Understanding Portfolio Efficiency with Conditioning Information

Transcription

1 Understnding Portfolio Efficiency with Conditioning Informtion Frncisco Peñrnd UPF, Rmon Tris Frgs 25-27, E Brcelon, Spin. Februry 2008 Abstrct This pper studies portfolio men-vrince efficiency when investors exploit return predictbility through ctive strtegies. We focus on the tension between the stndrd menvrince frontiers tht drive empiricl work nd the stndrd men-vrince preferences tht re used in finnce theory. We show tht such preferences choose portfolios tht re locted onfrontierththsnotbeenstudiedinthelitertureyet. Asresult,wefindnewbet to construct portfolio efficiency tests from the perspective of stndrd preferences. Finlly, welsostudytheimplictionsofsfessetndderivenewshrpertio. Keywords: Bet-pricing, Dynmic portfolio strtegies, Men-vrince frontiers, Representing portfolios, Shrpe rtios. JEL:C12,G11,G12 Wewould liketo thnk Enrique Sentn for helpfulcomments nd suggestions. Finncilsupportfrom the Spnish Ministry of Eduction nd Science through grnt SEJ is grtefully cknowledged.

2 1 Introduction Men-vrince nlysis continues to be widely used in economics nd finnce since Mrkowitz (1952) defined the return men-vrince frontier to chrcterize the risk-return trde-offs tht n investor fces. Its pplictions cover such key issues s portfolio choice, sset pricing tests nd performnce evlution. For instnce, Gibbons, Ross, nd Shnken (1989) is clssic referenceontestingportfolioefficiencyinsuchcontext. Ontheotherhnd,itisnempiricl fct tht sset returns re predictble, nd investors cn exploit this fct to their dvntge by using conditionl distributions s opposed to unconditionl ones in designing their portfolio strtegies. 1 For instnce, n investor cn not only choose pssive buy nd hold portfolio strtegy whose weights re fixed over time, but lso define dynmic trding strtegy s function of the voltility level of the stock mrket. The seminl pper of Hnsen nd Richrd (1987) developed the theoreticl frmework to study men-vrince efficiency of ctive strtegies. They nlyzed both frontier bsed on conditionl mens nd vrinces nd frontier bsed on their unconditionl counterprts, nmed s the conditionl nd unconditionl frontiers respectively, showing tht ctive returns on the unconditionl frontier re lso on the conditionl frontier, but not vicervers. Their min gol ws to study the tension between the conditionl implictions of sset pricing theory 2 nd the use of unconditionl moments in empiricl work. However, mny ppers hve used their unconditionl frontier to guide portfolio choice. See e.g. Ferson nd Siegel(2001) when there is men predictbility nd Ferson nd Siegel(2006) for portfolio efficiency test in such context. Other ppers such s Brndt nd Snt-Clr(2006) nd Bnsl, Dhlquist, nd Hrvey(2004) pproximte tht object by mens of pssive strtegies of mnged portfolios. Generl men-vrince preferences choose portfolios locted on the conditionl frontier. However, Ferson nd Siegel (2001) show tht the unconditionl frontier cn be rtionlized by qudrtic utility, but not by e.g. CARA utility plus conditionl normlity of returns(conditionl on signl), which my look s strong drwbck given the implusible properties of qudrtic utility. This fct represents tension between tht empiricl literture bsed on unconditionl frontiers nd the theoreticl literture tht uses men-vrince preferences tht cn be rtionlized by CARA utility. Ares such s mrket microstructure nd rtionl expecttions 1 SeeCochrne(2001)forsummryoftheempiriclevidenceonmenpredictbility,ndSentn(2005)for recent exmple of the link between regression forecsts nd optiml portfolios. 2 Whichoftentrnslteintosomeportfoliobeingloctedontheconditionlfrontier. 1

3 equilibri rely often on those stndrd men-vrince preferences, see e.g. Brunnermeier(2001) for survey of sset pricing theory under symmetric informtion or Esly nd O Hr(2004) s recentreference. DybvigndRoss(1985)lsousethisset-uptostudythecomplexityof performnce evlution of n informed mnger by n uninformed gent. Our pper focuses on tht tension, nlyzing the specific frontier tht corresponds to the stndrd preferences in the theoreticl literture. Obviously, the chosen portfolio strtegies belong to the conditionl frontier like the ones on the unconditionl frontier. But we cn chrcterize their properties more deeply nd hence we first show tht the stndrd preferences chose ctive returns tht re locted on frontier tht hs not been studied in the literture yet. We cll it the residul frontier becuse it minimizes the unconditionl second moment of the portfolio pyoff residul (the finl pyoff minus its conditionl men) insted of its totl pyoff for given unconditionl men. We lso study the sset pricing implictions of such frontier, which re importnt to construct portfolio efficiency tests from the perspective of stndrd preferences. We show tht new type of bet should be used, which we cll the residul bet becuse it is bsed on unconditionl second moments of portfolio pyoff residuls. We lso study the connection between the residul frontier, i.e. stndrd preferences, nd the unconditionl frontier, i.e. stndrd frontiers. Both frontiers re subsets of the conditionl frontier, but we show tht the residul nd unconditionl frontiers do not shre ny point unless we re in two quite specil cses, where both frontiers re equl. The connection between the three frontiers hs importnt implictions for portfolio efficiency tests since the relevnt bets differ. Using n unconditionl bet tests the loction of portfolio on the unconditionl frontier, which is not the interesting object for stndrd men-vrince gent, who should rely on residul bet insted. On the other hnd, conditionl bet tests the loction of portfolio on the conditionl frontier, where both the residul nd unconditionl frontier(nd other portfolios) belongto. Thtis,itisrelevnttestforstndrdmen-vrincegent,butitisnotsshrp or powerful s possible. Finlly, we lso nlyze the presence of sfe sset, i.e. n sset tht is either conditionlly or unconditionlly riskless. The conditionl frontier hs the conventionl properties of becoming stright lines nd the existence of tngency portfolio between the frontiers with nd without the sfe sset. We show tht the residul frontier becomes two stright lines, but tngency portfolio does not exist. The unconditionl frontier does not hve tngency portfolio either; 2

