Optimal Asset Allocation with Factor Models for Large Portfolios

Transcription

1 Opmal Asse Allocaon wh Facor Models for Large Porfolos M. Hashem Pesaran Paolo Zaffaron Unversy of Cambrdge and USC Imperal College London July 15, 2008 Absrac Ths paper characerzes he asympoc behavour, as he number of asses ges arbrarly large, of he porfolo weghs for he class of angency porfolos belongng o he Markowz paradgm. I s assumed ha he jon dsrbuon of asse reurns s characerzed by a general facor model, wh possbly heeroskedasc componens. Under hese condons, we esablsh ha a se of appealng properes, so far unnoced, characerze radonal Markowz porfolo radng sraeges. Frs, we show ha he angency porfolos fully dversfy he rsk assocaed wh he facor componen of asse reurn nnovaons. Second, wh respec o deermnaon of he porfolo weghs, he condonal dsrbuon of he facors s of second-order mporance as compared o he dsrbuon of he facor loadngs and ha of he dosyncrac componens. Thrd, alhough of crucal mporance n forecasng asse reurns, curren and lagged facors canno be exploed o forecas he lm porfolo reurns. These properes follow snce angency porfolos are asympocally bea-neural as he number of asses ges large. Our heorecal resuls also shed lgh on a number of ssues dscussed n he leraure regardng he lmng properes of porfolo weghs such as he dversfably propery and he number of domnan facors. JEL Classfcaons: C32, C52, C53, G11 Key Words: Asse Allocaon, Large Porfolos, Facor Models, Dversfcaon, Bea euraly. A prelmnary verson of hs paper was presened a he Cambrdge Fnance Workshop, he Gersenzee CEPR Conference, he Imperal College Fnancal Economercs Conference, and a he Bendhem Cener a Prnceon, Unversy of Pennsylvana, LSE and he Unversy of Brunel. We are graeful o he parcpans, and n parcular o he dscussans of our paper Janqng Fan and Erc Renaul for mos helpful commens. 1

2 1 Inroducon Facor models represen a parsmonous, ye flexble way of modellng he condonal jon probably dsrbuon of asse reurns when here are a large number of asses under consderaon. Promnen use of facor models nally focused on parameerzng he condonal mean, followng he hghly nfluenal capal asse prcng model of Sharpe 1964 and Lnner 1965, and he arbrage prcng heory of Ross In fac parsmony plays an even a more mporan role when modelng condonal covarance marx of a large number of asse reurns. Gven ha he man raonale for usng facor models s o deal wh porfolos wh a large number of asses, hs paper characerzes he dsrbuon of porfolo weghs, as he number of asses,, ncreases whou bounds, n he case of he commonly used mean-varance effcen porfolos hereafer MV. Our analyss s confned o myopc asse allocaon rules, all parcular cases of Markowz 1952 heory, whch are opmal only for a consan nvesmen opporuny se. Focusng on myopc radng sraeges s jusfed from a praccal perspecve n he case of large porfolos where applcaon of dynamc asse allocaon sraeges can be prohbve and s rarely red n pracce. The leraure on dynamc asse allocaon s ofen confned o a few broad asse classes, such as Treasury Blls, long erm bonds and eques see, for example, Campbell and Vcera A number of papers have already examned he lmng behavor of MV porfolos when here are a counably nfne number of prmve asses under consderaon. Chamberlan 1983 and Chamberlan and Rohschld 1983 suded he mplcaons of no arbrage for he MV effcen froner as ends o nfny. They hen consdered facor models and exended he arbrage prcng heory APT resul of Ross 1976 o he case where asse reurns follow an approxmae facor srucure. The laer exends he exac facor model by permng ceran lmed degree of correlaon across he dosyncrac componen of asse reurns. Hansen and Rchard 1987 exended he sac framework of Chamberlan 1983 and Chamberlan and Rohschld 1983, bu dd no focus on facor srucures. Subsequenly, Green and Hollfeld 1992 clarfed he relaonshps ha exs beween dversfcaon and MV effcency n a general seng. Employng a facor srucure, hese auhors provded a furher generalzaon showng ha even he approxmae facor srucure s oo srngen for he APT o hold. Whereas Chamberlan 1983 characerze dversfably by lookng a 2

3 he rae a whch he square norm of he porfolo weghs converge o zero as ends o nfny, Green and Hollfeld 1992 characerze dversfably n erms of sup-norm crera. Senana 2004 compares he sascal properes of sac and dynamc facor represenng porfolos, usng a dynamc verson of he APT. By and large all of he above papers focus on varous aspecs and generalzaons of he APT under he mananed assumpon of an underlyng facor srucure as. However, once one absracs from he APT, a number of oher neresng ssues arse ha have hhero been negleced n he leraure. For nsance, he precse behavor of he MV porfolo weghs as has been surprsngly overlooked. Lkewse, o our knowledge he sascal properes of he lm porfolo reurn have no been spelled ou. I urns ou ha neresng, and n fac somewha couner-nuve, resuls arses from hese nvesgaons, n parcular regardng he role played by he condonal dsrbuon of he facors. In hs paper we do no make use of no-arbrage assumpon and, herefore, do no nvesgae mplcaons of he APT n he case of large porfolos, unlke Hubermann 1982, Chamberlan 1983, Chamberlan and Rohschld 1983, Sambaugh 1983, Connor 1984, Ingersoll 1984, Grnbla and Tman 1987, Green and Hollfeld 1992, Senana 2004 among ohers. Lkewse, our goal here s no o nvesgae ssues relaed o parameer uncerany when esmang MV radng sraeges see Brand 2004 for a complee survey on he economerc ssues assocaed wh porfolo choce problems. We make he mananed hypohess ha he vecor of asse reurns s dsrbued accordng o a dynamc facor model, wh a specfcaon of he condonal varance marx of he dosyncrac componens whch s more general han he approxmae facor srucure of Chamberlan and Rohschld Under hs assumpon, he paper esablshes hree man resuls: a In he lm he MV porfolos fully dversfy he nnovaons n he common facor componens of asse reurns. I s well known ha MV porfolos do fully dversfy he dosyncrac componen of asse reurns nnovaons, bu o our knowledge s no recognzed ha he same apples o he facor nnovaons. Ths s an mporan feaure of MV porfolos wh praccal mplcaons whch we dscuss below. b The lm MV porfolo weghs he frs-order lm approxmaon for large of he MV porfolo weghs are funconally ndependen of he condonal dsrbuon of he facors. oce ha hs does no mply ha he facors hemselves are no mporan bu only ha her condonal 3

