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UnitedStatesEquityUnitedStatesEquityRISKMHANDBOOKregardingtheUnitedStatesEquityRiskMorregardingtheUnitedStatesEquityRiskMoranyresultstobeobtainedfromtheuseoftheUnitedStatesEquityRiskM.BARRAEXPRESSLYDISCLAIMSALLWARRORIMPLIED,UNITEDSTATESEQUITYRISKMESOFMERCHANTABILITYANDFITNESSFORAPARTICULARPURPOSEORUSEORTHEIREQUIVALENTSUNDERTHELAWSOFANYJURISDICTION.AlthoughBARRAintendstoobtaininformationanddatafromsourcesitconsiderstobereasonablyreliable,theaccuracyandcompletenessofsuchinformationanddataarenotguaranteedandBARRAwillnotbesubjecttoliabilityforanyerrorsoromissionstherein.Accordingly,suchinformationanddata,theUnitedStatesEquityRiskMM,andtheiroutputarenotwarrantedtobefromerror.BARRAdoesnotwarrantthattheUnitedStatesEquityRiskbefromunauthorizedhiddenprogramsintroducedintotheUnitedStatesEquityRiskMwithoutBARRA'sknowledge.CopyrightBARRA,Inc.1998.0111O02/98.ContentsAboutBARRA 1Apioneerinriskmanagement 1Introduction 3InthisContentsAboutBARRA 1Apioneerinriskmanagement 1Introduction 3Inthishandbook 3Furtherreferences 5Books 5SectionI:Theory1.WhyRiskisImportant 7Thegoalofriskanalysis 82.DefiningRisk 11Somebasicdefinitions 11Riskmeasurement 13Anexample 13Riskreductionthroughdiversification 14Drawbacksofsimpleriskcalculations 16Evolutionofconcepts 163.MingandForecastingRisk 21WhatareMFMs? 21HowdoMFMswork? 21AdvantagesofMFMs 22AsimpleMFM 23Mmathematics 25RiskpredictionwithMFMs 26i4.ModernPortfolioManagementandRisk 31Portfoliomanagement—twotypes 31Passivemanagement 31Activemanagement 32Decomposingrisk 34TotalRisk 34Systematic-ResidualRisk4.ModernPortfolioManagementandRisk 31Portfoliomanagement—twotypes 31Passivemanagement 31Activemanagement 32Decomposingrisk 34TotalRisk 34Systematic-ResidualRiskDecomposition 35ActiveRiskDecomposition 36ActiveSystematic-ActiveResidualRiskDecomposition 37Summaryofriskdecomposition 38Performanceattribution 38Summary 395.BARRAMultiple-FactorMing 41Overview 41Descriptorselectionandtesting 44Descriptorstandardization 44Riskindexformulation 45Industryallocation 45Factorreturnestimation 46Covariancematrixcalculation 46Exponentialweighting 47Computingmarketvolatility:ExtendedGARCHms 48Specificriskming 49Overview 49Methodology 50MMingtheaveragelevelofspecificrisk 50ingtherelativelevelofspecificrisk 51Estimatingthescalingcoefficients 52Finalspecificriskforecast 52Updatingthem.....................................52Comparisonofriskmfeatures 53iiU.S.EquityMVersion3(E3)SectionII:US-E3ModelSectionII:US-E3ModelDetails6.AdvantagesofUS-E3OverUS-E2 55Overview 55Industries 56Reclassification 56Flexibleindustries 57Increasedsizeoftheestimationuniverse 57Riskindices 58SizeNonlinearityfactor 58Leverage 58Simplervolatilitycalculation 58EliminationofUS-E2’sLaborIntensityandForeignIncome 59Improvedindependencebetweenriskindices 59Riskforecasting 59ImprovedGARCHmodel 59Specificriskmodel 60Modelfit-relatedissues 60Factorreturnestimation 60Ongoingdiagnostics 60Scheduledrefittingofmodelparameters 60Assetclassissues 61REITs 61Comingsoon 617.TheUS-E3EstimationUniverse 63Overview 63Selectionprocess 63S&P500membership 64Compustatdatapresent 64Capitalization 64Minimumprice 64Industryfill-in 64Grandfathering. 65ComparisonwiththeUS-E2estimationuniverse 65Contentsiii8.US-E3RiskIndicesandDescriptors 67DifferencesbetweenUS-E2and8.US-E3RiskIndicesandDescriptors 67DifferencesbetweenUS-E2andUS-E3riskindices 67Generaldifferences 67Specificdifferences 67Volatility 67Size 68SizeNonlinearity(US-E3) 68Growth 68FinalLeverage(US-E2)andLeverage(US-E3) 69ForeignIncome(US-E2)andCurrencySensitivity(US-E3) 69LaborIntensity(US-E2) 69US-E3andUS-E2riskindicesataglance 70Riskindexdefinitions 74Volatility 74Momentum 74Size 74SizeNonlinearity 74TradingActivity 74Growth 75EarningsYield 75Value 75EarningsVariability 75Leverage 75CurrencySensitivity 75DividendYield 75Non-EstimationUniverseIndicator 76Descriptordefinitions 769.US-E3Industries 77Overview 77Industryclassificationscheme 77Mini-industriesandindustries 77Sectors 78Industryweights 791.Thevaluationms 792.Assetindustryweightsforeachm...............79ivU.S.EquityMVersion3(E3)3.Finalweights 80Historicalassignmentofindustryandweights 80Ongoingassignmentofindustryandweights 80Industryevolution 81Sectormap3.Finalweights 80Historicalassignmentofindustryandweights 80Ongoingassignmentofindustryandweights 80Industryevolution 81SectormapofUS-E3industries 8210.FactorReturnEstimation 85Overview 85Estimations 85GLSweights 86Factorreturnsforhistoric“newborn”industries 86TheNONESTUfactorreturn 86Testing 8711.EstimatingtheFactorCovarianceMatrixinUS-E3 89Overview 8912.US-E3SpecificRiskMing 91Overview 91AppendixA:US-E3DescriptorDefinitions 93Volatility 93Momentum 95Size 96SizeNonlinearity 96TradingActivity 96Growth 98EarningsYield 100Value 101EarningsVariability 101Leverage 102CurrencySensitivity 103DividendYield 104Non-EstimationUniverseIndicator 104ContentsvAppendixB:US-E3Industries,Mini-Industries,ExampleAppendixB:US-E3Industries,Mini-Industries,ExampleCompanies,andCodes 105AppendixC:US-E3FrequencyDistributionsforPredictedBeta,SpecificRisk,RiskIndices 115Predictedbeta 115Specificrisk 115Riskindices 116AppendixD:US-E3RiskIndexFactorReturns 121Glossary 127Index 137Contributors143viU.S.EquityMVersion3(E3)AboutBARRAInrecentyearstheinvestmentmanagementindustryhasadjustedtocontinuingchanges—theoreticaladvances,technologicaldevelop-ments,andmarketgrowth.Toaddressthesechallenges,investmentAboutBARRAInrecentyearstheinvestmentmanagementindustryhasadjustedtocontinuingchanges—theoreticaladvances,technologicaldevelop-ments,andmarketgrowth.Toaddressthesechallenges,investmentmanagersandfinalinstitutionsrequirethemostadvancedandpowerfulanalyticaltoolsavailable.ApioneerinriskmanagementAstheleadingproviderofglobalinvestmentdecisiontools,BARRAhasrespondedtotheseindustrychangesbyprovidingqutativeproductsandservicesthatarebothflexibleandefficient.Sinceourfoundingin1975,BARRAhasbeenaleaderinmodernfinresearchandtechniques.alInitially,ourservicesfocusedonriskanalysisinequitymarkets.OurU.S.EquityMsetastandardofaccuracythatBARRAcontinuestofollow.BARRAusesthebestdataavailabletodevelopeconomet-ricfinalms.Inturn,thesemsarethebasisofsoftwareproductsdesignedtoenhanceportfolioperformancethroughreturnsforecasting,riskanalysis,portfolioconstruction,tran costanalysis,andhistoricalperformanceattribution.In1979,BARRAexpandedintothefixedincomeareawiththereleaseofU.S.bondvaluationandriskms.Inthemid-1980swedevelopedaglobaltacticalassetallocationsystem:TheBARRAWorldMarketsM .Morerecently,theTotalPlanRiskapproachwasdevelopedtoprovidemulti-asset-classvalue-at-risk(VAR)analyses.BARRAnowhasofficesaroundtheworldandproductsthatcovermostoftheworld’stradedsecurities.By1997,ourscomprisedapproximately1,200finalinstitutionsworldwidemanagingover$7trillioninassets.TheyrelyonBARRA’sinvestmenttechnologyandconsultingservicestostrengthentheirfininvestmentdecision-making.alanalysisand12U.S.Equity2U.S.EquityMVersion3(E3)IntroductionInthishandbookSectionI:Theorycontainsageneraldiscussionofequityriskandreturn,andthemethodsBARRAusestomters1through5compriseIntroductionInthishandbookSectionI:Theorycontainsageneraldiscussionofequityriskandreturn,andthemethodsBARRAusestomters1through5comprisethissection.portfoliorisk.Chap-Chapter1.WhyRiskisImportantgivesanoverviewofwhyfinprofessionalsshouldcareaboutrisk.alChapter2.DefiningRiskoutlinesthebasicstatisticalconceptsunder-lyingriskanalysis,andtracesthehistoryofequityrisktheory.Chapter3.M ingandForecastingRiskdiscussestheapplicationofmultiple-factormlem.ing(MFM)totheequityriskanalysisprob-Chapter4.ModernPortfolioManagementandRiskrelatesthevari-oustypesofactiveandpassiveequitymanagementtotheuseofariskm.stheprocessofChapter5.BARRAMultiple-FactorMingcreatingandmaintainingaBARRAequityMFM.sdiscussestheconstructionofourSectionII:US-E3Mthird-generationU.S.equityriskmindepth.Chapters6through12andAppendicesAthroughDcomprisethissection.Chapter6.AdvantagesofUS-E3OverUS-E2summarizesthereasonsforupdatingUS-E2andtheparticularadvancesoverUS-E2.withUS-E3Chapter7.TheUS-E3EstimationUniversediscussestheexpansionoftheUS-E3equityportfoliowhichisusedtoestimatethemainparam-etersoftheriskm.Thiswasdonetoensureamoreaccuratereflectionoftheinvestingactivitiesofours.Chapter8.US-E3RiskIndicesandDescriptorsdescribesdifferencesfromtheUS-E2treatmentofthesefactorsandimprovementsthathavebeeninthispartofthem.Chapter9.US-E3Industriesdiscussesthecompletereclassificationoftheindustryassignmentsandotherenhancementsintendedtokeep3themodelcurrentovertime.Onethemodelcurrentovertime.OneoftheleastsatisfactoryaspectsofUS-E2wastheunchangingnatureoftheindustryfactorspecification.Chapter10.FactorReturnEstimationcontainsabriefdescriptionofUS-E3’sparticularimplementationofthisprocess.AmoregeneraldiscussioniscontainedinChapter5.Chapter11.EstimatingtheFactorCovarianceMatrixinUS-E3alsobrieflydiscussesonlythoseaspectsofthissubjectwhichareparticu-lartoUS-E3,withamoregeneraltreatmentdetailedinChapter5.Chapter12.US-E3SpecificRiskModelingalsobrieflydiscussesonlythoseaspectsofthissubjectwhichareparticulartoUS-E3,withamoregeneraltreatmentdetailedinChapter5.AppendixA:US-E3DescriptorDefinitionsisthecompletelistofriskindicesandtheirunderlyingdatadescriptors.AppendixB:US-E3Industries,Mini-Industries,ExampleCompanies,andCodescontainsthecompletelistofindustries,theirconstituent“mini-industries,”andselectedexamplecompanies.AppendixC:US-E3FrequencyDistributionsforPredictedBeta,Spe-cificRisk,RiskIndicescontainsgraphicaldepictionsofthedistribu-tionofthesemodeloutputs.AppendixD:US-E3RiskIndexFactorReturnsgivesafullmodelhis-toryofthereturnstotheUS-E3riskindices.Finally,theGlossaryandIndexareusefulresourcesforclarifyingter-minologyandenhancingthehandbook’susefulness.4U.S.EquityModelVersion3(E3)BARRAhasacompalsBARRAhasacompalsdescribingthemensivecollectionofarticlesandothermateri-sandtheirapplications.Tolearnmoreaboutthetopicscontainedinthishandbook,consultthefollowingreferencesorourextensivePublicationsBibliography,whichisavailablefromBARRAofficesandfromourWebsiteat.BooksAndrewRuddandHenryK.Clasing,ModernPortfolioTheory:ThePrinciplesofInvestmentManagement,Orinda,CA,AndrewRudd,1988.RichardC.GrinoldandRonaldN.Kahn:ActivePortfolioManage-ment:Qu