在风险资产和无风险资产间的资产配置

在风险资产和无风险资产间的资产配置InternationalInvestmentsChapter6CAPITALALLOCATIONBETWEENRISKYANDRISK-FREEASSETS1GuidelinesI.在风险资产和无风险资产间的资产配置1.风险资产和无风险资产2.在风险资产和无风险资产间的资产配置II.一个风险资产和一个无风险资产的组合的风险和收益III.最佳资产分配决策IV.小结I. AssetAllocationbetweenRiskyandRisk-freeAssets1.TheRisk-freeAssetandtheRiskyAsset2.AssetAllocationbetweenRiskyandRisk-freeAssets

II. RiskandReturnsonaPortfolioofOneRiskyandOneRisk-freeAssetIII.OptimalAssetsAllocationChoiceIV.Summary2I.Allocationbetweenriskyassetsandrisk-freeassetsRisk-freeasset:Theinstrumentsareshortterm,highlyliquiddebtsecuritiessuchasU.STreasurybills,banker’sacceptances(BAs),commercialpaper(CP),andbankcertificatesofdeposit(CDs).Riskyasset:Anyformsoffirm’slong-termborrowing,likebondsandstocks,tradedinthecapitalmarket.Thechoiceoftheassetclassesinwhichtoinvestcanbebetweentherisk-freeassetsandtheriskyassets.3Allocationbetweenriskyassetsandrisk-freeassetsCapitalallocationdecisionisconcerningachoiceacrossassetclasses,notoneclassorsecuritiesofonefirm.Regardlesshowmanysecuritiesweinvestineachassetsclass,wetreateachassetsclassasasingleasset.Aportfolioofoneriskyassetandonerisk-freeassetislessriskythanholdingriskyassetsonly.4Allocationbetweenriskyassetsandrisk-freeassets

IfweformaportfoliowithWr,theproportionoftheinvestmentfundintheriskyasset,andWfor(1－Wr),intherisk-freeasset,then:E(RP)=WrE(Rr)+(1－Wr)Rf (6.1)whereE(Rr):theexpectedrateofreturnofriskyasset.SupposethatE(Rr)=10%,σr=18%,Rf=4%:E(RP)=0.10×Wr+0.04×(1－Wr) E(RP)=Rf+[E(Rr)－Rf]×Wr=0.04+(0.10－0.04)×Wr

thetotalreturnontheirinvestmentincreaseddependsonthemagnitudeoftheriskpremiumandinvestor'spositionintheriskyasset.5Allocationbetweenriskyassetsandrisk-freeassets

σP=Wr×σr (6.2)whereσrdenotesthestandarddeviationontheriskyasset’sexpectedreturn. IfWr=50%,σr=18%,then,

σP=0.5×0.18=0.096CapitalAllocationLine

Accordingtothelinearnatureofportfolio’sexpectedrateofreturnanditsriskasinequation(6.1)and(6.2),wecanillustrateitusingadiagramwithstandarddeviationonthehorizontalaxis,andexpectedrateofreturnontheverticalaxis.7Figure6.1 CapitalAllocationLine

Rf

=4%E(Rr)=10%100%inriskyassetσr

=18%

CAL(CapitalAllocationLine)σpE(RP)BA100%inrisk-freeasset8CapitalAllocationLineIfweconnectallportfoliopointswithdifferentweightsbeingthefractionofthetotalportfolioinvestedineachasset,wegotastraightlinewhichislimitedbetweenpointsAandB.Thislineiscalledthecapitalallocationline(CAL).Theslopeofthelineis:

(6.3)

9TheriskfreerateandriskpremiumRearrangeequation6.2toget:Wr=σP/σr.Thenwesubstituteitforwrinequation6.1toobtaintherelationshipbetweentheexpectedreturnandstandarddeviationoftheportfolioas:E(RP)=Rf+Wr[E(Rr)－Rf]=Rf+σP/σr[E(Rr)－Rf] (6.4)Equation6.4showstheexpectedreturnoftheportfolioisalinerfunctionofitsstandarddeviation,withaninterceptofRf

andaslopeof Theslopeofthelineiscalledthereward-to-variabilityratio10CapitalAllocationLinethereward-to-variabilityratio

standsfortheincreaseintheexpectedreturnperunitofadditionalstandarddeviationofthecompleteportfolio.Otherthingsequal,aninvestorwouldpreferasteeper-slopingCAL,becausethatmeanshigherexpectedreturnsforanygivenlevelofrisks.Therisk-expectedreturnsonanyinvestor’spositioncanbecalculatedpreciselyaccordingtoequation6.1and6.2. E.g.,ifaninvestorput0.2ofhiscompleteportfolioinriskyassets,then:E(RP)=0.10×0.2+0.04×0.8=5.2%

σP=0.18×0.2=3.6%11RiskToleranceinvestorswithhighrisktolerancebeinglowlevelsofrelativeriskaversion.Recalltheutilityfunction:U=E(RP)－0.5RAσP (6.5)Andequation6.1and6.2: E(RP)=Rf+[E(Rr)－Rf]Wr

σP=Wrσr

SubstitutingtheexpressionofWrinequation6.1togive:12Nowtheportfolioexpectedreturnandvarianceontherighthandsideoftheutilityfunctioncanbesubstitutedbyaboveexpressions:III.OptimalAssetAllocationChoice13Sincetheinvestorisattemptingtomaximizeutility,U,bychoosingthebestpositioninriskyasset,Wr,wefindthederivativeofthefunctionwithrespecttoWr,andsettingtheexpressiontozero.ThenweobtaintheequationforWr*,whichistheoptimalpositionfortherisk-averseinvestorintheriskyassetthatmaximizesU:

(6.6)OptimalAssetAllocationChoice14ConsideraninvestorwithRA=3,isconfronting:one-monthTreasurybillswithrf=4%,andariskyassetwithE(Rr)=10%,andσr=18%,hewoulddecidethechoiceofWr*toconstructanoptimalportfoliothatmaximizesU:Therefore,place62%ofhistotalfundintheriskyassetand38%intheTreasurybills,toachieveapossiblemaximumutility. E(RP)=0.62×0.10+0.38×0.04=7.72%

σP =0.62×0.18=11.16%OptimalAssetAllocationChoice15Table6.1 UtilitylevelsforvariouspositionsinriskyassetsRA=3

asyoumaysee,around0.6,makingtheutilityatitsmaximum.16Figure6.3Differentportfoliosfordifferentinvestors4%9.56%6.78%16.67%8.33%A=4A=2HLIndifferencecurvefortheinvestorwithlowRA.IndifferencecurvefortheinvestorwithhighRA.E(R)17OptimalAssetAllocationChoiceE.g.,forRA=4,giventhat,E(Rr)=10%,σr

2=0.0324andRf=4%,

Thisimpliesthattheinvestorwouldchoosetheportfoliowithaproportionof46.3%intheriskyassetand53.7%intherisk-freeasset.

E(RP,H)=0.463×0.10+0.537×0.04=6.78%

σP,H =0.463×0.18=8.33%ThisportfolioissymbolizedbyHinthegraph.18OptimalAssetAllocationChoiceForalessrisk-averseinvestorwithRA=2,thetangencypointLrepresentstheportfoliohewouldchooseinwhichalargeproportionoftheportfolioconsistsofriskyasset. E(RP,L)=0.9259×0.10+0.0741×0.04=9.56%

σP,L=0.9259×0.18=16.67%19IV.SummaryWeoftenrefertomoneymarketfundsasrisk-freeassets.Whenweformaportfolioconsistingoneriskypo