4 momens are no relevan n so far as he calculaon of he MV porfolo weghs s concerned. For example, esmaon of he facors and her loadngs are requred for a conssen esmaon of he dosyncrac componens. c In he lm he MV porfolo reurns are funconally ndependen of he curren and lagged values of he common facors. The facors could play a cenral role for forecasng asse reurns bu, as ges larger, her role vanshes n erms of her conrbuon o he lm porfolo reurns. In oher words, a any pon n me n he lm as, he condonal dsrbuon of he lm reurns on MV porfolos are funconally ndependen of he condonal dsrbuon of he common facor componen. eher of he above fndngs srcly mples he oher and are all of ndependen neres. Inally hese resuls may seem raher couner nuve snce one would expec he facor componen, beng domnan, wll be he mos mporan elemen n he deermnaon of asse reurns. Bu MV porfolos are funcons of he nverse of he covarance marx of asse reurns, and he common facor par of asse reurns ha generae srong cross reurn dependence wll urn no weak cross dependence when he nverse of he varance marx s consdered. By comparson, he dosyncrac componens of asse reurns ha exhb weak cross secon dependence wll begn o play a cenral role n deermnaon of MV porfolos as sars o become suffcenly large. Conceps of weak and srong cross secon dependence are developed n Pesaran and Tose In parcular he concep of weak cross secon dependence allows he maxmum egenvalue of he covarance marx of he dosyncrac componen of asse reurns o rse lke o. Formally, our resuls follows from a form of asympoc orhogonaly, newly esablshed n hs paper, beween he nverse of he condonal covarance marx of asse reurns and he marx of facor loadngs. Ths form of orhogonaly drecly mples ha all MV radng sraeges, and n fac all radng sraeges ha employ he nverse of he asse reurns covarance marx n he same way, are asympocally bea-neural. Fnally, noe ha none of hese resuls sugges ha he facors hemselves are unmporan for he lm MV porfolo weghs and he porfolo reurn, raher ha s he conrbuon of facors condonal dsrbuon o he MV porfolo reurn ha s asympocally neglgble. In fac, knowledge of he facors s essenal n pracce when dervng he oher pars of he facor model, namely he facor loadngs and he predcable and non-predcable dosyncrac componens of he asse reurns, ha are of frs-order mporance for he MV radng sraeges and her assocaed reurn. 4

5 The above fndngs also have a number of furher mplcaons of neres ha we summarze below: d- The lm MV porfolos are me-nvaran unless, dependng on he radng sraeges, he rsk free rae s me-varyng and he dosyncrac componen feaures me-varyng condonal heeroskedascy. d- The lm MV porfolo weghs are nvaran o any orhogonal roaon of he facors. d- Prmve condons requred for full-dversfcaon n he supnorm sense of Green and Hollfeld 1992 are esablshed. d-v Analycal characerzaons of he occurrence of negave porfolo weghs and of he relaed ssue of facor domnance, n he sense of Green and Hollfeld 1992 and Jagannahan and Ma 2003, are provded. The remander of he paper s organzed as follows. Secon 2 nroduces he conceps, ses ou he dynamc facor model, and dscusses s properes. The man fndngs are llusraed, as an example, wh respec o a sngle facor model n Secon 3. Secon 4 presens he man resuls n he general wh respec o he commonly used radng sraeges: he global mnmumvarance and he maxmum expeced quadrac uly porfolos. Secon 5 elaboraes and dscusses he mplcaons of he heorecal resuls. Secon 6 exends he resuls o wo oher angency porfolos, namely he mnmumvarance and he maxmum expeced reurn porfolos. Secon 7 concludes. Mahemacal proofs are colleced n an appendx. 2 Facor model: defnons and assumpons We assume he -dmensonal vecor r = r 1, r 2,..., r of asse reurns can be characerzed by he followng lnear dynamc facor model r = α 1 + Bf + ε, 1 where f s he k 1 vecor of possbly laen common facors, B = β 1,..., β s an k marx of facor loadngs, ε s an 1 vecor of dosyncrac 5

6 componens, and he 1 vecor α 1 represens he par of he condonal mean of he r ha does no depend on he common facors. Throughou wll be assumed ha k remans fxed as. We denfy he facor model by means of he followng assumpons: Assumpon 1 condonal mean reurns The vecor of laen facors f can be decomposed no s predcable componen, ν 1, and he remander u as f = ν 1 + u, 2 where ν 1 = Ef Z 1, wh Z 1 beng he sgma-algebra nduced by a g marx of observed varaes {Z s, s > 0}. α 1 = Er Bf Z 1, 3 α and u s are ndependenly dsrbued for all and s. 4 Under hs assumpon he condonal mean of asse reurns s gven by Er Z 1, B µ 1 = α 1 + Bν 1, 5 and he nnovaon n he common componens u s a marngale dfference process wh respec o Z 1. I s worh nong ha he decomposon n Assumpon 1 can also be defned wh respec o he sgma-algebra spanned by he unobserved nformaon se f s, s > 0. Ths wll no affec our man conclusons, bu wll rase a number of addonal dffcules wh respec o he emprcal mplemenaon of he model. Assumpon 1 rules ou a dynamc facor represenaon of asse reurns see Forn, Halln, Lpp, and Rechln 2000 and Sock and Wason 2002 such as r = α, 1 + β 1 c L 1 u + ε, where L s a lag operaor and c dffers across. Ths does no seem a parcularly mporan lmaon n he case of asse prcng models where he reurns are only enuously serally correlaed. In pracce, specfcaon and esmaon of µ 1 could be a major emprcal underakng, parcularly n he case of large porfolos. Bu gven he focus of our analyss, n wha follows we ake he specfcaon of µ 1, especally s α 1 componen, as gven. I s mporan, neverheless, o separae 6

7 α 1 from ν 1 snce he laer, as we shall see, does no ener he lm MV porfolos as ges large. Condons 2, 3 and 4 ogeher mply r = µ 1 + Bu + ε. 6 Hereafer, we shall refer o 6 as he facor model wh u beng he k 1 vecor of facor nnovaons whou furher reference o f. Regardng he facor loadngs, we consder he case where he elemens of B are random varaes sasfyng he followng lm condon: Assumpon 2 facor loadngs As B e p β 0, 7 where e = 1,..., 1 s an 1 vecor of ones, and p denoes convergence n probably. From 7 follows ha β represens he mean vecor Eβ. Assumpon 2 s an ergodcy assumpon over he cross secon. I s much weaker han he..d. assumpon ypcally made when consderng random facor loadngs. For nsance, a srong suffcen condon for 7 o hold s when he facor loadngs have fne second-order momens and absoluely summable cross covarances bu, n fac, Assumpon 2 s compable wh a much more subsanal degree of cross-seconal dependence among he elemens of β. The resuls presened n hs paper can be generalzed furher o he case of heerogeneous ye non-random β. Assumpon 3 nnovaons A any gven pon n me u Z 1 0, Ω 1, ε Z 1 0,, 8 ε and u are muually ndependen, 9 where Ω 1 and are posve defne marces, respecvely, of dmenson k k and for a fxed k and any fne. The resuls ha follow do no depend on a parcular specfcaon of he volaly model characerzng he asse reurns. Moreover, he facors can eher be observable or non-observable. As a consequence, Ω 1 and could belong o he mulvarae sochasc volaly class as well as o he 7

8 generalzed auoregressve condonal heeroskedascy class of volaly models. Parcular examples, o whch Assumpon 3 apples, are dscussed below. To derve he lmng behavor as of he varous angency porfolo weghs o be consdered below, we furher requre he followng assumpon: Assumpon 4 lm condons A any gven pon n me as and B e β B e β p A > 0, 10 B B p C 0, 11 e e e α α H 1 α p a > 0, 12 p c, 13 p d > 0, 14 where hereafer > 0 and 0 means, respecvely, posve defnve and posve sem-defne. Moreover a, c, d, A, C are O p 1 elemen by elemen such ha and d a c 2 > 0 almos surely 15 B s ndependenly dsrbued from boh H and α. 16 The common feaure of he lms presened n Assumpon 4 s ha hey nvolve, possbly weghed, averages of he elemens of. In parcular, hey mpose mplcly an upper bound on he speed wh whch he maxmum egenvalue of could dverge o nfny. Recall ha he larges egenvalue of concde wh he smalles egenvalue of H, by consrucon. Ths s clearly seen from condon 12: assumng for llusrave 8