4 butitisnotevencomposedbytwostrightlineswhenthesfessetisunconditionllyrisky, which renders unconditionl Shrpe rtios useless. We study the trnsltion of bet-pricing results tothesfe ssetcsend, moreimportntly, theuseof Shrpertios to testportfolio efficiency. We find gin tht stndrd men-vrince gent is minly interested in new type of Shrpe rtio, which we cll residul becuse the denomintor is bsed on residul stndrd devition. The rest of the pper is orgnized s follows. We introduce the generl theoreticl set-up in Section 2. Then, we review nd derive some useful results regrding the conditionl frontier in Section 3. Next, we develop the residul frontier s the relevnt object for stndrd menvrince preferences in Section 4, nd we discuss its connection to the unconditionl frontier insection5. Finlly, wenlyzethepresenceofrisklessssetinsection6,ndpresentour conclusions in Section 7. Proofs re gthered in the ppendix. 2 Theoreticl Bckground This section defines theoreticl set-up similr to Peñrnd nd Sentn(2007). They closely follow Hnsen nd Richrd(1987), where further detils cn be found. In ddition, we trnslte some concepts developed by Chmberlin nd Rothschild(1983) in different set-up. Finlly, we describe simple exmple tht will illustrte the concepts developed in the following sections 2.1 Informtion Structure nd Active Portfolio Strtegies Wewillconsidern economywithn riskyssets whoserndompyoffs x=(x 1,...,x N ) re defined on n underlying probbility spce. We ssume tht there re three importnt dtes in this economy: 0, 1 nd 2. We identify 0 s the decision dte, 1 s the trding dte nd 2 s the pyoff dte. Investors design ex nte portfolio strtegies t 0 tht my depend on the informtion tht they will observe t 1, when trding tkes plce, nd they finlly receive pyoffs t2. LetG 1 denotetheinvestors informtiontdte1,typicllycontiningsignlsobserved t1thtreinformtiveboutfuturessetpyoffs. Wedenotethesetofllrndomvribles thtremesurblewithrespecttog 1 byi 1,whileG 2 ndi 2 hvesimilrinterprettionwith reference to dte 2. We denote the first two conditionl moments of the primitive pyoffs nd their conditionl costs by E(x G 1 )=ν 1, E ( xx ) G 1 =Γ1, C(x G 1 )=c 1, (1) 3

5 respectively,llofwhichbelongtoi 1. Tovoidtriviluninformtivesetup,wessumetht not ll these rndom vribles re degenerte, i.e. equl to their unconditionl counterprts. They re denoted by E(x)=E(ν 1 )=ν, E ( xx ) =E(Γ 1 )=Γ. (2) We lso mke some dditionl ssumptions on(1) to void degenerte cses where the menvrincefrontiersreverysimple. WessumethtthedigonlelementsofΓ 1 reuniformly bounded with probbility one (.s.), so tht ll the elements of x belong to L 2, which is the collection of ll rndom vribles defined on the underlying probbility spce with bounded (unconditionl) second moments. Hence, we cn obtin the covrince mtrix of x s Σ 1 = Γ 1 ν 1 ν 1,whosesmllesteigenvluewessumeisuniformlyboundedwyfrom0.s.,which implies tht none of the primitive ssets is either conditionlly riskless or redundnt. Regrding therndomvectorsν 1 ndc 1,wessumethtthevectorc 1 hstlestoneentrydifferentfrom 0.s. nd not ll expected pyoffs re conditionlly proportionl to their prices with common fctor of proportionlity. Investors cn condition their portfolios weights on the informtion they know they will hve tthetimeoftrding,whichisgivenbyg 1. Consequently,theycnconstructctiveportfolio strtegieswithpyoffsp=x w 1,wheretheportfolioweightsw 1 I 1. Wewillrefertothepyoff spce defined by P = { p I 2 :p=x w 1, w 1 I 1 } s thectivepyoffspce. Trivilly, theconditionlmomentsndcostsoftheelementsofp will be E(p G 1 )=w 1ν 1, E ( p 2 ) G 1 =w 1 Γ 1 w 1, C(p G 1 )=w 1c 1, llofwhichbelongtoi 1. Similrly,theirunconditionlmomentsre E(p)=E ( w 1ν 1 ), E ( p 2 ) =E ( w 1Γ 1 w 1 ). Although we llow sset prices c 1 to depend on the vlues of the signls, there re two importnt exmples of pyoffs whose costs re non-rndom: gross returns, which re pyoffs with unit prices, nd rbitrge portfolios, or zero-cost pyoffs. They will be relevnt in the 4

6 nlysisthtfollowsndhencewewilldefinethecorrespondingtwosubsetsofp, R ={p P :C(p G 1 )=1}, A ={p P :C(p G 1 )=0}. Inprticulr,ifninvestorisendowedwithsomepositivewelthtdte1,whichwecn normlize to 1 without loss of generlity, then she will only be interested in portfolio strtegies thtcost1tdte1foreverypossiblevlueofz. Therefore,themen-vrincefrontierstht wewillstudyrecomposedbyreturnsinr. 2.2 Centred nd Uncentred Representing Portfolios Hnsen nd Richrd (1987) introduce conditionl nlogue to stndrd Hilbert spce bsed on the men squre inner product, E(xy G 1 ), nd the ssocited men squre norm E 1/2 (x 2 G 1 ),wherex,y L 2 2,ndL 2 2 istheconditionlnloguetol 2. Inthiscontext,wecn formllyunderstndc( G 1 )nde( G 1 )sconditionllycontinuouslinerfunctionls 3 tht mptheelementsofp ontoi 1. AconditionlversionoftheRieszrepresenttiontheoremthen impliesthtthereexisttwouniqueelementsofp thtrepresenttheseconditionlfunctionls over P. 4 In prticulr, the (ctive) uncentred men nd cost representing portfolios, p + nd p,respectively,willbesuchtht: 5 E(p G 1 )=E ( p + p G 1 ), C(p G1 )=E(p p G 1 ), p P, nd it is strightforwrd to show tht p + =x Γ 1 1 ν 1, p =x Γ 1 1 c 1. (3) Now wecn explin the role of our previous ssumptions on (1), which void cses where these RPs re too simple. We cn interpret p + s the conditionl projection of sfe pyoff 1 onto P. Our ssumption on Σ 1 implies tht it is not possible to generte conditionlly risklessportfoliofromxotherthnthetrivilone. Thtis,wessumethereisnotsfesset 3 TheexpectedvluefunctionlislwysconditionllycontinuousonL 2 2 byconditionlversionthemrkov inequlity. Similrly, our full rnk ssumption on Σ 1 implies tht Γ 1 hs full rnk too, nd consequently, tht thecostfunctionlislsoconditionllycontinuousonp,whichistntmounttothelwofoneprice. 4 ChmberlinndRothschild(1983)introducedmenndcostrepresentingportfoliostostudymen-vrince nlysis in infinite dimensionl pyoff spces in which conditionl informtion plys no role. Hnsen nd Richrd (1987) extended their results to conditioning informtion. 5 It is worth mentioning tht uncentred representing portfolios cn lso be defined in terms of unconditionl moments. Specificlly,thelwofitertedexpecttionsimpliesthtp + ndp lsorepresentunconditionlmens ndvergecostsonthectivepyoffspcep. 5

