PORTFOLIO ANALYSIS AND INVESTMENT PAIlecture07 投资组合分析.docx
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PORTFOLIO ANALYSIS AND INVESTMENT PAIlecture07 投资组合分析.docx
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PORTFOLIOANALYSISANDINVESTMENTPAIlecture07投资组合分析
Lecture7:
TheMVapproach:
thecaseofnassets,feasibleportfolioset,efficientportfoliofrontier,optimumportfolio,diversification,idiosyncraticandsystematicrisk,lendingandborrowingopportunitiesandthecapitalmarketline,separationfundtheorem,portfoliotheorybenefits
TheMean-VarianceApproach-Thecaseofnassets(illustration)
TheriskreturnillustrationofindividualsharesfromtheLondonStockExchangeshowsthatthereexistsharesthataresuperiortoothers,inthesensethattheyawardinvestorswithhigherreturnsandlowerrisk.However,ifweexcludedthecaseofsharesthatunderperform,thenthereexistsapositiverelationshipbetweenriskandreturn,whichmeansthatshareswithhigherreturnsembedalsohigherrisk.
Figure2.10LondonStockExchange
Suppose,weareinvestigatingthecaseofsixshares(simulation)withthefollowingmeanandstandarddeviation,respectively:
Figure2.11ReturnandStandarddeviationof6shares
stock
return
St.dev
1
5%
7%
2
6%
9%
3
9%
14%
4
4%
7%
5
3%
6%
6
7%
13%
Fromthisfigureweobservethatthereisatrendaccordingtowhich,higherriskisawardedwithhigherreturns.Supposenow,thatthereexist15investors,eachofwhichconstructsaportfolioconsistingofthesesharesbasedonhisinformationalsetandhisrisktolerance.Thefollowingfigureillustratestheshareweightsthateachinvestorhasutilizedinordertoconstructhisportfolio(i.e.the1stinvestorhasinvestedmostofhismoneyonthe3rdstock,whilethe12thonthe5thstock).
Figure2.12Simulatedportfolios–portfolioweights
Feasibleportfolioset
Itisobviousthattheconstructionofaportfolioisamorecomplicatedissue,sincethereexist,manyshares(notonlysix).Inthefollowingfigurewehaveillustratedthe15portfolio’srisk-returnrelationship,aswellasadashedcurve.Theareabelowthedashedcurverepresentsthesetofallfeasiblerisk-returncombinations(allpossibleportfolios)andiscalledfeasibleportfolioset(FPS).Asitisobvious,theFPScontainsalsothe15investor’schoices.Thus,investorsbasedontheirexpectations,constructportfoliosthatcontributetotheformulationoftherequiredreturnswhichwouldcompensatethemfortherisktheyundertake.Asitisobvious,thereexistsharesthereturnsofwhichdonotaccountforthehighriskleveltheyembed,sufficiently.
Figure2.13FeasiblePortfolioSet
EfficientPortfolioFrontier
Rationalinvestors,whoareriskaverters,wouldpreferportfoliosthereturnofwhichismaximizedforaspecificlevelofrisk,orinversely,wouldpreferportfoliostheriskofwhichisminimizedforaspecifictargetreturn.Thus,rationalinvestors’choicesarerepresentedbythenorth-westpointsoffigure2.13.
Inthecaseofthe6shareswecouldconstruct20differentportfolioswithrespecttothecomponentsoftheportfolio.However,ineachcaseofthe20portfoliostherearemanydifferentcombinationswithrespecttotheshareweightsaninvestoriswillingtoapply.Inthecaseweinvestigatenshareswemayconstructmanyportfoliosofdifferentsizeandforeachoftheseportfoliosthereexistmanyotherchoicesregardingtheshareweights.Allpossiblecombinationsofnsharesthatformulateaportfolioofsizek(k Thus,areasonablequestioniswhetheraninvestor,shouldconsiderallthesecombinationsbeforeconstructinghisportfolio.TheModernPortfolioTheoryanswersthisproblem,sinceinvestorsshouldnotinvestigateallpossibleportfolios,butinsteadonlythoseportfoliosthatforaspecificlevelofriskofferthemaximumreturn,orinversely,thoseportfoliosthatforaspecifictargetreturnembedthelowerlevelofrisk.Theportfoliosthatfollowthispropertyarecalledefficient,andthecombinationofalltheseefficientportfolios,formstheefficientportfoliofrontier(EPF). Thus,rationalinvestorswouldformulateanewsetofportfolios,theEfficientPortfolioFrontier,whichisderivedbythefeasibleportfoliosetandsatisfiesthefollowingtwoprinciples: -foraspecifictargetportfolioreturn,embedsthelowerlevelofrisk(west) -foraspecificlevelofrisk,offersthehighestportfolioreturn(north) Morespecifically,investorsinordertoreachtheEPFshouldmoveinthenorth-westdirectionontherisk-returnillustrationofthefeasibleportfolioset. AllportfoliosthatbelongtotheEPFcontainthehighestleveloftheratioreturn/riskandconsequentlyrepresenttheoptimumchoicesforinvestors. Figure2.14FeasiblePortfolioSetandtheEfficientPortfolioFrontier