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0.20% 0.40% 0.60% 0.80% 1.00% 1.20% 1.40% 1.60% 1.80% 2.00% 2.20% 2.40% 2.60% 2.80% 3.00% 3.20% 3.40%
“Edge of DOS”
Results of PCA with DOS analysis
-- 4 significant eigenvectors/eigenvalues -- first Eigen-state explains about 44% of the correlation -- total explained variance= 51%
Conjecture: The significant eigenvectors are coherent with respect to either size of sector
Random Matrix Theory
X tn , t = 1,2,.., T , n = 1,2,..., N
T
Top 10 GENZ CEPH HSIC CELG GILD BIIB XRAY AMLN CTAS ESRX
Bottom 10 MRVL NVDA FWLT BRCM SNDK JOYG RIMM BIDU LRCX ALT R
CHKP
MICC
SIRI
YHOO
HANS
VRSN
AMAT
BRCM
MSFT
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.01 0.63 1.25 1.87 2.49 3.11 3.73 4.35 4.97 5.59 6.21 6.83 7.45 8.07 8.69 9.31 9.93 10.5 11.2
Bulk spectrum
X ~ N (0,1)
t =1
Wmn = ∑ X tm X tn ,
W = Xt X eigenvalues of W
λn , n = 1,2,...N
What are the statistical properties of the eigenvalues as N, T tend to infinity? What are the fluctuations of the eigenvalues for large N, T?
Nasdaq-100 Components of NDX/QQQQ
Data: Jan 30, 2007 to Jan 23, 2009 502 dates, 501 periods 99 Stocks (1 removed) MNST (), now listed in NYSE
Marcenko-Pastur Theorem
# {k : λk / N ≤ E} F (E ) = lim N →∞ N T →∞
N / T →γ
integrated DOE
f (
f (E ) =
(E+ − E )(E − E− )
E
E± = 1 + γ
One large mass at 0.44, Some masses near 0.025 Nearly continuous density for lower levels
18 16 14 12 10 8 6 4 2 0 0.00%
Zoom of the DOS for low eigenvalues
DOS for a random Gaussian ensemble
Bouchaud, Cizeau, Laloux, Potters (PRL, 1999)
The bulk distribution is described approximately by Marcenko Pastur properly normalized
FW L D T IS C G A O O G G IL D IN FY D TV H O LX R IM M D EL L JN P R N TA P LO G N I VD A C SC O EB AY FA ST C O S C T H R W XL N X PE TM
Top 10 FWLT LEAP JOYG STLD TEVA DISCA DISH CEPH ATVI LBTYA
Bottom 10 BRCM FWLT DELL FMCN AKAM BIDU ALT R FLEX AMAT
Third Eigenvector: Manufacturing vs. Chips
0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25
Lecture 2: PCA and Risk Factors
Marco Avellaneda G63.2936.001 Spring Semester 2009
Principal components analysis for financial markets
-- Define a universe, or collection of stocks corresponding to the market of interest (e.g. US Equities, Nasdaq-100, Brazilian equities components of S&P 500) -- Collect as much data as possible -- On any given date, perform PCA on the correlation matrix, going back for T periods (days). The analysis is on a T by N matrix -- Estimate the number of significant components -- Analyze the corresponding eigenvectors and eigenportfolios -- Dynamics: evolution of the spectrum in time.
20 15
F(E)
10 5
0 0.03 0.07 0.10 0.14 0.18 0.21 0.25 0.28 0.32 0.36 0.39 0.43 0.46 0.50 0.54 0.57 0.61 0.64 0.68 0.72 0.75 0.79 0.82 0.86 0.93 0.97 0.9
0
E
DISH
ADBE
LRCX
IACI
QCOM
GENZ
SNDK
AMZN
AMLN
TEVA
LBTYA
AMAT
PAYX
FMCN
CELG
NTAP
LRCX
SHLD
STLD
ERTS
HSIC
FISV
INTC
SIRI
12 10 8 6 4 2 0 -2 -4 -6 -8
Top 10 GENZ BBBY BIIB GILD CEPH CELG ESRX CTAS AMGN
BBBY LLTC DELL MRVL INFY ISRG CMCSA CTXS QCOM CSCO
Ticker
Second Eigenvector: Biotech vs. Chips
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25
GENZ BIIB HOLX SRCL SPLS FAST PAYX
Top 10 LEAP FWLT DISCA FMCN NIHD ATVI GILD CEPH GOOG JOYG
Bottom 10 KLAC ALT R BBBY PETM LRCX AMAT LLTC SHLD XLNX BRCM
Bottom 10 BBBY LLTC SHLD AMAT ALT R PETM MRVL LRCX BRCM KLAC
10 5 0 -5 -10 -15
B C IIB M C SA DE LL HS IC AD B E M S FT EX PE VR TX NV DA SY M C SN D K W FM CO I ST XL NX BR C M AM AT LE A P AT VI BI DU XR A Y
``Coherence’’
First Eigenvector: Market
Sorted Eigenvector
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
CSC O INTC ORCL ADBE DTV SIAL MSFT SPLS INFY PAYX
Ticker
Sorted Weights
7 6 5 4 3 2 1 0
(
)
2
Marcenko Pastur (N/T=2)
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.01 0.63 1.25 1.87 2.49 3.11 3.73 4.35 4.97 5.59 6.21 6.83 7.45 8.07 8.69 9.31 9.93 10.5 11.2
significant eigenvalues
50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0%
1 4 7 1 0 1 3 1 6 1 9 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97
Density of States (from previous data)
Definition: If an eigenvector is such that stocks with a given property (size, industry sector) have entries with the same sign, then the eigenvector is said to be coherent (with respect to the given property).
