELL
TGT
T
VZ
CVX
EBAY EXC
SLB
CVX
MO
GD
OXY
COP
LMT
EMC
RTN
DVN APA
COP SLB
XOM
EBAY
2007-2008
Thresholded LVDN.
Results
33/43
LVDN - centrality
2000-2013
BAC Bank of America Corp.
JPM JP Morgan Chase & Co.
WFC Wells Fargo
C Citigroup Inc.
USB US Bancorp
APA Apache Corp.
SLB Schlumberger
COP Conoco Phillips
-
2007-2008
USB US Bancorp
BAC Bank of America Corp.
COF Capital One Financial Corp.
AIG American International Group Inc.
C Citigroup Inc.
WFC Wells Fargo
BA Boeing Co.
CVX Chevron
MCD McDonalds Corp.
IBM International Business Machines
Eigenvector centrality in the thresholded LVDN.
Results
34/43
LVDN - connectivity
Sector
Cons. Disc.
Cons. Stap.
Energy
Financial
Health Care
Industrials
Inf. Tech.
Materials
Telecom. Serv.
Utilities
Total degree
non-thresholded
2000-2013
2007-2008
from
to
from to
4.32
2.37 26.31 26.41
3.98
4.65 27.47 22.65
5.52
7.92 21.91 33.72
4.74
6.22 24.42 35.56
5.00
2.51 28.06 22.36
4.43
3.21 27.26 25.81
5.03
4.89 29.90 19.98
3.24
4.62 26.86 27.01
6.50
7.26 27.44 16.52
5.15
8.74 29.49 21.54
4.73
26.54
thresholded
2000-2013
2007-2008
from to
from to
0.24
0.24 1.96
3.29
0.00
0.44 2.80
1.36
1.90
1.54 4.01
6.10
0.77
0.77 5.23
6.69
0.00
0.00 4.27
1.83
0.21
0.21 2.43
3.53
0.00
0.00 3.35
1.58
0.00
0.00 2.99
0.81
1.91
1.91 0.00
0.00
0.00
0.00 4.94
2.38
0.47
3.40
From- and To-degree sectoral averages in LVDN.
Results
35/43
Summary
We determine and quantify the different sources of variation driving a
panel of volatilities of S&P100 stocks over the period 2000-2013;
increased connectivity during Financial Crisis;
key role of the Financial sector, particularly during the Financial Crisis;
other sectors such as Energy seem to have an important role too;
a “factor plus VAR” approach motivated by
financial interpretation: global vs. idiosyncratic risk;
existence of common factors is at odds with sparsity;
data structure as shown by partial spectral coherencies;
results are robust to
other VAR estimations as (i) group lasso, (ii) adaptive lasso;
different forecasting horizons h;
other identifications strategies as (i) centrality of PCN when signs of
correlations are accounted for, (ii) generalised variance decomposition.
Results
36/43
Thank you!
Questions?
Results
37/43
Partial spectral coherence
It is the analogous of partial correlation but in the frequency domain
−[Σ(θ)]ij
PSCij (θ) = p
[Σ(θ)]ii [Σ(θ)]jj
directly related to VAR coefficients;
Davis, Zang, Zheng, 2015
compare PSC of volatilities σn and idiosyncratic volatilities ξσ;n ;
difference between a sparse VAR on σn vs. sparse VAR on ξσ;n .
Results
38/43
Partial spectral coherence
percentiles
|PSCσn (θ = 0)|
|PSCξσ;n (θ = 0)|
|PSCσn (θ = 0) − PSCξσ;n (θ = 0)|
50th
0.06
0.06
0.02
90th
0.16
0.16
0.06
95th
0.18
0.19
0.08
97.5th
0.22
0.22
0.12
99th
0.25
0.25
0.15
max
0.35
0.34
0.24
Distribution of absolute value of PSC s’ entries.
Results
39/43
Partial spectral coherence
1
1
1
45
45
45
90
1
45
PSCσn (θ = 0)
90
90
1
45
PSCξσ;n (θ = 0)
90
90
1
45
90
|PSCσn (θ = 0) − PSCξσ;n (θ = 0)|
Left and middle panels: weights in absolute values below the 90th percentile in grey,
weights above the 90th percentile in red, and below the 10th percentile in blue. Right
panel: weights below the 90th percentile in grey, between the 90th and 95th percentiles
in blue, and above the 95th percentile in red.
Results
40/43
Eigenvector centrality in the signed PCN
2000-2013
BAC Bank of America Corp.
C Citigroup Inc.
WFC Wells Fargo
JPM JP Morgan Chase & Co.
AIG American International Group Inc.
USB US Bancorp
MS Morgan Stanley
AXP American Express Inc.
ORCL Oracle Corp.
COF Capital One Financial Corp.
2007-2008
BAC Bank of America Corp.
USB US Bancorp
JPM JP Morgan Chase & Co.
MS Morgan Stanley
AIG American International Group Inc.
C Citigroup Inc.
COF Capital One Financial Corp.
WFC Wells Fargo
GS Goldman Sachs
AXP American Express Inc.
Eigenvector centrality in the signed PCN.
Results
41/43
LVDN - obtained with signed PCN
HD
APADVN
PG
TGT
MRK
PFE
BAC
WFC
HON
T VZ
NSC
UNP
HALNOV
GS
COP
JNJ
UTX
MMM
MDT
COF
TXN
WMT
BAX
AMZN
ALL
LMT
ABT
SBUX
BRK.B
SLB
INTC
MSFT
ORCL
QCOM
WAG
UPS
MCD CVS
PEP
LOW
MDT
COF
NKE
BMY
HD
HAL
VZ
NOV
LLY
INTC
AMZN
BK
DIS
PG
2000-2013
FCX
UNP
NSC
BRK.B
SLB
JNJ
GILD
FDX
MMM
F
DIS
T
PFE
WMT
LMT
IBM
TGT
TXN
CAT
CMCSA
HON
CL
RTN
MRK
AAPL
KO
MSFT
GE
ORCL
GILD
WMB
TWX
NKE
SO
QCOM
CSCO
BA
COP
UTX
APC
BMY
AEP
SPG
AEP
SO
PEP
EXC
HPQ
DVN
CSCO
BA
WMB
UNH
RTN
ALL
AIG
DOW
AMGN
MS
EXC
EMC
GD
XOM
CVX
TWX
APC GS
EBAY
WAG
AMGN
DD
F
BK
CL
IBM
JPM
C
DOW
CVS
AIG
SPG BAX
MO
WFC USB
BAC
EMR
KO
AXP
CAT
UNH
EMC
DELL
MS
AAPL
MO
CVX
XOM
OXY
FDX
COST
UPS
USB
AXP
OXY
CMCSA MCD
GE
SBUX
FCX
LLY
JPM
C
EMR COST
GD
LOW
APA
EBAY HPQ
ABT
DELL
DD
2007-2008
Thresholded LVDN.
Results
42/43
LVDN - centrality obtained with signed PCN
2000-2013
BAC Bank of America Corp.
WFC Wells Fargo
JPM JP Morgan Chase & Co.
C Citigroup Inc.
USB US Bancorp
-
2007-2008
BAC Bank of America Corp
USB US Bancorp
WFC Wells Fargo
AXP American Express Inc.
UNH United Health Group Inc.
AIG American International Group Inc.
C Citigroup Inc.
MCD McDonalds Corp.
JPM JP Morgan Chase & Co.
OXY Occidental Petroleum Corp.
Eigenvector centrality in the thresholded LVDN.
Results
43/43