《金融学》（第二版） 课件 （英文） Chapter12 Portfolio Opportunities and Choice

Chapter12:

PortfolioOpportunities andChoiceObjectiveTounderstandthetheoryofpersonalportfolioselectionintheoryandinpractice1Chapter12Contents12.1Theprocessofpersonalportfolioselection12.2Thetrade-offbetweenexpectedreturnandrisk12.3Efficientdiversificationwithmanyriskyassets2ObjectivesTounderstandtheprocessofpersonalportfolioselectionintheoryandpractice3IntroductionHowshouldyouinvestyourwealthoptimally?PortfolioselectionYourwealthportfoliocontainsStock,bonds,sharesofunincorporatedbusinesses,houses,pensionbenefits,insurancepolicies,andallliabilities4PortfolioSelectionStrategyTherearegeneralprinciplestoguideyou,buttheimplementationwilldependsuchfactorsasyour(andyourspouse’s)age,existingwealth,existingandtargetlevelofeducation,health,futureearningspotential,consumptionpreferences,riskpreferences,lifegoals,yourchildren’seducationalneeds,obligationstoolderfamilymembers,andahostofotherfactors512.1TheProcessofPersonalPortfolioSelectionPortfolioselectionthestudyofhowpeopleshouldinvesttheirwealthprocessoftradingoffrisk&expectedreturntofindthebestportfolioofassets&liabilitiesNarrowerdfn:consideronlysecuritiesWiderdfn:housepurchase,insurance,debtBroaddfn:humancapital,education6TheLifeCycleTheriskexposureyoushouldacceptdependsuponyourageConsidertwoinvestments(rho=0.2)Security1hasavolatilityof20%andanexpectedreturnof12%Security2hasavolatilityof8%andanexpectedreturnof5%7PriceTrajectoriesThefollowinggraphshowthethepriceofthetwosecuritiesgeneratedbyabivariatenormaldistributionforreturnsThemoreriskysecuritymaybethoughtofasashareofcommonstockorastockmutualfundThelessriskysecuritymaybethoughtofasabondorabondmutualfund89InterpretationoftheGraphThegraphisplottedonalogscaleinsothatyoucanseetheimportantfeaturesThemagentabondtrajectoryisclearlylessriskythanthenavy-bluestocktrajectoryTheexpectedpricesofthebondandthestockarestraightlinesonalogscale10InterpretationoftheGraphRecallthelogscale:thevolatilityincreaseswiththelengthoftheinvestmentYoubegintoformtheconjecturethatthechancesofthestockpricebeinglessthanthepricebondishigherinearlieryears11GeneratingMoreTrajectoriesThiswasjustoneofaninfinitenumberoftrajectoriesgeneratedbythesame2means,2volatilities,andthecorrelationIhavenotcheatedyou,thiswasindeedthefirsttrajectorygeneratedbythestatisticsthefollowingtrajectoriesarenotreorderednoreditedInstructor:Onslowercomputerstheremaybeadelay121314…andLotsMore!15FromConjecturetoHypothesisYouareprobablyreadytomakethehypothesisthattheprobabilityofthehigh-risk,high-returnsecuritywillout-performthelow-risk,low-returnincreaseswithtime16But:Ipromisedtobeperfectlyfrankandhonest(pfah)withyouabouttheorderingofthesimulatedtrajectoriesThenexttrajectorytrulywasthenexttrajectoryinthesequence,honest!1718ExplanationThebondandthestockendupataboutthesameprice,whentheexpectedpricesaremorethanamagnitudeapartThereiseitheraverygoodexplanationforthis,orthereisaveryhighprobabilitythatIhavebeenmuchlessthanperfectlyfrankandhonestwithyou19AnotherViewoftheModelAlittlemathematics,andweareabletogeneratethefollowingpricedistributionsforthestockandthebondfor2,5