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1、BARRA’sRiskModelsAamirSheikhCopyright©1996BARRA,Inc.BARRA’SRISKMODELSInvestmentdecisionsboildowntopickingarisk-returncombinationwithwhichoneiscomfortable.Atoneendofthespectrumlienominallyrisklesssavingsaccounts,whereasattheotherendlieexoticderivativesecuritieswhosestructures,letalonetheirrisks,ar
2、edifficulttounderstand.Itisnaturaltothinkthatgreaterrisksarerewardedwithgreateraveragereturns.This,however,isanoversimplification.Greaterriskisrewardedonlytotheextentthattheeconomyasawholeisconcernedaboutthesourceofgreaterrisk.Properinvestmentdecisions,therefore,shouldbeginbyconsideringthesetofin
3、vestmentopportunitiesthatprovideagivenlevelofreturnforthesmallestlevelofrisk.Thissetisreferredtoastheefficientset.Withintheefficientset,greaterreturnmaybeobtainedonlybybearinggreaterrisk.Keytodefiningtheefficientsetisadefinitionandmeasurementofrisk.Acommonlyusedandeminentlyjustifiabledefinitionof
4、riskisthedispersionofactualreturnsaroundtheexpectedoraveragereturn.Thisdispersionismeasuredviathestandarddeviationofreturns.Althoughthisdispersionisreadilyquantifiedforindividualsecurities,thedispersionofportfolioreturnsiscruciallydependentonthedegreeofcomovementinsecurityreturns.Forexample,consi
5、dertwosecuritieswhosereturnsmoveinlockstep.Whenonesecurityreturns15percent,theotherreturns15percent,andsoon.Bothsecu-ritiesarerisky,butbysellingoneandbuyingtheother,wecanobtainaguaranteedreturn!Thus,indefiningtheefficientset,weneedmeasuresofsecuritydispersionandcomovement.Thesearecontainedintheco
6、variancematrixofsecurityreturns.Thereareanumberofwaysofestimatingthecovariancematrixofsecurityreturns.Substantialgainsaremadebyrecognizingthatcovariancesaredrivenbycommonsourcesofreturnsacrosssecurities.Thesecommonsourcesofreturnsarecalledcommonfactors.Estimatingthecovariancematrixofsecuri-tyretu
7、rnsthusdependsonestimatingafactormodelforsecurityreturns.Inthisarticle,wewilldiscussthebenefitsandcostsofthedifferentapproach-estoestimatingfactormodelsofsecurityreturns.Wewillbeginbydiscussinghowportfoliostandarddevia
