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HowtodoBlackLitterman
AStep-By-StepGuidetotheBlack-LittermanModel
ThomasIdzorek
PreliminaryCopy:
January1,2002
ThisCopy:
February10,2002
TheBlack-Littermanassetallocationmodel,createdbyFischerBlackandRobertLittermanofGoldman,Sachs&Company,isasophisticatedmethodusedtoovercometheproblemofunintuitive,highly-concentrated,input-sensitiveportfolios.Inputsensitivityisawell-documentedproblemwithmean-varianceoptimizationandisthemostlikelyreasonthatmoreportfoliomanagersdonotusetheMarkowitzparadigm,inwhichreturnismaximizedforagivenlevelofrisk.TheBlack-LittermanModelusesaBayesianapproachtocombinethesubjectiveviewsofaninvestorregardingtheexpectedreturnsofoneormoreassetswiththemarketequilibriumvector（thepriordistribution）ofexpectedreturnstoformanew,mixedestimateofexpectedreturns.Theresultingnewvectorofreturns（theposteriordistribution）isdescribedasacomplex,weightedaverageoftheinvestor’sviewsandthemarketequilibrium.
HavingattemptedtodecipherseveralarticlesabouttheBlack-LittermanModel,IhavefoundthatnoneoftherelativelyfewarticlesontheBlack-LittermanModelprovideenoughstep-by-stepinstructionsfortheaveragepractitionertoderivethenewvectorofexpectedreturns.Addingtothedifficulty,thefewexistingarticleslackconsistencyintheirmathematicalnotationsandnoonearticleprovidessufficientdetailforestablishingthevaluesofthemodel’sparameters.Inadditiontotouchingonthe“intuition”behindtheBlack-LittermanModel,thispaperconsolidatescriticalinsightscontainedinthevariousworksontheBlack-LittermanModelandfocusesonthedetailsofactuallycombiningmarketequilibriumexpectedreturnswith“investorviews”togenerateanewvectorofexpectedreturns.
TheIntuition
ThegoaloftheBlack-LittermanModelistocreatestable,mean-varianceefficientportfolios,basedonaninvestor’suniqueinsights,whichovercometheproblemofinput-sensitivity.AccordingtoLee（2000）,theBlack-LittermanModelalso“largelymitigates”theproblemofestimationerror-maximization（seeMichaud（1989））byspreadingtheerrorsthroughoutthevectorofexpectedreturns.
Themostimportantinputinmean-varianceoptimizationisthevectorofexpectedreturns;however,BestandGrauer（1991）demonstratethatasmallincreaseintheexpectedreturnofoneoftheportfolio'sassetscanforcehalfoftheassetsfromtheportfolio.Inasearchforareasonablestartingpoint,BlackandLitterman（1992）andHeandLitterman（1999）exploreseveralalternativeforecastsofexpectedreturns:
historicalreturns,equal“mean”returnsforallassets,andrisk-adjustedequalmeanreturns.Theydemonstratethatthesealternativeforecastsleadtoextremeportfolios–portfolioswithlargelongandshortpositionsconcentratedinarelativelysmallnumberofassets.
TheBlack-LittermanModeluses“equilibrium”returnsasaneutralstartingpoint.EquilibriumreturnsarecalculatedusingeithertheCAPM（anequilibriumpricingmodel）orareverseoptimizationmethodinwhichthevectorofimpliedexpectedequilibriumreturns（Π）isextractedfromknowninformation.Usingmatrixalgebra,onesolvesforΠintheformula,Π=δΣw,wherewisthevectorofmarketcapitalizationweights;Σisafixedcovariancematrix;and,δisarisk-aversioncoefficient.,Iftheportfolioinquestionis“well-diversified”relativetothemarketproxyusedtocalculatetheCAPMreturns（orifthemarketcapitalizationweightedcomponentsoftheportfolioinquestionareconsideredthemarketproxy）,thismethodofextractingtheimpliedexpectedequilibriumreturnsproducesanexpectedreturnvectorverysimilartotheonegeneratedbytheSharpe-LittnerCAPM.Infact,BestandGrauer（1985）outlinethenecessaryassumptionstocalculateCAPM-basedestimatesofexpectedreturnsthatmatchtheImpliedEquilibriumreturns.