7 inthesenseofsomep P suchthtp=e(p G 1 )=1ndhencewevoidp + =1. Obviously, thisisrelevntcsendwewillddsfessetinsection6. Wecnlsointerpretp stheconditionlprojectionofnyvlidstochsticdiscountfctor ontop. Weexcludedthepossibilitythtllprimitivessetsrerbitrgeportfoliosbyssuming thtthevectorc 1 hstlestoneentrydifferentfrom0.s. Withthisssumption,wevoid thesimplecsep =0. We lso ssumed tht not ll expected pyoffs re conditionlly proportionl to their prices with common fctor of proportionlity. In this wy, we implicitly rule out those situtions in whichllconditionllyexpectedreturnsrethesme. Thisssumptionvoidsthtp + ndp re conditionlly proportionl. Finlly, in the spce of pssive strtegies, Chmberlin nd Rothschild(1983) show tht n lterntive vlid topology with unconditionl moments cn be defined with the covrince s theinnerproductndthestndrddevitionsthenormwhenthereisnotsfesset. We cn trnslte such topology to P, i.e. work with the conditionl covrince nd stndrd devition,sincewererulingoutsfessetuntillter. Wecnrepresentthemenndcost functionlsbymensoftwolterntivecentredrps,p ndp++ inp,suchtht E(p G 1 )=Cov(p ++,p G 1 ), C(p G 1 )=Cov(p,p G 1 ), p P, whichdefinethedditionlpirofrpss where p ++ =x Σ 1 1 ν 1 = p +, (4) p =x Σ 1 1 c 1=p p +, 1 =ν 1Γ 1 1 c 1 =E(p G 1 )=C ( p + G 1 ) =E ( p + p G 1 ), 1 =ν 1Γ 1 1 ν 1 =E ( p + G 1 ) =E ( p +2 G 1 ). Notetht(4)iswelldefinedbecuse1 1 0inthecurrentset-up. Ifthereisnotsfe sset then Vr(p + G 1 ) = E(p + G 1 )[1 E(p + G 1 )] 0. Similrly, it will be convenient to define 2.3 A Binomil Exmple 1 =c 1Γ 1 1 c 1 =C(p G 1 )=E ( p 2 G 1 ). Fromtheperspectiveofdte0,thepyoffspcetdte2willgenerllybeinfinitedimensionl, even though investors only hve ccess to finite set of primitive sset pyoffs, becuse 6

8 they cn use ny piece of informtion known t dte 1 in designing their investment strtegies. For pedgogicl resons though, we will illustrte the next sections with n exmple where the dimension of P is finite, specificlly n informtion set chrcterized by binomil rndom vrible. We strt from2 1 vector x tht represents nnul gross returns, i.e. nonrndom c 1 equltovectorofones,ontwofinncilssets. Weconsiderninvestorthtobservessignl tdte1thtmytketwovlues,revelingoneoftwopossibleexpectedreturnvectorsnd twopossiblecovrincemtrices. Thtis,wecnunderstndtheinvestor sinformtionsetg 1 scontiningbinomilrndomvriblezthtcntkethefollowingtwovluesz=1,2. Ifz=1thentherelevntmenndvrincesregivenbytherelnumbers ν 1 1= 1.12, Σ 1 1= , whileifz=2thenthoserelnumbersbecome ν 1 1= 1, Σ 1 1= Those numbers try to cpture simple sitution where the investor chooses n optiml portfolio of stocks nd hedge funds, which re represented by the first nd second entry of x respectively. The signl reliztion z = 1 represents"good times", i.e. high return nd low risk, wherethehedgefundsshowlowcorreltionwithstocksndhighrtioofmenoverstndrd devition. The signl reliztion z = 2 represents "bd times", i.e. low return nd high risk, where the hedge funds show higher correltion with stocks nd perform poorly. We ssign probbility of 0.8 to"good times", which gives the unconditionl moments ν= , Σ= Let us define the dummy vribles ξ 1 =I(z=1), ξ 2 =1 ξ 1, where I( ) is the usul indictor function. The key feture of binomil set-up is tht ny p P cnberepresenteds p=x ( ξ 1 w 1 1 +ξ2 w 2 1) = ( ξ 1 x ) w ( ξ 2 x ) w 2 1, w 1 1,w2 1 RN. 7

9 Asresult,thepyoffspceP isindeedfinitedimensionlfromthepointofviewofdte0 since it could be generted by pssive strtegies on n ugmented but finite dimensionl set of mngedportfolioswhosepyoffstketheformξ 1 xndξ 2 x. Wesythtportfoliostrtegy ispssiveifthetwovectors ( w1 1 1),w2 reequl;otherwise,wesythtitisctive. 3 Efficient Portfolios for Generl Men-Vrince Preferences This section reviews the frontier where generl men-vrince preferences choose ctive returns, which ws studied by Hnsen nd Richrd(1987), nd lso develops some new results tht will be convenient lter. First we introduce some useful portfolios relted to the RPs. 3.1 Some Relevnt Returns nd Arbitrge Portfolios We define two portfolios relted to the uncentred RPs. First, the return ssocited to the uncentred cost RP, R = 1 p, (5) 1 which is the return with minimum conditionl second moment, i.e. the solution to the problem mine ( p 2 ) G 1. p R Second,theresidulfromtheconditionlprojectionofp + ontotheconditionlspnofp A + =p p, (6) whichcnbeinterpretedstheuncentredmenrpinthespceofrbitrgeportfolios,i.e. the unique rbitrge portfolio tht stisfies E ( A + p G 1 ) =E(p G1 ), p A. Similrly, we define two portfolios relted to the centred RPs. First, the return ssocited tothecentredcostrp R = 1 C(p G 1 ) p =R+ E(R G 1) 1 E ( A + )A +, (7) G 1 which is the return with minimum conditionl vrince, the solution to the problem min p R Vr(p G 1 ). 8

10 Second,theresidulfromtheconditionlcovrinceprojectionofp ++ onto the conditionl spnofp A ++ =p ++ E(p G 1) C(p G 1 ) p = whichisthecentredmenrpinthespceofrbitrgeportfolios, 1 1 E ( A + G 1 )A +, (8) E(p G 1 )=Cov(A ++,p G 1 ), p A. NotethtR R nda ++ reconditionllyproportionltoa +. ThefctorofproportionlityiswelldefinedbecusethesfessetdoesnotbelongtoA. 3.2 The Conditionl Return Frontier(CRF) Hnsen nd Richrd(1987) define the Conditionl Return Men-Vrince Frontier(CRF) s the ctive returns with minimum conditionl vrince for given profile of conditionl expected returns. Tht is, the set of ctive returns tht solve the optimiztion problem mine ( p 2 ) G 1 p R s.t. E(p G 1 )= ν 1, (9) which cn be represented s p C ( ν 1 )=R +ω 1 ( ν 1 )A +, ω 1 ( ν 1 )= ν 1 E(R G 1) E ( A + G 1 ). (10) They lso mention tht there is conditionl two-fund spnning on the CRF, i.e. ny element on the CRF cn be replicted by n ctive portfolio of two other elements on the CRF. The conditionl second moment of portfolios on the CRF is given by E[p 2 C( ν 1 ) G 1 ]=E ( R 2 G 1 ) +ω 2 1 ( ν 1 )E ( A + G 1 ), ndhencethecrfwillbehyperbolonthe[vr 1/2 (p G 1 ),E(p G 1 )]spceforprticulr vlueoftheconditioningvriblesing 1. BothreturnsofthecostRPs,R ndr,relocted on thecrf.figure1shows thecrftbothvlues of z forthebinomilexmpledescribed insection2. TheloctionofR isonlyshownwhenz=1becuseitsmenisverylowwhen z=2. TheloctionofR isonlyshownwhenz=2,butitsloctionwhenz=1issimplythe corresponding minimum vrince. 9

