PORTFOLIO ANALYSIS AND INVESTMENT PAIlecture07 投资组合分析.docx

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