Variety of Behavior of EquityReturns in Financial Markets

Fabrizio Lilloin collaboration with Rosario Mantegna

Observatory of Complex Systems

Society for Computational Economics, Yale, June 28-29, 2001

http://lagash.dft.unipa.it

Palermo University - Italy

OutlineOutline

• Longitudinal vs. cross-sectional quantities• Variety of a financial portfolio• Asymmetry of a financial portfolio• One-factor model• High frequency analysis• Conclusions

Longitudinal Longitudinal vsvs. cross-sectional quantities. cross-sectional quantities

0 1000 2000 3000trading day

AA

GE

JPM

KO

daily

ret

urn

longitudinal quantitieslongitudinal quantitiesmean return of asset i mean return of asset i µµii

volatility of asset i volatility of asset i σσii

cross-sectional quantitiescross-sectional quantitiesmean return at day t mean return at day t µ(µ(tt))

variety at day t variety at day t σσ(t)(t)

cross-correlation cross-correlation ρρijijbetween asset i and jbetween asset i and j

Database and investigated variableDatabase and investigated variable• The investigated market is the New York Stock

Exchange during the 12-year period (3032 trading days) January 1987 - December 1998

• The number of stocks n continuously traded in this timeperiod is 1071.

• The variable investigated is the daily price returndaily price return, whichis defined as

where is the closure price of the i-th stock at day t(t=1,2,..)

(((( )))) (((( )))))(

)(11tY

tYtYtRi

iii

−−−−++++====++++

( )tYi

What is the distribution of returnsof N stocks in a givengiven day ?

The answer depends on the day !

For each trading day we extract theensemble return distribution

• The ensemble return distribution is non Gaussian and the central part is conserved for long period of normal activity.• The ensemble return distribution changes dramatically during highly volatile periods. For example, during the Black Monday crash we have

Can we better quantify the statistical properties of the ensemble return distribution ?

-100 -50 0 50 100trading day

-0.2

-0.1

0.0

0.1

0.2R

Market averageMarket average• The simplest quantity useful to quantify the behavior ofthe set of stocks returns is the mean

• This quantity is just a notweighted index andquantifies the general trendof the market at day t.

=

=n

iim tR

ntr

1)(1)(

One-factor modelOne-factor model• We compare our empirical results with numerical and

analytical results based on the one-factor model

where and are two constant parameters, is azero mean idiosyncratic term characterized by avariance equal to

• is the market factor. We choose it as the Standardand Poor’s 500 index.

• We estimate the model parameters and generate anartificial market.

( ) ( ) ( )ttRtR iMiii εβα ++=

iα iβ ( )tiε

2iεσ

( )tRM

Non-Non-Gaussian Gaussian one-factor modelone-factor model

• We consider two possible choices for thestatistics of the idiosyncratic terms

1) Gaussian noise terms2) Student’s t noise terms

where

wii εσε =

( ) 1212

1)1(

)( ++ ≈+

= κκκ

κ wwCwP

3=κ

−0.06 −0.04 −0.02 0.00 0.02 0.04rm(t)

10−1

100

101

102

P[r

m(t

)]real dataone−factor model

• The one factor model well reproduces the statistical properties ofthe market average

VarietyVariety• We introduce the variety as the root mean squareof the stocks returns on a given day

( )( )=

−=n

imi trtR

ntV

1

2)(1)(

• The variety quantifies the different behaviorobserved in a market (or portfolio) on a given day.• The variety is not the volatility of the index.

VarietyVariety

1987 1990 1993 1996trading day

0

0.02

0.04

0.06

0.08

0.1

V(t

)

0 25 50 75 100trading day

0

0.5

1

auto

corr

elat

ion

• The time evolution of thevariety presents slowdynamics and severalbursts.

• The temporal correlationfunction of the variety has aslow decay with time

• There is a correlationbetween variety and marketvolatility (Pearson r =0.68;Spearman r =0.37) .

• The one-factor model actually predicts that the variety increaseswith which is a proxy of the market volatility. We prove that

where is the variety of idiosyncratic part.

( ) ( ) ( )trtvtV m2

2

222

ββ∆+≅

( ) ( )[ ]=

=N

iiN ttv

1

212 ε

2mr

−0.2 −0.1 0 0.1 0.2rm(t)

0

0.02

0.04

0.06

0.08

0.1

V(t

)

• The one-factor model assumes that the idiosyncratic terms areindependent of the market return.

• As a consequence, the variety of idiosyncratic terms isconstant in time and independent from the mean

• The empirical results show that a significant correlation between and indeed exists

•• The degree of correlation is stronger for rallies than for crashes.The degree of correlation is stronger for rallies than for crashes.

( )tv

( )tv

( )trm

( )trm

−0.05 0 0.05rm(t)

0

0.02

0.04

0.06

v(t)

−0.15 0 0.15

0 1000 2000 3000

trading day0

0.05

0.1

V(t

)

real datastudent one−factor model

0.01 0.02 0.03 0.04 0.05V(t)

10−1

100

101

102

103

P[V

(t)]

real datagaussian one−factor modelstudent one factor model

• The one-factor model is unable to reproduce thestatistical properties of the variety of returns of real

financial markets.

AsymmetryAsymmetry• What fractionWhat fraction ƒƒ of stocks did actually better than the market ?of stocks did actually better than the market ?

• A balanced market would have ƒ = 50%

• If ƒ > 50%, then the majority of the stocks beat the market, buta few ones lagging behind rather badly, and vice versa

asymmetry=mean-median

• We quantify this information by introducing the asymmetry ofthe ensemble return distribution, defined as

Is the asymmetry also correlated with the market average ?Is the asymmetry also correlated with the market average ?

• There is a strong correlation between asymmetry and market averageThere is a strong correlation between asymmetry and market average

• We observe that the ensemble return distribution is

Negatively skewed crashesPositively skewed rallies

High frequency analysisHigh frequency analysis• We perform a high frequency analysis of one of

the largest financial crashes of last 20 years,specifically the August 31, 1998 crash at NYSE• We use the high frequency data recorded in theTrade and Quote database and the investigated

period isAugust 20, 1998 - September 10, 1998

• We consider 3 time horizons∆t = 5 min, 55 min, 1 trading day

000

time (trading day)

0

0.01

0.02

0.03

0.04

V(t

)−0.03

−0.015

0

0.015

0.03

r m(t

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

• There is a strong correlation between variety and market averageThere is a strong correlation between variety and market averagealso at a also at a intradayintraday time horizon.time horizon.

• This effect is stronger at the opening and at the closure of thetrading day

∆t=55 min

−0.05 −0.03 −0.01 0.01 0.03 0.05−0.01

−0.005

0

0.005

0.01

A∆t

1 day

−0.05 −0.03 −0.01 0.01 0.03 0.05

µ∆t

−0.01

−0.005

0

0.005

0.01

A∆t

5 minutes

−0.05 −0.03 −0.01 0.01 0.03 0.05

55 minutes

• The symmetry change in the presence of large movements also at aintraday temporal scale.

• The asymmetry effect is stronger at shorter time scales.

ConclusionsConclusions• We study the cross sectional quantities in a financial

market at different time horizons.• We introduce the variety and we find that it has a long

time memory.• We find that the symmetry properties of the ensemble

return distribution change in extreme market days.• A simple one-factor model is unable to explain the

statistical properties of the variety and the change ofsymmetry in extreme days.