ThemajorityofarticlesontheBlack-LittermanModelhaveaddressedthemodelfromaglobalassetallocationperspective;therefore,thisarticlepresentsadomesticexample,basedontheDowJonesIndustrialAverage（DJIA）.Table1containsthreeestimatesofexpectedtotalreturnforthe30componentsoftheDJIA:
Historical,CAPM,andImpliedEquilibriumReturns.RatherthanusingtheBestandGrauer（1985）assumptionstoderiveaCAPMestimateofexpectedreturnthatexactlymatchestheImpliedEquilibriumReturnVector（П）,arelativelystandardCAPMestimateofexpectedreturnsisusedtoillustratethesimilaritybetweenthetwovectorsandthedifferencesintheportfoliosthattheyproduce.TheCAPMestimateofexpectedreturnsisbasedona60-monthbetarelativetotheDJIAtimes-seriesofreturns,arisk-freerateof5%,andamarketriskpremiumof7.5%.
Table1:
DJIAComponents–EstimatesofExpectedTotalReturn
Symbol
HistoricalReturnVector
CAPM
Return
Vector
Implied
Equilibrium
Return
Vector（П）
aa
17.30
15.43
13.81
ge
16.91
12.15
13.57
jnj
16.98
9.39
9.75
msft
23.95
14.89
20.41
axp
15.00
15.65
14.94
gm
4.59
13.50
12.83
jpm
5.31
15.89
16.46
pg
7.81
8.04
7.56
ba
-4.18
14.16
11.81
hd
29.38
11.59
12.52
ko
-0.57
10.95
10.92
sbc
10.10
7.76
8.79
c
24.55
16.59
16.97
hon
-0.05
16.89
14.50
mcd
2.74
10.70
10.44
t
-1.24
8.88
10.74
cat
7.97
13.08
10.92
hwp
-4.97
14.92
14.45
mmm
9.03
10.43
8.66
utx
13.39
16.51
15.47
dd
0.44
12.21
10.98
ibm
21.99
13.47
14.66
mo
10.47
7.57
6.86
wmt
30.23
10.94
12.77
dis
-2.59
12.89
12.41
intc
13.59
15.83
18.70
mrk
8.65
8.95
9.22
xom
11.10
8.39
7.88
ek
-17.00
11.08
10.61
ip
1.24
14.80
12.92
Average
9.07
12.45
12.42
Std.Dev.
10.72
2.88
3.22
High
30.23
16.89
20.41
Low
-17.00
7.57
6.86
TheHistoricalReturnVectorhasamuchlargerstandarddeviationandrangethantheothertwovectors.TheCAPMReturnVectorisquitesimilartotheImpliedEquilibriumReturnVector（П）（thecorrelationcoefficient（ρ）is85%）.Intuitively,onewouldexpecttwohighlycorrelatedreturnvectorstoleadtosimilarlycorrelatedportfolios.
InTable2,thethreeestimatesofexpectedreturnfromTable1arecombinedwiththehistoricalcovariancematrixofreturns（Σ）andtheriskaversionparameter（δ）,tofindtheoptimumportfolioweights.