11 1,1 z=1 MEAN 1,01 R + R* z=2 R** 0,92 0 0,1 0,2 SD Figure 1: Conditionl Frontier. The following lemm gives n lterntive representtion of the CRF. Lemm 1 TheCRFreturns(10)cnlsoberepresenteds p C ( ν 1 )=R +w 1 ( ν 1 )A ++, w 1 ( ν 1 )= ν 1 E(R G 1 ) E ( A ++ ), (11) G 1 This is strightforwrd impliction of(7) nd(8), [ ] [ ] R +w 1 ( ν 1 )A ++ = R + E(R G 1 ) 1 E ( A + )A + 1 +w 1 ( ν 1 ) G 1 1 E ( A + )A + G 1 [ ] =R+ E(R G 1)+w 1 ( ν 1 ) 1 E ( A + ) A + =R+ω 1 ( ν 1 )A +, G 1 whichshowsthelinkbetweenw 1 ( ν 1 )in(11)ndω 1 ( ν 1 )in(10). In our set-up of conditioning informtion, we understnd generl men-vrince preferences srepresentedbyu ( E(p G 1 ),Vr 1/2 (p G 1 ) ) forsomenonrndomfunctionu( )stisfyingsome stndrd properties: strictly incresing in the first rgument nd decresing in the second, nd strictlyconcve. 6 IfweplotthecorrespondingindifferencecurvesonFigure1thentngency with the "good" nd "bd" men-vrince frontier defines the optiml profile ν 1 for given U( ). Obviously,theoptimlprofilestisfies ν 1 E(R G 1 )ndwecnrefertothtprtof thecrfstheefficientset. Finlly, we cn lso represent the problem(9) by mens of criteri bsed on possibly rndom risk-return trde-offs. 6 The justifiction of men-vrince preferences ws linked to ellipticl distributions by Chmberlin (1983) nd Owen nd Rbinovitch(1983) in the context of pssive strtegies nd unconditionl moments. 10

12 Proposition 1 The following criteri choose portfolios on the CRF: 1. The optiml portfolio of problem mxe(p G 1 ) θ 1 p R 2 Vr(p G 1), givensomestrictlypositiveθ 1 I 1 isequlto(11)ifwechooseprofile ν 1 suchtht w 1 ( ν 1 )= 1 θ The optiml portfolio of problem mxe(p G 1 ) b 1 p R 2 E( p 2 ) G 1, givensomestrictlypositiveb 1 I 1 isequlto(10)ifwechooseprofile ν 1 suchtht ω 1 ( ν 1 )= 1 b Conditionl Bet-Pricing Portfolio efficiency tests re often bsed on the bet-pricing implictions of the relevnt frontier. Hnsen nd Richrd (1987) show the corresponding bet-pricing of the conditionl frontier, whichisbsedontheconditionlbetofreturnp R withrespecttoreturnp β R β 1 = Cov(p,p β G 1 ) Vr(p β G 1 ). Ech return p C on the CRF, prt from R, hs zero-bet counterprt on the CRF. Moreover, such return hs prticulr geometry on the [Vr 1/2 (p G 1 ),E(p G 1 )] spce for prticulr vlue of the conditioning vribles in G 1. Similrly to the geometry of pssive strtegies, return on the CRF nd its zero-bet counterprt re relted by tngency. For instnce, the CRF return with constnt 0 conditionl expected zero-bet return hs n interesting interprettion. It is the return of the uncentred men RP R + = 1 1 p + providedp + hsnonzerocostorequivlentlyp hsnonzeromen,i.e Theloction of R + when z = 2 is shown in Figure 1, while its loction when z = 1 is very close to the correspondingminimumvrince. Moreover,becusep ++ ndp + reconditionllyproportionl, thtreturnisequltothecorrespondingreturnofthecentredmenrp R ++ = 1 C ( p ++ )p ++. G 1 11

13 Thebet-pricingimplictionsoftheCRFrethefollowing: Areturnp β R differentfrom R isonthecrfifndonlyif E(p G 1 ) E 1 =β 1 [E(p β G 1 ) E 1 ], p R, forsomee 1 I 1. Obviously,thtrndomvribleisinterpretedstheconditionlmenofthe corresponding zero-bet return. Note two difficulties to implement n efficiency test bsed on the previous eqution. First, it isfullybsedonconditionlmoments. Second,givenreturnp β,wemustcheckthteqution p R,whichmybeduntingtsk. Fortuntely,thererecseswherethisisnotsodifficult suchsthebinomilexmplethtwerestudying,ndmorerelevntone,theexistenceof sfessetthtwewillstudylter. Inthtcse,wecnrelyontheoptimlShrpertios. 4 Efficient Portfolios for Stndrd Men-Vrince Preferences Now we study portfolio efficiency from the perspective of more specific men-vrince preferences, in prticulr the stndrd men-vrince preferences in finnce theory. We know tht the corresponding optiml portfolios will be subset the CRF, but we wnt to chrcterize their properties more deeply. Finlly we study the sset pricing implictions of those optiml portfolios, which re importnt to construct portfolio efficiency tests from the perspective of stndrd preferences. 4.1 The Portfolio Choice Problem The stndrd men-vrince preferences in finnce theory when conditioning informtion is tken into ccount re given by mxe(p G 1 ) θ p R 2 Vr(p G 1). (12) forsomestrictlypositiveθ R. AscommentedintheIntroduction,thisisstndrdfrmework in finnce theory which is usully justified by CARA utility E[ exp( θp) G 1 ] plus conditionl normlity of p. Given Proposition 1, we cn esily chrcterize the specific subset of the CRF tht this criterion chooses. 12

14 Corollry 1 The optiml portfolio tht solves problem(12) is R + 1 θ A++. Bnsl, Dhlquist, nd Hrvey (2004) mentioned the solution of the problem but did not use it. Insted, their finl portfolio exercise is bsed on pssive strtegies of mnged portfolios, which will be commented lter. Now we will study the specific men-vrince frontier where these portfolios re locted becusethisisthekeyfeturetodevelopnefficiencytest,whileithsnotbeenstudiedinthe literture. We will show tht non-stndrd frontier nd bet-pricing is indeed linked to the stndrd men-vrince preferences. 4.2 The Residul Return Frontier(RRF) Letusdefinetheresidulofportfoliopyoffwithrespecttoitsconditionlmens p=p E(p G 1 ), which cn be interpreted s the pyoff shock nd stisfies Vr( p)=e ( p 2) =E[Vr(p G 1 )]. In ddition, we define the residul inner product between rndom vribles x nd y by E( xỹ)=e[cov(x,y G 1 )] nd its corresponding norm s E 1/2 ( p). A priori, this my not be proper norm in the sense tht E ( p 2) = 0 implies p = E(p G 1 ) but not necessrily p = 0. However, t this stge we ssume there is not sfe sset, i.e. there is no p such tht p = E(p G 1 ) 0,ndhenceE ( p 2) =0impliesp=0. 7 We define the Residul Return Men-Vrince Frontier (RRF) s the ctive returns tht minimize the residul vrince(or second moment) for given trget of expected return. Thus, therrfwillbegivenbythesetofctivereturnsthtsolvetheproblem mine ( p 2) s.t. E(p)= ν. (13) p R The next proposition points out the relevnce of this non-stndrd frontier, it is linked to stndrd men-vrince preferences. 7 Even if there ws sfe sset, the residul inner product would define proper norm in A if there were no rbitrge opportunities. In tht context, sfe sset cnnot belong to A nd hence E ( p 2) = 0 implies p=e(p G 1)=0onthtspce. 13

