Rosario N. Mantegna Cross-sectional (ensemble) analysis of asset return dynamics in generic and specialized stock portfolios work done in collaboration with Fabrizio Lillo and Adriana Prest Mattedi 09/03 Next 2003, Villasimius
Overview We investigate statistical properties of ensemble of stocks simultaneously traded in a financial market. Cross-sectional statistical properties are investigated: (i) in a wide portfolio of stocks covering the entire New York Stock Exchange; (ii) in a specialized portfolio composed of stocks belonging to the aerospace sector; Discussion. 09/03 Next 2003, Villasimius 2
T In a series of studies, we have been focusing on the statistical properties of returns of an ensemble of stocks simultaneously traded in a stock market. Statistical properties of a time evolution Statistical properties of a statistical ensemble see our website: http://lagash.dft.unipa.it 09/03 Next 2003, Villasimius 3
New stylized facts from cross-sectional investigation daily return KO JPM GE AA cross-sectional quantities: mean return at day t µ(t) variety at day t σ(t) 0 1000 2000 3000 trading day A brief review can be found in Lillo, Mantegna, Bouchaud, Potters, Wilmott Magazine,December issue, 98-102, 2002. 09/03 Next 2003, Villasimius 4
Database Database and investigated variable The investigated market is the New York Stock Exchange during the 12-year period Jan 1987- Dec 1998. The number of stocks n is not constant and ranges from 1128 (in Jan 1987) to 2788 (in Dec 1998), 09/03 Next 2003, Villasimius 5
Daily data The variable investigated is the daily price return, which is defined as R ( t + 1) = i Y ( t i + 1) Y Y ( t) ( t) Where Y i (t) is the closure price of the i-th stock at day t (t=1,2, ). When the investigation of high-frequency data is necessary, we use the Trade and Quote database of NYSE. i i 09/03 Next 2003, Villasimius 6
Cross-sectional analysis Is an ensemble of stocks a homogeneous ensemble? Strictly speaking it is not Indeed, almost any ensemble of stocks is hierarchically structured with respect to stock capitalization and with respect to the sector of economic activity of the firm 09/03 Next 2003, Villasimius 7
Common behavior However, a degree of common behavior is detected in spite of the huge diversity presents among several stocks. The return distribution is non-gaussian and leptokurtic. 09/03 Next 2003, Villasimius 8
Ensemble return pdf We perform statistical analyses of ensemble return distribution Specifically, we consider the returns R i (t) of the i=1,..,n stocks composing the stock ensemble at each day t At each day t, the n returns are then used to obtain the ensemble return probability density function (pdf) P(R(t)) 09/03 Next 2003, Villasimius 9
Analysis over a wide time period Jan 1987- Dec 1998 A large number of trading days shows a typical ensemble return distribution but there are notable exceptions!! 09/03 Next 2003, Villasimius 10
Contour lines The time evolution of the ensemble return distribution can be tracked in a direct way by considering the contour lines of P(R,t) 09/03 Next 2003, Villasimius 11
Central moments In order to characterize the ensemble return distribution at day t, we extract the first two central moments for each of the 3032 trading days of the investigated period σ( t) = µ ( t) = 1 n( t) 1 n( t) 1 The mean µ(t) quantifies the general trend of the market at day t. n( t) i= 1 σ(t) quantifies the width of the ensemble return distribution and gives a measure of the variety of behavior observed at a given day. n( t) i= 1 R i ( t) ( R ( t) µ () t ) i 2 09/03 Next 2003, Villasimius 12
Statistical properties of Variety ACF(τ) 10 0 10 1 0.03 1987 1991 1995 1999 0.01 10 0 10 1 10 2 τ (trading day) The variety of the ensemble return distribution is a long-range correlated stochastic variable 09/03 Next 2003, Villasimius 13
Ensemble return pdf What is the ensemble return pdf in crash and rally days? 09/03 Next 2003, Villasimius 14
Ensemble return pdf The answer depends on the day! 5 June 1997 19 October 1987 09/03 Next 2003, Villasimius 15
Something peculiar occurs around crashes A blow up of the October 1987 crash 09/03 Next 2003, Villasimius 16
Rare event analysis We select the 9 trading days of our database in which the S&P500 has negative extreme returns and the 9 days of positive extreme returns. CRASHES Date S&P500 Return 19 10 1987-0.2041 26 10 1987-0.0830 27 10 1997-0.0686 31 08 1998-0.0679 08 01 1988-0.0674 13 10 1989-0.0611 16 10 1987-0.0513 14 04 1988-0.0453 30 11 1987-0.0416 RALLIES Date S&P500 Return 21 10 1987 0.0908 20 10 1987 0.0524 28 10 1997 0.0511 08 09 1998 0.0509 29 10 1987 0.0493 15 10 1998 0.0418 01 09 1998 0.0383 17 01 1991 0.0373 04 01 1988 0.0360 09/03 Next 2003, Villasimius 17