9 purposes ha s dagonal, wh h 1, n he, h enry, hen 12 allows max 1 h 1, = o p. Condon 11 requres a furher consran on he speed of dvergence of max 1 h 1, whch can now be a mos o p 1 2. Even hs case s much weaker han max 1 h 1, C <, for some consan C, mpled by he defnon of approxmae facor srucure see Chamberlan and Rohschld Green and Hollyfeld 1992 were he frs o noe ha, nsofar as opmal asse allocaon s concerned, a degree of cross-seconal dependence sronger han he one mpled by he approxmae facor srucure s permed. When s non-dagonal, he prevous dscusson apples o s larges egenvalues. Condons 10 and 11 requre he exsence of he second-order momens of he facor loadngs and mpose ceran consrans on he degree of cross-seconal dependence of he β. oe ha when 7, 12 and 16 hold, hen 10 s equvalen o sayng ha 1 B B has a posve defne lm. When β are..d. and H s dagonal, hen A = a covβ. Concernng 16, noe ha H and α need no be, and n general wll no be, muually ndependen. Condons 13 and 14 also requre he elemens of α no o grow, f a all, oo fas as compared wh. The lm c n condon 13 s bounded, n absolue value, by a d 1 2. The lm d n condon 14 s fne whenever 12 holds and α α / has a fne lm. Condon 15 s no needed n he case where α s a non-degenerae random varable. For some resuls, n parcular o derve he lm dsrbuon of he MV porfolo weghs, a sronger verson of Assumpon 4 s needed as se ou below: Assumpon 5 furher lm condons For any and a any gven pon n me, as B e p ξ 1, 17 e e p ξ 2, 18 α H 1 e p ξ 3, 19 wh ξ j = O p 1, for j=1,2,3, where e s he h column of he deny marx I and denoes he Eucldean norm. 9

10 1 2 1 vechb B vecha + a β β 1 B e a β 1 B α c β 1 e e a d 0, V, 20 for some posve sem-defne marx V, where d denoes convergence n dsrbuon and vecha sacks he dverse elemens of a symmerc marx A no a column vecor. Condons , mpose a fne upper bound o each of he columns of and are herefore much sronger han ha are expressed n erms of averages. In parcular, 18 s sasfed by an approxmae facor srucure. Condon 20 s somewha weaker han he oher pars of Assumpon 5 alhough, agan, allows for a smaller degree of crossseconal dependence han he one permed by Assumpon 4. In parcular, noe ha 20 rules ou ha α conans a common facor srucure. Ths s no resrcve as appears, snce any common componens of α can be ncluded n ν 1. In vew of 8, he facor srucure 6 mples he well-known form of he asse reurn condonal varance-covarance marx: E [r µ 1 r µ 1 Z 1, B] = Σ 1 = BΩ 1 B Thus model 6 ness he varous facor models wh me-varyng condonal second momen proposed n he economercs leraure see among many ohers Debold and erlove 1989, Kng, Senana, and Wadhwan 1994, Chb, ardar, and Shephard 2002, Forenn, Senana, and Shephard 2004, Connor, Korajczyk, and Lnon 2006, Doz and Renaul These papers, whch focus on esmaon of volaly facor models, n parcular when u s no observable, all assume consan condonal frs-order momens. On he oher hand, he fnance leraure dealng wh facor models-based asse allocaon assumes homoskedasc facors whereby Ω 1 = Ω, ofen normalzed o be equal o he deny marx see among many ohers Pesaran and Tmmermann 1995 and Kandel and Sambaugh A few conrbuons analyze asse allocaon problems allowng for volaly dynamcs bu mpose consan condonal means see for nsance Agular and Wes 2000 and Flemng, Krby, and Osdek Only recenly, a lmed number 10

11 of sudes have consdered me varaons n boh he frs and second condonal momens of asse reurns see for nsance Johannes, Polson, and Sroud 2002 and Han Model 6 ness all of he above specfcaons. 3 The sngle facor case Here we llusrae our resuls usng a sngle facor model k = 1 where 6 becomes r = µ 1 + β u + ε, 22 and Assumpons 1, 2 and 3 hold. Therefore now u s a scalar marngale dfference process wh condonal varance ω 1 > 0, and β s a 1 vecor of facor loadngs wh mean βe 0, and he varance marx σ 2 β I > 0, where β and σ 2 β are scalar. Le us also assume ha he dosyncrac errors ε are cross-seconally uncorrelaed, mplyng a dagonal, wh condonal varances h, 1 > 0, a.s. The condonal covarance marx of r wll hen be Σ 1 = ω 1 ββ +. oe ha an observaonally equvalen model can be obaned f one assumes homoskedasc facor, vz. Eu 2 Z 1 = 1, bu allow me-varyng facor loadngs of he form, β = ω 1 2 β. Consder now wo ypes of porfolos: he global-mnmum-varance henceforh gmv porfolo and he maxmum expeced uly henceforh meu porfolo. 3.1 The lm behavour of he gmv porfolo The gmv porfolo weghs, w gmv problem: yeldng = w gmv 1,..., w gmv, are he soluon o he w gmv = argmn w w Σ w, such ha w e = 1, 23 w gmv 1 = Σ 1 1e e Σ 1e. 24 I s well known ha hs porfolo does no belong o he effcen froner, excep when he condonal expeced reurns µ, 1 are he same across 11

12 bu, wh some abuse of noaon, we wll vew as belongng o he se of MV radng sraeges. everheless, hs porfolo s sll of neres snce s mplemenaon does no requre he esmaon of expeced reurns. Jagannahan and Ma 2003 show ha, n erms of asse allocaon, s ou-ofsample performance s comparable wh he performance of oher angency porfolos. In he case of he sngle facor model he opmal porfolo wegh for he h asse s gven by w gmv = 1 h 1, ϕ 1, 1 β κ,, where ϕ, = 1 κ, = j=1 h 1 jj, 1 ω 1 1 j=1 β jh 1 jj, j=1 β2 j h 1 jj,, 25 1 j=1 β jh 1 jj, 1 ω j=1 β2 j h 1 jj, 26 Under Assumpon 4, n parcular gven 16, and snce ω 1 e e = β β = j=1 h 1 jj, j=1 β2 j h 1 jj, = O p 1 we have ϕ, p p a, β H 1 e = p a σ 2 β + β 2, a σ 2 β σ 2 β + β 2, κ, p j=1 β jh 1 jj, β σ 2 β + β 2. p a β, 27 Hence, readly follows ha w gmv h 1, p 1 β β a σβ 2 β. 28 whch clearly shows ha he lm gmv porfolo weghs are funconally ndependen of he facor condonal varance, ω. The lm gmv porfolo 12

13 weghs are also asympocally bea-neural n he sense ha β w gmv 1 = β w gmv = 1 β h 1, 1 β β σ 2 β 1 + o p 1 p 0. =1 =1 β 29 I s also worh nong ha he lm porfolo weghs are me-varyng only f h, 1 are me-varyng. Moreover, noce ha f we assume, for some posve sochasc process θ > 0, h, = h θ > 0, 30 hen despe he me varyng naure of h,, he lm gmv porfolo weghs wll be me nvaran snce w gmv h 1 θ 1 p aθ 1 1 β β σβ 2 β = h 1 1 β β a σβ 2 β, where a = p lm 1 j=1 h 1 jj. Consder now he lm properes of he porfolo reurn ρ gmv By 22 hs can be wren as = r w gmv 1. ρ gmv = µ 1w gmv 1 + β w gmv 1 u + ε w gmv Snce we ake he lm as, we can drecly use he lm approxmaon yeldng, for he hrd erm n ρ gmv, 1 + ε w gmv β2 /σβ 2 1 = O 1 2 p = o a 1 p 1, esablshng ha he gmv porfolo does dversfy away he dosyncrac rsk. In vew of 29 he second erm of 31 also vanshes as, and hence here wll be no conrbuon from he common source of rsk u o he lm porfolo reurn. Therefore, he only erm of ρ gmv ha s no vanshng s µ 1w gmv 1 = α 1w gmv 1 + β w gmv 1 ν. However, we have jus seen ha β w gmv 1 = o p 1 mplyng ha here s no conrbuon fromhe predcable common componen ν 1 o he lm 13