15 Proposition 2 The optiml portfolio tht solves problem(13) is p R ( ν)=r +ω R ( ν)a ++, ω R ( ν)= ν E(R E ( A ++ ), (14) nditislsothesolutionoftheportfolioproblem(12)ifwechoose ν suchtht ω R ( ν)= 1 θ. ) Obviously, stndrd men-vrince gent chooses ω R ( ν) 0, i.e ν E(R ), which defines the efficient prt of the RRF. There is unconditionl two-fund spnning on the RRF, in the sense tht ny element on the RRF cn be replicted by pssive portfolio of two other elementsontherrf.theresidulvrinceofreturnsontherrfisgivenby Vr( p R ( ν))=e[vr(p R ( ν) G 1 )]=E[Vr(R G 1 )]+ω 2 R( ν)e ( A ++ ), ndhencetherrfwillbehyperbolonthe[vr 1/2 ( p),e(p)]spce. The minimumresidul vrince is given by R becuse minimizing Vr(p G 1)lso minimizese[vr(p G 1 )]=Vr( p)ndthefrontiersymptotesregivenbytheresidulvrince ofa ++ /E(A ++ ) [ [ Vr( p R ( ν)) lim ν ± E 2 (p R ( ν)) = 1 E ( A ++ ) =E Vr A ++ E ( A ++ ) G 1 ]] Figure 2 illustrtes the RRF for the binomil exmple described in Section 2. 1,09. p 1 MEAN 1,06 R ** 1,03 0 0,04 0,08 RESIDUAL SD p 2 Figure 2: Residul Frontier. 14

16 On the other hnd, R nd R + = R ++ re locted on the CRF but not on the RRF. Note tht the ctul difference between problems (13) nd(9) is the men constrint not the criterion to minimize, being stronger in the ltter. The first problem only constrins E(p) while theltterconstrinsthewholeprofilee(p G 1 ). Thecriteritominimizereequivlentbecuse the criterion cn lso be expressed s Vr(p G 1 ) for the CRF, nd s E[Vr(p G 1 )] for the RRF. Given Lemm 1 nd Proposition 2, we cn esily chrcterize the connection between the RRFndtheCRF. Corollry2 The returns on the RRF re lso onthe CRF. Specificlly, CRF portfolio will lsobeloctedontherrfifndonlyifwechoosetheconditionlmenprofiles ν 1 =ω R ( ν)e ( A ++ G 1 ) +E(R G 1 ), so tht w 1 ( ν 1 )=ω R ( ν). Notethtω R ( ν) 0, i.e. ν E(R ), implies ν 1 E(R G 1). Returnsontheefficient prtoftherrfrelsoloctedontheefficientprtofthecrf. 4.3 Residul Bet-Pricing This section develops the bet-pricing ssocited to the RRF, which is the relevnt betpricingforninvestorwithpreferences(12). Letusdefinetheresidulbetofreturnp R withrespecttoreturnp β R s β R = Cov( p, p β) Vr( p β ) = E( p p ( β) ) = E[Cov(p,p β G 1 )] E E [ Vr ( )]. p β G 1 p 2 β Ech return p R on the RRF, prt from R, hs zero-bet counterprt on the RRF. Specificlly,tworeturnsontheRRF,syp 1 =R +ω 1A ++ ndp 2 =R +ω 2A ++,stisfy E[Cov(p 1,p 2 G 1 )]=0 ifndonlyif(ω 1,ω 2 )rereltedby E[Vr(R G 1)]+ω 1 ω 2 E ( A ++ ) =0. Moreover,thereisninterestinggeometryintheirlink. Thereturnp 1 isthesolutionto E(p) E(p 2 ) mx p R Vr 1/2, ( p) 15

17 which represents the tngency from E(p 2 ) to the RRF on the [Vr 1/2 ( p),e(p)] spce. See Figure2forprticulrp 1. ThereturnR cnnothvezero-betcounterprtbecuse Cov(p,R G 1)=Vr(R G 1), p R. The following proposition cn be interpreted s the RRF counterprt of the results in Roll (1977) for pssive strtegies nd Hnsen nd Richrd(1987) for different ctive frontiers. Proposition3 Areturnp β R differentfromr isontherrfifndonlyif forsomee β R. E(p) E β =β R [E(p β ) E β ], p R, This proposition defines the hypothesis to test for portfolio efficiency tests tht re relevnt forstndrdmen-vrincegents,i.e. thosedefinedby(12). Givenreturnp β,wemustcheck tht p R itscorrespondinge(p)stisfiesthepreviouseqution. Theequtionisstndrd in the sense of using unconditionl mens, which cn be estimted by historicl verges, but it is not in the sense of not using unconditionl bets, neither conditionl ones. Moreover, we find gintheneedofchecking p R. Conditionl bet-pricing is comptible with residul bet-pricing in the sense tht the ltter imply the former, but not vicevers. Conditionl bet-pricing is relevnt for the stndrd menvrince preferences of(12) becuse her optiml portfolio is locted on such frontier. However, othernon-optimlportfoliosfromthepointofviewofthisgentrelsoloctedonthecrf. Therefore, n efficiency test bsed on conditionl bets my hve low power. 5 Stndrd Men-Vrince Frontiers in Empiricl Work Now we turn to portfolio efficiency from the perspective of the stndrd men-vrince frontiers in empiricl work. We will briefly review the ctive frontiers tht hve been considered intheliterturesofrndstudytheirconnectiontotheresidulfrontier. Thtis,wewillstudy the connection between stndrd frontiers nd stndrd preferences. 5.1 The Unconditionl Return Frontier(URF) Hnsen nd Richrd(1987) define the Unconditionl Return Men-Vrince Frontier(URF) s the ctive returns with minimum vrince for ech level of expected return. Hence, the URF 16

18 willbegivenbythesetofctivereturnsthtsolvetheproblem mine ( p 2) s.t. E(p)= ν. (15) p R Theyshowthtthegrossreturnsthtsolve(15)cnberepresenteds p U ( ν)=r +ω U ( ν)a +, ω U ( ν)= ν E(R ) E ( A + ). (16) As in the cse of the RRF, there is unconditionl two fund spnning on the URF. The unconditionl second moment of portfolios on the URF will be given by E[p 2 U( ν)]=e ( R 2 ) +ω 2 U ( ν)e ( A + ) ndhencetheurfwillbehyperbolonthe[vr 1/2 (p),e(p)]spce. TheURFlookssimilr totherrfinfigure2,butnowthex-xisistheunconditionlstndrddevitioninstedof the residul one. Its minimum is locted t ν= E(R ) 1 E ( A + ), while the minimum uncentred second moment is given by R becuse minimizing E ( p 2 ) G 1 lso minimizes E [ E ( p 2 G 1 )] =E ( p 2 ). The frontier symptotes re given by the vrince of A + /E(A + ) Vr(p U ( ν)) lim ν ± E 2 (p U ( ν)) =1 E(A+ ) E ( A + ) =Vr [ A + E ( A + ) The bet-pricing implictions of the URF re lso shown in Hnsen nd Richrd(1987). Let usdefinetheunconditionlbetofreturnp R withrespecttoreturnp β R s β U = Cov(p,p β) Vr(p β ). Echreturnp U ontheurf,prtfromtheminimumvrinceone,hszero-betcounterprtontheurf.moreover,thereisninterestinggeometryintheirlinkgivenbytngency sitwsthecsewiththerrf,seefigure2,butnowonthe[vr 1/2 (p),e(p)]spce. Hnsen nd Richrd (1987) show the following bet-pricing implictions of the URF: A returnp β R differentfromtheminimumvrinceoneisontheurfifndonlyif E(p) E β =β U [E(p β ) E β ], p R, forsomee β R. Thispricingequtionissimilrtothecseofpssivestrtegieswithoutconditioning informtion, since it is fully bsed on unconditionl moments, with the only(importnt) differenceoftherelevntreturnspce, p R. 17 ].