tativeTheoryandApplications,ProbusPublishing,Chi-cago,IL,1995.Introduction56U.S.Equity6U.S.EquityMVersion3(E3)1.WhyRiskisImportantSuperiorinvestmentperformanceistheproductofcarefulattentionto1.WhyRiskisImportantSuperiorinvestmentperformanceistheproductofcarefulattentiontofourelements:forreasonablereturnexpectationscontrollingrisksothatthepursuitofopportunitiesremainstem-peredbyprudencecontrollingcostssothatinvestmentprofitsarenotdissipatedinexcessiveorinefficienttradingcontrollingandmonitoringthetotalinvestmentprocesstomain-tainaconsistentinvestmentprogramThesefourelementsarepresentinanyinvestmentmanagementproblem,beitastrategicassetallocationdecision,anactivelyman-agedportfolio,oranindexfund—managedbottom-uportop-down,viatraditionalorqutativemethods.Figure1-1ThePerformancePyramidSuperiorPerformanceReturnForecastsProcessControlRiskControlCostControlInasimplerview,returnandriskaretheprotagonistandantagonistofinvesting.Accordingtoanoldadage,thetradeoffbetweenreturnandriskisthetradeoffbetweeneatingwellandsleeClearly,riskdoesn’tjustmattertoquants!well.7Ignoringriskishazardoustoyourportfolio.TheoptimalstrategyignoringriskplacestheIgnoringriskishazardoustoyourportfolio.Theoptimalstrategyignoringriskplacestheentireportfolioinonestock.Butnoinstitu-tionalinvestorfollowsthisstrategy.Henceriskconsiderationsmustimpacteveryinstitutionalportfolio.Unfortunately,theysometimesdonotimpactthemenough.Weneednotlookfartofindexamplesoffinaldisastersthatarosethroughlackofsufficientriskcontrol.ThedebaclesofOrangeCounty,BaringsBank,andthePiperJaffrayInstitutionalGovern-mentIncomeFundalltestifytothedangersofignoringorpoorlyunderstandingrisk.Butriskanalysisismorethanavoidingdisasters—itcaninfactenhanceopportunities.PeterBernsteinhasarguedthatalackofunderstandingofriskholdsbackeconomicdevelopment.1Moderneconomicgrowthrequiresunderstandingrisk.Whataretheexpectedreturnstoanewventure?Whataretherisks?Dothereturnsoutweightherisks?CanIhedgetherisks?Inmoderneconomies,thefutureisnotbeyondmanagement,notsimplysubjecttothewhimsofmanygods.Infact,theperiodwhichmarkedthedevelopmentofprobabilityandstatistics(duringandaftertheRenaissance)alsomarkedatimeofprofoundgrowthintrade,explo-ration,andwealth.Theideasofriskmanagementenabledthemod-erneconomicworld,accordingtoBernstein.Riskanalysisenhancedopportunities.WhileBernstein’sargumentmayseeminspiring—thoughnotofday-to-dayrelevance—infactthegoalofriskanalysisisnottominimizeriskbuttoproperlyweighrisksisleadstotakingmorerisk.return.Sometimesriskanaly-ThegoalofriskanalysisRiskisimportant.Itisacriticalelementofsuperiorinvestmentper-formance.Goodriskanalysisshouldprovidenotonlyanumber—aquficationofrisk—butinsight,especiallyinsightintothe“Perfor-mancePyramid.”Wehaveillustratedsuperiorperformanceasathree-dimensionalobject.Asinglerisknumberisonlyone-dimensional.Sowhatdowemeanbyinsight?1.SeePeterL.Bernstein,theGods:TheRemarkableStoryofRisk,JohnWiley&Sons,NewYork,1996.8U.S.EquityMVersion3(E3)RiskanalysisshoulduncovernotjustoverallRiskanalysisshoulduncovernotjustoverallrisk,butthelargestandsmallestbetsintheportfolio.Dothelargestbetscorrespondtothehighestexpectedreturns?Theyshould.Iftheydonot,theportfolioisn’tproperlybalngreturnandrisk.Arethebetstoolargeortoosmall?Whatisthe“worstcase”scenario?Howwilltheportfoliocomparetoitsbenchmark?Robustriskanalysiscanprovideanswerstoallthesequestionsaswellasinsighttoallinvestors.Inthisvolumewewilldiscussthehis-toryandcurrentpracticeofequityriskmingforsinglecountrymarkets.Othermethodsareusedfordifferentsecurities,suchasbondsorcurrencies,andfordifferentmarketstructures,suchastheglobalstockmarket.Theunderlyingmessageisclear:Theinvestorarmedwithsuperiormethodsofassessingandcontrollingriskpos-sessesasignificantcompetitiveedgeinmoderncapitalmarkets.1.WhyRiskisImportant910U.S.Equity10U.S.EquityMVersion3(E3)2.DefiningRiskdefinitionsInanuncertaininvestmentenvironment,investorsbearrisk.2.DefiningRiskdefinitionsInanuncertaininvestmentenvironment,investorsbearrisk.Riskisdefinedasthetotaldispersionorvolatilityofreturnsforasecurityorportfolio.Further,riskreflectsuncertaintyaboutthefuture.Wewilldefineriskasthestandarddeviationofthereturn.Riskisanabstractconcept.Aneconomistconsidersrisktobeexpressedina’spreferences.Whatisperceivedasriskyforoneindividualmaynotberiskyforanother.1Weneedanoperationalandthereforeuniversalandimaldefi-nitionofrisk.Institutionalmoneymanagersareagentsofpensionfundtrusteesandotherassetsponsors,whoarethemselvesagentsofthesponsoringorganizationand,ultimately,thebeneficiariesofthefund.Inthatsettingwecannothopetohaveaalviewofrisk.Weneedasymmetricviewofrisk.Institutionalmoneymanagersarejudgedrelativetoabenchmarkorrelativetotheirpeers.Themoneymanagersuffersasmuchifhedoesnotholdastockanditgoesupasifheheldalargerthadown.ageamountofthestockanditgoesWeneedaflexibledefinitionofrisk.Ourdefinitionofriskshouldapplytoindividualstocksandtoportfolios.Weshouldbeabletotalkaboutrealizedriskinthepast,andforecastriskoveranyfuturehorizon.Thedefinitionofriskthatmeetsthesecriteriaofbeinguniversal,symmetric,andflexibleisthestandarddeviationofreturn.2,3IfRPisaportfolio’stotalreturn,thentheportfolio’sstandarddeviationofThereisavastliteratureonthissubject.ThebooksofArrow,Raiffa,andBorchareagoodintroduction.Aneconomistwouldcallthestandarddeviationameasureofuncer-taintyratherthanrisk.Thereissomethingofadebatecurrentlyoverusingmeasuresof“downside”riskinsteadofvolatility.Giventhesymmetricnatureofac-tivereturns,U.S.institutionalinvestors’avoidanceofstrategieslikeportfolioinsurancewhichskewportfolioreturns,andthepracticallim-itationsofanalyzinglargeportfolios(>100names),downsideriskisin-appropriateorirrelevantforactivemanagement.Foradiscussion,seeRonaldKahnandDanStefek,“Heat,Light,andDownsideRisk,”1997.11StdP.Aportfolio’sexcessStdP.Aportfolio’sexcesseturnPreturnisdenotedbyPdiffersfromthetotalreturnRPbyaconstantRF,sotheriskoftheexcessreturnisequaltotheriskofthetotalreturn.Wewilltypicallyquotethisrisk,orstandarddeviationofreturn,onapercentperyearbasis.Wewillalsooccasionallyrefertothisqu tyasvolatility.1Theroughinterpretationofstandarddeviationisthatthereturnwillbewithinonestandarddeviationofitsexpectedvaluetwo-thirdsofthetimeandwithintwostandarddeviationsnineteentimesoutoftwenty.Figure2-1graphicallyillustratesthisfact.Figure2-1Risk:TheDispersionofReturnsThestandarddeviationisastatisticalmeasureofdispersionaroundanexpectedvalue—inthiscase,zero.1.Foramoreeddiscussionoftheseconcepts,pleaseseeRichardC.GrinoldandRonaldN.Kahn,ActivePortfolioManagement:Qu-tativeTheoryandApplications,ProbusPublishing,Chicago,IL,1995.12U.S.EquityMVersion3(E3)ExpectedValue–1Std.Dev. +1Std.Dev.