The pdf profile during crashes The shape of the ensemble return distribution changes during Date CRASHES S&P500 Return 19 10 1987-0.2041 26 10 1987-0.0830 27 10 1997-0.0686 31 08 1998-0.0679 08 01 1988-0.0674 13 10 1989-0.0611 16 10 1987-0.0513 14 04 1988-0.0453 30 11 1987-0.0416 a b c d e f g h i 09/03 Next 2003, Villasimius 18
The pdf profile during rallies and rally days RALLIES Date S&P500 Return 21 10 1987 0.0908 20 10 1987 0.0524 28 10 1997 0.0511 08 09 1998 0.0509 29 10 1987 0.0493 15 10 1998 0.0418 01 09 1998 0.0383 17 01 1991 0.0373 04 01 1988 0.0360 a b c d e f g h i 09/03 Next 2003, Villasimius 19
Skewness 2 1.5 1 0.5 We observe that the ensemble return distribution is Negatively skewed during crashes Positively skewed during rallies We quantify the asymmetry of the ensemble distribution by extracting the mean and the median of the distribution. Negatively skewed mean < median 0 0 0.5 1 1.5 2 2.5 2 1.5 1 0.5 0 Positively skewed mean > median 0 0.5 1 1.5 2 Advantage: the median depends weakly on rare events 09/03 Next 2003, Villasimius 20
Asymmetry of the pdf We quantify the degree of asymmetry of the ensemble return pdf by computing the mean-median parameter of each pdf A daily analysis shows that symmetry changes are continuously detected as the absolute mean return varies 09/03 Next 2003, Villasimius 21
One-factor model We compare our empirical results with numerical and analytical results based on the one-factor model R i () t = α + β R () t + ε () t i where α i and β i are two constant parameters, ε i is a zero mean idiosyncratic term characterized by a variance equal to σ εi R M is the market factor. We choose it as the Standard and Poor s 500 index. We estimate the model parameters and generate an artificial market. i M i 09/03 Next 2003, Villasimius 22
Non-Gaussian one-factor model We consider two possible choices for the statistics of the idiosyncratic terms 1) Gaussian noise terms 2) Student s t noise terms P w ε i = σ ε i 1 w κ ( ) = 2 + κ + where κ = 3 (1 + C w κ ) ( κ 1) 2 1 w 09/03 Next 2003, Villasimius 23
Statistical properties of market average The one factor model well reproduces the statistical properties of the market average real data --- one-factor model 09/03 Next 2003, Villasimius 24
Time evolution of variety The variety is not well described by the one-factor model µ(t) µ(t) σ(t) σ(t) real data one-factor real data one-factor 09/03 Next 2003, Villasimius 25
Modeling variety with a one-factor model The one-factor model actually predicts that the variety increases with r m2 (t), which is a proxy of the market volatility. We prove that V 2 1 where v () t = [ i () N ε t ] 2 N β () 2() 2 t v t + r () t i= 1 2 β 2 2 m is the variety of idiosyncratic part. 09/03 Next 2003, Villasimius 26
Variety during rare events V(t) The variety is not well described by the one factor model during rare events 09/03 Next 2003, Villasimius 27
Excess variety The variety of the idiosyncratic terms (excess variety) is clearly correlated with the R M (t) and a different behavior is observed for crashes and rallies (another source of asymmetry) Variety of the idiosyncratic term Market mean 09/03 Next 2003, Villasimius 28
Variety in a specialized portfolio What about a portfolio of stocks belonging to a well-defined economic sector? Here we present some preliminary results 09/03 Next 2003, Villasimius 29
The aerospace sector By using daily data we compute the Comprehensive AeroSpace Index (CAS Index) The CAS index is computed by considering 146 stocks belonging to this economic sector We also compute three sub-indices by considering explicitly stocks of firms whose main activity concerns (I) structures (II) components (III) services 09/03 Next 2003, Villasimius 30
09/03 Next 2003, Villasimius 31 Cas index return 1987-1998 Cas index The CAS index
09/03 Next 2003, Villasimius 32 CASI Varierty time evolution
09/03 Next 2003, Villasimius 33 CASI Variety acf Variety has a long time-memory!!!
Power-law behavior? The acf time-lag decay is compatible with a power-law decay 09/03 Next 2003, Villasimius 34
CASI variety during rare events Preliminary results suggest that the variety of a specialized portfolio shows and excess variety component which is significant and asymmetric with respect to crashes and rallies 09/03 Next 2003, Villasimius 35
Conclusion New stylized facts are discovered in financial markets Their knowledge is instrumental to achieve a satisfactory modeling of this fascinating complex system Cross-sectional investigation of an hierarchical complex system can be performed in generic and specialized portfolios providing relevant information for the modeling 09/03 Next 2003, Villasimius 36
09/03 Next 2003, Villasimius 37 The OCS website http://lagash.dft.unipa.it