Investment strategy due to the minimization of portfolio noi

Using a recently developed method of noise level estimation that makes use of properties of the coarse grained-entropy we have analyzed the noise level for the Dow Jones index and a few stocks from the New York Stock Exchange. We have found that the noise

arXiv:cond-mat/0412754v1 [cond-mat.stat-mech] 30 Dec 2004

Investmentstrategyduetotheminimizationofportfolionoiselevelbyobservationsof

coarse-grainedentropy

KrzysztofUrbanowiczandJanuszA.Ho lyst

FacultyofPhysicsandCenterofExcellenceforComplexSystemsResearch

WarsawUniversityofTechnology

Koszykowa75,PL–00-662Warsaw,Poland

1Introduction

Althoughitisacommonbelievethatthestockmarketbehaviourisdrivenbystochasticprocesses[1,2,3]itisdi culttoseparatestochasticanddeterminis-ticcomponentsofmarketdynamics.Infactthedeterministicfractionfollowsusuallyfromnonlineare ectsandcanpossessanon-periodicorevenchaoticcharacteristic[4,5].Theaimofthispaperistostudythelevelofdeterminismintimeseriescomingfromstockmarket.Wewillshowthatournoiselevelanalysiscanbeusefulforportfoliooptimization.

Weemployhereamethodofnoise-levelestimationthathasbeendescribedindetailsin[6].Themethodisquiteuniversalanditisvalidevenforhighnoiselevels.Itmakesuseofthefunctionaldependenceofcoarse-grainedcorrelation

PreprintsubmittedtoElsevierScience

2February2008

Using a recently developed method of noise level estimation that makes use of properties of the coarse grained-entropy we have analyzed the noise level for the Dow Jones index and a few stocks from the New York Stock Exchange. We have found that the noise

entropyK2(ε)[7]onthethresholdparameterε.SincethefunctionK2(ε)dependsinacharacteristicwayonthenoisestandarddeviationσthusonecanestimatethenoiselevelσobservingthedependenceK2(ε).Thevalidityofourmethodhasbeenveri edbyapplyingitforthenoiselevelestimationinseveralchaoticmodels[7]andfortheChuaelectroniccircuitcontaminatedbynoise.Themethoddistinguishesanoiseappearingduetothepresenceofastochasticprocessfromanon-periodicdeterministicbehaviour(includingthedeterministicchaos).Analyticcalculationsjustifyingourmethodhavebeendevelopedforthegaussiannoiseaddedtotheobserveddeterministicvariable.IthasbeenalsocheckedinnumericalexperimentsthatthemethodworksproperlyforauniformnoisedistributionandatleastforsomemodelswithdynamicalnoisecorrespondingtotheLangevineequation[6].

2Calculationsofnoiselevelinstockmarketdata

Wede nethenoiselevelastheratioofstandarddeviationofnoiseσtothestandarddeviationofdataσdata

NTS=

σ

Pi 1

.(2)

Fig.1presentstheplotofthenoiselevelNTSforacorrespondingtimeseriesxiwherevaluesofNTShavebeencalculatedasafunctionofatradingday.Thenoiselevelhasbeendeterminedinwindowsofthelength3000daysandispointedinthemiddleofeverywindow.Asonecanseethelevelofnoiserangesfrom60%to90%whatmakesanypointtopointforecastingimpossible.WeshouldmentionthatsincetherelativenoisevarianceisNTS2,thusinourcasethenoisevarianceis40 80%ofthedatavariance.Itfollowsthattherearetimeperiodswhenthepercentofanunknowndeterministicpartapproachesthelevel60%ofthesignal.

SimilarestimationsofthenoiselevelhavebeenperformedforselectedstocksoftheNYSE.ResultsforthemeanvaluesofcorrespondingNTSparametersarepresentedintheTable1.Asonecanexpectthenoiselevelofasingle

2

Using a recently developed method of noise level estimation that makes use of properties of the coarse grained-entropy we have analyzed the noise level for the Dow Jones index and a few stocks from the New York Stock Exchange. We have found that the noise

NTS

4

3

2

Fig.1.ThenoiselevelNTScalculatedforDowJonesindex(1896-2002).

stockismuchlargerthanfortheDJIA.Thisisbecausedeterministicpartsofdi erentstockpricesareusuallypositivecorrelatedwhatislesscommonforstochasticcomponents.

Thecrucialpointforourinvestmentstrategyarecorrelationsbetweenatem-poraryvalueofnoiselevelandatemporaryvalueofpricechanges.Wehavefoundthatforthemajorityofconsideredstocksthecorrelationcoe cientρismuchlargerthanzero(seetheTable1)inthetimeperiodwhentrendsofthesestockwerenegative.Anegativecorrelationcoe cienthasbeenobservedforonesharewithapositiveshare.Followingtheseobservationswehaveformedthefollowingheuristicrule:temporarypricechangesaremostlyconsistentwiththetrendforasmallnoiselevelbuttheyarefrequentlyopposedtothetrendforalargenoiselevel(thenoiselevelshouldbemeasuredlocally).Onecansaythatwhenpricechangesaremorestochasticinvestorsaremoredis-orientedthanforamoredeterministicpricemotionandtheymorefrequentlytradeagainstthegeneraltrend.

3Investmentstrategy

Usingthefactthatthelevelofnoiseiscorrelatedwiththestockpricechanges,onecancreateaportfoliowhichcanmaximizethepro t.Inthe rststepweconstructaportfoliowiththeminimalvalueofthestochasticvariable.We

3

Dow JonesIndex

Using a recently developed method of noise level estimation that makes use of properties of the coarse grained-entropy we have analyzed the noise level for the Dow Jones index and a few stocks from the New York Stock Exchange. We have found that the noise

StocksrecordedattheNYSEfortheperiod01.01.1999-31.12.2000.TableofthevaluesofthenoiselevelsNTS,correlationcoe cientρbetweenpriceschangesandthenoiselevelsandreturnsinthestudiedperiod.

Apple(Ap)

BankofAmerica(Boa)Boeing(Bg)Cisco(Ci)Compaq(Cq)Ford(Fo)

GeneralElectric(Ge)GeneralMotors(Gm)Ibm(Ibm)Mcdonald(Md)TexasInstrument(Te)

77%93%94%85%93%91%86%91%75%93%76%

0.530.22 0.450.57 0.09 0.160.750.440.0230.20.77

63% 24%98% 59% 66% 60% 53% 28% 55% 57% 45%

σj,Dσi

Using a recently developed method of noise level estimation that makes use of properties of the coarse grained-entropy we have analyzed the noise level for the Dow Jones index and a few stocks from the New York Stock Exchange. We have found that the noise

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