英國帝國理工課件PPT（計算金融單周期馬科維茨模型）Computational FinanceSingle Period Markowitz Model

Panos

ParpasSinglePeriodMarkowitzModel381ComputationalFinanceImperialCollegeLondonTopicsCoveredNotationandTerminologyRandomvariables,mean,(co)variance,correlationAssetreturn,portfolioreturnandriskPortfolioOptimisationOptimalassetallocation,riskmanagementShortsale,diversificationMean-Variancemodel,efficientfrontierTerminology

RandomVariable

yisarandomvariable,takesfinitenumberofvalues,yjforj=1,2,…,m.aprobability(associatedwitheachevent)representstherelativechanceofanoccurrenceofyj.suchthat

ExpectedValue(Meanvalueormean)

averagevalueobtainedbyregardingprobabilitiesasfrequencies

Variance

measureofpossibledeviationfromthemeanrelativefrequencyfinitenumberofpossibilitiesExpectedvalueofsquaredvariablehowmuchytendstovaryfromitsmeanTerminologyCovarianceoftworandomvariablesCorrelationbetweentworandomvariablessigndefinesdirectionoftherelationshipAssetReturnsAsset:investmentinstrumentthatcanbeboughtandsolduncertainassetprices–thereturnisrandomuncertaintydescribedinprobabilistictermsAssetreturn:youpurchaseanassettodayandsellitnextyearTotalreturnonthisinvestmentisdefinedasRateofreturn:

RateofreturnactsmuchlikeaninterestrateReturnsPortfolioReturnConsidernriskyassetstoformaportfolioandinvestamongnassetsSelectanamountinvestedintheithassetTheamountsinvestedcanbeexpressedasfractionsofthetotalinvestmentisfractionorweightofassetiinportfolioLetthetotalreturnofassetibe.Theamountofmoneygainedattheendoftheperiodonassetiis

Thetotalreturnoftheportfolio

TherateofreturnofportfolioPortfolioReturnBoththetotalreturnandtherateofreturnoftheportfolioofassetsareequaltotheweightedsumofthecorrespondingindividualassetreturns,withtheweightofanassetbeingitsrelativeweightintheportfolio.ExpectedreturnoftheportfolioistheweightedsumoftheindividualexpectedratesofreturnExampleConsideraportfolioofarisk-freeassetandtworiskystocksandthreeequallylikelystatesFindtheexpectedreturnoftheportfolioformedby3assetsScenariotreewith3states

rootrepresentstodayfutureuncertaintyisdiscretised

by3events(states)Example:ExpectedReturnsTheexpectedreturnofriskfreeassetisForstockAandB,expectedreturnoftheportfolioTheexpectedreturnoftheportfolioForequallyweightedportfolioRiskRisk:

achancethatinvestment’sactualreturnwillbedifferentthanexpected–includeslosingsomeoralloforiginalinvestmentriskaverseandrisk-seeking(loving)Systematicrisk:

influencesalargenumberofassets,suchaspoliticalevents–impossibletoprotectyourselfagainstthisrisk

Unsystematicrisk:

(specificrisk)

affectsverysmallnumberofassets.Forexample,newsthataffectsaspecificstocksuchasasuddenstrikeDiversificationistheonlywaytoprotectyourself

Others:

credit(default)risk,foreignexchangerisk,interestraterisk,politicalrisk,marketrisk

DiversificationRiskmanagementtechniqueMixesawidevarietyofinvestmentswithinaportfolioObjectiveistominimizetheimpactthatanyonesecuritywillhaveonoverallperformanceoftheportfolioForthebestdiversification,

portfolioshouldbespreadamongmanydifferentassets;cash,stocks,bondssecuritiesshouldvaryinriskandsecuritiesshouldvarybyindustrytominimizeunsystematicrisktodifferentcompaniesPortfolioRiskFornsecurities,portfoliorisk–varianceoftheportfolioreturn

Expectedvalueofsquaredvariable-howmuchrateofreturnofportfoliotendstovaryfromitsmeanExample:VarianceofPortfolio

Thevarianceandcovariancearecalculatedas

Example:VarianceofPortfolio

Theportfolioriskis

ForequallyweightedportfolioShortSale

Itispossibletosellanassetthatyoudonotownthroughtheshortselling(shorting)theasset.Inordertoimplementshortsellingborrowtheassetfromtheownersuchasbrokeragefirmselltheborrowedassettosomeoneelse,receivinganamountx0atalaterdatepurchasetheassetforx1,andreturntheassettolender

TheshortsellingisprofitableiftheassetpricedeclinesRiskysincethepotentialforlossisunlimited–ifassetvalueincreasesAlthoughprohibitedwithincertainfinancialinstitutions,notuniversallyforbiddenExample:Short-saleAssumethatcompanyXhasapooroutlooknextmonth.Thestockisnowtradingat￡65,butyouseeittradingmuchlowerthanthispriceinthefuture.Youdecidetotakeriskandtradeonthisstock.Twothingscanhappen;stockpricecangoupordownMakeMoneyLoseMoneyRISKY:noguaranteethatthepriceofashortstockwilldropOptimalPortfolioPortfoliotheoryeffectsofinvestordecisionsonsecuritypricesrelationshipthatshouldexistbetweenthereturnsandrisk.possibletohavedifferentportfoliosvaryinglevelsofrisk&returndecidehowmuchriskyoucanhandleandallocate(ordiversify)portfolioaccordingtothisdecisionHarryMarkowitz:1990NobelPrizewinnerinEconomicSciences

publishedin1952JournalofFinancetitled“PortfolioSelection”formalizedanintegratedtheoryofdiversification,portfoliorisks,efficient&inefficientportfoliosMaximize(expectedreturnofportfolio)(weightedsumofindividualexpectedratesofreturn)Minimize(portfoliorisk)(expectedvalueofsquaredvariable-howmuchrateofreturnofportfoliotendstovaryfromitsmean)TheMarkowitzModelConstructportfoliounderuncertaintySupposethattherearenassetswithrandomratesofreturn