mean and variance of portfolio return

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1、1.1MeanandvarianceofportfolioreturnSingle-periodinvestmentmodel{AssetreturnSupposethatyoupurchaseanassetattimezero,and1yearlateryouselltheasset.Thetotalreturnonyourinvestmentisde¯nedtobeamountreceivedtotalreturn=:amountinvestedIfX0andX1are,respectively,t

2、heamountsofmoneyinvestedandreceivedandRisthetotalreturn,thenX1R=:X0rateofreturnisamountreceived¡amountinvestedX1¡X0r==:amountinvestedX0ItisclearthatR=1+randX1=(1+r)X0:2²Amountreceived=dividendreceivedduringtheinvestmentperiod+terminalassetvalue,bothquant

3、itiesareuncertain.²Wetreatrasarandomvariable,characterizedbyitsprobabilitydistributionrateofreturnr1r2¢¢¢rnprobabilityofoccurrencep1p2¢¢¢pn²Twoimportantstatistics(discreterandomvariable)Xnmean=r=ripii=1Xn22pvariance=¾(r)=(ri¡r)i:i=13Statementoftheproblem

4、²Aportfolioisde¯nedbyallocatingfractionsofinitialwealthtoindividualassets.Thefractions(orweights)mustsumtoone(someoftheseweightsmaybenegative,correspondingtoshortselling).²Returnisquanti¯edbyportfolio'sexpectedrateofreturn;Riskisquanti¯edbyvarianceofport

5、folio'srateofreturn.Goal:Maximizereturnforagivenlevelofrisk;orminimizeriskforagivenlevelofreturn.(i)Howdowetodeterminetheoptimalportfolioallocation?(ii)Whatwouldbetheresultingportfolio'sexpectedrateofre-turnusingtheoptimalportfolioallocationwithagiventol

6、-eranceofrisk?4Limitationsinthemeanvarianceportfoliotheory²Onlythemeanandvarianceofratesofreturnsaretakenintoconsiderationintheportfolioanalysis.Thehigherordermo-ments(liketheskewness)oftheprobabilitydistributionoftheratesofreturnareirrelevantintheformul

7、ation.²Indeed,onlytheGaussian(normal)distributionisfullyspeci¯edbyitsmeanandvariance.Unfortunately,theratesofreturnofassetinvestmentarenotGaussianingeneral.²Calibrationofparametersinthemodelisalwayschallenging.1Pn{Samplemean:rb=rt.nt=121Pn2{Samplevarianc

8、e¾b=(rt¡rb),n¡1t=1wherertisthehistoricalrateofreturnobservedattimet;t=1;2;¢¢¢;n.5Shortsales²Itispossibletosellanassetthatyoudonotownthroughtheprocessofshortselling,orshorting,theasset.Youthenselltheborrowedassettosomeoneel