-2% 0 2%1outof6yrs. 2outof3yrs. 1outof6yrs.RiskmeasurementArelatedriskmeasureisvariance,thestandarddeviationsquared.Theformulaeare:RiskmeasurementArelatedriskmeasureisvariance,thestandarddeviationsquared.Theformulaeare:Va??r2where:=return,=expectedormeanreturn,rStdxVarxEx=standarddeviationofx,=varianceofx,and=expectedvalueofx.Thestandarddeviationisthemorecommonriskindicatorsinceitismeasuredinthesameunitsasreturn.Ofcourse,ifthestandarddeviationisknown,thevariancecanbeeasilycomputedandviceversa.Othermeasures,includingvalue-at-riskandshortfallrisk,canbeeasilycomputedfromthestandarddeviation.AnexampleThestandarddeviationhassomeinterestingcharacteristics.Inpar-ticularitdoesnothavetheportfolioproperty.Thestandarddevia-tionofastockportfolioisnottheweightedaverageofthestandarddeviationsofthecomponentstocks.Forexample,supposethecorrelationbetweenthereturnsofStocks1and2is2,then:12.Ifwehaveaportfolioof50%Stock1and50%Stock0.5120.52220.510.5212PandP0.510.522.DefiningRisk13withsuchequalitybeingmaintainedonlyifthetwostocksareper-fectlycorrelated(12=1).Forrisk,thewholeislessthanwithsuchequalitybeingmaintainedonlyifthetwostocksareper-fectlycorrelated(12=1).Forrisk,thewholeislessthanthesumofitsparts.Thisisthekeytoportfoliodiversification.Figure2-2RiskReductionBenefitsofDiversification:ATwo-StockExampleFigure2-2showsasimpleexample.TheriskofaportfolioupfromandGeneralElectricisplottedthefractionofGEstockintheportfolio.Thecurvedlinerepresentstheriskoftheportfolio;thestraightlinerepresentstheriskthatwewouldobtainifthereturnsonandGEwereperfectlycorrelated.TheriskofGEis27.4%peryear;theriskofis29.7%peryear;andthetworeturnsare62.9%correlated.Thedifferencebetweenthetwolinesisanindicationofthebenefitofdiversificationinreducingrisk.RiskreductionthroughdiversificationWecanseethepowerofdiversificationinanotherexample.GivenaportfolioofNstocks,eachwithriskanduncorrelatedreturns,theriskofanequal-weightedportfolioofthesestockswillbe:N PNotethattheaverageriskis,whiletheportfolioriskisN.14U.S.EquityMVersion3(E3)Risk0.3000.2950.2900.2850.2800.2750.2700.2650.2600.2550% 25% 50% 75% 100%GEHolding62.9%correlated 100%correlatedForamoreusefulinsightintodiversification,assumenowthatthecorrelationbetweenthereturnsofallpairsofstocksisequalto.Thentheriskofanequallyweightedportfoliois:1N1N Foramoreusefulinsightintodiversification,assumenowthatthecorrelationbetweenthereturnsofallpairsofstocksisequalto.Thentheriskofanequallyweightedportfoliois:1N1N PInthelimitthattheportfoliocontainsaverylargenumberofcorre-latedstocks,thisbecomes:PTogetafeelforthis,considertheexampleofanequal-weightedportfolioofthe20MajorMarketIndexconstituentstocks.InDecember1992,thesestockshadaageriskof27.8%,whiletheequal-weightedportfoliohasariskof20.4%.1Equation2-6thenimpliesaagecorrelationbetweenthesestocksof0.52.Risksdon’taddacrossstocksandrisksdon’taddacrosstime.How-ever,variancewilladdacrosstimeifthereturnsinoneintervaloftimeareuncorrelatedwiththereturnsinotherintervalsoftime.Theassumptionisthatreturnsareuncorrelatedfromperiodtoperiod.Thecorrelationofreturnsacrosstime(calledautocorrela-tion)istozeroformostassetclasses.Thismeansthatvarianceswillgrowwiththelengthoftheforecasthorizonandtheriskwillgrowwiththesquarerootoftheforecasthorizon.Thus,a5%annualactiveriskisequivalenttoa2.5%activeriskoverthefirstquarterora10%activeriskoverfouryears.Noticethatthevarianceoverthequarter,year,andfour-yearhorizon(6.25,25,and100)remainsproportionaltothelengthofthehorizon.Weusethisrelationshipeverytimewe“annualize”risk—i.e.,stan-dardizeourrisknumberstoanannualperiod.Ifweexaminemonthlyreturnstoastockandobserveamonthlyreturnstandarddeviationofmonthly,weconvertthistoannualriskaccordingto:annual 