14 porfolo reurn. Insead, one smply needs o consder he lm of [ α 1w gmv 1 1 = h 1 jj, 1 1 β ] β a 1 σ 2 j β α j, o p 1 j=1 β = = 1 a 1 1 a 1 where by Assumpon 4 Summarzng, we have j=1 j=1 α H 1 e h 1 jj, 1 α j, 1 h 1 jj, 1 α j, 1 = 1 j=1 β a 1 σ 2 β j=1 h 1 jj, 1 β j βα j, o p o p 1 = c 1 a o p 1, h 1 jj, α j, p c. 32 ρ gmv p c 1 /a 1, 33 namely ha he lm gmv porfolo reurn s Z 1 -adaped snce boh he dosyncrac and he common nnovaon are fully dversfed. Moreover, n he lgh of he asympoc bea-neuraly propery, boh he predcable and he non-predcable common componens have an asympocally neglgble conrbuon o he oucomes. The only pars of he asse reurn dsrbuon relevan o he lm gmv porfolo reurn are va he erms a and c, ha nvolve averages of asse-specfc means and volales, α and h 1,, over. Fnally, f h, are muually ndependen, hen he lm gmv porfolo reurn s gven by he lm of he sample mean of he α, namely n he lm only he asse-specfc reurns maer. 3.2 The lm behavour of he meu porfolo Suppose now ha besdes he rsky asses, nvesors can also allocae her funds o a rsk free asse wh a me-varyng rae of reurn, r 0, whch s known a he sar of radng perod. The maxmum expeced uly meu porfolo, based on a mean-varance uly funcon, s defned by he soluon o he followng opmzaon problem w 1 meu = argmax w w µ w er 0 1 γ 1 2 w Σ 1 w, 34 14

15 where w meu = w1 meu,..., w meu, 0 < γ 1 < s, possbly me-varyng, coeffcen of rsk averson. The soluon s w meu 1 = 1 γ 1 Σ 1 1µ 1 er In he case of he sngle facor model, he meu porfolo wegh for he h asse s gven by where w meu = γ 1 = γ 1 = γ 1 δ, = ω 1 h 1, [µ β δ, r 0 1 β κ, ], h 1, [α + β ν δ, r 0 1 β κ, ], h 1, [α r 0 β δ, ν r 0 κ, ], j=1 β jh 1 jj, µ j + j=1 β2 j h 1 jj, p c β a σ 2 β + β 2 + ν, wh c defned as before by 32. Recall also ha under Assumpon 4 1 j=1 β j h 1 jj, α j p c β. Hence, usng he above resuls and recallng ha κ, p have [ w meu p γ 1 h 1 βc /a r 0, α r 0 β σβ 2 + β 2 [ = h 1, γ b 1 + β 2 σ 2 β α r 0 ββ c r σβ 2 0 a β/σ 2 β + β 2 we where b = 1 + β 2 /σβ 2. Thus, he lm meu porfolo weghs are funconally ndependen of boh ν and ω. In conras o he prevous case, when 30 holds he meu porfolo weghs wll sll be me varyng and, n fac, wll be so even for homoskedasc ε. Furher, noce ha each porfolo wegh wll n general be dfferen from zero even asympocally n. Consder now he porfolo reurn ρ meu = r w 1 meu +1 e w 1 meu r 0 1 = µ 1w 1 meu + β w 1 meu 15 ] ], u +ε w meu 1 +1 e w meu 1 r 0 1

16 When normalzng ρ meu by 1 for he fourh erm one obans { [ = = e w meu 1 = 1 γ 1 b β c 1 r σβ 2 0, 1 1 a 1 [ 1 + β 2 σβ 2 j=1 1 j=1 h 1 jj, 1 β j } h 1 jj, 1 α j, 1 r 0, o p 1, j=1 h 1 jj, β 2 c γ 1 b σβ 2 1 r 0 1 a 1 c 1 r 0 1 β ] 2 a a 1 σβ o p 1 c 1 r 0 1 a 1 + o p γ 1 b For he hrd erm of ρ meu / ] ε w meu 1 = O p 2 j=1 h jj, 1 w meu 2 j, [ a 1 d = O p 1 c b 1 c 1 a 1 r 0 1 2] 1 2 = o a 1 p 1, and for he second erm β w 1 meu =O 1 p h 1 jj, 1 γ 1 α j 1 r 0 1 β j β c 1 a 1 r 0 1 σβ 2 γ 1 a 1 b j=1 Therefore, even he meu porfolo s bea-neural snce boh he dosyncrac and common sources of rsk n ρ meu / are dversfed away as ges large. I remans o consder µ 1w 1 meu α = 1 w 1 meu β w 1 meu + ν, bu, agan, snce β w meu 1 / = o p 1 here s no conrbuon of he predcable common componen ν 1 o he lm porfolo reurn. One s lef o consder j=1 h 1 jj, 1 β2 j =o p 1. 16

17 he lm of = = α 1w meu bγ 1 1 bγ 1 j=1 = 1 bγ β 2 σβ 2 h 1 jj, 1 α j 1 r 0 1 α j, 1 ββ j α j, 1 σ 2 β c 1 a 1 r o p 1, 1 + β 2 d σβ 2 1 c 1 r 0 1 β [ 2 c ] 1 r σβ c 1 + o p 1, a 1 d 1 c 1 r β 2 a a 1 σβ 2 1 d 1 c o p 1, 38 where by Assumpon 4 α H 1 α = 1 j=1 h 1 jj, α2 j, p d > 0. Thus, akng he dfference beween 38 and 36 yelds ρ meu p e 1 γ 1 b, where e = d 2c r 0 + a r β 2 σ 2 β d c2 a. Therefore, lke he gmv lm porfolo reurn, he lm of ρ meu / s Z 1 - adaped snce boh he dosyncrac and he common nnovaon are dversfed away. Moreover he predcable common componen has an asympocally neglgble conrbuon o porfolo reurns. In he followng secon we esablsh hese resuls n he general mulfacor seng, wh k > 1, and non-dagonal H, where we also show ha e 1 > 0, a.s. 4 The general mul-facor case We begn wh he gmv porfolo weghs defned by 24 and n wha follows we suppose ha r s generaed accordng o he mul-facor model 6, 17

18 Assumpons 1,2 and 3 hold, and all he lms are aken for each and as. In hs secon we also allow he dosyncrac errors, ε, o be weakly cross-seconally correlaed, namely we allow for a non-dagonal marx. Theorem 1 global mnmum-varance porfolo Le ẘ gmv 1 e [ ] = e + a e β BA 1 β, 39 a and recall ha e s a 1 row vecor of zeros excep for s h elemen whch s uny. When Assumpons 7, 10, 12 and 16 w gmv ẘ gmv p When, n addon o he assumpons made n, hold w gmv = ẘ gmv + 3/2 z gmv + 41 [ 2 b e B A + a β 1 β Ω 1 A + a β ] 1 β β + o p 2, n whch b = 1 + a β A 1 β, 42 and z gmv s a mxure of normally dsrbued random varables ha are only funcons of B and H. When, n addon o he assumpons made under, relaons 11 and 13 hold: ρ gmv 1 2 = r w gmv c 1 1 p, 43 a 1 µ gmv ρ, 1 σ gmv ρ, 1 p c 1 a 1, 44 where µ gmv ρ, 1 = Eρ gmv Z 1, and σ gmv ρ, 1 = varρ gmv Z 1. Remark 1a The gmv porfolo wegh of he h asse s, asympocally n, equvalen o ẘ gmv. Inspecng 39 emerges ha ẘ gmv s funconally ndependen from he facors covarance marx, Ω. Insead, s a funcon 18

