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1、VOLUME83,NUMBER7PHYSICALREVIEWLETTERS16AUGUST1999NoiseDressingofFinancialCorrelationMatricesLaurentLaloux,1,*PierreCizeau,1Jean-PhilippeBouchaud,1,2andMarcPotters11Science&Finance,109-111rueVictorHugo,92532LevalloisCedex,France2ServicedePhysiquedel’ÉtatCondensé,Centred’étudesdeSaclay,Or
2、medesMerisiers,91191Gif-sur-YvetteCedex,France(Received15December1998)Weshowthatresultsfromthetheoryofrandommatricesarepotentiallyofgreatinteresttounderstandthestatisticalstructureoftheempiricalcorrelationmatricesappearinginthestudyofmultivariatetimeseries.Thecentralresultofthepresentst
3、udy,whichfocusesonthecaseoffinancialpricefluctuations,istheremarkableagreementbetweenthetheoreticalprediction(basedontheassumptionthatthecorrelationmatrixisrandom)andempiricaldataconcerningthedensityofeigenvaluesassociatedtothetimeseriesofthedifferentstocksoftheS&P500(orothermajormarkets)
4、.Inparticular,thepresentstudyraisesseriousdoubtsontheblinduseofempiricalcorrelationmatricesforriskmanagement.PACSnumbers:05.45.Tp,02.10.Sp,05.40.Ca,87.23.GeP2NEmpiricalcorrelationmatricesareofgreatimportanceintotalvariancesPi,j1piCijpj,whereCistheco-dataanalysisinordertoextracttheunde
5、rlyinginformationvariancematrix.Theoptimalportfolio,whichminimizescontainedin“experimental”signalsandtimeseries(e.g.,theriskforagivenvalueofRP,caneasilybefoundexperimentaldatade-noising,patternrecognition,weatherintroducingaLagrangemultiplierandleadstoalinearforecast,econometricdata,mul
6、tivariateanalysis,etc.).InproblemwherethematrixChastobeinverted.Inpar-additiontothedirectmeasureofcorrelations,variousticular,thecompositionoftheleastriskyportfoliohasaclassesofstatisticaltools,suchasprincipalcomponentlargeweightontheeigenvectorsofCwiththesmallestanalysis,singularvalued
7、ecomposition,andfactoranalysis,eigenvalues[1,2].stronglyrelyonthevalidityofthecorrelationmatrixinHowever,areliableempiricaldeterminationofacorrela-ordertoobtainthemeaningfulpartofthesignal.Thus,ittionmatrixturnsouttobedifficult.ForasetofNdifferentisimportanttounderstandquantitat
