::February20071UniverstityofChicagoGraduateSchoolofBusiness,,ChicagoIL60636,@://
Contents1Introduction.......................................32Choosingpayoffsinone-periodportfolioproblems...................................................................................................-quadraticapproximationandmean-varianceanalysis.........-variancefrontier..........................................................-fundtheorem.............................................,multifactormodels,andfourfundtheorem............233Choosingpayoffsinintertemporalproblems..................................-lognormalproblem..........................-periodproblems...........................................................-runmeanlong-runvariancefrontier324Portfoliotheorybychoosingportfolioweights....................,power-lognormal...........................ffapproach......................385Dynamicintertemporalproblems...........................-variableMertonproblem..............................................-lognormaliidmodelwithconsumption................................fficiencyandKfundtheorems...............................................................511
.........................536Portfolioswithtime-varyingexpectedreturns....................ffapproach................................................................................................577WackyweightsandBayesianportfolios....................................................................................-section..........................“solutions”...........................................................................................................................................................................................................................................................748Bibliographicalnote...................................759Commentsonportfoliotheory.............................7510Problems.........................................8011References........................................8212Problemanswers......................................................:Horizonsandallocations...................8713ICAPManddiscountfactorestimates.........................8914Dynamicportfolioswithforecastablereturns.....................9015Notes...........................................942
1IntroductionNowweturntooneoftheclassicquestionsoffinance—,(subjective)distributionoftheirpayoffs,whatistheoptimalportfolio?Thisisobviouslyaninterestingproblem,,wehavemodeledtheconsumptionprocess,andthenfoundpricesfrommarginalutility,followingLucas’(1978)“endowmenteconomy”,,implicitlyspecifyinglineartechnologies,andderivetheoptimalquantities,,ff.Thisisoftenaveryeasywaytoapproachtheproblem,andittiesportfoliotheorydirectlyintothep=E(mx),unsurprisingly,thesamethingasstaticportfoliochoiceofmanagedportfolios,“standardapproach”toportfoliotheory,inwhichwechoosetheweightsinagivensetofassets,
2Choosingpayoff,-marizepricesandpayoff−1u(c)=λmfortheoptimalportfolioc=u(λm).Ifconsumptionisdrivenby0−1anassetpayoffxˆandoutsideincomee,thenxˆ=u(λm)−ffoutsideincome,,positivestochasticdiscountfactororcontingentclaimspricemsuchthatp=E(mx)foranypayoff:ratherthanfacetheinvestorswithpricesandpayoffs,wesummarizetheinformationinpricesandpayoffbyadiscountfactor,,consideraninvestorwithutilityfunctionoverterminalconsumptionE[u(c)],initialwealthWtoinvest,,butIthinkit’sreallyimportant,andit’,thoughtheremaybetradedassetswithsimilarpayoff,ofcourse,thereareassetsthatcanperfectlyreplicatethepayoff’’scallthepayoffofhisportfolioxˆ,soitspriceorvalueisp(xˆ)=E(mxˆ).Hewilleatc=xˆ+,hisproblemismaxE[u(xˆ+e)](mxˆ)=W(1){xˆ}Maxmeans“choosethepayoff,this{xˆ}meansXXmaxπu(xˆ+e).πmxˆ=Wiiiii{xˆ}(c)=λm(2)0u(xˆ+e)=λm.(3)−1xˆ=u(λm)−e.(4)4
WefindtheLagrangemultiplierλ,(4),socondition(2),,orcurvatureoftheutilityfunction,(4)saysthattheoptimalassetportfolioxˆfirstsellsoff,hedgesorotherwiseaccommodateslaborincomeeoneforoneandthenmakesupthediff(2),thenthemarginalutilityoffirstperiod00consumptionequalstheshadowvalueofwealth,λ=u(c).Addingadiscountfactorβfor0futureutility,so(2)becomesouroldfriend0u(c)β=(c)0Wedidn’,andratherthanfixconsumptionandsolveforprices(andreturns,etc.),wearefixingpricesandpayoffs,−γForpowerutilityu(c)=candnooutsideincome,thereturnontheoptimal11−1−ˆγγportfolioisR=m/E(m)Usingalognormaliidstockreturn,thisresult12(1−α)r+ασμ−r1()αˆ2specializestoR=eRwhereisthestockreturnandα≡.2TγσTheinvestorwantsapayoffwhichisanonlinear,powerfunctionofthestockreturn,“habit”or0−γ“subsistencelevel”,u(c)=(c−h).’strythisideaoutonourworkhorseexample,,thefirstordercondition,equation(2),is−γxˆ=λmsotheoptimalportfolio(4)is11−−γγxˆ=λm5
UsingthebudgetconstraintW=E(mxˆ)tofindthemultiplier,11−−γγW=E(mλm)1W−γ³´λ=,11−γEmtheoptimalportfoliois1−γmxˆ=W.(5)11−γE(m)1−γThemtermistheimportantone—ittellsushowtheportfolioxˆ,giventhisinvestor’,payoff—,thereturnontheoptimalportfolio1−γxˆmˆR==(6)11−WγE(m),wehavetospecifyaninterestingsetofpayoffsandtheirprices,’sconsidertheclassicBlack-Scholesenvironment:,themarketis“complete,”atleastenoughforthisexercise.(Thenextsectiondiscussesjusthow“complete”themarkethastobe.)Thestock,bond,anddiscountfactorfollowdS=μdt+σdz(7)SdB=rdt(8)BdΛμ−r=−rdt−dz(9)Λσ(Thesearealsoequations()fromChapter17,,(dΛ/Λ)=−rdtandE(dS/S)−rdt=−E(dΛ/ΛdS/S)).Thediscrete-timediscountfactorfortimeTpayoffsism=Λ/Λ.Solvingtheseequationsforwardandwithabitofalgebrabelow,wecanTT0evaluateEquation(6),12(1−α)r+ασα()ˆ2R=eR(10)TwhereR=S/Sdenotesthestockreturn,andTT01μ−rα≡.2γσ6
(αwillturnouttobethefractionofwealthinvestedinstocks,iftheportfolioisimplementedbydynamicstockandbondtrading.)Theoptimalpayoffμ−r=8%andσ=16%forafewvaluesofriskaversionγ.−γ==,thefunctionislinear—,theinvestorexploitsthestrongrisk-returntradeoff,takingapositionthatismuchmoresensitivetothestockreturnatR=(verticaldistancepastR=1),,theinvestoracceptsdrasticallylowerpayoffsinthegoodstates(ontheright)inordertogetasomewhatbetterpayoffinthemoreexpensive(highm)ff,hebuysacomplexsetofcontingentclaims,tradesdynamically,orbuysasetofoptions,inordertocreatethenonlinearpayoff,thisbehaviorderivesfromthenonlinearityofmarginalutility,γ = 1 γ = 2 γ = γ = 5 γ = 10 return R−γFigure1:(c)=%meanand16%standarddeviation,anda1%(7)-(9)are(see()formoredetail),µ¶2√σlnS=lnS+μ−T+σTε(11)T027pPortfolio payoff x
"#µ¶2√1μ−rμ−rlnΛ=lnΛ−r+T−Tε(12)T02σσwithε˜N(0,1).Wethushave("#)µ¶2√1μ−rμ−rm=exp−r+T−TεT2σσ""##µ¶µ¶µ¶µ¶µ¶22211−11μ−r11μ−rγEm=exp−1−r+T+1−TTγ2σ2γσ("#)µ¶µ¶µ¶µ¶2211μ−r11μ−r=exp−1−r+−1−Tγ2σ2γσ("#)µ¶µ¶µ¶2111μ−r=exp−1−r+T.γ2γσUsingR=S/Stosubstituteoutεin(12)TT0("#)µ¶∙µ¶¸212−11μ−r1μ−rσγm=expr+T+lnR−μ−TTT2γ2σγσ2("#)µ¶µ¶2211μ−rμ−rσ1μ−r=expr+−μ−T+lnRT22γ2σσ2γσ("#)µ¶µ¶µ¶22211μ−rμ−rμ−rσ1μ−r=expr−+−μ−T+lnRT22γ2σσσ2γσ("#)µ¶∙µ¶¸2211μ−rμ−rσ1μ−r=expr−+μ−r−μ−T+lnRT22γ2σσ2γσ("#)µ¶µ¶2211μ−rμ−rσ1μ−r=expr−−r−T+lnRT22γ2σσ2γσThen("#"#)µ¶µ¶µ¶µ¶µ¶22211μ−rμ−rσ111μ−rˆR=expr−−r−T+1−r+T2γ2σσ2γ2γσ½¾1μ−r×explnRT2γσ()µ¶µ¶½¾221μ−rσ11μ−r1μ−r=expr−r−−TexplnRT222γσ22γσγσ∙µ¶¸21σ22=expr−σα−αr−T×exp{αlnR}T22∙µ¶¸12α=exp(1−α)r+ασT×RT28
ImplementationThisexamplewillstillfeelemptytosomeonewhoknowsstandardportfoliotheory,inwhichthemaximizationisstatedoverportfoliosharesofspecificassetsratherthanoverthefinalpayoff.Sure,wehavecharacterizedtheoptimalpayoffs,butweren’twesupposedtobefindingoptimalportfolios?Whatstocks,bondsoroptionsdoesthisinvestoractuallyhold?,indexedbythestockreturn,,wehavemadetheportfolioproblemveryeasybycleverlychoosingasimplebasis—contingentclaims—:supposeyouwantedtoimplementthispatternofcontingentclaimsbyexplicitlybuyingstandardputandcalloptions,orbydynamictradinginastockorbond,oranyoftheinfinitenumberofequivalentrepackagingofsecuritiesthatspanthecompletemarket,
Now,theoptimalpayoffis−γ(xˆ−h)=λm11−−γγxˆ=λm+hEvaluatingthewealthconstraint,³´11−1−−rTγγW=E(mxˆ)=λEm+he0−rT1W−he−0γ³´λ=11−γEm1−γ¡¢m−rT³´xˆ=W−he+h011−γEmThediscountfactorhasnotchanged,-Scholesexamplewehavebeencarryingalong,thisresultgivesus,correspondingto(10),¡¢12−rT(1−α)r+ασTα()2xˆ=W−heeR+,theinvestorguaranteesthepayoffh.¡¢−rTThen,wealthleftoverafterbuyingabondthatguaranteesh,W−ffffsomeperformanceingoodstatesoftheworldtomakesurehisportfolioneverpaysoff−ˆ=u(λm)−estillgivestheoptimalportfolio,butingeneraltherearemanymandwedon’tknowwhichonelandsxˆ∈X,,whatifmarketsarenotcomplete?,let’,