19 of he facor loadngs B, of her frs momens β, of he mxed momen A and of he nverse of he dosyncrac componen covarance marx, H. Remark 1b From 41 s also easly seen ha he effec of Ω on he dsperson of he w gmv around ẘ gmv vanshes a a suffcenly rapd rae such ha even he asympoc dsrbuon of w gmv does no depend on Ω as ends o nfny. Remark 1c The gmv porfolo becomes fully dversfed wh respec o he dosyncrac as well as he facor componens of asse reurn nnovaons as. Moreover, he lm porfolo reurn s Z 1 -adaped as well as ndependen of he facor componen of asse reurns condonal mean ν 1. Remark 1d The ex ane Sharpe rao, defned by µ gmv ρ, 1/σρ, 1, gmv dverges a he rae of 1 2, unless c 1 = 0. Bu s no guaraneed ha he ex ane Sharpe rao n he case of gmv wll dverge o plus nfny. The oucome depends on sgn of c 1 whch s no guaraneed o be posve. Ths arses snce gmv porfolo does no make use of expeced mean reurns. Consder now he meu porfolo gven by 35, and as before suppose ha r follows he mul-facor model 6, and Assumpons 1,2 and 3 hold. Then we have Theorem 2 maxmum expeced uly porfolo Le ẘ meu = e γ b { α er 0 + [a α er 0 β c a r 0 B]A 1 β }. When condons 7, 10, 11, 12, 13, 16, 17 and 19 hold: w meu 45 ẘ meu p When, n addon o he condons n, 20 also holds: w meu 1 { = ẘ meu γ 1 e +o p 1, + 1/2 z meu + 47 B A + a β β [ 1 Ω 1 1 ν + A + a β β ]} 1 βc a r 0 where z meu s a mxure of normally dsrbued random varables ha are only funcons of γ, r 0, α, B, and H. 19

20 When, n addon o he condons n, 14also holds sasfes ρ meu 1 2 = r w meu e w meu 1 r 0 1, 1 ρ meu e 1 p, 48 γ 1 b 1 µ meu ρ, 1 r 0 1 σρ, 1 meu p e 1, 49 where µ meu ρ, 1 = Eρ meu Z 1, σρ, 1 meu = varρ meu Z 1, and e 1 > 0 almos surely. e = d 2r 0 c + a r0 2 + a d c 2 β A 1 β, 50 Remark 2a A a gven pon n me, he meu porfolo wegh of he h asse s asympocally equvalen o ẘ meu and does no converge o zero. Moreover, ẘ meu s funconally ndependen from he facors covarance marx, Ω, as well as from he facors condonal mean, ν. Remark 2b There s no conrbuon from eher Ω and ν o he asympoc dsrbuon of w meu around ẘ meu. Remark 2c The meu porfolo does no acheve dversfcaon of he dosyncrac and he facors componen of asse reurn nnovaons. Moreover, he par of he porfolo reurn nvolvng he facors componen s of he same order of magnude, n, as he par nvolvng he dosyncrac componen. Dversfcaon of boh componens s acheved f one consders 1 w meu. For he same reasons, convergence of he porfolo reurn ρ meu s acheved when normalzng by and s lm s Z 1 -adaped. In parcular, he lmng value of 1 ρ meu wll be a funcon of α 1, bu no of ν 1. Remark 2d The ex ane Sharpe rao dverges a he rae 1 2, and he lm s always posve. oe ha lm of he normalzed Sharpe rao s ndependen of he coeffcen of rsk averson, γ 1. Analog resuls can be derved for he mnmum-varance mv and he mean expeced me angency porfolos, as dscussed n Secon 6. 20

21 5 Dscusson of resuls 5.1 Conrbuon of facors o porfolo reurn The above heorems her par esablsh he lm porfolo reurn, normalzed wh a suable scalng facor, for varous MV radng sraeges. In parcular, ρ gmv has a well defned lm whereas ρ meu requres he scalng facor 1. The scalng facor s necessary snce he meu porfolo weghs do no converge o zero bu are n fac O p 1. Inspecng he resuls, s evden ha he lm MV porfolo reurns are Z 1 -adaped, ha s hey are neher funcons of he dosyncrac nnovaons, ε, nor he common nnovaons, u. The frs resul s well known, namely ha he conrbuon of he dosyncrac nnovaons o he porfolo reurn vanshes n mean square as. One of he novel resuls of hs paper s o show ha MV radng sraeges also succeed n dversfyng he effecs of he common nnovaons, u. Ths resul s drven by he fac ha he MV radng sraeges make use of he nverse of he condonal covarance marx Σ 1 n a convenen way. In parcular, he MV porfolo weghs have he form Σ 1 1δ 1, for some 1 vecor δ 1 = δz 1, meanng ha s funcon of Z 1, he exac form of whch depends on he ype of radng sraegy under consderaon. As a consequence, he porfolo reurn can be decomposed as: δ 1Σ 1 1r = δ 1Σ 1 1α 1 + δ 1Σ 1 1Bν 1 + δ 1Σ 1 1Bu + δ 1Σ 1 1ε. Lemma A n he appendx esablshes ha Σ 1 1B 2 = O p 1, so ha Σ 1 1 and B are asympocally orhogonal, and herefore he conrbuon of he common facor nnovaon, δ 1Σ 1 1Bu, o he reurn porfolo δ 1Σ 1 s of smaller order han he mean erm δ 1Σ 1 1α 1, as ges large. In oher words, by lemma A any porfolo of he form Σ 1 1δ 1 s asympocally beaneural. Obvously, he conrbuon of he dosyncrac erm, δ 1Σ 1 1ε, s also of smaller order. Therefore δ 1Σ 1 1r = δ 1Σ 1 1α o p 1 as. 1r Ths mples ha, subjec o a suable normalzaon, he conrbuons of u and ε o he lm porfolo reurn converges o zero, he only dfference beween he wo beng ha convergence occurs n frs mean n he case of he erms nvolvng u, and n mean square n he case of he erms n ε. 21

22 Gven he asympoc orhogonaly of Σ 1 1 and B also happens ha he conrbuon of he facors o he reurns condonal mean, namely δ 1Σ 1 1B ν 1, ypcally nvolvng lagged facors f 1, f 2,..., s also of smaller order. Therefore he lm porfolo reurn wll be gven smply by he lm of δ 1Σ 1 1α 1, where hs lm s Z 1 -adaped. oe ha our focusng on MV radng sraeges s less resrcve han mgh appear a frs snce our resuls apply o oher radng sraeges as long as hey can be wren as Σ 1 1δ 1. Ths for nsance holds for ceran dynamc radng sraeges where he porfolo weghs can be wren as he sum of he MV componen and an ner-emporal hedgng componen, boh of whch employ he nverse of he covarance marx n he suable way see Campbell and Vcera 2001, Campbell, Chan, and Vcera 2003 among ohers. The above resuls, on he vanshng mporance of he common componen of boh he nnovaon and he condonal mean as ges large, can also be undersood by consderng he followng approxmaon of Σ 1 Σ 1 ow defne he class of MV porfolo weghs BB B 1 B. 51 w 1 Σ 1 1 δ 1, 52 seng δ 1 = δ Z 1, 0 where Z 1 = Z 1 ν 1 mplyng δ 1 = δ Z 1, ν 1 = δz 1. oe ha, by consrucon w 1 s funconally ndependen of boh Ω 1 and ν 1. Also I s also easly seen ha Σ 1 B = 0 for any fne, and Σ 1 Σ 1 = O p 1. Hence, no only w 1 s approxmaely equvalen o Σ 1 1δ 1 n he sense jus descrbed, bu more mporanly yelds he porfolo reurn w 1r = w 1α 1 + Bν 1 + w 1Bu + w 1 ε = w 1α 1 + w 1 ε, whch s funconally ndependen of boh ν 1 and u due o bea-neuraly of w 1, for any fne. In shor he wo porfolos w 1 and w 1 are equvalen asympocally, namely w 1r = w 1α o p 1 and w 1α 1 w 1α 1 = o p 1 as We have seen ha dfferen MV radng sraeges mply dfferen raes a whch he correspondng porfolo weghs converge, f any, o zero. However, 22