Table2:
DJIAComponents–PortfolioWeights
Symbol
HistoricalWeight
CAPMWeight
ImpliedEquilibriumWeight
MarketCapitalizationWeight
aa
223.86%
2.67%
0.88%
0.88%
ge
-65.44%
9.80%
11.62%
11.62%
jnj
-70.08%
6.11%
5.29%
5.29%
msft
3.54%
3.22%
10.41%
10.41%
axp
-15.38%
5.54%
1.39%
1.39%
gm
5.76%
3.44%
0.79%
0.79%
jpm
-213.39%
1.94%
2.09%
2.09%
pg
92.00%
-1.33%
2.99%
2.99%
ba
-111.35%
4.71%
0.90%
0.90%
hd
280.01%
0.11%
3.49%
3.49%
ko
-151.58%
5.70%
3.42%
3.42%
sbc
17.11%
-4.28%
3.84%
3.84%
c
293.90%
5.11%
7.58%
7.58%
hon
15.65%
2.71%
0.80%
0.80%
mcd
-61.68%
1.32%
0.99%
0.99%
t
-86.44%
4.04%
1.87%
1.87%
cat
-70.67%
5.10%
0.52%
0.52%
hwp
-163.02%
6.60%
1.16%
1.16%
mmm
56.84%
4.73%
1.35%
1.35%
utx
-23.80%
4.38%
0.88%
0.88%
dd
-131.99%
1.03%
1.29%
1.29%
ibm
36.92%
5.57%
6.08%
6.08%
mo
136.78%
1.31%
2.90%
2.90%
wmt
21.03%
0.89%
7.49%
7.49%
dis
5.75%
-2.35%
1.23%
1.23%
intc
97.81%
-1.96%
6.16%
6.16%
mrk
144.34%
4.61%
3.90%
3.90%
xom
218.75%
4.10%
7.85%
7.85%
ek
-148.36%
2.04%
0.25%
0.25%
ip
-113.07%
4.76%
0.57%
0.57%
High
293.90%
9.80%
11.62%
11.62%
Low
-213.39%
-4.28%
0.25%
0.25%
Notsurprisingly,theHistoricalReturnVectorproducesanextremeportfolio.However,despitethesimilaritybetweentheCAPMReturnVectorandtheImpliedEquilibriumReturnVector（П）,thevectorsproducetworatherdistinctportfolios（thecorrelationcoefficient（ρ）is18%）.TheCAPM-basedportfoliocontainsfourshortpositionsandalmostalloftheweightsaresignificantlydifferentthanthebenchmarkmarketcapitalizationweightedportfolio.Asonewouldexpect（sincetheprocessofextractingtheImpliedEquilibriumreturnsgiventhemarketcapitalizationweightswasreversed）,theImpliedEquilibriumReturnVector（П）leadsbacktothemarketcapitalizationweightedportfolio.IntheabsenceofviewsthatdifferfromtheImpliedEquilibriumreturn,investorsshouldholdthemarketportfolio.TheImpliedEquilibriumReturnVector（П）isthemarket-neutralstartingpointfortheBlack-LittermanModel.
TheBlack-Littermanformula
Priortoadvancing,itisimportanttointroducetheBlack-Littermanformulaandprovideabriefdescriptionofeachofitselements.Throughoutthisarticle,kisusedtorepresentthenumberofviewsandnisusedtoexpressthenumberofassetsintheformula.
（1）
Where:
E[R]=New（posterior）CombinedReturnVector（nx1columnvector）
τ=Scalar
Σ=CovarianceMatrixofReturns（nxnmatrix）
P=Identifiestheassetsinvolvedintheviews（kxnmatrixor1xnrowvectorinthespecialcaseof1view）
Ω=Diagonalcovariancematrixoferrortermsinexpressedviewsrepresentingthelevelofconfidenceineachview（kxkmatrix）
П=ImpliedEquilibriumReturnVector（nx1columnvector）
Q=ViewVector（kx1columnvector）
（’indicatesthetransposeand-1indicatestheinverse.）
InvestorViews
Moreoftenthannot,aninvestmentmanagerhasspecificviewsregardingtheexpectedreturnofsomeoftheassetsinaportfolio,whichdifferfromtheImpliedEquilibriumreturn.TheBlack-LittermanModelallowssuchviewstobeexpressedineitherabsoluteorrelativeterms.BelowarethreesampleviewsexpressedusingtheformatofBlackandLitterman（1990）.
View1:
Merck（mrk）willhaveanabso
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