2γ = 1 γ = 2 γ = γ = 5 γ = 10 return RFigure2:ffffx∈Xtheinvestorknowsthepricep(x).Returnshaveprice1,-saleconstraintsortransactionscosts(that’sanotherinterestingextension),sothespaceXofpayoffsisclosedunderlineartransformations:x∈X,y∈X⇒ax+by∈XIassumethatthelawofonepriceholds,(ax+by)=ap(x)+bp(y).Asbefore,let’sfollowtheinsightthatsummarizingpricesandpayoff,weknowthatthelawof∗onepriceimpliesthatthereisauniquediscountx∈Xsuchthat∗p(x)=E(xx)(13)∗forallx∈,ifthepayoffspaceisgeneratedasallportfoliosofafinitevectorofbasispayoffsxwithpricevectorp,0∗00−1∗∗X={cx},thenx=pE(xx)xsatisfiesp=E(xx)andx∈(9),,∗thentherearemanydiscountfactorsandanym=x+ε,withE(εx)=0∀x∈Xis11pPortfolio payoff x
∗,x=proj(m|X)∗∗∗∗∗∗2∗correspondingtothepayoffxisR=x/p(x)=x/E(x).Ristheglobalminimum∗secondmomentreturn,∗∗apositivediscountfactorm=x+ε,butthepositivemmaynotlieinX,-periodportfolioproblemisnowmaxE[u(c)].(14){xˆ∈X}c=xˆ+e;W=p(xˆ).Thisisdifferentfromourfirstproblem(1)onlybytherestrictionxˆ∈X:marketsareincomplete,andtheinvestorcanonlychooseatradeablepayoff.ffsX={cx}.Then,theconstrainedportfolio0choiceisxˆ=αxandwecanchoosetheportfolioweightsα,respectinginthiswayxˆ∈"Ã!#XXmaxEuαx+=α{α}iiiThefirstorderconditionsare∂0:pλ=E[u(xˆ+e)x](15)ii∂αiforeachasseti,whereλ(15)isouroldfriendp=E(mx).ItholdsforeachassetinXifandonlyif0u(xˆ+e)/λisadiscountfactorforallpayoffsxˆ∈,0u(xˆ+e)=λm(16)wheremsatisfiesp=E(mx)forallx∈’sprettier,=E(mx)∀x∈X,,wecanstatetheconstraintasW=E(mxˆ)(14)isexactlythesameastheoriginalproblem,sowecanfindthefirstorderconditionbychoosingxˆineachstatedirectly,withnomentionoftheoriginalpricesandpayoff−1xˆ=u(λm)−
∗Ifmarketsarecomplete,asabove,thediscountfactorm=−1payoffistraded,sobothλmandu(λm)−,,wealsohavetopayattentiontotheconstraintxˆ∈,butnotyetasuffi,andforonlyoneofthemistheinverse∗’seasytoconstructx∈X,forexample,=x*+ε−1u'(m)aXˆxbx*Figure3:Portfolioprobleminincompletemarkets.∗Figure3illustratestheproblemforthecasee=ff∗=x+ε∗isdrawnatrightanglestoXsinceE(εx)=0∀x∈−1∗u(λm)=:0−1u(λm)isnotinthepayoffspaceX,soitcan’−1∗optimalportfolio:wehavechosentherightmsothatu(λm)isinthepayoff−1∗isthewrongchoiceaswell,sinceu(λx)takesyououtofthepayoff−1∗Asthefiguresuggests,marketsdon’thavetobecompletely“complete”forxˆ=u(λx)ffspaceXisclosedunder(some)nonlineartransforma-0−10−1∗∈X,wealsohaveu(x)∈X,thenxˆ=u(λx)willbetradeable,andwecanagainfindtheoptimalportfoliobyinvertingmarginalutilityfromtheeasy-∗to-computeuniquediscountfactorx∈“completemarkets,”,evenless“completeness”
Whatcanwedo?Howcanwepicktherightm?Ingeneral,,wecansearchoverallpossiblem,−1fortheonethatsendsu(λm)∈’tnecessarilyhard,sincewecansetupthe0−1searchasaminimization,minimizingthedistancebetweenu(m),wecaninventpricesforthemissingsecurities,solvethe(nowunique)completemarketsproblem,,wecanattachLagrangemultiplierstotheconstraintxˆ∈Xandfind“shadowprices”,,,weoftendoportfolioproblemswithastockandabond,,wehavetoresulttonumericalanswers,−1Third,wecansimplifyorapproximatetheproblem,sothatu(·)-quadraticapproximationandmean-varianceanalysis0bIfmarginalutilityislinear,u(c)=c−c,thenwecaneasilysolveforportfolios£¤bb∗ˆ=cˆ−eˆ−p(cˆ)−p(eˆ)−WR,wherecˆandeˆaremimickingpayoffsforastochasticblisspointandoutsideincome,Wis∗initialwealth,,afterhedgingoutsideincomerisk,andthenacceptinglowerconsumptioninthehighcontingentclaimspricestates.∗WeknowhowtofindadiscountfactorinthepayoffspaceX,,whilethepayoff:Withquadraticutility,marginalutility∗∈XisalsointhespaceXofpayoffs,,supposeutilityisquadratic1b2u(c)=−(c−c)(c)=c−
Thefirstordercondition(16)nowreadsbc−xˆ−e=λ,wecanprojectbothsidesontothepayoffspaceX,andsolvefortheoptimalportfolio.∗Sinceproj(m|X)=x,thisoperationyields¡¢∗bxˆ=−λx+projc−e|X.(17)Tomaketheresultclearer,weagainsolvefortheLagrangemultiplierλˆandcˆthemimickingportfoliosforpreferenceshocksandlaborincomerisk,eˆ≡proj(e|X)¡¢bbcˆ≡projc|X(ffsthatareclosest,inmeansquaresense,tothelaborincomeandblisspoints.)Thewealthconstraintthenstates∗bW=p(xˆ)=−λp(x)+p(cˆ)−p(eˆ)bp(cˆ)−p(eˆ)−W=λ∗p(x)Thus,theoptimalportfoliois£¤bb∗xˆ=cˆ−eˆ−p(cˆ)−p(eˆ)−WR,(18)∗∗∗∗∗2whereagainR=x/p(x)=x/E(x)isthereturncorrespondingtothediscount-factor∗payoff,,hewillbuyaportfoliothatisclosesttothisideal—,dependingoninitialwealthandhenceriskaversion(riskaversiondependsonwealth∗forquadraticutility),(forallbinterestingcases)wealthisnotsufficienttobuyblisspointconsumption,W+p(eˆ)<p(cˆ).∗∗Therefore,-variance∗frontier,sowhenyoushortR,(inabsolutevalue)∗ˆ,,,eachinvestor’soptimalportfolioisacombinationofamimickingportfoliotohedgelaborincomeandpreferenceshockrisk,plusaninvestmentinthe(mean-varianceefficient)minimumsecondmomentreturn,
-variancefrontierWithnooutsideincomee=0,wecanexpressthequadraticutilityportfolioproblemintermsoflocalriskaversion,¡¢1ff∗ˆR=R+R−R.γThisexpressionmakesitclearthattheinvestorholdsamean-varianceefficientportfolio,-varianceanalysisfocusesonaspecialcase:theinvestorhasnojob,solaborincomeiszero,theblisspointisnonstochastic,(18)specializestoµ¶bcb∗xˆ=c−−WR(19)fRb¡¢xˆc∗f∗ˆR==R+R−R(20)fWRWInChapter5,weshowedthatthemean-variancefrontieriscomposedofallportfoliosofthe∗f∗formR+α(R−R).Therefore,investorswithquadraticutilityandnolaborincomeallbholdmean-varianceeffi,-pointconsumptionforsure,WR=c,,theseglobalimplications—risingriskaversionwithwealth—areperversefeaturesofquadraticutility,,itisinterestingandusefultoexpresstheportfoliodecisionintermsofthelocalriskaversioncoeffi(20)asµ¶b¡¢cff∗ˆR=R+−1R−R(21)fRWLocalriskaversionforquadraticutilityisµ¶−100bcu(c)ccγ=−==−(c)c−ccNowwecanwritetheoptimalportfolio¡¢1ff∗ˆR=R+R−R.(22)γ16
fwhereweevaluatelocalriskaversionγatthepointc=-varianceefficientportfolio,withlargerinvestmentinthe∗riskyassetthelowerhisriskaversioncoeffi,Risonthelowerpartofthe∗mean-variancefrontier,,ifyouhadenoughwealthtobuyblisspointconsumptionRW=c,¡¢f1f∗ˆIevaluatethemean-varianceformulaR=R+R−RforthecommonγfecaseofariskfreerateRandvectorofexcessreturnsRwithmeanμandcovariancematrixΣ.Theresultisµ¶fRf∗0−1eR−R=μΣR0−11+μΣμThetermsarefamiliarfromsimplemean-variancemaximization:findingthemean-variancefrontierdirectlywefindthatmean-varianceefficientweightsare0−10−1alloftheformw=λμΣandthemaximumSharperatioisμΣμ.Formula(22)isalittledry,soit’ffspaceconsistsofariskfreerateRandNassetswithexcessreturnsR,sothatpf0eeeportfolioreturnsarealloftheformR=R+μ=E(R)andΣ=cov(R).∗Let’sfindRandhence(22),wecanfind11∗0−1ex=−μΣ(R−μ).ffRR∗∗f∗e∗f(Checkthatx∈X,E(xR)=1andE(xR)=0,orderiveitfromx=αR+0ew[R−μ].)Then11∗∗20−1p(x)=E(x)=+μΣμf2f2RRso∗0−1ex1−μΣ(R−μ)∗fR==R∗20−1E(x)1+μΣμandffRRf∗f0−1eR−R=R−+μΣ(R−μ)0−10−11+μΣμ1+μΣμfRf∗0−1eR−R=μΣR(23)0−11+μΣμ17
Togiveareferencefortheseformulas,σ(R)(R)=E00minwΣμ=E{w}Thefirstorderconditionsgive−1w=λΣμwhereλ,theportfoliosonthemean-variancefrontierhaveexcessreturnsoftheformep0−1eR=λμΣR0−1Thisisagreatformulatoremember:μΣgivestheweightsforamean-varianceeffi(23)-variancefrontierisep0−1pE(R)μΣμ0−1=p=μΣμep0−1σ(R)μΣμf∗Thus,youcanseethatthetermscalingR−-variancefrontierfromtheriskfreerateandany0−1eeffi,justusingμΣRmightseemsimplerthanusing(23),-varianceefficientportfoliof∗R−Rin(23)hasthedeliciouspropertythatitistheoptimalportfolioforriskaversionequaltoone,andtheunitsofanyinvestmenthavedirectlytheinterpretationasariskaversioncoeffi-fundtheoremInamarketofquadraticutility,e=0investors,wecanaggregateacrosspeopleandexpresstheoptimalportfolioasm¡¢γifmfˆR=R+R−RiγThisisa“two-fund”theorem—
∗,,conventionalmean-varianceanalysisusesthe“marketportfolio”’(21)as¡¢1iff∗ˆR=R+R−R.(24)iγmˆThemarketportfolioRisthewealth-weightedaverageofindividualportfolios,orthereturnonthesumofindividualpayoffs,PPNNiiiˆxˆWRmi=1i=1ˆR≡P=PNNjjWWj=1j=1Summing(24)overindividuals,then,PNi1W¡¢ii=1γmff∗ˆR=R+PR−=1Wecandefinean“averageriskaversioncoefficient”asthewealth-weightedaverageof(in-1verse)riskaversioncoefficients,PNi1W1ii=1γ≡PmNjγWj=1so¡¢1mff∗ˆR=R+R−γmff∗UsingthisrelationtosubstituteR−RinplaceofR−Rin(24),weobtainm¡¢γifmfˆR=R+R−R(25)iγ“marketportfolio”“netsupply”thentheyareincludedinthemarketportfolio,andtheremainingriskfreerateisinii“zeronetsupply.”Sincex=c,“Marketriskaversion”isalsothelocalriskaversionofaninvestorwiththeaverageblisspointandaveragewealth,PN1bic1Ni=1=−γRWNi=119