19 5.2 Reltionship with the CRF nd RRF Perhps the best known result of Hnsen nd Richrd (1987) is tht while unconditionl frontier portfolios lwys lie on the conditionl frontier, the converse is not generlly true. Note tht the ctul difference between problems(15) nd(9) is the men constrint not the criterion tominimize. Morespecificlly,CRFportfoliowilllsobeloctedontheURFifndonlyif we choose the conditionl men profile s ν 1 =ω U ( ν)e ( A + ) G 1 +E(R G 1 ), whichwecninterpretstheoptimlmenprofilegivene(p)= ν,sotht ω 1 ( ν 1 )=ω U ( ν). We cn lso interpret the connection between the CRF nd URF by mens of men-vrince preferences. Ferson nd Siegel(2001) show tht the optiml portfolio of n gent with qudrtic utility E [p b2 ] p2 G 1 for somestrictlypositiveb Risctullyreturn ontheurf.in ourset-up, Proposition 1 shows tht the solution of problem mxe(p G 1 ) b p R 2 E( p 2 ) G 1 (17) is the portfolio R +1 b A+, whichisequltothesolutionofproblem(15)ifwechoose ν suchtht ω U ( ν)= 1 b. Therefore, we cn rtionlize the URF by mens of qudrtic utility. However, such preferences re not used in finnce theory becuse they re not plusible description of investor behvior, showing decresing mrginl utility nd incresing bsolute risk version. WeknowthtboththeRRFndtheURFresubsetsoftheCRF,butwestilldonotknow thosesubsetsshrenyelement. IfwestrtfromtheRRF, [ ] [ ] R +ω R ( ν)a ++ = R + E(R G 1) 1 E ( A + )A + 1 +ω R ( ν) G 1 1 E ( A + )A + G 1 [ ] =R+ E(R G 1 )+ω R ( ν) 1 E ( A + ) A + =R +η 1 A +, G 1 18

20 then wecnsee thttherrfhs rndomweightη 1 on A +. However, theurfrequires nonrndomweightona +, R +ω U ( ν)a +, ndhencebothfrontierscnnotshrepointingenerl. Forinstnce,R belongstotherrf but not to the URF, 8 while R is locted on the URF but not on the RRF, except in very specilcsesthtwewillstudylter. Ontheotherhnd,R + =R++ isnotloctedontherrf ortheurf,e.g. thisreturncorrespondstoω 1 ( ν 1 )= 1 1 in(10),whichisrndomndhence cnnot correspond to(16). Echfrontiergivesthebestreturnfordifferentcriteri,Vr(p)ndVr( p),ndhencethe otherfrontierwillbeloctedtotherightonthecorrespondingspcesfigure3shows. 9 1,09 1,09 RRF URF MEAN 1,06 URF MEAN 1,06 RRF 1,03 0,04 0,06 0,08 RESIDUAL SD 1,03 0,06 0,08 0,1 SD Figure 3: Residul nd Unconditionl Frontiers. 8 Notetht [ ] E(R E(R )=E G 1) 1 E ( ) A +, G 1 whiletheexptectedreturnttheminimumoftheurfis E(R) 1 E ( ). A + 9 Theunconditionlsecondmomentofp R( ν)is E ( p 2 R( ν) ) =E ( R 2) +ω 2 R( ν)e ( A ++2 ) ( +2ωR( ν)e R wheree ( ) E ( ) ( ) 0. A ++2 A ++ nde R A ++ TheresidulvrinceofpU ( ν)is A ++ E[Vr(p U( ν) G 1)]=E[Vr(R G 1)]+ω 2 U( ν)e [ Vr ( A + G 1 )] +2ωU( ν)e [ Cov ( R,A + G 1 )], wheree [ Vr ( A + G 1 )] E ( A + ) nde [ Cov ( R,A + G 1 )] 0. ), 19

21 Ontheotherhnd,thenextpropositionsttestwouniquendspecilcseswheretheRRF ndurfreequl. Proposition4 TherereonlytwocseswheretheRRFndtheURFreequl: 1. A + =0ndhence bothr R nda ++ re zero. Both frontiers collpse to the sme singletonr =R. 2. R R nda ++ reunconditionllyproportionltoa + whena + 0. The first cse imposes tht E ( A +2 G 1 ) =E ( A + G 1 ) = =0 nd single fesible expected return. It is extreme cse where p nd p + re conditionlly proportionl, i.e. p + =k 1 p, k 1 I 1, becuse ll expected pyoffs re conditionlly proportionl to their prices, with common sclr fctor of proportionlity. This sitution cn be ssocited with the equilibrium of n economy withrisk-neutrlgentorsimplyhvingn=1. The second cse imposes tht E(R G 1)= 1 R, E ( A + G ) 1 = R, 1 1 when p nd p+ re not conditionlly proportionl.10 This cse llows different expected returns but the optiml conditionl mens re degenerte in the sense tht E(p R ( ν) G 1 )=E(p U ( ν) G 1 )= ν forny ν,ndhencethecriterione ( p 2) doesnotmkedifferencewithrespecttoe ( p 2) for givene(p)= ν. Indeed,wererulingoutthefirstcseinourset-upbecuseitisdegenertecsewhere bet-pricing is not well defined. In the second cse bet-pricing is well defined nd n unconditionl or residul bet does not mke difference given tht the conditionl men of optiml returns is nonrndom. However, such condition does not depend exclusively on sset pyoffs 10 Given tht E(R G 1) is the coefficient of the conditionl projection of p + onto the conditionl spn of p, n exmple of the second cse is p nd p + being orthogonl ( 1 =0)nd p + hving nonrndom conditionl men( 1 R). 20

22 hving constnt conditionl mens becuse the conditionl covrince mtrices lso mtter in the construction of ctive strtegies. Therefore, prt from two quite specil cses, the RRF nd URF re different objects. Both belong to the CRF nd hence conditionl bet-pricing is comptible with both residul nd unconditionl bet-pricing in the sense tht they imply the former, but not vicevers. But portfolio tht is on the RRF cnnot be on the URF, i.e. residul nd unconditionl bet-pricing (nd their portfolio efficiency implictions) re incomptible. Testing portfolio efficiency from the URF perspective, which is the usul context of empiricl work, is ctully relevnt for gents with qudrtic utility, which hs the problems tht we commented. 5.3 Pssive Frontiers nd Mnged Portfolios We mentioned tht the unconditionl frontier is the min object of interest in current empiricl work, but we cn distinguish two different pproches in the literture. There re ppers tht work directly with the URF, but there re lso ppers tht use conditioning informtion in wythtisctullypproximtingtheurf(butnotothersubsetsofthecrf).wereferto ppers such s Brndt nd Snt-Clr(2006) nd Bnsl, Dhlquist, nd Hrvey(2004) tht work with pssive strtegies of mnged portfolios. Mnged portfolios is common pproximtion in the empiricl literture to del with conditioning informtion. Insted of using ctive strtegies nd conditionl moments, the empiricl literture often relies on pssive strtegies nd unconditionl moments, i.e. pssive frontiers. However, those pssive strtegies re defined on enlrged pyoff spce constructed with mnged portfolios, which scle the originl pyoffs(usully returns) with vribles in the informtionset. Inourbinomilexmple,P cnbegenertedbypssivestrtegiesonnugmented setofmngedportfolioswhosepyoffstketheformξ 1 xndξ 2 x. Peñrnd nd Sentn(2007) comment tht the use of mnged portfolios with constnt cost, i.e. scling excess returns insted of returns, in pssive return frontier pproximtes the URF.Thisisnotsurprisingsincethecriterionfunctionndtheconstrintrethesme,E ( p 2) nd E(p) respectively, nd the only difference is the fesible set, R vs. n pproximtion bsed on mnged portfolios. We could think of similr pproch to pproximte the RRF. However, its construction requirese ( p 2) =E[p E(p G 1 )] 2 =E[Vr(p G 1 )],whichcnnotbedirectlycomputedfrom unconditionl moments. Tht is, we cn use constnt cost mnged portfolios to pproximte 21