12monthly1.ThesearepredictedvolatilitiesfromBARRA’sU.S.EquityM.2.DefiningRisk15DrawbacksofsimpleriskcalculationsThemathematicalcalculationofriskusingstandarddeviationofreturnsDrawbacksofsimpleriskcalculationsThemathematicalcalculationofriskusingstandarddeviationofreturnsisthereforestraightforwardandcanbeextendedtoanynum-berofsecurities.However,thisapproachsuffersfromseveraldraw-backs:Estimatingarobustcovariancematrixofreturnsrequiresdataforasmanyperiodsaswehavesecuritiestoanalyze.Forlargemarkets,suchastheU.S.stockmarket,theselongreturnshisto-riesmaysimplynotbeavailable.■Estimationerrormayoccurinanyoneperiodduetospuriousassetcorrelationsthatareunlikelytorepeatinasystematicfash-ion.■Asimplecovariancematrixofreturnsofferslittleinthewayofeconomicanalysis.Inotherwords,itislargelya“blackbox”approachwithlittleintuitivebasisorforecastability.■Forallthesereasons,finaleconomistshavesoughtformanyyearstominvestmentriskinmorenuancedways.Wewillnowturntoabriefhistoryoftheseefforts.EvolutionofconceptsThedevelopmentofequityriskconceptshasevolvedfromthemod-estandunscientificguessworkofearlyinvestmenttheorytothequtools.tativeanalysisandtechnicalsophisticationofmodernfin alBeforethe1950s,therewasnoconceptofsystematic,ormarket-related,return.Returnwasariseinthevalueofastockandriskwasadropinthevalueofastock.Theinvestor’sprimaryinvestment“Buyastock.Ifitgoesup,sellit.Ifitgoesdown,don’tbuyit.”WillRogers,1931toolswereintuitionandinsightfulfinalanalysis.Portfolioselec-tionwassimplyanactofassemblingagroupof“good”stocks.16U.S.EquityMVersion3(E3)Figure2-3DiversificationandRiskAsaportfoliomanagerincreasesthenumberofstocksinaportfolio,residual—ornon-market-related—riskisFigure2-3DiversificationandRiskAsaportfoliomanagerincreasesthenumberofstocksinaportfolio,residual—ornon-market-related—riskisdiversi-fied.Marketriskisundiversifi-able.Finaltheoristsbecamemorescientificandstatisticalintheearly“Diversificationisgood.”HarryMarkowitz,19521950s.HarryMarkowitzwasthefirsttoqufyrisk(asstandarddeviation)anddiversification.Heshowedpreciselyhowtheriskoftheportfoliowaslessthantheriskofitscomponents.Inthelate1950s,BreimanandKellyderivedmathematicallytheperilofignor-ingrisk.Theyshowedthatastrategythatexplicitlyriskoutperformedallotherstrategiesinthelongrun.1edforWenowknowhowdiversificationaffectsriskexposures.Itaveragesfactor-relatedrisk,suchasindustryexposures,andsignificantlyreducessecurity-specificrisk.However,diversificationdoesnoteliminateallriskbecausestockstendtomoveupanddowntogetherwiththemarket.Therefore,systematic,ormarket,riskcannotbeeliminatedbydiversification.Figure2-3showsthebalancebetweenresidualriskandmarketriskchangingasthenumberofdifferentstocksinaportfoliorises.Atacertainportfoliosize,allresidualriskiseffectivelyremoved,leavingonlymarketrisk.Asinvestmentmanagersbecamemoreknowledgeable,therewasapushtoidentifytheconceptualbasisunderlyingtheconceptsofrisk,diversification,andreturns.TheCapitalAssetPricingM(CAPM)wasoneapproachthatdescribedtheequilibriumrelation-shipbetweenreturnandsystematicrisk.WilliamSharpeearnedtheNobelPrizeinEconomicsforhisdevelopmentoftheCAPM.1.See,forexample,LeoBreiman,“InvestmentPoliciesforExpandingBusinessesOptimalinaLong-RunSense,”NavalResearchLogisticsQuarterly,Vol.7,No.4,December1960,pp.647–651.2.DefiningRisk17RiskofPortfolio(Standar