mfSinceanytwomean-varianceefficientportfoliosspanthefrontier,RandRforexample,,,theonlywaypeopledifferisbytheirriskaversion,soallinvestors’portfolioscanbeprovidedbytwofunds,a“marketportfolio”-variancefrontierdiagram,(R)Less risk aversemMarket RMore risk aversefRFrontiers(R)Figure4:Mean-varianceeffi,theoverallportfolio,includingthehedgeportfolioforoutsideincome,,wecoulduseclassicanalysistodeterminetherightoverallportfolio—keepinginmindthattheoverallmarketportfolioincludeshedgeportfoliosfortheaverageinvestorsoutsideincometoo—andthensubtractoffthehedgeportfolioforindividualoutsideincomeinordertoarriveattheindividual’,wecanexpresstheindividual’sportfolioas1)themarketassetportfolio,adjustedforriskaversionandthecompositionofwealth,2)theaverageoutsideincomehedgeportfolioforallotherinvestors,adjustedagainforriskaversionandwealthandfinally3)thehedgeportfoliofortheindividual’
Themean-variancefrontierisabeautifulandclassicresult,butmostinvestorsdoinfacthavejobs,,“totalportfolio”.Then,thetotalportfolioisstillmean-varianceefficient,,keepanonstochasticblisspoint,,equation(18)becomes£¤bb∗xˆ=c−eˆ−p(c)−p(eˆ)−WRWecanrewritethisas£¤bb∗eˆ+xˆ=c−p(c)−(W+p(eˆ))RThelefthandsideisthe“totalpayoff”,consistingoftheassetpayoffxˆandthelaborincomehedgeportfolioeˆ(Consumptionisthispayoffplusresiduallaborincome,c=xˆ+e=xˆ+(e−eˆ)+eˆ.)Wedefinearateofreturnonthe“totalportfolio”asthetotalpayoff—assetportfolioplushumancapital—dividedbytotalvalue,andproceedasbefore,∙¸bbeˆ+xˆcctp∗ˆR==−−1RfW+p(eˆ)W+p(eˆ)R[W+p(eˆ)]¡¢1tpff∗ˆR=R+R−RγNowb1c=−1fγR[W+p(eˆ)]bisdefinedasthelocalriskaversioncoefficientgivencandusingthevalueofinitialwealthandthetradeableportfolioclosesttolaborincome,,wecansaythatthetotalportfolioismean-varianceeffi,toexpressm¡¢γtp,iftp,mfˆR=R+R−R(26)iγmwhereRisnowthetotalwealthportfolioincludingtheoutsideincomeportfolios,—thethingtheinvestoractuallybuys—ˆisapayoff,tofigureouttheassetmarketpayoff,youhavetosubtractthe21
laborincomehedgeportfoliofromtheappropriatemean-varianceefficientportfolio.µ¶iiiiiixˆp(eˆ)+Weˆ+xˆeˆiˆR==−(27)iiiiiiWWp(eˆ)+Wp(eˆ)+Wµ¶µ¶iiip(eˆ)p(eˆ)eˆtp,iˆ=1+R−(28)iiiWWp(eˆ)µ¶µ¶iip(eˆ)p(eˆ)tp,iz,iˆˆ=1+R−R(29)iiWWwhereIuseieˆz,iˆR=ip(eˆ)todenotethereturnonthemimickingportfolioforoutsideincome.(Ican’tusethenaturale,ieˆnotationRsinceRstandsforexcessreturn.),intp,mˆthisrepresentationthecorresponding“marketportfolio”Rincludeseveryoneelse’,“total”returntothetwocomponents,a“mimickingportfolioreturn”andtheassetportfolioreturn,iieˆ+xˆtp,iˆR=iip(eˆ)+Wiiiip(eˆ)eˆWxˆ=+iiiiiip(eˆ)+Wp(eˆ)p(eˆ)+WWiz,iiiˆˆ=(1−w)R+wRHereiiieˆWp(eˆ)z,iiiˆR=;w=;1−w=.iiiiip(eˆ)p(eˆ)+Wp(eˆ)+Wtp,,substitutingin(26),m³´γiz,iiifmz,mmmfˆˆˆˆ(1−w)R+wR=R+(1−w)R+wR−Riγandhencemm³´mmm³´³´iγwγw(1−w)(1−w)ifmfz,mfz,ifˆˆˆˆR−R=R−R+R−R−R−R(30)iiiimiγwγwwwThisrepresentationemphasizesadeeppoint,youonlydeviatefromthemarketportfoliototheextentthatyouarediff-ual’sactualportfolioscalesupordownthemarketportfolioaccordingtotheindividual’-isidewealthrelativetototal,wislower,’slaborincomeThe22
nexttermsdescribehowyoushouldchangeyourportfolioifthecharacterofyouroutsideincomeisdifferentfromeveryoneelse’,supposethatoutsideincomeisnonsto-zfchastic,soR=,andweareleftwithmm³´γwifmfˆˆR−R=R−RiiγwThisistheusualformulaexceptthatriskaversionisnowmultipliedbytheshareofassetwealthintotalwealth,iWiiiγw=γ.iiW+p(e)iAnindividualwithalotofoutsideincomep(e),hisassetmarketportfolioshouldbeshiftedtowardsriskyassets;“effectiveriskaversion”fortheassetmarketportfolioisin(30),supposethattheinvestorhasthesamewealthandrelativewealthasthemarket,imimγ=γandw=w,(30)simplifiesto³´hi(1−w)ifmfz,me,iˆˆˆˆR−R=R−R+R−RwThisinvestorholdsthemarketportfolio(thistimetheactual,traded-assetmarketportfolio),plusahedgeportfolioderivedfromthedifferencebetweenhisincomeandtheaveragein-vestor’,’soutsideincomeejfˆisabond,R=R,,theinvestor’-variancesensebyprovidingthis“outsideincomeinsurance”,multifactormodels,andfourfundtheorem23
3ChoosingpayoffsinintertemporalproblemsOne-periodproblemsarefunandpedagogicallyattractive,.,,,,individualinvestors’outsideincomesvarywithtime,-livedagents,,manydynamicsetupsgiverisetoincompletemarkets,,,withalittlereinterpretationofsymbols,’toolkit,,t0βu(xˆ+e)=λm(31)ttt0−1txˆ=u(λm/β)−e.(32)tttwherewenowinterpretxˆtobetheflowofdividends(payouts)oftheoptimaltportfolio,;hisutilityfunctionis∞XtEβu(c).tt=1HehasinitialwealthWandhehasastreamofoutsideincome{e}.Hisproblemistopicktastreamofpayoffsordividends{xˆ},whichhewilleat,c=xˆ+,,theproblemis∞TXXtmaxEβu(xˆ+e)=Emxˆtttt{xˆ∈X}tt=1t=124
Heremrepresentsadiscountfactorprocess,ffx,mgeneratespricesptttby∞Xp==1Asbefore,absenceofarbitrageandthelawofonepriceguaranteethatwecanrepresentthepricesandpayoff(∂/∂xˆinstateiattimet)itt0βu(xˆ+e)=λm(33)tttThus,onceagaintheoptimalpayoffischaracterizedby0−1txˆ=u(λm/β)−e.(34)ttttTheformulaisonlydifferentbecauseutilityofconsumptionattimetismultipliedbyβ.Ifmisunique(completemarkets),,thenagainwehavetochoosetheright{m}sothat{xˆ}∈X.(Wehavetothinkinsomemoredetailwhatthispayoffspacetlookslikewhenmarketsarenotcomplete.)Asbefore,(ortofindwhatwealthcorrespondstoachoiceofλ),weimposetheconstraint,X0−1Emu(λm)=∞∞−ρtmaxEeu(xˆ+e)=Emxˆdtttttt=0t=0givingrisetotheidenticalconditions−ρt0eu(xˆ+e)=λm(35)ttt0−1−ρtxˆ=u(λm/e)−e.(36),µ¶αS1μ−rtxˆ=(const.)×;t2Sγσ025
Toseethisanalysismoreconcretely,andforitsowninterest,let’,livesforeverandwantsintermediateconsumption,andhaspowerutilityZ∞1−γxˆ−γt=0Hecandynamicallytrade,resultingin“complete”,(36)1¡¢1−−ρtγγxˆ=λemttAsbefore,wecansolveforλ,Z∞1¡¢1−−ρtγγW=Emλemdtttt=0Z∞11ρ1−−−tγγγW=λEemdttt=0sotheoptimalpayoffis1ρ−−tγγxˆemtt=(37)1Rρ1−W∞−tγγEemdttt=0Theanalogytotheone-periodresult(6),the“return”isnowadividendattimetdividedbyaninitialvalue,,wewillfirstfacethetechnicaljoboffindthediscountfactorthatrepresentsagivensetofassetpricesandpayoffs,sotomaketheanalysisconcreteandtosolveaclassicproblem,let’,astockandbondfollowdS=μdt+σdz(38)SdB=rdt.(39)B(ThinkofSandBasthecumulativevalueprocesswithdividendsreinvested,ifyou’=μdt+σdzandbondreturnrdt.),,we’vealreadyfoundthediscountfactor,bothinchapter17andinequation(9)above,m=Λ/Λwherett0dΛμ−r=−rdt−dz.(40)Λσ26
WecansubstitutedlnSfordzandsolve(38)-(40),(algebrabelow)resultinginμ−rµ¶−2σΛ1μ−rStt−1(μ+r)t()22σ=e×.ΛS00Andthus,forpowerutility,(37)becomesµ¶αStxˆ=(const.)×tS0whereagain1μ−rα=2γσ,andasI’llshowbelow,onewaytoimplementthisruleistoinvestaconstantlyrebalancedfractionofwealthαinstocks,,—thedenominatorof(37)takesalittlemorealgebraandisnotveryrevealing,buthereisthefinalanswer:∙µ¶¸µ¶α11xˆ11St−tρ+(γα−1)(μ+r)t22[]γ2=ρ+(γ−1)r+γασe(41)Wγ2S0Algebra:"#µ¶22dΛ1dΛ1μ−rμ−rdlnΛ=−=−r+dt−dz2Λ2Λ2σσµ¶2dS1dS12dlnS=−=μ−σdt+σdz2S2S2Forthenumerator,,"#µ¶∙µ¶¸21μ−rμ−r12dlnΛ=−r+dt−dlnS−μ−σdt22σσ2"#µ¶21μ−rμ(μ−r)1μ−rdlnΛ=−r−+−(μ−r)dt−dlnS222σσ2σ"#µ¶21μ−rμ(μ−r)1μ−rdlnΛ=−r−+−(μ−r)dt−dlnS222σσ2σ∙¸1μ−rμ−rdlnΛ=−1(μ+r)dt−dlnS222σσ∙¸1μ−rμ−rlnΛ−lnΛ=−1(μ+r)t−(lnS−lnS)t0t0222σσ27
μ−r µ¶−1μ−r2σΛS−1(μ+r)ttt22σm==ΛS00μ−r µ¶1ρ2ρ11μ−r−γσS−t−−1(μ+r)t−ttγ2γγ2γσem=etS0μ−rk lµ¶211μ−rγσS−ρ+−1(μ+r)tt2γ2σ=eS0Forthedenominator,it’seasiertoexpressΛintermsofanormalrandomvariable."#µ¶2√1μ−rμ−rlnΛ−lnΛ=−r+t−tεt02σσ kl √1211μ−rμ−r11−−1−r+t−1−tε()γγ2σσγm=et klk lµ¶2211μ−r1μ−r11−1−r++1−t1−()γ2σ2σγγEm=et qr2111μ−r−1−r+t()γ2γσ=eZµ¶Z qr∞1∞2ρρ111μ−r1−−1−r+t−t()γ−tγ2γσγγeEmdt=eedtt00Zqklr∞2111μ−r−ρ+(γ−1)r+t()γ2γσ=edt0γ=hi¡¢2μ−r11ρ+(γ−1)r+2γσThus,hi¡¢2μ−rμ−r11k lµ¶ρ+(γ−1)r+11μ−r2γσxˆ2γσS−ρ+−1(μ+r)ttt2γ2σ=eWγS0µ∙¸¶µ¶α11xˆ11St−ρ+(γα−1)(μ+r)tt22[]γ2=ρ+(γ−1)r+γασγ-periodproblemsIintroducealittlenotationthatmakesevenclearertheanalogybetweenP∞(x)≡Eβxtreatstimeandtt=1¡P¢∞,wewritep=Eβmx=E(mx).Inthisttt=1notation,