23 he ex ane Sharpe rao µ s ρ, 1 r 0 1 /σ s ρ, 1, correspondng o he MV sraegy of ype s, dverges as ends o nfny, and a he same rae of. The man dfference s ha whereas for he meu sraegy can only dverge o plus nfny, hs s no guaraneed by he gmv sraegy, for whch dvergence owards mnus nfny can occur. Ths parly reflecs he subopmal naure of he gmv sraegy ha does no make use of he reurn predcons, α Conrbuon of facors o porfolo weghs The condonal dsrbuon of he facors, f, s rrelevan, as far as he form of he lmng MV porfolo weghs w s s concerned. In fac, he facors condonal mean ν 1 and condonal covarance marx Ω 1 do no appear n he frs-order lm approxmaons se ou n 39 and 45. Ths oucome s a drec consequence of lemma A proved n he Appendx. An mmedae mplcaon s ha when evaluang he MV porfolo weghs emprcally one can avod specfyng, le alone esmang, he condonal mean and he condonal covarance marx of he common facors. Esmaes of hese quanes are needed only n so far as hey help n esmaon of α 1 and. For a fne, hs esmaon sraegy clearly would nvolve an approxmaon error snce he fne- expresson of he MV weghs wll necessarly be a funcon of Ω 1 and ν 1. However, such approxmaon error decreases o zero as ncreases and, a he same me, usng eher he lm porfolo formulae or he approxmae expresson 52, s lkely o be robus o he consequences of ncorrecly specfyng, or poorly esmang Ω 1 and ν 1. Par of Theorems 1 and 2 can be nerpreed as a conssency resul, showng he form of he lm approxmaons, as, of he MV porfolo weghs. Par of hese heorems consders f he condonal dsrbuon of f plays a role wh respec o he dsperson of he fne- porfolos around her lm approxmaon. Under suable regulary condons, he MV porfolo weghs have an asympoc dsrbuon, cenered around he lm porfolo weghs, whch s dsrbued ndependenly of he condonal momens of f. In oher words, he conrbuon of hese momens o he fne- MV porfolo weghs vanshes a a suably fas rae, faser han he rae requred o oban he asympoc dsrbuon of he porfolo weghs. The resul n par of he above heorems hold no only pon-wse for 23

24 each = 1, 2,..., bu also jonly for he enre vecor of porfolo weghs. In fac, can be shown ha w gmv ẘ gmv = o p 1 and w meu ẘ meu = o p 1. Anoher mporan consequence of par of hese heorems s ha he lmng porfolo weghs wll no be me-varyng unless H s, ha s only f he dosyncrac componen ε feaures dynamc condonal heeroskedascy. The mean-varance porfolo meu wll be me-varyng boh due o possble me varaon n H, and n he rsk-free rae r 0. If we relax our assumpons, say allowng B o be me-varyng B, hen for nsance he gmv porfolo weghs 39 become, under regulary condons smlar o he ones spelled ou n Theorem 1, w gmv 1 a e [ ] H e + a e β B A 1 β p 0 as. For hs case o be genunely neresng, B needs o be ndependen from he facors f hough. Ths rules ou he case B = BΩ 1 2, whch, as far as he dynamcs of r s concerned, s observaonally equvalen o 1. If nsead one alernavely assumes he parameer-free form Ω = I k, our resul connues o apply snce he lm porfolos connue o be funconally ndependen of any paramerc aspec of Ω. Facor models are nherenly undeermned snce 6 yelds he same vecor r gven a non-sngular k k marx C and replacng B and u by BC and C 1 u, respecvely. Deermnaon of C s crucal for denfcaon and esmaon of model 6. Ths s parcularly relevan n our conex snce besdes he facor loadngs, he marx C nduces also a roaon of Ω and ν and, due o her me-varaon, he rsk of possble lack of denfcaon s even more pronounced. However, hs ssue s of second-order mporance snce he lm porfolo weghs do no dependen on he condonal mean and covarance marx of f. One can easly verfy hs by replacng B, A 1 and β wh BC, C 1 A 1 C 1, and C β, respecvely no 39 and Porfolo dversfcaon Under our assumpons w gmv ẘ gmv p 0 as, where ẘ gmv = O p 1, for gven and, and are dfferen from zero almos surely. Therefore, he gmv porfolo s dversfed n he sense ha each 24

25 coeffcen w gmv becomes arbrarly small as grows. More formally, f sup 1 w gmv = o p 1 for each, hen we acheve full dversfcaon n he sup-norm sense of Green and Hollfeld Usng he lm approxmaon ẘ gmv urns ou o be much easer o fnd suffcen condons for full dversfcaon. For nsance, usng resuls of Theorem 1, one obans k h e + h β j = o p 53 where β j = Be k j sup 1 j=1 and h =. If full dversfcaon a rae 1 s requred, he lef hand sde of he prevous expresson mus be O p 1. In urn hs s sasfed whenever sup 1 sup 1 j k β j = O p 1 and h e = O p 1. In conras, he meu porfolo s no fully dversfable n he sense ha e s weghs do no converge o zero and nsead ẘ meu = O p 1. Therefore, as a consequence, he lm porfolo ρ meu requres he normalzaon 1 n order o oban a well-defned lm. Thus, he common pracce of buldng opmal porfolos mposng he resrcon ha he porfolo weghs are smaller han a gven predeermned quany, appears jusfed for he meu porfolo. In fac, here s no guaranee ha he weghs wll be smaller he larger he number of asses under consderaon. On he oher hand, under condons such as 53 or varaons of, he gmv porfolo weghs ges arbrarly small, for a suffcenly large. The defnon of complee dversfably of Chamberlan and Rohschld 1983 nsead requres, for he s radng sraegy, =1 ẘs 2 = o p 2 for each, and suffcen condons can be easly derved. For nsance, for he gmv porfolo s requred e e = o p 2, B B = o p 2. oce ha he second condon s mpled by 11. Ths defnon of complee dversfably requres sronger condons han he noon based on he sup-norm dscussed earler. 25

26 5.4 Shor-sellng and facor domnance When H s dagonal, easly follows ha A = a Σ β, where Σ β s he covarance marx of he β, yeldng for he gmv porfolo weghs w gmv h 1, [ p 1 a β Σ 1 β β β ]. 54 Moreover, f Σ β s dagonal, wh σ βj beng s j, j h enry, 54 smplfes furher o [ w gmv h 1, β1 p 1 a σ β1 2 β1 β 2 1 βk βk... β ] k, β 1 σ βk β k 55 where β j and β j are he j h elemen of β = β 1,..., β k and β = β 1,..., β k, respecvely. Green and Hollfeld 1992 argue ha he possbly of shor-sellng, n he sense of a repeaed fndng of negave opmal porfolo weghs, s relaed o he presence of one domnan facor. Our resul sheds some lgh on hs. One can see from 55 ha he lm porfolo weghs only depend on facor loadngs f he mean of hese laodngs s non-zero.e. f β 0. Such facors are regarded as domnan by Jagannahan and Ma 2003 More generally, a negave wegh can arse whenever he facor loadng assumes values smaller han her cross-seconal average. Ths effec s magnfed, he larger s he Sharpe rao of he facor loadng, defned by β j /σ βj. A large dsperson mples a smaller chance of fndng negave weghs, corroborang he fndngs based on smulaons repored by Jagannahan and Ma On he oher hand, noe also ha he larger he number of domnan facors under consderaon n he sense of Jagannahan and Ma 2003, he less lkely s ha a negave wegh would be encounered. Smlar oucomes oban for non-dagonal H. Ths renforces Green and Hollfeld 1992 s conjecure abou he presence of a sngle domnan facor whenever large negave weghs are observed. Under he same condons as above, for he meu porfolo weghs one 26