,10,and40yearsintothefuture2021Thereisalotgoingonhere,sowewillfurtherconstrainourviewFirstlookatstockpricesoveraperiodof10yearsThepricesaredistributedaccordingtothelognormaldistribution2223Notethescaleis$0to$800thedistributiondiffusesanddriftstowardshigherpriceswithtimethediffusionismorepronouncedintheearlieryearsthaninthelateryearsyoumayseethatthemode,median,andmeanappeartodriftapartwithtime24BondinTimeYouwillrecallthatifyouinvestina5-yeardefault-freepurediscountbondfor5years,thereturnisknownwithcertaintyToavoidthiseffect,assumeweinvestinshorttermbonds,androllthemoverastheymature2526Notethescaleisnow$0to$400(not$0to$800asinthecaseofthestock)weobservethesamekindofdiffusionanddriftbehavior,andthereislessofeach(remembertoadjustforthescale)27ContrastofTrajectoriesandDistributionsThepricedistributionsandthetrajectoriesweregeneratedfromthesamedistribution.ButTheydonotseemtoagreeThedistributionsappeartoproducemuchloweraverages(expectedreturns)thanthetrajectories28MeatyTailsTheresolutionisthatthedistributionshavemuchmeatiertailsthanyourintuitionallows,pushingthemedianandmeanfurtherandfurtherfromthemodewithtimeTheregionwherethelefttailappearstohavedriftedintoinsignificancehasaprofoundaffectonthemean29StockandBondsDistributionsComparedattheSameTimesThenextsequenceofslidescontraststhedistributionofstockandbondpricesat1,2,5,10,and40intothefutureSomeoftheslideshavedifferentmeasuresofcentraltendencyindicatedNotethebehaviorofthesestatisticsastimeincreases30Mode=104Mode=106Median=104Mean=104Median=111Mean=1133132Mode=122Mode=135Median=126Mean=128Median=165Mean=1823334Mode=503Mode=1,102Median=650Mean=739Median=5,460Mean=12,15135SlideSequenceSummaryThenexttablesummarizesthedriftsofthemeasuresofcentraltendencyNotethatthemeansdoinfacttiebacktothetrajectoriesThelast(anomalous?)trajectorynotanuncommonoccurrence,andIwaspfahwithyou3637ImplicationforInvestorsIfyouareolder,theaverageremaininglifeoftheinvestmentisrelativelyshort,andthereisalargerprobabilitythataninvestmentintheriskysecuritywillresultinalossThisisnotseriousifyouhavesubstantialassets,inwhichcaseyoucanaffordtotaketherisk,andenjoyhigherexpectedreturns38ImplicationforInvestorsIfyouareyounger,theaverageremaininglifeofretirementinvestmentislonger,andthereisonlyasmallprobabilitythataninvestmentintheriskysecuritywillbelessthanthe“safer”oneInvestinginthelessriskysecuritywillalmostalwaysresultinasignificantlysmallerretirementincome39ImplicationforInvestorsRelativelyearlyduringatypicallifecycle,theremaybeaneedtoliquidatesomeinvestedfunds,perhapsforahousedeposit,achild’seducation,oranuninsuredmedicalemergencyInthecasewhereliquidatinganinvestmentearlymaydamagelong-termgoals,someprecautionaryfundsshouldbekeptinlower-risksecurities40TimeHorizonsPlanninghorizonThetotallengthoftimeforwhichoneplansDecisionhorizonThelengthoftimebetweendecisionstoreviseaportfolioTradinghorizonTheshortestpossibletimeintervaloverwhichinvestorsmayrevisetheirportfolios41ComputingLifeExpectancyMortalitytablesmaybeorganizedasthreecolumns:actuaryage,deaths/yearper1000livebirths,andremaininglifeexpectation.Note:ifyousurvivefrom60to65,forexample,theexpecteddateofyourdeathadvancesby3to4yearsyoungwomenhaveahigherlifeexpectationthanmen,butthisislostwithadvancingage42UsefulInternetAddressTheSocietyofActuariesmaintainawebsitethatprovidesdetailedmortalitytables,interactivecomputermodels,mortgageexperiences,careerinformation,andcurrentresearchpapers434445LifeExpectancy05101520256065707580859095AgeRemainingExpectedLifeMExLifeFExLife46RiskToleranceYourtoleranceforbearingriskisamajordeterminantofportfoliochoicesItisthemirrorimageofriskaversionWhateveritscause,wedonotdistinguishbetweencapacitytobearriskandattitudetowardsrisk47RoleofProfessionalAssetManagersMostpeoplehaveneitherthetimenortheskillnecessarytooptimizeaportfolioforriskandreturnProfessionalfundmanagersprovidethisserviceasindividuallydesignedsolutionstothepreciseneedsofacustomer($$$$)asetoffinancialproductswhichmaybeusedtogethertosatisfymostcustomergoals($$)4812.2Trade-OffbetweenExpectedReturnandRiskAssumeaworldwithasingleriskyassetandasinglerisklessassetTheriskyassetis,intherealworld,aportfolioofriskyassetsTherisk-freeassetisadefault-freebondwiththesamematurityastheinvestor’sdecision(orpossiblythetrading)horizon49Trade-OffbetweenExpectedReturnandRiskTheassumptionofariskyandrisklesssecuritysimplifiestheanalysis50TheRisk-RewardTrade-OffLine51CombiningtheRisklessAssetandaSingleRiskyAssetAssumethatyouinvestW1proportionofyourwealthinsecurity1andproportionW2ofyourwealthinsecurity2Youmustinvestineither1or2,soW1+W2=1Let2betherisklessasset,and1betheriskyasset(portfolio)52CombiningtheRisklessAssetandaSingleRiskyAssetYourstatisticsbackgroundtellsyouhowtodeterminetheexpectedreturnandvolatilityofanytwo-securityportfolio1.Formanewrandomvariable,thereturnoftheportfolio,RP,fromthetwogivenrandomvariables,R1andR2RP=W1*R1+W2*R253CombiningtheRisklessAssetandaSingleRiskyAssetTheexpectedreturnoftheportfolioistheweightedaverageofthecomponentreturnsmp=W1*m1+W2*m2mp=W1*m1+(1-W1)*m254CombiningtheRisklessAssetandaSingleRiskyAssetThevolatilityoftheportfolioisnotquiteassimple:sp=((W1*s1)2+2W1*s1*W2*s2+(W2*s2)2)1/255CombiningtheRisklessAssetandaSingleRiskyAssetWeknowsomethingspecialabouttheportfolio,namelythatsecurity2isriskless,sos2=0,andspbecomes:sp=((W1*s1)2+2W1*s1*W2*0+(W2*0)2)1/2sp=|W1|*s156CombiningtheRisklessAssetandaSingleRiskyAssetInsummarysp=|W1|*s1,And:mp=W1*m1+(1-W1)*rf,So:IfW1>0,mp=[(rf-m1)/s1]*sp+rf

Elsemp=[(m1-rf)/s1]*sp+rf

57ReflectionTherisk-freerate,rf,theriskysecurity’sexpectedrateofreturn,m1,andvolatility,s1,areconstants,sowehavea“ray”that“reflects”fromtheexpectedreturnaxesatmp=rf58IllustrationConsiderthesetofallportfoliosthatmaybeformedbyinvesting(longandorshort)inariskysecuritywithavolatilityof20%andanexpectedreturnof15%arisklesssecuritywithavolatilityof0%andaknownreturnof5%5960Sub-OptimalInvestmentsInvestmentsonthehigherpartofthelinearealwayspreferred(bynormalfolk)toinvestmentsonthelowerpartoftheline,soforourcurrentpurposeswemayignorethelowerlineThatis,wewillnotselltheriskyassetshortandinvesttheproceedsintherisklesssecurity61Longriskyandshortrisk-free