,,letusdefineanexpectationoperatorthataddsovertimet−ρtusingβ,defineXoneperiod:E(x)≡E(x)=π(s)x(s)11s∞∞XXXttinfiniteperiod,discrete:E(x)≡Eβx=βπ(s)x(s)ttttt=1t=1stZ∞−ρtinfiniteperiod,continuous:E(x)≡Eexdtt0Itisconvenienttotakeβastheinvestor’sdiscountfactor,,∞Xtp(x)=Eβmx=E(mx).ttt=1tHereitisconvenienttostartwithadiscountfactorthatisscaledbyβinordertothentt0multiplybyβ.Inthecanonicalexamplewhichwasexpressedm=βu0(c)/u(c)wenowtt00havem=u0(c)/u(c).tt0(Oneproblemwiththisdefinitionisthattheweightsovertimedonotadduptoone,P(1−β)tE(1)=β/(1−β).OnecandefineE(x)=βE(x)torestorethisproperty,butthentβwemustwritepricingasp(x)=β/(1−β)E(mx).Ichoosethesimplerpricingequation,atthecostthatyouhavetobecarefulwhentakinglongrunmeansEofconstants.)Theinvestor’sobjectiveisXtmaxEβu(c)=maxE[u(c)]ttc=xˆ+etttTheconstraintisXtW=Eβmxˆ=E(mxˆ)tttInsum,weareexactlybacktomaxE[u(xˆ+e)]=E(mxˆ)ttThefirstorderconditionsare0u(xˆ+e)=λm0−1xˆ=u(λm)−etexactlyasbefore.(Werescaledm,whichiswhyit’snotm/βasin(34).)29
Withpowerutilityandnooutsideincome,wecanevaluatetheconstraintas11−−γγW=E(mλm)soagainthecompleteproblemis1−γxˆmtt=11−WγE(m)Allthepreviousanalysisgoesthroughunchanged!,“Returns”x/p(x)arenowdividendsattimetdividedbyinitialprice,andthe“long-runmean-variancefrontier”,however,ˆ/,apparentlyistherightgeneralizationof“return”,foranypayoffstreamIthinkitisbettertocallthe“return”a“yield,”xxtty==.tp({x})E(mx),wecandefine“excessyields”,whicharethezero-priceobjectsase12y=y−ffisthusoneinallstatesanddates,aperpetuityfx==tp({1})Thisis,infact,ff-styleorperiodtoperiodanalysis,,,oranindexedperpetuityistherisklessassetforaninvestorwithaninfinitehorizon,ffs,it’
Inplaceofourusualportfoliosandpayoffspaces,wehavespacesofyields,Y≡{y∈X:p(y)=1},eeeY≡{y∈X:p(y)=0}.It’snaturaltodefinealong-runmean/long-runvariancefrontierwhichsolves2minE(y)(y)=μ.{y∈Y}“Longrunvariance”,wefindthatthelong-runfrontierisgeneratedasmv∗e∗y=y+wy.(42)∗Here,yisthediscount-factormimickingportfolioreturn,∗∗xx∗y==.(43)∗∗2p(x)E(x)e∗Ifariskfreerateistraded,yissimplyf∗y−ye∗y=.(44)fyThelong-runmean-variancefrontierofexcessreturnsise2eminE(y)(y)=μ.ee{y∈Y}Thisfrontierisgeneratedsimplybyemve∗y=wyw∈<Ourpayoff,ifyouseeavariablez(,cay)thatforecastsreturnsR,youwanttoincludeinyourport-tt+,-variationinfuturereturnsorpayoffs,andifxisaprice-zeropayoff,thenwejustt+1includepayoffsoftheformf(z)xinthepayoffspaceX,andinspiredbyaTaylorapprox-tt+12imation,zx,zx,ffs(dividendstreams)fromtt+1t+1trealmanagedportfolios,,dynamictradingmeansthatfundsormanagedportfolioscansynthesizepayoff,,,thediscountfactorandhenceoptimalportfoliostwilltypicallydependonshockstoz,,
-runmeanlong-runvariancefrontierWithquadraticutilityandnooutsideincome,along-runinvestorinady-namic,interetemporal,ffsbetweenanindexedper-petuityandthemarketpayoff,-comenowapplyaswell,,withincompletemarketswefacethesameissueoffindingtheoneofmanypossiblediscountfactorsmwhichleadstoatradeablepayoff.Again,however,wecanusethequadraticutilityapproximation¡¢12bu(c)=−c−c2∙¸µ¶X¡¢112btb2U=E−c−c=Eβ−(c−c),¡¢1ff∗yˆ=y+y−y.γWerecognizealong-runmean/long-runvarianceeffi¡¢γifmfyˆ=y+yˆ−γThus,theclassicpropositionshavestraightforwardreinterpretations:-runmean/-runmean/’sportfoliocanbespannedbyarealperpetuityyandaclaimtoaggregatemconsumptionyˆIntheabsenceofoutsideincome,a“long-run”versionoftheCAPMholdsinthiseconomy,sincethemarketis“long-run”effi
Keepinmindthatallofthisapplieswitharbitraryreturndynamics—wearenotas-sumingiidreturns—anditholdswithincompletemarkets,-variancetheorygaveausefulapproximatecharacterizationofopti-malportfolioswithoutactuallycalculatingthem—findingthemean-variancefrontierishard—soherewegiveanapproximatecharacterizationofoptimalportfoliosinafullydynamic,∗∗intertemporal,—findingx,y,thelongrunmean-longrunvariancefrontier,orsupportingapayoffxˆwithdynamictradinginspecificassets—
4PortfoliotheorybychoosingportfolioweightsThestandardapproachtoportfolioproblemsisquitediffff,,wecansolveaone-periodprobleminwhichtheinvestorchoosesamongfreturnsRandR£¡¢¤f0emaxEuWR+αR0{α}Thefirstorderconditionisouroldfriend,0eE[u(W)R]=0T£¡¡¢¢¤0f0eEuWR+αR=00Theobviouseasycasetosolvewillbequadraticutility,£¡¢¤bf0eeE(c−WR+αR)R=00¡¢bfe0ee00=c−WRE(R)−WαE[RR]=000µ¶bc−1fee0eα=−RE[RR]E(R)W01−1fee0eα=RE[RR]E(R)γThisisthesamemean-varianceefficientportfoliowe’,supposethereisasignal,zthatpredictsreturnsR=a+bz+ε,ttt+1t+1andsupposewewanttomaximizeE(U(W))Now,theweightschangeeachperiod,,-hoctradingrules,forexampleα=a+bz(portfoliostwillgiveusarbitrarylinearfunctionsoftheserules,thuschangingtheinterceptandslopeasneeded),,inprinciple,equivalenttothefullydynamicportfoliotheory,justasinChapter8unconditionalmomentswithmanagedportfolioscould,inprinciple,,thelimitationofthisapproachisthatwedon’,wecanappealtotheuniversalpracticeinstaticproblems:Wedonotincludeallindividualstocks,bonds,currencies,etc.,
chosenportfoliosofassetsareenoughtocapturethecross-section,,ifonereallywantstheexactoptimum,,,power-lognormalIre-solvetheoneperiod,powerutility,,μ−r1constantly-rebalancedshareα=γσThisisaclassictheorem:ff,,theinvestorputsafractionα(c).{α}t∙¸dStdW=Wα+(1−α)rdtttttStc=W;WgivenTT0Istartwiththecanonicallognormaliidenvironment,dSt=μdt+σdztStdB=,wealthevolvesasdWt=[r+α(μ−r)]dt+ασdz.(45)ttWtWefindtheoptimalweightsα(W,t)=maxEV(W,t+dt)tt+dt{α}t35
andhence,usingIto’slemma,½¾120=maxEVdW+VdW+VdttWWWt{α}2t12220=maxWV[r+α(μ−r)]+WVασ+V(46)WtWWtt{α}2tThefirstorderconditionforportfoliochoiceαleadsdirectlytotVμ−rWα=−(47)t2WVσWWWewillendupproving1−γV(W,t)=k(t)Wtandthus1μ−rα=.(48)t2γσTheproportioninvestedintheriskyassetisaconstant,,thehigherthestockexcessreturn,lowervariance,;theyoungshouldinvestinstocks,:manyyoungpeopleinvestinbondsuntiltheybuildupasafe“nestegg,”,,,orsothatthe“safety-first”,time-varyingexpectedreturnscanraisetheSharperatiooflong-horizoninvestments,(Actually,thequantity−istheriskaversioncoeffi’ffectyourwealth,notyourconsump-tion,,riskaversionisalsoequaltothelocalcurvatureoftheutilityfunctionγ,andthereforeriskaversionisindependentoftimeandwealth,eventhoughV(W,t).)Withtheoptimalportfolioweightsinhand,investedwealthWfollowsµ¶α12S(1−α)r+σαT()2W=We(49)α=1,weobtainW=W(S/S),andif0T0rTa=0weobtainW=We,
AlgebraThealgebrafor(49)=(1−α)rdt+αWSt2dW1dWdS1tt22dlnW=−=(1−α)rdt+α−ασdtt2W2WS2tt2dS1dSdS1tt2dlnS=−=−σdtt2S2SS2ttµ¶11222dlnW=(1−α)rdt+αdlnS+σdt−ασdttt22∙¸12dlnW=(1−α)r+σα(1−α)dt+αdlnStt2µ¶12dlnW=(1−α)r+σαdt+αdlnStt2µ¶12lnW−lnW=(1−α)r+σαT+α(lnS−lnS)T0T02µ¶α12ST(1−α)r+σαT()2W=WeT0S0ThevaluefunctionItremainstoprovethatthevaluefunctionVreallydoeshavetheformV(W,t)=1−γk(t)W/(1−γ),andtofindk(t).tSubstitutingtheoptimalportfolioαinto(88),thevaluefunctionsolvesthedifferentialtequation12220=WV[r+α(μ−r)]+WVασ+VWtWWt2∙¸µ¶22V(μ−r)1WVV(μ−r)VWWWWt20=r−(μ−r)+σ+22WVσ2WVWVσWVWWWWWW2V(μ−r)1V(μ−r)VWWt0=r−(μ−r)++22WVσ2WVσWVWWWWW21V(μ−r)VWt0=r−+,(50)22WVσWVWWWsubjecttotheterminalconditionu(W)=V(W).TTTheusualmethodforsolvingsuchequationsistoguessthesolutionuptounderterminedparametersorsimplefunctions,,guessasolutionoftheformη(T−t)1−γV(W,t)=eW37
Hence,η(T−t)1−γV=−ηeWtη(T−t)−γV=(1−γ)eWWη(T−t)−γ−1V=−γ(1−γ)eWWWV1W−=WVγWWVηt=−WV1−γWPluggingintothePDE(90),thatequationholdsiftheundeterminedcoefficientηsolves211(μ−r)η0=r+−22γσ1−γHence,∙¸211(μ−r)η=(1−γ)r+22γσand 2(μ−r)11(1−γ)r+(T−t)22γσ1−γV(W,t)=eWSinceourguessworks,theportfolioweightsareinfactasgivenbyequation(48).Youmight1−γhaveguessedjustW,-marketproblem,wedon’’vealreadysolvedtheproblemandfoundthefinalpayoffs,ffapproachHavingboththediscountfactorapproachandtheportfolioweightapproachinhand,αtogettoxˆ=(const)×R,,,isthatwesolvedforalotofstuffwedidn’ff,butwefoundaspecificdynamictradingstrategytosupportthatpayoff.,youmightwanttoimplementtheoptimalpayoffffffspace,,yetwecan−γ0easilycharacterizethepayoff(c)=(c−h)aboveisonesuch38
ff,butyouwillfail,(89),however,theriskaversioncoeffi=h,youbecomemuchmoreriskaverse!fferentialequationforthevaluefunction(90),wehaveaterminalcondition1−γV(W,T)=(W−h).η(T−t)1−γOfcourse,ouroriginalguessV(W,t)=eWwon’η(T−t)1−γnaturalguessV(W,t)=e(W−f(t)h),alas,doesnotsolvethedifffferentialequation Z2(μ−r)11(1−ξ)r+(T−t)22γ1−ξσV(W,t)=a(ξ)eWdξtandthenfinda(ξ)—thetimeTpayofforthenumberofcontingentclaimstobuy—,itissimplerfirsttocharacterizetheoptimalpayoffxˆ,andthentochoosehowtoimplementthatpayoffbyaspecificchoiceofassets,,dynamictrading,purecontingentclaims,digitaloptions,,ingeneralincompletemarketsproblems,choosingportfolioweightsmeansyouknowyoualwaysstayintheassetspacexˆ∈
