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1、InvestmentManagementDivisionTheIntuitionBehindBlack-LittermanModelPortfolios■Inthisarticleandasourtitlesuggests,wedemonstrateamethodforunderstandingtheintuitionbehindtheBlack-Littermanassetallocationmodel.■Todothis,weuseexamplestoshowthedifferencebetweenthetraditionalmean-varianceoptimizationproce
2、ssandtheBlack-Littermanprocess.Weshowthatthemean-varianceoptimizationprocess,whileacademicallysound,canproduceresultsthatareextremeandnotparticularlyintuitive.Incontrast,weshowthattheoptimalportfoliosgeneratedbytheBlack-Littermanprocesshaveasimple,intuitiveproperty:-Theunconstrainedoptimalportfoli
3、oisthemarketequilibriumportfolioplusaweightedsumofportfoliosrepresentinganinvestor’sviews.-Theweightonaportfoliorepresentingaviewispositivewhentheviewismorebullishthantheoneimpliedbytheequilibriumandotherviews.-Theweightincreasesastheinvestorbecomesmorebullishontheviewaswellaswhentheinvestorbecome
4、smoreconfidentabouttheview.December1999FRONTSGoldmanSachsInvestmentManagementGoldmanSachsInvestmentManagementAppendixC1.GiventheexpectedreturnsmandthecovariancematrixS,theunconstrainedmaximizationproblem-1maxww¢m-d¢Sw2hasasolutionofw*=()dSm.w()m--112.GiventhecovariancematrixS,theminimumvariancepor
5、tfolioisw=SSiii()¢,whereiisavectorwithallelementsbeingone.23.Thesolutiontotheriskconstrainedoptimizationproblem,maxw¢m,subjecttoww¢£Ss,canbeexpressedInvestmentManagementResearch(r)-1GoldmanSachsQuantitativeResourcesGroupasw=sw*w*¢Sw*,wherew*=()dSmisthesolutionoftheunconstrainedproblem.24.Theriskan
6、dbudgetconstrainedoptimizationproblemcanbeformulatedasmaxw¢m,subjecttoww¢£SsGuangliangHe(212)357-3210()bm()andw¢i=1.Itssolutionhastheformwa=+wb*w,whereaandbarechoseninthewaybothriskRobertLitterman(212)902-1677andbudgetconstraintsaresatisfied.5.Therisk-,budget-,andbeta-constrainedoptimizationproble
7、mcanbeformulatedasmaxw¢m,subjectto2ww¢£Ss,w¢i=1,andww¢=Seqwweq¢Seq,whereweqisthemarketportfolio.Thesolutiontotheproblem(b)(m)hastheformofwa=+wb*wc+w,wherea,b,andcarechoseninthewayallthreeconstraintsareeqsatisfied