23 R, but we need some extr computtions for E(p G 1 ). On the other hnd, ppers such s Ferson nd Siegel (2006) point to significnt inefficiencies when using mnged portfolios nd dvocte the direct use of model of conditionl moments insted. 6 The Riskless Asset Cse Now we will study the relevnt sitution in which sfe sset exists. This cse simplifies the study of portfolio efficiency since it my give single optiml risk-return trde-off defined by n optiml Shrpe rtio. We will describe the corresponding implictions for the different frontiers tht we reviewed nd developed, with specil emphsis on bet-pricing nd Shrpe rtios given its relevnce in portfolio efficiency tests. 6.1 TheNewPyoffSpce Imgine tht investors hve ccess to set of ssets tht includes not only the originl risky ssetpyoffsinx,butlsothesfepyoffx 0 =1,withcostc 01 I 1. Inthiscontext,Q will denote the corresponding enlrged pyoff spce constructed by ctive strtegies on both x nd x 0 suchthtq P. WewilldenotebyS R thesubsetofreturnsinq. We cn define the conditionlly sfe return s R 0 = 1 c 01 I 1, ndwesythtthesfessetisunconditionllyrisklesswhenc 01 isnonrndom,sothtr 0 R. Sometimeswewilluseittodefineexcessreturnsse=p R 0 givenreturnp S. Inour binomilexmpleof"goodtimes"nd"bdtimes",wewillusesfessetdefinedbytherel numbers which imply the unconditionl moments R 1 0=1.045, R 2 0=1.02, E(R 0 )=1.04, Vr(R 0 )=0.0001, whileinthespecilcseofnunconditionllyrisklesssfesset,wesimplysetr 0 =1.04. TheconditionlmenndcostctiverepresentingportfoliosinthepyoffspceQ re q + =1, q =p +ψ 1 ( 1 p + ), ψ1 = c , (18) respectively. Withthem,wecnconstructthecounterprtsofR nda + onq. 22

24 However,wecnnotdefineq ++ ndq inq swedidin(4)forp,i.e. wecnnotrepresent conditionlmenndcostbymensofconditionlcovrincesonq. Fortuntely,wecnstill work with mny of our previous expressions becuse other relevnt concepts re still well defined: TheminimumconditionlvrincereturnistrivillyR 0 ndthereiswelldefinedcentredmen RP in the spce of rbitrge portfolios (where the sfe sset does not belong) tht plys the smerolesa The Conditionl Frontier Whensfepyoffisvilble,theelementsoftheCRFsolvethesmeproblems(9),except thtpisllowedtobelongtotheenlrgedsets. Therefore,eqution(11)lsodefinestheCRF ons ifweinterpretr sr 0 nda ++ sthenewcentredmenrpinthespceofrbitrge portfolios. TheelementsoftheCRFlielongtwostrightlinesonthe[ Vr(p G 1 ),E(p G 1 )] spceforechpossiblesignlvlueing 1,ndthosetwolinesintersectontheverticlxist R 0. Moreover, there is conditionl men profile ν 1 such tht the weight of the CRF on the conditionlly sfe pyoff x 0 will be identiclly 0 for every possible signl reliztion, which implies tht it will be equl to the CRF without sfe sset t tht point, i.e. there is tngency portfolio irrespective of whether R 0 is rndom or not. The risky component of the elements of the ugmented CRF is conditionlly proportionl to n element on the originl CRF. Usingtheexpressionsin(18),wecnwritethtreturnontheCRFs R ψ 1 1 ψ 1 1 A +. (19) Figure 4 illustrtes the previous fetures for the generl cse of n unconditionlly risky sfe sset. It the sfe sset ws unconditionlly riskless then there would be single intersection on the verticl xis. 23

25 1,1 z=1 MEAN 1,01 R 0 z=2 0,92 0 0,07 0,14 SD Figure 4: Conditionl Frontier with nd without sfe sset. Thelineritytrnsltesintosingleoptimlrisk-returntrde-offonthe[ Vr(p G 1 ),E(p G 1 )] spce for ech signl vlue nd hence we cn work with conditionl Shrpe rtios risky returns p S defineds SR 1 = E(e G 1) Vr 1/2 (e G 1 ), which hs mximum vlue for risky returns on the efficient side of the RRF (E(e G 1 ) 0). The squre of tht rndom vrible is equl to the conditionl expecttion of the counterprt ofa ++ onq ndwecnuseittotestportfolioefficiency. The conditionl bet-pricing result hs unique zero-bet return, the sfe sset itself. A returnp β S differentfromr 0 isonthecrfifndonlyif E(p G 1 ) R 0 =β 1 [E(p β G 1 ) R 0 ], p S, whichcnlsobeexpressedintermsofexcessreturnswithrespecttothesfesset 6.3 The Residul Frontier E(e G 1 )= Cov(e,e β G 1 ) Vr(e β G 1 ) E(e β G 1 ). When sfe pyoff is vilble, the elements of the RRF solve the sme problems (13), exceptthtpisllowedtobelongtotheenlrgedsets. Therefore,eqution(14)lsodefines therrfons ifweinterpretr sr 0 nda ++ sthenewcentredmenrpinthespceof rbitrgeportfolios. TheRRFbecomestwostrightlinesonthe[Vr 1/2 ( p),e(p)]spcewith 24

26 zeroresidulvrinceminimumtr 0 (i.e. t ν=e(r 0 )),independentlyofthesfessetbeing unconditionlly riskless or not, s Figure 5 shows. 1,09 MEAN 1,06 R 0 1,03 0 0,04 0,08 RESIDUAL SD Figure 5: Residul Frontier with nd without sfe sset. Thelineritytrnsltesintosingleoptimlrisk-returntrde-offonthe[Vr 1/2 ( p),e(p)] spcendhencewecnworkwithresidulshrpertiosofriskyreturnsp S defineds SR R = E(e) Vr 1/2 (ẽ), whichhsmximumvlueforriskyreturnsontheefficientsideoftherrf(e(e) 0). The squreofthtvlueisequltotheexpecttionofthecounterprtofa ++ onq ndwefind thefollowingreltionshipbetweensr R ontherrfndsr 1 onthecrf SR 2 R =E[ SR 2 1]. However, contrry to conventionl wisdom on men-vrince frontiers with sfe sset, there is no tngencyportfolioirrespectiveof whether R 0 is rndomor not, see Figure5. Therisky component of the elements of the ugmented RRF is conditionlly proportionl to n element ontheoriginlcrf(19)thtdoesnotbelongtotheoriginlrrf. The residul bet-pricing result hs unique zero-bet return, the sfe sset itself, s the following corollry sttes. Corollry3 Areturnp β S differentfromr 0 isontherrfifndonlyif E(p) E(R 0 )=β R [E(p β ) E(R 0 )], p S. 25