5DynamicintertemporalproblemsNowweremovetheiidassumptionandallowmeanreturns,-variableMertonproblemWeallowmeanreturns,μ−rttα=+ηβttdy,dR2γσttwhereγandηareriskaversionandaversiontotheriskthatthestatevariablettchanges,definedbycorrespondingderivativesofthevaluefunction,andβisdy,dRtheregressioncoeffiffects:1)“Markettiming.”Theallocationtotheriskyassetmayriseandfallovertime,forexampleifthemeanexcessreturnμ−rvariesttandγandσ)“Hedging”“bad”ttrealizationsofthestatevariable,’,notterminalwealth;∞−ρtmaxEeu(c).(51)t0dR=μ(y)dt+σ(y)dz(52)ttttdy=μ(y)dt+σ(y)dz(53)ttyttyTheobjectivecanalsobeorincludeterminalwealth,ZT−ρtmaxEeu(c)dt+EU(W).tT0InthetraditionalMertonsetup,,wecaneasilyextendthemodeltothinkofthemasstatevariablesforlabororproprietaryincomeeandincludec=x+,whichsimplifiesthealgebraandgives40
ffαintheriskyasset,wealthevolvesasdW=WαdR+W(1−α)rdt+(e−c)dtdW=[Wr+Wα(μ−r)+(e−c)]dt+Wασdze(reallye(y)),sotheBellmanequationis£¤−ρdtV(W,y,t)=maxu(c)dt+EeV(W,y,t+dt),tt+dtt+dt{c,α}usingIto’slemmaasusual,0=maxu(c)dt−ρVdt+Vdt+VE(dW)+VE(dy)tWtyt{c,α}1122+VdW+Vdy+,=maxu(c)−ρV(W,y,t)+V+V[Wr+Wα(μ−r)+e−c]+Vμ(54)tWyy{c,α}112222+VWασ+Vσ+WVασσ.WWyyWyyy22Now,fferentiating(54),∂0:u(c)=VW∂,wefindthefirstorderconditionforportfoliochoice:∂22:WV(μ−r)+WVσα+WσσV=0WWWyWy∂αV(μ−r)σVWyWyα=−−2WVσσWVWWWWThisistheall-importantanswerwearelookingfor:theweightsoftheoptimalportfolio.σσ=cov(dR,dy)isthecovarianceofreturninnovationswithstatevariableinnovations,soy2σσ/σ=βistheregressioncoefficientofstatevariableinnovationsonreturninnova-ydy,,wecanwritetheoptimalportfolioweightintheriskyassetasVμ−rVWtWytα=−−β(55)dy,dR2WVσWVWWWWt1μ−rηtt=+β(56)dy,dR2γσγt41
Inthesecondline,Ihaveintroducedthenotationγforriskaversionandηwhichmeasures“aversion”γ≡−;η≡.VVWWTheγ,,themeanandvariancechangeovertime—that’,Investorswill“timethemarket,”:Investorswillincreasetheirholdingoftheriskyassetifitcovariesnegativelywithstatevariablesofconcerntotheinvestor.“Ofconcern”“hedging”-termbondisanexcellenthedgefortheriskthatinterestratesdecline,,-reverttoo,weshouldexpectimportantquantitativeresultsfromtheMertonmodel:mean-reversioninstockpriceswillmakestocksevenmoreattractive.(Thislastconclusiondependsonriskaversion,ffff“wealtheffect.”Theinvestorwillbeabletoaff“substitutioneffect.”Athigherexpectedreturns,itpaystheinvestortoconsumelessnow,andthenconsumeevenmoreinthefuture,,equaltointertemporalsubstitution,ishigh,,soV=u(c)declinesnow,<,WWyifriskaversionisverylow,thesubstitutioneff,=u(c)rises,andV>ffectsoffset,soV=,sothatcaseapplies.)Ofcourse,riskaversionandstatevariableaversionarenotconstants,,
Conceptuallythisstepissimple,asbefore:ff,evenabrieflookattheproblemwillshowyouwhysolittlehasbeendoneonthiscrucialstep,fferentialequationis,from(54)(algebrabelow)£¤10−10−120=uu(V)−ρV+V+WVr−Vu(V)+Ve+Vμ+VσWtWWWWyyyyy2112−[V(μ−r)+σσV].WyWy22σVWWThisisnotapleasantpartialdifferentialequationtosolve,andanalyticsolutionsareusually0−10−(u(V))andu(V)areespeciallytroublesome,WWwhichaccountsforthepopularityofformulationsinvolvingtheutilityofterminalwealth,:1−γ,infinitehorizon,,V(W)=,powerutilityinvestors,,,V=0,,(noconsumption),AR(1)statevariable,nolaborincome,power(ormoregenerallyHARA)utility.(KimandOmberg1996).Herethe1−γ2naturalguessthatV(W,y,t)=Wexp[a(T−t)+b(T−t)y+c(T−T)y]works,thoughsolvingtheresultingdiff:Pluggingoptimalconsumptioncandportfolioαdecisionsinto(54),0=u(c)−ρV+V+V[Wr+Wα(μ−r)+e−c]+VμtWyy112222+VWασ+Vσ+WVασσWWyyWyyy2210−10−120=u(u(V))−ρV+V+WVr−Vu(V)+Ve+Vμ+VσWtWWWWyyyyy21222+VWασ+W(V(μ−r)+Vσσ)αWWWWyy243
10−10−120=u(u(V))−ρV+V+WVr−Vu(V)+Ve+Vμ+VσWtWWWWyyyyy2∙¸21V(μ−r)σσVWyWy22+VWσ+WW222WVσσWVWWWW∙¸V(μ−r)σσVWyWy−W[V(μ−r)+Vσσ]+WWyy22WVσσWVWWWW10−10−120=u(u(V))−ρV+V+WVr−Vu(V)+Ve+Vμ+VσWtWWWWyyyyy2112+[V(μ−r)+σσV]WyWy22σVWW1−[V(μ−r)+Vσσ][V(μ−r)+σσV]WWyyWyWy2σVWW10−10−120=u(u(V))−ρV+V+WVr−Vu(V)+Ve+Vμ+VσWtWWWWyyyyy2112−[V(μ−r)+σσV].WyWy22σ,,nooutsideincomeandiidreturns,thedifferentialequation(54)specializesto1−(1−γ)γ1V−11Wγ20=−ρV+V+WVr−VV−[V(μ−r)]tWWWW21−γ2σVWWTosolveit,weguessafunctionalform1−γWV=−γPluggingin,wefindthatthedifferentialequationholdsif∙¸21ρ1−γ1(μ−r)−γk=−r+.2γγ2γσHence,wecanfullyevaluatethepolicy:Optimalconsumptionfollows∙µ¶¸12−11(μ−r)γc=V=ρ−(1−γ)r+W(57)W2γ2γσ44
(γ=1)wehavec=ρffγ>1,higherreturns(eitherahigherriskfreerateorthehighersquaredSharperatiointhesecondterm)ffectsaregreaterthansubstitutioneffects(highγresistssubstitution),sothehigher“wealtheffect”γ<1,theoppositeistrue;theinvestortakesadvantageofhigherreturnsbyconsuminglessnow,,from(55),1μ−rα=.(58)2γσWealreadyhadtheoptimalconsumptionstreamin(41).Whatwelearnhereisthatwecansupportthatstreambytheconsumptionrule(57)andportfoliorule(58).TheAlgebra1−γWV=k1−γ−γV=kWW−γ−1V=−γkWWW1−(1−γ)(1−γ)γ2kW1−γ−γ21¡¢W1(kW)(μ−r)1−−γ−γγ0=−ρk+WkWr−kW+−γ−121−γ1−γ2γkWσ11−γk21ρk1(μ−r)k1−1−γ1−γ1−γ1−γ1−γγ0=W−W+rkW−kW+W21−γ1−γ2σγ1−γk21ρ1(μ−r)−γ0=−+r−k+21−γ1−γ2γσµ¶21γρ1(μ−r)−γ0=k−+r+21−γ1−γ2γσ∙¸21ρ1−γ1(μ−r)−γk=−r+2γγ2γση−10α=Σ(μ−r)+βdy,dRγγincludeamean-varianceefficientportfolio,butalsoincludemimickingportfoliosforstate-variablerisks45
Now,let’’schoiceamongassetsmaybeaff,Z∞−ρtmaxEeu(c).(59)t0dR=μ(y)dt+σ(y)dz(60)ttttdy=μ(y)dt+σ(y)dz(61)ttyttyde=μ(y)dt+σ(y)dz(62)ttetteNowIusedRtodenotethevectorofNreturnsdS/S,soμ(atleast)N+Kdimensionalvectorofindependent0shocks,E(dzdz)=,σisanN×(N+K)dimensionalmatrixandσisaK×(N+K)’llexaminethecaseinwhichoneassetisariskfreerate,rSinceitvariestovertime,,butsincelaborincomeisimportantandalltheresultswewillgettoaccommodateiteasily,,iftheinvestorputsweightsαontheriskyassets,wealthevolvesas¡¢00dW=WαdR+W(1−1α)rdt+(e−c)dt£¤00dW=Wr+Wα(μ−r)+(e−c)dt+Wασ−ρdtV(W,y,t)=maxu(c)dt+EeV(W,y,t+dt),tt+dtt+dt{c,α}andusingIto’slemmaasusual,0=maxu(c)dt−ρVdt+Vdt+VE0(dW)+VE(dy)tWtyt{c,α}1120+VdW+dyV0dy+,£¤00=maxu(c)−ρV(W,y,t)+V+VWr+Wα(μ−r)−c+V0μ(63)tWyy{c,α}11200000+VWασσα+Tr(σV0σ)+Wασσ(dzAdz)=dzAdz=A=Tr(A).iijjiii,ji46
Then,00000dyV0dy=(σdz)V0(σdz)=dzσV0σdz=Tr(σV0σ).yyyyyyyyyyyyyyWecandotheothertermssimilarly,0000dWV0dy=(Wασdz)V0(σdz)=WdzσαV0σdzWyWyyWyy00000=WTr(σαV0σ)=WTr(ασσV)=WασσVWyyWyWyyy20002000VdW=V(Wασdz)(Wασdz)=WVdzσαασdzWWWWWW¡¢¡¢200200200=WVTrσαασ=WVTrασσα=WVασσαWWWWWW0000(IusedTr(AA)=Tr(AA)andTr(AB)=Tr(AB).Thesefactsabouttracesletmecondensea(N+K)×(N+K)matrixtoa1×1quadraticforminthelastline,andletmetransformfrom00anexpressionforwhichitwouldbehardtotakeαderivatives,Tr(σαασ),toonethatiseasy,00ασσα).Now,fferentiating(63),weobtainagain∂0:u(c)=VW∂cDifferentiatingwithrespecttoα,∂200:WV(μ−r)+WVσσα+WσσV=0WWWWyy∂α¡¢¡¢VVW−1−1Wy000α=−σσ(μ−r)−σσσσyWVWVWWWWThisistheall-importantanswerwearelookingfor:.σσ=cov(dR,dR)=Σisthereturninnovationcovariancematrix.σσy0=cov(dR,dy)=σ0isthecovarianceofreturninnovationswithstatevariableinnovations,anddR,y−100−10(σσ)σσ=Σσ0=βisamatrixofmultipleregressioncoefficientsofstatevariabledR,yydy,,wecanwritetheoptimalportfolioweightsasVVWWy−10α=−Σ(μ−r)−β(64)dy,dRWVWVWWWWThefirsttermisexactlythesameaswehadbefore,−1inΣ(μ−r)theweightsofamean-varianceeffi:Inaniidworld,investorswillholdaninstantaneouslymean-varianceeffi’reusingdiffusionprocesseswhicharelocallynormal,,whichIusedabovetostartthinkingaboutmean-varianceeffi,notethatevenifαisconstantovertime,thismeansdynamicallytradingandrebalancing,sothatportfolioswillnotbemean-varianceeffi,theriskyassetshareαwillgenerallychangeovertime,givingevenmoreinterestingandmean-varianceineffi:Investorswillshifttheirportfolioweightstowardsassetsthatcovarywith,andhencecanhedge,fferintheirdegreeofriskaversionand“aversiontostatevariablerisk”sowecanwritetheoptimalportfolioas1η−10α=Σ(μ−r)+β(65)dy,dRγγ47
whereagainWVVWWWyγ≡−;η≡.VVWWIfapositivereturnonanassetisassociatedwithanincreaseinthestatevariabley,andifthisincreaseisassociatedwithanincreaseinthemarginalvalueofwealth,<0,,bychangingastatevariable,isasimportantasincreasingitdirectly,ffi-varianceproblem:ffi-varianceportfoliotheory,follow-ingFama(1996).TheMertoninvestorminimizesthevarianceofreturnsubjecttomeanreturn,’sformportfoliosp00dR=αdR+(1−α1)rdtThesuggestedmean,variance,covarianceproblemispppminvar(dR)=E;cov(dR,dy)=ξttt00000minασσα+α(μ−r)=E;ασσ=ξy{α}IntroducingLagrangemultipliersλ,λ,thefirstorderconditionsare1200σσα=λ(μ−r)+σσλ12y¡¢¡¢−1−1000α=λσσ(μ−r)+σσσσλ(66)12y−10α=λΣ(μ−r)+βλ(67)12dy,dRThisisexactlythesameansweras(64)!-variancefrontierisahyperbola,fficientportfolios.(Covariancewithastatevariableisalinearconstraintonreturns,,thefrontieristherevolutionofaparabolainmean-variance-covariancespace,,thefrontierisacone.)Asshowninthepicture,wecanthinkoftheinvestorasmaximizingpreferencesdefinedovermean,varianceandcovarianceoftheportfolio,