27 obans w meu p h 1, γ [ β1 α r 0 + σ β1 2 α r 0 c a r 0 β 1 a β1 2 βk + α r 0 c a r 0 β k σ βk a βk... ] 56 Therefore, as wh he gmv porfolo weghs one can see ha a negave wegh s more lkely for he asse for whch α r 0 < 0. Assumng c > a r 0, a negave wegh s more lkely o arse whenever he facor loadng assumes values smaller han her cross-seconal average and hs effec s magnfed, he larger s he Sharpe rao of he facor loadng. Fnally, he larger he number of domnan facors under consderaon, he less lkely ha a negave wegh would be encounered. 6 Oher opmzaon sraeges We now presen resuls for wo oher MV angency porfolos consdered n he leraure, namely he mnmum varance and he maxmum expeced reurn porfolos. We presen wo correspondng heorems whou proof, and commen on he resuls aferwards. The mv porfolo weghs w mv = w1 mv,..., w mv are defned by w mv 1 = argmn w w Σ 1 w, such ha w µ w er 0 1 = µ ρ, where µ ρ s he argeed expeced porfolo reurn assumed o exceed r 0 1 µ ρ > r 0 1, yeldng w mv 1 = µ ρ r 0 1 µ 1 er 0 1 Σ 1 1µ 1 er 0 1 Σ 1 1µ 1 er Theorem 3 mnmum varance porfolo Le ẘ mv = 1 µ ρ r 0 e e { α er 0 + [a α er 0 β c a r 0 B]A 1 β }

28 When condons 7, 10, 11, 12, 13, 14, 16, 17 and 19 hold: w mv ẘ mv p 0. When, n addon o he condons n, 20 holds: w mv = ẘ mv + 3/2 z mv { + 2 µρ r 0 e B A + a β 1 β Ω 1 e [ ν + A + a β ]} 1 β βc a r 0 +o p 2 where z mv s a mxure of normally dsrbued random varables ha are only funcons of µ ρ, r 0, α, B, and H. When he condons n hold ρ mv = r w mv e w mv 1r 0 1, sasfes 1 2 ρ mv p µ ρ, 59 µ mv ρ, 1 σρ, 1 mv p e 1, 60 r 0 1 seng µ mv ρ, 1 = Eρ mv Z 1 and σρ, 1 mv = varρ mv Z 1. For he maxmum expeced reurn porfolo, w me have = w me 1,..., w me, we w me 1 = argmax w w µ w er 0 1, such ha w Σ 1 w = σ 2 ρ, where σ 2 ρ s he argeed porfolo varance, yeldng he maxmum expeced reurn porfolo [ σ w 1 me ρ 2 = µ 1 er 0 1 Σ 1 1µ 1 er 0 1 ] 1 2 Σ 1 1µ 1 er

29 Theorem 4 maxmum expeced reurn porfolo Le ẘ me = 1 2 σ ρ e b e { α er 0 + [a α er 0 β c a r 0 B]A 1 β }. 61 When condons 7, 10, 11, 12, 13, 14, 16, 17 and 5.2due hold: 1 2 w me ẘ me p 0. When, n addon o he condons n, 20 hold: w me = ẘ me + 1 z me /2 σ ρ e B A + a β 1 [ β Ω 1 ν + A + a β ] β βc a r 0 e +o p 3 2, where z me s a mxure of normally dsrbued random varables, funcon of σρ, 2 r 0, α, B, and H, only. When he condons n hold: sasfes 1 2 ρ me = r w me e w me 1r 0 1, 1 2 ρ me p σ ρ e 1, 63 µ me ρ, 1 r 0 1 σρ, 1 me p e 1, 64 where µ me ρ, 1 = Eρ me Z 1, and σρ, 1 me = varρ me Z 1. Remark 3-4a The mv and me porfolo weghs of he h asse are, asympocally n, equvalen o ẘ mv and ẘ me, respecvely. Moreover, he laer are funconally ndependen of Ω and ν. Remark 3-4b The asympoc dsrbuons of w mv and w me, respecvely around ẘ mv and ẘ mv, do no depend on Ω and/or ν. Remark 3-4c The mv and me porfolos acheve full dversfcaon of boh he dosyncrac and common facor componens of asse reurn nnovaons, he laer when normalzed by 1 2. Under he same condons, he correspondng lm porfolo reurns are Z 1 -adaped. 29

30 Remark 3-4d The ex ane Sharpe rao dverges o plus nfny a rae 1 2. The lm of he normalzed Sharpe rao s ndependen of µ ρ and of σρ 2 for mv and me radng sraeges, respecvely. In parcular, once normalzed by 1 2, he lm s he same and concde wh he one obaned for he meu porfolo reurn. Ths follows snce all he hree MV angency porfolo weghs are proporonal o one anoher. Ths mporan proper s no shared by he he gmv porfolo. Remark 3-4e enre vecor of porfolo weghs, ha s w mv Par of he above heorems hold also jonly for he ẘ mv = o p 1 and w me ẘ me = o p 1 2. Remark 3-4f As for he oher opmzaon sraeges, boh 58 and 61 do no depend on any parcular roaon of he facors and facor loadngs. Remark 3-4g Boh he mv and he me porfolos are fully dversfable, alhough a dfferen raes of 1 and 1/2, respecvely, acheved whenever sup 1 h α = O p 1 and he lef hand sde of 53 s O p 1. 7 Fnal remarks In hs paper we have provded a number of heorecal resuls for he MV porfolos as he number of asses n he porfolo ges large. These resuls are a consequence of he asympoc bea-neuraly sasfed by MV porfolos. In parcular, under farly general condons we have shown ha o a frs order approxmaon he porfolo weghs and he assocaed ex ane Sharpe raos do no depend on he means and he varance-covarances of he common facors. Ths resul has a number of mporan praccal mplcaons. I s well know ha under he assumpon of correc model specfcaon, facor model-based opmal porfolos weghs leads o more effcen esmaes of he correspondng porfolo varance, as compared o he famlar sample momen plug-n esmaes see he emprcal resuls of Chan, Karcesk, and Lakonshok 1999 and he heorecal resuls of Fan, Fan, and Lv However, he asympoc ndependence of MV porfolo weghs from he common facors condonal dsrbuon, esablshed n hs paper, suggess ha n he case of large porfolos mgh be pruden o sde-sep he asks of specfcaon and esmaon of he condonal dsrbuon of he facors and nsead use he formulae for he porfolo weghs advanced n hs paper. In hs way mgh be possble o avod he adverse effecs of 30