Inthefollowingfigure(Figure2.15)weobservetheEPFandtwootherchoices,pointsKandK’.K’isnotafeasibleportfolio,whileKisnotefficient,sincethereexistotherportfolios(i.e.M,N)thatforthesameportfolioreturnembedlowerrisk,orforthesamelevelofrisktheyofferhigherreturns,respectively.Thus,rationalinvestorsthatareriskaverterswillneverchooseaportfoliointheareabelowtheEPF.Thefinallydecisionoftheinvestorwillbechosensoastomaximizehissatisfaction. Figure2.15EfficientPortfolioFrontier-EPF TheEfficientPortfoliosofouranalysisconsistonlyofequities,i.e.financialproductsthatembedrisk.Thus,theslopeofthetangencyoftheEPFcurverepresentstheextrariskthatinvestorsarewillingtoundertakeforamarginalincrementontheirportfolios’returns.AllportfoliosontheEPFhavethemaximumreturn/riskratio. Optimumportfolio Asitisknowtheutilityfunctionofariskaverterexpressedonthefirsttwomomentsisconvex,andeveryinvestoriswillingtomaximizehisutility. Figure2.16Riskaverterprofile(convexfunction) Byconsiderationoftheutilityfunctionintheinvestigationofaportfoliochoice,thereexistaportfolioontheEPFforwhichinvestor’sutilityismaximized.Thisportfolio(A)iscalledoptimum,sinceitofferstheinvestorthemaximumexpectedutilityamongallEPFportfoliosandisidentifiedatthepointatwhichtheslopeoftheutilityfunctionisequaltotheslopeoftheEPF,asshownonFigure2.17. Figure2.17OptimumPortfolio However,theutilityfunctionisalocusthatexpressestheindividualinvestor’sexpectationsandasaresulttheoptimumportfolioshouldbedifferentamonginvestors. Foramoreriskaverterinvestor(U1)theoptimumportfolioisAwhileforalessriskaverterinvestor(U2)theoptimumportfolioisB,asillustratedbelow: Figure2.18Optimumportfoliosfortworiskaverterinvestors(withdifferentrisktolerance) -AswehavealreadyseentheconcavityoftheEPFdependsonthecorrelationstructureoftheequityreturns,inversely,andasaresultlowercorrelationstructureofassetreturnswouldmaximizeinvestor’ssatisfaction. Diversification Theportfoliovarianceisgivenbythefollowingequation: Alternatively,byisolatingtheelementsofthemaindiagonalofthevar-covariancematrixweget: Assumingthatallshareshavethesamevariance(σ2),thateachpairofshareshasthesamecovariance(cov)andthattheallsharesintheportfoliohavethesameweight(w=1/n)thenweget: Thisrelationshipimpliesthatasthesizeoftheportfolioisincreased(n∞)thecontributionofindividualsharesontheportfoliovariance,isdecreased,sincetheportfoliovariancedependsmainlyonthecovariancebetweenshares’returns. Figure2.19Portfoliosizeanddiversification Asthenumberofsharesontheportfolioisincreasedtheidiosyncraticriskofeachshareisomitted(diversified),whilethesystematicriskisnotaffected. -idiosyncraticrisk: Itisthepartoftheriskofashare,duetothespecificcharacteristicsofthecorrespondinglistedfirm,suchasthemanagement,thetechnologicalfactors,theequipment,thesectorandmanyotherfactorsandiscalledidiosyncraticorspecificornonsystematicrisk.Whenashareisincludedinthemarketportfolio,partoftheshare’sriskisgoingtobeomittedgiventhatthecorrelationcoefficientofthereturnsoftheshareandthemarketportfolioislow.Thisportionofshareriskisnotofinterestwhendealingwithwelldiversifiedportfolios,inthesensethatunanticipatedlossesfromoneshareontheportfolioarehedgedbythegainsofanotherone. -systematicrisk Itisthepartoftheriskofashare,duetothecovariancebetweentheportfolio’sshares.Morespecifically,itisthepartoftheriskthatisundiversifiablebecauseitisassociatedwithmanyfinancialandmacroeconomicvariables(thenationalorinternationalpoliticalregime,theinflation,themonetarypolicy,thetaxpolicy,thelevelofinterestrates,theinvestors’expectationsaboutfutureeconomicstates)andiscalledsystematicormarketrisk.Thispartoftheriskisundiversifiableandallinvestorsshouldundertakethiswhenincludingthissharetotheirportfolios. Thus,financialmarketsdorewardinvestorsonlyforthesystematicriskofshares,sincethespecificriskcouldbeeliminatedinawelldiversifiedportfolio.Inadevelopedandcompletefinancialmarket,investorsshouldconsideronlythesystematicriskintheformulationoftheirportfolios,becauseonlythispartofshares’riskisnotomittedinawelldiversifiedportfolio. (totalrisk)=(systematicrisk)+(specificrisk) LendingandBorrowingopportunitiesandtheCapitalMarketLine Supposenow,thataninvestorcouldinvestonequity(shares)andrisklessassets,suchasgovernmentbonds(rf).Thefixedincomesecuritiesembednoriskwhichmeansthattheirstandarddeviationaswellastheircorrelationorcovariancewithotherrandomvariablesisequalwithzero. Now,s
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