E(R)σ(R)cov(R,y)Figure5:Multifactoreffcientportfolioand“indifferencecurve.”Thefirsttermin(64)and(67)isthemean-variancefrontier,oratangencyportfolio.(Setλ=0andequation(67)derivesthisresult.)Thus,weseethattypicalinvestorsdonotholdmean-2varianceeffi-varianceeffi
,butalsotocovarianceswithstatevariables,ormimickingportfolioreturnsmm0mμ−r=cov(dR,dR)γ−cov(dR,dy)ηmmz0mμ−r=cov(dR,dR)γ−cov(dR,dR)ηIt’ff,weightedbywealthα=PPiiiWα/,summing(65)overinvestors,iim1ηm−10α=Σ(μ−r)+β(70)dR,dymmγγHere,PPiηi1imWWii1ηiiγγ=PP;=mjmjγWγWjjTheICAPMsolvesthisexpressionforthemeanexcessreturnmmmμ−r=γΣα−σ0η(71)dR,dy0−1(Iusedβ=Σσ0.)ThemarketportfolioreturnisdR,dydR,dymm0dR=rdt+α(dR−r)dtThus,werecognizem0mmΣα=cov(dR,dR)α=cov(dR,dR)andwehaveTheICAPM:mm0mμ−r=cov(dR,dR)γ−cov(dR,dy)η.“state-variableaversion”coeffi(69)aretheprojectionsofstatevariablesonthe0z0spaceofexcessreturns,cov(dR,dy)=cov(dR,dR).Directly,zdR=β(dR−rdt)dy,dR00−1=cov(dy,dR)cov(dR,dR)(dR−rdt)z000−10cov(dR,dR)=cov(dR,dR)cov(dR,dR)cov(dR,dy)50
Thus,wecanexpresstheICAPMintermsofcovarianceswithmimickingportfolios,mmz0mμ−r=cov(dR,dR)γ−cov(dR,dR)η.Sincethestatevariablesareoftennebulousorhardtomeasure,thisformisusedwidelyinpractice.(Historically,“ICAPM”onlyreferstomodelsinwhichtheothervariablesarestatevariablesforinvestmentopportunities,notstatevariablesforoutsideincome,sinceMerton’soriginalpaperdidnotincludeoutsideincome.“Multifactormodels”,sinceit’sclearlysotrivialtoincludestatevariablesforoutsideincomeatleastthisfar,I’lluse“ICAPM”anyway.)Thisexpressionwithcovarianceontherighthandsideisnice,sincetheslopesarerelatedtopreference(well,valuefunction),it’straditionaltoexpresstherighthandsideintermsofregressionbetas,andtoforgetabouttheeconomicinterpretationoftheλslopecoefficients(especiallybecausetheyareoftenembarrassinglylarge).Thisiseasytodo:μ−r=βmλ−β0λ(72)mdydR,dRdR,dymcov(dR,dR)βm=;dR,dR2σmdR0−100β0=(cov(dy,dy)cov(dy,dR))dR,dy2σmdRλ=;mmγ0mλ=cov(dy,dy)ηdyNowwehaveexpectedreturnsasalinearfunctionofmarketbetas,andbetasonstatevariableinnovations(ortheirmimickingportfolios).Don’tforgetthatallthemomentsareconditional!’ttheICAPMabout“marketequilibrium?”Howdowejumpfroma“demandcurve”toanequilibriumwithoutsayinganythingaboutsupply?TheansweristhattheimplicitgeneralequilibriumbehindtheICAPMhaslineartechnologies:investorscanchangetheaggregateamountineachsecuritycostlessly,withoutaff,,inadynamicmodel,,ifeverybodyislikethis,
Relativetothemarketportfolio,ratherthanthetangencyportfolio,m¡¢γ1imi0m0zdR=rdt+(dR−rdt)+η−ηdRiiγγAninvestorholdsmoreorlessofthemarketportfolioaccordingtohisriskaversion,andthenmoreorlessofthemimickingportfoliosforstatevariablerisk,ashis“aversion”-varianceinvestorwillthusshadehisportfoliotowardthemimickingport-folioofaveragestatevariablerisk,,theremustbeotherinvestorswhohedgeevenmorethanaverage,,mforwhichη=’ff,use(71)toeliminate(μ−r)ontherighthandsideoftheindividualportfolioweights(65),toobtaini£¤1ηi−1mmm0α=ΣγΣα−σ0η+βdR,dydy,dRiiγγ¡¢mimη−ηγim0α=α+βdy,dRiiγγIfyoulikelookingattheactualportfolioreturnratherthanjusttheweights,ii0dR=rdt+α(dR−rdt)m¡¢γ1imi0m0zdR=rdt+(dR−rdt)+η−ηdRiiγγzwhereagaindR=β(dR−rdt)arethereturnsonthemimickingportfoliosforstate-variabledy,,heholdsmoreorlessofthemimickingportfoliosforstatevariableriskaccordingtowhethertheinvestor“feels”diff,returntothemarketportfolioin(70).Thefirstterm—andonlythefirstterm—givesthemean-varianceeffi,themarketportfolioisnolongermean-varianceeffi,youcanseethattheoptimalportfoliohassliddownfromtheverticalaxisofthenose-coneshapedmultifactoreffi,andhencethemarketportfolio,givesupsomemean-varianceeffi,forexample,whytheICAPMinterpretationoftheFama-French3factormodel,-varianceinvestor,aninvestorwhodoesnotfearthestate52
ivariablechangesandsohasη=0;-coneshapedmultifactorefficientfrontierinFigure5,ineffectsellingstate-variableinsurancetootherinvestors,γηimzdR=rdt+(dR−rdt)−dRiiγγ,quantitatively,fficients(ofaveragemmreturnsoncovariances)γ,η.Ontheotherhand,,say,thevaluepremiuminthisway,theremustbeaninvestorwhoseηmisforexampletwiceη,thushisoptimalportfolioreturnismm0γηimzdR=rdt+(dR−rdt)+dRiiγγHeseesthegreatnewsoftheSharperatioofvalueportfolios,butwantstosellnotbuy,,thisformulationincludesidiosyncraticstatevariablerisk,perhapsthemostimportant(andoverlooked)riskofall.η=V/Wisavectoranddifferentfordiff’sproblemstatevariablesforhisindividualoutsideincomerisk,eventhoughη=,forexample,thisinvestorisnodifferentfromeverybodyelseabouthismfeelingstowardsaggregatestatevariables,thenhisoptimalportfoliowillbemγ1imi0zdR=rdt+(dR−rdt)+ηdRiiγγThisinvestorholdsthemarketportfolio,plusaportfolioofassetsthatbestoffsetshisindividualoutsideincomerisks.(Statevariablesforinvestmentopportunitiesarebydefinitioncommontoallinvestors.)miThehedgeportfoliosforindividualrisks,withη=0obtainnoextrapremium;ηdoesnotentertheICAPM(71).Thus,γandη-formsolutionforcasesotherthanthesimpleonesstudiedabove,so,alas,
Again,westillneedtocomputethelevelsofriskaversionand“statevariableaversion”fromtheprimitivesofthemodel,,asbefore:,theresultingpartialdiff(54),theequationis£¤10−10−100=uu(V)−ρV+V+VWr−Vu(V)+V0μ+Tr(σV0σ)WtWWWyyyyyy2£¤1∗002∗00∗+Wα(μ−r)V+σσV+VWασσαWWyWWy2where¡¢¡¢VVW−1−1Wy∗000α=−σσ(μ−r)−σσσσyWVWVWWWW54
6Portfolioswithtime-varyingexpectedreturnsGiventhatmarketreturnsareforecastable,,let’
µ¶∙¸μ−rμ¯−rμ−rσσμzμwd=φ−dt+dz+dwσσσσσdx=−φ(x−x¯)dt+σdz+σdwttxztxwtdΛt=−rdt−xdz−ηdwtΛtNowwecanexpressthecurrentSharperatioasanAR(1)ZZTT−φs−φs−φTx−x¯=σedz+σedw+e(x−x¯)txzt−sxwt−s000andthenthediscountfactorisµ¶2dΛ1dΛ12dlnΛ=−=−r+xdt−xdzttt2Λ2Λ2Zµ¶Ztt12lnΛ−lnΛ=−r+xds−xdzt0sss200µ¶2dΛ1dΛ1122dlnΛ=−=−r+x+ηdt−xdz−ηdwttt2Λ2Λ22Zµ¶Ztt1122lnΛ−lnΛ=−r+x+ηds−xdz−η(w−w)(73)t0ssT0s2200Now,the“complete”marketscaserequiresthatthereisasingleshockdz,soshockstoμareperfectlycorrelatedwithshockstodR,“incomplete”whenthisisnotthecase,,“complete”marketscase,then,wehaveZT−φs−φTx−x¯=σedz+e(x−x¯)txzt−s00Zµ¶Ztt12lnΛ−lnΛ=−r+xds−xdz(74)t0sss2001∙¸−γΛtρtc=eλ(75)tΛ0Thissystemisnotquitesopleasant,thoughitisawell-studiedclass.(It’sa“stochasticvolatility”modeloftenusedtodescribestockprices.)Evenintheabsenceofaclosed-formsolution,however,youcanstraightforwardlysimulateitandwatchthediscountfactorandthenoptimalpayoffsrespondtothestatevariablesInthe“incomplete”marketscase,we’rebackto(73)togetherwith(75).Ourtaskistopickthechoiceofηsothatthefinalcisnotdrivenbyshocksdw,η=0twilldothetrick,sincethentheexplicitdependenceofΛonwinthelasttermof(73),
7WackyweightsandBayesianportfoliosPortfoliomaximizationdependsonparameters,,ratherthanmaximizingconditionalonparametersθ,Z0maxu(αRW)p(R|θ)dRt+10t+1t+1{α}Weintegrateovertheuncertaintyaboutparametersaswell,Z∙Z¸0maxu(αRW)p(R|θ)p(θ)dθdR(76)t+10t+1t+1{α}Thisapproachcanusefullytamethewildadviceofmostportfoliocalculations,:’tknowexactlywhattheequitypremiumis,wedon’tknowexactlywhattheregressionofreturnsondividendyieldsandotherpredictorvariableslookslike,andwedon’tknowexactlywhatthecrosssectionofmeanreturnsorthe“alphas”,inastatisticalframework,’sinterestingtotrackdownhowmuchuncertaintywehaveaboutoptimalportfolioweights,butthatconsiderationdoesn’,itmakesintuitivesensethatonewilltakelessadvantageofapoorlyestimatedmodel,,,,whenwehavemadeaportfoliocalculation,wemaximizedexpectedutility,-periodproblem,wesolvedZ0maxu(αRW)p(R|θ)dRt+10t+1t+1{α}whereθ,weshouldintegrateoveralltheparametervaluesaswell,∙Z¸0maxu(αRW)p(R|θ)p(θ)dθdR(77)t+10t+1t+1{α}The“predictivedensity”Zp(R)=p(R|θ)p(θ)dθt+1t+158
,,Bayesianportfoliotheorycancapturethreeeff,itcancapturetheeff,wetakea“diffuseprior”,sothatp(θ),itcanletusmix“prior”informationwith“sample”,wehaveanewmodelorideasuggestingaportfoliostrategy,(θ)nowweightsour“prior”information,reallysummingupalltheotherdatawe’velookedat,relativetothestatisticalinfor-mationinanewmodel,,aninvestorkeepslearningastimegoesby;eachnewdatapointisnotonlygoodandbadluck,butitalsoisextrainforma-tionthatchangestheinvestor’,theinvestor’sshiftingprobabilityviewsbecomea“statevariable.”Ionlyconsiderthefirsttwoeffectshere,,let’,,howmuchshouldanoptimalportfoliomarket-time,