Longbothriskyandrisk-free100%Risky100%Risk-less62ObservationsAninvestorwithalowrisktolerancemayinvestinaportfoliocontainingasmall%ofriskysecurities,andacorrespondinglyhigher%ofrisklesssecuritiesAninvestorwithahightoleranceforriskmaysellrisk-freesecuritieshedoesnotown,andinvesttheproceedingintheriskyinvestmentTheybothusethesametwosecurities63ObservationsThegraphhasbeenlabeledthe“capitalmarketline”alittleprematurelyWewillsoondiscoverthatiftheriskysecurityisthemarketportfolioofriskysecuritiesinvestorshavesimilarexpectationsandtimehorizonsAllinvestorswillinvest(longorshort)inthemarketportfolioandrisk-freesecurityThelinejoinsthecapitalmarketsforriskyandrisk-lesssecurities64AchievingaTargetExpectedReturn(1)Yourbosshasjustreadanad’thatincludedthedatafortheJanusTwentyFund(ScientificAmerican,Sept1998,page6)“Youbeatthem,orI’llfindanotherportfoliomanager”,shequips“Wrongwaytocomputereturn?”youventure,asyourushforthedoor65MutualFundAverage%TotalReturns66Toobtaina20%ReturnYousettleona20%return,anddecidenottopursueonthecomputationalissueRecall:mp=W1*m1+(1-W1)*rf

Yourportfolio:s=20%,m=15%,rf=5%So:W1=(mp-rf)/(m1-rf)=(0.20-0.05)/(0.15-0.05)=150%67Toobtaina20%ReturnAssumethatyoumanagea$50,000,000portfolioAW1of1.5or150%meansyouinvest(golong)$75,000,000,andborrow(short)$25,000,000tofinancethedifferenceBorrowingattherisk-freerateismoot68Toobtaina20%ReturnHowriskyisthisstrategy?sp=|W1|*s1=1.5*0.20=0.30Theportfoliohasavolatilityof30%69ImportantObservationItdoesn’trequiremuchskilltoleverageaportfolio;stockbrokerswillletmostinvestorstrade“onmargin”Whenevaluatinganinvestment’sperformance,youmustexamineboththeriskandtheexpectedreturn70ReturningtotheExampleAdvertisementsformutualfundsdonotgenerallydiscloseaquantifiablemeasureofrisk,andJanusisnoexceptionTheadvertised“JanusTwentyFund”returnsarecompletelymeaninglessfromafinancialpointofviewMoreinformationisneeded71ReturningtotheExampleYoucanleveragethefundsexpectedreturnsupordownIfyouwantanexpectedreturnsof10%,or,20%,30%,40%,50%,60%…youcanhaveit(undertheconditionyoucancontinuetoborrowattherisk-freerate)72HowShouldmyBossJudgemyFund’sPerformance?ItisalittleearlytoanswerthisquestionIftheriskysecurityisthemarketportfolio,thengivenyourportfolio’srisk,consistentreturnsabovetheCMLlinemayappearappealing73PortfolioEfficiency74PortfolioEfficiencyAnefficientportfolioisdefinedastheportfoliothatofferstheinvestorthehighestpossibleexpectedrateofreturnataspecificriskWenowinvestigatemorethanoneriskyassetinaportfolio7512.3EfficientDiversificationwithManyRiskyAssetsWehaveconsideredInvestmentswithasinglerisky,andasingleriskless,securityInvestmentswhereeachsecuritysharesthesameunderlyingreturnstatisticsWewillnowinvestigateinvestmentswithmorethanone(heterogeneous)stock76PortfolioofTwoRiskyAssetsRecallfromstatistics,thattworandomvariables,suchastwosecurityreturns,maybecombinedtoformanewrandomvariableAreasonableassumptionforreturnsondifferentsecuritiesisthelinearmodel:77TheRisk-RewardTrade-OffCurve:RiskyAssetsOnly78EquationsforTwoSharesThesumoftheweightsw1andw2being1isnotnecessaryforthevalidityofthefollowingequations,forportfoliosithappenstobetrueTheexpectedreturnontheportfolioisthesumofitsweightedexpectations79EquationsforTwoSharesIdeally,wewouldliketohaveasimilarresultforriskLaterwediscoverameasureofriskwiththisproperty,butforstandarddeviation:80MnemonicThereisamnemonicthatwillhelpyourememberthevolatilityequationsfortwoormoresecuritiesToobtaintheformula,movethrougheachcellinthetable,multiplyingitbytherowheadingbythecolumnheading,andsumming81Variancewith2Securities82Variancewith3Securities83Note:ThecorrelationofawithbisequaltothecorrelationofbwithaForeveryelementintheuppertriangle,thereisanelementinthelowertrianglesocomputeeachuppertriangleelementonce,andjustdoubleitThisgeneralizesintheexpectedmanner84CorrelatedCommonStockThenextslideshowsstatisticsoftwocommonstockwiththesestatistics:meanreturn1=0.15meanreturn2=0.10standarddeviation1=0.20standarddeviation2=0.25correlationofreturns=0.90initialprice1=$57.25initialprice2=$72.6258586ObservationThestatisticsindicatethatonesecurityappearstototallydominatetheotherSecurity1hasalowerriskandhigherreturnthansecurity2Inanefficientmarket:Wouldn’teverybodyshort2,andbuy1?Wouldn’tsupplyanddemandthencausetherelativeexpectedreturnsto“flip”?87DoesitHappen?ThepurposeofselectingtwoshareswiththisparadoxicalformistoillustrateanimportantpointlaterThiskindofrelationshipdoesoccurintherealworld88APairofPriceTrajectoriesThenextgraphshowsatrajectoryoftwosharepriceswiththestatisticswehaveselected8990ObservationIfyouwereto“cutapiece”fromonetrajectory,re-scaleitforrelativepricedifferences,andslideitovertheother,youwouldobservethatbothtrajectoriesbehaveinabroadlysimilarmanner,buteachhasindependentbehavioraswellQuickconfirmationisseenintheregion1to4yearswherepricesareclose91CorrelationThetwosharesarehighlycorrelatedTheytrackeachotherclosely,butevenadjustingforthedifferentaveragereturns,theyhavesomeindividualbehavior9293ObservationShortingthehigh-risk,low-returnstock,andre-investinginthelow-risk,high-returnstock,createsefficientportfoliosShortinghigh-riskby80%ofthenetwealthcratesaportfoliowithavolatilityof20%andareturnof19%(c.f.15%onsecurity1)Shortingby180%givesavolatilityof25%,andareturnof24%(c.f.10%onsecurity2)94ObservationInordertogenerateaportfoliothatgeneratesthesamerisk,butwithahigherreturnComputetheweightsoftheminimumportfolio,W1(min-var),W2(min-var)(Formulaegivenlater)UsetherelationshipWi(sub-opt)+Wi(opt)=2*Wi(min-var)95ObservationAnotherwaytogeneratethetwosecuritiesistoformtwoportfoliosconsistingofariskyandarisklesssecuritythateachmeettheefficientfrontierResult:twoportfoliosthatarelongtheriskysecurity,andshorttherisklesssecurityShortoneoftheportfoliosandinvestintheothertogenerateoneofthedesiredefficientportfoliosRepeattogeneratetheotherProvethattherisklesssecuritybecomesirrelevant96OptimalCombinationofRiskyAssetsThefollowingslidesaresamplesofthecomputationsusedtogeneratethegraphs97FragmentsoftheOutputTable98SampleoftheExcelFormulae99FormulaeforMinimumVariancePortfolio100SelectionofthePreferredPortfolio101FormulaeforTangentPortfolio102Example:What’stheBestReturngivena10%SD?103SelectingthePreferredPortfolioTheprocedureisasfollowsFindtheportfolioweightsofthetangentportfoliooftheline(CML)through(0,rf)DeterminethestandarddeviationandexpectationofthispointConstructtheequationoftheCMLApplyinvestmentcriterion