27 Note tht the pricing eqution cn lso be expressed in terms of excess returns with respect tothesfesset E(e)= Cov(ẽ,ẽ β) Vr(ẽ β ) E(e β), which looks very similr to stndrd bet-pricing equtions, with the novelty of computing bet with returns residuls in the sense of substrcting their conditionl mens. Ontheotherhnd,nowitisesiertotestportfolioefficiencybecusewecnrelyonSR R. Asfrsweknow,thisShrpertiohsnotbeenusedintheliterturesofreventhoughitis the relevnt one for n investor like(12). 6.4 The Unconditionl Frontier When sfe pyoff is vilble, the elements of the URF solve the sme problems (15), exceptthtpisllowedtobelongtotheenlrgedsets. Thesolution(16)isstillvlidfter theintroductionofsfepyoffifwesimplyreplcep + ndp in(3)withq + ndq in(18), respectively. In this context, Peñrnd nd Sentn (2007) point out two fcts bout the URF tht contrdict conventionl wisdom on men-vrince frontiers with sfe sset. First, s it ws thecsewiththerrf(seefigure5),thereisnotngencyportfolioirrespectiveofwhetherr 0 is rndom or not. The risky component of the elements of the ugmented URF is conditionlly proportionltonelementontheoriginlcrf(19)thtdoesnotbelongtotheoriginlurf. Second, there is feture thtdiffers from therrf.theelements of the URF do not lie longtwostrightlines onthe[vr 1/2 (p),e(p)]spceifr 0 is rndom, 11 whichmenstht thereisnotuniqueoptimlrisk-returntrde-off,norisr 0 frontierportfoliointhtcse(see HnsenndRichrd(1987)). TheunconditionlShrpertioofriskyreturnp S,defined s SR U = E(e) Vr 1/2 (e), is not unique on the URF nd hence SR U is not very useful to test portfolio efficiency. The unconditionl bet-pricing result is the sme s in the cse without sfe sset becuse the zero-bet return will depend on the chosen element of the URF. Figure 6 compres the RRF ndtheurfinthiscontext. 11 Theminimumvrincereturnstisfiesgin ν= ( 1 E ( )) A + 1 E(R )forthecorrespondingcounterprts ofr nda + onq. 26

28 1,09 1,09 MEAN 1,06 RRF MEAN 1,06 URF URF RRF 1,03 0 0,02 0,04 RESIDUAL SD 1,03 0 0,02 0,04 SD Figure 6: Residul nd Unconditionl Frontiers with sfe sset tht is unconditionlly risky. However, if the conditionlly sfe return is lso unconditionlly riskless becuse the price of x 0 is constnt, then theurf will indeed consist of two stright lines thtintersecton the verticlxistr 0. Figure7illustrtesthiscse,wherethereissingleoptimlSR U forrisky returnsontheefficientsideoftheurf(e(e) 0)ndwecnrelyonSR U totestportfolio efficiency. GiventheexpecttionofthecounterprtofA + onq,thesqureofthemximum SR U isthertioofthtexpecttiondividedbyoneminusthtexpecttion. 12 In the cse of n unconditionlly riskless sset, Jgnnthn(1996) relted the Shrpe rtio oftheurfwiththeshrpertioofthecrf 1 1+SR 2 U [ =E 1 1+SR 2 1 There is unique zero-bet return on the URF, the sfe sset itself. A return p β S differentfromr 0 isontheurfifndonlyif ]. E(p) R 0 =β U [E(p β ) R 0 ], p S, whilethtthepricingequtioncnlsobeexpressedintermsofexcessreturnswithrespectto the sfe sset E(e)= Cov(e,e β) Vr(e β ) E(e β). 12 IfwedenotetheconditionlmenofthtcounterprtofA + byk 1 I 1 thenthelinkbetweenthemximum ShrpertioontheRRFndtheURFis [ ] SR 2 k1 R=E, SRU 2 = E(k 1) 1 k 1 1 E(k 1 ). 27

29 1,09 1,09 MEAN 1,06 RRF MEAN 1,06 URF URF RRF 1,03 0 0,02 0,04 RESIDUAL SD 1,03 0 0,02 0,04 SD Figure 7: Residul nd Unconditionl Frontiers with sfe sset tht is unconditionlly riskless. TheliterturehsfocusedonnonrndomR 0 sofrndhsnotpointedoutthestrnge feturesoftheurfwhenthtisnotthecse. ThismybeduetothefctthtTresury-Bills historicl vrince is much lower tht other ssets historicl vrince, e.g. stocks. Nevertheless, Tresury-Bills cnnot be considered unconditionlly riskless. 7 Conclusion In context of ctive strtegies, this pper studies the reltionship between stndrd menvrince frontiers tht drive empiricl work nd the stndrd men-vrince preferences tht re used in finnce theory. The conditionl frontier represents the generl set-up where optiml portfolios of generl men-vrince gents re locted. We distinguish two subsets of the conditionl frontier, the unconditionl nd the residul frontier. The former is the min object regrding empiricl work but does not hve plusible connection to men-vrince preferences. Theresidulfrontierhsnotbeenstudiedinthelitertureyetbutitiswherestndrdmenvrince preferences choose portfolios. In ddition, we show tht both subsets of the conditionl frontier do not shre ny ctive return unless the dt stisfies two very specil cses, where both frontiers re equl. On the other hnd, the ctul difference between both frontiers nd the importnce of the previous cvet is n empiricl question tht will depend on the prticulr cse(i.e. ssets nd informtion set) under study. 28

30 For ech frontier, we reviewed or developed the corresponding bet-pricing to construct portfolio efficiency tests. Given the previous comments, it is cler tht unconditionl betpricing is not the relevnt tool for stndrd men-vrince gent, while conditionl bet-pricing my be helpful. Nevertheless, the ltter is not s powerful s residul bet-pricing. We did not study the trnsltion of these sset pricing implictions to stochstic discount fctors becuse such dulity nlysis is developed in Peñrnd nd Sentn (2007). In ddition, note tht bet-pricing results refer to pricing returns but not portfolios with generl cost. Bsu nd Stremme (2006) relte the pricing of portfolios with generl cost to Shrpe rtios, but those rtios refer to portfolio frontiers tht re not composed by returns. We lso studied the implictions of the existence of sfe sset, which my simplify portfolio efficiency tests by mens of Shrpe rtios. Not surprisingly, the resercher my use three different Shrpe rtios nd the previous comments on bet-pricing pply here gin. We lso point out tht conventionl fetures with pssive strtegies such s frontier linerity nd existence of tngency portfolio do not necessrily hold s we move from the conditionl frontier to the two subsets. 29

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