issuspiciouslylarge,especiallyfortheγ=2−=−+ 50 100 150 200012345678910d/p ratio (%)eE(R)1tFigure6:α=t2γσ.Expectedexcessreturnscomefromthefittedvalueofaregressionofreturnsondividendeyields,R=a+b(D/P)+ε.TheverticallinesmarkE(D/P)±2σ(D/P).ttt+1t+1Inthiscaseespeciallythough,fficientsof5orsointhemid1980s,theboomofthe1990sdespitelowdividendyieldscutthecoeffi,andthecoeffi“structuralshift”,,,whichismuchlongerthanthe5-10yearsamplesthatareconsidered“long”, to stocks (%)
250γ=2γ=251025200150100γ=550γ=10γ=250 19301940195019601970198019902000timeFigure7:(R)1tα=.Expectedexcessreturnscomefromthefittedvalueofaregressionofreturnst2γσeondividendyields,R=a+b(D/P)+ε.ttt+1t+-sectionIcalculateaverysimplemean-variancefrontier,andweseethatthemaximizerrecommends“wackyweights”-varianceproblemmightworkinpractice,,andhedgefundsconsiderhundredsofassets,alongwithtime-varyingmeansandcovariancematrices,,−1w=ΣμγwhereμandΣγ,Ichoosethescaletoreportsothatthevarianceoftheresultingportfolioisthesameasthevarianceofthemarketindex,2σ(rmrf)−1w=Σμ0−1μΣμ61allocation to stocks (%)
rmrf0min var σ (%/mo)Figure8:mean-varianceoptimizationwithexcessreturnsoftheFamaFrench25sizeandbook/marketportfolios,togetherwiththe3FamaFrenchfactors.“Optimal”isthemean-varianceoptimalportfolio,atthesamevarianceasthemarketreturn.“FF3F”,σ(wR)=wΣw=σ(rmrf).Thisportfolioisgraphedatthe“optimal”“FF3F”pointgivesthemean-varianceoptimalcombinationofthe3Fama-Frenchfactors,andthe“rmrf”pointgivesthemarketreturn,’%/%/monthisavailablewithnoincreaseinvolatility?However,let’stakealookattheactualportfoliotheoptimizerisrecommend-ing,inTable1below:62E (%/mo)
low234highsmall-149516996522-19-57190-13-60329-34-31-93414116-39-4235-2large87-198-22rmrfhmlsmb-9477-692σ(rmrf)−=ΣμintheFama-French0−1μΣμ25sizeandbook/“wackyweights!”149%shortpositioninsmallgrowth,then+190%inthe(2,3)portfolio,and-93%inthe(3,4)portfolio,-99%,−1Σ1w=0−11Σ1Thisportfolioisusefultoseetroubleswiththecovariancematrixbyitself,,bycleverlongandshortpositionsamongtheFamaFrench25returnsandthe3factors,onecansynthesizeanearlyriskfreeportfolio!Amoment’,small,andperhapseveninsignificant,differencesinaveragereturnscanshowupashugediff,inthattheyhavethesametruemean,standarddeviation,,,themean-variancefrontierconnectingthemisveryflat,-varianceportfolioisaninstanceofthesameproblem,inwhichAandBdon’“solutions”Again,(“traderskill”),mostportfolio63
Apparent optimal portfolioMeanVery long A, short BA in sampleTrue (equal)B in sampleRisk freeStandard deviationFigure9:(orareforcedtoadopt)alonglistofadditional,seeminglyarbitraryrules:don’tinvestmorethanxpercentinasinglesecurity,makesurethequantitiesarediversifiedacrossindustrycategories,%longAand0Bisbetterthantheextremelong-shortposition,butitdoesnotrecoverthetrueanswer50%Aand50%,,itseemsbettertofigureoutwhytheportfoliomaximizerisn’’,itseems,,let’sadapttheideassurroundingequation(77)
,wesimplyaddthemeanreturnstandarderrortothevarianceofreturns;theinvestoractsasifreturns£¤222areR∼Nμ¯,σ+σwhereμ¯isthemeanofhisestimateofthemeanreturns,andσisthet+1μμ,andfindthattheallocationtotherisky2assetisα=(¯μ−r)/γσ[1+(γ−1)h/T]wherehistheinvestor’,,power,iidworld,theallocationtostocksis1μ−rttα=2γσtFortypicalnumbersμ−r=8%andσ=16%,,==
Thus,wehave£¤£¤222R∼Nμ¯,σ+σ=Nμ¯,σ(1+1/T)(80)t+1μ’,-variancecalculation,(80),uncertaintyaboutthemeanismuchmoreimportantthanuncertaintyaboutvariances,,atshorthorizons,riskcomesalmostentirelyfromthevarianceofreturns,whileat10yearorlongerhorizons,,iftheannualizedmeanmarketreturnhasa5percentagepointstandarderror,thisaddsonly5/365=,atinycontributioncomparedto√a20/365=,ata10yearhorizon,10×5=50percentagepointsisamuchlargeramountofriskcomparedto20×10=200%,ffecttosomeextentin(80).Asweapplytheformulatolongerhorizons,,toagoodapproximation,wecanuse£¤2R∼Nμ¯,σ(1+h/T).(81)t+hIt’smoresatisfyingtobeabitmoreexplicitbothaboutthehorizonandabouttheportfoliooptimization,’’llpresumeaconstantlyrebalancedfractionαintheriskyasset,whichweknowfromaboveistheanswerwhenweallowαtobefreelychosen.(It2amignoringthelearningeffectthatσdeclinesovertheinvestmenthorizon.)Giventheportfolioμchoiceα,wealthevolvesasdW=rdt+α(μ−r)dt+ασdzW∙¸122dlnW=r+α(μ−r)−ασdt+ασdz2√122r+α(μ−r)−ασh+σαhε[]2W=We;ε˜N(0,1)h0Theinvestor’sproblemisÃ!1−γWTmaxEα1−γ1−γ³√´W1220(1−γ)r+α(μ−r)−ασh+(1−γ)ασhε[]2maxEeα1−γNow,theinvestorisunsureaboutthereturnshockε(ofcourse),butalsoaboutthemeanμ.Youcanseedirectlythatμandε
sourcesofriskareindependent—uncertaintyaboutthemeancomesfromthepastsample,anduncertaintyaboutreturnscomesfromthefuture.(Again,Iamignoringthefactthattheinvestorlearnsabitmoreaboutthemeanduringtheinvestmentperiod.)Thus,theproblemisZ1−γ√W1220(1−γ)r+α(μ−r)−ασh+(1−γ)ασhε[]2maxef(ε)f(μ)dεdμα1−γDoingtheconventionalεintegrationfirst,Z1−γW12212220(1−γ)r+α(μ−r)−ασh+(1−γ)ασh[]22maxef(μ)dμα1−γZ1−γW1220(1−γ)r+α(μ−r)−γασh[]2maxef(μ)dμα1−γLet’smodeltheinvestor’suncertaintyaboutμalsoasnormallydistributed,withmeanμ¯(samplemean)andstandarddeviationσ(standarderror).Thenwehaveμ1−γW122122220(1−γ)r+α(μ¯−r)−γασh+(1−γ)ασh[]μ22maxeα1−γα,and1−γ(..)cancelingWewehave0£¤2222(1−γ)(μ¯−r)−γασh+(1−γ)ασh=0죤22γασ−(1−γ)ασh=(¯μ−r)μandthusfinallyμ¯−rα=.22γσ+(γ−1)σhμ√Ifthevarianceofμcomesfromastandarderrorσ=σ/TinasampleT,thenμμ¯−rhiα=(82)γ−1h2γσ1+γT¡¢2Youcanseeintheseformulasthespecialcaseα=(μ−r)/γσ,,almostexactlyasinthesimplecalculation(81).Youalsocanseethattheeffectdisappearsforlogutility,γ=1,,IreproducethesetupinBarberis’s(2000),ina43yearanda11yearlongdataset,,calculatedusing(82)andusingthemeanandstandarddeviationofreturnsgiveninBarberis’’’calculations,67
γ = 5, T = 43γ = 10, T = 438040357030602550204015301002468100246810horizonhorizonγ = 5, T = 9γ = 10, T = 98040357030602550204015301002468100246810horizonhorizonFigure10:,andthedashedlinesignoreparameteruncertainty.μ−rTheallocationtostocksisα=.μ=,σ==43andμ=2σ[γ+(γ−1)h/T)],σ==9withr=(θ)fromapriorandalikelihoodfunction;theyincludeestimationuncertaintyaboutvariancesaswellasmeans,£¤22ApplyingtheruleR∼Nμ¯,σ+σinaveryback-of-theenvelopemanner,Ishowthatpara-t+1μmeteruncertaintysubstantiallytamesthestrongmarkettimingpredictionfromreturnforecastabil-2I’mstillnotexactlysurewhy(81)and(82)differatall,however,onereasonforthe“preliminary”disclaimer.¡¢32Idon’texactlymatchtheresultswithoutlearning,whichisabitofapuzzlesinceα=(μ−r)/γσ,Ihadtosetr=0togetevenavaguelysimilarcalculationwhenusingtheμandσfromBarberis’ to stocksallocation to stocksallocation to stocksallocation to stocks
fferentfromtheunconditionalmeanarelesswellmeasured,¡¢eˆER=αˆ+b(D/P)tttt+1Therefore,thesamplinguncertaintyabouttheconditionalmeanreturnis£¡¢¤2e222ˆˆσER=σ(aˆ)+σ(b)(D/P)+2cov(aˆ,b)D/P(83)tttttt+,theoptimalallocationthatincludesparameteruncertaintywilllikelyplacelessweightonthosemeans, 10 20012345678910dpFigure11:Expectedexcessreturnsasafunctionofdividendyield,,marked“none”presentsthestandardallocationresultforγ=5,basedontheregression(78)eˆE(R|D/P)11aˆ+b(D/P)ttttt+1α==.22γσ(ε)γσ(ε)69E(R|dp)
Thisisthesameastheγ=ˆ1aˆ+b(D/P)ttα=£¡¢¤e22γσ(ε)+h×σERtt+1using(83)tocalculatethebottomvarianceforeachinvestmenthorizonh,(2000).(Inparticular,compareittothepresentationofBarberis’resultsinFigure3ofCochrane(2000)“Newfactsinfinance”)Insum,Figure12saysthatinvestorswithdecentlylonghorizonsshouldsubstantiallydiscountthemarkettimingadviceofdividendyieldregressions,becauseofparameteruncertainty,,“buyandholdinvestor”whodoesnotchangehisallocationtostocksasthedividendyieldvariesthroughhisinvestmentlife,Iignoredthe“learning”effect,(1996)studyamonthlyhorizonandcometotheoppositeconclusion:“economic”significanceofdividendyieldforecastabilityismuchlargerthanitsstatisticalsignificance,,theapparentdiff,andperhapsmoreimportant,useofBayesianportfoliotheory,,,onewantstomergethesetwopiecesofinfor-mation,,“wackyweights”byshadingtheinputsbacktoa“prior”thatalphas(relativetoanychosenassetpricingmodel”³´³´ααˆp11E(α|αˆ,α)=+/+whereαˆisanestimate,andαisaprior(typicallyp2222pσ(αˆ)σ(α)σ(αˆ)σ(α)pp2zero)withconfidencelevelexpressedbyσ(α).Sensibly,oneweightsevidenceαˆ