31 model and parameer unceranes ha surround he specfcaon of he unobserved common facor models. Bu before hs ssue can be examned one also needs o consder he exen o whch he properes of he lm porfolos are sll vald when he remanng unknown parameers are replaced by her esmaes. Double asympoc resuls wll need o be esablshed, where boh he cross-secon and he me seres dmenson dverge o nfny, unlke hs paper whose resuls hold a each pon n me. Anoher naural drecon s o nvesgae he mplcaon of no-arbrage on our resuls, and as a consequence her relaonshp wh he APT of Ross In fac, when APT hold even he asse-specfc predcable componen of asse reurns wll be, approxmaely, an affne funcon of he facor loadngs. Therefore, n vew of he asympoc bea-neuraly of he MV sraeges, our resuls wll need o be modfed accordngly, n parcular when lookng a he behavour of he lm porfolo reurns. The basc message of our paper would be unchanged hough, n parcular regardng he vanshng mporance of he facors condonal dsrbuon for MV porfolo weghs and reurn. Appendx A: mahemacal proofs We sar wh a Lemma where we show ha for a gven and as, Σ 1 1 and B are asympocally orhogonal. Ths resul urns ou o be crcal for characerzng he behavor of opmal porfolos as ges large. Lemma A Le P be a sequence of random posve defnve marces such ha B B p P > 0 as. 65 Recallng ha e s he h column of he deny marx I, hen for any, and j e Σ 1 β j p 0 as, 1 j k, 66 where β j denoes he j h column of B = β 1... β k. Under 65 and B B p Q 0, 67 where Q denoes a sequence of random posve sem-defnve marces, for any Σ 1 β j 2 = O p 1, 1 j k, as

32 Proof of Lemma A. The resuls follow from he deny Σ 1 = B 1 Ω B B 1 1 B. 69 Pre-mulplyng boh sdes by e and pos-mulplyng boh sdes by β j yelds 66. We deal wh 68 more explcly. Frs noe ha e k j denoes he j h column of he deny I k marx 1 Ω 1 = 1 Ω B + 1 B = 1 [ 1 Ω 1 1 g j, B 1 1 B β j e k j B 1 1 B β j 1 B B 1 1 B + 1 B B 1 Ω 1 1 B B 1 1 B where noce ha g j s a k 1 vecor wh a fne norm. Therefore, subsung he laer expresson no 69 and recallng ha Be k j yeldng = β j follows ha Σ 1 Σ 1 β j 2 = β j 1 g j β j = 1 g j B β j = β j Be k j β j + 1 g j β j 1 β j β j + β j 1 B B g j + 1 g j, B Bg j β j + 1 β j = O p 1 g j β j β j 2 g j Bg j Q g j. Proof of Theorem 1. All he lms below are based on. For <, se w gmv = C, /D, where and C, = D, = e e BΩ 1 e e BΩ 1 whch easly follow from he deny B B 1 B e, + B B 1 B e, B β j] β j β j Bg j

33 For B = B e β 1 B B = 1 β β e e + 1 B B + 1 βe B + 1 B e β, so ha collecng erms 1 B B p A + a β β, snce 1 B e p 0 by E β = 0. Smlarly Hence, usng he deny A + a β β 1 = 1 B e p a β. A 1 a 1 + a β A 1 β A 1 β β A 1, whch yelds β A + a β β 1 β = β A 1 β/b, by Slusky s heorem, 1 D, p a 1 a β A 1 bu he rgh hand sde smplfes yeldng 1 D, p a b 1 a A 1 β β A a β A 1 By he same argumens, snce A + a β β 1 βa = a b 1 e For 41 C, = b 1 C, = e + = + e e BΩ 1 BB B 1 B BB B 1 B e e + a e β BA 1 + B B 1 B e BΩ 1 β β A 1 β, hen β + o p 1. e = + B B 1 B e e BB B 1 B e 70 BB B 1 Ω 1 Ω 1 + B B 1 B e

34 In relaon o 70, we seek he asympoc dsrbuon of where w gmv ẘ gmv C,, 1 2 w gmv e = 1 = 1 1 D, = 1 e e e 1 D, b 1 = 1 = 1 e e ẘ gmv BB e + a e β BA 1 a b 1 e e a b 1 e e BΩ 1 + B 1 D, B 1 B e + O p 2, β BA + a β β 1 a β. B 1 B e By he connuous mappng heorem 1 B B 1 B e 2 1 D, D, a b 1 A + a β β 1 a β a b 1 d ζ gmv 1, d ζ gmv 2, where ζ gmv 1, ζ gmv 2 are a k 1 and a scalar normally dsrbued random varable, respecvely, wh zero mean. Therefore by sandard resuls 1 2 w gmv ẘ gmv d ξ 1,ζ gmv 1 + ξ 2, ζ gmv 2 = z gmv,, whch s a mxure of normal random varables, unless ξ 1,, ξ 2, are boh non-random. Concernng erm 71 ha nvolves Ω e BB B 1 Ω 1 Ω 1 + B B 1 B e = a e BA + a β β 1 Ω 1 A + a β β 1 β1 + op 1. To esablsh 43, snce: w gmv 1 r = 1 [ e e Σ 1 Σ 1 1Bν 1 + u + e Σ 1 1α 1 + ε ], 1e 34

35 hen by Lemma A he frs erm on he rgh hand sde sasfes 1 e Σ 1 1e e Σ 1 1Bν 1 + u = O p 1. The covarance marx of he erm nvolvng ε s snce by easy calculaons e Σ 1 1e 2 e Σ 1 1 Σ 1 1e = O p 1, e Σ 1 1 Σ 1 1e e Σ 1 1e p 0, yeldng ε w gmv 1 = O p 1 2. Therefore, collecng erms ρ gmv = 1 e Σ 1 1α 1 1 e Σ 1 1e + O p 1 2, whch s asympocally equvalen, by par of hs proof, o 1 e a 1α 1 + a β 1 A 1 1B βe c 1 1α 1 p. 1 a 1 44 follows easly snce σ gmv 2 ρ, 1 = e Σ 1 1e 1 and where he lm of ρ gmv jus esablshed, concde wh he lm of s condonal mean µ gmv ρ, 1. Proof of Theorem 2. All he lms below are based on. By deny 69 w meu = 1 γ Σ 1 µ er 0 = 1 γ H 1 for <. Snce e Σ 1 µ er 0 BΩ 1 µ er 0 = e Σ 1 α e, + B B 1 B µ er 0, Σ 1 er 0 + e Σ 1 Bν, we jus need o deermne he behavor of he frs erm on he rgh hand sde. In fac, he second erm can be wren as e C, r 0 wh C, defned n he proof of Theorem 1 and he hrd erm, e Σ 1 Bν, goes o 35

36 0 by Lemma A. Thus, usng he same argumens used n proof of Theorem 1, e Σ 1 α = e snce A + a β β 1 βc = c b 1 e For 47 w meu + = + Σ 1 = BB α = 1 b e α e BA + a β β 1 βc + o p 1, A 1 β, sraghforward manpulaon yelds α + a µ β Bc A 1 β + o p 1. µ er 0 BB B 1 B µ er 0 B 1 B µ er 0 BΩ 1 + B B 1 B µ er 0 I BB B 1 B µ er 0 72 BB B 1 Ω 1 Ω 1 + B B 1 B µ er In relaon o 72, we seek he asympoc dsrbuon of where w meu = 1 γ = 1 γ = 1 γ ẘ meu = 1 γ e e e e 1 2 w meu µ er 0 e µ er 0 e α er 0 e α er 0 e ẘ meu BΩ 1 + B B 1 B µ er 0 BB B 1 B µ er 0 + O p 1 BB B 1 B α er 0 + 2O p 1, BA + a β β 1 c a r 0 β. By he connuous mappng heorem 1 2 B B 1 B α er 0 A + a β β 1 c a r 0 β d ζ meu where ζ meu yeldng, s a k 1 normally dsrbued random vecor wh zero mean 1 2 w meu ẘ meu d γ 1 ξ 1,ζ meu = z meu,, whch s a mxure of normal random varables, unless ξ 1, s non-random. 36