160140120100none1 yr80605 yr4010 yr200 20 40012345678910dpFigure12:Optimalallocationtostocksgiventhatreturnsarepredictablefromdividendyields,andincludingparameteruncertainty,forγ=,α,withstandarderrorσandαwithstandarderror112σ.Youcancallαthe“prior”andσyourconfidenceintheprior,andαthe“estimate,”21α21withσα?Theansweris(algebrabelow)αα12+22σσ12E(α|α,α)=(84)1211+22σσ12Sensibly,youcreatealinearcombinationofthetwosignals,αisavector,thegeneralcasereads¡¢¡¢−1−1−1−1−1E(α|α,α)=Σ+ΣΣα+Σα.12121212Anaturalapplicationofcourseisthatthe“prior”isα=0,withacommonconfidencelevel,1Σ=σ“frequentist”waytoderive(84)istothinkofthesignalsαandαasgeneratedfrom12thetrueα,α=α+ε11α=α+ε2271weight in stocks
withεandεindependentbuthavingdifferentvariancesσandσ.Thisisasimpleinstanceof1212GLSwithtwodatapoints—estimatealphawithtwodatapointsαandα.TheGLSformulais12¡¢−10−10−1XΩXXΩYÃ!−1∙¸∙¸∙¸∙¸−1−122£¤£¤σ01σ0α1111111220σ10σα222µ¶µ¶−111αα12++2222σσσσ1212Ifα,α,αareN×1vectors,thesameGLSformulawith2Ndatapointsis12Ã!−1∙¸∙¸∙¸∙¸−1−1£¤£¤Σ01Σ0α11111110Σ10Σα222¡¢¡¢−1−1−1−1−1Σ+ΣΣα+Σα121212Ifyou’renotfamiliarwithGLS,it’(wα+(1−w)α−α)12w2222minwσ+(1−w)σ12λ22wσ=(1−w)σ12122σσ21w==2211σ+σ+2212σσ12Naturally,(2000,,Eq25-28)considersastandardfactormodeleR=α+βF+utttewhereRisanexcessreturn,andFisafactor()α˜=wα+(1−w)αsampleprior12(σ/T)u³´w=2E(F)111++22σvar(F)(σ/T)αu2Here,E(F),var(F)aresamplemomentsofthefactor,σisthe(priorexpectedvalueofthe)u2residualvariance,andσisthevarianceoftheprioraboutα.Yourecognizethesameformulaαasin(84)butwithasquaredsharperatioofthefactoradjustment,,thiswillbesmallintypicalmonthlydata,,
-correlation,,includingenhancingthediagonalelements,,’thavemorethanTdegreesoffreedomsoa200×200covariancematrixestimatedinanythinglessthan201datapointsissingularbyconstruction,,samplecovariancematricestendtoshow“toomuch”correlation,“Bayesian”solutiontothisproblemistoemphasizethediagonals,ˆΣ=Σ+λ,weusuallyestimatelargecovariancematricesbyimposingafactorstructure,R=βf+εtt(N×1)=(K×N)(K×1)+(N×1);K<<N00cov(RR)=βΣβ+ΣfThen,moresensiblereturncovariancematricesemergebydownweightingΣ,imposingΣ=D,,“noninformativeprior”,theyconsidervarianceestimationaswell,withtheresultthatthereturndistributionhasatdistributionwithT−(numberofassets)ffffectmakestheparameterestimatesa“statevariable”inaMertoniancontext,(1998)studiesthiseffectinastaticcontextandXia(2001),Iturnedthiseffectoffbyspecifyingthattheinvestor“learns”onlyattheterminaldate,(2003,2004)
(79)—thoughtheremustbeaneasierway!22(r−μ)(μ−μ¯)+22σσμ2222r−2rμ+μμ−2μμ¯+μ¯+22σσ쵶µ¶µ¶2211rμ¯rμ¯2+μ−2+μ++222222σσσσσσμμμ⎡⎤³´³´³´⎛⎞22μ¯μ¯μ¯µ¶rrrµ¶+++2222222211σσσσσσrμ¯⎢μμ⎥μ2⎝⎠³´³´³´+μ−2μ+−++⎣⎦2222σσ111111σσμμ+++222222σσσσσσμμμ³´³´⎡⎤22μ¯μ¯µ¶rrµ¶++22222211σσσσrμ¯μμ⎣⎦³´³´+μ−−++2222σσ1111σσμμ++2222σσσσμμ⎡#$⎤⎡#$⎤22μ¯μ¯rr ++⎢22⎥⎢22⎥σσ22σσ1μrμ¯111μ⎢#$⎥⎢#$⎥−−++−+μ−2222Z2⎣⎦2⎣⎦σσσσ11μμ11++12222σσσσμμqeedμ√222πσ2πσμ⎡#$⎤2μ¯r+⎢⎥22⎢σσ⎥μ⎢⎡$⎥#$⎤#μ−2⎢⎥⎣⎦μ¯11rr ++22³´⎢22⎥σσσσ22−11μrμ¯1μ⎢#$⎥−−−++#$11−12222⎣⎦σZσ2π+2211μ11σσ++μ12222σσσσμμqeredμ√³´−1222πσ2πσ11μ2π+22σσμ⎡⎤22μ¯μ¯rr++2⎢4422σσσσ22⎥1μμrμ¯⎢#$⎥−−++222⎣⎦σσ11μ+122σσμre³´11222πσσ+22μσσμ⎡⎛⎞⎛⎞⎤μ¯r11⎢⎜⎟22⎜2⎟2σ2⎥σσ21rμμμ¯⎢⎜σ#$⎟#$⎜#$⎟⎥−1−−2+1−2⎣⎝⎠2⎝⎠2⎦σσ111111μ+++1222222σσσσσσμμμqe¡¢222πσ+σμ⎡⎛⎞⎛⎞⎤μ¯1r1⎢⎜2⎟22⎜⎟2⎥σ2σσμrμ21μ¯⎢⎜#$⎟#$⎜σ#$⎟⎥−−2+2⎣⎝⎠2⎝⎠2⎦σσ111111μ+++1222222σσσσσσμμμqe¡¢222πσ+σμ2(r−μ¯)1−1222σ+σ()μqe¡¢222πσ+σμ74
8Bibliographicalnote9CommentsonportfoliotheoryPortfoliotheorylookslikealotoffun,,computetheoptimalportfolio,,thereareanumberofconceptualandpracticallimitations,inadditiontotheadviceofBayesianportfoliotheory,thatdampenone’,,theremustbesomeoneelseouttherewhoseconstellationofoutsideincome,riskaversion,,generatingthereturnanomaly?,Ihaveemphasizedexpressionsofoptimalportfoliosintermsofeachinvestor’,theinvestorcanask“amImoreorlessriskaversethantheaverageinvestor?”,ofcourse,,onreflection,manyinvestorsmayrealize“thatlookedgood,butIhavenoreasontoreallythinkI’mdifferentthanaverage,”(locally)mean-varianceinvestor(,nooutsideincome)howtoprofitfrom“anomalies”,“alpha”tosuchinvestors,
ThisconundrumillustratesadeeperCatch-22:portfoliotheoryonlyreallyworksifyoucan’,thenpricesandreturnswillchangewhentheydoso,,itispossiblethatthe“anomalies”donotrepresent“equilibriumcompensationforrisk.”Theymightrepresent“mispricing.”Ifsothough,“mispricing”thatisnoteasytoarbitragebecausemarketfrictionsorinstitutionalconstraintsmakeitdiffi’ttradeontheanomalies,(thesmall-capfund),mostinvestorsthinktheyaresimplysmarterthantheaverage,
wronglyspecified(habits,forexample).Butifthatistrue,theportfolioadvicechangesaswell,ff“payoff”approachwiththemoretraditional“portfolio”,theformerhasanattractivesimplicity,elegance,“payoff”approachisthatitleavesyouwithoptimalpayoffs,-varianceefficient,butwestillreallydon’tknowhowtocalculateamean-varianceeffifferentideasfordoingthat,:ffstrategydoesthesamething:itstopsandcharacterizestheoptimalpayoffswithoutsolvingtheengineeringproblemofsupportingthosepayoff-vestorsare,intheend,interestedonlyinthepayoff-managementindustryshouldprovidethepayoff,,andwhydobondspaycoupons?-andstate-contingentstreamsarethesecuritiesconsumerswantintheend,,’,therearelotsofdifferentwaystoimplementanoptimalpayoff,,onehastostartalloveragainforeveryimplementation,’talwaysinterestedin(dynamictradingstrategies)onthewaytogivingtheeconomiccharacterizationofthepayoff,ff.ff-varianceapproximationismeanttoadvancethiscausesomewhat,bypointingoutanicecharacterizationthatisvalidinincompletemarkets,,in-tertemporalorincompletethemarket,theinvestorsplitshispayoffsbetweentheindexedperpetuityandthemarket,
eachinvestorwantstoinvestmoreinriskyassets,“buythemarketportfolio”wearesimplysaying“dowhateveryoneelsedoes.”Forexample,whenμ−rrisesinthesemodels,therewillbeawaveofshareissues;your“marketindex”willbuythesenewshares,soyouwillendupdoingthesame“dynamictrading”’,tailoredportfolios,,’sthemostimportantandmostoverlookedcomponentofportfoliotheory,,,’tshortyourowncompanystock,atleastdon’tbuymorethanyouhaveto,,thefirstthinginvestorsshoulddoisbuyhomeinsurance,eventhoughit’“tailoredportfolios.”Thefamoustwo-fundtheoremofmean-varianceanalysisdealtaseriousblowtotheonce-commonideathatinvestorsneededprofessionaladvicetopickastockportfolioappropriatefortheirindividualcharacteristics—-fundworld,-coststockindexfundsandmoneymarketmutualfundswereborn,andweareallbetteroff,“funds,”,,wehaveaneedforacademicresearchtoidentifyportfoliosthatmatchtypicaloutsideincomerisks,andanindustrytohelpinvestorschoosetherightones,,,-varianceefficientinvestmentor“alpha.”Astheformulasseparatethesetwofunctions,-incomehedgeportfolios,nonpricedfactorsarejustas,ormoreinter-estingthanthepriced(orpricing),asetoflow-costeasilyshortedindustryportfoliosmightbeveryusefulforhedgingoutsideincomerisk,eventhoughtheymayconformperfectlytotheCAPMandprovideno“alpha”’sinterestingthatacademicre-searchhasfocusedsoexclusivelyonfindingpricedfactors,’tobservethepresentvalueoflaborincomeoroutsidebusinesses,sotheusualapproachthatfocusesonreturnsandreturncovariancesishard78
,however,sothepayoff
$$130ordownto$90,withprobability1/2ofeachevent.(Callthetwostatesuandd.)Youcanalsoinvestinabond,whichpayszerointerest—A$100investmentgives$100forsure.(a)+1(b)Aone-periodinvestorwithlogutilityu(W)=ln(W)hasinitialwealth$+1t+1thisinvestor’:ffintheoneperiodBlack-Scholesexample,1W12−(1−α)r+ασα()γ2xˆ=m=WeR1T1−γE(m)Supposeinsteadofsupportingthispayoffbydynamictrading,youchoosetosupportitbyaportfolioofputandcalloptionsattimezero.(a)Findthenumberofputandcalloptionstobuy/:Graphthepayoffofbuyingacallwithstrikek,sellingacallwithstrikek+∆,sellingaputwithstrikek+∆,andbuyingaputwithstrikek+2∆.Takethelimitofthispayoffas∆→0insuchawaythattheintegralofthepayoffffintermsofhowmanyoftheseportfoliosyoubuyateachk,−γ(b)Similarly,implementthepayoffforhabitutility(c−h)−αWTU(W)=eT0andxandhenceW=wxarenormallydistributed,thenTTT22α2α00−αEW+σ(W)−αwE(x)+wVwTTT22E[U(W)]=e=eTthefirstorderconditiongives1−1w=VE(x)TαAgain,theoptimalportfolioismean-varianceeffi-yearhorizon(nointermediateconsumption)wakesupinthelognormaliidworld.(a)Mirroringwhatwedidwithpowerutility,findthereturnonhisoptimalportfolioin³´−1=,fRWγˆyoucanexpressthisreturnR=xˆ/Wintermsofthelocalriskaversionatinitialwealth,bwithoutcorWWearelookingherefortheanalogueto1−γxˆmˆR==11−WγE(m)80
(b)Addingthelognormaliidassumption,
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